(27)The Schaffer’s f6 problem represented in (28) is 2-dimensional, continuous, multimodal, and nonseparable with unknown number of many local minima. Its global minimum f5(x��)=0 is obtainable at x�� = 0:f5(x��)=��i=1d?1(0.5+sin2(xi+12+xi2)?0.5(0.001(xi+12+xi2)+1)2).(28)The Sphere problem also known as the first De Jong function is continuous, convex, unimodal, scalable, AZD9291 lung cancer and separable. It is one of the simplest test benchmark problems. Its global minimum f6(x��)=0 is obtainable at x�� = 0, and the problem is represented byf6(x��)=��i=1dxi2.(29)4.2. Parameter SettingsThe limits of particle velocity could negatively affect the performance of PSO algorithm if it is not properly set. As a result, different work has been done to determine the velocity limits of particles in order to improve on the performance of PSO.

Researches in this direction are [4, 24, 30] the three major methods that appear in the literature, for computing the velocity clamping (Vmin and Vmax ) are (i) multiplying the search space range with certain percentage (��); that is, Vmax = ��(Xmax ? Xmin ) and Vmin = ?Vmax ; (ii) multiplying both the minimum and maximum limits of the search space separately with certain percentage (��); that is, Vmax = ��(Xmax ) and Vmin = ��(Xmin ); (iii) assigning the search space upper limit to Vmax . It is obvious from (i) and (ii) that the parameter �� is very important. As a result, different values have been used by different authors [5, 6, 13, 30] for �� to determine velocity clamping for particles.

In trying to substantiate the fact that LDIW-PSO is not as weak or inferior as many authors claimed it to be, an experiment was conducted to investigate the effect of the parameter �� on the performance of LDIW-PSO using the benchmark problems described previously. The results were used as a guide to set �� in LDIW-PSO before embarking on some experimental comparison, between it and some other PSO variants described previously to prove that LDIW-PSO is superior to many of the variants that have been claimed to be better that it. The results of the experiments are listed in the Appendix. Using the results as guide, the value of �� was set in LDIW-PSO for the various test problems as listed in Table 1. However, �� was set to 0.015 for f2 in Experiment 2 and 0.25 for f3 in Experiments 2 and 5.Table 1Settings for parameter �� in LDIW-PSO.4.3. Experimental SetupThe settings for the different experiments carried out for the comparisons are described next one after Batimastat the other. In all the experiments, LDIW-PSO was subjected to the settings of its competitors as recorded in the literature. For LDIW-PSO, c1 = c2 = 2.0, ��max = 0.9, ��min = 0.4, Vmin = ��Xmin , and Vmax = ��Xmax .