A computational study of exact approaches for the adjustable robust resource-constrained project scheduling problem
Introduction
The resource-constrained project scheduling problem (RCPSP) is an outstanding and challenging operations research problem, with applications in a variety of applicative contexts (Bruni, Beraldi, Guerriero, Pinto, 2011, Gourgand, Kellert, 1991, Schwindt, Trautmann, 2003). In the traditional RCPSP the scheduling of project activities is addressed in presence of renewable resources and precedence constraints, with the aim of minimizing the project completion time (makespan). In most of the scientific contributions on the RCPSP, the main model parameters have been assumed to be deterministically known. However, a growing attention on projects execution and control has fostered a flourishing stream of literature incorporating the uncertainty in problem parameters. Numerous attempts have been made to address on the RCPSP under uncertainty in which activity durations are assumed to be random variables. Two main different approaches have been proposed in the last two decades. The problem can be viewed as a stochastic dynamic optimization problem, where decisions are made each time new information becomes available, or as a problem where a tentative plan, which can be changed during project execution, should be determined and agreed before knowing the realization of uncertainty. For an extensive review of research in this field, the reader is referred to Bruni et al. (2015); Herroelen, Leus, 2004b, Herroelen, Leus, 2005.
In the stochastic programming framework, for a special case with a budget constraint, a two-stage integer linear stochastic program has been proposed in Zhu et al. (2007). Building a baseline schedule which is protected against possible disruptions is the main research question addressed in Bruni et al. (2011a), where a chance-constrained based heuristic has been proposed. More recently, a two-stage stochastic programming model for the RCPSP has been proposed in Bruni et al. (2018) assuming that the random variables are discretely distributed.
The robust optimization (RO) approach (Ben-Tal et al., 2004) has been barely applied as a modeling framework for the RCPSP under uncertainty. A minimax absolute-regret robust RCPSP formulation has been proposed in Artigues et al. (2013). Assuming that scenarios represent different realizations of the activity durations, the objective is to find a schedule that minimizes the maximum absolute regret over all scenarios. The approach follows the risk-averse view of optimizing the worst-case deviation to scenario-dependent optimal solutions, while ensuring feasibility in all scenarios.
In the last ten years adjustable robust optimization has become a viable tool in dealing with optimization under uncertainty (Ben-Tal, Goryashko, Guslitzer, Nemirovski, 2004, Chen, Zhang, 2009). These approaches overcome the over-conservativeness of the static robust models and enables adaptive decision-making modelling paradigms, where the decision maker is able to adjust his strategy to information revealed over time. The most common type of the robust multi-stage models is the two-stage one, with a simple chronology of decisions and observations. In this case, in fact, here-and-now decisions are taken in face of uncertainty, whereas wait-and-see decisions can be determined when the uncertain data become known. Following the two-stage adjustable robust framework, which has quickly emerged as a cutting-edge research area to handle dynamic robust optimization problems (Ben-Tal, El Ghaoui, Nemirovski, 2009, Bertsimas, Brown, Caramanis, 2011, Chen, Zhang, 2009), an adjustable robust model for the RCPSP under general polyhedral uncertainty sets has been proposed in Bruni et al. (2017). The model fits the nature of the project scheduling decision-making process, where sequencing decisions, concerning the order of the project activities, can be taken in advance, whereas the activity starting times can be determined once the activity durations become known.
Motivated by this contribution and by the limited literature on the robust project scheduling problem, despite its high potential in a variety of applications, in this paper, we carry out a computational study aimed at investigating and comparing the computational benefits of two decomposition approaches for the robust RCPSP under budgeted uncertainty.
The remainder of the paper is organized as follows. In Section 2, we introduce a formal definition of the two-stage RCPSP, where the activity durations are subject to interval uncertainty and the level of robustness is controlled by a protection factor. In Sections 3 and 4, we present the decomposition approaches. An extensive computational experimentation, carried out on standard benchmark instances, is discussed in Section 5. Finally, some conclusions are drawn in Section 6.
Section snippets
The two-stage robust RCPSP under budgeted uncertainty
In this Section, the model development for the robust RCPSP is introduced. Different mathematical programming formulations have been proposed in the literature for the RCPSP, and recently they have been compared computationally (Koné et al., 2013). Among others, we mention formulations such as the discrete time formulation (Pritsker et al., 1969) where binary variables are indexed by both activities and time, the event based formulation, where variables are indexed by events (Koné et al., 2011)
The dual method
In this Section, we present a solution approach based on the method used by Thiele et al. (2009). Benders decomposition was introduced as a method to solve deterministic structured mixed-integer programs (Benders, 1962). It applies to problems where it can be easily identified a subset of complicating variables that, once fixed, considerably reduce the complexity of the problem. The algorithm alternates between a master problem (usually a relaxation of the problem) and a sub-problem. The
The primal algorithm
In this Section, we present a solution approach based on the method used by Zeng and Zhao (2013). Any two-stage robust problem with polyhedral uncertainty can be reduced to a possibly large-scale mixed integer program, where copies of the second stage variables are introduced for each extreme point of the uncertain polyhedron. The plain MIP, that could be (in principle) solved by any MIP solver, is reported hereafter, for the sake of completeness.
Computational results and discussion
This Section reports on the computational results carried out to assess and compare the performance of the proposed decomposition approaches. The algorithms have been coded in Java and run on a PC with 16GB RAM and 2.50 GHz Intel Core i7 - 4710 HQ CPU. The Cplex 12.5.1 library has been used to solve the master problem (with a time limit of 1200 s), whereas the subproblems have been solved by using the dynamic programming approach proposed in Bruni et al. (2017).
The numerical results have been
Conclusions
In this paper, we have proposed two exact methods for the robust RCPSP under budgeted uncertainty, where the two-stage robust optimization framework has been used to address the inherent uncertainty in the activity durations.
The two approaches provides quite different master structures. The master of the DM contains many mild constraints, while the PM has less constraints but with a combinatorial structure. As a result, solving the dual master requires significantly less effort resulting in
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