A modified harmony search for the T-single machine scheduling problem with variable and flexible maintenance

doi:10.1016/j.eswa.2022.116897

Expert Systems with Applications

Volume 198, 15 July 2022, 116897

https://doi.org/10.1016/j.eswa.2022.116897 Get rights and content

Highlights

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We address a complex single machine scheduling problem.
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We consider flexible and variable maintenance activity to be scheduled.
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We develop a novel self-adaptive harmony search algorithm.
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We compare different heuristic and meta-heuristic algorithms.
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We perform a sensitivity analysis involving several influencing factors.

Abstract

A common assumption in the scheduling literature provides that machines are always available, while often they have to be stopped because of preventive maintenance operations. Inspired by a real-life manufacturing context, this paper deals with the single machine scheduling problem with a flexible/variable maintenance to be executed within a specific time window. Differently from other studies in the literature, a set of non-resumable jobs with sequence dependent setup-times and non-zero release dates have to be scheduled along with a maintenance activity whose duration is a non-decreasing function of its starting time, with the objective of minimizing the total tardiness (T-problem). A mathematical model of the problem is proposed and, due to the high computational complexity, several heuristics have been implemented and tested with the aim of using some of them to make metaheuristic algorithms more effective and computationally efficient. Hence, a modified harmony search, powered by several reinforcement strategies and even equipped with a self-adaptive calibration mechanism, has been developed and compared with a set of additional algorithms proposed by the literature. Numerical results reveal that the modified harmony search outperforms the competing algorithms and no performance loss arises by adopting the self-adaptive mechanism. Finally, a proper sensitivity analysis allows assessing the impact of several influencing factors on the total tardiness minimization.

Introduction

A popular assumption in the scheduling literature concerns that machines or manufacturing resources in general are always available, while often they have to be stopped because of breakdowns or maintenances. Scheduling jobs when preventive maintenance activities are needed on machines has been receiving considerable attention from both academics and practitioners. Machines might require several kinds of preventive maintenance operations, e.g., cleaning, tool replacement, refilling, recharging, and other planned actions, so that job processing in that period is not allowed (Luo et al., 2015). In general, two major information are needed to define the unavailability interval related to maintenance, namely its starting time and its duration. Depending on the way these data can be handled by the decision-maker, a twofold perspective on the scheduling problem with maintenance activities comes out. In fact, a maintenance operation can be classified as fixed or flexible, on the basis of the following considerations. In scheduling with fixed maintenance, both starting time and duration are deterministic and known in advance. This topic has been extensively investigated by literature since 1990s and the seminal work of Qi et al. (1999) demonstrates that the problem of scheduling processing of jobs and preventive maintenance simultaneously in a single machine system, with the objective to minimize total completion time, is NP-hard in the strong sense. Different exact approaches are described in the study by Kacem et al. (2008). The last survey paper can be consulted for further information (Ma et al., 2010). Conversely, in scheduling with flexible maintenance, the starting time can be decided by the scheduler, while respecting a certain time limit on the task execution. In some cases, if the maintenance duration depends on the working conditions, e.g., it is a positive non-decreasing function of its starting time, the maintenance task is denoted as variable. Actually, the problem of processing jobs and maintenance simultaneously can be further distinguished in two research streams, the former being related to scheduling with a single unavailability interval and the latter concerning periodic availability intervals.

Although most recent studies focus on periodic maintenance issues, motivated by a real-life context, this paper tackles the single machine scheduling problem involving a single maintenance operation. In detail, this research has been inspired by a semi-conductor manufacturing company wherein every week a severely skilled and expensive operator of the machine supplier company has to perform a variable maintenance task. Since that kind of machine is dedicated to produce high added-value products, an effective scheduling strategy is required. From a formal point of view, the problem at hand can be modeled as a single-machine scheduling problem with flexible-variable maintenance. Differently from the state-of-the-art contributions provided so far, the maintenance task has to be executed within a certain time window (according to the operator’s working shift). In addition, jobs dynamically arrive from the upstream manufacturing stage (non-zero release dates) and require sequence-dependent setup times before being processed. To the best of our knowledge, this paper represents the first study on such a challenging scheduling problem.

Heuristic and metaheuristic algorithms have been encountering considerable attention in the literature coping with complex single machine scheduling problems, e.g., those involving sequence-dependent setup times and release times with timeliness objectives. (Valente and Gonçalves, 2009, Sioud et al., 2012, Subramanian et al., 2014, Fernandez-Viagas and Costa, 2021). Hence, a major objective of this study is to assess the effectiveness of several heuristic and metaheuristic algorithms for the total tardiness minimization in a complex single machine environment including machine availability constraint. However, the novelties of the present research can be summarized in the following:

i)
Job release dates, sequence-dependent setup times (SDST) and a flexible-variable preventive maintenance to be executed within a specific time-window have to be simultaneously handled in a single-machine scheduling problem.
ii)
A mixed-integer linear programming model is presented for the problem under investigation.
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A series of heuristic algorithms based on static and constructive procedures are tested.
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A new modified harmony search (MHS) algorithm powered by heuristic solutions, a harmony memory improvement procedure and an adaptive local search, is designed and tested against several algorithms from the relevant literature.
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The proposed metaheuristic is even equipped with a self-adaptive mechanism (SAHS), which avoids any time-consuming parameter calibration. The validity of the self-adaptive feature is statistically demonstrated.
vi)
A sensitivity analysis on the problem parameters concerning job and maintenance descriptors is performed.

The remainder of the paper is organized as follows. To support the originality of the proposal, Section 2 deals with a comprehensive review of the literature. Section 3 deals with the problem statement. Section 4 is dedicated to the Mixed-Integer Linear Programming (MILP) model. Section 5 presents a number of heuristic algorithms for the problem at hand. Section 6 deals with the structure of the proposed modified harmony search algorithm, also explaining the self-adaptive mechanism. Section 7 introduces the procedure for the design of experiments. Section 8 deals with the computational experiments, while Section 9 infers on the impact of the problem parameters by means of a proper sensitivity analysis. Conclusions and future research proposals are discussed in Section 10.

Section snippets

Background of literature and contributions

In this section the review of literature is organized in two sub-sections, the former being related to the papers studying the single machine scheduling problem with a single flexible maintenance operation, the latter being focused on the papers that consider periodic maintenance activities. Table 1 aims at favoring the classification of most literary contributions mentioned in the following paragraphs and allows further detecting the added-value of the present research.

Problem statement

The single machine scheduling problem at hand can be formalized as follows. A set of N independent non-resumable jobs has to be processed on a single machine. Each job j=(1,…,n) is available at time r_j. Setup times are non-anticipatory and sequence dependent, i.e., s_ij is the setup time of job j being processed after job i. Anticipatory setups as well as preemption are not allowed. The machine must be subject to a mandatory maintenance activity, which has to be flexibly executed within a

MILP model

In this section, we propose a MILP model to solve the SMFVM problem under investigation. The model is described below. It is worth pointing out that the maintenance activity is handled as an adding job, so n + 1 operations have to be scheduled.

Notations and parameters

n	number of jobs. Note: the (n + 1)-th job is the maintenance operation
i = 0,…,n + 1	preceding job index (including dummy job 0)
j = 1,…,n + 1	job index
s_ij, i = 0,…,n; j = 1…,n	setup time of job j processed after job i
p_j, j = 1…,n

Heuristic solutions for the SMFVM problem

It is known that heuristic algorithms are capable of obtaining a good solution with limited computational effort. Indeed, heuristics are often employed as staring solutions of metaheuristic algorithms in order to enhance the quality of solutions and the time to convergence as well. To this end, preliminarily to the development of metaheuristic algorithms, a series of heuristics have been tested on the SMFVM problem at hand; namely, four well-known static dispatching rules such as Shortest

The proposed metaheuristic algorithm

So far, the Harmony Search (HS) algorithm has been successfully applied to various optimization problems, and the historical developments of diverse algorithm structures instead of its applications can be examined in a recent survey by Zhang and Geem (2019). This section aims at describing the main features of the proposed harmony search algorithm. Specifically, the first sub-section deals with the structure of the conventional harmony search (HS) while, subsequently, the proposed modified

Generating the numerical experiments

To the best of our knowledge, this is the first time the tardiness minimization problem (also called T-problem) concerning a dynamic single machine with variable/flexible maintenance is addressed. Although Ovacikt and Uzsoy (1994) proposed a well-known method to build a benchmark of problems for the dynamic single machine scheduling problem with sequence-dependent setup times, the introduction of a variable/flexible maintenance issue, also featured by the restrictions mentioned in Section 3,

Computational experiments

This section presents several numerical investigations. First, the different heuristics introduced in Section 5 are compared also to select those to be inserted in the initial population of metaheuristics. Then, a series of numerical experiments is used to design the structure of the proposed metaheuristic. Finally, an extended comparison analysis involving several metaheuristics from the relevant literature is performed with the aim of emphasizing the effectiveness of the proposed modified

Sensitivity analysis

Once the outperformance of SAHS and MHS has been demonstrated, numerical results obtained by SAHS have been employed to develop a sensitivity analysis on the total tardiness objective function that is the response variable. Hence, the numerical results deriving from the comparison analysis, which in turn are connected to the design of experiments in Section 7, have been used to arrange an analysis of variance (ANOVA) (Rushing et al. 2014) in which the problem parameters, i.e., n, s, e, h, g, l

Conclusions

In this paper, a complex single machine scheduling problem is addressed in which a flexible-variable maintenance has to be scheduled with the planning horizon. The complexity of such problem, which makes that different from the rest of contributions proposed by literature so far, is mainly due to non-simultaneous job arrival times, sequence-dependent setup times and maintenance activity to be executed within a fixed time window. The minimization of the total tardiness is the goal to be pursued.

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.

Acknowledgments

The authors wish to thank the referees for their comments on the earlier versions of the manuscript. This research has been funded by the Spanish Ministry of Science and Innovation, under the project “ASSORT” with reference PID2019-108756RB-I00, by the Junta de Andalucia under the project “‘EFECTOS”, with reference US-1264511, and by the University of Catania under the project - PIACERI 2020-2022 - GOSPEL project, with reference number 59722022261.