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Nevertheless, ADMM method can obtain a satisfied solution for the non-convex PR problem [14, 19].
ADMM technology was utilized for solving the resulting problem, and a satisfied solution is obtained.
Nevertheless, ADMM technique, which can obtain a satisfied solution to the PR problem [13, 14], is utilized in this paper.
Since both of the variables x and z are updated at each inner step, it seems justified to conclude that a satisfied solution will be obtained after just a few steps.
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Suppose that when the reference value of membership function is ( μ 1, μ 2 ) = ( 1, 0.90 ), the decision maker is satisfied, then the interactive process is stopped, so we obtain that the satisfied solutions are x 121 = 1.34, x 122 = 7.16, x 131 = 5.51, x 212 = 10.51.
There are not satisfied solutions to solve the problem.
It could reach the satisfied solutions with high convergence speed in an acceptable computational time.
In the lower model, the objective is to minimize the total transfer time of passengers in every connection stop, and the synchronization coefficient describing the cases of lines-crossing in one connection stop and the satisfaction criteria, which can select a set of the satisfied solutions for the upper model to decide, are defined in this article.
Here we solve Model-1, to get the satisfied solutions, which are listed in Table 1.
Thus in the case ϕ ≤ θ (i.e., if condition C is not satisfied), the solution of problem (61) obtained by using bracketing method[30, 31] lies at most away from the optimal solution.
As in [12], we suppose that the following conditions are satisfied: (1) The solution set of the MSSFP is nonempty.
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