A new adaptive genetic algorithm for fixed channel assignment

Information Sciences 177 (2007) 2655–2678 www.elsevier.com/locate/ins

A new adaptive genetic algorithm for fixed channel assignment L.M. San Jose´-Revuelta

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Dept. de Teorı´a de la Sen˜al y Comunicaciones e I.T., E.T.S.I. Telecomunicacio´n, Universidad de Valladolid, 47011 Valladolid, Spain Received 2 June 2005; received in revised form 28 December 2006; accepted 3 January 2007

Abstract This paper presents a new genetic algorithm (GA) with good convergence properties and a remarkable low computational load. Such features are achieved by on-line tuning up the probabilities of mutation and crossover on the basis of the analysis of the individuals’ fitness entropy. This way, a brand new method to control and adjust the population diversity is obtained. The resulting GA attains quality solutions, thus offering an interesting alternative to other global search techniques, such as simulated annealing, Tabu search and neural networks, as well as to standard GAs. The new algorithm is applied to solve the problem of frequency reuse in mobile cellular communication systems, where the main aim is to obtain a conflict-free channel assignment among the cells such that the resulting bandwidth is close to the minimum channel span required for the whole network. The algorithm performance has been tested and compared by making use of a selection of the most well-known benchmark instances; optimal bandwidth solutions have been achieved within a reasonable computation time.  2007 Elsevier Inc. All rights reserved. Keywords: Micro genetic algorithm; Channel allocation; Diversity control; EMC

1. Introduction During the last years, the technological advances and the fast development of mobile communication terminals have assisted the progress of both wireless communications and mobile computing. Intensive research has been undertaken to develop sophisticated systems with increased network capacity and performance. Several tasks, such as resource allocation, design, planning, estimation and equalization, must be faced up. Most of these problems are characterized by search spaces whose complexity increases exponentially with the size of the input, being, therefore, intractable for solutions using analytical approaches or simple deterministic algorithms [23,26]. Genetic algorithms (GAs) are inspired by the principle of natural selection and survival of the fittest, and constitute an alternative method for finding solutions to highly-nonlinear problems, characterized by a multimodal solutions space. GAs efficiently combine the exploration/exploitation sense of the search so as to

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Tel.: +34 983423660; fax: +34 983423667. E-mail address: [email protected]

0020-0255/$ - see front matter  2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.01.003

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avoid getting trapped into suboptimal local minima. Unlike the steepest descent approaches, GAs require no gradient calculation and are much less susceptible to local optima, since global search techniques avoid such traps by providing the ability to selectively accept successive points even if they have a higher cost than the current solution [30]. This paper presents a low complexity GA whose internal parameters are in-service adapted in order to control the diversity of the population (DGlGA, diversity-guided micro-genetic algorithm). The diversity of the population is monitored by analyzing the Shannon entropy of the population individuals’ fitness. This diversity is closely related to the performance of evolutionary algorithms, though, traditionally, diversity measures have only been used to analyze algorithms rather than guide them. For instance, the diversity-control-oriented GA [36] calculates a survival probability by means of a diversity measure based on the Hamming distance between the individual and the current best individual. Therefore, diversity is preserved through the selection procedure. Another approach is the shifting-balance GA [34], where a containment factor between two subpopulations – based on Hamming distances between all the members of the two populations – determines the ratio between the individuals selected according to their fitness and the individuals selected to increase the distance between the two populations. A third approach is the Forking GA [42], which uses specialized diversity measures to turn a subset of the population into a subpopulation [43]. Finally, in [43], R.K. Ursem makes use of an algorithm that applies diversity-decreasing operators (selection and recombination) or diversity-increasing operators (mutation), depending on the mean diversity of the population. Once the DGlGA is described, the paper focuses on the application of this algorithm to solve the channel allocation problem (CAP) found in cellular radio systems. In this kind of systems, the frequency reuse by which the same frequencies/channels are reused in different cells becomes a crucial concept [13]. Each such location or cell is allocated a set of channels according to the expected traffic demand in that cell. The entire spectrum is allocated to a cluster of cells arranged in shapes that allow for uniform reuse patterns. The channels must be located in such a way as to satisfy certain frequency separation constraints to avoid channel interference using as small a bandwidth as possible. Thus, the CAP fits into the category of multimodal and NP-complete problems [11,16,20,23]. The economic consequences of this issue are evident. From the cost-of-service point of view, the number of required base stations to provide service to a given geographical area is a very important factor, and a reduction in the number of base stations can be achieved by an efficient reuse of the radio spectrum. The fixed CAP has been thoroughly studied during the past decades. A comprehensive summary of the work done before 1980 can be found in [16]. When only the co-channel constraint is considered, the CAP is equivalent to an NP-complete graph coloring problem [38]. In this case, various graph-theoretic approaches have been proposed [4,16,21,38]. Recently, specific assignment algorithms have been developed for use with the high-rate and variable-rate multimedia data transmissions that are peculiar to the third and fourth generation (3G/4G) communications systems [35,37]. From the point of view of evolutionary computation, some procedures based on neural networks (NNs) [8,18,24,27] and simulated annealing (SA) [7,22,29] have been equally considered. The simulated annealing (SA) approach solves the drawback of easy convergence to local optima found in NNs, though its rate of convergence is rather slow, and a carefully designed cooling schedule is required. A comparison between SA and the Tabu search (TS) methods shows that the TS algorithm is not only capable of matching, even outperforming SA, in locating the minimal number of frequencies for channel allocation, but it also constitutes a faster procedure [17]. In the 90’s, several GA-based approaches have been applied to solve the CAP: For instance, in [6], Cuppini defined and used an asexual crossover and a special mutation. A disadvantage of such crossover is that it can easily destroy the structure of the current solution and, thus, makes the algorithm harder to converge. On the other hand, in [25], Lai and Coghill represented the channel assignment solution as a string of channel numbers (instead of a binary string) that are grouped in such a way that each cell has a specified number of channels and, hence, the traffic requirement is satisfied. The evolution is then proceeded via a partially matched crossover operator (PMX) – this type of crossover has also been used in [12] – and basic mutation. Two years later, Ngo and Li [31] suggested a GA that used the so-called minimum separation encoding scheme, where the number of 1’s in each row of the binary assignment matrix corresponds to the number of channels allocated to the corresponding cell. To satisfy the demand requirement this would normally be constant. Each chromo-

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some is a binary string that represents the assignment matrix through a concatenation of its rows, and genetic operators are defined so as to preserve the number of 1’s in each chromosome. Authors stated that this algorithm outperforms the NN-based approach described in [8]. In this paper a new DGlGA is developed and applied to solve the CAP. It offers an extremely low computational load and provides a good quality solution (optimal, minimum-span, solution) while maintaining good convergence properties. The probabilities of mutation and crossover of the GA are on-line adjusted by making use of an individuals’ fitness dispersion measure based on the Shannon entropy. This way, the diversity of the population is monitored and controlled at every iteration of the algorithm. The remainder of this paper is organized as follows: First, Section 2 explains the main concepts relating to the DGlGA, with special emphasis on the differences that have been introduced with respect to the standard GA, in both the genetic operators and the population structure. The notion of entropy-guided genetic operators is also introduced. Next, Section 3 illustrates the problem of the entropy ambiguity and describes a simple method for its control. Section 4 describes the channel assignment problem introducing the different interference constraints and the channel demand concepts. Finally, Section 5 outlines the numerical simulation results. Notice that the references section at the end of the paper contains a quite large list of references, since the work presented in this paper deals with several different disciplines, such as genetic algorithms, diversity control and channel assignment techniques. 2. Diversity-guided l-genetic algorithm 2.1. Fundamentals, encoding and objective function The principle of GAs consists in representing a set of potential solutions (population) with a predetermined encoding rule. At every iteration k, each potential solution (chromosome) is associated to a figure of merit, or fitness value, ki(k), in accordance to its proximity to the optimal solution, i.e., each chromosome is evaluated for its fitness in solving a given optimization task. Considering the CAP, the goal is to avoid violating any of the adjacent-channel, co-channel and co-site constraints, while satisfying the required number of channels needed by each cell and minimizing the required bandwidth.1 When no prior knowledge of the solution is available, the initial set of potential solutions, P½0 (population np at k ¼ 0), is randomly generated.2 Let us denote P½k ¼ fui gi¼1 to the population at iteration k, with np being the number of individuals ui per generation. In our problem, ui consists of a string structure containing all the channels required for each base station (see Fig. 1). In this way, each string represents a particular assignment for all the base stations. By assigning the number of elements in each string so as to satisfy the required number of channels for each base station (or cell), we can effectively ignore such computations in the objective function. Notice that the length of each string in the population would essentially be the same and would not change during the assignment. The determination of the population size np is an important point to be addressed. In [25], Lai and Coghill suggest that a reasonable choice should be in the range 30–110 in order to have a large variability within the population. Since we propose a l-GA, the population size would be much smaller, in the range 10–20. 2.2. Genetic operators Once individuals are generated and given a fitness value, the next step consists in applying the genetic operators, mainly mutation and crossover, to those individuals selected using the fitness-based selection algorithm shown in Fig. 2. This selection procedure is based on a biased random selection, where the fittest individuals have a higher probability of being selectedPthan their weaker counterparts. This way, the probability of any np individual to be selected is P i ðkÞ ¼ ki ðkÞ= j¼1 kj ðkÞ, with ki(k) being the fitness of the ith individual in the

1 2

These concepts are described in Section 4. In our specific CAP, some a priori information is available and will help to generate the initial population (see [25] and Section 5).

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Fig. 1. Schematic representation of the chromosome or string structure for the channel assignment problem.

Fig. 2. Schematic representation of the global DGlGA structure.

population during the kth iteration of the algorithm. The simplest way to implement this concept is based on a roulette wheel, where the size of each slot in the wheel is proportional to the individual’s fitness [30]. Thereafter, the mutation operator modifies specific individuals with probability pm, changing the value of some concrete positions in the encoding of ui. The number of positions within the encoding of the channels of each cell that are candidates for possible mutation is half the number of channels present in that base station. Both the specific positions and the new values are randomly generated, and mutation is effectively performed with probability pm. Notice that this genetic operator promotes the exploration of different areas of the solutions space. A low level of mutation serves to prevent any element in the chromosome from remaining fixed to a single value in the entire population. However, a high level of mutation will essentially result in a random search. Hence, the value of pm must be chosen carefully in order to avoid excessive mutation. To maintain a good balance between such extremes a good initial value for pm is 0.01–0.05 [15,25]. In our application, pm ð0Þ ¼ 0:05. On the other hand, the crossover operator requires two operands (parents) to produce two new individuals (offspring). These new individuals are created when merging parents by crossing them at specific internal points. This operation is performed with probability pc. Since parents are selected from those individuals having a higher fitness, the small variations introduced within these individuals are intended to also generate high fit individuals. Simulation results showed that with the simple crossover operator, a significant number of the generated configurations has the same frequency assigned to a group of base stations that interfere with each other. To alleviate this problem we have used the partially matched crossover operator as proposed in [25]. This operator partitions each string into three randomly chosen portions. When this operator encounters that the same frequency has been assigned more than once, it solves this conflict by rearranging the conflicting elements in each string (see [25] for a detailed description). The range for pc in the DGlGA when applied to the CAP is [0.35, 0.85].

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2.3. Elitism and termination criteria The proposed DGlGA also implements an elitism strategy, where the elite Ek for P½k þ 1 is formed by selecting those individuals from both the elite of P½k and the mutated elite of P½k having the highest objective value in the population. The mutation of the elite is performed with a probability pm;e considerably smaller than pm – so as to avoid the destruction of good solution guesses. In our simulations we used pm;e ¼ xpm , with x ¼ 0:25 (heuristical trials showed little differences for values of x in the range [0.1, 0.5]). No crossover is performed on the elite since it implies, in most cases, abandoning the exploitation of these good potential solutions. This procedure, whose flowchart is outlined in Fig. 2, is iterated until a termination criterion is satisfied. For instance, the search can be terminated when a predetermined number of iterations (ng generations) has been processed or, using a different criterion, when there are no significant changes between the maximum and minimum values of the objective function in any two successive generations, i.e. TERMINATION CRITERIA = TRUE, IF { ½1  efmaxðk in P½kÞ  minðk in P½kÞg 6 maxðk in P½k þ 1Þ  minðk in P½k þ 1Þ 6 ½1 þ efmaxðk in P½kÞ  minðk in P½kÞg } where the parameter e determines how significant the changes must be between two successive iterations in order to finish the GA. Normally, e 2 ½0:01; 0:05. In all our simulations e ¼ 0:02 has been used. At the end, the string ui corresponding to the highest fit chromosome is finally chosen as the channel allocation problem solution. This elitist model of the genetic algorithm presents some convergence advantages over the standard GA. Using Markov chain modelling, it has been proved that GAs are guaranteed to asymptotically converge to the global optimum – with any choice of the initial population – if an elitist strategy is used, where at least the best chromosome at each generation is always maintained in the population [3]. However, Bhandari et al. [3] provided the proof that no finite stopping time can guarantee the optimal solution, though, in practice, the GA process must terminate after a finite number of iterations with a high probability that the process has achieved the global optimal solution. Note that, in our problem, the optimal string is not necessarily unique. There may be many strings that provide the optimal value. It has been shown that if the number of chromosomes having the optimal value is larger, then the probability of fast convergence becomes higher [3]. 2.4. Diversity and convergence in GAs As mentioned in Section 1, the novelty of the DGlGA comes from its very low computational load, and the introduction of an on-line procedure to adjust the parameters in order to achieve and maintain a good population diversity which leads to notable convergence properties. This diversity is a crucial issue in the performance of any evolutionary algorithm, including GAs: standard GAs have a tendency to converge prematurely to local optima, mainly due to selection pressure and too high gene flow between population members [43]. A high selection pressure will fill the population with clones of the fittest individuals, since they have the highest survival probability. Diversity declines after a short while, and, because the population consists of similar individuals, it may result in convergence to local minima. On the other hand, high gene flow is often determined by the population structure. In simple GAs any individual can mate with any other individual. Therefore, genes spread fast throughout the population, and diversity quickly drops. These facts point out the key role of maintaining a suitable diversity in the population in order to appropriately converge to the optimal solution, thus avoiding the presence of very similar individuals, as well as getting trapped into local minima. Two extremely simple strategies to deal with diversity have been proposed in [25,12]. In the first paper, Lai proposes to increase the mutation by a factor of two when the genetic material of both parents is the same in

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order to encourage greater diversity in the population as well as the propagation of good assignments. On the other hand, in [12], Ghosh et al. vary the mutation probability in n equally-length intervals along the ng iterations of the GA, while the probability of crossover is maintained constant (see Fig. 3 in [12, p. 863], for an example). The authors remark that this scheme increases the diversity of the population when the probability of mutation is increased and, conversely, as the optimal string is approached, the probability of mutation is reduced. However, though interesting as a first approach to diversity control, these two schemes are too simple and do not analyze the real diversity of the population at every iteration of the GA in order to proceed consistently. 2.5. Entropy-dependent genetic operators and convergence cycle The standard GA suffers from an excessive computational load: the application of the genetic operators is often costly and the evaluation of fitness can also be a very time-consuming task. Population sizes np normally are 100, 200 or even much higher (for instance, [43] uses np ¼ 400 individuals). In this paper, we propose a DGlGA that works with much smaller population sizes, in the order of 10–20 individuals. An elite of three individuals is kept and the crossover and mutation probabilities depend on the Shannon entropy of the population’s (excluding the elite) fitness, which is calculated as HðP½kÞ ¼ 

np X

ki ðkÞ log ki ðkÞ

i¼1

ð1Þ

Pnp ki ðkÞ. When all the fitness valwith ki ðkÞ being the normalized fitness of individual ui, i.e., ki ðkÞ ¼ ki ðkÞ= i¼1 ues are very similar, with small dispersion, HðP½kÞ becomes high and pc is decreased – it is not worthwhile wasting time merging very similar individuals. This way, the exploration character of the DGlGA is boosted, while, conversely, exploitation decreases. On the other hand, when this entropy is small, there exists a high diversity within the population, a fact that can be exploited in order to increase the horizontal sense of search. Following a similar reasoning, the probability of mutation is increased when the entropy is high, so as to augment the diversity of the population and escape from local suboptimal solutions (exploitation decreases, exploration becomes higher). Therefore, we have that probabilities pm and pc are directly/inversely proportional to the population fitness entropy, respectively. Some exponentially dependence on time k must also be included in the model – making use of exponential functions – in order to relax (decrease), along time, the degree of dependence of the genetic operators’ probabilities with the dispersion measure. This avoids abandoning good solution estimates when very low fitness individuals are sporadically created, specially when most of the population’s individuals have converged to the global optimum. A typical estimation convergence process will involve – supposing the GA converges properly – the following three different phases (see Fig. 3):

Fig. 3. Typical convergence cycle of the estimation DGlGA.

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• Phase I: Population P½0 has been just randomly generated. Individuals ui share a very few bits in common. Their fitness values are very similar since all of them are, probably, wrong solution estimates and, thus, the entropy is high. The probability of mutation is kept high so as to explore new potential solutions while the probability of crossover is small. • Phase II: Some individuals begin to converge to good solution estimates. Their fitness increases while the remaining individuals of the population still present low values. The entropy of the fitness considerably decreases. The probability of mutation is decreased gradually while the one associated to crossover becomes higher so as to exploit the genetic information of the good estimates. • Phase III: Most of the individuals in the population have converged to good solutions. The fitness of all the individuals is very close and all of them tend to be equiprobable solutions. The entropy is again high and the exploitative behavior prevails over the explorative one. As an example, since concrete values greatly depend on the randomly generated initial population, the approximate range in which pc and pm evolve along the execution of a satisfactory-convergence typical cycle is: pm ðIÞ ¼ 0:05 ! pm ðIIIÞ  0:001 and pc ðIÞ ¼ 0:35 ! pc ðIIIÞ  0:85. Keeping these three different phases in mind, the determination of the form of functions pc(k) and pm(k) is achieved as follows: • Probability of mutation: during Phase I, where most of the individuals are wrong estimates and the entropy is high, pm is required to be high so as to increase the explorative sense of the search. Thus, pm is directly proportional to the entropy HðP½kÞ and to another function gm ðkÞ ¼ expðbm kÞ, that, for low values of k and bm < 1, is close to 1. Thus, we can write pm ðkÞ ¼ nm ðkÞ  gm ðkÞ  HðP½kÞ

ð2Þ

where nm(k) stands for a normalization parameter, which ensures that pm 2 ½0; 1. Once the GA has converged to some good solution estimates, HðP½kÞ becomes high again (see Phase III in Fig. 3(a)), but now pm must be lowered so as to switch from exploration to the exploitation of new individuals. This is achieved selecting gm such that its decreasing character prevails over the entropy values. In our particular system, exponential functions gm(k) with bm 2 ½0:1; 0:3 were used. • Probability of crossover: in this case, pc should maintain low values in Phase I in order to allow an explorative search. During this phase HðP½kÞ is high and pc is chosen to be inversely proportional to the entropy HðP½kÞ: pc ðkÞ ¼

nc ðkÞ  gc ðkÞ HðP½kÞ

ð3Þ

where function gc(k) must be defined in order to guarantee that: (i) during Phase I, the product of gc(k) and the inverse of HðP½kÞ adopts low values (see Phase I in Fig. 3(a)), and (ii) in the third phase, the value of gc(k) must prevail over the low values of the inverse of the entropy, leading to high probabilities, in accordance to the desired exploitative sense of the search. A function gc(k) that achieves this goal is:  ak þ b in Phase I gc ðkÞ ¼ ð4Þ log10 ðk=k 0 Þ in Phases II and III where k 0 ¼ 4, a ¼ 0:0263 and b ¼ 0:3237 (these values are easily determined based on pc(0) and pc ðng Þ, with ng being the total number of generations). In both cases, probabilities must be properly normalized in the range ½0; 1 making use of the parameters nm(k) and nc(k). The weight of this updating of both probabilities also decreases with time, so that when Phase II is widely surpassed, the mutation and crossover probabilities remain mainly constant. In short, the resulting complexity of the DGlGA is notably decreased since crossover is applied with a very low probability (and only on individuals not belonging to the elite), and the diversity control allows the algorithm to work properly with a much smaller population size.

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3. Eliminating the entropy ambiguity The use of the entropy function as defined in Eq. (1) with the aim of classifying populations can lead, if not properly corrected, to ambiguous situations since a population with all its individuals being very similar and with high fitness values will show the same high entropy as another population with very similar individuals taking on, all of them, very low fitness values. Obviously, the dispersion in both cases is low, and the associated entropy will be high. In order to detect and differentiate these situations, the following simple strategy has been devised. Let us define the function, " # k length 1 np X 1 X 1 Hp ðk last ; k length Þ ¼ ki ðk last  jÞ ð5Þ k length j¼0 np i¼1 which represents the averaged mean value of the individuals’ fitness in klength successive generations, with the last one being klast. i.e. the mean value in the iterations ½k last  ðk length  1Þ; k last  ðk length  2Þ; . . . ; k last  1; k last . Once f þ a  1 generations have been processed, Hp ðk; aÞ and Hp ðk  f; aÞ are compared. The first one represents the averaged mean value of the individuals’ fitness in a successive generations – the current one, k, and the a  1 generations before – while the second parameter represents this same value but averaging the mean fitness values in a group of a iterations, f generations before. The simulations in this paper have been carried out with a ¼ 5 and f ¼ 15, thus the mean value of the fitness values averaged over the last 5 successive iterations is compared to this value 15 generations before. For instance, at generation k ¼ 31, the algorithm compares the averaged value of the mean fitness in generations 27, 28, 29, 30 and 31, with the averaged value of the fitness in generations 12, 13, 14, 15 and 16. In cases of good convergence (or, at least, when the DGlGA has begun to converge), the difference between Hp ðk; aÞ and Hp ðk  f; aÞ will be higher than a predetermined threshold Hthreshold (heuristically determined). On the other hand, when the population contains low fitness individuals after a þ f generations, it is a clear indicator of unsatisfactory convergence. In this case, two solutions have been tested: (i) to keep the probability of mutation with a high value and the probability of crossover close to zero, and (ii) to randomly re-initialize the whole population. Simulations show that the first approach was not able to re-conduct the search since very high mutation in most of the genes should be required and the number of remaining generations is too low to change considerably the whole population. Thereby, the DGlGA has been implemented with scheme (ii). According to this, the difference between both averaged values is compared to a threshold Hthreshold , and whenever condition Hp ðk; aÞ  Hp ðk  f; aÞ < Hthreshold

ð6Þ

is satisfied, the whole population will be randomly re-initialized. The values of the parameters a and f can be selected depending on the number of available channels, n, and the number of cells, c. An heuristically determined rule is to adjust them such that a þ f  ng =2 so as to allow a second estimation cycle if convergence was detected to be wrong. Obviously, this strategy is only applied when k P a þ f. The global convergence of this particular strategy has been numerically demonstrated with satisfactory results in the application proposed (CAP). A theoretical analysis and discussion lies beyond the scope of this paper and will be the object of future work. It is important to note that the proposed diversity control mechanism can be performed with a small computation increase. However, in the case of fixed channel allocation, the on-line implementation is of minor importance, whereas simulations show that on-line detection is perfectly feasible in the application, for instance, of the DGlGA to CDMA multiuser detection (one of the author’s current research lines). 4. Application: the channel assignment problem As mentioned in Section 1, frequency reuse is a key issue in current mobile communication systems. It is well known that the co-channel interference caused by frequency reuse is the most restraining factor on the

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overall system capacity in wireless networks. The main purpose is the simultaneous use of a given radio spectrum while maintaining an acceptable received radio signal. Specifically, each hexagonal cell is assigned a set of channels according to the expected traffic demand. This assignment of channels must satisfy the following EMC constraints: (1) Co-channel constraint (CCC): the same channel cannot be simultaneously assigned to certain pairs of cells. The co-channel reuse distance is defined as the minimum distance at which co-channels can be reused with acceptable interference [20]. (2) Co-site constraint (CSC): adjacent channels in the same cell must be separated by some minimum spectral distance. (3) Adjacent channel constraint (ACC): adjacent channels cannot be assigned to adjacent cells, i.e., any pair of channels in different cells must have a specified distance [9]. Besides these considerations, the channel assignment algorithm must also take into account the specified traffic profile (number of channels) required in each cell. These non-uniform cell demand requirements imply that those cells with a higher traffic demand will need the assignment of more channels. Channel assignment schemes can be classified according to different comparison criteria. For example, when the algorithms are compared based on the method used to assign channels, they can be essentially divided into two kinds of schemes: fixed and dynamic. In fixed channel assignment (FCA) the channels are permanently assigned to each cell, while in dynamic channel assignment (DCA) the channels are placed in a pool and dynamically assigned upon request. Although DCA is desirable, FCA outperforms most known DCA schemes under high traffic load conditions [13,16,19]. Since heavy traffic conditions are expected in future generations of cellular networks, efficient FCA schemes become more important. Another group of developed channel assignment methods can be classified in a third category: the hybrid CA (HCA), which is a mixture of the FCA and the DCA schemes. This strategy is used, for instance, in currently operating GSM networks. A comprehensive and exhaustive comparison between these approaches can be found in [20]. Let us consider the problem where a set of c channels must be assigned to n arbitrary cells. In our problem formulation we assume that the total number of available channels is given – this number can be determined by either the available radio spectrum or the lower bound estimated by a graph-theoretic method. Without loss of generality, the channels can be assumed to be evenly spaced in the radio frequency spectrum. Thus, using an appropriate mapping, channels can be represented by consecutive positive integers. Therefore, the interference constraints are modeled by an n  n compatibility matrix C, whose diagonal elements cii represent the co-site constraint, i.e., the number of frequency bands by which channels assigned to cell i must be separated. The non-diagonal elements cij represent the number of frequency bands by which channels assigned to cells i and j must differ. When this compatibility matrix is binary, the constraints can be expressed more simply: if the same channel cannot be reused by cells i and j, then cij ¼ 1, and, otherwise, cij ¼ 0. For example, the matrix shown in Fig. 4 concerns the channel assignment problem with n ¼ 5 cells. In Fig. 4 we can see, for instance, that the co-site constraint for the cell 2 is c22 ¼ 4, indicating that the gap in frequency domain between any two channels assigned to the cell 2 should be at least 4. On the other hand, the adjacent channel constraint between the cell 2 and the cell 4 is c24 ¼ 2, indicating that gap in frequency between any two channels assigned to the cell 2 and the cell 4 should be at least 2. The traffic requirements are modelled by means of an n-length demand vector d, whose elements represent the number of channels required in each of the cells. The assignment to be generated is denoted by an n  c

Fig. 4. Example of compatibility matrix for a network with 5 cells.

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Fig. 5. Structure of the allocation matrix A.

binary matrix A, whose element aij is 1 if channel j is assigned to cell i, and 0 otherwise – see Fig. 5. This implies that the total number of 1’s in row i of matrix A must be di. The cost due to the violation of interference constraints can be written as J 1 ¼ J CSC þ J ACC ;

ð7Þ

where JCSC and JACC represent, respectively, the costs due to the violations of the co-site and the adjacent channel constraints. The first one can be written as J CSC ¼ W CSC

ny nx X n X X i;i6¼j

x

CSCðfix ; fjx Þ;

ð8Þ

j

where parameter WCSC weighs the relative importance of CSC and CSCðfix ; fjx Þ is a measure of the co-site constraint satisfaction. This parameter equals 0 only if the difference between channels i and j of cell x is jfix  fjx j P cxx , and 1 otherwise. fab represents the assigned frequency for the ath channel of cell b, and na is the number of channels in the ath cell. On the other hand, the cost due to the adjacent channel constraint violation can be expressed as J ACC ¼ W ACC

ny nx X n X n X X x6¼y

y

i

ACCðfix ; fjy Þ;

ð9Þ

j

where ACCðfix ; fjy Þ

 ¼

0

if jfix  fjy j P cxy

1

otherwise:

ð10Þ

Parameter WACC in Eq. (9) is fixed to weigh the relative importance of the adjacent channel constraint. Finally, the cost due to the violation of the traffic demand requirements is modeled as n X X ðd i  aij Þ2 : ð11Þ J TRAFF ¼ W TRAFF i

j

Gathering together all the costs, the final cost function to be minimized is J ¼ J CSC þ J ACC þ J TRAFF :

ð12Þ

If the traffic demand requirements are incorporated implicitly by only considering those Passignments which satisfy them, then the cost function can be expressed by J ¼ J 1 ¼ J CSC þ J ACC , subject to j aij ¼ d i 8i. Therefore, the fitness function to be used in the DGlGA is given by k ¼ 1=J . 5. Numerical results In this section, transmitters are considered to be located at cell centers and the traffic is assumed to be inhomogeneous, with each cell having a different and a priori known traffic demand (this situation is frequently known as the static model of the CAP). Following the ideas shown in [25], the initial population is constructed

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Table 1 Specifications of benchmark problems 14, 15 and 16 Problem

No. of cells

Lower bound

Compatibility matrix

Demand vector

14 15 16

4 21 25

11 221 73

C1 C3 C2

d1 d2 d4

using the available a priori information, i.e., the algorithm assigns a valid string of frequencies to all the cells following a simple approach: first, the algorithm attempts to assign a set of valid frequencies to as many base stations as possible. In the event that valid frequencies cannot be located to some of the cells, they are then randomly assigned. The objective is to obtain a conflict-free assignment among the cells such that the resulting bandwidth is close to the minimum channel span required to satisfy a given demand. 5.1. The benchmark problems In order to evaluate the performance of the DGlGA algorithm and compare it to other approaches, performance is analyzed using the set of 13 benchmark problems defined in [38] (also used in [9]) as well as problems 1, 2 and 4 from [31] (we will refer to these problems with numbers 14, 15 and 16). In fact, problem 1 (our problem 14) is taken from Example 1 in [38] and problem 4 (our problem 16) is from [24]. The characteristics of these 13 benchmark instances can be found in [38] (or in Table I in [9]). The definition of problems 14, 15 and 16 is summarized in Table 1,3 where all the channel demand vectors and compatibility matrices are given in Figs. 6 and 7 and the total number of frequencies vary from 11 to 221. Benchmark problem 15 belongs to a particular set of useful benchmark tests for cellular assignment problems called Philadelphia problems. Notice that [38] presents some variations from the original Philadelphia problems, which were first presented by Anderson [1] in the early 70’s. These problems constitute, by far, the most common set of benchmark problems for channel assignment algorithms, i.e., given a cellular network, the demand in each cell and a number of interference constraints, the problem consists in determining the minimum span (difference between the maximum and the minimum assigned frequencies) of any valid channel assignment. Notice that problems 1–4 and 9–14 consider the ACC, CCC and CSC constraints, while problems 5–8, 15 and 16 consider only the CCC and CSC constraints. As an example, Fig. 8 shows the cellular geometry of the Philadelphia problem with n ¼ 21 cells (the cluster size for CCC is N c ¼ 7) [10]. This special example has been considered by many authors working in the field of frequency assignment. Therefore, it makes it possible to compare the derived solutions with previously published results. 5.2. Convergence performance Let us first study the convergence properties of the proposed method. The results shown in Table 2 are average values over 100 trials for each problem. The parameters to be set are: the number of iterations ng, the initial mutation and crossover probabilities, the population size np and the parameters of functions pc(k) and pm(k). Although the settings of these parameters are generally quite heuristic, Goldberg suggests in [14] a general rule for using a relatively small population size, high pc, and low pm, which must be adjusted to our DGlGA. After several trials that helped to fine tune the parameters ensuring that the computation is manageable, the optimal values were found to be: • Number of fitness evaluations: 100 (for problem 14), 25,000 (problems 5–8 and 11), 50,000 (problems 12, 15 and 16), 75,000 (problem 13), 100,000 (problems 1–4), 150,000 (problem 10) and 300,000 (problem 9). If the values corresponding to problems 10, 12, 14 and 16 are compared to those shown in [31] (where a different 3

Our demand vector d2 corresponds to the demand vector m2 used by [12,38], where the original value proposed by [10] was modified. On the other hand, what we have defined as problems 14, 15 and 16 correspond to Funabiki’s problems 1, 6 and 2, respectively [8].

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Fig. 6. Compatibility matrices of the benchmark problems considered. (a) C1, (b) C2 and (c) C3.

Fig. 7. Traffic demand vectors for the benchmark problems considered.

Fig. 8. Cellular geometry for the Philadelphia benchmark problem with n ¼ 21 cells.

GA-based approach is used to solve these instances) it can be seen that these values mean a reduction of 75% (in problems 15, 12 and 16) and 62.5% (in problem 10) with respect to the number of iterations needed in [31]; in problem 14 both approaches require 100 iterations. Notice that, since not every offspring needs to be evaluated in each generation, the number of fitness evaluations is a more representative parameter of the performance than the number of generations. • Initial crossover probability, pc(0): this parameter is set to 0.35 in problems 5–8 and 11-16, while instances 1–4, 9 and 10 showed better results with 0.25.4 • Initial mutation probability, pm(0): 0.05 for all the problems. • Population size, np: 10 individuals, except in problems 1–4, 9 and 10, which required 20.

4

Thereby, for this case (problems 1–4, 9 and 10), parameters a and b in Eq. (4) are: a ¼ 0:03157, b ¼ 0:2184.

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Table 2 Summary of the comparison between convergence results Problem no.

Frequency of convergence

10 12 14 15 16

NN [8]

MGA [31]

DGlGA

– 24 100 79 9

20 80 100 92 100

25 85 100 89 98

• Simulations show that W CSC ¼ 1 and W ACC ¼ 1:25 leads to faster convergence as compared to W ACC ¼ 1 (This result is in accordance to [25], where W ACC-optimal ¼ ð1:1Þ=2 was obtained. See Section 5.4, as well). • In addition, to encourage greater diversity in the initial population, the strings created at this stage are restricted to just a single copy so as to avoid duplicates. In order to compare the results with those of other related approaches, the performance is measured using the average frequency of convergence to the solutions, defined as the ratio of the total number of successful convergence to the total number of runs. Table 2 shows the results for problems 10, 12, 14, 15 and 16, whose convergence properties have been previously studied by Ngo and Li using a GA-based scheme [31] and by Funabiki and Takefuji, who applied a neural network-based algorithm to solve these instances [8]. The results show that both GA-based procedures outperform the convergence results of the neural network for solving the fixed CAP. The three approaches converge properly in 100% of cases in problem 14. In problems 12, 15 and 16, both GA-based methods converge more frequently than the NN approach. In problems 15 and 16, the DGlGA shows a slightly worse convergence results than [31] (only in about 3–5% fewer cases than [31]). In spite of that, the DGlGA needs fewer computational load than that required by [31] (see Table 3). In contrast, the DGlGA presents notably better convergence in problems 10 and 12. In essence, in problems 12, 15 and 16 both algorithms exhibit very similar results, with the DGlGA being much less complex. Next, Table 3 shows the execution time required to solve these problems. Bold figures show the CPU time normalized to the time required to solve problem 15 using the proposed DGlGA. It can be seen how the computational burden of the DGlGA is about 20% lower than that of the previous GA-based approach by Ngo and Li [31] (18% in problem 15, 23% in problem 12, and 20% in problems 10 and 16). Notice that this reduction in the computational load is reached maintaining a very similar – or even better – percentage of convergence (Table 2) and with both approaches getting optimal conflict-free solutions. If one compares the values given in Table 3 for [31] with the specific values reported in the original author’s paper, a small difference can be observed. The reason is that the algorithm has been programmed and run in a different platform and language. In order to get the comparative figures shown in Table 3, both methods were similarly programmed and run in the same computer environment. 5.3. Performance without time constraints In this subsection, different search techniques are compared when they run without any time constraint and an optimal solution is guaranteed. Table 4 shows the execution times for three different algorithms: (i) the IDA Table 3 Execution times (in seconds) for benchmark problems 10, 12, 14, 15 and 16 MGA [31] 10 12 14 15 16

4.129 0.959 0 0.192 0.284

CPU: AMD Athlon XP 2100+ 1.8 GHz.

DGlGA 26.12 6.07 0 1.22 1.80

3.285 0.738 0 0.158 0.226

20.78 4.67 0 1 1.43

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Table 4 Execution time performance comparison between three different methods for FCA No. of channels per cell (c) 3

No. of cells (n)

5 7 9

4

5

BDFS

IDA

DGlGA

BDFS

IDA

DGlGA

BDFS

IDA

DGlGA

0.23 5.81 41.01

1.18 33.25 254.71

1 6.32 33.36

1.12 149.75 259.94

6.89 1045 2225.13

1.6 118.04 211.56

37.10 491.74 1360.30

245.61 3540.54 13,834,29

24.06 403.12 1079.60

All values are normalized to the execution time of DGlGA with n ¼ 5, c ¼ 3.

(iterative deepening A) algorithm [33], which offers a quite simple algorithm that can solve large problems with a small computer memory, (ii) the so-called BDFS (block depth-first search) real-time heuristic search method proposed in [28], and (iii) the DGlGA. For the sake of comparison, we have chosen the same number of cells and number of channels than in [28]. It can be seen first that the BDFS algorithm produces an increasing average speedup over the IDA method. This is explained in [28] as follows: the IDA algorithm first explores the search tree in depth so long as the heuristic value of a node in a path is below a threshold value set for that iteration. If no goal is found within that threshold, this threshold is increased to the minimum heuristic value of all the nodes, which are generated but not expanded in the last iteration, and a new iteration begins. On the other hand, the proposed DGlGA outperforms BDFS (and, hence, IDA) whenever the complexity of the problem becomes considerable. In these cases, the running time of the DGlGA is about 20% smaller than the BDFS. Only in the three simplest cases (n = 5, c = 3), (n = 5, c = 4) and (n = 7, c = 3), the minimum computational load required to implement the DGlGA is larger than the BDFS, though still much better than the IDA. This is due to the fact that all the steps outlined in Sections 2 and 3 must be implemented even to deal with these simple configurations. A minimum population size as well as fitness function evaluations must be performed over that minimum population. However, the DGlGA shows a better efficiency in medium and high complex CAP configurations. 5.4. Determination of WCCC and WACC In order to get efficient execution times, a proper design of the algorithm and determination of the parameters WCSC and WACC in Eqs. (8) and (9) becomes an essential task, since their value greatly affects the number of generations required to converge. In this subsection, this influence is quantified using the same inhomogeneous 25-cell network used by Kunz and Lai in [24,25], respectively. The number of generations (iterations) required for a proper convergence is shown in Fig. 9 for different pairs of WCSC and WACC. The optimal values for the weights WCSC and WACC were found to be 1 and 1.25, respectively. Notice that the final assignment solution may be optimal in all the cases – in terms of ACC and CSC – though the assignment to specific cells may vary. The most important difference between these configurations (pairs of WACC and WCSC) is the required computational load for each of them, since parameter ng proportionally affects the execution time. This way, a precise calculus of WACC and WCSC is vital to get an efficient allocation algorithm. 5.5. Optimality of the solution In order to evaluate the performance of the different methods proposed to solve the CAP, it is extremely important to analyze the quality of the derived solutions, i.e. their proximity to the optimal lower bounds.5

5

The problem of finding the lower bound for the CAP lies aside the scope of this paper, though we use it in order to quantify the quality of the obtained solutions. A detailed description and more references can be found in [10,12,28,39,40].

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Fig. 9. Number of generations (ng) required for certain pairs of the weights WCSC and WACC.

For the sake of comparison, results from earlier works6 along with that from the DGlGA are shown in Table 5. The lower bound figures correspond to the lower bound for each of the problems as reported in [2,9,12]. Table 5 shows that the proposed scheme can solve each of the 16 benchmark problems here considered, finding an optimal solution for each one. Even in problem 10, which is a well-known more complex problem, the span result is optimal. A few of the other methods can also achieve optimal solutions for all of the problems, but their computational load is much higher and the number of iterations required (in the case of GAbased approaches) is also higher. As a consequence, the computation time of the proposed DGlGA is far more competitive. For instance, Ghosh et al. state in [12] that their method takes a few seconds to solve problems No. 4, 6, 8 and 12 (1, 3, 4 and 8 as referenced in their paper) and between 12 and 80 h for problems 2 and 10 (our problem 10 is Ghosh’s problem 6). In contrast, the DGlGA requires, at most, about 30% of the time required by [2,12] (of course, with all the simulations being run in the same computer). The neural network-based approaches involve much more complexity, as well. For instance, considering problem 16, Funabiki’s NN [8] requires at least 200 iterations in order to find a good solution, while the approach by Kunz [24] needs 2450. These figures imply a much longer execution time than that required by the DGlGA. Results in Table 5 indicate that only both the DGlGA and the method proposed in [9] achieve the lower bound solutions in all the benchmark problems considered. Besides, although not shown in the table, numerical results prove that the DGlGA saves about 30–45% of computational complexity w.r.t. [9], constituting, this way, a more efficient method as far as implementation is concerned. Methods by Nguyen and Wong [32] and Shimodaira [36] have been included for comparison, as well. The first one attains good quality solutions – only in problem 2 it was not able to achieve the optimal lower bound – though its computational load is about 90% higher than that of the DGlGA. The main reason is that the required number of iterations is much higher and the minimum population size required for proper convergence is, also, higher. A similar situation was observed with [36]. In this particular case, the lower bound was not achieved in problems 1, 2 and 9; besides, the global complexity is for the same reasons about 75% higher w.r.t. DGlGA. The full assignment of channels for a sample of the 16 benchmark problems considered in our simulations is presented in the Appendix. Due to space restrictions, results are shown for two problems (instances 1 and 2) with demand vector d3 (referred as m1 in [38, p. 849], and in [9, p. 400]), other two (instances 9 and 10) with d2 (m2 in [9,38]), and problem 16 from Table 1, with demand vector d T4 (defined in Fig. 7), as well. 6

These results have been taken, in part, from the comparative analyzes shown in [2,9,12].

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Problem no.

Lower bound

DGlGA

Funabiki [9]

Chakraborty [5]

Sivarajan [38]

Box [4]

Ghosh [12]

Nguyen [32]

Funabiki CAP3 [8]

Wang [44]

Ngo [31]

Shimodaira [36]

Beckmann [2]

Sung, GSP2 [41]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

427 427 533 533 381 381 533 533 258 253 309 309 529 11 221 73

427 427 533 533 381 381 533 533 258 253 309 309 529 11 221 73

427 427 533 533 381 381 533 533 258 258 309 309 529 – – 73

– 463 – 533 – 381 – 533 – 273 – 309 – – – –

460 447 536 533 381 381 533 533 283 270 310 310 529 11 – –

442 442 533 533 381 381 533 533 270 – 309 – 529 – – –

– 427 – 533 – 381 – 533 – 253 – 309 – – – –

427 440 533 533 381 381 533 533 258 – 309 – 529 – – –

– – – 533 – 381 – 533 – – 309 309 – 11 221 73

– 433 – 533 – 381 – 533 – – 309 – 529 – – –

– – – – – – – – – 268 – 309 – – 221 –

440 437 533 533 381 381 533 533 270 – 309 – 529 – – –

– 427 – 533 – 381 – 533 – 253 – 309 – – – –

450 444 533 533 381 381 533 533 273 – 309 – 529 – – –

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Table 5 Minimum frequencies by which the benchmark problems can be optimally completed

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Fig. 10. Evolution of the population fitness with and without parameter control (p.c.).

To conclude this analysis of the performance of the proposed method, two more difficult cases have been studied: • Problem A: n ¼ 21 cells, demand vector d A ¼ 2d 3 (d3 is shown in Fig. 7). ACC, CCC and CSC (cii ¼ 5) constraints are considered. • Problem B: same as A, but with demand vector d B ¼ 4d 3 . Our DGlGA is going to be compared with that method in Table 5 that obtained the best results in the previous sections, i.e. the GA-based approach by Funabiki et al. [9]. Using the DGlGA the best assignments we were able to find required 855 and 1713 channels, for problems A and B, respectively, while the method in [9] required slightly higher values, 858 and 1724. On the other hand, the computation times for the DGlGA were 11.86 and 23.76 s, for problems A and B, respectively, while the other GA-based algorithm took about 16.73 and 32.8 s, respectively. This means a reduction in time of 38–41% in favour of the proposed DGlGA method. So, the simulations carried out in this paper show that the proposed strategy makes it possible to achieve very good solutions with a minimum number of frequencies with very short computing times, outperforming, in all cases, the different methods chosen for comparison. 5.6. Entropy-guided parameter tuning Finally, some experiments were carried out in order to study the influence of the entropy-based tuning up process. Fig. 10 represents the evolution of the mean population fitness (mean value after 100 runs) of: (i) the individuals of the elite, (ii) all those individuals not belonging to the elite. For both cases, the evolution is drawn with and without entropy-guided adjustment for the first 30 iterations of the algorithm. For both sets of individuals, the mean fitness approaches the maximum before when the entropy is monitored and the probabilities pc and pm are dynamically adjusted. Furthermore, redundant individuals are more efficiently eliminated when the fitness entropy is used to adjust the genetic operators. This way, the diversity of the population increases, and, simultaneously, the probability of getting trapped into suboptimal solutions is reduced. This capability also affects the population size required to get an optimal frequency assignment, since: (i) very similar individuals do not represent any advantage at the time of searching the solutions space, and (ii) almost no improvement (in terms of mean population fitness) occurs during the exploration phases. Thus, many fitness evaluations can be saved during these periods. Making use of this fact, the number of evaluations can be lowered about 23%. 6. Conclusions A novel DGlGA with both good convergence properties and low complexity has been developed. The proposed entropy-guided procedure for adjusting the probabilities of the genetic operators in order to control the

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population diversity has been proven to lead to both a reduction of the population size required to attain a certain performance and a high probability of achieving an optimal result. The algorithm has been satisfactorily applied to solve the fixed channel assignment problem. Making use of several well-known benchmark instances, its performance has been shown to be superior to that of the existing frequency assignment algorithms in terms of computation time, convergence behavior and quality of the solution. Appendix This appendix shows the full assignments for benchmark instances No. 1, 2, 9, 10 and 16 (see Tables 6–10). Table 6 Channel assignment for benchmark problem 1 Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2 7 12 17 22 27 37 47 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 9 14 19 24 29 39 49 59 69 79 89 99 109 119 129 139 149 159 169 179 189 301 317 392 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

308 322 333 344 364 399 406 416 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

60 65 70 100 125 130 135 140 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15 20 105 155 160 165 170 175 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

34 44 54 64 74 84 94 229 296 368 393 400 405 410 415 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

57 67 77 87 97 107 117 127 137 147 157 167 177 182 187 196 202 208 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

33 43 53 63 73 83 93 103 113 123 133 143 153 163 173 193 199 205 211 217 223 230 236 242 253 259 265 271 277 283 289 295 306 311 321 327 334 339 345 351 357 363

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 197 203 209

184 194 200 206 212 218 224 231 237 243 248 254 260 266 272 278 284 290 310 350 356 362 374 380 386 408 418 424 0 0 0 0 0 0 0 0 0 0 0 0 0 0

35 40 45 55 75 80 85 90 95 120 145 150 228 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 9 24 29 49 59 69 99 109 115 129 180 189 301 392 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

16 21 26 31 36 41 46 51 56 61 66 71 76 81 215 221 227 234 240 246 275 287 293 298 325 330 336 341 347 353 396 0 0 0 0 0 0 0 0 0 0 0

184 194 200 206 212 218 224 231 237 243 248 254 260 266 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 10 15 20 25 30 35 40 45 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 228 0 0 0 0 0 0

3 8 13 18 23 28 38 48 58 68 78 88 98 108 118 128 138 148 158 168 178 183 188 195 201 207 213 219 225 232 238 244 249 255 261 267 273 279 285 291 300 309

34 44 54 64 74 84 94 104 114 124 134 144 154 164 174 229 296 302 307 312 328 338 368 393 400 405 410 415 0 0 0 0 0 0 0 0 0 0 0 0 0 0

47 57 67 77 87 97 107 117 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

32 42 50 62 72 82 92 102 112 122 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

52 132 142 152 162 172 185 190 198 204 210 216 222 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 7 12 17 22 27 37 127 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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Table 6 (continued) Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

369 375 381 387 394 401 409 414 419 425 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

215 221 227 234 240 246 251 257 263 269 275 281 287 293 298 304 314 319 325 330 336 341 347 353 359 366 371 377 383 389 396 403 412 421 427

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

316 323 332 343 349 355 361 373 379 385 391 398 407 417 423 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Table 7 Channel assignment for benchmark problem 2 Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

12 2 7 27 47 37 22 17 0 0 0 0 0 0

4 9 14 19 24 29 39 49 59 69 79 89 99 109

344 308 322 333 364 399 406 416 0 0 0 0 0 0

140 135 100 130 70 65 60 125 0 0 0 0 0 0

15 20 165 105 175 160 170 155 0 0 0 0 0 0

54 84 64 34 44 94 74 415 410 296 393 229 400 368

127 137 182 167 177 157 147 87 77 67 97 107 117 208

33 43 53 63 73 83 93 103 113 123 133 143 153 163

1 6 11 16 21 26 31 36 41 46 51 56 61 66

356 362 350 310 290 254 243 248 272 284 278 266 260 206

80 40 85 145 90 75 35 45 95 55 150 228 120 0

4 24 49 29 59 189 129 69 392 180 115 109 99 9

298 26 56 61 51 46 31 21 36 66 81 246 240 76

212 224 218 194 184 200 206 254 260 272 266 243 237 248

5 10 15 20 25 30 35 40 45 228 55 60 65 70

3 8 13 18 23 28 38 48 58 68 78 88 98 108

34 44 54 64 74 84 94 104 114 124 134 144 154 164

87 32 97 42 117 50 107 62 67 72 57 92 77 82 47 112 0 122 0 102 0 0 0 0 0 0 0 0 (continued on

19

20

21

162 17 142 12 52 7 204 2 210 22 222 37 216 27 190 127 185 0 198 0 172 0 152 0 132 0 0 0 next page)

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2674 Table 7 (continued) Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

119 129 139 149 159 169 179 189 301 317 392 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

405 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

196 187 202 57 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

173 306 193 199 205 211 217 223 419 236 242 425 253 259 265 271 277 283 289 295 321 311 230 327 334 339 345 351 357 363 369 375 381 387 394 401 409 414 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 304 191 197 203 209 215 221 227 234 240 246 251 257 263 269 275 281 287 293 298 366 186 314 319 325 330 336 341 347 353 359 427 371 377

200 184 194 218 231 224 212 237 374 418 408 380 386 424 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

301 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

16 71 234 353 336 341 325 396 330 221 227 215 287 293 347 275 41 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

231 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

118 128 138 148 158 168 178 188 195 201 207 213 219 225 232 238 244 249 255 261 267 273 279 285 291 300 309 316 323 332 343 349 355 361 183 373 379 385 391 398 407 417 423 0 0 0 0 0 0 0 0 0 0 0 0 0 0

174 302 296 312 328 338 393 400 405 410 415 368 229 307 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

L.M. San Jose´-Revuelta / Information Sciences 177 (2007) 2655–2678

2675

Table 7 (continued) Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

383 389 396 403 412 421

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

19

20

21

Table 8 Channel assignment for benchmark problem 9 Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

27 20 129 111 245 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

203 142 15 250 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

148 159 104 173 136 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

256 126 218 31 80 213 1 70 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

108 124 134 139 82 72 162 36 188 215 210 170 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

59 222 206 195 190 185 177 169 239 159 101 140 128 120 164 81 75 148 69 54 48 32 22 258 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

236 230 24 219 214 209 249 198 172 167 156 44 16 122 105 114 99 93 87 77 67 62 57 52 255 35 30 150 10 4 0 0 0 0 0 0 0 0 0 0 0 0 0

95 119 127 217 47 13 257 205 243 233 223 8 212 194 186 181 176 160 89 139 134 144 84 74 252 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

157 115 50 192 97 109 55 208 102 91 131 39 17 65 227 166 183 146 45 60 151 197 6 171 247 237 34 220 25 178 0 0 0 0 0 0 0 0 0 0 0 0 0

37 53 112 107 253 63 123 244 83 3 19 58 118 211 128 239 88 14 28 73 143 94 224 229 42 68 133 258 78 154 163 169 234 189 200 48 216 100 138 9 0 0 0

174 5 103 16 236 246 40 46 33 179 110 11 61 66 56 76 92 21 226 121 26 98 51 116 130 141 152 158 165 147 184 86 196 207 191 221 231 241 251 202 0 0 0

13 8 54 155 49 29 113 254 18 149 64 59 212 194 74 79 84 89 233 95 101 106 243 119 38 132 137 69 23 160 168 176 181 186 43 199 205 238 223 228 127 248 2

143 136 131 123 115 109 104 94 85 78 71 63 237 45 39 34 25 17 12 204 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

118 244 234 224 216 211 200 193 187 182 174 161 154 145 138 133 125 253 112 107 96 90 83 73 7 42 37 28 19 14 0 0 0 0 0 0 0 0 0 0 0 0 0

103 241 26 246 226 1 221 207 202 196 191 184 179 21 165 158 152 147 110 130 141 80 70 40 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

113 254 124 36 155 210 188 137 168 162 29 149 106 82 72 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

75 120 195 185 32 206 22 232 85 180 12 140 175 222 242 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

90 121 35 116 135 213 193 251 255 98 204 92 156 86 44 76 167 66 172 61 7 56 182 51 187 46 214 126 71 31 81 218 209 256 125 231 219 11 177 5 105 0 96 0 145 0 161 0 249 0 24 0 30 0 150 0 198 0 114 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (continued

132 129 18 164 238 201 248 225 153 111 59 20 215 62 101 122 170 41 38 240 54 99 69 93 199 87 49 77 79 67 228 117 64 57 108 52 43 235 2 245 0 27 0 230 0 190 0 10 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 on next page)

L.M. San Jose´-Revuelta / Information Sciences 177 (2007) 2655–2678

2676 Table 8 (continued) Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

144 217

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

Table 9 Channel assignment for benchmark problem 10 Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

230 235 240 245 250 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

53 58 63 68 73 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

231 236 241 246 251 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

49 54 59 64 69 74 80 86 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 12 19 26 33 40 47 52 57 62 67 72 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 9 16 23 30 37 44 78 84 90 95 101 107 113 119 140 146 152 198 203 208 213 218 223 228 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 13 20 27 34 41 48 75 81 87 92 98 104 110 116 122 128 134 158 164 170 175 180 185 190 195 200 205 210 215 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 10 17 24 31 38 45 50 55 60 65 70 77 83 89 96 102 108 114 120 172 177 182 187 192 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 14 21 28 35 42 79 85 91 127 133 139 145 151 157 163 169 174 179 184 189 194 214 219 224 229 234 239 244 249 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 11 18 25 32 39 46 51 56 61 66 71 76 82 88 94 100 106 112 118 124 130 136 142 148 154 160 166 171 176 181 186 191 196 201 206 211 216 221 226 0 0 0 0 0

1 8 15 22 29 36 43 78 84 90 96 102 108 114 120 126 132 138 144 150 156 162 168 173 178 183 188 193 198 203 208 213 218 223 228 233 238 243 248 253 0 0 0 0 0

3 10 17 24 31 38 45 50 55 60 65 70 75 81 87 93 99 105 111 117 123 129 135 141 147 153 159 165 170 175 180 185 190 195 200 205 210 215 220 225 231 236 241 246 251

49 54 59 64 69 93 99 105 111 117 171 176 181 186 191 196 200 206 211 216 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 11 18 25 32 39 46 51 56 61 66 71 124 130 136 142 148 154 160 166 173 178 183 188 193 231 236 241 246 251 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 8 15 22 29 36 43 94 100 106 112 118 126 132 138 144 150 156 222 227 233 238 243 248 253 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 12 19 26 33 40 47 52 57 62 67 72 123 129 135 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 9 16 23 30 37 44 199 204 209 232 237 242 247 252 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 13 20 27 34 41 48 53 58 63 68 73 92 98 104 110 116 122 128 134 140 146 152 158 164 230 235 240 245 250 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

97 103 109 115 121 171 176 181 186 191 196 201 206 211 216 230 235 240 245 250 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

75 81 87 111 117 125 143 149 155 161 167 173 178 183 188 193 213 218 223 228 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

50 55 60 65 70 77 83 89 95 101 107 113 119 131 137 170 175 180 185 190 197 202 207 220 225 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

L.M. San Jose´-Revuelta / Information Sciences 177 (2007) 2655–2678

2677

Table 10 Channel assignment for benchmark problem 16 Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

37 35 30 32 22 20 18 24 26 28 0

64 66 70 72 68 60 58 62 56 52 54

36 34 44 46 50 48 40 42 38 0 0

35 56 54 37 52 0 0 0 0 0 0

65 71 73 69 67 61 59 57 63 0 0

9 13 15 11 0 0 0 0 0 0 0

57 65 63 59 61 0 0 0 0 0 0

21 23 27 25 31 33 29 0 0 0 0

1 5 7 3 0 0 0 0 0 0 0

16 6 2 4 12 14 10 8 0 0 0

28 32 30 20 18 22 24 26 0 0 0

51 45 49 47 39 41 43 53 55 0 0

19 7 5 1 3 11 17 9 13 15 0

27 33 29 31 23 21 25 0 0 0 0

38 40 44 42 34 36 46 0 0 0 0

45 39 41 37 35 43 0 0 0 0 0

14 12 16 18 0 0 0 0 0 0 0

48 54 56 52 50 0 0 0 0 0 0

10 2 6 8 4 0 0 0 0 0 0

51 53 49 47 14 12 16 0 0 0 0

19 17 9 15 11 13 0 0 0 0 0

3 1 5 7 0 0 0 0 0 0 0

27 25 21 23 29 0 0 0 0 0 0

13 15 19 17 9 11 34 0 0 0 0

4 8 6 2 10 0 0 0 0 0 0

References [1] L.G. Anderson, A simulation study of some dynamic channel assignment algorithms in a high capacity mobile telecommunications system, IEEE Transactions on Communications 21 (11) (1973) 1294–1301. [2] D. Beckmann, U. Killat, A new strategy for the application of genetic algorithms to the channel assignment problem, IEEE Transactions on Vehicular Technology 48 (4) (1999) 1261–1269. [3] D. Bhandari, C.A. Murthy, S.K. Pal, Genetic algorithm with elitist model and its convergence, International Journal on Pattern Recognition and Artificial Intelligence 10 (6) (1996) 731–747. [4] F. Box, A heuristic technique for assignment frequencies to mobile radio nets, IEEE Transactions on Vehicular Technology 27 (2) (1978) 57–64. [5] G. Chakraborty, An efficient heuristic algorithm for channel assignment problem in cellular radio systems, IEEE Transactions on Vehicular Technology 50 (6) (2001) 1528–1539. [6] M. Cuppini, A genetic algorithm for channel assignment problems, European Transactions on Telecommunications and Related Technology 5 (2) (1994) 285–294. [7] M. Duque-Anton, D. Kunz, B. Ruber, Channel assignment for cellular radio using simulated annealing, IEEE Transactions on Vehicular Technology 42 (1) (1993) 14–21. [8] N. Funabiki, Y. Takefuji, A neural network parallel algorithm for channel assignment problems in cellular radio networks, IEEE Transactions on Vehicular Technology 41 (4) (1992) 430–437. [9] N. Funabiki, N. Okutani, S. Nishikawa, A three-stage heuristic combined neural-network algorithm for channel assignment in cellular mobile systems, IEEE Transactions on Vehicular Technology 49 (2) (2000) 397–403. [10] A. Gamst, Some lower bounds for a class of frequency assignment problems, IEEE Transactions on Vehicular Technology 35 (1) (1986) 8–14. [11] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, New York, 1979. [12] S.C. Ghosh, B.P. Sinha, N. Das, Channel assignment using genetic algorithm based on geometric symmetry, IEEE Transactions on Vehicular Technology 52 (4) (2003) 860–875. [13] J.D. Gibson (Ed.), The Mobile Communications Handbook, CRC Press, 1996. [14] D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA, 1989. [15] J.J. Grefenstette, Optimization of control parameters for genetic algorithms, IEEE Transactions on Systems, Man and Cybernetics 16 (1) (1986) 122–128. [16] W.K. Hale, Frequency assignment: theory and applications, Proceedings of the IEEE 68 (12) (1980) 1497–1514. [17] J.K. Hao, R. Dorne, P. Galiner, Tabu search for frequency assignment in mobile radio networks, Journal of Heuristics 4 (1) (1998) 47–62. [18] J.J. Hopfield, D.W. Tank, Neural computation of decisions in optimization problems, Biological Cybernetics 52 (3) (1985) 141–152. [19] J.Y. Jeng, Y.B. Lin, W. Chen, Dynamic resource assignment strategies for PCS data services, Information Sciences 109 (1998) 211– 225. [20] I. Katzela, M. Naghshineh, Channel assignment schemes for cellular telecommunication systems: a comprehensive survey, IEEE Personal Communications 3 (3) (1996) 10–31. [21] S. Kim, S.L. Kim, A two-phase algorithm for frequency assignment in cellular mobile systems, IEEE Transactions on Vehicular Technology 43 (3) (1994) 542–548. [22] S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Optimization by simulated annealing, Science 220 (4598) (1983) 671–680. [23] B. Krishnamachari, S. Wicker, Global Search Techniques for Problems in Mobile Communications, Wireless Multimedia Lab., School Elec. Eng., Cornell Univ., 1999. [24] D. Kunz, Channel assignment for cellular radio using neural networks, IEEE Transactions on Vehicular Technology 40 (1) (1991) 188–193.

2678

L.M. San Jose´-Revuelta / Information Sciences 177 (2007) 2655–2678

[25] W.K. Lai, G.G. Coghill, Channel assignment through evolutionary optimization, IEEE Transactions on Vehicular Technology 45 (1) (1996) 91–96. [26] Z.J. Lee, C.Y. Lee, A hybrid search algorithm with heuristics for resource allocation problem, Information Sciences 173 (1–3) (2005) 155–167. [27] G.D. Lochite, Frequency channel assignment using artificial neural networks, in: Proc. of the 8th IEE International Conference on Antennas and Propagation, vol. 2, 1993, pp. 948–951. [28] S. Mandal, D. Saha, A. Mahanti, A real-time heuristic search technique for fixed channel allocation (FCA) in mobile cellular communications, Microprocessors and Microsystems 28 (8) (2004) 411–416. [29] R. Mathar, J. Mattfeldt, Channel assignment in cellular radio networks, IEEE Transactions on Vehicular Technology 42 (4) (1993) 647–656. [30] M. Mitchell, An Introduction to Genetic Algorithms, MIT Press, Cambridge, MA, 1996. [31] C.Y. Ngo, V.O.K. Li, Fixed channel assignment in cellular radio networks using a modified genetic algorithm, IEEE Transactions on Vehicular Technology 47 (1) (1998) 163–172. [32] D.H.M. Nguyen, K.P. Wong, Controlling diversity of evolutionary algorithms, in: Proceedings of the 2nd International Conference on Machine Learning and Cybernetics, Xian, 2003, pp. 775–780. [33] N.J. Nilsson, Artificial Intelligence: A New Synthesis, Morgan Kaufmann, Los Altos, CA, 1998. [34] F. Oppacher, M. Wineberg, The shifting balance GA: improving the GA in dynamic environment, in: Proceedings of the Genetic and Evolutionary Computation Conference, vol. 1, 1999, pp. 504–510. [35] T.L. Sheu, Y.J. Wu, A preemptive channel allocation scheme for multimedia traffic in mobile wireless networks, Information Sciences 176 (2006) 217–236. [36] H. Shimodaira, A diversity control oriented GA (DCGA): development and experimental results, in: Proceedings of the Genetic and Evolutionary Computation Conference, vol. 1, 1999, pp. 603–611. [37] F.F. Shueh, W.S.E. Chen, Optimum OVSF code reassignment in Wideband-CDMA forward radio link, Information Sciences 174 (2005) 81–101. [38] K.N. Sivarajan, R.J. McEliece, J.W Ketchun, Channel assignment in cellular radio, in: Proceedings of the 39th IEEE Vehicular Technology Conference, 1989, pp. 846–850. [39] D.H. Smith, S. Hurley, Bounds for the frequency assignment problem, Discrete Mathematics 167/168 (1997) 571–582. [40] D.H. Smith, S. Hurley, S.M. Allen, A new lower bound for the channel assignment problem, IEEE Transactions on Vehicular Technology 49 (4) (2000) 1265–1272. [41] C.W. Sung, W.S. Wong, Sequential packing algorithm for channel assignment under cochannel and adjacent channel interference constraint, IEEE Transactions on Vehicular Technology 46 (3) (1997) 676–686. [42] S. Tsutsui, Y. Fujimoto, A. Ghosh, Forking genetic algorithms: GAs with search space division schemes, Evolutionary Computation 5 (1) (1997) 61–80. [43] R.K. Ursem, Diversity-guided evolutionary algorithms, in: Proceedings of the International Conference on Parallel Problem Solving from Nature VII (PPSN VII) 2002, Granada, Spain, pp. 462–471. [44] W. Wang, C.K. Rushforth, An adaptive local-search algorithm for the channel assignment problem (CAP), IEEE Transactions on Vehicular Technology 45 (3) (1996) 459–466.