Ž .Computers in Industry 36 1998 125–132

A solution to the facility layout problem usingsimulated annealing

Leonardo Chwif, Marcos R. Pereira Barretto ), Lucas Antonio MoscatoDepartment of Mechanical Engineering, Polytechnic School, UniÕersity of Sao Paulo, AÕ. Prof. Mello Moraes 2231, 05508-003 Sao Paulo,˜ ˜

SP, Brazil

Abstract

Ž .In this paper a solution in the continual plane to the Facility Layout Problem FLP is presented. It is based on SimulatedŽ .Annealing SA , a relatively recent algorithm for solving hard combinatorial optimization problems, like FLP. This approach

Ž .may be applied either in General Facility Layout Problems GFLP considering facilities areas, shapes and orientations or inŽ .Machine Layout problems MLP 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 opti-mum relative location of facilities on a planar site.

w xKouvelis et al. 1 pointed out that the optimumlocation of facilities is one of the most importantissues that must be resolved in early stages of themanufacturing system design. Besides that, the oper-ating expenses may be reduced, increasing productiv-ity. It has been remarked that 20–50% of the totaloperating expenses in manufacturing are related to

w xmaterial handling and layout 2 .Several formulations have been addressed in liter-

Ž .ature. Quadratic Assignment Problem QAP was

) Corresponding author. E-mail: mrpbarre@usp.br

w xfirst proposed by Koopmans and Beckman 3 . Sanhiw xand Gonzales 4 have shown its NP-completeness.

For details about QAP formulation refer to Lawlerw x w x w x5 , Hillier and Connors 6 , Ligget 7 , and Francis

w xand White 8 .A large number of methods or techniques have

been extensively proposed; they can be roughly clas-sified into optimum methods or suboptimum meth-ods. Some authors have attempted to solve thisproblems using search tree techniques, like Branch

w x Ž .and Bound 9,5 . Beam-Search BS and derivedŽ .techniques like FBS, DCBS were also proposed

w x10,11 .Others have been proposed methods based on

w xgraph theory 12,13 . Hierarchical approaches alsow xhave been discussed 14,15 . Due to the computa-

tional unfeasibility of many formulations severalheuristics methods have been developed. These

0166-3615r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0166-3615 97 00106-1

( )L. Chwif et al.rComputers in Industry 36 1998 125–132126

heuristics methods may be classified into two groups:constructive methods and iterative improvementmethods; refer to the surveys of Levary and Kalchikw x w x16 and Kusiak and Heragu 17 for a list.

Recently some attentions have been focused on aspecial class of search methods called extendedneighborhood search which may be considered as

Žgeneric heuristics methods they may be applied to.many optimization problems . The great advantage

of this methods is to avoid being caught in localoptima by sometimes accepting moves that worsenthe objective function. The three most known meth-

Ž . Ž w x.ods in literature are Tabu Search TS see Ref. 18 ,Ž . Ž w x.Genetic Algorithm GA see Ref. 19 and Simu-

Ž w x. w xlated Annealing see Ref. 20 . Heragu and Alfa 21compared the performance of SA and TS in FLP and

Ž .they proposed a Hybrid Simulated Annealing HSAw xusing a modified penalty algorithm. Tam 22 used

w x w xGA and Wilheim and Ward 23 , Tam 24 , Jajodia etw x w xal. 25 , Kouvelis et al. 26 and Suresh and Sahu

w x27 applied SA with encouraging results.From geometric point of view, FLP may be clas-

sified into discrete layout problems and continuallayout problems. The former approach divides thesite into a rectangular grid where each grid cell isassigned to a facility. If the facilities have unequalareas they could be divided in blocks and each blockshould have the same area and shape of an individualcell. Although unequal areas may be modeled, irreg-ular shapes are many times generated. In fact like

w xHeragu and Kusiak 28 reported there is still a lackof continual planes approaches when comparing todiscrete ones. The latter approach recently has beenreceived more attention and all the facilities may beplaced anywhere within the planar site. Refer to

w x w xMontreuil and Ratliff 29 , Heragu and Kusiak 30 ,w x w x w xHeragu 31 , Van Camp et al. 32 , Tam and Li 15 ,

w x w x w xTam 22 , Tam 24 , Banerjee et al. 33 , Chhajed etw x w xal. 34 and Welgama and Gibson 35 for continual

plane approaches.Besides the geometric constraints of facilities

Ž .area, shapes and orientations , Welgama and Gibsonw x35 also considered pick-up and drop-off points thatare capable of modelling a special case of the GFLPŽ .General Facility Layout Problems usually known

Ž .as MLP Machine Layout Problem .This paper is organized as follows: Section 2

Ždescribes the proposed problem formulation objec-

.tive function and the geometric constraints ; Section3 briefly reviews the simulated annealing algorithmand presents the choices made for its implementa-tion; Section 4 discusses the experimental resultsbased on tests problems; finally, Section 5 concludesthe paper, discussing its major contributions.

2. Problem formulation

Each facility may be represented as a rectangularblock, considering its size orientation and aspect

Ž w x.ratio. see Ref. 15 . The aspect ratio of block i isdefined:

height of facility i hia s si width of facility i wi

ŽHence block i may be modeled by its area A si. Ž . Ž .h w , lower a and upper a bound of ai i i l iu i

Ž w x. Ža g a , a . Blocks may be fixed when theyi i l iu.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 sa sa ; otherwise if its movable it will bei i l iu

considered as a free-orientation block. Thus the as-pect ratio must satisfy at least one of the two condi-tions shown below:

a Fa Fai l i iu

1 1Fa Fia aiu i l

The area within the facilities will be located at isbounded by a polygon which edges are paralleleither to the x-axis or y-axis.

Ž .In case of MLP, all blocks machines are consid-ered as rigid blocks. Additionally, it considers pick-up and drop-off points, following the approach taken

w xfrom Welgama and Gibson 35 . Basically six con-figurations of machines with different relative posi-tion of pick-uprdrop-off points may be specified.Therefore a block can be placed according to differ-

( )L. Chwif et al.rComputers in Industry 36 1998 125–132 127

Fig. 1. Different orientations with respect to each configuration.

ent orientations with respect to each configuration ofpick-up and drop-off points can be seen in Fig. 1.

The distance measure between block i and blockj is adopted rectilinear, thus:

d s x yx q y yyi j i j i j

Ž . Ž .Where x , y and x , y are the measuringi i j j

points of block i and j, respectively. In GFLP themeasuring points are block’s centroids, whereas in

MLP are pick-up or drop-off points. Fig. 2 summa-rizes the geometric modelling described in this sec-tion.

Ž .The occupied space ratio OSR is defined as:n

AÝ kks1OSRsCIA

Where n is the number of existing facilities andCIA is the inner area defined by contour.

Fig. 2. Summary of geometric modeling.

( )L. Chwif et al.rComputers in Industry 36 1998 125–132128

2.1. ObjectiÕe function

Ž .The objective function f contains basically thetotal transportation costs that must be minimized. Apenalty function is added to the transportation coststo avoid block overlapping. It is necessary since theoptimization method adopted solves only uncon-

Ž .strained optimization problems see Section 3 . Thefixed costs are discarded because in a long term theyare negligible regarding the transportation costs. Thusthe objective function takes the form shown below:

n n n n

fsa w d q2vb IÝ Ý Ý Ýi j i j i jis1 js1 is1 js1, i/j

A qB A , B )0i j i j i j i jI si j ½ 0 otherwisew qwi j

< <A s y x yxi j i j2

h qhi j< <B s y y yyi j i j2

n n

wÝ Ý i jis1 js1, j/i

vsn ny1Ž .

where: w is the flowrinteraction between facility ii j

and facility j; v is the mean flowrinteraction; d si j< < < < Ž .x yy q y yy where x , y , ks i, j are thei j i j k k

coordinates of centroid of block k in case of GFLP.< < < < Ž .d s x yx q x yy where x , y andi j p i d j p i p j pk pk

Ž .x , y , ks i, j are respectively, the coordinatesd k d k

of pick-up and drop-off points of machine k, in caseof MLP. w is the width of facility k, ks i, j; h isk k

the height of facility k, ks i, j; x , y are thek k

coordinates of the centroid of block k, ks i, j; a isthe weight of cost function; b is the weight ofpenalty function; n is the number of facilities.

Ž .Notice that the penalty function I is equal toi j

zero if block i and block j are not overlapped andpositive otherwise. Fig. 4 shows the configuration ofthe blocks with respect to overlapping.

As can be seen from Fig. 4a, A )0 and B )0i j i j

and thus I sA qB . Fig. 4b indicates that A -0i j i j i j i j

and B -0 so I s0.i j i j

The factor v, that multiplies the penalty function,was used to avoid the dependence between the weightb and the magnitude of the flowrinteraction.

3. Proposed procedure

3.1. Simulated Annealing

Simulated Annealing is a method based on MonteCarlo Simulation allowing to solve difficult combi-natorial optimization problems. The name comesfrom the analogy to the behavior of physical systemsby melting a substance and lowering its temperature

Žslowly until it reaches the freezing point physical.annealing . This physical annealing was first simu-

w xlated by Metropolis et al. 36 and Kirkpatrick et al.w x37 made the analogy with optimization problems.

Hence let’s suppose that there is a solution spaceŽ . Ž .S set of all solutions and an objective function f

Ž .real function defined on members of S . The pur-Ž .pose is to find a solution or state ,

igS

that minimizes f over S. SA makes use of aniterative improvement procedure which is deter-mined by a neighborhood generation. So first ainitial state is generated. Then SA generates neigh-borhoods at each temperature which is graduallylowered, until the stop criteria is true. SA avoidsbeing trapped in local optima by accepting some-

Žtimes uphill moves move that increases the objec-.tive function value . This acceptance is decided

yd r T Žstochastically with probability of e Metropolis.criterion , where d is the difference of costs between

the current state and its neighbour, and T is theŽ .current temperature control parameter . The higher

the temperature, the higher the probability of uphillmoves.

Some choices must be made for any implementa-tion of SA which determines the cooling schedule.This consists in finding basically the initial value oftemperature T , the temperature function which de-o

termines how the temperature is lowered, the numberŽ .of iteration N Epoch length to be performed at

each temperature or each epoch and the stop criteriato terminate the method. A geometric temperature

Ž . Ž .function is used, i.e., T tq1 sRT t where R isŽ .the cooling ratio 0-R-1 . The number of itera-

tions N is proportional to the size of problem; in thisŽ .case the number of existing facilities n . So NsLn,

( )L. Chwif et al.rComputers in Industry 36 1998 125–132 129

where L is an integer proportionality constant. Thealgorithm proceeds until temperature reaches the fi-nal temperature T , which corresponds in the analogyf

to the frozen temperature. The simulated annealingalgorithm in pseudo code, with the adopted coolingschedule can be seen next:

� 4Generate_state_i; initial stateTsT ;o

RepeatRepeat

Generate_state_j;Ž . Ž .ds f j y f i ;

If d-0 then i:s jŽ . Žyd r T .else if random 0,1 -e then

i:s j;k:skq1;

until ksNsLPn;T :sRPT ;until TFT .f

w xJohnson et al. 38 have pointed out that thetemperature value has direct relationship with theneighborhood acceptance rate. The acceptance rateŽ .AR of a temperature T is defined as:

NrAR T s1yŽ .

N

Where N is the number of the rejected neighbor-r

hoods during the same temperature T. Hence theinitial and final temperature may be chosen by estab-

Ž .lishing their respective acceptance rates AR T andoŽ .AR T . In Section 3.2 it will be shown the specificf

choices in order to apply SA in FLP. For furtherdetails about SA refer to Van Laarhoven and Aartsw x w x20 , Johnson et al. 38 and Eglese.

3.2. Specific choices

A solution in this case is any configuration offacilities on the planar site. The objective function fhas been already discussed in Section 2. The initialsolution may be given or determined at random.Now let’s concentrate how the neighborhood of asolution is determined. Two procedures are proposedthat works interchangeably, i.e., if the previous

neighborhood was generated by one then the nextgeneration will be determined by the other, andvice-versa. One procedure makes a pairwise-ex-change between facilities and it is similar to ‘pair-wise interchange’ procedure in CRAFT program.Note that in this case the aspect ratio and orientationare not changed.

The other procedure makes random moves on theŽplanar site at the four main directions upwards,

.downwards, leftwards and rightwards , or makes ran-dom changes of aspect ratios in case of GFLP ororientations in case of MLP. These two proceduresare described in pseudo-code below:

Choose_random_movable_facility_A;Choose_random_movable_facility_B;xsxb;ysyb;xbsxa;ybsya;xasx;yasy;

Ž . Ž .rr xa, ya , xb, yb are the coordinatesrrof centroid of facilities A and B, respectivelyChoose_random_movable_facility_A;Ž .if changesmove then

Ž .case direction of‘up’: y sy qp;A A

‘down’: y sy yp;A A

‘left’: x sx yp;A A

‘right’: x sx qp;A A

endelse begin

Ž .if problemsGLPF thenŽŽ . .a s randaspect a ,a ,1ra ,1ra ;A Al Au Au Al

rrchoose random aspect ratio within thegivenrrintervalsŽ .if problemsMLP then

Ž .case rotation of908:rotate_908_machine_A;1808:rotate_1808_machine_A;

end;end;

Ž .Observe that in case of procedure 2 on the left , thechoice between a move or change of aspectratiororientation is made with probability of 50%which is the same to the choice between rotation of908 and 1808.

( )L. Chwif et al.rComputers in Industry 36 1998 125–132130

Fig. 3. Example of configurations with respect to overlapping.

Ž .The move step p is defined as a percentage k ofŽ .the maximum polygon height H plus the maxi-

Ž .mum polygon width W —refer to Fig. 3. So:

kps HqwŽ .

100

4. Test and real-life results

4.1. Test problems

Basically six continual plane problems weretested. Problems 1 and 2 were taken from Tam and

w xLi 15 consisting of 12 and 15 facilities problems,respectively. Problems 3 and 4 with respectively, 20

w xand 30 facilities were chosen from Tam 24 in orderto evaluate the algorithms performance in a generalcase of FLP: existence of polygonal contour and

Žfixed facilities in this case as the geometric mod-elling is quite different than that was adopted, all thefacilities areas were reduced by 30% of Tam’s origi-

.nal problems . Problems 5 and 6 were provided byw xWelgama and Gibson 35 that consists of two MLP

problems with 6 and 12 machines, respectively.The algorithm was coded in C language and all

the tests were performed on a Pentium 133 MHzcompatible computer. Results are shown in Table 1,where CF denotes the cost function of the objectivefunction.

w xFor problems 1 and 2, Tam and Li 15 proposedŽan optimum procedure sequential augmented La-

.grangian method , and thus their values are opti-

mum, although it is computationally unfeasible forŽproblems more than 15 facilities a hierarchical ap-

.proach was needed to overcome this problem . Asthey employed a different cost function from here,the layout was generated trough the cost functionpresented in Section 3, and the cost function of thefinal layout was calculated with the one from Tam

w xand Li 15 .When comparing directly the results, it is clear

w xthat those obtained by Tam and Li 15 are better;but analyzing visually the obtained layouts two ten-

Ž .dencies should be noted: 1 alignment of facilitiescentroids, naturally leading to aisles, important to thetransportation of materials in shop floor. Note that nopre-reservation of spaces for aisles was made but SA

Ž .seems to create them when align centroids; 2 mini-Ž . Ž w x.mization of dead-space ratio DSR see ref. 35 ,

providing a more compact layout.Actually the DSR for problem 1 was 23% against

37% of Tam and Li, and for problem 2, 22.5%against 46.5%.

Unfortunately the results cannot be directly com-pared for problems 3 and 4 since the geometric

w xmodelling and cost function proposed by Tam 24were different from the adopted ones. In spite of thisfact, it may clearly be used for further comparisons.These two tendencies that were pointed out previ-ously, could also be confirmed in these two prob-lems.

It should be noted that for problem 5, the obtainedresult is practically the same from that of Welgama

w xand Gibson 35 , with the only difference of a gen-eral rotation of 908. For problem 6, SA provided a

Žsolution with better quality measured in terms of.CF for the same DSR than the construction proce-

w xdure proposed by Welgama and Gibson 35 . This

Table 1Compared results for the test problems

Ž . Ž .P no. CF with SA CF from Refs. Difference % Time s

1 3684 3162 y16.5 2152 7466 5862 y27 2243 9202 y y 3494 18603 y y 5245 417.5 421.5 1 356 5563 5903 6 250

( )L. Chwif et al.rComputers in Industry 36 1998 125–132 131

result points in the same direction of the speculationw xof Ligget 7 , that iterative improvement type algo-

rithms show better performance than constructionalgorithms.

ŽThe computational time for problem 4 30 facili-.ties was less than 3 min in a Pentium 133 MHz

microcomputer, which demonstrates the computa-tional feasibility for larger problems.

4.2. Real-life results

The technique has been applied to the GFLP of atruck plant in Brazil. It has helped the company’sarchitects to define the general layout, studying:Ø general locationØ 1 story vs. 2 story situations

Ž .Ø external gates issues quantity and locationAll studies were performed using the same programused for tests and described above, running in a P133microcomputer in less than 1 min.

5. Conclusions

In this paper a continual plane approach usingsimulated annealing is presented. It addresses somepractical aspects which are infrequently explored inliterature, such as:Ø Facilities with different areas, shapes and orienta-

tions;Ø Any Polygonal format for the border:Ø Fixed facilities;Ø Pick-up and drop-of points, in case of MLP.

The major weakness faced by the proposed SAbased algorithm is its difficulty to avoid overlapping

Ž .in problems with occupied space ratio OSR over75%. This fact however cannot be viewed as adrawback since the OSR seldom reaches 80% inpractical facilities layout problems due to the pres-ence of aisles, dead-spaces, etc.

Although the SA based algorithm is reasonablecomputationally and it shows good results, somemodifications should be proposed as future researchin order to increase its efficiency.

The program developed was only intended to testthe algorithm and to be used as a tool for casestudies. Therefore, it was not integrated to CADtools or other CIM tools, although it could be done

easily. It is planned to include this algorithm in ageneral optimization tools, to be used in layout,scheduling and other applications.

Acknowledgements

The authors would like to acknowledge Mer-cedes-Benz of Brazil for supporting this research for2 years. The authors are also grateful to the Depart-ment of Mechanical Engineering, Polytechnic School,University of Sao Paulo, by providing all facilities˜necessary to this research, and also to FAPESP,CNPq and CAPES for their support.

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Leonardo Chwif graduated in Mechani-Žcal Engineering Mechatronics Special-

.ization in 1992 at the University of Sao˜Paulo and got his MSc Degree in 1994from the same University. He is cur-rently a PhD student at the same Univer-sity. Upon graduation, Mr. Chwif joinedthe Brazilian branch of Mercedes-Benztruck manufacturer. Currently he worksat the Brazilian branch of Whirlpool.His research interests include manufac-turing simulation and optimization.

Marcos Ribeiro Pereira Barretto grad-uated in Electronics Engineering in1983. He got his MSc Degree in 1988and his PhD in 1993, all from the Uni-versity of Sao Paulo. He is currently˜Assistant Professor at the MechanicalEngineering Department of the Poly-technic School of the University of Sao˜Paulo. His research interests includemanufacturing integration and CIMtechnology.

Lucas Antonio Moscato graduated inElectronics Engineering in 1969. He gothis MSc and PhD degrees at the Poly-technic School of the University of Sao˜Paulo in 1971 and 1980, respectively.He is, since 1988, full Professor at theMechanical Engineering Dept. of theUniversity of Sao Paulo. His research˜includes automated manufacturing sys-tems and robotics.