Project scheduling using optimized financing

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PROJECT SCHEDULING USING OPTIMIZED FINANCING

BY SEYYED MOHAMMADREZA ALAVIPOUR

DEPARTMENT OF CIVIL, ARCHITECTURAL, AND ENVIRONMENTAL ENGINEERIGN

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering in the Graduate College of the Illinois Institute of Technology

Approved _________________________ Adviser

Chicago, Illinois July 2017

ACKNOWLEDGEMENT I would like to express the deepest appreciation to my advisor Dr. David Arditi, who dedicated his time to my research. This research would not have been possible without his valuable advice and persistent help. I would like to express my heartfelt appreciation to him, not only because of his guidance during the preparation of this dissertation, but also for his support in many different ways during my Ph.D. study. I would like to thank my committee members Dr. Ivan Mutis, Dr. Jamshid Mohammadi, Professor Raymond Lemming, and Dr. Ricky Cooper for giving their precious time and serving on the Ph.D. Committee. I would like to express my special appreciation to Professor Raymond Lemming, not only because of his invaluable advice and suggestions, but also for giving the motivation and invaluable inspiration during my Ph.D. study. I also thank Dr. Navid Sabbaghi for his first guidance to start this research. I would also like to thank Haniyeh Tabatabyi for her great assistance that helped me to address many critical problems during this research. My special thanks are due to my lovely family members, Ahmad Alavipour, Seddigheh Mohammadkhan, and Azadeh Alavipour for their endless support, guidance, patience and encouragement not only during the entire study, but also during my whole life. I could not have gotten to this point in my life without my parents` love and support.

S. M. Reza Alavipour

iii

TABLE OF CONTENTS Page ACKNOWLEDGEMENT

.......................................................................................

iii

LIST OF TABLES

...................................................................................................

vi

LIST OF FIGURES

.................................................................................................

x

LIST OF ABBREVIATION ABSTRACT

xii

............................................................................................................. xiv

CHAPTER 1. INTRODUCTION 1.1 1.2 1.3 1.4

....................................................................................

..................................................................................

1

Significance of the Study ................................................................ Objective of the Study ..................................................................... Anticipated Impact of the Study ..................................................... Organization of the Dissertation .....................................................

4 10 11 11

2. LITERATURE REVIEW

.......................................................................

13

Introduction to Construction Scheduling Problems ........................ Heuristic Methods ........................................................................... Mathematical Methods .................................................................... Metaheuristic Methods .................................................................... Hybrid Methods ..............................................................................

13 32 34 37 43

3. FINANCING COST, METHODS AND THE EFFECTS ON PROJECT PROFIT ........................................................................................................

45

2.1 2.2 2.3 2.4 2.5

3.1 Investigating the Behavior of Financing Cost and the Effects on the Project Profit ........................................................................................ 3.2 Reviewing Financing Terms and Methods for Construction Companies ....................................................................................... 4. FIRST STAGE OF THE RESEARCH

...................................................

4.1 Methodology and Computational Process of the First Stage Model .............................................................................................. 4.2 Project Schedule Creation for the First Stage Model ...................... 4.3 Creation of Project Cash Flow Forecast for the First Stage Model .............................................................................................. iv

45 52 64 64 65 71

4.4 Optimizing Financing Cost in the First Stage Model ..................... 80 4.5 Testing the First Stage Model ......................................................... 119 4.6 Conclusion of the First Stage .......................................................... 139 5. SECOND STAGE OF THE RESEARCH ................................................... 141 5.1 Project Schedule Creation for the Second Stage Models ................ 5.2 Creation of Project Cash Flow Forecast For the Second Stage Models ............................................................................................. 5.3 Model Optimization ........................................................................ 5.4 Computational Process of the Second Stage Models ...................... 5.5 Testing Models 1 and 2 in the Second Stage .................................. 5.6 Conclusion of the Second Stage ......................................................

144 146 150 153 165 188

6. THIRD STAGE OF THE RESEARCH....................................................... 190 6.1 Methodology and Computational Process of the Third Stage Models ............................................................................................. 192 6.2 Testing the Third Stage Models ...................................................... 193 6.3 Conclusion of the Third Stage Models ............................................ 214 7. FOURTH STAGE OF THE RESEARCH ................................................... 216 7.1 Methodology and Computational Process of the Fourth Stage Models ............................................................................................. 220 7.2 Testing the Fourth Stage Models .................................................... 221 7.3 Conclusion of the Fourth Stage ....................................................... 241 8. CONCLUSION

....................................................................................... 243

5.1 Summary ......................................................................................... 243 5.2 Conclusion ...................................................................................... 245 5.3 Future Research ............................................................................... 247 APPENDIX A. A CLASSIFICATION OF CONSTRUCTION SCHEDULING OPTIMIZATION RESEARCH ............................................................ 248 B. PROJECT SCHEDULE DATA AND ACTIVITY ACCELERATION METHODS FOR LARGE NETWORK USED TO TEST SECOND STAGE MODELS ................................................................................. 259 C. FINANCING DATA FOR LARGE NETWORK USED TO TEST SECOND STAGE MODELS..................................................................... 263 BIBLIOGRAPHY

.................................................................................................... 265 v

LIST OF TABLES Table

Page

2.1 Classification of Finance-based Scheduling Publications Based on the Assumptions Made in Calculating Costs and Payments .............................

28

3.1 Different Cases of Total Project Profit Regarding Total Financing Cost

...

47

4.1 Alternative Financing Methods (Page 1 of 2)

.............................................

82

4.1 Alternative Financing Methods (Page 2 of 2)

.............................................

83

4.2 Project Schedule Data Inputs and Model Output of CPM Calculations 4.3 The Inputs of Cost Data and the Contractual Terms of the Project

..... 121

............ 122

4.4 The Model Output of Contract Bid Price Calculations of the Project

........ 122

4.5 Model Output of the Project Cash Flow Calculations Excluding Financing Flow (Page 1 of 2) ....................................................................................... 126 4.5 Model Output of the Project Cash Flow Calculations Excluding Financing Flow (Page 2 of 2) ....................................................................................... 126 4.6 Financing Data of Three Different Cases

.................................................... 128

4.7 Model Output of Optimal Financing Results for Case 1

............................ 129


............................ 129


............................ 130

4.10 Model Output of Optimized Financing Inflow Schedule (Borrowed Money) for Each Case .............................................................................................. 131 4.11 Model Output of Optimized Financing Outflow Schedule (Repaid Money Including Interest) for Each Case ................................................................ 132 4.12 Summary of Project Financial Parameters Considering Each Case 4.13 Sensitivity Analysis of the Best and Worst Financing Cases

..................... 138

5.1 Project Schedule Data and Activity Acceleration Method Inputs

vi

............ 134

.............. 167

5.2 The Inputs of Cost Data and the Contractual Terms of the Project 5.3 Financing Data of Two Different Cases

............ 168

...................................................... 169

5.4 The Project Costs and Contract Price Calculation Using Method 1

........... 170

5.5 Optimum Results Obtained by Testing Model 1 on Small Network for Financing Cases 1 and 2 .............................................................................. 171 5.6 CPM Calculations for Optimal Activity acceleration methods Using Activity-On-Node Method, Topological Sorting, and Improved Dijkstra`s Algorithm .................................................................................................... 175 5.7 Optimal Financing Results Using Financing Case 1 for Optimal Project Completion Time ......................................................................................... 176 5.8 Optimal Financing Results Using Financing Case 2 for Optimal Project Completion Time ......................................................................................... 176 5.9 Cumulative Net Balance of the Cash Flow Excluding Financing Flow for Normal, Optimum, and Crash Project Completion Times .......................... 179 5.10 Optimized Financing Inflow Schedule (Borrowed Money) for Normal, Optimum, and Crash Project Completion Times ........................................ 180 5.11 Optimized Financing Outflow Schedule (Repaid Money Including Interest) for Normal, Optimum, and Crash Project Completion Times 5.12 Information of Each Test

...... 180

............................................................................ 182

5.13 The GA Parameters for the Small and Large Networks

............................. 183

5.14 The GA Stopping Rules for the Small and Large Networks 5.15 Optimal GA Parameters and Methods for Each Test

....................... 183

.................................. 184

5.16 Outputs Obtained in Optimal GA Parameters Identification Process for Test 1 Considering the Normal Project Completion Time .......................... 185 5.17 Outputs Obtained in Optimal GA Methods Identification Process for Test 1 Considering the Normal Project Completion Time .......................... 186 5.18 Validation of Models and Comparison of Computational Time between Tests ............................................................................................................ 187 6.1 Project Schedule Data and Activity Acceleration Method Inputs vii

.............. 195

6.2 The Inputs of Cost Data and the Contractual Terms of the Project

............ 196

6.3 The Project Costs and Contract Price Calculation Using Method 1

........... 196

6.4 Financing Data of Two Different Cases

...................................................... 197

6.5 Information of Each Alternative in Model 1

............................................... 198

6.6 Optimum Results Obtained by Testing Model 1 Considering Four Alternatives ................................................................................................. 199 6.7 Construction Schedule for Optimal Project Completion Time Considering Alts 1 and 2 ................................................................................................. 205 6.8 Optimized Financing Inflow Schedule (Borrowed Money) for Normal, Optimum, and Crash Project Completion Times Considering Alt 1 (Constant Start Times of Activities) ............................................................ 206 6.9 Optimized Financing Inflow Schedule (Borrowed Money) for Normal, Optimum, and Crash Project Completion Times Considering Alt 2 (Variable Start Times of Activities) ............................................................ 207 6.10 Optimized Financing Outflow Schedule (Repaid Money Including Interest) for Normal, Optimum, and Crash Project Completion Times Considering Alt 1 (Constant Start Times of Activities) .............................. 208 6.11 Optimized Financing Outflow Schedule (Repaid Money Including Interest) for Normal, Optimum, and Crash Project Completion Times Considering Alt 2 (Variable Start Times of Activities) .............................. 209 6.12 Information of Each Test Considering Models 1 and 2

.............................. 212

7.1 Project Schedule Data and Activity Acceleration Method Inputs

.............. 222

7.2 The Inputs of Cost Data and the Contractual Terms of the Project

............ 223

7.3 The Project Costs and Contract Price Using Method 2 7.4 Financing Data of Three Different Scenarios

.............................. 223

............................................. 225

7.5 Results of the Schedules That Lead to Optimal Profit Obtained by Testing Model 1 Considering Three Scenarios for Every Completion Time .......... 227 7.6 Results of Optimum Costs Obtained by Testing Model 1 Considering Three Scenarios for Every Completion Time ........................................................ 228

viii

7.7 Results of Optimum Profits Obtained by Testing Model 1 Considering Three Scenarios for Every Completion Time ............................................. 229 7.8 Optimized Financing Inflow Schedule (Borrowed Money) for Optimum Project Completion Times Considering Scenarios 1 to 3 ........................... 237 7.9 Optimized Financing Outflow Schedule (Repaid Money Including Interest) for Optimum Project Completion Times Considering Scenarios 1 to 3 ...... 238 7.10 Information about Tests of Models 1 and 2

................................................ 239

7.11 Optimal GA Methods Obtained for Tests 1 to 6

ix

......................................... 241

LIST OF FIGURES Figure

Page

2.1 Time-Cost Tradeoff Considering Realistic Assumptions for the Costs

.....

24

........

26

2.3 Time-Cost Tradeoff Considering the Assumptions Made in Category 1 Research .....................................................................................................

27

2.4 Optimum Decision Considering the Assumptions Made in Category 1 Research .....................................................................................................

29

2.5 Time-Cost Tradeoff Considering the Assumptions Made in Category 2 Research .....................................................................................................

30

2.6 Optimum Decision Considering the Assumptions Made in Category 2 Research .....................................................................................................

31

3.1 Time-Cost Tradeoff Considering Financing Cost for Case 1

....................

48


....................

48


....................

49


....................

49

2.2 Optimum Decision Considering Realistic Assumption for Payment

3.5 Optimum Decision Considering Financing Cost and Payment for Case 1

.

50


.

51


.

51


.

52

.....................................................................

66

4.1 First Stage Model Algorithm 4.2 Network of the Example Project

................................................................ 120

4.3 Cumulative Net Financing Cost NFCt of Financing Cases

...................... 133

4.4 Cumulative Cash Flow Including Financing Cost Nt of Financing Cases . 134 5.1 Second Stage Algorithm

............................................................................ 155 x

5.2 Hybrid GALP Algorithm for the Second Stage Models 5.3 A Sample of Encoding Chromosome 5.4 Small Network of the Example Project

............................ 156

........................................................ 157 ..................................................... 166

5.5 Project Financing Cost for Every Project Completion Time Considering Financing Cases 1 and 2 ............................................................................. 173 5.6 Project Total Cost Including Financing Cost and Profit for Every Project Completion Time Considering Financing Cases 1 and 2 ........................... 173 5.7 Required Credit for Line of Credit for Every Project Completion Time Considering Financing Cases 1 and 2 ........................................................ 174 6.1 Third Stage Algorithm

............................................................................... 194

6.2 Network of the Example Project

................................................................ 194

6.3 Financing Cost for Every Project Completion Time Considering Four Alternatives ................................................................................................ 202 6.4 Profit for Every Project Completion Time Considering Four Alternatives . 202 6.5 Average of the Cumulative Net Balance of the Cash Flow for Every Project Completion Time Considering Four Alternatives ......................... 203 6.6 Sum of the Activities` Total Floats for Every Project Completion Time Considering Four Alternatives ................................................................... 203 7.1 Network of the Example Project

................................................................ 221

7.2 Financing Cost for Every Project Completion Time Considering Three Scenarios with Early and Variable Activity Start Times ................................ 230 7.3 Profit for Every Project Completion Time Considering Three Scenarios with Early and Variable Activity Start Times ............................................ 230 7.4 Sum of Activities` Total Floats for Every Project Completion Time Considering Three Scenarios with Early and Variable Activity Start Times .......................................................................................................... 231

xi

LIST OF ABBREVIATION Abbreviation

Term

ACO

Ant Colony Optimization

APR

Annual Percentage Rate

CIP

Constraint Integer Programming

CP

Constraint Programming

CPM

Critical Path Method

DP

Dynamic Programming

EA

Evolutionary Algorithm

FBSP

Finance-Based Scheduling Problem

GA

Genetic Algorithm

GAs

Genetic Algorithms

ILP

Integer Linear Programming

IP

Integer Programming

LOB

Line of Balance

LP

Linear Programming

MLGAS

Machine Learning Genetic Algorithms

MIP

Mixed Integer Programming

NPV

Net Present Value

NSGA

Non-dominated Sorting Genetic Algorithm

PIP

Pure Integer Programming

PSO

Particle Swarm Optimization

xii

Abbreviation

Term

RCPSP

Resource-Constrained Project Scheduling Problem

RPSP

Resource Project Scheduling Problem

RWS

Roulette Wheel Selection

SA

Simulated Annealing

SFLA

Shuffled Frog-Leaping Algorithm

SPEA

Strength Pareto Evolutionary Algorithm

SSGS

Serial Schedule Generation Scheme

TCTP

Time-Cost Tradeoff Problem

TF

Total Float

TRS

Tournament Selection

TS

Tabu Search

xiii

ABSTRACT Contractors need financing throughout a project, mainly due to retainage, which is the money that the owner withholds to make sure that the project is performed properly by the contractor. Even if an owner does not withhold retainage, financing is still necessary because the periodic payments made by the owner are usually delayed. All pertinent studies conducted so far have considered only one source of financing without any consideration of different sources and types of financing, times of cash provisions, interest rates, and repayment options. Actually, if one assumes a predetermined credit limit and one source of financing, as past researchers have done, the optimal financing cost and the schedule that satisfies all constraints may be different than when several sources of financing and undetermined credit limit are considered. The main objective of this research is to focus on the optimization of financing cost by developing a financing optimization model based on different financing alternatives. Far lower financing cost and higher profit are obtained by using the proposed model compared to all models developed in past research. The research is conducted in four stages of development. In Stage 1, a financing optimization model is developed for a schedule that uses normal (not accelerated) activities, and early activity start and finish times. This model can be used before the contract is signed to offer the lower bid or can be used after the contract is signed to obtain higher profit. In Stage 2, the time-cost tradeoff algorithm is added to the model developed in Stage 1, considering accelerated activities between the crash and normal durations. In Stage 3, the model developed in Stage 2 is augmented by considering variable activity start times. Finally, in Stage 4, the model developed in Stage 3 is further improved by expanding time-cost xiv

tradeoff to allow for time extensions beyond the contract duration. All models developed in these four stages provide not only minimum financing cost, but also the ideal work schedule that achieves minimum total cost and maximum profit. Moreover, these models provide specific timings for borrowing and repaying funds.

xv

1 CHAPTER 1 1. INTRODUCTION Construction management decisions are made based on schedules that should be developed before starting a project. To have a useful and realistic plan, a construction schedule should be a forecast of the best way to carry out the process of construction. Therefore, it should be a realistic representation of the process to be controlled. Since the late 1950s, CPM has been widely used by project managers to coordinate construction projects (Icmeli 1993; Tsai and Chiu 1996). The objective of CPM is to minimize total project duration. Therefore, CPM handles only the time aspect without consideration of resource availability and cash flows (Elazouni and Gab-Allah 2004; Icmeli 1993). Partly due to fundamental defects in CPM, many contractors have failed to fully use CPM (Birrell 1980). Thus, techniques including resource management and time-cost tradeoff analysis have been developed to customize CPM schedules in order to fulfill users` concerns about resources, cost, and time. Considering the fact that choosing the best solutions for a scheduling problems is extremely important for failure and loss avoidance, the characteristics of scheduling problems must be recognized before making any decisions. Hence, Icmeli (1993) classified construction scheduling problems into three categories: (1) resourceconstrained project scheduling problem (RCPSP), (2) time-cost tradeoff problem (TCTP), and (3) payment scheduling problem (PSP). The use of resources is one of the important aspects of construction scheduling. CPM scheduling procedures begin with an assumption of unlimited availability of resources for each project activity (Abeyasinghe et al. 2001). However in many real life

2 situations, delays occur when resources required by activities are not available in adequate quantities during the time that they are scheduled to take place (Icmeli 1993). So, problems arise when activities need resources which are available only in limited quantities, not enough to satisfy demand. Consequently, project activities must be scheduled by considering the effect of limited resources (Abeyasinghe et al. 2001). This particular problem of resource-constrained scheduling (resource allocation) and resource leveling is known in the literature as the resource project scheduling problem (RPSP). Although many researchers have investigated the problem of scheduling under resource constraints and leveling, the major constraining factor for acquiring project resources is the availability of cash to finance the project (Abeyasinghe et al. 2001; Smith-Daniels et al. 1996). The time-cost tradeoff problem involves normal durations and accelerated durations of activities in which the accelerated durations are obtained by allocating more resources to them. This leads to shorter project durations and lower indirect costs at the expense of higher direct costs. Another construction scheduling problem is to schedule the project activities in such a way as to maximize the net present value (NPV) of cash flows by using discounted cash flow calculations. This particular problem is known as the payment scheduling problem (PSP). However, very few papers have been published that consider cash availability constraints. It must be noted that common construction schedules consider the solution for all these three scheduling problems based on the assumption that unlimited cash is available during the life of a project. Actually, due to retainage, which is the money that the owner withholds to make sure that the project is performed properly by the

3 contractor, the cash inflows and outflows are not balanced in every period during the project. Therefore, the contractor encounters deficits, and financial problems may arise if the contractor does not consider cash availability constraints while preparing the schedule. A schedule may not simulate reality if contractors do not consider cash availability constraints in all three mentioned scheduling problems. In addition to considering cash availability constraints in the payment scheduling problem, one has to incorporate financing costs also into the calculation since the use of money costs money. While a few researchers considered cash availability constraints in their models (e.g., Smith-Daniels et al. 1996), incorporating financing costs was neglected until 2004. The most likely reason for the absence of financing costs in the development of schedules is that it is the responsibility of the owner to finance the construction project. Moreover, schedulers generally consider project scheduling and financing as two independent areas of construction project management that are germane to production planning and business management, respectively. In 2004, Elazouni and Gab-Allah introduced finance-based scheduling, which integrates the scheduling and financing functions of a construction project and involves the scheduling of construction activities subject to precedence and financial constraints. The objective of finance-based scheduling is to determine a feasible schedule of activities to achieve certain predefined goals such as shortest project completion time, lowest cost, or highest profit subject to cash constraints while considering financing costs (Elazouni 2009; Elazouni and GabAllah 2004). This particular problem is known in the literature as the finance-based scheduling problem (FBSP).

4 The development of FBSP was achieved over three generations of research: (1) Scheduling of payments considering neither cash availability constraints nor financing costs; (2) Scheduling of payments considering cash availability constraints but no financing costs; and (3) finance-based scheduling considering both cash availability constraints and financing costs. The objective of this research is taking this process to its fourth generation by adding a financing optimization model based on different financing alternatives.

1.1

Significance of the Study Many contractors go bankrupt every year partly because of the high level of

uncertainty in the construction industry. Although many factors could be the cause of business failure, financial and budgetary factors are the most common causes of failure (Arditi et al. 2000). Over 60% of contractor failures have financial causes. The lack of finance causes 77 to 95% of contractor failures (Russell 1991). The absence of the linkage between financing and project scheduling results creating non-executable schedules which lead contractors to a high rate of failure. Having financing problems not only affects cash flow, but also influences other contract terms and interactions between project participants: friction between parties increases, total project cost increases, bids are unbalanced, more change orders and claims are filed, and contract terminations due to delay increase. Thus, the integration of financing and scheduling is of vital importance in developing executable schedules. Since many owners withhold a portion of the periodic payments as a retainage, the contractor has a deficit at the end of each period. This situation results in performing

5 less work in subsequent periods compared to the work that is scheduled. If the contractor insists on using only the money received from the owner, this situation worsens as time goes on and eventually forces the contractor to completely stop the work. Thus, a crucial factor for the construction contractor is to procure sufficient cash with minimum financing costs in a timely manner to execute construction operations on schedule (Elazouni and Gab-Allah 2004). However because the execution of construction projects demands large investments, contractors seldom finance the construction from their own savings (Elazouni and Metwally 2005). The credit line account is one of the most common and prevalent methods of financing construction projects (Ahuja 1976). The contractor can borrow money on an as-needed basis up to the credit limit which is set by the bank and then pays interest on the amount of funds borrowed (Afshar and Fathi 2009). Setting a limit on the credit allocated to a credit line is a common practice for bankers (Elazouni and Metwally 2005). A contractor`s total indebtedness to the bank including interest must not exceed the credit limit at any time. If the negative cumulative balance including financing costs exceeds the credit limit, finance-based scheduling modifies the start time of project activities (or maybe their durations) and if needed, extends the total project completion time without violating credit limits (Fathi and Afshar 2010). Most researchers who investigated finance-based scheduling, have assumed a predetermined credit limit and have modified the initial schedule of the project accordingly. However all pertinent studies have considered just one source of financing without any consideration of different alternatives of financing in terms of sources and types of financing, times of cash provisions, interest rates, and repayment options. In fact,

6 if one assumes a credit limit that is not predetermined (Fathi and Afshar 2010) and just one source of financing, the optimal credit limit and a schedule that satisfies all constraints may be different than when undetermined credit limit and only one source of financing (line of credit) are considered. Below is discussed how the results could be different if one considers different alternatives of financing (i.e., more sources and more types of financing, different times of cash provisions, different interest rates, and different repayment options). 

If the total indebtedness of the contractor to the lender does not exceed the credit limit based on the current schedule, the resulting schedule will be the same as the schedule developed with no financing considerations, but the profit for the contractor may not be optimal. Typically, a line of credit has a lower interest rate than a loan of comparable size. However, if the contractor is late with a repayment and postpones the payments (which is common for contractors), the interest paid for the credit line may increase substantially due to compound interest unlike a term loan where the interest rate stays the same for the life of the loan, meaning that a credit line can be a cost-effective alternative to month-tomonth financing (Lesonsky 2014; Simpson 2015). In addition, an unsecured credit line is more expensive than a secured loan, where in most cases, the interest on the credit line is not tax deductible (Simpson 2015). Furthermore, in reality a long-term loan is charged a lower interest rate. Therefore, it could be desirable for the contractor to use a long-term loan to reduce interest charges and to avoid a large overdraft on its primary account (Au and Hendrickson 1986). Given the aforementioned reasons, the selection of the optimum financing method may

7 depend on each specific cash flow and condition. In other words, considering just a credit line account as a single source of financing may result in higher financing costs. 

If the negative cumulative balance (including financing costs) tends to violate the credit limit, the model revises the early start time of project activities and extends the total project duration to bring back the project`s negative cash flow to within the available credit limit (Afshar and Fathi 2009; Elazouni and Gab-Allah 2004; Fathi and Afshar 2010). In this case, neither the schedule nor the profit are optimum. If one considers different sources of cash procurement, the schedule may not need to be extended. Therefore, the resulting schedule and the related profit are not optimal if one extends the schedule where it is not needed. There is no doubt that by procuring more money, the contractor has to pay more for financing, but it may be better to pay more for financing than extending the schedule since liquidated damages and late completion penalty (if any) could sometimes be higher than the additional financing cost. Furthermore, the early completion bonus (if any) should be taken into consideration in decisions related to scheduling and financing. Ultimately, considering more sources of cash (even if they have higher financing cost), liquidated damages, late completion penalty, and early completion bonus are essential in reaching a more realistic schedule and profit.



Although the schedule can be extended, an inherent goal of finance-based scheduling is to minimize the extension (Afshar and Fathi 2009). A low credit limit causes inevitable extensions of total project duration which results in

8 increased indirect costs (Elazouni and Gab-Allah 2004; Elazouni and Metwally 2005) and liquidated damages. On the other hand, a higher credit limit strengthens the contractor`s ability to provide more cash, which results in earning higher profits considering a particular execution time (Ali and Elazouni 2009; Elazouni and Metwally 2005; Fathi and Afshar 2010). Thus, if the contractor is inclined to earn a higher profit and incur lower total cost, the firm needs to consider more sources of financing to increase the credit limit and avoid extending the schedule. 

Financing models are more flexible when different alternatives of financing are considered. The contractor should be able to choose between different banks, different types of financing, different interest rates, different repayment options, and different project completion times in such a way as to reach an optimum decision. A financing optimization tool is needed to make decisions about various

parameters of financing and scheduling such as alternative sources, optimum amount of required money, financing costs, types of financing, interest rates, repayment options, and other provisions. The tradeoff between these parameters should be evaluated to decide which combination of parameters is more profitable. Without financing optimization, there is no way to figure out which decision is optimal. This is an area that has been neglected by researchers when developing finance-based scheduling. Below is discussed why the optimization of financing parameters can be advantageous for the contractor. 

When the contractor wants to get loans, it is difficult to persuade banks and potential lenders that the insufficiencies in the cash flow are ephemeral, especially when the contractor wants to increase the credit limit (Barbosa and Pimentel

9 2001). By providing an optimized financing schedule for the chosen cash flow, the contractor should be able to assure the banks that there is a reason why the optimal combination of parameters is selected and why all debts will be paid on the scheduled date. An optimized financing schedule gives the contractor more negotiation power when dealing with lenders. 

If a contractor does not know exactly how much credit is required for the project, securing a higher credit limit would be desirable for them. If banks and potential lenders accept to provide a higher credit limit, they will set a higher interest rate to reduce their risks. On the other hand, even if the contractor does not use the credit line, some banks will charge the contractor on the unused credit either monthly or annually (Elazouni and Metwally 2005). Therefore, holding unnecessary cash in liquid assets can imply a loss, while optimized financing prevents the contractor from incurring this type of loss by calculating a more realistic amount of credit for a given cash flow.



Since the credit line is for the ongoing cash needs of the business, it can be drawn on and repaid on an unscheduled basis. This unplanned process results in complicated interest calculations for contractors who may be quite surprised when they end up paying too much interest (Simpson 2015). One of the reasons why a credit line is appealing to contractors, is that many contractors are uncertain about when and how much to borrow and repay. Most of the time, they think that a credit line is quite profitable since they do not have a financing schedule. With financing optimization, contractors not only could input the banks` offers into the model to figure out which financing alternative or combination of financing

10 alternatives is the best, but also they can produce an optimal financing schedule for the optimum financing alternatives.

1.2

Objective of the Study Since financing optimization with respect to different financing alternatives has

been neglected in past project scheduling optimization research, the main objective of this research is to focus on financing optimization. The objectives of the proposed study are: 

To consider different alternatives of financing in terms of sources and types of financing, times of cash provisions, interest rates, and repayment options for two reasons: 1. To find a realistic amount of credit needed for the project by considering more sources and types of financing, hence preventing the possible extension of the work schedule. 2. To find the optimal total duration of the project which has the minimum total cost (direct, indirect, and optimal financing costs) and maximum profit, meaning the project is performed faster with no or minimal extension.



To incorporate financing optimization to finance-based scheduling for two reasons: 1. Financing decisions can be made by financial managers irrespective of the work schedule. This model unifies financing and scheduling decisions to achieve optimum arrangements of financing and scheduling.

11 2. This model ensures an optimum construction project schedule based on optimized financing.

1.3

Anticipated Impact of the Study The study introduces a significant improvement to the FBSP. This study considers

several financing alternatives in terms of sources and types of financing, times of cash provisions, interest rates, and repayment options. In addition, by considering several financing alternatives, the contractor earns higher profits by adjusting the work while paying less financing cost. It is also possible to prevent an extension in project duration compared to when only one financing alternative is considered, as was the case in all past studies. However, if the extension occurs, the proposed model finds the solution that yields higher profit and lower financing cost. Moreover, the proposed model provides optimum financing schedules, a related project schedule, and an analytical report. More importantly, the most common reason of failure in construction projects is financial problems. Therefore, improving financial arrangements is of particular importance. This research improves financing decisions, for contractors, and therefore the financial outcomes of the project. It should also be noted that this research makes an improvement to the CPM algorithm by using the activity-on-node method, topological sorting, and improved Dijkstra`s algorithm for solving CPM.

1.4

Organization of the Dissertation The study is conducted in four stages and is presented in eight chapters. After the

introduction in Chapter 1, a literature review is presented in Chapter 2, where project

12 scheduling problems and past studies are discussed. In Chapter 3, the financing cost and its effects on profit are discussed where alternative financing methods are reviewed. In Chapter 4, the Stage 1 model is proposed for completion in the normal duration. In Chapter 5, the Stage 2 models are proposed in which the time-cost tradeoff algorithm is added to the developed model in Stage 1, where the optimal result is obtained between the crash and normal project completion durations. In Chapter 6, Stage 3 models are proposed in which the variable activity start times are also considered to see whether the optimal results are improved by changing the start times of activities. In Chapter 7, the Stage 4 models are proposed based on the assumption that the contractor cannot obtain the necessary financing, and the project may need to be extended. In Chapter 8, conclusions are drawn and recommendations are made for future work.

13 CHAPTER 2 2. LITERATURE REVIEW In this chapter, first construction scheduling problems are discussed in the light of previous research. Then, each research study is reviewed based on its assumptions and methodology. The summary of the reviewed papers is presented in tabular format in Appendix A.

2.1

Introduction to Construction Scheduling Problems Construction scheduling involves sequencing activities while taking a number of

parameters into consideration. To have a realistic and executable schedule and to minimize the chances of schedule failure, a wide range of factors should be considered. The defects of CPM in scheduling construction projects were mentioned in Chapter 1, where it was also stated that the construction scheduling problem could be categorized into RPSP, TCTP, and FBSP. The resource project scheduling problem (RPSP) contains both resource leveling and resource-constrained scheduling. A plethora of methods and algorithms have been developed to address the deficiencies and also enhance the usefulness of CPM schedules. Despite the fact that money is of great significance, most of these techniques do not consider financing factors while modifying CPM schedules.

2.1.1

Resource Project Scheduling Problem (RPSP). In dealing with the resource

project scheduling problem (RPSP), practitioners have used two techniques, namely resource-constrained scheduling (also referred to as resource allocation) and resource leveling. In some construction projects, minimum consumption of resources could be

14 more desirable than minimum project duration (Birrell 1980). However, the early and late start and finish dates calculated with CPM are based on activity durations and the relationships between activities; this schedule produces the minimum project duration regardless of resource availability constraints (Abeyasinghe et al. 2001; Hong et al. 2001). There is a tradeoff between limited available resources and project duration. Therefore, effectively allocating the limited resources is critical in preparing a realistic schedule. Resource-constrained scheduling or resource allocation attempts to reschedule the activities subject to given resource capacities to minimize the extension of the project duration while allocating available resources effectively (Anagnostopoulos and Koulinas 2010; Hegazy 1999b). Resource leveling also plays a key role in RPSP to avoid the day-to-day fluctuation in resource demands, maintain an even flow of resource usage on each day, and satisfy the physical and technological limits of construction resources. Resource leveling is used to minimize variations among resource loading peaks while maintaining the original project duration (Anagnostopoulos and Koulinas 2010; Hegazy 1999b; Leu et al. 1999b). Contrary to constrained-resource scheduling where the duration of a project could be changed, in resource leveling, the project completion time of the original schedule remains fixed since the process is achieved by shifting noncritical activities within their available floats (Easa 1989). Research on the resource project scheduling problem (RPSP) has concentrated on resource leveling (i.e., Anagnostopoulos and Koulinas 2010; Easa 1989; Mattila and Abraham 1998; Xiong and Kuang, 2006), and on resource-constrained scheduling (i.e., Abeyasinghe et al. 2001; Berthold et al. 2010; Chen and Shahandashti 2009; Elmaghraby

15 1993; Gonçalves et al. 2008; Hong et al. 2001; Huang and Halpin 2000; Jaśkowski and Sobotka 2006; Jeffcoat and Bulfin 1993; Kim and Elis Jr 2010; Kim 2013; Leu et al. 1999a; Lova and Tormos 2001; Mika et al. 2008; Patterson and Huber 1974; Pritsker et al. 1969; Tsai and Chiu 1996; Thomas and Salhi 1998; Valls et al. 2003; Zhang et al. 2005; Zhang et al. 2006a; Zhang et al. 2006b; Zhang and Tam 2006). There are also publications that proposed models considering both resource leveling and resourceconstrained scheduling (i.e., Hegazy 1999b; Leu et al, 1999b; Pan et al. 2008). In all aforementioned papers, neither time-cost tradeoff nor payment and finance-based scheduling are considered.

2.1.2

Time-Cost Tradeoff Problem (TCTP). Construction scheduling is concerned

with optimally allocating resources and sequencing activities over the duration of project. The duration of the project is also another concern of the parties, specifically of the owner. Therefore, liquidated damages and/or late completion penalty is set by the owner to prevent delays. There may also be another contract clause about early completion bonus to encourage contractors to finish the project earlier, especially in the highway and transportation construction sector. However, the contractors should bear in mind that minimizing the project duration can be achieved by accelerating activities by allocating more resources to expedite the execution of activities, which invariably increases the direct cost (equipment and labor cost) for related activities and the total project, accordingly. On the other hand, reducing a project`s duration leads to decrease in indirect costs. Thus, schedulers can perform time-cost tradeoff analysis to find the most cost effective way in terms of total cost (direct + indirect costs) to complete a project;

16 however, the result does not represent the most profitable decision, unless liquidated damages and/or late completion penalty, and/or early completion bonus, and financing cost are also considered simultaneously. Using time-cost tradeoff, many researchers have developed diverse methods and algorithms to solve TCTP (i.e., Afshar et al. 2009; Ammar 2011; De et al. 1995; Elbeltagi et al. 2007; Ezeldin and Soliman 2009; Feng et al. 1997; Hui et al. 2013; Ipsilandis 2006; Ipsilandis 2007; Jiang and Zhu 2010; Lakshminarayanan et al. 2010; Li and Love 1997; Li et al. 1999; Liu et al. 1995; Leu et al. 2001; Ng and Zhang 2008; Que 2002; Robinson 1975; Zheng et al. 2004; Zheng et al. 2005). However, these research works did not consider resource availability constraints in their models. It should also be noted that some researchers attempted to integrate time-cost tradeoff and resource project scheduling in their models (i.e., Adeli and Karim 1997; Ahmed and Eldin 2004; Burns et al. 1996; Chen and Weng 2009; Dawood and Sriprasert 2006; Demeulemeester et al. 1996; Elazouni and Metwally 2007; Hegazy 1999a; Heinz and Beck 2012; Jalali and Shirvani 2011; Kandli and El-Rayes 2006; Leu and Yang 1999; Long and Ohsato 2009; Senouci and Al-Derham 2008; Shrivastava et al. 2012; Sunde and Lichtenberg 1995; Talbot 1982; Wiest 1967). Only few of these studies considered liquidated damages or late completion penalty (i.e., Ezeldin and Soliman 2009; Kandil and El-Rayes 2006; Ipsilandis 2006; Ipsilandis 2007; Wiest 1967), and both late completion penalty and early completion bonus (i.e., Elazouni and Metwally 2007; Hegazy 1999a; Jalali and Shirvani 2011) in their models while considering resource availability constraints. In addition, discounted cash flow is another factor which has been neglected by almost all of these researchers.

17 2.1.3

Finance-Based Scheduling Problem (FBSP). Although, RPSP and TCTP have

been researched extensively, payment scheduling and finance-based scheduling have received less attention from researchers even though cost and profit are most important. Traditional accounting practices and organizational constraints could be the reasons why project costs are not truly considered in making a schedule. The cost control objective is not considered in RPSP (Rusell 1970). In addition to the costs of activities, payment scheduling (Rusell 1970) and financing method could also affect the financial assessment. As it was mentioned in the Introduction, the development of FBSP was achieved over three generations of research in such a way that the first two generations are related to payment scheduling and the third generation is pertinent to finance-based scheduling. Below are discussed these generations and how they were developed separately. 1. Payment scheduling considering neither cash availability constraints nor financing costs 2. Payment scheduling considering cash availability constraints but no financing costs 3. Finance-based scheduling considering both cash availability constraints and financing costs Since the time value of money impacts achieving a realistic result, discounted cash flows should be considered while maximizing the net value of a cash flow. Moreover, because one of the objectives of the project is to maximize project Net Present Value (NPV), creating a schedule that balances the early receipt of interim payments with the postponement of specifically large disbursements, could be important (Smith-Daniels

18 et al. 1996). So, the first generation was introduced by Russell (1970) considering both the costs and payments received to schedule the project in such a manner as to maximize the net present value of cash flows (Russell 1970). Subsequently, Grinold (1972) and He et al. (2009) followed Russell`s work to solve PSP. However, none of them considered cash and resource availability constraints, and financing costs. However some other researchers (e.g., Chiu and Tsai 2002; Józefowska et al. 2002; Mika et al. 2005; Russell 1986) proposed models under resource availability constraints while still not considering cash constraints or financing costs. Usually at the end of the month, the contractor receives payment as a reimbursement for the accomplished works during that month. The owner withholds a portion of the payment as retainage. This retainage is kept by the owner until final completion and can be used by the owner to correct mistakes/omissions made by the contractor. Retainage causes cash constraints for the contractor and may limit the number and value of activities that can be scheduled simultaneously. As a result, NPV maximization that does not take cash availability constraints into consideration, is not realistic. In response to this problem, Doersch and Patterson (1977) studied the maximization of NPV by considering constraints in cash availability. Smith-Daniels et al. (1996) followed their work by presenting a heuristic approach. However, cash inflows to contractors occur at the end of fixed periods set forth by the owner, but Doersch and Patterson`s (1977) and Smith-Daniels et al.`s (1996) models assumed that cash inflows take place at the realization time of events during the project (Elazouni 2009). In addition, not considering financing costs is another shortcoming in these studies.

19 At the end of the month, the contractor uses cash mainly to cover expenses on direct costs of activities as well as indirect costs of the project during that month. Therefore, because of the retainage withheld by the owner, the indebtedness of the contractor increases gradually until the end of the project. Consequently, the contractor often needs additional funds from external sources to avert any deficits. Because most research has been conducted assuming availability of cash, these studies entirely discard financing costs, despite the fact that financing costs constitute a crucial concern. Financing costs must be considered and added to the model, otherwise the schedule cannot be executed. Even though a few researchers identified financing costs as a significant factor (e.g., Hegazy and Ersahin 2001; Li 1996), no research had been proposed that considered both cash availability constraints and financing costs until 2004. By adopting a mathematical method, Elazouni and Gab-Allah (2004) proposed a new model of finance-based scheduling for construction projects in such a way as to keep scheduled activities in balance with available cash while considering financing costs. Since this concept of scheduling is based on cash availability, it is referred to as financebased scheduling (Elazouni and Gab-Allah 2004). Finance-based scheduling tries to balance the activities` disbursements and the cash available through the owner`s payments and additional funds from external resources. The technique of finance-based scheduling does not allow negative cumulative balance of cash flows to violate the credit limit constraints (Elazouni and Gab-Allah 2004; Elazouni 2009). First of all, the method starts by shifting the early start time of each activity within its total float (TF) seeking an optimum solution which satisfies all constraints including available cash. If no feasible

20 solution exists, the current schedule is extended to get the most optimum result to satisfy the constraints and minimize the delay (Afshar and Fathi 2009). The typical way of obtaining additional cash in a construction project is to establish bank overdraft system with a specified credit limit (Ahuja 1976; Elazouni and Gab-Allah 2004). The credit limit specifies the maximum amount of cash the contractor is allowed to withdraw at any time during the project (Elazouni and Gab-Allah 2004). Also, bankers often require that the contractor deposits the owner`s progress payment into the credit line account for the purpose of reducing the indebtedness of the constructor and mitigating the banks` risk (Elazouni 2009). Accordingly, the value of the negative cumulative balance at the end of each period excluding interest is the total borrowed money. In addition, the financing costs at the end of each period are calculated by applying the specified interest rate to the negative cumulative balance (Elazouni 2009). The financing costs and the negative cumulative balance amount to the total indebtedness of the contractor to the bank at the end of each period. However if the contractor cannot pay the total indebtedness at the end of each period, the financing costs increase substantially due to compound interest. Considering a credit line account and calculating the financing costs and total indebtedness of contractors, Au and Hendrickson (1986) modeled the contractor`s cash inflows and outflows and determined the values at the end of a month. Mobilization costs, bond payment, direct costs, indirect costs, taxes, and financing costs are considered in this model as cash outflows. Advance payments, interim payments, and the final payment are considered as the cash inflows. The model determines the contractor`s indebtedness at the end of each period. Finally, the expected profit of the contractor is

21 determined after consecutively adding cash inflows and subtracting outflows throughout the project. Although Au and Hendrickson`s (1986) model suffers from the drawback of calculating approximate interest when it is used on intermediate balances instead of at the end of a period (Hendrickson and Au 2000), most finance-based modeling publications used this model on intermediate balances (i.e., Afshar and Fathi 2009; Ali and Elazouni 2009; Elazouni and Gab-Allah 2004; Elazouni and Metwally 2005; Elazouni and Metwally 2007; Elazouni 2009; Elazouni and Abido 2011; Fathi and Afshar 2010; Liu and Wang 2008; Liu and Wang 2009; Liu and Wang 2010). Elazouni and Gab-Allah (2004) used Au and Hendrickson`s (1986) model to propose finance-based scheduling for the first time. Although very few researchers have studied finance-based scheduling, most of them did not consider resource availability constraints (e.g., Afshar and Fathi 2009; Alghazi et al. 2012; Alghazi et al. 2013; Elazouni and Gab-Allah 2004; Elazouni and Metwally 2005; Elazouni 2009; Elazouni and Abido 2011; Fathi and Afshar 2010; Liu and Wang 2010). Whereas some researchers have developed solutions to the financebased scheduling problem by incorporating resources and time-cost tradeoff analysis in their models (e.g., Ali and Elazouni 2009; Elazouni and Metwally 2007; Liu and Wang 2008), not one of them considered financing optimization with different alternatives of financing. The line of credit method imposes limits on contractors and leads to a financing model that is quite rigid. According to Elazouni and Metwally (2005), bankers are always cautious when establishing overdraft accounts for contractors. As a result, a conflict is created between contractors and bankers. While contractors want larger credit limits to avoid extending the duration and increasing the cost of the project, bankers want to

22 minimize their risk of reserving big amounts of cash for a specific contractor (Elazouni and Metwally 2005). If the contractor has a good reputation and wants to increase the credit limit, sometimes bankers agree to raise the credit limit by increasing the interest rate, resulting in higher financing costs compared to a loan of comparable size (Lesonsky 2014). In Chapter 1 (Introduction), the reasons of why more financing alternatives should be considered, were discussed widely. Consequently, considering other sources and types of financing is essential. Managers in charge of finances must be able to obtain sufficient cash using different financial arrangements to reduce financing costs. Realistic direct costs, indirect costs, and payments are critical in finding the optimum financing cost, expected profit, and related schedule. In other words, the objective function of profit is the summation of total project completion cost (direct and indirect costs), total project completion payment, total financing cost, liquidated damages, and the bonus. In addition, the financing cost is dependent on the total project completion cost and payment. Therefore, considering unrealistic direct costs, indirect costs, and payments will lead to incorrect financing costs, profits, and a non-optimal schedule. First, realistic assumptions are discussed below, and then the unrealistic assumptions used in previous research studies and their impact on an optimum and realistic outcome are reviewed. First of all, the assumptions about costs and payments for finance-based scheduling are about the bidding time when the contractor wants to prepare a bid for the contract. In general, there is a trade-off between the time and a cost of a project. The change in indirect, direct, and total cost of a project (excluding financing costs) can be seen in Figure 2.1 for different project completion times. In Figure 2.1, three boundaries

23 of time-cost tradeoff are shown (i.e., crash point, normal point, and extended point). The crash point represents the most accelerated methods for all activities. The total cost is high at this point because more resources are used to accelerate the work. In the normal point, only normal methods are used in all activities. The total cost at this point represents the contractor`s bid. The time at this point is the contract time specified by the owner. The extended point represents the most decelerated methods for all activities. The total cost at this point is higher than the normal cost because of the indirect cost and the liquidated damages to be paid by the contractor for the delay (see Figure 2.2). When project completion time increases, the contractor uses fewer equipment and fewer workers to perform the work. Therefore, the direct cost goes down between the two boundaries of crash and normal points. However the direct cost goes down at a lower rate between normal point and extended point because the contractor uses fewer resources at a lower productivity. Although the direct cost changes between normal point and extended point, past finance-based scheduling publications consider that there is no change in direct cost. It should be noted that the concept of past finance-based scheduling models is not to change the resources allocated to activities and hence the duration of activities when the project is extended beyond the contract time (i.e., normal point). In other words, it is assumed in these studies that the extension time occurs when the maximum negative balance of cash flow is bigger than the amount of available cash to finance the project. Therefore, after the normal point, the project completion time is extended to reduce the maximum negative balance of cash flow and satisfy cash constraints. Thus, to extend the project completion time beyond the contract time (i.e., normal point), instead of changing the duration of activities, the available total float of activities increases to bring back the

24 maximum negative balance of cash flow below cash constraints while changing the start time of activities to minimize the financing cost. In this case, direct cost will remain constant regardless of how long completion takes. Disregarding the fact that direct cost will change between normal point and extended point will result in a different outcome. If one considers that the direct cost decreases if the project completion time is extended, the total cost of the project will be lower than the assumption which past finance-based scheduling publications have used.

Figure 2.1. Time-Cost Tradeoff Considering Realistic Assumptions for the Costs Figure 2.1 does not reflect liquidated damages. Since liquidated damages occur at the end of the project (if any), they do not affect the financing costs during the project; therefore, liquidated damages are not shown in Figure 2.1. Although liquidated damages do not affect financing costs during the project, they influence the profit or loss of the project; thus, they are considered in Figure 2.2.

25 As shown in Figure 2.1, the indirect cost of the project increases when the project completion time increases. Indirect cost consists of two parts, i.e., indirect cost of overheads and indirect cost of mobilization and bonding. The reason why the cost of mobilization and bonding should be calculated and added separately is that bonding and mobilization costs occur early in a project where some owners do not pay for these costs (Hinze 2012). Therefore, at the beginning of a project, the contractor should finance these costs to prevent any deficits. These costs can affect the negative balance of cash flow at the beginning of a project and increase the financing cost; thus, they should be considered individually in the cash outflow. Moreover, in reality, some indirect costs are constant, whereas most are variable (Hinze 2012). Therefore, according to Afshar and Fathi (2009), Fathi and Afshar (2010), and Peterson (2013), indirect cost of overheads consist of fixed overhead and variable overhead costs (Fathi and Afshar 2010; Peterson 2013). This is discussed in detail in Chapter 5. If one considers variable overhead costs in addition to fixed overhead costs, the indirect cost becomes non-linear (see Figure 2.1). This realistic assumption of indirect cost would change the amount of total cost (direct cost + indirect cost) and the financing costs where financing costs depend on the total cost. It is in the interest of the contractor to find the optimum schedule to acquire the highest profit by keeping the total cost down. However, financing costs should be considered in order to decide the optimum schedule. In Chapter 3, the behavior of financing cost is discussed. The assumptions made about the costs incurred by the contractor and the payments made by the owner are expected to change the calculation of

26 financing costs and consequently profit, which leads to the selection of different schedules that may not be realistic.

Figure 2.2. Optimum Decision Considering Realistic Assumption for Payment Most research on finance-based scheduling make different assumptions for calculating costs and payments. As seen in Table 2.1, these publications can be divided into five categories based on these assumptions. Below is discussed what these assumptions are and how these assumptions impact the optimum and realistic decision. As shown in Table 2.1, the publications in Category 1 consider just two boundaries, i.e., the normal point and the extended point. This model would not result in the optimum decision in terms of maximum profit since the model does not check before the normal point. As shown in Figure 2.3, these researchers considered indirect cost just as a constant cost per day. This assumption makes indirect cost linear and leads to an unrealistic total cost (i.e., direct cost + indirect cost). Also, researchers in Category 1 did not consider liquidated damages or late completion penalty which results in an unrealistic

27 profit or loss (if any). Moreover, these studies consider only a cost-plus contract in their research. This means that the owner reimburses all costs and pays a percentage of the cost as a fee to the contractor. Since the contractor`s fee is a percentage of cost, the contractor is motivated to jack up the cost to get a higher fee (see Figure 2.4). This type of contract is forbidden for public owners (i.e., local, state, and federal government) in many countries such as the US unless there is an emergency situation. This is why Category 1 models of finance-based scheduling cannot be used in public contracts. Also, the use of “cost + percentage of cost” contracts in Category 1 research changes the way financing cost is calculated and leads to unrealistic outcomes. As shown in Figure 2.4, regardless of the financing cost, the optimum decision is unrealistic and counterintuitive.

Figure 2.3. Time-Cost Tradeoff Considering the Assumptions Made in Category 1 Research

28

Indirect cost

-

Normal and extended points*

Fixed

Constant cost per day**

Yes

Cost + percentage of cost**

Yes

Yes

Crash, normal, and extended points

Variable

Percent of direct cost**

Yes

Cost + percentage of cost**

Yes

-


Fixed

Constant cost per day**

Yes

Unit price or lump sum

No**

-


Fixed

No**


Yes

No*

Crash, normal, and extended points

Variable

Constant cost per day + percent of direct cost Constant cost per day**

Yes


Elazouni (2009) 2

Elazouni and Metwally (2007) Ali and Elazouni (2009)

3

Elazouni and Metwally (2005) Elazouni and Abido (2011)

Contract type

Direct cost

No**

Separate cost of bond and mobilization

Boundaries

Elazouni and Gab-Allah (2004)

Early completion bonus

References

1

Liquidated damages and/or late completion penalty

Categories

Table 2.1. Classification of Finance-based Scheduling Publications Based on the Assumptions Made in Calculating Costs and Payments

Alghazi et al. (2012) Alghazi et al. (2013) 4

Afshar and Fathi (2009) Fathi and Afshar (2010)

5

Liu and Wang (2008)

* Incomplete model ** Unrealistic assumption

29

Figure 2.4. Optimum Decision Considering the Assumptions Made in Category 1 Research Category 2 research considers all three boundaries, i.e., crash, normal, and extended points. Unfortunately, these studies consider indirect cost to be a percentage of direct cost, quite an unrealistic assumption. As shown in Figure 2.5, both direct and indirect costs go down with increasing project completion time. As a result, the total cost goes down as well, an unrealistic and counterintuitive outcome. In addition to the unrealistic presentation of total cost, Category 2 research also assumes cost-plus contracts in their finance-based scheduling model. Because of this assumption, the optimum solution occurs when total cost is maximum. As shown in Figure 2.6, the crash point is the optimum point regardless of financing cost. This unrealistic result occurs because of the incorrect assumptions for both indirect cost and contract type.

30

Figure 2.5. Time-Cost Tradeoff Considering the Assumptions Made in Category 2 Research The research in Category 3 considers only two boundaries, i.e., the normal point and the extended point. Although, these researchers made realistic assumptions for payment, liquidated damages, bonding and mobilization costs, they made an unrealistic assumption for indirect cost. Since the calculation of financing cost is based on the total cost (direct + indirect costs), the selection of the optimum decision would also be unrealistic. In Category 4, studies do not consider bonding and mobilization as a separate cash outflow. These researchers made a realistic assumption for indirect cost, but they did not consider liquidated damages or late completion penalty that leads to an unrealistic result. Studies in Category 4 consider indirect cost as a constant daily cost in addition to the percentage of direct cost (i.e., daily cost + percentage of direct cost). The research in this category also considers only two boundaries, i.e., the normal point and the extended point which causes an incomplete model.

31

Figure 2.6. Optimum Decision Considering the Assumptions Made in Category 2 Research The study in Category 5 considers all three boundaries, i.e., crash, normal, and extended points, but unfortunately, it makes an unrealistic assumption for indirect cost. In addition, although a mobilization and bonding cost and late completion penalty are considered, disregarding a bonus may result in an incomplete model. Considering financing costs in construction scheduling results in estimating realistic costs, and project profit. Thus, this research proposes a sixth category whereby finance-based scheduling incorporates the optimization of financing alternatives. Financing alternatives are usually analyzed by financial managers independently of the construction schedule. Therefore, after considering all scheduling parameters, financing optimization based on different alternatives (expressed in terms of different sources and types of financing, times of cash provisions, interest rates, and repayment options) allows contractors to develop an optimum construction schedule. In RPSP, TCTP, and FBSP, it is critical to use appropriate optimization methods to augment the effectiveness and accuracy of the solution. In addition to developing

32 solutions to construction scheduling problems by using realistic assumptions, one needs to use optimization methods and algorithms. These methods can be classified into heuristic, mathematical, metaheuristic, constraint programming, and hybrid methods. The heuristic approach is the easiest and most practical one but there is no guarantee that the solution will be optimal. Mathematical programming including integer programming (IP), linear programming (LP), mixed integer linear programming (MILP), and dynamic programming (DP) is quite inflexible and results in the optimum but sometimes not feasible solution. The metaheuristic approach is based on evolutionary algorithms which are more flexible compared to mathematical programming and leads to a near optimal result, but there is no guarantee that the solution will be optimal. Finally, the most recent and powerful approach is a hybrid method which improves the quality of the solutions in terms of efficiency and computational time. A classification of construction scheduling research studies based on these optimization methods regarding different assumptions is shown in the table in Appendix (A). According to the information in the preceding sections, RPSP and TCTP have been sufficiently addressed by researchers over the years. The concern of this research is to minimize the financing cost and maximize the profit of the project. Therefore, the next sections of this chapter are devoted to reviewing the studies that have been conducted to solve FBSP or both FBSP and TCTP.

2.2

Heuristic Methods Heuristic methods are systematic approaches which are based on past experience.

The heuristic approach has the advantage of less computational time as compared to

33 mathematical methods. Heuristic methods adopt a practical approach by using manual means in which the results are not guaranteed to be optimal or perfect, but simple enough to be adopted to solve problems (Zhou et al. 2013). Even though this simplicity is a boon for large projects with many activities, a pool of possible solutions is not provided by heuristic methods since only one objective can be optimized with heuristic methods (Wiest 1967; Zhou et al. 2013). Inefficiencies in solving multi-objective scheduling problems constitute another deficiency of heuristic methods (Zhou et al. 2013). In the context of the payment scheduling problem (PSP), Russell (1986) proposed a heuristic model to maximize the net present value (NPV) of a project subject to schedule and resource constraints. Russell (1986) developed the problem of NPV maximization to include resource constraints while changing the start time of activities. Further, Chiu and Tsai (2002), proposed an efficient heuristic model to solve the resource-constrained problem and to maximize the NPV with regard to multiple projects where project delay penalty and early completion bonus are considered. Even though these studies (i.e., Chiu and Tsai 2002; Russell 1986) considered neither cash availability constraints nor financing cost, Smith-Daniels et al. (1996) further developed their models (i.e., Chiu and Tsai 2002; Russell 1986) of NPV to include cash availability constraints but not financing cost. In 2009, Elazouni proposed a new heuristic model to schedule multiple projects subject to considering both cash constraints and financing cost. Using this model, Elazouni (2009) was able to identify and rank possible schedules of activities, and select the best schedule in such a way as to minimize project delay while satisfying cash availability constraints and taking financing cost into consideration. Even though Elazouni (2009) considered both cash availability constraints and financing cost, since

34 the financing optimization using different alternatives is not considered, this model results in higher financing cost and a schedule that is not optimal because the credit limit is predetermined. Moreover, fixed activity durations in the extended period and no consideration of liquidated damages and/or late completion penalty, in considering only “cost + percentage of cost” contracts cause Elazouni`s (2009) model to generate unrealistic results (see Table 2.1 and Figures 2.3 and 2.4). Although these studies (i.e., Chiu and Tsai 2002; Elazouni 2009; Russell 1986; Smith-Daniels et al. 1996) focused on FBSP, they did not consider TCTP.

2.3

Mathematical Methods According to Zhou et al. (2013), mathematical methods, with regard to

construction scheduling problems, have been widely adopted because of their accuracy and guarantee of optimality. To use the mathematical method, an algebraic model including all objective functions and constraints should be formulated. While the efficiency and precision of mathematical methods increase their usefulness, formulating constraints and objective functions is time-consuming and difficult for construction schedulers who do not have enough mathematical expertise. Since few construction schedulers are trained to develop this type of formulation, especially for large projects, the application of mathematical methods to construction project scheduling has been limited to date (Zhou et al. 2013). A number of mathematical algorithms have been used to address construction scheduling problems, such as linear programming (LP), integer programming (IP), mixed integer programming (MIP), pure integer programming (PIP), and hybrid IP/LP

35 algorithm. LP is a mathematical method which assumes continuity of the solution region to optimize linear objective functions subject to linear equality and inequality constraints (Williams 2013; Zhou et al. 2013). If fractional solutions are not acceptable, then, the LP problem is transformed into an IP problem (Zhou et al. 2013). Integer programming falls into several classes: (1) pure integer programming (PIP) in which all variables are integer, (2) mixed integer programming (MIP) in which some variables are integer and some are real variables (i.e., fractional variables), and (3) zero-one integer programming in which all variables are zero-one (Williams 2013). According to Williams (2013), since construction scheduling problems involve networks, most of these problems can be modeled using LP or IP, but IP is not recommended except in very simple cases. Although IP has an attractive feature of modeling construction scheduling problems, reviewing past studies shows that IP models can be very difficult to solve and the computational difficulties of solving a complex problem can be very high. Therefore, there is another approach that is called constraint programming (CP) to solve problems that can be also solved by IP. The reason that CP makes problems easier to model and solve is because CP has representational advantages. In CP, a number of finite possible values are considered for each variable where constraints connect the possible combinations of values to make the constraints of CP a richer variety as compared to the linear constraints of integer programming. Although CP is easier to use for modeling and solving problems, the results may not be useful where one seeks an optimal solution (Williams 2013). In 1970, Russell introduced NPV concepts and mathematical programming for first time to determine event times that maximize the NPV, but with no consideration

36 given to cash availability and financing cost. The author proposed a network flow approach to the scheduling of a project by maximizing a nonlinear function subject to linear constraints. Similarly, Grinold (1972) developed Russell`s (1970) NPV model by suggesting two algorithms: (1) a fixed deadline algorithm to maximize NPV with a given project deadline and (2) a parametric algorithm to maximize the NPV for all possible project deadlines. Although Grinold (1972) tried to improve Russell`s (1970) NPV model, cash availability and financing cost are not considered in their model either. Afterwards, Elazouni and Gab-Allah (2004) introduced an IP-based finance-based scheduling model that considers both cash availability constraints and financing cost while minimizing project delay, but with no consideration given to financing optimization. Even though this model revises activity start times to satisfy cash availability constraints while minimizing project delay and financing cost, Elazouni and Gab-Allah`s (2004) model is unrealistic in that it does not treat mobilization and bonding cost separately, it does not consider liquidated damages, it assumes fixed activity durations in the extended period, and it considers only “cost + percentage of cost” contracts (see Table 2.1 and Figures 2.3 and 2.4). Moreover, the predetermined consideration of the credit limit results in a schedule that may not be optimal. It should be noted that TCTP is not considered in these studies (i.e., Elazouni and Gab-Allah 2004; Grinold 1972; Russell 1970). In 2008, Liu and Wang studied FBSP by using constraint programming while proposing their model in two scenarios. In the first scenario both RPSP and FBSP are considered in minimizing the net value of a cash flow that includes financing cost while there is a single resource limitation. In the second scenario, two types of resources and

37 resource combination selections are considered in the model alongside time-cost tradeoff. Although Liu and Wang (2008) developed finance-based scheduling models, these models suffer from deficiencies such as an unrealistic consideration of indirect cost (see Table 2.1), no consideration of financing optimization with different alternatives, and predetermined credit limits. In 2009, Liu and Wang focused on three different scenarios while using constraint programming to maximize project profit. The primary objective of their research was to maximize project profit. They considered repetitive activities, interruption time, crew availability, and financing cost in finding the optimum schedule. In the first scenario, the total interruption was set as zero to obtain maximum work continuity. In the second scenario, the project duration was set not to exceed an assigned duration. In the third scenario, both an assigned duration and a credit limit were considered as constraints. Subsequently, Liu and Wang (2010) adopted CP to solve multi project scheduling problems by maximizing project profit while considering financing cost. Both cash flow and financial requirements were considered in this research. Again, two scenarios were considered : (1) maximization of overall project profit where there is no credit limit and no due dates, and (2) maximization of overall project profit with credit limit and due dates. Even though more scenarios are considered in Liu and Wang`s (2009 and 2010) models, financing optimization with different alternatives and TCTP were not considered, and an unrealistic assumption was made about indirect cost.

2.4

Metaheuristic Methods Metaheuristic methods can be applied to a wide range of problems compared to

heuristic methods. Metaheuristic methods have been developed from the natural world

38 and behaviors that guide the search process to efficiently discover the search space by using search experience in order to find near-optimal solutions (Blum and Roli, 2003; Johnson 2008). However a candidate solution can be enhanced by using the metaheuristic methods and iterative computation while too many assumptions are not considered (Zhou et al. 2013), the algorithms are approximate and usually non-deterministic (Blum and Roli, 2003).

2.4.1

Genetic Algorithms (GAs). GAs have been the most popular method used to

address construction scheduling problems (Zhou et al. 2013). Genetic algorithms (GAs) are search algorithms that are based on natural selection and population genetics mechanisms to search for optimal solutions (Que 2002). GAs use a random but directed search for locating the globally optimal solution in which this characteristic is inspired by the process of natural evolution (Que 2002; Zhou et al. 2013). Based on its random searching feature, a wide range of optimization problems could be solved by using GAs while searching a larger solution space, especially in construction scheduling (Que 2002; Zhou et al. 2013). According to Zhou et al. (2013), these characteristics allow GAs to be one of the best methods to solve construction scheduling problems. Although GAs have been effectively used in construction scheduling problems, they have some deficiencies in their computation time, reliability, and practicality (Lee et al. 2015). Computation time can go up if the parent population is selected inappropriately (Zhou et al. 2013), and/or if the number of generations is specified arbitrarily, and/or if the values of GA parameters (i.e., population size, crossover percentage, mutation percentage, mutation probability, and stopping rule) are reset manually to perform successive iterations to enhance

39 reliability (Lee et al. 2015). In addition, according to Lee et al. (2015), the reliability of the GAs suffers if the GA is terminated after searching for only one cycle. The reliability of GAs can be increased if the GA runs using multiple sets of GA parameters without resetting the values of GA parameters manually. Therefore, to increase the reliability and practicality of GAs, the optimal values of GA parameters should be identified based on controlled experiments (Lee et al. 2015). Furthermore, the GA methods (i.e., parent selection method and crossover operation method) can also affect the reliability and computational time. Even though Lee et al. (2015) proposed a GA model to solve TCTP by eliminating the deficiencies of previous GA models, the process of identifying optimal GA methods were not considered in their model. In 2005, Elazouni and Metwally developed Elazouni and Gab-Allah`s (2004) model by using a genetic algorithm (GA) to overcome the shortcomings of integer programming where Elazouni and Gab-Allah (2004) could not sufficiently model all expenditures and income cash flows while extending the schedule to satisfy cash availability constraints and minimize project delay (Elazouni and Metwally 2005). In 2007, Elazouni and Metwally used genetic algorithms to expand finance-based scheduling by including time-cost tradeoff analysis into the finance-based scheduling problem to maximize project profit where Elazouni and Gab-Allah (2004) and Elazouni and Metwally (2005) did not consider. In 2009, Ali and Elazouni proposed a new finance-based scheduling model by using genetic algorithm for a project of repetitive non-serial activities to maximize the project profit while considering time-cost tradeoff analysis. Although they (i.e., Ali and Elazouni 2009; Elazouni and Metwally 2005; Elazouni and Metwally 2007) developed FBSP, their model suffered from deficiencies

40 including fixed of activity durations in the extended period, unrealistic indirect cost (see Table 2.1), no financing optimization with different alternatives, and predetermined credit limits. In addition, although Elazouni and Metwally (2007) and Ali and Elazouni (2009) performed time-cost tradeoff analysis, considering only “cost + percentage of cost” contracts resulted in an unrealistic financing cost and project profit (see Table 2.1 and Figures 2.5 and 2.6). To overcome the problem of assuming predetermined credit limit and then revise the initial schedule of the project accordingly, Afshar and Fathi (2009) and Fathi and Afshar (2010) proposed an elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) model with multiple objectives of minimizing financing cost, required credit, and project duration by using fuzzy-sets theory. Even though Afshar and Fathi (2009) and Fathi and Afshar (2010) developed their finance-based scheduling models by considering an undetermined credit limit and a correct representation of indirect cost, they did not consider liquidated damages and/or late completion penalty, they did not separate mobilization and bonding costs, and they assumed fixed activity durations in the extended period (see Table 2.1). in addition to these deficiencies, they did not consider time-cost tradeoff analysis, nor financing optimization with different alternatives. According to Jaśkowski and Sobotka (2006), evolutionary algorithms include evolution strategies, evolutionary programming, classifier systems genetic algorithms, and genetic programming. Evolutionary algorithms perform based on the rules of evolution of live organisms (Jaśkowski and Sobotka 2006). To develop the past models of FBSP, Elazouni and Abido (2011) took advantage of the EA algorithm and proposed a Strength Pareto Evolutionary Algorithm (SPEA) for multiple projects considering

41 multiple objectives where project profit maximization is an ultimate goal while the profit values of the individual projects conflict with each other. Afterwards, Alghazi et al. (2013) used a repair algorithm to improve the performance of past GA models with regard to FBSP in terms of the quality of solutions and computational time while maximizing profit. The results proved the superior performance of the repairedchromosome GA in terms of the computational cost and quality of solutions (Alghazi et al. 2013). The models developed by Alghazi et al. (2013) and Elazouni and Abido (2011) suffered from deficiencies including fixed activity durations in the extended period, unrealistic indirect cost (see Table 2.1), no financing optimization with different alternatives, and predetermined credit limits, and no time-cost tradeoff analysis.

2.4.2

Tabu Search (TS). According to Thomas and Salhi (1998), Tabu search (TS) is

designed to enhance the performance of local search by using memory structures and accepting infeasible solutions while driving the search away from local optima. This process starts from an initial solution which could be either feasible or infeasible. Then it uses a suitable neighborhood structure and a proper objective function evaluation. Tabu search employs intelligent uses of memory to iteratively move from solution 1 to solution 2 in the neighborhood of solution 1; the cycle continues until some stopping criterion is met. A proper data structure is needed to avoid recomputing the already compiled information especially in a large number of iterations (Thomas and Salhi 1998). Józefowska et al. (2002) applied a tabu search to solve the multi-mode resourceconstrained project scheduling problem in addition to considering time-cost tradeoff

42 analysis while maximizing the net present value of the project. However, neither cash availability constraints nor financing cost are considered in this research.

2.4.3

Simulated Annealing (SA). Simulated annealing (SA) is a variation of

neighborhood search and a computer simulation of the physical annealing process (Anagnostopoulos and Koulinas 2010; Jeffcoat and Bulfin 1993). The idea of simulated annealing was inspired by the annealing process in physics that involves the controlled cooling and heating of a material until a state of minimal energy is met (Anagnostopoulos and Koulinas 2010). SA searches for a local solution by modifying the current solution using a generation function (Alghazi et al. 2012). The benefit of SA is to prevent being trapped in local optima (Anagnostopoulos and Koulinas 2010). Mika et al. (2005) adopted both simulated annealing and tabu search to propose a model to solve the multi-mode resource-constrained project scheduling problem in addition to considering time-cost tradeoff analysis while maximizing net present value of the project considering four different payment models. In 2009, He et al. showed that SA performs better than TS, especially when the project scheduling problem is larger. It should be noted that in both research studies (i.e., He et al. 2009; Mika et al. 2005) neither financing cost nor cash availability constraints are considered.

2.4.4

Shuffled Frog-Leaping Algorithm (SFLA). The shuffled frog-leaping algorithm

(SFLA) is the most recent metaheuristic algorithm that is inspired by natural memetics. SFLA has been developed for solving combinatorial optimization problems by having an advantage in local search and global information exchange (Eusuff et al. 2006).

43 According to Alghazi et al. (2012), SFLA was designed by considering frogs that are partitioned into different memeplexes. The frogs are allowed to interact with each other to enhance their memes by sharing information. A meme is any kind of information that can pass from mind to mind. The search in the SFLA starts by selecting the populations of frogs randomly which are considered as the first solutions and then local search and shuffling processes continue until the entire search space is covered (Alghazi et al. 2012). For the first time, Alghazi et al. 2012 proposed a finance-based scheduling model using the shuffled frog-leaping algorithm (SFLA) while comparing the model against GA and SA meta-heuristics methods. The results prove that the computation time and the quality of solution obtained by the SFLA is better than those obtained by the GA and SA algorithms in a 210-activity project. However, fixed activity durations in the extended period, unrealistic indirect cost (see Table 2.1), no consideration of financing optimization with different alternatives, and predetermined credit limits result in a deficient model where time-cost tradeoff analysis is not considered.

2.5

Hybrid Methods According to Malek et al. (1989), the idea behind the design of hybrid algorithms

is to combine two or more different algorithms while solving the same problem. The purpose of hybrid methods is combining the strengths of the individual algorithms to: (1) increase the quality of the solutions, (2) decrease the computation time, and (3) effectively handle large scale problems (Malek et al. 1989). These advantages are obtained without creating new disadvantages if one uses hybrid algorithms (Malek et al. 1989). Hybrid algorithms can be categorized into: (1) hybrid mathematical algorithms,

44 (2) hybrid metaheuristic algorithms, and (3) hybrid mathematical and metaheuristic algorithms.

45 CHAPTER 3 3. FINANCING COST, METHODS AND THE EFFECTS ON PROJECT PROFIT In this chapter, first the behavior of financing cost and its effects on getting the maximum profit are discussed. Then, financing terms, methods, and their features that are available for the construction companies are reviewed.

3.1

Investigating the Behavior of Financing Cost and the Effects on the Project Profit As it was discussed in the Chapter 2, the minimum cost solution can be reached at

a point where all activities are scheduled with their normal durations, and all activities display their crashed durations. However, financing cost can change the optimum point. Therefore, the financing cost should be considered in the determination of the optimum schedule. On the other hand, since the total payment is based on the contract price, the payment will not change if methods of executing the activities are changed and project completion time is altered, unless the project completion time is extended because of owner-related issues such as extra work. As a result, the total payment to the contractor is considered to be constant for different project completion times. The parameters which affect financing cost are (1) the amount of required financing, (2) the required duration of keeping the money, and (3) the interest rate. Normally, the required financing depends on (1) the total cost incurred by the contractor, (2) the total payment received by the contractor, and (3) the schedule of construction activities for each project completion time.

46 Both duration and the amount of financing depend on the schedule of construction activities. According to Fathi and Afshar (2010), if the volume of work scheduled for execution in the early periods of the project is higher than the volume of work at the late periods, the need for the big portion of the required financing will come up early in the project (Fathi and Afshar 2010). Therefore, the length of time the borrowed money is kept for a big portion of the required financing is longer when activities are scheduled at their early start time compared to a schedule when activities are scheduled at their late start time; this situation (i.e., keeping a large sum of borrowed money) requires the contractor to pay more financing cost. On the other hand, if the contractor requires to pay the interest periodically (not at the end of the project), the required financing should also cover the periodic interest within the project. As a result, since early time schedule for activities imposes more interest at the early time of the project, the required financing increases to also cover more interest. Since the financing cost depends on required financing and since required financing depends on total cost and owner payments, it can be said that financing cost depends on total cost and owner payments of the project. Thus, to create a framework for the behavior of financing cost, the total cost and payment for different project completion times in addition to different interest rates and different schedules of construction activities should be considered simultaneously. As shown in Table 3.1, four cases can occur regarding the parameters that affect the financing cost and the optimum profit. Changes in the parameters change total financing cost, project profit, and the decision for an optimum schedule. Parameter combinations are limited to four cases. As it was mentioned earlier, the amount of total financing cost depends on the total cost

47 incurred by the contractor and payments made by the owner. While the total payment made by the owner is constant for different project completion times, the required financing increases or decreases if the total cost respectively increases or decreases. Whereas the required financing behaves according to total cost, total financing cost may not behave according to the total cost only, because of the effect of the required duration of keeping the money. If the curve of total cost is dipped, this means the influence of the magnitude of the required financing is more than the influence of time. Therefore, the curve of total financing cost moves similar to the curve of total cost (see Figures 3.1 and 3.2). In contrast, if the curve of total cost is flat, this means the influence of time is more than the influence of the magnitude of required financing. Therefore, the curve of total financing cost shows consistent increases as the project completion time increases (see Figures 3.3 and 3.4). The slope of the curve of total financing cost depends on the interest rate. If the interest rate is high, the slope of the financing cost curve is steep (see Figures 3.2 and 3.4).

Table 3.1. Different Cases of Total Project Profit Regarding Total Financing Cost Cases 1 2 3 4

Type of total cost distribution (excluding financing cost) dipped curve dipped curve flat curve flat curve

Interest rate low high low high

48

Figure 3.1. Time-Cost Tradeoff Considering Financing Cost for Case 1


49


Figure 3.4. Time-Cost Tradeoff Considering Financing Cost for Case 4 When total cost includes financing cost, one can find a point of maximum profit. Since the total owner payment is based on the contract price and does not change with changing project completion time, the maximum profit is the point where the total cost including financing cost is minimum (see Figures 3.5 and 3.7). However, as shown in Figures 3.6 and 3.8, there is no guarantee that the contractor achieves profit at this point,

50 because if the interest rate is high, he/she may get a loss. In addition, as shown in Figures 3.5 and 3.7, even if the contractor completes the project at its normal time, he/she may get a loss due to the amount of financing cost for the project. These Figures (i.e., Figures 3.5, 3.6, 3.7, and 3.8) prove the necessity of incorporating financing cost while calculating the anticipated profit in the project. Disregarding the financing cost or even optimal financing may result in generating loss regardless of which schedule is selected. Hence, it is vital for the contractor to consider the financing cost based on optimized financing before making an optimum decision for the schedule.

Figure 3.5. Optimum Decision Considering Financing Cost and Payment for Case 1

51

Figure 3.6. Optimum Decision Considering Financing Cost and Payment for Case 2

Figure 3.7. Optimum Decision Considering Financing Cost and Payment for Case 3 It should also be noted that if the curve of total cost excluding financing cost is very dipped, the financing cost pushes the optimum decision toward the optimum point of total cost excluding financing cost (see Figures 3.1 and 3.5). On the other hand, if the curve of total cost excluding financing cost is very flat, the financing cost pushes the optimum decision toward the crash point (see Figures 3.3 and 3.7). Therefore, if the

52 contractor has sufficient credit, there is no chance that an optimum profit is achieved near the normal point or after that.

Figure 3.8. Optimum Decision Considering Financing Cost and Payment for Case 4 The reason why different schedules were not considered in the four cases presented in Figures 3.1 to 3.8 is because different schedules may change the magnitude of the financing cost, but not the overall trend of financing cost. Fathi and Afshar`s (2010) calculation of the financing cost proves that increased total cost (excluding financing cost) results in an increase in the slope of the overall trendline of total financing cost.

3.2

Reviewing Financing Terms and Methods for Construction Companies Financial institutions like to offer a financing alternative as well as a package of

several alternatives including checking accounts, savings accounts, lines of credits, and short and long-term loans (Peterson 2013). Contractors prefer to obtain financing from a

53 single source. Regardless of whether the financing is obtained from one or multiple sources, contractors are inclined to borrow money in a way that best achieves their overall goals. Therefore, the selection of the optimal financing schedule out of the alternatives offered by available financial institutions is critical for the profitable existence of construction firms; thus, it is in the interest of contractors to consider a combination of alternatives offered by financial institutions. One of the most remarkable problems for construction companies is to maintain liquidity to support day-to-day activities (Nesan 2012). Lack of credit to perform day-today activities has resulted in many construction companies going out of business rather than the lack of technical capability to execute the project (Singh and Lokanathan 1992). Other construction companies have gone bankrupt due to poor cash flow prediction and being unable to anticipate the required financing for the project, which results in seeking unnecessary credit from banks or other lenders. Therefore, difficulty in predicting and obtaining the required financing has caused many construction contractors to fail more often than businesses in other industries, and resulted in the construction industry being classified as a high-risk industry (Strischek 2005). Borrowing a large amount of funds, especially from one single source of financing is difficult for contractors. This is even more difficult for small contractors since there are limited alternatives offered by banks or other lenders in the absence of sufficient collateral (Nesan 2012). Normally, bankers or other lenders want to see the cash flow projection and make sure that the cash generated by the operations is enough to pay off the debt along with its interest (Peterson 2013; Strischek 2005). In addition, it is important for a construction company to specify the required financing rather than leaving the lending institution guessing as to what credit is

54 required (Peterson 2013). To solve the problems of predicting and obtaining enough credit from banks or other lenders, first the contractor should explore more sources and methods of financing, and second the contractor should secure the necessary credit by presenting to the lenders a reliable cash flow prediction, including financing cost. In addition to reliable cash flow predictions, some banks may also require a collateral from contractors as a guarantee for the repayment. According to ProfitCrew and Andrews Hooper Pavlik PLC (2010), bankers often ask contractors to pledge personal assets in support of the borrowed money. Before a bank agrees to provide credit for a contractor, the bank often considers both financial and non-financial factors when deciding the amount of credit to be issued to the contractor. In its financial analysis, a bank will look at four main factors, namely profitability, liquidity, solvency, and leverage. As liquidity, the contractor should be able to show sufficient cash to cover at least four weeks of incomes and one month of trade and other payables. Regarding solvency, there is a difference between liquidity and solvency. Solvency is the ability to meet long-term obligations, whereas liquidity is the ability to meet short-term obligations. The bankers or other lenders will examine the ability of the contractors to pay both debt and equity holders. Therefore, the cash flow should predict enough funds to cover principal and interest payments, capital expenditures, and dividends. In addition, bankers and other lenders are cautious of lending a large credit to contractors that are highly leveraged with heavy short-term debt (ProfitCrew and Andrews Hooper Pavlik PLC 2010). In the end, after considering both financial and non-financial factors, a banker’s credit analysis determines the contractor’s ability to repay the required credit with its

55 interest. To reduce the risk of loss, and to enhance the ability of monitoring the financial performance of contractors, some terms may be included in the agreement, such as including a minimum current ratio (Equation 3.1), working capital (Equation 3.2), and tangible net worth (Equation 3.3) (ProfitCrew and Andrews Hooper Pavlik PLC 2010).

Current Ratio = Current Assets / Current Liabilities

(3.1)

Working Capital = Current Assets - Current Liabilities

(3.2)

Tangible Net Worth = Total Assets – Liabilities – Intangible Assets

(3.3)

To incorporate the financing cost into the cash flow, first the contractors should identify the potential banks and lenders, and then based on their terms of interest rate, repayment options, and credit limit try to predict project cash flow including financing cost. It should be noted that choosing banks and potential lenders necessitates the contractors to know about alternative of financing methods and their differences. About the methods of financing for construction, there could be both long and short term aspects. In the long term, financing should cover the mobilization, bonding, and some other costs which occur at the beginning of the project. In the short term, financing should cover the negative cash flow which occurs due to retainage and late payments of the owner. While line of credit is the most common form of construction financing for small or medium size projects, other forms include business credit cards, loans, equipment financing, trade credit, refinancing, and personal savings (Au and Hendrickson 1986; Fathi and Afshar 2010; Hendrickson and Au 2000; Nesan 2012). These mechanisms

56 depending on which one is selected can be collateral intensive (secured) or unsecured. Methods of financing are discussed in the following sections.

3.2.1

Line of Credit (Overdraft Accounts). A line of credit is flexible financing

obtained from a bank or other alternative lender. The line of credit is a limited amount of funds with two options of repaying, immediately or over a pre-specified period of time. With a line of credit, the contractor can access the specified credit as needed at any time, as long as he/she does not exceed the maximum set in the agreement (Peterson 2013; Prakash 2015a; Simpson 2015). A line of credit charges interest as soon as money is borrowed. However, some banks charge interest even for unused credit (Elazouni and Metwally 2005; Peterson 2013). In addition, lenders may charge the contractor drawing fees, maintenance fees, commitment fees, and other fees for a line of credit (Peterson 2013; Prakash 2015a; Simpson 2015). Moreover, although lines of credit are considered to be lower-risk methods of financing for short periods of time, compared to loans, they can be more complicated since money can be drawn and repaid on an unscheduled basis, without knowing how much money will be borrowed over the life of the line of credit (Peterson 2013; Simpson 2015). The line of credit can be one of the ideal forms of meeting short-term needs. When the contractor is on an uncertain schedule and needs ongoing cash procurements, this may be a good option. This type of financing should not be used for long-term financing (Peterson 2013; Lesonsky 2014; Simpson 2015).

57 In most cases, the interest on a line of credit is not tax deductible (Simpson 2015). In addition, a line of credit may have a variable interest rate which means that the total amount to be paid back will change over time with market conditions (Peterson 2013; Prakash 2015a). This line of credit can be renewed as long as the contractor is in good standing (Prakash 2015a). The line of credit can be used as overdraft protection when linked to a business account with a maximum overdraft limit imposed. The features of a line of credit can be summarized as follows: 

Needs approval (harder to qualify compared to loans) (Prakash 2015a; Simpson 2015)



In most cases unsecured (Prakash 2015a; Simpson 2015)



Lower interest rate compared to unsecured loans (Prakash 2015a; Simpson 2015)



Typically variable interest rate (changes with the market) (Peterson 2013; Prakash 2015a)



Maintenance, transaction, commitment fee, and other fees (some but not all banks and lenders) (Peterson 2013; Prakash 2015a; Simpson 2015)



In some cases, charge for an unused credit (Elazouni and Metwally 2005; Peterson 2013)



Different credit limits (higher credit limit results in higher interest rate) (Elazouni and Metwally 2005)



Suitable for short-term, ongoing, and unexpected needs (Peterson 2013)



Repayment immediately or over a pre-specified period of time (Prakash 2015a)



In most cases monthly interest payment (Peterson 2013; Prakash 2015a; Simpson 2015)

58 3.2.2

Business Credit Card. A credit card for use by a business rather than for

personal use. Banks try to entice business customers by offering them some benefits which could be different from one to another. For instance, some business credit cards offer cash back on purchases from suppliers with which contractors are likely to interact frequently. As an example, Ink Cash business credit card issued by Chase bank offers 5% cash back, 0% introductory annual percentage rate (APR) for 12 months (13.5%-20% variable APR afterwards) on purchases and balance transfers, rewards and incentives, and no annual fee. In addition, in contrast to a personal credit card, business credit limits tend to be higher. However business credit cards can also have some disadvantages. If the bill is not paid in full within the grace period (the period with no late fees, typically 15 days up to a month), the interest is charged in the same manner as lines of credit. Therefore, if a business needs to borrow money for more than the grace period, it should consider another source of financing (Adams 2008; Peterson 2013). Although business credit cards typically have a lower interest rate compared to personal credit cards, they are generally an expensive financing option compared with other types of financing such as loans or lines of credit. Particularly, a secured loan will probably have a lower interest rate since it requires a collateral, which decreases the risk of the bank (Adams 2008). The features of business credit cards can be summarized as follows (Adams 2008): 

Needs approval (Easier to qualify than loans or lines of credit)



Unsecured (usually needs personal guaranties instead of a collateral)



Higher interest rate compared to loans or lines of credit



Variable interest rate (changes with the market)

59 

Annual fees (some but not all banks), rewards and incentives



Different credit limits



Suitable for short-term needs, quick access, and repaying within grace period



Repayment before the statement comes



No interest payment within grace period, afterwards compound interest for late payment



Less protection

3.2.3

Loans. With a business term loan, a lump sum of money is borrowed either once

or in a series of loans with a negotiated schedule of repayments of principal and interest (Au and Hendrickson 1986). Typically, the amount and time of repayments are specified in the loan agreement (Hendrickson and Au 2000). The advantages of loans are different fixed repayment periods (unlike lines of credit) with fixed (typically) or variable interest rates, secured (collateralized) with lower interest rates as compared to line of credits or unsecured with easy and quick access to the cash (Lesonsky 2014; Peterson 2013; Prakash 2015a). If a loan has fixed interest rate and fixed repayment periods, then the monthly payments are the same until the loan is paid off which makes scheduling and managing the financing easier for contractors (Prakash 2015a). In many cases, it is easier to qualify for a loan than for a line of credit since the lines of credit can be unsecured and are riskier for banks and other lenders (Prakash 2015a). The longer contractors have been in business, the easier it is to get a loan, as the bank can better check the records (Lesonsky 2014). In addition, contractors who are engaged in large projects often own large assets; therefore, they can use their assets as

60 collateral and make use of loans that have lower interest charges than lines of credit (Hendrickson and Au 2000). According to Peterson (2013), a third-party guarantee is another common provision in obtaining a loan. In this method, the assets of a company or person are used to guarantee the loan under a third-party guarantee. This provision is common for a small company or a company without a good credit history. However, a third-party guarantee should be avoided where possible since assets other than the company`s assets are at risk (Peterson 2013). Loans can be obtained for periods of three months to twenty years, and unlike lines of credit that are usually renewed every 6 months to 2 years (as long as the contractors are in good standing), a term loan is fixed for the specified period (Lesonsky 2014). Typically, loans can be divided into short term loans and long term loans. Unlike long term loans, which commonly have repayment schedules of a year or more than a year, a short term loan should be used to finance short-term financial needs and must be repaid in a much shorter period in one year or less (Peterson 2013). Even though financial institutions are normally willing to lend money for a shorter period of time, typically, long-term loans have been used to finance large projects (Kramer and Fusaro 2010). The features of a loan can be summarized as follows: 

Needs approval (Easier to qualify as compared to line of credit as long as one has acceptable collateral) (Prakash 2015a)



In most cases, secured (Lesonsky 2014; Peterson 2013; Prakash 2015a)



Lower interest rate for secured loans as compared to lines of credit (Hendrickson and Au 2000; Simpson 2015)



Typically fixed interest rate (Peterson 2013; Prakash 2015a; Simpson 2015)

61 

No maintenance and no transaction fees but other fees such as application fees, processing fees, courier fees, and other title charges known as closing costs (some but not all banks and lenders) (Peterson 2013; Prakash 2015a)



Different credit limits



Suitable for longer period and scheduled needs and repayments (Lesonsky 2014; Peterson 2013)



Prearranged and negotiated series of loans (Au and Hendrickson 1986)



Prearranged and negotiated repayment schedule of principal and interest payment but typically monthly interest payment with the repayment of principal at the end of period (Au and Hendrickson 1986; Lesonsky 2014)

3.2.4

Equipment Financing. There are both banks and equipment finance companies

that offer special programs to contractors. There are two types of equipment financing: equipment loans, and equipment leasing. In the equipment loans, the equipment serves as collateral and usually contractors do not need any other collateral to secure the loan. Most equipment loans usually are preapproved and fixed-rate term loans with the same monthly and repayment schedule. However, the choice of variable rates is also available. Equipment loans constitute a lowrisk method of financing for banks. Banks usually approve contractors within hours for 100% financing with no down payment, and with the advantage of tax benefits and financing of soft costs up to 25% which may be covered in the financing (i.e., sales tax, shipping, software, training, maintenance and installation) (Chase Bank 2016.; Equipment Leasing and Finance Association 2016). Most equipment loans allow

62 contractors to deduct the monthly payments from their taxes as an operating expense (U.S. Bank 2016.). In addition, payments can be scheduled to help a business to forecast its monthly cash flow more accurately. The payments can also be scheduled to suit the seasonal needs of the contractors such as monthly, quarterly, semi-annually or annually (Chase Bank 2016; Equipment Leasing and Finance Association 2016; U.S. Bank 2016). Moreover, multiple pieces of equipment can be financed with one application. Many banks allow the contractors to make multiple purchases within 12-2 years and lock the interest rate and terms at the time each purchase is completed (U.S. Bank 2016). These offers make this type of financing safe and flexible for contractors and low risk for banks due to the collateral. The APR for equipment loans can vary from 7% to 30% for the expected life of the equipment. The APR depends on the client`s banking relationship, credit history, collateral, and the term of the loan (Chase Bank 2016; Equipment Leasing and Finance Association 2016; U.S. Bank 2016). The features of an equipment loan can be summarized as follows (Chase Bank 2016; U.S. Bank 2016): 

Needs approval (easier to qualify compared to loans and lines of credit)



Secured



Varying interest rate



100% of financing with no down payment and up to 25% of soft costs



Prearranged schedule of payments



Financing of multiple pieces of equipment in one application



Multiple purchases within the period



Tax benefits

63 3.2.5

Trade Credit. This type of financing is an agreement between a contractor and

suppliers or subcontractors. According to Peterson ( 2013), this type of financing can be used whenever there is a delay between the delivery of services and/or materials and the contractor`s payment for services received or materials supplied. Even though a trade credit is given for specific periods, some suppliers or subcontractors may offer a discount if construction companies pays their bills early (Peterson 2013). This type of financing may void discounts offered by suppliers or subcontractors. For instance, if a contractor pays within 15 days of the purchase, the contractor might get 3% discount, whereas if the payment is received in 60 days, the contractor might pay 3% more on the price of purchased goods. This type of financing is short term, and allows the contractor to use the freed up cash for other purposes.

3.2.6

Refinancing. It is the replacement of an older loan with another loan under better

terms, usually of the same size, and using the same collateral. Refinancing of debts could have some advantages for a contractor. The significant advantage of refinancing is that it allows a contractor to refinance at intermediate stages of a project to save interest charges. In other words, the agreement allows a contractor to refinance at a lower interest rate if the borrowing agreement is made during a period of high interest rate (Hendrickson and Au 2000).

64 CHAPTER 4 4. FIRST STAGE OF THE RESEARCH In the first stage, the proposed model optimizes financing considering a work schedule prepared with normal activity durations. The idea at this stage is not to extend the schedule, assuming that there is a sufficient amount of money to perform the project by optimizing financing alternatives. Therefore, extending the project duration is not considered in this stage. Less financing cost and more profit will be obtained by using the proposed model compared to all models developed in past research, without a need to extend the schedule and change the start time of activities as was the case in all past studies. Therefore, this model not only maximizes the profit of the contractor, but also avoids liquidated damages and does not harm the contractor`s business image. It also reduces the risk of ending up with an all critical time schedule typically caused by using total float to satisfy cash availability constraints. In this chapter, first, the methodology and computational process of the first stage model are discussed. Then, the process and equations are presented in detail. Subsequently, the model is tested in three different financing cases to prove the enhancement compared to past research. Finally, sensitivity analysis is preformed and a conclusion is drawn.

4.1

Methodology and Computational Process of the First Stage Model Integrating financing optimization and construction scheduling begins with

creating a time-based schedule. It then goes to financing optimization to create the optimal financing schedule. In this stage, the durations of activities are taken as

65 deterministic inputs. In addition, the early start times of the activities are taken as the actual start times of activities. This information is utilized to prepare the cash flow forecast. Then, financing optimization is performed to find the optimum financing cost. Ultimately, the schedules of financing and repayments are generated based on the optimum financing methods and considering the project schedule. The proposed computational method that leads to optimizing financing is presented in Figure 4.1. It is executed by an automated system using MATLAB 2013a.

4.2

Project Schedule Creation for the First Stage Model A project schedule should be in place to create a cash flow forecast that can be

used for optimizing financing cost. Whereas the working day is the most frequently used time unit to estimate activity durations (Elazouni 2009), the use of weeks may be more appropriate in projects with longer duration (Hinze 2012). Optimizing financing cost is quite important especially in projects with a longer duration. The cash flow forecast in these longer projects is created in weeks. This research assumes the working week as the time unit of activity duration. The working week is used in time-cost tradeoff analysis because the cash flow forecast is created in weeks. CPM is a scheduling technique that is widely used. It can handle thousands of activities efficiently. For creating a cash flow schedule, the start and finish times of activities are needed. CPM calculates the early start, early finish, late start, and late finish of activities.

66

1. Reading project schedule information 2. Topological sorting of activities

4.1 Calculating early starts [ESj] of activities 5.1 Calculating late finishes [LF i] of activities

4.2 Calculating early finishes [EFj] of activities

5.2 Calculating late starts [LS i] of activities

5.3 Calculating total floats [TF i] of activities

Project schedule creation

3. Saving schedule data in structure arrays

6. Computing start times and finish times [STs, FTs] and project duration [W, M] 7. Inputting cost data [OF, OV, OM, OMP, OB] 8.2 Calculating [FOC]

8.3 Calculating [VOC]

8.4 Calculating [MOB]

8.7 Computing contract bid price [CBP] and bid price factor [BPF]

8.5 Calculating [MP]

8.6 Calculating [BD]

8.8 Computing bid price factor [BPF]

9. Inputting contract terms [OAP, R, LS, LP, LR] and computing final payment time [TF] 10. Calculating cash outflow [Et] and cash inflow [Pt]

Creation of project cash flow

8.1 Calculating [DC]

11. Computing cumulative net balance of cash flow excluding financing flow [Nt] 12. Reading financing data [financing alternatives, APRs, interest payment times, total credit limits, credit limits in each period] 13. Setting up financing inflow [Bt] and financing outflows [Rt, Ft] 14. Setting up net financing flow [NFt] 15. Setting up constraints of financing

17. Setting up constraints of cumulative net balance of cash flow including financing flow [N 't] 18. Optimizing financing cost using linear programming 19. Generating the table of financing parameters [Bt, Rt, Ft, NFt, NFCt, N 't] 20. Computing total required financing considering each financing alternative 21. Generating financing inflow schedule (borrowed money) 22. Generating financing outflow schedule (repaid money including interest)

Figure 4.1. First Stage Model Algorithm

Optimizing financing

16. Inputting minimum cumulative net balance of cash flow excluding financing flow [MN]

67 The first step, Step 1 in Figure 4.1 is reading the CPM data. The data consists of activity ID, name, predecessors, duration, and direct cost. Then, the CPM algorithm is used to analyze the network and calculate the start and finish times of activities. Most researchers use activity-on-arrow networks (e.g., Anderson and Hales 1986; Shankar and Sireesha 2010), whereas activity-on-node networks were used in this research to create a cash flow forecast because of the popularity of activity-on-node over activity-on-arrow. The conventional CPM algorithm uses linear programming or dynamic programming to analyze the network and to find the critical path. These mathematical approaches can be inefficient (Hendrickson and Au 2000; Shankar and Sireesha 2010) whereas the modified Dijkstra`s algorithm can be applied to solve CPM more efficiently (Shankar and Sireesha 2010). It is well known that Dijkstra`s algorithm is a method used for finding the shortest path, but Shankar and Sireesha (2010) proposed a modified Dijkstra`s algorithm for CPM. The CPM is the longest path problem which is NP-Hard for a general network, but because the schedule network is a directed acyclic graph, this problem has a linear time solution. Therefore, the idea of linear time solution for shortest path can be used for CPM too. Even though Shankar and Sireesha (2010) proposed a modified Dijkstra`s algorithm for CPM, a CPM algorithm is proposed in this study that adopts the activity-on-node method, topological sorting, and an improved Dijkstra`s algorithm for CPM. The difference between Shankar and Sireesha`s (2010) “modified” algorithm and the proposed “improved” Dijkstra`s algorithm for CPM is that Shankar and Sireesha`s (2010) algorithm uses every arrow regardless of whether activities are connected or not, whereas the proposed method uses the STRUCTURE array of MATLAB to avoid calculating unnecessary paths and just compute those that are jointed.

68 For example, in Shankar and Sireesha`s (2010) modified algorithm, the connection between activity i and j is checked even if there is no connection between them, and if i and j are not connected by an arrow, then Di   . Thus, for a project with many activities, the computational time could be high. This research adopts the STRUCTURE array of MATLAB to avoid checking those paths that are not connected in order to reduce the computational time. In Step 2 (Figure 4.1), the activities are numbered by performing topological sorting of the activity-on-node network. Numbers are assigned to n activities by assigning one to the starting activity, and the next number to any unnumbered activity whose predecessors are numbered, and repeat this until all activities are numbered. In other words, v1  1, v2  2, v3  3, , vn  n where n is the last activity. In Step 3 (Figure 4.1), the STRUCTURE array of MATBLAB is used to save the connections between activities based on the schedule data provided by the user. Identifying the connections allows the system to avoid calculating unnecessary paths for those activities that are not connected. In Step 4 (Figure 4.1), the forward pass is performed to find the early start times of activities, denoted by ES j . A permanent label of ES1  0 is assigned to the first activity on node V1  1 and a temporary label equal to zero to the remaining n  1 activities. The forward pass is performed for all nodes j that are not yet permanently labeled and have nodes i as the predecessor activities. A new temporary label of ES j for j  2, 3, 4, , n (where n is the last activity) is computed using Equation 4.1 for those

activities that are connected.

69

ES j  max[label of j, (label of i  Di )], where ES1  0 and i  j

(4.1)

where i is the predecessor activity permanently labeled, and Di is its duration. The next activity receives a permanent label when ES j is computed considering all predecessor activities i that are connected to activity j . Then, this process is repeated until the end node Vn  n gets a permanent label. Therefore, ES1  0 and the permanent values of ESj are the earliest start times of activities. Then, the early finish times of activities, denoted by EFj , are computed using Equation 4.2.

EFj  ES j  D j ; j  1, 2, 3, ..., n

(4.2)

In Step 5 (Figure 4.1), the backward pass is performed to find the late finish times of activities, denoted by LFi . A permanent label of LFn  EFn is assigned to the last activity on node Vn  n and a temporary label equal to EFn , to the remaining n  1 activities. The backward pass is performed for all nodes i that are not yet permanently labeled and have nodes j as the successor activities. A new temporary label of LFi for i  n  1, n  2, n  3, , 1 (where n is the last activity) is computed using Equation 4.3

for those activities that are connected.

LFi  min[label of i, (label of j - D j )], where LFn  EFn and i  j

(4.3)

70 where j is the successor activity permanently labeled, and D j is its duration. The next activity receives a permanent label when LFi is computed considering all successor activities j that are connected to activity i . Then, this process is repeated until the starting node V1  1 gets a permanent label. Therefore, LFn  EFn and the permanent values of LSi are the latest finish times of activities. Then, the late start times of activities denoted by LSi are computed using Equation 4.4.

LSi  LFi  Di ; i  n, n  1, n  2, ..., 1

(4.4)

Afterwards, the total floats of activities denoted by TFi are computed using Equation 4.5.

TFi  LSi  ESi ; i  n, n  1, n  2, ..., 1

(4.5)

A cash flow forecast is prepared by first preparing a project schedule and then assigning the estimated direct costs to each of the activities (Peterson 2013). To create the cash flow forecast, the start and finish times of activities are needed. However it is recommended to compare the early and late times of activities to find which schedule satisfies the cash constraints and minimize the financing costs of the project (Russell 1970). As it was stated in Chapter 3, this comparison may result in a highly critical work schedule for the project. Therefore, as shown in Equations 4.6 and 4.7, this research

71 considers only the early start and finish times to minimize the risk of a critical time schedule. In Step 6 (Figure 4.1), the start times denoted by ST j are computed by Equation 4.6, and the finish times denoted by FTj are computed by Equation 4.7. At the end, the project duration is computed in weeks and months.

ST j  ES j ; j  1, 2, 3, ..., n

(4.6)

FT j  EFj ; j  1, 2, 3, ..., n

(4.7)

4.3

Creation of Project Cash Flow Forecast for the First Stage Model The cash flow forecast is a significant and sensitive tool that is used to compute

financing costs and profit. The creation of a cash flow forecast necessitates a separate analysis of the cash outflows (expenditures) and cash inflows (incomes) during the project. As discussed in Chapters 2 and 3, the frequency and magnitude of the costs incurred by the contractor and of the payments made by the owner affect the contractor`s financing costs and profit. Therefore, the cash outflows and cash inflows should be analyzed separately.

4.3.1

Project Cash Outflows. The total cost of a project, excluding financing cost, is

divided into direct cost and indirect cost. The direct cost of the project is the sum of the costs incurred by the contractor to pay for materials, labor, equipment, and subcontractors (Fathi and Afshar 2010; Hinze 2012; Peterson 2013). According to Hinze (2012), these costs are highly dependent on the scope of the project. Materials are generally purchased through purchase orders from suppliers. The arrangements for the payment to purchase

72 materials vary from supplier to supplier. In most cases, the cost of materials is expected to be paid within a month from the time of delivery. The cost of labor could be approximately one-third of the total cost of a project. The timing of the payments is an important factor in cash flow analysis, and can be weekly, biweekly or monthly. With regard to equipment examples, equipment can be leased/rented or owned by the contractor, but this does not significantly change cash flow analysis. It is common that the payments of leased or rented equipment are made monthly unless the equipment is rented for shorter periods (Hinze 2012; Peterson 2013). Regarding the cost of subcontractors, some work items are subcontracted when the tasks need specialized skills. The payments to the subcontractors are typically made on a monthly basis or when the contractor gets paid by the owner (Hinze 2012; Peterson 2013). As mentioned earlier, duration is expressed in work weeks in this research. The direct cost of activities is calculated in terms of $/week. However, the contractor pays at the end of each month. The net direct cost in week K is calculated by using Equation 4.8.

nk

DCk   Cak ; k  1, 2, , W

(4.8)

a 1

where DCk is the net direct cost in week k in which nk is the number of activities whose durations overlap with week k , Cak is the direct cost of activity a in week k , and W is the project duration computed in weeks. Then, DC , which is the direct cost of a project, is computed by using Equation 4.9.

73 W

DC   DCk

(4.9)

k 1

The indirect cost of a project is the sum of fixed overhead, variable overhead (Afshar and Fathi 2009; Fathi and Afshar 2010; Peterson 2013), and mobilization and bonding costs. Fixed overhead includes administrative expenses such as office rent, office insurance, office supplies, office furniture, salaries of employees in the administrative office, etc. (Clough et al. 2005; Peterson 2013). These costs tend to be constant over the duration of a project since, for instance, the rent of the office does not change with the volume of work at a construction site (Fathi and Afshar 2010; Peterson 2013). Fixed overhead costs should be estimated using historical costs to accurately project the cash flow (Peterson 2013). The fixed overhead cost of a project is denoted by FOC and is computed by multiplying the fixed overhead cost per week with the number of weeks to complete the project as in Equation 4.10.

FOC  OF  W

(4.10)

Variable overhead includes utility costs on the job site, personnel scheduling and management costs, taxes, project insurance, etc. (Afshar and Fathi 2009; Fathi and Afshar 2010; Peterson 2013). Variable overheads tend to vary with the volume of work performed (Fathi and Afshar 2010; Peterson 2013). For example, if the project is accelerated by allocating more resources, more manager-hours are required to manage the job site. In addition, power consumption, taxes related to direct costs, and project

74 insurance increase. According to Peterson (2013), the profit and variable overhead should not be used to cover fixed overhead, but several past studies use a percentage of direct cost to cover both fixed and variable overhead costs (e.g., Ali and Elazouni 2009; Elazouni and Metwally 2007; Liu and Wang 2008). Based on the definition presented by Afshar and Fathi (2009) and Fathi and Afshar (2010), variable overhead cost in week k is denoted by VOCk and is computed by using Equation 4.11 where OV is the variable overhead percentage of the direct cost. Then, the variable overhead cost is denoted by

VOC and is computed by using Equation 4.12.

VOCk  OV  DCk ; k  1, 2, , W

(4.11)

W

VOC  VOCk

(4.12)

k 1

The reason why mobilization and bonding costs should be considered separately, is because the calculation of a realistic financing cost depends on it, as discussed in Chapter 2. Mobilization cost includes those costs that are necessary for the movement and communication of personnel, movement of equipment, supplies, and other facilities required for work to begin on the job site. As per Elazouni and Metwally (2005, 2007), Elazouni (2009), Elazouni and Abido (2011), and Alghazi et al. (2012, 2013), the mobilization cost is denoted by MOB and is computed by using Equation 4.13 where

OM is the mobilization cost percentage of (direct cost+variable overhead cost).

MOB  OM  ( DC  VOC )

(4.13)

75 The factors that affect the cost of purchasing contract bonds are the duration of the project, the class of construction, the applicable bond rates, the risks involved in the project, and the total contract amount (Clough et al. 2005). The cost of bonding can be calculated only if the total project cost (including direct+indirect costs+markup) is known. The markup of a project is denoted by MP and is computed by using Equation 4.14 where OMP is the markup expressed as a percentage. Then, the cost of bonding is denoted by BD and is computed by using Equation 4.15 where OB is the bond premium the contractor pays for the bond expressed as a percentage. Given the fact that the cost of bonding could have a significant impact on cash outflow, choosing an unrealistic premium rate affects the calculation of the financing cost. The premium rate varies between 0.5% and 2% (Clough et al. 2005).

MP  OMP  ( DC  FOC  VOC  MOB)

(4.14)

BD  OB  ( DC  FOC  VOC  MOB  MP)

(4.15)

The contract bid price is denoted by CBP and is nothing but the sum of the direct cost (DC), the fixed and variable overheads (FOC+VOC), the mobilization cost (MOB), the cost of bonding (BD), and the markup (MP) (Equation 4.16). Dividing the contract bid price ( CBP ) by the direct cost (DC) of the project, gives a bid price factor, denoted by BPF. This is an important factor because the amount bid for an activity is obtained by multiplying the direct cost of this activity by this factor (Equation 4.17). If the contract is a unit price contract, then dividing the amount bid for an activity by the quantity results

76 in the unit price of that bid item (Clough et al. 2005). The BPF factor will be used in the later stages of this model.

CBP  DC  FOC  VOC  MOB  MP  BD

(4.16)

BPF  CBP / DC

(4.17)

In order to calculate the contract bid price, and the bid price factor, the cost data should be input by the user in Step 7 (Figure 4.1). In Step 8 (Figure 4.1), the contract bid price and the bid price factor are computed using Equations 4.8 to 4.17. In order to calculate cash outflows and create a cash flow profile, construction disbursement should be calculated for each period separately. The terminology presented by Afshar and Fathi (2009) and Fathi and Afshar (2010) is modified in this study to separate mobilization and bonding costs from other overhead costs. The construction disbursement at the end of a typical project period t , which is denoted by Et , is computed by using Equation 4.18. The mobilization and bonding costs occur at the beginning of the project at t  0 . It should be noted that usually the contractor makes disbursements at the end of each month. Indeed, as it was discussed earlier, the expenditures relative to direct cost (i.e., materials, labor, equipment, and subcontractors) are usually incurred at the end of each month. In addition, the contractor usually pays fixed overhead and variable overhead costs at the end of each month (Peterson 2013). Therefore, this research assumes that contractor payments are made at the end of each month.

77

 MOB  BD  Et   4t   ( DCk  OF  VOCk ) k  (4t  3)

t 0 t  1, 2, , M

(4.18)

where M is the project duration expressed in months.

4.3.2

Project Cash Inflows. Cash inflows are the owner payments that are due on

completion of designated phases of the work. According to Clough et al. (2005), progress payments during a prescribed period of time represent the value of the work done, including those that are implemented by subcontractors. In most construction contracts, the owner withholds a certain percentage of progress payments, which remains in the possession of the owner until the final payment is made, with no interest on these funds (Clough et al. 2005). This retainage may be 5, 10, or even 20% of the monthly payments (Hinze 2012), although it is typically 10% in public projects (Clough et al. 2005; Hinze 2012). According to Hinze (2012), the payments made by the owner are usually made on a monthly basis. The contractor usually requests payment at the end of each month, but some owners may require the contractor to submit pay requests at the end of every two months or more. It should also be noted that payment, after deduction of retainage, is made to the contractor after a month or more from the time the payment request is submitted. Moreover, the final payment, which also includes the release of the retainage, is made 1 to 3 months after the last progress payment is made (Hinze 2012). The retained percentage of pay requests, the frequency with which pay requests are submitted, the owner`s lag in responding to payment requests, and the lag to return the retained money

78 at the last payment should be included in the contract. This research allows contractors to use the terms of their particular contract. One of the significant factors in determining the financing needs for a project is the schedule of owner payments. The final payment often has a different payment schedule than progress payments (Peterson 2013). Therefore, in order to calculate cash inflows and create a cash flow profile, progress payments excluding the last progress payment, and the final payment which is the release of all of the retained money, should be computed separately. The number of months between submitting pay requests, the lag in responding to pay requests (in months), and the retained percentage of pay requests are denoted by LS , LP , and R , respectively. Since the contractor submits pay requests every

LS months regularly, T is the time when the requests for the owner payment are submitted, and T  LP  is the time when the progress payment is made. The progress payment at the end of each period T  LP  is denoted by PT  LP . Progress payments excluding the last progress payment are computed by using Equation 4.20 where d is the last pay request submission prior to the pay request submission for the last progress payment. The last progress payment is computed by using Equation 4.21. The reason why the last progress payment is calculated separately from other progress payments is that the number of months before the last progress payment can be less than LS months; therefore the calculation is different. It should be noted that in some contracts, the owner makes an advance payment based on the percentage of the contract price. If there is such a term in a contract, advance payment is denoted by AP and is computed by using Equation 4.19 where OAP

79 is the advance payment percentage of the contract bid price. As a result, the payments are adjusted to model an advance payment using Equations 4.20 to 4.22.

AP  OAP  CBP 

PT  L   (1  R )  BPF  P



(4.19) 4T



k  ( 4  ( T  LS )  1)



AP  LS



M

DCk  

;T  (1  LS ),(2  LS ),,(d  LS )  M (4.20)

4T   AP  ( M  d  LS ) PT  LP  (1  R)  BPF  DCk   ; T M  M k  (4 d  LS 1)  

(4.21)

The request for the final payment is submitted after the last progress payment is made. The final payment at T  LR  , which is denoted by PT  LR  , is calculated by using Equation 4.22 where LR is the lag in making the final payment and returning the retained money (in months).

PT  LR  R  BPF  DC ; T  M  LP

(4.22)

In order to create a cash flow forecast, the contract terms should be input by the user in Step 9 (Figure 4.1). For better representation, the time when the final payment is made, which is denoted by TF , is computed by using Equation 4.23. In Step 10 (Figure 4.1), the cash outflow and the cash inflow are computed using Equations 4.18 to 4.22.

TF  M  LP  LR

(4.23)

80 Then, the net operating cash flow in period t , which is denoted by At , is computed by using Equation 4.24.

At  Pt  Et ; t  0, 1, 2, , TF

(4.24)

In Step 11 (Figure 4.1), the cumulative cash flow excluding financing flow at the end of period t, which is denoted by Nt , is computed by using Equation 4.25 where Nt 1 is the cumulative cash flows from period 0 to period (t  1) .

 At ; t  0 Nt    Nt 1  At ; t  1, 2, , TF

4.4

(4.25)

Optimizing Financing Cost in the First Stage Model

In addition to considering construction disbursements, the financing costs should also be considered because it is a meaningful concern of contractors. Financing should be considered to be a business concern rather than a project concern. This research not only ensures that negative cash flows do not occur during the project, but also allows the contractor to increase profits by providing optimal financing. A contractor needs to prepare a cash flow forecast for every project. This effort involves plans for financing. Contractors need to show exactly what they plan to use the money for and how the indebtedness will be paid including the interest. If a contractor`s financial projection persuades banks and other lenders that these borrowed funds will eliminate the temporary deficits and be paid back according to a solid financing schedule,

81 lenders will feel confident that the contractor will be able to pay off the loan or extended credit (Lesonsky 2014). On the other hand, because different financing alternatives are possible, the financing structure of a construction project can take many shapes; therefore, the practice of financial planning is often esoteric (Hendrickson and Au 2000). Since it is essential that the agreement between lenders and borrowers reflect exactly how the money will flow, this research aims not only to optimize financing, but also to provide a schedule of financing. To optimize financing cost, there should be multiple alternatives to compare the relative costs of different financing schedules and to ensure sufficient financing to perform and complete the project (Hendrickson and Au 2000). The proposed financing optimization model would answer three questions in order to maximize the cash balance at the end of the project: (1) which alternative or combination of alternatives should be considered, (2) what amount of money should be taken in each alternative in each month, and (3) what amount of money including interest should be repaid in each alternative in each month. As shown in Table 4.1, these alternatives are divided into three categories, short-term loans, long-term loans, and lines of credit. Each alternative can be offered by different lenders (e.g., twenty two different banks offer twenty two alternatives), or all alternatives can be offered by one lender (i.e., one bank offers twenty two alternatives). This model is prepared such that it considers nearly all possible offers of lenders.

82

Every 3 months

6 7

Short-term loans

8 9

Every 6 months

10 11 12 13

Every 9 months

14 15 16 17 18 19 20

Every 12 months

After 12 months

Monthly or after 12 months

After 3 months After 6 months After 9 months After 12 months

Monthly or after 3 months Monthly or after 6 months Monthly or after 9 months Monthly or after 12 months

Compounded interest rate (%)

5

Monthly interest rate (%)

4

Credit limit in each period ($)

3

Monthly or after 3 months Monthly or after 6 months Monthly or after 9 months Monthly or after 12 months Monthly or after 3 months Monthly or after 6 months Monthly or after 9 months Monthly or after 12 months Monthly or after 3 months Monthly or after 6 months Monthly or after 9 months Monthly or after 12 months Monthly or after 3 months Monthly or after 6 months Monthly or after 9 months

Total credit limit ($)

2

After 3 months After 6 months After 9 months After 12 months After 3 months After 6 months After 9 months After 12 months After 3 months After 6 months After 9 months After 12 months After 3 months After 6 months After 9 months

Financing code

Every month

Interest payment time

Time of taking money

1

Time of repaying principal

Alternative

Financing method

Table 4.1. Alternative Financing Methods (Page 1 of 2)

A3

CLA3

CL'B3

i A3

î =(1+i ) 3 -1 A3 A3

A6

CLA6

CL'B6

i A6

î =(1+i ) 6 -1 A6 A6

A9

CLA9

CL'B9

i A9

î =(1+i ) 9 -1 A9 A9

A12

CLA12

CL'B12

i A12

î =(1+i )12 -1 A12 A12

B3

CLB3

CL'B3

i B3

î =(1+i ) 3 -1 B3 B3

B6

CLB6

CL'B6

i B6

î =(1+i ) 6 -1 B6 B6

B9

CLB9

CL'B9

i B9

î =(1+i ) 9 -1 B9 B9

B12

CLB12

CL'B12

i B12

î =(1+i )12 -1 B12 B12

C3

CLC3

CL'C3

i C3

î =(1+i ) 3 -1 C3 C3

C6

CLC6

CL'C6

i C6

î =(1+i ) 6 -1 C6 C6

C9

CLC9

CL'C9

i C9

î =(1+i ) 9 -1 C9 C9

C12

CLC12

CL'C12

iC12

î =(1+i )12 -1 C12 C12

D3

CLD3

CL'D3

iD3

î =(1+i ) 3 -1 D3 D3

D6

CLD6

CL'D6

i D6

î =(1+i ) 6 -1 D6 D6

D9

CLD9

CL'D9

i D9

î =(1+i ) 9 -1 D9 D9

D12

CLD12

CL'D12

i D12

î =(1+i )12 -1 D12 D12

E3

CLE3

CL'E3

i E3

î =(1+i ) 3 -1 E3 E3

E6

CLE6

CL'E6

i E6

î =(1+i ) 6 -1 E6 E6

E9

CLE9

CL'E9

i E9

î =(1+i ) 9 -1 E9 E9

E12

CLE12

CL'E12

i E12

î =(1+i )12 -1 E12 E12

83

Monthly or after 1 month Monthly or after 2 months Monthly or after 3 months Monthly or after 4 months Monthly or after 5 months Monthly or after 6 months Monthly or after 7 months Monthly or after 6 months Monthly or after 9 months Monthly or after 10 months Monthly or after 11 months Monthly after 12 months Monthly or end of project

Any time

Compounded interest rate (%)

After 1 month After 2 months After 3 months After 4 months After 5 months After 6 months After 7 months After 8 months After 9 months After 10 months After 11 months After 12 months End of project

22

Monthly interest rate (%)

Monthly or end of project

Credit limit in each period ($)

Monthly or end of project

Total credit limit ($)


Beginning of project

21

LP

-

CL'LP

i LP

î =(1+i )TF -1 LP LP

LC1

CLLC

CL'LC

i LC

i LC

Financing code

Time of repaying principal

Alternative

Time of taking money

Line of credit

Long-term loan

Financing method

Table 4.1. Alternative Financing Methods (Page 2 of 2)

LC2

î =(1+i ) 2 -1 LC2 LC

LC3

î =(1+i )3 -1 LC3 LC

LC4

î =(1+i ) 4 -1 LC4 LC

LC5

î =(1+i ) 5 -1 LC5 LC

LC6

î =(1+i )6 -1 LC6 LC

LC7

î =(1+i ) 7 -1 LC7 LC

LC8

î =(1+i )8 -1 LC8 LC

LC9

î =(1+i )9 -1 LC9 LC

LC10

î =(1+i )10 -1 LC10 LC

LC11

î =(1+i )11 -1 LC11 LC

LC12

î =(1+i )12 -1 LC12 LC

LCP

î =(1+i )TF -t -1 LCP LC

*

* t is the current time It is possible that lenders do not offer some or many of these proposed alternatives, or even some of these alternatives do not exist in a specific location, but the proposed financing model is flexible enough, such that a contractor can simply disregard those alternatives that are not available, and consider only those alternatives that are

84 offered by lenders. In addition, since many lenders give the contractor the opportunity to negotiate, the contractor may want to propose to the lenders any of the alternatives in Table 4.1 and negotiate with lenders. As long as a contractor can provide a profitable cash flow forecast including financing costs, the contractor has the power to negotiate with lenders for a series of loans. Therefore, a series of prearranged loans can be predefined considering a series of prearranged repayments of both interest and principal for a predefined number of repayment periods (Hendrickson and Au 2000). As per Au and Hendrickson (1986), in this study, in the category of short-term loans, it is assumed that the contractor can use a lump sum (i.e., credit limit of each short term loan) in a series of scheduled loans with a predefined schedule of repayments of principal and interest during the project. According to Kramer and Fusaro (2010), banks are inclined to lend money for short periods of time to better finance their own operations. The prearranged schedule of financing for a series of loans can provide banks with a way to treat such loans as short-term loans. As a result, bank capacity is enhanced by lending without charging high premiums for the principal they would have to retain for long-term loans (Kramer and Fusaro 2010). For example, for Alternative 1 in Table 4.1 (i.e., A3), the contractor can take a series of loans every month and pay off the principal of each loan after three months. In addition, the interest can also be paid monthly or compounded over three months. Another example can be Alternative 18 (i.e., E6) in Table 4.1. In this alternative, the contractor can take a series of loans every year and pay off the principal of each loan after six months from the time each loan is taken. In addition, the interest of this alternative (i.e., E6) can also be paid monthly or compounded over six months. The reason why this type of model is

85 considered for short-term loans, is that some lenders allow both principal and interest to be paid off at the maturity of the loan, while others require the contractor to pay the interest at regular intervals with the principal being paid at the end of the period (Peterson 2013). It should be noted that the total sum of a series of loans should not violate the total credit limit of each alternative in the category of short-term loans for the whole project (e.g., for Alternative 1, it should not violate CLA3). It is worth mentioning that the total credit limit for short-term loans is fixed and cannot be reused. In terms of the long term loan, only a lump sum can be taken at the beginning of the project. In addition, most lenders require the contractor to pay back a fixed amount monthly to cover both interest and principal, while others allow the contractor to pay off both the principal and compounded interest at the end of the project (Peterson 2013). Therefore, the long-term loan is not taken as a series of loans. The line of credit is considered as one account with different modes. Contrary to a financing alternative that has a different rate of interest and a different credit limit (as discussed in the preceding two paragraphs), in the line of credit there is one interest rate and one credit limit. Below are discussed the reasons why different cases (e.g., different principal and interest payment times) are considered in the line of credit. 

A line of credit does not need to have a prearranged schedule if one considers only a line of credit as the only source of financing. The contractor can withdraw as much as necessary any time as long as the sum is below the credit limit, and can pay back as much of the borrowed sum as necessary. However, if there are additional financing alternatives, the line of credit should be considered alongside these alternatives to find the optimum schedule. The desirability of a line of credit

86 is different depending on the interest payment time. If a lender requires the contractor to pay the interest monthly, then comparing different modes of the line of credit (say, LC1 vs. LC2) is useless because it is clear that LC1 is better than LC2 if they both have the same rate of interest. In such a case, the purpose of the different modes that are considered for the line of credit in Table 4.1 is to compare them with loan alternatives. For example, consider that a monthly interest rate of A3 is lower than the interest rate of a line of credit; in such a case, it is believed that A3 is preferable to the line of credit, but this is not true in all situations. For example, even though the monthly interest rate of A3 is lower, A3 is better only when it is compared to LC3. If the balance of the cash flow is negative just for one or two months, LC1 or LC2 can be a better choice depending on the difference in the interest rate between line of credit and A3. However, if a lender allows the contractor to decide the timing of interest payments, in addition to comparing different modes of the line of credit with loan alternatives, different modes of the line of credit should also be compared with each other to find the best time of interest payment. Regardless of the modes used (i.e., LC1, LC2, LC3, LC4, LC5, LC6, LC7, LC8, LC9, LC10, LC11, LC12, and LCP); the money borrowed minus money paid back plus interest must never violate the credit limit (i.e., CLLC) at any time during the project. In addition, there is a difference between line of credit and short-term loans. In the line of credit, the amount of money paid back can be reused as long as the credit limit is not exceeded, whereas once a short-term loan is borrowed, no more funds can be withdrawn from this account.

87 

Despite the complexity and unscheduled nature of a line of credit, this research provides a schedule for managing a line of credit. Considering all proposed modes for the line of credit in this research, financing optimization results in a schedule that shows what amount of money should be withdrawn each month from the line of credit account and what amount of money including its interest should be paid back each month to the line of credit account. The reason why the long term loan is considered (i.e., LP) is that depending on

the situation, some banks offer a long-term loan with lower interest rate compared with short-term loans and lines of credit. If the contractor is not able to remove the deficit, it will be necessary to postpone payments and therefore extend project duration. In this case, the comparison between long-term loan, short-term loans, and line of credit should be considered. The type of loan is important at the beginning of the project when the contractor should pay for bonding and mobilization costs especially if the owner does not make advance payments. Moreover, mobilization and bonding costs can be high, and if the interest rate of a long-term loan is lower than other alternatives, and if the cash flow is negative for a long period of time, a long-term loan may be a better alternative compared to other alternatives at the beginning of the project. In addition to a total credit limit, some lenders may impose a limitation on the amount of a loan or the amount of withdrawal from a line of credit in each period. For example, considering Alternative 1 in Table 4.1, as long as the sum of a series of loans for Alternative 1 does not violate the total credit limit (i.e., CLA3), each loan in each period should not violate the credit limit in each period either (i.e., CL'A3). This might also be the case for a line of credit. In addition to the total amount withdrawn from the

88 line of credit including its interest not violating the total credit limit during the project (i.e., CLLC in Alternative 22, Table 4.1), some banks or lenders may require the contractor to withdraw just a percentage of the total credit limit in each period (i.e., CL'LC in Alternative 22, Table 4.1). As a result, if there is a credit limit in each period, the proposed model is also able to consider that. As it was mentioned in Chapter 3, there are often some fees added to a loan or a line of credit to cover the cost of setting up the loan or the line of credit and to provide a profit for the financial institution (Peterson 2013). Therefore, in addition to the interest rate, these costs should also be considered when comparing alternatives. Typically, the interest rate is stated such that it includes compounding and lending fees (i.e., the fees for a loan or a line of credit) (Prakash 2015b). When the contractor wants to choose between alternatives, APR that includes fees will help the contractor to compare the total financing cost in terms of compounding and fees, and this is important especially when the contractor is looking to borrow over the long term (Prakash 2015a, 2015b). As a result, in this research, it is assumed that a lender quotes the contractor the APR (including fees), and then the monthly interest rate in Table 4.1 (which covers the lending fees) that is denoted by ialt is calculated by using Equation 4.26. In Table 4.1, the compounded interest rate for a period n , which takes both compounding and lending fees into consideration is denoted by iâlt and is calculated by using Equation 4.27.

ialt  (1  APR (including fees))(1/12)  1; alt  A3, A6, …, LC

(4.26)

iâlt  (1  ialt )n  1; alt  A3, A6, …, LC

(4.27)

89 The reason why banks are not considered by name is because this research does not differentiate between banks. Instead, this research tends to give more flexibility to the contractors to seek financing from different sources. In Step 12 (Figure 4.1), once contractors choose their sources, and after they negotiate with them, they can input the lenders` offers (i.e., financing alternatives, APRs, interest payment times, total credit limits, and credit limits in each period) into the model to generate the different alternatives, and then find the optimum combination of financing (i.e., schedule of financing inflow (borrowed money) and financing outflow (repaid money including interest)).

4.4.1

Project Financing Flow Creation. Below are presented the equations for all

proposed alternatives in Table 4.1. There is no requirement to use all the proposed equations. The analyst can consider the alternatives that are available to the contractor, and disregard the others. In order to find financing variables, the financing flow should be created to be added to the cash flow for optimization. To create the financing flow, the equations are divided into three parts: borrowed money (financing inflows), repaid money (financing outflows), and financing costs (financing outflows). In Step 13 (Figure 4.1), first, the financing inflows, which are the amount of money to be borrowed (i.e., cash provisions) at the end of each period, are computed by Equations 4.28 to 4.61 considering each alternative separately. It should be noted that if the period of a financing alternative exceeds the time when final payment ( TF ) is made, the financing alternative cannot be used and is equal to zero. In addition, the last time of taking money should be adjusted so that the time of repayment does not violate the time

90 of the final payment ( TF ) since debts and interest cannot be paid after final payment. Therefore, B3 , B6 , B9 , B12 , C 3 , C6 , C 9 , C12 , D3 , D6 , D9 , D12 , E 3 ,

E6 , E 9 , and E12 in Equations 4.32 to 4.47 are the last increments of time in which their repayment times do not violate the final payment`s time.

0  BA3t   A3t 0  0  BA6t   A6t 0 

0  BA9t   A9t 0 

if TF  3 if TF  3 ; t  0, 1, 2, , TF  3 if TF  3 ; t  TF  2, , TF if TF  6 if TF  6 ; t  0, 1, 2, , TF  6

0  BB6t   B 6t  0 0  BB9t   B9t 0 

(4.29)

if TF  6 ; t  TF  5, , TF

if TF  9 if TF  9 ; t  0, 1, 2, , TF  9

(4.30)

if TF  9 ; t  TF  8, , TF

if TF  12 0  BA12t   A12t if TF  12 ; t  0, 1, 2, , TF  12 0 if TF  12 ; t  TF  11, , TF 

0  BB3t   B3t 0 

(4.28)

(4.31)

if TF  3 if TF  3 ; t  0, 3, 6, , B3  TF  3 if TF  3 ; t  0, 3, 6, , B3  TF  3

(4.32)

if TF  6 if TF  6 ; t  0, 3, 6, , B6  TF  6 if TF  6 ; t  0, 3, 6, , B 6  TF  6

(4.33)


(4.34)

91 0  BB12t   B12t 0 

0  BC 3t  C 3t 0 


if TF  3 if TF  3 ; t  0, 6, 12, , C 3  TF  3 if TF  3 ; t  0, 6, 12, , C 3  TF  3

0  BC 6t  C 6t 0 

if TF  6

0  BC 9t  C 9t 0 

if TF  9

0  BC12t  C12t 0 

if TF  12

0  BD3t   D3t 0 

(4.35)

if TF  6 ; t  0, 6, 12, , C 6  TF  6 if TF  6 ; t  0, 6, 12, , C 6  TF  6

if TF  9 ; t  0, 6, 12, , C 9  TF  9 if TF  9 ; t  0, 6, 12, , C 9  TF  9

if TF  12 ; t  0, 6, 12, , C12  TF  12 if TF  12 ; t  0, 6, 12, , C12  TF  12

(4.36)

(4.37)

(4.38)

(4.39)

if TF  3 if TF  3 ; t  0, 9, 18, , D3  TF  3 if TF  3 ; t  0, 9, 18, , D3  TF  3

0  BD6t   D6t 0 

if TF  6

0  BD9t   D9t  0

if TF  9

0  BD12t   D12t 0 

if TF  12

if TF  6 ; t  0, 9, 18, , D6  TF  6 if TF  6 ; t  0, 9, 18, , D6  TF  6



(4.40)

(4.41)

(4.42)

(4.43)

92 0  BE 3t   E 3t 0 

if TF  3 if TF  3 ; t  0, 12, 24, , E 3  TF  3 if TF  3 ; t  0, 12, 24, , E 3  TF  3

0  BE 6t   E 6t 0 

if TF  6

0  BE 9t   E 9t 0 

if TF  9

0  BE12t   E12t 0 

if TF  12

0  BLPt   LPt 0 

0  BLC1t   LC1t 0 

if TF  6 ; t  0, 12, 24, , E 6  TF  6 if TF  6 ; t  0, 12, 24, , E 6  TF  6

if TF  9 ; t  0, 12, 24, , E 9  TF  9 if TF  9 ; t  0, 12, 24, , E 9  TF  9

if TF  12 ; t  0, 12, 24, , E12  TF  12 if TF  12 ; t  0, 12, 24, , E12  TF  12

(4.44)

(4.45)

(4.46)

(4.47)

if TF  12 if TF  12 ; t  0

(4.48)

if TF  12 ; t  0

if TF  1 if TF  1 ; t  0, 1, 2, , TF  1

(4.49)

if TF  1 ; t  TF

if TF  2 0  BLC 2t   LC 2t if TF  2 ; t  0, 1, 2, , TF  2 0 if TF  2 ; t  TF  1, TF 

0  BLC 3t   LC 3t 0 

if TF  3

0  BLC 4t   LC 4t 0 

if TF  4

if TF  3 ; t  0, 1, 2, , TF  3

(4.50)

(4.51)

if TF  3 ; t  TF  2, , TF

if TF  4 ; t  0, 1, 2, , TF  4 if TF  4 ; t  TF  3, , TF

(4.52)

93 0  BLC 5t   LC 5t 0 

if TF  5

0  BLC 6t   LC 6t 0 

if TF  6

0  BLC 7t   LC 7t 0 

if TF  7

0  BLC 8t   LC 8t 0 

if TF  8

0  BLC 9t   LC 9t 0 


if TF  6 ; t  0, 1, 2, , TF  6

(4.54)

if TF  6 ; t  TF  5, , TF

if TF  7 ; t  0, 1, 2, , TF  7

(4.55)

if TF  7 ; t  TF  6, , TF

if TF  8 ; t  0, 1, 2, , TF  8

(4.56)

if TF  8 ; t  TF  7, , TF if TF  9 if TF  9 ; t  0, 1, 2, , TF  9

(4.57)

if TF  9 ; t  TF  8, , TF

if TF  10 0  BLC10t   LC10t if TF  10 ; t  0, 1, 2, , TF  10 0 if TF  10 ; t  TF  9, , TF 

0  BLC11t   LC11t 0 

(4.53)

(4.58)

if TF  11 if TF  11 ; t  0, 1, 2, , TF  11

(4.59)

if TF  11 ; t  TF  10, , TF

0  BLC12t   LC12t 0 

if TF  12

0  BLCPt   LCPt 0 

if TF  13

if TF  12 ; t  0, 1, 2, , TF  12

(4.60)

if TF  12 ; t  TF  11, , TF


(4.61)

94 Then, the total borrowed money (financing inflow) at the end of period t , which is denoted by Bt , is computed by Equation 4.62.

Bt  BA3t  BA6t  BA9t  BA12t  BB3t  BB6t  BB12t  BC 3t  BC 6t  BC 9t  BC12t  BD3t  BD6t  BD9t  BD12t  BE 3t  BE 6t  BE 9t  BE12t  BLPt  BLC1t  BLC 2t  BLC 3t  BLC 4t  BLC 5t  BLC 6t  BLC 7t

(4.62)

 BLC8t  BLC 9t  BLC10t  BLC11t  BLC12t  BLCPt ; t  0, 1, 2, , TF

Now, the financing outflow, which consists of repaid money and financing costs, should be calculated based on borrowed money at the end of each period. The amount of money to be repaid (i.e., cash provisions) at the end of each period, are computed by Equations 4.63 to 4.96 considering each alternative separately. It should be noted that most of the banks offer long-term loans to be repaid in fixed monthly payments, which are first used to pay off the interest on the outstanding principal at the end of the previous month and then the remaining proceeds are used to reduce the borrowed money (Peterson 2013). Therefore, the Equation 4.83 of long-term loan includes monthly repayments or repayment at the end.

0  RA3t  0  BA3 t 3 

0  RA6t  0  BA6 t 6 

if TF  3 if TF  3 ; t  0, 1, 2

(4.63)

if TF  3 ; t  3, 4, 5, , TF

if TF  6 if TF  6 ; t  0, 1, 2, , 5 if TF  6 ; t  6, 7, 8, , TF

(4.64)

95 0  RA9t  0  BA9 t 9 

if TF  9 if TF  9 ; t  0, 1, 2, , 8 if TF  9 ; t  9, 10, 11, , TF

if TF  12 0  RA12t  0 if TF  12 ; t  0, 1, 2, , 11  BA12 if TF  12 ; t  12, 13, 14, , TF t 12  0  RB3t   BB3t  3 0 

(4.65)

(4.66)

if TF  3 if TF  3 ; t  3, 6, 9, , (B3  3)  TF if TF  3 ; t  3, 6, 9, , (B3  3)  TF

0  RB6t   BB6t  6  0

if TF  6

0  RB9t   BB9t  9 0 

if TF  9

if TF  6 ; t  6, 9, 12, , (B6  6)  TF if TF  6 ; t  6, 9, 12, , (B6  6)  TF

if TF  9 ; t  9, 12, 15, , (B9  9)  TF if TF  9 ; t  9, 12, 15, , (B9  9)  TF

if TF  12 0  RB12t   BB12t 12 if TF  12 ; t  12, 15, 18, , (B12  12)  TF  if TF  12 ; t  12, 15, 18, , (B12  12)  TF 0

0  RC 3t   BC 3t  3 0 

if TF  3

0  RC 6t   BC 6t  6 0 

if TF  6

0  RC 9t   BC 9t  9 0 

if TF  9

if TF  3 ; t  3, 9, 15, , (C 3  3)  TF if TF  3 ; t  3, 9, 15, , (C 3  3)  TF



(4.67)

(4.68)

(4.69)

(4.70)

(4.71

(4.72)

(4.73)

96 if TF  12 0  RC12t   BC12t 12 if TF  12 ; t  12, 18, 24, , (C12  12)  TF 0 if TF  12 ; t  12, 18, 24, , (C12  12)  TF 

0  RD3t   BD3t  3  0

if TF  3

0  RD6t   BD6t  6  0

if TF  6

0  RD9t   BD9t  9 0 

if TF  9

if TF  3 ; t  3, 12, 21, , (D3  3)  TF if TF  3 ; t  3, 12, 21, , (D3  3)  TF



if TF  12 0  RD12t   BD12t 12 if TF  12 ; t  12, 21, 30, , (D12  12)  TF 0 if TF  12 ; t  12, 21, 30, , (D12  12)  T  0  RE 3t   BE 3t  3 0 

if TF  3

0  RE 6t   BE 6t  6 0 

if TF  6

0  RE 9t   BE 9t  9 0 

if TF  9

0  RE12t   BE12t 12  0

if TF  12

if TF  3 ; t  3, 15, 27, , (E 3  3)  TF if TF  3 ; t  3, 15, 27, , (E 3  3)  TF



if TF  12 ; t  12, 24, 36, , (E12  12)  TF if TF  12 ; t  12, 24, 36, , (E12  12)  TF

(4.74)

(4.75)

(4.76)

(4.77)

(4.78)

(4.79)

(4.80)

(4.81)

(4.82)

97 0 0    i  (iˆ  1)  RLPt  BLP0   LP LP   FLPt ˆ i LP    BLP 0  0 0  RLC1t  0  BLC1 t 1 

if TF  12 if TF  12 and monthly repayments; t  0 if TF  12 and monthly repayments; t  1,,TF (4.83) if TF  12 and repayment at the end; t  TF if TF  12 and repayment at the end; t  TF

if TF  1 if TF  1 ; t  0

(4.84)

if TF  1 ; t  1, 2, 3, , TF

if TF  2 0  RLC 2t  0 if TF  2 ; t  0, 1  BLC 2 if TF  2 ; t  2, 3, 4, , TF t 2 

(4.85)

if TF  3 0  RLC 3t  0 if TF  3 ; t  0, 1, 2  BLC 3 if TF  3 ; t  3, 4, 5, , TF t 3 

(4.86)

if TF  4 0  RLC 4t  0 if TF  4 ; t  0, 1, , 3  BLC 4 if TF  4 ; t  4, 5, 6, , TF t 4 

(4.87)


(4.88)


(4.89)


(4.90)

98 if TF  8 0  RLC 8t  0 if TF  8 ; t  0, 1, , 7  BLC 8 if TF  8 ; t  8, 9, 10, , TF t 8 

(4.91)


(4.92)

if TF  10 0  RLC10t  0 if TF  10 ; t  0, 1, , 9  BLC10 t 10 if TF  10 ; t  10, 11, 12, , TF 

(4.93)


(4.94)


(4.95)

if TF  13 0  TF  RLCPt    BLCPi 13 if TF  13 ; t  TF i 13 0 if TF  13 ; t  TF

(4.96)

Then, the total repaid money (financing outflow) at the end of period t , which is denoted by Rt , is computed by Equation 4.97.

Rt  RA3t  RA6t  RA9t  RA12t  RB3t  RB6t  RB12t  RC 3t  RC 6t  RC 9t  RC12t  RD3t  RD6t  RD9t  RD12t  RE 3t  RE 6t  RE 9t  RE12t  RLPt  RLC1t  RLC 2t  RLC 3t  RLC 4t  RLC 5t  RLC 6t  RLC 7t  RLC 8t  RLC 9t  RLC10t  RLC11t  RLC12t  RLCPt ; t  0, 1, 2, , TF

(4.97)

99 The financing costs (financing outflow) based on borrowed money at the end of each period, considering each alternative separately, are computed by Equations 4.98 to 4.164. However, if the interest payment is made monthly, the monthly interest rate is used, where the financing costs are computed by Equations 4.98 to 4.130. If the interest payment is made at the same time as the time of principal repayment, the compounded interest rate is used, where the financing costs are computed by Equations 4.131 to 4.164.

0 0   t FA3t   (iA3  BA3t  i )  i 1  3  (iA3  BA3t  i )  i 1

0 0   t FA6t   (iA6  BA6t  i )  i 1 6  (iA6  BA6t  i )  i 1 0 0   t FA9t   (iA9  BA9t  i )  i 1 9  (iA9  BA9t  i )  i 1

if TF  3 if TF  3 ; t  0 if TF  3 ; t  1, 2

(4.98)

if TF  3 ; t  3, 4, 5, , TF

if TF  6 if TF  6 ; t  0 if TF  6 ; t  1, 2, 5

(4.99)

if TF  6 ; t  6, 7, , TF if TF  9 if TF  9 ; t  0 if TF  9 ; t  1, 2, 8 if TF  9 ; t  9, 10, , TF

(4.100)

100

0 0   t FA12t   (iA12  BA12t  i )  i 1  12  (iA12  BA12t  i )  i 1 0 0  FB3t  iB 3  BB3i  t 1   iB 3  BB3i i  t  3

0 0   t 1 FB6t   iB 6  BB6i  i 0  t 1   iB 6  BB6i i  t  6 0 0   t 1 FB9t   iB 9  BB9i  i 0  t 1   iB 9  BB9i i  t  9 0 0   t 1 FB12t   iB12  BB12i  i 0  t 1   iB12  BB12i i  t 12

if TF  12 if TF  12 ; t  0 if TF  12 ; t  1, 2, 11

(4.101)

if TF  12 ; t  12, 13, , TF if TF  3 if TF  3 ; t  0 if TF  3 ; t  1, 2, 3 ; i  0

(4.102)

if TF  3 ; t  4,5,TF

if TF  6 if TF  6 ; t  0 if TF  6 ; t  1, 2, , 6

(4.103)

if TF  6 ; t  7, 8,TF if TF  9 if TF  9 ; t  0 if TF  9 ; t  1, 2, , 8

(4.104)

if TF  9 ; t  9, 10,TF if TF  12 if TF  12 ; t  0 if TF  12 ; t  1, 2, , 11 if TF  12 ; t  12, 13,TF

(4.105)

101 0 0  FC 3t  iC 3  BC 3i  t 1   iC 3  BC 3i i  t  3

0 0   t 1 FC 6t   iC 6  BC 6i  i0  t 1   iC 6  BC 6i i  t  6 0 0   t 1 FC 9t   iC 9  BC 9i  i 0  t 1   iC 9  BC 9i i  t  9 0 0   t 1 FC12t   iC12  BC12i  i 0  t 1   iC12  BC12i i  t 12 0 0  FD3t  iD 3  BD3i  t 1   iD 3  BD3i i  t  3

if TF  3 if TF  3 ; t  0 if TF  3 ; t  1, 2, 3 ; i  0

(4.106)

if TF  3 ; t  4,5,TF

if TF  6 if TF  6 ; t  0 if TF  6 ; t  1, 2, , 6

(4.107)


(4.108)


(4.109)

if TF  12 ; t  12, 13,TF if TF  3 if TF  3 ; t  0 if TF  3 ; t  1, 2, 3 ; i  0 if TF  3 ; t  4,5,TF

(4.110)

102

0 0   t 1 FD6t   iD 6  BD6i  i 0  t 1   iD 6  BD6i i  t  6 0 0   t 1 FD9t   iD 9  BD9i  i 0  t 1   iD 9  BD9i i  t  9 0 0   t 1 FD12t   iD12  BD12i  i 0  t 1   iD12  BD12i i  t 12 0 0  FE 3t  iE 3  BE 3i  t 1   iE 3  BE 3i i  t  3

0 0   t 1 FE 6t   iE 6  BE 6i  i 0  t 1   iE 6  BE 6i i  t  6

if TF  6 if TF  6 ; t  0 if TF  6 ; t  1, 2, , 6

(4.111)


(4.112)


(4.113)

if TF  12 ; t  12, 13,TF if TF  3 if TF  3 ; t  0 if TF  3 ; t  1, 2, 3 ; i  0

(4.114)

if TF  3 ; t  4,5,TF

if TF  6 if TF  6 ; t  0 if TF  6 ; t  1, 2, , 6 if TF  6 ; t  7, 8,TF

(4.115)

103

0 0   t 1 FE 9t   iE 9  BE 9i  i 0  t 1   iE 9  BE 9i i  t  9 0 0   t 1 FE12t   iE12  BE12i  i0  t 1   iE12  BE12i i  t 12

if TF  9 if TF  9 ; t  0 if TF  9 ; t  1, 2, , 8

(4.116)


(4.117)

if TF  12 ; t  12, 13,TF

if TF  12 0  FLPt  0 if TF  12 ; t  0 (4.118)  t ( FLP  (BLP BLP   iLP  (iˆLP  1)   (t 1))  i if T  12 ; t  1,,T  i 1 LP F F 0 0   iˆLP    i 1

0  FLC1t  0 i  BLC1 t 1  LC 0 0  FLC 2t  iLC  BLC1t 1  2  iLC  BLC 2t  i  i 1

0 0   t FLC 3t   (iLC  BLC 3t  i )  i 1  3  (iLC  BLC 3t  i )  i 1

if TF  1 if TF  1 ; t  0

(4.119)

if TF  1 ; t  1, 2, 3, , TF if TF  2 if TF  2 ; t  0 if TF  2 ; t  1

(4.120)

if TF  2 ; t  2, 3, , TF

if TF  3 if TF  3 ; t  0 if TF  3 ; t  1, 2 if TF  3 ; t  3, 4, 5, , TF

(4.121)

104

0 0   t FLC 4t   (iLC  BLC 4t  i )  i 1  4  (iLC  BLC 4t  i )  i 1 0 0   t FLC 5t   (iLC  BLC 5t  i )  i 1  5  (iLC  BLC 5t  i )  i 1 0 0   t FLC 6t   (iLC  BLC 6t  i )  i 1  6  (iLC  BLC 6t  i )  i 1 0 0   t FLC 7t   (iLC  BLC 7t  i )  i 1 7  (iLC  BLC 7t  i )  i 1 0 0   t FLC8t   (iLC  BLC8t  i )  i 1 8  (iLC  BLC8t  i )  i 1

if TF  4 if TF  4 ; t  0 if TF  4 ; t  1, 2, 3

(4.122)

if TF  4 ; t  4, 5, 6, , TF if TF  5 if TF  5 ; t  0 if TF  5 ; t  1, 2, , 4

(4.123)


(4.124)


(4.125)

if TF  7 ; t  7, 8, 9, , TF if TF  8 if TF  8 ; t  0 if TF  8 ; t  1, 2, , 7 if TF  8 ; t  8, 9, 10, , TF

(4.126)

105

0 0   t FLC 9t   (iLC  BLC 9t  i )  i 1 9  (iLC  BLC 9t  i )  i 1 0 0   t FLC11t   (iLC  BLC11t  i )  i 1  11  (iLC  BLC11t  i )  i 1 0 0   t FLC12t   (iLC  BLC11t  i )  i 1  12  (iLC  BLC11t  i )  i 1 0 FLCPt  0  t  (i  BLCP ) LC t i  i 1

if TF  9 if TF  9 ; t  0 if TF  9 ; t  1, 2, , 8

(4.127)


(4.128)


(4.129)

if TF  12 ; t  12, 13, 14, , TF if TF  13 if TF  13 ; t  0

(4.130)

if TF  13 ; t  1, 2,TF

The financing costs are calculated considering compound interest in Equations 4.131 to 4.164. If the bank and the contractor agree that the contractor pays the interest when the principal is paid, Equations 4.131 to 4.164 are used.

106 0  FA3t  0 ˆ iA3  BA3t  3 0  FA6t  0 ˆ iA6  BA6t  6

0  FA9t  0 ˆ iA9  BA9t  9

if TF  3 if TF  3 ; t  0, 1, 2 if TF  3 ; t  3, 4, 5, , TF if TF  6 if TF  6 ; t  0, 1, 2, , 5

if TF  9 if TF  9 ; t  0, 1, 2, , 8

0  FB3t  iˆB 3  BB3t  3 0 

if TF  3

0  FB12t  iˆB12  BB12t 12 0 

(4.133)

if TF  9 ; t  9, 10, 11, , TF if TF  12

0  FB9t  iˆB 9  BB9t  9 0 

(4.132)

if TF  6 ; t  6, 7, 8, , TF

0  FA12t  0 ˆ iA12  BA12t 12

0  FB6t  iˆB 6  BB 6t  6 0 

(4.131)

if TF  12 ; t  0, 1, 2, , 11

(4.134)

if TF  12 ; t  12, 13, 14, , TF

if TF  3 ; t  3, 6, 9, , TF

(4.135)

if TF  3 ; t  3, 6, 9, , TF if TF  6 if TF  6 ; t  6, 9, 12, , TF

(4.136)


(4.137)

if TF  9 ; t  9, 12, 15, , TF if TF  12 if TF  12 ; t  12, 15, 18, , TF if TF  12 ; t  12, 15, 18, , TF

(4.138)

107 0  FC 3t  iˆC 3  BC 3t  3 0  0  FC 6t  iˆC 6  BC 6t  6 0 

if TF  3 if TF  3 ; t  3, 9, 15, , TF if TF  3 ; t  3, 9, 15, , TF if TF  6 if TF  6 ; t  6, 12, 18, , TF

if TF  9

0  FC12t  iˆC12  BC12t 12 0 

if TF  12

0  FD6t  iˆD 6  BD 6t  6 0  0  FD9t  iˆD 9  BD9t  9 0  0  FD12t  iˆD12  BD12t 12 0 

(4.140)

if TF  6 ; t  6, 12, 18, , TF

0  FC 9t  iˆC 9  BC 9t  9 0 

0  FD3t  iˆD 3  BD3t  3 0 

(4.139)

if TF  9 ; t  9, 15, 21, , TF

(4.141)

if TF  9 ; t  9, 15, 21, , TF

if TF  12 ; t  12, 18, 24, , TF

(4.142)


(4.143)


(4.144)


(4.145)

if TF  9 ; t  9, 18, 27, , TF if TF  12 if TF  12 ; t  12, 21, 30, , TF if TF  12 ; t  12, 21, 30, , T

(4.146)

108 0  FE 3t  iÊ 3  BE 3t  3 0  0  FE 6t  iÊ 6  BE 6t  6 0  0  FE 9t  iÊ 9  BE 9t  9 0  0  FE12t  iÊ12  BE12t 12 0  0  FLPt  iˆLP  BLP0 0 

0  FLC1t  0 ˆ iLC  BLC1t 1 0  FLC 2t  0 ˆ iLC1  BLC 2t  2 0  FLC 3t  0 ˆ iLC 3  BLC 3t  3

if TF  3 if TF  3 ; t  3, 15, 27, , TF

(4.147)


(4.148)


(4.149)


(4.150)

if TF  12 ; t  12, 24, 36, , TF if TF  12 if TF  12 ; t  TF

(4.151)

if TF  12 ; t  TF

if TF  1 if TF  1 ; t  0

(4.152)

if TF  1 ; t  1, 2, 3, , TF if TF  2 if TF  2 ; t  0, 1

(4.153)

if TF  2 ; t  2, 3, 4, , TF if TF  3 if TF  3 ; t  0, 1, 2 if TF  3 ; t  3, 4, 5, , TF

(4.154)

109 0  FLC 4t  0 ˆ iLC 4  BLC 4t  4 0  FLC 5t  0 ˆ iLC 5  BLC 5t  5 0  FLC 6t  0 ˆ iLC 6  BLC 6t  6 0  FLC 7t  0 ˆ iLC 7  BLC 7t  7 0  FLC 8t  0 ˆ iLC 8  BLC 8t 8

0  FLC 9t  0 ˆ iLC 9  BLC 9t  9

if TF  4 if TF  4 ; t  0, 1, , 3

(4.155)

if TF  4 ; t  4, 5, 6, , TF if TF  5 if TF  5 ; t  0, 1, , 4

(4.156)


(4.157)


(4.158)


(4.159)

if TF  8 ; t  8, 9, 10, , TF

if TF  9 if TF  9 ; t  0, 1, , 8

(4.160)

if TF  9 ; t  9, 10, 11, , TF

0 if TF  10  FLC10t  0 if TF  10 ; t  0, 1, , 9 ˆ iLC10  BLC10t 10 if TF  10 ; t  10, 11, 12, , TF

(4.161)

0 if TF  11  FLC11t  0 if TF  11 ; t  0, 1, , 10 ˆ iLC11  BLC11t 11 if TF  11 ; t  11, 12, 13, , TF

(4.162)

110 0 if TF  12  FLC12t  0 if TF  12 ; t  0, 1, , 11 ˆ iLC12  BLC12t 12 if TF  12 ; t  12, 13, 14, , TF

(4.163)

if TF  13 0  TF  FLCPt    (iˆLCP  BLCPi 13 ) if TF  13 ; t  TF i 13 0 if TF  13 ; t  TF

(4.164)

The total financing cost (financing outflow) at the end of period t , which is denoted by Ft , is computed by Equation 4.165. The financing flow is created by calculating Bt , Rt , and Ft .

Ft  FA3t  FA6t  FA9t  FA12t  FB3t  FB6t  FB12t  FC 3t  FC 6t  FC 9t  FC12t  FD3t  FD6t  FD9t  FD12t  FE 3t  FE 6t  FE 9t  FE12t  FLPt  FLC1t  FLC 2t  FLC 3t  FLC 4t  FLC 5t  FLC 6t  FLC 7t

(4.165)

 FLC 8t  FLC 9t  FLC10t  FLC11t  FLC12t  FLCPt ; t  0, 1, 2, , TF

The net financing outflow, which is the total money to be repaid including financing cost at the end of period t , is denoted by TRt and computed by Equation 4.166. In Step 14 (Figure 4.1), the net financing flow, which is denoted by NFt , is computed by Equation 4.167.

TRt  Rt  Ft ; t  0,1,2,, TF

(4.166)

NFt  Bt  TRt ; t  0,1,2,,TF

(4.167)

111 The cumulative net balance of the financing cost (i.e., interest), which is denoted by NFCt , is computed by using Equation 4.168 where NFCt 1 is the cumulative net financing costs from period 0 to period (t  1) . The total financing cost of the project is equal to the cumulative net financing costs at the time of final payment which is denoted by NFCTF . .

 Ft NFCt    NFCt 1  Ft

t0 t  1, 2,, TF

(4.168)

The total required financing for each alternative of loans (i.e., total amount of a series of loans for each alternative) is computed by Equations 4.169 to 4.189.

TF

TA3   BA3t

(4.169)

t 0 TF

TA6   BA6t

(4.170)

t 0

TF

TA9   BA9t

(4.171)

t 0

TF

TA12   BA12t

(4.172)

t 0

TF

TB3   BB3t

(4.173)

t 0 TF

TB6   BB6t t 0

(4.174)

112 TF

TB9   BA9t

(4.175)

t 0

TF

TB12   BB12t

(4.176)

t 0

TF

TC 3   BC 3t

(4.177)

t 0 TF

TB6   BC 6t

(4.178)

t 0 TF

TC 9   BC 9t

(4.179)

t 0

TF

TC12   BC12t

(4.180)

t 0

TF

TD3   BD3t

(4.181)

t 0 TF

TD6   BD6t

(4.182)

t 0 TF

TD9   BD9t

(4.183)

t 0

TF

TD12   BD12t

(4.184)

t 0

TF

TE 3   BE 3t

(4.185)

t 0 TF

TE 6   BE 6t

(4.186)

t 0 TF

TE 9   BE 9t t 0

(4.187)

113 TF

TE12   BE12t

(4.188)

TLP  LP0

(4.189)

t 0

The required credit for a line of credit is the maximum cumulative net balance at the end of each period considering total credit is withdrawn and total credit is repaid including financing costs. It should be noted that even if the contractor is allowed to compound the interest of the line of credit and is not required to pay the interest monthly, the accumulated compound interest should be covered at the end of each period while calculating the required credit for the line of credit. However, monthly interest or periodic compound interest may be paid out of the line of credit to increase the ability of borrowing more money. To calculate the required credit for the line of credit, the total withdrawal from the account at the end of each period t , which is denoted by TBLCt is computed by Equation 4.190. The total repayment to the account at the end of each period t , which is denoted by TRLCt is computed by using Equation 4.191. The total paid financing costs considering the line of credit account at the end of each period t , which is denoted by TFLCt is computed by using Equation 4.192. The accumulated balance of the line of credit at the end of each period excluding paid financing costs, which is denoted by NLCt , is computed by using Equation 4.193. The accumulated financing cost, which is not paid yet, is denoted by AFLCt and calculated by using Equation 4.194. Therefore, the required credit at the end of each period t , which is denoted by CLCt , is calculated by using Equation 4.195. Finally, the required credit for

114 the line of credit account, which is denoted by RCLC , is computed by using Equation 4.196.

TBLCt  BLC1t  BLC 2t    BLCPt ; t  0,1,, TF

(4.190)

TRLCt  RLC1t  RLC 2t    RLCPt ; t  0,1,, TF

(4.191)

TFLCt  FLC1t  FLC 2t    FLCPt ; t  0,1,, TF

(4.192)

TBLCt  TRLCt NLCt    NLCt 1  TBLCt  TRLCt

(4.193)

t 0 t  1, 2,, TF

0  AFLCt   t t i  ( NLCi 1  iLC  TFLCi )  (1  iLC )  i 1

t 0 t  1,2,, TF

(4.194)

CLCt  NLCt  AFLCt ; t  0, 1, , TF

(4.195)

RCLC  max(CLCt )

(4.196)

The constraints are divided into two categories: (1) constraints of financing, and (2) constraints of cumulative net balance of cash flow (including financing flow). In Step 15 (Figure 4.1), the constraints of financing are created. With regard to financing constraints, there could be two cases: there is predetermined credit limit relative to each alternative or there is not. The predetermined credit limit can be set in negotiations in two ways: (1) total credit limit, and (2) credit limit in each period. The proposed model is flexible enough to allow the contractor to choose whether there is a limitation for borrowing money or not. If there is a limitation on the total amount of borrowed money relative to each alternative, Equations 4.197 to 4.218 are applied. If there is a limitation

115 on the amount of money to be borrowed in each period relative to each alternative, Equations 4.219 to 4.241 are applied.

TA3  CLA3

(4.197)

TA6  CLA6

(4.198)

TA9  CLA9

(4.199)

TA12  CLA12

(4.200)

TB3  CLB3

(4.201)

TB6  CLB 6

(4.202)

TB9  CLB9

(4.203)

TB12  CLB12

(4.204)

TC3  CLC 3

(4.205)

TC 6  CLC 6

(4.206)

TC9  CLC 9

(4.207)

TC12  CLC12

(4.208)

TC12  CLC12

(4.209)

TD3  CLD3

(4.210)

TD6  CLD6

(4.211)

TD9  CLD9

(4.212)

TD12  CLD12

(4.213)

116

TE3  CLE 3

(4.214)

TE 6  CLE 6

(4.215)

TE9  CLE 9

(4.216)

TE12  CLE12

(4.217)

CLCt  CLLC ; t  0, 1, 2, , TF

(4.218)

BA3t  CLA3 ; t  0, 1, 2, , TF

(4.219)

BA6t  CLA6 ; t  0, 1, 2, , TF

(4.220)

BA9t  CLA9 ; t  0, 1, 2, , TF

(4.221)

BA12t  CLA12 ; t  0, 1, 2, , TF

(4.222)

BB3t  CLB 3 ; t  0, 1, 2, ,TF

(4.223)

BB6t  CLB 6 ; t  0, 1, 2, , TF

(4.224)

BB9t  CLB 9 ; t  0, 1, 2, ,TF

(4.225)

BB12t  CLB 12 ; t  0, 1, 2, ,TF

(4.226)

BC3t  CLC 3 ; t  0, 1, 2, ,TF

(4.227)

BC 6t  CLC 6 ; t  0, 1, 2, , TF

(4.228)

BC9t  CLC 9 ; t  0, 1, 2, , TF

(4.229)

BC12t  CLC 12 ; t  0, 1, 2, ,TF

(4.230)

BC12t  CLC 12 ; t  0, 1, 2, ,TF

(4.231)

BD3t  CLD 3 ; t  0, 1, 2, , TF

(4.232)

BD6t  CLD 6 ; t  0, 1, 2, ,TF

(4.233)

117

BD9t  CLD 9 ; t  0, 1, 2, , TF

(4.234)

BD12t  CLD 12 ; t  0, 1, 2, ,TF

(4.235)

BE3t  CLE 3 ; t  0, 1, 2, ,TF

(4.236)

BE 6t  CLE 6 ; t  0, 1, 2, ,TF

(4.237)

BE9t  CLE 9 ; t  0, 1, 2, , TF

(4.238)

BE12t  CLE 12 ; t  0, 1, 2, ,TF

(4.239)

 ; t 0 BLPt  CLLP

(4.240)

 ; t  0, 1, 2, , TF BCL1t , BLC 2t , BLC3t ,, BLCPt  CLLC

(4.241)

4.4.2

Project Cash Flow and Financing Flow Integration. The model proposed by

Hendrickson and Au (2000) is modified to integrate project cash flow and financing flow to prepare the model for optimization in Step 17 (Figure 4.1). The financing flow is integrated into the cumulative cash flow to get the cumulative net balance. The cumulative net balance of the cash flow (including financing flow), in period t , which is denoted by N t  , is computed by using Equation 4.242.

 At  NFt ; t  0 N t    Nt1  At  NFt ; t  1, 2, , TF

(4.242)

The cumulative net balance of the cash flow (including financing flow) must not be negative during the project. In addition, the contractor may want to include a contingency in the financing schedule to cover unpredicted expenses, or unexpected

118 changes in the cash flow forecast (Hendrickson and Au 2000). However, this requires more financing cost. Therefore, in Step 16 (Figure 4.1), the proposed model in this research allows the contractor to select the minimum cumulative net balance of the cash flow (including financing flow) at the end of each period whether it is zero with lower financing cost or it is larger than zero with higher financing cost. The minimum cumulative net balance of the cash flow (including financing flow) is denoted by MN and the constraint is provided in Equation 4.243.

N t  MN ; t  1, 2, , TF

4.4.3

(4.243)

Model Optimization. The optimization model is discussed below. The decision

variables, the objective function, and the constraints are defined separately. The decision variables are A3t , A6t , A9t , A12t , B3t , B6t , B9t , B12t , C 3t , C6t ,

C9t , C12t , D3t , D6t , D9t , D12t , E3t , E6t , E9t , E12t , LPt , LC1t , LC 2t , LC3t , LC 4t , LC5t , LC6t , LC 7t , LC8t , LC9t , LC10t , LC11t , LC12t , LCPt at the end of each period t  1, 2, 3, , TF . In Step 18 (Figure 4.1), the model is optimized. The objective function is the minimization of the total financing cost, which is denoted by Oobj , and is represented by Equation 4.244.

TF

Minimizing Oobj   Ft 0

(4.244)

119 In Step 19 (Figure 4.1), the financing outputs are obtained using Equations 4.62, 4.97, 4.165, 4.167, 4.168, and 4.242. In Step 20, the total required financing considering each alternative is computed using Equations 4.169 to 4.196. In Step 21, the monthly financing schedule of borrowed money is provided in the matrix using Equation 5.245. In Step 22, the monthly financing repayment schedule including interest is also provided in the matrix using Equation 5.246. The amount of repaid money and financing cost for each alternative are summed up at the end of each period to show the total repayment at the end of each month.

 A3   BA30  BA31  FPS   BA32  BA33     BA3 TF 

B3



C3





D3



E12

LP

 BB 30  BC 30  BD 30  BE12 0  BB 31  BC 31  BD 31  BE121

BLP0 TBLC 0 BLP1 TBLC1

 BB 32  BC 32  BD 32  BE12 2  BB 33  BC 33  BD 33  BE12 3

BLP2 TBLC 2 BLP3 TBLC 3

           BB 3TF  BC 3TF  BD 3TF  BE12TF BLPTF TBLCTF

 A3  B3  C3  D3  E12  RA3  FA3  RB3  FB3  RC3 FC3  RD3  FD3  RE3 FE3 0 0 0 0 0 0 0 0 0  0  RA31  FA31  RB31  FB31  RC31 FC31  RD31  FD31  RE31 FE31  FRS  RA32  FA32  RB32  FB32  RC32  FC32  RD32  FD32  RE32  FE32  RA33  FA33  RB33  FB33  RC33 FC33  RD33  FD33  RE33 FE33            RA3  FA3 RB3  FB3 RC3 FC3 RD3  FD3 RE3 FE3 T T T T T T T T T  T F

4.5

F

LC

F

F

F

F

F

F

F

F

 Codes  t  0 t  1  t  2 t  3   t  T F 

(4.245)

Codes  RLP0  FLP0 TRLC0 TFLC0 t 0  RLP1  FLP1 TRLC1 TFLC1 t 1  RLP2  FLP2 TRLC2 TFLC2 t  2 (4.246) RLP3  FLP3 TRLC3 TFLC3 t 3     RLPT  FLPT TRLCT TFLCT t TF LP

F

LC

F

F

F

Testing the First Stage Model

The network of an example project is shown in Figure 4.2. It is used to demonstrate the procedure explained in the preceding sections and carry out the application of the financing model. The schedule data of this project is read from an

120 Excel sheet by MATLAB 2013a. The schedule data are used to calculate the CPM by using the activity-on-node method, topological sorting, and the improved Dijkstra`s algorithm which was described in Section 4.2.

Start

A

D

G

J

M

B

E

H

K

N

C

F

I

P

O

L

Q

S

R

Figure 4.2. Network of the Example Project After running the model, the schedule data and output values of CPM calculations are displayed in tabular format in MATLAB (see Table 4.2). For this project which lasts 54 weeks (14 months), the cost data and contract terms dictate the information presented in Table 4.3. After the user enters the cost data, the output values of direct cost, total price, and weekly price of each activity are calculated and displayed in tabular format in MATLAB (see Table 4.4).

121 Table 4.2. Project Schedule Data Inputs and Model Output of CPM Calculations

Predecessor 3

Duration (weeks)

Direct cost ($/week)

Early start

Early finish

Late start

Late finish

Total float

Start time

Finish time

Activity name

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Start A B C D E F G H I J K L M N O P Q R

1 1 1 2 3 3 5 6 7 8 7 10 11 12 13 14 16 13

20

S

17

18

CPM calculations

Predecessor 2

Activity ID

Predecessor 1

Predecessors

3 4 6 9 9 12 12 14 16

7 15 -

0 10 6 12 4 3 5 5 8 7 6 5 9 8 5 4 6 7 5

0 50,000 100,000 60,000 60,000 40,000 90,000 20,000 30,000 60,000 90,000 100,000 60,000 40,000 70,000 70,000 90,000 60,000 50,000

0 0 0 0 10 6 12 17 9 17 22 17 24 28 22 36 36 40 40

0 10 6 12 14 9 17 22 17 24 28 22 33 36 27 40 42 47 45

0 3 5 0 13 11 12 17 14 20 22 22 27 28 31 36 41 40 42

0 13 11 12 17 14 17 22 22 27 28 27 36 36 36 40 47 47 47

0 3 5 0 3 5 0 0 5 3 0 5 3 0 9 0 5 0 2

0 0 0 0 10 6 12 17 9 17 22 17 24 28 22 36 36 40 40

0 10 6 12 14 9 17 22 17 24 28 22 33 36 27 40 42 47 45

19

7

60,000

47

54

47

54

0

47

54

122 Table 4.3. The Inputs of Cost Data and the Contractual Terms of the Project Data type Cost data

Contract terms

Item

Amount

Weekly fixed overhead cost OF

$25,000/week

Variable overhead percentage OV of (DC)

10%

Mobilization cost percentage OM of (DC+VOC)

5%

Markup percentage OP of (DC+FOC+VOC+MOB)

6%

Bond premium percentage OB of (DC+FOC+VOC+MOB+MP)

1%

Advance payment percentage OAP of contract bid price

0%

Retained percentage R of pay requests

10%

Number of months between submitting pay requests Lag in paying payment requests

LS

1 month

LP (months)

1 month

Lag to make the final payment and return the retained money

LR (months)

0 month

Table 4.4. The Model Output of Contract Bid Price Calculations of the Project Activity

Duration (weeks)


Total direct cost ($)

Total price ($)

Weekly price ($/week)

A B C D E F G H I J K L M N O P Q R

10 6 12 4 3 5 5 8 7 6 5 9 8 5 4 6 7 5

50,000 100,000 60,000 60,000 40,000 90,000 20,000 30,000 60,000 90,000 100,000 60,000 40,000 70,000 70,000 90,000 60,000 50,000

500,000 600,000 720,000 240,000 120,000 450,000 100,000 240,000 420,000 540,000 500,000 540,000 320,000 350,000 280,000 540,000 420,000 250,000

713,987.40 856,784.90 1,028,141.90 342,714.00 171,357.00 642,588.70 142,797.50 342,714.00 599,749.40 771,106.40 713,987.40 771,106.40 456,951.90 499,791.20 399,832.90 771,106.40 599,749.40 356,993.70

71,398.70 142,797.50 85,678.50 85,678.50 57,119.00 128,517.70 28,559.50 42,839.20 85,678.50 128,517.70 142,797.50 85,678.50 57,119.00 99,958.20 99,958.20 128,517.70 85,678.50 71,398.70

S

7

60,000

420,000

599,749.40

85,678.50

7,550,000

10,781,210

Total

123 The prices in Table 4.4 do not include the financing cost, and this table is prepared after the contract is signed. The direct cost (DC) of $7,550,000 is obtained from Table 4.4 using Equations 4.8 and 4.9. The fixed overhead cost (FOC) of $1,350,000 is calculated using Equation 4.10 by multiplying the project duration (in weeks) by weekly overhead cost of the project:

FOC  54  $25,000  $1,350,000

The variable overhead cost (VOC) of $755,000 is calculated using Equations 4.11 and 4.12 by multiplying the variable overhead percentage of (OV) by DC:

VOC  0.1  $7,550,000  $755,000

The mobilization cost (MOB) of $415,250 is obtained using Equation 4.13 by multiplying the mobilization cost percentage of (OM) by (DC+VOC):

MOB  0.05  ($7,550,000  $755,000)  $415, 250

The markup (MP) of $604,215 is calculated using Equation 4.14 by multiplying the markup percentage of (OMP) by (DC+FOC+VOC+MOB):

124 MP  0.06  ($7,550,000  $1,350,000  $755,000  $415, 250)  $604, 215

The cost of bonding (BD) of $106,745 is calculated using Equation 4.15 by multiplying the bond premium percentage of (OB) by (DC+FOC+VOC+MOB+MP):

BD  0.01  ($7,550,000  $1,350,000  $755,000  $415, 250  $604, 215)  $106,745

A contract bid price (CBP) of $10,781,210 is obtained using Equation 4.16 by summing up the direct cost (DC), the fixed overhead cost (FOC), the variable overhead cost (VOC), the mobilization cost (MOB), the markup (MP), and bonding cost (BD):

CBP  $7,550,000  $1,350,000  $755,000  $415,250  $604,215  $106,745  $10,781,210

The bid price factor (BPF) of 1.4280, which is used to calculate the price of activities and the payments that are made by the owner, is calculated using Equation 4.17 by dividing the contract bid price (CBP) by the direct cost (DC):

$10,781, 210 $7,550,000  1.4280

BPF 

After the contract bid price and bid price factor are calculated, the project cash flow forecast is computed as shown in Table 4.5, to be integrated with the project

125 financing flow in the next process. In Table 4.5, the cumulative net balance of the cash flow excluding financing flow (Nt) is constantly negative from the beginning of the project until the last payment of the owner. In terms of calculating financing cost and proving that this model is far more optimal than all past finance-based scheduling models, three different financing cases are considered based on the information presented in Table 4.6. In Case 1, all proposed financing alternatives are considered. The interest payment times of short-term and longterm loans are considered to be monthly and the interest payment time for the line of credit is considered to be based on the optimum time. In addition, no credit limit is considered for Case 1. In Case 2, the long-term loan is removed and only short-term loan alternatives and a line of credit are considered. In addition, in Case 2, a total credit limit of $100,000 is considered for the line of credit whereas no credit limit is considered for short-term loans. The interest payment times for both short-term loans and line of credit are the same as in Case 1. In Case 3, as it was the case in all past finance-based scheduling research, only a line of credit with no consideration of total credit limit is considered. It should also be mentioned that for the purpose of comparing the three cases, the APRs are considered to be the same in all three cases.

126 Table 4.5. Model Output of the Project Cash Flow Calculations Excluding Financing Flow (Page 1 of 2) End of Month

0 1

2

3

4

5

6

7

Disbursements

Mobilization and bonding Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E)

Amount ($)

Owner payment

521,995

Advance payment

0

521,995 840,000 100,000

Payment (P) Earned value Deduction of advance payment

0 0 0

84,000

Deduction of retainage

0

1,024,000 720,000 100,000


Amount ($)

0 1,199,499 0

72,000


892,000 590,000 100,000


59,000


102,814

749,000 600,000 100,000


925,328 842,505 0

60,000


84,251

760,000 660,000 100,000


758,255 856,785 0

66,000


85,678

826,000 800,000 100,000


771,106 942,463 0

80,000


94,246

980,000 810,000 100,000


81,000


991,000

Payment (P)

Nt ($)

-521,995

-1,545,995

119,950 1,079,549 1,028,142 0

848,217 1,142,380 0

-1,358,446

-1,182,118

-1,183,863

-1,238,757

-1,370,540

114,238 1,028,142

-1,333,398

127 Table 4.5. Model Output of the Project Cash Flow Calculations Excluding Financing Flow (Page 2 of 2) End of Month

8

9

10

11

12

13

14

15

Disbursements

Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E) Direct cost Fixed overhead cost Variable overhead cost Disbursement (E)

Amount ($)

Owner payment

Amount ($)

400,000 100,000

Earned value Deduction of advance payment

1,156,660 0

40,000


540,000 220,000 100,000


22,000


57,119

342,000 640,000 100,000


514,071 314,154 0

64,000


31,415

804,000 620,000 100,000


282,739 913,904 0

62,000


91,390

782,000 290,000 100,000


822,513 885,344 0

29,000


88,534

419,000 240,000 100,000


796,810 414,113 0

24,000


41,411

364,000 120,000 50,000


372,701 342,714 0

12,000


34,271

182,000 -


308,443 171,357 0

-


17,136

-

Release of retainage

1,078,121

-

Payment (P)

1,232,342

Nt ($)

115,666 1,040,994 571,190 0

-832,404

-660,333

-1,181,594

-1,141,081

-763,271

-754,570

-628,127

604,215

128 Table 4.6. Financing Data of Three Different Cases

No

Yes

Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Optimum time

No No No No No No No No No No No No No No No No No No No No -

No No No No No No No No No No No No No No No No No No No No No

Yes

Yes

Optimum time

Total credit limit

Yes

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No


15

No No No No No No No No No No No No No No No No No No No No No

Selected alternative

22

Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Optimum time

Case 3 Total credit limit


Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes



23 20 18 17 22 19 17 8 21 18 10 9 20 17 11 10 19 16 12 11 7


APR (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Case 2 Total credit limit

Alternative

Case 1

No

The resulting total borrowed money (Bt), total repaid money (Rt), total financing cost (Ft), net financing flow (NFt), cumulative net financing cost (NFCt), and cumulative cash flow (including financing flow) (N 't) are presented in Tables 4.7 to 4.9 for each of these cases separately. Moreover, the schedule of borrowed money and the schedule of repaid money including interest are presented in Tables 4.10 and 4.11, respectively.

129 Table 4.7. Model Output of Optimal Financing Results for Case 1 Month

Bt

Rt

Ft

NFt

NFCt

Nt

($)

($)

($)

($)

($)

($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1,377,532 264,507 72,594 351,126 0 0 230,086 62,996 0 0 212,345 158,599 0 91,437 37,106

0 91,835 254,034 266,738 91,835 91,835 91,835 91,835 154,832 91,835 91,835 189,663 364,951 91,835 154,511

0 4,209 6,109 7,470 6,468 6,468 6,468 8,303 9,041 8,303 8,303 9,449 12,859 8,303 9,037

1,377,532 168,463 -187,549 76,918 -98,303 -98,303 131,783 -37,142 -163,872 -100,138 112,207 -40,513 -377,810 -8,701 -126,443

0 4,209 10,317 17,788 24,255 30,723 37,191 45,494 54,534 62,837 71,139 80,588 93,447 101,750 110,786

855,537 0 0 253,246 153,197 0 0 0 337,121 409,054 0 0 0 0 0

15

0

738,914

9,415

-748,329

120,202

484,013

Table 4.8. Model Output of Optimal Financing Results for Case 2 Month

Bt

Rt

Ft

NFt

NFCt

Nt

($)

($)

($)

($)

($)

($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1,456,818 100,000 0 0 0 0 140,158 56,927 0 0 0 0 834,821 71,893 0

0 0 100,000 167,668 0 0 0 84,385 72,542 52,559 0 0 1,236,592 68,688 99,675

0 10,824 11,995 10,824 8,375 8,375 8,375 9,684 9,731 8,696 8,276 8,276 8,276 11,906 12,598

1,456,818 89,176 -111,995 -178,492 -8,375 -8,375 131,783 -37,142 -82,273 -61,254 -8,276 -8,276 -410,047 -8,701 -112,273

0 10,824 22,819 33,642 42,018 50,393 58,768 68,453 78,183 86,879 95,155 103,432 111,708 123,614 136,213

934,824 0 75,554 73,390 63,269 0 0 0 418,721 529,537 0 32,237 0 0 14,170

15

0

778,510

11,101

-789,611

147,314

456,901

130 Table 4.9. Model Output of Optimal Financing Results for Case 3

Bt

Rt

Ft

NFt

NFCt

Nt

($)

($)

($)

($)

($)

($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

521,995 1,093,258 383,785 301,449 305,139 325,653 399,449 320,242 98,444 102,495 682,928 327,901 188,688 304,689 284,285

0 68,456 563,466 467,102 294,595 261,945 258,364 344,700 577,798 259,794 152,319 356,016 547,113 298,783 393,706

0 802 7,869 10,675 8,799 8,815 9,301 12,684 21,640 14,772 9,348 12,398 19,384 14,607 17,022

521,995 1,024,000 -187,549 -176,328 1,745 54,894 131,783 -37,142 -500,994 -172,071 521,261 -40,513 -377,810 -8,701 -126,443

0 802 8,671 19,345 28,144 36,959 46,260 58,944 80,583 95,355 104,703 117,101 136,485 151,092 168,114

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15

0

796,241

32,924

-829,165

201,038

403,177

Month

4.5.1

Analysis of Financing Results of the First Stage Model

According to Tables 4.7 to 4.9, the results show that in Cases 1 and 2, the contractor retains more money at the end of each period since the cumulative net balance of the cash flow (including financing flow) ( Nt ) is frequently positive, whereas in Case 3, this cumulative cash flow (including financing flow) is positive only at the end of the project. Even though the contractor keeps more money in his/her hand in Cases 1 and 2, the cumulative net financing cost (NFCt) of these cases at the end of the project is less than Case 3 where just the line of credit is considered. Therefore, by considering the proposed model of financing, the contractor not only pays less financing cost, but also has more money for use as contingency funds.

Table 4.10. Model Output of Optimized Financing Inflow Schedule (Borrowed Money) for Each Case Month

Case 1

Case 2

Case 3

B12 ($)

C9 ($)

LP ($)

LC ($)

B12 ($)

C9 ($)

E3 ($)

LC ($)

LC ($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 0 0 351,126 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 230,086 0 0 0 0 0 0 0 0

1,377,532 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 264,507 72,594 0 0 0 0 62,996 0 0 212,345 158,599 0 91,437 37,106

1,236,592 0 0 0 0 0 0 0 0 0 0 0 0 0 0

52,559 0 0 0 0 0 40,158 0 0 0 0 0 0 0 0

167,668 0 0 0 0 0 0 0 0 0 0 0 738,351 0 0

0 100,000 0 0 0 0 100,000 56,927 0 0 0 0 96,470 71,893 0

521,995 1,093,258 383,785 301,449 305,139 325,653 399,449 320,242 98,444 102,495 682,928 327,901 188,688 304,689 284,285

15

0

0

0

0

0

0

0

0

0

351,126

230,086

1,377,532

899,582

1,236,592

92,717

906,019

425,290

5,640,399

Total required financing

131

Table 4.11. Model Output of Optimized Financing Outflow Schedule (Repaid Money Including Interest) for Each Case Month

Case 1 B12 ($)

C9 ($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 0 0 0 2,259 2,259 2,259 2,259 2,259 2,259 2,259 2,259 2,259 2,259 2,259

0 0 0 0 0 0 0 1,835 1,835 1,835 1,835 1,835 1,835 1,835 1,835

15

353,385

231,921

LP ($)

Case 2

Case 3

LC ($)

B12 ($)

C9 ($)

E3 ($)

LC ($)

LC ($)

0 96,044 96,044 96,044 96,044 96,044 96,044 96,044 96,044 96,044 96,044 96,044 96,044 96,044 96,044

0 0 164,098 178,164 0 0 0 0 63,734 0 0 98,974 277,672 0 63,410

0 7,956 7,956 7,956 7,956 7,956 7,956 7,956 7,956 7,956 7,956 7,956 1,244,548 0 0

0 419 419 419 419 419 419 739 739 53,298 320 320 320 320 320

0 2,448 2,448 170,116 0 0 0 0 0 0 0 0 0 10,781 10,781

0 0 101,171 0 0 0 0 85,373 73,577 0 0 0 0 69,493 101,171

0 69,258 571,334 477,777 303,394 270,759 267,666 357,384 599,437 274,566 161,667 368,414 566,498 313,390 410,728

96,044

66,979

0

40,479

749,132

0

829,165

132

133 As shown in Figure 4.3, Case 1 where all financing alternatives are considered, results in the minimum cumulative financing cost at the end of the project, whereas Cases 2 and 3 result in more financing cost. It should be noted that by adopting Case 3 as it was the case in all past studies, one incurs far more financing cost as compared to Case 1 which is proposed in this research (see both Figure 4.3 and Table 4.12). As shown in Table 4.12, if the contractor uses Case 1, the total financing cost of the project amounts to $120,202, whereas by adopting Cases 2 and 3, the total financing cost of the project amounts to $147,314 and $201,038, respectively. It should also be noted that in Case 1, the profit amounts to $484,013, whereas in Cases 2 and 3, the profit amounts to $456,901 and $403,177, respectively (see Figure 4.4 and Table 4.12). Therefore, the contractor achieves far more profit compared to past finance-based scheduling models by using the financing model proposed in this study. The proposed method could be particularly advantageous in a long-term project with a large contract price.

Cumulative Net Financing Cost NFCt ($)

225,000 200,000 175,000 150,000 125,000

Case 1

100,000

Case 2

75,000

Case 3

50,000 25,000 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time (month)

Figure 4.3. Cumulative Net Financing Cost NFCt of Financing Cases

134 Table 4.12. Summary of Project Financial Parameters Considering Each Case Financial Parameters Total borrowed money for the project ($) Total financing cost of the project ($) Profit of the project ($)

Cumulative Cash Flow Including Financing Cost Nt ($)

Required credit for line of credit ($)

Case 1

Case 2

Case 3

2,858,326 120,202 484,013

2,660,617 147,314 456,901

5,640,399 201,038 403,177

274,457

100,000

1,552,110

950,000 900,000 850,000 800,000 750,000 700,000 650,000 600,000 550,000 500,000 450,000 400,000 350,000 300,000 250,000 200,000 150,000 100,000 50,000 0

Case 1 Case 2 Case 3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time (month)

Figure 4.4. Cumulative Cash Flow Including Financing Cost Nt of Financing Cases Interestingly, as shown in Table 4.10, in Case 2 where only the long-term loan is removed and a total credit limit is considered for the line of credit, not only the optimum alternatives are changed compared to Case 1, but also the schedule of borrowed money is changed as well. Regarding Case 1, the optimum financing alternatives are B12, C9, LP, and LC, whereas considering Case 2, the optimum financing alternatives are B12, C9, E3,

135 and LC. This situation occurs even though the interest rate does not change from case to case. It should be taken into consideration that in Cases 1 and 2, not only the financing cost is less compared to Case 3, but also the contractor can avoid a large overdraft on its primary account. For example, as shown in Table 4.12, regarding Case 1, the required credit for the line of credit is $274,457, whereas this amount increases drastically to $1,552,110 in Case 3. It should also be noted that lenders may ask a higher interest rate where the required credit limit increases to this large number, which worsens Case 3 with increased financing cost. As shown in Table 4.10, in Case 1, in contrast to the belief that the long-term loan is not a good idea for temporary needs, the big portion of borrowed money should be taken as a long-term loan (i.e., $1,377,532 out of 2,858,326). The reason is that if the money is withdrawn from a line of credit and is not repaid for a period of time, the interest is compounded; therefore, when the cash flow is negative during the project, there is a need for another source to repay both the interest and the money withdrawn from the line of credit to prevent the financing cost from being compounded. As a proof, in Case 1, all withdrawn money from the line of credit is repaid after one or maximum two months and it is the reason why short term loans of 3 to 6 month duration are not used. For example, in Case 2 where the long-term loan does not exist, alternative E3 is another optimum alternative whereas it is not the optimum alternative in Case 1. In addition, as shown in Table 4.10, considering Case 1, the short-term loans of 12 and 9-month periods (i.e., alternatives B12 and C9) in months 3 and 6, are better than the long-term loan. It is true that the interest rate of alternatives B12 and C9 are higher than the interest rate of the long-term loan, but because of the lower duration and the

136 condition of the cash flow in those months, alternatives B12 and C9 are more advantageous. Another aim of this research is to provide a schedule for managing a line of credit despite the complexity and unscheduled nature of a line of credit. As shown in Tables 4.10 and 4.11, this objective is achieved. In addition, the proposed financing model is also capable to find the optimum time when the interest and withdrawn money from the line of credit should be repaid (see Table 4.11). Moreover, even though there is a belief that borrowing more money results in paying more financing cost, as shown in Table 4.12, this is not true. In Case 1 the total borrowed money is more than in Case 2, whereas the financing cost of Case 1 is less than in Case 2. It is also important to know that considering a different cash flow, different APRs, and different credit limits can change the optimum result.

4.5.2

Sensitivity Analysis and Validation of the First Stage Model. In addition to the

fact that changes in financing input data change the total financing cost of the project, changes in the contract terms also can change the financing cost. As shown in Table 4.13, a sensitivity analysis of the best and worse financing cases (i.e., Cases 1 and 3) is performed considering changes in the contract terms of retained percentage of pay requests (R), advance payment percentage of contract bid price (OAP), and lag in owner payments (LP) (months) to see the effects of contract terms on the amount of total financing cost. As shown in Table 4.13, in sensitivity analysis Number 1, the owner not only pays the contractor without any lag (i.e., LP  0 ), but also does not withhold any

137 retainage (i.e., R  0 ). There is a financing cost even in this case, because the contractor does not receive any advance payment at the beginning of the project where the contractor should pay for the mobilization and bonding costs. In sensitivity analysis Numbers 5, 6, 8, and 9 where the owner makes an advance payment and does not withhold any retainage, the contractor still has to pay financing cost due to the lag in owner payments. In sensitivity analysis Numbers 4, 7, 13, and 16, no financing cost is incurred when the owner makes an advance payment, and makes the intermediate payments without any delay. However, in sensitivity analysis Numbers 13 and 16, even if the owner withholds retainage, the total financing cost amounts to 0 because the markup covers the amount of retainage in each period. On the other hand, in sensitivity analysis Number 25, the owner again makes an advance payment in addition to paying the intermediate payments without any delay, but the retained percentage is more than profit; therefore, the contractor pays financing cost. The importance of the retainage can be identified in sensitivity analysis Numbers 19 and 28 where the owner makes payments without delay, but withholds retainage of 10 and 15 percent, respectively. As a result, the total financing cost jumps from $42,111 to $82,550 just because of the increase in the percentage of retainage.

138 Table 4.13. Sensitivity Analysis of the Best and Worst Financing Cases Contract terms

Case1

Case 3

Total financing cost of the project ($)

Total financing cost of the project ($)

Number

R (%)

OAP (%)

LP (months)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15

0 0 0 5 5 5 10 10 10 0 0 0 5 5 5 10 10 10 0 0 0 5 5 5 10 10 10 0 0 0 5 5 5 10 10

0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1

14,373 67,354 143,472 0 40,273 112,269 0 13,328 81,066 18,348 87,650 167,161 0 59,769 135,515 0 31,966 103,869 42,111 120,202 197,695 18,320 92,198 166,083 9,105 64,194 134,471 82,550 154,300 230,872 62,186 126,297 199,263 42,369 98,758

14,753 105,059 232,441 0 54,699 166,522 0 14,211 109,365 23,662 149,324 281,807 0 89,692 214,657 0 39,747 154,789 70,089 201,038 333,520 18,733 141,406 266,370 9,106 88,972 206,503 121,803 252,752 385,234 69,602 193,119 318,084 42,370 140,685

36

15

10

2

167,685

258,217

139 It should also be noted that regardless of changes in the contract terms, financing cost in Case 3 is bigger than in Case 1, and proves that the financing model proposed in this research always performs better than Case 3 regardless of contract terms. The significance of the proposed model can be more visible when the project duration is longer or the contract terms are unfavorable. For example, in sensitivity analysis Number 30, where the conditions are quite extreme (with 15% retainage, 0% advance payment, and 2 months delay in owner payments), the difference between Case 1 and Case 3 is $154,362 (i.e., $230,872 subtracted from $385,234), whereas considering the contract terms of the project in Number 20, where the conditions are quite average (with only 10% retainage, no advance payment, and only 1 month delay in owner payments), the difference is only $80,836 (i.e., $120,202 subtracted from $201,038).

4.6

Conclusion of the First Stage

three significant improvements are achieved by using the model of this stage, compared to all past research that has considered a work schedule at the normal point: (1) the total financing cost calculated by the first stage model is far less than the financing cost in the past research; therefore, contractors achieve more profit by using this model; (2) since more financing alternatives are included in this model, contractors do not need to change the start time of activities to satisfy cash constraints; therefore, financing needs are satisfied without using the total float of activities which results in reducing the risk of a critical work schedule; and (3) the schedule does not need to be extended due to problems in cash availability since more financing alternatives are provided, an optimum set of alternatives is selected and an optimum financing schedule is produced. In addition

140 to these three enhancements, the financing model proposed in this research provides financing schedules for both borrowed money and repaid money including interest. This model even provides schedules for a line of credit that has an unscheduled nature. Moreover, this model avoids imposing a large overdraft on the primary account. Usage of the model can be particularly advantageous when the project duration is long or the contract terms are unfavorable. During the project, a new financing schedule can be obtained by updating the schedule for those financing alternatives that can be changed. It should also be noted that even though the proposed model in this stage is used to maximize the profit of the contractor after the contract is signed, it can also be used at the bidding stage to reduce the bidding price to increase the chance of winning the contract for the contractor.

141 CHAPTER 5 5. SECOND STAGE OF THE RESEARCH In the second stage, both the time-cost tradeoff problem (TCTP) and the financebased scheduling problem (FBSP) are addressed. Since the few researchers who have solved the FBSP problem considered TCTP in their models with no regard to financing optimization, the integration of TCTP and FBSP using financing optimization would be a marked improvement. In the second stage, the proposed models optimize the profit by minimizing the total cost of the project including optimal financing cost (i.e., minimizing the summation of direct, indirect and optimal financing cost) where the early bonus is considered (if any) and the total payment is constant for different project completion times. In contrast to the first stage of this study where the optimal result was achieved by considering only normal durations, in the second stage, different activity acceleration methods are considered for each activity. Therefore, in the second stage of this study, activity acceleration methods with durations between normal and crash durations are considered as another variable to be optimized while all activities start at their early start time. In addition, since the early completion bonus (if any) can change the optimal project completion time and project schedule, it is added to the second stage models. Still, the idea at this stage is not to extend the scheduled completion, assuming that there is a sufficient amount of money to perform a project where financing alternatives have been optimized. Therefore, extending the project duration is not considered in this stage as was done in the first stage model. Two models are proposed for the second stage. Model 1, which is proposed for academic research, takes longer computation time because the optimal schedule (i.e.,

142 optimal activity acceleration methods) that results in the optimal profit and total cost including optimal financing cost, are computed for every project completion time between crash and normal points. The reasons why the optimal results are computed for every project completion time between crash and normal points in Model 1 are: (1) to draw the curves of the financial indicators (e.g.., profit, total cost excluding financing cost, financing cost, total cost including financing cost) of a project between crash and normal points for different cases, (2) to see the difference between the optimal results obtained for different project completion times, to perform a sensitivity analysis, and to make a decision. Model 2, which is proposed as practice-oriented research, takes less computation time compared to the Model 1 since it only searches for one optimal schedule (i.e., optimal activity acceleration methods) that results in an optimal profit rather than finding an optimal schedule for every project completion time between crash and normal points. The second stage models differ from all past studies that addressed both TCTP and FBSP in three respects. First, the second stage models achieve better financing cost and profit compared to past studies. Second, since the optimal profit depends on the total cost including financing cost, the profit can be increased if the financing cost is optimal. Financing cost is optimal if financing optimization is used considering different financing alternatives (it was proved in Chapter 4). Therefore, the total cost including financing cost should be calculated using the optimized financing cost, not just using financing cost as it was the case in all past studies. As a result, there are two types of variables in the second stage models: (1) activity acceleration methods of each activity which specify the durations and costs of each activity and (2) financing variables. The second stage models

143 are proposed by combining the genetic algorithm and linear programming and introduces a new hybrid GALP algorithm. The financing variables and financing cost are optimized using linear programming inside a genetic algorithm which is used to find the optimal schedule (i.e., optimal activity acceleration methods) that results in the optimal profit. Third, as it was discussed in Chapter 2, genetic algorithms have deficiencies in terms of computational time, reliability and practicality. Lee et al. (2015) proposed a new GA model that eliminates the deficiencies of past GA algorithms. The hybrid GALP algorithm used in the second stage not only uses the methodology of Lee et al. (2015) to create an initial parent chromosome and identify the optimal values of the GA parameters (i.e., population size, crossover percentage, mutation percentage, mutation probability, and stopping rule), but also adopts a controlled experiment method to identify optimal GA methods (i.e., parent selection method and crossover operation method) that affect the computational time and reliability of the results. Since no research study that has used GA algorithms to address both TCTP and FBSP eliminated the deficiencies of GA, removing these deficiencies adds another advantage in the second stage of this study compared to past studies. In this chapter, first, the differences from the first stage model in terms of project schedule creation and creation of cash flow forecast are discussed. Then, the computational process of the second stage model are presented. Subsequently, the second stage models are tested for two different construction schedules: (1) a small network and (2) a large network. For the small network, two financing cases (i.e., financing Case 1 including all financing alternatives and financing Case 2 including just line of credit) are tested using Model 1 (i.e., the academic model) to show the improvement in the optimal

144 results (i.e., the optimal profit and financing cost) by using the second stage model over past studies that considered just line of credit in their financing model. In addition, the large network is tested using both Models 1 (i.e., academic model) and 2 (i.e., practical model): (1) to validate the hybrid GALP, (2) to compare the computational time between Models 1 and 2, and (3) to prove the practicality of the second stage models for a large network.

5.1

Project Schedule Creation for the Second Stage Models

The main purpose of the second stage models are to find the optimal construction schedule by finding optimal activity durations and costs (i.e., activity acceleration methods, normal durations and costs, and accelerated durations and costs) to obtain the maximum profit. To calculate the profit of the project, a cash flow forecast should be created. To create cash flow forecast, the start and finish times of activities should be computed considering different durations and costs for each activity. To calculate the start and finish times of activities, the early start, early finish, late start, and late finish times of activities should be calculated. To calculate early times and late times of activities, project scheduling is performed using the activity-on-node method, topological sorting, and improved Dijkstra`s algorithm for CPM which was proposed in Chapter 4. However, for the second stage models, the alternative activity acceleration methods (i.e., activity accelerations indexes, normal durations and costs, and accelerated durations and costs) should be integrated to the schedule data (i.e., activity ID, name, predecessors). Therefore, Equations 4.1 to 4.7 presented in Chapter 4 are modified to integrate the

145 alternative activity acceleration methods. In the second stage models, the project schedule is created using Equations 5.1 to 5.8.

ES j  max[old label of j , (old label of i  D mi i )], where ES1  0 and i  j

(5.1)

EFj  ES j  D mi j ; j  1, 2, 3, ..., n

(5.2)

LFi  min[old label of i, (old label of j - D

mj j

)], where LFn  EFn and i  j

(5.3)

LSi  LFi  Dmi i ; i  n, n  1, n  2, ..., 1

(5.4)

TFi  LSi  ESi ; i  n, n  1, n  2, ..., 1

(5.5)

ST j  ES j ; j  1, 2, 3, ..., n

(5.6)

FT j  EFj ; j  1, 2, 3, ..., n

(5.7)

W  max EFj

(5.8)

where; Dmi i = duration of predecessor activity i using acceleration method mi for activity

i; D

mj j

= duration of successor activity j using acceleration method m j for activity j ;

EFj = early finish time of activity i ; ES j = early start time of activity j ; FT j = finish

time of activity j ; LFi = late finish time of activity i ; LSi = late start time of activity i ;

n = number of activities; ST j = start time of activity j ; TFi = total float of activity i ; and W = project duration computed in weeks.

146 5.2

Creation of Project Cash Flow Forecast for the Second Stage Models

As it was discussed in Chapter 4, the cash flow forecast should be created to find the optimal financing cost and variables. However, since the second stage models are to be used after the contract is signed, the contract price (CP ) and cost of bonding ( BD ) are considered to be constant over the changes in project completion time (W ) . Therefore, using Equations 5.9 to 5.17, the contract price and cost of bonding are computed by considering normal activity acceleration methods for all activities (i.e.,

mi  1 ).

nk

DCk   C mi ik ; k  1, 2, , W

(5.9)

i 1

W

DC   DCk

(5.10)

FOC  OF  W

(5.11)

VOCk  OV  DCk ; k  1, 2, , W

(5.12)

k 1

W

VOC  VOCk

(5.13)

MOB  OM  ( DC  VOC )

(5.14)

MP  OMP  ( DC  FOC  VOC  MOB)

(5.15)

BD  OB  ( DC  FOC  VOC  MOB  MP)

(5.16)

CP  DC  FOC  VOC  MOB  MP  BD

(5.17)

k 1

where; BD = cost of bonding; C mi ik = direct cost of activity i in week k using acceleration method mi for activity i ; CP = contract price considering normal

147 acceleration method mi  1 for all activities; DC = direct cost of project; DCk = net direct cost in week k ; FOC = fixed overhead cost of project; MOB = mobilization cost; MP = markup of project; n K = the number of activities whose duration overlaps with week k ;

OB = bond premium percentage; OF = fixed overhead cost per week; OM = mobilization cost percentage; OMP = markup percentage; OV = variable overhead percentage of the direct cost; VOC = variable overhead cost; and VOCk = variable overhead cost in week k . The cash outflow is also computed using Equations 5.9 to 5.14 and 5.18 where the cost of bonding ( BD ) is considered to be constant, and the mobilization cost ( MOB ) , net direct cost in week k ( DCk ) , variable overhead cost in week k (VOCk ) , and project completion time computed in weeks (W ) and months ( M ) are changed by using different activity acceleration methods for different activities, hence changing the duration and cost of activities. Therefore, the cash outflow changes when different sets of activity acceleration methods are adopted. The cash outflow, Et which is the construction disbursement at the end of each period t , is calculated using Equation 5.18.

 MOB  BD  Et   4t   ( DCk  OF  VOCk ) k  (4t  3)

t 0 t  1, 2, , M

(5.18)

By choosing different sets of activity acceleration methods, not only the costs, durations, project completion time, project completion cost excluding the financing cost (TC ) , and the cash outflow are changed, but also the cash inflow is changed, whereas the

148 total payment is constant over the changes of the project completion time. The reason why the cash inflow also changes is because the start times and finish times of activities are changed choosing different sets of activity acceleration methods. To calculate the cash inflow, first, the contract factor (CF ) should be calculated each time when the sets of activity acceleration methods are changed. Using Equation 5.19, where the contract price is fixed and the direct cost of the project changes by choosing different sets of activity acceleration methods, the contract factor (CF ) is calculated. Then, the cash inflow is calculated using Equations 5.20 to 5.23.

CP DC

(5.19)

AP  OAP  CP

(5.20)

CF 



PT  L   (1  R )  CF  P



4T



k  ( 4  ( T  LS )  1)



AP  LS



M

DCk  

;T  (1  LS ), (2  LS ), , ( d  LS )  M

(5.21)

4T   AP  ( M  d  LS ) PT  LP  (1  R)  CF  DCk   ; T M  M     (4 1) k d L S  

(5.22)

PT  LR  R  CF  DC ; T  M  LP

(5.23)

where; AP = advance payment; d = penultimate pay request submission prior to the pay request submission for the last progress payment; LR = lag to make the final payment and return the retained money (in months); LS = number of months between submitting pay requests; LP = the lag in paying payment requests (in months); OAP = advance payment percentage of contract bid price; PT  LP = progress payment at the end of each period

149

T  LP  ;

PT  LR = final payment; R = retained percentage of pay requests; T = the time

when the requests for the owner payment should be submitted; and T  LP = the time when the progress payment is made. After the cash outflow and inflow are created, the cash flow forecast can be created using Equations 5.24 to 5.26.

TF  M  LP  LR

(5.24)

At  Pt  Et ; t  0, 1, 2, , TF

(5.25)

 At ; t  0 Nt    Nt 1  At ; t  1, 2, , TF

(5.26)

where; At = net operating cash flow in period t ; Nt = cumulative cash flow excluding financing flow at the end of period t ; Nt 1 = cumulative cash flows from period 0 to period (t  1) ; and TF = time of final payment. To have the cash flow including financing flow, the financing flow should be integrated to the cash flow. The financing equations and model do not change over the stages of this research. Therefore, all equations of Chapter 4 for financing parameters are similar from stage to stage. As a result, the cumulative net balance of the cash flow (including financing flow), in period t , which is denoted by N t  , is computed by using Equation 5.27 where NFt is the net financing flow at the end of period t .

150

 At  NFt ; t  0 N t    Nt1  At  NFt ; t  1, 2, , TF

5.3

(5.27)

Model Optimization

The optimization model is formulated in three steps: (1) decision variables, (2) constraints, and (3) objective functions. Below, first the decision variables are discussed, and then the constraints and objective functions are presented.

5.3.1

Decision Variables. Two types of variables are considered in the second stage

models: (1) activities` acceleration methods and (2) financing variables. Each acceleration method represents the duration and cost of the activity. An activity with a different acceleration method has a different activity duration and cost. Optimization consists of selecting an optimal acceleration method from the available set of alternative activity acceleration methods for each activity to maximize the profit of the project after including the optimized financing cost. In other words, the financing model including different alternatives should be optimized for different sets of activity acceleration methods to find out what schedule results in maximum profit. The decision variables of activity acceleration methods and financing variables are presented in Equations 5.28 and 5.29, respectively.

X 1  mi  ; i  1, 2, , n

(5.28)

X 2  a3t , a6t , a9t , a12t , LPt , LCbt , LCPt  ; t  1, 2, , TF ; a  A, B, C , D, E; b  1, 2, ,12

(5.29)

151 5.3.2

Constraints. The constraints are formulated in two categories: (1) constraints of

financing, and (2) constraints of cumulative net balance of cash flow (including financing flow). For the financing constraints, there may be a predetermined credit limit relative to each alternative or there may not. If a predetermined credit limit is considered, two cases can exist: (1) total credit limit, and (2) credit limit in each period. The user can simply input the information about limitations for borrowing money, if any. Considering the second category of constraints, the cumulative net balance of the cash flow including financing flow must be positive at the end of each period. If the user wants to consider a contingency in the financing schedule, the minimum cumulative net balance of the cash flow including financing flow at the end of each period should be bigger than the minimum amount that is input by the user. Both these two categories of constraints were presented in Chapter 4.

5.3.3

Objective Functions. Although two objective functions are formulated for the

second stage models (i.e., financing cost, and profit), the main objective function is to maximize profit. Profit depends on the summation of total cost excluding the financing cost, optimized financing cost, and bonus. The total cost excluding the financing cost depends on the selection of activity acceleration methods. The financing cost depends on the cash flow and the cash flow depends on the selection of activity acceleration methods as well. Although both total cost and financing cost depend on the selection of the acceleration method for each activity, the financing cost is highly dependent on the cash flow. In other words, there can be multiple sets of activity acceleration methods that result in the same minimum total cost, whereas each of them results in different

152 financing cost since they create different cash flows. Therefore, finding the maximum profit requires optimizing financing for different sets of activity acceleration methods regardless of the total cost. To find the optimal profit, first, for the selected activity acceleration methods, the total indirect cost is calculated using Equation 5.30 where IDC is the indirect cost of the project that will vary according to changes in project completion time. Then, the total cost excluding financing cost which is denoted by TC is calculated using Equation 5.31. Afterwards, using Equation 5.32, the financing cost should be minimized using different financing alternatives. Finally, using Equation 5.33, where OBS represents early completion bonus per week, the maximum profit is selected where the summation of total cost excluding financing cost, optimal financing cost, and bonus is minimum. Since the contract price is constant regardless of the project completion time, the contract price does not change the optimal result. It should be noted that if the financing cost is not considered or if the cash flow is positive throughout the project, the second stage models would be converted to a time-cost tradeoff problem (TCTP).

IDC  MOB  BD  VOC  FOC

(5.30)

TC  DC  IDC

(5.31) TF

Minimize Obj1 ( FC )   Ft

(5.32)

Maximize Obj2 ( PR)  CP  OBS  W  (TC  Obj1 ( FC ))

(5.33)

0

153 5.4

Computational Process of the Second Stage Models

As it was discussed in the preceding section, the second stage models use the hybrid GALP algorithm to optimize the objective functions under specified constraints. The computational method developed in this stage integrates improved Dijkstra`s algorithm for CPM and the hybrid GALP algorithm into a single algorithm which is shown in Figure 5.1. The second stage models are performed in four main stages: (1) preparation of input data, (2) identification of optimal values of the GA parameters, (3) identification of optimal GA methods, and (4) execution of hybrid GALP. Although two models are presented in the second stage (i.e., Models 1 and 2), the algorithm shown in Figure 5.1 represents both models. In addition, the hybrid GALP algorithm is performed in four phases: (1) initializing the population, (2) evaluating the fitness function, (3) performing the process of evolution, and (4) initializing the next generation population. These four phases of the hybrid GALP algorithm are shown in Figure 5.2.

5.4.1

Input Data Preparation. In Step 1 (Figure 5.1), the schedule information (i.e.,

activity ID, name, predecessors, activity acceleration methods, normal durations and costs, and accelerated durations and costs) is read by an automated system using MATLAB from the Excel sheet. In Step 2 (Figure 5.1), two extreme boundaries are calculated using activity-on-node method, topological sorting, and the improved Dijkstra`s algorithm. In other words, the most accelerated project completion time (crash duration) and the normal project completion time (normal duration) are calculated. Although there are several intermediate alternatives between these two boundaries, these two extreme boundaries are calculated because they are used for the creation of the

154 financing flow and initialization of parent chromosomes. In Step 3 (Figure 5.1), the project cost data and contract terms are sought from the user. The contract price and cost of bonding are calculated in Step 4 (Figure 5.1). In Step 5 (Figure 5.1), the financing data are read from the Excel sheet, and in Step 6 (Figure 5.1) the financing flow, constraints, and variables are set up for every project completion time between crash and normal points, and are saved for use by the hybrid GALP algorithm for optimization. Therefore, when the project completion time is calculated by the hybrid GALP algorithm, the related financing flow, constraints, and variables should be used to be optimized. Step 6 (Figure 5.1) is performed separately from the hybrid GALP algorithm to enhance the computational time of the algorithm. In Figure 5.3, a sample chromosome representing a set of activity acceleration methods for a project is shown. Generally, the chromosome has a string of N genes. The number of genes in a chromosome is equal to the number of activities (e.g., six genes for a chromosome in Figure 5.3 for a schedule that has six activities). The boxes representing activity genes indicate the activity acceleration methods used in those activities. Each activity can be executed by three activity acceleration methods in which each activity acceleration method is associated with a unique activity duration and activity cost. The integer written in the boxes (i.e., 1, 2, and 3) represents the activity acceleration index used in each activity. Activity acceleration 1 means normal duration and cost, activity acceleration 2 means accelerated duration and associated cost, whereas activity acceleration 3 represents crashed duration and cost.

155

1. Reading project schedule information 2. Calculating crash and normal boundaries using improved modified Dijkstra`s algorithm

4. Computing contract price [CP] and cost of bonding [BD] 5. Reading financing data [financing alternatives, APRs, interest payment times, total credit limits, credit limits in each period] 6. Creating net financing flow [NFt] and constraints for every project completion time between crash and normal points

Preparation of input data

3. Inputting cost data [OF, OV, OM, OMP, OB] and contract terms [OBS, OAP, R, LS, LP, LR]

7. Creating an initial parent chromosome using an initialization method [Model 1: accelerated or normal parent, Model 2: matching parent]

8. Identifying the optimal values of GA parameters and methods? Yes 9. Setting the ranges and increments of GA parameters and stopping rules 10. Performing optimal GA parameters identification operation [Model 1: for normal point, Model 2: for an optimal solution between crash and normal points] 14. Resetting GA parameters

11. Saving outputs obtained in each GA cycle 12. Identifying the optimal values of GA parameters

No

13. Are GA parameters optimal? Yes 15. Setting optimal GA parameters and stopping rule

17. Saving outputs obtained in each GA cycle 18. Identifying the optimal GA methods 19. Setting optimal GA methods and updating stopping rule 20. Identifying best solution 21. Initializing parent chromosome using best solution 22. Executing hybrid GALP [Model 1: for every project completion time, Model 2: for an optimal solution] 23. Obtaining optimal results [PR, W, IDC, TC, FC, construction schedule, financing schedule]

Figure 5.1. Second Stage Algorithm

Identification of optimal GA methods Execution of hybrid GALP

16. Performing optimal GA methods identification operation [Model 1: for normal point, Model 2: for an optimal solution between normal and crash points]

Identification of optimal values of the GA parameters

No

156

Initializing the population

1. Reading GA parameters and methods 2. Generating an initial parent chromosome using an initialization method 3. Calculating the project completion time in weeks [W] and months [M] 4.2. Optimimzing financing variables and cost [FC] using linear programming

4.3. Calculating the early completion bonus

5. Calculating the profit [PR] and store the solution

6. All parent population [Np] evaluated?

No

7. Generating a random parent chromosome

Evaluating the fitness function

4.1 Calculating the total cost excluding financing cost [TC]

Yes No

8. All children population [Nc] evaluated?

11. All mutant population [Nm] evaluated? 10. Applying crossover using a crossover operation [single point, or double point, or uniform, or combined crossover]

No

Yes 12. Selecting one parent chromosome randomly 13. Applying mutation using the mutation operation

Performing the process of evolution

Yes 9. Selecting two parent chromosomes using a selection method [tournament or roulette wheel]

14. Merging parent, children, and mutant population [Np, Nc, Nm] to form the new combined population [Ng]

16. Truncating the new combined population [Ng] by keeping the top [Np] population to create the parent population of the next generation [Ng+1]

No

17. Is the stopping rule met?

Yes 18. Selecting the first parent chromosome as the best solution

Figure 5.2. Hybrid GALP Algorithm for the Second Stage Models

Initializing the next generation population

15. Sorting the new combined population [Ng]

157 A

B

C

D

E

F

1

1

3

2

1

3

Activity ID Activity Acceleration Index

Figure 5.3. A Sample of Encoding Chromosome In Step 7 (Figure 5.1), an initial parent chromosome is created using an initialization method. Lee et al.`s (2015) method is adopted to initialize the initial parent chromosome instead of initializing all parent chromosomes randomly. Lee et al. (2015) proved that their method to generate the initial parent chromosome enhances computational time. Three alternative initialization methods can be used in the second stage models. If the Model 1 is used, both “accelerated parent” (i.e., initializes the parent chromosome at an arbitrary point between normal and crash points) and “normal parent” (i.e., initializes the parent chromosome by allocating normal acceleration methods to all activities) can be used. However, in the next section, it is proved that using a normal parent results in a better computational time for Model 1. Since Model 1 runs the hybrid GALP algorithm for every project completion time between crash and normal points, and starts from normal project completion time and moves to the crash project completion time, the initial parent chromosome should be created for each project completion time separately. For each project completion time, the critical activities are identified, then, a critical activity is picked, and is accelerated (i.e., the number in the chromosome cell is increased by one digit). This is done to lead the GA to seek the best solution for each project completion time faster. This approach is created for Model 1 by this research. Considering Model 2, just a “matching parent” (i.e., initializes the parent chromosome by allocating normal duration and cost (i.e., activity acceleration index=1) to noncritical

158 activities, and the crashed duration and cost (i.e., activity acceleration index=3) to the critical activities) can be used as an initial parent chromosome. Lee et al. (2015) proved the enhancement in computational time when using this approach.

5.4.2

Identification of the Optimal Values of the GA Parameters. Basically, the GA

is executed by allocating values to the GA parameters (i.e., parent population ( N p ), crossover percentage ( C p ), mutation percentage ( M p ), and mutation probability ( M pb )). Although past studies selected GA parameters arbitrarily, Lee et al. (2015) proposed a new GA algorithm to identify the optimal values of GA parameters without resetting the GA parameters manually when other experiments are needed to be performed to validate an optimal GA result. Lee et al. (2015) proved that the proposed model enhances the GA algorithm in terms of reliability and practicality where the GA runs for different sets of GA parameters to see what happens to the computational time and the optimal result (Lee et al. 2015). Although Lee et al. (2015) proposed the model to find the optimal values of the GA parameters, the optimal GA methods (i.e., parent selection method and crossover operation method) that affect the computational time and reliability of the results were not discussed. In the proposed algorithm for the second stage models, the optimal GA methods are found after identifying the optimal values of GA parameters. However, in Step 8 (Figure 5.1), the user can decide whether to use the strategy to identify the optimal values of GA parameters and methods. If the user does not want to use this strategy, the hybrid GALP algorithm runs in Step 22 (Figure 5.1) where the GA parameters are allocated arbitrarily by the user. If the user adopts the strategy, the minimum, maximum, and the increment of each GA parameter (i.e., N p , C p , M p , M pb ) and stopping rules are

159 set by the user in Step 9 (Figure 5.1). Similar to Lee et al.`s (2015) method, three stopping rules (i.e., “TimeLimit”, “StallGenLimit”, and “StallTimeLimit”) are used for the termination of GA iterations. First, if the algorithm runs more than a certain period of time, the iterations are terminated regardless of the results. This certain period of time is called “TimeLimit”. If the “TimeLimit” is not satisfied, and there is no improvement in the output (i.e., the change in the result is less than the tolerance value of 1), the stall generations and stall time are counted. Then, the GA iterations stop as soon as any one of stall generations or stall time is satisfied by “StallGenLimit” and “StallTimeLimit”, respectively. In Step 10 (Figure 5.1), hybrid GALP experiments are conducted using different combinations of the parameters based on controlled experiments to see the changes to the outputs (i.e., computational time and optimal result). All GA parameters (i.e., N p , C p , M p , M pb ) alongside the computational time, optimal result, the result of stopping rules, and number of generations ( It gen ) that were necessary to reach the optimal solution are saved in Step 11 (Figure 5.1). It should be noted that if the Model 1 is used, the algorithm is conducted for the normal point (i.e., normal project completion time) using different combinations of parameters to identify the optimal values of GA parameters. If the Model 2 is used, the algorithm is conducted for identifying an optimal solution for the whole problem between crash and normal points. Concerning Model 1, the reason why these parameters are optimal for other points (i.e., other project completion times) between crash and normal points is because the GA parameters are problem-dependent; since the number of genes (variables) and the ranges of genes (number of acceleration methods) are not changed, then, the problem and the identified values of GA parameters will not change. In Step 12 (Figure 5.1), the optimal values of

160 the GA parameters are identified (i.e., N p , C p , M p , M pb , Itmin ). Similar to Lee et al.`s (2015) method, if the optimal result is found at the maximum boundaries of the GA parameters, the upper boundaries of GA parameters are relaxed and the GA parameters are reset in Step 14 (Figure 5.1) to repeat the process of identifying the optimal values of the GA parameters (i.e., repeating Steps 10-13) using new combinations of the parameters. It should be noted that the reason why minimum generation ( Itmin ) to reach the optimal solution is identified is because it ensures the GA result reaches adequate maturity while holding the computational time down (Lee et al. 2015). After identifying the optimal values of the GA parameters, these values along with the stopping rule are set to be used for identifying the optimal GA methods.

5.4.3

Identification of the Optimal GA Methods. Concerning the crossover operation,

it is necessary to use a method to select two parent chromosomes to create two children for the succeeding generation. The common methods to select the population are “roulette wheel selection” (RWS) and “tournament selection” (TRS). A “roulette wheel sampling” is a simple method of selection in a way that each individual has a slice of a circular roulette wheel in which the area of each slice is equal to the individual`s probability (Mitchell 1998). However, if the RWS is selected, the probability of each individual should be calculated based on discrete distribution using their fitness value. In order to calculate the probability of each individual, the “Boltzmann selection” method is adopted using the concepts mentioned in Gen and Cheng (2000), and Mitchell (1998). The probability of each individual i which is denoted by Pi is calculated using Equation 5.34, where fi is the fitness value of the individual i and fW is the worst fitness value

161 among all individuals` fitness values. In addition,  is the selection pressure, meaning if   0 then it means the probability of selecting each individual is equal and the

selection would be converted to a random selection. If    , the probability of the best individual is equal to one and the probabilities of other individuals are equal to zero, which means the selection is converted to an elitism selection. The selection pressure (  ) should be selected by the user. However, there is a rule of thumb using Equation 5.35 to use an appropriate value for  .

e

Pi 

Np

  (

e

fW ) fi

  (

fW ) fi

(5.34)

j 1

Np 2

 P  0.8 i 1

i

(5.35)

Using Equation 5.34, those individuals that have better fitness values would have more probability and occupy a bigger area in the roulette wheel, and have a higher chance to be selected; however, the selection is random. According to Mitchell (1998), although “roulette wheel selection” (RWS) is commonly used in GA models, “tournament selection” (TRS) and “rank selection” have been increasingly used in recent years. Even though TS is similar to “rank selection” in some respect, the TS is more efficient and more capable in terms of parallel performance (Mitchell 1998). For using TS, m individuals are selected randomly from the population. Then, among the m individuals that are selected, one individual that has the best fitness

162 value is selected to be the first candidate for the crossover operation. Since for each crossover operation two candidates are required, the m  1 individuals are returned to the population and TS is used again to select the second candidate for the crossover operation. Just like different selection methods exist in GAs, different crossover operations can be adopted as well. Deciding which crossover operations should be used is important. According to Mitchell (1998), the simplest form of crossover is “single point crossover” where a single crossover position is selected randomly and the genes of two candidates (selected parents) are exchanged to create two offsprings. Single point crossover has some deficiencies: (1) it cannot combine all possible chromosomes, (2) the long chromosome is likely to be destroyed under a single point crossover, and (3) the genes that are exchanged between the two parents always include the endpoints of the strings. To eliminate the second and third problems, “double point crossover” can be used where two positions are selected randomly and the genes between them are exchanged. The long chromosome is less likely to be destroyed by double point crossover. In addition, the genes that are exchanged do not include the endpoints of the strings necessarily. However, there are some solutions that cannot be achieved when double point crossover is adopted. Although under the use of “uniform crossover” all exchanging positions in the parents can be selected randomly and an exchange occurs at each gene, this feature can be highly disruptive (Mitchell 1998). It should be noted that no one knows which crossover operations perform better since the success or failure of these operations depends on the problem (i.e., fitness function, encoding, and GA algorithm) (Mitchell 1998). Noteworthy is that a fourth

163 crossover operation can be created and used which is called “combined crossover” where all these three operations are adopted based on certain percentages of usage that the user selects. In other words, the single point crossover is used to create CRsp percentage of the children population ( Nc ), double point crossover is used to create CRdp percentage of the children population ( Nc ), and the uniform crossover is used to create CRu percentage of the children population ( Nc ). The summation of these percentages should be equal to 100% (i.e., CRsp  CRdb  CRu  100% ). In the third part of the algorithm (Figure 5.1), the combinations of selection methods and crossover operations are tested to identify the best selection method and best crossover operation based on controlled experiments to see what happens to the computational time and the optimal result. In Step 16 (Figure 5.1), the identification of the optimal GA methods can be used for both Models 1 and 2. If Model 1 is used, the identification method is performed for an optimal solution for the normal point (normal project completion time). If Model 2 is used, the identification method is implemented for an optimal solution between the crash and normal points. The outputs are saved in Step 17 (Figure 5.1) and the optimal GA methods are identified in Step 18 (Figure 5.1). In Step 19 (Figure 5.1), the optimal GA methods along with the optimal values of the GA parameters are set and the stopping rule is updated. In Step 20 (Figure 5.1), the best solution, which has been found so far, is identified and is introduced as the initial parent chromosome in Step 21 (Figure 5.1).

164 5.4.4

Execution of Hybrid GALP. The hybrid GALP algorithm is executed in Step 22

of Figure 5.1 to find the optimal solution that is provided for further analysis in Step 23. The detailed description of the hybrid GALP algorithm is provided in Figure 5.2. Every time the algorithm runs, the GA parameters and methods are read in Step 1 (Figure 5.2). Then, using an initialization method, the initial parent chromosome is generated in Step 2 (Figure 5.2) and is evaluated through Steps 3-5 (Figure 5.2). After the initial parent chromosome is evaluated and stored, the parent chromosomes are generated randomly and tested through the fitness function evaluation process (i.e., Steps 3-5 in Figure 5.2) until all the parent population ( N p ) is generated and evaluated in Step 6 (Figure 5.2). If all parent chromosomes have not been generated, they are generated through the loop created in Step 7. When all parent chromosomes have been generated and evaluated, then the process of evolution starts in Step 9 (Figure 5.2), based on the chosen selection method (i.e., RWS or TS). Two parents are selected to generate two offsprings in Step 10 (Figure 5.2) using a selected crossover operation method (i.e., single point, double point, uniform, or combined crossover). The crossover process continues until all the children population ( Nc ) is generated and evaluated. The children population ( Nc ) is calculated using Equation 5.36. If all children chromosomes have been generated and evaluated in Step 8 (Figure 5.2), then the mutation process starts in Step 12 (Figure 5.2). In the mutation process, a parent chromosome is selected randomly to be mutated in Step 13 (Figure 5.2). In Step 13 (Figure 5.2), the mutation probability ( M pb ) is used to specify how many genes should be flipped into something else randomly. For example, if the chromosome has 100 genes and if the mutation probability is 1%, it means 1 out of 100 genes are picked randomly to be changed. The mutation process continues until all the

165 mutant population ( N m ) have been generated and evaluated. The mutant population ( N m ) is calculated using Equation 5.37. In Step 11 (Figure 5.2), if all mutant chromosomes are generated and evaluated, then the population of the next generation is initialized using Steps 14-16 (Figure 5.2). All three populations of parents, children, and mutants are merged in Step 14 (Figure 5.2) to be sorted in Step 15 (Figure 5.2). This new merged population which is denoted by N g is calculated using Equation 5.38. In Step 16 (Figure 5.2), the population of the next generation ( N g 1 ) should be equal to the parent population ( N p ). The population used in the next generation ( N g 1 ) should consist of the top solutions obtained in the previous generation. Then, if the stopping rule is not satisfied in Step 17 (Figure 5.2), the crossover and mutation would be performed on the population of the next generation until the stopping rule is met. The best solution is selected in Step 18 (Figure 5.2).

Cp  N p  Nc  2    2  

(5.36)

N m   M p  N p 

(5.37)

N g  N p  Nc  Nm

(5.38)

5.5

Testing Models 1 and 2 in the Second Stage

Model 1 (academic model) is tested for two different schedules: 1) a small network and 2) a large network. The small network is tested for two different financing cases: 1) all financing alternatives are considered, and 2) only line of credit is considered.

166 In addition, when testing the small network, considering all financing alternatives, two different initial parent settings were used to observe the differences in computational time by using the normal initial parent setting versus accelerated initial parent setting for Model 1. Finally, Models 1 and 2 are validated using a large network. All results and analysis are presented and discussed in the following sections.

5.5.1

Model Validation in Terms of Obtaining better Optimal Results Than

Earlier Studies. The small network of an example project is shown in Figure 5.4. It is

used to demonstrate the procedure explained in the preceding sections and prove the reliability and improvement of the optimal results than earlier studies by using Model 1. The schedule data and activity acceleration methods of this project, which are shown in Table 5.1, are read from an Excel sheet by MATLAB 2016a. The cost data and contractual terms of the project, which are shown in Table 5.2, are sought from the user while running the program inside MATLAB. Then, the financing data, which are shown in Table 5.3, are read from another Excel sheet by MATLAB.

A

Start

B

C

E

D

H

J

F

G

L

I

K

Figure 5.4. Small Network of the Example Project

M

N

167 The schedule data are used to calculate the CPM by using the activity-on-node method, topological sorting, and improved Dijkstra`s algorithm which was described in Chapter 4 (Section 4.2) and 5 (Section 5.1). The cost data and contractual terms are used to calculate the costs of the project and contract price in addition to preparing the cash flow for every different set of activity acceleration method. The cash flow is prepared by using the methodology that was proposed in Section 5.2. Moreover, the financing data are used to create the financing flow for every project completion time between crash and normal points. The financing flow is created using the financing model that was proposed in Chapter 4.

Table 5.1. Project Schedule Data and Activity Acceleration Method Inputs Duration (weeks) Acceleration Method 2

Acceleration Method 3




Start A B C D E F G H I J K L M N


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Predecessor 3

Activity name

Predecessor 2

Activity ID


Predecessor 1

Predecessors

1 1 1 3 2 5 4 6 7 7 10 9 11 12

5 8 13

14

11 4 8 3 9 8 10 9 11 12 11 9 10 12

10 3 5 2 8 7 7 8 8 11 8 8 9 11

9 4 6 7 10 7 8 10

70,000 40,000 45,000 50,000 70,000 60,000 55,000 70,000 60,000 85,000 80,000 60,000 100,000 70,000

90,000 60,000 70,000 85,000 90,000 70,000 75,000 80,000 80,000 100,000 110,000 80,000 115,000 80,000

110,000 85,000 90,000 90,000 115,000 130,000 130,000 100,000

168 To investigate whether this model generates more optimum results than the results in past studies that considered financing cost in their models, two financing cases are considered based on the information presented in Table 5.3. In financing Case 1, all proposed financing alternatives in Table 5.3 are considered. The interest payment times of short-term and long-term loans are considered to be monthly, and the interest payment time for line of credit is considered to be the optimum time. In Case 2, as it was the case in all past finance-based scheduling research, only a line of credit is considered. In addition, no credit limit is considered neither for Case 1, not in Case 2. It should also be mentioned that for the purpose of comparing the two cases, the APRs are considered to be constant in both cases.

Table 5.2. The Inputs of Cost Data and the Contractual Terms of the Project Data type Cost data

Contract terms

Item

Amount $100,000/week

Weekly fixed overhead cost OF Variable overhead percentage OV of (DC)

5%


4%


6%


1% $0/week

Early completion bonus OBS

0%

Advance payment percentage OAP of contract bid price Retained percentage R of pay requests Number of months between submitting pay requests Lag in paying payment requests

10% 1 month

LS

1 month

LP (months)


LR (months)

0 month

In order to compare the optimal profits for different project completion times between crash and normal points, the contract price should be calculated and considered

169 to be constant regardless of the changes of project completion time. In order to calculate the contract price, first, the project costs should be calculated. These costs and their calculations are shown in Table 5.4 where the normal acceleration method (i.e., Method 1) is used in all activities.

Table 5.3. Financing Data of Two Different Cases Case 1 Alternative

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

APR (%)

22 19 18 17 21 19 17 10 21 17 9 9 20 17 11 10 18 16 12 11 7 16

Case 2





Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes


No No No No No No No No No No No No No No No No No No No No No Yes

Optimum time

As it was discussed at the beginning of this Chapter, the second stage models achieve lower financing cost and higher profit compared to all past studies that addressed

170 both TCTP and FBSP. To prove this statement, the small network (Figure 5.4) is tested by using Model 1 (i.e., academic model) to: (1) achieve the detailed information of costs and optimal results for every project completion time between normal and crash points for further analysis, (2) to prove that the obtained financing cost for every project completion time between crash and normal points are far less than comparable studies if the second stage models are used, and (3) to prove that both the obtained optimal profit for every project completion time between normal and crash points and the obtained optimal profit of the project are higher than the case where only a line of credit is considered (as it was the case in all past studies). The results of this test on the small network are shown in Table 5.5.

Table 5.4. The Project Costs and Contract Price Calculation Using Method 1 Type of cost/price Direct cost (DC) Fixed overhead cost (FOC) Variable overhead cost (VOC) Mobilization cost (MOB) Markup (MP) Cost of bonding (BD) Contract price (CP) Indirect cost (IDC) Total cost excluding financing cost (TC)

Summation of activities` direct cost using Method 1

Amount ($) 8,670,000

52  $100, 000

5,200,000

Calculation

0.05  $8, 670, 000

433,500

0.04  ($8, 670, 000  $433, 500)

364,140

0.06  ($8, 670, 000  $5, 200, 000  $433, 500  $364,140)

880,058 155,477

0.01  ($8, 670, 000  $5, 200, 000  $433, 500  $364,140  $880, 058)

$8, 670, 000  $5, 200, 000  $433,500  $364,140  $880, 058  $155, 477 $5, 200, 000  $433, 500  $364,140  $155, 477 $8, 670, 000  $5, 200, 000  $433, 500  $364,140  $155, 477

15,703,175 6,153,117 14,823,117

Table 5.5. Optimum Results Obtained by Testing Model 1 on Small Network for Financing Cases 1 and 2 Definition of result IDC ($) DC ($) TC ($) FC-Case 1 ($) FC-Case 2 ($) TC including FC-Case 1 ($) TC including FC-Case 2($) PR-Case 1 ($) PR-Case 2 ($) Required credit for line of credit-Case 1 ($) Required credit for line of credit-Case 2 ($)

Project completion time (weeks) 52 6,153,117 8,670,000 14,823,117 162,300 257,994 14,985,417

51 6,051,277 8,650,000 14,701,277 153,193 246,124 14,854,470

50 5,946,217 8,595,000 14,541,217 127,461 216,637 14,668,678

49 5,847,137 8,605,000 14,452,137 125,698 213,316 14,577,835

48 5,750,817 8,645,000 14,395,817 123,303 207,130 14,519,120

47 5,660,017 8,745,000 14,405,017 123,465 207,176 14,528,482

46 5,571,057 8,865,000 14,436,057 119,982 205,319 14,556,039

45 5,482,097 8,985,000 14,467,097 122,329 208,177 14,589,426

44 5,394,977 9,125,000 14,519,977 126,033 215,481 14,646,010

43 5,306,017 9,245,000 14,551,017 134,642 225,270 14,685,659

15,081,111

14,947,401

14,757,854

14,665,453

14,602,946

14,612,193

14,641,376

14,675,274

14,735,458

14,776,287

717,759 622,064 1,147,094

848,706 755,775 903,318

1,034,498 945,322 587,983

1,125,340 1,037,722 486,636

1,184,056 1,100,229 686,667

1,174,693 1,090,983 569,895

1,147,137 1,061,799 480,565

1,113,750 1,027,902 495,312

1,057,166 967,717 724,171

1,017,516 926,889 442,365

1,882,822

1,831,845

1,843,572

1,844,019

1,845,806

1,914,385

1,885,839

1,891,135

2,042,682

2,073,671

171

172 As shown in Table 5.5 and Figures 5.5 to 5.7, two financing cases are tested using Model 1 for a small network. The results for each project completion time are obtained for a set of optimal activity acceleration methods which results in optimal profit for each project completion time. The normal duration of this small network is 52 weeks whereas the crash duration is 43 weeks. The obtained financing cost, considering optimal activity acceleration methods for each project completion time, is shown in Table 5.5 and Figure 5.5 where all obtained financing costs using financing Case 1 (all alternatives are considered) are far less than financing Case 2 (just line of credit is considered) for all project completion times. It should be mentioned that the slope of the curve of the total financing cost depends on the interest rate. If the interest rate is high, the slope of the financing cost curve is steep. This statement can be proved by the results of Figure 5.5, where the overall interest of financing Case 2, because of using just line of credit, is higher than the interest of financing Case 1 where all financing alternatives are considered. Therefore, the slope of financing Case 2 (i.e., 3,673.2) is steeper than the slope of financing Case 1 (i.e., 2,774.2). This fact can indicate that by increasing the project completion time, the difference of financing cost becomes bigger between financing Cases 1 and 2. Therefore, for a longer project, the importance of using financing Case 1 can be significant. In addition, because of the importance of financing cost, the obtained profit for every project completion time is also higher when financing Case 1 is used compared to financing Case 2 (see Table 5.5 and Figure 5.6). As shown in Table 5.5 and Figure 5.6, the optimal project completion time is where the project is completed in 48 weeks. The final optimal profits are $1,184,056 and $1,100,229 considering financing Cases 1 and 2, respectively.

173

Project Financing Cost FC ($)

275,000 250,000

y = 1836.6x2 - 170987x + 4E+06

225,000 200,000

Financing Case 1

175,000

Financing Case 2

150,000

Trendline for financing Case 1

y = 1387.1x2 - 128859x + 3E+06

Trendline for financing Case 2

125,000 100,000 42 43 44 45 46 47 48 49 50 51 52 53 Project Completion Time (weeks)

Figure 5.5. Project Financing Cost for Every Project Completion Time Considering Financing Cases 1 and 2 15,800,000 Crash Point

Normal Point

15,200,000 15,000,000 14,800,000 14,600,000

Project Profit (Case 1)

Price ($)

15,400,000

Project Profit (Case 2)

15,600,000 Total cost including financing cost (Case 1) Total cost including financing cost (Case 2) Total payment (contract price) Trendline for financing Case 1 Trendline for financing Case 2

14,400,000 42 43 44 45 46 47 48 49 50 51 52 53 Project Completion Time (weeks)

Figure 5.6. Project Total Cost Including Financing Cost and Profit for Every Project Completion Time Considering Financing Cases 1 and 2

174 In addition, using financing Case 2 results in a much higher required credit limit for a line of credit for every project completion time (see Table 5.5 and Figure 5.7), whereas it is desirable for the contractors to avoid a large overdraft on their primary account (Au and Hendrickson 1986). Therefore, the contractors can avoid a large

Required Credit for Line of Credit ($)

overdraft by using financing Case 1.

2,500,000 Crash Point 2,000,000

Normal Point

1,500,000 Financing Case 1

1,000,000

Financing Case 2 500,000 0 42 43 44 45 46 47 48 49 50 51 52 53 Project Completion Time (weeks)

Figure 5.7. Required Credit for Line of Credit for Every Project Completion Time Considering Financing Cases 1 and 2 In this small network, the selections of optimal activity acceleration methods (normal, accelerated, crashed) and optimal project completion time are very sensitive to the difference between the total costs for different project completion times rather than the difference between financing costs. The reason is because the differences between financing costs are less than the differences between total costs for this small network. If the differences between financing costs become bigger than the differences between total costs, the selections of the optimal activity acceleration methods (normal, accelerated,

175 crashed) and the optimal project completion time become more sensitive to the amount of financing cost. Therefore, if this case occurs, the optimal activity acceleration methods and project completion time may be different depending on the adoption of financing Case 1 or 2. In addition to the resulting schedule for the optimal point (i.e., Table 5.6), the resulting total borrowed money (Bt), total repaid money (Rt), total financing cost (Ft), net financing flow (NFt), cumulative net financing cost (NFCt), and cumulative cash flow including financing flow (N 't) are presented in Tables 5.7 and 5.8 for each case separately.

Table 5.6. CPM Calculations for Optimal Activity acceleration methods Using ActivityOn-Node Method, Topological Sorting, and Improved Dijkstra`s Algorithm

Activity ID

Activity name

Optimal activity acceleration index*

Optimal duration (weeks)

Optimal direct cost ($/week)

Early start

Early finish

Late start

Late finish

Total float

CPM calculations

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Start A B C D E F G H I J K L M N

1 1 3 1 1 1 2 2 3 1 2 1 1 2

11 4 4 3 9 8 7 8 7 12 8 9 10 11

70,000 40,000 85,000 50,000 70,000 60,000 75,000 80,000 90,000 85,000 110,000 60,000 100,000 80,000

0 0 0 4 11 7 7 20 15 15 22 28 27 37

11 4 4 7 20 15 14 28 22 27 30 37 37 48

0 0 11 4 11 7 15 20 22 15 29 28 27 37

11 4 15 7 20 15 22 28 29 27 37 37 37 48

0 0 11 0 0 0 8 0 7 0 7 0 0 0

* 1=Normal acceleration method, 2=Accelerated acceleration method, and 3=Crashed acceleration method

176 Table 5.7. Optimal Financing Results Using Financing Case 1 for Optimal Project Completion Time Month

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Bt

Rt

Ft

NFt

NFCt

Nt

($)

($)

($)

($)

($)

($)

2,362,031 0 0 0 19,054 323,929 237,343 231,292 40,099 0 0 256,426 554,636 0

0 181,695 181,695 181,695 181,695 185,853 340,476 383,415 531,975 318,472 181,695 181,695 307,713 866,739

0 7,272 7,272 7,272 7,272 7,324 9,298 10,841 14,521 11,403 7,272 7,272 8,841 17,441

2,362,031 -188,967 -188,967 -188,967 -169,913 130,752 -112,430 -162,964 -506,397 -329,875 -188,967 67,459 238,082 -884,180

0 7,272 14,545 21,817 29,090 36,414 45,711 56,552 71,073 82,476 89,749 97,021 105,862 123,303

1,843,464 435,497 557,350 34,204 0 0 0 0 0 0 34,289 0 0 1,184,056

Table 5.8. Optimal Financing Results Using Financing Case 2 for Optimal Project Completion Time Month

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Bt

Rt

Ft

NFt

NFCt

Nt

($)

($)

($)

($)

($)

($)

518,567 1,284,749 438,270 613,441 400,610 486,009 375,695 333,080 182,029 65,514 71,159 314,542 583,785 0

0 64,941 738,613 272,641 523,019 343,477 471,396 476,418 659,895 375,507 275,397 199,412 329,006 937,727

0 808 10,477 6,622 13,299 11,780 16,729 19,626 28,531 19,882 19,017 13,382 16,697 30,280

518,567 1,219,000 -310,820 334,179 -135,709 130,752 -112,430 -162,964 -506,397 -329,875 -223,256 101,748 238,082 -968,006

0 808 11,285 17,907 31,206 42,986 59,715 79,341 107,872 127,754 146,771 160,153 176,850 207,130

0 0 0 0 0 0 0 0 0 0 0 0 0 1,100,229

177 5.5.2

Analysis of Financing Results. The financing optimization answers three

questions: (1) which financing alternative or combination of financing alternatives should be considered, (2) what amount of money should be taken each month in each alternative, and (3) what amount of money including interest should be repaid each month in each alternative. Generally, the selection of optimal alternatives and the amount of money to be borrowed in each optimal alternative depend on two parameters: (1) the average of the cumulative net balance of the cash flow and (2) the differences between interest rates calculated based on the duration of keeping the money. The proposed financing model contains three types of financing methods: (1) short-term loan, (2) long-term loan, and (3) line of credit. Although the interest rate of line of credit is higher than long-term loan, a line of credit is usually used to borrow money in addition to a long-term loan. The reason is because even though the interest rate of a line of credit is higher than a long-term loan, the money borrowed from a line of credit can be paid off in a shorter period, whereas the money borrowed using a long-term loan cannot be repaid in full until the end of the project. The line of credit is desirable where the debt is repaid in the shortest time (e.g., one month) not allowing the interest being compounded. Therefore, whenever the cumulative net balance of the cash flow is negative for several months (sometimes until the end of the project), another financing source that has a lower interest rate is needed to pay off the line of credit`s debts. As a result, the best choice is to use a long-term loan that has the lowest interest rate between financing methods. A question may arise as to why the short-term loans appear in the optimal financing solution in addition to a long-term loan and a line of credit. There is a difference between the performance of short-term loans and long-term loans. The long-

178 term loan requires a fixed amount of both principal and interest to be paid by the borrower each month until the end of the project, whereas in short-term loans, just the interest should be paid off monthly and the principal is repaid at the maturity of the shortterm loans. Therefore, when the average of the cumulative net balance of the cash flow becomes significantly negative, the negativity of the cash flow increases when only a long-term loan is used since the portion of the principal should be repaid each month. Then, even though the long-term loan has a low interest rate, more money is needed for the long-term loan to counterbalance the negativity of the cash flow and this additional borrowing results in a higher financing cost. Thus, the optimal decision may include the use of short-term loans in addition to the long-term loan. It should be mentioned that the amount of money to be borrowed using short-term loans and a long-term loan also depends on the interest rates calculated based on the duration of keeping the money. To prove the aforementioned analysis, the resulting cumulative net balance of the cash flow excluding financing flow, the schedule of borrowed money, and the schedule of repaid money including interest are presented in Tables 5.9 to 5.11, respectively. In Table 5.9, crash duration and normal duration generate a negative average cost of $1,201,340 and $1,208,553, respectively, whereas the optimum duration results in a negative average cost of $997,496. Therefore, the schedule of borrowed money (i.e., Table 5.10) shows that for the optimum duration, the long-term loan and line of credit are optimal alternatives (i.e., LP and LC), whereas for the normal and crash durations, shortterm loans need to be added to the long-term loan and line of credit (i.e., C12, LP, and LC). This occurs exactly where the cumulative net balance of the cash flow becomes more negative. In addition, it should be mentioned that if the average of the cumulative

179 net balance of the cash flow becomes even more negative, a longer amount of money should be borrowed using short-term loans. This is apparent in Tables 5.9 and 5.10 where the average of the cumulative net balance of the cash flow for crash duration is $1,201,340 and the optimal borrowed monies using the long-term loan LP and the shortterm loan C12 are $2,276,681 and $316,663, respectively. On the other hand, the average of the cumulative net balance of the cash flow for normal duration is -$1,208,553, and the borrowed money for LP and C12 is $1,981,539 and $364,720, respectively. Therefore, if the average of the negativity of the cash flow increases, more money is needed to be borrowed using short-term loans; however, the duration of keeping money also affects the optimal amounts.

Table 5.9. Cumulative Net Balance of the Cash Flow Excluding Financing Flow for Normal, Optimum, and Crash Project Completion Times Month

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Average

Cumulative Net Balance of the Cash Flow Excluding Financing Flow Nt ($) Normal Optimum Crash duration duration duration -519,617 -518,567 -543,767 -1,570,617 -1,737,567 -1,930,767 -1,663,463 -1,426,747 -1,870,269 -1,748,304 -1,760,926 -1,996,193 -1,745,290 -1,625,217 -1,993,699 -1,790,773 -1,755,969 -1,845,111 -1,708,198 -1,643,539 -1,717,628 -1,625,073 -1,480,575 -1,629,432 -1,596,247 -974,178 -985,892 -1,455,664 -644,302 -666,224 -857,780 -421,047 -761,651 -671,533 -522,794 -828,941 -909,108 -760,877 1,152,158 -1,146,684 1,307,358 880,058 -1,208,553

-997,496

-1,201,340

180 Table 5.10. Optimized Financing Inflow Schedule (Borrowed Money) for Normal, Optimum, and Crash Project Completion Times Month

Normal duration

Crash duration

Optimum duration

C12 ($)

LP ($)

LC ($)

LP ($)

LC ($)

C12 ($)

LP ($)

LC ($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

364,721 0 0 0 0 0 0 0 0 0 0 0 0 0

1,981,539 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 195,726 151,711 164,401 247,411 239,374 6,789 0 752,539 662,300

2,362,031 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 19,054 323,929 237,343 231,292 40,099 0 0 256,426 554,636 0

316,663 0 0 0 0 0 0 0 0 0 0 0 0 -

2,276,681 0 0 0 0 0 0 0 0 0 0 0 0 -

0 0 0 0 196,556 143,759 177,430 271,087 0 0 173,860 358,178 0 -

14

0

0

0

-

-

-

-

-

Table 5.11. Optimized Financing Outflow Schedule (Repaid Money Including Interest) for Normal, Optimum, and Crash Project Completion Times Month

Normal duration

Optimum duration

Crash duration

C12 ($)

LP ($)

LC ($)

LP ($)

LC ($)

C12 ($)

LP ($)

LC ($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0 2,629 2,629 2,629 2,629 2,629 2,629 2,629 2,629 2,629 2,629 2,629 367,349 0

0 147,614 147,614 147,614 147,614 147,614 147,614 147,614 147,614 147,614 147,614 147,614 147,614 147,614

0 0 0 0 0 0 84,043 97,284 125,995 229,714 454,431 36,005 0 277,111

0 188,967 188,967 188,967 188,967 188,967 188,967 188,967 188,967 188,967 188,967 188,967 188,967 188,967

0 0 0 0 0 4,210 160,806 205,289 357,529 140,908 0 0 127,587 695,213

0 2,282 2,282 2,282 2,282 2,282 2,282 2,282 2,282 2,282 2,282 2,282 318,945 -

0 196,768 196,768 196,768 196,768 196,768 196,768 196,768 196,768 196,768 196,768 196,768 196,768 -

0 0 0 0 0 93,296 105,863 160,232 444,490 0 0 91,837 447,870 -

14

0

147,614

1,161,369

-

-

-

-

-

181 It is noteworthy to mention that although the APRs of the short-term loans C9 and C12 are equal (i.e., both have a 9% APR), and C9 is paid back in 9 months rather than 12 months for C12 (which results in less financing cost for C9), C9 is not among the optimal alternatives for any project completion time because the cash flow does not let the contractor pays off the loan in 9 months (see Table 5.10).

5.5.3

Validating Hybrid GALP Models 1 and 2. The small network is tested by using

two different initial parent settings adopting Model 1 (i.e., academic model): (1) to validate Model 1, and (2) to investigate whether the computational time is shorter when normal parent setting is used. In addition, the large network is tested by both Models 1 (i.e., academic model) and 2 (i.e., practical model): (1) to validate Model 2, (2) to investigate whether the computational time is shorter when Model 2 is used, and (3) to confirm the practicality of using the second stage models with large networks. The second stage models are tested in four cases. These cases are described in Table 5.12, according to which, Test 1 is performed using the small network, Model 1 (i.e., academic model), and accelerated parent setting. Test 2 is performed using the small network, Model 1 (i.e., academic model), and normal parent setting. Test 3 is performed using the large network, Model 1 (i.e., academic model), and normal parent setting. Test 4 is performed using the large network, Model 2 (i.e., practical model), and matching parent setting. The small network and its schedule information, activity acceleration methods, cost and contractual data, and financing data were presented in the preceding section. Although, for the large network, the cost and contractual data are identical to the small network, the network and its schedule information, activity acceleration methods,

182 and financing data are different. The schedule and activity acceleration methods for a large project are presented in Appendix B and the financing data are presented in Appendix C. In addition, the GA parameters and stopping rules are presented in Tables 5.13 and 5.14, respectively for each test. For Tests 1 and 2, 36 combinations exist

(36  N P  C p  M p  M pb  2  2  3  3) at the beginning. For Tests 3 and 4, 54 combinations exist (54  N P  C p  M p  M pb  3  2  3  3) at the beginning. It should be noted that the GA parameters are relaxed if the optimal result is obtained at the maximum boundaries of the GA parameters (in Tests 2, 3, and 4 this relaxation occurs that increases the number of combinations to 60, 78, and 70, respectively). In addition, in Table 5.14, three different stopping rules are considered. The GA process stops as soon as any one of these rules is met.

Table 5.12. Information of Each Test Test

Type of network

1

Small

2

Small

3

Large

4

Large

Type of model

Model 1 (academic) Model 1 (academic) Model 1 (academic) Model 2 (practical)

Initial parent setting

Total direct cost ($)

Total indirect cost ($)

Contract price ($)

Accelerated parent

8,670,000

6,153,117

15,703,175

Normal parent

8,670,000

6,153,117

15,703,175

Normal parent

11,040,000

6,196,470

18,259,811

Matching parent

11,040,000

6,196,470

18,259,811

The second stage models identify the optimal values of the GA parameters before performing the hybrid GALP algorithm. The optimum GA parameters and methods are obtained and presented in Table 5.15 to show that depending on different problems, not

183 only the optimum parameters are changed, but also the optimal methods are changed. In addition, it is worthwhile to mention that the “Combined” crossover operation that is introduced by this research, is the optimum crossover operation for both models (i.e., Models 1 and 2) when they are used for the large network.

Table 5.13. The GA Parameters for the Small and Large Networks

Min

Max

3 and 4

Max

GA parameters

Increment

Min

1 and 2

Increment

Test

Np

30

10

40

150

50

250

Cp

0.6

0.1

0.7

0.7

0.1

0.8

Mp

0.6

0.1

0.8

0.7

0.1

0.9

M pb

0.12

0.06

0.24

0.08

0.02

0.12

Table 5.14. The GA Stopping Rules for the Small and Large Networks

Test

1 and 2

3

4

GA stopping rule

TimeLimit (sec) StallGenLimit StallTimeLimit (sec)

1,800 15 900

7,500 25 4500

8,0000 25 4500

Moreover, to show how the processes of identifying the optimal GA parameters and methods are performed, the results are presented in Tables 5.16 and 5.17 for Test 1 considering the normal project completion time. Table 5.16 presents the optimal GA

184 parameters for the 36 combinations of GA parameters. Table 5.17 presents the optimal GA methods for the 8 combinations of the GA methods.

Table 5.15. Optimal GA Parameters and Methods for Each Test Test

1

2

3

4

Np

30

30

150

250

Cp

0.6

0.6

0.8

0.7

Mp

0.7

0.6

0.8

0.8

M pb

0.18

0.24

0.12

0.12

Itmin

26

16

33

54

Roulette Wheel Single Point

Tournament Uniform

Roulette Wheel Combined

Tournament Combined

GA parameters

Selection method Crossover operation

As shown in Table 5.16, the optimal GA parameters are obtained in GA cycle 17 where the GA cycle is satisfied in 1,347 seconds where the TimeLimit is 1,800 seconds. Since the optimal result is not obtained at the maximum boundaries of the GA parameters, there is no need to relax the boundaries of the GA parameters for Test 1. As a result, the GA parameters are set based on the values that are obtained in GA cycle 17 for the identification of optimal GA methods. As shown in Table 5.17, the optimal result is obtained in GA cycle 1 (i.e., Time=1,359 seconds) where the best methods are selected as “Roulette Wheel Selection” and “Single Point” crossover. The reason that in both Tables 5.16 and 5.17 some optimal results are negative is because a penalty is defined for a genetic algorithm if the constraints are not satisfied. Therefore, the negative results represent infeasible solutions.

185 Table 5.16. Outputs Obtained in Optimal GA Parameters Identification Process for Test 1 Considering the Normal Project Completion Time GA cycle number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Np

Cp

Mp

M pb

Itgen

Time (sec)

Stall Gen

Stall Time (sec)

Optimal profit ($)

30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40 30 40

0.6 0.6 0.7 0.7 0.6 0.6 0.7 0.7 0.6 0.6 0.7 0.7 0.6 0.6 0.7 0.7 0.6 0.6 0.7 0.7 0.6 0.6 0.7 0.7 0.6 0.6 0.7 0.7 0.6 0.6 0.7 0.7 0.6 0.6 0.7 0.7

0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8

0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24

34 28 34 26 31 27 32 22 32 18 27 21 33 26 34 27 26 25 28 25 33 25 22 23 32 29 31 27 28 27 33 25 26 25 30 23

1,660 1,800 1,800 1,800 1,610 1,800 1,800 1,645 1,800 1,349 1,656 1,677 1,560 1,640 1,800 1,800 1,347 1,721 1,580 1,800 1,800 1,800 1,347 1,822 1,518 1,800 1,630 1,800 1,425 1,800 1,800 1,800 1,437 1,817 1,800 1,800

15 10 13 9 15 10 9 12 14 12 15 12 15 14 14 2 15 13 15 2 11 11 15 7 15 9 15 2 15 12 12 12 15 8 13 1

737 661 703 631 791 703 524 909 793 908 933 973 724 901 757 139 788 907 865 151 620 828 929 564 721 577 803 138 773 824 677 895 841 596 794 80

717,759 717,759 717,759 717,759 710,074 717,759 717,759 717,759 717,759 717,759 717,759 717,759 698,058 717,759 717,759 698,058 717,759 717,759 707,144 717,759 717,759 690,472 710,074 717,759 717,759 717,759 717,759 -5,005,440 717,759 707,144 699,240 698,058 717,759 717,759 710,074 717,759

186 Table 5.17. Outputs Obtained in Optimal GA Methods Identification Process for Test 1 Considering the Normal Project Completion Time GA cycle number

Selection method

Crossover operation

It gen

Time (sec)

Stall Gen

Stall Time (sec)

Optimal profit ($)

1 2 3 4 5 6 7 8

Roulette Wheel Tournament Roulette Wheel Tournament Roulette Wheel Tournament Roulette Wheel Tournament

Single Point Single Point Double Point Double Point Uniform Uniform Combined Combined

26 30 27 30 30 28 30 27

1,359 1,555 1,395 1,535 1,558 1,436 1,555 1,408

15 15 15 15 15 15 15 15

794 787 792 771 792 778 789 794

717,759 690,472 717,759 -10,264,227 717,759 -10,249,100 717,759 717,759

According to Table 5.12, Tests 1 and 2 are performed for different initial parent settings to validate Model 1 and prove that normal parent setting results in a shorter computational time for Model 1. As shown in Table 5.18, the optimal results of Tests 1 and 2 are exactly the same (i.e., 48 week duration and a cost of $1,184,056) that validates Model 1 by showing that different initial parent settings result in the same outcome. In addition, Test 2 performs far better in terms of total computational time compared to Test 1 (i.e., 74,605 seconds versus 91,475 seconds) because the normal parent setting is used in Test 2, whereas accelerated parent setting is used in Test 1 as the initial parent chromosome. In addition, the processes used in Test 2 (i.e., optimal values of GA parameters, optimal GA methods, and hybrid GALP processes) achieve a shorter computational time compared to Test 1. Also, Test 2 achieves a shorter time per GA cycle (i.e., 956 seconds versus 1,694 seconds).

187 Table 5.18. Validation of Models and Comparison of Computational Time between Tests

Test

1 (Small network) (Model 1) (Accelerated parent)

2 (Small network) (Model 1) (Normal parent)

3 (Large network) (Model 1) (Normal parent)

4 (Large network) (Model 2) (Matching parent)

Run time (sec)

Number of GA cycle

Total time (sec)

Time per a GA cycle (sec)

Optimal profit ($)


Optimal values of GA parameters

62,778

36

91,475

1,694

1,184,056

48

Optimal GA methods

12,075

8

Hybrid GALP

16,622

10


56,906

60

74,605

956

1,184,056

48

Optimal GA methods

6,374

8

Hybrid GALP

11,325

10


558,565

78

764,210

7,348

1,839,712

33

Optimal GA methods

45,444

8

Hybrid GALP

160,200

18


494,791

70

572,508

7,247

1,839,766

33

Optimal GA methods

68,867

8

Hybrid GALP

8,850

1

Process of algorithm

To validate Model 2, prove that Model 2 results in shorter computational time, and show that Model 2 can deal with a large network, both Models 1 and 2 are tested for

188 a large network. The results in Table 5.18 show that both Models 1 and 2 produce a near optimal solution (i.e., 33 week duration and cost of $1,839,712 in Test 3, and 33 week duration and cost of $1,839,766 in Test 4) where the optimal GA parameters and methods are used. In addition, not only the total computational time for Model 2 is far shorter compared to Model 1 (i.e., 572,508 seconds for Model 2 versus 764,210 seconds for Model 1), but also a single cycle of hybrid GALP is performed in shorter computational time for Model 1 (i.e., 7,247 seconds for Model 2 versus 7,348 seconds for Model 1). Although Tests 3 and 4 take a long time to process, a single cycle of hybrid GALP, is performed in Model 2 in approximately 2 hours (i.e., 7,247 seconds) for a real-world problem such as a large network. However, it should be mentioned that no method and model exists currently that performs both time-cost tradeoff and financing optimization considering different financing alternatives simultaneously. Optimizing all financing alternatives using linear programming in each function evaluation in the GA algorithm leads to longer computational time.

5.6

Conclusion of the Second Stage

The second stage is presented in two models (Models 1 and 2) and tested in four cases to prove that the setup in the second stage has advantages over all past studies that addressed both TCTP and FBSP. The second stage models obtain better optimal results for (1) financing cost, and (2) profit. In addition, a new hybrid GALP algorithm is introduced that combines genetic algorithms and linear programming. Moreover, the hybrid GALP algorithm not only eliminates the deficiencies of GA algorithms by adopting the methodology of Lee et al. (2015), but also adopts the controlled experiment

189 operation to identify optimal GA methods. The effects of identifying optimal GA methods and introducing a new crossover operation (i.e., combined crossover) are demonstrated.

190 CHAPTER 6 6. THIRD STAGE OF THE RESEARCH In Chapter 4, a model was proposed in the first stage only for a normal project schedule to solve only the finance-based scheduling problem (FBSP), whereas the second stage involved two models (i.e., Models 1 and 2) that considered the finance-based scheduling problem (FBSP) and the time-cost tradeoff problem (TCTP), respectively. In both the first and second stages of this study, the start times of activities were kept constant while searching for an optimal solution to maximize profit. In this chapter, two models are proposed for the third stage of this study. These models expand the second stage models by considering the variable activity start times. In other words, in the third stage, activity acceleration methods, financing variables, and variable activity start times are optimized to find the project schedule that leads to optimum profit. To find the optimal profit, two factors are important, the total cost (excluding financing cost) and the financing cost. The total cost (excluding financing cost) does not change when the start times of activities are changed within the available total floats, whereas the financing cost does change. However, the question is: when should the activities start to minimize the financing cost? In Chapter 3, it was discussed that financing cost depends on the schedule of construction activities. It was also mentioned that according to Fathi and Afshar (2010), if the volume of work in the early periods of the project is higher than the volume of work at the late periods, the need for the big portion of the required financing will come up early in the project, which results in a longer duration of keeping money and more financing cost. Therefore, the duration of keeping the money and financing cost go down

191 when the start times of activities are pushed forward. On the other hand, in Chapter 5, it was also proved that the optimum financing solution depends on the average of the cumulative net balance of the cash flow. As a result, the financing cost depends on two parameters when different schedules of construction activities are considered: (1) the duration of keeping the money and (2) the average of the cumulative net balance of the cash flow. It should be noted that when the start times of activities are pushed forward, one of two cases occurs: (1) the duration of keeping the money goes down and the average of the cumulative net balance of the cash flow becomes less negative or (2) the duration of keeping the money decreases whereas the average of the cumulative net balance of the cash flow becomes more negative. If the average of the cumulative net balance of the cash flow becomes more negative, more money should be borrowed, which results in a higher financing cost, whereas reducing the duration of keeping the money results in a lower financing cost. Therefore, the effect of (1) the duration of keeping the money and (2) the average of the cumulative net balance of the cash flow on financing cost and profit should be considered simultaneously. However, regardless of whether the impact of the duration of keeping the money and the average of the cumulative net balance of the cash flow conflict or not, if the optimal financing cost is obtained in different start times rather than the early start times of activities, the schedule becomes more critical since the total floats of the activities are used. If the impact of these two factors (i.e., the duration of keeping the money and the average of the cumulative net balance of the cash flow) conflict with each other, the impact of the average of the cumulative net balance of the cash flow on the financing cost prevents postponing the start times of activities. This situation occurs because the average of the

192 cumulative net balance of the cash flow becomes more negative and results in a higher financing when the start times of activities are postponed if the impact of these two factors conflict. In this case, the tradeoff between the effects of these two factors on the financing cost is vital to find out when the activities should start to reach the minimum financing cost. The third stage models have the ability to find the optimal financing cost and profit by considering the effect of both the duration of keeping the money and the average of the cumulative net balance of the cash flow on the financing cost and profit. The third stage models also provide an optimal schedule without changing the start times of activities. In other words, both conditions (i.e., early activity start times and variable activity start times) are analyzed in the third stage models where the reports of the percentage of the critical activities, the total float of the project, and the average of the cumulative net balance of the cash flow in each case (i.e., early activity start times and variable activity start times) are provided.

6.1

Methodology and Computational Process of the Third Stage Models

Two models are proposed for the third stage. Model 1, which is proposed for academic research, takes longer computation time because the optimal solutions are computed for every project completion time between crash and normal points to see the difference of the optimal outcomes for every project completion time and to do a sensitivity analysis. Model 2, which is proposed for practical use, takes less computation time compared to Model 1 since it only searches for one optimal solution rather than

193 finding an optimal solution for every project completion time between the crash and normal points. As shown in Figure 6.1, the third stage models consist of two processes. The steps in Process 1 (i.e., Steps 1, 2, 3, 4, and 5) are exactly similar to the steps of the second stage models where the summation of total cost excluding financing cost and optimal financing cost is minimized to maximize the profit by optimizing activity acceleration methods and financing variables. In the Process 1, the start times of activities are constant. In Process 2, the resulting activity acceleration methods, that are obtained and saved in Process 1 (i.e., Step 5), are used (i.e., in Step 6) to see whether if there is a better solution by changing the start times of activities (i.e., Step 7). In Process 2, in addition to financing variables, the start times of activities are variable as well. Finally, the outputs of Process 2 are saved in Step 8. It should be noted that the same equations are used in the second stage models and the third stage models.

6.2

Testing the Third Stage Models

The third stage models (i.e., Models 1 and 2) are tested using an example project that is shown in Figure 6.2. First, Model 1 (academic model) is used to verify the structure and performance of the third stage models, and also analyze the results for every project completion time between crash and normal duration. Then, Model 2 (practical model) is used to validate Model 1 and to investigate whether the computational time is shorter when Model 2 is used.

194

Process 1: Optimization considering early activity start times

1. Prepare input data 2. Identify optimal values of the GA parameters 3. Identify optimal GA methods 4. Execute hybrid GALP to maximize profit by optimizing activity acceleration methods and financing variables [Model 1: for every project completion time, Model 2: for an optimal solution] 5. Save the outputs and the report for the optimization considering constant start times for activities

Process 2: Optimization considering variable activity start times

6. Use the schedule and the outputs obtained in the Process 1 7. Execute hybrid GALP to maximize profit by optimizing start times of activities and financing variables [Model 1: for every project completion time, Model 2: for an optimal solution] 8. Save the outputs and the report for the optimization considering variable start times for activities

Figure 6.1. Third Stage Algorithm

A

E

I

M

Q

B

F

J

N

R T

Start C

G

K

O

D

H

L

P

Figure 6.2. Network of the Example Project

S

195 The schedule data and activity acceleration methods are read from an Excel sheet by MATLAB 2016a. They are shown in Table 6.1. The cost data and contractual terms of the project are shown in Table 6.2.

Table 6.1. Project Schedule Data and Activity Acceleration Method Inputs Duration (weeks)

Predecessor 3







Start A B C D E F G H I J K L M N O P Q R S T

Predecessor 2

Activity name

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21


Predecessor 1

Activity ID

Predecessors

1 1 1 1 2 3 4 5 6 7 8 9 10 11 11 13 14 15 15 18

9 12 16 19

17 20

11 9 9 6 5 8 6 9 11 7 6 9 8 7 5 9 8 9 11 10

9 7 8 5 4 6 5 8 9 6 5 8 7 6 4 8 7 8 10 9

8 6 7 5 7 8 7 6 7 7 9 -

20,000 20,000 35,000 55,000 40,000 30,000 75,000 40,000 45,000 75,000 65,000 65,000 45,000 55,000 55,000 60,000 65,000 40,000 60,000 80,000

35,000 35,000 45,000 70,000 50,000 55,000 90,000 50,000 60,000 90,000 80,000 75,000 55,000 70,000 70,000 70,000 80,000 50,000 70,000 90,000

60,000 60,000 80,000 80,000 70,000 80,000 100,000 70,000 90,000 80,000 110,000 -

The CPM is calculated using topological sorting and the improved Dijkstra`s algorithm using the schedule information and activity acceleration methods that are shown in Table 6.1. The project costs and contract price are calculated using the

196 information shown in Table 6.2. The project costs and contract price are calculated where the normal acceleration method (i.e., Method 1) is used in all activities (see Table 6.3).

Table 6.2. The Inputs of Cost Data and the Contractual Terms of the Project Data type Cost data

Contract terms

Item

Amount $110,000/week

Weekly fixed overhead cost OF Variable overhead percentage OV of (DC)

5%


4%


5%


1% $0/week

Early completion bonus OBS

0%

Advance payment percentage OAP of contract bid price Retained percentage R of pay requests Number of months between submitting pay requests Lag in paying payment requests

10% 1 month

LS

1 month

LP (months)


LR (months)

0 month

Table 6.3. The Project Costs and Contract Price Calculation Using Method 1 Type of cost/price Direct cost (DC) Fixed overhead cost (FOC) Variable overhead cost (VOC) Mobilization cost (MOB) Markup (MP) Cost of bonding (BD) Contract price (CP) Indirect cost (IDC) Total cost excluding financing cost (TC)


Amount ($) 8,190,000

54  $110, 000

5,940,000

Calculation

0.05  $8,190, 000

409,500

0.04  ($8,190, 000  $409, 500)

343,980

0.05  ($8,190, 000  $5, 940, 000  $409, 500  $343, 980)

744,174 156,277

0.01  ($8,190, 000  $5, 940, 000  $409, 500  $343, 980  $744,174)

$8,190, 000  $5, 940, 000  $409, 500  $343,980  $744,174  $156, 277 $5,940, 000  $409, 500  $343,980  $156, 277 $8,190, 000  $5,940, 000  $409,500  $343,980  $156, 277

15,783,931 6,849,757 15,039,757

197 Models 1 and 2 are tested for two financing cases based on the financing data that are shown in Table 6.4. In financing Case 1, all proposed financing alternatives are considered, whereas in financing Case 2, only a line of credit is considered. As shown in Table 6.4, the interest payment times of short-term and long-term loans are monthly, whereas the interest payment time for the line of credit depends on the optimum time, where no credit limit is considered for both Cases 1 and 2.

Table 6.4. Financing Data of Two Different Cases Case 1

Case 2

Alternative

APR (%)





1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

22 19 18 17 21 19 17 10 21 17 9 9 20 17 11 10 18 16 12 11 7 16




Optimum time

198 6.2.1

Results and Analysis of Testing Model 1 in the Third Stage. In order to

investigate the effects of changing the start times of activities on the financing cost and the profit, four alternatives are considered that are shown in Table 6.5. The reason why four alternatives are considered is because two different optimization conditions are considered in addition to two financing cases (i.e. two financing cases  two optimization conditions= four alternatives). As shown in Table 6.5, in Condition 1, the optimal results are achieved considering early activity start times. In Condition 2, the optimal results are obtained considering variable activity start times. Although the models used in past studies are unable to provide the outputs for Conditions 1 and 2 in one test, the proposed models provide the outputs for both conditions in one test, even though the results are presented and discussed separately. It should be noted that the results presented in Table 6.6 represent the outcome of using Model 1 (Model 1 is proposed to investigate the results for every project completion time between crash and normal points). The results of the tests are shown in Table 6.6 for every project time considering each alternative separately.

Table 6.5. Information of Each Alternative in Model 1 Alternative

Financing case

Optimization condition

Test 1 Alt 1 Alt 2

Alt 3 Alt 4

Case 1 (All financing alternatives) Case 1 (All financing alternatives) Test 2 Case 2 (only a line of credit) Case 2 (only a line of credit)

Condition 1 (early start times of activities) Condition 2 (variable start times of activities) Condition 1 (early start times of activities) Condition 2 (variable start times of activities)

Table 6.6. Optimum Results Obtained by Testing Model 1 Considering Four Alternatives Definition of result

Alts 1 to 4 Alt 1 Alt 2 Alts 3 and 4 Alt 1 Alt 2 Alts 3 and 4

Project completion time (weeks) 51 50 49 48 47 46 45 44 Total cost excluding financing cost ($) 15,039,757 14,940,677 14,847,057 14,758,897 14,714,417 14,669,937 14,669,137 14,706,557 14,776,737 14,896,057 15,146,417 54

161,470 160,453 281,405

53

52

159,559 158,542 279,535

168,270 168,256 263,718

162,726 162,726 258,360

Financing cost ($) 159,310 158,486 159,310 158,481 255,044 253,986

156,655 156,396 247,683

157,766 157,766 252,143

165,101 165,101 263,075

175,891 175,891 281,226

181,544 180,982 285,456

Total cost including financing cost ($) 15,201,227 15,100,236 15,015,327 14,921,622 14,873,727 14,828,422 14,825,791 14,864,322 14,941,838 15,071,947 15,327,961 15,200,210 15,099,219 15,015,313 14,921,622 14,873,727 14,828,418 14,825,533 14,864,322 14,941,838 15,071,947 15,327,398 15,321,162 15,220,211 15,110,775 15,017,256 14,969,461 14,923,922 14,916,819 14,958,699 15,039,811 15,177,283 15,431,872

Alt 1 Alt 2 Alts 3 and 4

582,704 583,721 462,769

683,695 684,712 563,719


-1,242,446 -1,248,649 -1,242,446

-1,226,740 -1,232,944 -1,226,740

768,604 768,618 673,156

862,308 862,308 766,674

Profit ($) 910,204 955,508 910,204 955,513 814,470 860,008

958,139 958,398 867,111

919,608 919,608 825,231

842,093 842,093 744,119

711,983 711,983 606,648

455,970 456,532 352,058

Average of cumulative net balance of the cash flow ($) -1,235,198 -1,201,042 -1,179,981 -1,170,495 -1,233,324 -1,245,922 -1,201,042 -1,179,981 -1,185,097 -1,238,702 -1,235,198 -1,201,042 -1,179,981 -1,170,495 -1,233,324

-1,258,419 -1,258,419 -1,258,419

-1,321,460 -1,321,460 -1,321,460

-1,425,381 -1,425,381 -1,425,381

-1,598,060 -1,600,955 -1,598,060

49 47 49

49 47 49

35 31 35

22 22 22

Sum of the activities` total floats 19 18 19 14 19 18

14 13 14

12 12 12

8 8 8

5 5 5

4 3 4


30 30 30

30 30 30

30 35 30

75 75 75

Critical activities (%) 75 75 75 75 75 75

75 80 75

75 75 75

75 75 75

75 75 75

80 85 80

199


200 As shown in Table 6.6 and Figures 6.3 to 6.6, the results of four alternatives are provided using Model 1. In Alt 1, all financing alternatives are considered where early start times of activities are used. The purpose of Alt 1 is to find the optimal profit by finding the optimal activity acceleration methods and the financing variables in case all financing alternatives are considered. In Alt 2, although all financing alternatives are considered, the start times of activities are variable. This way, the effects of changing the start times on the optimal financing cost and profit are investigated. As shown in Table 6.6 and Figures 6.3 to 6.6, the results of Alts 1 and 2 are similar to each other and there are negligible changes only in some optimal results. Comparing Alts 1 and 2, the maximum changes occur at the project completion times of 54 weeks where the financing costs differ by only $1,017 (i.e., $160,453 subtracted from $161,470) amounting to a difference of only 0.6%. Although these differences are not sufficiently large to be apparent in Figures 6.3 and 6.4, the effects on the average of the cumulative net balance of the cash flow and on the total float of the project (i.e., the sum of the activities` total floats) are larger and can be seen in Figures 6.5 and 6.6. For example, the average of the cumulative net balance of the cash flow and the sum of the total floats for a project completion time of 54 weeks differ by (-$1,248,649)-($1,242,446) = -$6,203 (0.5%) and 49-47 = 2 weeks (4%), respectively. Table 6.6 and Figure 6.5 show that the average of the cumulative net balance of the cash flow is always equal to or more negative than in Alt 2 (variable activity start times) compared to Alt 1 (early activity start times), whereas Table 6.6 and Figure 6.6 show that the sum of the total floats is always equal to or lower than in Alt 2 (variable activity start times) compared to Alt 1 (early activity start times). Table 6.6 and Figure

201 6.6 also show that the sum of total floats goes down as project duration goes down from the normal 54 weeks to the crash 44 weeks. These are anticipated results and constitute evidence that the proposed model works well. The results of Alts 1 and 2 completely prove the statement at the beginning of this chapter that regardless of whether both the duration of keeping the money and the average of the cumulative net balance of the cash flow conflict or not, if the optimal financing cost is achieved in different start times rather than the early start times of activities, the schedule becomes more critical since the total floats of activities are used (see Table 6.6 and Figure 6.6). Considering Alts 1 and 2, the results obtained in Table 6.6 demonstrate that the benefits of the optimal results are negligible in terms of the financing cost and profit when the start times of activities change, whereas changing the start times of activities causes a more critical condition since more total float is used and the average of the cumulative net balance of the cash flow becomes more negative. The reason why changes in the start times of activities do not affect the optimal results significantly is because the duration of keeping the money and the average of the cumulative net balance of the cash flow conflict with each other. It is true that by changing the start times of activities the duration of keeping the money decreases, but in this project example, the average of the cumulative net balance of the cash flow increases, which results in borrowing more money. This tradeoff between the duration of keeping the money and the average of the cumulative net balance of the cash flow results in changing the start times of activities for some project completion times (i.e., project completion times of 54, 53, 52, 49, 48, and 44 weeks), whereas the start times of activities for other project completion times (i.e.,

202 project completion times of 51, 50, 47, 46, and 45 weeks) do not change (they remain in early start times)

283,000

Crash Point

Normal Point

Financing Cost ($)

273,000 263,000 253,000 243,000 233,000 223,000

Alt 1 Alt 2 Alt 3 Alt 4

213,000 203,000 193,000 183,000 173,000 163,000 153,000

43

44

45

46 47 48 49 50 51 52 53 Project Completion Time (weeks)

54

55

Profit ($)

Figure 6.3. Financing Cost for Every Project Completion Time Considering Four Alternatives 960,000 Crash Point 925,000 890,000 855,000 820,000 785,000 750,000 715,000 680,000 645,000 610,000 575,000 540,000 505,000 470,000 435,000 400,000 365,000 330,000 43 44 45

Normal Point



54

55

Figure 6.4. Profit for Every Project Completion Time Considering Four Alternatives

203

43

Project Completion Time (weeks) 44 45 46 47 48 49 50 51

52

53

54

55

Average of the Cumulative Net Balance of the Cash Flow ($)

-1,160,000 -1,210,000 -1,260,000 -1,310,000

Alt 1 Alt 2

-1,360,000

Alt 3

-1,410,000

Alt 4

-1,460,000 -1,510,000 -1,560,000 -1,610,000

Crash Point

Normal Point

Figure 6.5. Average of the Cumulative Net Balance of the Cash Flow for Every Project Completion Time Considering Four Alternatives 50 Crash Point

Normal Point

Sum of the Activities`s Total Floats

45 40 35 30


25 20 15 10 5 0 43

44

45


54

55

Figure 6.6. Sum of the Activities` Total Floats for Every Project Completion Time Considering Four Alternatives

204 Considering Alts 3 and 4, the tradeoff between the duration of keeping the money and the average of the cumulative net balance of the cash flow does not change the optimal results at any project completion time. In other words, the optimal results of Alt 3 and 4 are completely similar where the line of credit is considered (see Table 6.6 and Figures 6.3 to 6.6). The reason why the results of Alts 3 and 4 are completely identical is because the interest rate of Alts 3 and 4 are much higher than Alts 1 and 2 which causes the financing cost to be more sensitive to the increment of the negativity of the cash flow. As shown in Table 6.6 and Figures 6.3 and 6.4, it should be noted that regardless of whether the start times of activities change or not, the optimal financing cost and profit are far better for Alts 1 and 2 where all financing alternatives are considered compared to the optimal results obtained for Alts 3 and 4 when only a line of credit is considered. It proves that when only a line of credit is considered, even changes of the start times of activities cannot result in achieving the better optimal results. It should also be noted that although in this example project, the conflict between duration of keeping the money, and the average negative balance of the cash flow results in negligible changes in the optimal results, the effects on the optimal results could have been higher if the duration of keeping the money and the average negative balance of the cash flow did not conflict. There is no way to predict these effects without using the third stage models. The optimal results are obtained when the project is completed in 48 weeks. Considering Alts 1 and 2 where the optimal results are far better than Alts 3 and 4, if the start times of activities are set as early starts (Alt 1 in Table 6.6), the optimal financing cost and profit are $156,655 and $958,139, respectively. Whereas if the start times of

205 activities are variable (Alt 2 in Table 6.6), the optimal financing cost and profit are $156,396 and $958,398, respectively. To get a very small benefit of $259, not only does the construction schedule change and become more risky, but also the financing schedules change. The optimal construction schedule is presented in Table 6.7 and the optimal financing schedules are presented in Tables 6.8 to 6.11 for the optimum point.

Table 6.7. Construction Schedule for Optimal Project Completion Time Considering Alts 1 and 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21


1 1 1 2 2 1 2 2 2 2 1 2 2 2 1 2 1 1 2 2

11 9 9 5 4 8 5 8 9 6 6 8 7 6 5 8 8 9 10 9

20,000 20,000 35,000 70,000 50,000 30,000 90,000 50,000 60,000 90,000 65,000 75,000 55,000 70,000 55,000 70,000 65,000 40,000 70,000 90,000

Total float

Optimal direct cost ($/week)

Finish time


Start time

Optimal activity acceleration index*

Total float

Activity name

Finish time

Activity ID

Schedule of Alt 2

Start time

Schedule of Alt 1

0 0 0 0 0 11 9 9 5 15 17 14 13 24 23 23 21 31 29 29 39

0 11 9 9 5 15 17 14 13 24 23 20 21 31 29 28 29 39 38 39 48

0 0 0 4 0 0 0 4 0 0 0 4 0 0 0 1 0 0 1 0 0

0 0 0 0 0 11 9 9 5 15 17 14 13 24 23 24 21 31 29 29 39

0 11 9 9 5 15 17 14 13 24 23 20 21 31 29 29 29 39 38 39 48

0 0 0 4 0 0 0 4 0 0 0 4 0 0 0 0 0 0 1 0 0

* 1=Normal acceleration method, 2=Accelerated acceleration method, and 3=Crashed acceleration method

Table 6.8. Optimized Financing Inflow Schedule (Borrowed Money) for Normal, Optimum, and Crash Project Completion Times Considering Alt 1 (Constant Start Times of Activities) Month

Normal duration

Optimum duration

Crash duration

B12 ($)

C9 ($)

LP ($)

LC ($)

C12 ($)

LP ($)

LC ($)

C12 ($)

LP ($)

LC ($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 0 0 434,020 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 205,410 0 0 0 0 0 0 0 0

1,788,166 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 162,254 342,156 0 76,345 0 0 97,627 0 176,040 393,311 162,548

776,122 0 0 0 0 0 0 0 0 0 0 0 0 0 -

1,501,400 0 0 0 0 0 0 0 0 0 0 0 0 0 -

0 0 0 0 303,638 278,205 52,293 171,103 0 46,861 17,215 0 1,103,121 0 -

1,253,050 0 0 0 0 0 0 0 0 0 0 0 0 -

1,518,498 0 0 0 0 0 0 0 0 0 0 0 0 -

0 0 0 0 442,595 130,268 9,354 0 71,601 136,723 0 341,561 0 -

15

0

0

0

0

-

-

-

-

-

-

206

Table 6.9. Optimized Financing Inflow Schedule (Borrowed Money) for Normal, Optimum, and Crash Project Completion Times Considering Alt 2 (Variable Start Times of Activities) Month

Normal duration

Optimum duration

Crash duration

B12 ($)

C9 ($)

LP ($)

LC ($)

C12 ($)

LP ($)

LC ($)

C12 ($)

LP ($)

LC ($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 0 0 434,020 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 199,892 0 0 0 0 0 0 0 0

1,788,166 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 162,254 351,641 1,805 0 0 0 102,105 0 180,495 401,473 207,915

774,618 0 0 0 0 0 0 0 0 0 0 0 0 0 -

1,509,325 0 0 0 0 0 0 0 0 0 0 0 0 0 -

0 0 0 0 299,710 296,130 51,197 254,451 0 45,638 17,110 0 1,102,239 0 -

1,263,381 0 0 0 0 0 0 0 0 0 0 0 0 -

1,455,007 0 0 0 0 0 0 0 0 0 0 0 0 -

0 0 0 36,922 444,788 146,217 0 0 82,228 137,999 0 336,148 0 -

15

0

0

0

0

-

-

-

-

-

-

207

Table 6.10. Optimized Financing Outflow Schedule (Repaid Money Including Interest) for Normal, Optimum, and Crash Project Completion Times Considering Alt 1 (Constant Start Times of Activities) Month

Normal duration

Optimum duration

Crash duration

B12 ($)

C9 ($)

LP ($)

LC ($)

C12 ($)

LP ($)

LC ($)

C12 ($)

LP ($)

LC ($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 0 0 0 3,461 3,461 3,461 3,461 3,461 3,461 3,461 3,461 3,461 3,461 3,461

0 0 0 0 0 0 0 1,480 1,480 1,480 1,480 1,480 1,480 1,480 1,480

0 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674

0 0 0 0 0 63,472 448,471 0 77,295 0 0 98,842 0 42,734 199,972

0 5,594 5,594 5,594 5,594 5,594 5,594 5,594 5,594 5,594 5,594 5,594 781,716 0 -

0 120,115 120,115 120,115 120,115 120,115 120,115 120,115 120,115 120,115 120,115 120,115 120,115 120,115 -

0 0 0 0 0 117,460 367,824 77,055 257,643 0 34,153 30,886 0 1,116,849 -

0 9,031 9,031 9,031 9,031 9,031 9,031 9,031 9,031 9,031 9,031 9,031 1,262,081 -

0 131,240 131,240 131,240 131,240 131,240 131,240 131,240 131,240 131,240 131,240 131,240 131,240 -

0 0 0 0 0 289,361 268,012 34,371 0 39,151 172,181 0 345,812 -

15

437,481

206,891

124,674

504,162

-

-

-

-

-

-

208

Table 6.11. Optimized Financing Outflow Schedule (Repaid Money Including Interest) for Normal, Optimum, and Crash Project Completion Times Considering Alt 2 (Variable Start Times of Activities) Month

Normal duration

Optimum duration

Crash duration

B12 ($)

C9 ($)

LP ($)

LC ($)

C12 ($)

LP ($)

LC ($)

C12 ($)

LP ($)

LC ($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 0 0 0 3,461 3,461 3,461 3,461 3,461 3,461 3,461 3,461 3,461 3,461 3,461

0 0 0 0 0 0 0 1,441 1,441 1,441 1,441 1,441 1,441 1,441 1,441

0 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674 124,674

0 0 0 0 0 72,957 444,757 5,587 0 0 0 103,376 0 50,937 245,379

0 5,583 5,583 5,583 5,583 5,583 5,583 5,583 5,583 5,583 5,583 5,583 780,201 0 -

0 120,749 120,749 120,749 120,749 120,749 120,749 120,749 120,749 120,749 120,749 120,749 120,749 120,749 -

0 0 0 0 0 134,762 423,856 64,383 292,821 0 33,424 30,263 0 1,115,957 -

0 9,106 9,106 9,106 9,106 9,106 9,106 9,106 9,106 9,106 9,106 9,106 1,272,487 -

0 125,753 125,753 125,753 125,753 125,753 125,753 125,753 125,753 125,753 125,753 125,753 125,753 -

0 0 0 0 7,605 310,723 319,895 0 0 45,840 177,594 0 340,331 -

15

437,481

201,332

124,674

508,704

-

-

-

-

-

-

209

210 In Table 6.7, the optimal activity acceleration methods and other optimal resulting schedule data are provided for optimal project completion time where the start time of activity 16 is changed from week 23 to week 24 to reduce the financing cost by only $258. This small change can occur in some projects while the change can be bigger in some other projects. In addition, the schedules of borrowed money for Alts 1 and 2 are provided in Tables 6.8 and 6.9, respectively, and the schedules of repaid money including the interest are provided in Tables 6.10 and 6.11 for Alts 1 and 2, respectively. Tables 6.8 to 6.11 show that the financing schedules change when the start times of activities change. In Chapter 5, it was mentioned that the selected optimal financing alternatives and the amount of money to be borrowed depend on the average of the cumulative net balance of the cash flow and the duration of keeping the money. Moreover, it was discussed that when the average of the cumulative net balance of the cash flow becomes significantly negative, the short-term loans may be included in the optimal decision in addition to the long-term loan. As shown in Tables 6.8 to 6.11, in all three project completion times (i.e., crash, optimal, and normal project completion times), both short-term loans and longterm loan are among the selected optimal financing alternatives since the average of the cumulative net balance of the cash flow is significantly negative for these three project completion times. However, as it was mentioned in Chapter 5, in addition to the average of the cumulative net balance of the cash flow, the interest rates calculated based on the duration of keeping money, is another important factor when the amount of money to be borrowed is calculated using short-term loans and a long-term loan. Therefore, since both the duration of keeping the money and the cumulative net balance of the cash flow are

211 changed by changing the start times of activities, the schedules of borrowed money are changed for Alts 1 and 2 (see Tables 6.8 and 6.9). Furthermore, as shown in Figure 6.3, when project completion times of 53 and 54 weeks are considered, the financing costs are less compared to the financing costs when project completion times are 51 and 52 weeks. This decrease occurs because the financing period and existing financing alternatives change when the project completion time changes. As a result, because the financing period is 15 months when the project completion times are 53 and 54 weeks, the existing financing alternatives are the same. However the optimal financing alternatives are obtained similarly for both project completion times of 53 and 54 weeks and the difference is in the amount of borrowing in each optimal financing alternative. The optimal financing schedule is presented just for project completion time of 54 weeks (normal duration) in Table 6.8. For project completion time of 54 weeks (and also 53 weeks), B12 and C9 are the optimal financing alternatives in addition to the long-term loan (LP) and the line of credit (LC). However, BC12 and C9 are not among the optimal financing alternatives for project completion time of 52 weeks or less since the optimal amount for B12 occurs in month 3 and should be repaid after 12 months and the optimal amount for C9 occurs in month 6 and should be repaid after 9 month, both of which (i.e., B12 and C9) are repaid in month 15. Therefore, because the financing periods are less than 15 months (i.e., only for 52 weeks or less), the optimal financing alternatives B12 and C9 cannot be used in situations when the project completion times are 52 weeks or less.

212 6.2.2

Validating Models 1 and 2. Model 2 is proposed not only to improve the

computational time compared to Model 1, but also to validate the results of Model 1. Both financing cases are tested using both Models 1 and 2. The results and information of Model 1 (i.e., Tests 1 and 2) were presented in the preceding section (i.e., Section 6.2.1). Tests 3 and 4 are performed using Model 2 to test the two similar financing cases. The information regarding each test (i.e., Test 1 to 4) is presented in Table 6.12.

Model 1 (academic)

Model 2 (practical)

Model 2 (practical)

Case 2 (only a line of credit)

Case 1 (All financing alternatives) Case 2 (only a line of credit)

Total time with optimal GA identification process (sec)

Condition 1 (early activity start times) Condition 2 (variable activity start times) Condition 1 (early activity start times) Condition 2 (variable activity start times) Condition 1 (early activity start times) Condition 2 (variable activity start times) Condition 1 (early activity start times) Condition 2 (variable activity start times)

Total time without optimal GA identification process (sec)

Case 1 (All financing alternatives)

Optimal profit ($)

4

Model 1 (academic)

958,139

94,682

349,977

68,433

307,096

7,935

273,975

3,849

192,991

Optimal financing cost ($)

3


2

Financing case

1

Type of model

Test

Table 6.12. Information of Each Test Considering Models 1 and 2

156,655 156,396

958,398

247,683

867,111

247,683

867,111

156,655

958,139

156,396

958,398

247,683

867,111

247,683

867,111

213 For all four tests in Table 6.12, the processes of identifying the optimal values of the GA parameters and optimal GA methods are used to find the optimal GA parameters (i.e., N p , C p , M p , M pb ) and methods (i.e., parent selection method and crossover operation method). The initial, increment, and maximum values of each GA parameter are set by the user. Considering Tests 1 to 4, Np=[75:25:125], Cp=[0.7:0.1:0.8], Mp=[0.7:0.1:0.9], Mpb=[0.15:0.05:0.25]. For all Tests 1 to 4, 54 combinations exist at the beginning (i.e., 54  N P  C p  M p  M pb  3  2  3  3 ). However in these four tests (i.e., Tests 1 to 4), there is no need to relax the boundaries since the optimum GA parameters are obtained within the ranges. In addition, two parent selection methods (i.e., roulette wheel and tournament) and four crossover operation methods (i.e., single point, double point, uniform, and combined) are considered concerning GA methods in the process of identifying the optimal GA methods. Since the selection of the optimal GA methods depends on the problem (i.e., number of variables, initial parent setting, fitness function, encoding, and GA algorithm), one needs to know which parent selection method and which crossover operation perform better in terms of finding the optimal solution in less computational time. Using the process of identifying the optimal GA methods, the roulette wheel selection (RWS) method and the double crossover operation perform better in this problem and are identified as the optimal GA methods for both Tests 1 and 2, whereas the tournament selection (TRS) method and the combined crossover operation perform better in this problem and are identified as the optimal GA methods for both Tests 3 and 4.

214 According to Table 6.12, the results of Tests 3 and 4 validate the results of Tests 1 and 2 respectively. The only difference between Tests 1 and 2, and also Tests 3 and 4 is the financing case where Case 1 (all financing cases) is used for Tests 1 and 3, and Case 2 (only a line of credit) is used for Tests 2 and 4. As shown in Table 6.12, the total computation time of Model 2 is shorter than Model 1 regardless of whether optimal GA identification is used or not. For example, if optimal GA parameters and methods are used, 273,975 and 192,991 seconds are used for Tests 3 and 4, respectively, versus 349,977 and 307,096 seconds used for Tests 1 and 2, respectively. Similarly, if the GA parameters are set by the user, the total computation time to run Model 2 is shorter than Model 1 (i.e., 7,935 and 3,849 seconds to run Tests 3 and 4, respectively, versus 94,682 and 68,433 seconds to run Tests 1 and 2, respectively). This better performance of Model 2 in terms of the computational time occurs because Model 2 does not find the optimal result for every project completion time between crash and normal points, whereas Model 1 searches and finds the optimal result for every project completion time. No model currently exists that performs both time-cost tradeoff and financing optimization considering different financing alternatives simultaneously, also considering two conditions for start times of activities in the same model. Current models do not provide an analytical report that investigates the effect of variable activity start times on both optimal financing cost and construction schedule.

6.3

Conclusion of the Third Stage

The third stage models (i.e., Models 1 and 2) investigate the effects of changing the start times of activities on optimal financing cost, optimal profit, and optimal

215 construction schedule. Regardless of whether the duration of keeping the money and the average of the cumulative net balance of the cash flow conflict with each other or not, the third stage models are capable to find the optimal results while providing analytical reports for two conditions, namely (1) early activity start times and (2) variable activity start times. These reports allow users to perform sensitivity analysis and make a decision whether to accept a more critical schedule with less financing cost or less critical schedule with more financing cost. Although the example project that was tested in this chapter did not display significant decreases in the financing cost, the optimal financing cost would have gone down significantly if the average of the cumulative net balance of the cash flow had gone down when the start times of activities were changed.

216 CHAPTER 7 7. FOURTH STAGE OF THE RESEARCH In the first, second, and third stages of this research, models were proposed based on the assumption that contractors can obtain the required financing for the project, if necessary from different lenders. However, borrowing a large amount of money can be difficult for small contractors. This problem can be even worse in the absence of sufficient collateral since the alternatives offered by banks or other lenders can be limited (Nesan 2012). If a contractor cannot obtain the required financing for project completion times between the crash and normal points, the duration of the project should be extended. In this case, the models proposed in Stages 1 to 3 cannot be used to solve the problem and find the optimum solution. As a result, another model is needed to cover the extension period if the credit is not enough to perform the project between the crash and normal durations. In this chapter, two models (i.e., Models 1 and 2) are proposed to find the optimum profit after optimizing three variables (i.e., construction acceleration methods, financing variables, and start times of activities) for project completion times between the crash and extended points while considering liquidated damages or late completion penalty during the extension time. Even though some past studies do consider the extension time, the models proposed in the fourth stage of this study differ in four respects. 

All past studies consider one financing alternative (i.e., line of credit) without financing optimization, whereas in the fourth stage models, short-term and longterm loans in addition to the line of credit are used, not only to provide more

217 flexibility for the contractors to seek financing from more financing sources, but also to pay less financing cost by optimizing financing variables. 

Past studies do not provide a financing schedule that states what amount of money should be taken and what amount of money including interest should be repaid in each month. The fourth stage models find and provide an optimum financing schedule for both borrowed money and repaid money including interest.



When banks do not offer low interest rates in the absence of adequate collateral, contractors can obtain the required credit at a higher interest rate from alternative lenders. Therefore, small contractors who do not have sufficient collateral are able to find short-term loans from alternative lenders, albeit at high interest rates. The fourth stage models search for an optimal solution between crash and extended project completion times and allow contractors to perform the project without extending the project completion time, whereas past studies do not consider shortterm loans, and consequently are not able to provide a solution without extending the project duration. Even when short-term loans with interest rates higher than the interest rate of the line of credit are considered, not only can the project be performed between the crash and normal durations (no need to extend), but also the optimal financing cost can be lower for durations between the crash and normal points compared to the financing cost for extended project durations. The lower optimal financing cost is obtained because both the total cost and the duration of keeping the money are smaller for durations between the crash and normal points. However, considering the short-term loans, if alternative lenders ask for very high interest rates compared to the interest rate of a line of credit, the

218 optimal financing cost may be higher for durations between the crash and normal points, depending on the average of the cumulative net balance of the cash flow and the credit limit. Even if the optimal financing cost becomes higher between the crash and normal points compared to the optimal financing cost during the extension time, the optimal profit between the crash and normal points is still higher than the optimal profit in the extension period, because between crash and normal points, the total cost is lower and liquidated damages do not exist. To wrap up, if the credit limit is not adequate to perform the project between the crash and normal durations, past studies not only are unable to find a feasible solution between the crash and normal durations, but they find a solution that may be non-optimal either. In contrast, the models of the fourth stage are capable to search for the optimal solution between the crash and extended points, even if a contractor cannot obtain the required line of credit. 

When the contractors do not have adequate credit to perform the project between the crash and normal project completion times, the project has to be extended to reduce the maximum negative balance in the cash flow and to satisfy cash constraints (Elazouni and Gab-Allah 2004; Elazouni and Metwally 2005). This has been done in previous studies by pushing forward the start times of activities instead of changing the duration of the activities in extension time. In other words, past studies consider the normal acceleration methods for both normal and extended points. In contrast to previous studies, the proposed models in this fourth stage consider the most decelerated methods for all activities for the extended point. In other words, project duration can be extended by using decelerated

219 durations instead of using the normal durations, as was the case in past research (i.e., Ali and Elazouni 2009; Elazouni and Metwally 2007; Liu and Wang 2008). When the decelerated durations are used for activities, fewer resources are allocated to the activities with lower costs. Therefore, considering the decelerated durations instead of using the normal durations for activities in the extension time reduces the total cost, and the financing cost that depends on the total cost. To prove the aforementioned points, three scenarios are considered and tested using two models in this stage. In the first scenario, as was the scenario in all past studies, just the line of credit with an insufficient credit limit is considered in such a way that the project completion time has to be extended. In the second scenario, in addition to the line of credit with the same credit limit similar to the credit limit of the first scenario, it is also assumed that the contractor can obtain some short-term loans with higher interest rates than the interest rate of the line of credit; no credit limits are considered for short-term loans. In this scenario, it is assumed that since the small contractor does not have sufficient collateral, the banks are reluctant to offer the short-term and long-term loans with the low interest rates, therefore the contractor obtains some short-term loans from alternative lenders with higher interest rates. The purpose of the second scenario is to prove that even by considering short-term loans with higher interest rates, not only can the project be executed between the crash and normal project completion times, but also more profit is obtained compared to the first scenario where just the line of credit is considered with an insufficient credit limit. In the third scenario, it is assumed that all financing alternatives can be considered and the contractor has access to all short-term loans, long-term loans, and line of credit with no limitation. The purpose of the third

220 scenario is to prove that when all financing alternatives are considered, the optimal financing cost, optimal total cost including lower financing cost, and higher profit are obtained than any other scenarios where all financing alternatives are not considered.

7.1

Methodology and Computational Process of the Fourth Stage Models

Two models are proposed in the fourth stage of the study. Model 1 takes longer computation time because the optimal solutions are computed for every project completion time between crash and extended points, whereas Model 2 takes less computation time because it only finds one optimal solution between crash and extended points. The reason why the optimal solutions are computed for every project completion time in Model 1 is to see the difference of the optimal outcomes for every project completion time, to do a sensitivity analysis, and to be able to compare the results of different scenarios, whereas in Model 2, the purpose is to find the best optimal solution where a sensitivity analysis is not required. It should be noted that the same equations used in the second stage and the same algorithm used in the third stage are used in the fourth stage models, but with some modifications in the MATLAB programming. These changes include the extension time, the liquidated damages, and the most decelerated methods. Thus, contrary to the second and third stages, four activity acceleration methods are considered in the fourth stage. In other words, activity acceleration method 1 means the most decelerated duration and associated cost, activity acceleration method 2 involves normal duration and cost, activity acceleration method 3 signifies accelerated duration and associated cost, and activity acceleration method 4 represents crashed duration and cost.

221 7.2

Testing the Fourth Stage Models

To investigate how the optimal schedule, optimal financing cost, optimal total cost including financing cost, and optimal profit differ when different financing scenarios are considered, the two fourth stage models are tested using an example project that is shown in Figure 7.1. This example network is created to reflect the conditions in a real project (i.e., not too few and not too many activities, numerous predecessors and successors, and activities in parallel and in series). Whereas Model 1 is used to verify the structure of the fourth stage models and to compare the optimal financial outcomes of every project completion time between crash and extended durations, Model 2 is used to validate Model 1 and to confirm that the computation time of Model 2 is shorter than the computation time of Model 1.

A

E

I

M

Q

B

F

J

N

R T

Start C

G

K

O

D

H

L

P

S

Figure 7.1. Network of the Example Project The schedule data including activity acceleration methods, and cost data including contractual terms are shown in Tables 7.1 and 7.2, respectively. The topological sorting and improved Dijkstra`s algorithm are adopted to solve the CPM network presented in Figure 7.1 using the schedule information and cost data presented in Tables 7.1 and 7.2,

222 respectively. The project costs and contract price are calculated using the normal acceleration method (i.e., Method 2: normal time and normal cost) for all activities and are shown in Table 7.3.

Table 7.1. Project Schedule Data and Activity Acceleration Method Inputs



9 7 8 5 4 6 5 8 9 6 5 8 7 6 4 8 7 8 10 9


11 9 9 6 5 8 6 9 11 7 6 9 8 7 5 9 8 9 11 10


13 11 10 7 6 9 7 10 13 8 7 10 9 8 6 10 9 10 12 11



17 20


9 12 16 19

Predecessor 3

Predecessor 2

Predecessor 1

1 1 1 1 2 3 4 5 6 7 8 9 10 11 11 13 14 15 15 18



Duration (weeks) Acceleration Method 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Activity name

Activity ID

Predecessors

8 6 7 5 7 8 7 6 7 7 9 -

16,000 16,000 30,000 48,000 33,000 25,000 63,000 35,000 37,000 65,000 55,000 58,000 39,000 47,000 45,000 53,000 55,000 35,000 54,000 72,000

20,000 20,000 35,000 57,000 40,000 30,000 75,000 40,000 45,000 75,000 65,000 66,000 45,000 55,000 55,000 60,000 65,000 40,000 60,000 80,000

30,000 30,000 45,000 70,000 52,000 50,000 95,000 50,000 57,000 88,000 80,000 75,000 55,000 70,000 70,000 68,000 80,000 50,000 70,000 90,000

50,000 50,000 60,000 70,000 65,000 75,000 90,000 70,000 85,000 75,000 90,000 -

223 Table 7.2. The Inputs of Cost Data and the Contractual Terms of the Project Data type Cost data

Contract terms

Item

Amount

Weekly fixed overhead cost OF

$40,000/week

Variable overhead percentage OV of (DC)

8%


5%


7%


1%

Early completion bonus OBS Liquidated damages

$0/week

OLQ

$40,000/week

Advance payment percentage OAP of contract bid price

0%

Retained percentage R of pay requests

10%

Number of months between submitting pay requests Lag in paying payment requests

LS

1 month

LP (months)


1 month

LR (months)

0 month

Table 7.3. The Project Costs and Contract Price Using Method 2 Type of cost/price Direct cost (DC) Fixed overhead cost (FOC) Variable overhead cost (VOC) Mobilization cost (MOB) Markup (MP) Cost of bonding (BD) Contract price (CP) Indirect cost (IDC) Total cost excluding financing cost (TC)


Amount ($) 8,211,000

54  $40, 000

2,160,000

Calculation

0.08  $8, 211, 000

656,880

0.05  ($8, 211, 000  $656,880)

443,394

0.07  ($8, 211, 000  $2,160, 000  $656,880  $443, 394)

802,989 122,743

0.01  ($8, 211, 000  $2,160, 000  $656,880  $443, 394  $802, 989)

$8, 211, 000  $2,160, 000  $656,880  $443,394  $802, 989  $122, 743 $2,160, 000  $656,880  $443, 394  $122, 743 $8, 211, 000  $2,160, 000  $656,880  $443, 394  $122, 743

12,397,006 3,383,017 11,594,017

224 As it was mentioned at the beginning of this chapter, three scenarios are tested using both models (see Table 7.4). In Scenario 1, only the line of credit with a total credit limit of $1,670,000 is considered. In Scenario 2, in addition to considering the same line of credit of $1,670,000 used in Scenario 1, some randomly selected short-term loans (i.e., B3, B6, C3, and C6) are also considered with no credit limit, but with higher interest rates than the interest rate of the line of credit. The reason why no limit is considered for the selected short-term loans is that usually lenders allow borrowers to specify the required credit even when borrowers do not provide enough collateral, but they seek higher interest rates. In Scenario 3, all financing alternatives are considered with no credit limit to compare the results when the contractors have the ability to negotiate and specify the required financing. The results of Scenario 3 highlights the significance of using the proposed financing model rather than letting the lending institution decide the required credit.

225 Table 7.4. Financing Data of Three Different Scenarios

7.2.1

No No No No Yes



Total credit limit

Monthly Monthly Monthly Monthly Optimum time


No No No No Yes Yes No No Yes Yes No No No No No No No No No No No Yes


Yes

Scenario 3 Total credit limit


Optimum time



Scenario 2 Total credit limit

22 19 18 17 21 19 17 11 20 18 10 10 20 17 11 10 18 16 12 9 7 15


APR (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22


Alternative

Scenario 1

No No No No No No No No No No No No No No No No No No No No No No

Results and Analysis of Testing Model 1 in the Fourth Stage. The results of

testing all scenarios, using Model 1, are shown in Tables 7.5 to 7.7 and Figures 7.2 to 7.4. The crash, normal, and extended durations are obtained as 44, 54, and 61 weeks, respectively. As shown in Table 7.5, in Scenario 1, the project cannot be executed in less than 55 weeks, whereas in Scenario 2, the project can be performed for all project completion times except the crash duration (i.e., 44 weeks). The reason why in Scenario 1

226 the project cannot be completed in less than 55 weeks is because only one financing alternative is considered (i.e., only a line of credit with a credit limit of $1,670,000). The project has to be extended by one week from 54 to 55 weeks. In contrast, in Scenario 2, the project can be completed any time between 45 and 54 weeks (cannot be completed in the crash duration of 44 weeks) since some short-term loans are considered in addition to the same line of credit used in Scenario 1. It should be noted that even when short-term loans with no credit limit are considered in Scenario 2, the project cannot be completed in 44 weeks (the crash duration). The reason why the project cannot be completed in 44 weeks in Scenario 2, is because the interest of short-term loans must be paid monthly, whereas the average of the cumulative net balance of the cash flow is significantly negative. In Scenario 2, since the crash point has the largest negative average of the cumulative net balance of the cash flow, the interest of short-term loans cannot be paid monthly, using any kind of financing. Thus, in terms of the feasibility of the project, Scenario 2 has the advantage over Scenario 1, and Scenario 3 has the advantage over both Scenarios 1 and 2 (see Table 7.5).

Scenario 2

Scenario 3

Variable start times

Early start times


Early start times


Early start times


Early start times


Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Scenario 1

Early start times

No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Percentage of critical activities


Scenario 3

No No No No No No No No No No No Yes Yes Yes Yes Yes Yes Yes

Sum of the activities` total floats

Early start times

Scenario 2

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

Feasibility of the project Scenario 1

Project completion time (weeks)

Table 7.5. Results of the Schedules That Lead to Optimal Profit Obtained by Testing Model 1 Considering Three Scenarios for Every Completion Time

Scenario 1

Scenario 2

Scenario 3

27 24 38 47 47 47 62

27 24 38 47 47 47 62

0 0 0 0 0 1 5 9 12 15 16 25 22 31 31 31 46

0 0 0 0 0 1 5 9 12 15 16 25 22 30 30 30 45

0 0 0 0 0 0 1 5 9 12 15 16 25 22 31 31 31 46

0 0 0 0 0 0 1 5 9 7 10 11 18 19 27 25 27 40

75% 75% 30% 30% 30% 30% 30%

75% 75% 30% 30% 30% 30% 30%

100% 100% 100% 100% 100% 95% 75% 75% 75% 75% 75% 55% 75% 55% 55% 55% 30%

100% 100% 100% 100% 100% 95% 75% 75% 75% 75% 75% 55% 75% 60% 60% 60% 30%

100% 100% 100% 100% 100% 100% 95% 75% 75% 75% 75% 75% 55% 75% 55% 55% 55% 30%

100% 100% 100% 100% 100% 100% 95% 75% 75% 85% 80% 80% 65% 75% 65% 70% 65% 40%

227

Early start times


Scenarios 1 to 3

Early start times


Total cost including financing cost and liquidated damages ($)


44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

Liquidated damages ($)

Financing cost ($)

Early start times -

-

-

-

174,050

174,050

-

-

-

-

-

12,063,208

12,063,208

-

-

270,256

270,256

178,404

178,404

-

-

-

12,017,974

12,017,974

11,926,122

11,926,122

-

-

254,225

254,225

169,133

169,133

-

-

-

11,901,327

11,901,327

11,816,236

11,816,236

-

-

245,017

245,017

161,714

161,714

-

-

-

11,800,576

11,800,576

11,717,273

11,717,273

-

-

238,068

238,068

157,872

157,872

-

-

-

11,737,237

11,737,237

11,657,041

11,657,041

-

-

236,621

236,621

160,227

160,227

-

-

-

11,690,739

11,690,739

11,614,346

11,614,346

-

-

233,185

233,185

157,108

157,108

-

-

-

11,653,594

11,653,594

11,577,516

11,577,516

-

-

232,147

232,147

157,577

157,577

-

-

-

11,654,000

11,654,000

11,579,430

11,579,430

-

-

229,835

229,835

157,174

157,174

-

-

-

11,659,935

11,659,935

11,587,275

11,587,275

-

-

239,286

239,286

144,006

143,334

-

-

-

11,684,438

11,684,438

11,589,159

11,588,487

-

-

240,297

240,297

145,615

144,283

-

-

-

11,701,636

11,701,636

11,606,953

11,605,622

244,894

244,894

240,077

240,077

146,851

145,589

40,000

11,807,778

11,807,778

11,761,003

11,761,003

11,667,778

11,666,516

243,801

243,801

240,466

240,466

147,862

146,118

80,000

11,850,397

11,850,397

11,825,516

11,825,516

11,732,912

11,731,169

255,341

255,341

255,014

255,014

157,957

156,551

120,000

11,931,732

11,931,732

11,901,921

11,901,921

11,804,864

11,803,457

258,318

258,318

256,679

256,291

159,899

158,472

160,000

12,012,441

12,012,441

11,967,710

11,967,321

11,870,930

11,869,503

258,671

258,671

257,025

256,636

160,296

158,902

200,000

12,081,453

12,081,453

12,036,716

12,036,327

11,939,987

11,938,593

259,072

259,072

257,421

257,032

160,734

159,305

240,000

12,152,783

12,152,783

12,108,040

12,107,651

12,011,352

12,009,924

273,078

273,078

270,994

270,601

169,731

168,038

280,000

12,244,521

12,244,521

12,199,344

12,198,951

12,098,081

12,096,389

Scenario 3 Variable start times

Scenario 2

Early start times

Scenario 1


Scenario 3

Early start times

Scenario 2 Variable start times

Scenario 1

Early start times

Project completion time (weeks)

Table 7.6. Results of Optimum Costs Obtained by Testing Model 1 Considering Three Scenarios for Every Completion Time

228

229 Table 7.7. Results of Optimum Profits Obtained by Testing Model 1 Considering Three Scenarios for Every Completion Time Project completion time (weeks)

Early start times


Early start times


Early start times


Profit ($)

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

589,228 546,608 465,274 384,565 315,553 244,223 152,485

589,228 546,608 465,274 384,565 315,553 244,223 152,485

379,032 495,678 596,430 659,769 706,266 743,412 743,006 737,071 712,568 695,370 636,002 571,489 495,085 429,296 360,290 288,966 197,661

379,032 495,678 596,430 659,769 706,266 743,412 743,006 737,071 712,568 695,370 636,002 571,489 495,085 429,685 360,679 289,355 198,055

333,797 470,884 580,770 679,733 739,965 782,660 819,489 817,576 809,731 807,847 790,052 729,228 664,093 592,142 526,076 457,019 385,653 298,924

333,797 470,884 580,770 679,733 739,965 782,660 819,489 817,576 809,731 808,519 791,384 730,490 665,837 593,548 527,503 458,413 387,082 300,617

Scenario 1

Scenario 2

Scenario 3

230 Crash Point

Normal Point

Extended Point

270,000

Scenario 1 with early start times of activities

260,000

Financing Cost ($)

250,000

Scenario 1 with variable start times of activities

240,000 230,000


220,000 210,000


200,000 190,000


180,000 170,000


160,000 150,000 140,000 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 Project Completion Time (weeks)

Figure 7.2. Financing Cost for Every Project Completion Time Considering Three Scenarios with Early and Variable Activity Start Times

Profit ($)

Crash Point 830,000 780,000 730,000 680,000 630,000 580,000 530,000 480,000 430,000 380,000 330,000 280,000 230,000 180,000 130,000

Normal Point

Extended Point Scenario 1 with early start times of activities Scenario 1 with variable start times of activities Scenario 2 with early start times of activities Scenario 2 with variable start times of activities Scenario 3 with early start times of activities

Extension Time


43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 Project Completion Time (weeks)

Figure 7.3. Profit for Every Project Completion Time Considering Three Scenarios with Early and Variable Activity Start Times

231

Sum of the Activity Total Floats

Crash Point 60 56 52 48 44 40 36 32 28 24 20 16 12 8 4 0

Normal Point

Extended Point Scenario 1 with early start times of activities Scenario 1 with variable start times of activities Scenario 2 with early start times of activities Scenario 2 with variable start times of activities Scenario 3 with early start times of activities

Extension Time


43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 Project Completion Time (weeks)

Figure 7.4. Sum of Activities` Total Floats for Every Project Completion Time Considering Three Scenarios with Early and Variable Activity Start Times Comparing the results of Scenario 2 to Scenario 1 proves that not only can the contractor complete the project without extending the project and without paying liquidated damages (see Table 7.5), but also achieve more profit while paying less financing cost (see Tables 7.6 and 7.7 and Figures 7.2 and 7.3). As shown in Table 7.6 and Figure 7.2, the optimal financing cost that is obtained for every project completion time between the normal and extended points are also lower in Scenario 2 compared to Scenario 1. Lower optimal financing cost and higher profit were obtained with short-term loans (i.e., B3, B6, C3, and C6) even with higher interest rates than the interest rate of the line of credit. The reason is because when only the line of credit is considered in Scenario 1, the contractor cannot repay the debts in less duration because of the negativity of the cumulative net balance of the cash flow, and as a consequence, the interest of the line of credit is compounded. In contrast, in Scenario 2, because short-term loans are available

232 with higher interest rates, the interest of the selected short-term loans can be lower than the compounded interest of the line of credit for longer durations. Meanwhile, when all financing alternatives are considered with no credit limits (i.e., Scenario 3), the optimal financing costs and profits are far better than any other scenario (see Tables 7.6 and 7.7 and Figures 7.2 and 7.3). Therefore, although Scenario 2 obtains better optimal financing costs and profits compared to Scenario 1, Scenario 3 has the advantage of obtaining better optimal financing costs and profits over both Scenarios 1 and 2 (see Tables 7.6 and 7.7 and Figures 7.2 and 7.3). To conclude, Scenario 3 is the best and Scenario 1 is the worst in terms of the optimality of the solutions. In Tables 7.5 to 7.7 and Figures 7.2 to 7.4, the results are shown for two conditions, the optimal results using early and variable start times of activities. As shown in Tables 7.6 and 7.7 and Figures 7.2 and 7.3, the optimal results are not affected by changing the start times of activities in Scenario 1, whereas changing the start times of activities affects the optimal financing costs and profits at some project completion times in Scenarios 2 and 3 negligibly (i.e. affects the optimal financing cost and profit of project completion times between 58 and 61 weeks for Scenario 2, and between 53 and 61 weeks for Scenario 3). Although the changes in the optimal financing costs and profits are negligibly small in both Scenarios 2 and 3 when variable activity start times are considered, the size of the changes is bigger in Scenario 3 compared to Scenario 2 (see Tables 7.6 and 7.7, and Figures 7.2 to 7.3). For example, in Scenario 2, the profits for a project completion time of 61 weeks (i.e., extended point) differ by $198,055-$197,661= $394 (0.2%), whereas in Scenario 3, they differ by $300,617-$298,924= $1,693 (0.4%). On the other hand, as shown in Table 7.5 and Figure 7.4, when variable activity start

233 times are considered while searching for the optimal results, compared to Scenarios 2, Scenario 3 uses more total float that results in less available total float for the project and makes the schedule more critical. For instance, in Scenario 2, the sum of the total floats for a project completion time of 61 weeks (i.e., extended point) differs by 46-45=1 (2%), whereas in Scenario 3, it differs by 46-40= 6 (15%). In summary, when variable activity start times are considered, more total float is used in Scenario 3 compared to Scenario 2 (see Table 7.5 and Figure 7.4) where the size of the changes of the optimal results are bigger in Scenario 3 (see Tables 7.6 and 7.7). Therefore, the comparison between these three scenarios when variable activity start times are considered, suggests that: (1) lower optimal financing costs and higher profits are obtained when the schedule is more critical, and (2) the effects of considering variable activity start times are more pronounced in terms of the optimal financing cost and profit when all financing alternatives are considered (i.e., in Scenario 3). In other words, more risk results in more profit. It should also be noted that considering different scenarios of financing alternatives not only results in achieving different optimal financing cost and profit, but also results in different construction schedules. For example, as shown in Table 7.7 and Figure 7.3, in Scenario 1, the optimum profit is obtained using the construction schedule that results in a 56 week project completion time, whereas in Scenarios 2 and 3, the optimum profit is obtained when the project duration is 50 weeks. However, sensitivity analysis is highly recommended before choosing the construction schedule that leads to the optimal profit. For instance, in Scenario 3, the highest optimum profit of $819,489 is obtained when the project is completed in 50 weeks (see Table 7.7 and Figure 7.3) with 95% of the activities being critical (see Table 7.5 and Figure 7.4), whereas the next

234 optimum profit of $817,576 is obtained when the project is completed in 51 weeks (see Table 7.7 and Figure 7.3) with 75% of the activities being critical (see Table 7.5 and Figure 7.4). The user will have to decide which solution agrees better with the company`s risk-taking policy. In Scenario 1, the optimum financing cost is $243,801 that is obtained in a duration of 56 weeks, whereas the optimum financing cost in Scenarios 2 and 3 are $229,835 and $143,334 that are obtained in project durations of 52 and 53 weeks, respectively. Therefore, the result of the analysis of these three scenarios indicate that the best financing cost is obtained in Scenario 3 where all financing alternatives are available to the contractor. Although the optimum financing cost obtained in Scenario 3 proves the significance of having access to loans with a lower interest rates, it should be noted that the contractor is not able to find out how and what amount of money to borrow each month to pay the minimum financing cost, unless the contractor uses the proposed financing optimization method. The proposed financing model has the ability to set optimum financing schedules and to provide both borrowing and repayment schedules considering different scenarios. Given the different financing alternatives, not only is the optimal financing cost different, but also the financing schedules are different. The optimum financing schedules are obtained and presented in Tables 7.8 and 7.9 for each scenario separately (i.e., Scenarios 1, 2, and 3). The results in Table 7.8 indicate how and what amount of money the contractor should borrow each month to acquire the most optimum financing cost, and Table 7.9 shows how and what amount of money the contractor should repay including the interest every month based on the optimum borrowing schedule. As shown in Table 7.8, the optimal financing alternatives are E12,

235 LP, and LC for the optimum scenario (i.e., Scenario 3). In Scenario 3, although the APR of LC is higher than the APR of both E12 and LP (i.e., 15% for LC versus 9% for E12 and 7% for LP), and the APR of E12 is higher than the APR of LP (9% for E12 versus 7% for LP), these three financing alternatives (i.e., E12, LP, and LC) constitute the optimal financing alternatives. Contrary to the belief that the alternative with lower interest rate is always the best financing alternative, the financing schedule of Scenario 3 in Table 7.9 proves that this belief is not true. This finding also proves that the optimum financing schedule and optimum financing cost cannot be achieved without using the proposed financing model. Therefore, the results demonstrate the importance of using the proposed financing model before negotiating with banks and other lenders to specify optimal financing alternatives, optimal required credits, and optimal financing schedules. The results of testing Model 1 in the fourth stage prove all four statements at the beginning of this chapter. First, when all financing alternatives are considered (i.e., Scenario 3), the contractor pays less financing cost compared to Scenario 1 where only a line of credit is considered. Second, optimal financing schedules are provided using the fourth stage models, whereas past studies are not capable of finding and providing optimum financing schedules. Third, when contractors obtain some short-term loans even with higher interest rates, not only can the project be executed without an extension, but also lower financing cost and higher profit are obtained mainly because the liquidated damages are avoided. Fourth, in the models proposed in the fourth stage, the most decelerated methods of construction (activity acceleration method 1) are considered for activities. Therefore, in the extension time, the optimal financing cost and profit are obtained not only by changing the start times of activities but also by changing the

236 duration of activities, whereas this has been done in previous studies only by changing and pushing forward the start times of activities using the normal acceleration methods for all activities.

7.2.2

Validating Models 1 and 2. Model 2 is proposed and tested to validate Model 1

and improve the computational time when sensitivity analysis is not required. In the preceding section (i.e., Section 7.2.1), the results of testing Model 1 were presented and discussed. In this section, Model 2 is tested for three scenarios using the same example project (i.e., same schedule data, construction cost and contractual information, financing data, and scenarios). Therefore, six tests are performed and the information of each test in addition to the optimal results and computational times are presented in Table 7.10. In each test, the optimal GA parameters (i.e., N p , C p , M p , M pb ) and methods (i.e., parent selection method and crossover operation method) are identified using the processes described in Chapter 5. For all six tests, the initial, increment, and maximum values of each GA parameter are Np=[100:50:200], Cp=[0.7:0.1:0.9], Mp=[0.7:0.1:0.9], Mpb=[0.15:0.05:0.25]. As a result, for all tests (i.e., Tests 1 to 6), 81 combinations exist at the beginning (i.e., N P  C p  M p  M pb  3  3  3  3  81 ), but the GA parameters can be relaxed and the number of combination increases if the optimal result is obtained at the maximum boundaries of the GA parameters. However, since in this example, the optimum GA parameters are obtained between the ranges of the minimum and maximum boundaries of the GA parameters, the maximum boundaries of the GA parameters are not exceeded in these six tests (i.e., Tests 1 to 6).

Table 7.8. Optimized Financing Inflow Schedule (Borrowed Money) for Optimum Project Completion Times Considering Scenarios 1 to 3 Month

Scenario 1 (optimal duration=55 weeks) LC ($)

Scenario 2 (optimal duration=50 weeks) B3 ($)

Scenario 3 (optimal duration=50 weeks)

B6 ($)

LC ($)

E12 ($)

LP ($)

LC ($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

560,845 810,623 285,208 322,451 334,371 253,682 229,211 189,909 264,276 154,140 201,178 249,609 162,282 254,359 239,011

0 0 0 64,140 0 0 0 0 0 0 0 0 0 0 0

0 0 0 266,925 0 0 0 0 0 0 0 0 0 0 0

565,489 903,510 490,930 328,334 530,106 568,336 478,944 295,206 252,018 432,918 404,582 279,996 442,791 373,604 0

768,081 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1,362,584 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 328,627 182,312 52,021 11,892 1,672 114,570 0 910,410 275,869 0

15

0

-

-

-

-

-

-

237

Table 7.9. Optimized Financing Outflow Schedule (Repaid Money Including Interest) for Optimum Project Completion Times Considering Scenarios 1 to 3 Scenario 2 (optimal duration=50 weeks)


LC ($)

B3 ($)

B6 ($)

LC ($)

E12 ($)

LP ($)

LC ($)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 41,503 365,658 148,518 169,567 253,902 265,612 300,658 215,922 341,790 238,852 235,857 331,047 301,836 305,718

0 0 0 0 1,027 1,027 65,167 0 0 0 0 0 0 0 0

0 0 0 0 3,898 3,898 3,898 3,898 3,898 270,823 0 0 0 0 0

0 138,710 560,708 507,927 234,662 341,825 498,508 603,932 460,744 296,096 405,947 513,561 407,503 577,510 1,005,848

0 5,536 5,536 5,536 5,536 5,536 5,536 5,536 5,536 5,536 5,536 5,536 773,617 0 0

0 101,505 101,505 101,505 101,505 101,505 101,505 101,505 101,505 101,505 101,505 101,505 101,505 101,505 101,505

0 0 0 0 0 0 163,900 257,604 117,474 28,631 8,894 126,524 0 378,270 828,265

15

1,239,609

-

-

-

-

-

-

Month


238

Model 2 (practical)


Scenario

Total time with optimal GA identification process (sec)

6

Model 2 (practical)

Scenario 1 (only a line of credit)

Total time without optimal GA identification process (sec)

5

Model 2 (practical)

Optimal project duration based on maximum profit (weeks)

4

Model 1 (academic)

Scenario 2 (some shortterm loans and line of credit) Scenario 3 (all financing alternatives)

early activity start times variable activity start times early activity start times variable activity start times early activity start times variable activity start times early activity start times variable activity start times

55

214,106

840,943

216,003

854,148

282,072

927,686

12,104

578,136

8,487

561,025

13,045

630,494

Maximum profit ($)

3

Model 1 (academic)


Optimal project duration based on minimum financing cost (weeks)

2

Model 1 (academic)

Minimum financing cost ($)

1

Type of model

Test

Table 7.10. Information about Tests of Models 1 and 2

243,801

56

589,228

243,801

56

589,228

55

229,835

52

743,412

50

229,835

52

743,412

50

143,334

53

819,489

50

144,006

53

819,489

50

-

-

589,228

56

-

-

589,228

56

Scenario 2 (some shortterm loans and line of credit)

early activity start times variable activity start times

-

-

743,412

50

-

-

743,412

50

Scenario 3 (all financing alternatives)

early activity start times variable activity start times

-

-

819,489

50

-

-

819,489

50

239

240 In the process of identifying the optimal GA methods, two parent selection methods and four crossover operation methods are considered. In terms of parent selection, roulette wheel, and tournament selection methods are considered. In terms of crossover operation, single point, double point, uniform, and combined crossover operations are considered. Each GA method performs differently for different problems. In the process of identifying the optimal GA methods, those methods that perform better in terms of the computational time and the optimal solution, are selected. The optimal GA methods, which are obtained for each test, are presented in Table 7.11. It is worthwhile to mention that the combined crossover operation that is proposed by this research is the most dominant optimal crossover operation in Tests 1 to 6. Since in each test (i.e., Tests 1 to 6), different numbers of variables (financing variables) and different fitness functions and GA algorithms are considered, the optimal selection method and the optimal crossover operation are obtained differently. The difference in the optimal GA methods obtained for each test, indicates the importance of identifying the optimal GA methods to find the optimal solution in less computational time. As shown in Table 7.10, Model 1 is validated. Indeed, the results of Tests 4, 5, and 6 validate the results of Tests 1, 2, and 3, respectively. Model 1 has the advantage of providing sensitivity analysis and a report for every completion time, whereas Model 2 has the advantage of less computational time. In addition, in Model 1, the duration that leads to the optimal result can be found based either on the minimum financing cost or the maximum profit (see Tests 1 to 3 in Table 7.10), whereas Model 2 can only find the duration that leads to the maximum profit (see Tests 4 to 6 in Table 7.10).

241 Table 7.11. Optimal GA Methods Obtained for Tests 1 to 6 Optimal selection method

Optimal crossover operation


Tournament

Combined

Model 1 (academic)

Scenario 2 (some short-term loans and line of credit)

Roulette wheel

Double point

3

Model 1 (academic)


Tournament

Combined

4

Model 2 (practical)


Roulette wheel

Combined

5

Model 2 (practical)

Scenario 2 (some short-term loans and line of credit)

Tournament

Combined

6

Model 2 (practical)


Roulette wheel

Combined

Type of model

Scenario

1

Model 1 (academic)

2

Test

As shown in Table 7.10, regardless of whether optimal GA identification is used or not, the total computation time of Model 2 is shorter than Model 1 (the computational times of Tests 4, 5, and 6 are shorter than the computational time of Tests 1, 2, and 3, respectively). For example, if the GA parameters are set by the user when all financing alternatives are considered, the total computational time to run Model 2 is 13,045 seconds, whereas the time to run Model 1 is 282,072 seconds. The reason why Model 2 performs faster is because Model 2 does not run to find the optimal results for every project completion time between crash and normal points, whereas Model 1 runs for every project completion time to find the optimal result for each duration separately.

7.3

Conclusion of the Fourth Stage

The fourth stage proves that if all proposed financing alternatives are considered when the required credit is specified by the contractor, the financing cost and profit are

242 far better compared to any other scenario. Even if the contractor cannot obtain all financing alternatives, alternative lenders lend money for short-term loans with higher interest rates. Therefore, even when short-term loans with higher interest rates are considered, not only can the project be performed without an extension, but also lower optimal financing cost and higher profit are obtained compared to the case when only a line of credit is considered. The models proposed in the fourth stage are able to find the optimal results between the crash and extended project completion times considering different financing scenarios even when contractors are limited by banks or lenders.

243 CHAPTER 8 8. CONCLUSION 8.1

Summary

This research is a significant improvement on past studies that considered finance-based scheduling problems and/or the time-cost tradeoff problem. This study considers several financing alternatives in terms of sources and types of financing, times of cash provisions, interest rates, and repayment options. No study has ever considered the integration of financing optimization and construction scheduling with respect to multiple financing alternatives. It should also be noted that this research improves the CPM algorithm by using the AON method, topological sorting, and an improved Dijkstra`s algorithm. This research involves four stages of development. Each stage can be used for different conditions. 

If the contractor wants to minimize the bid price and increase the chance of winning the contract, the first stage model can be used to minimize the financing cost and the project bid price. If the first stage model is used before the contract is signed, the bid price is calculated based on the normal activity durations and time-cost tradeoff is not performed.



If the contract is signed and the contractor wants to minimize the financing cost and obtains more profit when normal activity durations are considered, the first stage model is used. If the contractor uses the first stage model after the contract is awarded, not only is the financing cost minimized and more profit is achieved, but also the project schedule is created based on early activity start times, and the project

244 is planned for completion without an extension to avoid liquidated damages. In this condition, time-cost tradeoff is not performed either. 

If the contractor wants to increase profits by considering time-cost tradeoff while minimizing financing cost, the second stage models can be used since these models integrate the time-cost tradeoff problem (TCTP) and the finance-based scheduling problem (FBSP). When the second stage models are used, the contractor is provided with a project schedule that is created by using early activity start times. This schedule leads to maximum profit while minimizing the sum of direct, indirect and optimal financing costs between the crash and normal points. As a result, the second stage models are indicated for those who want to obtain maximum profit between the crash and normal points considering early activity start times while sufficient credit can be obtained from financing resources.



If the contractor prefers to perform a sensitivity analysis to see whether the optimal results change significantly when variable activity start times are considered, they can use the third stage models. The contractor is able to consider a more critical schedule with higher profit, or a less critical schedule with less profit.



If a small contractor who does not have sufficient collateral and cannot obtain the required credit or has no access to many financing alternatives to execute the project between the crash and normal durations, the fourth stage models can be used where the optimal solution can be found between the crash and extended points. The fourth stage models are used to see which schedule leads to the maximum profit according to the specific financing scenario even if there are some limitations on obtaining the required credit or the financing alternatives.

245 It should be noted that all proposed models provide optimum financing schedules, a related project schedule, and an analytical report. The financing schedules, which are provided in two different reports, answer three questions in order to maximize the cash balance at the end of the project: (1) which financing alternative or combination of alternatives should be considered, (2) what amount of money should be taken each month in each alternative, and (3) what amount of money including interest should be repaid each month in each alternative. New financing schedules and a new analytical report can be obtained by updating the schedule whenever there is a change in those alternatives.

8.2

Conclusion

Financing is a significant factor for contractors since 77 to 95% of contractor failures caused by financing problems. Even though the financing cost needs to be considered when the schedule of a project is created, consideration of financing cost formally was neglected until 2004. Although the integration of financing cost into scheduling was investigated by some researchers after 2004, no research has been undertaken on financing optimization with respect to different financing alternatives with undetermined and predetermined credit limits. To fill this gap, the main objective of this research is to focus on financing optimization considering several financing alternatives in terms of sources and types of financing, times of cash provisions, interest rates, and repayment options. This research is a distinct improvement over past models due to the following reasons: 1. Far less financing cost is incurred and higher profit is obtained compared to past research if one uses the models proposed in this research.

246 2. The proposed models accommodate as many financing alternatives as available. As a result, lower financing cost is generated and the likelihood is increased of obtaining the required cash for on-schedule project completion. 3. Short-term and long-term loans are considered in the proposed models (never done before in past research) to avoid a large withdrawal from the line of credit and to reduce the interest paid on the contractor`s primary account. 4. The optimal financing cost and the schedule that satisfies all constraints may be different than when several sources of financing and undetermined credit limit are considered. The proposed models can consider either predetermined or undetermined credit limits, whereas past studies did not. 5. The optimum schedules of borrowed money and repaid money are provided. This is convenient for contractors as these schedules add credibility and strength to their application to lenders. No such schedules are provided in any of the past studies. 6. The contractor can set the minimum cumulative net balance of the cash flow at the end of each period as a contingency. Given uncertainties inherent in construction projects, the ability to set buffer funds is a distinct advantage that has never been exploited in past studies. 7. The contractor can perform a sensitivity analysis to see which schedule leads to the maximum profit according to the specific financing scenario. “What if” analysis helps with decision making in situations where model parameters fluctuate. 8. This research speeds up the CPM calculations by using an improved Dijkstra`s algorithm.

247 Given the points mentioned above, this research improves financing decisions for contractors and is expected to reduce the possibility of failure that can occur due to financial problems. If financing decisions are improved using the models proposed in this research, not only does the contractor benefit, but so does the owner too because: (1) there may be less friction between parties, (2) contractors may be less tempted to unbalance bids, (3) fewer change orders and claims may be filed, (4) the contract may not be terminated due to delay caused by financial problems, and (5) the project is not extended because of unavailability of cash.

8.3

Future Research

Future research may involve three areas. Since the proposed research is performed considering a deterministic project schedule, it is worthwhile to explore incorporating financing optimization and stochastic time-cost tradeoff. In addition, since construction companies may run multiple projects simultaneously, it would be beneficial to expand the models to be applicable to multiple projects. Moreover, the resource project scheduling problem (RPSP) can be solved while solving the time-cost tradeoff problem (TCTP) and the finance-based scheduling problem (FBSP) using financing optimization considering different financing alternatives. A very long computational time is a serious challenge that needs to be overcome in this research. Finally, the performance of the proposed model can be assessed under the effects of macroeconomic factors such as changes in government policies, and changes in market conditions caused by periodic recessions.

248

APPENDIX A A CLASSIFICATION OF CONSTRUCTION SCHEDULING OPTIMIZATION RESEARCH

Resource-constrained scheduling

 

 

   

   

Hong et al. (2001)

Single

Single

   





 

 

Abeyasinghe et al. (2001) Single

Single

Non Fixed Non Fixed Non Fixed Non Fixed Fixed Fixed Fixed

   





 

 

   





 

 



  





 

 

             

  

  

     

     

Wiest (1967)

Multi

Single

Sunde & Lichtenberg (1995) Russell (1986) Chiu & Tsai (2002) Smith-Daniels et al. (1996)

Single

Single

Single Multi Single

Single Single Single

Cash availability constraints

       

Financing costs

Fixed Fixed

Financing optimization using different financing alternatives

Single Single

Resource leveling

Multi Multi and Single


Tsai & Chiu (1996) Lova & Tormos (2001)

References


Maximizing NPV or profit

Minimizing total project duration

Finance-Based scheduling (FBSP)

Minimizing total project cost

Resource project Time-Cost scheduling tradeoff (RPSP) (TCTP) Resource availability constraints

Heuristic

Other issues

Fixed or non-Fixed

Heuristic

Activity duration

Single or multi objective

Methodologies

Objective

Single or multi project

Project type

249


Fixed Fixed Fixed

            

  

  

           

Single

Single

   





 

 

Easa (1989) Mattila & Abraham (1998) Ipsilandis (2006)

Single Single

Single Single

NonFixed Fixed Fixed

 

       

 

 

   

   

Single

Multi

    





 

 

Ipsilandis (2007)

Single

Multi

    





 

 

Ammar (2011)

Single

Single



   





 

 

Talbot (1982)

Single

Single

NonFixed NonFixed NonFixed NonFixed



  





 

 



Financing costs

Multi Multi Single


Elazouni (2009) Pritsker et al. (1969) Patterson & Huber (1974) Huang & Halpin (2000)

Resource leveling

Heuristic Heuristic Mathematical IP or LP


References


Methodologies






Other issues

Fixed or non-Fixed

Activity duration


Objective


Project type

250


Fixed Fixed Fixed

              

  

  

           

Single

Single

   





 

 

Robinson (1975)

Single

Single



   





 

 

De et al. (1995)

Single

Single



   





 

 

Demeulemeester et al. (1996) Adeli & Karim (1997)

Single

Single



  





 

 

Single

Single



  





 

 

Jaśkowski & Sobotka (2006) Elazouni & Abido (2011)

Single

Single

NonFixed NonFixed NonFixed NonFixed NonFixed Fixed

   





 

 

Multi

Multi

Fixed

    





   

DP

Metaheuristic

GA



Financing costs



Russell (1970) Grinold (1972) Elazouni & Gab-Allah (2004) Elmaghraby (1993)

Resource leveling

Mathematical IP or LP


References


Methodologies






Other issues

Fixed or non-Fixed

Activity duration


Objective


Project type

251

Single

Multi

Li & Love (1997)

Single

Single

Que (2002)

Single

Single

Zheng et al. (2004)

Single

Multi

   

       

 

 

   

   

   





 

 

    





 

 



   





 

 



   





 

 

    





 

 


Feng et al. (1997)

   

Financing costs

Single

 


Single


Kim (2013)

 

Resource leveling

Single Single


Multi Single


       


Gonçalves et al. (2008) Kim & Ellis Jr (2010)

Fixed Non Fixed Fixed Non Fixed NonFixed Non Fixed Non Fixed Non Fixed Non Fixed


Single Multi

Resource project Time-Cost scheduling tradeoff (RPSP) (TCTP)


Single Single

Fixed or non-Fixed

Leu et al. (1999b) Hegazy (1999b)

Other issues


GA

References

Activity duration

Resource availability constraints

Metaheuristic


Methodologies

Objective


Project type

252

Single

Multi

Single

Multi

Single

Multi

Single

Multi

Chen & Weng (2009)

Single

Multi

 

   





 

 

   





 

 

   





 

 

   





 

 

   





 

 

   





 

 

   





 

 


Kandil & El-Rayes (2006) Dawood & Sriprasert (2006) Senouci & Al-Derham (2008) Long & Ohsato (2009)

 

Financing costs

Single




Single


Hegazy (1999a)



Resource leveling

Multi


Single


    


Leu & Yang (1999)

Non Fixed Non Fixed Non Fixed Non Fixed Non Fixed Non Fixed Non Fixed Non Fixed


Multi



Single

Fixed or non-Fixed

Zheng et al. (2005)

Other issues


GA

References

Activity duration


Metaheuristic


Methodologies

Objective


Project type

253




   

Single Single

Multi Single

Fixed Fixed

         

 

 

       

Elazouni & Metwally (2007) Leu et al. (1999a)

Single

Single

   





   

Single

Single

   





 

 

Lie et al. (2001)

Single

Single



   





 

 

Afshar & Fathi (2009)

Single

Multi

NonFixed NonFixed NonFixed Fixed

    





   

Ali & Elazouni (2009)

Single

Single

   





   

NonFixed




Single

Financing costs

    

Elazouni & Metwally (2005) Fathi & Afshar (2010) Alghazi et al. (2013)


 

Resource leveling



Multi




Single


   

Single

Non Fixed Fixed

Jalali & Shirvani (2011)

References

Fixed or non-Fixed


Resource project Time-Cost scheduling tradeoff (RPSP) (TCTP) Minimizing total project cost

Other issues


Activity duration


GA


Metaheuristic


Methodologies

Objective


Project type

 

254

TS

Multi

Zhang et al. (2005) Zhang & Tam (2006)

Single Single

Single Single

Zhang et al. (2006b) Zhang et al. (2006a) Thomas & Salhi (1998) Valls et al. (2004) Pan et al. (2008)

Single Single Single Single Single

Single Single Single Single Single

    





 

 

    





 

 

   





 

 

       

 

 

   

   

    

    

    

    

    

    

    

    

    


Single

 

Financing costs

Multi

 


Single



Resource leveling

Lakshminarayanan et al. (2010) Shrivastava et al. (2012)


Multi




Single


    


Afshar et al. (2009)

NonFixed NonFixed NonFixed NonFixed Fixed NonFixed Fixed Fixed Fixed Fixed Fixed


Multi

Fixed or non-Fixed

Single



PSO

Ng & Zhang (2008)

Other issues


ACO

References

Activity duration


Metaheuristic


Methodologies

Objective


Project type

    

255

Single

Single

SFLA

Anagnostopoulos & Koulinas (2010) Elbeltagi et al. (2005)

Single

Single

CP

Alghazi et al. (2012) Liu & Wang (2008)

Single Single

Single Single

CP

 

 



  





 

 

       

 

 

   

   



   





 

 



   





 

 

Non Fixed Fixed NonFixed



   





 

 

        

 

 

       


Single



Financing costs

Single




He et al. (2009)


Single Single


Single Single


Jeffcoat & Bulfin (1993) Mika et al. (2005)

SA

   

Resource leveling

Single


Single


NonFixed NonFixed Fixed NonFixed NonFixed Fixed


Józefowska et al. (2002)



Single

Other issues Minimizing total project duration

Single

TS

References

Activity duration

Fixed or non-Fixed


Mika et al. (2008)

Methodologies

Metaheuristic

Objective


Project type

256

Single

Single

CIP

Berthold et al. (2010)

Single

Single

MIP-CIP

Heinz & Beck (2012)

Single

Single

Single

Single

Single

Single

Hybrid MLGAS Li et al. (1999) Metaheuristic SSGS Xiong & Kuang (2006) ACO



   

 

       

 

 

       



  





 

 



   





 

 

   





 

 



  





 

 



   





 

 



   





 

 


Jiang & Zhu (2010)



Financing costs

Single

  


Single


Burns et al. (1996)


Hybrid Mathematical


Single Single




Resource leveling

Multi Single


LP-IP

Liu & Wang (2010) Liu et al. (1995)

NonFixed Fixed Non Fixed Non Fixed Non Fixed Non Fixed Non Fixed Non Fixed Fixed


Single



Single

Other issues Minimizing total project duration

Liu & Wang (2009)

References

Activity duration

Fixed or non-Fixed


CP

Methodologies

CP

Objective


Project type

257




 

 

Single

Multi

    





 

 

Single

Multi

Non Fixed Non Fixed

   





 

 

Multi

Single

Non Fixed

    





 

 




Financing costs

   


Fixed

Resource leveling

Single


Multi






Hybrid GA-Math Ahmed & Eldin (2004) Mathematical & Metaheuristic GA-DP Ezeldin & Soliman (2009)


Hybrid GA-SA Chen & Shahandashti Metaheuristic (2009) ACO-GA Hui et al. (2013)

Other issues

Fixed or non-Fixed

References

Activity duration


Methodologies

Objective


Project type

258

259

APPENDIX B PROJECT SCHEDULE DATA AND ACTIVITY ACCELERATION METHODS FOR LARGE NETWORK USED TO TEST SECOND STAGE MODELS

260

Duration (weeks)

Activity name

Predecessor 3

Method 1

Method 2

Method 3

Method 2

Method 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Start A1 B1 C1 D1 E1 F1 G1 H1 I1 J1 K1 L1 M1

1 1 1 1 1 2 3 4 5 6 7 8 8

9

-

4 2 2 5 4 2 3 5 3 3 2 5 2

3 1 1 4 3 1 2 4 2 2 1 4 1

3 1 1 4 3 1 2 4 1 2 1 3 1

20,000 20,000 30,000 40,000 40,000 40,000 80,000 90,000 50,000 40,000 30,000 70,000 40,000

30,000 50,000 70,000 55,000 60,000 90,000 130,000 115,000 80,000 65,000 70,000 90,000 90,000

30,000 50,000 70,000 55,000 60,000 90,000 130,000 115,000 170,000 65,000 70,000 130,000 90,000

15

N1

10

-

-

3

2

1

30,000

50,000

110,000

16

O1

11

-

-

2

1

1

50,000

110,000

110,000

17

P1

12

-

-

2

2

2

40,000

40,000

40,000

18 19 20 21 22 23 24 25 26 27 28 29 30

Q1 R1 S1 T1 U1 V1 W1 X1 Y1 A2 B2 C2 D2

12 14 14 16 17 18 19 20 20 22 23 24 25

13 15 21 -

-

2 2 2 2 1 2 3 2 3 2 2 1 3

1 1 1 1 1 2 2 1 2 1 1 1 2

1 1 1 1 1 2 1 1 1 1 1 1 1

30,000 40,000 60,000 30,000 60,000 90,000 80,000 80,000 50,000 40,000 30,000 130,000 40,000

70,000 90,000 125,000 60,000 60,000 90,000 120,000 170,000 80,000 90,000 70,000 130,000 65,000

70,000 90,000 125,000 60,000 60,000 90,000 240,000 170,000 190,000 90,000 70,000 130,000 150,000

31

E2

26

-

-

3

2

1

60,000

100,000

210,000

32

F2

27

-

-

2

1

1

50,000

110,000

110,000

33

G2

28

-

-

2

2

2

40,000

40,000

40,000

Method 1

Activity ID

Predecessor 2


Predecessor 1

Predecessors

261

Duration (weeks)

Activity name

Predecessor 3

Method 1

Method 2

Method 3

Method 2

Method 3

34 35 36 37 38 39 40 41 42 43 44 45 46

H2 I2 J2 K2 L2 M2 N2 O2 P2 Q2 R2 S2 T2

29 30 31 32 33 33 35 36 37 37 39 39 41

34 38 40 -

-

3 2 2 2 1 2 3 2 3 2 2 1 3

2 1 1 1 1 1 2 1 2 1 1 1 2

1 1 1 1 1 1 1 1 1 1 1 1 1

40,000 40,000 60,000 40,000 60,000 40,000 80,000 90,000 50,000 40,000 30,000 90,000 40,000

65,000 90,000 125,000 90,000 60,000 90,000 120,000 190,000 80,000 90,000 70,000 90,000 90,000

140,000 90,000 125,000 900,000 60,000 90,000 240,000 190,000 190,000 90,000 70,000 90,000 200,000

47

U2

42

-

-

3

2

1

60,000

100,000

210,000

48

V2

43

-

-

2

1

1

40,000

90,000

90,000

49

W2

44

-

-

2

2

2

20,000

20,000

20,000

50 51 52 53 54 55 56 57 58 59 60 61 62

X2 Y2 A3 B3 C3 D3 E3 F3 G3 H3 I3 J3 K3

45 45 47 48 49 50 51 52 53 54 55 56 57

46 -

-

3 2 2 2 1 2 3 2 3 2 2 1 3

2 1 1 1 1 1 2 1 2 1 2 1 2

1 1 1 1 1 1 1 1 1 1 2 1 1

40,000 40,000 60,000 30,000 60,000 40,000 80,000 50,000 50,000 40,000 30,000 80,000 40,000

65,000 90,000 125,000 70,000 60,000 90,000 120,000 110,000 80,000 90,000 30,000 80,000 70,000

135,000 90,000 125,000 70,000 60,000 90,000 240,000 110,000 170,000 90,000 30,000 80,000 150,000

63

L3

58

-

-

3

2

1

30,000

50,000

110,000

64

M3

58

59

-

2

1

1

50,000

110,000

110,000

65

N3

60

-

-

2

2

2

40,000

40,000

40,000

66

O3

61

-

-

3

2

1

40,000

65,000

140,000

Method 1

Activity ID

Predecessor 2


Predecessor 1

Predecessors

262

Duration (weeks)

Activity ID

Activity name

Predecessor 2

Predecessor 3

Method 1

Method 2

Method 3

Method 1

Method 2

Method 3


Predecessor 1

Predecessors

67 68 69 70 71 72 73 74 75 76 77 78

P3 Q3 R3 S3 T3 U3 V3 W3 X3 Y3 A4 B4

62 62 64 64 66 67 68 69 70 70 72 73

63 65 71 -

-

2 2 2 1 2 3 2 3 2 2 1 3

1 1 1 1 1 2 1 2 1 2 1 2

1 1 1 1 1 1 1 1 1 2 1 1

40,000 60,000 30,000 60,000 40,000 80,000 50,000 50,000 40,000 30,000 100,000 40,000

90,000 125,000 70,000 60,000 90,000 125,000 120,000 80,000 90,000 30,000 100,000 90,000

90,000 125,000 70,000 60,000 90,000 260,000 120,000 170,000 90,000 30,000 100,000 190,000

79

C4

74

-

-

3

2

1

30,000

50,000

110,000

80

D4

75

-

-

2

1

1

50,000

110,000

110,000

81

E4

76

-

-

2

2

2

40,000

40,000

40,000

82 83 84 85 86 87 88 89 90 91 92 93

F4 G4 H4 I4 J4 K4 L4 M4 N4 O4 Q4 R4

77 78 79 80 81 82 83 83 85 86 87 89

84 88 -

-

2 2 2 2 1 4 3 2 2 2 2 3

1 1 1 1 1 3 2 1 1 2 1 2

1 1 1 1 1 2 1 1 1 2 1 1

30,000 40,000 60,000 20,000 60,000 40,000 80,000 50,000 50,000 40,000 50,000 50,000

70,000 90,000 125,000 60,000 60,000 55,000 120,000 110,000 110,000 40,000 110,000 80,000

70,000 90,000 125,000 60,000 60,000 85,000 240,000 110,000 110,000 40,000 110,000 170,000

94

S4

89

90

-

2

1

1

20,000

50,000

50,000

95

T4

91

-

-

2

2

2

30,000

30,000

30,000

96 97 98 99

W4 X4 Y4 Z

92 94 94 96

93 95 97

98

3 4 3 1

2 3 2 1

2 2 2 1

40,000 30,000 50000 40000

65,000 45,000 80,000 40,000

65,000 70,000 80,000 40,000

263

APPENDIX C FINANCING DATA FOR LARGE NETWORK USED TO TEST SECOND STAGE MODELS

264

Alternative

APR (%)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

22 19 18 16 21 18 17 10 19 17 10 9 20 16 11 9 19 16 12 10 6

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly Monthly

22

16

Yes

Optimum time

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