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The main purpose is to minimize the total inventory cost of chain, whereas the stochastic constraints are satisfied. In other words, the goal is to find optimum agreed stockpiles and period length for products to minimize the total inventory cost of the chain while the stochastic constraints are fulfilled.

This paper designs and optimizes an integrated inventory model in a four-echelon supply chain that contains a supplier, a producer, a wholesaler and multiple retailers. All four levels agree with each other to make an integrated inventory system. Products in this model have a multi-stage production process, and the model is bounded by multiple stochastic constraints. The problem model is nonlinear and large. So, the interior point method as an effective algorithm is used for solving the recent convex nonlinear model. Two numerical examples are solved to demonstrate the application of this methodology and to evaluate the performance of the proposed approach.

The findings showed the model is applicable for real-world supply chain problems in the cases that echelons are going to do executive external integration. Also, the Interior Point algorithm has a satisfactory performance and a high efficiency in terms of optimum solution for solving nonlinear and large models.

The authors designed and optimized the inventory cost in a four-level integrated supply chain in stochastic conditions. The new decision variables, number of chain levels, multi-products, stochastic constraints and multi-stage products in four-level integrated supply chain are other novelties of this paper. The authors provided an efficient algorithm for solving a large-scale and nonlinear model in this research, too.