This paper develops a stochastic Bat algorithm based on Monte Carlo simulation for solving stochastic multi-objective problems, where model parameters are random variables follow any arbitrary continuous distribution. The traditional Bat algorithm requires the deterministic equivalence of the problem which is only possible if the random variables follow specific distributions. However, in the developed algorithm, the stochastic model is solved directly without obtaining the deterministic equivalence. The feasibility of the stochastic constraints is checked using Monte Carlo simulation, and the developed algorithm is used to obtain the optimal solution. The developed algorithm combined the advantageous of the Bat algorithm and Monte Carlo techniques to solve complex stochastic multi-objective problems without the deterministic equivalent, which is difficult in most cases, and is tested on a numerical example that was previously solved by other algorithms and the obtained results are compared and showed that the proposed algorithm is more efficient and robust.

The max-min separable optimization problem is a special type of extremal algebra. It focuses on worst case for minimizing separable objective function with satisfied all max-min separable constraints to find the optimal solution. This problem was solved in previous studies by deterministic variables but in real situations some or all of the parameters could be described by stochastic variables. The aim of this paper is solving the problem under max-min stochastic linear constraints while keeping the decision variables as deterministic variables. The proposed algorithm solves that problem based on the concept of Monte Carlo method. The proposal algorithm can exclude almost unfeasible values of stochastic variables without need to enter the optimization process to avoid time consuming. The algorithm is effective in solving stochastic maxmin optimization problems with max-min stochastic constraints, given any random variables distribution for all constraints or any individual distribution of each constraint. The results present the optimal solution in case of environmental stochastic using proposed algorithm. The best results from stochastic model are better than the optimal result from deterministic model. The present algorithm is efficient for different applications such as electricity network problems, transportation problems, supply chain and logistics problems which can be formulated as max-min separable optimization problem under max-min stochastic constraints.

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