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A N E X A C T B R A N C H A N D B O U N D A L G O R I T H M F O R T H E G E N E R A LQ U A D R A T I C A S S I G N M E N T P R O B L E M

b yP E E R A Y U T H C H A R N S E T H I K U L , B . E . , M . S . i n I . E .

A D I S S E R T A T I O NIN

I N D U S T R I A L E N G I N E E R I N G

S u b m i t t e d t o t h e G r a d u a t e F a c u l t yo f T e x a s T e c h U n i v e r s i t y i nP a r t i a l F u l f i l l m e n t o ft h e R e q u i r e m e n t s f o rt h e D e g r e e o fD O C T O R O F P H I L O S O P H Y

A p p r o v e d

D e c e m b e r , 1 9 8 8

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TABLE OF CONTENTS

ACKNOWLEDGMENTS iiABSTRACT iiiLIST OF TABLES viLIST OF FIGURES viiCHAPTER

I. INTRODUCTION 11.1. The Koopmans-Beckmann Formulation 21.2. Difficulty in Solving the QAP Optimally ... 31.3. Branch and Bound Methods: AnIntroduction 41.4. Motivation for the Proposed Exact

Algorithm 5II. LITERATURE REVIEW 8

2.1. Exact Procedures 82.2. Linear Integer Programming Formulations ... 9

2.2.1. Lawler's Integer ProgrammingFormulation 92.2.2. The Mixed Integer Program of

Bazaraa and Sherali 102.2.3. The Mixed Integer Program of

Kaufman and Broechx 12

1 1 1

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2.2.4. Integer Program for the QAP withRectangular Distances 142.2.5. Branch and Bound Methods 16

2.3. Heuristic Procedures 232.3.1. Construction Methods 242.3.2. Improvement Procedures 242.3.3. Hybrid Methods 272.3.4. Simulated Annealing 28

2.4. Extensions of the QAP 29III. EXACT PROCEDURES FOR THE GENERAL QAP 31

3.1. The Gilmore-Lawler Method 313.2. The Kaku-Thompson Method 393.3. The Land-Gavett-Plyter Method 453.4. Pair-Exclusion Algorithm 543.5. Computation Issues 57

IV. A NEW EXACT ALGORITHM FOR THE QAP 594.1. The Bounding Linear Pair Assignment

Problem 614.2. A Generalized Lower Bounding Equation 644.3. The Branch and Bound Procedure 664.4. Numerical Example 68

V. COMPUTATIONAL EXPERIENCES 72VI. CONCLUDING REMARKS AND EXTENSIONS 93

REFERENCES 96APPENDIX

A DESIGN STRATEGY FOR A BRANCH AND BOUNDCODE FOR THE QAP 102

iv

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LIST OF FIGURES

1. Tree structure with single assignment, with levelcorresponding to locations [1] 172. Tree structure with single assignment, with levelcorresponding to objects [2] 183. Flow and distance for illustrative problem 3 34. Tree completed for the problem of Figure 3 using

the Gilmore-Lawler procedure 385. Tree elaborated for the problem of Figure 3 (by

the Kaku-Thompson method) 466. Data for the problem of Figure 3 represented interms of pairs of assignments 4 87. Tree elaborated for the problem of Figure 3 by

the Gavett-Plyter algorithm 528. Partial tree elaborated by the pair exclusionalgorithm 569. The complete search tree determined by theproposed algorithm 70

10. Ti versus n for GQAP 7911. Ti versus n for KBP 8 012. T2 versus n for GQAP 8113. T2 versus n for KBP 8214. N versus n for GQAP 8315. N versus n for KBP 8416. The illustrated tree 104

VI

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ABSTRACTThis research is concerned with the development of an

exact algorithm for a general quadratic assignment problem(QAP), of which the Koopmans-Beckmann formulation, in thecontext of an analysis of the location of economic activities or facilities, is a special case. The algorithm isbased on the linearization of a general QAP of size n intoa linear assignment problem of size n(n-l)/2. The objective value and the dual solution of this subproblem areused to compute the lower bound used in an exact branch andbound procedure. Computational experience and comparisonsto other well known methods are discussed.

Vll

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2 Xij = 1 for all j

2 Xij = 1 for all ijXij = (0,1) for all i and j

1.1. The Koopmans-Beckmann FormulationKoopmans and Beclcmann (1957) were the first to formu

late the QAP in assigning a number of plants or indivisibleoperations to a number of different geographical locations.Their formulation of Cij] L ^^ problem PI, which is somewhatless general than that given above, is as follows.

CijkL = fik

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Each assignment of plants to locations is represented byXij whereXij = 1 , if facility i is assigned to location j,

= 0, otherwise.Lawler (1963) illustrated that the traveling salesman

problem is a special case of the above formulation, inwhich bij = 0 for all i,j, djL is the distance traveledfrom city j to city L andf i k = 1 ^ i f k = i + l an d i < n

= 1 , i f i = n an d k = 1

= 0, otherwise,whereXij = 1, if the salesman visits city j on the i-th position

of the tour sequence,

= 0, otherwise.

1.2. Difficulty in Solving the QAPOptimally

The discrete nature and the complexity of the quadratic objective function do cause difficulty in verifyingthe optimal solution of the QAP. Horowitz and Sahni (1984)

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proved that the QAP is NP-hard, which requires an exponential time algorithm to solve the problem optimally. Evidently, the implicit enumeration method which is usuallyreferred to as "branch and bound" is the only appropriatestrategy in this case. From a computational point of view,all branch and bound methods previously developed areprobably not computationally feasible for n > 15 in thecase of the Koopmans-Beclcmann formulation and for n > 10 inthe general case. Nevertheless, some strategies such aspremature termination, upper bounding and stepped fathoming, which are investigated by Bazaraa and Elshafei (1979)and Bazaraa and Kirca (1983), can be used in order toobtain suboptimal solutions within a reasonable time (i.e.,one hour by a mainframe computer) for larger problems(n > 15) .

1.3. Branch and Bound Methods;An IntroductionAmong the most general approaches to the solution of

integer constrained optimization problems is that of"branch and bound." Like dynamic programming, branch andbound is an intelligently structured tree search of thespace of all feasible solutions. Most commonly, this spaceis repeatedly decomposed into smaller subspaces, and alower bound (in case of minimization) is computed for thecost of the solutions within each subspace. After each

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partitioning, those subspaces with a bound that exceeds thecost of a known feasible solution (an upper bound) areexcluded from all further partitioning. The search continues until a feasible solution is found such that its costis no greater than the bound for any subspace. Generally,a branch and bound algorithm relies on two elements:branching and bounding. Branching is concerned with choosing a reset positioning of the solution space to elaborateand evaluate. Bounding deals with the lower bound of theobjective function of each partitioning. The completedescription of a branch and bound algorithm requires aspecification of the rule that determines which of thecurrently active subspaces is to be chosen for branching,as well as the method for computing the lower bound of eachactive subspace (each node in the search tree). Historically, branch and bound methods have been applied to avariety of problems in integer programming, nonlinearprogramming, scheduling, decision processes, etc. A complete survey was reported by Lawler and Wood (1966).

1.4. Motivation for the Proposed ExactAlgorithm

Although several heuristic procedures have been suggested as the only appropriate way to solve the QAP, it isstill necessary to find a lower bound to evaluate thequality of the solution. Basically, the easiest way is to

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find the ratio between the solution found and the lowerbound obtained. A good lower bound is usually obtainedfrom an exact procedure. With the recent development ofparallel processing used in computers, it is possible tosolve a QAP by the exact and inexact procedure simultaneously. Both routines can provide an updated lower boundand upper bound to each other. According to this scheme, agood solution may be detected more quickly and the optimalsolution may be found in a reasonable time for some largeproblems.

Recently, it has been suggested that one of the mostsuccessful ways to solve a combinatorial optimization is tocombine both exact and inexact methods together. This isreferred to as the "hybrid method" by Bazaraa and Kirca(198 3). Other desirable attributes of branch and boundmethods which have received attention from researchers overthe past 25 years are as follows.

First, there is a possibility of obtaining usablesolutions and terminating problem-solving prior to theultimate completion of the problem-solving process(premature termination).

Second, when a feasible solution (an upper bound) isknown from experience or has been derived with the aid of aheuristic procedure (branch and bound methods alwayssubsequently search for solutions with an objective value

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better than the best known so far and will terminate if asolution is discovered attaining a known lower bound), theuse of a priori knowledge of upper and/or lower boundsserves to reduce the regions that need be searched. Therefore, in contexts where good heuristic procedures areavailable, a good upper bound may be found helpful inreducing time spent in the branch and bound process.

Third, all exact branch and bound algorithms can bemodified and employed to find not an optimal solution, butall optimal solutions or a specified number of most preferred solutions, or all solutions having a value within aspecified interval of the optimal objective value, and soon. Such possibilities may be of interest in contexts ofmultiple objective QAP.

The purpose of this research is to present a newbranch and bound algorithm for the QAP. Some previousworks will be reviewed in Chapter 2 and Chapter 3. Chapter 4 will present the proposed method in full detail whilecomputational experiences will be discussed and analyzed inChapter 5. Finally, concluding remarks and some extensionswill be described in Chapter 6.

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CHAPTER IILITERATURE REVIEW

In this chapter, a brief literature review of the QAPis discussed. In solving the QAP, a number of both exactand heuristic procedures have been reported in the literature over the past twenty-five years. Recently, morecomplicated models incorporating more realistic characteristics such as dynamic QAP, stochastic QAP and multipleobjective QAP have been investigated by many researchers.Reviews of previous research are enumerated as follows.

2.1. Exact ProceduresExact procedures for determining optimal solutions of

problem PI have been presented by Gilmore (1962), Lawler(1963), Land (1963), Gavett and Plyter (1966), Cabot andFrancis (1970), Bazaraa and Sherali (1980), Bazaraa andElshafei (1979), Edwards (1980), Kaufman and Broechx(1978), Burkard and Derigs (1980) and Kaku and Thompson(1986). For a review of exact algorithms, see Pierce andCrowston (1971), Sherali (1979) and Obata (1979). Most of

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the exact algorithms are based on two approaches: linearinteger programming formulations or branch and boundmethods.

2.2, Linear Integer ProgrammingFormulationsDue to the complexity of the quadratic objective

function in problem PI, several researchers have suggestedlinearizations, resulting in formulations of linear (mixed)integer programs, in order to develop some appropriatesolution techniques. Unfortunately, these approaches areusually impractical for even small n. Nevertheless, theseformulations are important contributions from the modelingviewpoint and can be illustrated as follows.

2.2.1. Lawler's Integer ProgrammingFormulationLawler (1963) showed that a quadratic assignment

problem with n^ variables X j can be linearized by definingn^ variables Yij3 L where, effectively, Y^JJ^L ~ ^ij^kL*Consider the following integer linear program in the n^ +n^ variables of Yij] L ^^^ ^ij Minimize

1 CijkL YijkL (P 2)i j k L

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10subject to2 Xij = 1 for all ji2 Xij = 1 for all ijXij + XkL - 2YijkL > 0 for all i,j,k,L

1 YijkL = n2ijkL -^Xij = {0,1} for all i,j

YijkL = (0/1) for all i,j,k,L.From the above model, the number of constraints and

variables increases from 2n to n^ + 2n + 1 and from n^ ton^ + n2, respectively. Evidently, the approach is impractical for all but the smallest problem.

2.2.2. The Mixed Integer Program ofBazaraa and SheraliBazaraa and Sherali (B&S) (1980) reformulated the QAP

as a 0-1 mixed linear integer program. The QAP model usedin their work was introduced by Graves and Whinston (197 0)and is slightly different from problem PI. In that model,it is assumed that there are p commodities that aretransported among n facilities. The first transformationis to transform the model to problem PI by comparing

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11coefficients between two models. The second transformationuses the following formula:

^ij -* bkLSijkL = + CijkL + CkLijn - 1 -

i = 1,...,n-l, K = i+1,...,n, L,j = l,...,n, L ^ j.

This transformation leads to the following equivalentproblem.M i n i m i z en-1 n n n1 1 Ir 1 SijkL Xij XkLi= l j^l K=i+1 i=lI^jsubject to all constraints in problem PI.

To solve the above model, B&S transform P2 into thefollowing 0-1 mixed linear integer program.Minimizen-1 n n n1=1 :)=i K=i+i 1=1

subject to1 Xii = 1 for all iL. ^IJ

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12I Xij = 1 for all ji

n n, I I YijkL - (n-k)Xij = 0 for i = l,...,n-lk=i+l L=l -"

1 3 j = 1,...,nk-1 n

I I YijkL - (n-k)XkL = 0 for k = l,...,n1 = 1 j=l37 L L = 1, . . . ,nXij = {0,1}

0 < YijkL < 1 for i = l,...,n-lk = i+1,...,nj,L = 1,...,n, j f L.

Note that problem P3 has n^ integer and n2(n-1)2/2continuous variables and 2n2 linear constraints. B&Sapplied Bender's decomposition to this formulation but wereunable to solve problems larger than n = 6 optimally due tostorage limitations.

2.2.3. The Mixed Integer Program ofKaufman and BroechxKaufman and Broechx (1978) have proposed a lineariza

tion method suggested by Glover (private communication).To define this method, the following two types of variablesare introduced.

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13Wij = Xij X X CijkL XkLk LUij = max [I I CijkL/0]k L

Then, problem PI can be reduced to the following mixedinteger program.

Minimize

I I Wij1 3 (P4)

subject toI Xi^ = 1 for all jI Xij = 1j for all i

Uij Xij + X X CijkL XkL - Wij < Uijk L for all i,j

Xij > {0,1} for all i,j

Wij > 0 for all i,j.

This formulation adds n2 new continuous variables andn*' new constraints. This gives a total of n2 binary variables plus n2 continuous variables, and n2 + 2n constraintsAlthough this is an improvement over problem P3, it wasreported by Kaku and Thompson (1986) that for a problem of

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14size n = 8, the core requirements become too large for themixed integer code they used and optimality could not beproved.

2.2.4. Integer Program for the QAPwith Rectangular DistancesLove and Wong (1976) proposed the following problem.

Suppose n locations are given as points in the first quadrant of a two-dimensional plane (given points can always berepresented on the first quadrant by translation of axes),and suppose the flow between each pair of facilities isgiven. If one uses rectangular distances, which sometimesare more appropriate in industrial settings than straight-line distances, then a binary mixed integer program tosolve the QAP is as follows.

Minimize

I I Wij(Rij + Lij + Aij + B i j )i = j j = i + ls u b j e c t t oRij - Lij = Xi - Xj i = l , . . . , n - lAij - Bij = Yi - Yj j = i + l , . . . , nXi + Yi = Z Sk Zik for a l l ik

(P5)

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16Bij = the vertical distance between facility i and facility

j if facility i is below facility j ; otherwise, Bijis zero,

(Xi, Yi) = the location of facility i, i=l,...,n, S-^ is thesum of coordinates of location k,

d ) ^ = the difference of coordinates of location k, values ofthe first coordinate minus value of the secondcoordinate, and

Zi]^ = 1 if facility i is assigned to location k= 0 otherwise.

The above formulation has n2 binary variables andn2 + 3n constraints. It involves ordered sets which can beprocessed by some integer programming codes. According tothe results presented in Love and Wong (197 6 ) , the approachcannot be applied successfully to QAPs with size n > 8.

2.2.5. Branch and Bound MethodsGilmore (1962) and Lawler (1963) developed a branch

and bound procedure based on the systematic considerationof single assignments Xij, as shown in Figure 1 and Figure 2.

Figure 1 illustrates a tree structure with the i-thlevel of nodes representing the permissible assignment for

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17

Level 1: OBJECT 1

Level 2: OBJECT 2

Q ) (T) . . . (n^ LOCATIONI >1 e

Level n: OBJECT n Figure 1. Tree structure with single assignment, withlevel corresponding to locations [1].

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19object i, while if we interchange objects and locations inthe cited procedure, we have a tree with level corresponding to locations rather than objects, as illustrated inFigure 2. The lower bound in each node can be computed bysolving n2 + i linear assignment problems (LAPs) of sizen - 1 . These LAPs are the result of positioning Cij) Lcorresponding with the restriction of Xij in each levelinto n2 minors of n2 elements each. Gilmore stated in hispaper that this algorithm is probably not computationallyfeasible for n much larger than 15.

Land (1963) and Gavett and Plyter (1966) have developed algorithms in which search proceeds on the basis of acontrolled enumeration of the variables YIJJTL ~ Xij X]^L,where as before each Xij denotes the locating of object iat location j. These authors also view the underlyingproblem as a linear assignment problem, but one of assigning a pair of object i and k to location j and L. In bothpapers, the algorithm developed applied to the symmetricQAP with CijkL = CiLkj- I^ general, there are n(n-l)/2pairs of objects and pairs of locations in the problemwhich represent the dimension of the associated linearassignment. However, there are many feasible solutions tothis assignment problem which are not feasible to theoriginal QAP. For example, it is entirely acceptable inthe linear assignment problem for objects A and B to be

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21that solving the QAP is one of the applications of theirmethod but no further investigation has been made.

Bazaraa and Elshafei (1979) developed an exact branchand bound procedure which is able to produce optimal solutions for problems with twelve facilities or less. Themethod incorporates the concept of stopped fathoming toreduce the effort expended in searching the decision trees.They use the binary search tree with the lower bound computed by the summation of cost of the partial assignment, thelower bound in the cost of interaction between assignedobjects and unassigned objects plus the fixed cost oflocating the unassigned objects, and the lower bound on thecost of interaction among the unassigned objects themselves.

Burkard and Derigs (1980) developed a computer programfor the binary search tree with the consideration of singleassignments in each node as described in Pierce andCrowston (1971). The lower bound of each node is computedby a row-column reduction (referred to as "the Perturbationmethod") developed by Burkard and Stratmann (1978). Theyreported some computational tests for the test examplespresented in Nugent et al. (1968). For the largest testproblem of 15 facilities, the CPU time for the CDC CYBER 76is 2947.32 seconds.

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22Edwards (1980) developed a binary branch and bound

algorithm for the QAP which has the property that at eachnode an associated solution can be obtained simultaneously,thereby rendering any premature termination of the algorithm less wasteful. They also developed another methodfor computing the lower bound by considering the minimalquadratic residual terms in the canonical form of the QAP.No computational test was reported.

Bazaraa and Kirca (1983) uses the Gilmore-Lawlermethod incorporated with the property referred to as"Mirror Images" to solve the QAP with Koopmann and BeckmannFormulation. They were able to solve a hospital layoutplanning of size n = 19 optimally within 110 seconds on theCDC-70.

Kaku and Thompson (1986) developed an exact algorithmthat is based on the linearization and decomposition of ageneral QAP of size n into n2 linear assignment problems ofsize n - 1 . The solutions of these subproblems are used tocalculate a lower bound for the original problem and thisbound is then used in an exact branch and bound procedure.These subproblems are similar to the minors defined byLawler (1963) but provide tighter bounds when the branchingrule uses the concept of alternate cost as suggested byPierce and Crowston (1971). Computational experience isgiven for solution to optimality of general QAPs of sizes

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23up to n = 10. A comparison with the Gilmore-Lawler methodand some integer programming approaches as described inthis section shows that this method performs better, especially as the size of the problem grows. The average timefor size n = 10 by a DEC-20 was reported as 1305.65 seconds. Nevertheless, this size of problem is still relatively small and further work is suggested.

It should be pointed out that some of the exact procedures previously described are designed to solve only theQAP with the Koopmann-Beclcmann formulation. In fact, onlythe Gilmore-Lawler procedure, the Land-Gavett-Plyter procedure, the pair exclusion method of Pierce and Crowston andthe method of Kaku and Thompson are designed and developedfor the general case of QAP.

2.3. Heuristic ProceduresDue to the complexity of the QAP, in general, as shown

in the previous discussion, none of the exact methods cansolve a problem with high dimension (n > 15) effectively.It is even worse in the general case of QAP for which Kakuand Thompson can solve the problems of size up to onlyn = 10. Thus, for larger problems, a considerable amountof effort has been given to the development of heuristicmethods that obtain a good quality solution with reasonablecomputational efforts. A comprehensive survey of these

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24heuristic procedures can be found in the works of Sherali(1979), Obata (1979), and Burkard and Stratmann (1978). Abrief summary of these methods is given as follows.

2.3.1. Construction MethodsStarting with a partial solution of the null assign

ment, a complete assignment is reached iteratively bylocating one or more objects at each iteration. Basically,these approaches have been based on biased random sampling,clustering methods (e.g., Hanan and Kurtzberg, 1972), rulessuch as nearest neighbor and pairwise assignment, Ai - Bjrule, and methods with increasing degree of freedom whichwere described and tested in Burkard and Stratmann (1978).Liggett (1981) developed a constructive procedure based onthe enumeration strategy presented by Graves and Whinston(1970). Liggett's results are compared favorably withother constructive procedures. However, it has been foundby Burkard and Stratmann (1978) that constructive approaches, in general, do not produce solutions that aresufficiently near optimal.

2.3.2. Improvement ProceduresStarting with a complete assignment of objects, an

improvement over the incumbent objective function value is

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25sought by exchanging assignments. The procedure is terminated when no improvements are possible. Generally, theinitial solution (a complete assignment) can be constructedby a constructive procedure. CRAFT (see Armour and Buffa,1963; and Armour, 1961), or the steepest-descent pairwise-interchange procedure (according to Francis and White,1974) , is considered to be one of the best improvementprocedures. The procedure examines all possible pairassignment exchanges and carries out the one which yieldsthe largest improvement, then starts the whole procedureagain until no improvement is found. Parker (197 6) examined the pairwise exchanges until an improvement of theobjective was found, carried out this exchange, and starteda new iteration. Three-, four- and five-way exchanges havebeen investigated by Liggett (1981). Generally, A-wiseexchanges have produced slight improvements in solutionquality, but computation times have been found to increaseat prohibitive rates. Heider (1972) examined the pairwiseexchanges in fixed order and carried out the exchange assoon as an improvement was found. Therefore, it is possible that there were several improvements in one iteration.Burkard and Strattmann (1978) tested the three alternativesof CRAFT and Parker and Heider, and found a variation ofthe last to give a superior solution. Vollmann, Nugent andZartler (1968) presented an alternative procedure which is

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26a modification of CRAFT, which requires less storage andhas a faster computation time. They also stated that theprocedure produces results that are not different withstatistical significance from CRAFT. The Hillier procedure(Hillier, 1963; Hillier and Connors, 1966) defines a"p-steps" neighborhood to search for improvement. A facility can be moved p-steps (grid-squares) horizontally,vertically, or diagonally. A move number is calculated toestimate the cost reduction that would result from movingfacility i from location j to location k. The largest movenumber is used to identify an exchange pair and the exchange is implemented, providing a reduced cost. Movenumbers are updated as necessary and p-steps changes aremade until no favorable exchanges can be identified. Theparameter p is then altered so that p is allowed to rangeall feasible solutions. Generally, p can be determined byconsidering the location layout or the site arrangement.Nugent et al. (1968) compared the Hillier procedure withCRAFT and biased random sampling. They concluded that thebiased random sampling provided the best solution on average, but the Hillier procedure was the "best buy" considering run time as well as the quality of solution produced.This result motivated Picone and Wilhelm (1984) to developa perturbation scheme called "the revised Hillier" toimprove Hillier's solution. This new heuristic combines

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27the Hillier procedure with 3-way and 4-way exchanges whichexpand the neighborhood that is searched by the Hillierprocedure. The revised Hillier procedure was compared withthe Hillier procedure and CRAFT using the standard testproblem shown in Nugent et al. (1968). In these tests,they found that the revised Hillier procedure consistentlygave the best solution ever published. Other improvementschemes can be found in Hitching and Cottam (1976), Sheraliand Rajgopal (1986), Gaschutz and Ahrens (1968), Tompkinsand Reed (1976), Foulds (1983), and Khalil (1973). Morecomparisons among CRAFT, the Hillier procedure, and someother random sampling schemes can also be found in Ritzman(1972) .

2.3.3. Hybrid MethodsMethods in this class combine several features of

exact and inexact procedures. According to several computational experiences reported in the literature, it seemsthat these methods are emerging as one of the most successful approaches for solving large QAPs. Examples of suchprocedures are the methods of Bazaraa and Sherali (1980)and Burkard and Stratmann (1978). Both methods use anexact solution scheme in conjunction with some improvementprocedures. Bazaraa and Sherali implement Benders' partitioning method to a mixed-integer formulation of theproblem and apply several improvement procedures to the

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29process seeks another neighborhood space randomly insteadof getting "stuck" on inferior solutions produced by thenormal "steepest-descent" algorithmic approach. Wilhelmand Ward (1987) presented an application of the simulatedannealing method to solve the QAP. It was shown that undercertain conditions simulated annealing can yield higherquality (lower cost) solutions at reasonable CPU timescompared to those from other traditional heuristics (CRAFT,biased random sampling, and the revised Hillier procedure) .

2.4. Extensions of the GAPA comprehensive review of applications of the QAP can

be found in Sherali and Rajgopal (1986). Elshafei (1977)discusses an application in the context of a hospital layout design. Other popular applications are the productionscheduling problem of Geoffrion and Graves (1976) and thebackboard wiring problem of Steinberg (1961), the problemof scheduling classes and examinations in an educationalsystem as formulated by Carlson and Nemhauser (1966) andthe problem of minimizing latency in magnetic drums on discstorage computers as discussed by Lawler (1963).

Recently, several extensions of the QAP have appeared,such as dynamic QAP by Rosenblatt (1986), stochastic QAP byRosenblatt and Lee (1987), multiple objective programming

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32Let

Lij = Min[X I CijkL XkL]/ XkL e S, for all k,L ,k Lwhere S is the set of all feasible assignments. Let

L*- = min[X I Lij Xij], Xij e S, for all i,j .i jThis is a linear assignment problem. Let X* be the optimalsolution to this problem with objective value L*, and Z* bethe objective value for the QAP with the solution X*. Nowlet L*

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(TO) (TO)

33

(FROM)D = B

A B D(FROM)

F = 2828 25 13

25 1513

15

23230

Figure 3. Flow and distance for illustrative problem.

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35Other Lij elements can be found in the same manner toproduce the matrix below.

L =

ABCD

297368244156

17425413182

293353245161

15221711673

Solving the linear assignment problem defined by thismatrix gives us L* = 792, which is the initial lower boundNext, the single assignment tree search process is startedat the first level. Let us consider A-1 as the partialassignment as illustrated below.

LOWER BOUND = 792

Now consider

B2C = [CB2kL]/ for all k,L.kL

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36In this case CB2kL = fBk * d2L/ for all k,L with theadditional constraint that k = A if and only if L = 1 andC B 2 A 1 = fBA * ^21. Therefore,

B-2kL

ABCD

168

0

7590

2024

Then we have L B 2 = 2 6 7 (the objective value of the linearassignment problem with the above cost matrix) + 168 ( C A I B 2due to partial assignment A-1) = 435. In a similar fashion, the other elements of this subproblem (node A-1) canbe determined and shown as follows.

L =

ABCD

0

435427151

505448213

29522599

Solving this linear assignment problem, we obtain a lowerbound for node A-1 of 873 (L*). In the same way, thelevel 1 of the enumeration tree is developed as follows.

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-> .

873 802 830 810

Then, node A-2 with the smallest lower bound is selected todevelop first at level 2. Now the partial assignments areA-2 and B-1 or B-3 or B-4. By repeating the procedure inthe same manner at levels 2, 3 and 4, the complete searchtree for this problem is as shown in Figure 4 with theoptimal assignment of A-2, B-4, C-3, D-1 and Z* = 806.

From the procedure previously described, the lowerbound in each node is obtained by solving at most n2 + ilinear assignment problems. However, in the special caseof the Koopmann-Beckmann formulation, both Lawler andGilmore have pointed out that the lower bound is moreeasily obtained by matching the largest value of t^y^ withthe smallest djL, the next largest fi^ with the secondsmallest djL and so on under restrictions of partial assignments. For fixed i and j , Lij can be obtained fasterand only one linear assignment problem (L) is solved ineach subproblem (node).

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38

(A-2)

(D-3) (h-y856 806

Figure 4. Tree completed for the problemof Figure 3 using the Gilmore-Lawler procedure.

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393.2. The Kaku-Thompson Method

Kaku and Thompson (1986) presented a binary treesearch for the general QAP. The initial lower bound iscalculated by the same method used by Gilmore (1962) andLawler (1963). The branching rule uses the concept ofalternate cost method as suggested by Little et al. (1963).Let us say that the linear assignment problem (L) is solvedresulting in an optimal cost of L* with some partial assignment restrictions. The dual variables of the optimalsolution (ui,vj) are used to reduce L, i.e., by replacingLij by Lij - ui - Vj. For every i and j which has not beenassigned and for which Lij = 0, the next smallest elementin row i and column j is found and their sum determined.The resulting amount is the minimum additional cost ifassignment (i,j) is not made. Also, this value plus thelower bound (L*) gives the alternate cost of this assignment. In the next level of the decision tree, the assignment which has the maximum alternate cost is made and thelower bound (L*) is computed by using the method presentedin Section 3.1. While backtracking, if the alternate costis greater than the best known solution (upper bound), thenode can be fathomed and the algorithm backtracked to thenext higher level. Otherwise, the next node in that levelis developed by blocking out the previous assignment in

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40this level and resolving the LAP (L) and using the alternate cost method to find the next assignment. The processterminates when the algorithm backtracks to level 0 (starting node).

From the above procedure, the algorithm can be presented as follows.

STEP 0. Initialization: set up subproblem Lij. k = 0,U B = =0.

STEP 1. Solve the subproblems to obtain the values Lij andsolve L to get the initial lower bound L*. Reduce[Lij] by the dual variables ui and vj and calculate the alternate cost. Choose an assignmentwith the maximum alternate cost, say P to Q.

STEP 2. Assign XpQ = 1 , k = k + 1 and set P -> Q as apartial assignment. Calculate the lower bound(L*) with the partial assignment restrictions. IfL* >: UB, go to Step 5. Otherwise, if k = n - 2,go to Step 4. Otherwise, go to Step 3.

STEP 3. Use the dual variables to reduce L and choose thenext assignment, say P and Q. If backtracking,choose the next assignment by blocking out theprevious assignment in this level. Calculate thealternate cost. Go to Step 2.

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41STEP 4. Calculate exact cost of two remaining completions.

If total cost < UB, update UB and state assignments .

STEP 5. If K = 0, stop. An optimal solution has beenfound. Otherwise, K = K - 1 and if the alternatecost > UB, repeat Step 5. Otherwise, determineall CijpQ, CpQij = 00 and solve relevant subproblems. Then set Lpg = . Solve L to get a tighter L*. If L* > UB, repeat Step 5. Otherwise, goto Step 3.

Again, the problem of Figure 3 is solved by the abovealgorithm. In this procedure, the alternate cost methodwill be reviewed.

The method begins with the first L matrix developedpreviously in Section 3.1 and the lower bound on all solutions = 792. The reduced L matrix (Lij - Ui - vj) obtainedfrom solving L by the Hungarian algorithm can be shownbelow.

ABC

D

1 42 04

0

01 50

3 5

5000

007

4 8

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43Solving the above linear assignment problem, L* = 795

with the following reduced L.

ABCD

0

01 2 8

0

28 7

0

00

2 2

Now, candidates for the next assignment are A-2, A- 4,B-4, C-2 and C-3. D-1 is not included because it is already a partial assignment. The assignment with the largest alternate cost is B-4 with 882. Therefore, the decision tree branch on B-4 is as follows.

Then, all possible alternatives can be enumerated atnode B-4

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46

D-ii 795

-41 796

& 806 838Figure 5. Tree elaborated for problemof Figure 3 (by the Kaku-Thompson method).

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47Xij denotes the locating of object i at location j. Theseauthors also developed the subproblem as a linear assignment problem, but one of assigning a pair of objects i andk to locations j and L, respectively. In both papers, thealgorithms developed apply to the symmetric Koopmann-Beckmann problem with CIJI^L = fik * ^jL- Nevertheless, itwill be shown that this method can be modified to handlethe general QAP.

In Figure 6, the relevant pair assignment for theproblem of Figure 3 is shown. In general, there aren(n-l)/2 pairs of objects and pairs of locations in theproblem.

However, there are many feasible solutions to theabove linear assignment problem which are not feasible tothe original QAP. For example, assigning A-B to 1-2 andA-C to 3-4 which is clearly infeasible for the QAP areacceptable in the above solution. For a feasible solutionto the QAP, the problem becomes a linear assignment problemwith the following additional constraints.

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49IF YijkL = 1 (3.1)

THEN Yibcd = 0, Ycbid = 0Yajcd = 0, Yadcj = 0

Yabkd = 0/ Ykbad = 0YabcL = 0/ YaLcb = 0 /

where i ? * a , j^^b, k?^c and L 5 d. Operationally, boththe algorithm of Land and that of Gavett and Plyter solvethe linear pair assignment problem and determine a reducedmatrix with non-negative entries of Cij] L ~ ' ik " ^i'L'where Cij] L ^ " ^ cost element of object pair k,L andlocation pair j,L, ui]

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50des cri bed in Sect ion 3.2, whi le Land always selects fromthe column having the least number of feasible elements inth e col umn- redu ced mat rix (a zero element having the largest alte rna te cos t based only on alternate cost in the samec o l u m n ) . Aft er committi ng a pair assignment to the solu tio n at a giv en node of the search tree, the conditi on ofYijkL = 1 is involved by setting Cabcd = " for all Yabcd =0 specified previously to obtain the resulting cost matrixused at the next level.

At this poin t, the Gavett-Plyter algorithm is illu stra ted by solv ing the probl em of Figur e 3. Fir st, thelinear pair assignment problem of Figure 6 is solved andthe optimal cost found is 3 89 with the following reducedma tr ix (due to the symmetry of Cijj^L i^ this case, thelow er bou nd ob tain ed is one-h alf of the exac t v a l u e ) .

A-BA-CA-DB-CB-DC-D

1-22-131553

1831

1-33-1163706

160'

1-44-1

0'9

3220'

44

2-33-2

01124

2-44-2

034

068110

89

3-44-3

8270"08

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51From the above matrix, candidates for the pair assign

ment at level 1 are (A-B,1-4), (A-B,2-3), (A-B,2-4),(A-C,2-4), (A-D,1-3), (A-D,3-4), (B-C,2-3), (B-C,3-4),(B-D,1-4), (B-D,2-3), (B-D,2-4), (C-D,1-2) and (C-D,1-3).The highest alternate cost is found in (A-C,2-4) with 3 98.Therefore, the level 1 of the search tree is shown asbelow.

398

By continuing in the same manner, the complete searchtree can be elaborated in Figure 7 as follows.

In comparison with other search strategies previouslydiscussed, this strategy may require a longer path in thedecision tree and a longer problem solution time to determine a first feasible solution. From a computationalviewpoint, both Land and Gavett and Plyter are able tosolve the QAP optimally up to size n = 8.

We now consider the extension of this class of algorithms to the general case in which CIJ^L ^ ^iLkj Forthis problem, two approaches can be used.

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53The first approach is to convert

I I I I CijkL Xij XkL1 D k Lto a scalar form of X^cx, where C is the assignment-to-assignment matrix of size n2. Then, this C-matrix is usedto compute the initial lower bound and the dual variablesin the search tree instead of the pair assignment matrix.

The second approach is to consider the followingassociated linear assignment problem.Minimize

I (CijkL * YijkL + CiLkj YiLkj)i j k Ls u b j e c t t on ( n - l ) / 2I I (YijkL + YiLkj) = 1

J /Ln ( n - l ) / 2

I I (YijkL + YiLkj) = 1i , kYijkL.YiLkj = (0.1}/ i,j,k,L.

By setting CijkL = i iri(CijkL/ iLkj } ^^^ "" ijkL ="" ijkL " iLkj / it follows that the lower bound to theoriginal QAP can be obtained by solving the above linearassignment problem of size n(n-l)/2 with cost elements

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55(B-C,3-4), (B-D,2-3) and (C-D,l-2) where (i-k,j-L) represents object pair i and k being assigned to location j andL. Next, n(n-l)/2 subproblems are generated by prohibitingeach pair assignment in this solution. The linear assignment problem associated with each subproblem (node) issolved and stored as a lower bound. This process proceedsuntil a feasible solution to the QAP found has the upperbound less than or equal to any lower bound of all activesubproblems (have not been branched or fathomed). For thisexample, the result in terms of a tree is partially shownin Figure 8.

In a similar manner, the process can now proceed byselecting any active node with the lowest bound, solve theassociated linear assignment problems and check feasibilityof the solution. Then, continue until the optimalitycondition (the upper bound > all lower bounds) is verified.

In general, it is difficult to anticipate the performance of this type of algorithm relatively to the types asdiscussed in the earlier section. For the related traveling salesman problem, this general approach has provedsignificantly more efficient than level-by-level assignmentalgorithms. Undoubtedly, this is basically due to the factthat the optimal traveling salesman problem is frequentlyquite close to the optimal linear assignment solution in

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57the respect that a large majority of assignments in theformer are present in the latter, so that a relativelysmall decision tree need be explicitly elaborated. In theQAP, it does not appear that the optimal quadratic assignment is close to the optimal linear pair assignment (Pierceand Crowston, 1971). Therefore, a much larger search treeis required (even to determine a first feasible solution).

3.5. Computation IssuesKaku and Thompson (198 6) reported a comparison between

their method (Section 3.2) and the Gilmore-Lawler method(Section 3.1). The results showed that their method performs better, especially as the size of the problem grows.Nevertheless, the largest size of problem reported is onlyn = 10 and the average run time is over 2 0 minutes on amainframe DEC-20. Burkard and Gerstl (1973) modified theLand-Gavett-Plyter method and reported that this method iscompetitive with the Gilmore-Lawler method until the sizeof the QAP is up to n = 8. So far, no computational comparison has been made for the pair exclusion method ofSection 3.4. Technically, this method has a memory management problem. As illustrated in Figure 8 of Section 3.4,the search tree of this method has no structure to aidmemory management in the computer program developed (i.e.,no certain value of the depth of the search tree) while the

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60lower bound. The lower bound in each node is computed byadding the dual variables (obtained in the first step) ofthe relevant partial assignments to the initial lowerbound. Mathematically, consider the following formula,

LB(I,MI)) = LBo + I I [Ci^(i)k^(k) - Uik -2.,4>{i) k,

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61bounds the objective function of the general QAP. Furthermore, equation 4.1 will be modified and illustrated thebounding condition for the QAP.

4.1. The Bounding Linear PairAssignment ProblemFor i,j = 1,2,...,n and k = i+1,i+2,...,n and L =

j+1,j+2,...,n denoted by CijkL/ ^^ element in the pair-assignment matrix P, let P* be the minimum value of theobjective function of the following problem:

Minimize

I I CijkL YijkL = P(Y) (PRl)ik jLsubject to

I YijkL = 1/ for all j,LikI YijkL = 1/ for all i,kJLYijkL = (0/1}

It is assumed that this LPAP has finite optimal solutions. Recall from Section 3.3 of Chapter 3 that YIJ^L =

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64which is a lower bound for a node of the search tree withpartial assignment [1,0(1)].

In this case, the symmetry of CIJ^L ( ijkL = ^kjiL ^"^CijkL - CkLij) is assumed. For a general CijkL/ two approaches suggested in Section 3.3 can be modified andobtained the same results. For instance, if CijkL ^ ^kLij/set CijkL = (CijkL + CkLij)/2 and if CijkL ^ Cj^jiL/ setCijkL = "iiri(CijkL/CkjiL) ' then substitute Cij^L to Cj j iLand analyze the results. Therefore, any QAP can be converted to a symmetric QAP.

4.2. A Generalized Lower BoundingEquation

In this section, we will introduce a new lower bounding equation bearing different from equation 4.1. It isdetermined by using the symmetry of CijkL/ the level of thesearch tree and partial assignment [1,(^(1)]. First, letus consider the following relationship,LBi[I,

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65ALB'i{I,

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68a partial assignment, set ^ = i + 1 and repeatStep 2. Otherwise, i = i - 1 and go to Step 4.

STEP 4: Backtracking: If i = 0, then an optimal solutionis found and verified. Otherwise, remove theprevious partial assignment assigned in level iand choose an available node with the next lowestbound (which is smaller than the upper bound) inthis level as the new partial assignment. Set t =/>>1 + 1 and repeat Step 2. If there is no suchnode, i = t - 1 and repeat Step 4.

4.4. Numerical ExampleTo illustrate how the algorithm presented in Sec

tion 4.3 works, the problem shown in Figure 3 will besolved. To accommodate larger problems, a computer programwas coded in QBASIC. More computational experiences usingthis program will be discussed in Chapter 5.

By applying the proposed algorithm step by step, thesolution can be obtained as follows. First let us recallthe corresponding pair assignment matrix for this problemas shown in Figure 6, Section 3.3. Next, solve the linearassignment problem by the Hungarian algorithm and obtainthe final cost matrix shown as follows.

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69

A-BA-CA-DB-CB-DC-D

2-11-231553

1831

3-11-3163706

16

4-11-409

3220

3-22-30

112400

4-22-400

68110

4-33-48

27008

0 0 44 34 89

The next step is to use the single assignment tree (inthis case, choose the search tree shown in Figure 2) andthe above cost matrix with the initial lower bound (778) tofind the optimal quadratic assignment. The complete enumeration tree of the branch and bound solution is illustrated in Figure 9 shown as follows.

From the above search tree, the optimal assignment isA-2, B-4, C-3, D-1 with the total cost of 806.

To explain how a lower bound can be computed, supposethe set of partial assignments is [(A,3},(B,4}]. From thefinal cost matrix resulting from solving the LPAP, Cp^2B4=8. Therefore, the lower bound in this case is 778 + 2 8 = 794. Another example is to compute the lower bound forthis set of partial assignment, [(A,2},(B,4},(C,3}].Again, from the final cost matrix, C A 2 C 3 = ii' C B 4 C 3 = 0.By using equation 4.2, the lower bound for this example is

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73of the problems are 5, 6, 7, 8, 9 and 10. Both algorithmswere coded and compiled by the QuickBASIC compiler version4.0 and run on the COMPAC 386/20 microcomputer. Evaluationof the solution in each replication of each case containsthree performance measurements:

1. Ti is defined as the time consumed until an optimal solution is verified. An optimal solution is verifiedwhen all branches are fathomed.

2. T2 is defined as the time consumed until an optimal solution is found. This does not include the time toconfirm that remaining branches will be fathomed.

3. N is defined as the number of nodes generated inbranch and bound processes.

Ti and T2 are estimated in the units of seconds. Forconvenience, the proposed method and the method of Kaku andThompson will be referred to as "PM" and "KT," respectively. Additionally, the complete enumeration method,which will be referred to as "CE," will be used to solveall generated problems of each size level to validatesolutions found by both methods. The total running timeswill be used to compare with average Ti values obtained byboth algorithms.

Results from the experimentation are shown in Table 1to Table 3. The results consist of the mean value (M),standard deviation (SD), the minimum (MN) and the maximum

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74Table 1

Statistical Results of Ti (sec.)(From 50 Replications Each)*

(n)Size5

6

7

8

9

10

MSDMNMXMSDMNMXMSDMNMXMSDMNMXMSDMNMXMSDMNMX

GQAPPM0.2120.0760.2000.3000.6700.0800.5000.8001.8200.3841.0002.000

12.2401.3059.00014.000

105.04011.34574.000133.000982.62085.024791.000

1173.000

KT0.3580.2190.2000.8002.3900.8501.0004.200

10.5603.4244.00019.000

67.26716.87040.000108.000

537.380150.552177.000850.0003886.773874.4961607.0005088.000

KBPPM0.1820.0860.1000.3000.4320.1570.2000.8000.9860.4691.0002.0004.6801.3332.0008.000

30.3009.21413.00060.000

205.23370.255116.000373.000

KT0.1720.1600.1000.6001.0820.5050.3002.5005.0602.5002.00013.000

24.08011.7935.00065.000

124.65081.46231.000362.000896.800437.868164.000

2260.000

CE0.5 1.0 4.0

25.0

230.0

2650.0^ ^

* CEGQAPKBPKTMMNMXPMSDTl

complete enumerationgeneral caseKoopman-Beckmann ProblemKaku-Thompson methodmean valueminimum valuemaximal valueproposed methodstandard deviationtime consumed until termination

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75

Table 2Statistical Results of T2 (sec.)*

(n)C -1 T -x-olZ

5

6

7

8

9

10

J C

MSDMNMXMSDMNMXMSDMNMXMSDMNMXMSDMNMXMSDMNMX

GQAPPM0.1340.0650.1000.3000.3740.1600.1000.6001.1200.6521.0002.0006.5203.7111.00013.000

53.18030.8101.000120.000

540.180345.3501.0001076.000

KT0.1460.1280.1000.5000.9940.9510.1003.3003.9204.0401.00015.000

33.26721.9411.00087.000

259.640189.6321.000667.000

1749.5001197.49747.0003968.000

KBPPM0.1220.0640.1000.2000.2640.1290.1000.1000.4400.4961.0001.0003.1401.6251.0006.000

17.98010.5421.00045.000

105.23369.5122.000263.000

KT0.0880.0890.0500.4000.3560.2890.1001.0001.5002.3001.00011.0007.5008.3901.00032.000

59.70072.2811.000269.000

395.800415.9001.0001426.000

*T2 = time consumed until an optimal solution isfound

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77(MX) of each performance measurement in each problem size.Note that the Koopmann-Beckmann problem and the generalCijkL case are referred to as "KBP" and "GQAP," respectively.

In Table 1, T^ values are compared. It is found thatPM has the best average T^ value and smaller standarddeviation as the size level grows for both the GQAP andKBP. For the case of the GQAP, an interesting resultobtained is that the KT cannot produce T^ on average betterthan CE. This means that CE is also a good alternative tosolve any QAP with n < 10. Nevertheless, the maximum Ti ofeach size level obtained from the PM is less than thatobtained from the CE. This implies that the PM outperformsthe CE for Ti values in every replication of each problemsize. For larger n (n > 10), it appears that computationaltimes required by all methods become prohibitive to collectsufficient information. Another result obtained shows thatthe KBP requires less Ti on average for comparable sizedproblems. This is true because of its restrictedformulation compared to the more general case of GQAP.

In Table 2, a comparison of T2 values demonstratesthe same conclusion obtained from Table 1, except forn = 5, for which KT performs slightly better. From Table3, the N values of both methods are compared. The results

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88populations have the same medians, where the observationscome in pairs with one element of each pair being from eachpopulation. The sign test was conducted and the resultsshown in Table 6 and Table 7 for the following hypothesisfor GQAP and KBP, respectively.

H Q : E(Xi) > E(Yi) for all i

H^: otherwise,where E(Xi) and E(Yi) represented the expected value of T^,T2 and N for PM and KT, respectively.

In Table 6 and Table 7, the results from the sign testare shown at the level of significance (a) 0.05. Let thetest statistic T equal the number of points which a performance measurement of PM is better (less) than that ofKT. Define n* as the total number of pairs excluding alltied pairs, and obtain

t = 0.5(n* - 1.645 Jn^)(round to positive integer). The decision rule is toreject H Q if T > n* - t. An example of applying the signtest to the results in Table 6 is as follows.

Consider the size of 6 for T, value of GQAP in Table6. There are 4 3 pairs for which PM gives better results.Therefore,T = 43.

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Table 6Results of GQAP from the Sign Test

(R = Reject H Q ; A = Accept HQ;OSL = Observed SignificanceLevel*)

89

Size

5

6

7

8

9

10

TlT2NTlT2NTlT2NTlT2NTlT2NTlT2N

T33460

43340

50330

50480

50400

50410

n*494750434850504550505050505050505050

t191820161820201720202020202020202020

n*-t

302830271630302830303030303030303030

ConclusionRRARRARRARRARRARRA

OSL< 0.050< 0.001> 0.999< 0.001< 0.001> 0.999< 0.001< 0.001< 0.999< 0.001< 0.001> 0.999< 0.001< 0.001> 0.999< 0.001< 0.001> 0.999

*OSL is defined as the smallest significance levelat which one can reject HQ (observing a more extreme test statistic value; H Q is true).

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91There are zero pairs for which KT gives better results, so

n* = 43(excluding all tied pairs) for Q = 0.05, t = 0.5 (43 -1.645743) - 16, n* - t = 27.

In this case, T > n* - t, so H Q is rejected, whichmeans that PM is preferred to KT for this size of n = 6 ofGQAP at ' = 0.05. From Tables 6 and 7, we can concludethat PM produces T^, T2 values on average at least as goodas or better than KT while KT produces a better averageN value, which supports the descriptive results from Figure 10 to Figure 15.

Another statistical test referred to as comparisonmean test with unknown variances (Montgomery, 197 6) wasconducted under normality assumption. The conclusion fromthis test is also the same as obtained from the sign test.

It should be pointed out that the run time values ofTl and T2 obtained from this experiment are dependent onthe efficiency of the computer code. Similar programmingtechniques, data structures as described in Appendix A anda common compiler (QuickBasic) are used for each algorithmin order to make these relative comparisons as reasonableas possible. Finally, we can summarize the results obtained from this chapter as follows:

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94solution and then verifying its optimality within a reasonable time is almost impossible in the case of large problems. Here, we must resort to suboptimal solutions.Bazaraa and Elshafei (1979) proposed two stepped fathomingmethods for obtaining controlled suboptimal solutions inthe context of branch and bound. The first method utilizesthe relation LB > QUB where a e (0,1] to fathom a partialassignment solution with the lower bound LB and the upperbound UB. The second method defines the relation a LB +(l-a)UB = K where as (0,1]. Since LB < UB, then K < UB.Two cases are possible. In the first case, we will be ableto find a complete solution with an objective functionvalue less than K. This new objective value becomes thenew UB. In the second case, there is no solution withobjective value less than K. K becomes the new LB. Theprocess repeats in the search tree until the differencebetween the lower and upper bound is smaller than a prescribed tolerance. In both cases, if a is close to 1, thesearch will not speed up considerably. On the other hand,if a is close to zero, then fathoming will be fast but itis likely that a good feasible solution will not be obtained. Bazaraa and Elshafei (1979) recommended a = 0.7 asa starting value; then as the search proceeded, a is increased until Q = 1.

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95The proposed method may be modified to solve a more

complicated model of the QAP such as dynamic QAP, multi-objectives QAP. Nevertheless, the modified method may beapplied to only small problems. Some branch and boundbased heuristics should be developed for the case of largeproblems. Finally, the computer programs developed for theproposed method and the method of Kaku and Thompson are notparticularly efficient codes; certainly there are improvements which can be made in this area to decrease executiontime. For instance, there are many methods to solve thelinear assignment problem. The particular procedure usedin both methods may not necessarily be the most efficient.Another factor is the efficiency of the compiler. TheQuickBASIC compiler may not be as efficient as a C-compileror a PASCAL compiler. Using Pascal or C would also allowthe use of data structures and dynamic memory allocation tostore the search tree.

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REFERENCES

Armour, G. C., "A Heuristic Algorithm and Simulation Approach to Relative Location of Facilities," Unpublished Doctoral Dissertation, UCLA, Los Angeles, CA,1961.Armour, G. C., and Buffa, E. S., "A Heuristic Algorithm andSimulation Approach to Relative Location of Facilities," Management Science. 9(2) (1963) 294-309.Aronson, J. E., "The Multiperiod Assignment Problem: AMulti-commodity Network Flow Model and SpecializedBranch and Bound Algorithm," European Journal ofOperational Research, 23 (1986) 367-381.Baker, K. R. , INTRODUCTION TO SEQUENCING AND SCHEDULING,John Wiley & Sons Inc., 1974.Bazaraa, M. S., and Elshafei, A. N., "An Exact Branch and

Bound Procedure for Quadratic Assignment Problems,"Naval Research Logistics Quarterly, 26 (1979) 109-121.Bazaraa, M. S., and Kirca, O., "A Branch-and-Bound-BasedHeuristic for Solving the Quadratic Assignment Problem," Naval Research Logistics Quarterly, 30 (1983)287-304.Bazaraa, M. S., and Sherali, H. D., "Benders' PartitioningScheme Applied to a New Formulation of the QuadraticAssignment Problem," Naval Research Logistics Quarter

ly, 27 (1980) 29-41.Bazaraa, M. S., and Sherali, H. D., "On the Use of theExact and Heuristic Cutting Plane Methods for theQuadratic Assignment Problem," Journal of OperationalResearch Society, 33 (1982) 991-1003.Buffa, E. S., Armour, G. C., and Vollmann, T. E., "Allocating Facilities with CRAFT," Harvard Business Review,42 (1964) 136-159.

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98Gavett, J. w., and Plyter, N. V., "The Optimal Assignmentof Facilities to Locations by Branch and Bound,"Opera tions Research. 14 (1966) 210-232.Geo ffr ion , A. M., and Graves, G. W., "Scheduling ParallelProduction Lines with Changeover Costs : PracticalAppli catio n of a Quadratic Assignment/ LP Approach,"Operations Research, 24 (1976) 595-610.Gilmore, P. c., "Optimal and Suboptimal Algorithms for theQuadratic Assignment Problem," Journal of Society ofIndustrial and Applied Mathematics. 10 (1962) 305-313.Graves, G. W., and Whinston, A. B., "An Algorithm for theQuadratic Assignment Problem," Management Science, 17(1970) 453-471.Hanan, M., and Kurtzberg, J., "A Review of the Placementand Quadratic Assignment Problems," SIAM Review, 14(1972) 324-342.Heider, C. H., "A Computationally Simplified Pair-ExchangeAlgorithm for the Quadratic Assignment Problem," PaperNo. 101 , Center for Naval Analys es, Arlington, VA(1972) .Heider, C. H. , "An N-Step, 2 Variables Search Algorithm forthe Component Placement Problem," Naval ResearchLogistics Quarterly. 20 (1973) 699-724.Hillier, F. S., "Quantitative Tools for Plant Layout Analysis," The Journal of Industrial Engineering. 14 (1963)33-40.Hillie r, F. S., and Conn ors, M. M., "Quadratic AssignmentProblem Algorithms and the Location of IndivisibleFacilities," Management Science. 13 (1966) 42-57.Hitching, G. G., and Cottam, M., "An Efficient HeuristicProcedure for Solving the Layout Design Problem,"Omega, 4 (1976) 205- 214.Horowitz, E., and Sahni, S., FUNDAMENTALS OF COMPUTERALGORITHMS, Computer Science Press, Rockville, MD(1984) .Kaku, B. K., and Thompson, G. L., "An Exact Algorithm forthe General Quadratic Assignment Problem," European

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99Kaufman, L. , and Broechx, F., "An Algorithm for the Quadratic Assignment Problem Using Benders' Decomposition," European Journal of Operational Research, 2(1978) 204-211.Khalil, T. M., "Facilities Relative Allocation Technique(FRAT)," International Journal of Production Research,11 (1973) 183.Khoshnevis, B., and Connelly, M., "Managing the Microcomputer Memory to Solve Larger Scale I.E. Problems,"Computer and Industrial Engineering. 10 (1986) 263-272.Koopmans, S., and Beckmann, M. J., "Assignment Problems andthe Location of Economic Activities," ECONOMETRICA, 25(1957) 52-75.Kuhn, H. W., "The Hungarian Method for the AssignmentProblem," Naval Research Logistics Quarterly. 2 (1955)83-97.Land, A. H., "A Problem of Assignment with Inter-RelatedCosts," Operational Research Quarterly. 14 (1963) 185-198.Lawler, E. L., "The Quadratic Assignment Problem," Management Science. 9 (1963) 586-599.Lawler, E. L., and Wood, D., "Branch-and-Bound Methods: ASurvey," Operations Research. 18 (1966) 699-719.Liggett, R. S., "The Quadratic Assignment Problem: AnExperimental Evaluation of Solution Strategies,"Management Science. 27 (1981) 442-458.Little, J. D. C., Murty, K. V., Sweeney, D. W., and Kard,C., "An Algorithm for the Traveling Salesman Problem,"Operations Research, 11 (1963) 972-989.Love, R. F., and Wong, J. Y., "Solving Quadratic AssignmentProblems with Rectangular Distances and Integer Programming," Naval Research Logistics Quarterly. 2 3(1976) 623-627.Malakooti, B., and D'Souza, G. I., "Multiple ObjectiveProgramming for the Quadratic Assignment Problem,"International Journal of Production Research, 25(1987) 297-312.

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100Montgomery, D. C., DESIGN AND ANALYSIS OF EXPERIMENTS, JohnWiley & Sons, Inc., 1976.Murtagh, B. A., Jefferson, T. R., and Sornprasit, V., "AHeuristic Procedure for Solving the Quadratic Assignment Problem," European Journal of Operational Research. 9 (1982) 71-76.Murty, K. G., LINEAR PROGRAMMING, John Wiley and Sons,Inc., 1983.Murty, K. G. , "Solving the Fixed-Charge Problem by Rankingthe Extreme Points," Operations Research, 16 (19 68)268-279.Nugent, C. E. , Vollmann, T. E., and Ruml, J., "An Experimental Comparison of Techniques for the Assignment ofFacilities to Locations," Operations Research, 16(1968) 150-173.Obata, T. , "The Quadratic Assignment Problem: Evaluationof Exact and Heuristic Algorithms," Unpublished Doctoral Dissertation, Rensselaer Polytechnic Institute(1979) .Parker, C. S., "An Experimental Investigation of SomeHeuristic Strategies for Component Placement," Operations Research Quarterly. 27 (1976) 71-81.Picone, C. J., and Wilhelm, W. E., "A Perturbation Schemeto Improve Hillier's Solution to the Facilities LayoutProblem," Management Science. 30 (1984) 1238-1249.Pierce, J. F., and Crowston, B. W., "Tree Search Algorithmsfor Quadratic Assignment Problems," Naval ResearchLogistics Quarterly. 18 (1971) 1-36.Ritzman, L. P., "The Efficiency of Computer Algorithms forPlant Layout," Management Science. 18 (1972) 240-248.Rosenblatt, M. J., "The Dynamics of Plant Layout," Management Science. 32 (1986) 76-86.Rosenblatt, M. J., and Lee, H. L., "A Robustness Approachto Facility Design," International Journal ol Production Research, 25 (1987) 479-486.Sherali, H. D., "The Quadratic Assignment Problem: Exactand Heuristic Methods," Unpublished Doctoral Dissertation, Georgia Institute of Technology (1979) .

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101Sherali, H. D., and Dickey, S. E., "An Extreme-Point-Ranking Algorithm for the Extreme Point MathematicalProgramming Problem," Computer and Operations Research. 13 (1986) 465-475.Sherali, H. D., and Rajgopal, P., "A Flexible Polynomial-Time Construction and Improvement Heuristic for theQuadratic Assignment Problem," Computer and OperationsResearch, 5 (1986) 587-600.Steinberg, L., "The Backboard Wiring Problem: A PlacementAlgorithm," SIAM Review, 3 (1961) 37-50.Tompkins, J. A., and Reed, R., "An Applied Model for theFacilities Design Problem," International Journal ofProduction Research, 14 (1976) 583.Tompkins, J. A., and White, J. A., FACILITIES PLANNING,John Wiley and Sons, New York (1984).Vollmann, T. E., Nugent, C. E. and Zartler, R. L., "AComputerized Model for Office Layout," The Journal ofIndustrial Engineering. 19 (1968) 321-329.Wilhelm, M. R., and Ward, T. L., "Solving Quadratic Assignment Problems by Simulated Annealing," H E Transac

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APPENDIXA DESIGN STRATEGY FOR A BRANCH AND

BOUND CODE FOR THE QAP

The purpose of this appendix is to describe a branchand bound scheme suitable for permutation problems such asthe QAP. The scheme is based on a backtracking strategydescribed by Baker (1974). This choice should make thingsa little bit easier for the computer programmer because thestorage demands of a backtracking code are not severe.Therefore, the programmer need not be overly concernedabout organizing his data in sophisticated ways in situations where core memory is a scarce resource.

Backtracking is the strategy of always branching fromthe node corresponding to the most nearly solved subproblem. Stated more explicitly, this means that backtrackingattempts to find a complete solution in the most directway, and then to retrace its steps only so far as necessaryto find the next closest solution in the tree. We haveutilized this algorithm for the QAP. In this case, thesingle assignment tree is used. Consider the followingefficient means of representing the tree internally.

102

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103Define TREE(I,J) to be an n x n matrix. Row i of thismatrix corresponds to level i (object i) of the search treeand contains the indices of locations that correspond tonewly generated nodes each time branching is executed. Toillustrate how this matrix is manipulated, consider thetree in Figure 16.

The corresponding TREE matrix takes the followingform.

TREE =

1324 19

2321

222423

The indices of row 1 in the upper half of this matrixcorrespond to the nodes at level 1 of the tree ranked bylower bound. Similarly, the indices of row i correspond tonodes at level i, ranked by their lower bound. Notice thatthe lower bounds corresponding to all active nodes arestored in the lower half of the TREE matrix. No otherlarge commitment of storage space is required, and relativelittle additional information is necessary to carry out thealgorithm.

Upon backtracking, the active nodes remaining atlevels 2 and 3 are fathomed since the complete solution has

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104

(l-2) 21 ( l ^ 19 1-4J 2 3

4-4 > 21

Figure 16. The illustrated tree.

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