2.2.1 Enhanced Jaya Algorithm with Two Groups

Initially the population is ranked in ascending order from the best to the worst based on the fitness f(x) of each candidate solution, and then divided into two groups. Note, the better group G_b contains n_b elements and the worse group G_w of n_w elements. n_b and n_w satisfy the following equation.

Then, the mean of the better group X_h,mbetter,j and the mean of the worse group X_h,mworse,j are calculated using the following equations:

Finally, the enhanced Jaya algorithm with two groups (E-Jaya1) is updated as follows:

Algorithm 2 describes the main part of E-Jaya1.

The complete E-Jaya1 can be obtained, if algorithm 2 is substituted for lines 4 through 6 of algorithm 1.

2.2.2 An adaptive method of dividing the two groups

The ratio of the better group r_b can be defined by the following:

Based on our experiments, a reasonable r_b is within the range of [0.5, 0.9]. A notable characteristic of the Jaya algorithm is that it does not need to set any algorithm-specific control parameters, thus, to have non algorithm-specific parameters in E-Jaya is also very necessary, which means that the parameter r_b should be adaptive. A new adaptive method was developed to generate the parameter r_b. The specific approach of this method is to produce a uniformly distributed random number between 0.5 and 0.9 for the parameter r_b.

Algorithm 3 describes the adaptive method of dividing the two groups.

repeat

 generate random numbers between 0.5 and 0.9 for rb

 calculate nb based on Equation (6)

 assess their fitness

 sort the fitness in ascending order

 choose nb and nw based on Equation (2)

 calculate Xh,mbetter,j and Xh,mworse,j based on Equations (3) and (4)

  calculate 
Xh,i,j′
 based on Equation (5)

  if 
Xh,i,j′<LB∨Xh,i,j′>UB
 then

   randomly select r from U(0,1)

   Xh,i,j′←LB+r·(UB−LB)

  endif

  if 
f(Xh,i,j′)<f(Xh,i,j)
 then

   Xh,i,j=Xh,i,j′

   fmin=f(Xh,i,j′)

  endif

until stop condition is reached

return the minimum fitness fmin and the corresponding solution Xh,i,j

2.2.3 Enhanced Jaya Algorithm with a Two Group Adaption

In E-Jaya, the adaptive method of generating the random number for the parameter r_b is aware of the adaptation. The sorted candidates X_h,i,j (i=1,2,…,n) are then divided into G_b and G_w according to their fitness, with n_b and n_w elements. Finally, the mean of the better group X_h,mbetter,j and the mean of the worse group X_h,mworse,j are used in Equation (5) for candidates updating.

Algorithm 4 describes the full version of the E-Jaya algorithm.

4.1.1 Enhanced Jaya with Two Groups

To evaluate the effectiveness of the E-Jaya1, the twelve functions were tested in different dimensions, such as 40, 60 and 100, for Jaya and E-Jaya1 with different parameters n_b.

The mean errors for Jaya and E-Jaya1 with different n_b while D=40, are presented in Table 2.

Once the group strategy was employed in Jaya (Table 2), the performance of Java can be greatly enhanced, as seen at F1, F2, F3, F4, F6 and F9. F5, F7, F10, F11 and F12 were also improved. When n_b=16, the best performance was achieved. As a result, the E-Jaya1 algorithm can improve Jaya’s performance.

To find why the group strategy is good for Jaya, the curves for different values of the best, the worst, the mbetter and the mworse were shown in Fig. 1. The function is F1 and n_b=18.

Fig. 1.

The curves for values of the best, the worst, the mbetter and the mworse.

It can be seen from the Figure 1, the values of the best and the worst change dramatically for the initial 50 iterations, but the values of the mbetter and the mworse quickly moves to the optimum zero point in just less than 20 iterations. Thus, E-Jaya1 can speed up the convergence rate and increase the accuracy of the solution.

The novel method to employ the behaviors of the better swarms and the worse swarms in E-Jaya1, can improve the performance of the Jaya algorithm, since Jaya only considers the behaviors of the two individuals that are best individual and the worst individual.

The mean errors for Jaya and E-Jaya1 with different n_b while D=60, are presented in Table 3.

The results from the Table 3 indicate that the E-Jaya1 clearly increases the performance of Jaya for all functions. The best performance was obtained when n_b=12.

The mean errors for Jaya and E-Jaya1 with different n_b while D=100, are presented in Table 4.

The results from the Table 4 show that even in the high dimensions of 100, the E-Jaya1 can still significantly enhance the performance of Jaya. The accuracy of the mean solution was improved from 10⁻² to 10⁻²⁶ for F1, from 10⁻¹ to 10⁻¹⁹ for F2, from 10⁻³ to 10⁻²⁷ for F3, from 10⁻¹ to 10⁻²⁵ for F4, from 10⁻¹ to 10⁻¹⁹ for F6, from to 10¹ to 10⁻¹ for F9 ,and from 10¹ to 10⁻⁹ for F10. The best performance was obtained when n_b=14.

The results in the Table 2, Table 3 and Table 4 indicate that the E-Jaya1 significantly increases the performance of Jaya. The best performance of E-Jaya1 for different dimensions will be chosen for later comparison. The reasonable parameter r_b is between 0.5 and 0.9.

4.1.2 Enhanced Jaya Algorithm with a Two Groups Adaption

To validate the adaptive method by dividing the two groups effectively, E-Jaya was tested on the twelve functions with dimensions of 40, 60 and 100. Comparisons among Jaya, E-Jaya1 and E-Jaya were also conducted.

The statistical results with D=40 for Jaya, E-Jaya1 and E-Jaya are presented in Table 5.

The statistical results indicate that all values of E-Jaya1 are much better than that of Jaya. Using the adaptive method of dividing the two groups, E-Jaya obtains the similar results to E-Jaya1, with the exception of F7 and F10. The best results of E-Jaya for F7 and F10 are slightly worse than that of E-Jaya1, but the mean results are similar. Thus, the adaptive method of dividing the two groups is effective.

The statistical results with D=60 for Jaya, E-Jaya1 and E-Jaya are presented in Table 6.

The results from Table 6 show that all statistical values of E-Jaya1 are also much better than that of Jaya, with the exception of the best value for F7. By using the adaptive method of dividing the two groups, E-Jaya obtains about the same results compared with E-Jaya1, with the exception of F9, F1, F2, F3, F4 and F6. The mean result of E-Jaya for F9 is much worse than that of E-Jaya1, but still outperforms Jaya. The results of E-Jaya for F1, F2, F3, F4 and F6 are superior to that of E-Jaya1. Thus, E-Jaya can significantly enhance the performance of Jaya.

The statistical results with D=100 for Jaya, E-Jaya1 and E-Jaya are also presented in Table 7.

The results from Table 7 also show that E-Jaya1 significantly improves the performance of statistical values for Jaya. By using the adaptive method of dividing the two groups, E-Jaya also achieves comparable results to that of E-Jaya1, with the exception of F7, F10 and F11. The best results of E-Jaya for F7 and F11 are worse than that of E-Jaya1, and the mean result of E-Jaya for F10 is worse than that of E-Jaya1, but they are still better than that of Jaya. The results of E-Jaya for F2, F3, F4 and F6 are superior to that of E-Jaya1. Thus, E-Jaya can significantly enhance the performance of Jaya.

The results from Table 5, Table 6 and Table 7 indicate that the E-Jaya can replace the E-Jaya1. And the adaptive method by dividing the two groups is effective.

4.1.3 Comparison with Jaya Algorithm

To evaluate the performance of the E-Jaya, the E-Jaya algorithm was compared with Jaya algorithm, and the statistical results were shown in Table 5 for dimensions of 40, Table 6 for dimensions of 60 and Table 7 for dimensions of 100. The results indicate that the E-Jaya algorithm can significantly enhance the performance of the Jaya algorithm.

The comparison curves of the mean values between Jaya and E-Jaya with different dimensions of 40, 60, and 100 are shown in Figure 2.

Fig. 2.

The comparison curves between Jaya and E-Jaya with different dimensions

Note, the blue line represents the Jaya curve, and the red line represents the E-Jaya curve. The solid line is use for curves with dimensions of 40, the dashed line is used for curves with dimensions of 60, and the dotted line is used for curves with dimensions of 100.

Fig. 2 shows that the E-Jaya algorithm is superior to the Jaya algorithm in terms of the solution accuracy.

4.2.1. Mean Errors Comparison

Additionally, E-Jaya was compared with other swarm intelligent algorithms, including a differential evolution (DE), a self-adapting control parameters in differential evolution²¹ (jDE), a FA, and a SPSO2011²². The NFE and population size are the same for each function. The parameters for these algorithms are shown in Table 8.

Setting the parameters of these algorithms is a challenge, taken from Ref. 21 for DE and jDE, and Ref.22 for SPSO2011. For FA, the reduction of randomness can accelerate the convergence rate. α^t+1 = (1 − δ)a^t in Ref.23 is used to decrease α gradually. δ is a small constant associated with the iteration.

Tables 9 presents the mean errors for the five algorithms with D=40.

Ranks for the five algorithms when D=40 is shown in Table 10, based on the results from Table 9.

The results from Table 10 show that E-Jaya and jDE both rank higher (1.83) among the five algorithms.

The mean errors for the five algorithms with D=60 is presented in Table 11.

Ranks for the five algorithms when D=60 is shown in Table 18, according to the results from Table 11.

The results in Table 12 indicate that E-Jaya algorithm ranks the highest (2.00) among the five algorithms.

The mean errors for the five algorithms with D=100 is described in Tables 13.

According to the data in Table 13, Table 14 presents the ranks for the five algorithms when D=100.

The results in Table 14 also indicate that E-Jaya ranks the highest (1.92) among the five algorithms, based on the solution accuracy.

4.2.2 Runtime Cost Comparison

The runtime cost for twelve function with D=40 is compared to the above algorithms, see Fig. 3.

Fig. 3.

Comparison for runtime cost of the five algorithms with D=40.

The results in Figure 3 indicate that the time cost of E-Jaya is the least and runtime cost of the jDE is the third most expensive among five algorithms. For solution accuracy, E-Jaya and jDE rank similarly, but E-Jaya has lower runtime costs when compared to jDE. Thus, E-Jaya is better than jDE in terms of solution accuracy and runtime cost.

Fig. 4 shows the runtime cost of the five algorithms for twelve function with D=60.

Fig. 4.

Comparison for runtime cost of the five algorithms with D=60.

The results from Fig. 4 show that the time cost of E-Jaya ranks the fourth most expensive, and jDE ranks the second most expensive among the five algorithms. The time cost of FA is the least, but the difference of time cost between FA and E-Jaya is extremely small. For example, the runtime cost of FA and E-Jaya are 13.93s and 14.11s for F12, respectively. Thus, they are considered to be the similar.

Fig. 5 depicts the runtime cost of the five algorithms for twelve function with D=100.

Fig. 5.

Comparison for runtime cost of the five algorithms with D=100.

The results from Fig. 5 also indicate that the time cost of jDE is the highest, and the time cost of FA and E-Jaya are the least and the second less, respectively. Thus, for the solution accuracy, the E-Jaya algorithm ranks the highest, and for the runtime cost, it ranks the second highest.

Overall, the E-Jaya algorithm ranks the highest according to the comparison of the solution accuracy and the runtime cost of the five algorithms among the different dimensions.