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where l is the one side length of domain D, δ (∈ R) is the one side Euclidean length of hyper-polygon in V defined as follows:
An aspiration function for a given target design vector is represented to decide the acceptance of inpidual as follows:
Let h (r) = e−γr, r = ||Xk − Xi||, where Xi is the position of target inpidual.
To set the γ , it is assumed that the ideal conditions are satisfied: (i) Sh is full, and (ii) all members of Sh are placed in the center of the NShmax sub-domains which are supposed to have same hyper-volume and not to have any cross set of each other and to fit the domain D absolutely. That is,
Then, set as fa = β , where β means the acceptance probability criterion.
The second term on the right side corresponds to the closest member of Sh to the target inpidual. The third term, R, is the residuals. The nature of h ( r), which is exponentially decreasing along with distances, makes R be much smaller than the first term.
The aspiration criterion is as follows:
• If rand > fa then accept, where rand = [0 1]
• If trial number > maximum trial number, where, set 50
If the target inpidual is not satisfied with above aspiration criterion, one crossover is generated again and the process is repeated. The procedure is summarized as follows: \
Step 1: Read N−1 inpiduals from selection process.
Step 2: Crossover N−2 inpiduals according to the crossover probability and go to step 5.
Step 3: One inpidual selected for tabu-list.
Step 4: If rand > fa, then go to step 5, otherwise return to step 3.
Step 5: Add generated inpiduals.
Part C shown detail in Fig. 1-3 represents an RSM region. It is largely pided into 3 parts. First, considering the boundary condition in the response surface for optimization, the upper and lower values of design variables can be considered in this calculation process. However, the merits of this method are diminished when additional constraints like natural frequency are considered, because it has to evaluate the objective function to get the results from external calculations. To overcome this problem, this study used Sh as the training data and inferred the satisfaction of constraint condition by using the radial basis function (RBF) neural network [13]. In this way, the calculation of actual problems could be avoided. Second, it makes a response surface from Sh by using the least square method (LSM). Finally, the optimum solution of the response surface is calculated by using TS. The gradient-based algorithm can be used to increase the convergent speed for optimization. However, the solutions satisfying constraint conditions cannot be guaranteed since the constraint conditions are difficult to define precisely. Also, we adopt TS which has an excellent initial convergent speed, because the implementation of the response surface concept is to search for the approximate candidate solution. The generated final solution is added with other existing GA’s inpiduals according to the sequence of Fig. 1 and the calculation of fitness is performed.
2.2 Implementation procedure of IEOA
The procedure of the proposed algorithm can be summarized as follows:
Step 1: The parameters (Psize, Pc, Pm, Ms and Mc) are set up.
where Pc, Pm are crossover probability and mutation probability. Ms and Mc are selection and crossover method, respectively.
Step 2: Generate the initial chromosome vk (k = 1, 2, …, Psize) randomly with n elements.
12 [ , , , ] k k k kn v x x x =
When the chromosomes are generated, the element value range of each chromosome should be satisfied as L U kj j kjx xx ≤≤. Each chromosome satisfies all constraints gi (vk) ≥ 0, ∀i. When a chromosome does not satisfy the conditions, then the chromosome has the lowest fitness. So it has a low possibility of selection to the next generation after all.