A satisfiability and workload-based exact method for the resource constrained project scheduling problem with generalized precedence constraints

Highlights

• An new exact method based on satisfiability and workload is proposed for the problem.

• The proposed approach also performs well on instances of the classical version.

• A number of open instances were solved for the first time.

• Several improved bounds are reported for instances that still remain open.

Abstract

This paper deals with the resource constrained project scheduling problem (RCPSP) with generalized precedence constraints (RCPSP/Max). We propose an exact method that tackles a relaxed version of the original problem by modeling it as a satisfiability problem (SAT). Several SAT formulations are introduced, as well as workload-based procedures that were developed in order to reduce the domain of the decision variables, and custom propagators that were implemented with a view of improving the efficiency of the SAT solver. Extensive computational experiments involving different configurations of the method were carried out on 2430 RCPSP/Max benchmark instances ranging from 10 to 500 activities and with 5 resources, and on 2040 RCPSP benchmark instances ranging from 30 to 120 activities and with 4 resources. The results obtained suggest that the proposed method is extremely competitive as almost all known optima were found and 86 instances were solved to optimality for the first time. Moreover, a number of bounds were improved for those instances that still remain open.

Introduction

Optimizing project scheduling is an important activity in virtually every field of knowledge because it can avoid a waste of time and resources, especially in big projects such as rocket launches, building hydro-electrical plants or oil and gas platforms (Pinedo, 2004). The resource constrained project scheduling problem (RCPSP) is a well-known combinatorial optimization problem that can be formally defined as follows. Let $V=\{0,1,\cdots ,N,N+1\}$ be the set of activities to be scheduled using a set $\mathcal{R}=\{1,\cdots ,m\}$ of resources throughout a time horizon [0, T). Each activity j ∈ V has an execution time p_j and it consumes a specific amount r_ij of resource $i\in \mathcal{R}$ at every time instant while it is being executed. For each resource $i\in \mathcal{R}$ there are R_i units available throughout the time horizon. Activities 0 and $N+1$ represent the beginning and end of the project, respectively, with $p_0=p_{N+1}=0$ and $r_{i0}=r_{i(N+1)}=0$ for all $i\in \mathcal{R}$. The schedule should also respect activity precedences, which are represented by a directed graph $G=(V,A)$. For each (j, l) ∈ A, let d_jl be the length of the arc and also the time lag from activity j to activity l. In this case, l must start at least d_jl time units after j starts and j must start at most $-d_{jl}$ time units after l starts. The objective is to minimize the makespan, that is, the time required to process all activities (or, alternatively, the completion time of the project).

Activities without predecessors will succeed 0, whereas those without successors will precede $N+1$. The precedence relations imply that each activity j ∈ V has an associated early start time (ES_j) and late finish time (LF_j), thus meaning that j must be executed within the interval [ES_j, LF_j). In the classical RCPSP, only strict precedences are allowed, i.e., if (j, l) ∈ A then $d_{jl}=p_j$. As for the RCPSP with generalized precedence constraints (RCPSP/Max), a.k.a. RCPSP with minimal and maximal time lags, which is the variant studied in this paper, $-\infty \le d_{jl}\le \infty$ and non-positive cycles are allowed in G.

RCPSP and its variants are usually $\mathcal{NP}$-hard as they generalize shop problems, such as open shop or job shop (Blazewicz, Lenstra, & Kan, 1983). Bartusch, Möhring, and Radermacher (1988) even showed that proving whether an instance has a feasible solution is $\mathcal{NP}$-hard for the RCPSP/Max.

In this paper, we are interested in the exact solution of the RCPSP/Max and its particular cases (including the classical RCPSP) for which we provide several contributions:

• We propose three novel formulations based on the satisfability problem (SAT), which are used to check whether a solution is feasible for a given makespan value. The first one extends the formulation suggested in Horbach (2010), whereas the second one incorporates the concept of ordered-sets as in the constraint programming (CP)-based formulation of Schutt, Feydy, Stuckey, and Wallace (2013b). With the observations made on those formulations, we were able to devise new ideas that were compiled in a third formulation.

• We implement custom propagators for an existing SAT solver in order to generate clauses on demand so as to improve the solver efficiency when dealing with large number of clauses in the formulation.

• We introduce a workload-based approach with a view of reducing the domain of the decision variables, thus potentially reducing the solution space.

• We show how to combine the SAT and workload-based approaches in order to build a highly competitive exact method for the RCPSP/Max.

Extensive computational experiments involving various settings of the proposed method were carried out on RCPSP/Max and RCPSP benchmark instances, where almost all known optimal solutions were obtained and a number of open problems were solved to optimality for the first time, thus illustrating the efficiency and competitiveness of our approach.

The remainder of the paper is organized as follows. Section 2 presents a literature review on the existing exact methods for resource constrained scheduling. Section 3 describes the main contributions of this work, including the novel SAT formulations, the custom propagators, and the workload-based approach. Section 4 contains the results of the extensive computational experiments performed on benchmark instances. Finally, Section 5 brings the concluding remarks.