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The main aim of the multi-objective design of a WDS is to find a solution which is optimal with respect to the single objective functions. Hence, it is necessary to select one of the solutions on the Pareto set to report it as the final result of the multi-objective design. In this regard, there are different conflict resolution methods to achieve this selection. This study implements the method of Young (1993), in which a desirability or utility function is distributed for each objective function and one of them is chosen by maximizing a mathematical equation based on the gradient of different points of the Pareto (Young 1993). By using this method, point C (Figure 4) is chosen and differentiated by a triangle in Figure 4. The minimum cost solution, maximum FRI solution, and the selected point by Young's (1993) model have been named A, B, and C, respectively, in Figure 4. Their corresponding pipe diameters, pressure head, and FRI values are reported in Tables 1 and 2. Also, the values of both objective functions are shown in these tables. By comparing these values it is seen that the reliability of solution C is 40 times that of solution A, while its cost is only 2.5 times of the cost associated with solution A. A similar comparison on points C and B indicates that solution C achieves 80% of the reliability of solution B while its cost is only a third (0.33) of that of B.

Table 1

The pipe diameters for solutions A, C, and B in the two-loop network

PointACB
Pipe no.D (cm)D (cm)D (cm)
45.72 45.72 45.72 
25.40 50.80 69.96 
40.46 50.80 69.96 
10.16 35.56 69.96 
40.64 45.72 69.96 
25.40 45.72 69.96 
25.40 50.80 69.96 
2.54 45.72 69.96 
Cost ($) 419,000 1,090,000 3,980,000 
PointACB
Pipe no.D (cm)D (cm)D (cm)
45.72 45.72 45.72 
25.40 50.80 69.96 
40.46 50.80 69.96 
10.16 35.56 69.96 
40.64 45.72 69.96 
25.40 45.72 69.96 
25.40 50.80 69.96 
2.54 45.72 69.96 
Cost ($) 419,000 1,090,000 3,980,000 
Table 2

The nodal pressure heads and FRI indices for solutions A, C, and B in the two-loop network

PointA
C
B
NodeFRIHead (m)FRIHead (m)FRIHead (m)
0.34 53.25 0.40 53.25 0.38 53.25 
0.04 30.46 0.76 42.41 0.79 42.93 
0.60 43.45 0.66 47.19 0.73 47.78 
0.10 33.81 0.36 51.88 0.37 52.74 
0.02 30.44 0.31 36.55 0.36 37.64 
0.02 30.55 0.63 41.55 0.69 42.64 
FRI obj. fun. 0.0234 0.9749 1.2055 
PointA
C
B
NodeFRIHead (m)FRIHead (m)FRIHead (m)
0.34 53.25 0.40 53.25 0.38 53.25 
0.04 30.46 0.76 42.41 0.79 42.93 
0.60 43.45 0.66 47.19 0.73 47.78 
0.10 33.81 0.36 51.88 0.37 52.74 
0.02 30.44 0.31 36.55 0.36 37.64 
0.02 30.55 0.63 41.55 0.69 42.64 
FRI obj. fun. 0.0234 0.9749 1.2055 

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