The discrete time-cost tradeoff problem revisited

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER

European Journal of Operational Research 81 (1995) 225-238

Invited Review

The discrete time-cost tradeoff problem revisited Prabuddha

D e *, E . J a m e s D u n n e , J a y B. G h o s h , C h a r l e s E . W e l l s

Department of MIS and Decision Sciences, University of Dayton, Dayton, OH 45469-2130, USA

Received June 1994

Abstract

In the management of a project, the project duration can often be compressed by accelerating some of its activities at an additional expense. This is the so-called time-cost tradeoff problem which has been studied extensively in the project management literature. However, the discrete version of the problem, encountered frequently in practice and also useful in modeling general time-cost relationships, has received only scant and sporadic attention. Prompted by the present emphasis on time-based competition and recent developments concerning problem complexity and solution, we reexamine this important problem in this paper. We begin by formally describing the problem and discussing the difficulties associated with its solution. We then provide an overview of the past solution approaches, identify their shortcomings, and present a new solution approach. Next, we present network decomposition/reduction as a convenient basis for solving the problem and analyzing its difficulty. Finally, we point to several new directions for future research, where we highlight the need for developing and evaluating effective procedures for solving the general time-cost tradeoff problem. To the best of our knowledge, the popular project management software packages do not include provisions for time-cost tradeoff analyses. Our work, we hope, will provide the groundwork and an incentive for alleviating this deficiency. Keywords: Project management; Networks; Mathematical programming

1. I n t r o d u c t i o n

Suppose that we are given a project network which represents a set of activities to be performed and their precedence relationships. An individual activity often may be performed in one of several ways, each with its unique time and cost requirements. Evidently, different decisions as to how the various activities are performed lead to different time-cost realizations for the overall network. The objective is to identify the

* Corresponding author.

sets of decisions that result in desirable time-cost realizations; this constitutes the so-called timecost t r a d e o f f p r o b l e m for a project network. The importance of the time-cost tradeoff problem was recognized over three decades ago, almost simultaneously with the development of project analysis techniques (Fulkerson, 1961; Kelley, 1961). Variations of the problem where the timecost relationships are assumed to be continuous have been addressed rather extensively in the literature (see standard texts such as Elmaghraby, 1977, and Moder, Phillips and Davis, 1983). In contrast, the literature for the case where the time-cost relationships are defined at discrete

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points (representing distinct alternatives) has been relatively sparse and somewhat disjoint. This is perhaps due to the inherent difficulty of solving the problem in this case. There are, however, at least two good reasons why the discrete version of the time-cost tradeoff problem should be considered important. First, discrete alternatives are quite common in practice (Harvey and Patterson, 1979; Hindelang and Muth, 1979). Second, discretization provides a convenient means for modeling any general time-cost relationship (Robinson, 1975; Panagiotakopoulos, 1977). Moreover, the current focus on time-based competition (Blackburn, 1991; Bockerstette and Shell, 1993) strongly suggests that interest in the problem be revived. There is evidence that this may already be happening, particularly with respect to some methodological aspects of the problem; see, for example, the following recent works: Bein, Kamburowski and Stallmann (1992), Elmaghraby (1993), Tufekci (1993), and Demeulemeester, Herroelen and Elmaghraby (1993). The case where activities have discrete alternatives has been subsumed in some of the early works on the general time-cost tradeoff problem. In Meyer and Shaffer (1965), a mixed integer programming formulation has been pursued for the problem. In Butcher (1967) and Robinson (1975), on the other hand, dynamic programming formulations have been provided but not developed to the point where they can be generally applied. At this time, none of these approaches appears to be practically viable from a computational standpoint. Panagiotakopoulos (1977) and Harvey and Patterson (1979) have explicitly considered the discrete time-cost tradeoff problem, and have provided solution approaches which appear to be more promising. The discrete problem has, in part, motivated the development of the Decision-CPM framework for analyzing project networks. In this context, the problem has first been solved as a general mixed-integer program in Crowston and Thompson (1967), which suffers from computational limitations similar to those of the Meyer and Shaffer approach. The specialized branchand-bound approach of Crowston (1970) is somewhat more promising. Encouraging results have

also been reported for the dynamic programming approach of Hindelang and Muth (1979). Subsequently, however, this latter approach has been shown to be flawed; a correction has been provided as well (De, Dunne, Ghosh and Wells, 1992). More recently, Tavares (1990) has looked at the problem under a very special restriction. De et al. have shown that the problem is very difficult in gerneral and have used network decomposition (Schwarze, 1980; Buer and M6hring, 1983; Muller and Spinrad, 1989) to establish precise complexities of various network structures. Tufekci (1993) has discussed the complexity of the problem for different time-cost relationships. Finnally, Elmagraby (1993) and Demeulemeester et al. (1993) have utilized the network reduction methodology of Bein et al. (1992) to propose hybrid algorithms (involving branch and bound and dynamic programming) for solving the general problem. The work of Demeulemeester et al. also includes a novel reduction-based approach and provides computational results. When applied to a general project network, the computational complexity of all of the exact solution approaches mentioned above would be exponential in the worst-case (i.e., the solution time would grow as an exponential function of the problem size). It has recently been shown (De et al., 1992) that any exact solution algorithm for the discrete time-cost tradeoff problem would very likely exhibit an exponential worst-case complexity. It follows that the search for exact algorithms which are also formally efficient is all but futile and that one should instead search for effective procedures (such as exact procedures which are good for special cases or heuristics which are good in general). Identification of such procedures would make it viable to include a time-cost analysis feature in standard project management software packages; such a feature is currently absent in these packages (Badiru, 1991). To facilitate the search for effective solution procedures, it is clear that a new and comprehensive overview of the discrete time-cost tradeoff problem is warranted. In this paper, we attempt to provide such an overview. We start out by formally describing the problem (using a bicrite-

P. De et al. / European Journal of Operational Research 81 (1995) 225-238

ria optimization perspective) and b y discussing the issues related to its complexity. We then summarize past research and provide an overview of the basic solution approaches available for the problem; we also propose a new solution approach. Subsequently, we discuss the use of network decomposition/reduction as a basis for solving the problem and analyzing its difficulty (decomposition of a network also yields several other advantages; see Parikh and Jewell, 1965, for example). The various perspectives on solving the problem discussed by us give rise to a number of interesting issues for future research, which we outline in the concluding section of the paper.

2. P r o b l e m a n d c o m p l e x i t y

A project network is essentially an acyclic directed graph. We adopt an activity-on-node representation for the graph, where there are n nodes numbered 1 through n (corresponding to n activities) such that each node has a lower number than all its successor nodes. Assume that activity i, i = 1. . . . . n, has a(i) alternatives of which alternative j, j = 1 , . . . , a ( i ) requires tii time and cij cost; further assume, without any loss of generality, that if k and r are two alternatives for activity i such that k < r, then tik < tir and cik > Cir. We introduce two dummy activities 0 and n + 1 for the start and finish nodes, respectively, and assume that the time and cost requirements for them are both zero. Clearly, the selection of a particular alternative for each activity results in a specific realization of the network. Let o--- {(i, j ) , i = 1 , . . . , n } be such a realization where alternative j is selected for activity i, and t(o-) and c(o-) be respectively the overall time and cost requirements for this realization. Note that t(-) is simply the sum of the times for the activities on the critical path from node 0 to node n + 1, and c(-) is the sum of the costs for all the n activities. Let O be the set of all l ~ l < i < n a(i) possible realizations. The objective often is to identify a realization from 0 which is optimal with respect

227

to either the time or the cost criterion. The following problems then become relevant: (P_T) Find o-t such that t ( o "t) = min {t(o-)). o-EO

(P_C) Find o-c such that c ( o -c) = rain {c(~r)}. o'EO

Both P T and P C are easily solved: to solve P _ T , select the least time alternative for each activity, and to solve P_C, select the least cost alternative. More realistically, however, it is necessary to consider time and cost simultaneously. One way to accomplish this is by identifying a realization which minimizes the time (cost) subject to a given budget b (due-date d). The resulting conditional time and cost minimization problems can be stated as follows: ( P _ T ]C) Find o-tic such that

t(o -tLc) =

rain {t(o-): c(o-) < b } .

( P _ C IT) Find o-clt such that c ( o -cl') = min {c(o-): t(o-) _
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is of a low order, the algorithm is considered

practically effective. A strongly NP-hard problem does not, in general, admit pseudo-polynomial algorithms. When the objective is to identify the entire time-cost tradeoff curve for the project network, one focuses on the parametric solution of either P _ T IC or P_ C IT. This is equivalent to the identification of a set of nondominated realizations. A realization cr is called nondominated if there does not exist another realization o- such that (a) t(o-') < t(o-) and c(o-') < c(~r), and (b) one of the above inequalities holds strictly. It is sufficient to identify a completely representative set 12 of nondominated realizations, which retains exactly one from among all realizations with the same t(-) and c(.). The time-cost tradeoff problem from a bicriteria viewpoint can thus be stated as follows: ( P _ T C ) Find 12. To properly ascertain the complexity of P _ T C , one needs to address two issues: the difficulty of

obtaining a specific nondominated realization, and the cardinality of 12. It should be evident that the problem of finding a specific nondominated realization is as difficult as solving either P _ T I C or P _ C I T , the complexity of which we have just discussed. As for the cardinality of 12, we now show that this can be exponentially large. Consider a series network where the set of time-cost alternatives for activity i, i = 1 . . . . , n, is given by { ( 2 2 n - i + 1 , 2 i ) , ( 2 i, 2 2 n - i + 1 ) } .

Note that t(tr) and c(o') for anyo- ~ O are unique by construction. (One easy way to see this is to look at the binary representations of t(o-) and c(tr) for all tr ~ 0 3 Further, note that t(o') + c((r) = 2 2n+1- 1 for any cr ~ O. Thus, given ~r and ~r' in O such that t(cr) < t(o"), we necessarily have c(o-) > c(o-'). This implies that any o-~ O is nondominated. The cardinality of 12 is therefore the same as that of O, which is 2 n.

Table 1 Summary of past research Problem context

Base formulation

Study

Problem focus

Solution method

Parametric ability

Remarks

General TimeCost Tradeoff

Mixed Integer Linear Programming (MILP)

MeyerShaffer HarveyPatterson

P C IT

General MILP algorithm Special 0-1 ILP algorithm

Multiplepass Multiplepass

Dynamic Programming (DP)

Butcher

P_T IC

Single-pass

Robinson

P_ ]C

De et al.

P_ C IT

Panagiotakopoulus CrowstonThompson Crowston

P_ C IT

HindelangMuth

P_ C IT

Incomplete detials Incomplete details Special DP algorithm Simplification; enumeration General MILP algorithm Simplification; general M I L P / branch & bound algorithm Special DP algorithm

May not be practical for solving the problem Appears promising based on reported results; needs further study Restricted to highly special networks; not implemented Good conceptual approach; implementation unclear Correction of HindelangMuth; not implemented Promising computational results; needs further study May not be practical for solving the problem Specific to Decision-CPM; simplification strategies need further study

Other DecisionCPM

Mixed Integer Linear Programming (MILP)

Dynamic Programming (DP)

P_CIT

P C IT P_ C IT

Single-pass Single-pass Multiplepass Multiplepass Multiplepass

Single-pass

First complete DP approach; insightful but flawed

P. De et aL / European Journal of Operational Research 81 (1995) 225-238

Thus, the solution to P TC may not in general be found in polynomial time. It can, however, be found in polynomial time for parallel networks and in pseudo-polynomial time for series and series-parallel networks (De et al. 1992). The solution can also be found in polynomial/ pseudo-polynomial time for special time-cost relationships (Tufekci 1993).

229

cost tradeoff problem and has been adopted by a number of researchers. Let xij be a 0-1 variable which is 1 if alternative j is selected for executing activity i and 0 otherwise; also, let s i (s i > 0) be the start time for activity i and S(i) be the set of immediate successors of i. The formulation for P _ C IT can be described as follows:

~

minimize

~

cijxij

(1)

l<_i
3. Overview of basic solution approaches

In this section, we provide an overview of the basic solution approaches for the discrete timecost tradeoff problem. We begin by summarizing the major features of the relevant work done in the past. We then briefly discuss the available approaches. We finally introduce a new dynamic programming formulation which has certain computational advantages. (Note that solution approaches based on network decomposition/reduction are discussed in Section 4).

3.1. Summary of past research Table 1 provides a quick look at the salient features of the various past studies, conducted in the context of either the general time-cost tradeoff problem or the Decision-CPM problem. For each study, the table shows what base formulation has been adopted (mixed integer linear programming, dynamic programming or other), which specific problem has been studied ( P _ T I C or P _ C IT), whether problem simplification techniques (such as the reduction of the set of alternatives for executing an activity) have been employed, which solution method (general purpose integer programming, specialized branch and bound, or specialized dynamic programming algorithm) has been used, and whether this method provides a solution to P _ T C in a single pass or multiple passes. The table also contains our remarks, primarily concerning the applicability of each of these approaches in practice.

subject to

~_, x u = 1 for all i = 1 , . . . , n ,

(2)

1 <_j < a ( i )

£ tijxij +S i~S k 1 <_j< a(i)

for all k ~ S( i)

and all i = 1 , . . . , n , Sn+ 1 ~

d.

(3) (4)

In the above formulation: (1) reflects the cost minimization objective; (2) ensures that exactly one alternative is chosen for each activity; (3) maintains the precedence relationships among the activities; and (4) guarantees that the project will complete by its due-date. The basic formulations used in Meyer and Shaffer (1965), Crowston and Thompson (1967), Crowston (1970), and Harvey and Patterson (1979) are quite similar to the one given above. Crowston, however, adopts a different formulation for the precedence constraints, which he exploits for problem simplification. Also, Harvey and Patterson eliminate si, i = 1 , . . . , n, from their formulation, and obtain a pure 0-1 integer program. To obtain their program: restrict s i to be an integer, define _si and gi to be respectively the early and late start times (relative to d) for activity i, let Yis be a 0-1 variable which is 1 if i starts at s and 0 otherwise, substitute Essyis for si, and add the constraints

~-,Yis=l

for a l l i = l , . . . , n

S

(note that the sums are computed over s

3.2. Outline of solution approaches

[_si, ~/]).

Mixed integer linear programming provides a convenient basis for modeling the discrete time-

Dynamic programming formulations provide another modeling approach. Butcher (1967) contains such formulations for P _ T I C when the

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project networks are pure series and pure parallel. Let z i be the budget allocated to activity i, hi(z i) be the least time in which i can be executed with zi, and & ( z ) be the least time to execute activities 1 through i with budget z. The recursive equations (where the minima are taken over z i ~ [0, z]) for the series and parallel cases are respectively

g i ( z ) = min{g i_ 1( z - zi) + hi(Zi)

}

for z = 0 , . . . , b ,

(5)

and

gi( z ) = min{max{g i_ 1( z - zi), h i ( z i ) }} for z = 0 , . . . , b .

(6)

Robinson (1975) presents a conceptual dynamic programming framework for solving P _ T [C in general project networks, but fails to provide any algorithm for implementing it. (We will see in Section 4 how a framework such as Robinson's can be implemented in practice.) Let q be the number of complete paths through the project network, L ( p ) be the set of nodes on path p, and g ( z ) ( = gn(z)) be the least time in which the project can be executed with budget z. We can then write g(z),taking the minimum over all {zi: 1 <_i <_n} such that ]~_,l<_i<_nZi=Z,as follows:

g ( z ) = min / max for all z = 0 . . . . . b.

(7)

Robinson provides a sufficient condition under which the problem in (7) will recursively decompose into single-dimensional problems as in (5) and (6). H e also recognizes the multi-dimensionality problem which arises when (7) does not decompose, and suggests a way (once again, only conceptually) as to how this might be handled. In Hindelang and Muth (1979), one finds for the first time a complete dynamic programming formulation for the discrete time-cost tradeoff problem; as already stated, their focus is on P _ C IT in the Decision-CPM context. Let ei(s) be the minimum cost of realizing node i and all of its successors such that node i starts at time s

and that all nodes complete by time d. The basic recursion is given by

ei(s)=

min {[ ~ ek(s+tij)]+Ciy } l <_j<_a(i) [ k ~S(i)

for s i < s < si.

(8)

Note that care needs to be taken while performing the recursion in (8) if an immediate successor k of i is a merge node (i.e., [ P ( k ) [ > 2, where P ( k ) is the set of immediate predecessors of node k). The true value of ek(') is used in the computation only if node i has the highest number among the immediate predecessors of k; ek(') is taken to be 0 otherwise. Recently, it is shown in De et al. (1992) that the H i n d e l a n g - M u t h approach is flawed because it allows two immediate predecessors of a merge node to assume different start times at that node. They correct it by freezing the start time of each nontrivial merge node (one that has multiple feasible start times) in the network to exactly one value in a given pass and by using multiple passes for all combinations of the feasible start times for such nodes. This leads to the multi-dimensionality problem that we have already discussed in relation to Robinson's work. (Recall that this difficulty is inevitable due to the complexity of the problem as discussed in Section 2.) De et al., however, show that effective solution is possible for a large class of networks using modular decomposition (which we discuss in Section 4). In an approach that is different from the others, Panagiotakopoulos (1977) focuses primarily on problem simplification. Given a due-date, this approach makes repeated passes through the network, updating estimates of the activity start and finish times, until the number of feasible alternatives for the activities cannot be reduced any further; an exhaustive enumeration is applied at this stage to solve the problem. Note that all of the solution approaches, which are both complete and correct, exhibit exponential worst-case complexities. For special network structures, however, some of the dynamic programs exhibit pseudo-polynomial complexities. For solving P _ T [C, Butcher's approach can easily be implemented to run in O(ndb 2) time for

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pure series and pure parallel networks, where Yt = maxl <_i<_n{a(i)}. The corrected HindelangMuth approach (De et al., 1992) can solve P_ C IT in O(n~d) time for pure series and pure parallel networks; its complexity in the general case is O(n2~dU), where u is the number of nontrivial merge nodes in the network.

{<2,40>,<4,20>} {<1,12>,<3,6>}

Start ~ /

, ~

Finish

3.3. A new dynamic program The corrected H i n d e l a n g - M u t h approach can be thought of as a decentralized approach (and will be referred to as such), since the cumulative cost information is distributed over a number of nodes in the network during its execution. We now present what we will be calling a centralized approach. While we describe the approach using an enumerative form (for the ease of exposition), it is basically a forward dynamic program. Let a (k + 1)-tuple q5k (which we will write as a list of k + 1 elements enclosed within ( and )) represent the state of the dynamic program when node k is being considered (call it stage k): for each node i, i < k, that has an immediate successor r such that r > k, let the finish time fi of i be recorded in the i-th element in q~z,; similarly, let the finish time fk of k be recorded in the k-th element and the minimum cost e k of executing activities 1 through k, given the first k elements of q~, be recorded in the (k + D-st element; all other elements of q5k are unspecified (and represented by -). Let g2k represent the set of all tuples @k at stage k. Initialize the dynamic program with

a 0 = {(0, 0)}, and for stages k = 1 , . . . , n + 1, do as follows: for each tuple q~k-1 E Ok_l, create re(k) new tuples such that the first k - 1 elements of the j-th new tuple q~k are the same as those of ¢~_1 and the k-th and (k + 1)-st elements are computed as

fk=

max {fi} +tkj

i~P(k)

and ek = e k - 1 -}- Ckj

respectively; for each newly created tuple @k, retain each element i, i = 1 . . . . . k - 1 , that is

j {<3,10>,<4,i>} {<2,6>,<3,3>} a-teTh=a~itt)2ei:~iStty v:fs 2 ~TC:2; Fig.1.Simpleexamplenetwork. necessary for the state description at stage k; once all new tuples are created at stage k, iteratively eliminate any tuple qs~ for which there is another tuple q~k such that f i < f " for all i, i = 1. . . . , k, at which the i-th element is specified, and e k _< e 'k (break ties arbitrarily). The set of tuples $2, + 1 retained at the end of stage n + 1 delivers g2, the solution to P _ T C . To illustrate this approach, we refer to the example network in Fig. 1. The set of tuples I2 k that has survived at the end of stage k in the example is shown in Table 2 for all k, k = 0 . . . . . n + 1. Consider the computations at stage 3 (k = 3). First, note that a tuple at stage 2 is given by ( -, fl, f2, e2) whereas one at stage 3 is given by ( -, -, fz, f3, e3)- Take the tuple ( -, 2, 3, 50) from stage 2 (see Table 2). At stage 3, from this t u n e , we will first create the tuples ( . , 2, 3, max{2, 3} + 1, 50 + i2)

= ( - , 2, 3, 4, 62) and ( . , 2, 3, max{2, 3} + 3, 50 + 6} = ( . , 2, 3, 6, 56) corresponding to the set of time-cost alternatives {(1, 12>, (3, 6)} for activity 3. Since f l is no longer needed for the state description at stage 3, we will rewrite the newly created tuples as ( . , • , 3, 4, 62) and ( - , -, 3, 6, 56). Similarly, from ( •, 4, 3, 30} at stage 2, we will later create the

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Table 2 Computations for example problem Stage Tuple representation Completelyrepresentative set of nondominated tuples 0 1

(f0, e0) (f0, f i e t )

2

(',fl, f2, ea)

3

( ' , ' , f 2 , f3, e3)

4

(', ", ",f>f4, e4)

5

( ' , ' , ' , ' , ' , f s, e5)

(0,0) (0, 2, 40) (0,4,20) (',2,3,50) (',2,4,46) (.,4,3,30) (.,4,4, 26) (-,',3,4,62) (-,',3,5,42) (',',3,7,36) (.,.,4,5,38) (',4,7,32) (.,.,-,4,5,68) (.,.,.,5,5,48) (.,.,.,7,5,42) (.,.,.,5,6,44) (.,.,.,7,6,38) (.,.,-,4,6,65) (.,.,-,5,7,41) (.,.,-,7,7,35) (.,.,-,-,-,5,48) (., . , . , . , . , 6 , 4 4 ) ( - , . , - , - , - , 7,35)

tuples ( . , . , 3 , 5 , 4 2 ) and ( - , . , 3 , 7 , 3 6 ) at stage 3. Notice that ( •, -, 3, 5, 42) thus created forces the elimination of ( . , . , 3, 6, 56) created earlier. Now, let v be the maximum over all stages k, k = 0 . . . . . n + 1, of the number of specified elements fi, i = 1 , . . . , k, in a (k + 1)-tuple at stage k. Also, let d = t(~rc). (Recall that o-c is the optimal realization for P_ C.) The complexities of the centralized approach for solving P _ C I T and P _ T C in general networks are seen to be O(nZad v) and O(ne-dd~), respectively. It is also evident that this approach performs poorly in pure parallel networks; this difficulty can, however, be easily remedied through the use of modular decomposition (see the next section). Note that the centralized approach has a computational advantage over the decentralized approach in certain network structures (particularly those that contain stages). Suppose that we are trying to solve P _ C I T for the complex network shown in Fig. 2. Here, v (which determines the compu-

Fig. 2. A complexnetwork.

tational viability of the centralized approach) is only 4, whereas u (which does the same for the decentralized approach) is 6. Also, note that both the centralized and decentralized approaches can be implemented either as a forward or a backward recursion and that the appropriate recursion can always be chosen a priori since it is readily estimated which is going to be advantageous in a given instance.

4. Solution through network d e c o m p o s i t i o n / reduction W e now discuss modular decomposition (Schwarze, 1980; Buer and M6hring, 1983; Muller and Spinrad, 1989) and its implications for the solution of the discrete time-cost tradeoff problem; we will also discuss briefly solution methodologies based on incremental reduction of networks (Bein et al., 1992). First, we describe what modular decomposition is and illustrate how it fits into the present context. We then discuss some of the advantages of using modular decomposition. We follow this by showing how a solution can be obtained through appropriate network modification and decomposition. Finally, we touch upon the solution approaches based on incremental network reduction. 4.1. Fundamentals o f modular decomposition Let N be the set of all nodes in a project network. Define Pie(i) and S'x(i) to be respectively the sets of all predecessors and all successors (not just the immediate ones) of node i that are also in some set X. We call M, a subset of N,

P. De et al. / European Journal of Operational Research 81 (1995) 225-238 Module I

@ © Module 2

Fig. 3. (a) A decomposable network.

Level [ 0,1,2,3,4,5,6,7,8,9,10 I S 0

1

I 1,2,3,4,5,6,7,8,9 I P

10

1

2

9 2

3

6

7

Reduced Graph

4 5

Fig. 3. (b) Composition tree for network in (a). Fig. 3. (c) Solution for Level 4. Module 3 Module 5

Module 4

Reduced Graph

Fig. 3. (e) Solution for Level 2.

_ReducedGraph Module 6

Reduced Graph

Fig. 3. (d) Solution for Level 3.

Fig. 3. (f) Solution for Level 1.

233

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a module of N if and only if for all i,k ~ M, we have

P~V\M( i) =-P~v\M( k ) and

S'N\ M( i ) =--S'u\ M( k ) . Refer to the network in Fig. 3a; here, the node sets {1, 2, 3, 4} and {5, 6, 7, 8, 9} serve as two examples of a module. Typically, a module can be partitioned into a number of submodules (which are smaller modules and can even be single nodes). Three types of modules are defined: a module is series (S) if it can be partitioned into submodules, all of which lie on the same unique path through the module; it is parallel (P) if it can be partitioned into a set of submodules, no two of which lie on the same path through the module; it is neighborhood (N) if it is neither series nor parallel. In Fig. 3a, {6, 7}, {2, 3} and {5, 6, 8, 9} serve as examples of series, parallel and neighborhood modules, respectively. For a more detailed and rigorous account of modular decomposition in an applied context, refer to Sidney and Steiner (1986). Recently, a number of sophisticated algorithms (see, for example, Schwarze 1980; Buer and M6hring, 1983; Muller and Spinrad 1989) have been proposed, which can identify all modules in a directed acyclic graph. Some of these algorithms (e.g., Buer and M6hring, 1983, and Muller and Spinrad, 1989) decompose the graph hierarchically into smaller and smaller modules (noting their types - S, P, or N - along the way) until the modules reduce to single nodes (or trivial modules), and produce a composition tree recording the results of the decomposition process. Fig. 3b shows the composition tree for the network in Fig. 3a. In the context of the time-cost tradeoff problem, it should be clear that a module is independent of the rest of the project network. Thus, in a network, a module can be conveniently replaced by a super-activity whose set of time-cost alternatives is given by the solution to P _ T C for the module. To solve P TC for the entire network, this process can be carried out for all modules in the associated composition tree by working in the

reverse hierarchical (bottom up) order. Figs. 3c through 3f illustrate the process for the network in Fig. 3a. At the lowest level (level 4) of the composition tree in Fig. 3b where we encounter a nontrivial module for the first time, nodes 2 and 3 form a parallel module and nodes 6 and 7 form a series module. Thus, we solve P TC independently for these two modules, obtaining the super-activities 21 and 61; these are in turn substituted in the original network to obtain the reduced network of Fig. 3c. We continue this process of solving P _ T C independently for the modules and reducing the network, until we reach level 1 in the composition tree and P _ T C is solved for the entire network (see Fig. 3d through 3f).

4.2. Advantages of modular decomposition Since the overhead due to modular decomposition is rather small (it can be achieved in O(n 2) time; see Muller and Spinrad, 1989), its application, in general, presents an obvious computational advantage for decomposable networks (where we solve a number of smaller problems). It helps us to analyze the difficulty of solving P _ T C for certain network structures and to guarantee superior computational performance for solution approaches based on dynamic programming. Decomposition also makes it easier to maintain and update a network, and facilitates understanding of the interactions that exist among the executions of the various activities. A composition tree is known to have O(n) modules (Buer and M6hring, 1983); thus, P _ T C for an entire network can be solved by solving it for the O(n) constituent modules. From the last section, we know that, for series and parallel modules, P _ T C can be solved in low-order pseudo-polynomial time (i.e., in a practically effective manner) by some of the dynamic programming approaches; for exam_pie, the decentralized approach takes only O(n~d) time for such modules. Thus, through modular decomposition, it becomes possible to effectively solve P _ T C for all series-parallel networks, viz., in O(n2ad) time with the decentralized approach. (Note here that the O(neffd) performance of the decentralized

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approach in the series-parallel case cannot be guaranteed without modular decomposition). For other decomposable networks (those that contain neighborhood modules), the application of modular decomposition may mitigate the multi-dimensionality problem suffered by the dynamic programming approaches. For example, u and v (see Section 3) of the decentralized and centralized approaches, respectively, now depend not on the entire network but rather on the neighborhood modules alone. Recall further that the centralized approach performs poorly in parallel modules. However, it follows from De et al. (1992) that P_TC for a parallel module can be solved in polynomial time; this can actually be done in O(n~) time. Thus, a modular decomposition framework could employ a polynomial-time approach for the parallel modules and the centralized approach for the others in order to obtain an effective overall approach for solving P TC. 4.3. Solution through series-parallel conversion

Clearly, P_TC can be effectively solved for series-parallel networks; this may not, however, be the case for networks containing the neighborhood module. Two approaches can be adopted in the latter case: in one, P_TC is solved directly through the application of general algorithms of the type discussed in Section 3; in the other, the network is first converted into an equivalent series-parallel network and P_TC is solved repeatedly on this network for different parameter values. The second approach, which we discuss now, can be thought of as one that strengthens Robinson's conceptual framework and provides it with an implementable basis. The implementation we propose performs the series-parallel conversion first. Alternative schemes where the conversion takes place incrementally are discussed in Section 4.4. The series-parallel decomposition of a neighborhood module is usually prevented by the presence of a few complicating nodes. If such a node can be executed using only one time-cost alternative, its presence does not pose any difficulty in terms of solving P_TC. We can simply remove

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the node from the original network and create a new network by placing its copies (repeating the time always but using the cost only once) on every path that exists between its immediate predecessors and immediate successors; the solution to P TC for the new network will be the same as that for the original one. If each complicating node has only one time-cost alternative and the above strategy is applied to all such nodes (creating node copies incrementally, starting with the lowest-numbered complicating node, and doing this repeatedly until each copy has exactly one incoming and one outgoing arc), the resulting network becomes series-parallel decomposable and thus can be solved effectively. Even when the complicating nodes have multiple time-cost alternatives, we can follow the same strategy by solving P_TC multiple times, once for each possible way of executing the complicating nodes. If w is the number of complicating nodes with multiple alternatives, then we will be required t o solve P_TC ~w times for the series-parallel network. Given the complexity of the dynamic programming approaches (when used along with modular decomposition) for solving P TC in series-parallel networks, it is easily seen that the approach outlined above will run in O(~2~ w+ ld) time, where is the number of nodes in the new network. Notice that pure series-parallel decomposition can be obtained in O(n) time (Valdes, Tarjan and Lawler, 1982). Consider once again the simple network in Fig. 1, where the complicating node is either node 2 or node 3. Taking 2 to be the complicating node, we replace it with two copies: 21 between 0 and 3; and 22 between 0 and 4. The resulting network, shown in Fig. 4a, is series-parallel decomposable; its composition tree is shown in Fig. 4b. This network needs to be solved twice, once with (3, 10) (and (3, 0)) and then with (4, 6) (and (4, 0)) as the time-cost alternative for nodes 21 and 22; combining the two solutions and retaining the nondominated time-cost realizations will solve P_TC for the original network. It is not difficult to identify a minimal set of complicating nodes, the removal of which is sufficient to produce a series-parallel network under our strategy. One possibility is to apply a proce-

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P. De et al. / European Journal of Operational Research 81 (1995) 225-238

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I 21 Fig. 4. (a) Series-parallel equivalent of example network. (b) Composition tree for network in (a). dure suggested by Robinson: find all pairs of nodes i and k such that there is a precedence relationship between them and that there are paths through the network on which i appears but k does not and vice versa; then find the minimal set of nodes, through matching techniques (Hopcroft and Karp, 1973) that cover all the pairs found above. This procedure can be executed in O(n 2"5) time. Note that a straightforward application of our construction strategy may lead to an exponential number of new nodes (this happens when a number of complicating nodes with multiple incoming and outgoing arcs appear contiguously in the network). To generate an equivalent series-parallel network with a polynomial number of nodes, a different strategy needs to be followed: for any two noncomplicating nodes i and k that have a precedence relationship, find if there is a subnetwork connecting i and k, which consists exclusively of complicating nodes; if so, place a superactivity node between i and k, and note that this node represents the subnetwork identified above

and that its time-cost pair in a given instance (i.e., for a specific combination of single time-cost alternatives assigned to the complicating nodes) is determined from the solution to P _ T (recall that P _ T has been introduced in Section 2) for this subnetwork. This procedure can easily be implemented to run in O(n 4) time. As an example, consider the complex network of Fig. 2, where one possible collection of the complicating nodes is {1,2,3,4,5,6}. In this case, our original strategy eventually leads to a network in which all six complicating nodes have 4 copies each. With the new strategy, however, we will replace the nodes in {1, 2, 3, 4, 5, 6} with only 3 super-activity nodes, one each between 0 and 7, 0 and 8, and 0 and 9; these new nodes will correspond to the subnetworks induced by {1, 2, 3, 4, 5}, {1, 2, 3, 5, 6}, and {1, 2, 3, 4, 6}, respectively.

4.4. Incremental reduction approaches The work of Bein et al. (1992) on optimal reduction of two-terminal directed acyclic networks provides an alternate way to implement Roinson's dynamic programming framework. Here, the project network is required to be in the activity-on-arrow representation. In this approach, a reduction plan is determined so as to minimize the number of reductions that are neither series nor parallel. This plan is implemented incrementally requiring the occasional fixing of an activity. Recently, Elmaghraby (1993) and Demeulemeester et el. (1993) have successfully used this approach. Demeulemeester et el. have also used a second reduction scheme and performed several computational experiments. For more details on this approach, see the cited papers.

5. Summary and conclusions The discrete time-cost tradeoff problem is clearly an important problem in project management, particularly in view of the current emphasis on time-based competition (Blackburn, 1991; Bockerstette and Shell, 1993). In this paper, we have revisited this problem and examined it in a

P. De et al. / European Journal of Operational Research 81 (1995) 225-238

comprehensive way. We have described different versions of the problem and discussed their complexities. We have reviewed the past solution approaches from the literature (identifying their limitations) and have presented a new approach. We have also advocated the use of modular decomposition as a convenient device for solving the problem and studying its complexity. In addition, we have touched upon solution methodologies that are based on incremental network reduction. A number of issues, however, remain open for future research. We have seen that solving the discrete timecost tradeoff problem is extremely difficult in general, but that it can be solved quite effectively for series, parallel, and series-parallel networks. For complex networks (i.e., those containing neighborhood modules), the solution approaches discussed earlier (viz., the decentralized and centralized dynamic programming approaches of Section 3 and the series-parallel conversion approach of Section 4.3) can be effective only as long as certain network parameters (viz., u, v, and w) have reasonably low values. Further research is needed to identify solution approaches which would be computationally viable over a wider class of networks. In this regard, research leading to a more complete understanding (beyond what has :been accomplished in Bein et al. 1992) of the relationship between various network structures and solution difficulty would be of significant value. In order to obtain an exact solution to the general problem, the use of modular decomposition/reduction becomes almost imperative. Further, in view of the inherent difficulty of solving the problem, other strategies to facilitate solution may need to be employed. A study of problem simplification techniques such as those explored in Crowston (1970), Robinson (1975), and Panagiotakopoulos (1977) may be worthwhile in this regard. Similarly, search for effective branch and bound approaches may prove beneficial. Such approaches have already been used successfully in conjunction with dynamic programming (Elmaghraby, 1993; Demeulemeester et al., 1993). Also, from a practical standpoint, research should perhaps focus more heavily on the devel-

237

opment of good heuristics (ones that will generally produce close-to-optimal solutions in reasonable time). Elmaghraby (1993) provides a lead in this direction. Another possibility emerges from our discussion of complicating nodes in Section 4.3; removing certain arcs associated with these nodes will render the resulting network seriesparallel and thus effectively solvable. This idea can be used in a Lagrangian relaxation framework (Fisher, 1985) to generate approximate solutions to P _ T IC or P C IT; a nice feature of this approach is that it delivers an a posteriori guarantee on the solution quality. Yet another possibility is to use generic heuristics such as those based on Artificial Intelligence concepts. Such heuristics, e.g., beam search and genetic search algorithms, have recently been used for solving bicriteria problems (De, Ghosh and Wells, 1992; Wu, Storer and Chang, 1993), and thus may as well be adapted to solve P_T C directly. Only limited computational results are available for the existing solution approaches. Additional computational studies, particularly with respect to the promising approaches, are thus called for. Furthermore, the performance of modular decomposition and problem simplification in various practical situations requires empirical evaluation. At the present time, none of the popular project management software packages includes provisions for time-cost tradeoff analyses (Badiru, 1991). The directions for future research suggested here should provide a basis for remedying this deficiency. /

References

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