A simulation–optimization model for solving flexible flow shop scheduling problems with rework and transportation

Highlights

• We propose a multi-objective harmony search algorithm with Gaussian mutation to solve flexible flow shop scheduling problems.

• We consider sequence-based setup time, transportation time, and probable rework in the proposed algorithm.

• We use response surface methodology to minimize both maximum completion time and mean tardiness concurrently.

• We evaluate the efficacy of the proposed algorithm using computational experiments based on six measures.

• The experimental results demonstrate the superiority of the proposed algorithm over the existing multi-objective algorithms.

Abstract

We propose an enhanced multi-objective harmony search (EMOHS) algorithm and a Gaussian mutation to solve the flexible flow shop scheduling problems with sequence-based setup time, transportation time, and probable rework. A constructive heuristic is used to generate the initial solution, and clustering is applied to improve the solution. The proposed algorithm uses response surface methodology to minimize both maximum completion time and mean tardiness, concurrently. We evaluate the efficacy of the proposed algorithm using computational experiments based on five measures of diversity metric, simultaneous rate of achievement for two objectives, mean ideal distance, quality metric, and coverage. The experimental results demonstrate the effectiveness of the proposed EMOHS compared with the existing algorithms for solving multi-objective problems.

Introduction

Developing efficient scheduling systems is an essential factor for the productivity of the manufacturing systems in today’s competitive marketplace. The flexible flow line environment is a relatively common scheduling system appearing in the flow industries such as steel, petroleum, chemical processing, and packaging ([1]). However, the current scheduling models do not consider the transportation time, probable rework, and sequence-dependent setup times in multi-objective scheduling problems. The classical flow shop problems involve only one machine at each stage. More than one machine at least at each stage turns the problem into a flexible flow-shop problem. Hence, the flexible flow shops are the generalized forms of simple flow shops. Chang and Liao [8] called the scheduling jobs in the flexible flow shops NP-complete problems. The focus of most research in the flow shop literature has been on the single-objective optimization. Kurz and Askin [34] proposed an integer-programming model for a hybrid flow shop (HFS) with sequence-dependent setup times (SDST), which is one of the studies that use a meta-heuristic approach. They developed a random key genetic algorithm to solve the problem against other heuristic rules due to the difficulty in developing a direct solution to the integer programming model. The authors introduced a genetic algorithm to resolve the problem. Lin et al. [37] provided a simulated annealing-based meta-heuristic for reducing the make-span in a flow-shop manufacturing cell with sequence-dependent family setup times. The algorithm proposed was compared in terms of effectiveness and efficiency with available heuristics on a benchmark problem dataset previously used in other studies. An efficient hybrid metaheuristic for scheduling jobs was presented by Behnamian et al. [4] in a hybrid flow-shop with sequence-dependent setup times. Their goal was to develop a schedule capable of minimizing the sum of the earliness and tardiness of jobs. Tardiness has a direct impact on customer satisfaction and profitability. In addition, there is an inverse relationship between makespan and throughput since minimizing makespan results in maximizing throughput. Naderi et al. [48] proposed hybrid flow shops where the setup times are sequence-dependent to minimize makespan and maximum tardiness. They introduced a novel simulated annealing with a migration mechanism and compared their algorithm with several high-performing metaheuristics. Readers can refer to Wang [63] for a comprehensive review of the scheduling problems with a flexible flow shop.

Recent studies have focused on the problem inherent in the single-machine scheduling with sequence-dependent setup times aimed at minimizing the total weighted tardiness of jobs. Chen [9] presented an iterated population-based variable neighborhood descent search algorithm designed to address a single machine with sequence-dependent setup times. This algorithm provided initial population solutions followed by the neighborhood descent search algorithm generated for each iteration. Subsequently, the solutions were enhanced by a greedy local search. Cheng et al. [12] developed a mixed-integer linear programming model for a no-wait flow shop scheduling problem with SDST and proposed a pairwise iterated greedy algorithm to solve the large size problems. Da Silva et al. [14] studied an online single machine scheduling problem with SDST. They proposed a mixed-integer linear programming formulation for the problem along with some heuristic algorithms. [31] proposed three metaheuristic algorithms (hybrid squirrel search algorithm, opposition based whale optimization algorithm, and discrete gray wolf optimization algorithm) to solve bi-criteria SDST hybrid flow shop scheduling problems. Appendix A presents a comprehensive list of all acronyms and abbreviations used in this paper.

Naderi et al. [47] reviewed the drawbacks of available models in the literature. Their work led to the development of four mixed-integer linear programming models, which were compared with each other in terms of size and computational complexity. Additionally, they proposed a novel hybrid particle swarm optimization algorithm. The research by Ribas et al. [56] and Ruiz and Vázquez-Rodríguez [57] can be used for further information and a better understanding of the hybrid flow shop scheduling problem. Hwang and Lin [27] proposed a two-stage flexible flow shop with four standard performance measures, namely, total completion time, maximum lateness, total tardiness, and the number of tardy jobs. They specified an optimum interleaving processing sequence of all the jobs associated with their starting times on the stage-2 bottleneck machine. Sioud and Gagné [59] proposed an enhanced migrating birds optimization algorithm to solve the flow shop problem with sequence-dependent setup times. They aimed to minimize the makespan using a novel approach based on the setup times and the new enhanced migrating birds optimization algorithm, namely a new heuristic based on setup times for the permutation flow shop with SDST, and an enhanced migrating birds optimization algorithm, respectively. Zhang et al. [69] proposed a multi-objective optimization model to solve the hybrid flow shop scheduling problems by minimizing total energy consumption and makespan. They proposed a decomposition-based metaheuristic algorithm for solving large-scale problems. Fernandez-Viagas et al. [21] examined the efficiency of different solution representations for solving hybrid flow shop scheduling problems. They found a trade-off between the ability to conduct an efficient search in this reduced solution space and the solution space reduction. Bargaoui and Driss [3] proposed a multi-stage model using the tabu search for solving the permutation flow shop scheduling problems. They used two multi-agent classes of supervisor and scheduler agents. The supervisor agent generates the initial solution with the tabu search core, and the scheduler agents are responsible for the evaluation of all neighborhood solutions and satisfaction of the constraints. Computational experiments on different benchmarks showed that their proposed model reaches a high-quality solution. The model proposed in this study is devised to minimize the makespan or the total duration of the schedule.

There have also been numerous studies in the scheduling literature focusing on multi-objective and hybrid flow shop scheduling problems. However, most of them have merely followed a single dimension pattern, i.e., either single objective hybrid flow shops, or a multi-objective optimization in scheduling such as single-machine or classical flow shops. Another study by Tran and Ng [62] resulted in a hybrid water flow algorithm for multi-objective flexible flow shop scheduling problems. It was designed to minimize the completion time of jobs and the total tardiness time of the jobs. According to Meng et al. [45], during the optimization, the lot-splitting is suitable in advance and fixed in the flexible flow shop problem. They used the lot-streaming flow shop scheduling problems and an improved migrating birds optimization for minimizing the maximum completion time or the make-span. They also presented a mathematical model to solve the n-job m-machine lot-streaming flow shop scheduling problem with separable sequence-independent setup times and equal-size sub-lots. A hybrid metaheuristic algorithm was proposed by Behnamian and Ghomi [5] designed to address a hybrid flow shop scheduling with the machine and resource-dependent processing times. The result of Yagmahan and Yenisey [65] was a multi-objective ant colony system algorithm, featured with combining the ant colony optimization approach and a local search strategy to resolve this scheduling problem. This was the first use of an ant colony optimization metaheuristic regarding multi-objective m-machine flow shop scheduling problems aimed at both objectives of make-span and total flow time. A hybrid multi-objective backtracking search algorithm to solve an energy-efficient permutation flow shop scheduling problem with controllable transportation time was developed by Lu et al. [38]. Behnamian et al. [6] described a multi-phase algorithm covering the Pareto optimal front hybrid metaheuristic to minimize the make-span and the sum of the earliness and tardiness of jobs. Yagmahan and Yenisey [64] investigated the flow shop scheduling problem with a focus on minimizing the make-span, total flow time, and total machine idle time. The ant colony optimization algorithm was employed to solve this problem. A limited monkey algorithm proposed by Marichelvam et al. [42] was used to solve the flow shop scheduling problem. It was demonstrated that the monkey algorithm is useful due to its simple structure and strong robustness. Recently, Nouri et al. [52] studied flexible job shop scheduling problems with transportation times and a single robot. They employed a new metaheuristic hybridization approach based on clustered holonic multi-agent models, and they used a local search with a set of cluster agents to improve the quality of the final population. Nouri et al. [51] presented a neighborhood-based genetic algorithm for a flexible job-shop scheduling problem with transportation time and multiple robots. They introduced a new metaheuristic hybridization approach based on a clustered holonic multiagent model. Nouri et al. [49] reviewed the job shop scheduling problems with transportation resources according to seven criteria, including transportation resource number, transportation resource type, job complexity, routing flexibility, recirculation constraint, optimization criteria, and the implemented approaches. Nouri et al. [50] proposed another interesting model for the simultaneous scheduling of jobs and robots in job shop scheduling.

Li et al. [36] presented a multi-stage HFS with single and batch processing machines to minimize the maximum completion time and the total weighted tardiness by developing a heuristic-search genetic algorithm. More recently, Dios et al. [17] compared the hybrid flow shop scheduling with missing operations (HFSMO) with the traditional HFS problems. The proposed algorithm turned out to be more effective in solving the HFSMO problem than the existing heuristics.

A Lorenz dominance based on the non-dominated sorting genetic algorithm (NSGAII) was presented by Dugardin et al. [18] to address a reentrant hybrid flow shop scheduling problem by maximizing the utilization rate of the bottleneck and minimizing the maximum completion time. According to the results, the Lorenz NSGAII can provide better solutions than the NSGAII and the strength Pareto evolutionary algorithm 2. Ebrahimi et al. [19] suggested a scheduling problem in an HFS environment, using a sequence-dependent family setup time to minimize the make-span and total tardiness. They used a normal data distribution and assumed uncertain due date in their model. They presented two metaheuristic algorithms (NSGAII and multi-objective genetic algorithm) and compared the quantitative and qualitative results of these two algorithms with a multi-phase genetic algorithm. Gholami-Zanjani et al. [24] considered a flow shop scheduling problem with sequence-dependent set-up times using robust and fuzzy optimization in an uncertain environment aimed at minimizing the weighted mean completion time. Considering a branch-and-bound algorithm for two-machine flow-shop scheduling problems with batch delivery costs, Mazdeh and Rostami [43] managed to minimize the maximum tardiness plus the sum of delivery costs. A mixed-integer linear programming model and a branch and bound algorithm were also developed to solve the problem.

Marichelvam and Geetha [41] used a harmony search algorithm with a real-industrial scheduling problem and proposed and analyzed a multistage HFS scheduling problem. Jabbarizadeh et al. [28] developed hybrid flexible flow shops with sequence-dependent setup times and machine availability constraints caused by preventive maintenance. The optimization criterion included the minimization of the make-span. Sokolov et al. [61] focused on a flexible flow shop problem for continuous production. Using uniform alternative machines, they analyzed the theories of the optimal scheduling approach. The proposed method can be used in the mathematics of functional spaces, including stability, robustness, controllability, adaptability. A multi-phase genetic algorithm developed by Karimi et al. [29] addresses the bi-objective group scheduling in a hybrid flexible flow shop. By incorporating flexible and diverse maintenance activities to minimize the total tardiness and maintenance costs, a permutation flow shop scheduling problem was proposed by Yu and Seif [66] as a mixed-integer linear program. A lower-bound-based genetic algorithm was presented that initially examines the parameters using a factorial experiment to identify the statistically significant ones. Zandieh and Karimi [68] employed a multi-objective group scheduling problem in a hybrid flexible flow-shop with sequence-dependent setup times by simultaneously minimizing the total weighted tardiness and maximum completion time. A multi-population genetic algorithm was proposed by them to search for a Pareto optimal solution, which was compared with two well-established benchmarks, a multi-objective genetic algorithm and NSGAII using three sizes of test problems, including small, medium and large to evaluate the performance of the proposed multi-population genetic algorithm.

Considering the literature reviewed above and to the best of our knowledge, the flexible flow shop scheduling problem with SDST, transportation time by conveyor, ready time, and probable rework have not been studied with two concurrent objectives. Thus, this research appears to be the first attempt in applying a new multi-objective algorithm called the novel harmony search algorithm with the Gaussian mutation. We chose five measures to assess the performance of our proposed algorithms. We used the performance measures to compare the results of the multi-objective algorithms quantitatively, followed by the normalization of both objective functions. The rest of the paper was organized as follows. In Section 2, the problem description was provided, and the assumptions were introduced in detail in Section 3. The multi-objective novel harmony search algorithm with the Gaussian mutation, generating an initial solution with the MCH algorithm, and the data-mining approach was described in Section 4. Section 5 focused on how to generate the test data, parameter tuning, and analyses of computational experiments. Finally, in Section 6, we presented the conclusions and some promising directions for future research in the area.