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at or near the bound of the constraint search space. And for the problem with constraints, it is proved to be very useful. If the individual vk mutates, then

vk = (vk1, vk2, . . . , vkm)

where vki is either left(vki) or right(vki) with same probability(where, left(vki), right(vki) denote the left, right bound of vki , respectively)

Selection abides by the principle: the efficient ones will prosper and the inefficient will be eliminated, searching for the best in the population. Consequently the number of the superior individuals increases gradually and the evolutionary course goes along the more optimization. There are many selection operators. We adopt roulette wheel selection since it is the simplest selection.


Termination criterions

The judgment of the termination is used to decide when to stop computing and return the result. We adopt the maximal iteration number as the judgment of the termination. The algorithm process using the GA is as follows:

S t e p 1 I n i t i a l i z a t i o n . g i v e t h e p o p u l a t i o n s c a l e M , c r o s s o v e r p r o b a b i l i t y P c , m u t a t i o n p r o b a - b i l i t y P m , t h e m a x i m a l i t e r a t i o n g e n e r a t i o n M A X G E N , a n d l e t t h e g e n e r a t i o n t = 0 ;

Step 2 Initialization of the initial population. M individuals are randomly generated in S, making up of the initial population.

Step 3 Computation of the fitness function. Evaluate the fitness value of the population according to the formula (1) .

Step 4 Generate the next generation by genetic operators. Select the individual by roulette wheel selection, crossover according to the formula (2) and mutate according to the formula (3) to generate the next generation.

Step 5 Judge the condition of the termination. When t is larger than the maximal iteration number, stop the GA and output the optimal solution. Otherwise, let t = t + 1, turn to Step 3


The Numerical Results

We proposed the following example to show the efficiency of the GA method solving the BLPP:


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