3D facility layout problem

Abstract

Facility layout aims to arrange a set of facilities in a site. The main objective function is to minimize the total material handling cost under production-derived constraints. This problem has received much attention during the past decades. However, these works have mainly focused on solving a 2D layout problem, dealing with the footprints of pieces of equipment. The obtained results have been then adapted to the real spatial constraints of a workshop. This research work looks to take account of spatial constraints within a 3D space from the very first steps of problem solving. The authors use a approach by combining a genetic algorithm with A*, 〈GA,A*〉 research. The genetic algorithm generates possible arrangements and A* finds the shortest paths that products must travel in a restricted 3D space. The application allows to converge to a layout minimizing the total material handling cost. This approach is illustrated by its application on an example inspired by a valve assembly workshop in Tunisia and the results are discussed from two points of view. The first one consists in comparing the effect of the choice of the distance measurement technique on the handling cost. For this purpose, the results of the application of 〈GA,A*〉 are compared with those obtained by combining the genetic algorithm and two of the most commonly used distance measurements in the literature of the discipline, namely the Euclidean distance, 〈GA,Euclidean〉, and the rectilinear distance, 〈GA,rectilinear〉. Our results show that the proposed approach offers better results than those of 〈GA,rectilinear〉 whereas they are not as good as those obtained by the 〈GA,Euclidean〉 approach. The effectiveness of the 〈GA,A*〉 approach is then studied from the perspective of the effect of the algorithm used for the generation of candidate arrangements. The final results obtained from the application of 〈GA,A*〉 are then compared with those of the approach combining particle swarm optimization and A*, 〈PSO,A*〉. This comparison shows that the 〈GA,A*〉 approach obtains better results. Nevertheless, its convergence speed is lower than that of 〈PSO,A*〉. The paper ends with some conclusions and perspectives.

Appendices

Appendix 1

See Figs.

Fig. 12

Best cost versus number of iterations, roulette operator (a) and tournament operator (b) obtained by 〈GA,A*〉

12 and

Fig. 13

Best cost versus number of iterations obtained by 〈PSO,A*〉

13.

Appendix 2

Interested readers can consult the Matlab program by coping and pasting the following link: https://drive.google.com/drive/folders/1ZLYG6uSmjZiuaD88feRzXWJtvH5Omf8c?usp=sharing

Appendix 3

See Table

Table 12 Raw data

12.

Appendix 4

Particle swarm optimization (PSO) algorithm-based on swarm intelligence and evolutionary computation, was originally developed by Kennedy and Eberhart (1995). PSO is a population-based search algorithm and it is inspired by the social behaviour of groups of fish and birds. It is initialized with a population of random solutions. Each solution is called a ‘particle’. Each particle in PSO is associated with a position vector and a velocity vector. The historical behaviours of each particle depends on three factors, which are the previous velocity, the best position of each particle ($P_{best}$), and the swarm best position ($G_{best}$). The population evolves by communicating the best global solution ever found by all particles and the personal best solution ever found by each particle. Therefore, all the particles tend to converge to the best solution quickly. Considering the simplicity and good convergence speed of PSO, we choose to compare our approach 〈GA,A*〉 with 〈PSO,A*〉. In this paper, the position of each particle represents a configuration.

In a D-dimensional search space, The position and the velocity of the $i^{th}$ particle is illustrated by $X_{I}$ = ($x_{i1}$, $x_{i2}$, …., $x_{iD}$) and $V_{i}$ = ($v_{i1}$, $v_{i2} , \ldots ,\,v_{iD}$).

Appendix 5

Interested readers can consult the Matlab program by coping and pasting the following link:

https://drive.google.com/drive/folders/1BsBqqeQ_-DH_JTid0gPRbkxpxpTM-REF?usp=sharing