These constraint violations derive from the intrinsic problems linked to the genetic representation/encoding of the decision variables, as the GA method is applied in its classical binary form. Each chromosome represents the 10 s/n of each dam's site (DLi = 1–61), the s/n of each chosen DH (DHi = 1, 2 or 3, meaning h = 1, 1.5 or 2 m, respectively), as well as the s/n of the chosen BP location. Each variable (DLi, DHi and BL) can receive a specific max value in the decimal numeral system. Specifically, 0 ≤ DLi ≤ 61, 0 ≤ DHi ≤ 3 and 1 ≤ BL ≤ 4. Table 1 presents all max values of the variables in decimal and binary representation, the number of digits (string length) of each chromosome part representing the respective variable, as well as the respective decimal value of the max binary number of a given length. The total length of a typical binary chromosome (Figure 4) is SL = 82 (SL1 = 60 digits for DL; SL2 = 20 for DH; SL3 = 2 for BL).
Table 1

Values/number of digits of unknown variables in decimal/binary numeral representation

VariableReal decimal
Binary
Binary used
Decimal used
MinMaxMinMaxMinMaxMinMax
DLa 61 000001 111101 000000 111111 63
DHb 01 11 00 11
BLc 01 11 00 11
VariableReal decimal
Binary
Binary used
Decimal used
MinMaxMinMaxMinMaxMinMax
DLa 61 000001 111101 000000 111111 63
DHb 01 11 00 11
BLc 01 11 00 11

aDL, dam location; DL, 0 means ‘no dam’ in a specific location.

bDH, dam height; DH, 0 means ‘no dam’.

cBL, BP location; BL = {0, 1, 2, and 3} means location {1,2,3,4} = {A, B, C, and D}.

Figure 4

Structure of a typical 82-digit-long full binary chromosome representing the example random solution of Figure 5 also deals with.

Figure 4

Structure of a typical 82-digit-long full binary chromosome representing the example random solution of Figure 5 also deals with.

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