Hybrid Metaheuristics for Solving the Blocking

International Journal of Engineering Research in Africa ISSN: 1663-4144, Vol. 36, pp 124-136 doi:10.4028/www.scientific.net/JERA.36.124 © 2018 Trans Tech Publications, Switzerland

Submitted: 2017-12-27 Revised: 2018-01-23 Accepted: 2018-01-24 Online: 2018-06-01

Hybrid Metaheuristics for Solving the Blocking Flowshop Scheduling Problem Ghita Lebbar1,a*, Abdellah El Barkany2,b, Abdelouahhab Jabri3,c and Ikram El Abbassi4,d 1,2,3

Mechanical Engineering Laboratory, Faculty of Sciences and Techniques, Sidi Mohammed Ben Abdellah University, Fez, Morocco

4

ECAM-EPMI, Research Laboratory in Industrial Eco-innovation and Energetics Quartz-Lab, Cergy-Pontoise, France

a*

[email protected], [email protected], [email protected] and [email protected]

Keywords: Blocking flowshop scheduling, makespan, SAGA, simulated annealing, genetic algorithm.

Abstract: This paper suggests two evolutionary optimization approaches for solving the blocking flow shop scheduling problem with the maximum completion time (makespan) criterion, namely the genetic algorithm (GA) and the simulated annealing genetic algorithms (SAGA) that combines the simulated annealing (SA) with the (GA), respectively. The considered problem and the proposed algorithms have some parameters to be adjusted through a design of experiments with exorbitant runs. In fact, a Taguchi method is presented to study the parameterization problem empirically. The performance of the proposed algorithms is evaluated by applying it to Taillard’s well-known benchmark problem, the experiment results show that the SA combined with GA method is advanced to the GA and to the compared algorithms proposed in the literature in minimizing makespan criterion. Ultimately, new known upper bounds for Taillard’s instances are reported for this problem, which can be used thereafter as a basis of benchmark in eventual investigations. Introduction In the manufacturing world, production scheduling is a decision-making and a branch of convoluted movement of information that design the operations planning and control processes. A sturdy production sequence impedes bottlenecks, alleviates idle time and curtails lead-time, thus, minimizing operating cost in one hand and achieving a considerable level of efficacy and productiveness in the other. The permutation flowshop scheduling problems (PFSP) is one of the most extensively researched combinatorial optimization scheduling problems in the literature and the practical applications. In the classical PFSP, a set of N jobs have to be performed sequentially, by a set of M serial machines in the same permutation and a work in process inventory is permitted since intermediate buffers have infinite capacity between consecutive machines. However, in real-life situations, there are many production factories, where the buffers capacity is zero or limited owing to the system features [1]. In similar situations, the basic PFSP becomes blocking flowshop scheduling problem (BFSP), for which, a job cannot leave a machine unless the next immediate downstream machine is available for processing. This problem is denoted as 𝐹𝐹𝑚𝑚 |𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏|𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 conforming to the standard notation introduced by [2]. With respect to the computational complexity of the problem, (Hall and Sriskandarajah, 1996) [3] proved, relying on the results obtained by [4], that the 𝐹𝐹𝑚𝑚 |𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏|𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 problem with m ≥ 3 is strongly NP-hard. Thereafter, (Allahverdi et al. 2008) [5] showed that the BFSP is NP-hard even for BFSP with m ≥ 2. As a result, exact methods such as branch and bound and dynamic programing are nots to solve the BFSP optimally, especially when it is concerning wide size problem. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.scientific.net. (#108616811, Sidi Mohammed Ben Abdellah University, Morocco-01/06/18,16:44:40)

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In view of the complexity of the considered problem, literature on the resolution of the BFSP with the makespan criterion shows a variety of approaches, namely constructive heuristics and improvement heuristics. Among constructive heuristics developed to solve the BFSP. (Mccornick et al.1989) [6], proposed an innovating heuristic, recognized as Profile Fitting (PF), so as to minimize cycle time in a blocking assembly line with m serial stations. (Leisten, 1990) [7] suggested another constructive technique to tackle the BFSP with permutation and non-permutation, in order to increase the exploitation of buffers and decrease the machine blocking times. He concluded that the developed heuristic was not apt to provide fitter solutions than results introduced by [8] heuristic produced originally for the traditional PFSP. On the basis of the afore mentioned heuristics, (Ronconi, 2004) [9] presented tree constructive heuristics, Known as a MinMax (MM) heuristic, (MME) heuristic that represents the combination of MM with NEH, and (PFE) heuristic which combines the PF with NEH heuristic, respectively. Their results indicated that the tree algorithms outperform the NEH heuristic in problems with up to 500 jobs and 20 machines. (Pan et al. 2010) [10] proposed recently two constructive heuristics namely, the weighted profile fitting (wPF) and Pan-Wang (Pw) algorithm relying on the PF method. Thereafter, they combined the NEH procedure with PF, wPF and Pw, and called heuristics, PF-NEH (x), wPF-NEH(x) and Pw-NEH(x), where 𝑥𝑥 ∈ [1, n] is the number of sequences to be generated. The computational results showed that the Pw-NEH provided promising solutions for the considered problem, compared to the existing algorithms. Regarding the meta-heuristics [11] proposed a genetic algorithm (GA) using the idea of deliberately slowing down processing of certain operations. Computational results indicated that the genetic algorithm performs notably better than the improvement heuristics presented by [12] . (Grawborski and pempra, 2000) [1] developed two tabu search algorithms, called (TS) and (TS+M) algorithms and compared them with the uppers bound of Taillard benchmark suite reported by [13]. They noted that tabu search algorithms outperform the branch and bound approach and Carrafa’s genetic algorithm. (Liu and reeves, 2001) [14] proposed a hybrid particle swarm algorithm (HPSO) to solve the BFSP. A hybrid discrete differential evolution algorithm (HDDE) was suggested by [15]. (Ribas et al. 2011) [16] introduced an iterated greedy (IG) algorithm. The author declared that the IG algorithm outperformed the HDDE and produced novel best solution for Taillard benchmark suite. (Han et al. 2012) [17] proposed an improved discrete artificial bee colony (IABC) algorithm for solving effectively the BFSP. A discrete artificial bee colony algorithm combining differential evolution (DE-ABC) was presented by [18]. (Sadaqa et al. 2015) [19] suggested a Meta-heuristic Randomized Priority Search (Meta-RaPS) on the basis on the NEH algorithm. Recently, (Han et al. 2016) [20] proposed a modified fruit fly optimization algorithm (MFFO) basing on the smell-based and vision-based search. The computational results of the MFFO indicated that the proposed algorithm is superior comparing to the algorithms suggested in the literature. The combination of the constituent from various metaheuristics is presently one of the most fruitful tendencies in optimization field. In fact, considerable algorithms that do not preserve the design of single classical metaheuristic have been developed ([21], [22], [23]). The principal incentive for the hybridization of those algorithms is to combine the qualities and the advantages of the individual combined approaches, minimize the effect of their corresponding disadvantages and subsequently enhance the systems performance. In this study, by combining the exploration ability of GA and The power to escape the local optimum of (SA), an effective genetic-simulated annealing algorithm (SAGA) algorithm is suggested for solving the blocking flow shop scheduling problem with makespan criterion. The rest of this paper is organized as follows. In Section 2, the description of Fm|blocking|Cmax is presented. Section 3 outlines the proposed SAGA algorithm. In section 4, algorithms parameters selection is discussed. Section 6 provides the experimental results. Finally, section 7 reports conclusion and future research areas.

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1. Formulation of the Blocking Flowshop Scheduling Problem In the 𝐹𝐹𝑚𝑚 |𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏|𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 problem, a set (𝑗𝑗1 , … , 𝑗𝑗𝑛𝑛 ) of independent n jobs have to be performed sequentially on a set (𝑀𝑀1 , … , 𝑀𝑀𝑚𝑚 ) of m machines in the same order without any intermediate buffers. Each job is composed of (𝑂𝑂𝑗𝑗1 , … , 𝑂𝑂𝑗𝑗𝑗𝑗 ) operations 𝑂𝑂𝑗𝑗𝑗𝑗 , where, each operation has a positive integer processing time. It is assumed that, the setup times are included in the processing times of operations. All jobs are ready for processing at the time zero. Each machine can process at most one job and each job can be processed on at most one machine. Further, Owing to the lack of the intermediate buffers between machines, jobs remains blocked until the next downstream machine is free. That is, the current job on the machine must be blocked after finishing its operation until the next machine is available for processing. The target is to search a sequence of jobs under the aforementioned restrictions for which the makespan is minimized. Let Π= (π1,π2,π3,…,πn) be a set of job permutations, where, π(j) represents the index of the job arranged at the j-th position of π. The Dπ(1),k represents the departure time of job π(1) in machine k. In accordance with [24], the departure time Dπ(j),k of each job in each machine can be calculated using the following recursive expressions: 𝑆𝑆π(𝑗𝑗),0 = 0 (1) Dπ(1),k = Dπ(1),k−1 + pπ(1),k

Sπ(j),0 = Dπ(j)−1,1

k = 1,2, . +. . , m − 1

j = 2,3, … , n

Dπ(j),k = max�Dπ(j),k−1 + pπ(j),k , Dπ(j)−1,k+1 � 𝑘𝑘 = 1, … , m − 1 j = 2, … , n Dπ(j),m = Dπ(j),m−1 + pπ(j),m

j = 1,2, . . . , n

(2) (3)

(4) (5)

In the recursive on top, pπ(j),k represents the processing time of the job π(j) in the sequence. 𝑆𝑆π(𝑗𝑗),0 denotes the start time Sπ(j),0 of the job π(j) on the first machine. Equation (2) calculates the departure time of the first job on machine k. Equation (4) calculates the departure time of the job π(j) on machine k, and Equation (5) computes the departure time of the job π(j) on machine m. According to [24], the makespan value of π can be then calculated as: Cmax (π) = Dπ(n),m

(6)

Hence, the aim of the 𝐹𝐹𝑚𝑚 |𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏|𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 problem is to find the best sequence π∗ in the set of all permutations Π, so that the completion time of the last job in the permutation is minimized, thus: Cmax (π∗ ) ≤ Cmax (π) π∗ , π 𝜖𝜖 Π

(7)

2. The Proposed SAGA for Solving the BFSP 2.1 The basic genetic algorithm

Genetic algorithms are stochastic search method that are operative to a large gamut of combinatorial optimization problems. GA have been inspired from the biological mechanism of reproduction and natural selection. They are founded on the durability of the fittest insight, yielding to find efficiently a new generation with superior fitness function. In scheduling context, GA consider sequences of job as candidate solution or chromosomes. Figure below depicts an example of a chromosome with permutation encoding. The set of chromosomes generated at the start of the genetic search are referred to as an initial population, within each individual is evaluated by its corresponding value of makespan. Further, before reaching a termination criterion, in every new generation, a new population that retains the same size PS is formed using the fundamental elements of the genetic search procedure namely, selection, crossover and mutation operators.

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Figure 1. Example of chromosome with permutation encoding Selection operator is arguably one of the most important operator since it directs the evolution, guides the search and eliminates the feeble chromosomes. GA uses a selection to choice a pair of candidate solutions from the population based on their fitness. Selected individuals are used to create new offsprings with which a new generation is formed. Various selection operators exist to select the parents, that is, roulette wheel selection, tournament, fitness proportionate Selection, rank selection, and so one. In this study, the deterministic tournament selection is adopted for which the best individuals that possess the best fitness value are randomly selected for crossover. The crossover is a process of producing one or more offsprings from one-selected parents. Various tools exist to apply crossover operation. In this study, the two-points crossover is used, in which two-points are randomly selected for sharing the parents. The genes outside the selected points are inherited from the parents to the child. The residual genes will be moved to the second parent in the same instruction from left to right. As a result, two offsprings are generated as shown in Fig.2. The ratio of how many individuals will be chosen for mating is called the crossover probability Pc. However, the higher this probability, the more the population undergoes change.

Figure 2. Crossover operation Mutation is the third operator used in GA process, it is applied to preserve genetic diversity from one generation to the next, to prevent premature convergence and provide a process to escape from local optima. In this work, shift change mutation is used, in this type of mutation, a job at one position is removed and implanted at another random position with a mutation probability Pm as described in Fig.3.

Figure 3. Mutation operation The termination criterion is when the program reaches an absolute number of generations without any improvement in the optimal solution. Algorithm 1 represents the more detailed pseudo-code of the genetic algorithms.

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Step 1: Initialize PS, Pc, Pm, max_iter, 𝛼𝛼 Step 2: Generate the initial population P (n) consisting of a random sequences of jobs Step 3: Calculate fitness value ‘makespan’ of each sequence in current population Step 4: select the best fitness g_oldbest.from the current population with the tournament selection approach Step 5: While number of maximum iterations has not been attained: For each sequence: 1 to PS: Select randomly two sequences from P (n) If random ≤ 𝑃𝑃𝑃𝑃 : Apply the two point’s crossover operator and produce two children If random ≤ 𝑃𝑃𝑃𝑃 : Apply the shift change mutation operator Copy the obtained offspring sequences to the new_population P (n+1) Evaluation of the New_population gi (P (n+1)) Step 6: If the stop criterion is satisfied, terminate de SAGA procedure and return the optimal value found, otherwise, go to the step 4

Algorithm 1. The pseudo-code of the GA approach 2.2 The basic simulated annealing Simulated annealing (SA) is a stochastic computational method for finding global optimum to a large combinatorial optimization problem. It is a Monte Carlo approach that simulates the process of slow cooling after heating studied in statistical mechanics systems. The concept of slow cooling is explained as slow diminishing in the probability of accepting worse solutions as the solution space is explored. An appropriate annealing schedule is defined to decrease regularly the control parameter of the optimization known as the temperature. Relying on the iterative progress, SA procedure allows escaping local minima, which are not global and explore globally the search space within a reasonable computing time. In the implementation of the process, a primary solution S is randomly generated, the move from the current solution S to a neighborhood solution 𝑆𝑆 ∗ depends on the probability P of the BoltzmanGibbs distribution of the objective function difference. Besides, this transition is accepted if the following condition is satisfied: 𝑃𝑃 = 𝑚𝑚𝑚𝑚𝑚𝑚 �1, 𝑒𝑒

−∆𝑓𝑓 𝑇𝑇

�≥Ω

(8)

Where, ∆𝑓𝑓 = 𝑓𝑓(𝑆𝑆) − 𝑓𝑓(𝑆𝑆 ∗ ) is the difference between the objective function of the two states, Ω is a randomly number generated in the interval [0, 1] and T is the current control parameter in the process. Starting from a high value of the temperature 𝑇𝑇0 , a search space is performed at each temperature for a certain number of iterance appointed as the markov chain length that depicts the number of moves performed for a fixed value of the temperature. In order to ensure Boltzmann annealing for finding the global minimum, the control parameter T have to decrease logarithmically with the time. Given that, 𝟎𝟎 < 𝛼𝛼 < 𝟏𝟏 is the cooling rate parameter, the annealing schedule is consequently defined as: 𝑇𝑇𝑛𝑛+1 = 𝛼𝛼 𝑇𝑇𝑛𝑛 (9) 2.3 The proposed SAGA algorithm

Genetic algorithms has substantiated to be an efficient and performing optimization approach in several combinatorial optimization problems. In spite of its strength to discover promptly the search space through the genetic operations of reproduction, mutation and crossover. The situation for which it could be stuck in a local optimum is still dissatisfying, in particular, concerning the amount of time absorbed to find an optimal solution. Thus, a more suitable strategy is the use of the integrated method of GA with other approach in which this fatal disadvantage will be avoided. The

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simulated annealing is also stochastic method that also search space by means of a series of iterations as mentioned above. Through the cooling process adopted, SA has a strong ability to avoid to be trapped in a local optimum by seeking the optimum around its initial solution, which tends to find local improvements efficiently. Aiming at combining the strength of the both strategies in one hand, and compensate the drawback of the genetic algorithm method in the other, this section introduces the suggested SAGA algorithm for solving the BFSP under study. Following the production of the initial population, a new set of children sequences are evolved through genetic operators of reproduction, crossover and mutation. Each individual is selected by the metropolis criterion of SA formulated in eq (8), given that, ∆𝑓𝑓 in the case of the BFSP represents the difference of makespan between the new population P(n+1) and the old one P(n), denoted as g_newbest and g_oldbest respectively. In other words, the proposed SAGA proceeds in two stages: in the first stage, a set of offsprings is developed by executing genetic implementation. Thereafter, the offsprings obtained are accepted to go into the next generation with use of the local selection of SA relying on the aforementioned probability that depends on the system Temperature T, such that the next generation contains solely the elements whose makespan value equals the highest value in terms of quality. The proposed in this paper admits the replacement of the old population if g_newbest is better than g_oldbest and in the contrary case with a Boltzmann probability superior to a random number generated between [0, 1]. This process is repeated iteratively until the maximum number of iterations is attained and the global optimal value of makespan is obtained. The specific steps of SAGA are illustrated in algorithm 2. Step 1: Initialize PS, Pc, Pm, max_iter, 𝛼𝛼, T=T0 Step 2: Generate the initial population P(n) consisting of a random sequences of jobs Step 3: Calculate fitness value ‘makespan’ of each sequence in current population Step 4: select the best fitness g_oldbest.from the current population Step 5: While number of maximum iterations has not been attained : For each sequence: 1 to PS: Select randomly two sequences from P (n) If random ≤ 𝑃𝑃𝑃𝑃 : Apply the two point’s crossover operator and produce two children If random ≤ 𝑃𝑃𝑃𝑃 : Apply the shift change mutation operator Copy the obtained offspring sequences to the new_population P (n+1) Evaluation of the New_population gi (P (n+1)) Shoot the best and the worst finesses: g _newbest = min [g] g _newworst= max [g] If g _newbest < g _oldbest : Provisory-population = New-population [g_newbest,: ] Else If g _newbest == g _oldbest : Go to the next iteration Else If g_newbest > g _oldbest : Calculate X= exp [-(g _newbest - g _oldbest ) / T] If X > Ω : Provisory_population=New_population [g_newbest , : ] New_population [: , g_ newworst ] = Provisory_population P (n+1) = New_population Reduce the temperature by setting T = 𝛼𝛼 *T Step 6: If the stop criterion is satisfied, terminate de SAGA procedure and return the optimal value found, otherwise, go to the step 5

Algorithm 2. The pseudo-code of the SAGA approach

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3. Taguchi Experimental Design The forcefulness and the efficacy of optimization approaches substantially depends on the adequate selection of parameters. In fact, concerning the genetic algorithms method, four parameters including population size, maximum number of iterations, crossover probability and mutation probability have to be neatened as design factors. Furthermore, to handle correctly the proposed SAGA strategy, two other factors are added, namely, the cool rate and the temperature parameter. As a result, to select the best combination of the afore-mentioned parameters, the Taguchi method is used in this section. The overall aim of this method is to explore the factors that influence the mean and the variance of the process performance. Based on the design of experiment, Taguchi method minimizes the impact of noise factor that cannot be controlled by designers and find the best level of the powerful controllable factors. To measure the amount of variation existing in the response variable, an indicator of variation is used (signal-to-noise ratio (S/N)). Usually, there are three classes of the performance characteristics to investigate the S/N ratio, nominal-the-best, larger-the-better, and smaller-the-better. As the present study revolves around a minimization problem, the smaller the fitness value the better. Thus, the S/N ration have to be measured using the smaller-the-better, as stated below: 𝑺𝑺

𝑵𝑵

1

= −10 log10 � ∑𝑛𝑛𝑖𝑖=1 𝑛𝑛

1

𝑦𝑦𝑖𝑖2

(10)

�

Where, 𝑛𝑛 and y represent the number of observations and the response variable, respectively. To direct parameter design, three levels have been attributed to each factor as illustrated in table (1-2). Moreover, relying on the orthogonal array distribution, 27 sets of experiment have been executed. Figures (4-5) show the response graphs obtained for the SAGA and the GA approaches. With respect to the SAGA method, the corresponding response graph indicates that the best combination parameter, which determines a robust parameter design for minimizing the makespan, requires a population size of 400, an iteration number of 600, a mutation probability of 0.1, a crossover probability of 0.9, a cool rate of 0.5 and a temperature of 700. Whereas, regarding the calibration of the classical genetic algorithm GA optimization approach, the four above-mentioned considered factors have to be adjusted such as : population size = 400, iteration number =400, mutation probability= 0.1 and the crossover probability =0.4. Table 3 gathers the selected level of each factor for the two-optimization tools. Table 1. Level range design of each factor of genetic algorithms Factors Level 1 Level 2 Level 3

Population size 100 200 400

Iteration number 200 400 600

Crossover probability 40 % 60 % 90 %

Mutation probability 10 % 40 % 70 %

Table 2. Level range design of each factor of simulated annealing genetic algorithm Factors

Population size

Level 1 Level 2 Level 3

100 200 400

Iteration number 200 400 600

Crossover probability 40 % 60 % 90 %

Mutation probability 10 % 40 % 70 %

cool rate

Temperature

0.3 0.5 0.7

500 700 1000

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Figure 4. Mean effect plot for SN ratio of GA parameters

Figure 5. Mean effect plot for SN ratio of SAGA parameters Table 3. Optimal parameters of the proposed optimization approaches Method SAGA GA

Population size 400 400

Iteration number 600 400

Crossover probability 90 % 40 %

Mutation probability 10 % 10 %

cool rate 0.5 -

Temperature 500 -

4. Computational Experiments 4.1 Test problem To evaluate the relative performance of the proposed algorithms, computational test were performed on the commonly known of Taillard’s benchmark suite [25]. The benchmark consists of 120 flowshop problems of different sizes, ranging from 20 jobs and 5 machines to 500 jobs and 20 machines, and each problem includes 10 instances. The proposed SAGA and GA approaches are written in python 3.5 and implemented in a computer microprocessor core i3, with 4,0 GB of RAM memory. The proposed algorithms are compared to other forcful existing metaheuristics in the literature, namely, IABC( han et al. 2012) [17]), DE-ABC( han et al. 2015) [18], IG( Ribas et al.2011) (16], and MFFO ( han et al. 2016) [20]. For each instance, every algorithm is run five independent replications, to assess the performance of each algorithm, a measure of quality denoted as ARDP is adopted. This measure computes the average percentage relative difference between the best makespan retained after the run of each method with the referenced one obtained by the branch and bound of (Ronconi, 2005). The ARDP of each algorithm is formulated as follows: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑖𝑖 =

1

50

𝟓𝟓 ∑10 𝑙𝑙=1 ∑𝒌𝒌=𝟏𝟏

𝒊𝒊 𝑪𝑪𝑹𝑹 𝒍𝒍 −𝑪𝑪𝒍𝒍,𝒌𝒌

𝑪𝑪𝑹𝑹 𝒍𝒍

∗ 100%

(11)

𝑖𝑖 Where, 𝐶𝐶𝑙𝑙,𝑘𝑘 and 𝐶𝐶𝑙𝑙𝑅𝑅 denote the makespan of the lth instance provided by the ith algorithm in the kth run and the referenced one got by Ronconi, respectively. Evidently, the greater the value of 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑖𝑖 , the best the result provided by the algorithm.

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4.2 Comparison of stat-of-the art algorithms Table 4 and figure 5 depict the comparisons results of ARDP obtained by IABC, DE-ABC, IG, MFFO, GA and SAGA algorithms taking into account the same experimental design. Concerning genetic algorithms method, it can be seen from Table 4 and figure 5 that for instances of 20*5, 20*20, 50*5 and 50*10, the GA is lightly superior to the compared algorithms, for the instance,20*10, the total average of ARDP obtained by GA is equal to that obtained by IABC algorithm. Conversely, for the remaining instances it is inferior to the five compared instances. As follows from the table and the figure shown below (4,6) the total average of ARDP, 4, 45 obtained by the proposed SAGA algorithm is larger than the respective values of 4.10, 4.24, 4.23, 4.38 and 4.26 yielded by IABC, DE-ABC, IG, MMFO and GA, respectively. It is clear that the proposed SAGA algorithm exceed the IABC, DE-ABC, IG, MFFP and GA algorithms through most of the 12 blocking flowshop scheduling problem sizes. To profoundly asses the effectiveness of the proposed SAGA algorithm, the convergence curves of different algorithms are analyzed as shown in figure 7. To outlines the convergence curves of the best makespan values produced by IG, MMFO, GA and SAGA algorithms for the selected instance Ta109 of Taillard’s benchmarks suite. It can be observed from the figure that, as the computation time increases, the convergence curve of the proposed SAGA achieves the undermost level compared to the state-of-the-art algorithms. In the final analysis, the computational results demonstrate the efficiency and the performance of the proposed SAGA approach for solving the 𝐹𝐹𝑚𝑚 |𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏|𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 problem. The principal reason for which the SAGA can be more performant than the traditional GA and the other powerful metaheuristic may be owing to the combination of the benefits of GA in the exploration ability and the cooling process of SA algorithm in escaping the local optimum in which, the GA may be trapped. 4.3 New upper bounds To get more perspicacity concerning the performance of the proposed SAGA algorithm, its conduct was investigated on the 120 instances introduced by Taillard for solving the blocking flowshop scheduling problem. Table (5-6) summarize the makespan values yielded by B&B developed by Ronconi, MMFO presented by Han2016 and those obtained by the proposed GA and SAGA algorithms. It can be clearly noted that in most of the Taillard instances presented above, the suggested SAGA provides better upper bounds (highlighted in bold) than those of all the other compared algorithms. 4.5

ARDP

4.4 4.3 4.2 4.1 4 3.9

IABC

IG

DE-ABC

MFFO

GA

SAGA

Methods

Figure 6. Benchmark results of the total avearge ARPD indicator

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Table 4. Comparison results of the ARPD measure

Makespan

Problems Instance 20*5 20*10 20*20 50*5 50*10 50*20 100*5 100*10 100*20 200*10 200*20 500*20 Average 16000 15800 15600 15400 15200 15000 14800 14600

IABC 0.41 2.38 3.30 4.68 6.15 6.29 2.31 5.97 5.39 4.14 4.46 3.7 4.10

DE-ABC 0.34 2.34 3.29 4.89 6.25 6.45 2.67 6.27 5.53 4.37 4.7 3.72 4.24

IG 0.43 2.37 3.30 4.91 6.33 6.38 2.89 6.10 5.48 4.25 4.7 3.7 4.23

MFFO 0.36 2.38 3.28 4.94 6.23 6.33 3.27 6.63 5.79 4.68 4.88 3.77 4.38

GA 0.4 2.39 3.30 4.96 6.26 6.3 3.01 6.07 5.61 4.57 4.7 3.6 4.26

SAGA 0.48 2.41 3.31 4.94 6.42 6.54 3.26 6.71 5.83 4.81 4.97 3.72 4.45

MMFO IG SAGA GA

0

20

40

60

Generation

80

100

Figure 7. Convergence curves of instance Ta109 Table 5. Comparison of best solutions for Taillard’s benchmark (new solutions are in bold) Instance Ta001 Ta002 Ta003 Ta004 Ta005 Ta006 Ta007 Ta008 Ta009 Ta010 Ta011 Ta012 Ta013 Ta014 Ta015 Ta016 Ta017 Ta018 Ta019 Ta020 Ta021 Ta022 Ta023 Ta024 Ta025 Ta026 Ta027 Ta028 Ta029 Ta030

B&B 1384 1411 1294 1448 1366 1363 1381 1384 1378 1283 1736 1897 1677 1622 1658 1640 1634 1741 1777 1847 2530 2297 2560 2399 2538 2467 2502 2411 2421 2407

MMFO 1374 1408 1280 1448 1341 1363 1381 1379 1373 1283 1698 1833 1659 1535 1617 1590 1622 1731 1749 1782 2436 2234 2479 2348 2435 2383 2390 2328 2363 2323

SAGA 1374 1408 1280 1448 1341 1363 1381 1379 1373 1283 1698 1833 1659 1535 1617 1590 1622 1731 1749 1782 2436 2234 2465 2348 2435 2372 2390 2328 2363 2323

GA 1374 1408 1280 1448 1341 1363 1381 1379 1373 1283 1698 1833 1659 1535 1617 1590 1622 1731 1749 1782 2436 2234 2479 2348 2435 2383 2390 2328 2363 2323

Instance Ta031 Ta032 Ta033 Ta034 Ta035 Ta036 Ta037 Ta038 Ta039 Ta040 Ta041 Ta042 Ta043 Ta044 Ta045 Ta046 Ta047 Ta048 Ta049 Ta050 Ta051 Ta052 Ta053 Ta054 Ta055 Ta056 Ta057 Ta058 Ta059 Ta060

B&B 3151 3395 3184 3303 3272 3400 3228 2360 3104 3264 3913 3798 3723 3885 3934 3831 3957 3774 3784 3928 4886 4668 4666 4650 4475 4521 4576 4688 4532 4846

MMFO 2997 3189 3007 3120 3159 3161 3013 3054 2908 3116 3638 3487 3481 6665 3628 3611 3681 3563 3532 3624 4500 4276 4266 4344 4268 4280 4308 4310 4310 4415

SAGA 2989 3192 2998 3117 3158 3156 3013 3050 2897 3094 3624 3475 3480 3666 3620 3607 3671 3562 3529 3619 4496 4276 4257 4338 4261 4275 4305 4310 4308 4415

GA 3005 3207 3014 3127 3169 3178 3115 3052 2908 3117 3642 3497 3491 3670 6332 3615 3685 3569 3531 3623 4503 4278 4274 4365 4269 4282 4308 4321 4322 4417

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Table 6. Comparison of best solutions for Taillard’s benchmark (new solutions are in bold) Instance

B&B

MMFO

SAGA

GA

Instance

B&B

MMFO

SAGA

GA

Ta061

6455

6130

6135

6141

Ta091

14113

13384

13357

13424

Ta062

6214

5992

5985

6008

Ta092

14127

13294

13280

13311

Ta063

6124

5913

5910

5921

Ta093

14416

13416

13416

13463

Ta064

5976

5710

5702

5723

Ta094

14435

13337

13335

13344

Ta065

6173

5938

5931

5949

Ta095

14113

13340

13311

13354

Ta066

6094

5808

5799

5811

Ta096

13309

13081

13026

13085

Ta067

6262

5959

5941

6002

Ta097

14563

13561

13571

13607

Ta068

6061

5865

5862

5881

Ta098

14329

13504

13502

13644

Ta069

6474

6106

6093

6102

Ta099

13923

13217

13201

13308

Ta070

6366

6112

6105

6141

Ta100

14435

13400

13389

13439

Ta071

7496

6988

6971

6998

Ta101

15579

14192

14889

14908

Ta072

7281

6714

6709

6729

Ta102

15728

14963

14952

14962

Ta073

7400

6834

6820

6892

Ta103

15915

15043

15026

15107

Ta074

7670

7102

7989

7125

Ta104

16039

14949

14930

14987

Ta075

7317

6804

6799

6935

Ta105

15938

14857

14857

14899

Ta076

7301

6615

6607

6639

Ta106

15911

14909

14925

14929

Ta077

7247

6770

6770

6811

Ta107

15898

14980

14972

15009

Ta078

7315

6806

6801

6859

Ta108

16022

15005

15000

15021

Ta079

7631

6997

6964

7055

Ta109

15817

14834

14782

15013

Ta080

7411

6910

6895

6928

Ta110

15969

14954

14921

14954

Ta081

8347

7774

7751

7799

Ta111

38334

35838

36504

36609

Ta082

8372

7838

7883

7892

Ta112

38642

35970

35946

36108

Ta083

8265

7777

7769

7802

Ta113

38163

35757

35615

36778

Ta084

8365

7818

7803

7836

Ta114

38625

36070

36045

36628

Ta085

304

7799

7769

7836

Ta115

38492

35869

35762

35869

Ta086

8450

7841

7870

7851

Ta116

38551

36099

36099

37006

Ta087

8507

7929

7973

7979

Ta117

38179

35760

35760

36805

Ta088

8584

7976

7975

7993

Ta118

38664

35921

35865

36792

Ta089

8341

7894

7832

7889

Ta119

38339

35739

35695

36885

Ta090

8489

7929

7906

7918

Ta120

38540

36120

35971

36184

5. Conclusions This paper suggests SAGA approach for solving the blocking flowshop scheduling problem (BFSP) with the maximum completion time criterion. The proposed algorithm combines the features of GA and the annealing mechanism of SA algorithm. The efficiency of the algorithm is evaluated on the commonly know 120 instances presented by Taillard. Compared to the results found by IABC, IG, DE-ABC, and GA, the computational results show the high-performance of the proposed SAGA approach for solving the𝐹𝐹𝑚𝑚 |𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏|𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 problem. Moreover, the results illustrates that for 82 out of 120 instances, SAGA has provided best upper bounds than MFFO and B&B methods.

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From the outcome of our investigation, it is possible to conclude that, due to the reasonable incorporation, the developed SAGA has a great potential for solving the BFSP, since it avails from the exploration ability of GA and The power to escape the local optimum of SA. As future direction, we intend to extend our research to solve the blocking flowshop scheduling problem with several criteria, as a single criterion is regarded as deficient for real-life and practical applications. Acknowledgements The authors would like to thank the editors and anonymous reviewers for their useful comments and their constructive suggestions, which helped to improve this paper in order to make it appropriate for publication in International Journal of Engineering Research in Africa. References [1]. Grabowski J, Skubalska E, Smutnicki C. On flowshop scheduling with release and due dates to minimize maximum lateness. Journal of Operational Research Society, 34 (1983) 615–20. [2]. Graham RL, Lawler EL, Lenstra JK, Rinnooy Kan AHG. Optimization and approximation in deterministic sequencing and scheduling: a survey. Annals of Discrete Mathematics, 5 (1979) 287–362. [3]. Hall NG, Sriskandarajah CA survey of machine scheduling problems with blocking and nowait in process. Operations Research, 44 (1996) 510–25. [4]. Papadimitriou CH, Kanellakis PC. Flowshop scheduling with limited temporary storage”. Journal of the Association for Computing Machinery, 27 (1980) 533–49. [5]. Allahverdi, A., C. T. Ng, T. E. Cheng, and M. Y. Kovalyov. A Survey of Scheduling Problems with Setup Times or Costs”. European Journal of Operational Research, 187(3) (2008) 985–1032. [6]. McCormick, S. T., M. L. Pinedo, S. Shenker, and B. Wolf. Sequencing in an Assembly Line with Blocking to Minimize Cycle Time. Operations Research, 37 (6) (1989) 925–935. [7]. Leisten, R. Flowshop sequencing problems with limited buffer storage, International Journal of Production Research, 28 (1990)2085–100. [8]. Nawaz, M., Enscore, E.E.J. and Ham, I. A heuristic algorithm for the m-machine, n-job flowshop sequencing problem, OMEGA – International Journal of Management Science, 11(1983) 91–95. [9]. Ronconi, D. P. A note on constructive heuristics for the flowshop problem with blocking. International Journal of Production Economics, 87 (1) (2004) 39-48. [10]. Pan, Q. K., and L. Wang. Effective Heuristics for the Blocking Flowshop Scheduling Problem with Makespan Minimization. Omega, 40 (2) (2012)218–229. [11]. Caraffa, V., S. Ianes, T. P. Bagchi, and C. Sriskandarajah. Minimizing Makespan in a Blocking Flowshop using Genetic Algorithms. International Journal of Production Economics, 70 (2) (2001)101–115. [12]. Abadi INK, Hall NG. Sriskandarajh C. Minimizing cycle time in a blocking flowshop. Operations Research, 48(2000)177–80. [13]. Ronconi, D.P. A branch-and-bound algorithm to minimize the makespan in a flowshop problem with blocking, Annals of Operations Research, 138 (1) (2005) 53–56. [14]. Liu JY, Reeves CR. Constructive and composite heuristic solutions to the P//Sci scheduling problem. European Journal of Operational Research, 132 (2) (2001) 439–52.

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