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The DFA was implemented in this problem with 100 fireflies, 15,000 iterations, 10 groups for categorizing the decision variables, and . The values of these parameters were chosen after a primary trial and error process. Table 2 lists the values of and the corresponding calculated after 1,500,000 functional evaluations using the obtained decision variables in the formulation of for different values of . It is found that as the value of decreases the values of and increase and decrease, respectively. In other words, the reduction of improves the desirability of the two objective functions. It is noted that the values of and reported in Table 2 are not desirable when . In this instance, the reliability and the vulnerability of the system are less than 1% and almost 60%, respectively. When it is seen that and are approximately 63 and 66 times greater and lower, respectively, than their values associated with the condition of , respectively. From Table 2, considering , the point (0.6333, 0.3959) is used as the right boundary of the OP that is calculated from the two-objective run. It is worth mentioning that this point (together with another point in the next sub-section) is used for calculating the point designated as in Figure 4. is applied with the performance criteria for evaluating the generated OP.

Table 2

Values of and the corresponding for different values of in the single-objective three-reservoir system considering as the objective function

Objective functionRate
10.750.5
 0.0084 0.1167 0.6333 
 0.6039 0.6240 0.3959 
Objective functionRate
10.750.5
 0.0084 0.1167 0.6333 
 0.6039 0.6240 0.3959 

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