Discrete Optimization
Multi-mode resource-constrained project scheduling using RCPSP and SAT solvers

https://doi.org/10.1016/j.ejor.2011.03.019Get rights and content

Abstract

This paper reports on a new solution approach for the well-known multi-mode resource-constrained project scheduling problem (MRCPSP). This problem type aims at the selection of a single activity mode from a set of available modes in order to construct a precedence and a (renewable and non-renewable) resource feasible project schedule with a minimal makespan. The problem type is known to be NP-hard and has been solved using various exact as well as (meta-)heuristic procedures.

The new algorithm splits the problem type into a mode assignment and a single mode project scheduling step. The mode assignment step is solved by a satisfiability (SAT) problem solver and returns a feasible mode selection to the project scheduling step. The project scheduling step is solved using an efficient meta-heuristic procedure from literature to solve the resource-constrained project scheduling problem (RCPSP). However, unlike many traditional meta-heuristic methods in literature to solve the MRCPSP, the new approach executes these two steps in one run, relying on a single priority list. Straightforward adaptations to the pure SAT solver by using pseudo boolean non-renewable resource constraints has led to a high quality solution approach in a reasonable computational time. Computational results show that the procedure can report similar or sometimes even better solutions than found by other procedures in literature, although it often requires a higher CPU time.

Highlights

► A novel approach to solve the multi-mode resource-constrained project scheduling problem. ► The solution approach splits the problem into a mode assignment step and a single mode project scheduling step. ► The solution approach use only one priority list, instead of using an activity and a mode list as normally done in literature. ► The solution approach has the potential to solve numerous extensions to the well-known MRCPSP. ► The results are comparable to state-of-the-art procedures and even outperforms them when run times are set long enough.

Introduction

Resource-constrained project scheduling has been a research topic for many decades, resulting in a wide variety of optimization procedures that differ in objective functions, activity assumptions, resource constraints and many more. The main focus on project makespan minimization has led to the development of various exact and (meta-)heuristic procedures for scheduling projects with tight renewable resource constraints where the well-known resource-constrained project scheduling problem (RCPSP) clearly took the lead. This problem type aims at minimizing the total duration or makespan of a project subject to precedence relations between the activities and the limited renewable resource availabilities, and is known to be NP-hard (Blazewicz et al., 1983). Extensions to other objective functions (see e.g. the vast amount of research in net present value optimization), resource constraints (to non-renewable and doubly-constrained resources), multiple activity modes, etc. often result in highly complex optimization problems and have been studied in literature by many authors. For an overview of resource-constrained project scheduling in general, we refer to excellent overview papers of Brucker et al., 1999, Herroelen et al., 1998, Icmeli et al., 1993, Kolisch and Padman, 2001, Kolisch and Hartmann, 2006, Özdamar and Ulusoy, 1995 and the research handbook by Demeulemeester and Herroelen (2002).

Many research efforts have extended the RCPSP to the presence of multiple activity modes where each activity can be executed under a different duration and a corresponding renewable and nonrenewable resource use. Due to the complex nature of the problem, only a few exact algorithms have been presented in literature (Slowinski, 1980, Talbot, 1982, Patterson et al., 1989, Speranza and Vercellis, 1993, Sprecher, 1994, Sprecher et al., 1997, Hartmann and Drexl, 1998, Sprecher and Drexl, 1998, Zhu et al., 2006). Due to the intrinsic hardness of the problem formulation, many heuristic (Boctor, 1993, Drexl and Grünewald, 1993, Özdamar and Ulusoy, 1994, Boctor, 1996a, Kolisch and Drexl, 1997, Knotts et al., 2000, Lova et al., 2006) and meta-heuristic (Slowinski et al., 1994, Boctor, 1996b, Mori and Tseng, 1997, Özdamar, 1999, Hartmann, 2001, Józefowska et al., 2001, Nonobe and Ibaraki, 2002, Alcaraz et al., 2003, Bouleimen and Lecocq, 2003, Zhang et al., 2006, Lova et al., 2009, Jarboui et al., 2008, Ranjbar et al., 2009, Van Peteghem and Vanhoucke, 2009, Van Peteghem and Vanhoucke, 2010) solution algorithms have been presented in the literature. A clear distinction can be made between algorithms incorporating both renewable and non-renewable resource constraints and algorithms limited to projects with only renewable resource constraints.

In this paper, we present a novel meta-heuristic approach to solve the non-preemptive multi-mode resource-constrained project scheduling problem (MRCPSP) within the presence of both limited renewable and non-renewable resource constraints. The paper is organized as follows. Section 2 introduces the notation and describes the problem formulation in detail. In Section 3 we present our approach to solve the scheduling problem type under study and give illustrative examples. Moreover, it is shown that the solution approach is very general and can be used for various other scheduling extensions. Section 4 enhances this solution approach to cope with excessive memory requirements. Section 5 reports comparative computational results and Section 6 contains the conclusions.

Section snippets

Model formulation

The multi-mode project scheduling problem with multiple renewable and non-renewable resources can be stated as follows. A set of activities N, numbered from a dummy start node 0 to a dummy end node n + 1, is to be scheduled without pre-emption on a set Rr of renewable resources and a set of Rn of non-renewable resources. Each renewable resource k  Rr has a constant availability akr per period while each non-renewable resource l  Rn is restricted to aln units over the complete planning horizon. Each

Solution approach

The MRCPSP can be easily modeled as an RCPSP instance where each multi-mode activity i is split into Mi single-mode sub-activities among which exactly one sub-activity needs to be selected for execution. Consequently, the project network of Fig. 1 can be transformed into an RCPSP network with i=1nMi non-dummy sub-activities as displayed in Fig. 2, where the first number below the node denotes the sub-activity duration and the two other numbers below the node the renewable and non-renewable

Adapted pseudo boolean solver approach

In the model presented earlier, each non-renewable constraint is the subject of the enumeration scheme of Section 3.1.1, which leads to a set of clauses that need to be stored as an input file for the SAT solver (i.e. the DPLL procedure) that is called for each activity list generated during the search. When the size of the project network instance become relatively large, both in terms of the number of project activities as the number of non-renewable resource constraints, the number of

Computational results

This section reports computational results to evaluate the performance of the algorithm. The algorithm has been coded in C++ and tests have been run on a Dell Dimension DM051 with a Pentium D with a 2.80 GHz processor. The first benchmark test set is the well-known PSPLIB dataset which contains multi-mode project network instances generated by ProGen (Kolisch et al., 1995) with 10, 12, 14, 16, 18, 20 and 30 activities and with 2 renewable and 2 nonrenewable resources. The set is available from

Conclusions

In this paper, a novel approach has been presented to solve the multi-mode resource-constrained project scheduling problem (MRCPSP). The algorithm splits the problem into a mode assignment step and a single mode project scheduling step. The mode assignment step is solved using a fast and efficient SAT solver. Due to excessive memory requirements, a number of small and straightforward adaptations to this solver have been implemented to solve the SAT problem instances in less memory. The single

Acknowledgements

We acknowledge the support given by the Fonds voor Wetenschappelijk Onderzoek (FWO), Vlaanderen, Belgium under Contract Number G.0463.04.

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      The classical RCPSP aims to generate an optimal schedule to minimize the whole project’s makespan satisfying the precedence relationship between activities and the resource constraints simultaneously (Afshar-Nadjafi et al., 2017). For example, Van Peteghem and Vanhoucke (2010), Coelho and Vanhoucke (2011) and Cheng et al. (2015) proposed to minimize the total makespan of the project for an RCPSP with consideration of the constraints for both renewable and non-renewable resources. These studies aim to deal with deterministic project structures, which means that the activities and the corresponding durations included in a project are fixed.

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