Cooperative hybrid evolutionary algorithm for large scale multi-stage multi-product batch plants scheduling problem

Highlights

• A novel cooperative hybrid evolutionary algorithm is designed for large scale MMSP.

• Hybrid assignment and selection strategies are proposed based on the CC framework.

• New evolutionary operations are adopted for the unit assignment and order sequence.

Abstract

As an important part of batch chemical industry scheduling problems, the multi-stage multi-product batch plant scheduling problem (MMSP) has been widely studied for decades. This problem is characterized by multiple stages with non-identical parallel units and operate based on customer orders. In this paper, we focus on the large scale MMSP and treat the minimization of make-span as the objective function. An efficient cooperative hybrid evolutionary algorithm is proposed based on the framework of cooperative co-evolution. First, a novel two-line encoding scheme is developed to represent the unit assignment and sequencing for orders respectively. Second, modified estimation of distribution algorithm (EDA) and differential evolutionary (DE) operations are proposed according to the feature of MMSP. EDA operation with a novel population-based incremental learning strategy is applied to handle the unit assignment variables. And novel DE operation based on a novel encoding method is adopted to deal with sequence variables. Then, two selection strategies are applied to preserve optimal and sub-optimal solutions for the proposed algorithm. The critical path based local search algorithm is adopted to further improve the efficiency of local optimization. The proposed algorithm has been tested by several instances with different sizes and characteristics. The numerical results and comparisons show that the proposed work is very competitive in solving large scale MMSP.

Introduction

The multi-stage multi-product batch plant scheduling problem (MMSP) is an important part of the batch chemical industry scheduling problems, and it has received increasing attention in the past two decades. Optimal short term scheduling of such plants is formidable because of alternative units for each order in each stage. And the production costs and organization benefits are affected by scheduling schemes. Therefore, developing good schedules for batch plant is very important. Different from the multi-purpose batch plants, multi-stage multi-product batch plants have multiple stages, and each stage has parallel production units. Such plants operate based on orders’ set with different batch sizes and due dates. The classic multi-stage multi-product batch plant topological structure is shown in Fig. 1. Customer orders are processed through production stages in sequence with the same recipe. If the batch size of each order keeps unchanged during the scheduling horizon, MMSP can be treated as a simplified flexible job-shop scheduling problem (FJSP).

Most of the methodologies for handling MMSP are focusing on solving mixed-integer linear programming (MILP), constraint programming (CP), heuristic, and meta-heuristic algorithms. As reviewed by Méndez et al. [1] and Franco et al. [2], MILP is still the most investigated formulation method. The majority of the MILP methods are consist of solving a discrete-time and continuous-time based formulations. Discrete-time method partition the time axis of each production unit into uniformly time slots [3]. Continuous-time models distribute time slots asynchronously for different production units and production tasks. It is first introduced by Pinto and Grossmann [4], which can handle the MMSP successfully. Afterward, the study on this problem is extended by imposing pre-ordering heuristic to reduce the computational time [5]. Another widely studied MILP formulation is the precedence-based formulation method. Gupta and Karimi [6] presented a novel MILP model that considers set times and release times of orders and units. Marchetti and Cerdá [7] introduced a new concept on this problem, namely the bottleneck stage. Based on this concept, a constant batch ordering rule (CBOR)-based formulation method is proposed, which result in the reduction of variables and superior solutions. Castro et al. [8] proposed a decomposition approach and based on the time-and sequence-based model to handle the large scale MMSP. The modeling method is further improved by considering changeover times [9]. Comparing with MILP formulation, CP is more efficient in solving minimizing the make-span [10]. Zeballos et al. [11] proposed a CP model and a search strategy that can handle MMSP with various features. Novara et al. [12] introduced a novel CP approach with the operating campaign condition. They further studied by taking the number limits of batches into account [2]. Except for MILP and CP based methods, there are many heuristic and meta-heuristic methods that been studied and designed for MMSP. He and Hui [13] address the scheduling problem by integrating the multiple heuristic rules into the genetic algorithm (GA). The search efficiency is rapidly increased by heuristic rules, and multiple rules help the GA keep the diversity. Shi and Yan [14] focused on a single-stage multi-product scheduling problem in batch plants with parallel units. They presented a novel heuristic rule-based scheduling method for solving MMSP. The lineup competition algorithm (LCA) was used to optimize order assignment rule among a group of order assignment rules. Each rule is assigned to select a suitable processing unit for each order. Later, they further extend the work to MMSP [15]. Similar to the previous work, LCA is applied with the optimal heuristic rules which can optimize MMSP.

As mentioned above, MMSP can be treated as a particular case of FJSP, which has been investigated through many theories and experiments. In this paper, the methodology of the proposed algorithm is based on the meta-heuristic algorithms that applied for FJSP. In recent years, this problem has been attracting searchers’ attention, and various optimization approaches are proposed. Comparing to other types of job-shop scheduling problems, FJSP is more close to the production environment, which plays an important role in scheduling problems [16]. Meta-heuristic algorithms are effective approaches to this problem [17]. Vilcot and Billaut [18] introduced two tabu algorithms to solve a multi-criteria FJSP from the printing and boarding industry. Na and Park [19] deal with a scheduling problem with multi-level job structures in a flexible job-shop environment. GA heuristics are applied with priority rules, and results outperformed MILP and CP optimizers. A modified discrete differential algorithm (DE) is introduced by Yuan and Xu [20]. By applying forward and backward conversion techniques, operation sequence and machine assignment vectors are both could evolve with DE algorithms. A critical-operations moving technique is also improvised to improve the efficiency of the whole algorithm. Shao et al. [21] proposed a hybrid discrete particle swarm optimization (DPSO) and simulated annealing algorithm (SA), where SA is used for local search in the framework of DPSO. Gao et al. [22] adopt a novel discrete harmony search algorithm for flexible job-shop scheduling problem. Wang et al.’ [23] work is applying the EDA with bi-population to solve the FJSP. And the machine assignment and operation sequence vectors evolve in two different population respectively. Ricardo and Arturo [24] introduced a generalized Mallows distribution based EDA to solve the FJSP with process plan flexibility. Other types of FJSP is also important researching area for researchers, such as local optimization searching strategies, multi-objective problems, and distributed scheduling problems. Amiri et al. [25] focused on the neighborhood structure of the FJSP and proposed a novel variable neighborhood search algorithm. They employed a shake procedure among the several neighborhood structures to perform the local optimization search strategies. Gao et al. [26] developed a novel hybrid genetic algorithm with a variable neighborhood descent strategy. Two local search procedures are performed to handle find assignable time intervals for the deleted operations. In order to improve the focus on the efficiency of local search, Oleh and Lars [27] solved the FJSP with the objective of total weighted tardiness. The novel iterative local search heuristic hybridized shifting bottleneck heuristic with a variable neighborhood search approach. Nicoara et al. [28] solved the multi-objective problems and reinforced the NSGA_II algorithm with a heuristic adaptive control strategy to fit more optimization problems. Jian et al. [29] addressed the multi-objective FJSP with a novel hybrid evolution approach. A modified crowding distance measure based on new chromosome representation and genetic operators is proposed to preserve the diversity of the population. Lu et al. [30] put attention on the distribute job-shop scheduling problem, and proposed a novel GA-JS algorithm. They developed a new chromosome representation by converting the 3-dimensional problem into a 1-dimensional scheme. The results of GA-JS show good performance in different cell environments. De and Pezzella [31] proposed an improved genetic algorithm with a concise chromosome representation considering the distributed manufacturing environment. The algorithm considers routing, sequencing, and job assignment problems simultaneously.

In this paper, we focus on the optimization of large scale MMSP. The standard and traditional meta-heuristic algorithms cannot be applied for large scale optimization scheduling problems. Because, as the problem scale and dimension increasing, the solution space and computational cost increase exponentially, which leads to unacceptable computing time. Motivated by the reasons, there are lots of studies are carried out to handle large scale problems. The cooperative co-evolutionary algorithm (CC) is an efficient framework for solving large scale optimization problems. CC algorithm can be classified into a decomposition algorithm, which is firstly proposed by Potter and De Jong [32]. It involves a static grouping strategy that decomposed all decision variables into various groups with an identical strategy. Frans and Andries [33] presented a cooperative algorithm CPSO-HK integrating particle swarm optimizer. CPSO-SK algorithm and PSO algorithm perform alternately in the evolving process to prevent trapping into suboptimal locations in searching space. As reported by the sub-sequent researches, grouping strategy is critical to the CC. Yang et al. [34] proposed a new CC framework, which could adapt the grouping scheme automatically by weighting strategies. Omidvar et al. [35], [36], [37] carried out a series of work on the grouping strategies. They proved that, as the number of interacting variables increasing, the probability of grouping interacting variables in one subpopulation would reduce significantly. And they also proved the inefficiency of the adaptive weighting technique. A novel technique for self-adaptation of the subcomponent sizes is demonstrated for cooperative evolution [35]. Later introduced a novel technique, namely the delta method. The interacting variables are divided into equal size component by the variation of each generation. If two variables have relatively small delta value among all the variables, they are put in the same component [36]. They also proposed an automatic decomposition strategy without prior knowledge, that grouping strategy is decided by differential condition [37]. Chen et al. [38] introduced a cooperative coevolutionary algorithm with a variable interaction learning strategy. They proposed a learning stage for the algorithm, which is used to detect the search space between decision variables. And then, these groups are optimized according to the classic CC framework. Li and Yao [39] proposed a CC particle swarm optimization algorithm to solve large scale problems. Random grouping strategy is introduced in their work, which decided the coevolving subcomponent dynamically. The algorithm outperforms the state of art algorithms, and shows highly competitive search efficiency. Recently, Lu et al. [40] optimized the FJSP by proposing a distributed cooperative algorithm (dcEA). The framework is built on the resilient distributed data set (RDD). And a hybrid of GA and PSO operations is used for the evolutionary process in each RDD. Many studies of CC algorithms have shown a great advantage in solving large scale non-separable function optimization problems and sequencing optimization problems.

As introduced above, there are several challenges for solving the large scale MMSP. First is that classic evolutionary operation cannot be applied to MMSP for the unique feature. Though MMSP and FJSP are very similar, there are still several unique features of MMSP that should be taken into consideration. As shown in Fig. 1, production units are stage-specific, which means that each unit only belongs to a single stage. Recipes of orders are all the same, and operations for orders can be classified according to stages. Moreover, various constraints also greatly affect the search efficiency of meta-heuristic algorithms, such as order due date constraints, topological constraints, and sequencing constraints. These features make schedules of MMSP following particular heuristic rules. And the searching efficiency of evolutionary algorithms is also highly influenced by these features. So the encoding scheme and evolutionary operations are important to improve the search efficiency of meta-heuristic algorithms. Second, for large scale optimization problems, increasing solution space results in numbers of local optimization. Grouping strategy based CC framework has been proved to be an efficient optimization method in solving large scale problems. However, they are not applied in large scale MMSP yet. The grouping strategy and evolutionary operations for non-separable functions optimization problems need to be modified to solve this problem. To deal with these challenges, a cooperative hybrid evolutionary algorithm (CHEA) is designed in this paper to handle the large scale MMSP. The contribution of this work is listed as follows:

• A novel cooperative hybrid evolutionary algorithm with the set-based grouping strategy is designed. A grouping strategy that decomposes high dimensional decision variables into lower dimensions sub-problems is proposed based on a novel encoding scheme.

• Hybrid individual assignment and selection strategies are proposed based on the CC framework. Two archives are introduced with different selection strategies which can improve the diversity of the population.

• Two modified evolutionary operations based on EDA and DE are adopted to handle the unit assignment and sequence of orders respectively, which also can balance the ability of exploration and exploitation effectively.

Experimental studies demonstrate the effectiveness and efficiency of the proposed algorithm in solving large scale MMSP. The organization of this paper is listed as follows: Section 2 introduces the formulation of MMSP. Section 3 introduces the detailed improvement of CHEA algorithm. Section 4 shows the results of experiments and analysis. Section 5 gives the conclusions and future work.