Research on Permutation Flow-shop Scheduling Problem based on

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Available online at www.sciencedirect.com Procedia Computer Science 00 (2017) 000–000

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Procedia Computer Science 112 (2017) 427–436

International Conference on Knowledge Based and Intelligent Information and Engineering Systems, KES2017, 6-8 September 2017, Marseille, France

Research on Permutation Flow-shop Scheduling Problem based on Improved Genetic Immune Algorithm with vaccinated offspring Fatima Benbouzid-Si Tayeba,*, Malika Bessedika, Mohamed Benbouzidb,c, Hamza Cheurfia, Ammar Blizaka a Laboratoire

Laboratoire des Méthodes de Conception de Systèmes (LMCS). École nationale Supérieure d’Informatique of Algiers (ESI). BP 68M Oued Smar 16270 Algiers (Algeria) b University of Brest, FRE CNRS 3744 IRDL, 29238 Brest, France c Shanghai Maritime University, 201306 Shanghai, China

Abstract This work proposes a hybrid of GA and immune algorithm for permutation flowshop scheduling problems to overcome the problem of GAs early convergence during the evolutionary processes. The proposed algorithm, called VacGA, introduces vaccination into the field of GAs based on the theory of immunity in biology. VacGA employs a GA to perform global search and an artificial immune system to perform local search. VacGA has been tested on Taillard’s benchmarks, and compared with standard GA and the best existing hybrid GAs. The obtained results shed light on the efficiency of our new hybrid method. Furthermore, the effects of some parameters are discussed. © 2017 The Authors. Published by Elsevier B.V. © 2017 The Authors. Published by B.V. Peer-review under responsibility ofElsevier KES International. Peer-review under responsibility of KES International Keywords: Permutation flowshop; Genetic algorithms; Artificial immune system; common subsequence; Vaccination.

* Corresponding author. Tel.: +213-23-939-130; fax: +213-23-939-134. E-mail address: [email protected] 1877-0509 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of KES International.

1877-0509 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of KES International 10.1016/j.procs.2017.08.055

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Fatima Benbouzid-Si Tayeb et al. / Procedia Computer Science 112 (2017) 427–436 Author name / Procedia Computer Science 00 (2017) 000–000

1. Introduction The permutation flowshop problem (PFSP) is a classic scheduling problem where n jobs must be processed on a set of m machines disposed in series and where each job must visit all machines in the same order. Many production scheduling problems resemble flowshops and hence it has generated much interest and had a big impact in industrial field as it is based on ideas gleaned from engineering [1]. The PFSP is one of the most studied problems in the OR literature. Since [2] showed the problem to be NPcomplete for more than two machines; most researchers have focused on implementing approximate methods to find high quality solutions for problems of medium or large size without excessive computation times. The reader may refer to the most recent review in [3], which covered heuristics and metaheuristics published in the literature from [4] compared with other non-efficient algorithms and/or under incomparable conditions. In the last decades, approaches based on genetic algorithms (GA) have received increased attention from the academic and industrial communities for dealing with optimization problems that have been shown to be intractable using conventional problem solving techniques. GA is different from most conventional calculus-based search algorithms in the following characteristics: no limitation on the continuity or discreteness of the search space, parallel computation of a population of solutions, using natural selection criteria, and no gradient information [5]. Nevertheless, one major drawback of GA is its premature convergence to local minima resulting in low accuracy. Also, since GA lacks hill-climbing capacity, it may easily fall in a trap and find a local minimum not the true solution. To avoid the premature convergence, one way is to use large population, but its computation time is a burden and the convergence speed to obtain results with reasonable precision is slow. Under the restriction of small population, GAs merged with other nature-inspired metaheuristics were developed resulting in more robust combinatorial optimization tools for solving complex problems. Hybridization, when properly applied, may further enhance effectiveness of the solution space search, and may overcome any inherent limitations of single metaheuristic algorithms [6]. In the literature, much research has gone into the hybridization of conventional GAs for PFSPs. In general, GA acting as a global search scheme is hybridized with a local search scheme to enhance both diversification and intensification [7-8]. The most recent hybrid GAs for PFSPs are surveyed in the following. Nearchou [9] proposed a metaheuristic that integrates a simulated annealing (SA) algorithm together with features borrowed from the fields of GAs and local search heuristics. Ruiz et al. [10] proposed a GA hybridized with a local search based in an insertion neighborhood for the flowshop scheduling problem (FSSP). Tseng and Lin [11] hybridized a GA and a novel local search scheme. It employs GA to do the global search and two local search methods to do the local. Zhang and Wang [12] addressed a hybrid GA named HGA. Their algorithm obtained 115 best results and 92 of which were newly discovered. Tseng and Lin [13] proposed the GLS algorithm which hybridizes GA and tabu search (TS). It used GA to do global search and the TS to do the local. Chang et al. [14] proposed ACGA (artificial chromosomes embedded in GA), which is a hybrid framework of EDAs and GAs, to tackle the PFSP to minimize makespan. The probabilistic model and genetic operators are used to generate new solutions. Their results show that ACGA performs better than GAs. Newly, Xu et al. [15] presented AGA, a hybrid GA where an enhanced variable neighborhood search (E-VNS) as well as a simple crossover operator is used on all pairs of individuals. For 120 benchmark instances, AGA obtains 118 best solutions reported in the literature and 83 of which are newly improved. Chen et al. [16] propose a self-guided GA for PFSP, which is based on a novel method to combine global statistical information with location information about individual solutions to deal with intractable combinatorial optimization problems. In 2012, Chen et al. [17] extend their previous study in ACGA and propose an extended ACGA (eACGA) to deal with the intractable combinatorial optimization problems by using both the univariate and the bivariate statistic information. Among the modern metaheuristic-based algorithms, the artificial immune systems (AIS) algorithm emerged as highly effective and efficient algorithmic approaches to NP-hard combinatorial optimization problems. Studies [18] have shown that it possesses several attractive immune properties, such as strong self-learning, long lasting memory, self-identity, fault tolerance and strong adaptability to the surroundings, that allow evolutionary algorithms to avoid premature convergence and improve local search due to global search and quick convergence ability. Many researchers have concluded that hybrid approaches for scheduling problems could end up with high quality results. In earlier researches, [19-22] proposed improved approaches hybridizing GA and AIS. The effects for hybrid


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algorithms are significant. Note, however, that hybrid of GA and AIS for the resolution of flowshop scheduling problems has been scarce. We can only quote work of Huang et al. (2009) and Tang et al. (2010). Huang et al. [23] developed a Hybrid Genetic-Immune algorithm (HGIA) to solve FSSP. GA and AIS cooperate with each other to search optimal solution by searching different objective functions. The experimental results validate the philosophy of HGIA in preventing the early convergence of GA. Tang et al. [24] proposed a Two-Phase Genetic-Immune Algorithm (TPGIA) to solve the FSSP. The GA is applied in the first-phase and when the processes are converged up to a pre-defined iteration then the AIS is introduced in the second phase. TPGIA performs well in preventing the early convergence of GA. In this study, by combining the respective advantages of GAs and AIS, an effective GA-AIS hybrid algorithm is proposed to overcome the shortcomings of each algorithm to enhance their exploration and exploitation abilities. The proposed algorithm uses vaccination [25] and receptor editing principles in GAs to move from the infeasible region to the feasible region of the problem. The remaining contents of the paper are organized as follows: section 2 presents the considered problem. In section 3, the proposed GA based hybrid AIS is presented. The results are discussed in section 4. Finally, section 5 gives some concluding remarks. 2. Permutation flow shop scheduling problem The problem under consideration is the PFSP to minimize the maximum completion time or makespan. The problem is formally defined in the following: n jobs {J1, J2,..., Jn} have to be processed with identical flow patterns on a series of m machines {M1, M2,..., Mm} where same job sequence is followed in all machines, i.e. processing order of the jobs on the machines is the same for every machine. The processing time of job Jj on machines Mi is denoted by pij and its completion time is cij (i = 1,...,m; j = 1,..., n). At any time, each job can be processed on at most one machine and each machine can process at most one job. Also, once a job is processed on a machine, it cannot be terminated before completion. The objective is to find the sequence of n jobs, which achieves the minimal makespan when all jobs are processed on the m machines of the shop. Following Pinedo’s notation [26], this problem is denoted by F/prmu/Cmax , where m is the number of machines, prmu denotes that only permutation schedules are allowed, and Cmax denotes the makespan minimization as optimization criterion. This problem has been proved to be NP-hard [2]. 3. The proposed hybrid algorithm The aim of leading immune concepts and methods into the field of GAs is to utilize local information to intervene in globally parallel process and restrain or avoid repetitive and useless work during the course, to overcome the blindness in action of crossover and mutation. In the following subsections, to describe the details of our proposed hybrid of GA and AIS for solving PFSPs, presently called vaccinated GA (hereafter VacGA), first the complete procedures of VacGA are depicted in Algorithm 1. Then, the solution representation (encoding scheme), which makes a solution recognizable for the algorithm is presented. Finally, the initial solutions and the operators used in VacGA are then explained. VacGA contains two major processes, which are GA and AIS. The GA acts as a global search method in our algorithm because it is good at searching the whole solution space globally. The GA searches the solution space with a population of chromosomes each of which is an encoded solution. An affinity is assigned to each chromosome per its solution quality. The better the solution quality of a chromosome is, the higher the affinity becomes. GA works iteratively. A single iteration is called a generation. The individuals of the new generation are obtained from the individuals of the previous one by applying reproduction and mutation procedures. Only the more fit individuals are selected. The standard implementation for solving PFSP using GA can be found in lines [5-10]. The idea of immunity is mainly realized through two steps based on reasonably selecting vaccines, i.e., an immune selection and a vaccination, of which the former is used for preventing the deterioration and the latter is for raising fitness [25]. Immune operators are shown in Lines [11-19], which represent the most remarkable characteristics of VacGA. The mechanisms and operators of VacGA are described below.


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Algorithm 1 The proposed VacGA 1: Input: Population size (Popsize), the crossover rate (Pc), the mutation rate (Pm), the vaccination rate (Pv), Elimination ratio (Relim), Renewal rate (α%), the frequency of elimination steps (a) and the termination conditions (Max_Gen, Improve_Gen); 2: Generate an initial population of Popsize chromosomes; 3: j=0; 4: x=0; 5: while j ≤ Max_Gen do 6: j=j+1; 7: x=x+1; 8: Select pairs of chromosomes from a current population per the selection probability to generate Parentset. 9: Perform crossover on each of the selected pairs in Step 2 with Pc. 10: Perform mutation on the selected chromosomes with Pm. 11: Update the memory set MS with the best chromosomes of the current population as detailed in Algorithm 2. 12: Select randomly two memory cells in MS to create a new vaccine 13: Perform vaccination on the selected chromosome with Pv. 14: If j=a (frequency of elimination steps) Then 15: Eliminate Relim% worst chromosomes in the current population with the highest makespan; 16: Create Relim% new random chromosomes; 17: Change the new created ones with the eliminated ones; 18: x=0 19: End if 20: Update the population to avoid entrapment in local optima and algorithm stall. 21: End while 22: Return best the best solution of the last generation

3.1. Encoding Scheme and Definition of Fitness Function In VacGA, we use a permutation π of jobs as a chromosome. For example, in a PFSP with six jobs and four machines, a permutation π = [2,3,1,6,5,4] is a chromosome that represents the sequence where job 2 gets processed first on all the machines followed by job 3 and so on. The objective function is to minimize makespan. Therefore, the definition of fitness function is just the reciprocal of the maximal makespan (section 2). 3.2.

Initial Population

Randomly generating an initial population is commonly applied in GA. The reason is that it may leads to a population diversification, which is in favor of the evolution. However, random individuals often have poor fitness and thus may slow down the convergence. To solve the problem, a feasible approach is to employ an effective constructive heuristic [10,27]. This approach not only provided a good deal of diversity in the population, but also performed better vis-à-vis the approach where the initial population consisted of only randomly generated solutions. Therefore, in this paper, we use a three-step initialization procedure as follows: (1) Generate one chromosome (schedule) by NEH heuristic [28], which is the best to date well-known construction heuristic for the PFSP; (2) Apply M times (M