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A methodology for solving facility layout problem considering barriers: genetic algorithm coupled with A* search

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Abstract

This work proposes a new methodology and mathematical formulation to address the facility layout problem. The goal is to minimise the total material handling cost subjected to production-derived constraints. This cost is a function of the distance that the products should cover within the facility. The first idea is to use the \( {\text{A}}^{ *} \) algorithm to identify the distances between workstations in a more realistic way. \( {\text{A}}^{ *} \) determines the shortest path within the facility that contains obstacles and transportation routes. The second idea is to combine a genetic algorithm and the \( {\text{A}}^{ *} \) algorithm with a homogenous methodology to improve the quality of the facility layouts. In an iterative way, the layout solution space is explored using the genetic algorithm. We study the impacts of the appropriate crossover and mutation operators and the values of the parameters used in this algorithm on the cost of the proposed arrangements. These operators and parameter values are fine-tuned using Monte Carlo simulations. The facility arrangements are all compared and discussed based on their material handling cost associated with the Euclidean distance, rectilinear distance, and \( {\text{A}}^{ *} \) algorithm. Finally, we present a set of conclusions regarding the suggested methodology and discuss our future research goals.

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Appendices

Appendix 1

See Table 7.

Table 7 Experiment layout and sample experimental data (RWS method)

Appendix 2

See Table 8.

Table 8 Parameter sets of the proposed GAs (tournament selection)

Appendix 3

The A* algorithm seeks a new measure of distance (see Fig. 9). It starts by placing the starting node in the open list called current node. The open list contains the list of nodes to be verified. It consists of nodes that have been visited but not explored yet. After inspecting its entire adjacent squares, the starting node is removed from the open list and placed in another list named closed list. The closed list contains all the nodes that, at one time or another, have been considered as part of the solution path. Before switching to the closed list, a node must first pass through the open list. Before being judged as good, it must first be studied by looking at all surrounding squares of the starting point while ignoring squares with obstacles. These studied nodes are also added to the open list. For each of these squares, the current node is saved as its ‘parent square’. The parent square is required when we want to trace the path. To determine if a node is susceptible to be part of the solution path, it is necessary to quantify its quality. To do this, we calculate the distance between the point studied and the last point that was judged as good. We also calculate the distance between the point studied and the point of destination. The sum of these two distances gives the quality of the studied node. This operation will be carried on until the target node is reached.

figure b
Fig. 9
figure 9

Different shortest paths between two facilities: rectilinear, Euclidean, A* Algorithm

Appendix 4

See Table 9.

Table 9 Advantages and disadvantages of different selection strategies

Roulette wheel selection. A chromosome is selected from the mating pool with a probability proportional to its fitness value. A chromosome with a high fitness value has a higher chance of being selected as a parent. Thus, the main idea is to prefer better solutions to worse ones. Consider a circle divided into n slices, where n is the number of individuals in the population. Each individual is assigned a slice of the roulette wheel, which is proportional to its fitness value. A pointer is chosen on the wheel circumference and the wheel is rotated in a repetitive way. The portion of the wheel that is in front of the fixed point is chosen as the parent.

Tournament selection. N individuals are randomly taken up from the population in a tournament selection. A copy of the best individual (based on fitness values) is kept in the mating pool as a parent. The number of individuals competing in each tournament is referred to as the tournament size.

Appendix 5

N point crossover. Almost all of the N point crossover processes use the same strategy. Hereafter, we define the single point crossover. A split line is randomly determined between 1 and (n − 1) genes (see Fig. 10), where n is the number of genes. Each parent is divided into two blocks of alleles. The portion before the split line is exchanged between the parents to create two new solutions, as illustrated in Fig. 10. This crossover operation is analogue to the binary crossover operation of GAs.

Fig. 10
figure 10

a Before single point crossover, b after single point crossover

See Fig. 10.

Appendix 6

Several types of mutation exchange and inversion operators are used at this step to permute two or more workstations. The exchange mutation performs a swap of two randomly selected genes. For instance, in Fig. 11, the two genes (x2, y2) and (x4, y4) are swapped. The inversion operator consists in selecting two genes randomly and inverting the position in the chromosome between these two blocks as shown in Fig. 12.

Fig. 11
figure 11

Exchange operator

Fig. 12
figure 12

Inversion operator

See Figs. 11, 12.

Appendix 7

Pre-treatment phase In this study, we try to determine the most efficient arrangement of equipment in an area of such company that produce four products {A-E-C-H} and five semi-final products {G-D-B-F-I}. The assembly of G and D gives the product A and the product E comes from the assembly of B, F and I. Data of the products of our problem are provided in Table 10.

Table 10 Data of the products of our problem

Appendix 8

See Fig. 13.

Fig. 13
figure 13

Best cost versus number of iterations, roulette operator (a) and tournament operator (b)

Appendix 9

SeeTable 11.

Table 11 Obtained distance among equipment units according to the roulette and tournament operators

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Besbes, M., Zolghadri, M., Costa Affonso, R. et al. A methodology for solving facility layout problem considering barriers: genetic algorithm coupled with A* search. J Intell Manuf 31, 615–640 (2020). https://doi.org/10.1007/s10845-019-01468-x

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