Robust scheduling and robustness measures for the discrete time/cost trade-off problem

Abstract

Projects are often subject to various sources of uncertainties that have a negative impact on activity durations and costs. Therefore, it is crucial to develop effective approaches to generate robust project schedules that are less vulnerable to disruptions caused by uncontrollable factors. In this paper, we investigate the robust discrete time/cost trade-off problem, which is a multi-mode project scheduling problem with important practical relevance. We introduce surrogate measures that aim at providing an accurate estimate of the schedule robustness. The pertinence of each proposed measure is assessed through computational experiments. Using the insights revealed by the computational study, we propose a two-stage robust scheduling algorithm. Finally, we provide evidence that the proposed approach can be extended to solve a complex robust problem with tardiness penalties and earliness revenues.

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 M_j. Each mode m ∈ M_j, is characterized by a processing time p_jm and a cost c_jm.

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.