Computers in Industry 36 Ž1998. 125–132
A solution to the facility layout problem using simulated annealing Leonardo Chwif, Marcos R. Pereira Barretto ) , Lucas Antonio Moscato Department of Mechanical Engineering, Polytechnic School, UniÕersity of Sao ˜ Paulo, AÕ. Prof. Mello Moraes 2231, 05508-003 Sao ˜ Paulo, SP, Brazil
Abstract In this paper a solution in the continual plane to the Facility Layout Problem ŽFLP. is presented. It is based on Simulated Annealing ŽSA., a relatively recent algorithm for solving hard combinatorial optimization problems, like FLP. This approach may be applied either in General Facility Layout Problems ŽGFLP. considering facilities areas, shapes and orientations or in Machine Layout problems ŽMLP. considering machine’s pick-up and drop-off points. It has been applied to real-life situations with useful results, indicating the effectiveness of this approach. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Simulated annealing; Facility layout problem; Optimization
1. Introduction The problem of facility layout is to find an optimum relative location of facilities on a planar site. Kouvelis et al. w1x pointed out that the optimum location of facilities is one of the most important issues that must be resolved in early stages of the manufacturing system design. Besides that, the operating expenses may be reduced, increasing productivity. It has been remarked that 20–50% of the total operating expenses in manufacturing are related to material handling and layout w2x. Several formulations have been addressed in literature. Quadratic Assignment Problem ŽQAP. was
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Corresponding author. E-mail:
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first proposed by Koopmans and Beckman w3x. Sanhi and Gonzales w4x have shown its NP-completeness. For details about QAP formulation refer to Lawler w5x, Hillier and Connors w6x, Ligget w7x, and Francis and White w8x. A large number of methods or techniques have been extensively proposed; they can be roughly classified into optimum methods or suboptimum methods. Some authors have attempted to solve this problems using search tree techniques, like Branch and Bound w9,5x. Beam-Search ŽBS. and derived techniques Žlike FBS, DCBS. were also proposed w10,11x. Others have been proposed methods based on graph theory w12,13x. Hierarchical approaches also have been discussed w14,15x. Due to the computational unfeasibility of many formulations several heuristics methods have been developed. These
0166-3615r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 6 - 3 6 1 5 Ž 9 7 . 0 0 1 0 6 - 1
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L. Chwif et al.r Computers in Industry 36 (1998) 125–132
heuristics methods may be classified into two groups: constructive methods and iterative improvement methods; refer to the surveys of Levary and Kalchik w16x and Kusiak and Heragu w17x for a list. Recently some attentions have been focused on a special class of search methods called extended neighborhood search which may be considered as generic heuristics methods Žthey may be applied to many optimization problems.. The great advantage of this methods is to avoid being caught in local optima by sometimes accepting moves that worsen the objective function. The three most known methods in literature are Tabu Search ŽTS. Žsee Ref. w18x., Genetic Algorithm ŽGA. Žsee Ref. w19x. and Simulated Annealing Žsee Ref. w20x.. Heragu and Alfa w21x compared the performance of SA and TS in FLP and they proposed a Hybrid Simulated Annealing ŽHSA. using a modified penalty algorithm. Tam w22x used GA and Wilheim and Ward w23x, Tam w24x, Jajodia et al. w25x, Kouvelis et al. w26x and Suresh and Sahu w27x applied SA with encouraging results. From geometric point of view, FLP may be classified into discrete layout problems and continual layout problems. The former approach divides the site into a rectangular grid where each grid cell is assigned to a facility. If the facilities have unequal areas they could be divided in blocks and each block should have the same area and shape of an individual cell. Although unequal areas may be modeled, irregular shapes are many times generated. In fact like Heragu and Kusiak w28x reported there is still a lack of continual planes approaches when comparing to discrete ones. The latter approach recently has been received more attention and all the facilities may be placed anywhere within the planar site. Refer to Montreuil and Ratliff w29x, Heragu and Kusiak w30x, Heragu w31x, Van Camp et al. w32x, Tam and Li w15x, Tam w22x, Tam w24x, Banerjee et al. w33x, Chhajed et al. w34x and Welgama and Gibson w35x for continual plane approaches. Besides the geometric constraints of facilities Žarea, shapes and orientations., Welgama and Gibson w35x also considered pick-up and drop-off points that are capable of modelling a special case of the GFLP ŽGeneral Facility Layout Problems. usually known as MLP ŽMachine Layout Problem.. This paper is organized as follows: Section 2 describes the proposed problem formulationŽobjec-
tive function and the geometric constraints.; Section 3 briefly reviews the simulated annealing algorithm and presents the choices made for its implementation; Section 4 discusses the experimental results based on tests problems; finally, Section 5 concludes the paper, discussing its major contributions.
2. Problem formulation Each facility may be represented as a rectangular block, considering its size orientation and aspect ratio. Žsee Ref. w15x.. The aspect ratio of block i is defined:
ai s
height of facility i s width of facility i
hi wi
Hence block i may be modeled by its area Ž A i s h i wi ., lower Ž a i l . and upper Ž a i u . bound of a i Ž a i g w a i l , a i u x.. Blocks may be fixed Žwhen they can only be placed at fixed locations. or movable Žwhen they can be placed in any location within the site area.. If block i is fixed, then it is rigid, so a i s a i l s a i u ; otherwise if its movable it will be considered as a free-orientation block. Thus the aspect ratio must satisfy at least one of the two conditions shown below: ai l F ai F ai u 1 ai u
F ai F
1 ai l
The area within the facilities will be located at is bounded by a polygon which edges are parallel either to the x-axis or y-axis. In case of MLP, all blocks Žmachines. are considered as rigid blocks. Additionally, it considers pickup and drop-off points, following the approach taken from Welgama and Gibson w35x. Basically six configurations of machines with different relative position of pick-uprdrop-off points may be specified. Therefore a block can be placed according to differ-
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Fig. 1. Different orientations with respect to each configuration.
ent orientations with respect to each configuration of pick-up and drop-off points can be seen in Fig. 1. The distance measure between block i and block j is adopted rectilinear, thus:
MLP are pick-up or drop-off points. Fig. 2 summarizes the geometric modelling described in this section. The occupied space ratio ŽOSR. is defined as: n
d i j s x i y x j q yi y y j
Ý OSR s
Where Ž x i , yi . and Ž x j , yj . are the measuring points of block i and j, respectively. In GFLP the measuring points are block’s centroids, whereas in
Ak
ks1
CIA
Where n is the number of existing facilities and CIA is the inner area defined by contour.
Fig. 2. Summary of geometric modeling.
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2.1. ObjectiÕe function
3. Proposed procedure
The objective function Ž f . contains basically the total transportation costs that must be minimized. A penalty function is added to the transportation costs to avoid block overlapping. It is necessary since the optimization method adopted solves only unconstrained optimization problems Žsee Section 3.. The fixed costs are discarded because in a long term they are negligible regarding the transportation costs. Thus the objective function takes the form shown below: n
fsa
n
n
is1 js1
Ii j s
½
Ai j s Bi j s
A i j , Bi j ) 0
0 wi q wj
otherwise
2 hi q h j 2
Ii j
is1 js1,i/j
A i j q Bi j
n
y< xi yxj < y < yi y y j
