An imperialist competitive algorithm with feedback for energy-efficient flexible job shop scheduling with transportation and sequence-dependent setup times

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Abstract

Flexible job shop scheduling problems have been extensively investigated in the past decade; however, transportation, sequence-dependent setup times (SDST) and energy efficiency are seldom incorporated together in flexible job shop. In this paper, energy-efficient flexible job shop scheduling problem (EFJSP) with transportation and SDST is considered and an imperialist competitive algorithm with feedback (FICA) is developed to minimize makespan, total tardiness and total energy consumption simultaneously. Assimilation and adaptive revolution are newly implemented by feedback and a new imperialist competition is presented by solution transferring among empires and the reinforced search. Extensive experiments are conducted and the computational results demonstrate that FICA provides promising results for EFJSP with transportation and SDST.

Introduction

Production scheduling is a decision-making process that plays an important role in manufacturing and production systems. Flexible job shop scheduling problem (FJSP) is a typical scheduling problem, which can be decomposed into routing sub-problem and scheduling sub-problem. FJSP has been extensively applied in many industries such as automobile assembly, textile, chemical material processing and semiconductor manufacturing (Li and Lin, 2016, Lei et al., 2018, Lee et al., 2017). It has attracted much attention since the pioneering works of Bruker and Schlie (1990), many methods including meta-heuristics (Jiang et al., 2014, Yin et al., 2017, Piroozfard et al., 2018b, Mokhtari and Hasani, 2017, Wu and Sun, 2018, Gong et al., 2019, Wang et al., 2018a, He et al., 2015, Lei et al., 2017, Liu et al., 2017a, Lei et al., 2019, Li et al., 2019b, Lei and Guo, 2015a, Nouiri et al., 2018, Jiang and Deng, 2018, Dai et al., 2019, Liu et al., 2019, Karimi et al., 2017, Nouri et al., 2016, Rossi and Dini, 2007, Defersha and Chen, 2010, Bagheri and Zandieh, 2011, Özgüven et al., 2012, Rossi, 2014, Mousakhani, 2013, Li et al., 2019c) and deep learning (Luo, 2020) are adopted and various problems including energy-efficient FJSP, FJSP with transportation and FJSP with sequence-dependent setup times (SDST) have been extensively investigated (Jiang et al., 2014, Yin et al., 2017, Piroozfard et al., 2018b, Mokhtari and Hasani, 2017, Wu and Sun, 2018, Gong et al., 2019, Wang et al., 2018a, He et al., 2015, Lei et al., 2017, Liu et al., 2017a, Lei et al., 2019, Li et al., 2019b, Lei and Guo, 2015a, Nouiri et al., 2018, Jiang and Deng, 2018, Dai et al., 2019, Liu et al., 2019, Karimi et al., 2017, Nouri et al., 2016, Rossi and Dini, 2007, Defersha and Chen, 2010, Bagheri and Zandieh, 2011, Özgüven et al., 2012, Rossi, 2014, Mousakhani, 2013, Li et al., 2019c) in the past decade.

As a scheduling problem with an objective of improving energy efficiency or energy related constraints, energy-efficient scheduling has gotten wide attention in recent years and a number of results have been obtained in various environments such as parallel machines (Wu and Che, 2019, Wang et al., 2018b), flow shop (Lei et al., 2018, Mansouri et al., 2016, Liu et al., 2017b, Lu et al., 2017, Li et al., 2018, Wang and Wang, 2020) and job shop (Jiang et al., 2014, Yin et al., 2017, Piroozfard et al., 2018b, Mokhtari and Hasani, 2017, Wu and Sun, 2018, Gong et al., 2019, Wang et al., 2018a, He et al., 2015, Lei et al., 2017, Liu et al., 2017a, Lei et al., 2019, Li et al., 2019b, Lei and Guo, 2015a, Nouiri et al., 2018, Jiang and Deng, 2018, Dai et al., 2019, Liu et al., 2019, Karimi et al., 2017, Nouri et al., 2016, Rossi and Dini, 2007, Defersha and Chen, 2010, Bagheri and Zandieh, 2011, Özgüven et al., 2012, Rossi, 2014, Mousakhani, 2013, Li et al., 2019c, Liu et al., 2014, May et al., 2015, Lei and Guo, 2015b, Zhang and Chiong, 2016, Salido et al., 2016, Gu et al., 2020).

EFJSP is an important energy-efficient scheduling problem and has been studied extensively. Jiang et al. (2014) proposed a blood-variation-based non-dominated sorting genetic algorithm-II (NSGA-II). Yin et al. (2017) provided a multi-objective genetic algorithm (MOGA) for FJSP with productivity, energy efficiency and noise reduction. Piroozfard et al. (2018b) presented a MOGA to minimize total carbon footprint and total late work criterion. A hybrid algorithm based on evolutionary algorithm and simulated annealing is studied by Mokhtari and Hasani (2017) to minimize total completion time and total energy cost and maximize the total availability of the system. Wu and Sun (2018) applied a NSGA-II for FJSP with energy-saving measures. Gong et al. (2019) presented non-dominated genetic algorithm (GA)-III for many-objective FJSP with five objectives under dynamic electricity pricing. Wang et al. (2018a) developed a two-phase method based on GA and particle swarm optimization (PSO). Besides the above works, He et al. (2015) introduced a nested partitions algorithm. Lei et al. (2017) designed a new shuffled frog-leaping algorithm for the problem with total energy consumption and workload balance. Liu et al. (2017a) provided a hybrid fruit fly optimization algorithm to minimize carbon footprint and makespan. Lei et al. (2019) presented a two-phase meta-heuristic based on imperialist competitive algorithm (ICA) and variable neighborhood search (VNS) for FJSP with total energy consumption threshold and Li et al. (2019b) developed an ICA with diversified operators. Other meta-heuristics are also applied to solve EFJSP, which include VNS (Lei and Guo, 2015a), PSO (Nouiri et al., 2018) and cat swarm optimization algorithm (Jiang and Deng, 2018).

Regarding FJSP with transportation, Dai et al. (2019) proposed an enhanced genetic algorithm (EGA) for EFJSP with transportation constraints. Liu et al. (2019) considered an integrated green scheduling problem of flexible job shop and crane transportation and developed an integrated algorithm based on GA, glowworm swarm optimization and green transport heuristic. Karimi et al. (2017) incorporated the transportation times between machines into FJSP and presented a hybrid ICA. Nouri et al. (2016) applied a neighborhood-based GA for FJSP with many robots. Rossi and Dini (2007) presented ant colony optimization algorithm for FJSP with SDST and transportation times.

FJSP with SDST also has been frequently investigated. Defersha and Chen (2010) presented a mathematic model and a parallel GA. Bagheri and Zandieh (2011) proposed a VNS based on integrated approach to minimize makespan and mean tardiness. Özgüven et al. (2012) constructed some mixed integer goal programming models. Rossi (2014) presented an ant colony optimization with reinforced pheromone relationships for FJSP with SDST and transportation. Mousakhani (2013) developed a mathematical model and an effective meta-heuristic. Li et al. (2019c) proposed an elitist non-dominated sorting hybrid algorithm to minimize makespan and total setup cost.

As stated above, energy-related objectives and constraints, transportation and SDST are often adopted into FJSP independently and only several works are related to FJSP with two of these elements; moreover, FJSP with three elements is hardly dealt with. Energy, transportation and SDST are the common conditions and often exist simultaneously in the real-life flexible job shop. The consideration on FJSP with these three conditions can result in good schedule with high value and should be important topic of FJSP. On the other hand, ICA has been applied to solve EFJSP and FJSP with transportation; however, It is hardly used to optimize EFJSP with SDST and transportation.

ICA has good neighborhood search ability, effective global search property, good convergence rate and flexible structure (Atashpaz-Gagari and Lucas, 2007, Hosseini and Khaled, 2014). In recent years, ICA has been extensively applied to solve various production scheduling problems (Lei et al., 2019, Li et al., 2019b, Karimi et al., 2017, Shokrollahpour et al., 2011, Goldansaz et al., 2013, Seidgar et al., 2014, Zandieh et al., 2017, Pan et al., 2018, Li et al., 2019a) and Lei and Cai (2020) provided a survey on the applications of ICA to production scheduling. Table 1 shows some related works on ICA for scheduling. As shown in Table 1, Zandieh et al. (2017) handled FJSP with condition-based maintenance by a hybrid ICA with SA based on assimilation and revolution of standard ICA. Many-objective FJSP is solved by using an ICA with the diversified assimilation and revolution (Li et al., 2019b). ICA is also used to deal with scheduling problems with SDST, assembly and maintenance in flow shop etc. The features of ICA and its extensive applications to scheduling motivate us to solve EFJSP with SDST and transportation by ICA.

In the previous works on ICA for scheduling, some optimization mechanisms are seldom introduced. For example, feedback often exists in control systems and seldom added into ICA. Feedback means that the optimization behaviors of ICA are affected or decided by the past optimization results. The inclusion of feedback will adjust dynamically the search process of ICA and search efficiency may be improved, so feedback is a possible path to construct ICA with promising performance.

In this study, EFJSP with transportation and SDST is investigated and a new imperialist competitive algorithm with feedback (FICA) is proposed to minimize makespan, total tardiness and total energy consumption. To obtain high quality solutions, assimilation and revolution are newly performed by feedback and a new imperialist competition is developed based on solution transferring among empires and the reinforced search. We compare FICA with other algorithms from the literature to test its performance by extensive experiments. The computational results show that FICA has promising performances in solving the considered EFJSP.

The outline of the remained part of the paper is organized as follows. Section 2 describes the problem followed by introduction to ICA in Section 3. FICA for EFJSP with transportation and SDST is shown in Section 4. The experimental results of FICA are shown in Section 5. Conclusions and some topics of the future research are provided in the final section.

Section snippets

Problem description

Table 2 shows the notations used in this section.

EFJSP with transportation and SDST is described as follows. There are a set of jobs J=J1,J2,,Jn and a set of machines M=M1,M2,,Mm. Job Ji consists of hi operations. Operation oij can be processed on any compatible machines in a set Sij, SijM. There is a set V of speeds (vel1,vel2,,veld) for each machine. pijkl=ηijkvell. Setup times sjik and s0ik are adopted. For jobs Ji,Jj,Jl processed on the same machine Mk, the relationship sjiksjlk+slik

Introduction to ICA

In ICA, a country represents a solution and its quality is measured by cost, which is defined as a function related to objectives. The better the solution is, the smaller the cost is. The detailed steps of ICA are shown in Algorithm 1. For imperialist k, its normalized cost is c̄k, its power is powk and NCk indicates the number of its colonies, Ncol is total number of colonies.

c̄k=maxvHcvck powk=|c̄kvHc̄v| NCk=round(powk×Ncol) Ncol=NNimwhere round(x) is a function that gives the

FICA for EFJSP with transportation and SDST

In the previous ICAs (Lei et al., 2019, Li et al., 2019b, Karimi et al., 2017), feedback is seldom applied to decide or change optimization behaviors or parameters of ICA by the past optimization results. The inclusion of feedback can adjust dynamically the search process and improve search efficiency. In this paper, feedback is added into ICA and FICA is presented for EFJSP with SDST and transportation.

Computational experiments

Extensive experiments are conducted on a set of problems to test the performance of FICA for the considered EFJSP. All experiments are implemented in Microsoft Visual C++ 2015 and run on 4.0G RAM 2.00 GHz CPU PC.

Conclusions

In this study, we addressed EFJSP with transportation and SDST and proposed a FICA to minimize three objectives simultaneously. Assimilation and adaptive revolution are performed by feedback and a new imperialist competition is developed by solution transferring among empires and a reinforced search on some worst solutions of population. The effectiveness of FICA is shown by extensive experiments and the computational results demonstrate that FICA can provide better results than its comparative

CRediT authorship contribution statement

Ming Li: Conceptualization, Methodology, Software, Data curation, Software, Validation. Deming Lei: Writing - original draft, Supervision, Writing - review & editing.

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 work is supported by the National Natural Science Foundation of China (61573264) and the Fundamental Research Funds for the Central Universities, China (2019-YB-030).

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