A two-phase evolutionary algorithm for multi-objective distributed assembly permutation flowshop scheduling problem

doi:10.1016/j.swevo.2022.101128

Swarm and Evolutionary Computation

Volume 74, October 2022, 101128

https://doi.org/10.1016/j.swevo.2022.101128 Get rights and content

Highlights

•
We study a multi-objective distributed assembly flowshop scheduling problem.
•
We present a two-phase evolutionary algorithm divided by time.
•
Two novel heuristics specific to different objectives are proposed.
•
Six crossover operators and two mutation operators are designed specifically.
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The effectiveness of our algorithm is demonstrated by 810 benchmark instances.

Abstract

In recent years, multi-objective optimization problems have received extensive attention. This paper proposes a two-phase evolutionary algorithm (TEA) to solve the multi-objective distributed assembly permutation flowshop scheduling problem with total flowtime and total tardiness criteria. The first phase of the TEA uses a novel two-population structure to simultaneously optimize the two criteria. According to the characteristics of the structure, two construction heuristics are designed to obtain solutions with both high quality and diversity. Four crossover and two mutation operators are presented. The interaction between the two populations increases the diversity within each population. The second phase integrates the previous two populations and uses the method of normalized objective function to enhance efficiency. Two new crossover operators are proposed to improve the performance of the algorithm. The TEA first finds several approximate front solutions to provide ideal feasible areas, and then continuously expands the solutions to the Pareto front. It embodies convergence, diversity and uniformity in different phases. Finally, through 810 benchmark instances, the TEA is compared with five classical and popular algorithms. The effectiveness of the two mechanisms in the TEA is analyzed. Experimental results demonstrate the superior performance of the TEA in terms of distribution and coverage of solutions.

Introduction

The distributed assembly permutation flowshop scheduling problem (DAPFSP) [1] is a subject that has been studied for many years. It is evolved from the classic distributed permutation flowshop scheduling problem (DPFSP), and is equivalent to adding the constraint of an assembly line on this basis [2, 3]. The problem starts with jobs being processed on the same serial machines in different factories, which are then turned into multiple products to be assembled on an assembly line.

In the past, researchers only considered a single criterion for the DAPFSP [5], [6], [7]. In reality, however, managers not only need to get jobs done as quickly as possible, but also consider indicators such as customer satisfaction. This leads to a multi-objective DAPFSP (MO-DAPFSP), which is more relevant and widely applied to manufacturing, service and information processing systems [4, 23]. The study of this problem is of practical value, however, there is very little literature on it. In this paper, we study an MO-DAPFSP based on minimizing total flowtime ( $T F$ ) and total tardiness ( $T T$ ), which belongs to a NP-hard problem [15]. A good scheduling of these two objectives can achieve a better optimal balance in reducing production costs, enhancing production efficiency and improving customer satisfaction and contract duration.

In this problem, the optimization of the $T F$ depends only on the completion time of internal products, while the calculation of the $T T$ involves the external delivery time promised to customers. Therefore, it is not possible to achieve the optimal values of both objectives at the same time, but only to trade-off them so as to achieve the best values possible. Recently, a two-phase algorithm has been shown to exhibit good performance in solving flowshop scheduling problems [13, 14] and other types of multi-objective optimization problems (MOP) [8, 9]. By combining the advantages of both algorithms, it is able to optimize the trade-off in a more hierarchical way using different operators. Thus, to better address this balancing motivation, we propose a two-phase evolutionary algorithm (TEA) to improve the diversity and convergence of the algorithm. The main contributions are as follows:

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A new heuristic algorithm for the two-population structure with targeted initialization for both objectives is proposed. The heuristic efficiently combines the RA and NEH, corresponding to jobs and products respectively, and is improved by a random operator.
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A two-population structure-based non-dominated sorting genetic algorithm-II (TPNSGA-II) is proposed as the first phase of the TEA. According to the different properties of the two population objectives and the different effects on product and job genetics, four kinds of crossover, two mutation operators and an integration interaction method are proposed, which reduce the repeatability within the population and increase the diversity between populations. A rough Pareto front can be obtained, which provides the ideal feasible area for the next phase.
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In the second phase, the decomposition-based multi-objective evolutionary algorithm (MOEA/D) is used to integrate the two populations, which eliminates the tedious individual improvement phase and designs the crossover and mutation operators. This phase expands the solution obtained in the first phase close to the Pareto front.
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A two-phase node segmentation method, including time and the degree of optimization of the non-dominant solutions, is proposed.
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Based on the benchmark instances, the superiority of the proposed TEA over the classical and advanced multi-objective algorithms is proved.

This paper is organized as follows. Section 2 reviews the literature on the DAPFSP and MOP. The problem is described in Section 3. Section 4 illustrates the TEA in detail. Besides, the parameter calibration and experimental results are analyzed in Section 5, and the performance of the proposed algorithm is evaluated. Finally, Section 6 summarizes and prospects the research.

Section snippets

Distributed assembly scheduling problem

As the economy grows, the assembly scheduling problem with single-factory has gradually evolved into the distributed assembly scheduling problem with multi-factories. This problem includes two stages of processing and assembly. From the perspective of configuration, it can be divided into four types. The first type is the distributed two-stage assembly scheduling problem (DTSAFSP), in which jobs are assigned to different factories for processing, all of which have the same parallel machines.

Basic multi-objective optimization problems

The MOP is to determine a decision variable vector in the feasible region to minimize an objective function vector [30]. In general, it can take the following form: $M i n i m i z e {f_{1} (x), f_{2} (x), \dots, f_{η} (x)}, x \in Θ$ where $x$ is the vector of the decision variables, which is distributed in the feasible solution space $Θ$ . And $f_{1}, f_{2}, \dots, f_{η}$ represents $η$ objective functions.

In this paper, the following concepts will be used to solve the MOP

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Dominant/Non-dominated solution: There are two feasible solutions $a$ and $b$ . Only if $\forall$

Two-phase evolutionary algorithm

In this section, we propose a TEA to better coordinate population diversity and convergence, as well as deal with the balance between production efficiency and customer satisfaction. The overall framework is shown in Algorithm 1, where $A S$ preserves the non-dominated solutions in the TEA. In the first phase, the objective value based TPNSGA-II is used to find a rough Pareto front as far as possible, and identify several approximate front solutions to provide ideal solutions for the next phase.

Computational evaluation

In this section, we calibrate the parameters and evaluate the effectiveness of the proposed TEA in solving the MO-DAPFSP with $T F$ and $T T$ . All experiments are implemented by using C++ language in Visual Studio 2019 and run on Intel (R) Xeon (R) CPU E5-2640 v4 @ 2.40GHz with 3.99 GB RAM in the Windows Server 2012 Operation System. $d_{r}$ is a data that needs to be adopted because the problem is related to tardiness, and its calculation formula is listed below: $d_{r} = r a n d [C_{s} (1 - Q - \frac{R}{2}), C_{s} (1 - Q + \frac{R}{2})]$ where $r a n d$

Conclusions

The paper is the first to solve the MO-DAPFSP with $T F$ and $T T$ criteria by using the TEA. The first phase of the TEA uses a two-population structure based on the NSGA-II framework. Each population uses different operators for different objectives in a targeted way. For these two populations, we design the corresponding initialization methods, four crossover operators and two mutation operators. Besides, the proposed two-population interaction method reduces the internal repeatability of the two

Author statement

Ying-Ying Huang: Conceptualization, Methodology, Software, Data curation, Writing- Original draft preparation. Quan-Ke Pan: Writing- Reviewing and Editing, Validation, Project administration, Funding acquisition. Liang Gao: Visualization, Investigation. Zhong-Hua Miao: Resources. Chen Peng: Supervision.

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

This research is partially supported by the National Key Research and Development Program 2020YFB1708200, the National Science Foundation of China 61973203, Program of Shanghai Academic/Technology Research Leader 21XD1401000, and Shanghai Key Laboratory of Power Station Automation Technology.

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View full text

A two-phase evolutionary algorithm for multi-objective distributed assembly permutation flowshop scheduling problem

Highlights

Abstract

Introduction

Section snippets

Distributed assembly scheduling problem

Basic multi-objective optimization problems

Two-phase evolutionary algorithm

Computational evaluation

Conclusions

Author statement

Declaration of Competing Interest

Acknowledgments

Eng. Appl. Artif. Intel.

Swarm. Evol. Comput.

Simul. Model. Pract. Th.

Comput. Oper. Res.

J. Manuf. Syst.

Appl. Soft. Comput.

Swarm. Evol. Comput.

Expert. Syst. Appl.

Comput. Ind. Eng.

Math.Comput. Simulat.

Eur. J. Oper. Res.

Swarm. Evol. Comput.

Eur. J. Oper. Res.

Swarm Evol. Comput.

Appl.Soft. Comput.

Expert. Syst. Appl.

Appl. Soft. Comput.

Swarm. Evol. Comput.

Appl. Soft. Comput.

Comput. Ind. Eng.

Expert. Syst. Appl.

The distributed assembly permutation flowshop scheduling problem

Int. J. Prod. Res.

A two-stage multiobjective evolutionary algorithm for multiobjective multi-depot vehicle routing problem with time windows

IEEE. T. Cybernetics.

A hybrid multiobjective memetic algorithm for multiobjective periodic vehicle routing problem with time windows

IEEE. T. Syst. Man. Cy-S.