Optimising project performance: the triangular

152

Int. J. Engineering Management and Economics, Vol. 3, Nos. 1/2, 2012

Optimising project performance: the triangular trade-off optimisation approach Baruch Keren Department of Industrial Engineering and Management, SCE-Shamoon College of Engineering, Bialik/Basel Sts., Beer Sheva 84100, Israel E-mail: [email protected] and Department of Management and Economics, The Open University of Israel, Ravutzki 108, Raanana 43107, Israel E-mail: [email protected]

Yuval Cohen* Department of Management and Economics, The Open University of Israel, Ravutzki 108, P.O. Box 808, Raanana, 43107, Israel Fax: +972-97780668 E-mail: [email protected] and Department of Industrial Engineering, Tel-Aviv Afeka College of Engineering, Tel-Aviv, 69107 Israel E-mail: [email protected] *Corresponding author Abstract: It is generally accepted that the three major dimensions of project success are time, budget, and quality. Most of the research in project planning is focused on the time-cost trade-off, and only a few papers have considered the three dimensions together. This paper describes the evolution of formulations for optimising project time-cost and quality, and continues by developing a new, non-linear optimisation formulation that better reflects the triangular trade-off structure between time, budget, and quality. In particular, the Cobb-Douglas formula is adopted and its use is illustrated. The model is discussed, and the structure of the trade-off is analysed and illustrated. Keywords: project management; project scheduling; project performance; project trade-off; Cobb-Douglas function. Reference to this paper should be made as follows: Keren, B. and Cohen, Y. (2012) ‘Optimising project performance: the triangular trade-off optimisation approach’, Int. J. Engineering Management and Economics, Vol. 3, Nos. 1/2, pp.152–170.

Copyright © 2012 Inderscience Enterprises Ltd.

Optimising project performance

153

Biographical notes: Baruch Keren is a Senior Lecturer in the Industrial Engineering and Management Department, Shamoon College of Engineering and Lecturer at the Open University of Israel. He received his BSc, MSc and PhD (Summa Cum Laude) in Industrial Engineering from Ben-Gurion University. His professional experience includes 13 years in the Israel Chemicals Ltd. and its subsidiaries in the areas of industrial engineering, economics and auditing. His current research interests are decision-making under uncertainty, production planning, project management and operations research. Yuval Cohen is the Head of the Industrial Engineering programme at the Open University of Israel and Senior Lecturer at Tel-Aviv Afeka college of Engineering. His areas of specialty are production planning, industrial learning, design of manufacturing control systems and logistics management. He has published many papers in these areas. He received his PhD from the University of Pittsburgh (USA), his MSc from the Technion – Israel Institute of Technology and BSc from Ben-Gurion University. He is a Fellow of the Institute of Industrial Engineers (IIE) and a full member of the Institute for Operations Research and Management Sciences (INFORMS).

1

Introduction

Planning and executing projects (and their goals) always involves elements of time (due-date), cost (budget) and quality (specifications). There is a natural trade-off between time, cost and quality, but these three critical objectives are not independent, but rather intricately related. The time-cost-quality trade-off problem (TCQP) is a multiple objective optimisation problem, which mainly focuses on selecting options with corresponding time, cost and quality to complete an activity in order to minimise project duration and cost, while maximising project quality (see Rwelamila and Hall, 1995; Wang and Feng, 2008). Obviously, the value and performance of a project increase as time and cost decrease, and as quality increases. While the importance of project quality has never been disputed, modern project management research began with emphasis on minimising the time needed for a project. The first two major project planning techniques focused on project duration (and its relationship to costs): 1

the critical path method (CPM) (Kelley, 1961)

2

project and evaluation review technique (PERT) (Malcolm et al., 1959).

Research on the time-cost trade-off followed using linear programming (LP) crashing techniques (Charnes and Cooper, 1962; Fulkerson, 1961; Prager, 1963). These crashing techniques were deterministic models of activities characterised by linear (or piece-wise linear) cost functions with respect to their duration. In particular, some models were presented both as network-flow problems and as the network-flow dual problems (Moder and Phillips, 1970). Other models focused on improving the algorithm efficiency (Goyal, 1975; Siemens, 1971). The research has continued in several directions:

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B. Keren and Y. Cohen

1.1 Non-linear and/or non-continuous trade-off functions Kapur (1973) presented an algorithm for project cost-duration analysis problems with quadratic and convex cost functions. Falk and Horowitz (1974) presented cost-duration analysis for critical path problems with concave cost-time functions. Demeulemeester et al. (1996) describe two algorithms, based on dynamic programming logic, for optimally solving the discrete time/cost trade-off problem (DTCTP) in deterministic activity-on-arc (AOA) networks of the CPM type. The costs are not given in a function form, but rather as discrete values related to certain durations. This structure turns the problem into a non-polynomial combinatorial problem. Later, Demeulemeester et al. (2000) presented an improved branch-and-bound technique for solving the DTCTP. Sunde and Lichtenberg (1995) extended the time-cost trade-off to include net present value (NPV) calculations.

1.2 Trade-off structure Berman (1964) studied the effect of resource allocation. Pulat and Horn (1996) extended the time-cost trade-off to the time-resource trade-off, making in possible to deal with duration taken as a function of several resources. Liberatore and Pollack-Johnson (2006) suggested extending the trade-off by removing precedence relationships and task-streaming. Another direction was to minimise project cost: Wu and Li (1994) proposed a new technique for minimising the project cost based on a cut set parallel difference method. Their work attracted some related feedback and corrections, such as Kamburowski (1995). Drezet and Billaut (2008) proposed a scheduling solution technique for projects with labour constraints and time-dependent activities. Only a few papers include a direct treatment of the trade-off between quality, time and budget together: Babu and Suresh (1996) suggested that project quality may be affected by project crashing and developed LP models to study the trade-offs between time, cost, and quality. Each of the three models developed optimises one of these entities by assigning desired bounds to the other two. Their model was validated by Khang and Myint (1999) using an illustrative case study for showing its practical value. Tareghian and Taheri (2006) developed three inter-related integer programming models for discrete values such that each model optimises one of the given entities (time, cost, or quality) by assigning desired bounds on the other two. In order to solve these models, they later used a scatter search technique (Tareghian and Taheri, 2007). El-Rayes and Kandil (2005) designed a multi-objective genetic algorithm (GA) model to transform the traditional two-dimensional time-cost trade-off analysis to a three-dimensional time-cost-quality trade-off analysis.

1.3 Project trade-off under stochastic scheduling Research considering the stochastic nature of activities has been dealing with the effect of resource limitations and extending the deterministic models described above in Section 1.2, such as Pulat and Horn (1996). Some of the papers in this field are: Golenko-Ginzburg and Gonik (1998), Lova et al. (2009), Van Dorp and Duffey (1999) and Van de Vonder et al. (2005). Al-Fawzan and Haouari (2005) proposed a bi-objective model for robust resource-constrained project scheduling.


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Deckro et al. (1995) and Deckro and Hebert (2002) presented a quadratic formulation based on minimising costs of normal durations plus costs related to quadratic deviations from these normal durations. Laslo (2003) proposed a stochastic extension of the CPM time-cost trade-off model which included four fundamental formulations that represent different assumptions of the effect of the changing performance speed on the distribution parameters of the activity duration, as well as the effect of the random activity duration on the activity cost. Cohen et al. (2007) suggested a model to provide a robust policy rather than optimal values.

1.4 Evolutionary computing methods for optimising the trade-off Due to the computational difficulty of solving the time-cost-quality trade-off problem as a combinatorial problem, researchers have been trying to apply evolutionary methods characterised by global search capability to solve the problem. These methods are: GA (e.g., Lambrechts et al., 2008; Xingfu et al., 2007), ant colony optimisation (Afshar et al., 2007), particle swarm optimisation algorithm (Zhiyong et al., 2007) and hierarchical subpopulation particle swarm optimisation algorithm (Wang and Feng, 2008). Kilic et al. (2008) used GA for minimising risk of two criteria. The quantifying of project quality is one of the major obstacles for the TCQP formulation. The questions that must be considered when doing so are: 1

How to measure or estimate the quality of each activity in a project when each one might have different properties and type?

2

How does allocation of more (less) time, budget and resources to an activity affects its quality?

3

How to quantify the overall project quality given that the quality of each component activity was determined?

4

How to combine time, cost and quality into one model?

El-Rayes and Kandil (2005) proposed a model that deals with some of those issues. For a discrete and given level of resources n, they suggested quantifying the overall project l

quality by the weighted average formula

K

∑ wt ∑ wt

i ,k

i

i =1

× Qin,k , where Qin,k is the

k =1

performance of quality indicator (k) in activity (i) using resource utilisation (n); wti and wti,k are the weight of activity (i) and the weight of quality indicator respectively. The model of El-Rayes and Kandil (2005) requires entering input for the quality of all activities in every feasible scenario. Their method also requires planners to identify two types of weights: wti that represents the importance and contribution of the quality of this activity to the overall quality of the project; and wti,k the weight of quality indicator in activity (i) that indicates the relative importance of this indicator to others being used to measure the quality of the activity. When solving their combinatorial TCQP formulation, El-Rayes and Kandil (2005) used an evolutionary method that, by definition, cannot guaranty optimality. In this paper, we use continuous functions for cost, quality, and time. This enables us to formulate the TCQP as mathematical programming problem and to solve it optimally. The Cobb-Douglas function, validated by Pendharkar et al. (2008) is the key that makes

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this continuous trade-off formulation between time, cost and quality possible. The rest of this paper is organised as follows: Section 2 presents the evolution of terminology and related formulations. Section 3 develops a linear formulation that includes a quality-time trade-off. Section 4 presents the Cobb-Douglas function and its relationship to the activity trade-off. Section 5 presents the proposed formulation, discusses it, and displays graphs to illustrate the triangular trade-off. Section 6 concludes the paper.

2

Terminology and basic related formulations

2.1 Fundamentals Many of the crashing problem formulations are presented as activities on arcs (AOA) networks. An AOA network is defined as a directed acyclic graph G(V,E), where V is a set of vertices (nodes) and E is a set of edges (arcs) connecting certain pairs of these nodes. The shortest completion time of an acyclic graph problem (AOA network) can be formulated and solved by a variety of methods. Formulation 1 (below) is a typical LP formulation for finding the shortest completion time of a graph. The beginning time of the project at node 1 is set to be V1 ≥ 0. In general we define Vi to be the time-length of the longest path from node 1 to the node i. We define ti,j as the time length of the arc from the node i to node j. For any node Vj, we assume that Vj ≥ Vi + ti,j, for all the nodes Vi which immediately (directly) precede Vj. The nodes are numbered such that the last node (Vn) occurs at the end of the project. In this simple CPM problem, the objective is to minimise the makespan of the project, and is conveniently written as: Min Vn, where Vn is the time of the last node of the project (sink). Thus, the CPM LP formulation for minimising the makespan is as follows: Formulation 1 Min Vn s.t. V j − Vi ≥ ti , j j = 2,..., n ; i = j − 1, j − 2,...,1 ; ti , j ∈ E

(1)

Vi ≥ 0 i = 1,.., n.

In formulation (1), an activity cannot begin before all its immediate precedence activities are completed. All values of ti,j are given constants. The number of the precedence constraints is the sum of the number of arcs (including the dummy arcs) + number of nodes n.

2.2 Example and data for a case study For illustration and comparison, this paper uses a simple project activity network described in Figure 1. This network and the data in Table 1 are the same data as in Babu and Suresh (1996). The example and the data in this subsection shall be used throughout the paper.

Optimising project performance Figure 1

157

A simple project activity on arcs (AOA) network 1

8

2

6

3

4

10

13

12

11

5

9

7

Table 1

Numerical data

Task no.

Task code

Task Slope Slope Precedence Tndays Tcdays CnUSD CcUSD Qn Qc Ei,j CT Q

1

E12

E1,2

-

2

1

2

E23

E2,3

E1,2

4

2

3

E34

E3,4

E2,3

10

7

6,200 7,300

1

0.8

367 0.067 415.4

4

E45

E4,5

E3,4

4

3

4,100 4,900

1

0.8

800 0.200

820

5

E46

E4,6

E3,4

6

4

2,600 3,000

1

0.7

200 0.150

390

6

E47

E4,7

E3,4

7

5

2,100 2,400

1

0.7

150 0.150

315

7

E57

E5,7

E4,5

5

3

1,800 2,200

1

0.3

200 0.350

630

8

E68

E6,8

E4,6

7

4

9,000 9,600

1 0.75 200 0.083

747

800

Pi,j

2,300

1

0.9 1,500 0.100

80

3,200 3,600

1

0.7

480

200 0.150

9

E79

E7,9

E4,7, E5,7

8

6

4,300 4,600

1

0.4

150 0.300

1290

10

E810

E8,10

E4,5, E6,8

9

6

2,000 2,500

1

0.5

167 0.167

334

11

E911

E9,11

E7,9

4

3

1,600 1,800

1

0.9

200 0.100

160

12

E912

E9,12

E7,9

5

3

2,500 3,000

1 0.95 250 0.025

62.5

13

E1013 E10,13

E8,10

2

1

1,000 1,500

1

500 0.100

100

14

E1213 E12,13 E9,11, E9,12

6

1

3,300 4,000

1 0.95 140 0.010

33

44,500 Source: Taken from Babu and Suresh (1996)

0.9

158


The critical path and minimal time of the network in Figure 1 has the following LP formulation: Example of Formulation 1 Min V13 s.t. V2 − V1 ≥ t1,2 ;

V3 − V2 ≥ t2,3 ;

V4 − V3 ≥ t3,4 ;

V5 − V4 ≥ t4,5 ;

V6 − V4 ≥ t4,6 ;

V7 − V4 ≥ t4,7 ;

V7 − V5 ≥ t5,7 ;

V8 − V5 ≥ t5,8 ;

V8 − V6 ≥ t6,8 ;

V9 − V7 ≥ t7,9 ;

V10 − V8 ≥ t8,10 ;

V11 − V9 ≥ t9,11 ;

V12 − V9 ≥ t9,12 ;

V12 − V11 ≥ t11,12 ; V13 − V10 ≥ t10,13 ; V13 − V12 ≥ t12,13 ;

(2)

Vi ≥ 0 i = 1, 2,...,13

For the normal durations in Table 1 applied to the activities in Figure 1, formulation (2) yields project duration of V13 = 44. The variable cost of the project (VC) is the sum of the normal cost of all activities: VC = USD44,500 (The notation ‘$’ is used in formulae for simplicity’s sake.) Assume also that a fixed cost of K is required for each day of the project. Hence the fixed cost is K ⋅ Vn. In our example it is assumed that K = USD400, so the fixed cost is FC = 400 × 44 = $17,600. Therefore, the total cost of the project in this solution is: TC = 44,500 + 17,600 = $62,100. The total cost formula for this case is: TC = K ⋅ Vn +

∑∑ Cn

i, j

i j i , j∈E

2.3 Crashing: cost/time trade-off problem and budget consideration Some variable definitions are required for introducing more data to the example. Definitions Tni,j

normal activity duration (i and j are the activity’s start and finish milestones)

Tci,j

crash activity duration

Cni,j

normal activity cost

Cci,j

crash activity cost

Qni,j

normal activity quality

Qci,j

crash activity quality

CTi,j

cost-time’ slope for an activity

QTi,j

quality – time’ slope for an activity

ai,j

amount of activity time reduced from its normal duration (Tni,j)

B

total expediting budget (for shortening activity times).

(3)


159

An extension of the previous problem is known as ‘the cost/time trade-off problem’. For this extension, a limited budget B for reducing the activities of the project is given. Each activity Ei,j has a given normal duration ti,j and can be accelerated to a given crash time value TCi,j by spending some expediting budget. A common assumption is linearity, i.e., in order to reduce the duration of activity Ei,j by ai,j units of time, a budget value of CTi,j × ai,j must be spent, where CTi,j is a given constant and ai,j is a decision variable. Since an activity cannot be performed in a shorter time than its crash value ai,j ≤ ti,j – Tci,j, the general formulation of this problem is the following: Formulation 2 Min Vn s.t.

(

V j − Vi ≥ Tni , j − ai , j

∑∑ CT

i, j

i

)

j = 2,..., n ; i = j − 1, j − 2,...,1 ; Tni , j ∈ E

⋅ ai , j ≤ B ∀i, j ∈ E

(4)

j

ai , j ≤ Tni , j − Tci , j ∀i, j ∈ E ai , j ≥ 0 ∀i, j ∈ E Vi ≥ 0 i = 1,.., n

The example project shown in Figure 1, with the corresponding data of Table 1 and a shortened budget over USD7,552, gives the minimal makespan V13 = 26. However, reducing the budget below USD7,552 increases the makespan (e.g., to B = USD3002 results in V13 = 32). Comparing the total cost of these two points, we have: B = $7,552 : V13 = 26

TC = 400 × 26 + 44,500 + 7,552 = $62,452

B = $3, 002 : V13 = 32

TC = 400 × 32 + 44,500 + 3, 002 = $60,302

The difference between the total cost (TC) of these two cases is due to the trade-off between expediting expenditure and the overhead cost saved by it: since we assumed that there is a fixed overhead cost of K for each day of the project, the fixed cost is K ⋅ Vn. The fixed cost can be reduced by performing the project activities faster. But accelerated (expedited) performance increases the variable cost of the project by

∑∑ CT

i, j

i

⋅ ai , j which may be limited by the budget B. Thus, saving a day reduces the

j

overhead by USD400. The expediting cost is either determined by the budget (B) or by a marginal cost of USD400 per day. The expediting cost is a piece-wise linear function of time. By minor modifications, the cost/time trade-off problem can be formulated as a LP problem that computes the optimal acceleration (expediting) budget and its allocation among the activities and the optimal project makespan. The general formulation of this problem is:

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Formulation 3 Min KVn +

∑∑ CT

i, j

i

⋅ ai , j

j

s.t.

(

V j − Vi ≥ ti , j − ai , j

)

j = 2,..., n ; i = j − 1, j − 2,...,1 ; ti , j ∈ E

(5)

ai , j ≤ ti , j − Tci , j ∀i, j ∈ E ai , j ≥ 0 ∀i, j ∈ E Vi ≥ 0 i = 1,.., n

The major difference between this formulation and previous formulations is that the objective function reflects cost and not the makespan. Applying the data of Figure 1 and Table 1 to Formulation 3, we get a solution to this formulation (with K = 400) that yields an optimal project-makespan of V13 = 30 days, a budget for expediting: B = $3,802 and the optimal shortening plan: a2,3 = 2, a4,6 = 2, a5,7 = 2, a7,9 = 2, a3,4 = 3, a8,10 = 3, a12,13 = 3. The optimal total cost of the project for this case is TC = 400 × 30 + 44,500 + 3,802 = $60,302 (this must be equal to or better than the TC of previous formulations). In our case any project makespan in the range [30, 32] gives the same optimal cost. Therefore, we must choose a minimum cost preferred solution according to secondary criteria (minimum makespan, minimum expediting cost, etc.).

3

Adding quality to the cost-time trade-off problem

This section presents a linear formulation that is consistent with the approach presented in Babu and Suresh (1996). Babu and Suresh (1996) and later Khang and Myint (1999) claimed that the quality of an activity decreases when its duration is reduced. When the duration of activity Ei,j is reduced by ai,j units of time and more money is consumed, there is also a loss of quality. In the formulation suggested in this section the amount of this quality loss is QTi,j ⋅ ai,j, where QTi,j is a given constant of quality/time trade-off and ai,j is a decision variable. The quality of an activity Ei,j is defined as Qi,j. The maximum quality of an activity is defined to be Qi,j = 1 (100%) and is achieved at the end of its normal duration. The actual quality of an accelerated activity is Qi,j = (1 – QTi,j ⋅ ai,j). However, the quality cannot be set to less than some limits since it may risk the entire project. Therefore, we must set constraints of the form Qci,j ≤ {Qi,j = (1 – QTi,j ⋅ ai,j)} ≤ 1. However, even if a project with low quality were acceptable, its value should decrease. We calculate the weighted loss of quality of the Pi , j QTi , j ⋅ ai , j , where Pi,j is the activity penalty for lack of quality for project by

∑∑ i

j

the entire project. The cost of lost quality in that project is:

∑∑ P

i , j QTi , j

i

we obtain,

j

⋅ ai , j . Thus


161

Formulation 4 Min KVn +

∑∑ CT

i , j ai , j

i

j

+

∑∑ P

i, j

i

⋅ QTi , j ⋅ ai , j

j

s.t.

(

V j − Vi ≥ ti , j − ai , j

)

j = 2,..., n ; i = j − 1, j − 2,...,1 ; ti , j ∈ E

ai , j ≤ ti , j − Tci , j ∀i, j ∈ E ai , j ≤

1 − Qci , j QTi , j

(6)

∀i, j ∈ E

ai , j ≥ 0 ∀i, j ∈ E Vi ≥ 0 i = 1,.., n.

The objective function in (6) can be written as KVn +

∑∑ (CT

i, j

i

+ Pi , j ⋅ QTi , j )ai , j . The

j

conclusion is that the Babu and Suresh (1996) model includes two cost components that increase linearly while expediting an activity: the expedition cost and the loss-of-quality cost. A practical method for estimating the value of Pi,j is to assume that it is proportional to activity Ei,j normal cost, e.g., Pi,j = Cni,j. According to this assumption, the cost of lost of quality in a project is: Cni , j ⋅ QTi , j ⋅ ai , j .

∑∑ i

j

Assuming that Pi,j = Cni,j, K = $400 and applying the other data for the example project shown in Figure 1 with the corresponding data of Table 1 to (6), we find: the optimal project makespan is V13 = 40 days; the optimal budget for acceleration is B = $560; the cost of quality loss is USD132; and the optimal duration shortening plan is: a12,13 = 4. The optimal total cost of the project for this case is TC = 400 × 40 + 44,500 + 560 +13 = $61,192. This time the cost of quality-loss is also added, resulting in higher total cost (TC) than the one found for the Formulation 3. As expected (to avoid large quality loss), the optimal duration is much higher (and the acceleration budget much smaller) than the one found for Formulation 3. However, Formulation 4 does not consider the trade-off between cost and quality, and therefore ignores part of the triangular trade-off. For this purpose, we introduce a new approach, which uses the Cobb-Douglas function, in Section 4.

4

Cobb-Douglas function and its relationship to the activity trade-off

The validity of the Cobb-Douglas function form for projects was tested and proved by Pendharkar et al. (2008). They empirically tested Cobb-Douglas functional form with respect to team size and software size for a real-world data set containing more than 500 software projects. Their results indicated that the hypothesised Cobb-Douglas function form is true. The Cobb-Douglas functional form of production functions is widely used to represent the relationship of inputs to an output (Felipe and Adams, 2005). In their

162


seminal paper, Cobb and Douglas (1928) expressed the relationship (i.e., trade-off formula) between: 1

output denoted by Q

2

labour denoted by L

3

capital denoted by B, as: Q = ALα ⋅ B β

(7)

where A, α, β are coefficients, determined by technology and traditionally estimated by linear regression over logarithmic transformation of (7) that gives (8): ln(Q) = ln( A) + α ln( L) + β ln( B )

(8)

For estimating the values A, α, β for the linear equation (8) the company has to acquire database of previous projects and operations so that a reliable regression could generate trustworthy values. The correlation coefficient of the regression gives an indication of the estimations’ reliability level. The database should include the throughput, the labour and the budget. The throughput could be measured in monetary values or quantities produced and labour by total labour cost or by invested hours. The variable A is known as ‘total-factor productivity (TFP)’ and reflects the ratio of outputs to inputs related to the relevant state of technology. The value of A is computed via the multiple regression of equation (8). α and β are coefficients that reflect the change of the throughput related to L and B respectively. In other words, α and β indicate the marginal throughput increase as result of increasing L and B by one unit respectively. The partial marginal throughput change can be increasing for values α or β greater than 1, decreasing for values between zero to 1, and for negative values of α or β the throughput decreases as resources are added. Empirically (e.g., Felipe and Adams, 2005) α and β are statistically significant, non-negative and α + β ≈ 1. So that: Q = ALα ⋅ B1−α

(9)

For an activity in a project Ei,j, we propose replacing the labour/work-load (L) by activity duration (ti,j) times the number of workers (Wi,j), and defining A as a conversion constant between output and quality. The capital (B) is translated into the investment in equipment and materials, or the activity cost, and instead of capital we will use the term budget. As mentioned, Cobb-Douglas production function is widely used in economic applications to represent the relationship of inputs to output. Since budget and time can be considered as inputs of a project while its quality (or performance) can be considered as an output, it is natural to extend Cobb-Douglas to project management. Moreover, Cobb-Douglas has three parameters A, α, β which provide flexibility for shaping the trade-off curves close to their empirical nature. The accurate values of Cobb-Douglas’s parameters can be estimated by linear regression for any project/activity based on analysis of historical data. The correlation coefficient which is computed in any linear regression can be a means for validating the trade-off correctness. Since the model uses the AOA structure, activities will be identified by their pair of origin-destination nodes (i, j). The actual quality (performance index) of each activity (i, j), in the project is expressed by Qi,j. Defining the upper boundary for the quality level


163

as 100%, the value range of Qi,j is: 0 ≤ Qi,j ≤ 1. The single activity adjusted formula becomes: αi , j

Qi , j = Ai , j (Wi , j ⋅ ti , j )

1−α i , j

⋅ Bi , j

(10)

Equation (10) holds for each project operation (i, j) and expresses its quality. Alternatively, by several simple computing manipulations the work-load and the budget can be expressed as: 1−α i , j

1 αi , j

(Wi, j ⋅ ti, j )

=

Ai , j ⋅ Bi , j

⇒ (Wi , j ⋅ ti , j )

Qi , j

−α i , j

1−α i , j

=

Ai , j ⋅ Bi , j Qi , j

(11)

That is, 1−α i , j

Ai , j ⋅ Bi , j

Wi , j ⋅ ti , j

(

)

Wi , j ⋅ ti , j

⎛ Qi , j =⎜ ⎜ Ai , j ⋅ Bi , j1−α i , j ⎝

−α i , j

=

(

⇒ Wi , j ⋅ ti , j

Qi , j

)

αi , j

=

Qi , j 1−α i , j

Ai , j ⋅ Bi , j

(12)

and 1

⎞α i , j ⎟ ⎟ ⎠

(13)

In the same manner:

Bi , j

⎛ Qi , j =⎜ ⎜ Ai , j ⋅ (Wi , j ⋅ ti , j )αi , j ⎝

1

⎞1−α i , j ⎟ ⎟ ⎠

(14)

It is reasonable to assume that during the planning phase of a project, the planner sets for every work package of the work breakdown structure (WBS), the necessary workforce and time Wi,j ⋅ ti,j, and appropriate equipment, materials and budget Bi,j in order to achieve the required performance level Qi,j. Any lack of budget, time, or workers for the activity would decrease the performance level Qi,j.

5

Minimum cost formulation for the triangular trade-off

This section presents our proposed formulation. For that purpose, the formulation in Section 3 will be converted to a Cobb-Douglass oriented formulation. For this purpose, we define the Cobb-Douglass somewhat differently: α

(1−αi , j )

Qi , j = Ai , j ⋅ ti , ij, j ⋅ Bi , j

(15)

In equation (15), Bi,j is the budget of activity Ei,j, ti,j is its time and Qi,j is its quality (performance level). According to equation (15), we can simultaneously set planning values for time, budget (cost), and quality for all project activities. Now time, budget, and quality are decision variables which get their values by optimisation.

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The cost minimisation of the following Formulation 5 includes the fixed cost (K ⋅ Vn), the cost of carrying out all activities Bi , j and the penalty for deviations from the

∑∑

normal quality:

i

∑∑ P

i , j (1 − Qi , j ).

i

j

The precedence constraints of the AOA network

j

preserve their main structure while using the above Cobb-Douglass formulation for tradeα

(1−α i , j )

offs. Since we assume that Qi , j = Ai , j ⋅ ti , ij, j ⋅ Bi , j

for all activities, we can also

conclude that: 1

ti , j

⎛ ⎞α i , j Qi , j ⎜ ⎟ =⎜ 1−α i , j ) ⎟ ( ⎜ Ai , j ⋅ Bi , j ⎟ ⎝ ⎠

(16)

Thus, the new form of Formulation 5 is: Formulation 5 Min K ⋅ Vn +

∑∑ P (1 − Q )

∑∑ B

i, j +

i

i, j

j

i

i, j

j

s.t. ⎛ Qi , j ⎜ V j − Vi − ⎜ (1−αi, j ) ⎜ ⎝ Ai , j ⋅ Bi , j V j − Vi ≥ 0

1

⎞αi , j ⎟ ≥0 ⎟ ⎟ ⎠

∀i, j ∈ E and known Ai , j , α i , j for any dummy activity Ei , j

(17)

1

⎛ ⎞αi , j Qi , j ⎜ ⎟ − ti , j (min) ≥ 0 ⎜ ⎟ 1−αi , j ) ⎟ ( ⎜ ⎝ Ai , j ⋅ Bi , j ⎠ Qci , j ≤ Qi , j ≤ 1 ∀i, j ∈ E

∀i, j ∈ E and known Ai , j , α i , j

Vi ≥ 0 i = 1,.., n

In practice, any activity has limited ranges for quality, duration and cost (budget). In order to avoid out-of-range or feasibility problems, it is important to verify by equation (15) that each activity Ei,j would at least obtain its minimum acceptable quality Qi,j when its maximum duration ti,j and maximum budget Bi,j are set. Another recommended check is to verify that both the minimum time duration and minimum budget for each activity, provide a quality level such that Qi,j ≤ 1. The budget constraints Bi , j (min) ≤ Bi , j ≤ Bi , j (max) and time constraints can be added to formulation (5). These constraints also facilitate the process of deriving an initial feasible solution used to solve the mathematical programming problem in (17). The numerical values of Ai,j and αi,j for our example are contained in Table 2. Since we have only two points for each activity with values for time, cost (budget) and quality,


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the values of Ai,j and αi,j for each activity were calculated by equation (18) and the rest of the data is as given in Table 1. For any case where equation (18) for activity Ei,j, yields: 1 – αi,j = 0, a constraint that Bi,j ≥ Bi,j(min) must be set. Otherwise, the optimisation process would drive the activity cost to zero (Bi,j = 0).

α i, j

Table 2

⎧ ⎪ ⎪ 0 if ⎪ ⎪ ⎪ ⎪ ⎛ Bn ⎞ ⎪ ⎛ Qn ⎞ ⎪ ln ⎜ Q ⎟ − ln ⎜ B ⎟ ⎪ ⎝ C ⎠ if =⎨ ⎝ C ⎠ ⎛ ⎞ ⎛ T ⎪ ln n − ln Bn ⎞ ⎜ ⎟ ⎪ ⎜⎝ TC ⎟⎠ BC ⎠ ⎝ ⎪ ⎪ ⎪ ⎪ 1 if ⎪ ⎪ ⎪⎩

⎛ ⎛ Qn ⎞ ⎛ Bn ⎞ ⎞ ⎜ ln ⎜ ⎟ − ln ⎜ ⎟⎟ QC ⎠ BC ⎠ ⎟ ⎝ ⎝ ⎜ 0≤ ⎜ ⎛T ⎞ ⎛B ⎞⎟ ⎜ ln ⎜ n ⎟ − ln ⎜ n ⎟ ⎟ ⎜ ⎟ ⎝ BC ⎠ ⎠ ⎝ ⎝ TC ⎠ ⎛ ⎛ Qn ⎞ ⎛ Bn ⎞ ⎞ ⎜ ln ⎜ ⎟ − ln ⎜ ⎟⎟ Q ⎝ BC ⎠ ⎟ < 1 0