An algorithm for the quadratic assignment problem using Benders' decomposition L. KAUFMAN Vri]e Universiteit Brussel, Pleinlaan 2, B.1050 Brussels, Belgium

F. BROECKX Rijksuniversitair Centru m Ant werpen, Middelheimlaan 1. B-2020 Antwerpen, Belgium

Received February 1977 Revised November 1977

A well known method used for solving quadratic assign- ment problems proceeds by the construction of an equivalent much larger linear assignment problem with many side con- straints. The disadvantage of this method lies in the weakness of the bounds obtained by solving the linear problem. An alternate linearization has been suggested using a general method of Glover. in this paper the mixed integer program obtained by Glover's method is discussed and a solution using Benders' decomposition is proposed.

1. Introduction

The assignment problem (AP) is defined as the problem of assigning n facilities to n locations at minimum cost. It is assumed that each facility must be assigned to exactly one location.

A mathematical formulation of this problem is given by:

H n

Min D ~ c o x 0 , i = 1 / :1

subject to ~ x 0 =1 , /

(1)

v i = 1, ..., n , (2)

~ x # = l , v / = 1 , . . . , n , (3) i

xiie {0, I } , (4)

© North-Holland Publishing Comapny European Journal of Operationa! :f~2search 2 (I 978) 207 - 211.

where xii represents a (0 -1) variable equal to 1 if and only if facility i is assigned to location ] and cii is the cost of such an assignment.

Constraints (2) express that each facility i must be assigned to exactly one location and constraints (3) that each location/must have one facility assigned to it. If the costs depend on the assignment of pairs of facilities, the objective becomes a quadratic func- tion of the assignment variables xik. It takes the following form:

Min ~ D ~ ~ CqktXikX#, (5) i / k t

where ci/m is the cost of assigning facilities i and ] to locations k and L

Problem (5), (2)-(4) is called the quadratic assign- ment problem (QAP). When the costs ci/kt are given by

¢ijkl = f i i " dk l (6)

the problem is called the Koopmans-Beckmamt problem (KBP). In this case the fii represent unit transportation costs for given flows to be sent from i to/, and the dm represent the distances between k and L

It was first stated mathematically by these two authors in 1957 [19] in the following way:

n - I n n n

i=l /=i+l k=l 1=I {7)

n

subject to ~ Xik = 1 , i= 1

vk = I . . . . . n , (8)

r/

D Xik = 1 , k = l

Vi = 1, ..., n , (9)

xik e {o, 1). (1o)

Many algorithms, both heuristic and exact, have been proposed for solving the QAP and its special case, the KBP.

Hillier [17], Hillier and Connors [I 8], Steinberg [24], Armour and Buffa [ 1 ], Gilmore [ 12], Gaschutz and Ahrens [9] proposed heuristic algorithms. Nugent, Vollman and Rural [221 besides proposing a new heuristic procedure made an interesting comparison between existing heuristics using a set of randomly generated test problems.

Among the exact algorithms those of Burkard [3], Elshafei [7], Gavel* and Plyter [ 10], Gilmore [ 12],

207

2 0 8 L. Kau#,a,n, F. Broeekx / An algorithm for the quadratic assignment problem

Hansen and Kauiman [15], Land [20], Lawler [21 ] and Pierce and Crowston [23] can be mentioned. In this last paper tree-search procedures for the QAP are discussed. As is well known using Karp's terminology the KBP is NP complete (see [6]). Hansen and Kauf- man [16] proposed a method based upon a lineadza- tion technique.

In thi~ paper computational experience is presented using the linearized problem and a new approach using Benders" decomposition method is proposed.

2. Linearization of the problem

We here consider the symmetric KBP:

w . z = ,

" i k t •

subject to: ~ X i k = 1 , Vk = 1, ..., n , i

(ll)

(12)

In order to write Glover's most general conditions, we introduce

c,+,~ = max(~ ~ ~idk,, O) and ] l

c ~ = rain( ~ ~fiftkt , 0). (19) / I

Then they are

c~ < wa, < o k , (20)

E) ~ 5 : ' : ~j, - ck( l - x~k)-< ~ k i :

< Z-'~'7~ ~ j~jdklX]l- c~( l - x i k ) . (21) i t

In (15) it turns out tha~ c~ -= 0, so that for intro- duction ,~7 each new variable wik, four new con- straints ~:e introduced:

~ X i k - 1 , V i = 1, ..., n , (13) k

xik G (0, 1). (14)

For such a quadratic objective function, several tech- niques have been proposed for iinearization (Fortet [8 ] and GIover and Woolsey [ 14J).

We will use Glovei's linearization technique [ 13], which iz extremely favourable in this type of problem, where one only has positive cost-coefficients

c~m = ~i" dka. (l 5)

Rewrite 7. as: z = ~ ~ xik(~-~ ~ fj/d~tx/l) (16) i k j t

and introduce continuous va ~iables

W~k = xik Z~ Z~ ~'#kt x/,. (17) i t

There are n 2 such new continuous variables, and the new objective is to minimize

= 23 i k

08)

It is clear Wik is tO be zero ifxt~ is zero, and wik = Z/Zz ~dktXp ifxik equals one.

4- 0 < Wik < CikXik, (22)

~ /OeklXjl - c?~( ~ - Xik) < wik i t

< ~. l ~. j f i i d k l X # . (23) / I

As the objective is to minimize the sum of the Wik,

and as the Wik do not appear in any other constraints, the two right hand sides of (22)-(23) can be dropped. The QAP then reduces to the following linear prob- lem:

min ~ ~ wik, (24) i k

subject t o

xik = 1 , ¥ i = 1, ..., n , (25) k

xik = I , Vk = l , . . . , n , (26) i

C +k Xik

+

i t

Xik ~. { 0 , 1 } , Wik >I O, cont. (28)

L. Kaufman, F. Broeekx ~An algorithm for the quadratic assignment problem 209

Table 1

n No. of var. (0, 1) Cont. No. of constr. M.C. Other constr.

6 66 36 30 42 12 30 8 120 64 56 72 16 56

12 276 144 132 156 24 132 20 780 400 380 420 40 380

n 2n 2 - n n 2 n(n - 1) n(n + 1) 2n n(n - 1)

We can now make the following comments: Glover's method finally introduces n 2 new continuous variables and n 2 new constraints. These variables are without sign restriction in the general QAP but posi- tive in this type of problem. The number of constraints is low compared with other linearization methods, and the problem can be solved by a code for mixed integer programming.

3. Computat ional experience

To test the algorithm IBM's mixed integer code MPSX was used to solve two of the problems pro- posed by Nugent, Vollman and Ruml [22]. A heuris- tic CRAFT-type method was used to compute a sub- optimal value for the economic function. Starting from an arbitrary solution this method inspects a reduced number of possible pair exchanges, carries out that one which yields the largest improvement, and starts a new iteration. It is that suboptimal value that is introduced in MPSX as a first upper bound.

Some of the (0 -1 ) variables can be considered as continuous. Indeed using the structure of the mul- tiple choice constraints (12) and (13), it is possible to make a variable continuous in 2n - 1 of these con- straints.

4- It is possible to reduce the vgues of the Cik coef-

ficiems in the convexity constraints (27). Indeed c+k can be considered in Glover's linearization method as an upper bound on the total value taken by the coefficients of this constraint (ZiY, t fiidkIXfl). How-

ever the variables xjt must satisfy the multiple choice constraints (12) and (13), which implies that the largest total value which the constraints can take is given by solving a linear assignment problem (LAP) with coefficients f q " dta.This makes it possible to tighten the constraints considerably.

As the input of MPSX becomes very large with growing n, a matrix-generator was written. The den-

sity of the complete matrix is approximately 25%. It can be reduced by n rows and n columns by a sim- ple sequential grouping of the quadratic terms, pre- paring for Glover's linearization. Table 1 gives an im- pression of the complexity of the problem for growing values of n.

Nugent, Vollman and Rumrs n = 6 problem was solved completely by MPSX in 2.63 rain. and for the n = 8 problem the optimal solution was |bund in 4 rain on an IBM 370/168. However this solution was not proven to be optimal as the core requirements were too large for MPSX. It is our opinion that a sophisticated use of MPSX should make it possible to solve problems approximately up to the size of n = 12. These results seem rather disappointing compared with those obtained by Burkard [4] and Burkard and Stratmann [5] obtained on a CDC CYBER 76. In the next section we will attempt to solve the linearized problem using the Benders' decomposition method.

4. Benders' decomposition

In this" paragraph a different approach is proposed to solve the linearized QAP. This method is based on the Benders decomposition technique [2].

By rewriting the variables Wik as variables Xk the linearized form of the QAP can be written as:

n 2

Min 2.~ x k , k=!

(29)

subject to

n

~ y i / = I , i ; l

V] = 1, ...,n ,

n

~ y i / = 1, ]--1

Vi= l, . . . , n , (31)

210 L. Kaufman, F. Broeekx / An algorithm for the quadratic assignment problem

~-J ~-I a ~ y i i - - x k < b k , V k = 1 , . . . , n 2, i i (32)

x >0, {o, 1}. (33)

A special property of this model is that if the vari- ables Yij are fixed the remaining optimization prob- lem is very easy to solve. ~ i s property suggests the use of Benders decomposition for solving the prob- lem. The application of this method leads to the fol- lowing algorithm:

Initialization

- Select a convergence tolerance parameter e >/0. Se tH=0 .

- Initialize UB (upper bound) = 0% and LB 0ower bound) = 0.

- If a feasible solution (yb) is known go to step 2. Otherwise go to step 1.

Steo 1 Solve the current master problem

Z H = Min Yo, (34)

yo>~ ~ u ~ ( ~ ~ a ~ . Y i j - b k ) h = I , 2 , . . . , H , t, i i (35)

Yii = 1 , V/=I .... , n , (36) i

¢..h'+l,~ Denote the optimal value by T u'i/ ,, and the opti- • -/]'~H+I~ real solution by (x H+l) The quantity • Left j is an

upper bound on the optimal solution of the QAP. It is easy to see that TOr~//+t) is the value of the solution

--- H+I, t H+I~ If l"O'ii j < UB, set UB = T(,Yii , . (44)

If UB ~ LB + e, terminate. (45)

(b) Determine an optimal dual solution of the sub- problem with (y ) = (yH+l). Call this solution (uH+l). Increase H by l and go to step 1.

Several remarks can be made about the use of Benders' decomposition:

(1) Problem (41)-(43)can be solved trivially. Indeed if X;i~i/a~/- bk is positive, set

Xk = ~ . ~ a ~ y ~ / / + l - bk (46) i !

and otherwise set

xk = 0 . 1,47)

To find a set of optimal dual variables it is sufficient to set

Uk = l (48)

y~/= 1, Vi= 1, . . . ,n, (37) /

yij E (0, 1 ), (38)

Yo~>0. (39)

At the first iteration there is no constraint (35) and Zn = O.

Call the optimal solution (y/~+t Yo) and set LB -- Z H.

Terminate if UB ~ LB + e . (40)

Step 2 (a) Solve the subprob!em

Mi, x , , ( ,1) k

5-' k u + l subject to .zJ a/jy~] - xk < bk, (42) i /

xk >o. (43)

aijyi/ - bk > 0 (49) i t

and otherwise

o . (5o)

(2) A variant which could be used here with suc- cess has been suggested by Geoffrion and Graves [ 11 ] in their application of Benders' decomposition to a location problem.

Instead of completely solving the problem of step 1 it is sufficient to search for a 0-1 solution which has a value less than UB - e. This implies of course that the problem does not yield a lower bound and the termination rules must be altered. The method can then only be terminated if the problem of step 1 has no feasible solution with value less than UB - e.

(3) The motivation behind the variant of Remark 2 is that it takes several Benders' cuts (constraints (35)) in order to obtain good bounds.

Therefore for the first iterations optimizing is a

L. Kaufman, t7. Broeckx / A n algorithm for the quadratic assignment problem 211

waste of time. Another motivation is that if only a 15 ! feasible solution is sought problem (34)-(371 can be made a pure integer problem instead of a mixed- [61 integer problem (variable Y0 is continuous).

Indeed the constraints

Y o < U B - e : (51)

and

y o > ~ u h ( ~ k h bk) h = 1 9, H a i jY i i - , . . . . , k ~ i (52)

can be replaced by the constraints

[71

181

[101 - a q y # - bk) h = 1, 2, ..., H

k ~ i (53) {Ill

using any objective. If a solution is found it is then immediately ased

to find another Benders' cut. If it is better than the 1121

previous upper bound it replaces it. (4) it is also possible to use a hybrid method" i.e.

to suboptimize for a number of iterations and subse- [131

quently to introduce a lower bound by optimizing

completely. [ 141 (5) As all coefficients of the KBP are integer,

taking e = 0.99 will always yield an optimal solution.

Computational experience with this method has 1151 led to rather disappointing results. These are due to several reasons, among which the lack of an efficient [161 program for O- 1 or mixed integer problems seems

the most important . However an improvement of the

use of Benders' decomposition will probably lead to a more efficient program. Fur ther computational I 17 I

results are forthcoming. [ 181

References [19]

[1] G.C. Armour and E.S. Buff a, A heuristic algerithm and simulation approach to the relative location of facilities, [20] Management Sci. 9 (1963) 294-309.

[21 J.F. Benders, Partitioning procedures for solving mixed- [211 variables programming problems, Numer. Math. 4 (I 962) 238-252. [22]

[3] R.E. Burkard, A perturbation method for solving quadratic assignment problems, Working Paper, Univer- sitiit zu K61n (August 1973).

[41 R.E. Burkard, Numerische Erfahrungen mit Summen- [23] und Bottleneck Zuordnungsproblemen, in: L. Collatz, G. Meinardus and H. Wernor, eds., Numerische Metho- den bei Graphentheoretischen und Kombinatorischen [ 24 ] Problemen (Birkh~'user, Basel, 1975) 9-25.

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