Fixed set search application for minimizing the makespan on unrelated parallel machines with sequence-dependent setup times

Highlights

• A novel Fixed Set Search metaheuristic is applied to a scheduling problem.

• The Fixed set search adds a learning mechanism to the Greedy Randomized Adaptive Search Procedure.

• The method has been evaluated on standardly used benchmark data sets.

• The Fixed set search significantly outperforms other population based metaheuristics for the problem of interest.

Abstract

This paper addresses the problem of minimizing the makespan on unrelated parallel machines with sequence-dependent setup times. The term unrelated machines is used in the sense that there is no correlation between the processing times for jobs between different machines. Due to the NP-hardness of this problem, a wide range of metaheuristics has been developed to find near-optimal solutions. Out of such methods, the ones based on constructive greedy algorithms like the Greedy Randomized Adaptive Search Procedure (GRASP), Ant Colony Optimization (ACO) and Worm Optimization (WO) proved to be most efficient. The Fixed Set Search (FSS) is a novel population-based metaheuristic of this type that adds a learning mechanism to the GRASP. The basic concept of FSS is to avoid focusing on specific high quality solutions but on parts or elements that such solutions have. This is done through fixing a set of elements that exist in such solutions and dedicating computational effort to finding near-optimal solutions for the underlying subproblem. In this work, the FSS is applied to the problem of interest. Computational experiments show that the FSS manages to significantly outperform the GRASP, ACO and WO on the standard benchmark instances when the quality of found solutions is considered without an increase in computational cost. This application of the FSS is significant as it shows that it can also be applied to scheduling type problems, in addition to covering and routing ones.

Introduction

One of the most important practical problems encountered in manufacturing is the optimal use of existing machines. In many cases it is related to scheduling different types of jobs on one or multiple machines. This basic setting can be related to a wide range of operational problems. This type of problem has been formalized using the concept of job scheduling problems, commonly by the parallel machine scheduling problem (PMSP). Due to the use of this model for diverse real-world systems, many variations have been defined to address their specific properties. Some settings are based on relations between different machines, precedence constraints between jobs [1], [2], resource constraints [3], [4], maintenance [5], job incompatibilities [6], limited information [7], multi-objective [8], etc.

In this paper the focus is on the setting with unrelated parallel machines with machine-dependent and job sequence-dependent setup times. In it, the time a job needs to be processed is dependent on the machine and on the preceding job. This problem setting has been used in combination with a variety of objective functions like minimization of the makespan [9], [10], [11], total completion time [12], weighted tardiness of the jobs [13], and many others. Mixed integer programs (MIP) have been proposed for finding optimal solutions for the PMSP [14], [15]. An alternative, more efficient, MIP formulation has been presented in [16], for which several variations are explored in [17]. Branch-and-bound algorithms have also been used to find optimal solutions for this type of problem [18], [19].

In this paper, the focus is on the PMSP with unrelated parallel machines and sequence-dependent times with the objective of minimizing the makespan for which the common notation is $R_M\left|S_{ijk}\right|C_{Max}$. This variation of the PMSP is an NP-hard problem, since even the simpler special case of identical parallel machine scheduling, with the objective of minimizing the makespan is NP-hard even in the case of only two machines [20], [21]. Due to this fact, a wide range of heuristic and metaheuristic methods have been designed to find near-optimal solutions. In [14], a basic heuristic approach has been proposed. Some of the earlier works on the application of metaheuristics to this problem are dedicated to tabu search [22] and simulated annealing [23], [24], [25]. Population-based metaheuristics like artificial bee colony [26] and genetic algorithms [15], [27] have been applied with great success to the PMSP. In case of the PMSP the most efficient approach has been the combined use of MIP and heuristics in a matheuristic setting [17]. Although this approach manages to outperform standard metaheuristics it necessitates an extensive and complex implementation.

The use of metaheuristics based on the greedy algorithm in combination with local searches has proven to be most promising for the PMSP in the sense of a good balance between the complexity of implementation and the quality of found solutions. In [9], an interesting method named Meta-RS has been presented. In practice this method is highly similar to the Greedy Randomized Adaptive Search Procedure (GRASP) [28]. An ant colony optimization (ACO) algorithm has been proposed in [29] and later improved in [10] which significantly outperforms the Meta-RS. The same authors applied the Worm Optimization (WO), another metaheuristic based on the constructive greedy algorithm, which improves the learning mechanism of ACO [11]. The WO produced further improvements in the quality of found solutions.

Recently, a new metaheuristic based on the repetitive use of greedy algorithms, the Fixed Set Search (FSS) has been introduced and successfully applied to the Traveling Salesman Problem (TSP) [30], the Minimum Weighted Vertex Cover Problem (MWVCP) [31] and the Power Dominating Set Problem [32]. The novelty in the design of the FSS, compared to ACO and similar methods, is that it assumes that a local search plus further simple modifications are used. In practice, it uses a simple approach to add a learning mechanism to the GRASP metaheuristic. The idea of the FSS is to exploit the fact that some elements frequently appear in high quality solutions. In this paper the FSS is applied to the PMSP. The importance of this application is in extending the number of problem families to which the FSS can successfully be applied. The FSS has previously been applied to routing problems [30] and to covering problems with limited [31] as well as unlimited distance of effect [32]. In this way it has been confirmed that the FSS can be applied to very different types of combinatorial optimization problems.

The performed computational experiments show that the FSS manages to outperform state-of-the-art methods in the quality of found solutions. The FSS achieves this at a similar computational cost. Current research on metaheuristics based on the greedy algorithm lacks the evaluation in view of optimal solutions and MIPs in general. Because of this we have conducted additional computational experiments to evaluate when the use of such metaheuristics is justified.

The paper is organized as follows. In the next section the problem formulation is given in a graph form suitable for use in the FSS. In Section 3 the greedy algorithm used as a base for the FSS is presented. In the following section, the used local search is presented and how the greedy algorithm is extended to the GRASP metaheuristic. In Section 5 details of the FSS and its application to the PMSP are given. The subsequent section is dedicated to the discussion of the performed computational experiments. Section 7 concludes.