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The trade-off between the net present cost of a project and the probability to complete it on schedule
JOURNAL
OF OPERATIONS
Vol. 6. No. 4. August
MANAGEMENT-SPECIAL
COMBINED
ISSUE
1986
The Trade-Off Between the Net Present Cost of a Project and the Probability to Complete It on Schedule AVRAHAM
SHTUB*
EXECUTIVE
SUMMARY
In this article we consider the probability of not completing a project on schedule (or the risk of delays) and its effect on the net present cost of the project. We propose an efficient frontier that points out to management the trade-off between low risk, early start schedules and high risk, late start schedules. Early start schedules minimize the risk of delays at the cost of early investment in project activities and material. Late start schedules delay capital outlays while increasing the risk of not completing the project on its due date. The methodology developed in this study is aimed at strategic level decision making. At this level, decisions are based on incomplete information that calls for stochastic analysis and the introduction of uncertainty. Uncertainty in project management is introduced through stochastic activity duration and stochastic lead times of resources required for the project. The commonly used CPM analysis ignores those aspects of uncertainty. PERT analysis does consider the stochastic nature of activity durations but computes only the probability to complete the project on a given date for a single schedule. A crucial decision high value-added. thus reducing the the probability of
at the strategic level of project management is when to schedule activities with The decision makers have to trade-off the advantages of delaying such activities, net present cost of the project, with the disadvantages associated with increasing not completing the project on time.
The number of feasible schedules in a real project is typically large and exact analysis of all possible schedules is difficult to perform, if not impossible. This article presents a heuristic procedure that generates an efficient frontier representing the risk of delays versus the net present cost of the project. The efficient frontier is a decision aid for the manager who has to choose the appropriate schedule for the project. Most computer packages for project management are based on CPM (especially packages for personal computers). Our heuristic procedure is designed to be used as an extension to CPM analysis. The procedure starts with the early start schedule developed by CPM and, using the computed slacks, tries to delay activities with high value-added one at a time. At each iteration a Monte-Carlo-type simulation is used to approximate the probability of not completing the project on time. This probability is stored along with the net present cost of the project. The result of the analysis is a set of points on the plane representing the probability of not completing the project on time versus the net present cost of the project. Each point corresponds to a specific schedule. Management can choose the most appropriate schedule for implementation based on its attitude towards risk and its financial policy.
* Tel Aviv University,
Department
Journal of Operations
of I.E., Ramat
Management
Aviv, Israel.
461
A simple example is used to illustrate the heuristic procedure. In the example, a project with six activities and two types of resources is analyzed. Five schedules are generated with net present cost ranging from $45,000 to $8,191,000 and the probability of not completing the project on time ranging from 0.0001 to 0.75. Our experience with a real project of 400 activities is reported as well. The heuristic procedure can be implemented easily on advanced “Fourth Generation” packages for project management such as IBM’s Application System (AS) or Metier’s Artemis system. The heuristic procedure can also be implemented on personal computers by processing the output of any CPM package by the special subroutine that is developed in this study.
INTRODUflION Project scheduling techniques such as PERT (Program Evaluation and Review Technique [7]) and CPM (Critical Path Method [9]) were developed in the late fifties to aid project managers in planning monitoring and control of projects’ time tables. PERT and CPM are still the basis of most project management tools used today. Since their development, additional techniques were added to the basic scheduling algorithms. These techniques were designed to handle resource constraints [3,4, 161, project net present value [5], project cash flow [2, 121, material management [ 1, 131, and situations in which activity duration is a function of the resources allocated to perform the activity [ 141. The CPM approach, which is used by most practitioners, schedules each activity to start as early as possible (subject to the project’s starting time and precedence relations) and calculates the slack of each activity that is not on the critical path. Thus the schedule generated by CPM is an “early start schedule.” An early start schedule has the advantage of minimizing the risk or probability of not completing the project on time. Since the performance time of most activities in real projects is a random variable, a project manager whose goal is to maximize the probability of finishing the project as early as possible may adopt this schedule. Most project managers, however, are concerned with the cost of the project and the timing of major capital outlay. High interest rates motivate a tendency to delay high cost activities. The two conflicting objectives-minimizing the risk of delays and minimizing net present cost of the project-have to be traded off and an appropriate tool is required for the trade-off analysis. In this study, we present an analysis methodology and a heuristic procedure that generates a set of alternative schedules. Each schedule is defined by its corresponding present cost and the probability of not completing the project on its given due date. In Figure 1 we illustrate the result of such analysis-an efficient frontier of schedules that minimizes the probability of not completing the project on time for a given present cost or minimizing the net present cost for a given probability not to complete the project on time. The project manager has to choose a schedule on the efficient frontier according to his risk-cost preferences. Our heuristic generates an approximation of the efficient frontier. It is based on the integration of existing knowledge in the area of project management. Specifically, it is possible to use any project management software that has a resource allocation module and the ability to read external files and to produce files in a machine readable form as the basic building block of the heuristic. The rest of this article is divided into three sections. In the section that follows we present the methodology and our heuristic procedure. Next, we illustrate the use of the procedure
FIGURE 1 The Efficient Frontier
Inefficient
5
x
x-
schedules The
efficient
frontier
x x
;i
:
Late stort schedule NET
~
Early stort schedule PRESENT
COST
via a small numerical example. In the last section we report the results of the analysis real project, and present our conclusions. DEVELOPING
ALTERNATIVE
of a
SCHEDULES
The basic scheduling algorithm (in CPM) is designed to schedule each activity at the earliest feasible time without violating precedence relations. In the schedule generated by CPM, activities are divided into two sets: 1. A set of critical activities that are members of the longest sequence of activities connecting the starting point of the project to its end. Critical activities have a zero slack, that is, they cannot be delayed without delaying the end of the project. 2. A set of noncritical activities. These activities can be delayed as long as the delay is no longer than the activity’s slack and the project can still finish on time. The major difference between PERT and CPM is in the way activity durations are treated. In CPM the assumption is that each activity has a known deterministic duration, and the result of the analysis is a schedule specifying the exact day on which the project will end. In PERT the assumption is that activity duration is a random variable with a known mean and variance, and the result of the analysis is the probability to finish the project on a given date. Most software packages for project management are using CPM for scheduling. Some packages apply Monte-Carlo simulation to the schedule generated by CPM to calculate the
Journal of Operations
Management
463
probabilities of completing the project on time (a common terminology is “risk analysis,” calculating the risk of not completing the project on time [8]). The project environment is either resource constrained or resources are assumed to be available in unlimited quantities. In a resource constrained environment a logic called “resource allocation” is applied (Wiest and Levy [ 151). Project activities are scheduled so that resource constraints are not violated and the project ends as early as possible. In the case that resources are assumed to be unlimited, a “project leveling” logic is applied, which tries to minimize fluctuations in resource consumption while ending the project on its due date. Procedures for project scheduling that handle resources are described by Wiest [ 161, Wiest and Levy [ 151, Elmaghraby [6], Davis [3, 41, and Patterson [ 10, 111. If case resources are purchased or subcontracted, the project manager has to consider two aspects of the problem: 1. Feasibility-each activity has to be scheduled subject to precedence constraints, and availability of material and resources required to perform the activity. Resources and materials on hand and on order have to be compared to requirements and orders should be issued based on net requirements. Furthermore, lead time has to be considered and time-phasing mechanisms used so that activities will not be scheduled while the required resource or material is not available on time. 2. Optimality-the investment in resouces and material, which in some cases is a major part of the project’s budget, is affected by the project schedule. The project manager has to trade off schedules in which activities are scheduled to start as early as possible to minimize the risk of delays against schedules in which the cost of capital is reduced to a minimum by scheduling activities with high investment (high value-added) as late as possible. The difference between an early start schedule in which each activity is scheduled to start as early as possible subject to the constraints, and a late start schedule in which each activity is scheduled to start as late as possible subject to a given due date, defines the slack of each activity. The project manager has to decide when to schedule each activity given its slack. The decision is based on the two contrasting objectives discussed earlier. Since the time value of money is a nonlinear function, a simple multi-objective linear program such as goal programming is not appropriate for the analysis. A possible approach is to generate a set of points, each representing a feasible schedule with its corresponding net present cost of the project and the risk, or probability of not completing the project on time. The project manager can then choose the schedule that fits his risk-cost preferences best. The value-added of each activity is composed of two parts-the cost of material and the cost of resources. In case the resources are fixed during the life cycle of the project (the “resource leveling” case), the value-added is a function of material cost only. Our procedure can handle any source of value-added-material or resources. The procedure is based on the assumption that payment for the project is received at the end of the project. In this case, to maximize the net present value of the project, costlier activities have to be moved to later dates. Under this assumption, minimizing the net present cost of the project is equivalent to maximizing its net present value. The logic used in the following heuristic procedure is in line with existing knowledge in this area [2, 5, 121. To describe the heuristic
procedure,
define:
DD = the given due date of the project
464
APICS
S = NPC = A = SIA = ES =
an index corresponding to the schedule under consideration net present cost of the project an activity in the project, A E Q slack of activity A early start schedule in which each activity is scheduled to start on the earliest feasible date LS = late start schedule in which each activity is scheduled to start as late as possible without violating the due date constraint.
The procedure follows: Input the project data including: Step 1: The activities and precedence relations among them. a) Material information including types of material required for each activity, b) quantities, lead time, and cost for each material type, current inventory status. Resource information including type and quantities of resources required for c) each activity, cost of each resource, lead time to acquire each resource and resource availabilities. At = Time interval used in a search subroutine when the current schedule is d) not feasible dt = the sensitivity factor-desired length of the interval by which an activity is e) delayed between two iterations DD = the project due date f) Step 2: Find a feasible early start schedule (i.e., each activity is scheduled to start as early as possible without violating precedence relations, resource or material constraints) and set S = 0. Step 3. If no feasible early start schedule exists, go to Step 4; otherwise, go to step 5. Step 4: Print “no feasible solution exists” and stop. Step 5: Find the probability of completing the project by its due date using MonteCarlo simulation and the net present cost if the current schedule S is implemented. Step 6: Find all the activities with positive slack. Choose activity A with highest valueadded among those activities. In case of a tie choose the activity with highest value-added and longest slack. Step 7: If all the activities have a zero slack go to Step 8; otherwise go to Step 9. Step 8: Plot the probability of not completing the project on its due date versus the NPC for each schedule S. Mark the points along the efficient frontier; stop. Step 9: Generate a new schedule S = S + 1 by delaying activity A by the minimum of its slack SIA and the sensitivity factor dt. Step 10: If the new schedule S is feasible, that is, material and resource constraints are not violated, go to Step 11. Otherwise, go to step 12. Step 11: Update the slack of all activities according to schedule S; return to Step 5. Step 12: Divide the time period between early start and early finish of A in the current schedule S into segments of length At (the last segment can be shorter). Try to find a feasible schedule in which A starts in the first segment. If no such schedule exists, try the next segment and so on. Since the late start of A is feasible a new feasible, schedule must be found. Go to Step 11.
Journal of Operations
Management
465
Explanations Since the purpose of the procedure is to generate an approximation of the efficient frontier, an effort is made to reduce the net present cost of the project by the maximum possible at each iteration. This is done by delaying (in Step 6) activities with highest value-added, and breaking ties based on maximum slack. The net present cost is inversly related to both the value-added of activities and the delay in their scheduled start; thus the selection rule in Step 6 is aimed at minimizing the NPC. The value of dt is a factor controlling the number of schedules generated by the procedure. Each activity A that has an initial slack SIA will cause approximateIy % iterations of the procedure. Thus a rough estimate of the number of points on the efficient frontier is $ X C SIA. The user should select the value of dt based on the trade-off between desired AEO
number of points on the efficient frontier and CPU time available for the analysis. In case the new schedule generated in Step 9 is not feasible due to lack of resources or material, a search subroutine is used in Step IO to find an alternative feasible schedule. The value of At determine the number of searches performed. In the worst case this number is slA At ’ The procedure terminates when, in the current schedule, all activities with a positive value-added are delayed to their late start. The set of feasible schedules generated by the procedure is plotted with the schedules on the efficient frontier marked. These are the schedules that for a given NPC yield minimum risk. The procedure’s flow diagram is presented in Figure 2. EXAMPLE To illustrate the heuristic procedure, a seven-activities network similar to the example of Smith-Daniles and Aquilano [ 131 is used. The network is presented in Figure 3. We assume that the duration of each activity is a normally distributed random variable. The Monte-Carlo simulation can handle stochastic lead times as well as stochastic durations but to keep the example simple deterministic lead times are assumed. One type of material and one type of resource are required. Activity duration, its standard deviation resource, and material requirements are shown in Table 1. Beginning material inventory is eight units, each unit cost $lO’/unit and the lead time, assumed deterministic, is ninety days. Twelve units of the resource are available every day. The project due date is after ten months {day 300) and the interest rate is 10% per month. The procedure was implemented on IBM Application System-AS [8] with At = dt = 300 days. Five schedules were found, as shown in Table 2. Schedules I, 2, 3, and 5 define the efficient frontier while Schedule 4 is inefficient since at the same risk, Schedule 5 yields a lower NPC. The scheduled start of each activity in each schedule is seen in Table 3. The five schedules are only a small sample of all possible schedules, the sample size can increase by setting At and dt to values smaller than 300. The result indicates that the project “net present cost” drops sharply when the risk of not completing in 300 days is
466
APES
FIGURE 2 The Heuristic Procedure
I. Data
Input:
a. Network b. Material
file
c.Resource d.Time
file
AT
difference
c.Sensitivity 1. Due
factor
data
dt
DD 4
2.
Find
a
early
feasible
start
schedule
s=o
NO
I 4. Print
“No
feasible
solution
exists” stop
5. Calculate on
P(not
time)
NPC
and
complete
save
calculated
the
schedule
with for
s.
b 6. Find
the
with
activity
hiqhest
added
and
A
value slack>0
_
II
No
I 8. Print
solution
-
stop 9. Delay
calculate
activity A dt of A I i slack the new
schedule
S’SII
up
II. Updote slack
the to
Min
the of
all
activities a
_
1 and
NO
resource I 12. For
values
n= 1.2.3.u Yes
Journal of Operations
Management
of such
that
wAt
the
first
in
which
at
time
find schedule
A starts <
n.Al
467
FIGURE 3 An Example Network
above 70%; thus, if the due date is not strictly possible. CONCLUSIONS
AND
enforced,
substantial
capital
savings
are
SUMMARY
The time value of money is an important factor in long projects. The project manager has to consider the trade-off between the net present cost of his project and the risk of not completing the project on time. The heuristic procedure presented in this article is designed to generate a set of schedules and to choose the most efficient schedules on the risk versus NPC curve. The manager can use the results to evaluate different time tables for his project. The basic building blocks of the procedure are CPM analysis and Monte-Carlo simulation. It is possible to implement the procedure on advanced project management software packages such as IBM’s Application System or Metier’s Artemis system. These packages include a built-in Monte-Carlo simulator and facilities for developing user subroutines. The implementation of the procedure with any other CPM software such as PMW, Project 6, or Pertmaster is possible if the output of the CPM analysis is readable by a user-written subroutine. TABLE
Expected Activity
Number 100 6 5 4 3 2 1
466
Duration
1
Standard Deviation of Duration
Resource Requirements
(days)
(days)
(per day)
0 30 30 30 30 30 0
0 5 5 5 5 5 0
0 2 10 3 4 5 0
Material Requirements 0 6 8 8 4 5 0
Remarks “start”
“end”
APICS
TABLE
Schedule
Number
2
Probability of Not Completing by Day 300
S
1 2 3 4 5
Net Present Cost
lo-4 0.55 0.7 0.75 0.75
8.191 7.216 6.879 0.214 0.045
x X X X x
lo6 lo6 lo6 lo6 lo6
TABLE 3 Schedule Scheduled Start of Activitv Number 1 2 3 4 5 6 100
Number:
1
2
3
4
5
150 120 120 120 90 1 1
300 270 90 91 240 1 1
300 270 90 270 240 1 1
300 270 1 270 240 210 1
300 270 270 270 240 210
The procedure was implemented on an IBM Application System (AS) little effort (about ten hours of programming). The analysis of a real project proved to be costly in terms of computer resources (on the average each required thirty seconds CPU time on an IBM 8033N computer), but analysis is not required very frequently and should be used as a decision strategic level decisions, the cost of computer resources is well justified.
I
[8], with relatively with 400 activities schedule iteration since this type of support system for
REFERENCES Aquilano, N.J. and D.E. Smith. “A Formal Set of Algorithms for Project Scheduling with Critical Path Method-Material Requirements Planning.” Journal of Operations Management, Vol. 1, No. 2 (November 1980). Bey, R.B., R.H. Doersch, and J.H. Patterson. “The Net Present Value Criterion: Its Impact on Project Scheduling.” Project Management Quarterly June 1981. Davis, E.W. “Project Scheduling Under Resource Constraints-Historical Review and Categorization of Procedures.” AIIE TRANSACTIONS, Vol. 5, No. 4 (December 1973). Davis, E.W. and J.H. Patterson. “A Comparison of Heuristic and Optimum Solutions in Resource
Journal of Operations
Management
Constrained Project Scheduling.” Management Science, Vol. 21, No. 8 (April 1975).
5. Doersch,
R.H. and J.H. Patterson. “Scheduling a Project to Maximize Its Present Value: A Zero-One Programming Approach.” Management Science, Vol. 23, No. 8 (April 1977).
6,
S.E. Activity Networks: Project Planning and Control by Network Models. New York:
Elmaghraby,
John Wiley and Sons, 1977. of the Navy. PERT, Program Evalua7. Department tion Research Task. Phase I Summary Report. Washington, D.C.: Special Projects Office, Bureau of Ordinance, 7, (1958). 8. International
Business
Machine
Corporation.
Ap-
469
9.
10.
11.
12.
470
plication System Project Control User Guide. 2d ed. August 1984, pp. 157-165. Kelley, J.E. and M.R. Walker. “Critical Path Planning and Scheduling.” Proceedings of the Eastern Joint Computer Conference. Boston, Mass., 1959. Patterson, J.H. “Alternative Methods of Project Scheduling with Limited Resources.” Naval Research Logistics Quarterly, Vol. 20, No. 4 (December 1973). Patterson, J.H. “Project Scheduling: The Effects of Problem Structure on Heuristic Performance.” Naval Research Logistic’s Quarterly, Vol. 23, No. 1 (March 1976). Russel, A.H. “Cash Flow in Networks.” Management Science, Vol. 16, No. 5 (January 1970).
13. Smith-Daniels, D.E. and N.J. Aquilano. “Constrained Resource Project Scheduling Subject to Material Constraints.” Journal of Operations Management, Vol. 4, No. 4 (August 1984). 14. Talbot, F.B. “Resource-Constrained Project Scheduling with Time-Resource Trade-offs: The Non Preemptive Case.” Management Science, Vol. 28, No. 10 (October 1982). 15. Wiest, J.D. and F.K. Levy. “A Management Guide to PERT/CPM.” Englewood Cliffs, N.J., PrenticeHall, 1977. 16. Wiest, J.D. “Some Properties of Schedules for Large Projects with Limited Resources.” Operations Research, Vol. 12, No. 3 (May-June 1964).
APICS
OF OPERATIONS
Vol. 6. No. 4. August
MANAGEMENT-SPECIAL
COMBINED
ISSUE
1986
The Trade-Off Between the Net Present Cost of a Project and the Probability to Complete It on Schedule AVRAHAM
SHTUB*
EXECUTIVE
SUMMARY
In this article we consider the probability of not completing a project on schedule (or the risk of delays) and its effect on the net present cost of the project. We propose an efficient frontier that points out to management the trade-off between low risk, early start schedules and high risk, late start schedules. Early start schedules minimize the risk of delays at the cost of early investment in project activities and material. Late start schedules delay capital outlays while increasing the risk of not completing the project on its due date. The methodology developed in this study is aimed at strategic level decision making. At this level, decisions are based on incomplete information that calls for stochastic analysis and the introduction of uncertainty. Uncertainty in project management is introduced through stochastic activity duration and stochastic lead times of resources required for the project. The commonly used CPM analysis ignores those aspects of uncertainty. PERT analysis does consider the stochastic nature of activity durations but computes only the probability to complete the project on a given date for a single schedule. A crucial decision high value-added. thus reducing the the probability of
at the strategic level of project management is when to schedule activities with The decision makers have to trade-off the advantages of delaying such activities, net present cost of the project, with the disadvantages associated with increasing not completing the project on time.
The number of feasible schedules in a real project is typically large and exact analysis of all possible schedules is difficult to perform, if not impossible. This article presents a heuristic procedure that generates an efficient frontier representing the risk of delays versus the net present cost of the project. The efficient frontier is a decision aid for the manager who has to choose the appropriate schedule for the project. Most computer packages for project management are based on CPM (especially packages for personal computers). Our heuristic procedure is designed to be used as an extension to CPM analysis. The procedure starts with the early start schedule developed by CPM and, using the computed slacks, tries to delay activities with high value-added one at a time. At each iteration a Monte-Carlo-type simulation is used to approximate the probability of not completing the project on time. This probability is stored along with the net present cost of the project. The result of the analysis is a set of points on the plane representing the probability of not completing the project on time versus the net present cost of the project. Each point corresponds to a specific schedule. Management can choose the most appropriate schedule for implementation based on its attitude towards risk and its financial policy.
* Tel Aviv University,
Department
Journal of Operations
of I.E., Ramat
Management
Aviv, Israel.
461
A simple example is used to illustrate the heuristic procedure. In the example, a project with six activities and two types of resources is analyzed. Five schedules are generated with net present cost ranging from $45,000 to $8,191,000 and the probability of not completing the project on time ranging from 0.0001 to 0.75. Our experience with a real project of 400 activities is reported as well. The heuristic procedure can be implemented easily on advanced “Fourth Generation” packages for project management such as IBM’s Application System (AS) or Metier’s Artemis system. The heuristic procedure can also be implemented on personal computers by processing the output of any CPM package by the special subroutine that is developed in this study.
INTRODUflION Project scheduling techniques such as PERT (Program Evaluation and Review Technique [7]) and CPM (Critical Path Method [9]) were developed in the late fifties to aid project managers in planning monitoring and control of projects’ time tables. PERT and CPM are still the basis of most project management tools used today. Since their development, additional techniques were added to the basic scheduling algorithms. These techniques were designed to handle resource constraints [3,4, 161, project net present value [5], project cash flow [2, 121, material management [ 1, 131, and situations in which activity duration is a function of the resources allocated to perform the activity [ 141. The CPM approach, which is used by most practitioners, schedules each activity to start as early as possible (subject to the project’s starting time and precedence relations) and calculates the slack of each activity that is not on the critical path. Thus the schedule generated by CPM is an “early start schedule.” An early start schedule has the advantage of minimizing the risk or probability of not completing the project on time. Since the performance time of most activities in real projects is a random variable, a project manager whose goal is to maximize the probability of finishing the project as early as possible may adopt this schedule. Most project managers, however, are concerned with the cost of the project and the timing of major capital outlay. High interest rates motivate a tendency to delay high cost activities. The two conflicting objectives-minimizing the risk of delays and minimizing net present cost of the project-have to be traded off and an appropriate tool is required for the trade-off analysis. In this study, we present an analysis methodology and a heuristic procedure that generates a set of alternative schedules. Each schedule is defined by its corresponding present cost and the probability of not completing the project on its given due date. In Figure 1 we illustrate the result of such analysis-an efficient frontier of schedules that minimizes the probability of not completing the project on time for a given present cost or minimizing the net present cost for a given probability not to complete the project on time. The project manager has to choose a schedule on the efficient frontier according to his risk-cost preferences. Our heuristic generates an approximation of the efficient frontier. It is based on the integration of existing knowledge in the area of project management. Specifically, it is possible to use any project management software that has a resource allocation module and the ability to read external files and to produce files in a machine readable form as the basic building block of the heuristic. The rest of this article is divided into three sections. In the section that follows we present the methodology and our heuristic procedure. Next, we illustrate the use of the procedure
FIGURE 1 The Efficient Frontier
Inefficient
5
x
x-
schedules The
efficient
frontier
x x
;i
:
Late stort schedule NET
~
Early stort schedule PRESENT
COST
via a small numerical example. In the last section we report the results of the analysis real project, and present our conclusions. DEVELOPING
ALTERNATIVE
of a
SCHEDULES
The basic scheduling algorithm (in CPM) is designed to schedule each activity at the earliest feasible time without violating precedence relations. In the schedule generated by CPM, activities are divided into two sets: 1. A set of critical activities that are members of the longest sequence of activities connecting the starting point of the project to its end. Critical activities have a zero slack, that is, they cannot be delayed without delaying the end of the project. 2. A set of noncritical activities. These activities can be delayed as long as the delay is no longer than the activity’s slack and the project can still finish on time. The major difference between PERT and CPM is in the way activity durations are treated. In CPM the assumption is that each activity has a known deterministic duration, and the result of the analysis is a schedule specifying the exact day on which the project will end. In PERT the assumption is that activity duration is a random variable with a known mean and variance, and the result of the analysis is the probability to finish the project on a given date. Most software packages for project management are using CPM for scheduling. Some packages apply Monte-Carlo simulation to the schedule generated by CPM to calculate the
Journal of Operations
Management
463
probabilities of completing the project on time (a common terminology is “risk analysis,” calculating the risk of not completing the project on time [8]). The project environment is either resource constrained or resources are assumed to be available in unlimited quantities. In a resource constrained environment a logic called “resource allocation” is applied (Wiest and Levy [ 151). Project activities are scheduled so that resource constraints are not violated and the project ends as early as possible. In the case that resources are assumed to be unlimited, a “project leveling” logic is applied, which tries to minimize fluctuations in resource consumption while ending the project on its due date. Procedures for project scheduling that handle resources are described by Wiest [ 161, Wiest and Levy [ 151, Elmaghraby [6], Davis [3, 41, and Patterson [ 10, 111. If case resources are purchased or subcontracted, the project manager has to consider two aspects of the problem: 1. Feasibility-each activity has to be scheduled subject to precedence constraints, and availability of material and resources required to perform the activity. Resources and materials on hand and on order have to be compared to requirements and orders should be issued based on net requirements. Furthermore, lead time has to be considered and time-phasing mechanisms used so that activities will not be scheduled while the required resource or material is not available on time. 2. Optimality-the investment in resouces and material, which in some cases is a major part of the project’s budget, is affected by the project schedule. The project manager has to trade off schedules in which activities are scheduled to start as early as possible to minimize the risk of delays against schedules in which the cost of capital is reduced to a minimum by scheduling activities with high investment (high value-added) as late as possible. The difference between an early start schedule in which each activity is scheduled to start as early as possible subject to the constraints, and a late start schedule in which each activity is scheduled to start as late as possible subject to a given due date, defines the slack of each activity. The project manager has to decide when to schedule each activity given its slack. The decision is based on the two contrasting objectives discussed earlier. Since the time value of money is a nonlinear function, a simple multi-objective linear program such as goal programming is not appropriate for the analysis. A possible approach is to generate a set of points, each representing a feasible schedule with its corresponding net present cost of the project and the risk, or probability of not completing the project on time. The project manager can then choose the schedule that fits his risk-cost preferences best. The value-added of each activity is composed of two parts-the cost of material and the cost of resources. In case the resources are fixed during the life cycle of the project (the “resource leveling” case), the value-added is a function of material cost only. Our procedure can handle any source of value-added-material or resources. The procedure is based on the assumption that payment for the project is received at the end of the project. In this case, to maximize the net present value of the project, costlier activities have to be moved to later dates. Under this assumption, minimizing the net present cost of the project is equivalent to maximizing its net present value. The logic used in the following heuristic procedure is in line with existing knowledge in this area [2, 5, 121. To describe the heuristic
procedure,
define:
DD = the given due date of the project
464
APICS
S = NPC = A = SIA = ES =
an index corresponding to the schedule under consideration net present cost of the project an activity in the project, A E Q slack of activity A early start schedule in which each activity is scheduled to start on the earliest feasible date LS = late start schedule in which each activity is scheduled to start as late as possible without violating the due date constraint.
The procedure follows: Input the project data including: Step 1: The activities and precedence relations among them. a) Material information including types of material required for each activity, b) quantities, lead time, and cost for each material type, current inventory status. Resource information including type and quantities of resources required for c) each activity, cost of each resource, lead time to acquire each resource and resource availabilities. At = Time interval used in a search subroutine when the current schedule is d) not feasible dt = the sensitivity factor-desired length of the interval by which an activity is e) delayed between two iterations DD = the project due date f) Step 2: Find a feasible early start schedule (i.e., each activity is scheduled to start as early as possible without violating precedence relations, resource or material constraints) and set S = 0. Step 3. If no feasible early start schedule exists, go to Step 4; otherwise, go to step 5. Step 4: Print “no feasible solution exists” and stop. Step 5: Find the probability of completing the project by its due date using MonteCarlo simulation and the net present cost if the current schedule S is implemented. Step 6: Find all the activities with positive slack. Choose activity A with highest valueadded among those activities. In case of a tie choose the activity with highest value-added and longest slack. Step 7: If all the activities have a zero slack go to Step 8; otherwise go to Step 9. Step 8: Plot the probability of not completing the project on its due date versus the NPC for each schedule S. Mark the points along the efficient frontier; stop. Step 9: Generate a new schedule S = S + 1 by delaying activity A by the minimum of its slack SIA and the sensitivity factor dt. Step 10: If the new schedule S is feasible, that is, material and resource constraints are not violated, go to Step 11. Otherwise, go to step 12. Step 11: Update the slack of all activities according to schedule S; return to Step 5. Step 12: Divide the time period between early start and early finish of A in the current schedule S into segments of length At (the last segment can be shorter). Try to find a feasible schedule in which A starts in the first segment. If no such schedule exists, try the next segment and so on. Since the late start of A is feasible a new feasible, schedule must be found. Go to Step 11.
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Explanations Since the purpose of the procedure is to generate an approximation of the efficient frontier, an effort is made to reduce the net present cost of the project by the maximum possible at each iteration. This is done by delaying (in Step 6) activities with highest value-added, and breaking ties based on maximum slack. The net present cost is inversly related to both the value-added of activities and the delay in their scheduled start; thus the selection rule in Step 6 is aimed at minimizing the NPC. The value of dt is a factor controlling the number of schedules generated by the procedure. Each activity A that has an initial slack SIA will cause approximateIy % iterations of the procedure. Thus a rough estimate of the number of points on the efficient frontier is $ X C SIA. The user should select the value of dt based on the trade-off between desired AEO
number of points on the efficient frontier and CPU time available for the analysis. In case the new schedule generated in Step 9 is not feasible due to lack of resources or material, a search subroutine is used in Step IO to find an alternative feasible schedule. The value of At determine the number of searches performed. In the worst case this number is slA At ’ The procedure terminates when, in the current schedule, all activities with a positive value-added are delayed to their late start. The set of feasible schedules generated by the procedure is plotted with the schedules on the efficient frontier marked. These are the schedules that for a given NPC yield minimum risk. The procedure’s flow diagram is presented in Figure 2. EXAMPLE To illustrate the heuristic procedure, a seven-activities network similar to the example of Smith-Daniles and Aquilano [ 131 is used. The network is presented in Figure 3. We assume that the duration of each activity is a normally distributed random variable. The Monte-Carlo simulation can handle stochastic lead times as well as stochastic durations but to keep the example simple deterministic lead times are assumed. One type of material and one type of resource are required. Activity duration, its standard deviation resource, and material requirements are shown in Table 1. Beginning material inventory is eight units, each unit cost $lO’/unit and the lead time, assumed deterministic, is ninety days. Twelve units of the resource are available every day. The project due date is after ten months {day 300) and the interest rate is 10% per month. The procedure was implemented on IBM Application System-AS [8] with At = dt = 300 days. Five schedules were found, as shown in Table 2. Schedules I, 2, 3, and 5 define the efficient frontier while Schedule 4 is inefficient since at the same risk, Schedule 5 yields a lower NPC. The scheduled start of each activity in each schedule is seen in Table 3. The five schedules are only a small sample of all possible schedules, the sample size can increase by setting At and dt to values smaller than 300. The result indicates that the project “net present cost” drops sharply when the risk of not completing in 300 days is
466
APES
FIGURE 2 The Heuristic Procedure
I. Data
Input:
a. Network b. Material
file
c.Resource d.Time
file
AT
difference
c.Sensitivity 1. Due
factor
data
dt
DD 4
2.
Find
a
early
feasible
start
schedule
s=o
NO
I 4. Print
“No
feasible
solution
exists” stop
5. Calculate on
P(not
time)
NPC
and
complete
save
calculated
the
schedule
with for
s.
b 6. Find
the
with
activity
hiqhest
added
and
A
value slack>0
_
II
No
I 8. Print
solution
-
stop 9. Delay
calculate
activity A dt of A I i slack the new
schedule
S’SII
up
II. Updote slack
the to
Min
the of
all
activities a
_
1 and
NO
resource I 12. For
values
n= 1.2.3.u Yes
Journal of Operations
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of such
that
wAt
the
first
in
which
at
time
find schedule
A starts <
n.Al
467
FIGURE 3 An Example Network
above 70%; thus, if the due date is not strictly possible. CONCLUSIONS
AND
enforced,
substantial
capital
savings
are
SUMMARY
The time value of money is an important factor in long projects. The project manager has to consider the trade-off between the net present cost of his project and the risk of not completing the project on time. The heuristic procedure presented in this article is designed to generate a set of schedules and to choose the most efficient schedules on the risk versus NPC curve. The manager can use the results to evaluate different time tables for his project. The basic building blocks of the procedure are CPM analysis and Monte-Carlo simulation. It is possible to implement the procedure on advanced project management software packages such as IBM’s Application System or Metier’s Artemis system. These packages include a built-in Monte-Carlo simulator and facilities for developing user subroutines. The implementation of the procedure with any other CPM software such as PMW, Project 6, or Pertmaster is possible if the output of the CPM analysis is readable by a user-written subroutine. TABLE
Expected Activity
Number 100 6 5 4 3 2 1
466
Duration
1
Standard Deviation of Duration
Resource Requirements
(days)
(days)
(per day)
0 30 30 30 30 30 0
0 5 5 5 5 5 0
0 2 10 3 4 5 0
Material Requirements 0 6 8 8 4 5 0
Remarks “start”
“end”
APICS
TABLE
Schedule
Number
2
Probability of Not Completing by Day 300
S
1 2 3 4 5
Net Present Cost
lo-4 0.55 0.7 0.75 0.75
8.191 7.216 6.879 0.214 0.045
x X X X x
lo6 lo6 lo6 lo6 lo6
TABLE 3 Schedule Scheduled Start of Activitv Number 1 2 3 4 5 6 100
Number:
1
2
3
4
5
150 120 120 120 90 1 1
300 270 90 91 240 1 1
300 270 90 270 240 1 1
300 270 1 270 240 210 1
300 270 270 270 240 210
The procedure was implemented on an IBM Application System (AS) little effort (about ten hours of programming). The analysis of a real project proved to be costly in terms of computer resources (on the average each required thirty seconds CPU time on an IBM 8033N computer), but analysis is not required very frequently and should be used as a decision strategic level decisions, the cost of computer resources is well justified.
I
[8], with relatively with 400 activities schedule iteration since this type of support system for
REFERENCES Aquilano, N.J. and D.E. Smith. “A Formal Set of Algorithms for Project Scheduling with Critical Path Method-Material Requirements Planning.” Journal of Operations Management, Vol. 1, No. 2 (November 1980). Bey, R.B., R.H. Doersch, and J.H. Patterson. “The Net Present Value Criterion: Its Impact on Project Scheduling.” Project Management Quarterly June 1981. Davis, E.W. “Project Scheduling Under Resource Constraints-Historical Review and Categorization of Procedures.” AIIE TRANSACTIONS, Vol. 5, No. 4 (December 1973). Davis, E.W. and J.H. Patterson. “A Comparison of Heuristic and Optimum Solutions in Resource
Journal of Operations
Management
Constrained Project Scheduling.” Management Science, Vol. 21, No. 8 (April 1975).
5. Doersch,
R.H. and J.H. Patterson. “Scheduling a Project to Maximize Its Present Value: A Zero-One Programming Approach.” Management Science, Vol. 23, No. 8 (April 1977).
6,
S.E. Activity Networks: Project Planning and Control by Network Models. New York:
Elmaghraby,
John Wiley and Sons, 1977. of the Navy. PERT, Program Evalua7. Department tion Research Task. Phase I Summary Report. Washington, D.C.: Special Projects Office, Bureau of Ordinance, 7, (1958). 8. International
Business
Machine
Corporation.
Ap-
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9.
10.
11.
12.
470
plication System Project Control User Guide. 2d ed. August 1984, pp. 157-165. Kelley, J.E. and M.R. Walker. “Critical Path Planning and Scheduling.” Proceedings of the Eastern Joint Computer Conference. Boston, Mass., 1959. Patterson, J.H. “Alternative Methods of Project Scheduling with Limited Resources.” Naval Research Logistics Quarterly, Vol. 20, No. 4 (December 1973). Patterson, J.H. “Project Scheduling: The Effects of Problem Structure on Heuristic Performance.” Naval Research Logistic’s Quarterly, Vol. 23, No. 1 (March 1976). Russel, A.H. “Cash Flow in Networks.” Management Science, Vol. 16, No. 5 (January 1970).
13. Smith-Daniels, D.E. and N.J. Aquilano. “Constrained Resource Project Scheduling Subject to Material Constraints.” Journal of Operations Management, Vol. 4, No. 4 (August 1984). 14. Talbot, F.B. “Resource-Constrained Project Scheduling with Time-Resource Trade-offs: The Non Preemptive Case.” Management Science, Vol. 28, No. 10 (October 1982). 15. Wiest, J.D. and F.K. Levy. “A Management Guide to PERT/CPM.” Englewood Cliffs, N.J., PrenticeHall, 1977. 16. Wiest, J.D. “Some Properties of Schedules for Large Projects with Limited Resources.” Operations Research, Vol. 12, No. 3 (May-June 1964).
APICS