Here, we deal with the constraints as follows: generate a group of individuals randomly, then retain the individuals satisfying the constraints Ax + By ≤ r as the initial population and drop out the ones not satisfying the constraints. The individuals generated by this way all satisfy the constraints. And, the off-springs satisfy the constraints by corresponding crossover and mutation operators.

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# Design of the Fitness Function

To solve the problem (P 3) by GA, the definition of the feasible degree is firstly introduced and the fitness function is constructed to solve the problem by GA. Let d denote the large enough penalty interval of the feasible region for each(x, y) ∈ S:

# De

nition 3: Let

∈ [0, 1] denotes the feasible degree of satisfying the feasible region, and

describe it by the following function:

=

1,

1 0,

k

y Y x) d

if

ky

# Y (x)k = 0

k, if 0 < ky

# Y (x)k ≤ d

if

ky

# Y (x)k > d

where k.k denotes the norm.

Further,the fitness function of the GA can be stated as:

eval(v_{k}) = (F (x, y)

F_{min }

)∗

where F_{min }

is the minimal value of F (x, y) on S.

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Genetic Operators

The crossover operator is one of the important genetic operators. In the optimization prob- lem with continuous variable, many crossover operators appeared, such as[23]: simple crossover, heuristic crossover and arithmetical crossover. Among them, arithmetical crossover has the most popular application. The paper uses arithmetical crossover which can ensure the off-springs are still in the constraint region and moreover the system is more stable and the variance of the best solution is smaller. The arithmetical crossover can generate two off-springs which are totally linear combined by the father individuals. If v_{1 }and v_{2 }crossover, then the final off-springs are:

v_{1 }=

∗ v_{1 }+ (1

) ∗ v_{2 }

v_{2 }=

∗ v_{2 }+ (1

) ∗ v_{1 }

where

∈ [0, 1] is a random number.

is,v_{1}, v_{2 }∈ S)

The arithmetical crossover can ensure closure (that

The mutation operator is another important genetic operator in GA. Many mutation operators appeared such as[23]: uniform mutation, non-uniform mutation and boundary mutation. We adopt the boundary mutation, which is constructed for the problem whose optimal solution is

4