managing time, cost & quality

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Applied Mathematics and Computation xxx (2006) xxx–xxx

www.elsevier.com/locate/amc

On the discrete time, cost and quality trade-off problem

Hamed R. Tareghian, Seyyed Hassan Taheri *

Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

Department of Mathematics, University of Khayam, Mashhad, Iran

Abstract

In this paper a solution procedure is developed to study the tradeoffs among time, cost and quality in the managementof a project. This problem assumes the duration and quality of project activities to be discrete, non-increasing functions ofa single non-renewable resource. Three inter-related integer programming models are developed such that each model opti-mizes one of the given entities by assigning desired bounds on the other two. Different forms of quality aggregations andeffect of activity mode reductions are also investigated. The computational performance of the models is presented using anumerical example.� 2006 Elsevier Inc. All rights reserved.

Keywords: Project management; Discrete time; Cost and quality trade-off problem

1. Introduction

The discrete time–cost trade-off problem, hereafter referred to as DTCTP, is a well-known problem fromthe project management literature (see e.g. [1,2]). The instances of this problem are the so-called projects thatconsist of a finite set A of activities together with a partial order � on A. The DTCTP occurs when the dura-tion of project activities are a discrete non-increasing function of a single non-renewable resource committedto them. Any activity, ij 2 A, may be executed in say, k different modes, where the rth mode (1 6 r 6 k) takestijr time and costs an amount cijr of money. Without loss of generality, it is assumed that all values of tijr and cijr

are non-negative integers. The executions of activities in A depend on their precedence relations. In otherwords, if ij � jk, then activity ij may not be started unless activity jk has been completed. All activities areavailable for processing at time zero.

A feasible solution ~s ¼ fr1; . . . ; rjAjg of such a DTCTP consists of assigning an execution mode, r,(1 6 r 6 k) to activity ij 2 A. The cost cð~sÞ of solution~s is the total amount of money spent on~s. The duration

dð~sÞ of solution~s is the finish time of the earliest start schedule that assigns duration tijr to activity ij 2 A. Inthis schedule considering the precedence relations, each activity starts at the earliest possible time. As such the

0096-3003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.02.029

* Corresponding author.E-mail addresses: [email protected] (H.R. Tareghian), [email protected] (S.H. Taheri).

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2 H.R. Tareghian, S.H. Taheri / Applied Mathematics and Computation xxx (2006) xxx–xxx

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duration of solution~s equals the length of the longest path in the partial order, where the length of the path isthe sum of the durations of the activities in the path.

The objectives of the solution procedures to DTCTP can be threefold. The so-called deadline problem aimsat minimizing the total cost of the project such that a given deadline is met, while the budget problem involvesminimizing the project duration without exceeding a given budget. There is a tradeoff between cost and time.Solutions with shorter durations usually cost more, while solutions with low costs usually take longer. By fix-ing either cost or duration, two related optimization problems are obtained with the objective of minimizingthe other parameter. A third objective is to construct the complete and efficient time/cost profile over the set offeasible project durations.

Several procedures have been proposed to optimally solve the DTCTP. Early optimal solution proceduresare mainly based on dynamic programming e.g. [2,3], or enumeration algorithms [4]. The current exact algo-rithms still rely on dynamic programming, but in addition exploit the decomposition structure of the projectnetwork (e.g. see [5] for one of the implementations of series-parallel decompositions). A Lagrangian relaxa-tion-based heuristic for activity on arc (AoA) networks is developed in [6]. It follows from [7] that both dead-line and budget problems are very difficult to solve; they are, in fact, strongly NP-hard for general projectnetworks.

It was recently suggested that the quality of a completed project may be affected by project crashing [8]. Assuch, in expedition of a project, its quality should also be taken into consideration along with the time and costtradeoffs. Considering the inter-twined effects of time, cost and quality in project management, it seems rea-sonable to develop a mathematical model, hereafter referred to as DTCQTP, which considers project’s time,cost and quality simultaneously. In the DTCQTP, like DTCTP, project’s activities are performed in one ofseveral alternatives. For each activity a set of time, cost and quality triplet, referred to as mode, are given.The quality of each activity is specified discretely at a scale of [0.75–0.99]. The overall project quality is a func-tion of quality attained at each activity. The form of this function depends on the nature of the problem andthe definition of quality adapted. The purpose of this problem is to complete the project at a given deadline,provided that its total cost is minimized while its overall quality is maximized. The literature on solution meth-ods for the DTCQTP is scant. The authors are not aware of any work in the literature that has studied theDTCQTP. The procedure developed in [8] which considers the tradeoffs among cost, time and quality in con-tinuous mode, was later used to study an actual cement factory construction project in Thailand [9].

In this paper, we develop three inter-related binary integer programming models that assist project manag-ers make trade-off decisions. In the following section we formally describe the problem and present the math-ematical models. In Section 3, we discuss the results using an illustrative example. A mechanism is proposedfor activity mode reductions and its effect on models performance is discussed. In the last section, a summaryis given and some conclusions are drawn.

2. Problem statement

Assume that the specifications of a project is given in the form of a directed acyclic graph G = (V,A), inwhich V is the set of nodes, representing events, and A is the set of arcs, representing activities, in the AoAmode of representation. For each activity, ij 2 A, a set Mij of modes, is given. For each mode, r 2Mij, let tijr,cijr and qijr denote the duration, the cost and the quality attained by performing activity ij in mode r, respec-tively. It is assumed that for each activity ij 2 A, tijr > tijr+1 implies cijr < cijr+1, but qijr > qijr+1. The nature ofeach activity determines the quality level (0 6 qijr 6 1) assigned to it. The objective is to construct the completeand efficient time, cost and quality profile to offer decision support in crashing a project. The following nota-tion is used to represent the relevant data in the DTCQTP problem:

V set of nodes, V = {1,2, . . . ,n}A set of activitiesB set of critical activitiesMij set of execution modes for activity, ij, ij 2 A

tijr duration of activity ij when executed in mode, r, r = 1, . . . , jMijjcijr cost of performing activity ij in mode, r, r = 1, . . . , jMijj

H.R. Tareghian, S.H. Taheri / Applied Mathematics and Computation xxx (2006) xxx–xxx 3

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qijr quality of performing activity ij in mode, r, r = 1, . . . , jMijjxi occurrence time of node, i, i 2 V

aij lower bound for the quality of activity, ij, ij 2 A, r = 1, . . . , jMijjCmax upper bound for project costTmax upper bound for project deadline

The three objective functions sought to be optimized in the DTCQTP, namely project’s quality, duration,and cost can be defined as follows:

f1 ¼1

jAjX

ij2A

X

k2Mij

qijkxijk; ð1Þ

f2 ¼X

ij2B

X

k2Mij

tijkxijk þX

i2V

xi; ð2Þ

f3 ¼X

ij2A

X

k2Mij

cijkxijk. ð3Þ

In order to assist decisions regarding project crashing, we have developed three mathematical models to opti-mize the above functions. The first model follows:

max f 1 ð4Þs:t:

X

r2Mij

xijr ¼ 1; ij 2 A; ð5Þ

xj � xi PX

r2Mij

tijrxijr; ij 2 A; i; j 2 V ; ð6ÞX

ij2A

X

r2Mij

cijrxijr 6 Cmax; ð7Þ

xn 6 T max; ð8Þx1 ¼ 0; ð9Þxi P 0; i 2 V ; integer; ð10Þxijr ¼ f0; 1g; ij 2 A; r 2 Mij: ð11Þ

The second model which shares constraints (5)–(7), and (9)–(11) from the first model is

min f 2 ð12Þs.t.X

r2Mij

qijrxijr P aij; ij 2 A. ð13Þ

And finally, the third model that shares constraints (5), (6), (8)–(11) from the first model is

min f 3 ð14Þs.t.X

r2Mij

qijrxijr P aij; ij 2 A. ð15Þ

Objective function (1) maximizes the project’s overall quality, while objective functions (2) and (3) minimizethe project’s total duration and costs respectively. Constraint (5) ensures that one and only one executionmode is assigned to each activity. Constraint (6) preserves the precedence relations between project activities.Constraint (7) defines the project’s cost threshold, while constraint (8) defines the project deadline. Constraint(9) forces the project not to start earlier than time zero. Constraint (10) ensures non-negativity of decision vari-ables, while constraint (11) is a binary mode indicator which is 1 when mode k is assigned to activity, ij, and 0,otherwise. Finally, constraint (13) enforces that each activity preserves a certain level of quality.


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3. Computational study

In order to illustrate our proposed method, we used DAGEN [10] to generate an instance of a project net-work. The project so generated consists of 45 activities whose coefficient of complexity equals 4.5 and its com-plexity index equals 9 (see Fig. 1). In this network, there are 256 paths that link source (node 1) to sink (node10).

The number of modes per activity is randomly chosen from jMijj 2 [2,10]. For each activity, the executionmode is generated as follows: first the number of modes for activity ij is generated, i.e. jMijj � DU (2, 10). Thenthe duration of each mode is randomly sampled from DU (1, 100). After durations for all the execution modesare generated, they are sorted in ascending order, i.e. tosij = {t1, t2, . . . , tjMijj; ti 6 ti+1 "i = 1, . . . , jMijj � 1}. Thecorresponding cost for t1 which is incidentally the maximum cost mode is sampled randomly fromDU(500, 800). The costs for other modes are determined as follows:

ci ¼ ci�1 � siðti � ti�1Þ; i ¼ 2; . . . ; jMijj;
where si is u(1,3), s1 = 1, and ci, ti are the cost and the duration for mode i respectively. The quality attained byeach activity under normal circumstances is assumed 99%. The quality for the other modes of execution is gen-erated as follows. As it is assumed that the activity duration compressions do not adversely affect activity qual-ities by the same magnitude, we have classified activities into five different categories. The quality attained byactivities in categories 1 to 5 is allowed to vary between 0.95–0.99, 0.90–0.99, 0.85–0.99, 0.80–0.99 and finally0.75–0.99 respectively.
The project as described above was used to study the performance of the models described in (1)–(15). Theresults obtained by solving the given models are shown in Tables 1–3 respectively. Version 3 of the optimizercalled LINGO was available and hence was used to solve the models. The shortest possible time for complet-ing the project is 189.

Table 1 shows the cost of the project as its deadline and quality varies. For model 3, T 2 {189,220,270,320,350,410,490,585,650,700,745,750}, and project quality varies between 0.75 and 0.99 in incre-ments of 0.05. It is evident that high quality and short deadlines can not be achieved simultaneously. As

Fig. 1. Project network.

Table 1Project costs when its quality and deadline is varied

Deadline 0.75 0.80 0.85 0.90 0.95 0.99

189 26,239 NF NF NF NF NF220 25,643 NF NF NF NF NF270 24,940 25,060 NF NF NF NF320 24,617 24,664 25,009 NF NF NF350 24,488 24,488 24,681 NF NF NF410 24,184 24,184 24,368 24,634 NF NF490 23,993 23,993 24,061 24,103 NF NF585 23,839 23,839 23,853 23,874 NF NF650 23,766 23,766 23,766 23,766 23,823 NF700 23,706 23,706 23,706 23,706 23,746 NF745 23,651 23,651 23,651 23,651 23,651 23,651750 23,651 23,651 23,651 23,651 23,651 23,651

NF: not feasible.

Table 2Project deadline when its quality and budget is varied

Budget 0.75 0.80 0.85 0.90 0.95 0.99

23,651 745 745 745 745 745 74523,670 735 735 735 735 745 74523,750 664 664 667 670 700 74523,850 583 583 593 594 639 74524,000 483 483 505 519 591 74524,200 403 403 445 471 588 74525,200 242 263 305 409 508 74525,700 212 256 305 409 588 74526,000 198 256 305 409 588 74526,200 195 256 305 409 588 74526,239 189 256 305 409 588 74526,300 189 256 305 409 588 745

Table 3Project quality when its deadline and budget is varied

Budget 189 220 270 320 350 410 490 585 650 700 745 750

23,651 NF NF NF NF NF NF NF NF NF NF 0.990 0.99023,670 NF NF NF NF NF NF NF NF NF NF 0.990 0.99023,750 NF NF NF NF NF NF NF NF NF 0.989 0.990 0.99023,850 NF NF NF NF NF NF NF 0.984 0.988 0.989 0.990 0.99024,000 NF NF NF NF NF NF 0.979 0.987 0.988 0.989 0.990 0.99024,200 NF NF NF NF NF 0.975 0.986 0.987 0.988 0.989 0.990 0.99025,200 NF NF 0.964 0.973 0.976 0.982 0.986 0.987 0.988 0.989 0.990 0.99025,700 NF 0.947 0.964 0.973 0.976 0.982 0.986 0.987 0.988 0.989 0.990 0.99026,000 NF 0.948 0.964 0.973 0.976 0.982 0.986 0.987 0.988 0.989 0.990 0.99026,200 NF 0.948 0.964 0.973 0.976 0.982 0.986 0.987 0.988 0.989 0.990 0.99026,239 0.933 0.948 0.964 0.973 0.976 0.982 0.986 0.987 0.988 0.989 0.990 0.99026,300 0.933 0.948 0.964 0.973 0.976 0.982 0.986 0.987 0.988 0.989 0.990 0.990

NF: not feasible.


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project deadlines extend, the project cost is reduced irrespective of its quality. However, the extension of pro-ject deadline beyond a certain limit does not affect the project costs for any level of quality.

Table 2 depicts how project deadline is affected by quality and level of budget. For model 2, project budgetvaries from 23,651 to 26,300, while its quality varies between 0.75 and 0.99 in increments of 0.05.

Table 4Project quality when its deadline and budget is varied

Budget 189 220 270 320 350 410 490 585 650 700 745 750

23,651 NF NF NF NF NF NF NF NF NF NF 0.990 0.99023,670 NF NF NF NF NF NF NF NF NF NF 0.990 0.99023,750 NF NF NF NF NF NF NF NF NF 0.989 0.990 0.99023,850 NF NF NF NF NF NF NF 0.982 0.988 0.989 0.990 0.99024,000 NF NF NF NF NF NF 0.979 0.987 0.988 0.989 0.990 0.99024,200 NF NF NF NF NF 0.974 0.986 0.987 0.988 0.989 0.990 0.99025,200 NF NF 0.963 0.969 0.975 0.981 0.986 0.987 0.988 0.989 0.990 0.99025,700 NF 0.941 0.963 0.972 0.975 0.981 0.986 0.987 0.988 0.989 0.990 0.99026,000 NF 0.946 0.963 0.972 0.975 0.981 0.986 0.987 0.988 0.989 0.990 0.99026,200 NF 0.946 0.963 0.972 0.975 0.981 0.986 0.987 0.988 0.989 0.990 0.99026,239 0.931 0.946 0.963 0.972 0.975 0.981 0.986 0.987 0.988 0.989 0.990 0.99026,300 0.931 0.946 0.963 0.972 0.975 0.981 0.986 0.987 0.988 0.989 0.990 0.990

NF: not feasible.


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When the budget is low, quality level does not affect the deadline. However, by increasing the budgetshorter deadlines are possible at all levels of quality. The effect of quality on project deadline is more evidentat higher budgets. For instance, when the budget is 23,651 irrespective of quality level, the deadline can not bereduced from 745. However, if the budget is increased by say 11% to 26,239, then the project deadline can beshortened by at least 74% to 189.

Table 3 displays how project quality is affected by project deadlines and level of budget. For model 1,T 2 {189,220,270,320,350,410,490,585,650,700,745,750}, while project budget varies from 23,651 to26,300. Allocating higher budgets make it possible to obtain higher level of quality and also shorter deadlines.At low levels of budget, and with project deadlines less than 745, no solution is feasible. However, if oneincreases the budget level by only 10% to 26,239, feasible solutions at different levels of quality are obtained.

In the above models, we have utilized the arithmetic mean for quality aggregation. In Table 4, we giveresults when other aggregation method such as geometric mean is used.

In Table 4 the effect of project budget and project deadline on project quality is depicted, when the projectquality is the geometric mean of individual activities quality. As expected, since the quality of activities are notvery dispersed, the effect of using geometric mean as the aggregation method, are not significantly differentfrom Table 3, where arithmetic mean has been used as the aggregation method.

3.1. Activity mode reductions

In this section we describe a mechanism that enables us to simplify the problem by eliminating some of theactivities execution modes. As most of the models variables are binary, one can reduce the number of variablesby eliminating some of the execution modes from consideration, and hence, simplify the solution process foran instance of the problem. Let

tosij ¼ ft1; t2; . . . ; tjMijj; ti > tiþ1; 8i ¼ 1; . . . ; jMijj � 1g

be the duration modes for activity ij sorted in ascending order. For each path, v, linking source to sink, wecompute its longest duration

lv2P ¼X

ij2v

tij1;

where P is the set of all source to sink paths, v = 1, . . . , jPj. From the above paths, we create a set, x, that holdsthose paths whose length exceed Tmax, i.e.

x ¼ fv 2 P jlv > T maxg.
We now define, w, a set holding activities that lie in at least one of the paths, say g 2 v, (v 2 x). For all theactivities in

Table 5Effect of activity mode reductions on the models performance

Deadline II 0.75 0.80 0.85 0.90 0.95 0.99

189 247 +6.3 NF NF NF NF NF220 245 �5.0 NF NF NF NF NF270 221 �7.0 �13.0 NF NF NF NF320 200 �4.8 +11.2 +6.6 NF NF NF350 173 +9.1 +16.1 �10.1 NF NF NF410 138 +10.3 +1.6 +10.9 �26.7 NF NF490 78 +19.7 +16.2 �15.2 +15.7 NF NF585 24 +38.8 +34.3 +34.5 +22.6 NF NF650 7 +36.7 +46.8 +53.2 +52.6 +35.5 NF700 1 +63.1 +63.5 +61.8 +65.1 +50.8 NF745 0 +99.5 +99.5 +99.4 +99.5 +99.5 +99.5750 0 +99.5 +99.5 +99.4 +99.5 +99.5 +99.5

NF: not feasible.II: no. of paths whose lengths exceed the given deadline (remember that there are 256 paths in the network).


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f ¼ A� w

only one single execution mode is considered, namely their longest duration mode. Mode reductions may alsobe possible for those activities that lie in paths which jointly belong to P � x and x. Obviously, the modereductions for these activities should not affect the feasibility of the solution. The solution remains feasibleif by utilizing the remaining duration modes of activities, the given deadline can be met. However, if thereis more than one such activity for mode reductions, some considerations should be given to their relative costs.For this, we sort activities of each path in ascending order according to the value of s calculated as

s ¼ tijk � tijkþ1

cijkþ1 � cijk; ij 2 v; k ¼ 1; . . . ; jMijj � 1.

As such, in each path, activities on top of the list are better candidates costwise for mode reductions. We haveapplied this mechanism to our solution procedure. Table 5 presents the effect of using this mechanism on thenumber of iterations used to solve the given problem. It shows the percentage of increase (�) or decrease (+) inthe number of iterations used to solve the problem. As it can be seen, the use of the mode reduction mecha-nism can reduce the number of iterations up to 99.5%. However, some increase in the number of iterations canalso be observed. This may be attributed to the nature of the algorithm that the optimizer, namely LINGOuses to solve the problem.

4. Summary and conclusions

In this paper we developed three binary integer programming models to assist decisions regarding the trade-offs among cost, duration and quality of a project, where cost and quality are discrete non-increasing functionsof project duration. The inter-twined effect of time, cost and quality in DTCQTP was shown. In addition,some quantitative results were also presented. Different methods of quality aggregation methods were studied.A mechanism for activity mode reduction was developed, and its effect on the performance of the models wasinvestigated. The authors are presently working on a hybrid metaheuristic combining scatter search and elec-tromagnetism ideas to solve large scale DTCQTP.

References

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[5] E. Demeulemeester, W. Herroelen, S.E. Elmaghraby, Optimal procedures for the discrete time/cost trade-off problem in projectnetworks, European Journal of Operational Research 88 (1996) 50–68.

[6] C. Akkan, A Lagrangian heuristic for the discrete time–cost tradeoff problem for activity-on-arc project networks, Working Paper,Koc University, Istanbul, 1998.

[7] V.G. Deineko, G.J. Woeginger, Hardness of approximation of the discrete time–cost tradeoff problem, Operations Research Letters29 (2001) 207–210.

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