Discrete OptimizationRobust scheduling and robustness measures for the discrete time/cost trade-off problem
Introduction
In project management, it is often possible to expedite the duration of some activities and therefore reduce the project duration with additional costs. This time/cost trade-off has been widely studied in the literature focusing on linear and continuous time/cost relationships. In this paper, we address the discrete version, namely the discrete time/cost trade-off problem (DTCTP), which is a multi-mode project scheduling problem having practical relevance. Project managers often allocate more resources to accelerate the activities and each resource allocation defines an execution mode. Thus, multiple alternatives usually exist to execute an activity. DTCTP utilizes only one single nonrenewable resource (money) and does not explicitly consider renewable resources (e.g. machines, equipment and staff), which are available at constant amounts in every instance of the planning period.
Formally, the DTCTP is defined as follows. Given a project with a set of n activities along with a corresponding precedence graph G = (N, A), where N is the set of nodes that refer to the activities of the project, and A ⊂ N × N is the set of immediate precedence constraints on the activities. It is noteworthy that G also includes two dummy “start” and “end” nodes indexed by 0 and n + 1, respectively. Each activity j (j = 1, …, n) can be performed at one of the modes chosen from the set Mj. Each mode m ∈ Mj, is characterized by a processing time pjm and a cost cjm.
Two basic versions of the DTCTP have been defined in the literature so far: the deadline problem (DTCTP-D) and the budget problem (DTCTP-B). In the deadline problem, given a project deadline δ, one of the possible modes is assigned to each activity so that the makespan does not exceed δ and the total cost is minimized. The budget problem, on the contrary, minimizes the makespan while not exceeding a maximum preset budget B. Despite its practical relevance, the research on DTCTP is rather sparse due to its inherent computational complexity (it has been shown to be strongly NP-hard for general activity networks (De et al., 1997)). In their comprehensive review papers, De et al., 1995, Weglarz et al., 2010 discuss the problem characteristics as well as exact and approximate solution strategies. We refer the readers to the papers of Demeulemeester et al., 1996, Demeulemeester et al., 1998 for exact algorithms and to Skutella, 1998, Akkan et al., 2005, Vanhoucke and Debels, 2007, Hafızoğlu and Azizoğlu, 2010 for approximate algorithms. Furthermore, Erengüç et al. (1993) apply Benders decomposition to solve the time/cost trade-off problem with discounted cash flows, which combines the DTCTP and the payment-scheduling problem.
The existing studies on DTCTP generally assume complete information and deterministic environment. However, in practice, projects are often subject to various sources of uncertainty that may arise from the work content, resource availabilities, project network, etc. A schedule that is optimal with respect to project duration or cost may largely be affected by these disruptions. Therefore, it is crucial to develop effective approaches to generate project schedules, which are less vulnerable to disruptions caused by these uncontrollable factors. To the best our knowledge, the only paper which addresses uncertainty on DTCTP is by Klerides and Hadjiconstantinou (2010); they used stochastic programming to model uncertain activity durations.
The contribution of our paper is threefold. First, we introduce a new version of the DTCTP under uncertainty with tardiness penalties and earliness revenues. Second, we propose some surrogate measures to evaluate schedule robustness. The quality of the proposed schedules is assessed through several performance measures. Finally, we develop a two-phase approach for generating robust schedules. The solution approach integrates an analytical tool to support the decision makers in budget allocation decisions and a robust scheduling algorithm. The developed scheduling algorithm addresses the crucial need to construct robust project schedules that are less vulnerable to disruptions caused by uncontrollable factors. Furthermore, it serves as a basis to develop decision support systems (DSS) to help project managers in planning under uncertain environments.
Section snippets
Discrete time/cost trade-off problem under uncertainty
Stochastic programming and robust optimization are two fundamental optimization approaches under uncertainty. Stochastic programming uses probabilistic models to describe uncertain data in terms of probability distributions. Typically, the average performance of the system is examined and expectation over the assumed probability distribution is taken. Robust optimization is a modeling approach to generate a plan that is insensitive to data uncertainty. Generally, the worst-case performance of
Measuring robustness
Developing quantitative metrics that provide a good estimate of schedule robustness is essential for building robust scheduling algorithms. The baseline schedules are execution plans prepared prior to the project execution. The schedules that are created by using these robustness measures could absorb unanticipated disruptions. Existing robust scheduling studies generally employ either direct measures, which are derived from realized performances, or heuristic approaches, which utilize simple
Robust scheduling of DTCTP
Using the insight obtained by the computational experiments, we generate the baseline schedule by maximizing the project buffer size (RM9), the robustness measure that has the highest correlation with the performance measures, so that the schedule involves sufficient safety time to absorb unanticipated disruptions. However, while maximizing robustness, the project cost should also remain within acceptable limits.
Conclusion
To address the crucial need to build robust project schedules that are less vulnerable to disruptions caused by uncontrollable factors, we have investigated the robust scheduling of a variant of the multi-mode discrete time/cost trade-off project scheduling problem. In this variant, the problem is to select a mode for each activity so that the project is completed within a preset deadline and the total cost is minimized. We describe and analyze the pertinence of several robustness measures. We
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