Optimizing of an Integrated Production-Distribution System with Probabilistic Parameters in a Multi-Level Supply Chain Network Considering the Backorder

One of the main arguments in the supply chain is integrated production-distribution planning. Integrated production and distribution of products in a supply chain plays an important role in reducing the costs of the chain. In this paper, a mathematical model for the integrated production-distribution problem in a three-level supply chain, including manufacturing plants, distribution centers and customers with the several product types and over several time periods is provided. The model objective is to minimize the total costs of the supply chain according to the exiting capacities. To consider the uncertainties involved in real problems, the investigated problem in this study, some parameters, such as model costs using the Markowitz model is assumed non-deterministic. Finally the model with probabilistic parameters using the genetic algorithm will be solved. The initial model is linear, but after becoming deterministic parameters to the stochastic parameters, the model will be nonlinear. In this study, to randomize the model parameters, first by considering a normal distribution with certain mean and variance, each of the existing costs became a stochastic function. Finally, using the Markowitz model, the deterministic model with definite parameters will become a stochastic model.