Input: The initial temperature Output: Pareto-solution and for all and Step 1. Randomly generate a feasible solution according to the fuzzy simulation for possibilistic constraints and take it as the initial parameter for the lower level; Step 2. Solve all the multiobjective optimization problems on the lower level by SMOSA based on the fuzzy simulation and we obtain the Pareto optimal solution for all . Put , into a Pareto set of solutions and compute all objective values of the upper and lower levels; Step 3. Generate a new solution in the neighborhood of by the random perturbation; Step 4. Check the feasibility by fuzzy simulation according to all the constraints on both levels. If not, return to Step 3; Step 5. Compute the objective values on both level, respectively. Compare the generated solution with all solutions in the Pareto set and update the Pareto set if necessary; Step 6. Replace the current solution with the generated solution if is archived and go to Step 7; Step 7. Accept the generated solution as the input solution for the lower level if it is not archived with the probability: , where . If the generated solution is accepted, take it into the lower level and solve them. Then we get a new solution and put it into the Pareto set. If not, go to Step 9; Step 8. Compare and according to the evaluation function based on the compromise approach proposed by Xu and Li [15]. If is more optimal than , let . If not Step 9. Periodically, restart with a randomly selected solution from the Pareto set. While periodically restarting with the archived solutions, Suppapitnarm et al. [16] have recommended biasing towards the extreme ends of the trade-off surface; Step 10. Periodically reduce the temperature by using a problem-dependent annealing schedule Step 11. Repeat steps 2–10, until a predefined number of iterations is carried out.
Algorithm 3: FS-ISA algorithm for bi-level multi-objective programming.