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- 1-butanol-water 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)-1-Butanol (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]-ethanol-1-butanol-water system. LDIW shows the high success rate as compared to GLbestIW.

LDIW is found to decrease linearly independent of gbest and pbest/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 gbest and pbesti 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.

 
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