Multi-operator communication based differential evolution with sequential Tabu Search approach for job shop scheduling problems

Highlights

• A decoding heuristic is developed to obtain active schedules.

• A common positional-based diversity check mechanism of a population is designed.

• Evolution process is enhanced with mixed selection and communication strategies.

• Multi-operator differential evolution and Tabu search are integrated sequentially.

• Performance of the proposed algorithm is validated against the tested algorithms.

Abstract

This paper develops a multi-operator based differential evolution with a communication strategy (MCDE) being integrated with a sequential Tabu Search (MCDE/TS) to solve the job shop scheduling problem (JSSP) with the objective of minimizing makespan. The three variants of DE which are implemented in the proposed algorithm evolve as independent sub-populations, which relate to a communication strategy that maintains the diversity and quality of each sub-population by employing a proposed mixed selection strategy to avoid premature convergence. The best solution order obtained from MCDE is then passed to Tabu Search (TS) and the evolution process is continued, creating neighbour solutions with N7 neighbourhood structure. This algorithm ensures the population diversity with curving the premature convergence but experiences faster convergence. The design of experiment for parameter tuning is employed for the best combination of the proposed algorithm’s parameter. The performance of the proposed MCDE/TS algorithm is evaluated against a number of state-of-the-art algorithms to show its competence in solving 122 standard benchmark instances.

Introduction

Production scheduling addresses many common problemssuch as ‘single machine’ scheduling, ‘flow shop’ scheduling, ‘open shop’ scheduling, and ‘job shop’ scheduling [1]. The ‘job shop’ scheduling problem (JSSP) is an intractable combinatorial optimization problem, and ’flow shop’ and ’parallel machine scheduling’ problems are constrained versions of the JSSP [1]. JSSP deals with scheduling [2] a set of $\left(n\right)$ jobs on a set of $\left(m\right)$ machines under a set of constraints [3], e.g. given order of operations of jobs on machines. The performance measures of JSSP are makespan, throughput time, earliness, tardiness, and due-date cost [4]. Makespan, as the time at which processing of all jobs are completed [5], is the most commonly used performance measure for JSSP. The JSSP has been prominent among the hardest NP-hard combinatorial optimization problems [6] having a solution space ${(n!)}^m$ [7] reflecting a high computational complexity [8].

Although a considerable amount of research has been dedicated to the development of efficient solving methods for JSSP, many of them were found to be unable to solve even relatively small-sized JSSP, e.g. 10*10 proposed by [9], for more than a quarter of a century [10]. In the past few decades, only a handful of exact and heuristic algorithms have been proposed in the literature. Among many exact approaches, the branch-and-bound technique is quite efficient but only for small-scale JSSP. The integer model was modified in [11] to significantly reduce computation times for solving JSSP. The dispatching rules approach is also widely used to solve JSSP, which was improved by developing a truncation procedure in [12]. Moreover, the shifting bottleneck procedure developed in [13] and then modified in [14] showed good performance for small-sized problems. However, those exact approaches are incapable of solving large scale problems due to unreasonable computational expenses [2], [15].

The understanding of the complexity of the JSSP has directed the research towards approximation approaches which include heuristics and meta-heuristics. The heuristic approaches are developed based on the swarm intelligent (SI) algorithms, physics-based (PB) algorithms, and evolutionary algorithms (EA) [15]. Among SIs, ant colony optimization (ACO) [16], artificial bee colony algorithm (ABC) [17], particle swarm optimization (PSO) [18] all mimic the collective intelligent behaviours in nature. Simulated annealing (SA) [19] is the prominent example of a PB algorithm inspired from basic physics. These heuristics can achieve better results for small- and moderate-sizes JSSP instances. In addition to those heuristic approaches, in the recent years, many meta-heuristics have been developed and applied for solving JSSP: PSO [20], ABC [21], bat algorithm (BA) [4], teaching-learning based optimization (TLBO) [22], bacterial forging algorithm (BFA) [23], firefly algorithm (FFA) [24], cuckoo optimization algorithm (COA) [25], whale optimization algorithm (WOA) [26], bio-geography-based optimization algorithm (BBO) [27], and beer froth artificial bee colony algorithm (BeFABC) [28]. The most prominent examples of EA are differential evolution (DE) [29] and genetic algorithm (GA) [30] those use biologically inspired evolution mechanisms such as mutation, crossover and selection. DE algorithm was first introduced in [31] for the continuous optimization domain and was subsequently implemented to solve the flow shop scheduling problem [32], permutation flow shop scheduling problem [33], and JSSP [34] for the first time. This algorithm has been used in wide range of optimization domains since it has strong searching ability with better convergence rate [29]. This does not mean that it can avoid becoming trapped in local optima. Only a few studies in the literature have been focused on this open problem while solving JSSP. This problem has been first addressed in [29] who developed two algorithms, the first of which utilized the benefits of multiple mutation operators while the second employed a switching strategy to improve solution quality. They also modified the local mutation operator to enhance the solution quality. The main target of their research was to balance the exploration and exploitation ability of DE, avoiding premature convergence and resulting in improved searching power. However, their approach suffered from solution quality. Despite such developments on meta-heuristic approaches, their application is limited by the lack of fine-tuning solutions around optima [35]. The premature convergence and getting stuck in local optima, which prevents JSSP from identifying global optima [26], are common problems for most of the mete-heuristics [36]. As a result, considerable research has been directed to overcome these drawbacks through changing evolutionary strategies [29], hybridizing multiple algorithms [15] and so on.

The evolutionary process for achieving global optima of an algorithm strongly depends on the combined effect of exploration and exploitation properties [7]. The lack of exploration property leads to premature convergence and becoming trapped in local optima, and the lack of exploitation property causes excessive computational expenses and poor solution quality [37]. Therefore, there is a need to have a balance between the diversity (that ensures global searching capability) and intensity (that allows accurate searching capability in near solution) of the population over the evolution process. Population-based algorithms (PBA) have a strong exploration property [7] as they have a multi-point searching ability. The principal issues with PBA are the low convergence rate and easy filling of similar individuals in the population, accelerating premature convergence [38]. On the other hand, many researchers consider the local search algorithms (LSA), as they have better exploitation ability. However, the main limitation of this approach is the single point searching process. Consequently, as both PBA and LSA have their own limitations, the adoption of hybridization approach is encouraged to exploit the benefit of multiple mechanisms and/or algorithms. For example, [39] proposed the combined multi-agent genetic algorithm and Tabu Search (TS) (MAGATS) for balancing intensification and diversification properties; [7] proposed a differential-based harmony search algorithm to improve the diversification of the population; and [40] generated the initial population through estimation of distribution algorithm to ensure the population diversity. [41] hybridized fast simulated annealing (FSA) with quenching where FSA was employed for global search whereas quenching was integrated for localized search in neighbours of current solution. [34] developed a hybridized approach in which DE and TS were employed together in a single loop, so that a certain proportion of best solution based on the fitness value was passed into TS and then back to the DE main loop. They suggested that, since the selection process of DE is essentially greedy and accelerates the premature convergence, a mechanism can be developed and integrated to DE to deal with stagnation in local optima. To balance between diversity and intensity, [40] developed a hybrid approach to combine the dual characteristics of DE and estimation of distribution algorithms as a single meta-heuristic is not capable of creating that balance. This approach utilized the chaotic strategy in DE to strengthen its searching power. Their proposed algorithm has difficulty in solving large-scale problems and therefore they recommended the integration of other types of combinatorial optimization algorithms to improve results. Moreover, many other algorithms have been used in combination with DE to improve their performance. To illustrate, the differential-based enhanced mechanism is used in [7] to maintain the diversity in the population of the harmony search algorithm. DE is employed to WOA in [15] to improve the exploitation and local search capability. As DE has multiple mutation mechanisms with individual search ability [42], an evolutionary structure based on DE may provide the best computational experience for solving complex JSSP. Therefore, a diversity check mechanism can be employed with DE to ensure diversity in the population. In addition, the multiple mutation operators that have individual characteristics can be run in parallel for different population sets. Dynamic communication can also be linked to update each sub-population.

Another approach to ensure better searching capability is to adopt a strategy changing scheme. For example, [43] proposed a self-adaptive phase strategy to balance the exploration and exploitation properties of the search process, and [44] implemented a dynamic migration strategy to make decisions while communicating a colony with its neighbours.

It is obvious that recent state-of-the-art algorithms either amalgamate several strategies or employ PBA. In solving JSSP, a strong local search has always been employed in recent state-of-the-art algorithms [2], although a local search with single solution based approach is computationally expensive [34]. Eventually, Tabu Search (TS) was introduced by [45] as a local search, which was proved to be effective for it exploitation property and to avoid being trapped in local optima [5]. The most successful applications of TS are TSAB [46], i-TSAB [47] and TS/SA [10]. As the run time and solution quality of TS are mostly affected by the initial population generation and their quality of initial solutions respectively [46], [48], [49], employing TS after a population-based algorithm, which provides a diverged and comparatively better fitness solution, can produce better solutions with lower computational expense.

Considering all the shortcomings highlighted in our review of the relevant literature, we propose an approach combining multi-operator differential evolution (MODE) with a communication scheme and TS that we will refer to as MCDE/TS. This novel approach employs a mixed selection strategy on the randomly generated population to ensure and characterize the initial population diversity and fitness. This generated initial population is then run in parallel applying several mutation operators in each sub-population. A dynamic communication scheme is also implemented to improve each sub-population in order to avoid premature convergence. The best schedule obtained from the evolutionary process of the MCDE is passed into TS as the strong local search technique. TS technique finds the critical path of the candidate solution and then divides the critical path into blocks. Then, neighbour solutions are generated by the well-known N7 neighbourhood structure. Tabu list with dynamic Tabu tenure is maintained to avoid revisiting the same schedule repeatedly and being stuck in local optima. At the same time, aspiration criteria are considered in TS to achieve the best offer from the evolutionary process. Finally, this proposed algorithm is utilized to solve $122$ standard benchmark instances for comparison with a number of state-of-the-art algorithms.

The proposed MCDE/TS algorithm is capable of solving real-world problems such as multi-product assembly of the apparel industries [50], customized products in furniture industries [51], vehicle production in automobile industries [21], and assembly fixtures in an aeronautic industries [52]. Several schedulers have been developed and used in industries, which mainly focus on customer order sequencing based on the demand season, material availability and due date. However, the proposed algorithm can optimize workflows for each job by a proper mapping between/among associated resources, resulting a holistic views of the production insight. This predictive visibility and insight provided by the JSSP solutions allow decision-makers enough time and maneuverability to enhance operational decision to leverage benefits. This also facilitates responsive integration with other necessary plannings such as material acquisition [53], due date assignment [54], order acceptance [55], and delivery planning problems [56]. In addition, since this problem is a highly constrained combinatorial problem, and many algorithms failed to converge to the best possible solutions as explained before, the proposed algorithm exploits multiple DE variants in a single algorithm frame by maintaining dynamic communication. This strategy ensures the diversity in population and the convergence towards the best possible solution [57], [58] by overcoming the aforementioned drawbacks. The performance of the algorithm is also improved by implementing a local search approach if the first strategy does not converge. Thus, this algorithm can be implemented as an effective scheduler to create a concrete operational planning with less used computational resources.

However, the key contributions of this paper are as follows:

• A decoding scheme is developed to obtain an active schedule in every case, which directly reduces the search space and computational expenses of JSSP.

• The proposed algorithm incorporates a diversity checkmechanism, based on the common positional-based similarity degree, to combine both fitness value and the diversity of population. This mixed selection strategy will ensure the population diversity and help avoid being trapped in local optima. This mixed selection strategy is also applied to generate a relatively diverged initial population with higher fitness value.

• Integration of parallel multiple mutation operations, a dynamic communication scheme, and a diversity check mechanism are developed to improve the current target vectors to enhance the evolutionary process, which can prevent filling more similar individuals in the population.

• TS, as a strong local search, is implemented to the MCDE and runs by creating neighbour solution using N7 neighbourhood structure. Neighbour solutions are obtained from a randomly selected critical path after dividing it into a number of critical blocks, ensuring non-cyclic and non-repeated moves.

• The performance of this proposed MCDE/TS is verified by simulations on $122$ standard benchmark instances of JSSP and compared with selected state-of-the-art algorithms.

The rest of this article is organized as follows: Section 2 describes the fundamental concept of the DE algorithm. Section 3 illustrates the detailed definition and problem formulation of the JSSP, followed by Section 4 describing the proposed algorithm including population initialization, multi-operator evolutionary process, communication scheme and local search technique. Parameter analysis is performed in Section 5. The results obtained from various standard benchmark instances are presented and discussed in Section 6 followed by statistical analysis in Section 7. Finally, concluding remarks and future research directions are presented in Section 8.