**4.4. Diversity and related genetic operators**

Convergence properties become notably improved with the introduction of procedures to adjust the parameters in order to achieve and maintain a good population diversity. 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 (Ursem, 2002). A high selection pressure will fill the population with clones of the fittest individuals and 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, genes spread fast throughout the population, and diversity quickly declines.

On the other hand, one of the main drawbacks of standard GAs is their excessive computational load: the application of the genetic operators is often costly and the fitness function evaluation is also a very time-consuming step. Besides, population sizes *np* normally are 100, 200 or even much higher −for instance, (Ursem, 2002) uses *np*=400 individuals. The method used in this paper works with much smaller population sizes, in the order of 10 to 20 individuals. An elite of 3 individuals is selected and the crossover and mutation probabilities depend on the Shannon entropy of the population (excluding the elite) fitness, which is calculated as

$$\mathcal{H}(\mathcal{P}[k]) = -\sum\_{i=1}^{n\_p} \rho\_i^\*(k) \log \rho\_i^\*(k) \tag{8}$$

with *ρi\**(*k*) being the normalized fitness of individual *ui*, i.e.,

58 Simulated Annealing – Single and Multiple Objective Problems

maintained in the population (Bhandari, 1996).

many strings that provide the optimal value (Bhandari, 1996).

the proposed test problem instances led to this situation.

**4.4. Diversity and related genetic operators** 

channel allocation problem solution.

reasonable amount of time.

**4.3. Elitism, termination criteria and convergence** 

assigned more than once, it solves this conflict by rearranging the conflicting elements in

The proposed algorithm also implements an elitism strategy, where the elite for **P**[k+1] is formed by selecting those individuals from both the elite of **P**[k] and the mutated elite of **P**[k] having the highest fitness value in the population. The mutation of the elite is

This elitist model of the GA 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

The whole procedure is iterated until a termination criterion is satisfied. In our simulations, the search is terminated when there are no significant changes between the maximum and minimum values of the objective function in any two successive generations. Notice that, it can not be guaranteed that a valid solution is found in a finite number of iterations. Besides, the time required to compute an optimal solution increases exponentially with the size of the problem (Beckmann, 1999; Kunz, 1991; Funabiki, 1992). Thus, it is necessary to develop approximate methods capable of finding at least a near-optimum solution within a

However, Bhandari et al. 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 and there may be

In our proposed GA, the coding scheme guarantees that the traffic demand is always satisfied. However, in hard assignment instances it can be impossible to minimize the cost function to zero, i.e., some of the interference constraints may be violated by the generated assignment. In those cases where the optimal solution is not achieved in a finite time, invalid assignments should be resolved by manually assigning more frequencies to the affected cells (thus yielding a span that is larger that the lower limit). Nevertheless, none of

At the end, the string *ui* corresponding to the highest fit chromosome is finally chosen as the

Convergence properties become notably improved with the introduction of procedures to adjust the parameters in order to achieve and maintain a good population diversity. This

each string −see (Lai, 1996) for a detailed description. Our range for *pc* is [0.35, 0.85].

performed with a probability *pm,e*=0.25*pm*. No crossover is performed on the elite.

$$\rho\_i^\*(k) = \frac{\rho\_i(k)}{\sum\_{j=1}^{n\_s} \rho\_j(k)} \tag{9}$$

When all the fitness values are very similar, with small dispersion, **H(P**[*k*]) becomes high and *pc* is decreased −it is not worthwhile wasting time merging very similar individuals. This way, the exploration character of the GA 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.

Finally, some exponentially dependence on time *k* is included in the model −making use of exponential functions− in order to relax, along time, the degree of dependence of the genetic operators' probabilities on the dispersion measure.

The complexity of the thus obtained GA 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 scheme allows the algorithm to work properly with a much smaller population size.
