QAP and solve it using optimal algorithms; for example: Lawler (1963), Kaufman and Broeckx (1978),. Bazaraa and Sherali (1980) and Burkard and Bonniger ...
European Journal of Operational Research 53 (1991) 1-13 North-Holland
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Efficient models for the facility layout problem Sunderesh S. Heragu Department of Management and Marketing, School of Business and Economics, State University of New York, Plattsburgh, N Y 12901, USA
Andrew Kusiak Department of Industrial Engineering, The University of Iowa, Iowa City, IA 52242, USA Received January 1989; revised July 1989
Abstract: In this paper, two new models of the facility layout problem are presented: linear continuous with absolute values in the objective function and constraints, and linear mixed integer. The linear mixed integer models have lesser number of integer variables than any other existing formulation for the facility layout problem. While most other linear mixed-integer models available in the literature have been obtained through a linearization of the quadratic assignment problem, the ones presented in this paper are not. The continuous models have an even more compact form. An advantage of the formulations presented in this paper is that the location of sites need not be known a priori. More importantly, two of the formulations model the layout problem with facilities of unequal area. Solving the models presented with an unconstrained optimization algorithm yields good quality suboptimal solutions in a relatively low computation time. The continuous models appear to be more useful for solving the facility layout problem than other models published in the literature. Keywords: Linear programming, non-linear programming, models, algorithms, facility layout problem
I. Introduction The facility layout problem involves the arrangement of a given number of facilities so that the total cost to move the required material between the facilities, is minimized. Heragu and Kusiak (1988) discussed application of the facility layout problem to layout of machines in automated manufacturing systems and identified four patterns of machine layout, namely: circular single-row, linear single-row, linear double-row and multi-row. Similarly, for the facility layout problem, two patterns of layout frequently occur in practice: single-row, in which facilities are arranged linearly in one row and multi-row, in which facilities are arranged linearly in two or more rows (Figure 1). In the above, the shape and dimension of the building are not considered. Also, the facilities may have empty space between them. The single-row and multi-row layout problem have also been known in the literature, as one-dimensional and two-dimensional space allocation problem, respectively (Simmons, 1969). A special case of the single-row layout problem, i.e. when all facilities are of the same length, is known as the linear ordering problem (Adolphson and Hu, 1973). A number of applications of the single-row layout problem have been identified. Examples are: the problem of assigning files to the cylinders of a disk, arrangement of books in a shelf, warehouse layout (Picard and Queyranne, 1981), layout of machines on one side of the AGV path on a factory floor (Heragu and Kusiak, 1988). A common application of the multi-row layout problem is 0377-2217/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
S.S. Heragu, A. Kusiak / Efficient rnodelsfor the facility layout problem
I (a)
II-(b) Figure 1. Samplepatterns for the facilitylayoutproblem.(a) Single-rowlayout,(b) multi-rowlayout the layout of machines in an automated manufacturing system. Other applications of the multi-row layout problem include, computer backboard wiring (Steinberg, 1961), typewriter keyboard design (Burkard, 1984), etc. A number of models, for example, the Quadratic Assignment Problem (QAP) (Koopmans and Beckmann, 1957), linear mixed-integer programming problem (Love and Wong, 1976a), nonconvex mathematical programming problem (Drezner, 1980), have been developed for the layout problem. All the formulations, except that of Drezner (1980) and Beghin-Picavet and Hansen (1982), require that the location of sites be known a priori. The formulations presented in this paper, are more general than the existing models because the location of sites is not required to be known a priori. The single-row and multi-row facility layout problems are known to be NP-complete (Beghin-Picavet and Hansen, 1982). The models and algorithms developed for these layout problems are briefly surveyed in the next section. In Section 3, new models (continuous linear with absolute variables in the objective function and constraints and linear mixed integer) are presented. Computational results for the models are provided in Section 4. Conclusions are drawn in Section 5.
2. Literature survey
Single-row facility layout problem. Karp and Held (1967) and Simmons (1969) developed a dynamic programming algorithm and a branch-and-bound algorithm, respectively, for the single-row facility layout problem. Love and Wong (1976b) presented a linear mixed-integer programming model and solved it using the IBM MIP code (IBM, 1974). Dynamic programming algorithms have also been developed by Picard and Queyranne (1981) and Beghin-Picavet and Hansen (1982). Picard and Queyranne (1981) extended the dynamic programming algorithm of Karp and Held (1967). All the above algorithms have relatively high computational time and memory requirements. Picard and Queyranne (1981) reported that an 11-facility layout problem required less than a second of central processing unit (CPU) time and 100k memory on an IBM 360/75. But for larger layout problems, for
S.S. Heragu, A. Kusiak / Efficient models for the facility layout problem
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example the 20-facility layout problem, they indicated that the dynamic programming algorithm would require excessively high computation time and memory. Heragu and Kusiak (1988) presented a simple heuristic for the single-row facility layout problem. Multi-row facility layout problem. Since the early 1960s a great deal of research on the multi-row facility layout problem has taken place. Kusiak and Heragu (1987) surveyed models and algorithms (optimal and heuristic) for the multi-row facility layout problem. Koopmans and Beckmann (1957) modeled the multi-row layout problem as a QAP. Since then, a number of researchers have attempted to linearize the QAP and solve it using optimal algorithms; for example: Lawler (1963), Kaufman and Broeckx (1978), Bazaraa and Sherali (1980) and Burkard and Bonniger (1983). Love and Wong (1976a) presented a linear mixed-integer program (which is not a linear transformation of the QAP), for the multi-row layout problem. Two classes of optimal algorithms have been used to solve the facility layout problem: - branch-and-bound algorithms; -- cutting-plane algorithms. The algorithms developed by Lawler (1963) and Kaku and Thompson (1986), are examples of optimal branch-and-bound algorithms and the algorithms developed by Bazaraa and Sherali (1980) and Burkard and Bonniger (1983) are examples of cutting-plane algorithms that have been developed for the linear transformations of the QAP model. The main disadvantage of optimal algorithms is that they all have rather high computational time and memory requirements. For example, to find an optimal solution for a 15-facility multi-row layout problem, almost an hour of CPU time was required on a CDC Cyber 76 computer (Burkard, 1984). As a result, a number of heuristic algorithms have been developed. The heuristic algorithms can be divided into four classes: - construction algorithms; - improvement algorithms; - hybrid algorithms; - graph-theoretic algorithms. The heuristic algorithms developed by Heragu and Kusiak (1988), Armour and Buffa (1963), Bazaraa and Kirca (1983), and Foulds et al (1985), are examples of construction, improvement, hybrid and graph-theoretic algorithms, respectively.
3. Models for the facility layout problem In order to model the single-row facility layout problem, the following assumptions are made: (i) facilities are to be arranged along a straight line as shown in Figure l(a); (ii) facilities are to be oriented in only one given direction; (iii) the shape of facilities is known in advance; (iv) there is no restriction on the shape of the building in which the facilities are to be located. rl
AclLI i Y
lj
li
FACILITY J
Xi
~
I dij
xj Figure 2. Illustration of parameters and decision variables for the single-row facility layout problem
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S.S. Heragu, A. Kusiak /Efficient models for the facility layout problem
The following notation is used in models M1, M l a and M l b : f,j cU l, dU x,
= = = = =
number of trips to be made between facilities i and j; cost per unit distance travelled between facilities i and j; length of facility i; minimum distance by which facilities i and j are to be separated; distance between center of facility i and reference line rl.
The parameters l i and dig, decision variable x i and the reference line rl are illustrated in Figure 2.
Model MI The objective function of model M1 minimizes the total cost involved in making the required trips between facilities. minimize
c, ifij Ix, - xj I
~
(1)
i=1 j = i + l
Ix~-xjl>~½(l,+lj)+d,;,
subjectto
i=l,...,n-1,
j=i+l
. . . . . n.
(2)
Constraint (2) ensures that no two facilities in the layout overlap. Model M1 cannot be solved optimally by a standard linear programming code, as it includes absolute values in the objective function and constraints. In order to transform model M1 into an equivalent linear mixed-integer programming model M l a , define: X,
xij
-
-
Xj)
0
z,j=
(3)
if x j - xj~< 0;
(,,,- x+) xTj=
if x , - x j > 0,
0
if x, - Xg ~ 0;
1
if x i < x j,
0
ifx~>lxj.
(5)
Based on the above, it is obvious that
Ix, - x, I = x,~ + x,~,
(6)
(~, -
(7)
x j) = x, 3 - x ~ .
Model Mla
minimize
Z
i=1 j = i + l
subject to
ci#,j(+,; + xT,+)
(s)
xi-xg+MZu>_-½(l,+l/)+d,j,
i=1 ..... +-1,
-(xi-xj)+M(1-zi/)>~½(li+l/)+dij, x~-xu=xi-x xi+,x~>lO, z,a=0orl,
j,
i=1 ..... n-l,
i=l,...,n-1, i=1 ..... n-l,
j=i+l
j=i+l j=i+l
i=1
n-t,
. . . . . n,
..... n, . . . . . n.
j=i+l
..... ,, j=i+l
(9) . . . . . n,
(10) (11) (12) (13)
s.s. Heragu, A. Kusiak / Efficient models for the facility layout problem
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Constraints (9) and (10) ensure that no two facilities in the layout overlap. Since z;j is a 0 or 1 variable, only one of the constraints (9) and (10) holds. Constraint (11) is identical to expression (7). Constraint (12) ensures nonnegativity and constraint (13) imposes integrality. In the above and other models presented in this paper, the letter M denotes an arbitrarily large positive number. Murty (1983) has shown that in any model, if the transformation similar to (6) is made in the objective function and transformation similar to (7) is made in the constraint, then at least one of x,,+, x,~ will always be zero, i.e. +
XIIXI
!
= O.
(14)
Observation. If transformation of the form (15), which is similar to (6), is made in the constraint Lx,-xjl
i.e.
>b,i,
xi-~ +x/y>~b,,,
(15)
where b,j is a real constant, then the solution to the model will not always satisfy (14). This can be easily verified using problem number 2 in Table 2, provided later in Section 4. This is why constraint (2) in model M1, which is similar to (15), has been replaced by constraints (9) and (10) in model Mla. It should be noted that the single-row facility layout problem can also be modeled as a non-linear continuous problem as shown below: Model M l b
minimize
E
c,,z,
+ x,,)
(16)
i=1 j = i + l
subject to
i = 1 ..... n - l ,
xi+j+xij>½(li+lj)+dij,
i = 1 ..... n - l ,
xj~-xT/=x,-x/, x;ixij+- = 0 , +
Xij,Xtj
i=1 .... n-l,
>~0,
i=l,
..
. , n-1
j=i+l
j=i+l ,
j=i+l
j=i+l
..... n,
..... n,
.... ,n, ,...,
(17) (18) (19)
(20)
n.
vrl
FACILITY i
xi
II
FACILITYj t y xj
,!
J hrl
Figure 3. Illustration of decision variables and referencelines for the multi-row layout problem with facilities of equal area
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S.S. Heragu, A. Kusiak / Efficient models for the faciliO' layout problem
Constraint (17) ensures that no two facilities in the layout overlap. Constraint (18) is identical to expression (7). Constraint (19) ensures one of the variables x,~, x~ is always zero. Constraint (20) imposes nonnegativity. Model M1 is used to formulate the single-row facility layout problem. In general, one finds that the facilities have to be located in two or more rows. To model this problem, either the QAP or its equivalent linear transformations have frequently been used. Below, a linear program (21)-(23) which can be used to model the layout problem in which the facilities are of equal area and square in shape, is presented. In addition to c~j, dis, f~s, defined in model M1, the following notation is used: x~ = horizontal distance between facility i and vertical reference vrl. y, = vertical distance between facility i and horizontal reference line hrl. The above decision variables and the reference lines vrl and hrl are illustrated in Figure 3.
Model M2 The objective function of mode! M2 is similar to that of model M1 and minimizes the total cost involved in making the required number of trips between the facilities. All the facilities are considered to have a size of one unit. n-1
Z
minimize
~ c,Jij(Ix,-xjl+ly,-yjl)
(21)
i=1 j = i + l
subjectto
Px,-xjl+[ys-yjl xi,y i integer,
>tl,
i=1 ..... n-I,
i = 1 . . . . . n.
j=i+l
. . . . . n,
(22) (23)
Constraints (22) and (23) ensure that no two facilities in the layout overlap. Constraint (23) imposes integrality. If the horizontal and vertical dimensions of the floor plan are denoted as h and v respectively, then by adding constraints (24) and (25) provided below, one can ensure that the facilities are located within the boundaries of the floor plan.
]x,-xj]
~
