a metaheuristic scheduling procedure for resource-constrained projects with cash flows

A Metaheuristic Scheduling Procedure forResource-Constrained Projects with Cash Flows

Dan Zhu,1 Rema Padman2

1College of Business Administration, Iowa State University, Ames, Iowa 50011

2The H. John Heinz III School of Public Policy and Management, Carnegie MellonUniversity, Pittsburgh, Pennsylvania 15213

Received March 1999; accepted 1 June 1999

Abstract: Resource-constrained project scheduling problems with cash flows (RCPSPCF) arecomplex, combinatorial optimization problems. Many heuristics have been reported in the liter-ature that produce reasonable schedules in limited project environments. However, the lack ofa heuristic that dominates under differing project conditions can lead to a suboptimal choice ofan appropriate heuristic for scheduling any given project. This may result in poor schedules andmonetary losses. This paper reports on the application of the tabu search metaheuristic procedurefor the RCPSPCF. Strategies for neighborhood generation and candidate selection that exploit thespecial features of the problem are combined with a simple multiheuristic start procedure. Ex-tensive experimentation, with multiple data sets and comparison with an upper bound, indicatesa significant improvement, both in project Net Present Value (NPV) as well as the number ofprojects, where the metaheuristic outperforms the best known heuristics in the literature. Morespecifically, this procedure produces the best schedules in over 85% of the projects tested, incontrast to the best single-pass heuristics which have been shown to dominate in at most 20% ofthe same cases. This iterative, general purpose heuristic is able to adapt significantly better to thecomplex interactions of the many critical parameters of the RCPSPCF than single-pass heuristicsthat use more specific information about each project environment. c© 1999 John Wiley & Sons, Inc.Naval Research Logistics 46: 912–927, 1999

Keywords: tabu search; constrained project scheduling; heuristics

1. INTRODUCTION

The problem of developing schedules for projects with resource constraints and cash flowsarises in many organizational settings such as construction planning and research and develop-ment. The resource constrained project scheduling problem is concerned with the scheduling ofa collection of precedence related activities subject to constraints on available resources. Giventhe significance of cash flows, which occur in the form of expenses for initiating activities and

Correspondence to: R. PadmanContract grant sponsor: National Science Foundation; Contract grant number: IRI-9634383.Contract grant sponsor: Central Investment Fund for Research Enhancement.Contract grant sponsor: Faculty Development Fund at Carnegie Mellon University.Contract grant sponsor: U.S. Army; Contract grant number: DAAH04-94-G-0239

c© 1999 John Wiley & Sons, Inc. CCC 0894-069X/99/080912-16

Zhu and Padman: Metaheuristic Scheduling 913

payments for completed work, the development of project schedules which maximize the NetPresent Value (NPV) of the cash flows is of considerable practical importance [1]. This is a diffi-cult combinatorial optimization problem which precludes the development of optimal schedulesfor projects of practical size [5].

Due to the intractable nature of this problem, many heuristic procedures have been developedto obtain good solutions [2, 3, 10, 12, 13, 15]. However, no one heuristic dominates in allproject environments. Extensive experimentation indicates that the best performing heuristicsdominate in at most 20% of the problems tested [12, 13]. The NPV criterion emphasizes thefinancial returns of a project in terms of the net dollar value. For large projects with billionsof dollars of investment, even small improvements in the schedule may result in a significantincrease in NPV. Conversely, the choice of an inappropriate heuristic could result in poor schedulesand monetary losses [18]. The search for a dominating heuristic is thus of considerable prac-tical value.

These earlier approaches have also mostly focused on the development and application of asingle heuristic that generates the project schedule through a single-pass, forward-sequencingapproach where activities are prioritized for the allocation of resources during the schedule gen-eration process [2, 12, 13, 15]. However, the complex interactions of the many critical parametersof the resource-constrained project scheduling problem with cash flows (RCPSPCF) argues formultipass procedures, such as the tabu search approach.

This paper addresses the design, implementation and testing of a tabu search procedure forthe RCPSPCF in which multiple general heuristics, acting as agents, are embedded as movegeneration strategies. They build on an initial schedule, also generated by a combination ofmany simple heuristics [16, 19]. Tabu search is a metaheuristic that helps other search proce-dures escape from local optima through the flexible use of adaptive memory and imposition oftabu restrictions. These restrictions constrain the solution search process by consulting the his-tory of past solutions [6–8]. Tabu search has been successfully applied to a variety of domainsfor solving difficult combinatorial optimization problems, as well documented in [8], amongothers.

The next section describes the RCPSPCF and presents an example used to illustrate the problem.Section 3 discusses the procedure and the major elements of the design. Section 4 details theexperimental data and testing, followed by a discussion of the results. Section 5 presents detailedinsights while Section 6 summarizes the conclusions of the study.

2. THE RESOURCE-CONSTRAINED PROJECT SCHEDULINGPROBLEM WITH CASH FLOWS

The resource constrained project scheduling problem with cash flows addresses the schedulingof a number of activities subject to constraints on precedence requirements and resource limita-tions. A series of cash flows occur over the duration of the project in the form of cash outflowsfor project expenditures and cash inflows as payments for completed activities.

Figure 1 depicts an example project network. Each node represents an activity and the directedarcs represent the precedence relationships among activities. Each activity i(i = 1, . . . , m) has afixed duration di, resource requirements (ri1, . . . , riK) of K types (in this example K = 3), andcash flows F out

i and F ini representing expenses and payments.

Given a project network with m activities, and a discount rate of α for the cash flows, theRCPSP with the NPV objective can be formulated, as shown below, as a mixed integer nonlinearprogram where xit(i = 1, . . . , m) are binary variables that determine the start time of activity i,i.e., xit = 1 if activity i starts at time t, and 0 otherwise.

914 Naval Research Logistics, Vol. 46 (1999)

Figure 1. Example project network.

Maximizem∑

i=1

Li∑

t=Ei

F outi e−α(t)xit +

m∑

i=1

Li∑

t=Ei

F ini e−α(t+di)xit

subject toLi∑

t=Ei

xit = 1, i = 1, . . . , m, (1)

Lj∑

t=Ej

txjt −Li∑

t=Ei

txit ≥ di, ∀(i, j) ∈ A, (2)

m∑

i=1

t+di−1∑

q=t

rikxiq ≤ Rkt, k = 1, . . . , K, t = 1, . . . , D, (3)

where

m = number of activities,F out

i = cash outflow for activity i,F in

i = cash inflow for activity i,α = discount rate,di = duration of activity i,Ei = earliest start time for activity i,Li = latest start time for activity i,K = number of resource types in the project,rik = requirement of resource type k by activity i,Rkt = amount of resource type k available in period t,

A = set of precedence relationships,D = deadline for the project.


Constraints (1) ensure a unique start time for each activity. Constraints (2) describe the prece-dence relationships. Constraints (3) represent the resource utilization requirements. The objectiveis to maximize the NPV of the cash flows.

Obtaining optimal schedules for projects of reasonable size is impractical since the problem isNP-complete [5]. Hence heuristic procedures have found extensive application. Many of theseheuristics use information generated by the Critical Path Method such as slack or early and latestart times [3]. Russell [15] and Padman et al. [12, 13] developed and tested several heuristics thatuse information from a transshipment network relaxation of the RCPSPCF, such as dual pricesand target scheduling dates. The heuristics are embedded in a single-pass, forward-sequencingprocedure. Once a heuristic is selected, it is applied throughout the scheduling process.

These previous studies show that no single heuristic dominates in all the project environments [3,12, 13, 15]. This is primarily due to the complexity of interactions of the many critical parametersthat characterize the RCPSPCF. Current best heuristics exploit specific features of the project suchas slack, dual values (describing cost of delaying activities), and cash flows. However, there hasbeen no investigation of the benefits of linking multiple strategies that explore interactions of allthese factors, guided by a metaheuristic search strategy such as tabu search. As the characteristicsof the project, such as pattern of cash flows and level of resource constrainedness, change overits duration, the search strategy needs to adapt to the new conditions to generate better solutions.Based on this premise, we embed multiple search procedures in this paper to generate feasiblesolutions within a tabu search framework and show empirically that the results are significantlybetter than those currently available.

3. TABU SEARCH

Tabu search is a metaheuristic procedure which facilitates embedded search procedures toescape from local optima [6–8]. Any search procedure can be embedded in the tabu searchframework to avoid explicit or implicit cycling behavior. This method has been well documentedin the literature; hence we include only a brief summary here. The general search algorithm startswith an initial solution to the problem. The procedure then iteratively selects a best move fromthe neighborhood of solutions to improve the solution, as measured by an evaluation function. Ifthe move results in a local optimum, restrictions are imposed on the moves to differentiate theadmissible moves from the tabu moves. The most common tabu moves are those that attempt toreverse or repeat previous moves. By restricting these moves for a fixed time period, called tabutenure, the search may be driven out of local traps. This restriction can be freed at critical values,called aspiration level, where this seemingly tabu move may actually lead to better solutions. Byenforcing the tabu restriction, the short term memory structure in tabu search procedure encouragesthe search to focus on a more promising region, while long-term memory allows the search tobecome more diversified.

Tabu search has been successfully applied to hard instances of Traveling Salesman Problem(TSP), quadratic assignment problem, and production scheduling such as job shop and flowshop, and manpower scheduling problems [8]. For the resource-constrained project schedulingproblem with the NPV objective, Icmeli and Erenguc [10] proposed two heuristics based on tabusearch, using short-term and long-term memory. Their tabu search procedure based on short-termmemory used the simple operation of shifting an activity one time period earlier or later to generateneighborhoods. Long-term memory used L stages to start with a different solution in each stageand the best solution from the L stages was chosen as the solution to the problem. The heuristicswere tested on a set of 50 small randomly generated projects from the literature. The results werecompared with those from the minimum slack heuristic as well as to upper bounds obtained from


a linear programming relaxation strengthened by valid cuts. The authors concluded that the newheuristics improved schedules and NPV with reasonable computational effort.

Our study differs from and extends this earlier research in three distinct ways—in the typeof strategies used for initial solution, neighborhood, and candidate list generation; the number,sizes, and variety of problems tested; and the examination of various tabu search parameterssuch as impact of initial solutions, tabu tenure, types of moves, termination criteria, and memorystructure on final solutions for this class of problems. In the Icmeli and Erenguc [10] study,the test set contained small projects ranging from 8 to 51 activities, with many projects havingonly 22 or 27 activities. Based on the results, it is not clear how the tabu search metaheuristicwill scale for problems of practical size, which are much larger than the problems tested in thisstudy. Furthermore, while shifting activities one time period improved NPV, there is significantpotential for further enhancement of local search using more intelligent neighborhood generationand candidate selection strategies. We report on such strategies in this paper and test them on 720randomly generated projects with 46 and 110 activities, as well as the data set used in previousresearch. We compare the results of our procedure to the best performing heuristics from theliterature and to an upper bound. Thus the practical value and difficulty in identifying a dominantheuristic for the RCPSPCF leads us to the design of a scheduling system described in the nextsection.

4. TSCPS—TABU SEARCH FOR CONSTRAINED PROJECT SCHEDULING

4. 1. Framework

The framework for our tabu search procedure incorporates efficient candidate move generation,evaluation, and selection to both intensify and diversify the search for a good project schedule.Figure 2 depicts a data flow diagram for applying tabu search to the RCPSPCF.

The search starts with an initial schedule that is generated using a combination of multiplesimple heuristics, described later, that are applied to the project data. This initial schedule, storedin working memory, is improved using a number of modification heuristics. These heuristics,working as move generators, generate candidates in the neighborhood of the current solution toserve as moves using insertion and swapping of activities such that the newly revised scheduleremains precedence feasible. To make the search process more efficient, only activities with highcash flow weights or net cash flows, defined later, are considered at this stage. In each iteration,an admissible move is selected from the candidate list that has no tabu restrictions and has highestNPV. Aspiration level criteria enables override of the tabu restrictions to improve the schedule.This new schedule is then returned to the working memory. The search stops after a specifiednumber of iterations. The best solution obtained over all the iterations is chosen as the schedulefor a given resource constrained project scheduling problem. Alternate starting solutions, alsogenerated using the combination procedure, are explored using schedules stored in long-termmemory.

4. 2. Design Elements

The critical elements of tabu search procedure that impact implementation and solution, andits application to the RCPSPCF are discussed in the following:

• Initial Solution: During the schedule generation process, a number of activitiesmay become eligible for scheduling when their predecessor activities are com-pleted. Given the constraints on availability of resources, some of the activitiescan start immediately while others have to be delayed. Heuristics are usedto prioritize activities in the eligible queue. Given that heuristic performance


Figure 2. TSCPS framework.

and project characteristics vary over the duration of the project across differentproject conditions, switching from one heuristic to another could potentiallygenerate better schedules than a single heuristic acting alone [4, 9, 16, 17, 19].Using this approach, an initial schedule is constructed by the application of sixsimple heuristics in a combination procedure that dynamically selects a heuristiceach time activities have to be prioritized. More specifically, in every iterationfor scheduling activities, one of the six heuristics is selected randomly to orderthe activities in the eligible queue. This ordering is used to schedule activitiesuntil resource limits are reached. The remaining activities in the eligible queueare delayed till the next point in time when resources will be made available.In this study, we consider the following heuristics to act as construction agents:

— Random Selection (Random): Choose activities randomly from the eligibleactivity queue and schedule them till available resources are exhausted.

— Shortest Processing Time (SPT): Select activities from the eligible activityqueue in ascending order of their durations.

— Longest Processing Time (LPT): Select activities from the eligible activityqueue in descending order of their durations.

— Minimum Slack (MINSLK): Select the activity with the minimum slack.The slack is the difference between the late finish time and early finish timecomputed using the critical path method. Slack is continuously updated.

— Cash Flow Weight (CFW): Select activity with the highest cash flowweight. The cash flow weight of each activity is computed as the sumof the cash flows of all its successors.


This combination procedure of simple heuristics produces a superior schedulein less time than many of the complex network relaxation based heuristics dis-cussed in [16, 17, 19]. The tabu search approach used three different solutionsas starting points, including the best one, from the schedules generated by thecombination rule.

• Neighborhood Generation Strategy: At each iteration, a candidate list of movesmust be identified. Previous studies have shown that the choice of a neighbor-hood can have a significant impact on the solution and generation time [8].While a large number of candidate moves generated from multiple heuristicsmay increase the likelihood of obtaining good solutions, the computational ef-fort for selecting and updating these move attributes will increase as well.The tradeoff between efficiency of search and effectiveness of the procedurecalls for a simple neighborhood structure and a good move selection strategy.There are a number of ways to choose a neighborhood of possible moves. Theobjective is to generate a small enough candidate list of alternatives that arelikely to be superior to the current solution. We tested two different candidatelist strategies, both motivated by the fact that NPV can be increased by earlyscheduling of those activities with high positive cash flows and delaying activ-ities with high negative cash flows.The first strategy thus restricts moves to those exchanges (insertion or swap)that occur only between activities with higher cash flow weights. The secondstrategy limits moves among activities with high net cash flows which is thenet of the positive and negative cash flows for each activity. The two strategiesthus explore a long-term and short-term view, respectively, of the cash flows inthe project.

• Neighborhood Structure: In the project scheduling problem, a schedule canbe modified by a pairwise interchange (swapping) of the assignment of starttime of two activities, or by inserting an activity in front of another activity.Previous studies have found that a combination of these two moves will ingeneral lead to good solutions [6–8]. We incorporate both types of moves ingenerating the neighborhoods. For a project with m activities, the swappingfunction Swap(Ai, Aj), where Ai and Aj are the two activities to be exchanged,results in m∗(m − 1)/2 neighborhoods, assuming all pairs of activities can beswapped. Similarly, an insertion move, Insert (Ai, Aj), where activity j isinserted in front of activity i, applicable to all the activities, leads to m∗(m −3) + 2 neighborhoods [11]. Given the precedence relations in the RCPSPCF,the number of neighborhoods are clearly less than these bounds. Furthermore,in order to limit the size of the candidate list, we do not allow violation ofprecedence constraints. Violations of resource constraints are repaired throughconstraint propagation methods.

• Evaluation Function: After the neighborhood is defined, a move needs to beselected from the set of admissible moves or candidate moves. The admissibilityof a move can be evaluated using tabu restrictions or evaluation functions. Acommon evaluation strategy is to use the objective function of the problem. Inthe case of the RCPSPCF, we use the NPV of the schedules on the candidatelist to choose the best possible move.


• Tabu Restrictions: A major feature of the tabu search procedure is the set ofrestrictions imposed on candidate moves to prevent cycling. Many successfulapplications recommend using a simple neighborhood strategy and appropriateevaluation functions with more complex candidate selection and evaluationcriteria [8]. A move is evaluated based on the tabu status in addition to theevaluation function described above. There are two types of tabu restrictionscorresponding to the two types of neighborhoods defined earlier. The basicidea is to try to avoid a reversal of moves. For example, suppose an activity Ai

is just interchanged with activity Aj , then a move which attempts to swap Ai

and Aj back in the very next iteration is abandoned. Similarly, if an activityAj is inserted ahead of activity Ai in one iteration, the attempt to move it backimmediately thereafter is a forbidden move.

• Aspiration Level: Tabu restrictions limit move selection to avoid cycling behav-ior. However, under certain circumstances, one might find the move which isclassified as tabu is actually an excellent move that may lead to a better solutionthan is currently available. An aspiration criterion allows the tabu status of amove to be overridden if a solution is better than the current best solution. Forthe RCPSPCF, a comparison of the NPVs of the schedules is sufficient to detectif aspiration level is achieved. The tabu restriction of the candidate schedule isremoved and the move is executed.

• Candidate List Size and Tabu Tenure: The number of moves recorded andevaluated in any iteration is not limited in our study. The size of tabu listrestricts the number of iterations that a particular move is prohibited. Glover[6–8] suggests 7 as a reasonable number. We tested three levels of 5, 7, and 20for tabu tenure.

• Termination Condition: Given that tabu search is an iterative procedure, manystopping criteria can be utilized to terminate the search process. We tested three:the first specifies the number of iterations; the second specifies the number ofiterations where there is no improvement; and the third applies a time limitequal to that taken by the benchmark heuristics from the literature for the sameproject environment.

• Memory Structure: Tabu search allows the use of both short-term and long-termmemory structure through the use of tabu lists and multiple starting solutions,respectively. As discussed earlier, we use both structures in our design andexperimentation.

5. DATA, EXPERIMENTS, AND RESULTS

In this section, we describe the experimental data, computational testing, and results.

5. 1. Experimental Data

The empirical testing was conducted using two data sets. The first is a subset of a largeexperimental data set, called the PSD data set, generated in [12, 13] from various parameters thatdescribe the resource-constrained project scheduling problem. The second data set is based on amodification of the Patterson problems [14]. A subset of these problems were tested by Icmeliand Erenguc [10] in an earlier study of tabu search applied to the RCPSPCF.


The PSD data set consists of 1440 randomly generated project networks with resource con-straints and cash flows. These 1440 problems were generated from six experimental parametersincluding project size (size), project network structure (shape), resource constrainedness (AUF),interest rate (CC), profit margin (PM), and frequency of progress payments (FP). These six pa-rameters form a critical subset of the project descriptors presented in [18]. All six parameterswere each varied over several levels in constructing the project networks.

Project size is characterized by the number of activities involved, i.e., all small size projectnetworks have 46 activities while the medium size problems have 110 activities. There arethree types of resources associated with each activity. The degree of resource constrainedness ischaracterized by AUF (average utilization factor). It varies from a low level of 1.0 to a mediumlevel of 1.5 and a high level of 2.0. Network shape varies over three levels: balanced, skewedto the right, and skewed to the left. There are three cash flow parameters: the interest rate,which varies over two levels from a low of 10% to a high of 20% annually; frequency of progresspayments, which is also set at two levels, where a low level indicates that on average, every seventhactivity yields a positive progress payment upon its completion, while a high level indicates thaton average every third activity yields a positive progress payment; and, profit margin, reflectedby the final payment for the completed project, which is set at two levels, where the low level isapproximately equal to the sum of the cash outflows for the project plus 30% and a high levelequal to the sum of the outflows plus 50%. A full factorial experiment was conducted whichresulted in 144 different scheduling environments leading to 1440 problems with 10 replicates ineach environment [13].

In this paper, we evaluate the tabu search metaheuristic on only 720 projects (360 small sizeand 360 medium size) from the above set by setting the cost of capital to only one level, at 20%for the small projects and 10% for the large projects.

The second data set, from the Icmeli and Erenguc study [10], consists of 50 project networkswith number of activities ranging from 8 activities to 51 activities. Cash flows were generatedrandomly from a uniform distribution on [−500, 1000]. Due to difficulties in conversion fromthis data set to our format, only a subset of 38 problems were evaluated using our approach.

5. 2. Computational Testing and Results

The system implementation includes the schedule generation and schedule modification pro-cesses within the tabu search framework. Tabu search controls the search process during schedulemodification to enhance the NPV performance. A number of project scheduling and tabu searchparameters were tested with the system such as different initial solutions, neighborhood generationstrategies, the candidate list size and tabu tenure, stopping condition, and memory structure.

The detailed results including NPV and percentage gain by tabu search over current best heuris-tics for the 720 projects can be found in [16]. The results are compared against two of the betterperforming heuristics from the literature. Two good heuristics (LTP/LAN, CFW-OCC) from[13] were selected as the benchmark heuristics. In an evaluation of sixteen heuristics using thesame PSD data set, Padman et al. [13] concluded that LTP/LAN and CFW/OCC were supe-rior in performance, with LTP/LAN producing the highest NPV in 12% of the networks andCFW/OCC in 20%. These rules, utilizing information from a transshipment network relaxationof the RCPSPCF, are summarized below.

• LTP/LAN: Activities are scheduled in ascending order of tardiness penalties.Tardiness penalties are the sum of the dual prices of the immediate predecessorsof any activity. Activities with zero tardiness penalties are scheduled accordingto lowest activity number.


• CFW/OCC: Cash flow weights are used in descending order for schedulingthose activities with positive tardiness penalties, and opportunity cost of cashflows, in ascending order, for scheduling those with zero tardiness penalties.

In addition to LTP/LAN and CFW/OCC, we compared the results from the tabu search pro-cedure with those from the best heuristic for each problem as identified in [13]. The averageNPV for 720 small and medium size projects generated from each heuristic is summarized inFigures 3 and 4, respectively. These results indicate that TSCPS is significantly better in averageperformance than the best performing heuristic from the literature. It also dominates LTP/LANand CFW-OCC.

We also compared the tabu search NPVs with those from a transshipment network relaxationof the RCPSPCF that provides an upper bound on the objective function value. This bound isobtained by solving the NPV project scheduling problem without the resource constraints [13].The results indicate that the NPVs from the tabu search approach are about 11% below theupper bound for medium size problems and 27.5% below for small size problems. In contrast,earlier research with the sixteen heuristics demonstrated that the benchmark heuristics are about15% below the upper bound for medium size problems and 35% below for small size problems.Differentiating by the level of resource constrainedness of the problems, we find that for themedium size problems, the % below upper bound are 21.0%, 10.6%, and 1.1% for the high,medium, and low levels, respectively, when compared to 25%, 15%, and 6% for the benchmarkheuristics. For the small size problems, the corresponding values are 43.0%, 34.0%, and 4.8%for tabu search and 49%, 42.0%, and 14% for the benchmark heuristics.

Figure 3. Performance comparison: average NPV on small size projects (48 activities).


Figure 4. Performance comparison: average NPV on large size projects (110 activities).

Furthermore, computing the percentage of projects where TSCPS performs better than otherheuristics, it can be seen that this procedure has the best performance on 80% of small size projectsand 94% of large size projects in the PSD data. This performance holds across all the projectenvironments represented by the test problems. In comparison, in [13], LTP/LAN was observedto be the best heuristic for 12% of the problems and CFW/OCC for 20%. The average runningtimes are approximately 150 s on small size projects (with 46 activities), and 600 s on large sizeprojects (with 110 activities), comparable to that of the benchmark heuristics.

We also compared our tabu search results with the Icmeli and Erenguc results [10] on thePatterson data [14]. This comparison indicates an improvement in NPV for 76% of the projects.The average improvement is about 5% when the discount rate is 0.01 and 3.5% for discount rateof 0.02. Given the small size of the problems and the large payment at the end of the project, itis expected that most heuristics will perform about the same on this set of projects.

5. 3. Effects of Neighborhood Strategies

Tabu search, as a local search strategy, attempts to identify improving solutions by searchingamong the neighbors of the current solution. The selection of a move may become very expensivefor very large problems if the neighborhood of a move is very large, or if the computation involvedin evaluating each move is large. The first question is how to find a small enough neighborhoodto make computation tractable, yet find good solutions. As mentioned earlier, we designed twocandidate list strategies, cash flow weight and net cash flow, that use cash flow information sincethe objective is to maximize project NPV.


Figure 5. Trace of TSCPS on a large size project.

As hypothesized, repeated experiments with a subset of the data set identified cash flow weight(CFW) criterion as the superior strategy. This may primarily be due to the fact that CFW considersthe long-term impact of choosing a specific activity to schedule from the set of schedulableactivities whereas net cash flow evaluates the local impact. CFW captures not only the casewhere the activity under consideration may have high positive cash flows, but also the casewhere a successor activity on the path to completion may generate high positive cash flows. Ifthe activity is on a path that leads to high cash flows, it is desirable to schedule it earlier toincrease NPV.

Thus, we employed this criterion to constrain the candidate lists in subsequent experiments.This constraint allows fast generation of a small set of superior candidates. Thus the candidatemoves can be ordered according to increasing values of cash flow weight. A concern associatedwith restricting neighborhood solutions is that many good solutions may be eliminated fromconsideration. To evaluate the significance of the loss, we tested the candidate list approachagainst the alternative of considering all possible moves at each iteration, running each approachfor the same number of iterations, on a limited set of test problems. The outcomes show that theuse of the candidate list did not cause solutions to deteriorate, indicating that the CFW approachindeed generates an effective subset of candidates.

5. 4. Effects of Different Tabu Sizes

An important feature of tabu search is its superimposition of tabu restrictions, where the durationof each restriction is a parameter. We experimented with three levels of tabu sizes, 5, 7, and 20,and observed that the performance of TSCPS was not affected by different tabu tenures, whichheld for the subset of problems tested. Based on this test, in all further experiments, we chose 7as the tabu size. A move is critical if all moves in the candidate list are tabu and none satisfy the


Figure 6. Trace of TSCPS on a small size project.

aspiration criteria. In this situation, our system selects one of the moves at random to break thedeadlock.

5. 5. Effects of Termination Criterion

Given the iterative nature of the procedure, an important parameter is how long the searchshould continue and when to stop. Figures 5 and 6 illustrate the trace of TSCPS on a specificexample of small and large size projects, respectively. These traces are illustrative of the behaviorof the set sample projects we tested.

It is interesting to observe that large and small projects exhibit significantly differing behavioras the iterations are traced. Large projects have a steadily improving pattern till they taper offwhile small projects show considerable variability in NPV from iteration to iteration. This can beattributed to two reasons. First, the overall duration of small projects is short, thus accentuatingthe variability in frequency and level of cash flows. Secondly, this variability in frequency andlevel of cash flows directly affects the neighborhood generation strategies and candidate movesused in the experiments, which are based on cash flow weight.

Using the number of iterations as the stopping criterion, we experimented with several iterationlimits between 100 and 1000, finally fixing it at 100 iterations.

5. 6. Effects of Initial Solution Quality and Memory Structure

Tabu search is a controlled search based on given starting solutions. Prior research has shownthat initial solution quality does not have a significant impact on the overall performance of tabusearch [8]. However, for large projects that are highly resource-constrained, the incorporation oflong-term memory can yield better performance [7, 10]. This can be accomplished by generating


Figure 7. Effects of memory structure.

multiple starting solutions to be processed by the tabu search procedure. Many solutions from theinitial schedule generation process using the heuristic-combination model can be saved for theuse here. In our experiments, the system improved the solution based on two different schedulesgenerated from the combination rule.

As shown in Figure 7, the darker line represents the trace for the initial run. The bestsolution is found at iteration 51 and NPV = $777.17. The dashed line represents the sec-ond attempt where it starts from a different initial solution drawn from the solution pool gen-erated by the combination model. The best solution is found at iteration 102 and NPV =$797.78. The solution obtained from the second search provides a chance for TSCPS to ob-tain a better solution than the previous search, even though it may be starting with a worse initialschedule.

5. 7. Discussion

The results of the extensive computational testing indicate that the tabu search approach thatexploits information about the critical parameters of the problem in an integrated manner signifi-cantly improves the schedules and NPV for the resource-constrained project scheduling problemwith cash flows. While they also provide many insights about the impact of some critical projectand tabu search parameters on the solution, we can draw several conclusions about the reasons asto why tabu search produces such notable improvements for the RCPSPCF.

First, the parameters that are critical to the RCPSPCF are many and complex, such as thelevel of resource constrainedness, project profitability, frequency and level of payments, size andtopology of the project, and so on. Most of these parameters are derived attributes that dependon multiple characteristics of the project such as duration, availability of resources, utilization ofresources, and critical path of the project. The successful problem-dependent, parameter-basedheuristics developed in the literature, such as CFW/OCC and LTP/LAN, are able to utilize only


limited information about the project characteristics. For example, LTP/LAN utilizes durationand discounted cash flow information to develop a priority ordering rule. Thus other importantcharacteristics such as resource constraints cannot be considered while ordering activities forscheduling. This constrains the quality of the solutions generated. In contrast, a metaheuristicprocedure such as tabu search allows for exploration of the search space of multiple interactingproject characteristics.

The second reason is that the traditional heuristics for the problem have been mostly single pass,forward sequencing methods which do not allow exploration of swapping, inserting or shiftingactivities once they are scheduled. In contrast, tabu search is an iterative procedure that combinesmultiple modes of switching activities over many iterations and the best schedule is selected.These advantages of the metaheuristic search procedure are domain-independent, allowing theinvestigation of good solutions to complex problems such as the RCPSPCF.

6. CONCLUSIONS AND EXTENSIONS

We have presented a meta-heuristic search strategy for the resource constrained project schedul-ing problem with the NPV objective. We have also designed and conducted several experiments toevaluate both the parameters within tabu search and those that are critical to the project schedulingproblem.

TSCPS is tested over 720 small size and medium size projects. The results show that the impactof tabu search is significant, especially for large projects. The TSCPS procedure dominates in94% of large size projects and 80% of small size ones. Note that while the overall performanceof TSCPS is superior to that of the best performing heuristic from the literature for small projects,the potential for improvements is even greater for large projects. Thus the TSCPS is a betterapproach for large projects than small ones.

The solutions can be further improved by employing additional long-term memory features.However, the success achieved with the current procedure provided a good balance between easeof implementation and desired improvement in solution. Some long-term elements are includedin this study by restarting the tabu search procedure using three of the best solutions from thecombination method that generates the initial feasible schedule. The solutions obtained by themultiple attempts are comparable in objective function value.

This approach allows for very general problem specification, and is effective over a wide rangeof problem sizes and levels of difficulty. The advantages of the local search algorithm are itsability to handle arbitrary objective functions and constraints, and its effectiveness over a range ofproblems. The tabu search approach presented in this paper can be embedded in decision supportenvironments for constrained project scheduling to guide the generation of better schedules.Some possible extensions include the incorporation of learning at several levels of the tabu searchprocedure. Identification of rules for switching from one heuristic to another during scheduleconstruction and modification, and the impact of learning on the selection of heuristics for thedifferent operations of generating neighborhoods, candidate lists and moves need to be furtherexplored.

ACKNOWLEDGMENTS

This research is partially supported by NSF Grant IRI-9634383, a grant from the CentralInvestment Fund for Research Enhancement, a grant from the Faculty Development Fund atCarnegie Mellon University, and a grant from the U.S. Army (DAAH04-94-G-0239). We wouldalso like to thank two anonymous referees and the Associate Editor for their helpful suggestionsand corrections.


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a metaheuristic scheduling procedure for resource-constrained projects with cash flows

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