Optimising the job-shop scheduling problem using a multi-objective Jaya algorithm

Highlights

• A multi-objective job-shop scheduling problem is studied.

• An effective multi-objective Jaya algorithm is proposed to solve this problem.

• Grey entropy parallel analysis is utilised to evaluate the solutions.

• Opposition-based learning is employed to improve solution quality.

• The proposed algorithm is shown to outperform other state-of-the-art algorithms.

Abstract

This paper presents an effective multi-objective Jaya (EMOJaya) algorithm to solve a multi-objective job-shop scheduling problem, aiming to simultaneously minimise the makespan, total flow time and mean tardiness. A strategy based on grey entropy parallel analysis (GEPA) is developed to assess and select solutions during the search process. To obtain a high-quality reference sequence for GEPA, an opposition-based learning (OBL) strategy is used in parallel. Additionally, the OBL strategy is incorporated into Jaya’s search operation and external archive to enhance the search ability and convergence rate of the algorithm. Computational experiments based on 30 benchmark instances with different scales confirm that GEPA and OBL can significantly improve the performance of our proposed EMOJaya. Experimental results also show that EMOJaya is able to outperform three state-of-the-art multi-objective algorithms in solving the problem at hand in terms of convergence, diversity and distribution. Further, EMOJaya can obtain more high-quality scheduling schemes, which provide more and better options for decision makers.

Introduction

The job-shop scheduling problem (JSP) is one of the most classical production scheduling problems. The aim of the JSP is to find sequences of jobs and machines to optimise specific production objectives. A good JSP solution can effectively help manufacturers improve production efficiency and reduce production costs [1], [2]. As a result, the JSP has been widely studied for decades. It has been demonstrated that the JSP is an NP-hard problem [3], which means that an optimal solution cannot be obtained in finite time. Initial studies on the JSP focused on a single objective—principally, the optimisation of makespan. Some exact solution methods, such as the dispatching rule [4] and branch and bound algorithm [5], were used to handle the problem. However, exact solution methods become inefficient when the scale of the problem increases. Population-based meta-heuristic algorithms can be used to solve NP-hard problems effectively [6], [7], [8], [9], especially large-scale optimisation problems. They can obtain an approximate solution in a reasonable computational time. Meta-heuristics have characteristics such as parallelism, diversity, robustness and good compatibility. Therefore, meta-heuristic algorithms have been widely adopted to address the JSP in recent years. These meta-heuristics include the genetic algorithm (GA) [10], [11], simulated annealing (SA) [12], tabu search (TS) [13], ant colony optimisation (ACO) [14], particle swarm optimisation (PSO) [15], bee colony optimisation [16], imperialist competitive algorithm [17], memetic algorithm (MA) [18], and differential evolution [19]. Additionally, some hybrid methods have also been used for the JSP, such as the GA with TS [20], ACO with TS [21] and PSO with SA [22].

In the real world, however, manufacturers often have to consider multiple factors to maintain the balance of multiple production performance indicators (e.g., makespan and tardiness) [23]. As a result, the multi-objective JSP (MOJSP) has gained increasing interest in recent years (e.g., see [24], [25]). The MOJSP is more difficult to solve than the JSP with a single objective, and is also an NP-hard problem. Popular meta-heuristic algorithms have been used to solve the MOJSP. For example, Lei and Wu [26] designed a crowding measure-based multi-objective evolutionary algorithm to solve the MOJSP, aiming to minimise the makespan and total tardiness of jobs. They used the crowding measure to adjust the external population and assign varying fitness for individuals. Kachitvichyanukul et al. [27] proposed a two-stage GA for the MOJSP, in which three objectives – makespan, total weighted earliness and total weighted tardiness – were dealt with via a weighted aggregating objective function. Wisittipanich et al. [28] presented an efficient PSO algorithm to obtain the Pareto front for the MOJSP, in which an elite group was employed to store updated non-dominated solutions found by the whole swarm, and those solutions were used as the guidance for particle movement. Zhao et al. [29] proposed an improved PSO algorithm with a decline disturbance index (DDPSO) to solve an extended MOJSP. In DDPSO, the ability of particles to explore the global and local optimisation solutions was improved, and the probability of being trapped in the local optima was reduced. Ariyasingha et al. [30] employed a multi-objective ACO algorithm to deal with an MOJSP with four objectives, in which the performance of the proposed algorithm was evaluated by changing the number of objectives and ants. Udomsakdigool and Khachitvichyanukul [31] presented an ACO algorithm for solving the MOJSP, aiming to minimise the makespan, mean flow time and mean tardiness. Niu et al. [32] developed an intelligent water drops algorithm to search non-dominated solutions for the MOJSP. Tavakkoli-Moghaddam et al. [33] proposed a new multi-objective Pareto archive PSO algorithm combined with genetic operators and variable neighbourhood search to handle a bi-objective JSP, aiming to minimise weighted mean flow time and total penalties of tardiness and earliness. Ripon et al. [34] addressed the MOJSP by using a non-dominated sorting GA (NSGA). They proposed an improved precedence preservation crossover for their algorithm.

Researchers have also hybridised meta-heuristics to solve MOJSPs. Meng et al. [35], for example, developed a GA that combines SA and TS, and proposed an MOJSP model that adopted a weighted sum method to optimise the makespan and total flow time. Yang and Gu [36] designed a novel quad-space cultural genetic tabu algorithm for the MOJSP; this algorithm deals with different levels of populations globally and locally by applying genetic and tabu searches separately, and exchanges information regularly to make the process more effective towards promising areas, along with modified multi-objective domination and transform functions. Zhao et al. [37] studied an MOJSP to minimise three objectives, including the makespan, total flow time and tardiness. An improved multi-objective evolutionary algorithm based on decomposition (IMOEA/D) was proposed by them to solve the MOJSP. In their IMOEA/D, several priority rules were presented to construct the initial population with a high level of quality. Gong et al. [2] presented an effective MA (EMA) to solve the MOJSP, considering the minimisation of makespan and total tardiness. A new effective local search approach was proposed and integrated into the MA to improve the speed of the algorithm and fully exploit the solution space. Kurdi [38] proposed an improved island-model MA with a new nature-inspired cooperation phase for the MOJSP. Three objective functions – makespan, total weighted tardiness and total weighted earliness – were aggregated using the weighting approach.

From the above literature, we can see that different types of meta-heuristic algorithms have been used for the MOJSP. Most of these existing meta-heuristics optimise a problem by executing a variety of search operations. For example, the GA searches for a satisfactory solution using three operations—selection, crossover and mutation. This requires a large number of fitness evaluations. Furthermore, most meta-heuristic algorithms have several parameters that need to be tuned. The performance of a meta-heuristic algorithm is significantly affected by specific parameters. Improper tuning of algorithm parameters may increase the computational effort and weaken the solution quality, or even cause the algorithm to fall into local optima [39].

The Jaya algorithm is a new type of meta-heuristic proposed by Rao [39]. It is based on a victorious concept that the solution obtained for a given problem should move towards the best solution and avoid the worst solution. Compared with the existing meta-heuristics, the merits of the Jaya algorithm are as follows. First, the optimisation procedure of the basic Jaya algorithm consists of only one search operator. It is straightforward to code and easy to implement for optimisation problems. This can reduce the computational complexity of the Jaya algorithm and help the method obtain satisfactory solutions with fewer fitness evaluations. Second, the Jaya algorithm was inspired by the idea that the solutions must approach the best solution and evade the worst. With this idea, the population is updated towards the best solution, which means that the convergence of the Jaya algorithm can be improved. Third, the Jaya algorithm does not have specific parameters, which can reduce parameter tuning effort. Without being influenced by many parameters, the algorithm can maintain a more stable performance.

Given the above advantages, there has been growing interest in applying Jaya to deal with complex optimisation problems, including multi-objective problems (MOPs). Yu et al. [40] presented a performance-guided Jaya algorithm to extract parameters of different photovoltaic models. Rao et al. [41] applied the Jaya algorithm for economic optimisation of shell-and-tube heat exchangers. To estimate the parameters of a Li-ion battery model, Wang et al. [42] proposed a parallel Jaya algorithm implemented on a graphics processing unit. Chen et al. [43] introduced a modified Jaya algorithm for parameter identification of chaotic systems. To address redundancy allocation problems, Ghavidel et al. [44] proposed an efficient improved hybrid Jaya algorithm based on time-varying acceleration coefficients and teaching–learning-based optimisation. Rao et al. [45] proposed a multi-objective Jaya algorithm to deal with modern machining processes. Warid et al. [46] introduced a novel quasi-oppositional modified Jaya algorithm to solve different multi-objective optimal power flow problems.

For MOPs, determining how to evaluate and select solutions is a challenging task. Three types of widely used fitness evaluation methods are: (1) the weighted sum method, (2) the Pareto dominance method, and (3) the performance indicator method. The weighted sum method is commonly used for evaluating and selecting solutions in an MOP [47]. However, the main problem with this method is that it is difficult to determine how appropriate weights should be assigned to each objective for a specific MOP. The Pareto dominance [48], [49] and performance indicator [50], [51] methods have proven efficient for evaluating solutions and can effectively improve the performance of multi-objective algorithms. Nevertheless, when the number of objectives or problem sizes increases, the optimisation performance of most algorithms based on these two methods decreases rapidly, and their search efficiency also decreases as a result of the complex calculations required [52], [53].

Recently, Zhu et al. [54] presented a new fitness assignment strategy based on grey entropy parallel analysis (GEPA) to evaluate solutions for MOPs. Here, grey relational analysis and information entropy theory are used to deal with the objective function value sequences. Each objective function value sequence corresponds to one solution with a grey entropy parallel relational grade. This grey entropy parallel relational grade is used as a fitness value to evaluate the quality of solutions in the current population. Via GEPA, a complicated MOP can be converted into a simple single-objective optimisation problem without any weight or much calculation effort. GEPA was integrated into a GA for tackling the multi-objective flow-shop scheduling problem, and was shown to have better performance than the weighted sum and Pareto dominance methods [55].

Most meta-heuristics, including the Jaya algorithm, are prone to the local optima trap [55]. Thus, determining how to achieve a balance between exploration (global search) and exploitation (local search) is a general issue to consider in designing an efficient meta-heuristic optimisation algorithm  [56], [57]. The use of machine learning to improve the performance of optimisation algorithms has become an emerging trend in recent years  [56], [57]. The main idea here is that machine learning methods can be used to analyse historical data (e.g., population evolution information and problem feature information) and guide the evolution of an algorithm  [56], [58]. Opposition-based learning (OBL) is a machine learning strategy proposed by Tizhoosh [59]. OBL can learn from the opposite direction of a current solution and explore some unknown search space to enhance the optimisation ability of an algorithm. By employing OBL, the diversity of solutions in a population can be enhanced, and the algorithm has a high probability of escaping from local optima  [60], [61]. Many studies have shown that OBL can boost the convergence rate of algorithms and enhance solution quality. For example, Ahandani et al. [61] used OBL to accelerate the shuffled frog leaping algorithm and diversify its search moves. Remli et al. [62] combined scatter search with OBL for large-scale parameter estimation in kinetic models of biochemical systems. To avoid local optima, Zhang [60] incorporated OBL into the harmony search algorithm to solve a power system’s economic load dispatch problem.

Inspired by the promising potential of the Jaya algorithm, as well as that of GEPA and OBL, we propose an efficient multi-objective Jaya (EMOJaya) algorithm to address the MOJSP for the first time in the relevant literature. The main contributions of this work are as follows:

• The GEPA method is adopted to evaluate the population’s solutions. To obtain a high-quality reference sequence for the GEPA, an OBL-based parallel, single-objective Jaya algorithm is applied.

• A novel EMOJaya algorithm is proposed for handling the MOJSP. GEPA is incorporated to develop a new fitness evaluation strategy to improve the population evaluation ability of EMOJaya. OBL is also adopted to enhance the quality of solutions and improve the local search ability of EMOJaya.

• The effectiveness of GEPA and OBL is validated by considerable computational and statistical experiments. The extensive comparison experiments show that the EMOJaya algorithm outperforms three state-of-the-art algorithms in solving the MOJSP.

The remainder of this paper is organised as follows. In Section 2, we present a multi-objective model for the MOJSP. Then, Section 3 introduces the basic Jaya algorithm. Section 4 describes the proposed EMOJaya algorithm, while Section 5 presents the experimental study and results. Finally, we draw conclusions and discuss future work in Section 6.