**1. Introduction**

46 Simulated Annealing – Single and Multiple Objective Problems

Journal Phys. Math. Gen. N°31, pp 8373-8385.

and Computing, 4, pp 21-32.

[18] Thompson J, Dowsland KA (1995) General cooling schedules for a simulated annealing based timetabling system. Proceedings of the first International Conference on the

[21] Lin C.K.Y, Haley K.B, Sparks C (1995) A comparative study of both standard and adaptive versions of threshold accepting and simulated annealing algorithms in three

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> This chapter focuses on a specific application of two Natural Computation (NC)-based techniques: Simulated Quenching (SQ) and Genetic Algorithms (GA): the problem of channels' assignment to radio base stations in a spectrum efficient way. This task is known to be an NP-complete optimization problem and has been extensively studied in the last two decades. We have decided to include both SQ and GA in the chapter so as o give an academic orientation to this work for readers interested in practical comparisons of NC methods.

> According to this point of view, our main interest is in describing and comparing the performances of these algorithms for solving the channel allocation problem (CAP). Therefore, the aim of our work is not the full theoretical description of SQ and GA. Surely, some other chapters in this book will cover this issue.

> There exists an important research activity in the field of mobile communications in order to develop sophisticated systems with increased network capacity and performance. A particular problem in this context is the assignment of available channels (or frequencies) to based stations in a way that quality of service is guaranteed. Like most of the problems that appear in complex modern systems, this one is characterized by a search space whose complexity increases exponentially with the size of the input, being, therefore, intractable for solutions using analytical or simple deterministic approaches (Krishnamachari, 1998; Lee, 2005). An important group of these problems −including the one we are interested in− belong to the class of NP-complete problems (Garey, 1979).

> Simulated Quenching (SQ) belongs to the family of Simulated Annealing (SA)-like algorithms. Simulated Annealing is a general method for solving these kind of combinatorial optimization problems. It was originally proposed by (Kirkpatrick, 1983) and (Černy, 1985). Since then, it has been applied in many engineering areas. The basic SA algorithm can be considered a generalization of the local search scheme, where in each step

<sup>© 2012</sup> San-José-Revuelta, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 San-José-Revuelta, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

of the iterative process a neighbour *s*' of each current solution *s* is proposed at random. Then *s* is only replaced by *s*' if cost does not rise. The main problems of SA are: (i) the possibility to get trapped in local minima, and (ii) the large computational load that leads to slow algorithms.

Simulated Quenching Algorithm for Frequency Planning in Cellular Systems 49

hand, several GA-based approaches have been applied to solve the CAP: For instance, (Cuppini, 1994) defines and uses specific genetic operators: 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, make the algorithm harder to converge. In (Lai, 1996), Lai and Coghill represented the channel assignment solution as a string of channel numbers grouped in such a way that 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 (Ghosh, 2003)− and basic mutation. Two years later, Ngo and Li (Ngo, 1998) 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. Authors stated that this algorithm outperforms

The particularities of the GA shown in section 5 are, mainly, a low computational load and the capability of achieving good quality solutions (optimal, minimum-span, solutions) while maintaining satisfactory 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 (San José, 2007). This way, the diversity of the population is monitored and the thus obtained method offers the flexibility and robustness peculiar to GAs.

Another closely related problem that we are not going to deal with is the determination of a lower bound for the *span* (difference between the largest channel used and the smallest channel used) in channel assignment problems. For instance, (Mandal, 2004) and (Smith,

• The first section (section 2) presents a brief introduction to the frequency (or channel) allocation problem. As mentioned above, there exist many good articles covering this problem, and our main purpose is to introduce the basic references of SQ and GAs applied to the CAP, to pose the problem as well as to describe a first motivation to

• Next, section 3 explains mathematically the interference and traffic constraints that

• Sections 4 and 5 are devoted to the detailed description of both the proposed SQ and the GA-based algorithms for frequency allocation, respectively. The first one is mainly based on the method proposed in (Duque, 1993), though incorporating some improvements concerning the weakest aspects outlined in the conclusions of the work of Duque et al. Important concepts such as the specific *cooling procedure* and the *neighbourhood production* are here described. On the other hand, the genetic algorithm included in section 5 is based on our previous work in the field of NC-based strategies for approaching not only the frequency allocation problem (San José, 2007), but also digital communications (San José,

2008) or image processing problems (San José, 2009), most of them using GAs.

• Finally, the last section of the chapter is devoted to the description of the numerical results, with special emphasis on the comparison between Neural Networks (NNs), SQ

the NN-based approach described in (Funabiki, 1992).

2000) address this problem when proposing solutions to the CAP.

The chapter is organized in the following sections:

research into this field.

define the CAP.

Both of the algorithms described in this chapter try to solve these drawbacks. The first one, SQ, speeds up the algorithm by quickly reducing the temperature in the system, though, in the process, the advantage of SA, i.e. convergence to the global optimum, is defeated (Ingber, 1993). Due to this, SQ is termed as a greedier algorithm in terms of computational load compromising for the global optimum. Besides, the proposed method, occasionally allows moves to solutions of higher cost according to the so-called Metropolis criterion (Metropolis, 1953) (see section 4.1). However, Duque et al. (Duque, 1993) point out that the main drawback they found was to get trapped to a local minimum (by a misplaced transition) with a low chance to get out of it.

The problem of convergence to suboptimal solutions can be efficiently addressed with another NC-based technique, the GAs. These algorithms are based on he principle of natural selection and survival of the fittest, thus constituting an alternative method for finding solutions to highly-nonlinear problems with multimodal solutions' spaces. GAs efficiently combine explorative and exploitative search to avoid convergence to suboptimal solutions. Unlike many other approaches, GAs are much less susceptible to local optima, since they provide the ability to selectively accept successive potential solutions even if they have a higher cost than the current solution (Mitchell, 1996).

This chapter focuses on the application of these methods to solve the channel allocation problem found in cellular radio systems. In this kind of systems, the frequency reuse by which the same channels are reused in different cells becomes crucial (Gibson, 1996). Every cell is allocated a set of channels according to its expected traffic demand. The entire available spectrum is allocated to a cluster of cells arranged in shapes that allow for uniform reuse patterns. Channels must be located satisfying certain frequency separation constraints to avoid channel interference using as small a bandwidth as possible. Considering this framework, the CAP fits into the category of multimodal and NP-complete problems (Krishnamachari, 1998; Garey, 1979; Hale, 1980; Katzela, 1996).

The *fixed* CAP −see section 3.2− has been extensively studied during the past decades. A comprehensive summary of the work done before 1980 can be found in (Hale, 1980). When only the co-channel constraint is considered, the CAP is equivalent to an NP-complete graph coloring problem (Sivarajan, 1989). In this simpler case, various graph-theoretic approaches have been proposed (Hale, 1980; Sivarajan, 1989; Box 1978; Kim, 1994).

From the point of view of NC, some procedures based on NNs (Funabiki, 1992; Hopfield, 1985; Kunz, 1991; Lochite, 1993) and SA or SQ (Duque, 1993; Kirkpatrick, 1983; Mathar, 1993) have already been considered. SA-SQ techniques generally improve the problem of convergence to local optima found with NNs, though their rate of convergence is rather slow (specially in SA), and a carefully designed cooling schedule is required. On the other hand, several GA-based approaches have been applied to solve the CAP: For instance, (Cuppini, 1994) defines and uses specific genetic operators: 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, make the algorithm harder to converge. In (Lai, 1996), Lai and Coghill represented the channel assignment solution as a string of channel numbers grouped in such a way that 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 (Ghosh, 2003)− and basic mutation. Two years later, Ngo and Li (Ngo, 1998) 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. Authors stated that this algorithm outperforms the NN-based approach described in (Funabiki, 1992).

The particularities of the GA shown in section 5 are, mainly, a low computational load and the capability of achieving good quality solutions (optimal, minimum-span, solutions) while maintaining satisfactory 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 (San José, 2007). This way, the diversity of the population is monitored and the thus obtained method offers the flexibility and robustness peculiar to GAs.

Another closely related problem that we are not going to deal with is the determination of a lower bound for the *span* (difference between the largest channel used and the smallest channel used) in channel assignment problems. For instance, (Mandal, 2004) and (Smith, 2000) address this problem when proposing solutions to the CAP.

The chapter is organized in the following sections:

48 Simulated Annealing – Single and Multiple Objective Problems

transition) with a low chance to get out of it.

higher cost than the current solution (Mitchell, 1996).

(Krishnamachari, 1998; Garey, 1979; Hale, 1980; Katzela, 1996).

have been proposed (Hale, 1980; Sivarajan, 1989; Box 1978; Kim, 1994).

algorithms.

of the iterative process a neighbour *s*' of each current solution *s* is proposed at random. Then *s* is only replaced by *s*' if cost does not rise. The main problems of SA are: (i) the possibility to get trapped in local minima, and (ii) the large computational load that leads to slow

Both of the algorithms described in this chapter try to solve these drawbacks. The first one, SQ, speeds up the algorithm by quickly reducing the temperature in the system, though, in the process, the advantage of SA, i.e. convergence to the global optimum, is defeated (Ingber, 1993). Due to this, SQ is termed as a greedier algorithm in terms of computational load compromising for the global optimum. Besides, the proposed method, occasionally allows moves to solutions of higher cost according to the so-called Metropolis criterion (Metropolis, 1953) (see section 4.1). However, Duque et al. (Duque, 1993) point out that the main drawback they found was to get trapped to a local minimum (by a misplaced

The problem of convergence to suboptimal solutions can be efficiently addressed with another NC-based technique, the GAs. These algorithms are based on he principle of natural selection and survival of the fittest, thus constituting an alternative method for finding solutions to highly-nonlinear problems with multimodal solutions' spaces. GAs efficiently combine explorative and exploitative search to avoid convergence to suboptimal solutions. Unlike many other approaches, GAs are much less susceptible to local optima, since they provide the ability to selectively accept successive potential solutions even if they have a

This chapter focuses on the application of these methods to solve the channel allocation problem found in cellular radio systems. In this kind of systems, the frequency reuse by which the same channels are reused in different cells becomes crucial (Gibson, 1996). Every cell is allocated a set of channels according to its expected traffic demand. The entire available spectrum is allocated to a cluster of cells arranged in shapes that allow for uniform reuse patterns. Channels must be located satisfying certain frequency separation constraints to avoid channel interference using as small a bandwidth as possible. Considering this framework, the CAP fits into the category of multimodal and NP-complete problems

The *fixed* CAP −see section 3.2− has been extensively studied during the past decades. A comprehensive summary of the work done before 1980 can be found in (Hale, 1980). When only the co-channel constraint is considered, the CAP is equivalent to an NP-complete graph coloring problem (Sivarajan, 1989). In this simpler case, various graph-theoretic approaches

From the point of view of NC, some procedures based on NNs (Funabiki, 1992; Hopfield, 1985; Kunz, 1991; Lochite, 1993) and SA or SQ (Duque, 1993; Kirkpatrick, 1983; Mathar, 1993) have already been considered. SA-SQ techniques generally improve the problem of convergence to local optima found with NNs, though their rate of convergence is rather slow (specially in SA), and a carefully designed cooling schedule is required. On the other


and GAs: advantages and drawbacks of each of them are here explained. For the sake of comparison, a set of well-known problem instances was selected since they have been used in most of the papers related to this problem, thus allowing a direct comparison.

Simulated Quenching Algorithm for Frequency Planning in Cellular Systems 51

<sup>1</sup> *CSC ACC JJ J* = + (1)

= Φ (2)

β

α

= Ψ (3)

*i j* <sup>Φ</sup> *<sup>f</sup> <sup>f</sup>* is a measure of

represents the assigned

The traffic demand is modelled by means of an *n*-length demand vector **d**=[*d1,d2,...,dn*]T, whose elements represent the number of channels required in each of the cells. For instance,

The assignment to be generated is denoted by an *n*×*c* binary matrix **A**, whose element *aij* is 1 if channel *j* is assigned to cell *i*, and 0 otherwise. This implies that the total number of 1's in

where *JCSC* and *JACC* represent, respectively, the costs due to the violations of the co-site and

,

the co-site constraint satisfaction. This parameter equals 0 only if the difference between

frequency for the αth channel of cell β, and *nα* is the number of channels in the αth cell.

On the other hand, the cost due to the adjacent channel constraint violation is obtained as

*ACC ACC i j xyy i j*

≠

*f f*

*CSC CSC i j x ii j j*

 *f f* ≠

weighs the relative importance of CSC and (,) *x x*

*i j xx <sup>f</sup>* − ≥ *f c* , and 1 otherwise. *<sup>f</sup>*

*<sup>y</sup> <sup>x</sup> <sup>n</sup> <sup>n</sup> n n x y*

(,)

(,)

*<sup>y</sup> <sup>x</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> x x*

Fig. 1 shows the four demand vectors that will be used in the simulations' section.

**Figure 1.** Traffic demand vectors for the benchmark problems considered.

The cost due to the violation of interference constraints can be written as

the adjacent channel constraints. The first one can be written as

*J*

*J*

λ

λ

row *i* of matrix **A** must be *di* (see Fig. 2).

**Figure 2.** Structure of the allocation matrix **A**.

λ

channels *i* and *j* of cell *x* is| | *x x*

where parameter *CSC*
