## 1 Introduction
On the other hand, it is possible to develop meta-heuristic algorithms for the solution of PDP mathematical model in complex SCS. Meta-heuristic algorithms, inspired by natural process and systems, are powerful methods for solving single or multi-objective engineering problems by means of using biological knowledge [38]. Recently, there are many successful meta-heuristic approaches that have been used to solve complex problems such as Genetic Algorithm (GA), Simulated Annealing Algorithm (SA), Ant Colony Optimization (ACO), Tabu search (TS), Particle Swarm Optimization (PSO), and etc. [39; 40; 41]. However, they are successful in obtaining feasible solutions and cannot guarantee the optimality of solutions. [PERSON] and et al. [42] proposed a mathematical model for integrated PDP and a solution algorithm using ACO. [PERSON] and et al. [43] developed a 0-1 integer mathematical model for PDP and ACO to solve the problem. [PERSON] and et al. [44] proposed an ACO approach for decentralized PDP. [PERSON] [45] proposed a mixed integer non-linear model for a PDP and solved the problem by using GA. [PERSON] and et al. [2] modeled the PDP as two nonlinear bi-level programming and proposed solution algorithms combining GA and PSO. [PERSON] and et al. [46] developed a multi-objective mixed integer mathematical model for PDP in a fuzzy environment and proposed two meta-heuristic algorithms based on GA and Concessive Variable Neighborhood Search (CVNS). [PERSON] et al. [47], proposed a 0-1 mixed integer model for PDP that aims to minimize the total cost and proposed two TS algorithms. [PERSON] [48] developed a GA for PDP in a multi plants SCS in order to allocate the jobs to the plants. [PERSON] [49] considered a PDP that handles inventory management, facility location and distribution process and proposed a GA based meta-heuristic algorithm. [PERSON] et al. [50] developed a steady-state GA to solve PDP for a SCS network design problem. In this paper, we considered a PDP for big size SCS by using centralized approach at tactical decision level in a fuzzy stochastic environment. The SCS has four echelons (suppliers, plants,warehouses, retailers) with multi-products, multi-transport paths, and multi-time periods. The PDP handled in this study has been investigated by only [PERSON] [1] in the literature according to our best knowledge. He developed a fuzzy stochastic 0-1 mixed integer mathematical model and converted it into a deterministic multi-objective mixed integer model by integrating fuzzy programming and chance-constrained programming techniques. Finally, he proposed a solution approach to solve deterministic multi-objective mix integer linear programming (MOMILP). The proposed solution approach can be successfully implemented into PDP for a small SCS. However, the mathematical model includes binary variables that select the transportation paths between the echelons of the SCS. Therefore, it cannot be solved optimally by using optimization packages when the numbers of the binary variables increase, which is a challenge. The main contribution of this study is developing meta-heuristic algorithms to deal with this challenge. Two meta-heuristic algorithms have been proposed for big size PDP in a fuzzy and stochastic environment; ACO and GA. The proposed meta-heuristic algorithms have been designed for determining routing problem (selecting transportation paths) in PDP. The solution procedure mentioned by [PERSON] [1] has been extended by integrating proposed meta-heuristic algorithms. The paper is organized as follows: the problem definition and mathematical model are given in Section 2. Section 3 presents the proposed solution approach. The computational experiment is performed in Section 4. Finally, in Section 5, we make some concluding remarks.