A differential evolution algorithm for the customer order scheduling problem with sequence-dependent setup times

https://doi.org/10.1016/j.eswa.2021.116097Get rights and content

Highlights

  • We address the COSP with sequence-dependent setup times.

  • We propose a mixed-integer linear programming model for the problem under study.

  • We develop a hybrid DDE metaheuristic with an innovative restart operator.

  • We propose two local search procedures based on heuristic dominance rules.

Abstract

Although the customer order scheduling problem to minimize the total completion time has received a lot of attention from researchers, the literature has not considered so far the case where there are sequence-dependent setups between jobs belonging to different orders, a case that may occurs in real-life scenarios. For this NP-hard problem we develop a novel efficient approximate solution procedure. More specifically, we develop an innovative discrete differential evolution algorithm where differential mutations are performed directly in the permutation space and that uses a novel, parameter-free, restart procedure. The so-obtained solutions are improved by two proposed local search mechanisms that employ problem-specific, heuristic dominance relations. We carry out an extensive computational experience with randomly generated test instances to compare our proposal with existing algorithms from related problems. In these experiments, the proposed algorithm obtains the best results in terms of their average relative percentage deviation and success rate. Furthermore, an analysis of variance test, followed by a Tukey’s test, confirms the excellent performance of the algorithm proposed.

Introduction

A lot of attention has been paid to assembly scheduling problems in the last few years (Framinan et al., 2019). Amongst the different assembly scheduling environments, the customer order scheduling arises in several real-world applications, such as the paper industry, the pharmaceutical industry, as well as assembly operations (Leung et al., 2005). In the customer order scheduling environment, a given number of customer orders, each of them composed of different products or services (jobs in the following), has to be processed in a set of dedicated parallel machines. The order is completed once its corresponding jobs have been completed, thus it can be seen as an special case of assembly scheduling with zero assembly processing time.

In most real-life settings, the machines in the shop cannot process different types of jobs without tool changes, adjustments in their settings, etc. More specifically, our work is inspired by the case of a laboratory for quality control in the pharmaceutical industry. The routine of the laboratory is to analyse raw materials, products in process, and finished products. Each order is usually composed of several products that require a specific type of chemical analysis. Depending on the sequence of the products to be analysed, the corresponding equipment requires a different setup, hence the need of explicitly considering the setup times, which in general constitutes an important topic in the scheduling literature (Abreu et al., 2020, Allahverdi et al., 2008, Moccellin et al., 2018). However, studies tackling the customer order scheduling problems including setup times are scarce, being (Prata et al., 2021a) the only contribution addressing the problem, in that case for makespan minimization. Therefore, to the best of our knowledge, the order scheduling problem with sequence-dependent setup times with the objective of minimizing the total completion time has not been previously researched.

Therefore, our paper addresses the customer order scheduling with sequence-dependent setup times to minimize total completion time. The main contributions of this paper are threefold. First, we present a mixed-integer linear formulation to tackle this problem. Second, we develop a discrete differential evolution (DDE) metaheuristic to provide high-quality solutions within feasible computational times. The proposed algorithm applies a new parameter-free restart procedure, a chaotic mechanism to self-adjust the crossover rate, and it incorporates two novel elements i.e. (1) two local search procedures based on dominance relations and (2) a novel restart procedure based solely on the fitness of the solutions. The extensive computational experiments carried out show that the proposed DDE algorithm yields excellent results for the problem and that the novel elements embedded in the DDE substantially improves its performance.

The remainder of the paper is organized as follows: in Section 2, we discuss the problem background. In Section 3, we present the Mixed-Integer Linear Programming (MILP) model for the problem, while in Section 4 we describe the proposed algorithm. In Section 5, the results from the computational experiments are presented and discussed and, finally, in Section 6 we draw some conclusions and suggestions for future works.

Section snippets

Literature review

As discussed in Section 1, to the best of our knowledge, the problem under consideration has not been addressed in the literature, even if the customer order scheduling problem is becoming a major topic for production scheduling researchers (Wu, Yang et al., 2019). Therefore, we here discuss related problem in order to investigate whether their solution procedures can be adapted to our problem. More specifically, we review the customer order scheduling problem with total completion time

Problem statement and MILP formulation

The problem under consideration can be described as follows: There are n customer orders, each one composed of m jobs, that must be completed in a shop. Each job i (i=1,,m) in the order j (j=1,,n) has to be processed on a dedicated machine requiring pij times unit to be completed. In addition, job k requires a setup time sijk if processed after job j, which is sequence-dependent (since these are dedicated machines, we can also say that sijk is the setup time of order k in machine i if

A new discrete differential evolution algorithm

Differential Evolution (DE) is a well-known metaheuristic originally proposed by Storn and Price (1997) for continuous optimization problems. In the last few years, several extensions of the standard DE algorithm have been proposed for discrete optimization problems, including manufacturing scheduling problems (Bai et al., 2017, Liu et al., 2020, Pan et al., 2008, Wang et al., 2010, Zhao et al., 2020, Zhou et al., 2021). DE algorithms are usually classified as evolutionary algorithms since they

Computational experiments

In order to establish the efficiency of the proposed algorithm, we conduct a comprehensive computational experience. In Section 5.1 we discuss the design of the experimentation, including the generation of the testbed instances and the indicators employed to evaluate the algorithms. Section 5.2 is devoted to the calibration of the proposed DDE, analysing its parameters and the contribution to its efficiency of the different local search mechanisms proposed. Next, we present in Section 5.3 the

Final remarks and perspectives

In this paper, we have addressed the customer order scheduling problem with sequence-dependent setup times using the total completion time as objective function. To the best of our knowledge, this problem has not been dealt with in the previous literature.

Since the problem considered is NP-hard, we have developed a Discrete Differential Evolution (DDE) algorithm to solve it. The operators work directly with the permutation space so the operations with float numbers – characteristic in

CRediT authorship contribution statement

Bruno de Athayde Prata: Conceptualization, Investigation, Software, Formal analysis, Writing – original draft, Visualization, Funding acquisition. Carlos Diego Rodrigues: Formal analysis, Methodology, Writing. Jose Manuel Framinan: Conceptualization, Writing – review & editing, Supervision, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was financed in part by the Coordination for the Improvement of Higher Education Personnel (CAPES), Brazil and the National Council for Scientific and Technological Development (CNPq), Brazil, through grant 303594/2018-7. The support of the Spanish Ministry of Science and Innovation, Spain via the ASSORT grant with reference PID2019-108756RB-I00, and the Andalusian Regional Government, Spain via grants DEMAND and EFECTOS with references P18-FR-1149 and US-1264511 respectively is

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