# The net present value criterion

## its impact on project scheduling

Roger B. Bey | Robert H. Doersch | James H. Patterson |

University of Missouri-Columbia | Stanley Works, Inc. | University of Missouri—Columbia |

The decision to organize on a project basis often is an indication that a firm is committing substantial portions of its financial resources to relatively few projects. This is in contrast to job shop manufacture or assembly line-mass production, where each product is a small portion of the total commitments of the firm. For example, in the case of launching a communications satellite, the total financial outlays easily can exceed tens or even a few hundreds of millions of dollars, with the time for completion of the entire project being several years. In these instances, the effective timing of cash receipts and outlays can have a significant impact on the ultimate profitability of the endeavor. And even in the case of a relatively small contractor, opportunities do exist for increasing profitability through the simple mechanism of efficiently timing cash receipts and outlays.

In this paper, we demonstrate how profitability in project activity can be increased through effectively timing cash outlays and receipts, and offer guidelines for organizing a project to take advantage of the savings available through the wise cash-flow scheduling of a project’s activities.

**I. Introduction**

A significant consideration for a project manager in sequencing the activities of a project should be the maximization of the value of the project subject to resource and financial constraints. Network techniques such as PERT and CPM are popular procedures in project scheduling for obtaining project schedules to assist a manager in obtaining a “time-feasible” sequence for performing each of the project activities. However, the application of network scheduling methods to minimizing a project’s cost or maximizing a project’s value, except for the limited consideration of the time-cost trade-off problem, has been almost completely ignored. The foregoing situation is ironic since the largest single inducement to a firm’s involvement in any non-mandatory project is the project’s expected financial potential. While a manager cannot consider only the pecuniary aspects of a project, these aspects should not be treated in a cavalier fashion. The limited acceptance, outside the academic community, of the application of network techniques to the financial aspects of project scheduling may be due to operation researchers’ failure to extend project scheduling techniques beyond the salient features of the problem.

The objective of this paper is to illustrate through a numerical example how project scheduling techniques can be utilized to minimize a project’s cost or maximize a project’s expected value. The basis of each of the foregoing objective functions is the project’s net present value. Some of the financial aspects of this scheduling problem have been addressed previously [4, 7, and 9]. However, we address the problem in a manner which allows for a type of objective function and constraints not previously considered. We recognize the importance and general acceptance of heuristic solution methods and the minimization of project duration solutions to this problem, and examine these solution procedures and analyze their associated financial implications. Furthermore, since many projects have an associated bonus/penalty reward structure for early/late project completion, we consider the implications of a bonus/penalty reward structure on optimal project schedules when the financial aspects of the problem are included. In addition, we show how the net present value criterion can be used to incorporate the bonus/penalty reward structure into the optimal time-sequencing of project activities. The remainder of the paper is organized in five sections. Section II explains the project scheduling finance problem; Section III presents the example problem used to illustrate the scheduling concepts developed with several net present value objective functions; Section IV discusses alternate objective functions and solution methodologies; and Section V presents our conclusions.

**II. The Project Scheduling Finance Problem.**

*A. Cash Flows and Investments in a Project*

*A. Cash Flows and Investments in a Project*

All projects entail a set of cash flows (expenditures and receipts) distributed in some pattern throughout the life of the project. Ideally, the sum of the cash flows is positive. The pattern of cash flows or the cumulative cash flow profile may vary from project to project. A typical cumulative cash flow profile for a small project is as depicted in Figure 1. In this example, expenditures occur periodically throughout the life of the project and receipts consist of a single cash flow at the completion of the project. Typical of the foregoing pattern of cash flows are the project cash flow profiles of small contractors. For example, contractors in the speculative housing industry often assume full responsibility for financing the project, finance 100 percent of the project, receive a single lump-sum payment when the project (house) is sold, and then repay all outstanding project loans. As shown in Figure 1, the amount of funds available for reinvestment or distribution at time t (time of sale) is the difference between the receipts and the cumulative expenditures including all financing charges.

A second typical cumulative cash flow profile, as illustrated in Figure 2, consists of a set of both periodic expenditures and receipts (progress payments) distributed throughout the life of the project.* The foregoing is typical of major contracts for long-term construction projects. Progress payments may be made either at the achievement of some prespecified milestones or on a routine basis.

Another consideration in the project scheduling finance problem is the project’s required level and timing of investments. Typically, activities constrained either to the beginning or to the end of a project are of a small dollar value relative to those activities which occur midway through the project. A representative project investment profile is given in Figure 3.

As illustrated in Figures 1-3, two problems that warrant serious consideration during the scheduling of a project’s activities are the required peak capital investment and the required time to realize a positive net cumulative cash flow. By appropriately scheduling expenditures and progress payments, the financing of peak capital investment requirements may be met more effectively. That is, the project’s cash receipts are obtained when they are most beneficial for financing the project. Thus a concept very relevant to effective project scheduling is the maximization of a project’s value or equivalently the maximization of the project’s net present value.*

*A project’s net present value (hereafter NPV) is defined as the sum of the discounted value of all receipts minus the sum of the discounted value of all expenditures. All discounting is to the beginning of the project. A rate frequently used for discounting is the firm’s cost of capital. The NPV criterion recognizes the advantage of receiving $1 today as opposed to receiving $1 at some future point in time. That is, the NPV criterion accounts for the time value of money.

*B. Traditional Treatment of the Cash-Flow Scheduling Problem*

*B. Traditional Treatment of the Cash-Flow Scheduling Problem*

Historically the problem of “optimally” scheduling expenditures and receipts has been approached by attempting to maximize the project’s cash flow. Most often, the objective of maximizing cash flows has been approached via the bidding mechanism by means of the “front loaded” or unbalanced bid. As described by Gates [5], “…unbalanced bidding… has greater short-term financial implications than any strategy ever likely to be developed.” The underlying motivation of the front loaded bidding strategy stems from the fact that the proposal for many contracted projects often consists of a list of the number of units of various items which must be bid on a per unit basis. The contract is awarded to the company with the total lowest bid. Hence, given a total project bid, it is to the contractor’s advantage to increase the unit price for work to be completed early and for which progress payments directly related to the magnitude of the investment required to perform the activity are available, and to decrease the price of work to be completed late in the project (without changing the total bid).

The front loaded bidding strategy increases the NPV of the project. However, the potential disadvantages of following a front loaded bidding strategy can be substantial. Since the proposal is only an estimate of the total *quantity* of work to be performed, the actual completion of the project may require a greater volume of an underbid item than was proposed and may result in substantial losses. Also, if a bid is blatantly unbalanced, it may be refused.

*C. Application of Resource-Constrained Scheduling Techniques*

*C. Application of Resource-Constrained Scheduling Techniques*

In contrast to the *ad hoc* front loaded bidding strategy, a systematic value maximizing project scheduling approach can be achieved by considering cash as any other resource in project scheduling. However, to do so the effects of past cash flows on the present supply of capital available to the firm must be considered. Once capital is included as a resource, the activities of the project can be constrained to a realistic level of financial involvement, and the possible project activity schedules can be determined. Then, the NPV for each attainable project schedule can be calculated. Hence, the maximum expected value of the project, subject to capital availability constraints, can be achieved by selecting the project schedule with the maximum NPV.

If only a time-critical objective function is considered, a non-value maximizing solution may result. Also, depending on the distribution of cash flows in the project network, the project manager may be either indifferent to interruptions in the time critical path or may be best served by postponing certain activities. For example, once the revenue from a project has been received, a contractor may not be overly concerned about how long it takes to, say, clean up or paint lines on a highway. In drafting a contract, the contractor’s client also should be mindful of these facts.

Other possible objective functions for project scheduling include minimizing the peak capital commitment or minimizing the payback period. The first is an extension of resource leveling and is treated partially below. The second, in general, is not as useful as the NPV concept since it considers neither cash flows occurring after the payback period nor the time value of money. The most discriminating criterion for evaluating investments is the project’s NPV. A general model for applying the NPV criterion is outlined in [8]. An example illustrating the benefits of the NPV criterion is presented in the following section.

**III. An Example Project Network Problem**

*A. Problem Specification*

*A. Problem Specification*

To illustrate the benefits associated with maximizing a NPV objective function through the judicious scheduling of a project’s activities, we consider an addition to a hospital, a construction project, as our example problem. A simplified network diagram for the hospital addition is illustrated in Figure 4, and the project’s associated activity durations, activity cash flows, and monthly progress payments for successful project completion are listed in Table 1.

Activity Number | Description | Duration (months) | Investment Required* | Cash Flows* For Month | ||||

1 | 2 | 3 | 4 | 5 | ||||

1 | Excavation & Foundation | 2 | 245 | 10 | 20 | |||

2 | Erect Steel Work | 5 | 2,200 | 10 | 15 | 15 | 15 | 20 |

3 | Utility Substation Extension | 4 | 690 | 15 | 20 | 20 | 25 | |

4 | Pour Floors & Walls | 3 | 1,900 | 30 | 40 | 90 | ||

5 | Install Plumbing | 1 | 595 | 40 | ||||

6 | Electrical Contracting | 2 | 830 | 20 | 40 | |||

7 | Heating & Air Conditioning | 1 | 680 | 50 | ||||

8 | Erect Roof | 1 | 510 | 30 | ||||

9 | Interior Finishing | 3 | 1,550 | 25 | 40 | 50 | ||

10 | Install Equipment & Furniture | 2 | 1,200 | 60 | 90 |

* ($1,000)

The cost of the project is $10,400,000 in direct expenses (material and labor), plus a $790,000 fee to the contractor for his participation in the project, plus financing charges incurred during construction. In addition, a consulting firm has advised the hospital trustees that as an inducement to the contractor to complete the project within an acceptable time limit, they should offer a bonus/penalty reward structure for early/late project completion. The bonus/penalty reward structure recommended by the consulting firm and adopted by the trustees is as listed in Table 2.

Time Period of Project Completion | Bonus (Penalty) $1000 |

17 | $ 50 |

18 | 20 |

19 | 0 |

20 | (25) |

21 | (75) |

22 | (175) |

Project financing consists of a construction loan during the project’s construction, and the issuance of long-term bonds just prior to the hospital addition’s opening. The hospital trustees have arranged the construction loan with a local bank to finance all direct costs and contractor fees. The terms of the construction loan are that no financing charges are due until the loan is repaid. Repayment must occur 22 months after the project begins, or whenever the addition is opened, whichever occurs first. In addition, the maximum amount of construction funds that can be outstanding on activity work at any point in time is $2,200,000, plus any reinvestment the contractor wishes to make from progress payments.* The borrowing ceiling constraint is the bank’s control mechanism for insuring that sufficient progress is made on project activities before more funds are released, and that activities are completed once they are started. As explained by Weglarz and Slowinski [12], the restrictions on cash for this project is an example of a *doubly-constrained resource.* That is, the amount of cash available is constrained both in total amount and in the amount available each period. Once an activity is completed, the amount of cash required to complete the activity is once more made available for investing in subsequent activities. If no activities are in progress, the amount of funds available for investing is $2,200,000.

The hospital trustees have arranged the long-term bond financing with an investment banking firm. Terms of the bond issue are that the investment banker will underwrite the amount of bonds necessary to cover all direct costs, fees, and construction financing charges associated with the addition. The bond’s coupon rate will be determined at the time the bonds are issued.

Unless otherwise stated the discount rate for all NPV calculations is assumed to be 2 percent per period (month). The critical path for this project consists of activities 1-2-4-6-9-10 and is 17 months in duration.

The question that we must now answer is: What is the optimal sequence of activities that maximizes the value of the project? The answer to this question, as illustrated in our example, is that the optimal sequence varies with the objective function selected and whether we are viewing this problem from the standpoint of the contractor or from the standpoint of the hospital trustees. Each of these alternatives is considered in the following sections. However, before determining the optimal sequence of activities, we must introduce the concept of *resource conflict resolution* and its impact on resource (cash) project scheduling.

*If the contractor invests (i.e., lends) some of the progress payments to the project, the opportunity cost on these funds is the same as the discount rate in the NPV.

*B. Resource Conflict Resolution*

*B. Resource Conflict Resolution*

The concept of resource conflict resolution is that when two or more activities are competing for resources and insufficient resources are available to schedule all of the competing activities, a resource conflict exists. Resolution of the conflict requires scheduling some of the activities and postponing the others. The question is, which activities should be scheduled and which activities should be postponed, or how should the conflict for resources be resolved?

In our hospital addition example, a resource conflict occurs early in the scheduling of activities. In month 1, Activity No. 1 (Excavation and Foundation Work), by virtue of the activity precedence constraints, is the only activity that can begin and requires $245,000 of capital and two months to complete. This capital requirement is far below the $2,200,000 per period ceiling. Hence, Activity 1 can be scheduled without a resource conflict. In month 3, after Activity No. 1 has been completed, the capital available for investment in project activities is $2,230,000 – the $2,200,000 borrowing ceiling plus the $30,000 progress payments from Activity No. 1. In month 3, two activities – No. 2 (Erect Steel Work) and No. 3 (Build Utility Substation Extension) can begin. However, the capital requirements of $2,890,000 ($2,200,000 + $690,000) for these two competing activities exceed the $2,230,000 available. Hence, a resource conflict exists and a decision must be made as to whether Activity No. 2 or Activity No. 3 should be scheduled first. The scheduling decision made in month 3 should be in terms of the impact on the overall value of the project. An examination of activities only in month 3 may be myopic and may result in a non-optimal solution.

The foregoing is a typical example of a resource conflict resolution decision that must be made by any method used for project scheduling under resource constraints. Several of the methods available for resolving this conflict are discussed in the following pages.

*C. Alternate Resource-Conflict Solution Considerations*

*C. Alternate Resource-Conflict Solution Considerations*

The sequence in which activities should be scheduled depends on whether we are viewing the problem from the viewpoint of the contractor or the hospital trustees. The best advantage to the contractor is to schedule the project for early completion and receive the early completion bonus, and to schedule the activities first that result in the earliest possible receipt of the largest cumulative progress payments. In this manner, the contractor can maximize the sum of the present value of his cash receipts plus the present value of an early completion bonus.* On the other hand, the best advantage to the hospital trustees is to schedule the activities as late as possible within the stipulated confines of finishing the project in 22 months, and to postpone as late as possible those activities requiring the largest investments.

*Present value (hereafter PV) is defined as the sum of the discount value of all receipts *or* the sum of the discount value of all expenditures.

Figure 5, a Gantt chart, shows the time performance of the activities resulting in (1) the minimum cost (direct costs plus contractor fees plus construction financing charges) to the hospital and (2) the maximum PV progress payments plus early completion bonus to the contractor. From Figure 5, it is clear that the different objective functions of the participants have created an immediate conflict between the objectives of the contractor and the objectives of the hospital trustees concerning the optimal sequence of activities to complete the project.

The investment profiles of the $10,400,000 in direct costs corresponding to the project schedules shown in Figure 5 are given in Figure 6. Profile (a) is optimal for the hospital. Profile (b) is optimal for the contractor. The corresponding cash receipt profiles for the contractor with and without the bonus/penalty reward structure are given in Figure 7. From Figures 6 and 7, which show the build-up of investments and cash receipts, it is clear why the two activity schedules are viewed differently by each of the parties involved, and why neither party prefers the other party’s optimal solution. The shaded area in Figure 6 (b) represents the investment in the hospital addition which is financed, but may not be being used, because of the difference in schedule lengths obtained. Careful planning of the project and the progress payment schedule may avoid the difference in schedule lengths and may result in more desirable solutions for both parties.

*D. Minimizing the PV of the Project’s Cost*

*D. Minimizing the PV of the Project’s Cost*

In this section we assume that the objective for the hospital trustees is to minimize the effective costs of the project. That is, minimize the PV of all costs of the project during construction, or, equivalently, minimize the amount of long-term bonds that must be issued. However, since the contractor does not finance the project, he is unconcerned with minimizing the PV of the costs, but has an objective of maximizing the PV of his cash receipts. Finally, we assume that even if the project is completed early, the addition will not open before its scheduled time of 22 months after the project begins.

As shown in Figures 6 and 7, the optimal schedules for the hospital and the contractor are 22 and 17 months in duration, respectively. The corresponding PVs for the contractor and for the hospital are given in Tables 3 and 4. In addition, the amount of long-term bonds required is shown in Table 4.

Project Duration | PV With Bonus/Penalty | PV Without Bonus/Penalty |

17 | $662, 880 | $627, 172 |

22 | 482, 701 | 595, 899 |

Project Duration | With Bonus/Penalty | Without Bonus/Penalty | ||

PV of Costs | Long-Term Bonds | PV of Costs | Long-Term Bonds | |

17 | $9,484,086 | $14,662,204 | $9,448,375 | $14,606, 995 |

22 | 8,756,602 | 13,537,528 | 8,869,801 | 13,712,532 |

The PV to the contractor represents the “value” of the project to the contractor at the start of the project. That is, the contractor should be indifferent to receiving the PV amount at the beginning of the project and receiving the specified progress payments over the life of the project. Equivalently, if the contractor could receive the amount of the PV at the start of the project and invest it at 2 percent per month, the amount he would have upon completion of the project would be identical to receiving the progress payments as outlined and reinvested when received at 2 percent per month.

The real “kicker” to the hospital is that if the contractor follows what is in his best interest and completes the project in 17 months versus the optimal 22 month schedule for the hospital, the PV of the costs increases with and without the bonus/penalty reward structure $727,484 and $578,574, respectively. The corresponding differences in the amount of long-term bonds that must be issued are $1,124,676 and $894,463. Similarly, when the time for completion increases from 17 to 22 months – the optimal project completion time for the hospital – the PV of the project to the contractor decreases by $31,273 without the bonus/penalty reward structure and decreases by $180,179 with the bonus/penalty reward structure. The changes in the PV for the contractor are small relative to the changes for the hospital. Also, it may be possible to revise the bonus/penalty reward structure so that both parties benefit. From the foregoing it is evident that a manager should be cognizant of the potential incongruence that can be created during the planning phases of a project while structuring the progress payments.

Another factor that a manager should consider is the effect of changes in the cost of capital on the optimal sequence of activities. That is, as the cost of capital changes, so too can the optimal sequence of the project activities. For our problem, the optimal solutions for both the contractor and for the hospital are rather insensitive to the cost of capital. For example, as the cost of capital decreases to 1 percent per month the optimal sequencing solutions do not change from those shown in Figure 5, but the PVs for the contractor and for the hospital change in magnitude from the values listed in Tables 3 and 4 to those given in Table 5. Quite often, a change in the cost of capital does have an effect on the optimal time sequencing of activities, and this possibility also should be investigated in the planning phase of a project.

Project Duration | PV of Cash Receipts | PV of Costs | ||

With Bonus/Penalty | Without Bonus/Penalty | With Bonus/Penalty | Without Bonus/Penalty | |

17 | $745,150 | $702,931 | $10,310,440 | $10,268,220 |

22 | 543,915 | 684,511 | 9,800,941 | 9,941,535 |

*E. Maximizing the NP V of the Project*

*E. Maximizing the NP V of the Project*

An alternate objective for the hospital trustees is to maximize the value of the addition by maximizing the project’s NPV. In addition to the constraints previously imposed on the problem, we assume that the addition is opened the month it is completed, and generates the following cash flow income stream (i.e., net of expenses): $100,000 the first month, $150,000 the second month, and $200,000 the third month and all months thereafter. The “stair step” cash flow pattern reflects the start-up costs inherent in any new venture.

Since the contractor does not participate in any of the future revenues of the hospital, the contractor’s optimal solution is unchanged and is as given in Figure 5 and Table 3. However, the optimal schedule for the hospital, without a bonus/penalty reward structure, as shown in Figure 8, is 17 months in duration. Although the optimal schedules for the contractor and the hospital are of the same duration, the sequence of activities is different. This difference as shown in Figures 5 and 8, consists of reversing the sequencing of activities No. 7 and No. 8 for the hospital and for the contractor. The sequencing reversal occurs because activity No. 7 requires more capital than activity No. 8 but activity No. 7 also has a larger progress payment than No. 8. Hence, since the difference in capital requirements is considerable greater than the difference in progress payments, it is advantageous for the hospital to schedule activity No. 8, the activity with the smaller capital requirements, first. On the other hand, due to the larger progress payments, it is advantageous for the contractor to schedule activity No. 7 first.

When the bonus/penalty reward structure is considered, the hospital’s optimal schedule changes quite dramatically to a duration of 21 months and the sequence of activities shown in Figure 9. The schedule change occurs because the hospital is able to capitalize on the $75,000 late completion penalty.

The NPVs of the addition with (Figure 9) and without (Figure 8) a bonus/penalty reward structure are -$2,363,063 and -$2,407,977, respectively. The negative NPVs represent the cost in current dollars beyond the projected income stream in having the facility available. Since the addition, if built, will not be self-supporting, it will require a subsidy from the community.

Alternatively, the community may want to consider the proposed addition only if its NPV is positive. This possibly can be achieved if: (1) a cost of capital less than 2 percent per month is available, (2) the revenues are increased through higher rates to patients, (3) the cost of the project is lowered, or (4) some combination of these factors can be obtained.

Given the negative NPVs associated with the previous two schedules, the hospital trustees may decide to delay the start of the project until more favorable financing terms are available. In this instance, it is interesting to determine the NPV of the project (for the associated optimal schedule) under several different discount rates to determine which rates yield a positive NPV. When the discount rate is 1.5 percent per month, the NPVs to the hospital of the project are $354,942 and $393,762 with and without a bonus/penalty reward structure, respectively. When the discount rate drops to 1 percent per month, the corresponding NPVs change to $6,454,051 and $6,496,273. Given this type of analysis, a manager can determine the sensitivity of the value of the project to the cost of capital, and “enter the market” when it appears to be a desirable venture. Furthermore, since the optimal activity schedule is a function of the cost of capital, it is critical in the *analysis* of a project that the cost of capital and the optimal activity schedule be considered simultaneously.

**IV. Alternate Solution Methods**

**A. Binary Programming**

**A. Binary Programming**

The solution technique used to obtain the schedules shown in Figures 5,8, and 9 in binary or zero-one programming [1, 4, and 6], a branch of mathematical programming, where the decision variables are constrained to either 0 or 1. For example, we might define our decision variable as follows:

The objective is to determine the values of the X_{it}s (i.e., the finish or completion times for each of the activities) which minimizes the NPV of the project’s costs, maximizes the NPV of the contractor’s cash receipts, or maximizes the hospital’s NPV, etc. Viewed in this manner, the importance of stating explicitly the criterion we wish to optimize through mathematical programming is clear. Also considered in this formulation are the constraints on the amount of cash available each period, and the precedence restrictions that must not be violated in any optimal assignment of activities. A more complete description of the mathematical formulation of this problem is given in [4]. In this paper, our intention is simply to demonstrate how the mathematical formulation can operate in practice, and not to spend a great deal of time on its operating details.

**B. Heuristic Solution Methods**

Quite often, heuristic solution procedures are used to solve for a feasible project schedule that is not necessarily optimal. In fact, the large commercial software packages that solve the resource constrained version of the project scheduling problem use heuristic solution techniques. A variety of reasons exist for doing this. Some of these include: (1) the difficulty of conceptualizing the mathematical formulation of the problem, (2) the difficulty in solving some problems once formulated, and (3) the cost of an optimal solution for very large problems.

One possible heuristic for solving the hospital addition problem from the viewpoint of the contractor is to schedule first those activities representing the largest PVs of cash flows. That is, when a conflict arises between the demands for funds and the amount of funds available, schedule first the activity with the largest PV of its cash flows. This rule is an example of a heuristic scheduling procedure — a rule of thumb for sequencing activities that is simple to implement and the intended logic of which relates to satisfying some measure of performance.

If the contractor follows this procedure, in time Period 3, Activity No. 3 is scheduled in lieu of Activity No. 2, since the demand for cash is greater than the amount available to schedule both activities. Continuing with this logic, the schedule depicted in Figure 10 results. The resulting schedule is 21 periods in duration, and is not an optimal schedule for *either* party under *any* of the criteria we have examined thus far!

The PVs of the cash flows associated with the schedule depicted in Figure 10 are given in Table 6. When the cost of capital decreases to 1 percent per month, the PVs of the project change to $6,199,992 and $6,139,137 with and without the bonus/penalty reward structure, respectively. These latter figures, although positive, are still approximately $255,000 less than what could be obtained by a different sequencing of the project activities, and point to the danger inherent in using any heuristic procedure for project scheduling.

Objective | With Bonus/Penalty | Without Bonus/Penalty |

PV of Cash Receipts | $ 550,370 | $ 599,854 |

PV of Costs | 8,867,383 | 8,916,797 |

NPV of Project | -2,365,875 | -2,415,359 |

*C. Minimizing Project Duration*

*C. Minimizing Project Duration*

In the past, when limitations on resources force the postponement of certain activities in order that other activities can be scheduled [3, 10, and 11], operations researchers have focused their attention on the problem of minimizing project duration. In Figure 11, we show the problem that develops in our example when the limitation on capital is held to a *constant* $2,200,000 per month, *and* the overriding criterion becomes minimizing the time required to complete the project (i.e., minimize project duration). With the $2,200,000 per period capital ceiling, sufficient capital does not exist in Time Period 13 to schedule Activities No. 3 and No. 9. This conflict results in the project requiring more than 17 time periods to complete. Thus, we examine the schedule which develops when we attempt to schedule the activities of the example project to minimize project duration, subject to this more restrictive resource limitation.

The minimum duration solution (Talbot [11]) with a capital limitation of $2,200,000 per month is 19 months, and since the bonus or penalty is zero for a 19 month solution, the PV for the contractor is $621,396 with or without the bonus/penalty reward structure. Given the inability of either the hospital or the contractor to increase the amount of cash available in periods 13-15, this represents the minimum time rerequired to complete this project under the cash limitations imposed. By examining the problem in this light, a manager can assess the effects of the cash ceiling, and can offer *quantifiable* indices of what will result if the ceiling is allowed to expand during certain periods. The costs of increasing the ceiling (for example, through paying a higher financing cost on premium funds) can be assessed on the resulting schedule length, or more importantly, on the project’s NPV or the minimum PV of the costs of the project. For example, for the data in question, and assuming a cost of capital of 1 percent per month, the 19-month solution results in a project NPV of $6,220,699. This compares to the 17-month solution with the higher capital ceiling of $6,454,051 and $6,496,273 with and without the bonus/penalty reward structure, respectively. The large difference between the NPVs for the 17-and 19-month solutions indicate that a considerable premium could be paid to obtain the additional $40,000 required for months 13-15. This type of analysis allows an assessment of the effects of using an adjunct objective function such as minimizing project duration when the actual desired goal is to maximize the project’s NPV and also allows a sensitivity analysis of the effects of relaxing the project’s constraints on the project’s schedule span, profitability, etc.

**V. Conclusions**

As illustrated, the application of the NPV criterion to schedule a project may result in a different “optimal” activity performance sequence than other criteria such as minimizing project duration. Under some conditions, when the contractee provides all of the financing for a project and the contractor receives a single lump sum payment at the end, the NPV criterion is equivalent to other scheduling solution techniques — in this instance the critical path or the minimum duration solution resumes its primary importance in scheduling project activity. However, since the NPV criterion always yields a solution which maximizes the expected value of the project, whereas the other solution techniques studied may or may not yield a maximum expected project value, we regard the NPV criterion to be a superior decision rule and recommend that it be adopted for project scheduling.

In addition to insuring a maximum expected value solution, the NPV criterion allows the project manager to carry out a sensitivity analysis of the impact of changing the input parameters not only on the optimal activity schedule, as the other solution techniques also do, but more importantly on the expected value of the project. Likewise, the sensitivity analysis can be used to determine an appropriate bonus/penalty reward structure, timing of capital expenditures, and acceptable rates to pay for financing. In particular, the NPV criterion is especially beneficial in determining the value of obtaining premium financing to expedite project completion.

The NPV approach also emphasizes the necessity of paralleling the rewards of one party (the contractee) to those of the second party (the contractor). When cash flows occur within a project, managers should be mindful in the planning stages to construct the rewards such that those schedules beneficial to one party are also beneficial to the other. Otherwise, dysfunctional behavior may be perceived, when the party in question is following the approach which optimizes his level of performance.

The NPV criterion can be employed for any set of investments in which the final state is known, but the timing sequence is optional. One possible application outside of the traditional project scheduling considered in this paper would be the scheduling of planned economic growth projects in developing countries.

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10. Stinson, Joel B., Edward W. Davis, and Basheer M. Khumawala, “Multiple Resource-Constrained Scheduling Using Branch and Bound,” *AIIE Transactions*, Vol. 10, No. 3 (September 1978), pp. 252-259.

11. Talbot, F. Brian and James H. Patterson, “An Efficient Integer Programming Algorithm With Network Cuts for Solving Resource Constrained Scheduling Problems,” *Management Science*, Vol. 24, No. 11 (July 1978), pp. 1163-1174.

12. Weglarz, Jan and Roman Slowinski, “Project Scheduling With Continuously Divisible, Doubly-Constrained Resources” and “Two Approaches to Problems of Resource Allocation Among Project Activities – A Comparative Study,” Working Papers, Institute of Control Engineering, Technical University of Poznan (Poznan, Poland), (May 1979).

It is requested that all materials submitted for consideration of publication maintain the following guidelines.

Submit three copies of the manuscript, typed on 8½× 11 paper, double spaced throughout, with one inch margins. Footnotes should be numbered consecutively and arranged at the end of the manuscript. References should be numbered and listed alphabetically by author at the end of the paper and should follow the form:

3. Wiest, J. D. and F. K. Levy, *A Management Guide to PERT/CPM*, Prentice Hall, Inc., 1969.

4. Patterson, J. H., “Alternative Methods of Project Scheduling with Limited Resources,” *Naval Logistics Quarterly*, December, 1973.

References referred to in the body of the text should be identified by numbers in parentheses, e.g. (1) or (2) p. 118. If more than one reference is referred to the following should be used: (2) (16) (25) pp. 214-9.

Tables and figures should be numbered in arabic and should be grouped at the end of the manuscript. Within the body of the manuscript indicate the preferred location. If possible, charts, illustrations and tables should be camera ready; all photographs should be black and white, glossy. The author is reminded that all tables, figures, charts, etc. will be printed either 3-1/8 or 6-3/4 inches in width; superfluous figures etc. will be excluded at the discretion of the editor.

*Here, by cumulative net cash flows, we mean the difference between cumulative cash receipts and cumulative cash outflows at any point in time.

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