A green scheduling algorithm for the distributed flowshop problem

doi:10.1016/j.asoc.2021.107526

Applied Soft Computing

Volume 109, September 2021, 107526

https://doi.org/10.1016/j.asoc.2021.107526 Get rights and content

Abstract

In recent years, sustainable development and green manufacturing have attracted widespread attention to environmental problems becoming increasingly serious. Meanwhile, affected by the intensification of market competition and economic globalization, distributed manufacturing systems have become increasingly common. This paper addresses the energy-efficient scheduling of the distributed permutation flowshop (EEDPFSP) with the criteria of minimizing both total flow time and total energy consumption. Considering the distributed and multi-objective optimization complexity, an improved NSGAII algorithm (INSGAII) is proposed. First, we analyze the problem-specific characteristics and designed new operators based on the knowledge of the problem. Second, four constructive heuristic algorithms are proposed to produce high-quality initial solutions. Third, inspired by the artificial bee colony algorithm, we propose a new colony generation method using the operators designed. Fourth, a local intensification is designed for exploiting better non-dominated solutions. The influence of parameter settings is investigated by experiments to determine the optimal parameter configuration of the INSGAII. Finally, a large number of computational tests and comparisons have been carried out to verify the effectiveness of the proposed INSGAII in solving EEDPFSP.

Introduction

In recent years, green manufacturing to reduce environmental pollution and energy waste has attracted more and more attention from the world [1]. Therefore, it is extremely important to consider and measure energy conservation as an important content while studying traditional economic criteria such as makespan, total flow time, etc. Green scheduling has become a hot research topic [2]. Generally, the machine can work in multiple different states, and it is a better choice in the energy-saving state. The early research of green scheduling mainly studied when and which state the machine should be switched to [3]. However, the research on green scheduling is still very limited. Efficient methods, especially meta-heuristic methods for large-scale problems, are worth studying.

In this paper, an algorithm based on NSGAII, which is a very useful algorithm in the field of multi-objective evolution, is proposed to solve the energy-efficient distributed permutation flowshop scheduling problem (EEDPFSP) with the criteria to minimize the total flow time and total energy consumption. We summarize the main contributions as follows. (1) Four initialization algorithms based on problem-specific characteristics are proposed to produce good initial solutions. (2) Seven operators for solutions, including SpeedUp, RandSpeedUp, SpeedDown, RandSpeedDown, RightShift, Insert and Swap, are proposed and efficient search algorithms are designed based on these operators. (3) The new population generation method fused with problem characteristics and solution representation is designed. (4) A local intensification algorithm is designed to enhance the local search ability of the algorithm. Finally, a large number of numerical tests prove the effectiveness of the above design and the superiority of the algorithm proposed in this paper over KCA (Knowledge-Based Cooperative Algorithm) [4], CMA (Competitive Memetic Algorithm) [5], MOEA/D [6] and NSGAII [7] algorithms.

The remainder of this paper is organized as follows. Section 2 presents a literature review of related works. In Section 3, the EEDPFSP with total flowtime criterion is formulated. Section 4 presents the proposed and improved NSGAII for solving EEDPFSP in detail. We report the computational results and comparisons in Section 5 following the parameter setting. Finally, Section 6 provides the concluding remarks and suggests some future work.

Section snippets

Literature review

The widespread application of distributed manufacturing has attracted research on the distributed permutation flowshop scheduling problem (DPFSP) [8], which is a generalization of traditional permutation flowshop scheduling problem (PFSP). The DPFSP is more difficult and tricky than PFSP because it not only deals with the job sequence, but also determines the factory assignment [9]. Researchers have proposed various algorithms for solving DPFSP with different optimization objectives. The paper

Multi-objective optimization problem

In this subsection, the basic knowledge of multi-objective optimization problems (MOOP) is introduced. Without loss of generality, there are m objective functions and n-dimensional decision variables. A Multi-objective optimization problem can be described as: $min f (x) = {f_{1} (x), f_{2} (x), \dots, f_{m} (x)}, x \in Ω$ where $x = (x_{1}, x_{2}, \dots, x_{n})$ is a decision variable in the decision space $Ω$ . $f (x)$ is the objective function which is composed of conflicting sub-objective functions such as $f_{1}$ , $f_{2}$ , $f_{3}$ , etc.

Let a and b be two

Basic NSGAII

NSGAII is one of the most superior multi-objective evolutionary optimization algorithms [7]. Fig. 5 shows the pseudo-code of the basic NSGAII algorithm.

NSGAII was originally used to solve the optimization problem of continuous functions. Fewer pieces of literature were using NSGAII to solve the DPFSP. Therefore, besides the coding method and operators, it is necessary to design corresponding improvement strategies according to the characteristics of the DPFSP. These improvement strategies will

Experimental setting

In the experiments and comparisons, 600 instances are used to test the performance of our algorithm. Following [4], [5], we set $F = {2, 3, 4, 5}$ , $n = {20, 40, 60, 80, 100}$ , and $m = {4, 8, 16}$ . For each combination of ${F, n, m}$ , there are ten instances. So, the number of instances is 4 × 5 $\times$ 3 × 10 $=$ 600. The standard processing time $t_{i, j}$ is generated uniformly within range [5,50] and the processing speed v can be set as ${1, 1.3, 1.55, 1.75, 2.10}$ . The Energy Consumption is set as PP $_{j, v}$ $=$ 4 $\times$ v²kW and SP $_{j}$ $=$ 1 kW.

Conclusions

In this paper, an improved NSGAII is proposed to solve the EEDPFSP with the minimization of both the total flowtime and the energy consumption criteria. A large number of numerical experiments are used to test the performance of the algorithm. The experimental results show that the new algorithm has better performance than existing algorithms in terms of solution quality and diversity. In addition, the proposed algorithm can obtain high-quality feasible solutions under different stopping

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

Thanks to anonymous reviewers for their comments to improve the quality of this article. This research is partially supported by the National Key Research and Development Program (No.2020YFB1708200), the National Science Foundation of China 61973203, and Shanghai Key Laboratory of Power station Automation Technology .