The present paper focuses on modeling and optimizing a four-echelon integrated supply chain, which consists of a supplier, a producer, a wholesaler and multiple retailers. These echelons interact and agree with each other on having the same period length and stockpiles for each product to make an integrated formation for minimizing the total cost of chain. Resources follow normal distributions with known means and variances. In this regard, stochastic constraints on costs of procurement or production, space and order quantity are considered, and in this, there are several singlestage products considered in shortage condition. The objectives set in this piece are to find both the number of agreed optimum stockpile and agreed optimum period length with the purpose of minimizing the total inventory cost of the chain while the stochastic constraints are met. The model used to clarify the problem is nonlinear and large, and so, the sequential quadratic programming (SQP) developed as more effective algorithm with less iteration adopted for solving the recent convex nonlinear model. A numerical example is also solved to demonstrate the application of this model and to evaluate optimum performance of the newly developed SQP algorithm. The results of the sensitivity analysis illustrate that the intended SQP algorithm has excellent performance in terms of optimality for solving research nonlinear model.
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Gharaei et al. (2017) studied this question.