heuristic scheduling of resource-constrained projects with cash flows

Heuristic Scheduling of Resource-Constrained Projects withCash Flows

Rema Padman,1 Dwight E. Smith-Daniels, 2 Vicki L. Smith-Daniels2

1Heinz School of Public Policy and Management, Carnegie Mellon University,Pittsburgh, Pennsylvania 15213

2Department of Management, College of Business, Arizona State University, Tempe,Arizona 85287

Received March 1995; revised March 1996; accepted 22 August 1996

Abstract: Resource-constrained project scheduling with cash flows occurs in many set-tings, ranging from research and development to commercial and residential construction.Although efforts have been made to develop efficient optimal procedures to maximize thenet present value of cash flows for resource-constrained projects, the inherent intractabilityof the problem has led to the development of a variety of heuristic methods to aid in thedevelopment of near-optimal schedules for large projects. This research focuses on the useof insights gained from the solution of a relaxed optimization model in developing heuristicprocedures to schedule projects with multiple constrained resources. It is shown that aheuristic procedure with embedded priority rules that uses information from the revisedsolution of a relaxed optimization model increases project net present value. The heuristicprocedure and nine different embedded priority rules are tested in a variety of projectenvironments that account for different network structures, levels of resource constrained-ness, and cash-flow parameters. Extensive testing with problems ranging in size from 21 to1000 activities shows that the new heuristic procedures dominate heuristics using informationfrom the critical path method (CPM), and in most cases outperform heuristics from previousresearch. The best performing heuristic rules classify activities into priority and secondaryqueues according to whether they lead to immediate progress payments, thus front loadingthe project schedule. q 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 365–381, 1997

1. INTRODUCTION

Project managers must schedule large projects subject to conflicting objectives and limitedresources. Project objectives may include minimizing project makespan, efficient utilizationof resources, and effective management of cash outlays and receipts. The constraints andparameters of the project scheduling problem include activity durations and precedencerelationships, and limits on the availability of labor, materials, facilities, equipment, andcapital. Project planners frequently use the network scheduling procedures such as theprogram evaluation and review technique (PERT) and the critical path method (CPM) tofind the duration of the longest path in the network (the critical path) and a schedule of

Correspondence to: Dwight E. Smith-Daniels.

q 1997 by John Wiley & Sons, Inc. CCC 0894-069X/97/040365-17

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366 Naval Research Logistics, Vol. 44 (1997)

activities. However, these techniques usually do not find feasible schedules when resourcesare limited in availability. In response to this deficiency, numerous authors have developedexact and heuristic methods for the resource-constrained problem, where a schedule isderived by allocating limited resources to competing activities such that a project’s makespanis minimized [4–6, 16]. Because resource-constrained project scheduling problems oftenconsist of thousands of activities, project managers have utilized heuristic procedures inprojects in manufacturing and service industries [4, 21].

Project managers must also consider the impact of cash flows on the project plan, schedule,and performance. Cash outflows include expenditures for labor, equipment, and materials,and cash inflows take the form of progress payments for completed work and a final paymentpaid upon completion of the entire project. Net present value (NPV) is an effective measureof project financial performance, because it balances the objectives of minimizing projectmakespan and maximizing project value [3] . In the first research using the NPV objective,A.H. Russell [19] models the unconstrained NPV project scheduling problem as flows ina network which may be solved as a series of transshipment problems. Elmaghraby andHerroelen [9] present an alternate optimal algorithm for the unconstrained NPV maximiza-tion problem.

The initial work incorporating resource constraints and an NPV objective [7] presents abinary integer programming model for scheduling a project with a capital constraint. Twoother exact solution procedures have since appeared in the literature. Patterson, Talbot,Slowinski, and Weglarz [17] present a backtracking algorithm for both the capital con-strained and renewable, nonstorable resource-constrained problems with both makespan andNPV objective functions. They conclude that it is much more difficult to find and verifyoptimal solutions for the NPV-maximization problem. Yang, Talbot, and Patterson [22]use an implicit enumeration branch-and-bound procedure to solve NPV resource-constrainedscheduling problems of up to 22 activities optimally.

Because of the intractability of the resource-constrained project scheduling problem, R.A.Russell [20] adopts a heuristic approach for maximizing the NPV of resource-constrainedprojects. He uses information from the solution to the network flow model of A.H. Russell[19] combined with several heuristics embedded in a greedy single-pass forward algorithmand compares their performance against heuristics whose objective is to minimize projectmakespan. Pinder [18] and Baroum and Patterson [2] both propose and test heuristicprocedures that prioritize activities for the assignment of constrained resources accordingto their cash flow weight (CFW), which is defined as the sum of the cash flows of successoractivities in the project network. Baroum and Patterson found in extensive tests that thesimple CFW heuristic outperformed more complex versions of CFW that used discountedcash flows.

In this article we present a heuristic algorithm with embedded priority rules to maximizethe NPV of cash flows for resource-constrained projects. The priority rules use updatedinformation from the solution to the unconstrained network model of A.H. Russell [19] toguide scheduling decisions when resource conflicts develop during the construction of aproject schedule. The heuristic algorithm exploits the capabilities of a dual simplex algorithmfor the minimum-cost network flow problem [1] that allows for the efficient reoptimizationof a partially completed schedule. We propose and test nine heuristic rules that use therevised activity start times and tardiness penalties with information on resource requirements,cash flows, and activity durations. An experimental design tests the performance of therevised schedule and tardiness penalty (RSTP) heuristic algorithm and embedded rules and

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367Padman et al.: Resource-Constrained Project Scheduling

compares their performance to that of existing procedures for project scheduling in a re-source-constrained environment.

Our study yields several unique contributions to the research on resource-constrainedproject scheduling with cash flows. First, the heuristic rules use interesting insights basedon the revised dual values obtained from reoptimizing the network formulation of A.H.Russell, in contrast to R.A. Russell’s research, which used only the initial solution to thenetwork throughout the scheduling process. The heuristic algorithm and RSTP priority rulessignificantly outperformed those tested by R.A. Russell. Second, the computational studyis performed with the use of a full factorial design, where the project networks are generatedrandomly based on six experimental variables that measure critical aspects present in proj-ects. Thus, effectiveness of the heuristic methods can be measured along these six dimen-sions. Third, we provide results for the first extensive tests using large problems with 500and 1000 activities. We note that the dominance of the best performing RSTP rules isclearly exhibited only as problem size increases. Finally, we note that priority rules usingearliness and opportunity costs, and tardiness penalties derived from the unconstrainedoptimal network solution successfully capture scheduling practices found in project manage-ment in services and manufacturing. These concepts can be generalized and extended toother scheduling and line-balancing problems.

The NPV project scheduling problem with multiple resource constraints is formulated asa nonlinear integer programming model in Section 2. The RSTP heuristic rules and algorithmare presented in Section 3, and the test experiments that are used in evaluating the perfor-mance of the RSTP heuristics are described in Section 4. An analysis of the results of theexperiments and computational experience is reported in Section 5. A summary and adiscussion of future research directions are presented in Section 6.

2. PROBLEM STATEMENT

The problem addressed in this article is one where a series of cash flows occur over thecourse of a project of makespan p as cash outflows for project expenditures and cash inflowsthat are progress payments for completed work. Each activity (arc) k is defined by a startingnode, labeled i(k) , and an ending node j(k) , with an associated cash flow, Fi for eachevent i Å 1, . . . , N . A cost of capital a is used in computing the NPV of the project.Without loss of generality, we assume that all cash outflows occur at i(k) and all cashinflows (progress payments) occur at j(k) nodes. There are m activities in the project, eachwith fixed duration dk , and no activity may be preempted once it begins. Resource usagefor activity k is predetermined and constant throughout the duration of the activity. Thereare q different resource categories, and rwk is the number of units of resource w Å 1, . . . ,q required by activity k . The maximum number of units of resource type w available duringany period is uw , and the resource limits vector RL is defined as (u1 , u2 , . . . , uq) . Thedecision variables are Ti , the time of each event i where the initial event is numbered 1and T1 Å 0. The set Zs contains all activities in progress during period s and the resourcerequirements vector RZs

, for period s is ((k√Zsr1k , . . . , (k√Zs

rqk) .As stated in R.A. Russell [20], the resource-constrained cash-flow problem can be formu-

lated mathematically as

(P1) Maximize ∑N

iÅ1

Fi exp(0aTi ) , i Å 1, . . . , N ,

subject to

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Ti (k ) 0 Tj (k ) ° dk , k Å 1, . . . , m ,

RZs° RL , s Å 1, . . . , p .

Optimal solutions for (P1) are impractical if not impossible to generate for large projects,because the resource-constrained scheduling problem is NP-complete [10]. We describe aheuristic algorithm in Section 3 that embeds information from the unconstrained optimalsolution to the NPV model in a series of priority rules.

3. ALGORITHM AND RSTP HEURISTIC PRIORITY RULES

Our research presents nine RSTP heuristic priority rules that use updated optimal activitystart times and dual prices from the minimum-cost network flow model of A.H. Russell[19]. They are embedded within a greedy single-pass parallel sequencing algorithm tomaximize project NPV subject to multiple resource constraints. Previous work in the areaof resource-constrained project scheduling where the objective is to minimize project dura-tion [5] demonstrates that single-pass procedures can derive near-optimal makespan sched-ules with a parallel sequencing approach where the activities are prioritized for the allocationof resources during the schedule generation process.

It is appropriate to first review the relevant features and characteristics of the A.H.Russell [19] model. A.H. Russell approximates the nonlinear objective function in (P1) byincorporating only the first-order linear terms of the associated Taylor expansion in a linearprogramming formulation of the NPV project scheduling problem. The dual formulation ofthis linear model is a transshipment type of network flow model. Computational experience[20] shows that this method converges very rapidly given an early start time initial schedule,and that it is computationally efficient for large-scale problems. The solution of the transship-ment problem is a system of flows that can be interpreted as dual prices lk representingthe reduction in NPV when the duration of activity k is extended by one period. Based onthe principle of complementary slackness, lengthening activity k’s duration by one periodresults in a schedule change where either (a) i(k) is started one period earlier or (b) i(k)begins at its scheduled time but j(k) occurs one period later. For case (a) to occur, slacktime must be available to allow i(k) to start one period earlier, implying that the dual pricesof all immediate predecessors are zero. In case (b) activity k’s predecessors do not haveslack, and the scheduled time of j(k) must be delayed along with all successor nodes.Furthermore, delaying j(k) will delay the start of all activities in the project whose optimaltime is greater than or equal to the late start time Lk found by CPM; that is, Ti (k ) ¢ Lk .Each of these cases of course is useful in determining the cost of delays due to resourceconstraints.

To estimate the reduction in project NPV due to activity delays caused by resourceconstraints, we define activity k’s tardiness penalty Pk as the sum of the dual prices of thepredecessor activities. In contrast, R.A. Russell’s [20] single-iteration heuristics assumethat the cost of delaying an activity when resource conflicts occur is represented by thedual price of the arc representing the potential delayed activity, ignoring the cash flowcorresponding to the i(k) node of a delayed activity. Our method of representing tardinesspenalties provides an accurate estimate of marginal changes in project NPV except in onecase. When Pk Å 0 and Ti (k ) ú Ek , where Ek is the early start time for activity k found by

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CPM, the tardiness penalty is only an approximation of the changes in project NPV, becausestarting this activity past its optimal start time will delay successor activities.

In summary, the single-pass parallel sequencing procedure considers the value of delaysin scheduling activities using revised tardiness penalties. Dual prices change as delays inproject activities occur as a function of resource constraints, and the unconstrained optimalschedule of activities may be different from the initial solution to the unconstrained networkmodel, particularly at increased high levels of resource constrainedness. The changes indual prices require the reoptimization of the unconstrained network problem to obtaincurrent optimal start times and revised activity dual prices given the partially generatedresource-constrained schedule. Dual prices in the unconstrained network solution representchanges in cash flows throughout the project as the result of lengthening an activity’sduration, not just the cash flows of the unscheduled activities. To overcome this limitation,as described in Section 4.2, the heuristic procedure treats nodes associated with activitiesthat have been completed whose timing cannot be changed as tied events [19] in theunconstrained network. The heuristic procedure exploits the capabilities of a dual algorithmfor the minimum-cost network flow problem [1] that efficiently reoptimizes a partiallycompleted schedule and solves the updated unconstrained network model after all currentlyfeasible scheduling decisions have been made.

3.1. RSTP Heuristic Priority Rules

The nine RSTP heuristic priority rules proposed in this article schedule activities withthe use of information on tardiness penalties, target schedule times, opportunity costs, andcash-flow weights. Each of the priority rules described below is embedded in the single-pass greedy forward algorithm described in Section 3.2. The first five heuristics use aprimary rule combined with a secondary, tie-breaking rule, and the latter four rules use theconcept of front loading, where activities that lead immediately to progress payments arescheduled as early as possible. Project managers often use front loading to maximize short-term financial performance.

3.1.1. Single-Queue Priority Rules

The five single-queue RSTP heuristics are presented in this section. Ties with respect tothe primary rule criteria are broken by the lowest-activity-number (LAN) rule, which selectsthe activity with the lowest i(k) node.

MTP/LAN (Maximum Tardiness Penalty/Lowest Activity Number): The MTP/LAN rule selects among competing activities according to the maximum tardiness penalty,Pk , thus locally minimizing the decrease in NPV caused by the delay of activities.

OCS/LAN (Opportunity Cost of Scheduling/Lowest Activity Number): The MTP/LAN heuristic does not consider the length of time that an activity consumes resources orthe tardiness penalties incurred by delaying the remaining schedulable activities. TheOCS/LAN rule is designed to measure the opportunity cost of scheduling an activity kon multiple, parallel resources by evaluating the cost of delaying all other activitiescurrently in the queue of schedulable activities during the duration of activity k . Activi-ties with the smallest opportunity cost are given priority in scheduling according to thefollowing value for activity k :

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∑k =√Ak

Pk =dk ,

where Ak is the set of remaining activities in the schedule queue after activity k is performed.OCR/LAN (Opportunity Cost of Resources/Lowest Activity Number): Scheduling

an activity simultaneously on multiple, parallel resources will not always delay the remainingschedulable activities, because resources may be available to allocate to additional activities.The OCR/LAN rule evaluates the resources consumed and with the opportunity cost ofscheduling, and selects the activity with the minimum opportunity cost of resources, definedfor activity k as

dk ∑k =√Ak

∑q

wÅ1

Pk =(rwk = /uw) .

NOC/LAN (Net Opportunity Cost of Scheduling/Lowest Activity Number): TheNOC rule attempts to balance the tardiness penalty of delaying an activity against theschedule cost of performing the activity. Although the OCS and OCR priority sequencingrules attempt to measure the opportunity cost of delaying the remaining activities in theschedule queue, these rules do not consider the reduction in NPV from delaying the startof an activity. Thus, the NOC rule selects the activity k with the maximum net opportunitycost of resources, defined as

Pk 0 dk ∑k =√Ak

∑q

wÅ1

Pk =(rwk = /uw) .

TSCFW (Target Scheduling Cash Flow Weight): Given the strong performance ofthe cash-flow weight rules in the study by Baroum and Patterson [2], this concept wasimplemented together with the revised target schedule dates derived by the network optimi-zation model. A potential shortcoming of dual price-guided priority rules is that dual pricesdo not consider the possibility that activity slack time will be eliminated as the resource-constrained schedule is generated by the forward-pass greedy scheduling algorithm. Morespecifically, if an activity k in the optimal spanning tree solution is connected to a successorwith positive slack, then the activity’s dual price Pk ignores the additional cash flows thatresult after the successor activity is completed. The TSCFW rule attempts to disregard thisslack time in the measurement of a cash-flow-guided priority index. The priority index foreach activity k can be computed as the sum of the cash flows for k and the cash flows forthe set of activities Bk that logically follow k in the project network:

∑k =√Bk

Fk = .

3.1.2. Front-Loading Priority Rules

A front-loading strategy gives priority to activities leading to immediate progress pay-ments without considering the impact on financial performance further downstream in theproject. One method of implementing a front-loading strategy is a two-queue schedulingapproach. The first three rules described below use a two-queue scheduling approach that

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classifies activities according to whether the immediate successor activities have zero dualprices. Because an activity with zero dual-price successors has activity slack time associatedwith its schedule, the activity’s tardiness penalty measures only its discounted cash flows.Placing activities with zero dual-price successors in the priority queue promotes the schedul-ing of activities with immediate progress payments. The fourth rule, LTP/LAN (lowesttardiness penalty/ lowest activity number) is a single-queue rule that takes an alternativeapproach to front loading by identifying activities that may lead to the earlier receipt ofprogress payments by means of their tardiness penalties.

MTP-ET (Maximum Tardiness Penalty–Earliest Finish Time): All activities in thepriority queue are scheduled in descending order of tardiness penalties. Activities in thesecond queue are scheduled in ascending order of minimum updated early finish times,found with the use of CPM and the partially generated schedule. Early finish time ties arebroken by random assignment.

CFW-CFW (Cash-Flow Weight–Cash-Flow Weight): The CFW-CFW priority ruleis a two-queue application of the CFW rule that uses information from the unconstrainednetwork flow solution in finding target schedule dates and queue membership. Activitiesin the priority queue are scheduled in descending order of cash flow weight (CFW). Afterall feasible front-loading alternatives are exhausted in the first queue, activities are selectedfrom the second queue according to highest cash-flow weight.

CFW-OCC (Cash Flow Weight–Opportunity Cost of Cash Flows): The CFW-OCCheuristic uses both the front-loading strategy and the opportunity cost of scheduling com-puted from cash-flow weights. Activities in the priority queue are scheduled in descendingorder of CFW, whereas activities in the second queue are selected in ascending order ofthe opportunity cost of cash (OCC) flow index, computed as

dk ∑k =√Bk

Fk = .

LTP/LAN (Lowest Tardiness Penalty/Lowest Activity Number): LTP/LAN at-tempts to schedule activities connected to nearby progress payments by giving priority inscheduling to those activities with the minimum nonzero tardiness penalty. An activity witha low tardiness penalty can represent an activity requiring or leading to large cash outflowsor an activity not connected in the optimal spanning tree to the payment at the end of theproject. The objective of LTP/LAN is to identify those activities not connected to the finalpayment in the project, but to an intermediate progress payment. Although this priority rulewould appear to yield lower project NPV, it is important to note that dual prices tend tobe higher for activities connected in the optimal spanning tree to the payment at the endof the project. Furthermore, if the total cash expenditures for a series of activities exceedsthe progress payments for those activities, then the optimal unconstrained schedule delaysthe start of these activities to the latest possible date and connects them to the final progresspayment. Activities with zero tardiness penalties are scheduled with the use of LAN.

3.2. Greedy Scheduling Procedure

The scheduling procedure is described below and by means of a flow chart in Figure 1.The greedy scheduling procedure begins with the generation of the unconstrained networksolution:

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Solve the unconstrained NPV project scheduling scheduling problem using the minimum cost network flow model. Set current time, t 5 0.

Find the set of precedence feasible activities whose scheduled start time in the unconstrained network solution is equal to t.

Re-optimize the unconstrained NPV network flow problem.

Are any of the activities resource

feasible at time t ?

Yes

Yes

No

Designate the set of precedence, time and resource-feasible activities to be the schedule queue. Sort the schedule queue according to the primary and secondary heuristic rules.

Find the next time when sufficient resources are available to schedule an activity: 1.≤Set t equal to that period. 2.≤Delay the minimum possible start

≤time for all precedence feasible ≤activities to that period.

Schedule the priority activity

Stop

Are there additional activities

remaining in the queue at time t ?

Have all activities in the project

been scheduled?

Yes

No No

Figure 1. Flow chart for the optimization-guided heuristic.

STEP 1: Solve the unconstrained NPV project scheduling problem with the use of theminimum-cost network flow model.1

This problem converges in a finite number of iterations [19].2 The computational com-

1 To guarantee a feasible solution when project NPV is negative, we tie the final node to the initialnode with the use of a reverse arc with a very long duration.2 In actual testing, the network flow problem converged to the optimal solution within, on average,three iterations.

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plexity of each iteration is that of solving a transshipment problem by the network simplexmethod, which is of O(n 3 log n) [13]. A preorder traversal of the network basis can bedone in O(n) time to generate the event times of the activities.

STEP 2: Schedule eligible activities with the use of a priority rule until the resourcerequirements of the activities remaining in the queue exceed the quantity of resources thatare currently available. If all activities have been scheduled, stop.

Eligible activities are defined as those activities whose predecessor activities have beenscheduled (precedence feasible) and scheduled start time in the current network flow solu-tion is equal to the current time in the scheduling process. Each of the priority decisionrules described in Section 3.1 is applied as the criterion for scheduling activities until thereare insufficient resources remaining to perform any of the activities remaining in the schedulequeue. Step 2 requires two scans of the queue of eligible activities, one to sort the set ofactivities by evaluating the heuristic rule for each activity, and the second to schedule theactivity after checking for the availability of resources. The time taken for this process isdominated by the first scan and is of O(m2 log m) .

STEP 3: Delay the start times of eligible activities that were not scheduled.

Activities that were not scheduled in the previous step remain in the schedule queue withtheir start times delayed to the instant at which additional resources become available.Precedence feasible activities whose start times are current at that time are then added tothe queue of eligible activities. This procedure examines activities and updates the networkin O(m) / O(n 2) time.

STEP 4: Resolve the network flow problem with the modified start times from step 3and go to step 2.

The modified unconstrained minimum-cost network flow problem has changed durationson the arcs (cost parameter in the dual) and changed present values at the nodes (right-hand-side parameter in the dual) . To derive the fair cost of delaying activities in a partiallycompleted schedule, events associated with activities that have been completed whose timingcannot be changed are treated as tied events [19]. The updated network is solved with theuse of an efficient dual simplex algorithm developed for the minimum-cost network flowproblem [1] by employing the optimization procedure for solving the modified problem andthe reoptimization procedure for convergence to optimality. Because the dominant step ineach iteration is the solution and convergence of the transshipment model, the well-knownadvantages of solving large transshipment problems with a relatively small amount ofexecution time are exploited in the efficient implementation of a greedy single-passalgorithm.

The number of iterations required to obtain the final schedule is equal to the number ofdelays encountered while scheduling the activities. This can be bounded by the sum of thedurations of all of the activities. In particular, when the problems are very highly resourceconstrained, the number of iterations is bounded by m , the number of activities in theproject network. In actual testing, however, the number of iterations was observed to be afraction of the bounds provided above.

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Table 1. Benchmark priority rules.

Rule Description

LFT/LAN Priority is given to the activities with the minimum late finish time (LFT) asdetermined by CPM. LAN is used to break ties.

MS/LAN Activities are sequenced according to the minimum slack rule, where the slack foractivity k is the difference between Lk and Ek , slack is updated during thescheduling process, and priority is given to the activity with minimum slackvalue. LAN is used to break ties.

CFW/LAN Activities are scheduled in descending order to cash-flow weight (CFW), as definedin Section 3.1.1. Ties are broken with the use of LAN.

TS/LAN The target scheduling rule (TS) assigns priority to the activity with the maximumdifference between the current revised early finish time in the resource-constrained schedule and the optimal unconstrained finish time, Tj (k ) . Ties arebroken with LAN.

TS/Dual Activities are sequenced based on the target scheduling (TS) rule. Ties are brokenby selecting the activity with the maximum dual price, as defined by [20].

Dual/TS Activities are sequenced in ascending order of dual price values, as defined byR.A. Russell. Ties are broken using the target scheduling (TS) rule.

RAND-50 Activities are selected at random from the queue of resource and precedencefeasible activities. Fifty schedules are generated for each problem and theschedule with the maximum NPV is reported.

4. EXPERIMENTAL DESIGN

We performed three sets of experiments, including problems from the literature, and twosets of experiments involving larger projects ranging in size from 48 to 1000 activities. Thefirst set of experiments were performed on 50 problems from the R.A. Russell study [20].These problems were derived from the 110 test problems collected by Patterson [16] andranged in size from 21 to 24 activities. R.A. Russell converted the Patterson problems toactivity-on-arc form and randomly assigned cash flows ranging between 0$10,000 and$10,000 to each node of the networks. The final payment was set such that each projectwould have a positive NPV.3

In both the first and second set of experiments, the nine RSTP heuristic procedures werecompared with the single-iteration procedures from R.A. Russell [20] and a set of heuristicsthat employ information derived from CPM (Table 1). Russell’s single-iteration heuristics(TS/LAN, TS/Dual, and Dual/TS) use the dual prices and target scheduling dates derivedat the beginning of the schedule generation process from the unconstrained network optimi-zation model. A cash-flow weight rule (CFW-LAN) [2, 18] was included as a benchmarkrule because of its good NPV performance in earlier studies.

To reflect a wide range of project scheduling environments, the second set of computa-tional experiments was generated from six experimental factors, including problem size,project structure, resource constrainedness, interest rate, frequency of progress payments,and profit margin. A full factorial experiment was performed, resulting in the schedulingof 1440 different project network problems including 144 scheduling conditions with 10replicates in each cell.

Because the NPV project scheduling heuristics exhibited significantly different perfor-

3 For the purposes of this research, each test problem is modified with dummy arcs such that eachcash-flow occurs at a unique node in the project; that is each cash outflow occurs on the i(k) node,and each cash inflow occurs on the j(k) node. All other parameters remained the same.

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mance depending on problem size [20], in this set of experiments problems of two sizeswere generated: 48 activities and 110 activities. Networks of three distinctly different shapes,balanced, skewed to the right, and skewed to the left were used to learn whether heuristicperformance was affected by the experimental variable, project structure. Because resourcerequirements and cash flows were randomly assigned to activities, the peak resource require-ments and cash flows occurred, on average, to the right or left in the skewed networks,whereas they were evenly distributed in the balanced networks. Activity durations weregenerated randomly from a uniform distribution with endpoints of one and nine periods.Precedence relationships between activities were randomly generated in activity-on-nodeform, such that the coefficient of network complexity, as defined by Pascoe [15] to representthe interconnectedness of the network, was set at an average of 1.85, and ranged from 1.7to 1.9. The networks were then converted to an activity-on-arc form for purposes of usingthe A.H. Russell network flow formulation.

Projects were scheduled subject to constraints on three different resources, and betweenone and nine units of each resource was randomly assigned to each activity. The degree ofresource constrainedness was implemented as an experimental factor using the averageutilization factor (AUF) [11], the ratio of total work content to total resources over themakespan of the unconstrained CPM schedule. As in R.A. Russell [20], AUF was calculatedas an aggregate over the project makespan, rather than by finding the average of the ratiosin each period of the project. Three factor levels were used, ranging from the low settingof AUF equals 1.0 to the medium setting of 1.5 and the high setting of 2.0.

Three cash-flow experimental variables were defined in this set of experiments, becauseof the lack of extensive computational testing with respect to cash-flow factors in previouswork [2, 20]: frequency of progress payments, profit margin, and cost of capital. A highfactor level of frequency of progress payments generated experiments with an averageprogress payment frequency of one for every third activity, while the average frequency atthe low factor level was once every seventh activity. Cash outflows were randomly selectedfrom a uniform distribution ranging between $1 and $100. Each progress payment was setequal to the cost of the activities preceding the progress payment and a 10% profit. Costof capital and profit margin were also assigned two factor levels. The cost of capital wasset at a low level of 10% annually, and a high level of 20% annually. The final paymentfor the project was set at a low level approximately equal to the sum of the cash outflowsfor the project plus 30% and a high level equal to the sum of the outflows plus 50%.

We tested a reduced set of RSTP heuristics in the third set of experiments to determinethe capabilities of the various heuristics in meeting the computational requirements ofscheduling large problems with 500 and 1000 activities. Given their performance in thefirst and second set of experiments, we used the MTP-ET, LTP/LAN, OCR/LAN, andCFW-OCC RSTP heuristic procedures for scheduling the large problems. The benchmarkheuristics chosen for this portion of the study included MS/LAN, TS/LAN, and CFW/LAN.

The cost of capital was set at 20% and profit margin was set at 130% of total cashoutflows. Problem structure was again set at three levels: centered, left-skewed, and right-skewed. Precedence relationships were randomly generated, such that the coefficient ofnetwork complexity averaged 1.75 across all projects, with a range of 1.5–1.9. Resourceutilization was set at AUF values of 1.2 and 1.8. Progress payment frequency was againset at a low level of one payment for every seven activities and a high level of one paymentfor every three activities. However, although the progress payments for the small problemswere generated based on a sequential ordering of the activities, the progress payments forthe large problems were generated in linked packages, such that each cash inflow takes

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Table 2. Results for the first two experiments: mean percent below maximum NPV schedule.

Projectsize

(activities) MTP/LAN OCS/LAN OCR/LAN NOC/LAN TSCFW MTP-ET CFW-CFW CFW-OCC

21–24 1.21 1.21 1.24 1.30 1.50 1.31 1.47 1.7648 8.72 7.00 6.70 6.78 6.51 3.50 3.27 2.56

110 2.67 2.49 2.56 2.93 2.54 1.83 1.73 1.58

Projectsize

(activities) LTP/LAN LFT/LAN MS/LAN CFW/LAN TS/LAN TS/Dual Dual/TS RAND-50

21–24 1.53 1.57 1.60 1.39 1.34 1.37 3.22 0.4248 3.09 10.21 10.86 3.22 6.84 8.26 6.52 4.64

110 1.86 6.60 6.04 4.55 3.20 2.98 6.66 4.61

place upon the completion of a linked set of activities. Upon completion of the package, aprogress payment was paid equal to 40% of the cash outflows for the package. Finally, fivereplicates were generated for each of the 12 scheduling conditions, resulting in 120 differentproject network problems.

5. RESULTS

A summary of the relative performance of the heuristic priority rules for first two experi-ments, including the fifty 21–24-activity problems from R.A. Russell [20] and the 48- and110-activity problems is displayed in Table 2. The average performance is listed for eachheuristic relative to a resource feasible upper bound, defined as the maximum NPV valuederived by a heuristic rule for a given test problem. A similar measure was used to establisha bound by Morton, Lawrence, Rajagopolan, and Kekre [12]. The NPV results in Table 2for the 21–24-activity problems are similar to those found by R.A. Russell, in that theRAND-50 rule provided the best average NPV performance, whereas Dual/TS was theworst-performing heuristic. The best-performing RSTP heuristics, MTP/LAN and OCS/LAN, only slightly out-performed R.A. Russell’s single-iteration rules (TS/LAN, TS/Dual,and Dual/TS). The two-queue RSTP heuristic procedures did not perform as well as thesingle-queue revised RSTP heuristic procedures. As in previous studies, CFW/LANprovided a higher average NPV value than the duration minimizing rules, LFT/LANand MS/LAN.

The small difference in NPV performance across the 16 heuristic rules for the 21–24-activity problems is consistent with the performance found by R.A. Russell. Perhaps thesmall differences in NPV can be attributed to the size, resource requirements, and cash-flow structure of the 21–24-activity problems. Because cash flows for these problems wereselected from a uniform distribution ranging from 0$10,000 to $10,000, many nodes re-ceived cash inflows; thus the benchmark heuristics that have been shown to provide minimalproject durations may have provided performance equal to that of heuristics designed tomaximize project NPV. For instance, if positive cash flows are assigned primarily to thecritical path, the MS/LAN and LFT/LAN rule may exhibit good NPV performance. Inaddition, the duration of these small projects may be sufficiently short that there is littlereduction in discounting factors when an activity is delayed by a short period as comparedto another activity.

The results for the 48- and 110-activity problems reported in Table 2 illustrate that

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Table 3. Heuristic performance for the 500- and 1000-activity projects.

Number ofactivities OCR/LAN MTP-ET CFW-OCC LTP/LAN MS/LAN CFW/LAN TS/LAN

500 11.02 4.23 1.69 8.33 26.57 27.29 8.911000 9.01 4.80 1.34 7.15 24.32 28.56 10.22Overall

mean: 10.01 4.51 1.51 7.74 25.44 27.93 9.56

revision of the unconstrained NPV model to account for delays in the partially generatedproject schedule significantly enhances project NPV, particularly if a front-loading schedul-ing rule is employed. The CFW-OCC priority rule generated higher NPV schedules thatwere, on average, 2.07% below the best-performing heuristic solution to each problem. TheLTP/LAN, CFW-CFW, MTP-ET, and CFW-LAN scheduling rules derived average NPVvalues that were 2.48%, 2.50%, 2.66%, and 3.89% below the average best performance,respectively. OCR/LAN outperformed the other two opportunity cost rules, and TSCFWprovided the worst performance of the cash-flow weight rules. Except for MTP/LAN, theRSTP heuristics produced higher average NPV than R.A. Russell’s single-iteration rules.In comparison to the new rules, the benchmark heuristics LFT/LAN and MS/LAN werethe worst performers. Although the RAND-50 rule yielded maximum NPV schedules forthe 21–24-activity projects, the results for the 48- and 110-activity problems suggests thatthe relative performance of the RAND-50 rule declines as project size increases. It couldbe hypothesized that although the RAND-50 rule provides an extensive, if not complete,enumeration of the possible solutions to the first set of problems (21–24 activities) , thiswas not the case for the larger problems, particularly for those with 110 activities or more.In fact, it was found for a limited set of the large problems used in the third set of experimentsthat when the number of random solutions was increased from 50 to 5000, the RAND-5000 rule failed to outperform the three to four best-performing rules. The execution timefor RAND-5000, in contrast, was approximately a factor of 100 times greater than thatrequired by the optimization-guided heuristic algorithm.

Duncan’s multiple range test [8] was performed at the 0.05 level of significance withthe use of the results from the 48- and 100-activity problems to identify statistical differencesin NPV generated by the heuristic rules. The results indicate that there are statisticallysignificant differences in project NPV between the front-loading scheduling rules, MTP-ET, CFW-CFW, CFW-OCC, and LTP/LAN and a group of the worse-performing heuristics:LFT/LAN, MS/LAN, and Dual/TS. Duncan’s multiple-range tests were also performedfor each of the 144 experimental factor combinations. There were statistical differences inNPV performance in 26 of the 72 scheduling environments for the 48-activity problems,and for only 13 of the 110-activity problems. Further details of these results are pro-vided in [14].

The results for the third set of experiments, which used a reduced set of RSTP andbenchmark heuristics, are shown in Table 3. CFW-OCC and LTP/LAN, the strongestperformers in the second set of experiments, were included in this set, along with MTP-ET, to further explore the performance of the front-loading concept. The opportunity-costconcept was tested with the use of the OCR/LAN rule because it had the strongest perfor-mance of the opportunity-cost rules. We included the TS/LAN single-iteration rule toprovide a benchmark to measure the value of revising the unconstrained network solution.The results for the large problems illustrate the strong performance of the front-loading

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Table 4. Relative performance of heuristics rules: project size, structure, and resource utilizationa.

Structure Activities Utilization OCR/LAN MTP-ET CFW-OCC LTP/LAN MS/LAN CFW/LAN TS/LAN

Right 48 andskewed 110 1.0 1.01 1.01 0.89 1.66 3.18 4.05 1.72

1.5 4.50 1.93 1.58 3.44 7.74 4.59 4.602.0 7.66 3.06 2.21 2.94 16.34 5.69 9.52

500 and1000 1.2 6.42 3.11 1.00 5.32 15.56 17.32 3.87

1.8 30.68 9.51 0.02 14.39 69.94 68.35 34.76

48 andBalanced 110 1.0 1.14 1.55 1.56 1.82 1.90 3.07 0.73

1.5 3.96 3.22 2.24 2.55 6.76 3.51 3.922.0 8.70 4.68 3.07 2.86 14.20 3.57 9.38

500 and1000 1.2 2.71 4.59 4.66 7.38 7.85 13.92 1.29

1.8 10.88 3.78 0.32 8.79 30.94 31.64 10.97

Left 48 andskewed 110 1.0 1.11 1.19 1.19 1.58 2.26 2.83 0.92

1.5 5.94 3.00 2.64 2.39 8.96 3.53 4.602.0 7.55 4.42 3.50 3.40 13.59 4.05 9.05

500 and1000 1.2 1.92 2.91 2.44 5.17 6.23 11.42 1.24

1.8 7.47 3.18 0.64 5.41 22.16 24.91 5.26a The tabled values represent the average performance of each rule relative to the best solution toeach problem.

concept. CFW-OCC and MTP-ET generated higher average NPV schedules that were 1.51%and 4.51% below the best heuristic solution to each problem. In contrast, the benchmarkheuristics MS/LAN and CFW/LAN performed poorly, yielding NPV values that were, onaverage, 25.44% and 27.93% below the best-performing heuristic.

Duncan’s multiple-range test was performed for each of the 12 experimental conditionsused for the third set of experiments. Results from this analysis also support the superiorNPV performance of CFW-OCC. In particular, CFW-OCC dominates MS/LAN and CFW/LAN in 71% of the scheduling environments, in addition to dominating the other front-loading rules, and MTP-ET and LTP/LAN dominate in 29% and 42% of the environments,respectively. However, for balanced, 1000-activity projects with low resource utilizationand low frequency of progress payments, TS/LAN and OCR/LAN derived statisticallyhigher NPV schedules than CFW-OCC.

To analyze the influence of the experimental variables, we examined differences inheuristic performance for combinations of project structure and resource utilization for thesecond and third sets of experiments. Table 4 illustrates that the front-loading rules obtainedsignificantly higher NPV performance under conditions of either high resource utilizationor right-skewed structure. We expected the front-loading rules to perform better for right-skewed projects, because most of the progress payments are positioned toward the beginningof the project, where differences in discounting factors are the largest. In addition, highresource utilization leads to extension of the project makespan, which increases the impor-tance of those scheduling decisions made in the earlier portion of the project’s duration andagain leads to better performance by front-loading rules, because they give priority toactivities leading to immediate progress payments.

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Further insights on the efficiency of CFW-OCC can be acquired by comparing its perfor-mance with the other front-loading rules. In comparison to the two-queue-rule MTP-ET,CFW-OCC performed significantly better at high resource utilization levels, particularly forthe 500- and 1000-activity projects. It is likely that CFW is a superior first-queue rule in highutilization environments, because it recognizes not only cash flows generated by immediateprogress payments, but also downstream progress payments. With respect to the second-queue rules, the OCC rule also uses cash flow weights, again giving priority in schedulingto the activity on the path leading to additional positive cash flows, whereas the ET rulepromotes early completion of the project.

LTP/LAN produced slightly higher average NPV solutions than CFW-OCC in the 48-and 110-activity problems for balanced or left-skewed projects under conditions of highresource utilization. This performance differential might be explained by the LTP/LANrule’s tendency to complete paths leading to immediate progress payments more quicklythan the CFW-OCC rule. However, this behavior may not be as critical to NPV performancein larger projects, where CFW-OCC significantly outperformed LTP/LAN. For large proj-ects, a larger number of longer, multiple paths of activities must be scheduled. Althoughthe LTP/LAN rule will tend to focus on paths leading directly to progress payments, theCFW-OCC rule tends to alternate in the selection of activities along multiple paths leadingto cash inflows, thus scheduling the larger number of progress payments available in largeprojects earlier than LTP/LAN. In conclusion, although LTP/LAN was the best-performingrule for the 48- and 110-activity high utilization projects, it performed significantly worsethan CFW-OCC in several environments, which suggests that CFW-OCC may be the pre-ferred scheduling rule. Furthermore, the results for the Duncan’s multiple range test forthe 500- and 1000-activity projects indicate that CFW-OCC statistically dominated theperformance of LTP/LAN in 10 settings, although LTP/LAN never statistically outper-formed CFW-OCC.

TS/LAN performed better than all other rules, including CFW-OCC for projects of allsizes with low levels of resource utilization that were balanced, or skewed to the left. Thissuperior performance may be explained by two characteristics of the TS/LAN rule devel-oped by R.A. Russell. First, because R.A. Russell’s implementation of TS/LAN releasesan activity to the schedule queue once it is precedence-feasible rather than at its updatedtarget schedule date, there are more activities in the schedule queue, thus promoting theexpedient completion of the project. Second, it is likely that TS/LAN performs well underconditions of low resource utilization because activities can be scheduled close to timesprovided by the initial unconstrained NPV-optimal schedule. In contrast, the RSTP heuristicsand algorithm do not consider activities for scheduling until they have reached their revisedtarget schedule times.

To assist the reader in selecting a rule given a particular experimental environment, wesummarize the results of the second and third sets of experiments in Table 5, where we listthe best-performing heuristic for each combination of problem size, structure, and utilization.As noted previously, CFW-OCC provided the highest mean project NPV for all rulesincluded in the study, and exhibited the best NPV performance for a greater number ofcombinations of resource utilization, project structure, and problem size than any otherheuristic rule. In particular, CFW-OCC exhibited the best NPV performance for right-skewed networks at all levels of resource utilization. In contrast, TS/LAN was the bestperforming rule under conditions of low resource utilization and balanced or left-skewedproject network structure. Finally, CFW-OCC usually provided the best performance underconditions of medium to high resource utilization, and in those cases where CFW-OCC

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Table 5. Best performing rule: project structure, size, and resource utilization.

Structure

Size Utilization Right skewed Balanced Left skewed

48 and 110 activities 1.0 CFW-OCC TS/LAN TS/LAN1.5 CFW-OCC CFW-OCC LTP/LAN2.0 CFW-OCC LTP/LAN LTP/LAN

500 and 1000 activities 1.2 CFW-OCC TS/LAN TS/LAN1.8 CFW-OCC CFW-OCC CFW-OCC

performance was exceeded by that of LTP-LAN, CFW-OCC was the second best performingrule by a relatively small margin. Therefore, in most cases characterized by medium tohigh resource utilization, we would recommend the use of CFW-OCC in addition to thatof LTP-LAN, as CFW-OCC still provided the best NPV value in for some replications inenvironments where LTP/LAN had the highest mean NPV performance.

All results were obtained using FORTRAN77 implementations of the various schedulingheuristics on a Cray II computer at optimization level 2. However, for purposes of practicalapplications, it was found that computation time for the most highly constrained 1000-activity problems with the use of an 80486-based 50-mHz personal computer were approxi-mately 10 minutes per project. In contrast, the benchmark single-pass algorithms requiredapproximately 30 s of computation time. Although the reoptimization algorithm requiredhigher computation time, it seems that practical-size problems may be solved in an accept-able amount of time with the use of small computers, with the resulting advantages ofsignificant improvements in project NPV.

6. CONCLUSIONS

This research has implications for both practitioners and academicians. From the prac-titioner’s perspective, this study demonstrates that the front-loading practice implementedin the context of a heuristic procedure yields high NPV schedules. In addition, the relativeperformance of these procedures increases with project size, particularly for the CFW-OCCrule, thus illustrating the value of their use for problems of the size frequently found inindustrial settings. A further advantage of the front-loading priority rules is that the availabil-ity of additional capital earlier in the project can result in more effective cash-flow manage-ment. The results indicate that heuristics whose objective is to minimize the makespan ofresource-constrained projects should not be used when the measure of project success isNPV. From the researcher’s perspective, the results suggest that NPV can be improved byrevising the network flow solution in the generation of a resource-constrained schedule.The best performing revised optimization rule benefited from the use of target scheduledates from the revised solution to the network model. Finally, the results illustrate that theperformance of single-pass heuristic procedures must be evaluated on larger resource-constrained projects than networks with 25 activities, as was previously reported in theliterature.

REFERENCES

[1] Ali, A.I., Padman, R., and Thiagarajan, H., ‘‘Dual Algorithms for Pure Network Problems,’’Operations Research, 37, 159–171 (1989).

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[2] Baroum, S., and Patterson, J.H., ‘‘A Heuristic Algorithm for Maximizing the Net Present Valueof Cash-Flows in Resource-Constrained Project Schedule,’’ Working Paper, Indiana University,1989.

[3] Bey, R., Doersch, R.H., and Patterson, J.H., ‘‘The Net Present Value Criterion: Its Impact onProject Scheduling,’’ Project Management Quarterly, 12, 35–45 (1981).

[4] Davis, E.W., ‘‘Project Scheduling Under Resource Constraints—Historical Review and Catego-rization of Procedures,’’ AIIE Transactions, 5, 297–313 (1973).

[5] Davis, E.W., and Patterson, J.H., ‘‘A Comparison of Heuristic and Optimum Solutions inResource-Constrained Project Scheduling,’’ Management Science, 21, 944–955 (1975).

[6] Demeulemeester, E., and Herroelen, W., ‘‘A Branch-and-Bound Procedure for the MultipleResource-Constrained Project Scheduling Problem,’’ Management Science, 38, 1803–1818(1992).

[7] Doersch, R.H., and Patterson, J.H., ‘‘Scheduling a Project to Maximize its Present Value: AZero-One Programming Approach,’’ Management Science, 23, 882–889 (1977).

[8] Duncan, D.B., ‘‘Multiple Range and Multiple F Tests,’’ Biometrics, 11, 1–42 (1955).[9] Elmaghraby, S.E., and Herroelen, W., ‘‘The Scheduling of Activities to Maximize the Net

Present Value of Projects,’’ European Journal of Operational Research, 49, 35–49 (1990).[10] Garey, M.R., and Johnson, D.S., ‘‘Computers and Intractability: A Guide to the Theory of NP-

Completeness,’’ W.H. Freeman and Co., New York, 1979.[11] Kurtulus, I.S., and Davis, E.W., ‘‘Multi-Project Scheduling: Categorization of Heuristic Rules

Performance,’’ Management Science, 28, 161–172 (1982).[12] Morton, T.E., Lawrence, S.R., Rajagopolan, S., and Kekre, S., ‘‘SCHED-STAR: A Price-Based

Shop Scheduling Module,’’ Journal of Manufacturing and Operations Management, 1, 131–181 (1988).

[13] Orlin, J.B., ‘‘Genuinely Polynomial Simplex and Non-Simplex Algorithms for the MinimumCost Network Flow Problem,’’ Sloan Working Paper No. 1615-84, Sloan School of Manage-ment, Massachusetts Institute of Technology, December 1984.

[14] Padman, R., Smith-Daniels, D.E., and Smith-Daniels, V.L., ‘‘Heuristic Scheduling of Projectswith Cash Flows: An Optimization-Guided Approach,’’ Working Paper, Heinz School of PublicPolicy and Management, 1993.

[15] Pascoe, T., ‘‘An Experimental Comparison of Heuristic Methods for Allocating Resources,’’unpublished Ph.D. thesis, Cambridge University, 1965.

[16] Patterson, J.H., ‘‘A Comparison of Exact Approaches for Solving the Multiple Resource Con-strained Project Scheduling Problem,’’ Management Science, 30, 854–867 (1984).

[17] Patterson, J.H., Talbot, F.B., Slowinski, R., and Weglarz, J., ‘‘Computational Experience witha Backtracking Algorithm for Solving a General Class of Precedence and Resource-ConstrainedScheduling Problems,’’ European Journal of Operational Research, 49, 68–79 (1990).

[18] Pinder, J.P., ‘‘A Hierarchical Project Management Model,’’ unpublished doctoral dissertation,University of North Carolina, Chapel Hill, 1988.

[19] Russell, A.H., ‘‘Cash Flows in Networks,’’ Management Science, 16, 357–373 (1970).[20] Russell, R.A., ‘‘A Comparison of Heuristics for Scheduling Projects with Cash Flows and

Resource Restrictions,’’ Management Science, 32, 1291–1300 (1986).[21] Willis, R.J., ‘‘Critical Path Analysis and Resource-Constrained Project Scheduling—Theory

and Practice,’’ European Journal of Operational Research, 21, 149–155 (1985).[22] Yang, K.K., Talbot, F.B., and Patterson, J.H., ‘‘Scheduling a Project to Maximize Its Net Present

Value: An Integer Programming Approach,’’ European Journal of Operational Research, 64,188–198 (1993).

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heuristic scheduling of resource-constrained projects with cash flows

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