Mean and standard deviation for the facilities design problem

Computersind. EngngVol. 19, Nos 1-4, pp. 313-317, 1990

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MEAN AND STANDARD DEVIATION FOR THE FACILITIES DESIGN PROBLEM

S. Allen Broughton and Veeravudhi Charumongkol Department of Mathematics and Department of Industrial Engineering Cleveland State University Cleveland, Ohio 44115

ABSTRACT An important problem in facilities design to find an assignment of n facilities to n locations so that total materials handling cost is minimized. For problems of moderate size, suboptimal solutions must be accepted since optimal algorithms are computationally infeasible. If the mean and standard deviation of the layout cost distribution is known, then statistical methods may be used to measure and compare the efficiencies of various suboptimal solutions as well as to monitor the efficiency of the same assignment under changing production environments. In this paper a new, simple algorithm to calculate the exact value of the standard deviation of the layout cost distribution is presented (the mean is easy). This algorithm has a computational efficiency of O(n 2) arithmetic operations for a problem of size n × n, an improvement over previous methods which are either inexact or have a computational efficiency of O(n4). Results of tests verifying the accuracy and claimed efficiency of this algorithm, as implemented on a microcomputer, are also presented (about 0.85 second for a 30 x 30 problem). INTRODUCTION A problem confronting the industrial plant designer is to assign n facilities to n locations in a plant so that the total material handling cost is minimized. Since there are n! possible assignments to consider, algorithms yielding optimal solutions, are computationally infeasible even for moderately large values of n. A number of computationally feasible algorithms, yielding only suboptimal solutions, have been devised, e.g., [AB], [BAV], [HC], [G]. The software package C R A F T ([AB], [BAV]), which is commonly uscd in facilities design, is based on some of these algorithms. Since we have to consider suboptimal solutions it is important to have a measure of the quality of a proposed solution. A commonly used statistical measurc is the efficiency of a proposed assignment which was introduced by Wallace et. al. in [WHT]. We now describe this measure as well as the cost distribution associated to the facilities design problem. Let ~r denote a typical assignment of n facilities to n locations in a plant, i.e., for each i = 1 , . . . , n facility i is assigned to location It(i). There is an associated materials handling cost which is denoted by C~, we defer the explicit definition of C , to formula (5). As we vary through all possible assignments we get a data set consisting of n! C~'s. Let p and a be the mean and standard deviation of this data set, they are defined by: 1

(1)

P ----~.WE g'~'

1 f

where the summations are over all n! possible assignments 7r. If the layout cost distribution is, for example, normal then about 95.4% of the layout costs C~ are with two standard deviations of the mean aand about 99.7% of the layout costs are within three standard deviations of the mean. Thus, it is statistically unlikely that many layout costs smaller than p - 2a or/~ - 3a will be found. This justifies the following measure of the quality of a proposed facility assignment. The efficiency e,~, of the assignment v is given by: ° 313

314

(3)

Proceedings of the 12th Annual Conference on Computers & Industrial Engineering

e~ = p + r a - C,~ 2ra × 100%,

where r, p, a are as above and r = 1,2,3... is a selected level of standard, usually r = 2 or 3. The quantity en is just a linear measure of efficiency in which the "statistically worst case" # + ra is rated at 0% and the "statistically best case" p - ra is rated at 100%. This measure of efficiency is only useful if we have good algorithms to compute the distribution parameters p and a. A formula for p, formula (6), was first given by R.E. Gomory and reported in fG]. The first algorithm to compute a was developed by Wallace et. al. [WHT], subsequent refinements have been given in [DS], [SS], [KKN1], [KKN2] and [MB]. In [WHT], exact formulae for n < 5 were obtained by algebraic manipulation of the formula for the variance and an approximate formula was proposed for larger values of n. The general equation of the variance, which has a form similar to formula (7) below, was also given in [WHT] but no exact algorithm was developed. From the form of the equations in [WHT] it is clear that about n ~ multiplications and additions were required for exact evaluation of the standard deviation. Therefore, Wallace et. al. had theoretically succeeded in reducing the complexity of calculation from worse than O(n!) arithmetic operations to O(n 4) arithmetic operations. The subsequent papers, cited above, offer refinements of the formulae given in [WHT], but the formulae are either inexact ([SS], [MB]) or exact algorithms are given, but without a substantial reduction of the computational complexity ([DS], [KKN1], [KKN2]). Moreover, in those papers which do present exact algorithms, no formal derivation of the general case is given. In this paper we give simple formulae for the exact calculation of a, which require only O(n 2 ) arithmetic operations (Theorem 1). The formulae may be derived by using the mathematical theory of group actions on networks. Due limitations of space we only give the formulae for computing the mean and standard deviation in this paper; the complete development of the formula and results, their statistical verification and a pseudocode algorithm for the computation of p and a are given in [B-C]. FORMULAE

FOR p AND

a

In the usual formulation of the facilities assignment problem we are given two n × n matrices F = (fij) and D = (dij) where flj is the rate of fiow of materials between locations i and j and dij is either the rectilinear distance or straightline distance between locations i and j. The fij often have a built-in cost factor so they actually represent per unit length materials handling cost rather than actual flow. The matrix D is generally assumed to be symmetric since the path travelled between location i and j is generally the same backwards and forwards. We do not assume that F is symmetric since backward and forward flow between facilities need not be equal. If we start from an initial layout in which facility i is put in location i, then the per unit time materials handling cost of the layout is clearly given by:

C =

y ~ f~,~d~,~. i-~-I j ~ - I

Since there is no transport cost for material handling at location j we have assumed fi,i = di,i equation. The cost equation may be rewritten:

C =~

~

-=

0 in this

(fi,j + fj,i)di,j.

i-~l j-~i+l

The quantity ¢i,j = fi,j + f£i is the total forward and backward flow between facilities i and j and is the (i,j)-entry of the symmetric matrix F + F t (t = transpose). Let us call • = (¢i,j) = F + F ~ the total flow matrix. In this notation, the layout cost is given by:

(4)

C=~ ~ ¢~,jd~,i. i l l j~i-bl

Now suppose that we make a reassignment of the facilities to the locations, say that facility i is moved to location lr(i). Consider the total cost C~ for the new assignment. Since facilities i and j will be moved to location ~r(i) and ~r(j) respectively there will now be a total flow of ¢i,j between locations 7r(i) and 7r(j). Thus, the contribution fbi,jdl,j in equation (4) should be replaced by ¢i,jd~(o,~(j ) since the distance

Broughton and Charumongkol: Facilities Design Problem

315

between locations ~r(i) and is ~r(j) is d~(0,r(j ). The formula for C,r is:

C~- = '~"~ ~

(5)

¢,,.id..(O,,(./).

iffil ,./----i+ l

Let E . denote the collection of all n! possible assignments. The collection {C~ : 7r E En} is the data set of layout costs for which we seek the mean and standard deviation. We state our results as a series of formulae in Theorem 1 below. A computer program or spread sheet program may be easily written from the formulae (cf. [B-C] for proofs and computer algorithm). T h e o r e m 1. Let the n x n flow and distance matrices, F = (fit) and D = (dii) be given. Define the total flow matrix ¢ by: q~ = (¢ii) = (fli + fji). Let {C,t : ~" E E , } be the cost distribution defined by (5) and let IJ and a be the mean and standard deviation of this distribution, defined by (1) and (2). Define the following quantities:

No - n(n - 1) 2

'

g,,, = n(n

1)(n - 2),

-

Na =

n(n - 1)(n - 2)(n - 3) 4

'

Sd=~dii, i=l jffii+l

i----1 j----i+l

iffil j f f i i + l

i----1 j - ~ i + l

Uid = E d~j, .iffil

if1 rt

u2¢ ~ ( v , ~ ) 2 , =

n

V'd

=

i----1

i----1

T,,,¢ = U2 ¢ - 2 T , ¢ ,

Ta¢

=

(s¢)

2 -

Toe - T,.¢,

Trod = U2 d - 2Tod, Tdd = (Sd) 2 - Tod - Trod.

Then, the distribution parameters p and a are ~ven by:

/z = ~.-~S~Sd,

(6) and

(7)

~2 =

T.~,T,d

N- Y + VERIFICATION

OF FORMULAE

T..¢Tmd

T~¢Tdd

~2.

+ AND COMPUTATIONAL

COMPLEXITY

A computer program based on the formula above has been developed to compute the standard deviation of layout cost distributions. The program was coded in TURBO BASIC using the pseudocode algorithm given in [B-C] as a model. As a verification that the computer program gives the correct standard deviation, all the sample problems appearing in the literature for which the standard deviation had been calculated exactly were sought out and the standard deviations recalculated. The trials were run on an 8 Mhz. IBM compatible microcomputer with an Intel 80286 processor and 80287 numeric coprocessor. The data obtained in these trials are given in Table 1. Most of the discrepancies between the results in the literature and the recalculated standard deviations, in relative terms, are much less than 0.1%. The worst case is the 20 x 20 sample problem where the relative difference is about 0.15%. By inspection of the formulae in Theorem 1 it appears that the number of calculations for a problem of size n x n has been reduced from O(n 4) to O(n2). To check the validity of this and also to extrapolate running times to larger problem sizes, the running times for the trials were recorded and a regression model of Y (running time) on X (number of facilities) was developed. A quadratic model Y = aX 2 + bX + c

316

Proceedings of the 12th Annual Conference on Computers & Industrial Engineering

was assumed because of the theoretical result on computational efficiency. The statistical analysis in [B-CJ shows that the data almost perfectly fits the quadratic model: Y = 0.000884X 2 + 0.00145X + 0.001278. By using the regression model above, running time for a 40-facility problem (the maximum size considered by CRAFT) is predicted to be 1.4736 seconds, as shown in the table. An O(n 4) algorithm would presumably take about 1600 times as long. SUMMARY AND DISCUSSION A new algorithm for calculating the standard deviation of the of the cost distribution associated to the facilities design problem has been developed. The algorithm calculates the exact value of the standard deviation and is more efficient than previous algorithms, O(n2) vs. O(n4). Algorithms, of a lower order of computational efficiency are not possible, since the data in this problem has O(n2) degrees of freedom. A computer program, written to test the algorithm, and running on a commonly available microcomputer is calculates the distribution parameters for a 30 × 30 problem in about 0.85 second and is predicted to calculate the parameters for 40 × 40 problem (the largest size considered by CRAFT) in less than 1.5 seconds. The algorithms are easily implemented on a microcomputer as a compiled program or a as spreadsheet program. The distribution parameters can be used in calculating the efficiency of proposed solutions in facilities design (formula (3)). The use of this measure generally assumes a normal or nearly normal distribution. Some experiments on the appraximate form of the cost distribution are reported in [H], [WHT] and [KKN2]. These experiments yielded distributions that were mainly normal, lognormal, two parameter gamma distributions and, alternatively, three parameter Weibull distributions. For these distributions a choice of level of standard r = 2 or 3 is appropriate. However, as noted in [WHT] there are reported cases of solutions lying as much as 9 (d. [NVRD and 6.13 (cf. [H]) standard deviations below average cost. Even with these examples, the efficiency using the level of standard r = 2 or 3 is still particularly useful in comparison of the efficiency of an assignment before and after plant design parameters have been altered or as a control variable in situations where there is a frequent change in production flows but it is difficult to seek or implement a new high--e~ciency layout after every production change. For example the second author [C] has used our method of calculating efficiency to compare the quality of a solution provided by CRAFT and the quality of the same solution after the plant layout provided by CRAFT had been "smoothed out", using software developed by him. Table 1: Standard Deviations and Running Times Number of Facilities 3 4 5 6 7 8 12 15 20 30 40

Computed Standard Deviation from literature current algorithm 2.494446 5.497447 6.980947 11.809590 10.793770 11.877200 22.885840 108.717400 146.182000 N/A N/A

2.494438 5.497474 6.980926 11.809601 10.793884 11.877149 22.886221 108.782082 146.390600 212.564703 N/A

Running Time (seconds) 0.0109 0.0219 0.0329 0.0439 0.0549 0.0659 0.1482 0.2197 0.3844 0.8403 1.4736

Reference [KKN2] [KKN2] [KKN2] [KKN2] [KKN2] [KKN2] [KKN2] [KK N2] [KKN2] [NVRl (predicted)

REFERENCES lAB] G. C. A r m o u r &~ E. S. Buffa A Heuristic Algorithm and Simulation Approach to the Relative Location of Facilities, Mgmt. Sci. Vol. 9 (1963), pp. 294-309. [B-C] A. B r o u g h t o n £r V. Charumongkol, Mean and Standard Deviation of the Layout Cost Distribution in F~cilities Design Problems, preprint, Cleveland State Univ. [C] V. Charumongkol, Interactive Microcomputer Graphic Methods for Smoothing CRAFT Layouts, Ph.D. thesis, Cleveland State Univ. (1990).

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[BAV] E.

S. Buffa, G. C. A r m o u r ~r T. E. Vollmann, Allocating F~ilities with CRAFT, Harvard Business Rev., Vol. 42 (1964), pp. 136-158.

[DS] K.N.

D u t t a & K. Sahu, Some Studies on Distribution Parameters for Facilities Design Problems, Int. J. Prod. Res., Vol. 19, No. 2 (1981), pp. 725-736.

[C] P.C.

Gilmore, A Solution to the Module Placement Problem IBM Research Report RC--430, April 26, 1961.

[HI G.G.

Hitehings, Analysis and Development of Techniques for Improving the Layout of Plant and Equipment, Ph.D. thesis, Univ. of Wales, Cardiff (1973).

[HC] F.S.

Hillier & M.M. Connors, Quadratic Assignment Problem Algorithms and the Location of Indivisible Facilities, Mgmt. Sci., Vol. 13 (1966), pp. 42-57.

[KKNI] V.K.

K h a r e , M.K. Khare, & M.L. Neema, Computer Aided Estimation of Distribution Parameters for Facilities Design Problems, Proc. XXVIII National Convention, IIIE.

[KKN2] V.K.

K h a r e , M.K. K h a r e , & M.L. Neema, Estimation of Distribution Parameters Associated with Facilities Design Problems Involving Forward and Backtracking of Materials, Computers Ind. Engng., Vol. 14, No. 1 (1988), pp. 63-75.

[MB] P.B.

Mahapatra & D. S. Bedi, FALSA--Facilities Allocation by Statistical Analysis Int. J. Prod. Res. Vol. 22 (1984), pp. 173-181.

[NVR] C.E. Nugent, T.E. Vollmann, & J. Rural, An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Ops. Res., Vol. 16 (1968), pp. 150-173.

[ss]

S. Sahu ~r K. Sahu, On the Estimation of Parameters for Distributions Associated with the Facilities Design Problem, Int. J. Prod. Res., Vol. 17, No. 2 (1979), pp. 137-142.

[WHT] H. Wallace, G.G. Hitchlngs & D.R. Towill, Parameter Estimation for Distributions Associated with the Facilities Design Problem, Int. J. Prod. Res., Vol. 14, No. 2 (1976), pp. 263-274.