A COMPARATIVE STUDY OF SOLVING QUADRATIC
ASSIGNMENT PROBLEMS USING SOME STANDARD
MINLP SOLVERS
Toni Lastusilta and Tapio Westerlund
Department of Chemical Engineering, Åbo Akademi University, Biskopsgatan 8, Åbo, Finland
Keywords: Combinatorial optimization, Quadratic assignment problem, Mixed-integer non-linear programming,
General algebraic modeling system.
Abstract: The Quadratic Assignment Problem (QAP) has important application areas, for example, facility layout
(Dickey and Hopkins, 1972) and electronic component placement (Rabak and Sichman, 2003). The NP-hard
problem already becomes difficult and time consuming to solve satisfactorily for small applications. It is
therefore of interest to investigate how well standard MINLP methods can provide good solutions within a
reasonable time, even though global optimality can not be guaranteed. In this study we focus on solving a
subset of 50 problems in the QAP library (Burkard et al., 1997). We use a standard Mixed-Integer Non-
Linear Programming (MINLP) formulation modelled in the General Algebraic Modeling System (GAMS)
(Rosenthal, 2010). The solution quality and solution time is evaluated for the solvers AlphaECP, Bonmin,
DICOPT and SBB. We compare the solvers when a 1 hour time limit per problem is used, where the solvers
are started from 3 random start points, i.e. initial variable levels. Furthermore, we investigate how well the
most promising solver DICOPT performs when started from 50 random start points for 22 problems for
which the global optimal solution is known.
1 INTRODUCTION
The Quadratic Assignment Problem is a general
model formulation and has been applied to many
different application areas, of which the following
could be mentioned: assignments of buildings in a
University Campus (Dickey and Hopkins, 1972);
locating hospital departments (Elshafei, 1977);
zoning forest for different uses (Bos, 1993);
electronic component placement on a printed circuit
card (Rabak and Sichman, 2003) and computer
motherboard design (Miranda et. al., 2005).
Originally the QAP formulation was introduced
for the facility layout problem where  facilities are
placed in  locations. The objective is to minimize
the sum of the products of flows and distances
between the facilities. Let 

define the distances
between locations  and  and let 

define the
material flows between the facilities  and , i.e. two
× constant matrices. Let 

be the decision
variable that facility  is placed at location . The
QAP problem was first presented by Koopmans and
Beckmann (1957) in the following form:
 



∙











(1)
Subject to

,
=1,


=1,…,
(2)

,
=1,


 = 1,…,
(3)


∈

0,1

, ,

=1,…,
(4)
The bilinear objective function (1) results in a non-
convex formulation, hence optimality can not be
guaranteed with convex optimization techniques.
Therefore the QAP problem is often reformulated in
convex or linear form, having the disadvantage that
the number of variables and constraints will increase
substantially. Exact algorithms based on
reformulation and convexification guarantee global
optimality but problems with the size ≥30 are
already very difficult to solve to proven optimality
(Çela, 1998). Heuristic algorithms can, however,
409
Lastusilta T. and Westerlund T..
A COMPARATIVE STUDY OF SOLVING QUADRATIC ASSIGNMENT PROBLEMS USING SOME STANDARD MINLP SOLVERS.
DOI: 10.5220/0003598604090412
In Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2011), pages
409-412
ISBN: 978-989-8425-78-2
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

often provide good solutions within a reasonable
time but not verify the quality of them. Among
heuristic methods one finds: Construction Methods
(CM), Limited Enumeration Methods (LEM),
Improvement Methods (IM), Tabu Search (TS),
Simulated Annealing (SA), Genetic Algorithms
(GA), Greedy Randomized Adaptive Search
Procedures (GRASP) and Ant Systems (AS). The
goal of this study is to compare how a set of
standard MINLP solvers perform, when the bilinear
QAP formulation is used.
The MINLP solvers are standard GAMS solvers
and uses the following solution techniques:
AlphaECP solves the problem by cutting plane
techniques; Bonmin (Basic Open-source Nonlinear
Mixed Integer programming) uses a simple branch
and bound algorithm and solves a Non-Linear
Programming (NLP) problem in each node;
DICOPT (Discrete and Continuous Optimizer) is an
outer-approximation method; SBB combines a
standard branch and bound method with some of the
NLP solvers in GAMS.
2 SETUP
The basic QAP formulation in (1-3) is modelled in
GAMS 23.6.2 and all solvers are used with default
parameter settings. We use here the following
abbreviations: ECP for AlphaECP, BON for Bonmin
and DOP for DICOPT. 50 problems from the QAP
library are solved: retrieved March 22nd, 2011, from
http://www.seas.upenn.edu/qaplib/. The problems
are selected from the QAP library (Hahn and Anjos,
2002) with the following criteria: 30 ≤  ≤ 90,
where  is the size of the square matrices (×).
Thus, the selected set contains problems that are
very difficult to solve to proven optimality with
exact algorithms. Two problems, lipa90a and
lipa90b, with =90 are, however, not included in
the comparison because all solvers stopped within 4
minutes because of a system or memory limitation.
In the first test the solvers are set to solve 50
problems with a 1 hour time limit per problem per
starting point, hence the total solution time may raise
to 3 hours when the solvers are started from 3
random points. The second test batch consists of the
22 problems from the first set, which are the
problems where the global optimal solution is
known. In this test DICOPT is called with a 1 hour
time limit per problem per start point from 50
random points. The test computer is an Intel core i7
with 4 cores of 2,8GHz and 6GB of memory.
3 RESULTS
Table 1 shows how well, in general, the four solvers
solved the problems. Table item “Avg. % from best
solution” describes the average (avg.) deviation in
percentage from the best solutions known reported
in the QAPLIB (Hahn and Anjos, 2002), for the 13
problems where all the solvers found a solution.
Table item “Number of best solutions” denotes for
how many problems a solver found a better solution
than the other three solvers. Note that SBB found a
better solution than the three other solvers for 26
problems, but could not find any solution for 8
problems. Furthermore, it is worth noting the
exceptionally short solution time for DICOPT to
find good solutions.
Table 1: Overall performance of the 4 solvers.
ECP DOP SBB BON
Problems solved 50 50 42 13
Avg. solution time (min) 36 4 35 43
Avg. % from best solution 4.7 5.6 3.7 4.3
Number of best solutions 8 5 26 4
In Table 2 the problem size is included in the
name. The table reveals, for each solver, the best
solution when the solver is started from the 3
starting points. The star in Table 2 indicates that the
best known solution is a global optimal one. None of
the solvers are able to find exceptionally good
solutions compared to the other solvers. Table 3
reveals the improvement for DICOPT when the
solver is started from 50 random starting points
instead of 3. The standard deviation denotes the
standard deviation in the obtained value of the
objective function.
4 CONCLUSIONS
In this study 50 challenging problems from the
QAP-library were solved with some standard
MINLP solvers from GAMS. The compared solvers
were: AlphaECP, Bonmin, DICOPT and SBB.
AlphaECP found good solutions for all the
problems, but typically used the total solution time
available before termination. Bonmin found a
solution only for 13 problems, but 4 of them were
better than any of the other solvers. DICOPT solved
the problems significantly faster than the three other
solvers, but was unable to significantly improve the
solution quality when the solver was started from 50
random start points instead of 3.
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
410

Table 2: Solution and average time usage when starting the solvers from 3 starting points for 50 problems.
Problem Best Solution value % from best Avg. time (min)
solution known ECP DOP SBB BON ECP DOP SBB BON ECP DOP SBB BON
esc32a 130 148 154 142 146 13.8 18.5 9.2 12.3 0.2 0.0 0.4 46.2
esc32b 168 192 192 184 184 14.3 14.3 9.5 9.5 0.1 0.0 1.8 28.0
esc32c 642 642 642 642 NA 0.0 0.0 0.0 NA 60.0 0.0 60.0 54.9
esc32d 200 210 200 200 NA 5.0 0.0 0.0 NA 0.9 0.0 1.4 60.3
esc32e 2* 2 2 2 2 0.0 0.0 0.0 0.0 0.0 0.0 1.9 39.1
esc32g 6* 6 6 6 6 0.0 0.0 0.0 0.0 0.0 0.0 2.0 21.6
esc32h 438 440 448 440 NA 0.5 2.3 0.5 NA 60.0 0.0 60.0 61.2
esc64a 116 116 116 116 NA 0.0 0.0 0.0 NA 60.0 0.1 60.0 0.0
kra30a 88900* 92610 90760 92810 91400 4.2 2.1 4.4 2.8 26.3 0.1 2.8 53.4
kra30b 91420* 94490 94490 92290 94400 3.4 3.4 1.0 3.3 60.0 0.0 6.0 45.6
kra32 88700* 91420 90470 89820 NA 3.1 2.0 1.3 NA 60.0 0.1 17.9 29.6
lipa30a 13178* 13413 13469 13438 13485 1.8 2.2 2.0 2.3 60.0 0.1 60.0 37.8
lipa30b 151426* 174540 174202 172718 174318 15.3 15.0 14.1 15.1 44.9 0.2 26.3 65.0
lipa40a 31538* 32104 32187 NA NA 1.8 2.1 NA NA 60.0 0.5 60.0 0.0
lipa40b 476581* 559251 559417 554548 NA 17.3 17.4 16.4 NA 60.0 0.8 38.2 0.0
lipa50a 62093* 63046 63275 NA NA 1.5 1.9 NA NA 60.0 1.4 60.0 0.0
lipa50b 1210244* 1422035 1423875 1421683 NA 17.5 17.7 17.5 NA 60.0 2.9 60.0 0.0
lipa60a 107218* 108560 108835 NA NA 1.3 1.5 NA NA 60.0 4.0 60.1 0.0
lipa60b 2520135* 2999484 2994479 2986313 NA 19.0 18.8 18.5 NA 60.0 5.9 60.1 0.0
lipa70a 169755* 171800 171954 NA NA 1.2 1.3 NA NA 60.0 7.9 60.2 0.0
lipa70b 4603200* 5527251 5506114 NA NA 20.1 19.6 NA NA 60.0 11.2 60.1 0.0
lipa80a 253195* 255986 256237 NA NA 1.1 1.2 NA NA 60.1 16.9 60.2 0.0
lipa80b 7763962* 9350196 9329120 NA NA 20.4 20.2 NA NA 60.0 24.4 60.1 0.0
nug30 6124* 6218 6280 6156 6154 1.5 2.5 0.5 0.5 5.7 0.1 4.5 61.4
sko42 15812 15960 15904 15838 15818 0.9 0.6 0.2 0.0 21.2 0.7 29.0 42.5
sko49 23386 23628 23738 23556 NA 1.0 1.5 0.7 NA 53.6 1.2 35.8 0.0
sko56 34458 35088 34688 34598 NA 1.8 0.7 0.4 NA 35.1 2.5 60.1 0.0
sko64 48498 49466 48798 48724 NA 2.0 0.6 0.5 NA 44.9 4.7 60.1 0.0
sko72 66256 67202 66852 66716 NA 1.4 0.9 0.7 NA 60.0 8.7 60.2 0.0
sko81 90998 92558 91396 91250 NA 1.7 0.4 0.3 NA 60.0 13.3 58.5 0.0
sko90 115534 117116 116260 116560 NA 1.4 0.6 0.9 NA 60.0 26.7 60.3 0.0
ste36a 9526* 9838 10228 9754 10402 3.3 7.4 2.4 9.2 1.7 0.1 11.4 41.2
ste36b 15852* 16212 16620 16316 15852 2.3 4.8 2.9 0.0 0.2 0.0 0.3 7.9
ste36c 8239110* 8287134 8720778 8306974 NA 0.6 5.8 0.8 NA 44.2 0.1 60.0 27.3
tai30a 1818146 1868648 1879624 1857106 NA 2.8 3.4 2.1 NA 1.9 0.1 3.5 0.0
tai30b 637117113 705935352 711585738 698531894 NA 10.8 11.7 9.6 NA 17.9 0.1 4.1 36.9
tai35a 2422002 2495070 2500438 2470970 NA 3.0 3.2 2.0 NA 5.2 0.3 11.0 0.0
tai35b 283315445 287808844 285343286 283334598 NA 1.6 0.7 0.0 NA 43.4 0.2 20.1 9.6
tai40a 3139370 3218910 3216932 3209658 NA 2.5 2.5 2.2 NA 1.4 0.7 13.6 0.0
tai40b 637250948 698738895 682857820 674921066 NA 9.6 7.2 5.9 NA 60.0 0.3 40.1 39.3
tai50a 4938796 5073328 5029626 5023310 NA 2.7 1.8 1.7 NA 3.8 1.5 13.8 0.0
tai50b 458821517 495201830 478598493 477360768 NA 7.9 4.3 4.0 NA 7.6 0.7 60.0 0.0
tai60a 7205962 7422590 7362494 7356088 NA 3.0 2.2 2.1 NA 12.8 5.5 30.1 0.0
tai60b 608215054 633529622 627096107 626306798 NA 4.2 3.1 3.0 NA 60.0 1.7 60.0 0.0
tai64c 1855928 1890604 1988186 1988186 NA 1.9 7.1 7.1 NA 0.2 0.0 0.0 6.0
tai80a 13499184 13874028 13745826 13738850 NA 2.8 1.8 1.8 NA 32.6 17.6 56.0 0.0
tai80b 818415043 880698860 850580839 NA NA 7.6 3.9 NA NA 60.0 11.1 60.1 0.0
tho30 149936* 151328 153210 152118 150810 0.9 2.2 1.5 0.6 1.1 0.1 0.9 39.6
tho40 240516 244676 244120 241234 NA 1.7 1.5 0.3 NA 2.4 0.2 1.7 0.0
wil50 48816 49138 49016 48920 NA 0.7 0.4 0.2 NA 60.0 2.7 60.1 0.0
A COMPARATIVE STUDY OF SOLVING QUADRATIC ASSIGNMENT PROBLEMS USING SOME STANDARD
MINLP SOLVERS
411

Table 3: The performance of DICOPT when starting the solver from 50 random start points.
Problem Solution Time (sec)
Optimal Best solution found % from optimal Improvement Standard deviation Avg. time/ problem
esc32e 2* 2 0.0 0 4 0.1
esc32g 6* 6 0.0 0 1 0.1
kra30a 88900* 90760 2.1 0 1842 3.0
kra30b 91420* 91910 0.5 2580 1462 2.8
kra32 88700* 88700 0.0 1770 2424 3.5
lipa30a 13178* 13469 2.2 0 36 7.9
lipa30b 151426* 174202 15.0 0 1277 12.2
lipa40a 31538* 32135 1.9 52 46 28.5
lipa40b 476581* 555183 16.5 4234 3148 50.2
lipa50a 62093* 63122 1.7 153 85 90.1
lipa50b 1210244* 1412355 16.7 11520 6067 175.1
lipa60a 107218* 108750 1.4 85 109 225.3
lipa60b 2520135* 2979395 18.2 15084 8800 449.0
lipa70a 169755* 171821 1.2 133 108 496.2
lipa70b 4603200* 5469943 18.8 36171 15075 929.9
lipa80a 253195* 255993 1.1 244 173 1015.8
lipa80b 7763962* 9283379 19.6 45741 21762 1817.1
nug30 6124* 6158 0.6 122 66 6.6
ste36a 9526* 9994 4.9 234 326 4.9
ste36b 15852* 16478 3.9 142 307 2.5
ste36c 8239110* 8472932 2.8 247846 121457 5.5
tho30 149936* 150724 0.5 2486 402 3.5
SBB found a better solution for 26 problems than the
other three solvers, but was unable to find a solution
for 8 problems. When any of the 4 solvers
terminated with a solution, then the best solution of
3 was always less than 21 % from the best solution
known.
ACKNOWLEDGEMENTS
Financial support from the Academy of Finland
(Project: 127992) is gratefully acknowledged.
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Applications
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