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Bozorg-Haddad et al. (2011) calculated the global optimal NLP solution to be 308.29. In this case, the constraints and the objective function are linear-based, which means that the NLP is truly a LP problem. Table 1 lists the results for the 500,000 objective function evaluations, from which it is deduced that the average objective-function values calculated with the SOS algorithm, WCA, and the GA equaled 306.50, 304.92, and 299.70, respectively. Thus, optimization with the SOS algorithm outperformed the WCA and the GA. (1) The superiority of SOS algorithm to the optimization methods was also true for the best performance of the three EAs. The best objective function values calculated with the SOS algorithm, WCA, and the GA were 307.85, 306.92, and 300.47, respectively. (2) The best performances of the WCA and the GA converged to 99.56% and 97.46% of the global optimum solution, respectively, while the SOS algorithm in its highest performance reached 99.86% of the global optimal solution. The convergence history of the SOS algorithm and the GA based on their average performance for the four-reservoir system is illustrated in Figure 5. Figure 5 clearly indicates the superiority of the SOS over the GA. Moreover, Figure 6(a) reveals monthly releases and storages in each reservoir, respectively, on the basis of the best optimal values from the SOS algorithm and the GA. Figure 6(a) depicts appropriate compatibility between the SOS and the NLP results. The incompatibility of optimal releases between the GA and NLP for all four reservoirs is clear from Figure 6(a). Figure 6(b) shows that the incompatibility of the optimal storage calculated with the GA and NLP for the third and fourth reservoirs is higher than that between the first and second reservoirs. A slight incompatibility is also observed between the SOS algorithm and NLP's results for the third and fourth reservoirs.
Table 1

Summarized results of 10 runs of the SOS algorithm, the WCA, and the GA for the four-reservoir system

Number of runGAaWCAaSOSNLPb
300.42 306.83 305.99 308.29 
298.89 302.40 306.45  
300.09 303.65 307.30  
300.47 303.60 306.11  
298.46 302.38 307.85  
300.00 306.01 305.67  
299.22 304.05 306.86  
299.87 306.75 307.28  
299.20 306.63 305.47  
10 300.35 306.92 305.97  
Best 300.47 306.92 307.85  
Worst 298.46 302.38 305.47  
Average 299.70 304.92 306.50  
Standard deviation 0.7060 1.8863 0.7914  
Coefficient of variation 0.0024 0.0062 0.0026  
Number of runGAaWCAaSOSNLPb
300.42 306.83 305.99 308.29 
298.89 302.40 306.45  
300.09 303.65 307.30  
300.47 303.60 306.11  
298.46 302.38 307.85  
300.00 306.01 305.67  
299.22 304.05 306.86  
299.87 306.75 307.28  
299.20 306.63 305.47  
10 300.35 306.92 305.97  
Best 300.47 306.92 307.85  
Worst 298.46 302.38 305.47  
Average 299.70 304.92 306.50  
Standard deviation 0.7060 1.8863 0.7914  
Coefficient of variation 0.0024 0.0062 0.0026  

aBozorg-Haddad et al. (2015c).

bBozorg-Haddad et al. (2011).

Figure 5

The convergence of the SOS and the GA for the four-reservoir system's problem.

Figure 5

The convergence of the SOS and the GA for the four-reservoir system's problem.

Figure 6

Comparison of the SOS algorithm, and the GA in their best performance for benchmark problem in terms of (a) monthly release of the four-reservoir system and (b) variations in storage volume.

Figure 6

Comparison of the SOS algorithm, and the GA in their best performance for benchmark problem in terms of (a) monthly release of the four-reservoir system and (b) variations in storage volume.

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