A nonlinear optimization approach for solving facility layout problems

174

European Journal of Operational Research 57 (1991) 174-189 North-Holland

A nonlinear optimization approach for solving facility layout problems Drew J. van Camp Department of Computer Science, University of Toronto, Canada Michael W. Carter Department of Industrial Engineering, University of Toronto, Canada Anthony Vannelli

Department of Electrical and Computer Engineering, University of Waterloo, Canada Received July 1990; revised June 1991

Abstract: Facility layout research has many subproblems for which optimization models can be devel-

oped. Although modeling these problems is often simple, the resulting large size of these models can make solution by decision makers very difficult. This paper is concerned with the development of new heuristics to be used as aids in finding good solutions to the facility layout problem. In particular, nonlinear programming (NLP) techniques are developed that minimize the material handling cost. Descriptions of the problems to be solved are presented along with the formulations of the NLP models to be optimized and the methods used to solve them. Also presented are computational results and a comparison of these results to those of other layout algorithms. The heuristics developed are shown to produce solutions comparable to, and in many cases better than, those of other algorithms. The approach is particularly effective for problems with unequal-size departments. As well, the models are shown to be computationally practical for solving real-world problems. Keywords: Facility layout, office layout, nonlinear optimization, quadratic penalty function, heuristics

I. Introduction

The facility layout problem is one of the most well-studied areas in production facilities analysis (for example, see [11,16,17]). The objective of the facility layout problem is to determine the placements for a group of departments, offices or machines within a facility, in order to minimize some objective function. The function that one tries to minimize may be quantitative or qualitative. Some of the quantitative measures that may be considered are: costs of transporting product between departments or costs of laying communications wiring. Qualitative measures may be things such as noise and vibration disturbances or flow of information between areas. In this paper we will restrict our attention to quantitative models of the problem. Optimal algorithms for solving the layout problem are NP-complete [25], and therefore, exact solution methods are only feasible for small or greatly restricted problems. Because of this, most layout approaches are heuristic in nature. This paper presents a heuristic algorithm to be used in the development of minimal-cost facility layouts. The algorithm, called NLT (Nonlinear optimization Layout Technique), is based on a nonlinear modeling formulation of the problem. The objective of this model is to develop a layout that minimizes 0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

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the cost of material flow between departments within a plant. The method can be adapted to solve any quantitative problem where cost is a function of the relative location of one or a pair of departments. Models with constraints similar to those in NLT have been developed by other authors [7,15,20]; however, the methods used to solve the models are very diverse. DISCON [7] is a hybrid algorithm that can be used to lay out circles in the plane. To do this the layout problem is modeled as a nonconvex mathematical programming problem. The problem is solved by using a two-phase algorithm called Dispersion-Concentration algorithm that first spreads all the circles far apart, and then brings them in as close together as possible. MATCH, by Montreuil et al. [20] uses a b-matching model in order to develop a block layout of rectangular departments. The objective function used is a weighted sum of adjacent departments. Heragu and Kusiak [15] developed two separate mathematical models for the facility layout problem. The first model was a linear continuous one with absolute values in the objective function and constraints. The second was a linear mixed-integer model. Both models were solved using an unconstrained optimization algorithm. NLT uses a nonlinear model to formulate the layout problem. Linear and nonlinear optimization models have been extensively used to model multi-facility location problems. An early treatment of the Euclidean distance multi-facility layout problem was given by Miele [19]. Francis [10] introduced the rectangular distance version of the multi-facility layout problem. Wersan, Quon and Charnes [28] and Wesolowsky and Love [29] applied a linear programming approach for rectilinear cost distance minimization,

~_~

m--lm Wlij{IXil--ajl[q-]xi2--aj2t} + E E

Min i=1 j - I

i-I

W2ij{IXil--Xrl[+lXi2--Xr2 I}

(1)

r=i+l

where m is the number of new facilities added to the location problem, n is the number of existing facilities, wii j is the unit distance cost between the new facility i and the existing facility j, Wzij is the unit distance cost between the new facility i and the new facility j, and a jl, a j2 is the (x, y) location of the existing facility j. Problem (1) was generalized by Morris and Verdini [22,23] so that the more general Ip distance measure is minimized to solve the multi-facility location problem M i n ~ ~Wlij(lXil--ajl i-1 j=l m-I

~.~

-F E i-I

]p Jr-[xi2--aj2lP) I/p

W2ij(lXil--XrllP+

Ixi2-xr2lP) 1/p.

(2)

r=i+l

The interested reader is referred to a more detailed discussion on the nonlinear programming approaches found in Love et al. [18]. 1.1. Nonlinear optimization layout technique A major shortcoming of most layout algorithms is the simplistic way in which they represent departments. Most of the algorithms represent departments as equal-sized squares [16] or as combinations of such squares [2,14]. Some algorithms also place restrictions on the shapes of these composite departments to make the size of the problem tractable [2]. Because of these restrictions, the dimensions and positions in which departments can be placed must often be determined by the user before the algorithm can be implemented. Another problem for improvement of algorithms that allow different size departments is that the final layout generated may contain irregular shapes that are not feasible for an actual facility. Both of these problems are overcome in NLT. The formulations of the models are continuous. This allows departments to be of any area with associated dimensions and to be placed in any position within the plant. Because of the nature of the nonlinear models in NLT, all departments are constrained to be

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rectangular in shape. This results in highly regular facility layout. Also, NLT incorporates an interchange procedure and user interface that allows the user to move departments or manipulate the solution models. NLT is based on the use of three related nonlinear optimization models. Section 2 presents a basic nonlinear mathematical model of the facility layout problem incorporating variable rectangular departments. We then describe our nonlinear optimization procedure, a modified version of a well-known reliable approach to general nonlinear problems. This basic model is very useful for solving the difficult problem of fitting rectangular departments into a given fixed plant, but the solution is highly dependent on the initial location of the individual departments. Therefore, in Section 3, we develop two 'relaxed' mathematical models. These models allow much more freedom of movement for the departments, and the solution of the relaxed problems produces a good starting solution for the basic model. In Section 4, we compare our nonlinear approach on a wide variety of test problems from the literature. We demonstrate that our solutions are comparable to, and in many cases better than those of other algorithms. Our approach appears to provide very reliable and consistently good solutions to all problems.

2. A basic mathematical model of the layout problem In this section, a model used to approximate the real layout problem is developed. As stated in the previous section, in NLT the departments are considered to be of fixed area but of variable dimensioned rectangular shape. The same is true of the facility as a whole. The model NLT is based on assumes that the total cost per unit distance of transporting items between departments is known. It also assumes that the same costs are known for each department and the outside wall (for example, shipping departments have flow to the outside wall). Given these costs, the objective is to minimize the weighted distances between the departments, and between the departments and the outside wall, where the weight is the transportation cost times the distance between the departments. In our models, we measure distances from the centers of the departments. Using this, the mathematical formulation of the cost of a layout is

f(x,

y)=

Ecijxdij+ Eg.ixdi ij

(3)

i

where the center of department i is at (xi, yi), and the following notation is used: cij = The (annual) cost of moving material between departments i and j per unit distance.

dij =

The Euclidean distance between the centers of departments i and j (dij = ~/(x i -xj)2(yi _ y j ) 2 ) . ci = The (annual) cost of moving material between the department i and the outside wall per unit distance. a~i = The Euclidean distance between the center of department i and the nearest exterior wall. Throughout this paper we will refer to this function as the material handling cost. In many practical problems a Euclidean measure is appropriate for calculating costs (e.g. in overhead material handling systems, wiring between office units, and many advanced or flexible manufacturing systems). Of course, in office layout, with walls and corridors, a rectilinear assumption is more realistic. However, Cheng and Kuh [6] and Weis et al. [27] have shown, based on computational experiments, that the two measures converge, especially as the number of items being laid out grows large. As well, Blanks [5] has shown probabilistically that as the number of items goes to infinity, the two measures converge. Thus, the use of the Euclidean measure is a reasonable assumption even for this case. It can be shown (cf. Eyster et al., 1973 and Love et al., 1988) that the rectilinear distance function can be approximated by the following hyperbolic approximation procedure (HAP):

di,=[(xi-xy)2+e]'/2+[(yi-y,)2+e]

t/2

(4)

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where E is small and goes to 0. Using this approximation, all orders of derivatives of f(x, y) in (3) would be continuous everywhere. It would not be difficult to implement the nonlinear function f(x, y) using the H A P in the presented N L T approach. The model used to approximate the layout problem, which we will call (P-I), may be written as Min

f(x, y)= Eco ×dij + E(ixd ij

s.t.

Ixe-xjl

i

i

-½(wi+

if [y, - y / I - ½(hi + h / ) < O ,

(5)

-x,l-½(w,+wi)
(6)

hi)>

0

>0

for a l l i

(7)

~-hi) > 0

for all i,

(s)

(x i -- ~wi) , 1 7, > 0 + ~w

for a l l i ,

(9)

for all i,

(10)

lYi

- Yil -½(hi

Wi)>O

5w - ( x i+4wi)_ ½hr-

(Yi +

1

+

1

1

( y i - ~hi) + 2hr> 0

if Ixi

min(wi, h i ) - I mi">O

for a l l i ,

lm a x - m i n ( w i , h i ) > 0

for a l l i ,

min( w r, hr) - l.rrain > 0, 1}' a x - m i n ( w r, hr) > 0

(11) (12) (13) (14)

where the following notation is used: x i = The x-coordinate of the center of department i. Yi -----The y-coordinate of the center of department i. wi = The width (length in the x direction) of department i, h i = The height (length in the y direction) of department i. W-r= The width of the enclosing facility. h r = The height of the enclosing facility. I mm= The minimum allowable length for the shortest side of department i. l m~X= The maximum allowable length for the shortest side of department i. l~ ~n = The minimum allowable length for the shortest side of the facility, 1.,'?~x = The maximum allowable length for the shortest side of the facility. The constraints in model (P-I) can be broken down into three categories. The first two categories are 'hard' constraints imposed by the structure of the problem. They are: no two departments may overlap; all departments must remain in the facility. The non-overlap constraints are given in (5) and (6). These constraints state that if the distance between the centers of two departments in the y direction is less than the sum of their heights, they cannot overlap in the x direction, and vice versa. The variable h is introduced only for ease of exposition. It is dependent on w by the relation h i = o i / w i where a i is the constant area of department i, while h i and w i are variable. Note that equations (5) and (6) are both non-differentiable and also introduce nonconvexities into the problem. The constraints to ensure that all departments remain inside the facility are given by (7)-(10). These constraints ensure that for each department, the right wall is to the left of the facility's right wall, and the left wall is to the right of the facility's left wall. Similarly, they ensure that each department's upper wall is below the facility's upper wall, and that the lower wall is aboce the facility's lower wall. As with the departments, h T = aT./w T where a r is the (constant) area of the facility. The third category is less rigidly defined and more strongly dependent on the particular layout problem being solved. It is that the dimensions of each department, and the facility as a whole, must be within acceptable limits as defined for a specific problem. These constraints are given in (11)-(14). These

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equations simply state that the shortest side of each department (and the facility) must be at least lmin (/~in) and at most l max (l~ax). Normally, there is no maximum limit on the width, since no limit implies that departments are allowed to be square (at maximum width). This is usually quite acceptable. Observe that the objective function and the constraints of (P-I) are in fact nonlinear, since the variables hi, h r and dij are dependent on xi, y~ and wi by relations that are nonlinear. The function gradient has discontinuities at any point where dgj = 0 (i.e. departments i and j have the same centers). Fortunately, these points are all outside of the feasible region for the problem. For n departments, the numbers of variables and constraints in this problem are 3n + 1 and n 2 + 5n + 2, respectively. Models with constraints similar to these have been developed by others, among them Montreuil [21] and Heragu and Kusiak [15].

2.1. Optimization procedure The model presented above is transformed from a constrained form into an unconstrained form by an exterior point quadratic penalty function method. This method works by including a penalty in the objective function for every constraint that is violated and then minimizing the transformed objective function. The penalty is iteratively increased and the objective function minimized again until, in the limit, a feasible point is generated. Given that the constraints of the model are nonconvex, the optimization procedure is guaranteed to find only a local minimum to the problem. In the quadratic penalty function method, the penalty added into the objective function is a constant /~ times the square of a measure of the degree to which a constraint is violated. This gives a continuous function with a continuous gradient. The specific method used to solve the problems is taken from Bazaraa and Shetty [3, pp. 340-343]. The method can be used to solve problems of the form min s.t.

f(z) & ( z ) > 0,

i = 1 , 2 . . . . . m,

h i ( z ) = 0,

j = 1 , 2 . . . . ,l

where f(z), gi(z) and hi(z) are general nonlinear functions. The constrained problem (P-I) is transformed into an unconstrained problem using the following method. Choose two values /~0 > 0 (the penalty multiplier), /3 > 1 (the rate at which /x is increased) and an initial point z 0. Starting with iteration k = 0: Step 1. Use the initial point z~ to solve the problem /

min

F(z) = f ( z ) +/z k ~'~ ~b[gi(z)] 2 q-I,Zk E h i ( z ) 2 i=1

s.t.

(15)

j=l

z ~Z.

where ~(~7) =

{'~, 0,

for ~7 < 0 , otherwise.

Let z, +1 be a locally optimal solution to this problem,

Step 2. If the penalty l

]J~k ~ ~)[gi( i--I

Zk+l)] 2

+ ]J~k E hj( Zk+l) 2 <

~-,

j-1

where E is some small number, stop; otherwise let /z~+ 1 =/3/x k, replace k by k + 1 and repeat.

(16)

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Although the above method can be applied to most nonlinear problems, it must be modified to solve the layout problem developed in the previous section. The need for modification is due to the constraints in (5) and (6), which ensure that no two rectangles overlap:

Ixi-xiL-~2(wi+wj)>~O ]Yi--YjI--~ '(hi+hj)>~O

if lYi-Yjl-5(h,+'

hj)
if Ix i -- xj] -- ~( 1 w, + wj)
If the algorithm is used directly on these constraints, not only does the transformed objective function's gradient become discontinuous on the boundaries where the constraints are first violated, the function itself also becomes discontinuous at these points. The method used to overcome this problem takes into account how the departments overlap each other. If there is an overlap (if x and y both overlap), then we only penalize the coordinate with the smallest amount of overlap. That is, letting Pij be the penalty term for the overlap of departments i and

J, if

l ~(wi + wj) - I x i - x j l

I < ~:(h~ + hi) - lY/-Yj I,

then

Pi~=p~4,[Ixe-xjl-½(wi+w~)]

otherwise

Pij=txJa[ l y , - y j l - ½ ( h

2,

(17)

i +hj)] 2.

The minimization algorithm used to optimize the unconstrained function created from problem (P-I) by the transformation in (15) at each iteration is a quasi-Newton method. Here the Hessian matrix is approximated using only first derivative information. The specific quasi-Newton method used for this problem consists of a BFGS ( B r o y d e n - F l e t c h e r - G o l d f a r b - S h a n n o ) update to approximate the Hessian [13, p. 119]. The update also uses a Cholesky decomposition to avoid taking the inverse of the approximate Hessian. The BFGS update was chosen because of its low requirement for restart and its lack of dependence on exact line searches at each iteration. The Cholesky factorization for any matrix can be calculated in O(n 3) operations; however, here it is not necessary to recalculate it at each iteration. It is possible to calculate the factorization of the initial matrix and update the factorization itself at each iteration. This update requires only O(n 2) operations. The method used to update the Cholesky decomposition has been presented by Gill et al. [12]. The line search used to minimize the function along the direction calculated at each stage of the quasi-Newton optimization is an inexact cubic line search [9].

2.2. Interchange procedure The optimization routine terminates with a feasible solution at a local optimum. We then consider all pairs of departments, looking for an interchange that will reduce the objective function value. If we find one, we make the exchange and then return to the nonlinear algorithm to see if further improvements can be made. Many authors have suggested the use of pairwise exchanges. One of the first algorithms to use an interchange procedure was C R A F T [1]. Since its introduction many other similar algorithms have been developed, but they are normally limited to interchanging departments of equal size (in a few cases, unequal-size exchanges of adjacent departments are permitted). One of the unique features of our approach is that we allow the exchange of any pair of departments. The unconstrained penalty function method will accept a solution that is 'infeasible' (where one department is too large for the available hole), and then use the penalties to force the departments to slide over and make room. Observe that this in itself offers a major advantage over exchange procedures like C R A F T when departments are mostly of unequal sizes. A major limitation of our approach follows from our method of estimating the associated change in the objective function value when a pair of departments is exchanged. We base the estimate on the

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exchange of the centers of the affected departments only. The estimate is exact when departments are of equal size, but they are only estimates for unequal departments. The actual change depends on the new feasible solution to the nonlinear program, and it could be either better or worse than expected. It is possible to construct examples where a particular exchange estimate indicates a decrease in the objective function, but the actual final solution is worse. Of course, when departments are close to the same size, the estimate will also be close. The approach taken in NLT to determine the set of allowable interchanges is to ask the user to input a tolerance 6 for comparing two departments. If the relative difference in the areas of the two departments is less than 6, i.e. 1---

ai a]

<6

(18)

where a i <~ at, then the two departments may be interchanged. In NLT, the default value used for 6 is 0.1. Once the set of interchanges has been determined, NLT iteratively attempts two-way interchanges between departmental centers for all those allowed. The two-way interchange that yields the greatest reduction in the overall material handling cost is carried out, and the optimization procedures for the model are restarted from this new initial position. If no two-way interchange causes a reduction in the cost, three-way interchanges are attempted. Three-way interchanges are attempted for all sets of departments i, j, k such that 1-

<6

(19)

where a i <.4 a t <~ a k. If the interchange is feasible, then there are two possible ways to perform it. These are to move departments i to j, j to k, and k to i, or to move departments i to k, k to j, and j to i. Both of these sets of interchanges are attempted by NLT. As with two-way interchanges, the best three-way is implemented. If no improving interchange is found, the procedure terminates. Three-way interchanges are tried only if two-way interchanges fail, because of their high computational expense. While the number of pairwise interchanges for two-ways is less than or equal to n:, the number of triples is O(n3). 2.3. L i m i t a t i o n s

of the primary

model

Even with the interchange procedure, there are difficulties with the primary layout model as presented in this section. The basic model can be solved using the methods described above; however, the problem is a tightly constrained one with many local minima and it is strongly dependent on the initial point used. The constraints are of a form such that if two departments are adjacent to each other in the initial solution, they tend to remain adjacent in the final solution. As well, the constraints may create a situation where no feasible solution can be found. To avoid these problems it was necessary to develop a method for creating a better initial point. It was decided to relax the model in problem (P-I) somewhat and solve this relaxed model. We then used the solution of the relaxation as an initial solution for the basic model. The following section describes the simplified models used to generate this initial point.

3. A three-stage approach

In this section, we will present two relaxed versions of model (P-I). We will begin by motivating the relaxation in simple terms and will then present a mathematical representation of these models together with a practical illustration.

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In the first relaxation, we decided to represent each department as a circle with an area proportional to that of the actual department. This model is an extension of the approach presented by Drezner [8] in which circles represented departments of equal area. The size of the circles is such that they can be inscribed in a square with the same area as the actual department (i.e. the diameter equals the root of the area). We then try to find a layout for the circles such that the circles do not overlap with each other and they all fit neatly inside the rectangular facility. The motivation behind using a circular representation for the departments was that we felt that it would be easier for circles to move 'around' each other in the nonlinear optimization problem than it would be for rectangles. In fact, it is important to realize that the penalty function approach actually allows circles to move through each other. In the early iterations we use a small value for the overlap penalty. The minimization process concentrates on pulling departments with high interaction together and virtually ignores the circle boundaries. As the overlap penalty increases, the circles are gradually pushed apart. The final solution of this circular model can be used as an initial solution to the basic model. We simply leave the centers of departments where they are and begin optimizing with all departments as squares. The optimization procedure will then determine the proper width, height and center for each department. A second relaxation went one step further: suppose we ignore the boundaries of departments altogether. In this model, we simply want a solution that evenly distributes the centers of departments over the facility and minimizes the material handling cost between these centers. In order to force departments to spread out evenly across the facility, we use a statistical definition of the distribution of the centers. Specifically, we will use the moments of an ideal, evenly balanced layout as constraints to force the layout to have evenly distributed centers. In the implementation of N L T we refer to the solution of each of the relaxed problems and the basic layout model itself as 'stages'. The relaxed problem, where the boundaries of the departments are ignored, is referred to as Stage One. The problem where the departments are modeled as circles is referred to as Stage Two, and the final problem, where the departments are modeled as rectangles, is referred to as Stage Three. The models developed in this section have been implemented in a microcomputer environment. The implementation has been written in the 'C' programming language and run on an Apollo DN3500 engineering workstation. The machine has a rating of approximately 5 MIPS. All computational results presented in this paper are for this machine.

3.1. Mathematical formulations for the relaxed problems In this section, we will describe the Stage One and Two models, and illustrate the solution procedure in detail, using a practical example. The data for this problem came from a real production plant that produces electronic components [4]. The problem consists of 10 departments of unequal areas. These areas, a i, in square meters, are given in Table 1. The cost per unit distance of material flow between the departments is given in Table 2. Since the layout was to be developed for an existing facility, the total shape of the plant was constrained to being rectangular, of dimensions 25 x 51 m. Thus, a T = 1275, and l~ in= l~ ax= 25. It was assumed that for a valid layout, no department could be narrower than 5 m (l rain = 5), and there was no restriction on the maximum width of a department (they can be square).

Table 1 Departmental areas: ten-department problem Department Area (a i)

1

2

3

4

5

6

7

8

9

10

238

112

160

80

120

80

60

85

221

119

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Table 2 Cost of material flow: ten-department problem Dept.

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10

-

0 -

0 0 -

0 0 28

0 0 70 0 -

218 148 0 28 0 -

0 0 0 70 0 0 -

0 0 0 140 210 0 0 -

0 296 0 0 0 0 0 0

0 0 0 0 0 0 28 888 59.2

3.2. The Stage One model The initial point used to start the optimization is shown in Figure 1. This is the default, infeasible layout used by NLT. It places all departments in an elliptical pattern about the center of the facility. The Stage One model attempts to find a solution that minimizes total material flow costs with the centers of all departments spread as evenly as possible over the plant. The model ignores the boundaries and areas of the departments. This is accomplished by first defining an 'ideal' layout where the centers are perfectly distributed. An ideal layout for our ten-department example is shown in Figure 2. Note that the numbers in the figures identify positions, not departments. Any single department may be placed at any of the ten positions. Although it is not possible to say a department should be placed at a particular position, one can say that a department should be located near one of the ten positions. To determine the ideal layout for a given number of departments, the algorithm first calculates the number of evely spaced columns to place the departments in, proportional to the dimensions of the plant. It then places the departments evenly within these columns. In order to produce an evenly spread layout, we construct a set of constraints on the centers of the departments. These constraints ensure that the moments about the x- and y-axes of both the desired

7 3

10 6

2

9 5

1

8

4

Figure 1. Default initial layout: ten-department problem

Figure 2. Ideal layout: ten-department problem

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layout and the layout being developed are the same. By the m-th moments about the axes of a layout we mean

M,,(k)

=

E Xi(,,-k) Yik ,

k = 0 ..... m

(20)

i

where there are m + 1 moments at the m-th level. The constraints for this model, then, are

M--m(k ) = M m ( k ) ,

for k = 0 . . . . . m and m = 1. . . . . N

(21)

where M--m(k), k = 0 . . . . . m, are the m-th moments of the actual layout being generated and Mm(k), k = 0 , . . . , m, are the constant moments of the ideal layout. As a simple demonstration of the moments, assume that the origin of the (x, y) plane is located in the center of the facility: The constraints for the first moments of the ideal layout specify that: /1(0 ) = EXi= i

0,

/ l ( 1 ) = ~ y , = (). i

(22)

In other words, the sum of the department centers to the right of (above) the origin should equal to sum to the left (below). The value of N in (21) determines the highest order of the moments to be considered in the problem. By requiring successively higher-order moments of the layout being developed to be equal to the ideal layout, one can come as close to it as desired. However, as one adds these constraints to the problem, it becomes harder to solve with minimal objective function improvements. By adding in higher moments, one develops a more regular layout, but the value of an accurate solution must be weighed against the fact that it is an initial solution that will be improved upon by a more complex model. In our tested examples, the solution to the problem with only the four lowest-order moments gives a layout where the departments are spread evenly, but not symmetrically, about the facility. The concept of using moments about the axes to develop a layout is not new. Blanks [5] and Drezner [8] have used moments as aids for solving location problems. Blanks deals with layouts for VLSI, while Drezner deals with the quadratic assignment problem. Both authors, however, consider only the lowest two order moments about the axes. A simplification used in this model is the replacement of the weighted Euclidean distance by the weighted square of the Euclidean distance. This is because of the discontinuities in the gradient of the Euclidean distance. The gradient is undefined at the point where dij = 0 (i.e. the departments have the same centers). Since there are no constraints in this model to ensure that the departments do not overlap, it is possible for d u to be zero. To avoid this problem, we square the distance measure. Using this simplification, the model for Stage One, which we call (P-II), is: E c i j ×d2j + E c i ×d~ /j i

min

f(x, y)=

s.t.

Mm(k ) = M m ( k ),

for k = 0 . . . . . m, and m = 1. . . . . N.

For a given problem with n departments using up to N moments, the problem will have 2n variables and 1 N ( N ~: 3) constraints. For the ten-department example with fourth-order moments, this gives 20 variables and 14 constraints. The Stage One model was solved with the two lowest orders of moments being constrained. The problem was run to the point where the penalty cost comprised less than one percent of the total function value. The layout obtained is shown in Figure 3. The CPU time needed to obtain this result was approximately 4 seconds. This solution was then used as an initial point for rerunning the Stage One model with the three lowest orders of moments being constrained. The solution to this problem is shown in Figure 4. This solution was then used as an initial point for rerunning the Stage One model with the four lowest orders of moments being constrained. The solution to this problem is shown in Figure 5.

D.J. t,an Camp et al. / Nonlinear optimization approach for soh,ing FL problems

184

Figure 3. Stage One solution, two moments: ten-department problem

Figure 4. Stage One solution, three moments: ten-department problem

Note that the Stage One model was run three times, adding constraints in for each successive run (the default strategy used by NLT).

3.3. The Stage Two model The final solution of the Stage One problem is used as the initial position for Stage Two. As described earlier, we now treat each department as a circle, and penalize overlapping departments. The model for this stage, (P-Ill), is very similar to model (P-I): min

f(x, y)=

Ec,jxd,j+ Ee, x4 O

s.t.

i

du - ( r i+rJ)>lO for alli, j, I

(23)

f o r a l l i,

(24)

½hT--(Yi +ri)>lO for alli,

(25)

~ W T - - ( X i + r i ) >10

oC

f7

A 4

)

3 5

) Figure 5. Stage One solution, four moments: ten-department problem

Figure 6. Stage Two solution: ten-department problem

D.J. t~an Camp et al. / Nonlinear optimization approach for soh'ing FL problems I (xi-ri)+2wT>~O

1 (yi-ri)+2hr>~O

185

for a l l i ,

(26)

for a l l i ,

(27)

min( w T, hT) - l~ in >/0,

(28)

l~ "x - min(w.r, hT) >/0

(29)

where r~ is the radius of the circle representing department i. The objective function uses the material handling cost from (3). In this model the distance measure is not squared, since the constraints do not allow overlap. As in (P-I), there are three categories of constraints in the model. The first group of constraints ensures that no two departments overlap. In this stage of the problem there is only one type of constraint in this category. It is that, for all pairs of circles, the distance between the centers must be greater than the sum of their radii. This is given by (23). Note that this constraint introduces nonconvexities in the problem. The problem with only this constraint has been considered by Drezner [7]. The second category of constraints, (24)-(27), requires that all departments remain inside the facility. These are similar to those of the basic model (P-I). As before, the final category of constraints, (28) and (29), ensures that the dimensions of the facility as a whole must be within acceptable limits. For a given number of departments, n, the number of variables in this problem is 2n + 1 and the 1 2 number of constraints is ~(n + 7n + 4). For a moderate-size layout problem of ten departments this gives 21 variables and 87 constraints. The solution of this problem alone has practical applications, for example, laying out instruments on a control panel; however, it is used here to find a starting solution for the Stage Three problem. The layout in Figure 5 (the final solution from Stage One) was used as the initial layout for running the Stage Two model where the overlapping of the circles representing the departments is penalized. The Stage Two problem was solved until the penalty cost was less than 1 percent of the total function value, and its solution is shown in Figure 6.

3.4. The Stage Three model Finally, the Stage Three model was run, using the solution from Stage Two as an initial layout. This model was solved to the point where the penalty cost was less than 0.01 percent of the total cost. After the solution of each model, interchanges were attempted to improve on the layout. The final layout developed is shown in Figure 7. This solution has a Euclidean cost of 24445. The total CPU time used to run all stages of the problems was 847 seconds. A total of 49181 iterations and 264616 function evaluations were required. A complete summary of the computational results is shown in Table 3. Note that the major part of the computations occurred in the solution of the Stage Three model. This is due to the accuracy required for the final solution. A large amount of this time is used for simply 'fine tuning' the layout and eliminating all the overlap. To illustrate this, Figure 8 shows the layout developed by NLT when the Stage Three model was solved only to the point where the penalty cost was less than 1 percent of the total cost. The time required to obtain this solution was 111 seconds. The cumulative iterations and function calls to produce this layout were 10430 and 31415, respectively. Note the similarity between this layout and the layout developed using the default parameters. The existing layout used by the company has Euclidean cost of 43305. The layout found by NLT produced a cost reduction of over 40% from this original cost.

4.

Computational results

The implementation of NLT recognizes that there are many factors in the layout design process that are not considered by its models. To deal with these factors, the program allows extensive user input

D.J. uan Camp et al. / Nl~nlinear optimization approach for soloing FL problems

186

5

5

7 10

10

8

4

3

9

2 1 6 Figure 7. Stage Three solution: ten-department problem

Figure 8. Interim solution: ten-department problem

throughout the optimization procedures. To simplify the user input, a graphical display of the current layout along with its cost is presented as the algorithm is running. Using a combination of graphical user interface and command line interface, the user may manipulate the positions of departments, force interchanges, or cause premature termination of the solution procedures. One also may choose the initial layout for starting the optimization for the model. NLT has been run on the standard problems presented by Nugent et al. [24]. Nugent presented 8 different problems varying in size from five to thirty departments. All the departments were of equal size and shape (unit squares). The shapes of the facilities were also fixed. Because of these restrictions, Nugent's problems could all be modeled as quadratic assignment problems. NLT produced good-quality results for these problems; however, because of the many constraints placed on the final layout (all departments of equal size and being restricted to square shape inside a fixed-dimension facility) the computational time required to generate the solutions was large. The strength of NLT is that it can not only solve problems with equal-size departments, but also those with different-size departments, without overly restricting their dimensions. We see the potential and flexibility of this method to be for this important class of practical problems and have concentrated on this class when comparing NLT to other algorithms. Few problems with departments of unequal areas appear in the literature [17]. Some that do are from Bazaraa [2], and from Montreuil, Ratliff, and Goetschalckx [20]. These were considered in order to compare NLT to other algorithms. The problems from Bazaraa contain twelve and thirteen unequal-area departments, respectively. The layouts generated by NLT are shown in Figures 9 and 10. In setting up these problems for NLT it was assumed that the minimum width for a department was one unit and that the dimensions of the facilities were fixed to those in Bazaraa's paper. Costs for these problems have been calculated by Bazaraa [2] and Hassan et al. [14] using rectilinear distances. Both authors break down each department into unit squares and then calculate the cost using the weighted flow between these squares. The rectilinear costs of the layouts

Table 3 Summary of computations: ten-department problem Model

Cumulative iterations

Cumulative function calls

Cumulative CPU seconds

Stage Stage Stage Stage Stage

93 336 3478 6607 49181

131 538 9344 19778 264616

4 12 42 53 847

1:2 moments 1:3 moments 1:4 moments 2 3

187

D.J. ran Camp et al. / Nonlinear optimization approach .for soh,ing FL problems

,5

10

,'<

,1

"2

I:f

:l

2

1

3

,5 L

Figure 9. Layout developed for Bazaraa's first test problem

Figure 10. Layout developed for Bazaraa's second test problem

generated by these techniques including NLT are shown in Table 4. Thus the costs are compared equitably although NLT minimizes the Euclidean cost and measures distances between centroids of departments. The CPU time used to solve the first problem was 376 seconds. The number of iterations performed was 75619, and the number of function evaluations was 115540. For the second problem the corresponding numbers were 461 seconds, 68685 iterations and 108579 function evaluations. Based on these two test problems, NLT produced layouts with lower cost than those found by Bazaraa's procedure and costs comparable to those reported by Hassan (The best results reported by Hassan are approximately 10% better than our results). The large reductions in cost from those reported by Bazaraa can be attributed to his algorithm requiring a much more restricted solution space than does NLT. For Bazaraa's method all dimensions are restricted to integer values and a set of allowable locations for each department must be given. These are not required in NLT. The slightly better results of Hassan are not surprising. In his computations, Hassan made no restrictions on the dimensions of the facility and allowed departments to take on nonrectangular shapes. The third problem NLT was tested on is from Montreuil et al. [20]. This example contained 12 departments with flow between the departments, and from the departments to the outside of the facility. The solution found by NLT for this problem is shown in Figure 11. It has a rectilinear cost of 512. The CPU time used to solve the problem was 927 seconds. As well, 96842 iterations and 330039 function evaluations were performed. The MATCH system used by Montreuil to develop a layout maximizes a weighted sum of adjacent departments; it does not minimize the rectilinear cost of the layout. The rectilinear cost of the layout found in [20] is found to be 511. Although the solution methods used by MATCH and NLT, as well as the layouts developed, are very different, they have almost identical costs. An important fact to note is that the objective function being minimized by NLT is calculated using Euclidean distances, while the cost with the algorithms are compared is calculated using rectilinear distances. Although there is a correlation between the optima of the two objective functions, the optimum for one is not necessarily the optimum of the other [6]. Despite this, the quality of the results for NLT is always comparable with the algorithms that do attempt to minimize the rectilinear distance.

5. Conclusions

The goal of this paper was to present a new construction algorithm for the facility layout problem based on nonlinear programming. To this end, the three models used in NLT to approximate the layout problem were developed. The final form of the algorithm is an iterative structure in which successively more difficult problems are solved, using the solution from the previous stage as a'n initial point.

188

D.J. van Camp et al. / Nonlinear optimization approach for solving FL problems

Table 4 Comparison of results: Bazaraa's test problems Algorithm

Problem 1

Problem 2

Bazaraa's procedure PLANET ~ SHAPE NLT

14079 11664-11808 10578-11140 11910

8170.5 6399-6480 6339-6462 6875

Costs for PLANET and SHAPE as reported by Hassan et al. [14].

WH

PA

MT

ST

OV

SE

FP

ES

()F

SH SC SS

Figure 11. Layout developed for Montreuil's problem T h e p e r f o r m a n c e of N L T on several test p r o b l e m s was p r e s e n t e d . A d e t a i l e d d e s c r i p t i o n o f the solution p r o c e d u r e for a r e a l - w o r l d p r o b l e m was p r e s e n t e d to illustrate the a l g o r i t h m s in N L T . A s well, N L T was run on s t a n d a r d p r o b l e m s from the l i t e r a t u r e , a n d its results w e r e c o m p a r e d to those from o t h e r algorithms. N L T i l l u s t r a t e d that it could p r o d u c e high-quality solutions at low c o m p u t a t i o n a l expense. T h e m o d e l f o r m u l a t i o n p r e s e n t e d in this p a p e r is not o p t i m i z e d for s p e e d or accuracy; however, it a l r e a d y o u t p e r f o r m s m a n y o t h e r a l g o r i t h m s in run times a n d solutions p r o d u c e d . A simplified, n o n g r a p h i c version o f N L T has b e e n i m p l e m e n t e d for use on t h e I B M - P C . R u n s of t h e p r o b l e m s p r e s e n t e d here, using t h e P C version, suggest that run time is still fast e n o u g h to be u s e d on r e a l - w o r l d problems.

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