Project managers frequently confront the problem of ... - Science Direct

Abstract ‘R-e problem of scheduling resoulie-constrained projects to maximize their Net Present Y&e (NPV) is studied in this The performance of seventeen scheduling heuristics. ten of which are original to this work, is evaluated, Heuristic performance is rated separately on maximization of project NPV and minimization of project duration. Two of the new heuristics are significantly better then the standard heuristics at yielding higher project NPVs. These two heuristics also perform well at reducing project dura?ion. Two heuristics, Mirtslack and Shortest Processing Time (SFT), that performed welt in previous studies are shown to be poor performers at maximizing project NPV. psper.

Keywords: Project management; Scheduling; Heuristics; Net present value 1, Introdluction

Project managers frequently confront the problem of scheduling project activities subject to resource constraints. This problem arises frequently in construction, aerospace, shipbuilding, software, health care, and manufacturing industries. Early research by Davis and Patterson (1975? and Patterson (1384) focuses on determining schedules that resolve the resource conflicts while minimizing project duration. Due to the high cost of capital and the issue of time versus cost trade-offs, recent research (Smith-Daniels and Aquilano, 1987; Smith-Daniels and SmithDaniels, 1987; Pinder, 1988; Pinder and Marucheck, 1989; Elmaghraby and Herroeien, 1990; Padman and Smith-Daniels, 1993) concentrates on the objective

* Corresponding author. 02X?-6963/96/$15.00 Copyright PII SO272-6963(96)OOOO3-4

of maximizing projec: net present value (NF J). In bidding for projects s?lch as highway construction and defense contracts, the bid is contingent on the project NPV. The NPV is dependent upon the timing of the cash flows associated with activity completion. Contractors often engage in a practice known as “front-end loading”, in which costs of activities early in fhe project are overstated and costs of activities near the end of the project are understated. This practice allows the contractor to receive higher positive cash flows earlier in the project, thereby increasing the NPV and allowing the contractor to submit a lower bid to increase the likelihood of winning the contract. Bey et al. (1981) recommend using the NPV objective for capital intensive projects in order to recover capital costs more quickly. This paper presents heuristics that provide schedules with superior project NPVs and are easily implemented by project managers.

0 1996 Elsevier Science B.V. Al9 rights reserved.

AH. Russell (1970) introduces the problem of maximizing project NPV. Consider a project with N activities, each with duration (cl,} (i = f ,2, . . . ,NL These activities have completion times {T,} and associated cash flows (CF,]. Also associated with each activiiy is a set of immediate predecessors (P,} that must be completed before the given activity can be started. Furthermore, there are NR different types of required resources that are utilized rather than consumed. For each type of resource, there is a fixed amount. R, (.j = 1,2,. . . , AIR) of the resource available, and each activity has resource requirements {r,). This problem, P, is formulated: 8:

Maximize NPV = t CF, exp( - CY~;), ;s,

(‘1

subject to the precedence constraints -? - ‘ji’j 2 cl,, 1,

for i== 1,2 ,..., N, ~IEP,,

for j = 1,2.. . . ,NR, t = 1,2,. . . *CP,,,,, ,

(33

where a is the dissount r&, A, is the set of ongoing activities at time t, and CPm, is the maximum possible critical path (i.e., the longest possible Table I Overview Opritnitcttion

of literature

on scheduling

to maximize

project

project duration). The nonlinear objective function (Eq. (1)) is the net present value of a series of continuollsly compounded cash flows. Constraints (Eq. (2)) maintain the activity precedence requirements of the network; while constraints (Eq. (3)) maintain the feasibility of the schedule with respect to resource availabihty at each point in time. Due to the combinatorial nature of this problem, formulations &U-I. Russell, ‘070; Grinold, 1972; Doerf+ and Patterson, 1977; ‘1awares, 1990a,b) tend to be intractable for projects with a large number of activities. As a consequence, many heuristic scheduling rules have been developed (Smith-Daniels and Aquilano, 1987; Pinder, 1938; R.H. Russell, 1986). Because the resource-constrained project scheduling problem subsumes the job shop and assembly shop scheduling problems (Bock and Patterson, 1990), these heuristic scheduling rules also apply to these environments. Resource-constrained research is quite extensive. Patterson ( 1984), Elmaghraby and Herroelen (1990) and Padman and Smith-Daniels (1993) provide thorough literature reviews and comparisons of techniques. Previous research is categorized as either an optimization or heuristic soiution method (see Table 1). The heuristic methods are further categorized as

NPV

merhod.v

Unconsfrained A.H. Russell 11970) Grinoid (1972) Do+rsch and Parterson (1977) Elmaghraby and Herr&en (I 990) Resource-constrained Smith-Daniels and Smith-Daniels (1987) Tavares (1990a) Tavares ( IY9Ob)

Transform Transform

nonlinear nonlinear

program program

to transshipment dual LP. into LP with restrictions.

O- 1 IP, capital rationing constraints. Partitioned “trees”; pay late, collect IP, maierials requirernants DP, critical path activities Sequential

early.

constraints. only.

stages formulation.

Heuristic methuds Optimization-bcised Patterson et al. (1990) Padman and Smith-Daniels (1993) lcmeli and Erenguc (1994) Priority rule-bused R.A. Russell (1986) Smith-Daniels and Aquilnno (1987) Pinder (1988) Pinder and Marucheck (1989)

Backtracking Transshipment Tabu search

algorithm w/heuristics. dual LP algorithm w/heuristics. heuristic.

Job-shop scheduling and LP-dual heuristics. “Right-shifted” (pay late) scheduling heuristic. Discounted Discounted

cash flow weight cash flow weight

heuristics. heuristics.

optimization-based or priority rufe-based. In optimization-based heuristics, a modified version of problem P is solved where resource constraints (Eq. (3)) are relaxed. Heuristic procedures are subsequently used to resolve the constraint violations. In co:?trast, priority rule-based methods use heuristics to cal, ,late and assign priorities to each activity in order to satisfy the resource constraints. Although optimization-based heuristics can provide good solutions, Patterson et al. (1990) showed that optimization-based heuristics do not always significantly improve the initial solutions, particularly for the max-NPV objective functicn. The major drawback of the optimization-based he.,,-;;tics is the computational b! :Jen for implementation. They require the formulatinr; and sfriuti~~ of the dual to the linear approximation of the relaxed optin.:;: :tion problem. This formulation requires N constraints and

i= I

(4)

variables, where lP,l is the cnrdinality of set {I’,). Thus, there is one constraint for each activity and one variable for each precedence requirement in the network. There are several advantages to using prioritybased heuristics. The most important advantage is the relative ease of calculating the priority rules compared to optimization-based methods. Thus, priority rules are widely used in many industry situations (e.g., construction projects that share heavy equipment) and can readily be implemented by practitioners. Another advantage of priority-based heuristics is their intuitive appeal based on their foundation in scheduling theory. This paper introduces several heuristics based on the discounted cash flow associated with each activity and evaluates the performance of seventeen scheduling heuristics. Historical usage of the Critical Path Method fCPM‘b has led project managers to focus on minimizing project duration in an attempt to receive payments as soon as possible and thereby maximize project NPV. Heuristic performance is evaluated separately on maximization of project NPV and minimization of project duration to determine if the heuristics lead to schedules which improve both

measures. Six standard heuristics. previously shown to perform well at minimizing project duration (Davis and Patterson, 1975; Grinold, 1972; IL&. Russell, f986; Kurtulus and Davis, 19S2) are included in this study. One optimization-based heuristic, the best overall performing rule of Padman and Smith-Daniels (19931, provides a basis of comparison of the discounted cash flow heuristics to optimization-based heuristics. Two heuristics, original to this work, are shown to perform as well as the optimization-based heuristic under many different project environments. The next section presents the scheduling heuristics tested in this work. Section 3 contains a discl!ssion of the projects and the experimental design used in this study. The results are presented and discussed in Section 4. The last section provides conclusions and recommendations for managerial implementation.

2. ITescription of schednfing rules In developing the scheduling heuristics to maximize project NPV, consider the objective function of the problem P. Priority-based heuristics are constructed from the parameters CF., (Y and T, (a function of ti,). Five pairs of rules (Table 2) based on these activity parameters are presented. Each pair of rules uses a different tveighting factor (w;), based upon these parameters (CF;, a and 7;), to calculate the scheduling priority of each activity. There are two variations of sch,eduling for each weighting factor. The first variation uses only the weighting factor and actit‘ties are prioritized in increasing order of I/w,. Thus, for two activities i and j, if l/wi < I/M;, activity i is assigned a higher priority. The second variation is based on the Shortest Weighted Processing Time Rule (SWPT), and activity priorities are set in increasing order of d&v;. Weighting by duration gives priority to activities with shorter durations to accelerate the receipt of potential positive cash flows. When a set of activities eligible to be scheduled have positive and negative cash flows, the opposing signs of the cash flows can invert the desired order of prioritization. The following procedure was used to resolve this problem. First, order the activities with positive cash flows according to the scheduling

Table 2 Summary

of rules

Description

--

~-----_I_xI

and wqhting

factor

AcraRym

1__1_1_~~_

Discounted cash tlow based on xlivity duration It’, = cry cxpc - * d,! Discotated cash flow at early-finish time

DCFD DCFDW DCFEF

IV, = CF, cxpt - u EI;, )

DCFlF8 DCFLF DCFLFW CDCFEF CDCFEFW

Discounted

cash flow at late-finish

w, = CF, exd

time

-- a LF, )

Discounted

cash flow of future activities w, = (23; G s CF, 1 exp( - o EF;j. where

Discounted

&sh

flo:y

of future

w, = LE., E ,y,CFJag:-n

“‘Sfmdurd” Minimum

nt early-finish

times

S, is the set: li nnd its succcssor~}

activities

at late-fmish

times

LFJ

-

I/M d,/,r, f jW,

d,/u

I

r/w, dijWi 1 ,A,

CDCFLF

WV, 1/‘c;

I33cFLFw

d, /s+*,

MINSLK EDD SIT GNS

‘V, iv,

kettrisrics w, = LF, - EF, vi, = LF,

slack

Ex!‘*w doe dnte Shortest processing time CaxItest number of successors Greatest succeeding processing

time

Greatest successive resource requirements Opportunity cost of scheduling, immediate .~“---_-.__(I. ___

release

w, = d, H’, = 13iES,11 u’, = -7 “i E s,d, T-.v.Q R 1”“ = c, E 5, L, -: , *, )ci =,f(dunl price)

heuristic. Thus, the eligible activities with positive cash flows will be scheduled before the eligible activities with negative cash flows. Second, transform the negative cash flows by ‘6

Rule StruCllire

transformed

=

MaxCF * MinCF/(

- CF,),

(5)

-.-

GSPT CSRR iocs

w,

f/T I /w, I /I)., I /IV,

activity and its predecessors/successors. Therefore, the weighting factors require more calcchttion,. The second pair of rules (DCFEF and DCFEi;Wj uses n*, = Cc exp( - LYEF, ) ,

(7)

where MaxCF and MinCF are the maximum and minimum cash flows, respectively, of the eligible activities with negative cash flows. This transformation preserves the magnitude and ratio scaling of the negative cash flows. Finally, use the transformed cash flows to order these activities according to the scheduling heuristic. The first pair of rules (DCFD and DCFDW) computes the discounted cash flow based on the duration of each activity:

where EF, is the unconstrained early-finish time of activity i. The motivation behind this rule is that the early-finish time represents the lowest possible value of ?;$ i.e., the most opportune timing for activities with positive cash flows. For projects where the resource constraints are not tight this should provide a good estimate of the actual T. The third pair of rules (DCFLF and DCFLFW) uses

wi = CF,exp( --adi).

where LF, is the unconstrained late-fittish time of activity i. The fourth and fifth pairs of rules (CDCFEF, CDCFEFW, CDCFLF and CDCFLFW) use estimares of the present value of the cash flows of succeeding activities in determining priorities. These rules estimate the importance of delaying an activity by considering the present value of the cash being blocked by not scheduling the activity. Thus, these

(6)

This is a simple weighting factor that does not require any network information other than activity duration and therefore requires the fewest calculations. It is similar to many static dispatching rules UWA.I in job shop scheduling. The second and third pairs of rules (DCFEF, DCFEFW, DCFLF and DCFLFW) incorporate more information about the

w, = C& exp( - a~!,(),

(8)

rules are forward-looking because they consider future cash flows In the fourth pair of rules (CDCFEF and CDCFEFW).

where (S,] is activity i and its successors. The summation term is the total cash flow of activity i and all of its successors. The fifth pair of rules (CDCFLF and CDCFLPW) uses

The ten disecunted cash flow ruies developed above arc benchn;rrked against seven rules presented in previous literature. The best overall perii.:;iting rule of Padman and Smith-Daniels !1993) was used as a benchmark for optimization-based heuristics. This rule, referred to as IOCS (Opportunity Cost of Scheduling, Immediate Release). initially requires the formulation and solution of the dual to the relaxed probEem P for each project. This heuristic attempts to measure the opportmzity cost incurred by scheduling the particular activity rather than other activities in the queue, based on ‘*tardiness” and “earliness” penalties computed from the solution to the dual of the relaxed optimization model. Davis and Patterson (1975) anti Kurtulus and Davis (1982) compare several scheduling rules for minimizing project duration. As a result of their performance in previous studies. six scheduling rules are used as benchmarks to eT/aluate the performance of the NPV-based rules. The standard rules are: f . Mi11imum Actiuity Slack: Priority in resolving resource conflicts is given to the activity with minimum slack. (The highest rated rule in (Davis and Patterson, 1975; PA. Russell, 1986) and third highest in (Davis, 19751.) 2. Earliest DzdeDate: Priority is given to the activity with the minimum late-finish time. 3. Shortest Processing Time: Activities with the shortest duration have priority. (The second highest rated rule in (Davis, 1975!.) 4. Greatest Number of Szzccessors: Priority is assigned to activities with the greatest number of successors. 5. Greatest Succeeditzg Processing Time: Activities

whose successors have the largest amount of required processing time receive priority. 6. Greatest Sz4ccessire Resmrce RequiremWs: I%ority is given to activities whose successors have the largest amount of resource requirements. (The highest rated rule in Davis. 1975.1 Thus, seventeen scheduling rules are evaluated; these are summarized in Table 2.

3. Experimentid

design

Several studies (Kurtulus and Davis, 1982; Davis, 1975; Patterson, 1976: Badiru, 1988) have developed summary measures to describe project network attributes. These measures are used to categorize various network conditions. This research uses three summary measures to identify the network auributes of size, shape and severity of resource constraints. Network size is measured by the number 0:’ ac-tivities in the network. Davis (1975) observes that the number of possible resource conflicts typically increases combinatorially as the number of activities in a network increases. Thus, the number of activities is used as a measure of problem size and comparative complexity. The Average Resource Loading Factor (ARLF) used in other studies (Kurtulus and Davis, 1982) to measure network shape is dependent on both network shape and the pattern of resource demand. Thus, in this study the centroid is preferred as a measure of network shape because it ailows for the indepenlent assessment of the effect of network shape on the project duration and NPV. The network centrcid is given by CP- I CP-I Centroid= c (t+ 1/2)n,/ c n,X I/U’. r=O r=O (11) where II, is the number of activities that are underway (active) at time r and CP is the length of the critical path in the unconstrained schedule. Division by the critical path length reduces the centroid to a dimensionless number. This avoids problems of scale created by different project sizes and magnitudes of activity durations. With this measure, a value of 0 indicates an extremely left-shifted project and a I

Factor

Mensure

Lcwi

Size

Numkrof

Shape

activities Centroid

Small < 25 Left < 0.30

AUF

Resource constraint

I

Low i 0.90

Level

2

Medium 25-m Urnform m30-0.60 Medium O.YQ- I .60

Lewi .‘I --Lnrpe 1 50 Right > 0.6 Hiph > 1.60

r,+

I -.ii

iI GO 0 -12 ii ;4

0L

‘4’:

n iii

(1‘2

(1‘4

Centmid

Fig. 2. Distribution

denotes a right-shtited project with most of the activities near the end of the project. Projects whose activities are evenly distributed from beginning to eLid b?ve a centroid of 0.5. Constraint severity measures the extent of “tightness‘ ’ of resource constraints. A measure used to determine constraint severit is the .4veravc 1 Utilizaticin Factor (XUFj 0: :(urtula”s and Davis i I982j. To compute the AUF. the Utihzation Factor (UF) for each resource j, is first calculated: UF, = t (R,,d,) /RfX,,,,i=l

( 12)

The AUF is then determined by AUF=

5 j-

WF,/NR.

(13)

I

Wi!h this measure. va!ues less than I typically indicate projects with resource constraints that are not tight. A value greater than 1.6 indicates a project with tight resource constraints (Grinold, 1972; R.A. RusselI, 1986). Two sets of projects were used in this experiment. The first is a set initially developed and analyzed by

1, Distribution

of number

of activities

of Pntterson

projects.

of Patterson

projects

Patterson (1976) and subsequently used by SmithDaniels and Aquilano 61987) and R.A. Russell (1986). The second set was generated to provide a full factorial experimental design. Based on previous studies (Smith-Daniels and Aquilano. 1987; Gsinold, 1972; R.A. Russell, 19861, the cut-offs for the different levels of the factors were determined and are presented in Table 3. The number of activities, centroid and AUF of each project in the Patterson set were calculated and the projects were categorized according to this experimental design. The distributions of the design factors are in Figs. 1-3. The frequencies of each cell of the experimental design are listed in Table 4. Since the Patterson projects do not provide full factorial experimental design, a set of 810 simulated projects (3 sizes X 3 shapes X 3 levels of constraint X 30 projects per cell) was generated. The distributions of the design factors for these additional projects are in Figs. 4-6. The number of activities for each project was generated from a uniform distribution. The distribu-

a3

Fig.

ofcentroids

0.4

CL.5 0.6

Fig. 3. Distribution

0.7

03 0.‘) AUF

of AUFs

1.0

1.1

of Patterson

1.2

1.3

projects.

1.4

235

Fig. 4. Distrihtion

of nun&x

ofnctivities

of additional

projects.

tion parameters depend upon the hour. :‘-.dictated by the size of the pi eject, wirh the large prajects ranging from 60 to 200 activities. Tne &rations (in months! for each project were generated from the dist;li::zt.ion of the activity durations of the Patterson projects (Fig. 7). The activity precedence relationships were then created subject to the specified shape of the project. To assure consistency with the Patterson datn, the number of successors for each activity ranged from one to five. Each activity can require up to three different resource types (like the Patterson set) and the amount of resources required by each activity is generated from a uniform distribution with the same bounds found in the Patterson data (0 to IS>. The AUFs for the projects are controlled by varying the amount of resources available for each project. The available resources are generated from a uniform distribution subject to the specified AUF. The original Patterson projects do not have cash flows associated with each activity, therefore cash flows were generated from uniform distributions. The bounds for these distributions are a function of the activity durations and resource requirements.

Fig.

5. Distributionof centroidsof rtdditlnnalprojects.

Fig. 6. Distribwion

uf AUFs

of additional

projects.

These bounds provide positive or negative cash flows for the activities. After scheduling ali projects using each of the heuristics at three levels of cx (S‘S! 12% and 20% APR) and finding no significant difference in heuristic performance, t2 = 12% APR wLasused for this study. Given that the experimental factors are expected to have a strong effect on the project durations aud NPVs, the objective of this study is to determine the ordinal performance of the heuristics. With this objective, the Friedman nonparametric method (Conover, 1980) is the appropriate test for significance. This nonparametric method is equivaient to an analysis of variance (ANOVA) where the dependent variable is the rank order statistic of tht: objective function value (NPV or project duration). After scheduling each project using each of the seventeen heuristics. the rank order statistics for the net present values (signified by NPV Rank) are determined by ranking ‘he NPVs for each rule within each project. The maximum NPV for each project receives the lowest (best) ranking and ties are zrgeraged. The rank order statistics of the project durations (critical path length) are denoted as CP Rank. The minimum

Fig. 7. LXstribution

of activity

dumtions

of the Pmxson

projects.

Table 4 Frequency

table of patterson

projects,

by SIX. shape and cons~rainl

-_

Medium

SlIXlll

hrpr

Constrnir

Left

Uniform

Right

Left

Uniform

Ri$

Low Medium High

3 1 0

32 I9 0

0 0 0

0 0 0

40 5 0

0 0

/

Left

Uniform

Right

0 0 0

I0 0 0

0 0 0

tive function value of the m-th observation, resulting from the use of rule i upon a project of size j, shape k, and constraint severity 1. p is the overall mean rank (number of rules tested/Z>. The other terms represent differences from this mean due to the effects of the rules and the interaction ivith the factors.

duration for each F:r)jecf receives the lowest rank. The average rank (pj for any project, or group (designated by blocking factor) of projects, is the nuY:iber of rules tested divided by 2. Hence, in an AIKWn. on the ranks, the treatmelrt factors (size, shape and constraint) do not explain any of the varia’ion in the rankings by themselves. Thus, the ktars are not included as ~na?heffects in tire mad-:; instead, they are treated as nested effects. The general model is

4. Results

Rank,,,hFl = ,u + Rule, + Size,[Rule,] + Shape,[Rule,]

The frequency table for the Patterson projects (Table 4) indicates that the only blocks of the experimental design with a significant number of projects are the blocks with small and medium size, uniform

+ Constraint,[Ruie,]

+ E,.

( 141

In this model, Rank ,,k,n, is the rank for the objec-

Table 5 Nonpammetric

multiple

comparison

test for the patterson

projects

h’f Y Runk

CP Rtrr~k

Rule

Mean

Grouping

IOCS ~~DcFLF CDCFEF GSRR GSFJT DCnFw GNS EDD DCFEFW DCFDW MINSLK DCFEF DWD DCFLF CDCFLFW -nxximv SPT

5.77 6.09 7.25 7.44 7.50 7.77 7.86 8.11 8.29 9.76 9.82 10.44 IO.49 IO.53 10.90 11.10 13.66

A AB ABC NC BC

*

c C c CD

- Means with the same letter nx not signiftcantly

DE DE E E E E E F different

Rule

Mean

Grouping

IOCS EDCFLF GSPT ZDCFEF EDD GNS GSRR MINSLK DCFLFW DCFEFW DCFDW DCFLF DCFEF DCFD 23DCFLFW CDCFEFW SPT

5.46 5.76 5.89 6.30 7.29 7.75 9.29 9.30 9.61 9.70 9.76 10.26 10.32 10.47 IO.52 10.83 13.87

A A AB ABC BC

nt a = 0.01.

*

CD DE DE EF EF EF EF EF EF EF F G

shape, and iow oonstmint severity. Thus, ANOVAs were conducted using all Patterson projects and Rule, as the independent variable. The results arc listed in Table 5. Th:se results show that, for these prqjects, the choice of scheduling rule has a significant effect on the project net present value and project duration. As expected, the computationally intensive IOCS has the minimum rankings for both objective functions. Two of the di:.counted cash flow rules. CDCFEF and CDCFLF. have rankings that are not significantly different from those of’ the IOCS rule. i’“tese two heuristics, as well 1s two of the best st?ndard rules (Greatest Succeedin+ Processing rime and Greatest Successive Resource Requirementsf are for,:;;. .Jlooking. These rules arc more effective because they use information from futttre activities in the network; CDCFEF and CDCFLF use future cash receipts and payments. while GSPT and GSRR use future resource requirements. Note that SPT, a heuristic commonly used in practice, is signiiicantly worse for both objective functions. The results of similar ANGVks for the larger set of projects (Table 5) show that the choice of scheduling rule has a significant effect on the project NPV and project duration. As with the Patterson projects,

Table 6 Nonparamctric

multiple

comparison

test for the additinnai

the overall performances of CDCFLF and CDCFEF arc not significantly different from that of IOCS. As for the structure of tnr cash flow-based heurisiics, the forward-looking discounted cash flow rules are significantly more effective when not weighted by duration. In contrast, weighting by duration had no significant effect on the discounted cash flow rules that wem not forward-looking. The results indicate no significant difference due to the use of the early-finish time sersus the late-finish time. Therefore. the early-finish rules were ehminated from further testing. Once again, the most effective standard rules arc the resource based rules: GSRR and GSFT. Two rules commonly used in practice, MINSLK and SPT were significantly the least effective heuristics under both criteria. On this basis, they were eliminated from further testing. Furthermore. these resillts indicate that even simple discounted cash flow hzutistics, such as DCFEF. offer a significant improvement in project NPV over rules that are currently commonl,y used in practice (GSPT, GNS, EDD. MINSLKI SPT). The Friedman test does not allow for interaction terms or more than a two-way design. Thus the main effect (Rule,) was tested first using a one-way de-

projects CP Rmk

NPV Rut& RUk

Mean

Grouping

IOCS YZDCtXF IlDCFEF DCFww DCFEFW DCFEF IX.3 .F DCFDW GSRR DCFD LDCFEFW IZDCFLFW GSPT GNS EDD MINSLK SPT

6.71 6.90 7.31 7.52 7.79 7.92 8.08 8.11 8.14 8.40 9.58 9.52 9.58 9.74 10.05 13.52 13.65

A AB ABC BC

r Means

*

c CD c D CD c 13 D

with the same letter at(+? nnt significtlntiy

E E E EF F c G differcnt

Kufc

M&n

Grouping

IOCS UXFLF UU%F GSRR GSPT DCFLFW EDD Gl’iS DCFLF XDCFEFW DCFEFW CDCFLF-W DCFEF DCFDW DCFD MlNSLK SPT

7.12 7.26 7.40 7.75 a.23 8.43 8.49 8.60 8.62 8.64 8.71 8.8-S 8.88 9 54 10.80 12.6p 12.69

A AB A B BC

at a = O.OI.

-

CD CDE CDE DE D E DE DE E EF FG G 1-I H

Table 7 Results of ANOVA

for nested

are not significantly differ2nr. Possible reasons for XDCFLF not performing as well on projects with medium constraint severity are that the projects with low constraint severity are relatively easy and have few possible combinations to consider, while the high t nstraint projects may have few alternative soiutitins. Notc that the performance of CDCFLF was not significantly differe than that of IOCS at the 3% Ievel. Another observation from the results is that there are several situations in which the performance of the “simple” rule, DCFLFW, is not significantly different from those of IOCS and XDCFLF. With its ease of implementation, DCFLFW becomes a good choice for specific project environments such as small or low constraint severity projects. Of the two standard heuristics. only GSRR is as effective at maximizing NPV as the cash flow-based rules; and then, only for smdh or low constraint severity projects. In addition to the nonparametric method used above, the relative performance of the heuristics were determined by comparing their performance to the corresponding unconstrained project schedules. The percent below unconstrained project NPV was computed for each project and rule by

models

Source

DF

Dependent vari:thle: Rule Size[RuleJ Shapc[RuIe] Constraint[Rule] Dependent variable: Rule Size[Ruie] Shape[Rulc] Constmint[Rulel

WV Rwk 4 262.713 IO 152.038 10 10.365 10 352.316 CP Rtrnb 4 62.699 10 37.022 10 4.070 !Q 125.943

Sum of squares

F ratio

-_I~

Prob > F

39.832 9.218 0.628 21.362

0.000 0.000 0.793 0.000

I I.562 2.362 0.3W 9.29l

0.000 0.006 0.978 0.000

Ggn. Since this factor is significant, the nested factors die tested because they are subsets of the data and retain the same rankings within a project. Based upon the results. the following rules: IOCS. DCELFW, EDCFLF. CSF Y atrd GSRR, wet% used to test the effects of size, shape and constraint scverity. This reduction helps discriminate between rules if there are significant differences in performance. These rules were tested using the nested model in Eq. (14) and the results are in Table 7. These tests show that the project network shape has no significant effect on the performance of the rules on either objective. The Friedman test was performed on each level of size and constraint separately because the two factors are significant. Table 8 lists the rules that are in the A-group of the multiple comparison test of these ANOVAs. These results indicate that for all cases, except for maximizing NPV for projects of medium constraint severity, the perfomlances of IOCS and CDCFLF Table 8 Results of ANOVAs

for size and constraint

o/obelow unconstrained NPV =

unconstrained NPV - NPV for the rule unconstrained NPV

Similar relative performance measures were calculated for the critical path duration. The ANOVA results using these measures were also calculated for

factors Group

“A“

rules

*

Factor

Level

NPV Rank

CP Rank

Size

Small

IOCS. CDCFLF, DCFLFW, GSRR IOCS, XDCFLF. DCFLFW WCS, CDCFLF IOCS, XDCFLF. DCFLFW. GSRR IOCS IOCS. ZDCFLP

IOCS. TDCFLF. DCFLFW, GSPT, GSRR IOCS, EDCFLF. DCFLFW, GSRR IOCS, CDCFLF, DCFLFW IOCS, XDCFLF, DCFLFW, GSPT. GSRR WCS. EDCFLF, GSRR IOCS, EDCFLF

Medium Large Low

Constraint

Medium High 4 Means

are not significantly

different

at a = 0.01

. (‘5)

comparative purposes. and. althr~ugh not prcsentcd it? this paper, are identical to the results for the nonparametric measures. This is because when ordinal a3d interval measures are functions of the same intcrvali measure, as is the case here, &hen statistica! significance determined by nonparametric measures implies statistical significance for the interval mcnsure (Conover. 1989).

5. Summary and conclusians This research presents several ncbv scbcduling heuristics. hascd ct.1 discounted cash flows, that are effective at maximizing project net present vaiui 2nd minimizing project duration. These heuristics are managerially significant because they arc simple to compute in the context af project management and are intuitively based on scheduling theory. Thus. the) require fess computaticnal effort than optimkation-based heuristics. which facilitates their implementation by project managera. The results of this research indicate that for many project environments the CDCFLF/CDCFEF rules are as effective as the optimization-based rule at improving project NPV, The optimization-guided rule (IOCSi does provide slightly (0.01 level of significance versus 0.05) better performance for the projects with medium constraint severity. Furthermore. these results indicate that even simple rfiscountcd cash flow heuristics, such as DCFEF, offer ;I significant improvement in project NPV over rules that are commonly used in practice. This research provides practitioners with hcuristics that are effective at improving project net yrwsent value and yet are readriy implementable.

COnOVer. W.S . 1980 Praclicnl NOnpar3meiric Statistics, ‘+Viley arid Sons. New York. pp. Y-305. 2nd Edition. Davis. E.W. 1975. “Fro&r network summ‘a.ry measures strained-K.x)urcr 2. pp. 132-l-22.

schedulinp”,

Badiru. A.B., 1988. “Towards the standardization of performance me,asures for project scheduling heuristics”, IEEE Transactions on Engineering Mnnagemenb vol. 35. no. 2, pp. 82-89. Bey, R.. R.W. Doersch and J.N. Patterson, 1981. “The net present value criterion: 11s impact on project scheduling”, Project Management Quarterly, vol. 12, no. 2. pp. 35-45. Bock. D.B. and J.H. Patterson, 1990. “A comparison of due date

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