Modeling and optimization of four levels integrated supply chain with the aims of determination number of optimum stockpile and period length: Sequential Quadratic Programming and Interior Point Algorithms

In this paper, we have modeled a four levels integrated supply chain, the first level of a supplier, the second level of a producer, the third level of a wholesaler, and the fourth level consists of multiple retailers. Four levels of mentioned supply chain cooperate with each other in order to make an integrated supply chain so be minimized total cost of supply chain. So, they have agreed with each other on having a same period length and same number of stockpile for each one of products. In this problem, there are several products as single stage and shortage is allowed for all products in mentioned model. The objectives are to find the number of optimum stockpile for each level and optimum period length so be minimized total cost of mentioned integrated model while the model constraints are satisfied. Mentioned model constraints, consist of limited procurement cost, limited space, limitation in maximum number of all products which levels can be ordered and number of orders limitation. Another realistic assumption, commonly ignored, is considering the limitation in number of stockpiles in any supply chain level. Mentioned problem model is nonlinear and large, Moreover, we have used sequential quadratic programming (SQP) and interior point methods which are powerful and effective solution methods for a wide range of nonlinear optimization problems. These methods with super-linear convergence rate help us to find optimum solution in large-scale nonlinear problems. Two numerical examples are solved in order to demonstrate the applicability of four levels integrated supply chain and to evaluate the optimum performance of SQP and interior point methods. The results show that SQP and interior point methods have satisfactory performance in terms of optimum solutions, number of iterations to achieve the optimum solution, infeasibility, optimality error, and complementarity for solving research nonlinear and large model. At the end, a sensitivity analysis is performed on the change rate of the obtained objective function based on the change rate of the products period length.