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A hybrid optimization algorithm with genetic and bacterial operators for the design of cellular manufacturing systems
9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC Conference on Manufacturing Modelling, Management and Control 9th IFAC IFAC Conference on Manufacturing Modelling, Modelling, Management Management and and 9th Available online at www.sciencedirect.com Control Conference on Manufacturing 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Control Control Berlin, Germany, August 28-30, 2019 Control Berlin, Berlin, Germany, Germany, August August 28-30, 28-30, 2019 2019 Berlin, Germany, August 28-30, 2019
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IFAC PapersOnLine 52-13 (2019) 1409–1414
A hybrid optimization algorithm with A hybrid optimization algorithm with A hybrid optimization algorithm with genetic and bacterial operators for the A hybrid optimization algorithm with genetic and bacterial operators for genetic and bacterial operators for the the design of cellular manufacturing genetic and bacterial operators systems for the design of cellular manufacturing systems design of manufacturing design Camilo of cellular cellular manufacturing systems systems Mej´ıa-Moncayo ∗∗ Olga Battaia ∗∗ ∗∗ Camilo Mej´ıa-Moncayo ∗ Olga Battaia ∗∗ ∗∗ Camilo Camilo Mej´ Mej´ıa-Moncayo ıa-Moncayo ∗∗ Olga Olga Battaia Battaia ∗∗ ıa-Moncayo Olga Battaia ∗Camilo Mej´ a , Colombia (e-mail: ∗ Universidad EAN, Bogot´ EAN, a ∗ ∗ Universidad Universidad EAN, Bogot´ Bogot´ a,,, Colombia Colombia (e-mail: (e-mail: [email protected]) Universidad EAN, Bogot´ a Colombia (e-mail: ∗ ∗∗ [email protected]) Universidad EAN,Toulouse, Bogot´ a, Colombia (e-mail: [email protected]) France (e-mail: [email protected]) ∗∗ ISAE-Supaero, Toulouse, France (e-mail: ∗∗ [email protected]) ∗∗ ISAE-Supaero, ISAE-Supaero, Toulouse, France (e-mail: [email protected] ) Toulouse, France )(e-mail: ∗∗ ISAE-Supaero, [email protected]
ISAE-Supaero, Toulouse, France ))(e-mail: [email protected] [email protected] [email protected] ) Abstract: We We consider aa design design problem problem for for cellular cellular manufacturing systems. systems. In In order to to improve Abstract: Abstract: Weofconsider consider a design design problem foroptimization cellular manufacturing manufacturing systems. In order order to improve improve the efficiency the designed system, three problems are considered jointly in the the Abstract: We consider a problem for cellular manufacturing systems. In order to improve the efficiency the designed system, three problems are considered jointly in Abstract: Weof consider a design problem foroptimization cellular manufacturing systems. Inlayout. order toAimprove the efficiency of the designed system, three optimization problems are considered jointly in the same optimization procedure: cell formation, workload balancing and cell hybrid the efficiency of the designed system, three optimization problems are considered jointly in the same optimization cell formation, workload balancing and cell layout. A hybrid the efficiency ofalgorithm the procedure: designed three problems considered in the the same optimization procedure: cell workload balancing and cell A metaheuristic withsystem, genetic and optimization bacterial operators is are developed to jointly optimize same optimization procedure: cell formation, formation, workload balancing and cell layout. layout. A hybrid hybrid metaheuristic algorithm with genetic and bacterial operators is developed to optimize the same optimization procedure: cell formation, workload balancing and cell layout. A hybrid metaheuristic algorithm with genetic and bacterial operators is developed to optimize the performances of the designed cellular manufacturing system. The results obtained on benchmark metaheuristic algorithm withcellular genetic and bacterialsystem. operators is developed to on optimize the performances of the manufacturing The obtained benchmark metaheuristic withcellular genetic and bacterialsystem. operators is developed to on optimize the performances ofalgorithm the designed designed cellular manufacturing system. The results results obtained on benchmark problems are are discussed. discussed. performances of the designed manufacturing The results obtained benchmark problems performances of the designed cellular manufacturing system. The results obtained on benchmark problems are discussed. problems are discussed. © 2019, IFAC problems are (International discussed. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Cellular Cellular manufacturing; manufacturing; Cell Cell formation; formation; Workload Workload balancing; balancing; Facility Facility layout; layout; Keywords: Keywords: Cellular manufacturing; manufacturing; Cell formation; formation; Workload balancing; balancing; Facility Facility layout; layout; Genetic Algorithm; Bacterial chemotaxis; Hybrid algorithm Keywords: Cellular Cell Workload Genetic Bacterial Hybrid Keywords: Cellular manufacturing; Cell formation; Workload balancing; Facility layout; Genetic Algorithm; Algorithm; Bacterial chemotaxis; chemotaxis; Hybrid algorithm algorithm Genetic Algorithm; Bacterial chemotaxis; Hybrid algorithm Genetic Algorithm; Bacterial chemotaxis; Hybrid algorithm 1. INTRODUCTION INTRODUCTION The advantages advantages of of balanced balanced production production systems systems were were exex1. The 1. The advantages of balanced production systems were extensively discussed as well in the literature. Among them, 1. INTRODUCTION INTRODUCTION The advantages of balanced production systems were extensively discussed as well in the literature. Among them, 1. INTRODUCTION The advantages of balanced production systems extensively discussed as the Among them, improved productivity andin better resources usagewere can be tensively discussed as well well inbetter the literature. literature. Among them, improved productivity and resources usage can be tensively discussed as well in the literature. Among them, improved productivity and better resources usage can be cited. A taxonomy of problems and solution methods for improved productivity and better resources usage can be Designing an efficient manufacturing system is a hard cited. A taxonomy of problems and solution methods for Designing an efficient manufacturing system is aa hard improved productivity and better resources can be A of and solution methods for balancing production lines is introduced introduced by usage Batta¨ ıa and and cited. A taxonomy taxonomy of problems problems and solution methods for Designing an efficient manufacturing is task involving involving the consideration of many manysystem different factors. Designing an the efficient manufacturing system is factors. a hard hard cited. balancing production lines is by Batta¨ ıa task consideration of different cited. A taxonomy of problems and solution methods for production lines is by ıa Designing efficient manufacturing is factors. a hard balancing Dolgui (2013). (2013). Although such manufacturing organizabalancing production lines is introduced introduced by Batta¨ Batta¨ ıa and and task the consideration many different This involving task an becomes even more of complex with increasing increasing task involving the consideration of manysystem different factors. Dolgui Although such aaa manufacturing organizaThis task becomes even more complex with balancing production lines is introduced by Batta¨ ıamass and Dolgui (2013). Although such manufacturing organizatask involving the consideration of many different factors. tion suits well for mass production, its adaptation for Dolgui (2013). Although such a manufacturing organizaThis task becomes even more complex with increasing product customization. Product customization suits better This task becomes even more complex with increasing tion suits well for mass production, its adaptation for mass product customization. Product customization suits better Dolgui (2013). Although such a manufacturing organization suits well for mass production, its adaptation for mass This task becomes even more complex with increasing customized production systems as cells manufacturing, tion suits well for mass production, itscells adaptation for mass product customization. better customers’ demand but butProduct makes it itcustomization challenging to tosuits attain the customized product customization. Product customization suits better production systems as manufacturing, customers’ demand makes challenging attain the tion well foroptimization mass production, adaptation for mass customized production systems as product customization. Product customization suits better needssuits adapted procedures asmanufacturing, described by customized production systems asitscells cells manufacturing, customers’ demand but makes it challenging to attain the performances of mass production systems, in this context customers’ demand but makes it challenging to attain the needs adapted optimization procedures as described by performances of mass production systems, in this context customized production systems as cells manufacturing, needs adapted optimization procedures as described by customers’ demand but makes it challenging to attain the Nallusamy (2016). needs adapted optimization procedures as described by performances of mass production systems, in this context cellular manufacturing systems are generally an efficient performances of mass production systems, in this context Nallusamy (2016). cellular manufacturing systems are generally an efficient needs adapted optimization procedures as described by (2016). performances of mass production systems, in this context Nallusamy Nallusamy (2016). cellular manufacturing systems generally an alternative. cellular manufacturing systems are are generally an efficient efficient Furthermore, the facility facility layout layout problem problem aims aims in in defining defining alternative. Nallusamy (2016). cellular manufacturing systems are generally an efficient Furthermore, the alternative. alternative. Furthermore, the facility layout problem aims in defining the best position for each machine. This problem has Furthermore, the facility layout problem aims in defining Cellular manufacturing systems are usually used for the alternative. the best position for each machine. This problem has Cellular manufacturing systems are usually used for the Furthermore, the facility layout problem aims in defining the best position for each machine. This problem has attracted significant attention in the literature e.g. (Drira the best position for each machine. This problem has Cellular manufacturing systems are usually used for the batch production of part families. This production archiCellular manufacturing systems are usually used for the attracted significant attention in the literature e.g. (Drira batch production of part families. This production archithe position for each machine. This problem has significant attention in the literature Cellular manufacturing systems usually used for the attracted et al., al.,best 2006; Singh and and Sharma, 2006; Romero ete.g. al., (Drira 2015). attracted significant attention in the Romero literature e.g. (Drira batch of part families. This production architectureproduction facilitates the administration and control of the batch production of part families.are This production archiet 2006; Singh Sharma, 2006; et al., 2015). tecture facilitates the administration and control of the attracted significant attention in the literature e.g. (Drira et al., 2006; Singh and Sharma, 2006; Romero et al., 2015). batch production of part families. This production archiIts formulation is based on the available space, which et al., 2006; Singh and Sharma, 2006; Romero et al., 2015). tecture facilitates the administration and control of the production system the andadministration at the the same same time time makes itofmore more tecture facilitates and makes controlit the Its formulation based on the available space, which production system and at et Singhis 2006; etway. al., In 2015). formulation is on available which tecture facilitates the administration and makes controlit ofmore the Its canal., be2006; considered inbased aSharma, continuous orRomero discretespace, the Its formulation isand based on the the available space, which production system and at the same time flexible to respond to demand changes. production system and at the same time makes it more can be considered in a continuous or discrete way. In the flexible to respond demand Its formulation is in based on the available space, which be considered aa continuous or discrete way. In the production system to and at thechanges. same time makes it more can first case, the problem is modeled in the way to provide can be considered in continuous or discrete way. In the flexible to respond to demand changes. flexible to respond to demand changes. case, the problem is modeled in the way to provide can be considered a continuous or discrete In the the In order ordertoto torespond optimizetothe the performances of the the designed designed celcel- first first case, the is modeled in the way to provide flexible demand changes.of the coordinates coordinates of in each machine. In the second case, first case, the problem problem ismachine. modeledIn inthe thesecond wayway. tocase, provide In optimize performances the of each the In order to optimize the performances of the designed celfirst case, the problem is modeled in the way to provide lular manufacturing system with the objective to achieve In order to optimize the performances of the designed celthe coordinates of each machine. In the second case, the solution of the model gives the relative position of all the coordinates of each machine. In the second case, the lular manufacturing system with the objective to achieve of the model gives the relative position of all In order to optimize the performances of the products, designed cellular manufacturing system with the to the coordinates of each machine. In themost second case, the mass production efficiency for customized we solution lular manufacturing system withcustomized the objective objective to achieve achieve solution of the model gives the relative position of all machines in the cell. In both cases, the used way to solution of the model gives the relative position of all mass production efficiency for products, we the cell. In both cases, the most used way to lular manufacturing system withcustomized the objective to achieve mass production efficiency for products, we solution ofin the model gives the relative position of all propose to consider consider jointly three three optimization problems: mass production efficiency for customized products, we machines machines in the cell. In both cases, the most used way to evaluate the layout solution is the transport cost related machines in the cell. In both cases, the most used way to propose to jointly optimization problems: the layout the transport cost related mass production efficiency for customized we evaluate propose to jointly three optimization problems: machines intransfer. the cell.solution In both is the most used way to cell formation formation problem, workload balancingproducts, and facility propose to consider consider jointly three optimization problems: evaluate the layout solution is the cost to product product evaluate the layout solution iscases, the transport transport cost related related cell problem, workload balancing and facility to transfer. propose to consider jointly three optimization problems: cell formation problem, workload balancing and facility evaluate the layout solution is the transport cost related layout. cell formation problem, workload balancing and facility to product transfer. to product transfer. layout. cell formation problem, workload balancing and facility to In order order to improve improve the efficiency efficiency of of aa cellular cellular manufacturmanufacturlayout. product transfer.the layout. In to Cell formation is the optimization problem aiming in In order to improve the efficiency of a cellular manufacturlayout. ing system, we propose in this study to consider its layout layout In order to improve the efficiency of a cellular manufacturCell formation is the optimization problem aiming in ing system, we propose this study consider its Cell formation is the optimization problem aiming in In order to improve the in efficiency of aato cellular manufacturgrouping machines into manufacturing cells and parts into Cell formation is the optimization problem aiming in ing system, we propose in this study to consider its layout design with the objective to achieve balanced work distriing system, we propose in this study to consider its layout grouping machines manufacturing cells and parts into with the objective achieve aa to balanced work distriCell formation is into the optimization problem aiming in design grouping machines into manufacturing cells and parts ing system, we the propose into this study consider its layout families on the basis basis of their their similarities in the the design and grouping machines into manufacturing cells anddesign parts into into design with the objective to achieve balanced work distribution among machines of the different cells in which design with the objective to achieve a balanced work distrifamilies on the of similarities in and among machines of the different cells in which grouping machines manufacturing cells and parts into families on the basis of similarities in design and design with thethe objective to achieve a producing balanced work distrimanufacturing processes (Selim et al., al., 1998; 1998; Papaioannou families on the processes basisinto of their their similarities in the the design and bution bution among the machines of the different cells in which the system works like a line system batches of bution among the machines of the different cells in which manufacturing (Selim et Papaioannou system works like aa line system producing of families on the basis This of their similarities in the design and the manufacturing processes (Selim et al., 1998; Papaioannou bution among theachieve machines of the we different cellsbatches inseveral which and Wilson, 2010). optimization problem has been manufacturing processes (Selim et al., 1998; Papaioannou the system works like line system producing batches of part families. To this goal, solve jointly the system works like a line system producing batches of and Wilson, 2010). This optimization problem has been part families. To achieve this goal, we solve jointly several manufacturing processes (Selim et al., 1998; Papaioannou and Wilson, 2010). This optimization problem has the system works like aincluding line batches of extensively studied in the the literature. For example, example, the part and Wilson,studied 2010). This optimization problem has been been families. To this goal, we solve several optimization problems cell formation, workload part families. To achieve achieve this system goal,cell weproducing solve jointly jointly several extensively in literature. For the optimization problems including formation, workload and Wilson, 2010). This optimization problem has been extensively studied in the literature. For example, part families. To achieve this goal, we solve jointly several study of (Selim et al., 1998) provides an overview of the extensively studied in the literature. For example, optimization problems including cell formation, workload balancing and andproblems inter and and intra cells cells layout. We Weworkload develop optimization including cell formation, study of (Selim et al., 1998) provides an of the balancing inter intra layout. develop extensively studied indescriptive the literature. Foroverview example, study of et 1998) provides an overview of optimization problems including cellbacterial formation, workload following techniques: methods, cluster analystudy of (Selim (Selim et al., al., 1998) provides an overview of the the balancing and inter and intra cells layout. We develop a new hybrid algorithm based on and genetic balancing and inter and intra cells layout. We develop following techniques: descriptive methods, cluster analynew hybrid algorithm on bacterial and genetic study of (Selim et al.,descriptive 1998) provides an overview of and the aabalancing following techniques: methods, cluster analyand inter and based intra cells layout. We develop sis, graph partitioning, mathematical programming following techniques: descriptive methods, cluster analynew hybrid algorithm based on bacterial and genetic algorithms for the defined optimization problem. a new hybrid algorithm based on bacterial and genetic sis, graph partitioning, mathematical programming and for the defined optimization problem. following techniques: descriptive methods, cluster analysis, graph partitioning, mathematical programming and aalgorithms new hybrid algorithm based on bacterial and genetic artificial intelligence. A taxonomy of coefficient coefficient methods sis, graphintelligence. partitioning, mathematical programming and algorithms algorithms for the defined optimization problem. for the defined optimization problem. artificial A taxonomy of methods sis, graphintelligence. partitioning, mathematical programming and The paper paper for is structured structured as follows. The The proposed algoalgoartificial A taxonomy of methods algorithms the definedas optimization problem. is proposed proposed by Yin Yin and and Yasuda (2006). Papaioannou and The artificial intelligence. A Yasuda taxonomy of coefficient coefficient methods is follows. proposed is by (2006). Papaioannou and The paper is structured as follows. The proposed algoartificial intelligence. A taxonomy of coefficient methods rithms are presented in Sections 2-4 and include a genetic The paper is structured as follows. The proposed algois proposed by Yin and Yasuda (2006). Papaioannou and Wilson (2010) discuss inYasuda particular heuristic and metametais proposed by Yin andin (2006). Papaioannou and rithms are presented in Sections 2-4 and a Wilson (2010) discuss particular heuristic and The paper is structured as follows. Theinclude proposed algoare presented 2-4 and include aa genetic genetic is proposed by Yin and Yasuda (2006). Papaioannou and rithms algorithm (Section 2),in discrete bacterial chemotaxis oprithms are (Section presented inaaSections Sections 2-4 and include genetic Wilson (2010) discuss in particular heuristic and metaheuristics algorithms developed for cell formation. Wilson (2010) discuss in particular heuristic and metaalgorithm 2), discrete bacterial chemotaxis opheuristics algorithms developed for cell formation. rithms are presented in Sections 2-4 and include a genetic algorithm (Section 2), a discrete bacterial chemotaxis Wilson (2010) discuss in particular heuristic and metaalgorithm (Section 2), a discrete bacterial chemotaxis opopheuristics heuristics algorithms algorithms developed developed for for cell cell formation. formation. algorithm (Section 2), a discrete bacterial chemotaxis opheuristics algorithms developed for cell formation.
2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 1426Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2019 IFAC 1426Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 1426 Copyright © 2019 IFAC 1426 10.1016/j.ifacol.2019.11.396 Copyright © 2019 IFAC 1426
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timization algorithm (Section 3) and a hybrid algorithm based on previous two (Section 4). The performances of algorithms on benchmark problems are discussed in Section 5. Finally, Section 6 presents the conclusions of this study and future research directions. 2. GENETIC ALGORITHMS Genetic algorithms, introduced by Holland (1975), are search algorithms based on the mechanics of natural selection and genetics. The GAs have been applied in various fields such as mathematics, engineering, biology, and social sciences Goldberg (1989). GA combine the concept of survival of the fittest with a structure of random information exchange to form a robust search algorithm. GA start with generating an initial population of possible solutions (individuals or chromosomes), and then evaluate the fitness of each individual in the population and select some of them to be ”parents”. Such parents are involved in a reproduction procedure through a crossover operator. The reproduction is the way of information sharing among individuals. Additionally, GA use a mutation operator to randomly alter the genetic composition of offspring with the aim to improve the diversity of the population and better explore the search space. The processes of evaluation, selection, reproduction, and mutation are repeated until a termination criterion is satisfied as described in Algorithm 1. However, GA converge often to a local optimum after a certain number of generations. This is caused by a low diversity in the populations or by the incapacity of the mutation process to avoid local optima.
in order to replace death bacteria and create new organisms which can be dispersed to increase the probability of survival of the species. Algorithm 2 DBCOA Generation of initial bacteria Evaluation of objective function for t = 1 to N ed do for u = 1 to N re do for v = 1 to N c do Chemotaxis end for Reproduction end for Elimination and Dispersion end for In bacterial algorithms, each artificial bacterium needs to search optimal value through the bacteria chemotaxis and realizes the swarming process to exploit sources of food, reproduction by duplication produces copies of fittest individuals. Elimination and dispersion processes have been defined to avoiding falling into premature convergence by a local optimal, creating new bacteria dispersed or located in other positions different to the originals which are replaced. Chemotaxis is the main process of this contribution since it allows exploring the search space in an efficient way. However, DBCOA lacks a procedure for sharing information among solutions. In order to fix this drawback, we propose to hybridize it with GA taking the advantages of two approaches. 4. DISCRETE GENETIC BACTERIAL ALGORITHM - DGBA
Algorithm 1 Genetic Algorithm Generation of initial population for i = 1 to M ax − generation do Fitness function evaluation Selection of parents Crossover Mutation end for
Many natural processes have been emulated in optimization procedures like Genetic Algorithms proposed by Holland Holland (1975) and explained in section 2. Nevertheless, the performances of GA can be improved due to the integration of an exploration process. At the same time, DBCOA (Mej´ıa-Moncayo et al. (2018) ) has an efficient exploration process, however, it cannot share information among individuals.
3. DISCRETE BACTERIAL CHEMOTAXIS OPTIMIZATION ALGORITHM Discrete Bacterial Chemotaxis Optimization Algorithm - DBCOA, is an optimization algorithm introduced by Mej´ıa-Moncayo et al. (2018) for the solution of cellular manufacturing layout. It is a discrete optimization algorithm based on Bacterial Foraging Optimization Algorithm (Passino (2002)) with which shares the structure, which by mean of a randomly hierarchical chemotaxis process emulates bacterial exploration in a discrete search space. Bacterial Foraging Optimization Algorithm (BFOA) proposed by Passino (2002) is a technique which models the food-seeking and reproductive behavior of common bacteria such as E. Coli in order to solve optimization problems in continuous search spaces. In nature, bacteria move in their environment by mean of flagella movements. Furthermore, bacteria use duplication for their reproduction
The idea of this paper is to use the symbiosis of GA and DBCOA in order to obtain an efficient solution procedure to address the problem of designing cellular manufacturing systems. Therefore, GA and DBCOA are used to create a hybrid metaheuristic named Discrete Genetic Bacterial Algorithm - DGBA. This is done in order to better explore the advantages of both methods. Thus, a chemotaxis process of DBCOA is followed by GA parents selection (binary tournament) for GA reproduction by crossing (OX order crossover explained in Gen and Cheng (2000)) and at the end, the elimination and dispersion procedures of DBCOA are used. The whole procedure is summarized in Algorithm 3. The used operators are explained here below. 4.1 Solution Coding The codification is the way by which variables that represent individuals and populations are encoded into
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l to Cell1 , in this example the fourth element of θmp . This gives the definition of the machines Cell2 = [2, 4]. In a similar way, we proceed with the parts permutation and limits of families vectors to fully define Family1 and Family2 . Finally, the quantity of machines is defined by l θqm = [2, 3, 1, 2, 1].
Algorithm 3 DGBA Generation of initial population for i = 1 to M ax − generation do Fitness function evaluation Chemotaxis Selection of parents Crossover Elimination and dispersion end for
4.2 Chemotaxis
data structures. We selected a coding based on the approach presented in Gen et al. (2009), where each solution or bacterium θl is encoded by vectors θl = l l l l l ). The first and second vectors , θcn , θcl , θfl l , θqm (θmp , θpp l are respectively machines permutation θmp and parts perl mutation θpp , the third vector represents the cells’ number l l θcn , the fourth and fifth vectors are the cells’ limits θcl l and families’ limits θf l , the last one encodes the quantity l of machines θqm . For example, if we consider a cellular manufacturing system with five machines and seven parts, one possible solution for the cell manufacturing layout design problem (see figure 1), i.e. a bacterium θl encoded as shown below: l l l l l , θpp , θcn , θcl , θfl l , θqm ) θl = (θmp
θl = [(5, 3, 1, 2, 4), (6, 3, 7, 2, 4, 1, 5), (2), (3, 5, 0, 0, 0),(4, 7, 0, 0, 0), (2, 3, 1, 2, 1)]
The implemented Chemotaxis has a hierarchical structure that imitates the movements carried out by bacteria in their natural environment through swimming and tumbling via flagella. This process has the objective of exploring the search space to find optimal solutions and avoid local optima. Given the nature of the cell manufacturing layout, the movements of swim and tumble performed by bacteria in a continuous search space have to be discretized. We chose to implement hierarchical changes into the encoded solutions, by changing the vector-based representation through the swap, insertions, changes of the number of cells and of their limits, and the quantity of machines. The modified individuals are then compared with the original ones (in terms of their objective function value) and retained if a fitness improvement occurs. This process, summarized in Algorithm 4, is repeated until no more change is possible. In Algorithm 4, the following shorthand notation is used: l • SP M () is a function that swaps two elements of θmp l to produce θspm . l • SP P () is a function that swaps two elements of θpp l to produce θspp . l • IP M () is a function that inserts one element in θmp l to produce θipm . l • IP P () is a function that inserts one element in θpp to l produce θipp . l • N LM () is a function that modifies the limits of θlm l to produce θnlm . l • N LP () is a function that modifies the limits of θlp to l produce θnlp . • QM () is a function that increase or decrease the l l quantity of machines θqm to produce θnqm .
4.3 Reproduction Fig. 1. Schematic layout of codification example This solution defines the following cells and families: Cell1 = [5, 3, 1], Cell2 = [2, 4], Family1 = [6, 3, 7, 2] and Family2 = [4, 1, 5]. To get this information out of θl it is necessary to check l the number of cells or families θcn , which for this example l is 2. Next, we look at the first element of vector θcl of the limits of cells, in this case (3). This number defines the number of elements belonging to the first cell, starting l from the first element of vector θmp , which gives as result Cell1 = [5, 3, 1]. l As for Cell2 , we use the second element of θcl , which is (5). l This number establishes the position in vector θmp , of the last element to be assigned to Cell2 . The first element of Cell2 is the element adjacent to the last element assigned
The reproduction is the process by which the fittest solutions produce other individuals by crossover and replace the least healthy solutions, which eventually disappear from the population. Reproduction by OX order crossover is applied to permutation vectors, limits of cells and families are swapped between parents. For the quantity of machines, a simple crossover is used. 4.4 Elimination and dispersion Elimination and dispersion are random processes, in which a solution is selected for elimination based on a random sampling: if bacterium θl obtains a random number pled that less than the threshold probability of elimination Ped , then θl is eliminated from the population, and a new bacterium is created according to the generation process that provides the initial population.
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Table 1. Average Values of the Objective Function
Algorithm 4 DBCOA chemotaxis l θspm ← SP M (θl ) l if J θspm < J θl then θl ← θl spm else l θspp ← SP P (θl ) l if J θspp < J θl then θl ← θl spp else l θipm ← IP M (θl ) l if J θipm < J θl then θil ← θl ipm else l θipp ← IP P (θl ) l if J θipp < J θl then θl ← θl ipp else l l θnlm ← l N LM (θ ) l if J θnlm < J θ then θl ← θi nlm else l θnlp ← N LP (θl ) l if J θnlp < J θl then
Problem 12x20 14x24 20x51 24x40 25x40 30x30 35x20 5x7 7x11 8x20
• for Problems 2,4,7,8 and 10: production sequences, operation times and production volumes were created randomly; • for Problems 1 and 3: operation times and production volumes; • for Problems 5,6 and 10: production volumes.
The process to create the above data was the following: production sequences were created by mean of random permutations, for operation times and production volumes, random numbers were used, from 10 to 100 and 10000 to 200000 respectively.
θl ← θl nlp else l θnqm ← QM (θl ) l if J θnqm < J θl then θl ← θl nqm end if end if end if end if end if end if end if
The three algorithms, GA (blue), DBGA (green) and DBCOA (red) were applied to the benchmark problem instances. The obtained results for the objective function and solution times are reported in Figures 2 and 3, respectively. In Figure 2, the proposed algorithm DBGA (green) achieves consistent results in terms of the value of objective function in comparison with GA and DBCOA, this is reinforced by the average values of the objective function in Table 1. As it can be seen, DBGA achieves seven of the ten best values (they are presented in bold). This is due to the combination of the advantages of both algorithms achieved by the hybridization.
5. PERFORMANCE EVALUATION To evaluate the performance of the proposed algorithm, the following benchmark problem instances from literature were used (missing parameters were created in a random way): (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Average Values of Objective Function GA DBCOA DGBA 1,36E+11 1,44E+11 1,35E+11 6,97E+08 6,75E+08 6,23E+08 2,96E+12 4,13E+12 2,52E+12 6,92E+10 2,74E+10 4,12E+10 1,32E+12 1,17E+12 9,98E+11 7,42E+11 4,34E+11 3,47E+11 8,46E+10 1,89E+10 1,75E+10 6,05E+02 1,64E+05 9,11E+03 7,62E+03 4,66E+06 3,11E+03 9,76E+10 2,17E+11 1,49E+11
Harhalakis et al. (1990) (12 machines and 20 parts) King (1980) (14 machines and 24 parts) Harhalakis et al. (1990) (20 machines and 51 parts) Chandrasekharan and Rajagopalan (1986) (24 machines and 40 parts) Nair and Narendran (1998) (25 machines and 40 parts) Krishnan et al. (2011) (30 machines and 30 parts) Burbidge (1975) (35 machines and 20 parts) Vitanov et al. (2008) (5 machines and 7 parts) Seifoddini and Djassemi (1996) (7 machines and 11 parts) Nair and Narendran (1998) (8 machines and 20 parts)
The data created randomly for the each benchmark problem included:
Figure 3 presents the comparison of time spent to reach each solution by the three algorithms. It is not possible to establish a clear ranking of algorithms, it depends on the evaluated instance. More details are given in Figure 4 where the runs for the ten instances are presented. For these runs, DGBA achieves the lowest values in the majority of cases. Nevertheless, better solutions require more time in some cases and that is the reason why in some instances the DGBA time is higher than GA and DBCOA. Moreover, Figure 4 exposes the behavior of DGBA where it can be observed an initial drop trend searching the minimum due to the exploration process and then the following trend due to the exploitation process in which artificial bacteria try to find better solutions in their neighborhood, until the achievement of a termination criterion. Finally, Figure 4 shows that the hybridization helps to use the efficient features of both GA and DBCOA to avoid local minima. In addition, it is possible to adapt the features of the hybrid approach to the specific characteristics of each instance.
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Fig. 2. Boxplots of values of the objective function obtained by GA, DBGA, and DBCOA for benchmark problem instances
Fig. 3. Boxplots time (sec) obtained by GA, DBGA, and DBCOA for benchmark problem instances
6. CONCLUSIONS
Chandrasekharan, M.P. and Rajagopalan, R. (1986). MODROC: an extension of rank order clustering for group technology. International Journal of Production Research, 24(5), 1221–1233. Drira, A., Pierreval, H., and Hajri-Gabouj, S. (2006). Facility layout problems: A literature analysis. IFAC Proceedings Volumes, 39(3), 389–400. Gen, M. and Cheng, R. (2000). Genetic Algorithms and Engineering Optimization. John Wiley & Sons. Gen, M., Lin, L., and Zhang, H. (2009). Evolutionary techniques for optimization problems in integrated manufacturing system: State-of-the-art-survey. Computers and Industrial Engineering, 56(3), 779–808. Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Longman Publishing Co., Inc. Harhalakis, G., Nagi, R., and Proth, J.M. (1990). An efficient heuristic in manufacturing cell formation for group technology applications. International Journal of Production Research, 28(1), 185–198. Holland, J.H. (1975). Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence., volume 8. MIT Press, Ann Arbor. King, R. (1980). Machine-Component Group Formation in Group Technology. 8(2), 2–8.
In this paper, we studied an aggregated design problem for cellular manufacturing systems where the following three optimization problems were solved jointly: cell formation, workload balancing and cell layout. A hybrid metaheuristic algorithm named Discrete Genetic Bacterial Algorithm with genetic and bacterial operators was developed to solve the aggregated optimization problem. Its performances have been evaluated on a series of benchmark problem instances. The obtained results show substantial improvements in comparison to the two basic algorithms: GA and DBCOA. Further development of the algorithm will include its application to other discrete optimization problems in manufacturing, e.g. design of complex manufacturing lines. The design problem of a cellular manufacturing system can be enriched by considering its environmental impact, for example energy consumption. REFERENCES Batta¨ıa, O. and Dolgui, A. (2013). A taxonomy of line balancing problems and their solutionapproaches. International Journal of Production Economics, 142(2), 259–277. Burbidge, J.L. (1975). The introduction of group technology. Wiley.
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Romero, G.A., Mej´ıa-Moncayo, C., and Torres, J.A. (2015). Modelos matem´ aticos para la definici´ on del layout de las celdas de manufactura. Revisi´ on de literatura. Revista Tecnura, 19(46), 135–148. Seifoddini, H. and Djassemi, M. (1996). A new grouping measure for evaluation of machine-component matrices. International Journal of Production Research, 34(5), 1179–1193. Selim, H.M., Askin, R.G., and Vakharia, A.J. (1998). Cell formation in group technology: Review, evaluation and directions for future research. Computers & Industrial Engineering, 34(1), 3–20. Singh, S.P. and Sharma, R.R.K. (2006). A review of different approaches to the facility layout problems. International Journal of Advanced Manufacturing Technology, 30(5-6), 425–433. Vitanov, V., Tjahjono, B., and Marghalany, I. (2008). Heuristic rules-based logic cell formation algorithm. International Journal of Production Research, 46(2), 321–344. Yin, Y. and Yasuda, K. (2006). Similarity coefficient methods applied to the cell formation problem: A taxonomy and review. International Journal of Production Economics, 101(2), 329–352.
Fig. 4. Objective Function Vs Time (sec) obtained by GA, DBGA, and DBCOA for benchmark problem instances Krishnan, K.K., Mirzaei, S., Venkatasamy, V., and Pillai, V.M. (2011). A comprehensive approach to facility layout design and cell formation. The International Journal of Advanced Manufacturing Technology, 59(58), 737–753. Mej´ıa-Moncayo, C., Rojas, A.E., and Mura, I. (2018). A Discrete Bacterial Chemotaxis Approach to the Design of Cellular Manufacturing Layouts. Lecture Notes in Computer Science, 10960 LNCS, 423–437. Nair, G.J. and Narendran, T.T. (1998). CASE: A clustering algorithm for cell formation with sequence data. International Journal of Production Research, 36(1), 157–180. Nallusamy, S. (2016). Productivity enhancement in a small scale manufacturing unit through proposed line balancing and cellular layout. International Journal of Performability Engineering, 12(6), 523–534. Papaioannou, G. and Wilson, J.M. (2010). The evolution of cell formation problem methodologies based on recent studies (1997-2008): Review and directions for future research. European Journal of Operational Research, 206(3), 509–521. Passino, K.M. (2002). Biomimicry of Bacterial Foraging for Distributed Optimization and Control. IEEE Control Systems, 22(3), 52–67. 1431
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A hybrid optimization algorithm with A hybrid optimization algorithm with A hybrid optimization algorithm with genetic and bacterial operators for the A hybrid optimization algorithm with genetic and bacterial operators for genetic and bacterial operators for the the design of cellular manufacturing genetic and bacterial operators systems for the design of cellular manufacturing systems design of manufacturing design Camilo of cellular cellular manufacturing systems systems Mej´ıa-Moncayo ∗∗ Olga Battaia ∗∗ ∗∗ Camilo Mej´ıa-Moncayo ∗ Olga Battaia ∗∗ ∗∗ Camilo Camilo Mej´ Mej´ıa-Moncayo ıa-Moncayo ∗∗ Olga Olga Battaia Battaia ∗∗ ıa-Moncayo Olga Battaia ∗Camilo Mej´ a , Colombia (e-mail: ∗ Universidad EAN, Bogot´ EAN, a ∗ ∗ Universidad Universidad EAN, Bogot´ Bogot´ a,,, Colombia Colombia (e-mail: (e-mail: [email protected]) Universidad EAN, Bogot´ a Colombia (e-mail: ∗ ∗∗ [email protected]) Universidad EAN,Toulouse, Bogot´ a, Colombia (e-mail: [email protected]) France (e-mail: [email protected]) ∗∗ ISAE-Supaero, Toulouse, France (e-mail: ∗∗ [email protected]) ∗∗ ISAE-Supaero, ISAE-Supaero, Toulouse, France (e-mail: [email protected] ) Toulouse, France )(e-mail: ∗∗ ISAE-Supaero, [email protected]
ISAE-Supaero, Toulouse, France ))(e-mail: [email protected] [email protected] [email protected] ) Abstract: We We consider aa design design problem problem for for cellular cellular manufacturing systems. systems. In In order to to improve Abstract: Abstract: Weofconsider consider a design design problem foroptimization cellular manufacturing manufacturing systems. In order order to improve improve the efficiency the designed system, three problems are considered jointly in the the Abstract: We consider a problem for cellular manufacturing systems. In order to improve the efficiency the designed system, three problems are considered jointly in Abstract: Weof consider a design problem foroptimization cellular manufacturing systems. Inlayout. order toAimprove the efficiency of the designed system, three optimization problems are considered jointly in the same optimization procedure: cell formation, workload balancing and cell hybrid the efficiency of the designed system, three optimization problems are considered jointly in the same optimization cell formation, workload balancing and cell layout. A hybrid the efficiency ofalgorithm the procedure: designed three problems considered in the the same optimization procedure: cell workload balancing and cell A metaheuristic withsystem, genetic and optimization bacterial operators is are developed to jointly optimize same optimization procedure: cell formation, formation, workload balancing and cell layout. layout. A hybrid hybrid metaheuristic algorithm with genetic and bacterial operators is developed to optimize the same optimization procedure: cell formation, workload balancing and cell layout. A hybrid metaheuristic algorithm with genetic and bacterial operators is developed to optimize the performances of the designed cellular manufacturing system. The results obtained on benchmark metaheuristic algorithm withcellular genetic and bacterialsystem. operators is developed to on optimize the performances of the manufacturing The obtained benchmark metaheuristic withcellular genetic and bacterialsystem. operators is developed to on optimize the performances ofalgorithm the designed designed cellular manufacturing system. The results results obtained on benchmark problems are are discussed. discussed. performances of the designed manufacturing The results obtained benchmark problems performances of the designed cellular manufacturing system. The results obtained on benchmark problems are discussed. problems are discussed. © 2019, IFAC problems are (International discussed. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Cellular Cellular manufacturing; manufacturing; Cell Cell formation; formation; Workload Workload balancing; balancing; Facility Facility layout; layout; Keywords: Keywords: Cellular manufacturing; manufacturing; Cell formation; formation; Workload balancing; balancing; Facility Facility layout; layout; Genetic Algorithm; Bacterial chemotaxis; Hybrid algorithm Keywords: Cellular Cell Workload Genetic Bacterial Hybrid Keywords: Cellular manufacturing; Cell formation; Workload balancing; Facility layout; Genetic Algorithm; Algorithm; Bacterial chemotaxis; chemotaxis; Hybrid algorithm algorithm Genetic Algorithm; Bacterial chemotaxis; Hybrid algorithm Genetic Algorithm; Bacterial chemotaxis; Hybrid algorithm 1. INTRODUCTION INTRODUCTION The advantages advantages of of balanced balanced production production systems systems were were exex1. The 1. The advantages of balanced production systems were extensively discussed as well in the literature. Among them, 1. INTRODUCTION INTRODUCTION The advantages of balanced production systems were extensively discussed as well in the literature. Among them, 1. INTRODUCTION The advantages of balanced production systems extensively discussed as the Among them, improved productivity andin better resources usagewere can be tensively discussed as well well inbetter the literature. literature. Among them, improved productivity and resources usage can be tensively discussed as well in the literature. Among them, improved productivity and better resources usage can be cited. A taxonomy of problems and solution methods for improved productivity and better resources usage can be Designing an efficient manufacturing system is a hard cited. A taxonomy of problems and solution methods for Designing an efficient manufacturing system is aa hard improved productivity and better resources can be A of and solution methods for balancing production lines is introduced introduced by usage Batta¨ ıa and and cited. A taxonomy taxonomy of problems problems and solution methods for Designing an efficient manufacturing is task involving involving the consideration of many manysystem different factors. Designing an the efficient manufacturing system is factors. a hard hard cited. balancing production lines is by Batta¨ ıa task consideration of different cited. A taxonomy of problems and solution methods for production lines is by ıa Designing efficient manufacturing is factors. a hard balancing Dolgui (2013). (2013). Although such manufacturing organizabalancing production lines is introduced introduced by Batta¨ Batta¨ ıa and and task the consideration many different This involving task an becomes even more of complex with increasing increasing task involving the consideration of manysystem different factors. Dolgui Although such aaa manufacturing organizaThis task becomes even more complex with balancing production lines is introduced by Batta¨ ıamass and Dolgui (2013). Although such manufacturing organizatask involving the consideration of many different factors. tion suits well for mass production, its adaptation for Dolgui (2013). Although such a manufacturing organizaThis task becomes even more complex with increasing product customization. Product customization suits better This task becomes even more complex with increasing tion suits well for mass production, its adaptation for mass product customization. Product customization suits better Dolgui (2013). Although such a manufacturing organization suits well for mass production, its adaptation for mass This task becomes even more complex with increasing customized production systems as cells manufacturing, tion suits well for mass production, itscells adaptation for mass product customization. better customers’ demand but butProduct makes it itcustomization challenging to tosuits attain the customized product customization. Product customization suits better production systems as manufacturing, customers’ demand makes challenging attain the tion well foroptimization mass production, adaptation for mass customized production systems as product customization. Product customization suits better needssuits adapted procedures asmanufacturing, described by customized production systems asitscells cells manufacturing, customers’ demand but makes it challenging to attain the performances of mass production systems, in this context customers’ demand but makes it challenging to attain the needs adapted optimization procedures as described by performances of mass production systems, in this context customized production systems as cells manufacturing, needs adapted optimization procedures as described by customers’ demand but makes it challenging to attain the Nallusamy (2016). needs adapted optimization procedures as described by performances of mass production systems, in this context cellular manufacturing systems are generally an efficient performances of mass production systems, in this context Nallusamy (2016). cellular manufacturing systems are generally an efficient needs adapted optimization procedures as described by (2016). performances of mass production systems, in this context Nallusamy Nallusamy (2016). cellular manufacturing systems generally an alternative. cellular manufacturing systems are are generally an efficient efficient Furthermore, the facility facility layout layout problem problem aims aims in in defining defining alternative. Nallusamy (2016). cellular manufacturing systems are generally an efficient Furthermore, the alternative. alternative. Furthermore, the facility layout problem aims in defining the best position for each machine. This problem has Furthermore, the facility layout problem aims in defining Cellular manufacturing systems are usually used for the alternative. the best position for each machine. This problem has Cellular manufacturing systems are usually used for the Furthermore, the facility layout problem aims in defining the best position for each machine. This problem has attracted significant attention in the literature e.g. (Drira the best position for each machine. This problem has Cellular manufacturing systems are usually used for the batch production of part families. This production archiCellular manufacturing systems are usually used for the attracted significant attention in the literature e.g. (Drira batch production of part families. This production archithe position for each machine. This problem has significant attention in the literature Cellular manufacturing systems usually used for the attracted et al., al.,best 2006; Singh and and Sharma, 2006; Romero ete.g. al., (Drira 2015). attracted significant attention in the Romero literature e.g. (Drira batch of part families. This production architectureproduction facilitates the administration and control of the batch production of part families.are This production archiet 2006; Singh Sharma, 2006; et al., 2015). tecture facilitates the administration and control of the attracted significant attention in the literature e.g. (Drira et al., 2006; Singh and Sharma, 2006; Romero et al., 2015). batch production of part families. This production archiIts formulation is based on the available space, which et al., 2006; Singh and Sharma, 2006; Romero et al., 2015). tecture facilitates the administration and control of the production system the andadministration at the the same same time time makes itofmore more tecture facilitates and makes controlit the Its formulation based on the available space, which production system and at et Singhis 2006; etway. al., In 2015). formulation is on available which tecture facilitates the administration and makes controlit ofmore the Its canal., be2006; considered inbased aSharma, continuous orRomero discretespace, the Its formulation isand based on the the available space, which production system and at the same time flexible to respond to demand changes. production system and at the same time makes it more can be considered in a continuous or discrete way. In the flexible to respond demand Its formulation is in based on the available space, which be considered aa continuous or discrete way. In the production system to and at thechanges. same time makes it more can first case, the problem is modeled in the way to provide can be considered in continuous or discrete way. In the flexible to respond to demand changes. flexible to respond to demand changes. case, the problem is modeled in the way to provide can be considered a continuous or discrete In the the In order ordertoto torespond optimizetothe the performances of the the designed designed celcel- first first case, the is modeled in the way to provide flexible demand changes.of the coordinates coordinates of in each machine. In the second case, first case, the problem problem ismachine. modeledIn inthe thesecond wayway. tocase, provide In optimize performances the of each the In order to optimize the performances of the designed celfirst case, the problem is modeled in the way to provide lular manufacturing system with the objective to achieve In order to optimize the performances of the designed celthe coordinates of each machine. In the second case, the solution of the model gives the relative position of all the coordinates of each machine. In the second case, the lular manufacturing system with the objective to achieve of the model gives the relative position of all In order to optimize the performances of the products, designed cellular manufacturing system with the to the coordinates of each machine. In themost second case, the mass production efficiency for customized we solution lular manufacturing system withcustomized the objective objective to achieve achieve solution of the model gives the relative position of all machines in the cell. In both cases, the used way to solution of the model gives the relative position of all mass production efficiency for products, we the cell. In both cases, the most used way to lular manufacturing system withcustomized the objective to achieve mass production efficiency for products, we solution ofin the model gives the relative position of all propose to consider consider jointly three three optimization problems: mass production efficiency for customized products, we machines machines in the cell. In both cases, the most used way to evaluate the layout solution is the transport cost related machines in the cell. In both cases, the most used way to propose to jointly optimization problems: the layout the transport cost related mass production efficiency for customized we evaluate propose to jointly three optimization problems: machines intransfer. the cell.solution In both is the most used way to cell formation formation problem, workload balancingproducts, and facility propose to consider consider jointly three optimization problems: evaluate the layout solution is the cost to product product evaluate the layout solution iscases, the transport transport cost related related cell problem, workload balancing and facility to transfer. propose to consider jointly three optimization problems: cell formation problem, workload balancing and facility evaluate the layout solution is the transport cost related layout. cell formation problem, workload balancing and facility to product transfer. to product transfer. layout. cell formation problem, workload balancing and facility to In order order to improve improve the efficiency efficiency of of aa cellular cellular manufacturmanufacturlayout. product transfer.the layout. In to Cell formation is the optimization problem aiming in In order to improve the efficiency of a cellular manufacturlayout. ing system, we propose in this study to consider its layout layout In order to improve the efficiency of a cellular manufacturCell formation is the optimization problem aiming in ing system, we propose this study consider its Cell formation is the optimization problem aiming in In order to improve the in efficiency of aato cellular manufacturgrouping machines into manufacturing cells and parts into Cell formation is the optimization problem aiming in ing system, we propose in this study to consider its layout design with the objective to achieve balanced work distriing system, we propose in this study to consider its layout grouping machines manufacturing cells and parts into with the objective achieve aa to balanced work distriCell formation is into the optimization problem aiming in design grouping machines into manufacturing cells and parts ing system, we the propose into this study consider its layout families on the basis basis of their their similarities in the the design and grouping machines into manufacturing cells anddesign parts into into design with the objective to achieve balanced work distribution among machines of the different cells in which design with the objective to achieve a balanced work distrifamilies on the of similarities in and among machines of the different cells in which grouping machines manufacturing cells and parts into families on the basis of similarities in design and design with thethe objective to achieve a producing balanced work distrimanufacturing processes (Selim et al., al., 1998; 1998; Papaioannou families on the processes basisinto of their their similarities in the the design and bution bution among the machines of the different cells in which the system works like a line system batches of bution among the machines of the different cells in which manufacturing (Selim et Papaioannou system works like aa line system producing of families on the basis This of their similarities in the design and the manufacturing processes (Selim et al., 1998; Papaioannou bution among theachieve machines of the we different cellsbatches inseveral which and Wilson, 2010). optimization problem has been manufacturing processes (Selim et al., 1998; Papaioannou the system works like line system producing batches of part families. To this goal, solve jointly the system works like a line system producing batches of and Wilson, 2010). This optimization problem has been part families. To achieve this goal, we solve jointly several manufacturing processes (Selim et al., 1998; Papaioannou and Wilson, 2010). This optimization problem has the system works like aincluding line batches of extensively studied in the the literature. For example, example, the part and Wilson,studied 2010). This optimization problem has been been families. To this goal, we solve several optimization problems cell formation, workload part families. To achieve achieve this system goal,cell weproducing solve jointly jointly several extensively in literature. For the optimization problems including formation, workload and Wilson, 2010). This optimization problem has been extensively studied in the literature. For example, part families. To achieve this goal, we solve jointly several study of (Selim et al., 1998) provides an overview of the extensively studied in the literature. For example, optimization problems including cell formation, workload balancing and andproblems inter and and intra cells cells layout. We Weworkload develop optimization including cell formation, study of (Selim et al., 1998) provides an of the balancing inter intra layout. develop extensively studied indescriptive the literature. Foroverview example, study of et 1998) provides an overview of optimization problems including cellbacterial formation, workload following techniques: methods, cluster analystudy of (Selim (Selim et al., al., 1998) provides an overview of the the balancing and inter and intra cells layout. We develop a new hybrid algorithm based on and genetic balancing and inter and intra cells layout. We develop following techniques: descriptive methods, cluster analynew hybrid algorithm on bacterial and genetic study of (Selim et al.,descriptive 1998) provides an overview of and the aabalancing following techniques: methods, cluster analyand inter and based intra cells layout. We develop sis, graph partitioning, mathematical programming following techniques: descriptive methods, cluster analynew hybrid algorithm based on bacterial and genetic algorithms for the defined optimization problem. a new hybrid algorithm based on bacterial and genetic sis, graph partitioning, mathematical programming and for the defined optimization problem. following techniques: descriptive methods, cluster analysis, graph partitioning, mathematical programming and aalgorithms new hybrid algorithm based on bacterial and genetic artificial intelligence. A taxonomy of coefficient coefficient methods sis, graphintelligence. partitioning, mathematical programming and algorithms algorithms for the defined optimization problem. for the defined optimization problem. artificial A taxonomy of methods sis, graphintelligence. partitioning, mathematical programming and The paper paper for is structured structured as follows. The The proposed algoalgoartificial A taxonomy of methods algorithms the definedas optimization problem. is proposed proposed by Yin Yin and and Yasuda (2006). Papaioannou and The artificial intelligence. A Yasuda taxonomy of coefficient coefficient methods is follows. proposed is by (2006). Papaioannou and The paper is structured as follows. The proposed algoartificial intelligence. A taxonomy of coefficient methods rithms are presented in Sections 2-4 and include a genetic The paper is structured as follows. The proposed algois proposed by Yin and Yasuda (2006). Papaioannou and Wilson (2010) discuss inYasuda particular heuristic and metametais proposed by Yin andin (2006). Papaioannou and rithms are presented in Sections 2-4 and a Wilson (2010) discuss particular heuristic and The paper is structured as follows. Theinclude proposed algoare presented 2-4 and include aa genetic genetic is proposed by Yin and Yasuda (2006). Papaioannou and rithms algorithm (Section 2),in discrete bacterial chemotaxis oprithms are (Section presented inaaSections Sections 2-4 and include genetic Wilson (2010) discuss in particular heuristic and metaheuristics algorithms developed for cell formation. Wilson (2010) discuss in particular heuristic and metaalgorithm 2), discrete bacterial chemotaxis opheuristics algorithms developed for cell formation. rithms are presented in Sections 2-4 and include a genetic algorithm (Section 2), a discrete bacterial chemotaxis Wilson (2010) discuss in particular heuristic and metaalgorithm (Section 2), a discrete bacterial chemotaxis opopheuristics heuristics algorithms algorithms developed developed for for cell cell formation. formation. algorithm (Section 2), a discrete bacterial chemotaxis opheuristics algorithms developed for cell formation.
2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 1426Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2019 IFAC 1426Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 1426 Copyright © 2019 IFAC 1426 10.1016/j.ifacol.2019.11.396 Copyright © 2019 IFAC 1426
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timization algorithm (Section 3) and a hybrid algorithm based on previous two (Section 4). The performances of algorithms on benchmark problems are discussed in Section 5. Finally, Section 6 presents the conclusions of this study and future research directions. 2. GENETIC ALGORITHMS Genetic algorithms, introduced by Holland (1975), are search algorithms based on the mechanics of natural selection and genetics. The GAs have been applied in various fields such as mathematics, engineering, biology, and social sciences Goldberg (1989). GA combine the concept of survival of the fittest with a structure of random information exchange to form a robust search algorithm. GA start with generating an initial population of possible solutions (individuals or chromosomes), and then evaluate the fitness of each individual in the population and select some of them to be ”parents”. Such parents are involved in a reproduction procedure through a crossover operator. The reproduction is the way of information sharing among individuals. Additionally, GA use a mutation operator to randomly alter the genetic composition of offspring with the aim to improve the diversity of the population and better explore the search space. The processes of evaluation, selection, reproduction, and mutation are repeated until a termination criterion is satisfied as described in Algorithm 1. However, GA converge often to a local optimum after a certain number of generations. This is caused by a low diversity in the populations or by the incapacity of the mutation process to avoid local optima.
in order to replace death bacteria and create new organisms which can be dispersed to increase the probability of survival of the species. Algorithm 2 DBCOA Generation of initial bacteria Evaluation of objective function for t = 1 to N ed do for u = 1 to N re do for v = 1 to N c do Chemotaxis end for Reproduction end for Elimination and Dispersion end for In bacterial algorithms, each artificial bacterium needs to search optimal value through the bacteria chemotaxis and realizes the swarming process to exploit sources of food, reproduction by duplication produces copies of fittest individuals. Elimination and dispersion processes have been defined to avoiding falling into premature convergence by a local optimal, creating new bacteria dispersed or located in other positions different to the originals which are replaced. Chemotaxis is the main process of this contribution since it allows exploring the search space in an efficient way. However, DBCOA lacks a procedure for sharing information among solutions. In order to fix this drawback, we propose to hybridize it with GA taking the advantages of two approaches. 4. DISCRETE GENETIC BACTERIAL ALGORITHM - DGBA
Algorithm 1 Genetic Algorithm Generation of initial population for i = 1 to M ax − generation do Fitness function evaluation Selection of parents Crossover Mutation end for
Many natural processes have been emulated in optimization procedures like Genetic Algorithms proposed by Holland Holland (1975) and explained in section 2. Nevertheless, the performances of GA can be improved due to the integration of an exploration process. At the same time, DBCOA (Mej´ıa-Moncayo et al. (2018) ) has an efficient exploration process, however, it cannot share information among individuals.
3. DISCRETE BACTERIAL CHEMOTAXIS OPTIMIZATION ALGORITHM Discrete Bacterial Chemotaxis Optimization Algorithm - DBCOA, is an optimization algorithm introduced by Mej´ıa-Moncayo et al. (2018) for the solution of cellular manufacturing layout. It is a discrete optimization algorithm based on Bacterial Foraging Optimization Algorithm (Passino (2002)) with which shares the structure, which by mean of a randomly hierarchical chemotaxis process emulates bacterial exploration in a discrete search space. Bacterial Foraging Optimization Algorithm (BFOA) proposed by Passino (2002) is a technique which models the food-seeking and reproductive behavior of common bacteria such as E. Coli in order to solve optimization problems in continuous search spaces. In nature, bacteria move in their environment by mean of flagella movements. Furthermore, bacteria use duplication for their reproduction
The idea of this paper is to use the symbiosis of GA and DBCOA in order to obtain an efficient solution procedure to address the problem of designing cellular manufacturing systems. Therefore, GA and DBCOA are used to create a hybrid metaheuristic named Discrete Genetic Bacterial Algorithm - DGBA. This is done in order to better explore the advantages of both methods. Thus, a chemotaxis process of DBCOA is followed by GA parents selection (binary tournament) for GA reproduction by crossing (OX order crossover explained in Gen and Cheng (2000)) and at the end, the elimination and dispersion procedures of DBCOA are used. The whole procedure is summarized in Algorithm 3. The used operators are explained here below. 4.1 Solution Coding The codification is the way by which variables that represent individuals and populations are encoded into
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l to Cell1 , in this example the fourth element of θmp . This gives the definition of the machines Cell2 = [2, 4]. In a similar way, we proceed with the parts permutation and limits of families vectors to fully define Family1 and Family2 . Finally, the quantity of machines is defined by l θqm = [2, 3, 1, 2, 1].
Algorithm 3 DGBA Generation of initial population for i = 1 to M ax − generation do Fitness function evaluation Chemotaxis Selection of parents Crossover Elimination and dispersion end for
4.2 Chemotaxis
data structures. We selected a coding based on the approach presented in Gen et al. (2009), where each solution or bacterium θl is encoded by vectors θl = l l l l l ). The first and second vectors , θcn , θcl , θfl l , θqm (θmp , θpp l are respectively machines permutation θmp and parts perl mutation θpp , the third vector represents the cells’ number l l θcn , the fourth and fifth vectors are the cells’ limits θcl l and families’ limits θf l , the last one encodes the quantity l of machines θqm . For example, if we consider a cellular manufacturing system with five machines and seven parts, one possible solution for the cell manufacturing layout design problem (see figure 1), i.e. a bacterium θl encoded as shown below: l l l l l , θpp , θcn , θcl , θfl l , θqm ) θl = (θmp
θl = [(5, 3, 1, 2, 4), (6, 3, 7, 2, 4, 1, 5), (2), (3, 5, 0, 0, 0),(4, 7, 0, 0, 0), (2, 3, 1, 2, 1)]
The implemented Chemotaxis has a hierarchical structure that imitates the movements carried out by bacteria in their natural environment through swimming and tumbling via flagella. This process has the objective of exploring the search space to find optimal solutions and avoid local optima. Given the nature of the cell manufacturing layout, the movements of swim and tumble performed by bacteria in a continuous search space have to be discretized. We chose to implement hierarchical changes into the encoded solutions, by changing the vector-based representation through the swap, insertions, changes of the number of cells and of their limits, and the quantity of machines. The modified individuals are then compared with the original ones (in terms of their objective function value) and retained if a fitness improvement occurs. This process, summarized in Algorithm 4, is repeated until no more change is possible. In Algorithm 4, the following shorthand notation is used: l • SP M () is a function that swaps two elements of θmp l to produce θspm . l • SP P () is a function that swaps two elements of θpp l to produce θspp . l • IP M () is a function that inserts one element in θmp l to produce θipm . l • IP P () is a function that inserts one element in θpp to l produce θipp . l • N LM () is a function that modifies the limits of θlm l to produce θnlm . l • N LP () is a function that modifies the limits of θlp to l produce θnlp . • QM () is a function that increase or decrease the l l quantity of machines θqm to produce θnqm .
4.3 Reproduction Fig. 1. Schematic layout of codification example This solution defines the following cells and families: Cell1 = [5, 3, 1], Cell2 = [2, 4], Family1 = [6, 3, 7, 2] and Family2 = [4, 1, 5]. To get this information out of θl it is necessary to check l the number of cells or families θcn , which for this example l is 2. Next, we look at the first element of vector θcl of the limits of cells, in this case (3). This number defines the number of elements belonging to the first cell, starting l from the first element of vector θmp , which gives as result Cell1 = [5, 3, 1]. l As for Cell2 , we use the second element of θcl , which is (5). l This number establishes the position in vector θmp , of the last element to be assigned to Cell2 . The first element of Cell2 is the element adjacent to the last element assigned
The reproduction is the process by which the fittest solutions produce other individuals by crossover and replace the least healthy solutions, which eventually disappear from the population. Reproduction by OX order crossover is applied to permutation vectors, limits of cells and families are swapped between parents. For the quantity of machines, a simple crossover is used. 4.4 Elimination and dispersion Elimination and dispersion are random processes, in which a solution is selected for elimination based on a random sampling: if bacterium θl obtains a random number pled that less than the threshold probability of elimination Ped , then θl is eliminated from the population, and a new bacterium is created according to the generation process that provides the initial population.
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Table 1. Average Values of the Objective Function
Algorithm 4 DBCOA chemotaxis l θspm ← SP M (θl ) l if J θspm < J θl then θl ← θl spm else l θspp ← SP P (θl ) l if J θspp < J θl then θl ← θl spp else l θipm ← IP M (θl ) l if J θipm < J θl then θil ← θl ipm else l θipp ← IP P (θl ) l if J θipp < J θl then θl ← θl ipp else l l θnlm ← l N LM (θ ) l if J θnlm < J θ then θl ← θi nlm else l θnlp ← N LP (θl ) l if J θnlp < J θl then
Problem 12x20 14x24 20x51 24x40 25x40 30x30 35x20 5x7 7x11 8x20
• for Problems 2,4,7,8 and 10: production sequences, operation times and production volumes were created randomly; • for Problems 1 and 3: operation times and production volumes; • for Problems 5,6 and 10: production volumes.
The process to create the above data was the following: production sequences were created by mean of random permutations, for operation times and production volumes, random numbers were used, from 10 to 100 and 10000 to 200000 respectively.
θl ← θl nlp else l θnqm ← QM (θl ) l if J θnqm < J θl then θl ← θl nqm end if end if end if end if end if end if end if
The three algorithms, GA (blue), DBGA (green) and DBCOA (red) were applied to the benchmark problem instances. The obtained results for the objective function and solution times are reported in Figures 2 and 3, respectively. In Figure 2, the proposed algorithm DBGA (green) achieves consistent results in terms of the value of objective function in comparison with GA and DBCOA, this is reinforced by the average values of the objective function in Table 1. As it can be seen, DBGA achieves seven of the ten best values (they are presented in bold). This is due to the combination of the advantages of both algorithms achieved by the hybridization.
5. PERFORMANCE EVALUATION To evaluate the performance of the proposed algorithm, the following benchmark problem instances from literature were used (missing parameters were created in a random way): (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Average Values of Objective Function GA DBCOA DGBA 1,36E+11 1,44E+11 1,35E+11 6,97E+08 6,75E+08 6,23E+08 2,96E+12 4,13E+12 2,52E+12 6,92E+10 2,74E+10 4,12E+10 1,32E+12 1,17E+12 9,98E+11 7,42E+11 4,34E+11 3,47E+11 8,46E+10 1,89E+10 1,75E+10 6,05E+02 1,64E+05 9,11E+03 7,62E+03 4,66E+06 3,11E+03 9,76E+10 2,17E+11 1,49E+11
Harhalakis et al. (1990) (12 machines and 20 parts) King (1980) (14 machines and 24 parts) Harhalakis et al. (1990) (20 machines and 51 parts) Chandrasekharan and Rajagopalan (1986) (24 machines and 40 parts) Nair and Narendran (1998) (25 machines and 40 parts) Krishnan et al. (2011) (30 machines and 30 parts) Burbidge (1975) (35 machines and 20 parts) Vitanov et al. (2008) (5 machines and 7 parts) Seifoddini and Djassemi (1996) (7 machines and 11 parts) Nair and Narendran (1998) (8 machines and 20 parts)
The data created randomly for the each benchmark problem included:
Figure 3 presents the comparison of time spent to reach each solution by the three algorithms. It is not possible to establish a clear ranking of algorithms, it depends on the evaluated instance. More details are given in Figure 4 where the runs for the ten instances are presented. For these runs, DGBA achieves the lowest values in the majority of cases. Nevertheless, better solutions require more time in some cases and that is the reason why in some instances the DGBA time is higher than GA and DBCOA. Moreover, Figure 4 exposes the behavior of DGBA where it can be observed an initial drop trend searching the minimum due to the exploration process and then the following trend due to the exploitation process in which artificial bacteria try to find better solutions in their neighborhood, until the achievement of a termination criterion. Finally, Figure 4 shows that the hybridization helps to use the efficient features of both GA and DBCOA to avoid local minima. In addition, it is possible to adapt the features of the hybrid approach to the specific characteristics of each instance.
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Fig. 2. Boxplots of values of the objective function obtained by GA, DBGA, and DBCOA for benchmark problem instances
Fig. 3. Boxplots time (sec) obtained by GA, DBGA, and DBCOA for benchmark problem instances
6. CONCLUSIONS
Chandrasekharan, M.P. and Rajagopalan, R. (1986). MODROC: an extension of rank order clustering for group technology. International Journal of Production Research, 24(5), 1221–1233. Drira, A., Pierreval, H., and Hajri-Gabouj, S. (2006). Facility layout problems: A literature analysis. IFAC Proceedings Volumes, 39(3), 389–400. Gen, M. and Cheng, R. (2000). Genetic Algorithms and Engineering Optimization. John Wiley & Sons. Gen, M., Lin, L., and Zhang, H. (2009). Evolutionary techniques for optimization problems in integrated manufacturing system: State-of-the-art-survey. Computers and Industrial Engineering, 56(3), 779–808. Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Longman Publishing Co., Inc. Harhalakis, G., Nagi, R., and Proth, J.M. (1990). An efficient heuristic in manufacturing cell formation for group technology applications. International Journal of Production Research, 28(1), 185–198. Holland, J.H. (1975). Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence., volume 8. MIT Press, Ann Arbor. King, R. (1980). Machine-Component Group Formation in Group Technology. 8(2), 2–8.
In this paper, we studied an aggregated design problem for cellular manufacturing systems where the following three optimization problems were solved jointly: cell formation, workload balancing and cell layout. A hybrid metaheuristic algorithm named Discrete Genetic Bacterial Algorithm with genetic and bacterial operators was developed to solve the aggregated optimization problem. Its performances have been evaluated on a series of benchmark problem instances. The obtained results show substantial improvements in comparison to the two basic algorithms: GA and DBCOA. Further development of the algorithm will include its application to other discrete optimization problems in manufacturing, e.g. design of complex manufacturing lines. The design problem of a cellular manufacturing system can be enriched by considering its environmental impact, for example energy consumption. REFERENCES Batta¨ıa, O. and Dolgui, A. (2013). A taxonomy of line balancing problems and their solutionapproaches. International Journal of Production Economics, 142(2), 259–277. Burbidge, J.L. (1975). The introduction of group technology. Wiley.
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Romero, G.A., Mej´ıa-Moncayo, C., and Torres, J.A. (2015). Modelos matem´ aticos para la definici´ on del layout de las celdas de manufactura. Revisi´ on de literatura. Revista Tecnura, 19(46), 135–148. Seifoddini, H. and Djassemi, M. (1996). A new grouping measure for evaluation of machine-component matrices. International Journal of Production Research, 34(5), 1179–1193. Selim, H.M., Askin, R.G., and Vakharia, A.J. (1998). Cell formation in group technology: Review, evaluation and directions for future research. Computers & Industrial Engineering, 34(1), 3–20. Singh, S.P. and Sharma, R.R.K. (2006). A review of different approaches to the facility layout problems. International Journal of Advanced Manufacturing Technology, 30(5-6), 425–433. Vitanov, V., Tjahjono, B., and Marghalany, I. (2008). Heuristic rules-based logic cell formation algorithm. International Journal of Production Research, 46(2), 321–344. Yin, Y. and Yasuda, K. (2006). Similarity coefficient methods applied to the cell formation problem: A taxonomy and review. International Journal of Production Economics, 101(2), 329–352.
Fig. 4. Objective Function Vs Time (sec) obtained by GA, DBGA, and DBCOA for benchmark problem instances Krishnan, K.K., Mirzaei, S., Venkatasamy, V., and Pillai, V.M. (2011). A comprehensive approach to facility layout design and cell formation. The International Journal of Advanced Manufacturing Technology, 59(58), 737–753. Mej´ıa-Moncayo, C., Rojas, A.E., and Mura, I. (2018). A Discrete Bacterial Chemotaxis Approach to the Design of Cellular Manufacturing Layouts. Lecture Notes in Computer Science, 10960 LNCS, 423–437. Nair, G.J. and Narendran, T.T. (1998). CASE: A clustering algorithm for cell formation with sequence data. International Journal of Production Research, 36(1), 157–180. Nallusamy, S. (2016). Productivity enhancement in a small scale manufacturing unit through proposed line balancing and cellular layout. International Journal of Performability Engineering, 12(6), 523–534. Papaioannou, G. and Wilson, J.M. (2010). The evolution of cell formation problem methodologies based on recent studies (1997-2008): Review and directions for future research. European Journal of Operational Research, 206(3), 509–521. Passino, K.M. (2002). Biomimicry of Bacterial Foraging for Distributed Optimization and Control. IEEE Control Systems, 22(3), 52–67. 1431