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 (*DL*_{i} = 1–61), the s/n of each chosen DH (*DH*_{i} = 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 (*DL*_{i}, *DH*_{i} and *BL*) can receive a specific max value in the decimal numeral system. Specifically, 0 ≤ *DL*_{i} ≤ 61, 0 ≤ *DH*_{i} ≤ 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 (*SL*_{1} = 60 digits for DL; *SL*_{2} = 20 for DH; *SL*_{3} = 2 for BL).

Table 1

Variable . | Real decimal . | Binary . | Binary used . | Decimal used . | ||||
---|---|---|---|---|---|---|---|---|

Min . | Max . | Min . | Max . | Min . | Max . | Min . | Max . | |

DL^{a} | 1 | 61 | 000001 | 111101 | 000000 | 111111 | 0 | 63 |

DH^{b} | 1 | 3 | 01 | 11 | 00 | 11 | 0 | 3 |

BL^{c} | 1 | 4 | 01 | 11 | 00 | 11 | 0 | 3 |

Variable . | Real decimal . | Binary . | Binary used . | Decimal used . | ||||
---|---|---|---|---|---|---|---|---|

Min . | Max . | Min . | Max . | Min . | Max . | Min . | Max . | |

DL^{a} | 1 | 61 | 000001 | 111101 | 000000 | 111111 | 0 | 63 |

DH^{b} | 1 | 3 | 01 | 11 | 00 | 11 | 0 | 3 |

BL^{c} | 1 | 4 | 01 | 11 | 00 | 11 | 0 | 3 |

^{a}DL, dam location; DL, 0 means ‘no dam’ in a specific location.

^{b}DH, dam height; DH, 0 means ‘no dam’.

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

Figure 4

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