**5.4. Optimal solutions**

62 Simulated Annealing – Single and Multiple Objective Problems

Problem No.

NN-based algorithm to solve these instances (Funabiki, 1992).

Funabiki & Takefuji's NN (Funabiki, 1992)

**Table 2.** Comparison between convergence results.

**5.3. Computational complexity** 

problem 12, and 20% in problems 10 and 16).

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 (Ngo, 1998) and by Funabiki and Takefuji, who applied a

> Ngo & Li's MGA (Ngo, 1998)

10 - 21 24 15 12 23 80 86 78 14 100 100 100 100 15 77 92 90 95 16 9 99 99 98

Results show that both GA or SQ based procedures outperform the convergence results of the neural network for solving the fixed CAP. The four approaches converge properly in 100% of cases in problem 14. In problems 12, 15 and 16, both genetic methods converge more frequently than the neural network-based approach, and SQ is slightly better than GA in problem 15, while marginally worse in problems 12 and 16. In problem 15 the GA shows a little bit worse convergence results than (Ngo, 1998) (only in about 2%) while SQ moderately improves the MGA. In spite of that, the proposed method involves fewer computational load than that required by (Ngo, 1998) (see Table 3) and the complexity of the SQ method is intermediate between that of the standard GA (MGA) and that of the proposed μGA. In contrast, the μGA presents notably better convergence in problems 10 and 12, where MGA and SQ offer very similar results. In essence, in problems 12, 15 and 16

Table 3 shows the execution times required to solve these problems. Bold figures show the

It can be seen how the computational burden of the proposed method is about 20% lower than that of the standard GA by Ngo and Li (Ngo, 1998) (18% in problem 15, 23% in

On the other hand, the SQ method shows larger execution times in order to obtain similar convergence figures (as noticed in previous sections). Only in problem No. 15 SQ requires less computational load than the MGA algorithm, although, even in this problem, the μGA obtained the results faster. Notice that this reduction in the computational load observed in

algorithms exhibit very similar results, with the μGA being less complex.

CPU time normalized to the time required to solve problem 15 using the μGA.

Percentages of convergence (%)

Proposed μGA

Proposed SQ

> Now, different search techniques are compared when they run without any time constraint and an optimal solution is guaranteed. Figure 7 shows the execution times for three different algorithms: (i) the IDA (Iterative Deepening A) algorithm (Nilsson, 1998), which offers a quite simple algorithm that can solve large problems with a small computer memory, (ii) the so-called BDFS (Block Depth-Fist Search) real-time heuristic search method proposed in (Mandal, 2004), (iii) the proposed GA, and (iv) the proposed SQ method. For the sake of comparison, we have chosen the same number of cells and number of channels than in (Mandal, 2004).

> It can be seen first that the BDFS algorithm produces an increasing average speedup over the IDA method. On the other hand, the proposed μGA outperforms BDFS (and, hence, IDA) whenever the complexity of the problem becomes considerable. In these cases, the running time of the μGA is about 20% smaller than the BDFS. Only in the three simplest cases (a: *n*=5, *c*=3), (b: *n*=5, *c*=4) and (c: *n*=7, *c*=3), the minimum computational load required to implement the μGA is larger than the BDFS, though still much better than the IDA.

> When SQ is used, results show that for simple configurations, computational load is approximately that of the GA-based method. However, as complexity (in terms of the number of channels) is increased, the computational load of the SQ procedure tends towards that of the IDA algorithm. These results are in accordance with those outlined in the other numerical simulations.

Simulated Quenching Algorithm for Frequency Planning in Cellular Systems 65

(Funabiki, 2000) SQ

(Funabiki, 2000) SQ

seconds, respectively, and, finally, the SQ approach took 15.20 and 37.15 seconds, respectively (see Table 6). This means a reduction in time of 38−41% in favour of the proposed μGA method, while the NN and SQ approaches showed very similar execution

> A 855 858 855 B 1713 1724 1715

> A 11.86 16.73 15.20 B 23.76 32.80 37.15

NC-based algorithms (GA and SQ) have been proven to fit very well for solving complex NP-complete problems such as the fixed channel allocation problem. Both of them show good convergence properties and reduced computational load. We have solved 18 different benchmark instances with successful results, proving, this way, the accuracy, flexibility and robustness of the proposed methods. Making use of several well-known benchmark instances, their performances have been shown to be superior to those of the existing frequency assignment algorithms in terms of computation time, convergence properties and quality of the solution. Even when compared to one of the best previous approaches −based on a NN-based scheme−, GA and SQ methods have been able to find better solutions to the

While both the μGA and SQ offer similar computational load, convergence properties and quality of the solution for simple and moderately-simple benchmark instances, the proposed μGA shows the most reduced computational load when applied to complex problems.

This work has been partially supported by Spanish project TEC2010-21303-C04-04.

<sup>μ</sup>GA NN

<sup>μ</sup>GA NN

**Table 5.** Best assignments for benchmark instances A and B.

**Table 6.** Computation time for benchmark instances A and B.

times.

**6. Conclusions** 

**Author details** 

Luis M. San-José-Revuelta *University of Valladolid, Spain* 

**Acknowledgement** 

most complex benchmarks tested.

**Figure 7.** Execution time performance comparison between three different methods for CAP.
