Discrete OptimizationProperties of multi-mode resource-constrained project scheduling problems with resource vacations and activity splitting
Introduction
Many real-world scheduling problems can be modeled as multi-mode resource-constrained project scheduling problems (MRCPSP). The MRCPSP consists in scheduling project activities to complete a project in the minimum possible time under the presence of precedence and resource constraints. In this paper, activities require resources and all resources are assumed to be renewable (no non-renewable resources are included), which are available in limited amounts, and each activity must be performed in one of several possible modes (with each mode possibly having different activity durations, and different resource requirements).
In actual scheduling problems that can be modeled as MRCPSPs resources will not be available at all times. In some cases the periods of resource unavailability are known in advance. This is particularly true when the resources are human resources. Human resources can become unavailable due to vacations, special projects, training, or an unlimited number of other reasons, and these types of absences are usually known in advance. For non-human resources predictable absences often take the form of scheduled maintenance, overhauls, or the use of a machine for a special activity. The sources of resource unavailability, as described in the prior examples, and which are typically known in advance, are referred to as resource vacations. The result of incorporating resource vacations is that the initial availability of resources varies over time, which is the same result as using time-varying resource capacities as defined in Drexl and Grünewald (1993) (there is a difference in how the varying resource capacities are generated). Real examples of MRCPSPs with resource vacations are the scheduling of engineering design activities in product development; the scheduling of activities in financial audits; and the assignment of coding activities in large software development.
In addition to resource vacations, which cause initial resource availability to vary over time, temporary resource unavailability is also caused by scheduling activities (when activities are scheduled serially). Unlike temporary resource unavailability caused by resource vacations this type of resource unavailability is not known in advance.
The most current heuristics and exact algorithms for the MRCPSP assume that an activity, once started, cannot be interrupted. In other words, each activity in a project can be scheduled only when the precedence constraints are satisfied and the required resources are available for the duration of the activity. This assumption may lead to schedules that can be significantly improved if activities can be split (i.e., suspended and restarted) around unavailable resources when scheduling. Examples of such work can be listed and classified as follows (for a thorough classification of RCPSPs, see Brucker et al., 1999, Herroelen et al., 1999):
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Exact methods—Hartmann and Drexl (1998), Kolisch et al. (1995), Heilmann (2003), Sprecher and Drexl (1998), Sprecher et al. (1997), among others;
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Heuristics (priority rule-based)—Boctor (1993), Kolisch, 1996a, Kolisch, 1996b, among others;
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Meta-heuristic methods—Alcaraz et al. (2003), Bouleimen and Lecocq (2003), Hartmann (2001), Heilmann (2001), Jozefowska et al. (2001), Nonobe and Ibaraki (2001), among others.
The focus and contribution of this work is to provide evidence, through computational experiments, into what types of project scenarios may result in significant makespan improvements with activity splitting, and how such scenarios can be characterized. Different project scenarios are investigated using test problem sets that are generated using parameters that characterize different resource requirement levels, resource capacities, and the periods and lengths of resource unavailability. The problem instances generated are solved to optimality with and without activity splitting using the branch-and-bound algorithm of Hartmann and Drexl (1998). Modifications are made to the algorithm so that it is capable of solving pre-emptive problems (details of the branch-and-bound algorithms used are in Appendix B). Additionally, splitting activities in many actual projects is real (especially when human resources are involved) and thus further understanding of the benefit and/or necessity of this practice will be of value.
The remainder of the paper is organized as follows: In Section 2, the mathematical model of the MRCPSP with activity splitting is presented. Section 3 contains a relevant literature review of the MRCPSP that consider activity splitting. In Section 4 the empirical approach used in this research is described, and in Section 5 the computational experiments and results are presented. Finally, conclusions are presented in Section 6.
Section snippets
Problem description
This section provides the precise description of the problem. The MRCPSP considered adheres to the following assumptions:
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A project consists of J activities represented using an activity-on-node (AON) representation, where nodes represent the activities and arcs denote the precedence relationships. Two dummy activities are introduced. Dummy activity 1 represents the start activity of the project and dummy activity J represents the end activity of the project, where each node is reachable from
Relevant literature review
RCPSP research with activity splitting (referred to as pre-emption) is found in Bianco et al. (1999), Demeulemeester and Herroelen (1996), and Valls et al. (1999), and has so far considered only the single mode problem.
Bianco et al. (1999) considered the problem of scheduling activities on dedicated resources, where each renewable resource can execute one activity at a time and each activity can be pre-empted at integer points of time and resumed later without additional duration. Their
Approach
The focus of this work is on developing evidence into what types of project scenarios may result in significant makespan improvements with activity splitting in the presence of known resource vacations. To this end the approach used was a designed computational experiment that includes the three factors found in ProGen (Kolisch et al., 1992, Kolisch et al., 1995, Kolisch and Sprecher, 1996) to characterize a RCPSP and two additional factors added to characterize resource vacations. Small
Experiment 1: Initial factor screening
The objective of experiment 1 is to determine the project parameters where activity splitting has a high probability of improving schedules. Factors (defining a project scenario) having no impact on makespan improvement are removed in subsequent experiments. In experiment 1, two levels of the five project factors were considered resulting in a 25 full factorial design (see Table 3). We conducted 20 replications (20 different problem instances) for each treatment. Factor levels were selected to
Conclusions
This study investigated potential makespan improvements obtainable from allowing activity splitting in the scheduling of the multi-mode resource-constrained project scheduling problem where renewable resources may be temporarily unavailable. Full factorial experiments are conducted where experimental factors are project parameters that characterize different resource scenarios. A branch-and-bound procedure is applied to solve small problem instances with and without activity splitting to
Acknowledgments
The authors would like to thank the referees for their comments and suggestions which have considerably improved this paper.
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