Normativa internacional más importante aplicable a los RPAS

Multi-objective optimization algorithms are non-deterministic techniques, hence different execu- tions of the same algorithm over the same problem instances, but with different random seed, are likely to bring different results. This can lean some inconvenient when the algorithm are assessed or even when comparing different algorithms.

Although there are works that cope with the theoretical analysis of many heuristic methods and problems [150, 151], some of them are often highly complex. Consequently, they only can be compared using empirical data from multiple experimental runs.

The comparison between two multi-objective optimization algorithms are in terms of qualities of their results, that is to say, we compare their outcomes, either directly or by means of measures that summarize multiple executions of the optimizers (a value of 31 executions is a commonly adopted and accepted amount).

In this thesis, we focus on multi-objective problems. The outcome of that class of algorithm is a Pareto set. On the one hand, Pareto sets can be partially ordered according to Pareto-optimality. Consequently, some Pareto sets can be said to be better than others. On the other hand, Pareto sets are often incomparable in terms of Pareto-optimality. Hence, the analysis of multi-objective algorithms often requires more advanced techniques.

To sum up, due to the tests we carried out, we used different quality indicator since, in some cases we wanted to evaluate the convergence or the uniform diversity of a metaheuristics or in other cases, we focused on measuring the performance of parallel algorithms, in terms of computing effort, their scalability.

3.5.1

Convergence and Uniform Diversity

In multi-objective optimization, there exists different quality indicators for measuring these criteria. Convergence and diversity, in the literature has been proposed the followings: Generational Dis- tance (IGD) [152], Inverse Generational Distance (IIGD), Hypervolume (IHV )[153], Epsilon(I1+)

[154], Spread or4(I4) [88], and others. However, in this thesis we focus inIGD andIHV. Some

of them are intended to measure only the convergence or diversity, and others take into account both criteria. Figure 3.6 depicts a classification of some indicators based on which aspect they measure.

• IGD. The goal of this indicator is to measure how far the elements in the computed approx-

imation from those in the optimal Pareto front [152] an it is defined as.

IGD= pPn i=1d 2 i n (3.1)

wherenis the number of solutions in the approximation anddi is defined as, the Euclidean

distance (measured in objective space) between each of these solutions and the nearest mem- ber in the optimal Pareto front. Fronts with lower values of IGD are better. A value of

IGD = 0indicates that all the generated elements are in the Pareto front.

54 3.5. EXPERIMENTAL METHODOLOGY

Figure 3.6: Classification of quality indicators.

• IHV. This indicator calculates the volume, in the objective space, covered by members of

a non-dominated set of solutionsQ, for instance, the region enclosed into the discontinuous in Figure 3.7, Q = {A, B, C}. Mathematically, for each solution i ∈ Q, a hypercube vi

is constructed with a reference point W and the solution i as the diagonal corners of the hypercube.

Assuming a minimization problem involving dobjectives the reference point W ∈ Rd can simply be found by constructing a vector of the same dimension as the number of objective functions, where each component is the worst value found for that objective. Thereafter, a union of all hypercubes is found and itsIHV is calculated:

IHV =volume   |Q| [ i=1 vi   (3.2)

Fronts with large values of IHV are desirable. Sometimes, the IHV takes a value equal to

0, meaning that the fronts obtained by the algorithms are outside the limits of the Pareto front.

CHAPTER 3. METHODS AND MATERIALS 55

3.5.2

Speedup and Efficiency

One of the most widely used indicators for measuring the performance of a parallel algorithm is the Speedup(SN). The standard formula of the speedup is represented in Equation 3.3 and calculates

the ratio ofT1 overTN, whereT1is the running time of the analyzed algorithm in 1 processor and

TN is the running time of the parallelized algorithm onN processors (cores) 7.

SN = T1 TN (3.3) EN = SN N ×100 (3.4)

A related measure is theEfficiencyof a parallel algorithm, which is calculated with the formula of Equation 3.4. An algorithm scales linearly (ideal) when it reaches a speedupSN =N and hence,

the parallel efficiency isEN = 100%. In the execution of an algorithm with linear speedup, doubling

the number of processors means doubling the speed.

3.5.3

Computational Effort

To measure the parallel computing performance of jMetalSP framework and the Big Data platform 3.5, we have executed the algorithms, with the same setup but, using different numbers of cores and workers in order to assess the scalability of the proposals. With this methodology, we can compare running time of the sequential algorithm (usually we estimate this value) with the parallel versions. In this thesis, we have configured the cluster as it is shown in Table 3.1. We always start the tests with 100 cores since we can assess first of all whether the proposed algorithm has good performance or not. Next steps are conducted by decrementing the number of cores, following two strategies: the first one decreases the number of cores per worker and no change in the number of workers are performed; the second one, decreases the number of worker and keep the number of cores per worker. The first strategy can show the overhead of communications, we still have 10 workers consequently this increase in the runtime has not change. However, in the second setting, we reduce the number of workers and consequently the overhead time of communications, but the workers have less cores so their running times are higher.

Table 3.1: Different configurations of cluster setting to explore or test No _{Total Cores N}o _{Cores per Worker N}o _Workers

100 10 10 80 10 8 8 10 50 10 5 5 10 20 10 2 2 10 10 10 1 1 10 1 1 1