A new approach for workshop design

Transcription

1 DOI /s A new approach for workshop design Naim Yalaoui Halim Mahdi Lionel Amodeo Farouk Yalaoui Received: 11 July 2008 / Accepted: 8 December 2009 Springer Science+Business Media, LLC 2009 Abstract In this paper we solve a combined group technology problem with a facility layout problem (FLP). This new approach is called T-FLP. We have developed a hybrid algorithm containing three main steps. The first one, called MPGV (Machine Part Grouping with Volume) is a decomposition method that can create families of product and machine groups based on a volume data matrix. The second one consists on assigning machines to fixed locations, using as a constraint, the solution of the MPGV. This problem is solved as a Quadratic Assignment Problem (QAP). In the third step, we make a global evaluation of all the solutions. A loop on cells is performed using a minimum and maximum number of cells. This loop can choose the appropriate number of cells based on the best solution of a global evaluation. The hybrid algorithm is implemented with two different rules for taking into account the constraint of the MPGV solution. This has generated two methods called YMAY1 and YMAY2. In the MPGV we use a data oriented genetic algorithm. The QAP is solved with an Ant Colony Optimization mixed with a Guided Local Search (ACOGLS). This method N. Yalaoui (B) L. Amodeo F. Yalaoui Institut Charles Delaunay-OSI, Universite de Technologie de Troyes, FRE CNRS 2848, 12, rue Marie Curie, Troyes, France naim.yalaoui@utt.fr L. Amodeo lionel.amodeo@utt.fr F. Yalaoui farouk.yalaoui@utt.fr N. Yalaoui H. Mahdi Caillau company, 28, rue Ernest Renan, Issy les moulineaux, France nyalaoui@caillau.fr H. Mahdi hmahdi@caillau.com has been used to solve a real industrial case. For estimating the efficiency of our method, we have compared our results with an optimal solution obtained by complete enumeration (an exact method). Keywords Group technology Genetic algorithm Ant colony optimization Guided local search Facility layout problem Quadratic assignment problem Introduction In this paper, we present an iterative combined FLP (Facility layout problem) with a group technology problem called T-FLP, which has never been treated in the literature. The objective function is the minimization of the cost travel (combining distances and volumes). This method consists of three steps presented in Fig. 1 and Algorithm 1. The first one is a decomposition step which combines groups of machines with product families, based on the product flows between machines. As a result, we obtain machine cells including product families. The second step assigns the obtained cells to a fixed location. As a constraint, all the machines of the same cell must be kept grouped. At Step 3, we evaluate all the solutions using a specific aggregation of both group technology and facility layout problems criteria. This evaluation represents the global one. Each iteration in the loop on cells determines the maximum number of cells to create. At the end, the global evaluation is sorted in non ascending order, and the best solution is selected. This represents the best compromise between the facility layout solution and the group technology solution. Its formulation will be presented later. In the Step 1, we generate product families belonging to groups of machines using a specific genetic algorithm.

2 Step 1: Loop on cells Group Technology Problem Data: Number of machines and products Volume matrix (Machine products) Number (Cells_nb) and maximum number (Max_cell)of cells to create Constraints: Maximum and minimum number of machine per cell. Each machine and product must be assigned in only one cell. Objectif function: Maximise volume transported inside the cells. Optimisation algorithm: Genetic algorithm. Step 2: Quadratic Assignement Problem Data: Number of machines Flow matrix (between machines) Volume matrix (between machines) Distance matrix (between locations) Constraints: Group Technology solutions Each machine must be assigned to only one location Each location can contain only one machine Objectif function: Minimise distance between machines. Optimisation algorithm: Ant Colony Optimisation mixed with a guided local search. Data: Solution of group technology problem. Step 3: YES Global evaluation Cells_nb<Max_cell Solution of the quadratic assignement problem. Coefficient w Objectif function Evaluate the combined solution (GT QAP) NO Choose the best solution Fig. 1 Algorithm of proposed method T-FLP The used encoding is an indirect one, i.e. a building procedure is necessary to evaluate and generate the corresponding solution. For generating the crossovers, we use a data oriented approach in order to improve the quality of the obtained solutions. The objective function consists in maximizing the quantities inside the groups of machine and reducing these quantities outside of it. Finally the machines and products are grouped following their data volume. In the second step, we have obtained solutions using an ant colony optimization mixed with a guided local search method, the particularity is that the results of the step 1 becomes a constraint for this step. In the third step, we evaluate a global solution which aggregates the objective functions of both step 1 and step 2, then a check is done. If the loop has reached the maximum number of cells then it stops and a sorting is done on the global evaluation of each iteration, otherwise it goes back to step 1 with an increased number of cells. Algorithm 1 Structure of the methods YMAY1 and YMAY2 1: for i=1 to number of cells do 2: Resolve Group Technology problem. 3: Resolve Facility layout problem with the constraint of group technology solution (using rule 1 or rule 2). 4: Evaluate the Global evaluation function. 5: end for 6: Print the best solution of global evaluation This paper is organized as follows: in Literature review an overview of the literature is presented. The Problem description and formulation describes our modeling problem. Resolution method describes the Genetic Algorithm and the Ant Colony Optimization. Computational experimentation presents the results with comments. In Industrial application, we present the application to a real industrial problem. A conclusion is done in the last section.

3 Literature review The aim of paper is to solve a facility layout problem using an initial step (Step 1) of decomposition. This decomposition is made by solving the machine part grouping problem with volume (MPGV) based on a group technology problem (GT). We present in this section a literature review about a group technology approach and its resolution methods, and then on the facility layout problem and its resolution methods. The group technology algorithms used for workshop design was initiated by Mc Auley (1972). The main objective is to obtain families of products and their assignment to machines cells. For this type of problems, a binary matrix values machine-products is used as input data to identify the products used by the machines. Other data may be included such as the order and duration of operations, the alternative plan flow, the machine capacity, the intercell transport costs, the acquisition costs of machines. The most treated criterion in literature is the minimization of travel between cells. This requires minimization of exceptions and voids in the matrix. The exceptions define the intercells flow, and voids penalize the density of cells. More criteria can be found as cost minimization in balancing the content of cells, or balancing flows intercells. As constraints, simple approaches do not consider any one, whereas other approaches impose limits on cells sizes or the number of cells, capacity of machines, budget constraints, or number of operators. According to these criteria and constraints, three types can be found in the literature. The first type is to group only products in families (King and Nakornchai 1982) or machines in cells (Murugan and Selladurai 2007). The second one is to assign products to families and then create machines cells. This type aims to obtain an effective grouping, bringing together at first products to groups with a similarity coefficients or dissimilarity and in second, groups machines in cells. The third type aims to group products and machines in cells at the same time. This last type is much studied, and there are several methods such as mathematical approaches, methods based on the coefficient of similarity or dissimilarity, and metaheuristics. The similarity and dissimilarity approach (Adenzo-Diaz et al. 2001) allow to identify groups of similar products and assign them to cells of machines. The advantage of this approach is to reduce the setup time, the smaller lots, reducing the time inventory. The grouping of machines allows to reduce the transfer of material between cells. This method was initiated by Mc Auley (1972). He uses it grouping pairs of machines. It was used later by Rajagopalan and Batra (1975), Waghadekar and Sahu (1984), Vannelli and Kumar (1986), Seifoddini and Wolfe (1986), Vakharia and Wemmerlov (1990). Even if this method was the base of many studies, its efficiency is limited and can not group machine and product at the same time. That is why it is ever used as the first phase of study (Rajamani et al. 1990). This approach is widely used for industrial problem cases. The analytical approach is most commonly used for grouping machines into cells. Jeon and Leep (2006) summarized the different studies of this type. Choobineh Jeon and Leep (2006) proposed a solution in two stages. The first one includes products according to their range, the second stage includes the machines in cells. Rajamani et al. (1990) use analytical approach to grouping machines and products simultaneously under the constraints of alternatives machines. Rajamani et al. (1992) has for the same problem developed a mathematical model that takes into account the number of products in each cell and maximum amounts that can be produced in each cell. Adil et al. (1993) have proposed a new approach that takes into account two very important aspects: investment and cost of operations simultaneously during the construction of cells. They designed for it a new mathematical approach. It is the only paper which considers also the setup time and the downtime. Jeon et al. (1998)have proposed an approach based on the Adil et al. work s. Adil et al. (1993), another mathematical approach in two phases. Adil and Gosh (2005) have developed an algorithm which includes similar machines in cells. There are several studies which used a metaheuristic approach in the literature. Genetic algorithms are the most used method. There are also neural networks or a particle swarm algorithms (Andrés and Lazano 2006), which are expanding. Chan et al. (1998), have developed a genetic algorithm, which treats the problem of grouping machines and products in cells. Kutub Uddin and Shanker (2002) were interested by the machine part grouping problem with an alternative operating range. Ravichandran et al. (2002)have demonstrated the interest of fuzzy and genetic algorithms for the grouping of products for optimizing sequences, taking as a constraint flexible workshop. Muruganandam et al. (2005) used a genetic algorithm for a group technology and incorporated in the last part of the algorithm a tabu search, to try to achieve or come as close as possible to the optimum. Mahapatra and Pandian (2008) have suggested to use genetic algorithms with a new objective function and use a specific coding of the initial population. Chan et al. (2006) have developed a genetic algorithm in two levels, with alternative machines constraints. The first level of the method treats the machine part grouping problem and the second one solves the sequential organization of cells problem. The facility layout problem usually occurs for a reorganization of an existing structure or a new one. Recently a full description of layout problem and resolution approach was made by Drira et al. (2007). The paper describes workshop characteristics on the layout resolution approach, and different formulation of layout problem. FLP is known to be NP-Hard (Hani et al. 2007), and is usually formulated

4 as a quadratic assignment problem (QAP). Loiola et al. (2000) have described all the problems formulated as QAP. Other formulations exist such as mixed integer programming (Amaral 2006; Konak et al. 2006; Tavakkoli-Maghaddam et al. 2007), and a graph theoretic model (Loiola et al. 2000). For solving the FLP, several approaches are used. Drira et al. (2007) present exact methods (Meller et al. 2007; Tavakkoli-Maghaddam et al. 2007) and approximated ones. For the approximate class method, we quote for example, (Chiang et al. 2006), they have developed a tabu search algorithm. They used a neighborhood search based on the exchanging of two locations of facilities using a long term memory structure, a dynamic tabu list size, an intensification criteria and diversification strategies. Chwif et al. (1998) used a simulated annealing algorithm to solve the layout problem with ratio facility sizes aspect. Genetic algorithm was used by different authors such as Wang et al. in Some other works using the genetic algorithm are cited in a survey of Drira et al Ant colony optimization has also been considered for solving FLP. Solimanpur et al. (2005) developed an ant algorithm for sequences dependent single row machine. Baykasoglu et al. (2006) proposed an ant colony algorithm for solving unconstrained and budget constrained dynamic layout problems. Hybrid metaheuristics has also been used. Hani et al. (2007) used an ant colony optimization mixed with a guided local search for solving an industrial layout problem. Mahdi (2000), has resolved the FLP using a simulated annealing mixed with an exact method for minimizing the material handling cost. Following the methods used to solve the facility layout problem, we notice that several formulations exist and the most used is the quadratic assignment problem. As constraint, it takes into account the size of workshop and machines, fixed or flexible location, the structure with one or multiple floor. To solve the facility layout problem, we can find exact and metaheuristic methods. The most used is the genetic algorithm and recently a very efficient ant colony algorithm was developed which done the optimum solution for a large number of problems (Hani et al. 2007). We notice in the literature that there is no method which dealing with a group technology for assigning machine in location in the workshop. So our contribution focuses on the study of a new approach that can give an efficient machine-product grouping and an interesting assignment of machine in a workshop. This has a various interest since in the practical way, the manager have less than difficulties to schedule and organize his production and for the technical way, we use a new resolution method to the FLP with a first decomposition method. For the two aims there are no method. Problem description and formulation The aim of our study is to operate on the strategic and tactical plan to reorganize a workshop. It is an iterative method with three evaluations in each iteration. The first evaluates the quality of the cells created in the first step. In the second step, we evaluate the quality of the assignment of the machines in the locations with taking into account those belonging to the same cell. And finally, the last evaluation at the third step is a global one which describes the quality of the solution. It combines the two evaluations opered before using a specific coefficient. The iterative aspect is made using a loop on cells which needs to fix the maximum number of cells at the beginning. This number inform the machines which number of cells are available. The solutions obtained can use the fixed number of cells or less. The limit of the loop is equal to the number of machines. This loop on cells allows ensuring a good exploration of the solution space regarding to the cell number. This section will define the mathematical models used to represent our problem. At first the mathematical model of the group technology problem will be described. The second subsection presents the quadratic assignment problem, and finally the global evaluation is developed. Group technology problem The goal of solving a group technology problem is to minimize inter-cell movements. A binary matrix is usually used as input data. In our problem, a new evaluation is applied with a volume matrix. The mathematical model used to solve our problem was proposed first by Chan et al. (2006). We have changed the objective function to take into account the intercell volume transported. Maximum and minimum number of machines per cells and the maximum number of cells has been integrated as constraints. The objective function value is: Max Z MPGV = C int Q int (1 C int )Q ext (1) where C int [0, 1] is the weight coefficient of internal volume of production which is generally considered between {0.5,...,0.7} (Adil and Gosh 2005; Chan et al. 1998, 2006; Yalaoui et al. 2008). The production volume inside the cells Qint: Q int = L P k=1 j=1 i=1 M Q ij x ik y jk (2) where: i, j, k: are the indices, respectively, of machines, products and cells. M, P, L: are respectively, the maximum number of machines, products, and cells.

5 x ik and y jk : are equal to 1 if machine i, respectively product j, is assigned to cell k, and 0 otherwise. Q ext calculates the inter-cell transported volume: Q ext = Mi=1 Pj=1 Q ij Lk=1 x ik y jk 2 Q ij represents the flow matrix of volume of products handled by machines. As constraints: i, L k=1 x ik = 1 each machine must be i, L k=1 y ik = 1 assigned only in one cell. each product must be Lk=1 Mi=1 x ik <(β 1 M) assigned only in one cell. number of machines per cell must not exceed i, L k=1 x ik >β 2 β 1 M. each cell must contain at least β 2 machines to ensure the cell density. β 1 [0, 1] and β 2 [1, M] which is proportional to the cell number L(to create). The value of β 1 is determined following the maximum number of machines which we want to obtain in each cell. For example, β 1 = 0.5 means that the maximum number of machines in each cells must not exceed 50% of the total number of machines. The value of β 2 is the minimum number of machine in each cell that it should be created. For example, β 2 = 3 means that the minimum number of machines in each cell must not be less than 3 machines. The following section presents the second step of our method which is a solution of assigning machines to locations solved as a quadratic assignment problem. Quadratic assignment problem The quadratic assignment problem has been traditionally used for a facility layout problem. Loiola et al. (2000) described the problem formulation. In a QAP, first introduced by Koopmans and Beckmann in 1957 (Hani et al. 2007), a set of machines is assigned to a set of locations in order to minimize the total cost associated to distances between locations and flows. The objective function is defined by Loiola et al. (2000): Min Z QAP = n n n i=1 j=1 k=1 p=1 (3) n f ij d kp x ik y jp (4) with: n the number of machines ni=1 x ik = 1, n j=1 y jp = 1 and locations. each machine i, j can be assigned only in one nj=1 x ij = 1, x ij {0, 1} location k,p. each location can contain only one machine. 1 i, j n f ij represents the flow between facilities i and j(number of products exchanged between i and j) d kp represents the distance between locations k and p (in meter) Global evaluation The global evaluation represents the combination of the solution of the group technology problem, which maximizes the volume inside the cells, and the solution of the QAP which minimizes the total distance between machines. The objective is to maximize the intracellular volumes while minimizing the distances expressed at the following equation: Max F = wf GT (1 w)f QAP (5) where w [0, 1] is the weight coefficient of the importance of group technology compared to the facility location, which is fixed by the decision maker. Z GT F GT = Z MAXGT (6) F QAP = Z QAP Z MAXQAP (7) where Z GT Z MAXGT Z QAP Z MAXQAP is the best solution obtained by the Z MPGV (Eq. 1), using the MPGV method for solving the GT problem. is the solution obtained for one cell, which is the maximum value of the solution. is the best solution obtained by Z QAP (Eq. 4), using the ACOGLS method for solving the QAP problem. is the maximum value of the QAP. It is calculated with: i, k, x ik = 1; j, p, y jp = 1 (8)

6 Equation 8 defines that each machine takes all places. This definition allows us to calculate a upper born, where we relax the second constraint of the Eq. 4. Fig. 2 Chromosome encoding step1 and step2 Resolution method This section is dedicated to the presentation of the implemented methods to solve our problem. At first a genetic algorithm used to solve the GT problem is presented, and after the ant colony optimization mixed with a guided local search used to solve the assignment of machines to the location is described. We have chosen a genetic algorithm since it has given very interesting results for solving a group technology problem and the machine part grouping problem (Chan et al. 2006; Yalaoui 2008). Ant colony optimization mixed with a guided search has (ACOGLS) proved its effectiveness for solving a facility layout problem (Hani et al. 2007). That is why we link the two algorithms. Genetic algorithm (GA) for solving GT problem The GA used for solving the problem is based on the works of Chan et al. (1998, 2006). Several changes have been done on the original algorithm (they are defined in bold in Algorithm 2). This section presents the genetic algorithm, the different changes, and its parameters. Algorithm 2 Modified GA algorithm 1: Step 1: Generate random initial population of n chromosomes. 2: Step 1.1: Assign randomly machine to cells. 3: Step 1.2: Sorting the products according to their volumes 4: decreasing. 5: Step 1.3: Assign the products to the cells following their 6: machines ranges. 7: for each iteration do 8: Step 2: Calculate the objective function `for each chromosome 9: Step 3: Check if all the solutions of the population are achievable, else penalize them: 10: Step 3.1: Check that each machine contains at least one product in his cell, which uses it. 11: Step 3.2: Check all products that contain in their cells at least one machine in its cell, part of their ranges. 12: Step 4: Penalize the infeasible solutions 13: Step 5: Create the new population: 14: Step 5.1: Random selection of the best chromosomes from the value of the fitness function 15: Step 5.2: Apply a cross using a data oriented approach on the chromosomes selected with a probability. 16: Step 5.3: Apply a mutation on chromosomes selected with a probability. 17: Use the new chromosomes in the new generation. 18: Print the best solution if the criterion for judgment is reached, else go to step 2. 19: end for Fig. 3 Chromosome encoding step 3 Encoding chromosome Each chromosome in the population represents a solution for the GT problem. Each chromosome is composed of two parts, the first for machines and the second one for products. The size (number of genes) of a chromosome is the sum of the number of machines and the products. Each gene of a chromosome contains an integer number that represents the indices of cell in which machines or products are assigned. We obtain the initial solution with the Algorithm 3: Algorithm 3 Algorithm for initial solution 1: for Each chromosome do 2: Step 1: Assign randomly machine to cells (Table1). 3: Step 2: Sort the product according to their volumes decreasing. 4: Step 3: Assign the product to the cells following their machines ranges (Table2). 5: end for The Fig. 2 shows that the machine 1 is assigned to the cell 2, machine 2 is assigned to the cell 1, and the machine 3 is assigned to the cell 1. The products are not yet assigned. Once the products are sorted according to their volumes decreasing, they are assigned to the cells following their machines ranges (Step 3). The final chromosome is obtained (Fig. 3) with product 1 assigned to the cell 2, product 2 assigned to the cell 1, product 3 assigned to the cell 1, product 4 assigned to the cell 3. Fitness evaluation The value provided by the fitness function where its formulation is presented in the Eq.1 aims to maximize the volume Table 1 Parameters of GA and ACOGLS GA ACOGLS Parameters Value Parameters Value Number of iterations 500 N 40 Populationsize 50 α 0.7 C int 0.7 max-iter 40 P croi 0.8 π 0 0 P mut 0.2

7 Table 2 Results with w = 0.5 ENUM YMAY2 z gt z flp z gt z flp GAP gt GAP flp Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr

8 Table 3 Results with w = 1 ENUM YMAY2 Fig. 4 Parents: select point for crossing Fig. 5 Childs: machine part Fig. 6 Childs: Products exchanged between the cells. This evaluation allows to selecting or rejecting a solution. Once the function is calculated for each chromosome, the solutions of the population are sorted in no decreasing order and the best solution will be saved. This one has the greater value than the other solutions which means that the volumes exchanged between the cells for this solution selected are less than the other ones. Reproduction The reproduction uses a crossing process which consists in obtaining two children by crossing one or more genes from parents. For solving our problem, data oriented approach has been used. A cross point is made on the machines, and the products are assigned according to their job sequences and their volume as made to create to initial solution. When the crossing is made on machine part of the chromosome, the Step 1.2 and 1.3 are applied on the product part of the chromosome. In the following example the values of the two solutions presented in the Figs. 4, 5, 6 are randomly generated. In the Fig. 4, the problem contains 5 machines and 7 products. Table 3 shows that the machine 4 and machine 5 are selected for crossing. In Fig. 5 the crossing operation was operated in the part of machine, and product is not changed yet. Fig. 6 shows the reassignment of products in cells following their machine range. For Child 1,theproduct 1 and the product 6 are assigned to cell 3, and the product 4 and 7 are assigned to cell 1. ForChild 2,theproduct 2 and 6 are assigned to cell 3, and the product 5 is assigned to cell 1. Mutation The objective of the mutation is to avoid the algorithm to fall into a local optimum. Two points are randomly selected on the chromosome. In our case, random mutation on the machine is z gt z gt z flp GAP gt Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr generated to disturb solutions. Products are assigned according the machines cells. Stopping criteria In the literature two types of stopping criteria exist. Either the system is fixed relative to the number of generations, otherwise the search process stops when the best objective function is not improved for a fixed number of generations. For our solution, the two criteria are considered together.

9 At each iteration, the algorithm verifies if the maximum number of generations (equal to 500) is reached, otherwise it verifies if the best objective function is not improved for a fixed number of iterations (equal to 100). Ant colony optimization Solution construction Consider a population of N ants. Each ant k (k = 1,..., N) constructs a solution by assigning resources to locations. The solution is constructed using a tabu list which takes into account the resources already allocated. For solving our problem, the initial solution must take into account the GT solution. When the resources are assigned to the location, verification must be made to confirm that all the resources belonging to the same cell are closed. The algorithm ACOGLS presents ant colony optimization combined with guided local search. This algorithm has been modified to check the conditions of the rule 1 and rule 2 for taking into account the GT solution. Rule 1 which is part of YMAY1 method, uses in its calculations the position of machines. As explained in the Algorithm 5, when the cells are created in the first step, the position of the machines in the cells are stored. In each solution obtained of each iteration, a test is made to verifiy if all the machines which belonging the same cells are always taken, otherwise, the solution is panalized. Rule 2 (Algorithm 6) which is part of YMAY2, uses in its calculations a counter. The rule starts with the machine that is in the first position, it detects a cell which it belongs. Knowing that the number of machines in each cell is saved, if the counter advance and can not find the same number successively, then the solution is rejected. Algorithm 4 Algorithm of ant colony optimization mixed with guided local search 1: Backup GT solution 2: Store the machine part of chromosome which obtained the best solution in group technology. 3: Store the number of machines in each cell. 4: Step 1: Initialization of parameters. 5: for Each iteration do 6: for all ants do 7: Step 2.1: Assign tasks to locations with the given assignment probability, 8: Step 2.2: Check the condition (rule1 or rule2) 9: end for 10: Step 2.3: Perform the guided local search GLSQAP, 11: Step 2.4: Update the pheromones, 12: Step 2.5: Return to Step2.1 until a stopping criterion is satisfied. 13: end for 14: Step 3: Store the best solution. Algorithm 5 Check condition with rule 1 1: for Each cells do 2: for Each machine belonging the cell do 3: store the position number of the machine 4: end for 5: end for 6: for Each cells do 7: for (i = 1 To Nbmachine) do 8: for ( j = i + 1 To Nbmachine) do 9: if position i position j )>(Nbmachine i) then 10: reject the solution 11: end if 12: end for 13: end for 14: end for Algorithm 6 Check condition with rule 2 1: // number_machine[c[i]] : number of machine in the cell i. 2: // machinecell[ j] : the cell of the machine j 3: number = 1, counter = 0 4: for Each machine do 5: C[number] = the cell of the machine number 1 6: for j = number to number_machine [C[number]] + number do 7: if machinecell[j] = C[number] then 8: counter = counter + 1 9: end if 10: end for 11: if counter = number_machine [C[number]] then 12: reject solution 13: number = maximum_number_of_machine 14: end if 15: number = number + number_machine [C[j+1]] 16: end for Selection procedure An ant k with k = (1,...,N) chooses resource i to be assigned to location 1 by the following probability: pil k = α τ il + (1 α) η il (9) i =tabu k (α τ il + (1 α) η il ) where α is the parameter which affects importance to quantity of pheromone and to the visibility. It allows to calculate the cost of assigning a resource i to the location l. The visibility is defined as: η il = 1 (10) 1 + C C defines the cost associated with the assignment (i, l) as: l 1 C(i, l) = ( f r(s)i d sl + f ir(s) d ls ) (11) s=1 where r denotes a permutation of resources under construction. This formula means that the assignments with smaller contribution to the objective function would be more desirable for selection. That means also that adding a resource i to a location l depends on all the resources assigned previously.

10 Pheromone updating The pheromone updating mechanism is represented by the following equation: τ il (t) = λτ il (t 1) + k τ k il (12) where τ il (t) is the quantity of pheromones associated with the assignment of the resource i to location l for each ant k for the iteration t. As an ant chooses this assignment, the quantity increases. The parameter λ is a scaling factor. A large λ results in quick convergence to a local optima solution. It s used to penalize the worst solutions of the iteration (t 1). its value varies between 0.7 and 0.9. τil k = Best f it k fit[k] denotes the magnitude of change in the trail level of an assignment through ant. As seen, the smaller is the fitness solution fit[k] obtained by ant k, the more would be the increment in trail levels selected by ant k. Local search The method 2-opt for local search which is adapted to QAP (Hani et al. 2007) has been chosen. This method applies to a given solution all possible permutations of pairs of machines. The permutation giving the minimal cost is selected as a local minimum next to the starting solution. This process is repeated until no improvement is observed. In order to limit computation time during the exchanges, we made the following simplification: if the exchange is done between the elements π i and π j of the permutation δ, the difference in the objective function value will be then: δ (π,i, j) = (d ii d jj )( f πi π i f π j π j ) + (d ij d ji )( f π j π i f πi π j ) + (d ki d kj )( f πk π j f πk π i ) k =i, j + (d ik d jk )( f π j π k f πi π k ) (13) Hani et al. (2007) cite that this simplification was made by (Gambardella et al. 1997) for implementing the HAS-QAP (Hybrid ant system for solving a quadratic assignment problem). Diversification Diversification consists on erasing all the informations contained in the pheromone trail by reinitializing the pheromone trail matrix and generating randomly a new current solution for all the ants. Guided local search The guided local search occurs when the local search reaches a local optimum. It changes the objective function by increasing the penalties, associated with features of s return the local optimum. The choice of GLS to penalize features of s is made by comparing the utility (util(s, f i )) of each one and penalize those who has the maximum value. It is defined below: util(s, f i ) = I i (s) c i(s) (14) 1 + p i with: I i (s) indicates if the features is present in the current solution or not. c i (s) gives the cost of having f i in s. p i initially set to 0, used to penalize the occurrence of f i in local minima. GLS uses an augmented cost function, to allow it to guide the local search algorithm out of the local minimum, by penalising features present in that local minimum. The augmented cost function is defined as: n h(s) = g(s) + λ I i (s).p i (15) with: g(s) λ i=1 the cost function parameter used to alter the diversification of the search of solutions Hani et al. (2007) present the corresponding GLS for the QAP. The features f i,πi of a solution s corresponds to the assignment of machine i to the location π i. The cost related to features f i,πi depends on the interaction of the task i with all other tasks of the solution s. The cost is given by: c(i,π i ) = n f ij D πi π j (16) j=1 The value λ adapted to QAP is below: ni=1 nj=1 λ f ij n nj=1 i=1 D ij = n 4 (17) The GLSQAP algorithm could be summarized as follows (Hani et al. 2007): Computational experimentation The method is implemented using Visual C on an Intel Core Duo with 1.6 Ghz CPU speed. The genetic

11 Algorithm 7 GLSQAP algorithm 1: Calculation of λ 2: The best solution s = initial solution s. 3: Perform a local search 2-opt with respect to the augmented cost function, s* is found as the solution having the lower augmented cost. If cost (s ) <cost(s ), replace s by s*. Find the assignment (feature) of s* having the maximum utility, let it be f i,πi for example. Increase the corresponding penalty: p i,πi = p i,πi : Return to step 3 until a given stopping criterion is satisfied. 5: s is the best solution found for the original problem. algorithm is tested on a population between 20 and 400 chromosomes, and fixed at 50. Number of iterations is tested between 100 and and fixed at 500. Probability of crossing is defined as 0.8(Chan et al. 1998), mutation as 0.2(Chan et al. 2006) and C int as 0.7. The ACOGLS is tested on a population between 10 and 100 and fixed as 40. α which contributes to make a balance between the choice adopted by the whole of the ants, is tested between 0.5 and 0.8 and set at 0.7. π 0 (Initial pheromone rate), and max-iter (maximum iteration) is tested between 10 and 100 and fixed at 40. The minimum value of the loop on cells is fixed to 2 cells and maximum value to maximum number of machines for each problem. To verify the quality of our method with the optimal solutions, the enumeration is implemented on 35 instances for solving the GT problem and 45 instances for the FLP. This exact method enumerates all the search space and selects the optimal one. The results are shown in Tables 2, 3 and 4 which use as parameters w = 0.5,w = 1 and w = 0.5 respectively. PR i j is a problem randomly generated where i defines the number of machines, and j defines the indices of the instance. The different matrix are generated using a uniform distribution flow between 0 and 10, volume between 4 and 1000 and distance between 2 and 45. z gt, defined in Eq. 1, maximizes the volume belonging to the cells and z flp, defined in Eq. 4, minimizes the distance between the machines. F 1 represents the value of global evaluation of our problem using the method YMAY1 and F represents the global evaluation of our problem using YMAY2 method. The two methods are compared on 90 instances. The w variation shows the interest of our method. So: If w = 0.5 the GT and layout problems have the same interest. If w = 1 the solution of the group technology is considered. If w = 0 the solution of the assignment machine is considered (FLP). Table 4 Results with w = 0 ENUM YMAY2 z flp z gt z flp GAP flp Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr

12 All results tables contain different gaps and defined as follow: GAP gt = z gt(enumeration) z flp (YMAY2) z gt (enumeration) (18) GAP flp = z flp(enumeration) z lfp (YMAY2) z flp (enumeration) (19) GAP F = F 1 F F (20) GAP nbc = nbc YMAY1 nbc YMAY2 nbc YMAY2 (21) GAP compt = compt YMAY1 compt YMAY2 (22) compt YAMAY2 The GAP gt allows us to compare the solution of the z gt of the enumeration method with the z gt of YMAY2 method. The GAP flp allows us to compare the solution of the z flp of the enumeration method with the z flp of YMAY2 method. The GAP F allows us to compare the solution of the F1 (the global evaluation) of the YMAY1 method with the F (the global evaluation) of YMAY2 method. The GAP nbc compares the average number of cells created by the YMAY1 versus YMAY2. The GAP compt compares the computational time of YMAY1 with YAMAY2. The Table 2 shows the solution obtained with w = 0.5. We notice that for GT, our method is optimal for the majority of the instances. The machine assignment solution has an average GAP value of 3.6%, and we obtain the optimal solution for 63% of the instances. For w = 1, the results in Table 3 show that the optimal solution is obtained for the majority of the problem instances. Due to the computational time constraint of the complete enumeration, the YMAY2 method is tested on 35 problem instances. The solutions obtained have an average GAP value of 1.5%, and we obtain the optimal solution for 70% of problem instances. The Table 4 with w = 0, where the assignment machine in YMAY2 is considered (FLP) and compared with the complete enumeration, shows that the optimal solution is obtained for the small problem instances. The solutions obtained have an average GAP value of 1.7%. GAP F in Tables 5 and 7, gives the GAP between the global evaluation of YMAY1 and YMAY2 which are compared on 90 instances. The results obtained by YMAY2 improve the ones obtained by YMAY1, with an average value of GAP equal to: For w = 0 in Table 5: we notice an improvement on the global evaluation of YMAY2 with an GAP equal to 1% and it is faster (regarding the GAP compt ) than YMAY1 with a GAP equal to 41%. This can be provide on the number of cells created by the two methods, we note a difference equal to 31% (Table 6). The average number of cells created by YMAY2 for each problem size (which contain 5 instances) varies between 2.2 and 4.8, and for YMAY1, the values vary between 2 and 7.4. This difference directly impacts the quality of the solution. For w = 0.5, the methods give the same results for ten instances (PR 3 j to PR 4 j ). Over all the instances, YMAY2 is better than YMAY1 with a GAP equal to 11% (Table 7). The Table 8 shows that YMAY2 is faster with a GAP equal to 43% and the number of cells is less than YMAY1 with a GAP nbc equal to 5%. We notice that we have a small difference in the number of cells but the result of the global evaluation shows that YMAY2 is the best, This is due to the rule 2 of taking into account the solutions of the GT problem. For w = 1, the results are not shown, because, the solution takes into account just the solution of the MPGV. Knowing that it is the same for the two methods (YMAY1 ans YMAY2), so the solutions are the same in the two cases. To verify the quality of our approach T-FLP compared to standard approaches Group technology or facility layout problem, we have tested values of w [0, 1] and analyzed solutions. Fig. 7 shows the evolution of the evaluation function according to w for the three methods: YMAY1, YMAY2, and ACOGLS. The third method (ACOGLS) solves just the layout problem. The methods are tested on PR10 with ten machines and products. Fig. 7 shows that for w = 0, where the methods tends to minimize the distances between machines without addressing the group technology, ACO- GLS method obtains the best solution. YMAY2 is close to results of ACOGLS. For w = 1 (Fig. 8), the solution obtained by YMAY2 is better than those proposed by YMAY1 and ACOGLS. For w = 0.5, (Fig. 7), where we combine the maximization of volumes and the minimization of distances, we see that YMAY2 gives very good results and better than the ant colony optimization and the YMAY1. If we analyze all the values of w () in details, we note that YMAY2 provides better solutions than YMAY1 and AC- OGLS methods for w = 0.1. For w [0.3; 0.5], there is a marked improvement compared to YMAY1 and it is far ahead ACOGLS. From w [0.5; 1], we notice that the YMAY2 and YMAY1 method is closer. This can be explained by the interest of bringing the solution to the group technology where we see that both methods give the optimal solution for w = 1. Industrial application With over 85 years experience in the design, manufacture and sales of hose clamps, CAILLAU company has earned a solid reputation in the automotive world. With the aim of continuous improvement and innovation, the company would expand its premises, and reorganize its entire structure

13 Table 5 Comparison between YMAY1 and YMAY2 (F vs F1) [w = 0] F1 F GAP F1 F GAP Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Average 0.01

14 Table 6 Comparison between YMAY1 and YMAY2 (CompT and Nbc) [w = 0] YMAY1 YMAY2 GAP nbc CompT nbc CompT GAP nbc GAP Comp Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Average Table 7 Comparison between YMAY1 and YMAY2 (F vs F1) [w = 0.5] F1 F GAP - F1 F GAP Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr

15 Table 7 continued F1 F GAP - F1 F GAP Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Average 0.11 Gloabel evaluation ACOGLS YMAY1 YMAY Fig. 7 Comparison w [0, 1] w value YMAY 2 method has been applied on this industrial case. The steps followed are as shown in the Fig. 1 with the data presented in the previous paragraph. The loop on cells varied between 3 and 36. The first step creates the cells using the formulation of the Eq. 1. In the second one the machine is assigned to the 36 locations selected using the Eq. 4, and finally, in the step 3, the solutions are evaluated using the formulation of the Eq. 5. The obtained solution of group technology shows three cells which is a very interesting with all volumes equal to pallets is transfered inside the cells and no transfer between cells. As shown in the Table 9, this solution provides an improvement of pallets transferred between cells compared to the current solution. Compared to the actual solution, our algorithm YMAY2 produces an improvement with an average GAP equal to 28% compared to the facility location and 34.5% on the total distance traveled by forklifts. machines. So the industrial problem consists of 36 machines and 47 products. The volumes used include two years of production shipped in pallets. The number of location is equal to the number of machines (Figs. 9,10; Table 9). Conclusion In this paper a new workshop design approach called T-FLP is developed. The main idea is to place machines

16 Table 8 Comparison between YMAY1 and YMAY2 (CompT and Nbc) [w = 0.5] YMAY1 YMAY2 GAP nbc CompT nbc CompT GAP nbc GAP Comp Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Average Fig. 8 Comparison with w [0, 1] details Global evaluation ACOGLS YMAY1 YMAY w value = [0,0.2] Global evaluation Global evaluation ACOGLS YMAY1 YMAY w value = [0.3,0.6] ACOGLS YMAY1 YMAY w value = [0.7,1]

17 Fig. 9 Initial matrix for the GT Fig. 10 Solution of grouptechnology Table 9 A new solution values Solution Current Gap (%) obtained solution Objectif function value Traveled distance busy by the forklifts (Km. palette) Traveled distance empty by the forklifts (Km. palette) Total Distance (Km. palette) Occupancy times of forklifts (hours) in a workshop while taking into account technology information. The new approach combines a Group Technology Problem (GTP) which is an initial step of decomposition and a Facility Layout Problem (FLP). The goal of this work is to come as close as possible to industrial conditions by considering resource assignment. The GT solution proposed to group similar entities according to their job sequences, and their volumes exchanged, which reduce certainly the storage area. A new iterative methods YMAY1 and YMAY2 were implemented to solve the problem. The results obtained are very interesting in comparison with the optimal configuration. According to the global evaluation, the comparison with another method is able to see the usefulness of the new approach and to check the quality of the developed methods. This global evaluation shows that managers can combine the technology informations, and