A memetic algorithm for the flexible flow line scheduling problem with processor blocking

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Abstract

This paper introduces an efficient memetic algorithm (MA) combined with a novel local search engine, namely, nested variable neighbourhood search (NVNS), to solve the flexible flow line scheduling problem with processor blocking (FFLB) and without intermediate buffers. A flexible flow line consists of several processing stages in series, with or without intermediate buffers, with each stage having one or more identical parallel processors. The line produces a number of different products, and each product must be processed by at most one processor in each stage. To obtain an optimal solution for this type of complex, large-sized problem in reasonable computational time using traditional approaches and optimization tools is extremely difficult. Our proposed MA employs a new representation, operators, and local search method to solve the above-mentioned problem. The computational results obtained in experiments demonstrate the efficiency of the proposed MA, which is significantly superior to the classical genetic algorithm (CGA) under the same conditions when the population size is increased in the CGA.

Introduction

A flexible flow line (FFL) consists of several processing stages in series, may be separated by finite inter-stage buffers. Each stage has one or more identical parallel processors. The line produces several different product types, and each product must be processed by at most one processor in each stage. A product, once its processing is completed on a processor in some stage, is transferred directly to either an available processor in the next stage (or another downstream stage depending on the product processing route), or a buffer ahead of that stage, when such an intermediate buffer is available [1]. When an intermediate buffer is unavailable, the product remains blocking the processor until a downstream processor becomes available. However, this blocking prevents processing of other products on the blocked processor. This type of problem is referred to as a flexible flow line problem with processor blocking (FFLB). The objective is to determine a production schedule for all products to be completed in a minimum time (i.e., minimize the makespan). Actually, a flexible flow line represents a special type of traditional flow shop, in which there is only one processor in each stage and unlimited intermediate storage between successive processors. The flexible flow line with unlimited intermediate buffers has also been termed a hybrid flow shop [2]. In general, blocking scheduling problems arise in modern manufacturing environments with limited intermediate buffers between processors, such as just-in-time production systems or flexible assembly lines, and those without intermediate buffers, such as surface mount technology (SMT) lines in the electronics industry for assembling printed circuit boards which includes three different stages in the following sequence: solder printing, component placement, and solder reflow [3].

The literature on scheduling problems of the traditional flow shop, flexible flow lines, and parallel processor shops is abundant and contains various optimization and approximation algorithms to solve these problems [2]. Studies on the development of scheduling algorithms for flexible flow lines (or shops) with finite capacity buffers or without in-process buffers are mostly restricted to heuristics that obtain good solutions in reasonable computational times [4], [5]. Sawik [4] first proposed an integer-programming formulation for scheduling flexible flow lines with blocking and limited buffers. He examined the model for small-sized problems (e.g., at most five stages and ten products). A main disadvantage of the above integer-programming approach for scheduling problems is the need for solving large mixed-integer programs, for which optimization is meaningful [6]. Kis and Pesch [7] provided the first comprehensive and uniform overview of exact solution methods for flexible flow shops with branching, bounding, and propagation of constraints under two different objective functions that minimize the makespan and mean flow time. Torabi et al. [8] addressed a common cyclic multi-product lot-scheduling problem in deterministic flexible job shops by proposing an efficient enumeration method to determine the optimal solution for the given model. Jungwattanakita et al. [9] presented a mixed-integer programming model for flexible flow shop problems with unrelated parallel processors that minimizes the makespan and number of tardy jobs. They also considered both sequence and processor-dependent setup times for the given problem. Various construction and improvement methods are proposed in [9] to solve the problem, and a special design of genetic algorithms is also considered. It is worth noting that the size and complexity of the integer-programming formulation increase when the introduction of finite capacity buffers results in a blocking scheduling problem. Although recent theoretical advances in integer programming and computer hardware have resulted in robust commercial software, large-sized problems cannot be solved optimally within a reasonable time on a personal computer. Thus, we must use meta-heuristic algorithms for solving such problems.

There are a few studies in the literature that apply heuristic and meta-heuristic algorithms to solve the FFLB. Kurz and Askin [10] developed a random-keys genetic algorithm to solve the problem of flexible flow lines with sequence-dependent setup times by minimizing the makespan. Tavakkoli-Moghaddam et al. [11] proposed a genetic algorithm (GA) with a novel, GA representation and operators to solve the FFLB by minimizing the makespan. Torabi et al. [12] proposed a hybrid genetic algorithm (HGA) to solve a lot-size and delivery scheduling problem in a simple supply chain, where a single supplier produces multiple components on an FFL and delivers them directly to an assembly facility (AF). Jenabi et al. [13] proposed two meta-heuristic algorithms, including the HGA and simulated annealing (SA), to solve a new 0–1 mixed-nonlinear mathematical model of the economic lot-sizing and scheduling problem in flexible flow lines with unrelated parallel processors over a finite planning horizon. The objective determines a cyclic schedule by minimizing the sum of setup and inventory holding costs per unit time without any stock-out. Akrami et al. [14] developed two heuristic approaches, including GA and an optimal enumeration method (OEM), to solve a new model of common cycle multi-product lot-sizing and scheduling problem in deterministic flexible flow shops with a finite planning horizon and limited intermediate buffers. The objective minimizes the sum of setup cost, inventory holding costs, and number of cycles. Kaczmarczyk et al. [15] proposed an improvement heuristic approach for scheduling of printed wiring board assembly in SMT lines. They considered the processor blocking and limited processor availability due to the scheduled downtimes. The heuristic approach, which is a combination of tabu search (TS) and set of dispatching rules, has a hierarchical structure based on the decomposition of the scheduling problem into two sub-problems: sequencing and assignment/timing solved sequentially.

Quadt and Kuhn [16] considered a batch scheduling problem for flexible flow lines, in which a number of jobs must be scheduled, and each job belongs to a specific product type. Setup costs are incurred when changing a processor from one product type to another. The objective is to minimize setup costs and the mean flow time. They presented a solution approach consisting of two (i.e., outer and inner) nested genetic algorithms. In an initialization step, they generated so-called “machine/time slots” for all production stages in order to assign a specific job to each of those slots. For all stages, the proposed approach implies four sub-problems determining: (1) how many processors to set up for each product type; (2) in what batch size; (3) when in the given time window; and (4) on which of the pre-determined processors to produce the jobs. Voβ and Witt [17] considered a real-world multi-mode multi-project scheduling problem, in which the resources form a hybrid flow shop consisting of 16 production stages. Furthermore, sequence-dependent setup states were arisen at two production stages leading to a batching problem by minimizing the weighted tardiness. They presented a mathematical model, based on the well-known resource constrained project scheduling problem, and applied a heuristic solution procedure based on dispatching rules. Janiak et al. [18] applied first three constructive algorithms, and then three metaheuristics, based on TS and simulated annealing (SA) algorithms, to solve a hybrid shop scheduling problem with three objectives: the total weighted earliness, total weighted tardiness, and total weighted waiting time. They experimentally showed that the quality of solutions obtained by the SA algorithm is rather poor compared with the solutions reported by the TS algorithm.

Memetic algorithms (MAs) [19], [20], [21] are population-based heuristic search approaches for optimization problems, and are similar to GAs. GAs are based on the concept of biological evolution. MAs, on the other hand, mimic cultural evolution. In nature, genes are usually not modified during an individual's lifetime; however, memes are modified [22]. A meme is defined as a unit of information that reproduces itself when people exchange ideas [22]. In contrast to genes, memes are typically adapted by people who transmit them before they are passed on to the next generation. According to Moscato and Norman [19], “memetic evolution” can be mimicked by combining evolutionary algorithms with local refinement strategies such as local neighbourhood search or simulated annealing. Thus, the genetic local search (GLS) proposed by More et al. [23] is a special case of a memetic algorithm, which has been shown to be very effective for several combinatorial optimization problems, such as binary quadratic programming (BQP) [24] and the cell formation problem [25]. An extended version of the classical MA was proposed by Berretta and Moscato [26], namely structured-MA, in which there is a structured and hierarchical relationship between individuals at each population. This relationship restricts the crossover possibilities such that all individuals cannot recombine with each other without limitation. In this case, recombination takes place based on a certain relationship between individuals.

In this paper, we extend the study conducted by Tavakkoli-Moghaddamet al. [27]. We investigate whether an efficient MA with a powerful local search engine can be used to solve the mathematical model introduced by Sawik [28]. In large-sized problems, the results obtained by the proposed MA are compared with the classical GA results and lower-bound solution embedded in Sawik's model as a constraint. The rest of this paper is organized as follows. A brief review of FFLB is presented in Section 3. The proposed MA for solving the scheduling FFLB is presented in Section 4. The computational results are reported in Section 5. We conclude the paper in Section 5.

Section snippets

Flexible flow lines with processor blocking

Fig. 1 shows a FFLB consisting of m processing stages in series, in which stage i (i=1,,m) is made up of ni1, i.e., identical parallel processors. The system produces various types of p products. Each product must be processed without pre-emption on exactly one processor in each stage sequentially. That is, each product must be processed in stage 1 through stage m in the same order. The order of processing products in each stage is identical and determined by an input sequence, in which

Memetic algorithm implementation

The proposed MA was implementation as described in the following subsections.

Computational results

In this section, 20 numerical examples are solved to illustrate the efficiency of the proposed MA, whose performance is compared with that of the classical GA under the same conditions. The various sizes we used for the test problems were taken from Wardono and Fathi [30], with P=10,20,30, and 40, and m=2,3,4,5, and 6. Each problem has integer processing times between 1 and 10 generated by uniform distribution. To standardize all test problems and make the comparison of runs simpler, we assumed

Conclusion

In this paper, a metaheuristic-based memetic algorithm (MA) is presented for the scheduling of a flexible flow line problem with processor blocking (FFLB). The proposed MA utilizes a powerful local search engine, namely nested variable neighbourhood search (NVNS), which is an extended version of the classical VNS. In the structure of the NVNS, operators are prioritized as nested loops in terms of their effectiveness, in which each solution is improved by each operator is checked by another

Acknowledgements

The authors would like to acknowledge the Iran National Science Foundation (INSF) for the financial support of this work. The authors would like to also thank our colleagues as well as the anonymous reviewers for their helpful comments and suggestions, which greatly improved the readability and presentation of this paper.

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