heuristic scheduling of capital constrained projects

Dwight E. Smith-Daniels ‘.*? Rema Badman b> Vi&i L. Smith-Daniels ’

Abstract

The movement to produe: and process development projects that involve joint ventures among strategic partners, as well as the increasing prevalence of projects within organizations has led to increased implementation of project scheduling methods. It is frequently the case that a capital constraint is placed on a project, thus limiting the number and value of activities that can be scheduled to occur simultaneously. FIowever, the quantity of capital nvailabie to schedule activities can increase as additional cash is received as progress payments for completed activities. Since the project manager’s objective is to maximize project Net Present Value (NPV), it is important for the manager to develop a schedule that balances the early receipt of progress payments (which improve NPV and increase the capital balance avai!able), with the delay of particularly large expenditures. Due to the intractability of optimal methods, the use of heuristic methods is required to solve problems of practical size. This paper presents the first test of heuristic methods for solving this problem. We use information from a relaxed optimization-guided model that employs information from the unconstrained NPV-optimal problem in a heuristic procedure for solving the capital constrained probiem. An experimental design is employed to test the heuristics that includes multiple factor levels for a number of project characteristics, includfng capital utilizatton, frequency of progress payments, and project network structure. The results indicate very good relative performance for the optimization-guided procedures as compared to two benchmark heuristics.

1. Introduction

Recent efforts to make organizations more com- petitive have moved the management of projects to the forefront of organizational change and redesign. Project-based work has been the notm for decades in industries such as construction, defense and enter- tainment. Now, as organizations restructure their processes and boundaries and transform from hierar- chical, functionally-oriented work to flattened, reengineered organizations, project-based work is

* Corresponding author.

emerging as the prevalent work structure throughout many organizations (Stewart, 1995a,b). Contributing to the trend toward projects has been the strategic realignment of people, products and resources around an organization’s core competencies. Rather than owning the entire value chain of activities, organizations are focusing on core competencies by identifying those value chain activities where they can be “‘world class”, and outsourcing or forming alliances and partnerships with other firms for those tasks that can be performed with ‘“best in world” capabilities by the partner (Bowen et al., 1994). These emerging organizational trends suggest that the project manager must place greater emphasis on acquiring and coordinating external project resources sources such

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as suppliers, joint venture partners, and suhcontrac- tars. Moreover. as organizations continue to fact intensified competition and low profit margins, the major constraining factor for managing projects and acquiring project resources will probably be the avail&ility of capital to finance new and continuing projects.

Today’s changing project environment challenges the project manager to balance not only the conflict- ing priorities engendered by time, resources, capital and budget, but also ihe impact of cash flows on the project plan, schedule and perfcnmance. In their groundbreaking paper, Doersch and Patterson (l97?) define the capital-coll;strained project scheduling problem (CCPSP) as one of scheduling a project v~ith Wh positive and negative cash flows that take place over the course of the project, where investment in project activities is constrained by a capital constraint. Pro-ject cashiloh j include cash outflows rttr pcr~mnel, L’L~I~~~MCEN a!ld materials, and cash inflows received as progress payments from internal or external customers and clients for completion of a portion of a project. Typically (Bey et al., 198 1) project investors will place a constraint on the amount of funds that can be outstanding on work on project aclivities at any point in time. The capital constraint is usually imposed on a project to limit the amount of capital that may be expended per period for internal resources, suppliers, joint venture partners, and subcontractors for project activities. In organizations wiih limited capital to invest in new and continuing projects, reinvesting progress payments provides for internal accountability for completing por- tions of the projects and a source of capital in addition to that provided by investors and partners that can be used for earlier scheduling of project activities. Thus, the amount of capital available in each period is a renewable resource, where the investors make the same amount of capitA available in each subsequent period of the project, plus any progress payments (cash inflows) that are reinvested in the project.

Bey et al. (1981) illustrate that the NPV objective function is the appropriate measure of performance for the CCPSP. The use of the NPV objective function captures the trade-offs between (a) the delay of activities incurring large capital requirements and/or cash outtl~ws, (bj early receipt of cash inflows

through the early complctior. of activities leading to cash inflows. and (ci the reduction or avoidance of any delay penalties or reductions in revenue associated with late completion of project activities.

While tile management uf projects with cash flows and limited capital appears to be emerging as a comma.. organizational work structure, to date there has hocn little research on the CCPSP. Rather. the majority of prqject schedulin; Isearch has focused on developing exact and heuristic methods for the classic resource constrained project scheduling problem (RCPSP) (Davis, 1973). In the RCPSP. activities are scheduted subject IO externally imposed constraints on the availability of non-storabfe renewable resources such as labor and equipment. It is assumed in the RCPSP that the objective in scheduling activities is to minimize project duration. In contrast. in the CCPSP the amount of capital available to schedule activities at any point in the schedule increases as a function ol‘ the selection of previously scheduled activities, since any profits from progress payments can be reinvested as capital in the project. By reinvesting progress payments in a project, the increased capital budget can be used to schedule activities earlier in the project. which rnajj lead to a shorter project duration and/or higher project NPV. More- over, the exact and heuristic algorithms designed for the RCPSP may be unable to find a feasible schedule for some instances of the CCPSP. For example, this situation can occur when a project begins with a tight capital constraint and the subsequent cash outflows for one or more activities tead to a reduction in the capital available to a level below that required to schedule succeeding activities.

In this paper we present three heuristic procedures for solving the CCPSP and test their performance in solving relatively large capital-constrained projects with cash flows. Two of the heuristic procedures tested in this research utilize information from the optimal solution to the unconstrained NPV project schedule in the generation of a solution. The other heuristic procedure is based on a simple cash flow priority rule embedded in a forward pass algorithm that considers project cash flow structure in resolv- ing resource constraints (Baroum and Patterson, 1993). An experimental design tests the performance of these heuristic procedures in maximizing project NPV for a number of project characteristics includ-

ing project structure. number of progress payments, and amount of capital. Solutions generated from the heuristic procedures are compared to a benchmark solution that is the best NPV schedule for 50 ran- dl+mly generated project schedules. The results of this computational study indicate that optimization- guided heuristics generate significantly higher NPV schedules than the cash flow priority heuristic, and the random scheduling rule outperforms the cash flow priority sequencing rule.

In the next section, we provide a mathematical formulation of the CCPSP and an example project. In Section 3 ,ve review the previous research. In Section 4 we present the revised-optL;:;zation and cash flow weight heuristic procedures. The test experiments used in evaluating the performance of the heuristic procedure sre described in Section 5. ‘&he results of these experiments are given in Section 6 and the conclusions implications of the study for researchers and managers are presented in Section 7.

2. Problem statement

The problem formulation presented in this section is based upon the binary integer programming formulation of Doersch and Patterson (1977). In contrast to their formulation. which utilized an activity- on-node network model, we use an activity-on-arc network. The activity-on-arc network formulation is required to facilitate the use of an efficient algorithm to solve the unconstrained NW project scheduling problem, thus we adopt that approach to formulate the constrained problem.

Notation and definitions used in the formufation include:

Co: total capital availqble at beginning of the project in period 0.

Zk: capital investment required by activity k, k = 12 , , . . ..m.

F;(,,:cash outflow at the beginning of activity r4 at node i, where each activity is defined by nodes i and j.

Fjt,,:cash inflow received upon completion of activity k at node j.

d,: duration of activity k, where k may not be preempted,

7;,,,: time at which node i of activity k is scheduled to occur.

Z~i(i):the set of activities, LS, that are scheduled to be active in period r.

Z,ril:the set of activities, p, completed prior to period r.

cr: opportunity cost of capital.

The terms in the objer live function represent cash outflows, cash inflows and capital costs, respec- tively, where each component is discounted back to the beginning of the project:

+ [I’,exp( -ad,) - j,fexp( - ojqcn)).

The activity precedence constraints are

T& - T,,, 2 d, 3 k= I,2,...,02.

As the project is enacted, the net capital balance reflects positive and negative cash flows associated with activities and nodes completed in previous periods. As in Doersch and Patterson (1977); we assume that capital is a renewable resource, where the initial capital availability ce is augmented or reduced by the cash flows that occur throughout the project. Thus, the capital constraints for each period of the project are

c I,~%+ c (qw+F;,,,)* k E z 4r, kEqw,

The formulation illustrates two distinct properties of the CCPSP as compared to the RCPSP. First, the implication of the capital constraint set is that the capital availability is dependent on previously scheduled activities. IJnlike the RCPSP, where the initial resource availability in any period is independent of the project schedule, the resource availability in this problem changes as a function of the cash flows associated with the schedule of activities. Second. since activity cash flows are added and deducted from the capital constraint as they occur over the life of the project, it is possible that a heuristic procedure may generate a partially complete schedule where it is impossible to complete the project due to insuffi- cient capital availability. This usually would not be

Fig. I. Project network and ttme-only schedule for the example.

true for the RCPSP except in cases where the re- ~:XUXY constraint varies over the course of the project.

fi!‘e ~wJ use an example adapted from the literature (Doersch and Patterson, 19771 to illustrate the two properties of the CCPSP discussed above. An activity-on-node diagram or rhe example is illustrated in Fig. 1 aiong with the time-only early and late start schedules for the project. The time-only critical path is l-3-5-7-8. In Table I we present the capital requirements and cash flow data for the project. The cost of capital is 2% per period, and an initial capital constraint of 80 is placed on the project.

The optimal solution is illustrated in the chart in Fig. 2(a). The top line in this graph illustrates the capital availability, including the additions and de- ductions during each period of the project. The boxes represent the capital requirements for each activity. The cash flows associated with each activity are added to or deducted from the capital constraint in the period following their occurrence. For instance, the cash flow of 4 occurring in the last period of the

first activity is added to the initial capital availability of 80 to yir:td capital availability of 84 in the third period. An obvious implication of the increasing capitat availability is that the pairs of activities scheduled in periods 5 through 9 could not be sched- uIed simultaneously if we followed the approach taken in the RCPSP and fixed the resource (capital) constraint at 80 throughout the project.

In contrast to the optimal solution in Fig. 2(a), the schedule in Fig. 2(b) was generated by prioritizing activities according to a minimum slack heuristic rule. where capital is allocated first to the activity with the minimum slack value. This rule was applied using a forward pass algorithm, where activities were considered in parallel for allocation of capital. Prece- dence feasible activities were prioritized for schedut- ing in ascending order of the slack values computed from the time-only project schedule. This schedule is

Fig. 2. Capitnl re~tuir~merits ~lnd availability for the e%unple. (a) Optimal capital constrained schedule. @a) Schedule generated using the minimum sltlck heuristic.

identical to the o;:imal schedule for the first three activities. However in period 5. \&cre activities 4, 5 and 6 are available For scheduling, the use <xi z.re minimum slack heuristic leads to the selection of activities 4 and 5 for scheduling rather than 4 and 6. The start of activity 6 is &en delayed until period 7 to be scheduled to start in month 5 instead of activity 6, which results in a project duration of 12 months, a reduction of two months. However, project NPV is reduced from 83.5 to 83. This illustrates a property that both the CCPSP and RCPSP have in common: the dur&on-optimal resource constrained schedule may have a shorter duration than the NPV-optimal solution. Thus, heuristics that performed well for the duration objective may not yield particularly good NPV performance.

Another feature of the CCPSP - the feasibility property - can be illustrated through a modification of the example above. Suppose that the capital requirements for activities 4, 5 and 6 are now as illustrated in Table 2. In this case. if activity 5 is scheduled before activities 4 and 6, a feasible solution to the problem is not possible. At the beginning of period 5, when activity 5 would be scheduled to begin, available capital is 100, while activity 5 requires 100. The available capital in period 6 would need to be at least 105 for activity 5 to continue to completion, since a deduction of 5 from capital takes place in the second period of activity 5. In contrast, if activities 4 and 6 are scheduled before 5, the capital balance upon the completion of 4 and 6 would be 137, which would be sufficient to schedule 5 upon the completion of both 4 and 6. The project duration would be 16 months, since activity 7 couid not begin until 5 were complete, due to the capital constraint.

3. Batkground

Optimal solutions to the CCPSP are impractical if not impossible to gt~neratc for large projects since the general class of resout~ constrained scheduling problems is NP-compiete (Carey and Johnson, 19791. As in the RCPSP, the ii?herent intractability of the CCPSP requires the development of heuristic procedures, yet the authors are not aware of any research devoted to the developrnent of methods that account for the special characteristics of the CCPSP. The CCPSP and the RCPSP are both difficult if not impossible to solve optimally for projects of the size of more than 100 activities that are found in practical applications.

Since the early work of D3vi.i (1973), numerous exact and heuristic algorithms have been developed for the RCPSP (e.g.. Davis and Patterscn, 1975; Patterson, 19X4; Demeulemeester and Herroelcn. 1992). In contrast, a number of studies (Smith- Daniel:; and Aquilano, 1987; Baroum and Patterson, 1993; Padman et al.. 1994; Padman and Smith- Daniels, I9931 have found that heuristic methods developed for the RCPSP ihat perform well with respect to a duration objective function generated solutions with lower NPV, on average, than heuristics developed to maximize NPV for resource constrained projects.

A number of previous studies have developed heuristic procedures for scheduling the RCPSP with an NPV objective (RCPSP-NPV). In the first work involving inaximization of project NPV, A.H. RUS- sell (I 970) represents the unconstrained NPV project scheduling problem as flows in a ncrwork that may be solved as a series of transshipment problems. He illustrates through an example the problem structure and the derivation of dual prices representing the cost of delaying project activities. Grinold (1972) adds a project deadline parameter to this problem and shows that scheduling a project with an NPV objective can be transformed into an equivalent linear program which has the structure of a weighted distribution problem. R.A. Russell (1986) developed heuristics for scheduling the RCPSP-NPV using in- format& from the network flow model of AH. Russell (19701. R.A. Russell developed several heuristic rules that used dual prices from the unconstrained network flow model to establish scheduling

priorities within a greedy single-iteration algorithm and tested the performance of there priority ruies against heuristics that performed welt with respect to the objective of minimizing makespan. Russell’s study indicates that in medium to tightly resource- constrained projects, higher NPV schedules were derived using a priority rule using target schedule dates from the unconstrained network llow solution and the accompanying dual prices as a tie-breaker. In a subsequent study, Baroum and Patterson (19931 proposed a heuristic procedure that is based upon the application of a cash %w weight (CFW) rule. They tested single-pass PI ,;edures using weights that are the sum of cumulative future cash Ilows in the project and that prioritize activities for scheduling by giving preference to those activities vvith the highest ~:nsf; flow weight. After generating an initial schedule with the cash flow weight rule, multi-pass en- hance, rents were t.hen used to improve project NPV. Earoum and Patterson (1393) found in extensive tests that the CFW priority scheduling rule outper- Iormcd the minimum slack :,chedulirng rule in all case.5 where progress payments were received throughout the project.

Padman et al. (1994) and Padman and Smith- Danieh (19931 solved the RCPSP-NPV by modify- ing R.i:. Russell’s project scheduling approach in a numbc: of ways. First, they enhanced the greedy singre-iteration scheduling algorithm by updating the uncons&tined network model when activities are delayed beyond their optimal scheduled start times in the unconstrained NPV solution. This modification exploits the capabilities of a dual simplex algorithm for the minimum cost network flow problems (Ali et al., 19891 that allows for the efficient reoptimization of a partially completed schedule. Second, new priority rules were developed using the revised activity start times and tardiness penalties from the updated network model along with information on resource requirements, cash flows, and activity durations. Third, to estimate the reduction in project NPV due to activity delays caused by resource constraints, Padmnn et al. used a different representation of an activity’s tardiness penalty than those used by R.A. Russell as discussed below. In extensive testing of heuristic procedures on resource-constrained projects with cash flows, Padman et al. found that their revised optimization-guided heuristics derived signif-

icantly higher NPV schedules than the procedures of R.A. RusselI (19Rhj and several other heuristics including the minimum siack and CFW priority rules.

The CCPSP presents an interesting scheduling challenge given that it: reinvestment of progress payments increases the capital resources available to schedule project activities. In projects with progress payments, project managers often use front-loading to maxtmize short-term financial performance. The concept of front-loading schedules activities ihat lead immediately to progress payments as early as possible, without considering the impact on financial performance further downstream in the project. For the CCPSP, the front-loading of activities with progress payments may improve project NPV and allow for the scheduling of additionai activities when capital is increased by reinvestment of progress payments. Of- tentimes increasing the capital available for scheduling activities early in the project allows for the implementation of the NPV unconstrained network solution.

The three heuristic procedures presented here for the CCPSP are modifications of heuristic procedures developed for the RCPSP-NPV. Two heuristics are revised optimization-guided heuristics adapted from Padman ct al. (19941, modified for the increasing capital constraint in thz CCPSP. The third heuristic is Baroum and Patterson’s CFW priority rule, modified for the CCPSP (Baroum and Patterson. 1993).

3.1. Rehsed optirllizatiorl-guided heuristic procedures

The two revised optimization-guided heuristic priority rules utilize a single-pass greedy algorithm and a priority index in deriving a capital-constrained project schedule that attempts to maximize project NPV. Priority indices are computed from the dual prices and activity start dates obtained from the unconstrained NPV network model. The heuristic algorithm exploits the capabilities of a dual algorithm for the minimum cost network flow problem CAli et al., 19891 that efficiently rcoptimizes a partially completed schedule and solves the updated

UflCOWtrililEd netw0rk model after al! currently fea- sihle scheduling decisions have hcen made. Each rtptimi/,ation-guided heuristic procedure schedules activities using a priority index that incorporates delay penalties within the greedy forward algorithm. Delays in the start of activities beyond the unconstrained NPV schedule necessitate the reoptimizstion of the unconstrained NPV network problem to obtain current optimal start times and revised activity dual price:; given the partially genera!& c@cal-constrained schedule. As such, the unconstrained optimal schedule of activities may he different from the initial solution to the x-constrained network model. particularly when capital is more highly constrained in availability. In the case where capital availability is more than adequate, either due to a large initial capital investment or by receiving progr: ;E payments early in the project’s duration, the revised optimization-guided heuristics will not need to update the NPV unconstraised problem as often to reii:+-c* changes in the project schedule due to limited capital resources.

In the next section, we describe how the delay penalties used as priority indices are computed. This discussion is followed by a presentation of the two revised optimization priority rules. A description of the greedy forward algorithm for the revised optimization heuristics is presented in Section 4.13. In Section 4.1.4, the cash flow weight scheduling heuristic is presented as a means of deriving a solution for the CCPSP.

41.1. Deluy penalties The scheduling heuristics proposed in this paper

utilize delay penalties to make scheduling decisions. These penalties are computed from the dud prices and activity start dates provided by the unconarained NPV network model of AH Russell (1970j, there- fore a discussion of his model is appropriate. AH. Russell approximates the nonlinear objective function by incorporating only the first-order linear terms of the associated Taylor expansion in a linear programming formulation of the NPV project scheduling prob’iem. The dual formuMon of this iinear model is a transshipment type of network flow model. Computational experience (R.A. Russell, 1986) shows that this method converges very rapidly given an early start time initial schedule and that it is

co~~putat~~~na~l~ efficient for iarpc-scale prci’r.icms. The solution of the transshipment problem is a sys- kin of flows that can he interpreted as dual prices A,, rcprcsenting the reduction in NPV when the duration of an activity k is extended hy one period. Based on the principie of complementary slackness. lengthening activity k’s ciurntion by one period results in a schedule change where either (a) i(k) is started one period earlier, or (h) i(k) begins at the scheduled time, but j(~?$ occurs one period later. For case (a) co occur, slack time must he available to allow i(k) to start one period earlier. implying that the dual prices of aii immediate predecessors are zero. In case (b), activity X-‘s predecessors do not have s!ack, and the scheduled time of j(X-j must be delayed along with aI successor nodes. Furthermore, delaying $k) will delay the start of all activities in the project whose optimal start time is greater than or equal to the lace start time L, found by the Critical Path Method (CPM), that is, q(P) 2 12,. Each of these cases of course are usefu! in determining the cost of delays caused by the capital constraint.

To estimate the reduction in project NPV due to activity delays caused by the capital constraim, we define activity k’s tardiness penalty. Pk as the sum of the dust prices of the predecessor activities. fn contrast. R.A. Russeil’s (1986) single-iteration heuristics assume that the cost of delaying an activity when resource conflicts occur can he represented by the dual price of the arc representing the potential delayed activity, ignoring the cash flow correspond- mg to the i(k) node of a delayed activity. Our mcchod of representing tardiness penalties provides an accurate estimate of marginal changes in project NPV except +I one case. When PA = 0 and 7;(k) > Ek. where E, is the early start time for activity k found by CPM, vile tardiness penalty is only an approxima- tion of the changes in project NPV, since starting this activity past its optimal start time wiIl delay successor activities.

Dual prices in the unconstrained network solution represent changes in cash flows throughout the project as the result of lengthening arcs in the network due to delays in activity start times caused by resource constraints, not just the cash flows of the unscheduled activities. To overcome this limitation, the heuristic procedure treats nodes associated with act:vitics that have been completed whose timing

cannot be changed as tied events in the unconstrained solution (AH. Russell, 1970; Padman et al., 1994; Pndman and Smith-Daniels, 19931.

4.1.2. Priority rides Both priority rules dcscrihcd below give rise to a

front-loading scheduling strategy, In that they give priority to activities leading to immediate progress payment,, while facilitating the scheduling of additional activities by raising the level of the capital constraint. One method of implementing a front- loading strategy is the two-queue scheduling approach that is implen tted in the MTP-ET priority rule. This approach ciassifies activities according to whether the immediate successor activities have zero dual prices. Since an activity with zero dual price successors has activity slack time associated with its schedule, the activity’s delay penalty measures only its ~*rscounted cash flows. Placing activities with zero dual price successors in the priority queue promotes the scheduling of activities with immediate progress paymrnts. After activities in the priority queue are scheduled, activities art> select& from a second quee:: according to a different priority index. LPT/LAN is a single-queue rule that takes another approach to front-loading by identifying activities by means of their delay penalties that may lead to the earlier receipt of progress payments.

MT&ET (Maximum tardiness penalty - Early finish time). As mentioned above, this priority rule implements a front-loading scheduling strategy by using a two-queue priority approach. Activities are placed in the priority queue if their immediate successor activities have zero dual prices. A zero dual price indicates that the j(k) node of an activity is associated with a positive cash flow (progress payment), thus it is scheduled as soon as possible. Activities in the priority queue are scheduled in ascending order of their delay penalties ( Pk). Activi- ties in the second queue are scheduled in ascending order of minimum updated early finish times, found using CPM and the partially generated schedule. Eariy finish time ties are broken by random assign- ment, Activities in the second queue are only nvail- able for scheduling if they are precedence feasible and their scheduled start time in the current network fbw solution is equal to the current time in the scheduling process.

LTP J LAN ~~~i~~~~~~~ bwdiness penalty / Lowest astivity number). The LTP/LAN rule im- picmcnts a front-!oadin g strategy by attempting to schedule actrvities first that are connected to nearby progress payments. Priority is given to those activi- tics with a minimum non-zero tardiness penalty. while activities with zero tardiness penaltics are scheduled using LAN. An activity with a low tardiness penalty can be an lctivity requiring or leading to large sash outflows or an activity not connected in the optimal spanning tree to the final payment in the project. The objective of LTP/LAN is to identify those nc”;vities not connected to the final payment in the project, but to an intermediate progress payment. While this priority rule would appear to result in lower project NPV, it is important to note that dual prices tend to be higher for activities connected in the optimal spanning tree to the final node and payment in the project. Furthermore, if the total cash expenditures for a series of activities exceed the progress payments for those activities, then the optimal unconstrained schedule delays the start of the activities to the latest possible date. thereby connect- ing them to the final progress payment. The use of this rule will also serve to delay activities that yield net reductions in the capital constraint to later periods in a project when additional capita! is available, thus avoiding a situation such as that found in the example, where it became infeasible to complete the project, The project NPV is improved through the early receipt of cash and the capital balance is increased, which should permit scheduling of additional work in later periods.

4.1.3. The greedy forward pass algorithm The scheduling procedure is summarized below,

as well as in the flow chart in Fig. 3. The greedy scheduling procedure begins with the generation of the unconstrained network solution:

Step 1. Solve the unconstrained NPV project scheduling problem using the minimum cost network flow model.

This problem converges in a finite number of iterations (AH RusseIl, 1930). ’ The computational

1 In testing, the netwark flow problem converged to the oprimill solution within, on average, three iterations.

xl lime when sufficient fallable lo schedule

slar! ftme fo; ail precedence !easible actitities to that period.

) Designate the sel ol precedence, lime and capital-leasibie acliv~’ es to be the scnedule queue. Sari lhe Schedule queue according lo the primary and secondary L reuristic rules.

--

Fig. 3. Flow chart for the optimization-guided heuristic algorithm.

complexity of each iteration is that of soiving a transshipment problem by the network simplex method, which is of O($log nl (Orlin, 19841, where n is the number of nodes in the network. A preorder traversal of the network basis can be done in O(rr) time to generate the eveni times of the activities.

Step 2. Schedule eligible activities using a heuristic priority rule until available capital is consumed or the capital requirements of the remaining activities exceed that which is currently available. Update the capital constraint in the appropriate future periods to reflect any progress payments received at the com-

pletion of ‘he activities scheduled in this step. If all activities have been scheduled, stop.

Eligible activities are defined as those activities whose predecessors haye been scheduled (precedence feasible) and the scheduled start times in the current network flow solution are equal to the current time in the scheduling process. Thus, precedence-feasible activities that are scheduled to occur later (as given by the network solution) are elinii- nated from the queue of activities currently cornpet- irg for resources. Each of the priority decision rules dez+bed in Section 4.1.2 is applied as the criterion

for srh&!ing activities 6lniil capital has been fully utilized, or the remaining capital is not sufficient to petform the remaining acFivities in the schedule queue, Step 2 requires two scans of the queue of eligible activities, once to sort the set of activities by evaluating the heuristic rule for each activity, and the second time to schedule the activity after checking for the availability of resources. The time taken for this proc~ is dominated by the first scan and is of O(m*log nr), whete m is the number of activities in the project. As in Dcersch and Patterson (197’71, our formulation assumes that any positive cash flows received from schedu!‘. activities are added to capital available in the period following their receipt.

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 with their start tizlcz delayed to the next possible time at which capital becomes available. Precedence feasible activities whose start times are current at that time are also :~&3 :o the queue of eligi.b!e YXivities. This procedure examines =rctivitics and tipdates the network i i i

O(m) + O(n’) time. Step 4. Resolve the NPV network flow problem

with the modified event times from Step 3 and go to Step 2.

The modified NPV unconstrained minimum cost network flow problem has changed durations on the arcs (cost parameter in the dual) as well as the present values at the nodes (right hand side parameter in the dual). To derive the fair cost of delaying activities in a partially completed schedule, events associated with activities that have been completed and whose timing cannot be changed are treated as tied events. The updated network is solved using an efficient dual simplex algorithm developed for the minimum cost network flow problem (Ali et al., 19891.

The number of iterations required to obtain the final schedule is equal to the number of delays encountered while scheduling the activit,ies. This can be bounded by the sum of the durations of all of the activities. In particular, when the problem is so constrained that no two activities can be scheduled concurrently, the number of iterations is bounded by nt, the number ‘of activities in the project network. In actual testin& however, the number of iterations was

ubserved to be a fraction of the bounds pro~idcd above.

The cash flow w:ight heuristic procedure (CFW/LAN) is a greedy forward pass algorithm that uses a cash flow priority index to derive a project schedule. An activity’s cash flow weight is computed as the sum ? :he cash flows of all the activities that logically succeed it in the precedence network. This heuristic is implemented in a parallel forward pass algorithm, where activities are added to Fhe que& of schedulable activities when they become precedence feasible. The activity with maximum CFW is scheduled first, while ties are broken using lowest activity number (LAN). As activities are scheduled, the capital constraint is modified to reflect any changes due to activity cash flows.

5, Experimentad design

The complexity of Qe CCPSP combined with the results of earlier studies involving the NPV objective with cash flows (R-A. Russell, 1986; Padman et al., 1994 Padman and Smith-Daniels, 19933 suggests that the relative performance of the heuristic rules, in terms of maximizing project hTV, may be a function of a number of environmental factors. A full factorial ex,periment is performed resulting in the scheduling of 60 different project network problems by each scheduling procedure, including three different scheduling conditions with five replicates in each cell.

The performance of the revised optimization- guided procedures and the cash flow weight scheduling rule are evaluated by measuring the percent below the best NPV schedule generated for a particular project. A Rand-50 scheduling rule generates a benchmark schedule for comparison. The Rand-50 rule schedules proceeds through the project and selects activities at random for scheduling from the queue of resource and precedence feasible activities. Fifty schedules are generated for each problem and the schedule with the maximum NPV is reported as the Rand-50 solution.

Due to the relatively inconclusive results found for the small problems (4 to 50 activities) in the

Patterson data set in previous studies (R.A. Russell, 1986; Padman et al., 19941, we chose not to use this data set in this study, and instead utilized problems of a much larger size, 230 activities. The computational experiments were generated from three experimental factors including project srructure, capital utilization, and average frequency of progress payments.

Project networks of three distinctly different shapes,, balanced, skewed to the right and skewed to the left were used to learn whether heuristic performance was affected by the experimental variable. project structure. Since capital requirements and cash flows were randomly assigned to activities, the peak capital requirements and cash flows occurred, on average, to the right or left in the skewed networks, while they were evenly distributed in th. balanced networks. Activity durations were generated randomly from a uniform distribution with endpoints of one and nine periods. Precedence relationships i,:-l tween activities were randomly generated in activity- on-node form such that the coefficient of network complexity, as defined by Pascoe (1965), to represent the interconnectedness of the network, was set at an average of 2.0 across the set of networks.

The networks were converted to an activity-on-arc form for purposes of utilizing the AH. Russell (1970) network flow formulation. Since ihe activity-on-arc network model requires that project cash flows occur at nodes, dummy activities are inco,rporated in the test problems to ensure that each activity is represented by a unique node representing the initial capital investment, while a progress payment, if any, occurs at an immediate successor node representing the completion of the activity. This approach ensures that the unconstrained solution procedure will not cycle, while also realistic representing actual projects (Elmaghraby and Herroelen, 1990).

The use of frequency of progress payments as a fixed experimental factor allowed for the study of a hypothesized difference the CFVY heuristic, which. on balance, selects activities leading to cash downstream, and the front-loading optimization-guided heuristics, which should tend to be greedy in search- ing for cash sources. At the high factor level of progress payment frequency a positive progress payment occurred, on average, every third activity, while at the law level a payment occurred, on average,

every seventh activity. As the frequency of prop,ress payments increases. the best priority rule may be one that attempts to maximize short term returns by seer,+w immediate payments. while it was hypothe- ‘-“‘Lb sized that the performance of the CFW approach would improve as the frequency of payments de- creased.

The degree of capital utilization was based on the average capital constrainedness of the time-only early start schedule. Since the capital availability increases over the course of the project. the average capital utilization factor (ACLF) was computed as the average level of capital utilization across the duration of each problem, using the time-only early start schedule found using the critical path method. The ACUF measure is based on the average utilization factor defined in Kurtulus and Davis (1977). Two factor levels are used in the experiments ranging from a tow setting of ACUF equal to 120% of capital requirements to the high level of 18OcTt ACUP was computed as the average ratio of capita! requirements to capital availability for the duration of the project:

The va!ue af cg was set to yield the desired level of ACUF for each of the project networks. It was hypothesized that the relative performance of MTP- ET, LTP/LAN and CFW/LAN would increase relative to the Rand-50 benchmark as capital utilization increased.

Activity capital requirements were randomly generated, and anged from $50 to $250. Progress payments Nere generated by randomly generating “packages” of activities based on precedence constraints, such that each cash inflow takes place upon the completion of a linked set of activities. Upon completion of a package, progress payments were set equal to a 25% profit above the sum of activity cash outflows for the package. The final payment for the project was generated to yield a 40% profit for the project. These profit levels insured a positive NPV for each problem at each level of utilization. The cost d capital was set at a level of 0.02 per period.

A comparative summary of the relative performance of the heuristic rules is displayed in Tables 3 and 4, The NPV resulting from the application of each scheduling procedure is reported relative to a resource feasible upper bound, defined as the maximum NPV value derived by a heuristic scheduling procedure for a given test problem. In terms of average performance, there was relatively little difference between the LTP/LAN and MTP-ET heuristics, while both rules outperformed the CFW/LAN heuristic by an averaL of IO%, and the Rand-50 rule to a lesser degree (8.4% below maximum NPV). The results of Duncan’s multiple comparison test (Duncan, 1955) indicate that there was a significant difference at the 0.01 level between the two best performing heuristics, LTP/LAN and MTP-ET, and SC. poorer performing heuristics, CFW/LAN and Ra.nd-50, Mile NPV was often equal for CFW/LAN and Rand-50, the mean performance for CFW/LAN never’ exceed& the performs-ce of Rand-50. Inter- estingly enough, in ;n earlier study of the performance of these heuristic rules for the RCPSP-NPV (Padman et al., 19941, the difference between CFW/LAN and the revised optimization-guided procedures MTP-ET and LiP/LAN was approximately 3% for problems of a similar size. This result suggests that revised optimization-guided heuristic procedures, in this case MTP-ET and LTP/LAN, perform particularly well for the CCPSP.

Table 3 Heuristic performance: Mean percent below maximum NPV schedule

The results presented in Tables 3 and 4 indicate that project characteristics influence the reiative performance of the heuristic procedures. In particular. for projects skewed to either the right or left with high frequency of progress payments, the optimization-guided heuristics derive significantly higher NPV schedules than CFW/LAN and Rand-50. The superior NPV performance of MTP-ET and LTP/LAN also occur” :rb the scheduling of projects with a centered project schedule and low capita1 utilization.

These results seem to suggest that the revised optim;&ation-guided procedures outperform CFW/LAN for several reasons. First, the front-loading strategy of identifying activities with progress payments and giving them priority in scheduling increases the capital available for scheduling more activities earlier in the project. Projects that are skewed to the right or left have a greater number of parallel paths in the network and as such offer a greater opportunity scheduling activities with immediate progress payments. Since the CFW/LAN heuristic prcccdure gives priority in scheduling to activities that generate the highest net cash flow from successor activities, this rule does not distin- guish between activities with immediate progress payments and those leading to future downstream progress payments. Second, the superior performance of the optimization-guided heuristics may be explained by the fact that they delay activities until their target schedule dates in the revised uncon-

Heuristic scheduling rule:

Project StNCNre

Righi

Centered

Left

Capital utilization

Law

High

Low

High

Low

High

Progress payment frequency LTP/LAN M-P-ET CFW/LAN Rand-50

Low 0.00 0.00 3.86 3.86 High 0.00 0.00 15.92 15.92 Low 0.00 0.05 4.98 4.98 High 0.00 0.26 10.87 9.49 LOW 0.00 0.09 11.39 11.39 High 0.01 0.00 14.24 14.24 Low 0.00 0.18 4.68 3.27 High 0.19 0.14 9.29 6.77 Low 0.00 0.00 4.43 4.34 High 0.00 0.22 16a27 14.72 Low 0.00 0.16 5.44 3.44 High 0.0 i 0.67 12.51 8.78

Factor Level LTP/LAN MTP-ET CFLl!jlAN Rand-5a

Projecl strwtur(: Right o.Mo 0.078 8.9I?9 8.564 Centered 0.049 bf.ia3 9.899 5.919 Lefl 0.002 0.263 9.663 7.817

Capital utilization Low O.OOi 0.052 11.021 10.746 High 0.033 0.243 7.960 6.120

Propiw payment frequency Low 0.001 0.08 1 5.797 5.214 High 0.033 0.215 13.184 11.652

strained NPV network, as compared to CFW/LAN which schedules activities when they become precedence feasible. This suggests that the optimization- guided heuristics improve PU’PV by delaying huge cash outflows and capital requirements t:; ;ster periods unless they lead to a profitable progress payment. It is more likely that a large uumber of precedence feasible activities will be available fsr scheduling in projects with activities skewed to the right or left of the network. This means that CFW/LAN may schedule activities earlier than the optimal start dates in the revised unconstrained NPV schedule, thus increasing the present value of cash outflows. Third, the revised optimization-guided procedures schedule activities according to their updated unconstrained NFV network solution when the capital available for the project exceeds activity capital requirements and cash outflows. Thus the revised optimization-guided procedures that front-load activities leading to immediate progress payments are more likely to increase the capital constraint more quickly than CFW/LAN, thereby allowin the implementation of the revised unconstrained network solution. In addition, in projects characterized by a centered netwark structure and low capital utilization, the revised optimization-guided heuristics also identify immediate progress payments for front loading and implement the unconstrained solution wheq capital availability permits.

All results were obtained using FORTRAN imple- mentations of the various scheduling heuristics with an interactive VAX computer. It was found that computation time for the most highly constrained 230 activity problems was approximately ten seconds. In contrast, the benchmark single-pass procedures required approximately three seconds of com-

putation time. Thus, while the reoptimization procedure resulted in higher computation time, it seems that practical size probiems may he solved in an aceeptabfe amount of time using small computers, including persoual computers.

The objective of this research was to present and test the performance of optimization-guided heuristics for solving the capital constrained project scheduling problem. This research has implications for both practitioners and researchers. While the initial work of Bey et al. (1981) showed the impor- tance of scheduling capital constrained projects with an NPV, rather than a duration objective, and the possible monetary benefits of such an approach, little additional research has been performed to develop feasible techniques for practitioner use. We have illustrated that two revised optimization-guided heuristics that implement front-loading strategies for scheduling project activities perform well with respect to a randomly derived bound and the CMI/LAr~ priority rule. The rules were tested in large projects in a number of different experimental environments.

Future research should be devoted to additional deveiopment of heuristic procedures and application of the heuristics to more robust problem environments. Two emerging problem areas are suggested by the literature and practice. First, the issue of both capital and workforce constraints should be considered. This initially would include consideration of the ,trade-off between the cost of human resources and activity capital requirements, although these may

have a laose correlation in capital intensive high-tech development environments. Procedures in this envi- ronmenf perhaps would be required to examine the trade-offs between the fixed nature of the human resource constraint as compared to the variable nature of the capital constraint. Extensions would con- sider the role of premium resources is increasing the availability of non-storable resources, The second problem for consideration is the muiti-project scheduling problem. In this case, the sponsoring organizations would b, concerned with maximizing the NW of multiple projects, again subject to either capital constraints, workforce constraints, or both. This particular problem has strategic implications, since the release of development projects and their resultl;rg completion dates would have an impact on the F%m’s competitiveness and return on capital in- vestmein. Rules for releasing projects as well as priority scheduling rules for activities would be im- portart determinants of perfonance.

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