Journal of the Chinese Institute of Industrial Engineers Vol. 27, No. 1, January 2010, 44–51
An approach for identifying the most likely activity sequence in a resource-constrained stochastic network Ching-Chih Tseng* and Po-Wen Ko Graduate Institute of Business Administration, Dayeh University, 168 University Road, Dacun, Changhua 51595, Taiwan, ROC
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(Received December 2008; revised June 2009; accepted June 2009) The biggest concern of a project manager is to identify the proper sequence of activities during project planning and execution so that he or she can appropriately arrange the necessary resources under limited resource availability. The purpose of this article is to develop an approach for identifying the most likely activity sequence in a resource-constrained stochastic network. The approach consists of two stages. The first is a procedure to construct a scenario tree. In the procedure, we integrate a large enough number of sets of feasible activity sequences into a scenario tree. Each set of feasible activity sequences is generated by resource-constrained, deterministic project-scheduling problems with a configuration of randomly generated activity durations. The second stage is a backward-pruning procedure subsequently adopted to eliminate the worse branches from the scenario tree according to a specific objective function. This procedure leads to a reduced scenario tree, on which basis the most likely activity sequence can be easily identified. For demonstration purposes, a simple network example is considered. Keywords: resource-constrained stochastic network; most likely activity sequence; scenario tree
1. Introduction The biggest concern of a project manager is to identify the proper sequence of activities during project planning and execution so that he or she can appropriately arrange the necessary resources under limited resource availability. In projects, in which constraints are only technological, using critical path method (CPM) analysis, a project manager can easily identify the critical path, consisting of a sequence of activity with zero slack, when activity times are certain. When activity times are uncertain, the critical path is not unique. In such a situation, it is meaningful to estimate the criticality of each activity rather than to identify the critical path. The criticality index (CI) of an activity is defined as the proportion of runs in which the activity was on the critical path [13]. Dodin and Elmaghraby [5] provided some theoretical background on this problem as well as extensive test results for large PERT networks. Adlakha and Kulkarni [1] have provided a bibliography that classifies the enormous number of contributions on this topic up until 1987. Instead of investigating the CI of an activity [15], some authors [9–11] provided wider perspectives on project complexity resulting from both time uncertainty and network complexity, characterized by PERT-state and PERT-path techniques. At first, *Corresponding author. Email:
[email protected] ISSN 1017–0669 print/ISSN 2151–7606 online ß 2010 Chinese Institute of Industrial Engineers DOI: 10.1080/10170660903513905 http://www.informaworld.com
Mummolo [9] proposed a new network method, the PERT-path network technique (PPNT). He defined a PERT-path network (PPN) as a directed graph with a tree topology, which facilitates deeper analysis of the possible (partial or complete) project evolutions. During the planning stage, the complete path state that a PERT system will actually follow is still unknown. During this stage, several evolutions are possible, each of which has a probability of occurrence that depends on the planner’s estimates of activity-time probability distribution functions (PDFs) and PERT network topology. Due to uncertain activity durations, more than one path, a sequence of activity-finished events, is probably generated. Thus, when a project is executing, a PERT-path tree network representing all different sequences of the activity-finished event is generated. Activities can be completed during the project time following a possible completion sequence according to the times and precedence constraints among the respective activities. Mummolo [10] evaluated the probability of each studied path on the basis of a PERT-path and represented the uncertainty of each schedule as an entropy value. Then, Pontrandolfo [11] evaluated the expected project duration by calculating the probability and the average length of each path. Since entropy is an objective value of risk evaluation, Bushuyev and Sochnev [4] simultaneously
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Journal of the Chinese Institute of Industrial Engineers used Monte Carlo simulation to evaluate project risk with entropy and subsequently recommended entropy measurement as an alternative approach to risk analyses and project control. However, resource availability is limited in most situations. When resource constraint is considered, CPM analysis is not appropriate. Wiest [14] suggested an amended definition of the critical path: the critical sequence of activities. This sequence reflects both the traditional technological dependencies, as depicted explicitly in the project network, and the dependencies implied by the sharing of scare resources. This concept was first implemented by Woodworth and Shanahan [16], while Bowers [2] described a more succinct algorithm and examined the attributes of the measure, comparing it to other possible measures of criticality. However, Bowers [2] suggested that the resource-constrained float is a reasonably robust measure and should provide a useful basis for identifying the critical activities in stochastic networks. The most complex problem considered is resource-constrained stochastic networks in which activity duration is uncertain. Thus, the complexity of the project-scheduling problem increases dramatically, even for relatively small projects. Fortunately, the increased availability of greater computing power has encouraged the use of simple Monte Carlo simulation rather than analytic approaches [3]. Bowers [3] may have first introduced and presented an alternative method of identifying the most important activities in a stochastic network with resource constraints, using the correlation of an activity’s duration with the project’s duration. However, a project manager cannot effectively manage and execute a project only on the basis of the information of the criticality and cruciality of an activity. The most important information a project manager wants to know is, which activity sequence has most likely occurred during a project execution in a resource-constrained project with uncertain durations. He or she can use this activity sequence as the control baseline. As Fernandez [6] advocated, ‘‘a solution to such a problem should provide the project manager with the guidance, to determine the proper sequencing of tasks during the course of a project so as to optimize the objective function, while respecting the precedence and resource constraints.’’ However, the ‘‘critical sequence of activities’’ is impossible to identify in a resource-constrained stochastic network, because it is not unique. A simple case is introduced to explain this situation. Suppose that there is a four-activity project, as shown in Figure 1, represented by an activityon-arc (AOA) network.
The durations and required amount of resources of the activities is given in Table 1. The duration of each activity is estimated by three times estimate methods, i.e., pessimistic, most likely, and optimistic time. There are two renewable resource types: Types 1 and 2. The availability of resources 1 and 2 are assumed to be one and two, respectively. In this case, there are only two feasible schedules (given in Figure 2): either begin activity (1, 2) first (Schedule A) or activity (1, 3) first (Schedule B). Two possible configurations of activity duration, randomly generated from its corresponding interval, are given in Table 2. From Figure 2, we can find that, in Configuration 1, the critical activity sequence is (1, 3) ! (1, 2) ! (2, 4) and Schedule B is the optimal schedule. Whereas, in Configuration 2, the critical activity sequence is (1, 2) ! (1, 3) ! (3, 4) and Schedule A is the optimal schedule.
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Figure 1. Four-activity AOA PERT network.
Table 1. Data of a project. Activity (1, 2) (1, 3) (2, 4) (3, 4)
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That is to say, for a specific instance (i.e. a deterministic resource-constrained project in which each activity duration is randomly generated from a specific probability distribution), a sequence of activities may be critical in such an instance, but it becomes noncritical in other instances. Consequently, we can only identify the most likely activity sequence instead of finding the critical sequence of activities. In this research, we attempt to develop an approach for identifying the most likely activity sequence in a resourceconstrained stochastic network. The remainder of the report is organized as follows. In Section 2, an approach for identifying the most likely activity sequence in a resourceconstrained stochastic network is proposed. In Section 3, a simple network example is used to illustrate the proposed approach. In Section 4, conclusions are presented.
2. Approach In the following, an approach for identifying the most likely activity sequence in a resourceconstrained stochastic network is proposed. The approach consists of two stages: Stage 1 is to construct a scenario tree. We use the concept of qualitative simulation graph methodology (QSGM), which was proposed by Ingalls and Morrice [8], to build a set of paths for each deterministic resource-constrained network, generated by a configuration of activity durations. Further discussion about QSGM is presented in the Appendix. A path is defined as a feasible sequence of activity events in the project. (Note that the term ‘‘feasible’’ means that the schedule meets both constraints of resource and precedence.) Then we integrate all feasible paths, generated by a specified number of configurations, into a scenario tree. A configuration is defined as a set of activity duration realizations, one realization for each activity in the project. A path from the root to an extremity of the event tree represents a schedule. With each path, a given probability is associated (the ratio of the occurrence number of the path over the total number of iterations). Hence, the resulting scenario tree, which consists of events, decision forks, and chance forks, represents all possible paths that a project execution can follow. As the project is executed, some forks will be encountered. Each fork in the tree is either a decision fork, whose branches consist of alternative courses of action to decide which activity (or activities) should be performed first when resource conflict is occurred or a chance fork, whose branches consist of the uncertainties resulting from an uncertain duration of activity.
After simulating and integrating all of the possible event sequences of activities in the project, the integrated scenario tree consists of three parts of information. The first is the decision fork, which resulted from the resource conflict; the second is the chance fork, which comes from the uncertain activity durations; the third is that with each path, a given probability and the length are associated. Once all feasible paths are generated by simulating a large enough number of configurations, then a scenario tree, which consists of all possible and approximately complete paths, can be presented. Based on the integrated scenario tree and a specified objective function, we can perform our proposed pruning procedure to get a reduced scenario tree. Stage 2 is a backward-pruning procedure, subsequently adopted to eliminate the worse branches of the scenario tree with the larger average project durations. The procedure leads us to the reduced scenario tree. The information embedded in the reduced scenario tree only includes (activity-start and -finish) events, chance forks, and with each path a given probability is associated. At each decision fork, a decision would be made when there are multiple activities that could start the process, but there is a lack of available resources to start all the activities. The scheduler would decide which activity (or activities) would be allocated the constrained resources. The backward-pruning procedure is described in detail as follows. Step 1: Examine each fork in the scenario tree from the right-most to the root. If it is a chance fork, go to Step 2; or if it is a decision fork, go to Step 3; otherwise, examine the nearby ones. If there is no fork and the root of the scenario tree is reached, go to Step 4. Step 2: Calculate the expected project duration of the chance fork by summing the expected project duration of each branch path. Step 3: Prune away the branch with the worse objective function. Step 4:
A reduced scenario tree is obtained.
By our proposed approach, the complexity of the problem can be simplified into a reduced scenario tree; the project manager can easily identify the most likely activity sequence in the reduced scenario tree.
3. Illustrated example As an illustrative example, a stochastic network consisting of five activities with one renewable resource is considered. A five-activity AOA PERT network is shown in Figure 3. The three values in
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parenthesis over an arrow represent the optimistic, most likely, and pessimistic time of an activity. The available quantity of resources in each period during project execution is 2. The required amount of resources of each activity is 1. Table 3, presents three possible configurations of activity durations. (Note that for simplicity, we assume that all the realizations of activity durations are integers.)
3.1 Constructing a scenario tree Applying the concept of QSGM, we can generate the corresponding possibly feasible paths (schedules) of each configuration shown in Figure 4. For Configuration 1, there are two feasible paths: Paths 1 and 2. The lengths (i.e. project duration) of Paths 1 and 2 are 15 and 17, respectively. Similarly, there are four and two feasible paths for Configurations 2 and 3, respectively. Note that Sxy (Fxy), shown in Figure 4, denotes the start (finish) event of activity (x, y) in each feasible path. We then integrate all the feasible paths of these three configurations into a scenario tree, as shown in Figure 5. The scenario tree is similar to a
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Figure 3. Five-activity AOA PERT network.
3.2 Backward-pruning procedure
Table 3. Three configurations of activity durations. Activity
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Configuration 1 Configuration 2 Configuration 3
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decision tree, which consists of events, decision forks, and chance forks. One merit of the resulting scenario tree is that any branch (i.e. any schedule) of the tree is feasible for resource and precedence constraints. The only thing for the project manager to do is to decide, which branch should be followed at each decision fork. In the following, we will explain how to integrate a scenario tree from the possible scenarios of each configuration. From the left-most of each path in Figure 4, the first two events, S12 and S13, are the same, so we link them with an arrow. But at the third event, two states have possibly occurred: either F12 or F13. As the project executes, we do not know which activity is completed first, unless their durations are realized. Therefore, we must set a chance fork, denoted as CF (F12, F13), which results from the uncertain durations of activities (1, 2) and (1, 3). If activity (1, 3) is completed first, we immediately face a decision over whether activity (3, 4) or activity (3, 5) should start first. At this time, a decision fork, DF (S34, S35), is generated. Finally, a scenario tree is constructed, as in Figure 5. Note that there are two paths, the same in the scenario tree: one is the second path of Configuration 1 and the other is the second path of Configuration 3. The integrated scenario tree provides information including possible paths (sequences of activity events), project execution, the occurrence number, and the length of each path. Therefore, the project manager can easily decide which activity should be chosen to start first according to his or her objective function.
5 6 5
In the following, we use the above instance to illustrate the backward-pruning procedure. From Figure 5, in the scenario tree, there are seven feasible paths, which resulted from four decision forks and two chance forks. In the backwardpruning procedure, we examine the fork in the
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Figure 4. Feasible paths of three possible configurations of project.
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scenario tree from the right-most to the root. First, a chance fork, CF (F24, F34), is encountered, therefore we performed Step 2. The expected project duration of the chance fork, CF (F24, F34), is 14.5 days, i.e. (15 1 þ 14 1)/2. Then, we check the next fork. Next three decision forks, DF (S34, S35), DF (S24, S35), and DF (S24, S34), are encountered, we perform Step 3. First, we proceed with the decision fork DF (S34, S35). Since the expected project duration of the branch following S34 (14.5 days) is smaller then that of the branch following S35 (17.5 days), according to minimizing project duration assumed, the branch following S35 is deleted. Similarly, we perform the other two decision forks: DF (S24, S35) and DF (S24, S34). Until the backward-pruning process is finished, the reduced scenario tree, shown in Figure 6, which only contains chance forks, will be obtained. The expected project duration of the reduced scenario tree is 15. From Figure 6, we know that there are three possible paths remaining after minimizing the expected project duration. With each path, the occurring probability of 1/3 is associated.
3.3 Identifying the most likely activity sequence If we simulate 10,000 configurations instead of three, then apply the proposed approach consisting of constructing a scenario tree and employing the backward-pruning procedure. The reduced scenario tree can be obtained and is shown in Figure 7. Based on this reduced scenario tree, a project manager can easily identify which path is most likely to occur. That is to say, the most likely activity sequence is S12 ! S13 ! F12 ! S24 !
F13 ! S34 ! F24 ! S35 ! F34 ! S45 ! F35 ! F45 with highest occurrence probability 0.5328 (i.e. 5328/10,000).
4. Conclusion and future research In this report, an approach for identifying the most likely activity sequence in a resource-constrained project with uncertain durations is proposed. The approach involves two stages. Stage 1 is to construct the scenario tree. We used the concept of QSGM, which is proposed by Ingalls and Morrice [8] to build a set of paths for each resource-constrained project-scheduling problem with deterministic activity durations, generated by a configuration of activity durations. In the first stage, a specified number of deterministic resourceconstrained networks are simulated, and the paths integrated into a scenario tree. In the second stage, a reduced scenario tree is obtained by performing the proposed backward-pruning procedure according to a specific objective function. Based on the proposed approach, a project manager can wholly understand all possible paths of project evolution and easily identify the most likely activity sequence in a resource-constrained project with uncertain durations. By using our proposed approach, the complexity of the SRCP problem can be simplified into a reduced scenario tree. As is known, generating all possible paths by QSGM is less efficient and the computational time will increase exponentially with the problem size. Future research may develop an algorithm to improve its efficiency. Project duration
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Figure 5. Integrated scenario tree.
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Figure 6. Reduced scenario tree from three configurations.
Expected project duration (15* 1/3 + 14* 1/3 + 16* 1/3) = 15
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Journal of the Chinese Institute of Industrial Engineers Expected project duration: 14.548 days The probability of the branch path
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Figure 7. Reduced scenario tree from 10,000 configurations.
Notes on contributors Ching-Chih Tseng is an Associate Professor in the Department of Business Administration Management at Dayeh University, Taiwan. He received his Bachelor’s degree and Master’s degree in Industrial Management from National Cheng Kung University, Taiwan, in 1985, and his PhD degree in Industrial Engineering and Management from Tokyo Institute of Technology in 1996. His research activities include project management, multi-criteria decision analysis, quality function deployment for product design and development, and e-commerce. Po-Wen Ko is currently a Senior Engineer in the TV strategy procurement department of the TV Procurement Division at AU Optronics Corporation, Taiwan. He received his Master’s degree in Business Administration from Dayeh University, Taiwan, in 2007.
References [1] Adlakha, V.G. and V.G. Kulkarni, ‘‘A classified bibliography of research on stochastic PERT networks: 1966–1987,’’ INFOR, 27, 272–296 (1989). [2] Bowers, J.A., ‘‘Criticality in resource constrained networks,’’ Journal of Operational Research Society, 46, 80–91 (1995). [3] Bowers, J.A., ‘‘Identifying critical activities in stochastic resource constrained networks,’’ International Journal of Management Science, 24, 37–46 (1996). [4] Bushuyev, S.D. and S.V. Sochnev, ‘‘Entropy measurement as a project control tool,’’ International Journal of Project Management, 17, 343–350 (1999).
[5] Dodin, M.B. and S.E. Elmaghraby, ‘‘Approximating the criticality indices of the activities in PERT networks,’’ Management Science, 31, 207–223 (1985). [6] Fernandez, A., ‘‘Understanding simulation solutions to resource constrained project scheduling problems with stochastic task durations,’’ Engineering Management Journal, 10, 5–13 (1998). [7] Ingalls, R.G., D.J. Morrice and A.B. Whinston, ‘‘The implementation of temporal intervals in qualitative simulation graphs,’’ ACM Transactions on Modeling and Computer Simulation, 10, 215–240 (2000). [8] Ingalls, R.G. and D.J. Morrice, ‘‘PERT scheduling with resource constraints using qualitative simulation graphs,’’ Project Management Journal, 35, 5–14 (2004). [9] Mummolo, G., ‘‘PERT-path network technique: a new approach to project planning,’’ International Journal of Project Management, 12, 89–99 (1994). [10] Mummolo, G., ‘‘Measuring uncertainty and criticality in network planning by PERT-path technique,’’ International Journal of Project Management, 15, 377–387 (1997). [11] Pontrandolfo, P., ‘‘Project duration in stochastic networks by PERT-path technique,’’ International Journal of Project Management, 18, 215–222 (2000). [12] Schruben, L.W., ‘‘Simulation modeling with event graphs,’’ Communications of the ACM, 26, 957–963 (1983).
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[13] Shtub, A., J. Bard and S. Globerson, Project Management: Engineering, Technology, and Implementation, Prentice Hall, New Jersey (1994). [14] Wiest, J.D., ‘‘Some properties of schedules for large projects with limited resources,’’ Operations Research, 12, 395–418 (1963).
[15] Williams, T., ‘‘Criticality in stochastic networks,’’ Journal of Operational Research Society, 43, 353–357 (1992). [16] Woodworth, B.M. and S. Shanahan, ‘‘Identifying the critical sequence in a resource constrained project,’’ International Journal of Project Management, 6, 89–96 (1988).
Appendix
purpose, the node will be labeled Hx, where x is the number of the AOA node.
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The QSGM that has been developed by Ingalls et al. [7] is a general-purpose qualitative discrete-event simulation framework, which can be used for any type of discrete-event simulation (DES) problem. Schruben [12] introduced event graphs (EGs) as a modeling paradigm for DES. The QSGM would then use the interval information to generate all possible feasible paths in the problem. The conversion of the AOA network with resources to a QSGM model is identical to conversion to an EG, which can be achieved by the following steps [8]. Step 1: Add an initialization node, called node 0, for initializing the available number of resources. Step 2: For each node in the AOA network, create a corresponding node. This node will be the hit node and count the number of previous completed activities associated with the corresponding AOA node. For our
Step 3: For each arc in the AOA network, there are two more nodes that are created in the EG. The first is the start event that signifies the start of the activity and the finish event that signifies the completion of the activity. For our purpose, the start node is labeled Sxy and the finish node is labeled Fxy, where x is the number of the start node and y is the number of the finish node for the given arc in the PERT network. Step 4: Each hit node is connected to its corresponding nodes with a scheduling condition and an execution condition. The scheduling condition checks to see if the hit node has been hit the proper number of times (i.e. all predecessor activities have been completed). The execution condition checks to see if resources are available. Based on converted EG, all feasible paths can be easily formed.
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(scenario tree)
(reduced scenario tree)
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