Elite guided steady-state genetic algorithm for minimizing total tardiness in flowshops☆
Introduction
Scheduling is a decision-making process that plays an important role in most manufacturing and service industries. It is used in procurement and production, in transportation and distribution, and in information processing and communication. The scheduling function in a company uses mathematical techniques or heuristic methods to allocate limited resources to the processing of tasks. A proper allocation of resources enables the company to optimize its objectives and achieve its goals (Pinedo & Chao, 1999). The scheduling in a flowshop situation where all the jobs pass through all the machines in the same order is one of the most important problems in the field of production planning (Zegordi, Itoh, & Enkawa, 1995).
Johnson (1954) was the pioneer in the research of flowshop problems. He proposed an easy algorithm for the two machine flowshop problem with minimizing makespan as the criterion. Since then, several researchers have focused on solving m-machine () flowshop problems with the same criteria. As these problems fall in the class of NP-hard, complete enumeration techniques must be used to solve these problems. As the problem size increases, these approaches are not computationally practical. For this reason, researchers have constantly focused on developing heuristics for the hard problems (Chen, Vempati, & Aljaber, 1995).
The scheduling problem becomes more difficult when due dates are involved. Hence, there are only a few papers that deal with objectives involving due dates of jobs in flowshops (Kim, 1995). Du and Leung (1990) have shown that the problem of minimizing total tardiness is NP-hard in the ordinary sense even when there is only one machine, which is a special case of the flowshop. Also, another special case in which all the due dates are equal to zero is also proved to be NP-hard when (Kim, 1995).
Currently available solution approaches for flowshop scheduling problems with the objective of minimizing total tardiness (FSPT) can be classified into three categories as follows: exact methods, constructive heuristics, and metaheuristics.
Due to the complexity of flowshop scheduling problems, using exact methods to solve them is impracticable for instances of more than a few jobs and/or machines (Vallada, Ruiz, & Minella, 2008). Though optimal solutions for FSPT can be obtained via exact methods including branch and bound algorithms (Chung et al., 2006, Kim, 1993a, Kim, 1995, Pan et al., 2002, Pan and Fan, 1997, Schaller, 2005, Sen et al., 1989), most problems found in industry are sufficiently large so as to preclude the use of the exact methods. For practical purposes, it is more appropriate to look for heuristic algorithms that generate a near-optimal solution at a relatively minor computational expense.
Constructive heuristics build a feasible schedule from scratch. In these heuristics, once a job sequence is determined, it is fixed and cannot be reversed. Gelders and Sambandam, 1978, Ow, 1985, Kim, 1993b, Raman, 1995, Kim et al., 1996, and Koulamas (1998) proposed this type of heuristic. Vallada et al. (2008) showed that, the best constructive heuristics for FSPT are proposed by Kim et al. (1996). These heuristic algorithms use an improvement procedure based on insertion and interchange of jobs.
Metaheuristics start from a previously generated initial solution or solutions and try to improve this solution until predetermined stopping criteria are met. To date, proposed metaheuristics for FSPT can be categorized as genetic algorithms (Onwubolu and Mutingi, 1999, Vallada, in press), simulated annealing (Adenso-Diaz, 1996, Hasija and Rajendran, 2004, Parthasarathy and Rajendran, 1998), tabu search (Adenso-Diaz, 1992, Armentano and Ronconi, 1999), and differential evolution algorithm (Onwubolu & Davendra, 2006). Vallada et al. (2008) also showed that one of the best performing methods is the simulated annealing algorithm proposed by Hasija and Rajendran (2004). The salient feature of this algorithm was the development of two new perturbation schemes that have been incorporated in the basic simulated annealing algorithm. In addition, an improvement scheme to improve the seed solution that was given as the input to the simulated annealing was presented, an archive containing the best ten sequences obtained from the simulated annealing was maintained, and all such sequences were subjected to the improvement scheme at the end of the simulated annealing algorithm to improve the final solution. Hasija and Rajendran (2004) showed the superior performance of their algorithm against the simulated annealing algorithm proposed by Parthasarathy and Rajendran (1998) and the tabu search algorithm proposed by Armentano and Ronconi (1999). In the most recent study on this problem presented by Vallada and Ruiz (in press), three genetic algorithm based heuristics were proposed. The algorithms include advanced techniques like path relinking, local search and a procedure to control the diversity of the population. Experimental results showed the superiority of the proposed algorithms over simulated annealing algorithms by Parthasarathy and Rajendran (1998) and Hasija and Rajendran (2004) and adaptations of other state-of-the-art methods originally proposed for other objectives, mainly makespan.
A detailed review and evaluation of heuristics and metaheuristics considering the FSPT can be found in a recent article by Vallada et al. (2008). In this research, a steady-state genetic algorithm (ssGA) based approach which we named as an elite guided steady-state genetic algorithm (ssGA-Elite) has been developed for minimizing total tardiness of jobs in a permutation flowshop environment. Using problem-specific knowledge, an efficient solution improvement scheme and an appropriate crossover operator are designed and integrated into the proposed method. Using benchmarking problems, the algorithm has been compared with the simulated annealing based algorithm of Hasija and Rajendran (2004), and the best of the genetic algorithm based algorithms proposed by Vallada and Ruiz (in press). Results of the comparison are presented. Efficiency of the algorithm of Hasija and Rajendran (2004) has been shown by Vallada et al. (2008). So we felt it was necessary to compare against it.
The rest of the article is organized as follows. In the next section we give the formulation of the permutation flowshop scheduling problem. In Section 4, we give the notation used in the rest of the article, and Section 3 introduces genetic algorithms (GA) and ssGA. Section 5 presents the detailed description of the proposed method including lateness-based crossover operator (LBCX), tardiness-based elite guided improvement scheme (TABES), and other components and parameters. In Section 6, the effectiveness of the proposed algorithm is demonstrated by computational results based on some benchmark problems. The conclusions are discussed in Section 7.
Section snippets
Problem Formulation
In a flowshop scheduling problem there is a set of n jobs, tasks or items (1, 2, ... , n) to be processed on a set of m machines or processors (1, 2, ..., m) in the same order, i.e. first on machine 1 then on machine 2 and so on until machine m. The objective is to find a sequence for the processing of the jobs on the machines so that a given criterion is optimized (Ruiz & Maroto, 2005). The machines are set-up in a series, and whenever a job completes its processing on one machine, it joins the queue
Notations
n number of jobs in a given problem m number of machines in a given problem Pij processing time of job i on machine j P(t) the population in the tth generation Pi[j] a job in position j of the solution Pi St similarity threshold value for generating the current population randomly E an elite list L length of E TRpop the tournament size ratio parameter of the selection mechanism TRcx The tournament size ratio parameter of the lateness-based crossover operator N the size of the population
Genetic algorithms and steady-state genetic algorithms
GA are search algorithms based on the mechanics of natural selection and natural genetics (Goldberg, 1989). GA emphasize genetic encoding of potential solutions into chromosomes and apply genetic operators to these chromosomes (Sarker, Mohammadian, & Yao, 2002). Each individual represents a potential solution to the problem at hand, and is evaluated to give some measure of its fitness. Some individuals undergo stochastic transformation by means of genetic operations to form new individuals.
Proposed steady-state genetic algorithm
The proposed ssGA-Elite method is based on the ssGA approach. In order to improve search performance, some modifications have been incorporated and new components have been developed and integrated into its overall procedure. Because of its structure, ssGA usually converges to a local minimum faster than the traditional GA. So, a mechanism is needed to diversify among different areas of solution space when the population converges. In the proposed method, this mechanism is developed as follows.
Computational experiments
ssGA-Elite is compared against the simulated annealing algorithm proposed by Hasija and Rajendran (2004) (namely SAH), and the best of the three genetic algorithms proposed by Vallada and Ruiz (in press) (named GAPR). ssGA-Elite, SAH and GAPR are programmed in C++, and implemented on the PC with P4 CPU 3.00 Ghz, 1 GB RAM. Care has been taken to code other two heuristics as efficiently as possible to avoid bias in the computational comparison.
For evaluating the heuristics, we use the benchmark
Conclusions
In this study, the permutation flowshop scheduling problem with the objective of minimizing total tardiness is considered. By introducing problem-specific knowledge, a new crossover operator and solution improvement scheme have been designed and integrated into the steady-state genetic algorithm in order to achieve better quality and robustness of solutions. Experimental results demonstrate the effectiveness of the proposed method on a set of 540 benchmark problem instances.
Further
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