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from Bard and Falk, Bard and Moore, Hansen, Jaumard and Savard [4, 15, 16], penalty- function approach method from White and Anandalingam, Aiyoshi and Shimizu [17, 18, 19], etc.

3. The evolutionary method. It mainly includes the GA solving the Stackelberg-Nash equilib- rium [20], and the GA of Mathieu, Pittard and Anandalingam [21], etc.

The paper studies and designs the genetic algorithms of BLPP by constructing the fitness function of the upper-level programming problem based on the definition of the feasible degree. This GA avoids the use of penalty function to deal with the constraints, by changing the ran- domly generated initial population into an initial population satisfying the constraints in order to improve the ability of the GA to deal with the constraints. Finally, the numerical results of some examples indicate the feasibility and effectiveness of the proposed method

# The Mathematical Model of the BLPP

Candler and Townsley proposed the common mathematical model of BLPP[13]. Later, Bard, Bard and Falk, Bialas and Karwan proposed various mathematical models of BLPP[1, 6, 7, 15, 22]. Here, we consider the following BLPP with the form:

=

aT x + bT y

=

cT x + dT y

r 0

(P 1)maxF (x, y) where y solves (P 2)maxf(x, y) subject to Ax + By x, y

where A Rm×n , B Rm×n2 , a, c, x Rn , b, d, y Rn2 , r Rm Once x is fixed, the term cT x in the objective function of the lower-level problem is a constant. So the objective function of the lower-level problem is simply denoted as:

f = dT y

Let S = {(x, y)|Ax + By r} denote the constraint region of BLPP. Here, we assume S is

nonempty and bounded.

# Let Q(x) = {y|By ≤ r

Ax, y 0} nonempty and bounded.

Let Y (x)

denote the optimal solution set of the problem

max

y

x)

f = dT y. We assume the element of the set

Y (x) exists and is unique, then the inducible region is: ψf (S) = {(x, y)|(x, y) S, y = Y (x)} Hence, the problem (P1) can be changed into:

2

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