a tabu search algorithm for frequency assignment

Annals of Operations Research 63(1996)301-319 301

A tabu search algorithm for frequency assignment

D.J. Castelino, S. Hurley and N.M. Stephens

Department of Computer Science, University of Wales Cardiff, P.O. Box 916, Cardiff, CF2 3XF, UK

E-maih [email protected]

This paper presents the application of a tabu search algorithm for solving the frequency assignment problem. This problem, known to be NP-hard, is to find an assignment of frequencies for a number of communication links, which satisfy various constraints. We report on our computational experiments in terms of computational efficiency and quality of the solutions obtained for realistic, computer-generated problem instances. The method is efficient, robust and stable and gives solutions which compare more favourably than ones obtained using a genetic algorithm.

1. Introduction

The radio link frequency assignment problem is a combinatorial optimisation problem which occurs in many military and civil applications [3,24]. The main objective is to assign radio frequencies to a number of transmitters subject to a number of constraints, such that minimum interference is suffered. Ideally zero interference is desired, but given the limited region within which the transmitters operate, a more reasonable goal is to minimise the interference. The problem is classified computationally as NP-hard. Hence, there is no known algorithm that can generate a guaranteed optimal solution in an execution time that may be expressed as a finite polynomial of the problem dimension. As optimal solutions can be extremely difficult to determine, a great deal of attention has been focused on heuristic procedures capable of generating near-optimal solutions.

Metzger [ 19] has shown that if the only constraints in the frequency assignment problem are co-channel constraints (see section 2.1), then the problem is equivalent to the classical graph cotouring problem. In this graph formulation, each node is represented by a transmitter and each colour by a frequency and so colouring the nodes of the graph leads to a frequency assignment plan.

© J.C. Baltzer AG, Science Publishers

302 D.J. Castelino et al., An algorithm for frequency assignment

A graph theoretical formulation of the frequency assignment problem has been proposed by Hale [17], to find the minimum span and minimum order assignment. Costa [4] has implemented simulated annealing and tabu search approaches for finding the minimum span T-colouring for randomly generated graphs up to 1000 nodes with various edge densities, and shows that tabu search yields superior solutions to that of simulated annealing. The T-colouring formulation restricts the problem to constraints between transmitter pairs, constraints between three or more transmitters cannot be modelled. The aim of this paper is to implement a tabu search approach and compare it with a parallel genetic algorithm and a local steepest descent search procedure on data which give rise to realistic sets of constraints.

Duque-Ant6n et al. [8] have implemented a simulated annealing algorithm for solving the channel assignment problem in cellular radio networks. The aim was to minimise interference while simultaneously assigning a certain prescribed number of channels per cell. In another study for this problem, Mathar and Mattfeldt [18] investigated the use of several algorithms based on the simulated annealing approach but using a different model. In a set of computational experiments, the authors showed all variants gave good quality solutions when compared to optimal solutions obtained by tailored algorithms.

De Werra and Gay [32] have developed a heuristic based on a generalization of a greedy algorithm, originally used for finding the minimum span T-colouring of randomly generated graphs and a type of Euclidean graph consisting of 1000 nodes.

However, such colouring concepts are not sufficient for handling other constraint types which arise in practical problems and in this paper, we consider real-life frequency assignment scenarios where additional constraints other than frequency distance separations are required (such as intermodulation products, see section 2.1), and present the application of a tabu search heuristic to determine such an assignment and compare the results with a genetic algorithm. Tabu search, introduced by Glover [10,11], is becoming increasingly recognised as an efficient way of finding high- quality solutions to hard combinatorial problems. It has been successfully applied to a number of combinatorial problems such as location and allocation [2,22, 26, 29], routing [21,25,31 ], scheduling [7, 12, 22, 23, 33] and telecommunications [ 1,4, 15, 27].

The paper is organised as follows. In the following section the frequency assignment problem is formulated, detailing the types of interference and their corresponding constraints. We show that it is NP-hard. In the next section the basic principles underlying the tabu search heuristic are outlined. Section 4 describes in detail our implementation of tabu search for solving the frequency assignment problem. Section 5 describes our implementation of a local steepest descent search strategy. In section 6 the parallel genetic algorithm implementation is discussed. Section 7 reports on computational results obtained in terms of computational efficiency and quality of the solutions obtained for computer-generated, realistic problem instances. Further, our results are compared to the parallel genetic algorithm for the same size problems. Finally, concluding remarks are discussed in section 8.

D.J. Castelino et al., An algorithm for frequency assignment 303

2. Radio link frequency assignment problem

We shall consider the problem where radio communication is required between a number of sites. Each site has transmitters and receivers and it is required to assign frequencies to links between certain transmitters and receivers. The available frequencies are normally separated by 50 kHz, equivalent to one channel, hence from a finite set. Not necessarily all channels are available for the system, however, since some are reserved for other purposes.

2.1. INTERFERENCE AND CONSTRAINTS

In order to reduce interference, constraints are imposed on the assignment. Interference occurs when certain pairs of links are assigned frequencies which are the same or close together. This can happen when a transmitter or a receiver of links are at the same site, within a few tens of metres of each other (co-site interference), or when equipment is at a distance of several kilometres or more ( far-site interference).

2.1.1. Co-site interference

This occurs when transmitters and receivers are at the same location. Technical limitations in the construction of receivers mean that certain combinations of receiver frequency and co-sited transmitter frequency are not permitted. The constraints which arise due to co-site interference include the following:

Co-site frequency separation: Any pair of frequencies at a site must be separated by a certain fixed amount, typically for a large problem, 500 kHz or 10 channels. If a channel is to be used by a high power transmitter then its frequency separation should be larger, say 2 MHz or 40 channels. The constraint can therefore be of the form

IJ~ -3~1 _>m,

where m refers to the number of channels separation required between radios i and j.

Intermodulation products: The second co-site constraints are aimed at intermodulation protection. These occur when two or more signals mix in a non-linear electrical system to form a third (unwanted) frequency. If it happens to be close to the wanted frequency it may not be filtered out and so will cause interference. The constraints

corresponding to the worst products are:

2 f - f j 4:3

3fi - 2 f j ve A

+ f j - A * f ,

2 fi + fj - 2 fk * ft

(two signal, third order),

(two signal, fifth order),

(three signal, third order),

(three signal, fifth order).


2.1.2. Far-site interference

This occurs between equipment that is separated by some distance. There are two kinds of constraint which arise from considering this type of interference.

Co-channel constraints: This is the most important factor in the consideration of farsite interference. A pair of communication circuits located at different sites must not be assigned the same frequency unless they are sufficiently geographically separated. This gives rise to constraints of the form

if,

Adjacent channel constraints: When a transmitter and a victim receiver are tuned to similar frequencies (normally within three channels of each other), there is still the potential for interference. Therefore a number of constraints arise of the following form,

-f l > m

for some value of m, where m is the number of channels separation.

We shall assume in this paper that interference can be avoided if there is sufficient channel separation between the frequencies assigned to the pairs of links. Of course, the required minimal separation may be zero if no possible interference can occur between two links whatever the assignment. For other pairs of links, the minimal separation depends on distances between transmitters and receivers and on the terrain. Problems which include constraints for intermodulation protection can be easily solved using the techniques described in this paper, without any significant changes. The omission of these constraints is for expositional convenience only.

2.2. MATHEMATICAL MODEL

We now describe the mathematical model of the radio link frequency assignment problem. We shall assume that there are N links, numbered from 1 to N, and that there are F frequencies. The set of available frequencies is denoted by

A = {al ,a 2 . . . . . aF}.

For each pair of links, (i, j ) , with 1 _< i < j < N there is a value cij<_ 0 which measures the minimal frequency separation required for links i and j. We denote by C the number of cii which are non-zero, where C is the number of non-trivial constraints and clearly

C <_ N ( N - 1) 2


The frequency assignment problem is to assign a frequency to each link so that there is no interference. Hence, it is to determine N values

(f l , f2 ..... fN) cAN

such that, for all i , j with 1 <i<j<_N

[ f i - f j l >-cij. (2.1)

There are F N possible frequency assignments and because it may be difficult, or even impossible, to determine one which imposes no interference we consider instead the problem of determining an assignment where the total interference is minimal. For the purposes of this paper, total interference for any assignment to the N links is measured by counting the number of violations of (2.1). Other measures include:

.

.

3.

,

5.

the difference between the largest and smallest frequency used; this is the span of the assignment;

the number of distinct frequencies used; this is the order of the assignment;

the weighting of constraints so that more critical constraints contribute more to the total interference;

the sum of the positive discrepancies, c U- ~ - f j [;

a combination of these four.

If f = ( f l , f2 ..... fN) denotes a frequency assignment and d ( f ) denotes the total interference, the problem is to determine an f ~ A N for which d ( f ) is minimal or near-minimal. A typical practical example might involve between N = 200 and N = 700 links with about C = N2/8 non-trivial constraints and 50 available frequencies.

2.3. COMPLEXITY OF THE FREQUENCY ASSIGNMENT PROBLEM

In this section we show that the radio link assignment problem is NP-hard. We consider the simpler problem when all constraints are of the form

I -fj l >-cij

with 1 < i < j <N, and the value of Cij = 0 or 1. We map the assignment problem on to a graph G = ( V , E ) for which V=

{1 . . . . . N} and E is the set of edges ( i , j ) for which cij = 1. The assignment problem for the graph is to allocate frequencies fj, 1 < j < N, to the vertices of the graph so that adjacent vertices do not have the same frequency. This problem is clearly that of colouring the graph, G, with F colours and is known to be NP-hard [9].


Costa [4] has implemented an exact algorithm for determining the minimum span of any graph of reasonable size, by enumerating implicitly all possible colourings of the graph under consideration. Results were presented for this branch and bound approach for graphs up to 36 nodes. The method is unrealistic for large-sized problems.

3. Tabu search

In this section we outline the basic principles and components of the tabu search (TS) approach. For a detailed description of the method and its refinements, the reader is referred to Glover [13,14] and Taillard et al. [30].

Tabu search is basically an iterative procedure which starts from an initial solution in a search space that constitutes all solutions (i.e. assignments of frequencies to radio links), and moves step by step in the search space towards, hopefully, an optimal solution. At each step a neighbourhood, or set of moves, is defined which is applied to a given solution to produce a new one. Amongst all neighbouring solutions, TS seeks one with a best heuristic evaluation in the manner of a steepest descent algorithm. However, to discourage the possibility of being trapped in locally optimal but not globally optimal solutions and to prevent cycling (i.e. returning to solutions previously visited), a certain subset of moves in a neighbourhood are classified as forbidden or tabu, termed tabu moves. It is this feature that distinguishes TS from other descent methods. The tabu moves are determined by one or more tabu conditions and are based on the long term and short term history of the sequence of moves.

An important distinction in the TS method arises by distinguishing between the short term and long term memory functions. The short term memory function is a recency based memory, designed to keep track of the solution attributes that compose tabu restrictions that have changed during the recent past and to determine when these restrictions are applicable. The goal of the tabu restrictions is to prevent certain solutions from the recent past from being re-visited. The long term memory function is a frequency based memory function, whose goal is to prevent long term cycling and diversify the search. This is often achieved by penalising certain features that have been highly frequent in the history of the search process. The benefit from this is hopefully to create new solutions by avoiding those features commonly used in the past and perhaps encouraging those that have occurred less frequently. A study of various diversification strategies in TS methods has been given in [28].

A tabu restriction is enforced when the attributes underlying its definition satisfy certain thresholds of recency or frequency. However, choosing the parameters of recency and frequency where it is hopefully a robust and stable parameter typically depends on the problem characteristics and therefore is necessarily an experimental procedure.

Flexibility in the search process is important to overcome the rigidity of the tabu restrictions, that may lead to attractive solutions being overlooked. This is


achieved by means of aspiration criteria. The aspiration criteria are measures designed to overrule the tabu status if a move is deemed favourable and sufficient to prevent cycling. Therefore, both the tabu restrictions and aspiration criteria play a dual role in constraining and guiding the search process. Early applications employed a simple type of aspiration criteria consisting of removing a tabu classification from a trial move when the solution yields a solution better than the best found so far, though other suggestions have been illustrated (see Glover and Laguna [16]).

4. Implementation of tabu search procedure

In this section we describe how the tabu search procedure has been implemented to the frequency assignment problem. We explain the representation of assignments, the notions of neighbourhood and move and how recency and frequency were measured. We also describe how all neighbours of an assignment were evaluated efficiently to enable sufficient iterations in a reasonable time.

4.1. REPRESENTATION OF AN ASSIGNMENT AND NEIGHBOURHOOD

A frequency assignment f = ( f l ..... fN) is represented using an array of indexes Ix1 .. . . ,XN] where j) = axj for I < j < N. The neighbours of f are those assignments where the array of indexes differs in precisely one component. Thus, i f f ' is represented by [x~ ..... X~v], then f ' = ( f { . . . . . f;v) is a neighbour of f if there exists j, 1 < j < N, such that x~ ~x j , and for all i = 1 ..... N with i ~ j we have x/'= xi. Any assignment f has N ( F - 1) neighbours. Each neighbour corresponds to a pair ( j , x~) with 1 < j < N,

, ( s p 1 _ xj < F, (xj ~ xj). Costa [4] uses the same type of move generation; however, only a fixed sample of the neighbours (which do not lead to tabu moves) are examined at each iteration.

4.2. METHOD OF EVALUATION

One method to evaluate the total interference for all the neighbours o f f is to count the number of constraint violations for each neighbour in turn. This would require the investigation of C constraints for each of the N ( F - 1) neighbours and so take time O(CNF). A more efficient technique which we adopted was to initialise to zero a two-dimensional array b, indexed by link number and frequency index. For each constraint value cij > 0 the values x~ for which

lax; - axil = l f f - f j l >- cij

and the values x~ for which

I % - a x ; I = - f ; I >


were determined and the values b[i,x~] and b[j,x}] incremented. The final values in the array b represented the total interference of the neighbours off . The computation time required by this method was O(CF), approximately a factor of N/2 faster than before.

4,3 DEFINITION OF A TABU MOVE

A move to a neighbour (i, x~) corresponding to changing the assignment of link i to x~ was said to be tabu if it did not satisfy the recency or frequency condition. These conditions were determined by two numbers: a positive integer r and a real number s, where 0 < s < 1. The recency condition specifies that the link number i is not the link number of any of the previous r moves. The frequency condition specifies that the proportion of the number of times link i had been changed over all iterations did not exceed s. Thus, if at iteration j, link ij has been changed, then a move at iteration k + 1 is tabu if either (recency)

ik+~ = it, where k - r < t < k,

or (~equency) 1 k Z l>,.

j=l,ij =ik+ I

At each iteration, the method selects from the non-tabu neighbours that neighbour f(k+b off(k~ for which d ( f C~+l~) is minimal. Note that it is possible that d ( f (k+b) > d(f(k~); this allows escape from local minima.

Tabu search allows a tabu move to be selected when certain aspiration criteria are satisfied. In our implementation, a tabu move was selected if the neighbour f(k+l~ satisfied

d ( f (~+l)) < d ( f ' )

for all neighbours f " o f f (kl and

d ( f (~+t~) < d ( f (j~) for all j, 1 <j <_ k.

To summarize, ifftabu denotes the best tabu neighbour o f f (kl and fnon the best non- tabu neighbour, then the rule for determining f(kl is

i f d( ftabu ) < d( fnon)

d(.~ab.) < d( f ( j ))

then f(k +l) = ft~b.

else f(t+ll = fnon"

and

for 1 <_j<_k

Notice that there are some implicit relations between the parameters r and s which have to be satisfied. If r > N or s < 1/N, then, after N non-tabu moves, every move


is tabu. Also, after k iterations of non-tabu moves, it follows from the recency condition that a link can have changed at most [k / r 7 times. Hence, the frequency value, to have any effect, should satisfy

ks < ~k/r~.

We obtain, therefore, a relation between r and s which is

In practice, we choose 1 / N < s < 1/r.

s = &/r + (1 - A)/N

for some value of ~ with 0 < 2~ < 1.

The TS algorithm is terminated if zero interference has been obtained or if we reach a pre-determined maximum number of iterations. We then produce an assignment with the lowest measure of interference.

5. Local steepest descent implementation

A simple, but effective, local steepest descent heuristic is presented here to allow comparison with the genetic algorithm and tabu search implementations. The starting point to the procedure is generated by assigning each link a randomly chosen frequency from A = {a l, a z ..... aF}. The interference of this assignment is then calculated. A neighbouring assignment is generated by assigning the first link the frequency al (provided it was not initially assigned this frequency) and the interference calculated. If it is lower than the previous assignment it is kept and the next frequency a2 is tried (provided it was not assigned initially). This is continued until frequency aF has been assigned and tested. At this point the second link is assigned the frequency al (again provided it was not assigned this frequency initially), and the interference calculated and compared. This procedure continues until finally the last link is tried with the frequency aF (provided it was not assigned initially).

The whole procedure, starting with the first link, is started again. Termination of the algorithm occurs when no improvement is made in the interference level on two successive passes through all the links, i.e. a local minimum has been located. Formally it can be described as follows:

Step 1. Randomly assign each transmitter, t i ( i = 1 ..... N), a frequency channel, 3~, that is uni formly distr ibuted f rom the set of available f requencies A = {al,a2 . . . . . a F } . Call his assignment Z and calculate I, the interference in terms of the number of constraints violated.

Step 2. If I = O, stop, the current assignment is feasible. Otherwise, set i = 1 and set INTERFERENCE = I.


Step 3.

Step 4.

Step 5.

Step 6.

Set j = 1.

Ifj~ =aj go to step 5. Otherwise generate an assignment Z ' by assigning transmitter t i the frequency aj. Calculate I', if I ' < I replace Z with Z ' and set 1 =I'.

If I = 0, stop, the current assignment is feasible. I f j < F, set j = j + 1 and go to step 4.

I f i < N , s e t i = i + l and go to step 3. I f i = N a n d I<INTERFERENCE go to step 2. Otherwise, stop: the current assignment is locally optimal.

6. Genetic algorithm implementation

In this section we outline the genetic algorithm implementation used in the comparison section. Full details can be found in [5].

6.1. REPRESENTATION OF CHROMOSOMES

The length of each chromosome is equal to N, which corresponds to the number of links to be assigned a frequency. The value of each element in the chromosome is an integer corresponding to a legal frequency. Therefore, if we have a problem composed of five links and two available frequency channels (14, 28), the chromosome (14,28, 14, 14,28) indicates that links 1,3 and 4 are assigned frequency channel 14 and that links 2 and 5 are assigned frequency channel 28.

6.2. PARALLEL POPULATIONS

The genetic algorithm (GA) is implemented as cooperating sequential GAs (more recently described as island genetic algorithms). Each processor of a parallel computer runs a sequential GA and is responsible for its own, different, population of chromosomes. Occasionally, an exchange of chromosomes takes place between the various populations; for example, a small number of the fitter chromosomes in one population replace the least fit chromosomes in another population at set intervals. This allows for the migration and interaction of chromosomes from distinct populations.

6.3. GENETIC OPERATORS AND OPERATOR RATES

The single representation used in section 6.1 for the chromosomes allows the use of the standard one-point, two-point and uniform crossover operators. Mutation is at random, according to a prescribed rate; an element of a chromosome is replaced by a random, allowable, frequency.

The parameter settings used by the parallel genetic algorithm were as follows:

maximum number of generations 1000


number of parallel populations population size crossover probability mutation probability migration rate scaling elitism duplication avoidance niche formation

4 50 0.75 0.015 2% of population at intervals of 25 generations yes yes n o

n o

Details of the performance of an alternative chromosome representation, on a different set of test problems, can be found in [6].

7. Test problems and results

The code for tabu search to obtain sub-optimal frequency assignments has been run many times for six computer generated realistic examples. The number of links for the six problems were 252, 282, 410, 450, 490 and 726. The number of constraints were 7,059 (252 links), 10,430 (282), 22,346 (410), 25,721 (450), 36,024 (490) and 75,306 (726). In all runs the number of available frequencies was F = 50. Initial solutions were made by randomly assigning frequencies to transmitters.

We pursue several objectives with our experimental plan.

1. to identify the most suitable recency parameter;

2. to identify the most suitable frequency parameter;

3. to apply these parameters to other problem settings.

Extensive analysis was performed on two random test problems to determine the best parameters of recency and frequency. The first test problem consisted of 252 links and the second of 450 links, with each to be assigned frequencies, taken from a distinct set of 50 frequencies. The search space sizes were 10428, and 10765 respectively. In all the problems considered, the constraints consisted of co-channel and adjacent channel far-site constraints and a co-site frequency separation constraint of 10 channels. The four types of non-trivial constraint are shown below.

c1: I >o,

C2: ] f i - 3 ~ l > l ,

C3: I > 2 ,

C4: [ f i -3~l > 10,

where fi,j~ correspond to the frequencies assigned to links i and j.

312 D.J, Castelino et al,, An algorithm for frequency assignment

O3

>o <

45

40

35

30

25

20

I I I i

"graph1 .dat" - -

I I I

50 100 150 Recency (r)

Figure 1. N = 252 links problem with no frequency condition.

1

2OO

7C

co

t 0

<

195

190

185

180

175 120

I 1 I 4 I

"graph3.dat" - -

140 160 180 200 220 Recency (r)

Figure 2. N = 450 links problem with no frequency condition.

I

240 260


40

35

30

25

20 0

I I I ............. [

0.2 0.4 0.6 0.8 Frequency (lambda)

Figure 3. N= 252 links problem with r = 100 fixed.

i

"graph2.dat" - -

c

Y~

< /

o

o

II Z oo c

E

<

200

195

190

185

180

175

170

J

t

"graph4,dat" - -

I I I

0.2 0.4 0.6 Frequency (lambda)

Figure 4. N= 450 links problem with r = 180 fixed.

I

0.8

314 D.J. Castetino et al., An algorithm for frequency assignment

7.1. TUNING THE PARAMETERS

In the first set of tests, the frequency condition (s) was suppressed and ten runs

were made for varying values of r from 0 to N in steps of 10. For both problems the

best values for r were found to be at about 2N/5 but the precise value was not critical.

In the second set of tests, the value of r was fixed at about 2N/5 and ten runs were made for varying values of 2, between 0 and 1. For both problems, the best values

for 7t. were found to be at about 0.5 but, again, the precise value was not critical in

the range between 0.3 and 0.7. Figures 1,2,3 and 4 give a diagrammatic view of some of the runs used in the

testing stage to determine good values for the parameters r and s. Figures 1 and 2 plot the average best interference from ten runs as the recency varies, for the 252

and 450 link problems respectively. Figures 3 and 4 plot the average best interference

from ten runs as t varies with r = 100 fixed for the 252 links problem and with r = 180 fixed for the 450 links problem.

Av. I.

45-

40

30

25

22° 0 0 o 160 ~

140 "~ . /'0.2 120

80 ~ / 0 . 6 , , , 60 ~ / ~ '~ lambda

40 20 0.8 1

Figure 5. 2D (r- / l ) plot, with contour map, for the 252 links problem.

D.J. CasteIino et al., An algorithm for frequency assignment 315

Figure 5 is a 2D ( r - &) plot of the average best interference found, from 10 runs (10,000 iterations each), for varying values of r and ~, for the 252 links problem. We see that it confirms the choice of recency and frequency values used for this problem instance.

7.2. E X P E R I M E N T A L EVIDENCE

Figure 6 shows how one run of 50,000 iterations for the N = 282 links problem progressed by plotting the best interference obtained against the iteration number. It can be seen that, while there is still improvement after a large number of iterations, the benefit from continuing has diminished.

150

145

140

135

< 130

125

120 ~ E u 0 5000 10000 15000

"graph5.dat" - -

\ \

I t

20000 25000 30000 Iterat ion Number

I I I

35000 40000 45000 50000

Figure 6. N = 282 links problem with r = 112, & = 0.5.

Table 1 gives some sample runs, with different random initial assignments, for the 252 and 450 link problems, with A, set at 0.5 for both problems and r = 100 and 180 respectively. We see that the TS procedure quickly reduces the number of constraint violations and produces fairly uniform final levels of interference with standard deviations of 1.58 and 3.39 respectively.

In twelve runs, each of 4,000 iterations, the best interference found was 136 and the worst 150, with a percentage reduction in initial and starting interference of 82% in both cases. Comparing this with the single run of 50,000 iterations, which


Table 1

Sample runs with ~.= 0.5 and 10,000 iterations giving percentage deviation of initial and final interference

for two problems of size N = 252 and N= 450.

Run Initial Final Percentage

interference interference interference 252 450 252 450 252 450

1 367 1309 27 181 92.6 86.1 2 408 1389 25 186 93 .8 86.6 3 409 1346 26 I83 93 .6 86.4 4 417 1345 28 185 93 .2 86.2 5 392 1302 24 190 93 .8 85.4

produced a final interference of 122 and a percentage reduction in interference of 85%, we see that good solutions can be found relatively quickly but a small amount of improvement (I 9%) in the final level of interference is obtained by continuing for 50,000 iterations.

7.3. TIME ANALYSIS

The major part of the work involved in each iteration of tabu search in our application was in the computation of the total interference of each of the N ( F - 1) neighbours of the current assignment. As indicated in section 4.2, this can be accom- plished in time which is O(CF) = O(NZF) for the practical problems we are considering here. This is considerably more efficient than the O(N3F) time that the naive evaluation gives.

Precisely how many iterations are required depends crucially on N ,F and the threshold of interference that is acceptable. From section 7.2 we see that tabu search obtains quality solutions quickly, but that a large number of iterations are necessary to cut these by say 20%. For example, in figure 6, a value of 150 for the interference was obtained after about 4,000 iterations but it required 50,000 iterations to obtain a value of 122. For the program, written in C and run on a SPARCServer 10/51 workstation (27.3 Mflops), this represents about 20 minutes for the former and 4 hours for the latter.

7.4. COMPARISON WITH OTHER TECHNIQUES

Since the problem is NP-complete, it is impossible to obtain an exact value for the minimum interference for the large practical problems we have considered. Comparisons, therefore, can only be made with other heuristic algorithms. To allow


comparison, we used the genetic algorithm results as a benchmark. In particular, we tested 200,000 assignments for each test problem and noted the wall clock time for the length of each run. The local steepest descent and tabu search procedures were then allowed the same time as that used by the genetic algorithm. The times allowed were 100 minutes for the 282 links problem, and 153 minutes, 176 minutes, 222 minutes and 475 minutes for the 410, 450, 490 and 726 test problems respectively. As the LSD procedure terminates when no improvement in interference is found on two successive passes through all links, multiple runs (with different random starting assignments) were performed until the required wall clock time had elapsed. Results, presented in table 2, show that TS compares very favourably with the results obtained for both the GA and the LSD procedure with respect to the best and average number of constraint violations found. Although each method is given an equal amount of time per test problem, the time per assignment is much lower for the LSD and TS methods as compared to the GA. This is mainly because the TS and LSD implementations can use more efficient function evaluation procedures than is used in the GA implementation.

Table 2

TS, LSD and GA comparison for test problems (10 runs each).

Best Average Standard Average number of N interference interference deviation assignments tested

TS GA LSD TS GA LSD TS GA LSD TS GA LSD

282 128 501 159 134.0 506.9 165.2 3.71 8.02 3.01 1.381 × 108 2 x 105 1.259 x 107

410 379 1160 431 400.8 I173~0 444.5 ]0.86 15.50 7.34 1,536 × 108 2 × 105 2.252 × 107

450 176 845 203 188.0 849.0 208.4 6.50 4.58 4.33 1.293 × 108 2 x 105 2.421 × 107

490 658 1828 726 67&9 1834.4 737.2 16.15 8.08 7.10 1.332 x 108 2 x l0 s 2~511 x 107

726 I535 3947 1660 1602.8 3960.8 1704.5 31.33 24.43 24.07 2.112 x t08 2 x 105 3.688 x 107

8. Concluding remarks

We have described an application of tabu search to the frequency assignment problem and presented a synopsis of the test results. These results indicate that good values of the parameters for recency and frequency have been found and that the technique is robust and stable providing uniformly good solutions in a reasonable time phase. While it is difficult to assess the quality of the solutions for practically sized computationally hard problems, there is, nonetheless, some evidence to believe that the method competes favourably with other heuristic methods such as genetic algorithms and local steepest descent search.


Acknowledgements

We would like to thank Steve Bracking, Ray Bradbeer and Major Rick Bar ford

o f the De fence Research Agency at Malvern and Dr. David Tamar ind and Capta in

Mike Griff i ths o f the A r m y E M C Agency at Blandford for d iscuss ions which enabled

us to genera te realistic test data for C o m b a t Net Radio ass ignment scenar ios which

fo rmed the basis o f the data sets used in this paper.

References

[1 ] H. Beltran and D. Skorin-Kapov, On minimum cost isolated failure immune networks, Telecommun. Syst.(1994), to appear.

[2] J. Chakrapani and J. Skorin-Kapov, Massively parallel tabu search for the quadratic assignment problem, Ann. Oper. Res. 41(1993)207-230.

[3] K. Chiba, E Takahata and M. Nohara, Theory and performance of frequency assignment schemes for carriers with different bandwidths under demand assignment SCPC/FDMA operation, IEICE Trans. Commun. E75-B(1992)476-486.

[4] D. Costa, On the use of some known methods for t-colourings of graphs, Ann. Oper. Res. 41(1993)343-358.

[5] W. Crompton, S. Hurley and NM. Stephens, Frequency assignment using a parallel genetic algorithm, Proc. IEE/IEEE Natural Algorithms in Signal Processing Workshop, vol. 2 (1993) pp. 26/1 - 26/8.

[6] W. Crompton, S. Hurley and NM. Stephens, A parallel genetic algorithm for frequency assignment problems, Proc. IMACS/IEEE Conf. on Signal Processing, Robotics and Neural Networks, Lille, France (1994) pp. 81-84.

[7] R.L. Daniels and J.B. Mazzola, A tabu-search heuristic for the flexible resource flow shop scheduling problem, Ann. Oper. Res. 41(1993)207-230.

[8] M. Duque-Anton, D. Kunz and B. Rtiber, Channel assignment for cellular radio using simulated annealing, IEEE Trans. Vehicular Technol. 42(1993)647-656.

[9] M.R. Garey and D.S. Johnson, Computers and Intractability, A Guide to the Theory of NP- Completeness (Freeman, 1979).

[ 10] E Glover, Heuristics for integer programming using surrogate constraints, Dec. Sci. 8(1977)156-166. [11] F. Glover, Future paths for integer programming and links to artificial intelligence, Comp. Opel-.

Res. 13(1986)543-549. [ 12] F. Glover and C. McMillan, General employee scheduling problem: An integration of management

science and artificial intelligence, Comp. Oper. Res. 15(1986)563-573. [13] E Glover, Tabu search - Part 1, ORSA J. Comp. 1(1989)190-206. [I4] F. Glover, Tabu search - Part 2, ORSA J. Comp. 1(1990)4-32. [15] F. Glover, Least cost network design for a new service, Ann. Oper. Res. 33(1992) 351-362. [I6] F. Glover and M. Laguna, Tabu search, in: Modern Heuristic Techniques for Combinatorial

Problems, ed. C.R. Reeves (Blackwell Scientific, 1993). [17] W.K. Hale, Frequency assignment: theory and applications, Proc. IEEE 68(1980)1497-1514. [18] R. Mathar and J. Mattfeldt, Channel assignment in cellular radio networks, IEEE Trans. Vehicular

Technol. 42(1993)647-656. [19] B.H. Metzger, Spectrum management technique, presented at 38th National ORSA Meeting, Detroit,

MI (1970). [20] E.L. Mooney and R.L. Rardin, Tabu search for a class of scheduling problems, Ann. Oper. Res.

41 (1993)253-278. [21] I.H. Osman, Metastrategy simulated annealing and tabu search algorithms for vehicle routing

problems, Ann. Oper. Res. 41(1993)421-451.

D.J. Castelino et at., An algorithm f or frequency assignment 319

[22] I.H. Osman and N. Christofides, Capacitated clustering by hybrid simulated annealing and tabu search, Int. Trans. Oper. Res. 1(1994)317-336.

[23] IH. Osman, Heuristics for the generalised assignment problem: simulated annealing and tabu search approaches, OR-Spektrum (1994), to appear.

[24] A. Raychaudhuri, Optimal multiple interval assignments in frequency assignment and traffic phasing, Discr. AppI. Math. 40(1992)319-332.

[25] F. Semet and E. Taillard, Solving real-life vehicle routing problems efficiently using tabu search, Ann. Oper. Res. 41(t993)469-488.

[26] J. Skorin-Kapov, Tabu search applied to the quadratic assignment problem, ORSA J. Comp. 2(1990)33-45.

[27] D. Skorin-Kapov and J. Skorin-Kapov, On tabu search for the location of interacting hub facilities, Euro. J. Oper. Res. (1992).

[28] E Soriano and M. Gendreau, Diversification strategies in tabu search algorithms for the maximum clique problem, Centre de Recherche sur les Transports, Publication 940, Universit6 de Montr6al (1993).

[29] E. Taillard, Robust taboo search for the quadratic assignment problem, Parallel Comp. 17(1991)443-455.

[30] E. Taillard, F. Glover and D. de Werra, A user's guide to tabu search, Ann. Oper. Res. 41(1993)3-28.

[31] SR. Thangiah, I.H. Osman, R. Vinayagamoorthy and T. Sun, Algorithms for vehicle routing problems with time deadlines, Amer. J. Math. Manag. Sci. 13(1993)323-355.

[32] D. de Werra and Y. Gay, Chromatic scheduling and frequency assignment, Discr. Appl. Math. 49(1994) 165-174.

[33] J. Wesley Barnes, E Glover and M. Laguna, Tabu search methods for a single machine scheduling problem, J. Int. Man. 2(1991)63-74.

a tabu search algorithm for frequency assignment

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