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Automated layout of facilities of unequal areas
Computersind. F~nf Vol. 24, No. 3, pp. 355-366, 1993 Printed in Great Britain. All rights reserved
AUTOMATED
0360-8352/93 $6.00+ 0.00 Copyright © 1993 Pergamon Press Ltd
LAYOUT
OF FACILITIES AREAS
OF UNEQUAL
M. HASAN IMAM and MUSTAtLSANMm College of Engineering, Umm AI-Qura University, Saudi Arabia
(Receivedfor publication 18 February 1993) Al~nt't---Common to the analytical techniques for automated layout of rectangular facilities of unequal areas is the problem of too ra#d movement of the representative Mocks to form a duster. This phenomenon makes the c o n ~ desisns too dependent on the initial layout and the order of movement of the blocks. A concept of "controlled conversence" is introclu__oed_to solve this problem, Convers~ce is controlled by carrying out the optimization with the "envelop blocks" of sizes much larger than the actual facilities. The sizes of the envelop blocks are gradually reduced to the actual sizes of the facilities through the optimization cycles. Test results are Oven to demonstrate the effectiveness of the presented technique.
1. INTRODUCTION
Numerous algorithms have been presented over the past three decades for interactive as well as automated layout design of facilities [1-3, 12-14]. Most of these algorithms were primarily developed for equal-area facilities to be located on a grid structure. Some recently published methods [4--6] are also based on the same assumption. Relatively little research work has been reported concerning the practical problem of optimizing the layout of rectangular facilities of unequal areas. The traditional approach for solving such problems is to consider each facility represented by a collection of small size "unit blocks" such that their total area is (approximately) equal to the area of the facility [7-10]. In this approach the problem size increases in accordance with the total number of unit blocks and also the shapes of the facilities cannot be pre-specified. Furthermore, the resulting layout can have odd shapes [11]. Blair and Miller [12] consider the sizes of facilities without dividing them into unit blocks, but the developed program FLING is extremely interactive and requires user's input in making decisions. Drezner and Wesolowsky [13] have presented analytical methods based on the "univariate" search and the steepest descent method to optimize the location of facilities. The facilities are represented by points and the dimensions are not considered. It requires the presence of some fixed points (i.e. pre-assigned fazilitles) and the layout obtained only gives relative locations of the facilities and must be adjusted to position the actual facilities without overlaps. Drezner [14] has used the Lagrangian differential gradient method to solve the facility layout problem representing the facilities by circular shapes. It is an improved method as compared to other methods which neglect the sizes of facilities during optimization process. The modified shape assumption, however, does not suit the facih'ties with aspect ratios much different from unity. Also, when the actual facilities replace the corresponding circles in the converged design, the facilities may overlap each other. Imam and Mir [15] have presented an analytical technique based on the univariate search method for the general topology optimization problem. It incorporates the actual dimensions of the rectangular blocks in the optimization process. Starting from a random initial design, ~ e blocks move freely, one at a time, in search of their optimal positions without any overlaps. Analytical techniques appear to be quite promising for automated layout design of facilities of unequal areas on a continuous plane. However, a problem common to such techniques is the formation of a cluster of the representative blocks during theinitial iterations of the optimization process. This phenomenon of rapid clusters formation biases the movement of other blocks and makes the converged design too dependent on the initial layout arid the order of movement of the blocks. A solution to this problem is presented by introducing the concept of "controlled convergence" which utilizes "envelop blocks" during the optimization process. In the initial layout the sizes of the envelop blocks are much larger than the sizes of'the actual facilities. Their sizes are gradually reduced to the actual sizes of the facilities through the optimization cycles. 355
356
M. H&s~'~ IMAMand MUSTAtl~N Mat
The presented analytical technique includes the desirable features of both the constructive and iterative improvement procedures. In constructive procedures, facilities are added, one at a time, to the partially formed layout. Since, all the facilities are not present in the partial layout, the "best" location of the incoming facility is determined only with respect to the already placed facilities. The incoming facility, however, has the freedom to locate its position in the entire layout plane. In contrast, in the iterative improvement procedures, all the facilities are present at every stage of the improvement process, but their movement is normally restricted to pairwise or n-way swapping only. In the technique presented here the envelop blocks for all the facilities are present during the optimization process and they also have the freedom to move in the entire plane. The technique has been implemented in a computer program FLOAT. Some test results are given which demonstrate the effectiveness of the technique in obtaining better layout designs. 2. PROBLEMFORMULATION The cost function C to be minimized is the total material handling cost given as [4], n-I I=l j--t+l
where, n = number of facilities, ftj-- flow between facilities i and j, du = distance between facilities i and j. The distance du is a function of the design variables [(x~,y~);(xj, yj)] which are the co-ordinates of the centroids of the rectangular facilities. Three different distance measurements are considered as follows:
Rectilinear distance
d,j= l x , - xjl + ly,- yjI
(2)
Euclidean distance d,j = [(x, -
xj) 2 + (y, - yj)2],/~
(3)
Squared Euclidean distance d,j = (xt - xj) 2 + (Y, - yj)2
(4)
The overlapping of facilities i and j is a function of the positions of the centroids and their dimensions. Facilities i and j will overlap, as shown in Fig. 1, only if both of the following two conditions are satisfied.
I x , - xj[ < ½(L, + LI)
(5)
<½(W,+ Wj)
(6)
[y,-yjl
Where L: and Wi represent the length and width of facility i along X- and Y-axes, respectively. The required working area around a facility may also be incorporated in its dimensions (L, W). Equations (5) and (6) represent constraints for the optimization problem which may be solved either as a constrained non-linear optimization problem or as an unconstrained non-linear optimization problem by introducing penalty function for the ¢,pnstraints. Different from these two commonly used approaches, an optimization strategy is presented in the following which gives the solution of the problem by unconstrained optimization without involving the penalty function. The use of a penalty function is avoided by explicit determination of the infeasible regions which is possible due to the nature of the problem. For other types of optimizatiODproblems the infeasible regions cannot be explicitly defined and therefore a penalty function approach is needed so that the solutions are driven outside the infeasible regions by the penalty on the cost function.
Automatedlayout of facilitiesof unequalareas
1J_
357
x I , Y=)
I
!----."-Z-:":.~-7"".'-~-"
(x, J
I..
yj)
L
I-
J
_1
-I
Fig. 1. Overlappingof facilities. 3. OPTIMIZATIONSTRATEGY The strategy developed for optimizing the initial layout is based on the univariate search method in which only one design variable is varied at a time [16]. This implies that only one fadlity is moved at a time either along X-axis or F-axis. An iterative procedure is used for plating each fadlity at its optimal position in the direction of movement. The steps of the procedure are given below. Step 1. An initial design is obtained by selecting a set of random values for the design variables such that the distance measure dij is much larger than the sizes of the facilities. Thus the initial design consists of facilities placed far from each other. This way the search space is large enough and therefore there are better chances for obtaining a layout closer to the optimal design. Step 2. T h e ordering of the 2n design ~variables, which correspond to the x- and y-coordinates of the facilities, is determined by one of the following two ordering criteria: (1) R a n d o m ordering: T h e ordering of the design variables is determined randomly at the beginning of each iteration cycle. (2) Semi-random ordering: The sequence of movement of facilities is determined in the descending Order of the "flow index" F~, defined for facility i as,
r,= y,/,j
(7)
J-! For each facility, the ordering of its two design variables is detennin~ randomly. Tlds way the ordering of the 2n design variables is completely specified at the beginning of the optimization process. Step 3. The value of the selected design variable xs which gives the minimum value of the cost function is calculated as x f by the following equation [17]: x* =
1/)23C I + b31C2 -I- bt2 C3 2 a23Ci -I- a31C2 + at2 C3
(8)
where, a F = xp - x,, b~,_ xp 2 - x , 2 and Ci, C2, C3 are the values of the cost function with the x-coordinates of the centroid of facility i at points xt, x2, x3 along the horizontal direction of movement. These points are selected such that C~,>~C2 ~<(73. For horizontal movement, the y-coordinate of the centroid of facility i remains unchanged~ Step 4. A check is made using equations (5) and (6) to determine if any overlaps occur for x~= x f If there are no overlaps, then x* is in the feasible region and gives the optimal position for the x-coordinate of the centroid of facility/. Otherwise, step 5 is carried out to move the facility i to its best position outside the infeasible region.
358
M. H ~
IMAMand MUSrAmAN Met
J
Ca)
k
M
(b) Fig. 2. (a)Optimal position of facility i ignoring overlaps. (b) Facility i at its best non-overlapping portion. Step 5. In case of an overlap, as shown in Fig. 2(a), facility i is moved out of the infeasible region to its best non-overlapping position. Since the cost function increase as facility i is moved along x-axis in either direction from its optimum position, the best non-overlapping position of facility i is that which corresponds to a boundary-to-boundary contact with the overlapped facilityj as shown in Fig. 2(b). If this move does not cause overlaps with other facilities, as is the case in Fig. 2(b), the facility i has achieved its optimal position. If overlap occurs, similar moves are made until a feasible solution is found. The procedure outlined above is applied to all facilities for their horizontal movement. A similar procedure is used for the vertical movement of each facility. One iteration cycle is complete when the optimal positions of all the facilities in the current layout have been determined along the horizontal as well as vertical direction. Iteration cycles are repeated until convergence.
4. LIMITATIONSAND DIFFICULTIES The strategy presented above does allow easier implementation as compared to the penalty function approach presented in [15] and is unique in a way since it solves the unconstrained optimization problem without penalty f ~ n . However, there is one peculiar drawback which is common to both techniques. This is due to the fact that in the first few moves two blocks come close together andin the subsequent moves other blocks form a cluster without any space between the blocks as shown in Fig. 3. These blocks in the cluster tend to hold their positions and the other blocks cannot find their optimum positions mainly because of the following reasons.
Automated layout of facilities of unequal areas
359
[]
, #Z,3SI
ir
I
]
D
D
Fig. 3. Formation of a cluster after few initial moves.
(a) If a block's optimum position is inside the cluster it is impossible to obtain it because it will cause overlaps. (b) The cluster, having a higher value of total flow index as compared to individual blocks, biases the moving blocks towards itself. Thus the converged design is too dependent on the initial layout and the ordering of the design variables. Solution to this problem requires controll~ convergence and avoidance of formation of a cluster of facilities during the initial phase of opUmizatton. A technique to carry out this is presented m the following and is a key in obtaining improved layout designs. 5. C O N T R O L L E D C O N V E R G E N C E
Cluster formation during the iteration cycles can be avoided by allowing the facilities to take their relative optimum positions while maintaining distances between them so that a facility may be placed between two other facilities. Thus free and relatively unbiased movement in the design space may take place. For this purpose the idea presented here is to envelop each facility by a larger size block called "envelop block". The dimensions of the envelop blocks arc determined by multiplying the dimensions of the facilities with a factor called the "magnification factor". The optimization is then carried out for these envelop blocks rather than the actual facilities. The converged solution, at the completion of the first phase of the optimization process, will therefore have a cluster formation of envelop blocks as shown in Fig. 4(a), but the facilities, shown hatched in the figure, will remain apart from each other depending upon the magnification factor. The sizes of the envelop blocks are then reduced by lowering the magnification factor. This causes free spaces between the envelop blocks as shown in Fig. 4(b). The optimization process is then repeated for the second phase using the reduced-size envelop blocks. This whole cycle is repeated until the sizes of the envelop blocks become equal to the actual sizes of the facilities. This results in controlled convergence and gives better results as shown in the following section. 6. C O M P U T E R I M P L E M E N T A T I O N A N D E V A L U A T I O N
A computer program FLOAT (Facilities Layout Optimization using Analytical Technique) was written to implement the proposed technique. The program flowchart is shown in Fig. 5. The input to the program includes the dimensions of the facilities (Li, Wi) and the flow matrix ~j]. The user is prompted to select the type of distance measurement, specify the ordering criterion, and to input CAIE 24/3---C
360
M. HASAr~IMAMand MUSTAH~NMm
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l []
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1
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I I l
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(b) Fig. 4. (a) Cluster of envelop blocks at the completionof phase I. (b) Free spaces betweenthe envelop blocks at the beginningof phase II. the ma~,nification factors for different phases of the optimization process. The output is a graphical display of the optimal layout design. Options such as printing the coordinates of the centroids of the facilities and viewing the layout at any stage of the optimization process are also included. Various test problems were run to evaluate the performance of FLOAT. Two of these test problems are described below.
Test problem No. 1 The program FLOAT has been developed for facilities of unequal areas to be optimally placed on a continuous plane. For equal-area facilities to be located on a grid structure, heuristic methods are more appropriate since the facilities can be simply swapped without resulting in any overlaps.
Automated layout of facilities of unequal areas
361
Read number of facilities (NF), their sizes and the flow matrix
Number of design variables NDVS --- 2*NF
Start
)
~Optimization) Generate random initial design
Input magnification factor (FMAG)
I
Build "envelop blocks" (calculate sizes)
]
Find costs C1 C2 and C3 for three values of the Ith D.V. such that C1 > C2 < C3 Find optimal value of Ith D.V. (Eq. 8)
Construct array for ordering of design variables
I
I
No
+
Find value of Ith D.V. for no overlaps
I=O
i i i (Ith D.V.)
~Return ) Call optimization routine
Yes
storedcsin/
on output file and display layout on option
No No
Yes
Yes =
Fig. 5. Flowchart for the computer program FLOAT.
f ~.~ Stop
)
[
M. H/dAN IMAM and MUSTAHSAN Mm
362
Table I. Comparison of results for test problem No. I Program Optimal cost Best design Mean Std. deviation Heuristic (4) 506.00 541.07 14.57 FLOAT 504.96 517.80 9.87
Furthermore, the layout remains compact and no space is wasted as "dead space" as is the case with analytical methods. In spite of these factors, the program FLOAT has been quite successful with equal-area facilities, as is shown by the following test problem. The data for this test problem of 12 facilities is taken from [4]. In [4] a multigoal cost is considered. For the purpose of comparison, the weight W 1 was set to zero so that the cost function value given in [4] corresponds to the total flow cost as used in this paper. The program was run ten times using different initial designs. Table 1 compares the results obtained by FLOAT with those given in [4] for the case of W 1 set to zero. The best design obtained by FLOAT has about the same cost as that of the best design in [4], but there is a noticeable reduction in the mean and standard deviation values. The coordinates of the centroids of the facilities for the initial and the optimal layout designs are given in Figs 6(a and b).
Test problem No. 2 A test problem of twenty facilities was designed to evaluate the performance of FLOAT for facilities of unequal areas. The flow matrix and the dimensions of the facilities are given in the Appendix. The initial design was obtained by randomly placing the envelop blocks whose dimensions were determined by using a magnification factor of 25. After 6 iteration cycles of the
COST= 334607.1
ITERATION NO,= 0 FACILITY
1 2 3 4 5 6 7 8 9 i0 ii 12
XCNTR
YCNTR
4515 396 1072 84 3795 683 1948 26 2767 008 2940 957 967,3434 4068,899 4566,882 1651,415 1028,188 1018,093
309.1094 2970.865 3514.736 738.1047 4703.929 44,25588 1208,392 4266.036 403,1773 2926.09 916.2166 3426.271
(a)
ITERATION NO.= 12 FACILITY 1 2 3 4 5 6 7 8 9 10 11 12
COST=
XCNTR 2515.004 2514,989 2514.004 2512.989 2513.989 2516.004 2516.004 2513.989 2512.989 2514,989 2515.004 2513.989 (b)
504,9562 YCNTR 1886.485 1885,485 1887.004 1887.485 1885.011 1887.485 1886.485 1888.004 1886.485 1888.485 1887.485 1886.011
Fig. 6. Coordinates o f the centroids o f the facilities for test problem No. 1. (a) Initial random layout. (b) Optimal layout design.
Automated layout of facilities of unequal areas
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(c) Fig. 7. (a) Ouster of envelopblocksat the completionof phase I for test problem No. 2. (b) Ouster of envelop blocks at the completion of phas¢ II. (c) Optimal layout design. optimization process a cluster of envelop blocks was formed as shown in Fig. 7(a). The actual facilities, shown hatcbed in the figure, are enclosed by the envelop blocks and therefore remain apart from each other. The magnification factor was then changed to 5 to reduce the sizes of the envelop blocks. This caused frvv spaces between the envelop blocks for starting the socond phase of the optimization process. After 6 more iteration cycles a cluster of envelop blocks was again formed, as shown in Fig. 7(b). The magnification factor was then lowered to one so that the envelop blocks took the actual sizes of the facilities. The optimization process was then repeated and continued until it converged after 7 more iteration cycles. The converged design is shown in Fig. 7(c). For the purpose of clarity, different scales have been used for plotting Figs. 7(a, b and c). Comparing the converged design of actual facilities as shown in Fig. 7(c) with the converged design of envelop blocks as shown in Fig. 7(a), it is obvious that controlled convergence has ¢uabled a number of facilities to move to their optimal positions inside the cluster formed by the envelop
ITERATION NO.= 0 FACILITY 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
COST=
870250.2
XCNTR
YCNTR
4263.3 2790,197 2500.406 1072.157 3712.112 396~.117 1575.736 3507.861 3560.227 3033.134 4243.195 1998.42 3625.444 3879.225 203.0029 3203.857 2696.655 26.71067 3368.972 2329.042 ~)
4124.346 530,5148 2582.593 3218.473 3795.124 3617.332 1213.111 2279.862 3982.834 1014.954 2854.586 75.28056 3039.081 832.4484 1091,359 769.6562 2402.192 2741.851 276.9908 3383.384
~g.B(a)
Automated layout of facilities of unequal areas
ITERATION NO.= 19
COST=
2544.409 2544.409 2543.909 2539.909 2541.909 2547.409 2545.909 2541.909 254~.409 2538.909 2541.909 2544.409 2546.409 2538.909 2541.909 2542.409 254~.409 2541.409 2539.409 2542.909
3251.69
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1264.933 YCNTR
XCNTR
FACILITY
365
3249. 482 3247.374 3247 278 3254 778 3249 482 3249 482 3243 278 3254 69 3252 177 3245 278 3252.69 3251.982 3243. 982 3251. 778 32~7.278 3246. 982 3249. 278 3249. 278 3249. 278 (b)
Fig. 8. Coordinates of the centroids of the facilities for test problem No. 2. (a) Initial random layout. (b) Optimal layout design.
blocks. The coordinates of the centroids of the facilities for the initial and the optimal layout designs are given in Figs. 8(a and b). The rectilinear distance meas~ement as given in equation (2) was used for calculating the cost function. The ordering of the design variables was determined using the semi-random ordering criterion as described earlier. Optimal designs were obtained for ten different initial designs to evaluate the performance of FLOAT. The same dam with the same random seeds for the initial layouts were utilized to obtain optimal designs using a general purpose topology optimization program TOPOPT [15]. A comparison of the results is summarized in Table 2. It is clear from the results that FLOAT can obtain better designs with reduced dependence on the initial layouts. The results are also displayed graphically in Fig. 9 by using "performance comparison graphs" (PCG's) to help visualize the superiority of FLOAT over TOPOPT. 7. CONCLUSIONS
An improved analytical technique for the automated layout design of rectangular facilities of unequal areas has been presented. The main objective is to minimize the effect of the initial layout and the order of movement of facilities on the converged designs. This is achieved by controlling the TDPrIPT
-p u1 o (J
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2
3
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Trlai No. Fig. 9. Performance comparison graphs (PCG's) for test problem No. 2.
366
M. HASANIMAMand MUSTAHSANM m Table 2. Comparison of results for test problem No. 2 Program Optimal cost Best design Mean Std. deviation TOPOPT [15] 1320.72 1395.6 45.67 FLOAT 1264.94 1333.8 27.51
convergence so that a cluster of facilities does not form in the early iterations of the optimization process. The convergence is controlled by starting the optimization with envelop blocks of sizes much larger than the actual facilities. The sizes of the envelop blocks are reduced at each iteration. This opens up the formed cluster and provides spaces between the blocks for further movement. The final optimization cycle is run with blocks' sizes equal to the actual facilities. This approach is in sharp contrast to some other analytical techniques which solve the optimization problem by considering the facilities as points only and disregard their sizes during the optimization process. Test results given in
the paper demonstrate the effectivenessof the presented technique in obtaining improved layout designs. REFERENCES 1. R. R. Levary and S. Kalchik. Facilities layout--a survey of solution procedures. J. Computers Indust. Engng 9, 141-148 (1985). 2. A. Kusiak and S. S. Heragn. The facility layout problem. Fur. J. Opl Res. 29, 229-251 (1987). 3. L. R. Foulds. Techniques for facilities layout: deciding which pairs of activities should be adjacent Mgmt Sci., 29, 1414-1426 (1983). 4. V. K. Khare, M. K. Khare and M. L. Neema. Combined computer-aided approach for the facilities design problem and estimation of the distribution parameter in the case of multignal optimization. J. Computers lndust. Engng 14, 465-476 (1988). 5. D. T. Connolly. An improved annealing scheme for the QAP. Eur. J. Opns Res. 46, 93-100 (1990). 6. A. Houshyar. Computer aided facility layout: an interactive multi-goal approach. J. Computers Ind. Engng. 20, 177-186 (1991). 7. M. R. Ziai and D. R. Sule. Microcomputer facility layout design. Y. Computers Ind. Engng 15, 259-263 (1990). 8. R. Allenbach and M. Werner. Facility layout problem. J. Computers Ind. Engng 19, 290-293 (1990). 9. J. A. Svestka. "MOCRAFT: a professional quality microcomputer implementation of CRAFT with multiple objectives. Y. Computers Ind. Engng 18, 13-22 (1990). 10. Y.A. Hosni, T. S. Atkins and G. E. Whitehouse. "MICROCRAFT (IIE software---plant layout package). Unpublished technical report. 11. V. Charumongkol. Interactive microcomputer graphic methods for smoothing CRAFT layouts. J. Computers Ind. Engng 19, 304-308 (1990). 12. E. L. Blair and S. Miller. An interactiveapproach to facilitiesdesign using microcomputers. J. Comput. Ind. Engng
9, 91-102 (1985). 13. Z. Drezner and G. O. Wesolowsky. Layout of facilities with some fixed points. J. Comput. Ops. Res. 12, 603-610 (1985). 14. Z. Drezner. A new method for the layout problem. J. Opns Res. 28, 1375-1384 (1980). 15. M. H. Imam and M. Mir. Nonlinear programming approach to automated topology optimization. J. Computer-Aided Design 21, 107-115 (1989). 16. R. L. Fox. Optimum Methods for Engineering Design. Addison-Wesley, Massachusetts (1971). 17. D. G. Luenberger. Introduction to Linear and Nonlinear Programming. Addison-Wesley, Massachusetts (1973).
APPENDIX The flow matrix and the dimensions of facilities for test problem No. 2 are given in the following: Flow matrix 0 3 0 0 4 2 0 0 4 0 0 5 3 0 5 0 0 1 0 0 3 0 1 0 1 2 5 0 3 0 0 0 2 0 3 0 3 1 2 3 0 1 0 4 0 0 3 0 0 0 1 0 0 0 0 0 5 0 2 3 0 0 4 0 4 0 0 l 5 3 0 2 0 0 4 5 0 1 0 0 4 1 0 4 0 0 0 0 I 4 1 5 0 0 3 2 0 5 0 4 2 2 0 0 0 0 3 0 0 5 0 0 3 0 0 0 2 0 0 0 0 5 3 0 0 3 0 0 0 0 0 0 4 0 2 0 3 2 0 1 000 I 0 0 0 0 0 0 2 0 0 5 0 4 0 1 0 0 4 3 0 5 1 0 0 0 0 3 0 5 0 0 0 2 0 0 0 0 0 0 0 3 4 5 0 0 3 0 0 5 0 1 2 4 0 3 4 0 0 0 1 0 1 0 0 2 0 0 0 0 0 5 5 4 0 4 3 1 5 0 0 2 5 0 0 0 5 5 0 0 5 0 2 0 0 1 0 0 3 2 0 0 0 3 4 0 0 0 0 5 0 0 3 0 2 0 0 0 0 0 0 0 0 0 0 5 0 1 5 0 0 0 0 5 0 5 1 0 5 3 0 4 3 0 2 0 0 2 5 2 3 0 0 0 1 4 3 3 0 0 0 5 2 0 0 4 2 4 4 0 0 5 0 0 4 5 0 0 0 3 5 0 0 2 3 0 0 0 0 0 2 0 1 4 0 0 1 5 I I 0 1 5 0 2 1 0 3 4 1 0 5 4 5 0 0 4 1 0 2 2 0 0 0 0 0 0 4 3 0 0 1 3 0 1 4 0 5 0 3 3 0 4 0 1 0 0 0 1 0 0 0 3 0 5 1 5 0
No.
Dimensions (L, W)
I 2 3 4 5 6 7 8 9 I0 II 12 13 14 15 16 17 18 19 20
(1,2) (2,2) (l,l) (2,3) (3,3) (2,2) (2,1) (2,3) (3,2) (3,3) (2,3) (1,2) (3,2) (3,3) (3,3) (2,2) (3,2) (2,2) (2,2) (2,1)
AUTOMATED
0360-8352/93 $6.00+ 0.00 Copyright © 1993 Pergamon Press Ltd
LAYOUT
OF FACILITIES AREAS
OF UNEQUAL
M. HASAN IMAM and MUSTAtLSANMm College of Engineering, Umm AI-Qura University, Saudi Arabia
(Receivedfor publication 18 February 1993) Al~nt't---Common to the analytical techniques for automated layout of rectangular facilities of unequal areas is the problem of too ra#d movement of the representative Mocks to form a duster. This phenomenon makes the c o n ~ desisns too dependent on the initial layout and the order of movement of the blocks. A concept of "controlled conversence" is introclu__oed_to solve this problem, Convers~ce is controlled by carrying out the optimization with the "envelop blocks" of sizes much larger than the actual facilities. The sizes of the envelop blocks are gradually reduced to the actual sizes of the facilities through the optimization cycles. Test results are Oven to demonstrate the effectiveness of the presented technique.
1. INTRODUCTION
Numerous algorithms have been presented over the past three decades for interactive as well as automated layout design of facilities [1-3, 12-14]. Most of these algorithms were primarily developed for equal-area facilities to be located on a grid structure. Some recently published methods [4--6] are also based on the same assumption. Relatively little research work has been reported concerning the practical problem of optimizing the layout of rectangular facilities of unequal areas. The traditional approach for solving such problems is to consider each facility represented by a collection of small size "unit blocks" such that their total area is (approximately) equal to the area of the facility [7-10]. In this approach the problem size increases in accordance with the total number of unit blocks and also the shapes of the facilities cannot be pre-specified. Furthermore, the resulting layout can have odd shapes [11]. Blair and Miller [12] consider the sizes of facilities without dividing them into unit blocks, but the developed program FLING is extremely interactive and requires user's input in making decisions. Drezner and Wesolowsky [13] have presented analytical methods based on the "univariate" search and the steepest descent method to optimize the location of facilities. The facilities are represented by points and the dimensions are not considered. It requires the presence of some fixed points (i.e. pre-assigned fazilitles) and the layout obtained only gives relative locations of the facilities and must be adjusted to position the actual facilities without overlaps. Drezner [14] has used the Lagrangian differential gradient method to solve the facility layout problem representing the facilities by circular shapes. It is an improved method as compared to other methods which neglect the sizes of facilities during optimization process. The modified shape assumption, however, does not suit the facih'ties with aspect ratios much different from unity. Also, when the actual facilities replace the corresponding circles in the converged design, the facilities may overlap each other. Imam and Mir [15] have presented an analytical technique based on the univariate search method for the general topology optimization problem. It incorporates the actual dimensions of the rectangular blocks in the optimization process. Starting from a random initial design, ~ e blocks move freely, one at a time, in search of their optimal positions without any overlaps. Analytical techniques appear to be quite promising for automated layout design of facilities of unequal areas on a continuous plane. However, a problem common to such techniques is the formation of a cluster of the representative blocks during theinitial iterations of the optimization process. This phenomenon of rapid clusters formation biases the movement of other blocks and makes the converged design too dependent on the initial layout arid the order of movement of the blocks. A solution to this problem is presented by introducing the concept of "controlled convergence" which utilizes "envelop blocks" during the optimization process. In the initial layout the sizes of the envelop blocks are much larger than the sizes of'the actual facilities. Their sizes are gradually reduced to the actual sizes of the facilities through the optimization cycles. 355
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The presented analytical technique includes the desirable features of both the constructive and iterative improvement procedures. In constructive procedures, facilities are added, one at a time, to the partially formed layout. Since, all the facilities are not present in the partial layout, the "best" location of the incoming facility is determined only with respect to the already placed facilities. The incoming facility, however, has the freedom to locate its position in the entire layout plane. In contrast, in the iterative improvement procedures, all the facilities are present at every stage of the improvement process, but their movement is normally restricted to pairwise or n-way swapping only. In the technique presented here the envelop blocks for all the facilities are present during the optimization process and they also have the freedom to move in the entire plane. The technique has been implemented in a computer program FLOAT. Some test results are given which demonstrate the effectiveness of the technique in obtaining better layout designs. 2. PROBLEMFORMULATION The cost function C to be minimized is the total material handling cost given as [4], n-I I=l j--t+l
where, n = number of facilities, ftj-- flow between facilities i and j, du = distance between facilities i and j. The distance du is a function of the design variables [(x~,y~);(xj, yj)] which are the co-ordinates of the centroids of the rectangular facilities. Three different distance measurements are considered as follows:
Rectilinear distance
d,j= l x , - xjl + ly,- yjI
(2)
Euclidean distance d,j = [(x, -
xj) 2 + (y, - yj)2],/~
(3)
Squared Euclidean distance d,j = (xt - xj) 2 + (Y, - yj)2
(4)
The overlapping of facilities i and j is a function of the positions of the centroids and their dimensions. Facilities i and j will overlap, as shown in Fig. 1, only if both of the following two conditions are satisfied.
I x , - xj[ < ½(L, + LI)
(5)
<½(W,+ Wj)
(6)
[y,-yjl
Where L: and Wi represent the length and width of facility i along X- and Y-axes, respectively. The required working area around a facility may also be incorporated in its dimensions (L, W). Equations (5) and (6) represent constraints for the optimization problem which may be solved either as a constrained non-linear optimization problem or as an unconstrained non-linear optimization problem by introducing penalty function for the ¢,pnstraints. Different from these two commonly used approaches, an optimization strategy is presented in the following which gives the solution of the problem by unconstrained optimization without involving the penalty function. The use of a penalty function is avoided by explicit determination of the infeasible regions which is possible due to the nature of the problem. For other types of optimizatiODproblems the infeasible regions cannot be explicitly defined and therefore a penalty function approach is needed so that the solutions are driven outside the infeasible regions by the penalty on the cost function.
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Fig. 1. Overlappingof facilities. 3. OPTIMIZATIONSTRATEGY The strategy developed for optimizing the initial layout is based on the univariate search method in which only one design variable is varied at a time [16]. This implies that only one fadlity is moved at a time either along X-axis or F-axis. An iterative procedure is used for plating each fadlity at its optimal position in the direction of movement. The steps of the procedure are given below. Step 1. An initial design is obtained by selecting a set of random values for the design variables such that the distance measure dij is much larger than the sizes of the facilities. Thus the initial design consists of facilities placed far from each other. This way the search space is large enough and therefore there are better chances for obtaining a layout closer to the optimal design. Step 2. T h e ordering of the 2n design ~variables, which correspond to the x- and y-coordinates of the facilities, is determined by one of the following two ordering criteria: (1) R a n d o m ordering: T h e ordering of the design variables is determined randomly at the beginning of each iteration cycle. (2) Semi-random ordering: The sequence of movement of facilities is determined in the descending Order of the "flow index" F~, defined for facility i as,
r,= y,/,j
(7)
J-! For each facility, the ordering of its two design variables is detennin~ randomly. Tlds way the ordering of the 2n design variables is completely specified at the beginning of the optimization process. Step 3. The value of the selected design variable xs which gives the minimum value of the cost function is calculated as x f by the following equation [17]: x* =
1/)23C I + b31C2 -I- bt2 C3 2 a23Ci -I- a31C2 + at2 C3
(8)
where, a F = xp - x,, b~,_ xp 2 - x , 2 and Ci, C2, C3 are the values of the cost function with the x-coordinates of the centroid of facility i at points xt, x2, x3 along the horizontal direction of movement. These points are selected such that C~,>~C2 ~<(73. For horizontal movement, the y-coordinate of the centroid of facility i remains unchanged~ Step 4. A check is made using equations (5) and (6) to determine if any overlaps occur for x~= x f If there are no overlaps, then x* is in the feasible region and gives the optimal position for the x-coordinate of the centroid of facility/. Otherwise, step 5 is carried out to move the facility i to its best position outside the infeasible region.
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(b) Fig. 2. (a)Optimal position of facility i ignoring overlaps. (b) Facility i at its best non-overlapping portion. Step 5. In case of an overlap, as shown in Fig. 2(a), facility i is moved out of the infeasible region to its best non-overlapping position. Since the cost function increase as facility i is moved along x-axis in either direction from its optimum position, the best non-overlapping position of facility i is that which corresponds to a boundary-to-boundary contact with the overlapped facilityj as shown in Fig. 2(b). If this move does not cause overlaps with other facilities, as is the case in Fig. 2(b), the facility i has achieved its optimal position. If overlap occurs, similar moves are made until a feasible solution is found. The procedure outlined above is applied to all facilities for their horizontal movement. A similar procedure is used for the vertical movement of each facility. One iteration cycle is complete when the optimal positions of all the facilities in the current layout have been determined along the horizontal as well as vertical direction. Iteration cycles are repeated until convergence.
4. LIMITATIONSAND DIFFICULTIES The strategy presented above does allow easier implementation as compared to the penalty function approach presented in [15] and is unique in a way since it solves the unconstrained optimization problem without penalty f ~ n . However, there is one peculiar drawback which is common to both techniques. This is due to the fact that in the first few moves two blocks come close together andin the subsequent moves other blocks form a cluster without any space between the blocks as shown in Fig. 3. These blocks in the cluster tend to hold their positions and the other blocks cannot find their optimum positions mainly because of the following reasons.
Automated layout of facilities of unequal areas
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(a) If a block's optimum position is inside the cluster it is impossible to obtain it because it will cause overlaps. (b) The cluster, having a higher value of total flow index as compared to individual blocks, biases the moving blocks towards itself. Thus the converged design is too dependent on the initial layout and the ordering of the design variables. Solution to this problem requires controll~ convergence and avoidance of formation of a cluster of facilities during the initial phase of opUmizatton. A technique to carry out this is presented m the following and is a key in obtaining improved layout designs. 5. C O N T R O L L E D C O N V E R G E N C E
Cluster formation during the iteration cycles can be avoided by allowing the facilities to take their relative optimum positions while maintaining distances between them so that a facility may be placed between two other facilities. Thus free and relatively unbiased movement in the design space may take place. For this purpose the idea presented here is to envelop each facility by a larger size block called "envelop block". The dimensions of the envelop blocks arc determined by multiplying the dimensions of the facilities with a factor called the "magnification factor". The optimization is then carried out for these envelop blocks rather than the actual facilities. The converged solution, at the completion of the first phase of the optimization process, will therefore have a cluster formation of envelop blocks as shown in Fig. 4(a), but the facilities, shown hatched in the figure, will remain apart from each other depending upon the magnification factor. The sizes of the envelop blocks are then reduced by lowering the magnification factor. This causes free spaces between the envelop blocks as shown in Fig. 4(b). The optimization process is then repeated for the second phase using the reduced-size envelop blocks. This whole cycle is repeated until the sizes of the envelop blocks become equal to the actual sizes of the facilities. This results in controlled convergence and gives better results as shown in the following section. 6. C O M P U T E R I M P L E M E N T A T I O N A N D E V A L U A T I O N
A computer program FLOAT (Facilities Layout Optimization using Analytical Technique) was written to implement the proposed technique. The program flowchart is shown in Fig. 5. The input to the program includes the dimensions of the facilities (Li, Wi) and the flow matrix ~j]. The user is prompted to select the type of distance measurement, specify the ordering criterion, and to input CAIE 24/3---C
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Test problem No. 1 The program FLOAT has been developed for facilities of unequal areas to be optimally placed on a continuous plane. For equal-area facilities to be located on a grid structure, heuristic methods are more appropriate since the facilities can be simply swapped without resulting in any overlaps.
Automated layout of facilities of unequal areas
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M. H/dAN IMAM and MUSTAHSAN Mm
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Table I. Comparison of results for test problem No. I Program Optimal cost Best design Mean Std. deviation Heuristic (4) 506.00 541.07 14.57 FLOAT 504.96 517.80 9.87
Furthermore, the layout remains compact and no space is wasted as "dead space" as is the case with analytical methods. In spite of these factors, the program FLOAT has been quite successful with equal-area facilities, as is shown by the following test problem. The data for this test problem of 12 facilities is taken from [4]. In [4] a multigoal cost is considered. For the purpose of comparison, the weight W 1 was set to zero so that the cost function value given in [4] corresponds to the total flow cost as used in this paper. The program was run ten times using different initial designs. Table 1 compares the results obtained by FLOAT with those given in [4] for the case of W 1 set to zero. The best design obtained by FLOAT has about the same cost as that of the best design in [4], but there is a noticeable reduction in the mean and standard deviation values. The coordinates of the centroids of the facilities for the initial and the optimal layout designs are given in Figs 6(a and b).
Test problem No. 2 A test problem of twenty facilities was designed to evaluate the performance of FLOAT for facilities of unequal areas. The flow matrix and the dimensions of the facilities are given in the Appendix. The initial design was obtained by randomly placing the envelop blocks whose dimensions were determined by using a magnification factor of 25. After 6 iteration cycles of the
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Fig. 6. Coordinates o f the centroids o f the facilities for test problem No. 1. (a) Initial random layout. (b) Optimal layout design.
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ITERATION NO.= 0 FACILITY 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
COST=
870250.2
XCNTR
YCNTR
4263.3 2790,197 2500.406 1072.157 3712.112 396~.117 1575.736 3507.861 3560.227 3033.134 4243.195 1998.42 3625.444 3879.225 203.0029 3203.857 2696.655 26.71067 3368.972 2329.042 ~)
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ITERATION NO.= 19
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2544.409 2544.409 2543.909 2539.909 2541.909 2547.409 2545.909 2541.909 254~.409 2538.909 2541.909 2544.409 2546.409 2538.909 2541.909 2542.409 254~.409 2541.409 2539.409 2542.909
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Fig. 8. Coordinates of the centroids of the facilities for test problem No. 2. (a) Initial random layout. (b) Optimal layout design.
blocks. The coordinates of the centroids of the facilities for the initial and the optimal layout designs are given in Figs. 8(a and b). The rectilinear distance meas~ement as given in equation (2) was used for calculating the cost function. The ordering of the design variables was determined using the semi-random ordering criterion as described earlier. Optimal designs were obtained for ten different initial designs to evaluate the performance of FLOAT. The same dam with the same random seeds for the initial layouts were utilized to obtain optimal designs using a general purpose topology optimization program TOPOPT [15]. A comparison of the results is summarized in Table 2. It is clear from the results that FLOAT can obtain better designs with reduced dependence on the initial layouts. The results are also displayed graphically in Fig. 9 by using "performance comparison graphs" (PCG's) to help visualize the superiority of FLOAT over TOPOPT. 7. CONCLUSIONS
An improved analytical technique for the automated layout design of rectangular facilities of unequal areas has been presented. The main objective is to minimize the effect of the initial layout and the order of movement of facilities on the converged designs. This is achieved by controlling the TDPrIPT
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convergence so that a cluster of facilities does not form in the early iterations of the optimization process. The convergence is controlled by starting the optimization with envelop blocks of sizes much larger than the actual facilities. The sizes of the envelop blocks are reduced at each iteration. This opens up the formed cluster and provides spaces between the blocks for further movement. The final optimization cycle is run with blocks' sizes equal to the actual facilities. This approach is in sharp contrast to some other analytical techniques which solve the optimization problem by considering the facilities as points only and disregard their sizes during the optimization process. Test results given in
the paper demonstrate the effectivenessof the presented technique in obtaining improved layout designs. REFERENCES 1. R. R. Levary and S. Kalchik. Facilities layout--a survey of solution procedures. J. Computers Indust. Engng 9, 141-148 (1985). 2. A. Kusiak and S. S. Heragn. The facility layout problem. Fur. J. Opl Res. 29, 229-251 (1987). 3. L. R. Foulds. Techniques for facilities layout: deciding which pairs of activities should be adjacent Mgmt Sci., 29, 1414-1426 (1983). 4. V. K. Khare, M. K. Khare and M. L. Neema. Combined computer-aided approach for the facilities design problem and estimation of the distribution parameter in the case of multignal optimization. J. Computers lndust. Engng 14, 465-476 (1988). 5. D. T. Connolly. An improved annealing scheme for the QAP. Eur. J. Opns Res. 46, 93-100 (1990). 6. A. Houshyar. Computer aided facility layout: an interactive multi-goal approach. J. Computers Ind. Engng. 20, 177-186 (1991). 7. M. R. Ziai and D. R. Sule. Microcomputer facility layout design. Y. Computers Ind. Engng 15, 259-263 (1990). 8. R. Allenbach and M. Werner. Facility layout problem. J. Computers Ind. Engng 19, 290-293 (1990). 9. J. A. Svestka. "MOCRAFT: a professional quality microcomputer implementation of CRAFT with multiple objectives. Y. Computers Ind. Engng 18, 13-22 (1990). 10. Y.A. Hosni, T. S. Atkins and G. E. Whitehouse. "MICROCRAFT (IIE software---plant layout package). Unpublished technical report. 11. V. Charumongkol. Interactive microcomputer graphic methods for smoothing CRAFT layouts. J. Computers Ind. Engng 19, 304-308 (1990). 12. E. L. Blair and S. Miller. An interactiveapproach to facilitiesdesign using microcomputers. J. Comput. Ind. Engng
9, 91-102 (1985). 13. Z. Drezner and G. O. Wesolowsky. Layout of facilities with some fixed points. J. Comput. Ops. Res. 12, 603-610 (1985). 14. Z. Drezner. A new method for the layout problem. J. Opns Res. 28, 1375-1384 (1980). 15. M. H. Imam and M. Mir. Nonlinear programming approach to automated topology optimization. J. Computer-Aided Design 21, 107-115 (1989). 16. R. L. Fox. Optimum Methods for Engineering Design. Addison-Wesley, Massachusetts (1971). 17. D. G. Luenberger. Introduction to Linear and Nonlinear Programming. Addison-Wesley, Massachusetts (1973).
APPENDIX The flow matrix and the dimensions of facilities for test problem No. 2 are given in the following: Flow matrix 0 3 0 0 4 2 0 0 4 0 0 5 3 0 5 0 0 1 0 0 3 0 1 0 1 2 5 0 3 0 0 0 2 0 3 0 3 1 2 3 0 1 0 4 0 0 3 0 0 0 1 0 0 0 0 0 5 0 2 3 0 0 4 0 4 0 0 l 5 3 0 2 0 0 4 5 0 1 0 0 4 1 0 4 0 0 0 0 I 4 1 5 0 0 3 2 0 5 0 4 2 2 0 0 0 0 3 0 0 5 0 0 3 0 0 0 2 0 0 0 0 5 3 0 0 3 0 0 0 0 0 0 4 0 2 0 3 2 0 1 000 I 0 0 0 0 0 0 2 0 0 5 0 4 0 1 0 0 4 3 0 5 1 0 0 0 0 3 0 5 0 0 0 2 0 0 0 0 0 0 0 3 4 5 0 0 3 0 0 5 0 1 2 4 0 3 4 0 0 0 1 0 1 0 0 2 0 0 0 0 0 5 5 4 0 4 3 1 5 0 0 2 5 0 0 0 5 5 0 0 5 0 2 0 0 1 0 0 3 2 0 0 0 3 4 0 0 0 0 5 0 0 3 0 2 0 0 0 0 0 0 0 0 0 0 5 0 1 5 0 0 0 0 5 0 5 1 0 5 3 0 4 3 0 2 0 0 2 5 2 3 0 0 0 1 4 3 3 0 0 0 5 2 0 0 4 2 4 4 0 0 5 0 0 4 5 0 0 0 3 5 0 0 2 3 0 0 0 0 0 2 0 1 4 0 0 1 5 I I 0 1 5 0 2 1 0 3 4 1 0 5 4 5 0 0 4 1 0 2 2 0 0 0 0 0 0 4 3 0 0 1 3 0 1 4 0 5 0 3 3 0 4 0 1 0 0 0 1 0 0 0 3 0 5 1 5 0
No.
Dimensions (L, W)
I 2 3 4 5 6 7 8 9 I0 II 12 13 14 15 16 17 18 19 20
(1,2) (2,2) (l,l) (2,3) (3,3) (2,2) (2,1) (2,3) (3,2) (3,3) (2,3) (1,2) (3,2) (3,3) (3,3) (2,2) (3,2) (2,2) (2,2) (2,1)