Results and Discussions
Tuning of Particle Swarm Optimization Parameters
Since PSO is a stochastic algorithm, its efficiency is tuned by different parameters like population size and inertia weight (Equations 6.17 through 6.19, 6.32 and 6.33). Initially the effect of these parameters on PSO efficiency is checked (Figure 6.3). The efficiency is analysed by the evolution of different parameters like best solution, success rate (SR), average number of iterations (AIT), mean of the best solutions (mean) and standard deviation (SD). We consider 50 individual runs of optimization for the [TDTHP][DCA]ethanol 1butanolwater system in order to tune the parameters. The SR is defined as the percentage of runs giving the best solution. AIT is defined as the mean of total iterations to achieve the best solution. Mean and SD are calculated from the solutions across all 50 runs. The number of populations (nop) considered
TABLE 6.2
Parameter Tuning and Efficiency Analysis for the [TDTHP][DCA] (l)Ethanol (2)1Butanol (3)Water (4) System at T = 298.15 K and p = 1 atm
Population Size 
GLbestIW 
LDIW 

AIT 
SR (%) 
AIT 
SR (%) 

10 
44 
34 
100 
70 
20 
41 
48 
100 
82 
30 
42 
64 
99 
86 
40 
39 
64 
100 
90 
50 
41 
82 
100 
94 
60 
40 
74 
100 
92 
70 
43 
70 
100 
94 
80 
37 
78 
100 
96 
90 
40 
80 
100 
98 
100 
40 
76 
100 
98 
for the study ranges from 10 to 100 with an increment of 10. Two inertia weight approaches (Equations 6.17 through 6.19) are studied for the optimization. Table 6.2 shows the SR and AIT for both inertia weight approaches with different populations for the [TDTHP][DCA]ethanol1butanolwater system. LDIW shows the high success rate as compared to GLbestIW.
LDIW is found to decrease linearly independent of g_{best} and p_{best/i} values (Equation 6.17), thereby giving the solution after a higher number of generations. From Figures 6.5 through 6.9, it can be seen that few local minima are evaluated far away from global minima at a higher number of generations. It indicates that a higher number of generations are required to achieve the termination criterion (Equation 6.22). GLbestIW shows very few local minima at the initial generations (Figures 6.10 through 6.14). Thus both GLbestIW and GLbestAC depend on g_{best} and p_{besti} values (Equations 6.18 and 6.19), which converge faster. AIT for LDIW was approximately 2.5 times higher than GLbestIW. Thus GLbestIW is chosen for further study considering lesser function evaluations. The population containing 10 and 20 particles shows SR less than 50% with GLbestIW, while the population of 30 and 40 particles has similar SR and AIT. Higher nop (>50) did not improve the SR and AIT significantly. So we choose a population of 30 particles for optimization as a fewer number of function evaluations are needed.