Determining an opt imum location for an undesi-rable facility in a workroom environment

E. M e l a c h r i n o u d i s

Northeastern University, Boston, Massachusetts, USA (Received July 1984)

This paper deals with the problem of placing an undesirable but necessary piece of equipment, process or facility into a working environment. Locating a piece of equipment that produces contaminants or creates stresses for nearby workers, placing a storage facility for flammable materials or locating hazardous waste in the workroom environment, are all typical examples of the undesirable facility location problem. The degree of undesirability between an existing facility or worker and the new undesirable entity is reflected through a weighting factor. The problem is formally defined to be the selection of a location within the convex region that maximizes the minimum weighted Euclidean distance with respect to all existing facilities. A 'Maximin' model is formulated and two solution procedures introduced. A geometrical approach and an algorithmic approach are described in detail. An example is provided for each solution procedure and the computational efficiency of the algorithm is discussed and illustrated.

Key words: environment (working), optimization, undesirable facility

Production engineers are often faced with the problem of determining the least dangerous or the most acceptable location in which to place an undesirable piece of equip- ment in a workroom area. Equipment, facilities, or processes that could potentially introduce hazards to the workplace (i.e. high noise levels, high temperatures, molten metals, projectiles, or radiant energy) are classified here as being undesirable facilities.

The actual position in which an undesirable piece of equipment or process is placed in the working environment can have a direct bearing on the degree to which other facilities and workers are threatened. Assuming that reflec- tion of noise, radiation or pollutants on the boundaries of the workroom area are insignificant or nonexistent, the best location for an undesirable entity could be considered to be a position in the working area that maximizes the minimum distance from the entity to the workers and other facilities in the same region. This class of location problem is termed a 'Maximin' problem, and mathematically it can be stated as follows:

Find a point (x ,y) on a convex two-dimensional bounded region which maximizes the minimum weighted

distance from the point (x ,y) to a given set of existing points Pi(ai, bi) in the region, or:

m a x ( . f ( x , y ) = min w i d i ( x , y ) : ( x , y ) E S ) (1) l ~ i ~ n

Where: f ( x , y ) a measure of system's effectiveness; S a convex feasible region in E 2 (workroom area); n number of existing machines or workers that are in S; di(x, y) Eucli- dean distance between new machine at (x, y ) and ith exist- ing worker or machine located at Pi(ai, bi), i.e. di(x , y) = [(x - -a i ) 2 + (y -- bi) 2 ] u2; wi weighting factor greater than zero which represents relative incompatibility between ith existing worker or machine and new undesirable facility.

The above formulation could also be used to solve the 'inverse' problem, i.e. the problem of locating an optimal site (x ,y) for a worker or a group of workers among un- desirable or hazardous machines; in the latter case the different weights, wi, assigned to the machines should be according to their pollutant intensities (e.g. their emitted sound or radiation intensities). On a large scale, the above formulation could be used to determine the optimal site for a house, office or laboratory in an environment containing

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Optimum location for undesirab/e facility: E. Melachrinoudis

unpleasant or hazardous sources of noise, chemical pollu- tion or other undesirable sources.

Within this paper a geometrical approach and an exact algorithm will be discussed and the conditions under which each of these approaches can be most effectively applied will be established.

P rev ious research

A considerable number of studies focus upon the problem of locating a new machine in an existing layout. Bind- schindler and Moore, 1 Moore 2 and Francis and White a have all presented solution procedures for selecting a location for a new piece of equipment such that some objective function is minimized. In a majority of the research reported it is assumed that locating a new piece of equip- ment or process close to a worker or an existing facility will not have a detrimental effect on the system's acceptability or operational effectiveness.

Very few procedures for solving bounded machine location problems that involve undesirable facilities appear in the literature. A paper that discussed a formulation similar to the Maximin problem given in (1)was published in 1978.4 In that work the problem of locating an un- desirable type of facility was solved using a criterion that maximizes the sum ('Maxisum') of the weighted distances from the undesirable facilities to all existing facilities.

Shames s defined a special case of the Maximin location problem referred to as the largest empty circle problem: 'Given n points in the plane, find the largest empty circle that contains no points of the set yet whose centre is interior to the convex hull of the points'. The problem is a special case o f ( l ) where w i = 1 for all i and the feasible region S is the convex hull of the facility points. It is solved very efficiently through construction of the Voronoi diagram.

Dasarathy and White 6 extended the previous unweighted problem (wi = 1 for all i) for a higher-dimensional space and a convex feasible region. They provided an algorithm for a three-dimensional space. Drezner and Wesolowski 7 proposed a solution to a mathematical programming prob- lem which has the same objective function as (1) but a different feasible region. Specifically, in reference 7 the location of the new facility was constrained to be within a pre-specified distance from each facility point.

G e o m e t r i c a l a p p r o a c h

Consider a simple case of three points PI, P2, P3, in a rectangular region S (Figure 1 ). Assume weights that are different for each point, specifically wl = 5, w 2 = 4 and w3 = 3. Given a value z for the objective function f ( x , y ) in (1):

d i ( x , y ) = [(x -- ai) 2 + (y -- bi)2] 1/2/> z/wi (2)

i = 1 , 2 , 3

which implies that for the selected value z the feasible points (x, y ) are outside circles having centres P/and radii d i = z /w i (i = 1,2, 3). The circles have been drawn in Figure I for z = 21.5 and the feasible region ( x , y ) has been shaded. The existence of feasible points outside the circles and within S is indicative of the potential increase of the objective function.

?

P, (6.5, 8)

/ 03

pz(o,o) p,,(t3,o)

Figure 1 Search f o r op t ima l po in ts when w~ = 5, w= = 4, w 3 = 3

l'f the shaded areas are reduced by increasing z and the radii d i the local maxima 01,02 and 03 are found: 02 corresponds to the largest value of z = 31.6 and therefore is the global maximum since any further increase does not leave feasible points within S, according to (2).

Starting from 0 and increasing z by a predetermined quantity Az, a step by step convergence to a global maxi- mum is guaranteed. A termination rule must be set such that when the shaded area is small enough to be considered a point, the procedure stops. Because of the convergence technique involved, the computation time increases very rapidly as the number of existing facility points exceeds 10. The need to use mathematical programming techniques, when the number of facility points is large, becomes obvious. In the subsequent sections the Maximin location problem is solved by the use of mathematical programming tools.

S t a n d a r d m a t h e m a t i c a l p r o g r a m m i n g f o r m u l a t i o n

The original problem formulation given in (1) is not con- venient to work with. Due to the general nature of this model it can be easily formulated by separating the two optimization operators and rewriting it in the following standard form:

max z

st z < wi[(x - a i ) 2 + (y -b~)2] 1~2

(x,y)eS

i = 1 , 2 . . . . ,n

(3)

(4)

(5)

and Pi(ai, bi) E S and S is convex.

For a region S which is a convex polygon the constraints A X <~ C can be used to identify the feasible region. Any shape of a convex region S can be approximated quite accurately by a convex polygonal region of a sufficient number of sides. The original problem for n existing facilities and a polygonal region of m sides after squaring constraint (4) can now be written as follows:

Problem P

max f ( X ) = z (6)

s tg i (X)=z2- -w~[(x- -a i )2+O,- -b i )2]<~O (7)

i = 1 , 2 . . . . , n

AX -- C < 0 (8)

366 Appl. Math. Modelling, 1985, Vol. 9, October

where:

A ( m x 3) matrix with only zeros in the third column C (m x 1) vector x ( x , y , z )

Although the objective function is linear, the equivalent formulation of the Maximin location problem given in (6), (7) and (8) is nonlinear and nonconvex. The nonlinearity and nonconvexity is found in the constraints (7) which define regions outside circular cones that have their vertices on the x-y plane and their axes of symmetry perpendicular to the plane. The linear constraints (8) define a right prismatic region open from above whose base is the feasible region S and whose lateral faces are generated by the sides of the region S in the direction of the z-axis.

Since the weights are f'mite numbers, the feasible space will be bounded in the z-direction by conic surfaces within a right prism, and therefore the value of the objective function z is bounded from above. Since z > 0 for every (x ,y ) except for the facility points where it becomes zero, a positive maximum always exists. Also, since the problem is not convex, it is possible to have more than one local maximum and the global maximum may not be unique.

S o l u t i o n p r o c e d u r e f o r t he m a t h e m a t i c a l p r o g r a m m i n g a p p r o a c h

The Kuhn Tucker (K-T) theorem states: Let a point X*(x*,y*, z*) be a local maximum for the Problem P. If the Constraint Qualification holds and f (X) , gi(X) have continuous first derivatives, there must exist vectors )t* ()tT,)t~,. )t*) and * * * • ", )t0()tn + l, )tn+ 2 . . . . . )t*+ ra), such that:

Fi

V / ( x * ) - Y . * • • r , _ )ti Vgi(X ) - - A )t o - 0 (9) i = 1

)t.*, x ~ > 0 (10)

X*gi(X* ) = 0 i = 1, 2 . . . . . n (11)

) t*+][AX*-C]i=O ] = 1 , 2 . . . . . m (12)

where:

A r transpose of matrix A lAX* -C]] ]th element of the vector containing the

left-hand sides of the inequalities (8).

Since 7 f ( X ) = (0, 0, 1) and Vgi(X) = (--2w~(x-ai) , -- 2w~(y -- bi), 2z) f (X) and gi(X) have continuous first derivatives.

Furthermore, it has been shown s that the Constraint Qualification holds for every feasible point of problem P. Based on the K-T conditions the following theorem can be used in locating the local maxima.

Theorem Local maxima which do not lie on the boundary of S

will lie within the convex hull of the facility points P/.

~oof For a local maximum not belonging to the boundary of

S, )t~ = 0 in vector equation (9). Furthermore, since w~ > 0 the same equation analysed for the first two com- ponents of the vectors becomes:

n n

y" ) t ,* . (x*-a0=0 E ) tT( .v*-b t )=0 i = 1 i = 1

Optimum location for undesirable facility: E. Melachrinoudis "

o r :

n

(x*,y*) = Z ui~,. 1 = 1

where:

Pi = )t 2 i = 1., 2 ..... n i=

which proves the theorem since:

/1

~ p i = l i = 1

According to the theorem the algorithm should involve two searches for local maxima, i.e., one within the convex hull of the facility points and one on the boundary of S. Since the objective function of Problem P is linear, all local maxima X*(x*,y*, z*) are K-T boundary points of the three-dimensional feasible space. Furthermore, all K-T boundary points can be generated by allowing one or more Lagrangian multipliers in (11) and (12) to be nonzero (all)ti being zero at the same time violates (9)). Consider the following two cases:

Case 1. )tk ~e 0 and )ti = 0 V i 4: k Substituting in (9) and (11) for 1 <~ k ~< n one obtains:

(0, 0, 1) = )tk(-- 2w~(x--ak), -- 2w~(y -- bk), 2z)

gk (X) = z 2 - w ~ [ ( x - a k ) ~ + O' - bk)2l = 0

Since w~ > 0 there is no value for z which satisfies both equations, implying in turn that X(x ,y , z) is not a K-T point.

Similarly for n + 1 ~< k ~< n + m the vector equation (9) is not satisfied because the third element of the left-hand side vector can never become zero.

Case2. ) tk¢O, X l ¢ O a n d X i = O V i e k , l Without loss of generality assume that 1 ~< k ~< 1 ~< n

and that X(x ,y , z) is the point identified by gi(X) = 0, i = k, l and is satisfying (7) with strict inequality for i 4: k, I. Then (11) suggests that X(x ,y , z) is a K-T point if its projection (x ,y) on the x-y plane lies on the linear segment PkPl. Such a point, however, is not a local maxi- mum since a movement by a distance e away from (x ,y) in a perpendicular direction to PkPi, small enough to maintain feasibility in (7), will improve the minimum weighted distance from Pk and Pi and therefore the objective function value.

If 1 ~< k < n and n + 1 ~< l ~< n + m then similarly, equa- tions (9), (11), and (12) suggest that the corresponding K-T point X ( x , y , z ) will have, as projection (x ,y) on the x-y plane, the projection of Pk on the lth side of the polygon S, corresponding to equation [~IX -- C]t = 0. Such a K-T point, however, is not a local maximum since a movement by a distance e along the side and away from (x,y) , small enough to maintain feasibility in (7), will increase the weighted distance from Pk and therefore the objective function value.

If n + 1 ~< k < ! ~ n + m the vector equation (9) can never be satisfied, since the third component of the left-hand side vector cannot become zero.

The above two cases imply that all local maxima should be included in the set of K-T points, generated by allowing

Appl. Math. Modelling, 1985, Vol. 9, October 367

Opt imum location for undesirable faci l i ty: E. Melachrinoudis

three or more Lagrangian multipliers in (11) and (l 2) to be nonzero. By allowing exactly three hi to be nonzero K-T points can be generated by solving the corresponding system of three simultaneous equations:

g i ( X ) = 0 i = k, I, m (13)

gi(X) = o i = k 1

[ A X - C ] i = O j = p , p + 1 ~ (14)

gi(x) = o i = k, t (15)

l A X - C ] i=o j=p Allowing more than three X i to be nonzero results in local maxima identified by more than three equations similar to the above. But these local maxima will have been already generated after solving all possible systems of equations (13), (14) and (15). All local maxima lying within the convex hull o f P i are solutions of (13) found by taking the points P/three at a time. All local maxima lying on the boundary of S, i.e. on the vertices and sides of the boundary, are solutions of all possible systems of equations (14) and (15), respectively. The best value of the objec- tive function z0 at the vertices E l , / = 1,2 . . . . . m, can provide a lower bound for global z* and at the same time take care of all systems of equations (14):

Zo = max (m!n wilPiE i l) (16) j I

AS illustrated in step 3 of the following algorithm, by performing a direct search for local maxima on the sides of the boundary of S, only a small fraction of the set of systems of equations of type (15) have to be solved. Also only noncollinear points Pi, i = k, 1, m, should be con- sidered for the systems of equations of type (13), other- wise point ( x , y ) will lie on the line defined by the three points, but such a point (x, y) cannot be a local maximum. The systems (13) have at most two solutions, each of which should be tested for feasibility using (7) and (8). The following algorithm starts with the lower bound Zo on z*, and the associated solution vertex (xo,Yo) found in (16), and considers candidate local maxima which are solutions of the equation types shown in (13) and (15). The value of the objective function z is updated until there is no further improvement while examining candidate local maxima.

An exact algorithm

(1) Start with the lower bound solution (xo,Yo, Zo). (2) Take all combinations of three noncolinear points

Pi, i = k, l, m. Solve the system of three simultaneous equations (13) each time for i = k, l, m. Check each solution for feasibility. If a solution is feasible and provides a value z greater than or equal to the best value found thus far, consider this solution(s) as the current best solution(s).

(3) Assume that zo was computed in (16) for ] = p and i = k. Consider the plane which is going through the pth side EpEp+ l and is parallel to the z-axis as shown in Figure 2. The intersection of that plane with the conic surface gk (X) = 0 is a hyperbola. Move along this hyperbola, until another feasible hyperbola(s) is encountered, by solving (n -- 1) systems (15) for 14= k. The point at which the hyperbolic path changes direction is next checked to see if its value z is at least equal to the best value found thus far. If yes, the

best solution(s) is/are updated accordingly. This is repeated until a new side plane p + 1 is encountered. Move in the same fashion along the new plane. Stop the clockwise motion along the boundary of S when the pth side is encountered. At the end, the current best solution(s) is/are the global maximum(a).

Computational expe r i ence

As mentioned earlier, the computer code based on the geometrical approach can handle efficiently up to 10 facility points. The limit of 10 occurs because the computa- tion time increases very rapidly as the number of facility points increases beyond this. An advantage of the geo- metrical approach is that it can be used for any type of feasible region S not necessarily convex.

The exact algorithm based on the mathematical pro- gramming approach was tested in a large variety of problems. The facility points were generated at random according to a uniform distribution. The weights w i were also selected

Figure 2 Direct search onpth boundary plane

200

I 0 0

I 0

50 40L- 3 0 -

A t~

20

U

5 4 -

2 -

I I I I I . I I 1 I L _ _ 20 40 60 80 100 120 140 160 180 200

Number of focilities (n)

Comput ing t ime versus number of facilities Figure 3

368 Appl. Math. Modelling, 1985, Vol. 9, October

Opt imum location fo r undesirable faci l i ty: Eo Melachrinoudis

Tab~ 1 Data on existingfacilities

i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

a i 10 30 60 0 40 10 20 60 80 90 50 10 30 50 60 40 50 70 40 70 b i 0 10 10 20 20 30 30 30 30 30 40 50 50 50 50 0 0 20 30 40 w i 1 2 2 3 2 1.5 2 2.5 4 2 1 1 3 2 3 2 1 2 5 2

0 I0 20 30 40 50 60 70 80 90 0 "

i i ! ! i ! : i

. . . . i . . . . i . . . . i . . . . i . . . . i . . . . ! . . . .

3 o . . . . i . . . . i . . . . i . . . . i . . . . i . . . . i . . . . i . . . . i . . . .

40 .... i .... : .... : .... : : i

Figure 4 Positions of existing machines and workers -- -- *, and location of global maximum -- -- G

at random in the range 1-5 according to a uniform distri- bution. In the computational testing as the number of sides increased from 8 to 32 the computing time did not increase substantially) Because the search for local maxima, within the convex hull of the facility points, is where most of the computing time is consumed, a longer search on the boundary does not have a significant effect on the overall time.

When varying the number of facility points from 5 to 200 the growth in computing time was proportional to n a as is shown in Figure 3. All the work was performed on a CDC CYBER 175 system.

Example p rob lem

To illustrate the use of the exact maximin algorithm, consider a situation where an undesirable facility is to be placed in a convex workroom area that contains 20 existing machines and workers. As illustrated in Figure 4, the machines and people can be represented as points on the x - y plane. The degree of undesirability that exists between the new machine and the existing points (people and machines) is reflected through the values assigned to the

weighting factors, w t. Table ] lists'the location of the points under study and the associated w t.

It should be noted that due to the general properties of the Maximin model, the weighting factors should be selected such that a small weighting factor identifies a high degree of incompatibility or unattractiveness and a high weighting factor identifies a low degree of incompatibility. At the present time research is being performed to determine a procedure for assigning weighting factors to specific combinations of machines, or machines and workers. For the purpose of the work performed thus far it has been found that the range 1 ~< w i ~< 5 for the w i has been most appropriate.

Figure 4 is a computer printout illustrating the positions of the existing machines and workers and the location of the global maximum corresponding to the optimal solutions:

x* = 84.38 y* = 10.25 z* = 28.71

A c k n o w l e d g e m e n t

The author would like to thank the referees for their valuable comments and suggestions.

References

1 Bindshindler, A. E. and Moore, J. M. 'Optimum location of new machines in existing plant layouts', J. Industrial Engineering, 1961, 12 (I)

2 Moore, J. M. 'Plant layout and design', MacMillan Company, 1962

3 Francis, R. L. and White, J. A. 'Facility layout and location: An analytical approach', Prentice-Hail, 1974

4 Church, R. and Garfinkel, R. "Locating an obnoxious facility on a network', Transportation Science, May 1978, 12 (2)

5 Shames, N. 'Computational geometry', PhD Dissertation, Yale University, 1977

6 Dasatathy, B. and White, L. 'A maximin location problem', Operations Research, November-December 1980, 28 (6)

7 Drezner, Z. and Wesolowski, G. 'A maximin location problem with maximum distance constraints', AIIE Transactions, Sep- tember 1980, 12 (3)

8 Melachrinoudis, E. 'The Maximin location problem using a Euclidean metric', PhD Dissertation, University of Massachu- setts, September 1980

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