A Connectionist Approach To The Quadratic Assignment Problem

Computers
ops Res. Vol. 19, No. 314, pp. 281-295, 1992 0305-0548/92 ss.oo + 0.00
Printedin Great Britain.All rightsreserved Copyright0 1992 PcrgamonPressplc
A CONNECTIONIST APPROACH TO THE QUADRATIC
ASSIGNMENT PROBLEM-f
JAISHANKAR CHAKRAPANI~ and JADRANKA SKORIN-KAPOV$~
Department of Applied Mathematics and Statistics and Harriman School for Management and
Policy, State University of NewYork at Stony Brook, Stony Brook, NY 11794, U.S.A.
Scope and Purpose-Quadratic assignment problem (QAP) is an NP-hard combinatorial optimization
problem arising in engineering, computer design, manufacturing and many other domains. Due to their
non-convex objective function, QAPs have a number of locally optimal solutions. Simulated annealing
and tabu search are strategies to escape from local optima and to guide the search beyond them.
Boltzrnann machines are connectionist models that use simulated annealing. In this paper we extend
and improve a connectionist model based on Boltzrnann machines to solve the QAP. We also compare
it with a related model employing tabu search. Computational results are given.
Abstract-The possibilities of applying a Boltzmann machine, and a related connectionist model in which
the escape from local optima is performed in a deterministic way using tabu search, are tested for the
quadratic assignment problem (QAP). Inefficiences with this approach led to an improved computational
model for the QAP which is based on connectionist architecture. Computational results for problems of
dimensions ranging from 5 up to 90 are given.
1. INTRODUCTION
The quadratic assignment problem (QAP) is an NP-hard problem that has been applied to many
different situations calling for optimization, including: minimizing total wire length in electronic
assemblies, determining the location of machines, departments or offices within a plant so as to
minimize transportation efforts and costs, ordering interrelated data on a disk, scheduling theory,
etc. (see e.g. [ 1I).
The problem is to find an assignment of n objects to n locations that minimizes the cumulative
product of flow between every two objects and distance between every two locations. Denoting by
dij the distance between the locations i and j, and by{,, the flow between the objects p and 4, the
problem can be written as
min i i i i zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML
dijfpsXipXj4
[=I p=l j=l g=l
subject to
i xip = 1 Vi
p=l
xip = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB
0, 1 ViVp.
The variable xip is non-zero if the object p is assigned to the location i and zero otherwise. The
objective function is quadratic and nonconvex implying the existence of a number of local optima,
and the feasible set contains n! distinct elements (the set of permutations). If the flow matrix
F = (f,,) is a cyclic permutation matrix, the QAP reduces to the traveling salesman problem (TSP).
t Research partially supported by NSF grant DDM-8909206.
$Jadranka Skorin-Kapov is an Assistant Professor in the Harriman School for Management and Policy, SUNY at Stony
Brook. She received her B.Sc. and M.Sc. in Applied Mathematics from the University of Zagreb, and her
Ph.D. in Operations Research from the University of British Columbia. Her research interests are in the area of
combinatorial optimization. She has published in Mathematical Programming, Operations Research Letters, ORSA
J
ournal of Computing and Discrete Applied Mathematics.
8Jaishankar Chakrapani is a Ph.D. student in the Department of Applied Mathematics, SUNY at Stony Brook. He received
his B.E. in Computer Science and Engineering from the Indian Institute of Science and his BSc. in Applied Sciences
from Bharathiar University, India. His research interests include neural networks, genetic algorithms and combinatorial
optimization.
c Author to whom all correspondence should be addressed.
287

288 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
JAISHANKAR
CHAKRAPANI
and JADRANKA
SKORIN-KAPOV
Connectionist models are constructed to follow the analogy with neural networks in the human
brain and are also referred to as artificial neural networks. They consist of nodes representing
neurons and arcs representing a pattern of connectivity among the neurons. An activity level is
associated with each node, and weights or connection strengths are associated with each arc. Activity
levels and connection strengths could change according to functions directing the systems’s behavior.
Connectionist models are classified into analog or binary depending on the values that an activity
level could take.
Hopfield and Tank [2] have proposed an analog connectionist model and Aarts and Korst [3]
have adapted and used a binary connectionist model (Boltzmann machine) for heuristically solving
the TSP. Reference [4] describes some combinatorial optimization problems solved suboptimally
by Boltzmann machines. Wilson and Pawley [S] have reported difficulties in the application of
Hopfield and Tank’s model to the TSP. In their paper, Aarts and Korst [3, p. 911 have demonstrated
the competence of Boltzmann machines as compared to Hopfield and Tank’s approach in solving
the TSP. They also remark on the ease of algorithmically simulating Boltzmann machines. We,
therefore, decided to base our study on Boltzmann machines.
A Boltzmann machine having n units can be represented by an undirected graph G = (U, E)
where the set of vertices U = {ui, . . ., u,} represents the set of units, and the set of edges zyxwvutsrqponmlk
E E U x U
represents the set of connections between the units. Each unit Uican be in one of the two possible
states: “on” (1) or “off (0). The states of all the units determine a configuration of the Boltzmann
machine. Each edge (ui, uj), including loops, has a connection strength associated with it. Connection
strengths may be positive (excitatory) or negative (inhibitory). A connection between any two units
is activated if both units are “on”. Associated with this structure is the so called consensus function
which can be viewed as an overall measure of “agreement” among units in a network. It can be
informally defined as the sum of activated connection strengths. In order to maximize the consensus
function, the units adjust their states to the states of the neighboring units, thereby activating (or
deactivating) connection strenghts. The choice of the connection strengths is a function of the
specific problem. Aarts and Korts [3] show that for a certain choice of connection strengths, the
maximization of the consensus function corresponds to the minimization of the objective function
for the TSP. The assignment type constraints (i.e. each city has to be visited exactly once) are
mapped onto the structure of a Boltzmann machine in view of inhibitory connections (decreasing
the consensus if a city is visited more than once) and bias connections (decreasing the consensus
if a city is not visited at all).
When a configuration of a Boltzmann machine is such that a change in any of the units only
decreases the consensus function, a local optimum is reached. In a Boltzmann machine, the way
to get out of a local optimum is governed by simulated annealing [6]. Under certain assumptions
about the annealing schedule, asymptotic convergence to a global optimum can be proved. In
practice, depending on the annealing schedule used, the method ends up in a local optimum. A
Boltzmann machine can be viewed as a massively parallel simulated annealing method.
Another approach to cope with local optimality, called tabu search, has been proposed and
extended by Glover [ 7,8]. As opposed to simulated annealing, in a simple tabu search the transition
from one feasible solution to another is performed deterministically. The whole neighborhood (as
defined) is searched, and the best solution (according to a given criterion) is taken as the current
solution. Suppose that the criterion to evaluate moves (i.e. changes from one feasible solution to
another) is the change in the objective function. It is clear that after reaching a local optimum, an
inferior move will be taken. In order to prevent cycling, i.e. falling back to the same local optimum,
reversal of a number of recent moves is forbidden (or tabu). The size of the tabu list determines
how many moves to forbid at each iteration. This list can then be updated circularly, thereby
releasing a move after a number of iterations. The move can be released even before its tabu status
expires if it leads to a solution better than any previously encountered. This is called the aspiration
criterion. Dynamic tabu list strategies that vary the size or composition of the list offer the potential
for interesting refinements. More details and some other components of tabu search such as long
term memory are broadly defined in [7] and [8]. A probabilistic variant of tabu search has also
been proposed [7] and has been shown by Faigle and Kern [9] to have mathematical convergence
properties analogous to those of simulated annealing, based on a broader foundation. However,
empirical studies have so far focused on the deterministic form of tabu search, and several have
established the competitiveness of this form with simulated annealing.

Connectionist approach to the QAP 289
In this paper we generalize the connectionist model proposed by Aarts and Korst [3] for the
TSP to solve the QAP. As Aarts and Korst [S] stated, a Boltzmann machine approach for the
TSP is not as successful as the same approach for some other ,graph problems due to the specific
construction of the machine (see [ 10, p. 1761). One of our objectives in extending their model was
to determine if a deterministic tabu search approach superimposed on a connectionist model could
improve upon a Boltzmann machine (a connectionist model on which simulated annealing is
superimposed). Further analysis of the generalized model led us to a new, improved, massively
parallel computational model based on connectionist architecture. Computational experience with
QAPs of dimension up to 90 clearly establishes the superiority of our new model.
In the sequel we will label the connectionist model with the same architecture as of Boltzmann
machine’s as Model 1. In the next section we describe Model 1 and the algorithms performed on
it. Inefficiencies with Model 1 led us to Model 2, a related connectionist architecture. Model 2 and
the corresponding algorithms are described in Section 3. The computational results are presented
in Section 4 and the conclusions in Section 5.
2. MODEL 1: A CONNECTIONIST MODEL BASED ON THE BOLTZMANN MACHINE
In Section 2.1 we provide a formal description of the connectionist model and the appropriate
choice of the connection strengths for the QAP. Section 2.2 describes the algorithms used to
maximize the consensus function. We then discuss the limitations of the model and propose some
changes which results in a modest improvement. This motivated the changes resulting in Model 2.
For the sake of consistency, we preserve most of the notation from [3].
Let us denote the set of units in a Boltzmann machine for the QAP by U = {u11,. . ., uln, . .. , u,,}.
The set of edges E is identified with the set of connections. Recall that any configuration of a
Boltzmann machine on the above structure can be represented by an n x n matrix of zeros and
ones. Of course, only matrices having exactly one non-zero element in each row and column, i.e.
permutation matrices, qualify as feasible solutions to the QAP. Let us denote the state of any unit
uip in a configuration zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
k by k(Uip). Denoting the set of configurations by K, the consensus function
c: K + 8 is defined as follows:
c(k) = 1 (s(uip, uj,)k(ui,fkfUj~)l(Uip, aj*)oE)
where s( Uip,Uj~)denotes the strength of the connection (Uip,Ujq)*
By choosing appropriate connection
strengths one can assure that local maxima of the consensus function are obtained with
configurations corresponding to permutation matrices. This is formally stated in Theorem 1. In the
context of the QAP there are n2(n - 1)2/2 distance-flow connections from C,, = {(UC,U,)(i # j
and p # 41, n2 bias connections from C, = ((Uc,Ujq)li =j and P = 41, and finally n2(n - 1)
inhibitory connections from Ci = ((ui,, Uj~)i(i=I and p # 4) or (i #j and P = 4)). Thus each unit
is connected to (n - 1)2 other units via distance-flow connections, has its bias connection, and is
connected to 2(n - 1) other units via inhibitory connections.
For each k E K, let us denote by N, the set of its neighboring configurations, i.e. the set of
configurations obtained from k by changing the state of exactly one unit. More formally, changing
the state of unit uip in configuration k, the neighboring configuration ki, is obtained as
kc(ui,) = 1 - k(Uip) and kip(tijg) = k(~~~)V~
# i or 4 # p. Let US denote by A(k, kipf the difference
in consensus functions for configurations k and ki,, i.e. A(k, kip) = C(k,,) - C(k). The same
derivation as in [3], results in the formula
A(k, kip) = ( 1 - 2kip(uip))6(k kip),
where
and
Eip = {Uj~Ij # i or 4 #P, (Uip,U,)fE}.
The following theorem, a generalization of Theorem 1 in [ 31, defines the appropriate connection

JAISHANKAR
CHAKRAPANI
and JADRANKA
SKORIN-KAPOV
strengths and establishes the mapping between consensus and objective function for the QAP. It
can be proved along the similar lines and the proof is therefore omitted. zyxwvutsrqponmlkjihgfedcbaZ
Theorem 1
Let the connection strengths of the Boltzmann machine for the QAP be given by
V(ai,,aig) ECd/: s(uip, ajq) = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE
-Wjfpq,
V(~i,,nj~)~Cb: s(Uip,Uip)>max(2C(dij,fp,q’lI= l,...,n;
j' f .se
Zj"; p’ # **- # p”; q1 # *** # q”)},
V( uip,ui,) E Ci : s(Uipyuiq) < - min {s(nip,Uip),s(Uiq,UC)}and
V(aip, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Ujp)E Ci : s(“ip* ujp) < -min(s(ui,, %p)v s(“jp, ujp)>*
( 1) Then there is a one to one correspondence between local maxima of the consensus
function and permutations (i.e. configurations of the Boltzmann machine which
are feasible for the QAP).
(2) Higher consensus is attained for permutations yielding smaller objective function
values.
Note:
(1) We assume that the matrices D and F are symmetric. If not, the distance-flow
connections change to zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB
-dijfpq - djif4p, and the other connections change
accordingly.
(2) Bias connections are positive and will therefore “encourage” units to take nonzero
values in order to maximize the consensus. On the contrary, the inhibitory
connections are negative and effective when more than one unit is nonzero in a
row or column.
(3) Bias connections as stated in Theorem 1 are very difficult to compute because
it involves finding the maximal value of a quadratic function over (n - l)-
dimensional permutation matrices. In order to get tractable bias connections we
replaced them by zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
V( zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
uip, ui,) E C,: S(Uip, Ui,) > (PI - 1)max {d, zyxwvutsrqponmlkjihgfedcba
fp4
1
Vj,VP,Vq}. The
validity of the theorem is preserved.
2.2. Consensus maximization on Model 1
We tried simulated annealing as well as tabu search as strategies to escape local maxima when
optimizing the consensus. Note that when using simulated annealing as a search strategy Model
1 corresponds exactly to a Boltzmann machine. Both algorithms were tested on Nugent’s problems
of dimensions 5, 6, 7, 8, 12 and 15. The computational results are given in Section 4. Due to poor
performance of the algorithms we did not try problems of higher dimension.
When using simulated annealing, a neighboring configuration is selected at random. Depending
on the temperature ?zthe configuration is accepted with probability l/( 1 + exp( - A(k, k,)/ ‘I))-the
typical probability used in Boltzmann machines. The cooling schedule used was T+ 1 = aT. The
cooling rate c1is a constant less than, but close to 1. We tried two values of a: 0.99 and 0.95.
Large bias connections require the use of very high starting temperatures in order to assure high
probability of accepting any configuration initially. This results in slower convergence and very
high computational times. We tried to overcome this problem by performing transitions from one
configuration to the other in a deterministic way using tabu search.
When using tabu search, all the n2 neighboring configurations are examined and a configuration
corresponding to the maximal increase in consensus (or minimal decrease when moving out of
local optima) is accepted deterministically. The tabu list, which we elected to update circularly,
contains units whose states had been changed in the most recent tabu size number of iterations.
Tabu size varied between n and 2n. Aspiration criterion was always used.
Large bias connection strengths assure that a locally optimal configuration k corresponds to a
permutation. The value of the consensus function can then be written as K -f(k), where K is a
constant depending on bias connection strengths, and f(k)
is the objective function value (as given

Connectionist approach to the QAP 291
Table I. Snapshot of the first four iterations of tabu search on Model I forNugent’s problem
of size 5
Iteration No. Configuration Tabu list zyxwvutsrqponmlkjihgfedcbaZYX
0 ooo10;COcOl; 1cOoo; 00100; 01ooo Empty
1 OoolO; ooool; oooo0; COloo; Olooo “31
2 OoolO; ooool; M)loo; 00100; Olaoo +I. Us3
3 OoolO; ooool; 00100; oocO0; Olooo u31. %s. %3
4 OoolO: oowl; 00100; loooo; Olooo U?!, u,a, UI1. u,,
in the QAP formulation) for the configuration zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH
k. Due to the magnitude of bias connection strengths,
the constant K is relatively large compared to f(k). This results in the inability of the model in
making a good distinction between permutations yielding lower versus those yielding higher objective
function values. We tried to improve our results by dynamically decreasing the values for the bias
and inhibitory connection strengths. Every time the search resulted in a better local maxima of the
consensus function, the bias connection strengths were reset to the corresponding objective function
value. The inhibitory connection strengths were changed accordingly. Now, Theorem 1 remains
valid only for neighborhoods of local optima with better consensus function values than previously
encountered. Clearly, this is the region of interest. This resulted in a certain improvement in the
algorithm, but still it could not compare favorably with other heuristic methods for the QAP, e.g.
with the algorithm in [ 111. Note that the model evolves passing through configurations that are
not permutations, i.e. that are not feasible for the QAP. If one starts with a permutation, the next
permutation can be obtained only after a minimum of four iterations and useful information about
the merit of such exchange might be lost along the way. This behavior can be best seen using the
following example.
Example
For Nugent’s problem of dimension 5 [ 121, let us start with the permutation 45132 for which
the objective function value equals 60. Though the tabu size for this example was set to 5, any
value greater than 3 would suffice. In each iteration a single unit changes its state and is placed
in the tabu list for tabu size iterations. Table 1 outlines the configuration of the system along with
the tabu list for the first four iterations. A configuration is specified by listing the state of all units
with a “;” separating rows. The permutation obtained after 4 iterations has an objective function
value 62. However, an optimal permutation 45123 is reachable from the starting permutation in
four iterations, i.e. in one pairwise exchange.
3. MODEL 2: CONNECTIONIST MODEL WITHOUT BIAS AND INHIBITORY
CONNECTIONS
In the previous section, large bias connections and infeasible configurations were identified as
possible sources of computational problems. Guiding the search to visit only feasible configurations
will eliminate the necessity of having bias and inhibitory connections at all.
Formally, the architecture of Model 2 can also be represented by an undirected graph G = (U, E’),
where U is the same as in Model 1, and E’ = {(ui,, uj,)li # j and p # 4). The neighborhood N,
for a configuration k is redefined by considering configurations derivable from k through a series
of state changes of four units. Let us denote by kipjq the configuration obtained from k by changing
the state of units uip, Uiqrujp and ujcl.
Suppose that the configuration k, with k(uip) = 1and k(ujq) = 1,corresponds to a feasible solution
to the QAP. The configuration kipjs results in a (feasible) configuration with k(uiq) = k(ujp) = 1
and k( ui,) = k( uj,) = 0. This corresponds to a pairwise exchange of objects p and q. Also, the value
of the consensus function equals the negative value of the objective function for the corresponding
solution of the QAP. Thus, the relevant part of Theorem 1 (i.e. the second part) is still valid. The
change in consensus A(k, kipjq) can be evaluated using the change in consensus due to the state
change for individual units in the following way (see Section 2.1):
A(k kip) = (1 - 2k,(u,))6(k kip) = -6(k kip) = - Cs(Ui,,ujq)
+ 1 {s(Uip,U~m)k(U~rn)l(Uip,U~rn)E~
- zyxwvutsrqponmlkjihgfedcbaZYXWVU
(UipUjq)Jl.

J
AISHANKAR
~~~K~~PA~~ and J
ADRANKA
SKORIN-KAWV
Similarly,
Now,
A(k kipjq)= A(k k<p)
f A(k kjq)+ A(k kiq)+ A(k kjp)+ s(uiprUjq)+ S(Uiq,
ujp).
From the above expression, it is clear that by knowing the change in the consensus due to the
individual units Afk, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
k,pjg) can be evaluated in time of the order required to compute A(k, k,,).
Thus, Model 2 achieves speedups of the same order as achieved by Model 1.
The following theorem establishes the efficiency of an algorithmic simulation of Model 2.
Theorem 2
Let k be the current configuration of Model 2 and let k’ be the previous configuration. Let
k = k:pqj. Then for any unit u,,,,,d(k, k,,) can be computed in constant time using 6(k’, k;,).
Proof. For notational simplicity we assume that bias and inhibitory connections exist with
strength zero.
Similarly, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
+ C (S(Utm,U,,)k(u,,)lu,~~El’” - sip - UjqI,
d(k k,,) = s(ui,, uiq) f S(Ulrn,
Ujp)+ 1 (s(&m, G)k(u,,)Iu,,EElm - U+- rJjp)*
Therefore, 6(k, k,,) can be computed by the following expression.
a(k ki,) = S(k’, 6,) - s(Ulm,Ui,) - s(ulmn,jq)+ s(~lm,Ui,) + s(Utmr
ajp)*
This requires only a constant effort.
Note:
0
(I) The change in the consensus function due to a neighboring configuration can be
computed in constant time. For Model 2 this corresponds to computing the
change in the objective function due to a pairwise exchange.
(2) An equivalent result holds for Model 1 also.
We performed both simulated annealing and tabu search on Model 2. The procedures are the
same as in Model 2, except that now a neighboring configuration and the corresponding change
in consensus function are redefined. In addition, tabu search employs a different tabu list. Recall
that in the context of Model 2, a move corresponds to four units changing their states. Units, say
Uipand Ujsare turned “o ff’, and uiqand ujp are turned “on”. The tabu list consists of pairs {Uip,uj4}
of units that are turned “off’.
4. COMPUTATIONAL RESULTS
The computational experiments were performed on SUN Microsystems at SUNY, Stony Brook.
The coding was done in C. In the tables below time refers to CPU seconds.
The data sets given by Nugent et al. [ 121, Steinberg [ 133, Krarup (see [ 1I]) and Skorin-Kapov
[ 111 were used. Optimal objectives for Nugent’s problems of size up to 15 are known from the
literature (see e.g. [ 11). For the other problems the best known objectives have not been proven
to be optimal. For Model 1, due to computational inefficiency, only smaller dimensional problems
by Nugent were attempted.
We first implemented a straightforward extension of Aarts and Korst [3] approach for TSP to
the QAP. The results are presented in Table 2. We tried two cooling rates corresponding to CI= 0.99
and CY
= 0.95. Starting temperature was the consensus function value for a random feasible

~onn~tionist approach to the QAP 293
Table 2. Model 1 with simulated annealing
Test problem Total time Starting temp. No. of cycles Cooling rate
Best objective
achieved
Best published
objective
N5 1.51 144 1250 0.99 50 zyxwvutsrqponmlkjihgfedcbaZY
50
0.57 0.95 50
N6 1.83 360 1800 0.99 86 86
0.78 0.95 92
N7 2.87 576 2450 0.99 148 148
0.86 0.95 IS6
N8 3.91 672 3200 0.99 224 214
0.92 0.95 248
N I2 7.64 1320 7200 0.99 664 578
1.78 0.95 690
N 15 19.11 2016 11250 0.99 1420 1150
3.97 0.95 1434
Table 3. Model 1 with tabu search, aspiration used
Test
problem
Time/
iteration
Max
iteration
Tabu
size
With fixed bias
Best objective
achieved
Best
iteration
Tabu
size
With dcc. bias
Best objective
achieved
Besi
iteration zyxwvutsrq
N5
N6
N7
NS
N 12
N 15 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
0.0005 100 6 50 42 5 SO 12
0.0005 200 7 86 63 6 86 172
O.M)IO 6cQ 12 148 533 7 148 450
0‘0010 loo0 13 214 980 11 214 40
0.0020 loo0 15 632 11 12 624 633
0.0035 zoo0 20 1340 14 15 1264 979
Table 4. Model 2 with simulated annealing, c(= 0.95
Test problem Totat time Starting temp. No. of cycles Best objective achieved
N5
N6
N7
N8
N 12
N 15
N 20
N 30
0.44 82 1250 50
0.63 106 1800 86
0.7 I 148 2450 148
0.84 328 3200 214
1.37 716 72CO 578
3.53 1620 11250 1150
11.18 3568 2oOoa 2570
40.62 8174 45ooo 6128
configuration. The number of cycles for each temperature was set to 50n’. Observe that the CPU
time increases significantly as a function of problem size and of cooling rate.
Next, we implemented Model 1 with tabu search as the strategy to overcome local optima. The
best results obtained using tabu search on Model 1 with static, as well as monotonically decreasing
bias and inhibitory connection strengths are presented in Table 3. Aspiration criterion was always
used. We performed the same number of iterations for both fixed and decreasing connection
strengths. The results show the clear superiority of tabu search when Model 1 is used, for problems zyxwvuts
of size fl greater than 7. Also, the strategy of decreasing bias and inhibitory connection strengths
results in a consistent improvement. Further, the time/iteration is O(n’) resulting in total time
significantly less than that for simulated annealing with ct= 0.99.
We then replaced Model 1 with our proposed alternative Model 2. Simulated annealing on
Model 2 was attempted for all Nugent’s problems. We used a cooling rate of a = 0.95. Other
parameters were set as for Model 1. The results are given in Table 4.
As shown, the new model enables simulated annealing to obtain generally better solutions than
obtained using Model 1. The new Model 2 likewise enables tabu search to obtain better solutions
than previously. As in the case of Model 1, the solutions obtained with tabu search are always as
good or better than those obtained with simulated annealing. In addition, the solution times do
not grow nearly as rapidly using tabu search, thus allowing us to examine larger problem sizes.

294 JAISHANKAR
CHAKRAPANI
and JADRANKA
SKORIN-KAPOV
Table 5. Model 2 with tabu search, No. iterations = lOOn
Test problem Time/iteration Aspiration used? Tabu size Best iteration
Best objective Best published
achieved objective
N5 0.0005
N6 0.0005
N7 0.0010
N8 0.0010
N 12 0.0020
N 15 0.0036
N 20 0.0070
N 30 0.0192
KI 30 0.0192
K2 30 0.0192
SI 36 0.0271
S2 36 0.0271
SK 42 0.0578
SK 49 0.1890
SK 56 0.2440
SK 64 0.3210
SK 72 0.4C00
SK 81 0.5290
SK 90 0.6620
no 5
no 6 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO
IlO 7
no 8
yes 12
ye= 15 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML
Ye S 20
yes 15
yes 20
ye= 15
yes 25
yes 30
yes 40
yes 20
yes 25
yes 45
yes 55
ye= 20 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI
Ye= 30
Yes 55
Yes 70
yes 25
yes 30
yes 60
yes 70
yes 30
yes 40
yes 30
yes 40
yes 40
yes 50
yes 45
yes 60
yes 50
ye= 65
yes 50
yes 60
yes 50
yes 60
9
2
22
3
89
378
431
194
363
205
2388
467
601
366
2745
2290
1130
2139
2408
2199
I556
62
2142
3052
3428
665
1074
827
1668
5555
2789
2618
652
2678
1026
4369
6281
8922
9ooo
50
86
148
214
578
1150
2570
6124
6128
90720
90920
88900
88900
91900
91590
91420
91490
10200
9602
9526
9576
18526
17630
15852
15852
15818
15846
23412
23398
34658
34662
48760
48790
66268
663 10
91362
91402
116134
116116
50
86
148
214
578
1150
2570
6124
88900
91420
9526
15852
15864
23536
34736
48964
66756
91432
116360
We set the number of iterations to lOOn.For problems of size greater or equal to 12 aspiration
cirterion was always used. The size of the tabu list was problem specific. For problems of size up
to 20, tabu size was set to n. For larger problems, we tried two tabu sizes which were random
multiples of five in the interval [(n/2), n]. The results were very good for all problems except those
due to Krarup and Steinberg. Therefore, for those problem we decided to try larger tabu sizes.
Again, we chose two random multiples of five in the interval [n, 2n]. Note that Nugent’s and
Skorin-Kapov’s problems were generated along similar lines whereas, Krarup’s and Steinberg’s
problems were generated in a different way. All the results are given in Table 5. The best published
objective is taken from zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
[1 11. The CPU time/iteration is 0(n2) and compares favorably with other
heuristics (see e.g. [ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
11, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
11).For all the test problems we report solutions as good or better than the best
published.
5. CONCLUSIONS
In this paper we have proposed a massively parallel heuristic algorithm for the QAP. The
algorithm is based on connectionist architecture and significantly improves over a related
connectionist model based on the architecture of a Boltzmann machine. Moreover, we propose the
use of tabu search in order to escape from local optima as an alternative to the stochastic simulated
annealing. For the problems tested, this approach seems to work very efficiently.
REFERENCES
1. R. E. Burkard, Quadratic assignment problems. Eur. J. Ops Res. 15,283-289 (1984).
2. J. J. Hopfield and D. W. Tank, Neural computation of decisions in optimization problems. Biol. Cybernet. 52, 141-152
(1985).

Connection& approach to the QAP 295
3. E. H. L. Aarts and J. H. M. Korst, Boltzmann machines for travelling salesman problems. Eur. J. Ops zyxwvutsrqponmlkjihgf
Rex 39,79-95 (1989).
4. E. H. L. Aarts and J. H. M. Korst, Simulated Annealing and Boltrmann Machines-A Stochastic Approach to Combinatorial
Optimization and Neural Computation. Wiley, New York (1989).
5. G. V. Wilson and G. S. Pawley, On the stability of the travelling salesman problem algorithm of Hopfield and Tank.
Biol. Cybernet. 63-70 (1988).
6. S. Kirkpatrick, C. D. Gellati and M. P. Vecchi, Optimizing by simulated annealing. Science 220, 671-680 (1983).
7. F. Glover. Tabu search--Dart i. ORSA J
. Comout. 1. 190-206 (1989).
8. F. Glover, Tabu search-part ii. ORSA J. Con&t. 2,4-32 (1990). ’
9. U. Faigle and W. Kern, Some convergence results for probablistic tabu search. Research Memorandum 882, Faculty
of Applied Mathematics, University of Twente, The Netherlands (1989).
10. E. H. L. Aarts and J. H. M. Korst, Simulated annealing and Boltzmann machines. In Discrete Mathematics and
Optimization. Wiley-Interscience, New York (1988).
11. J. Skorin-Kapov, Tabu search applied to the quadratic assignment problem. ORSA J
. Comput. 2, 33-45 (1990).
12. C. E. Nugent, T. E. Vollmann and J. Ruml, An experimental comparison of techniques for the assignment of facilities
to locations. Ops Res. 16, 150-173 (1968).
13. L. Steinberg, The backboard wiring problem: a placement algorithm. SIAM Rev. 3, 37-50 (1961).

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