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The NLP solution was equal to 1.213 (Bozorg-Haddad et al. 2015c). The superiority of an EA over other ones can be obtained based on the similarity of its near-optimal solution to the global optimal solution. Each run of the SOS algorithm lasted approximately 30 seconds. Table 2 demonstrates the performance of the SOS algorithm vs. GA and WCA based on 10 runs. The main results to emerge from Table 2 are as follows. (1) The GA in its best performance converged to 1.535, while the SOS algorithm reached the value 1.240 in its best performance. The SOS also outdid the WCA, which converged to 1.260 in its best run. (2) It is noteworthy that even the worst performance of the SOS algorithm was better than the best performance of the GA. (3) The coefficient of variation of the GA's solutions was almost five times greater than that of the SOS algorithm. Yet, on the basis of standard deviation and coefficient of variation, the WCAs' runs exhibited smaller variability than the SOS algorithm. The coefficients of variation of the SOS algorithm and WCA were close to zero.

Table 2

Summarized results of 10 runs of the SOS algorithm, the WCA, and the GA for the Karun4 problem

Number of runGAaWCAaSOSNLPa
1.673 1.289 1.257 1.213 
1.549 1.269 1.253  
1.865 1.287 1.240  
1.752 1.260 1.291  
1.987 1.289 1.242  
1.753 1.285 1.245  
1.931 1.281 1.248  
1.57 1.279 1.314  
1.842 1.286 1.272  
10 1.535 1.262 1.248  
Best 1.535 1.260 1.240  
Worst 1.987 1.289 1.314  
Average 1.746 1.279 1.261  
Standard deviation 0.162 0.010 0.024  
Coefficient of variation 0.093 0.008 0.019  
Number of runGAaWCAaSOSNLPa
1.673 1.289 1.257 1.213 
1.549 1.269 1.253  
1.865 1.287 1.240  
1.752 1.260 1.291  
1.987 1.289 1.242  
1.753 1.285 1.245  
1.931 1.281 1.248  
1.57 1.279 1.314  
1.842 1.286 1.272  
10 1.535 1.262 1.248  
Best 1.535 1.260 1.240  
Worst 1.987 1.289 1.314  
Average 1.746 1.279 1.261  
Standard deviation 0.162 0.010 0.024  
Coefficient of variation 0.093 0.008 0.019  

aBozorg-Haddad et al. (2015c).

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