An integrated reliable four-level supply chain with multi-stage products under shortage and stochastic constraints

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Abstract
In this paper, we design and optimise an integrated four-level Supply Chain (SC), which contains a supplier, a producer, a wholesaler and multiple retailers. The levels cooperate to make an Integrated SC (ISC) so that the inventory cost is minimised and the reliability is maximised, simultaneously. The model is constrained by real stochastic constraints on total space, number of orders, procurement cost, shortage cost, setup cost and production capacity. An Lp-Metric function converts the reliability function and cost function into a single-objective function to optimise the number of stockpiles and period lengths. The designed ISC is a large-scale Nonlinear Programming (NLP) and hard to solve by generic methods. Accordingly, two algorithms, entitled ‘Sequential Quadratic Programming (SQP)’ and ‘Interior Point (IP)’ with super-linear convergence rates are applied for finding the optimum solution. The performance of proposed algorithms is compared based on optimality criteria. Findings showed that the obtained solutions by SQP algorithm have better performance than IP algorithm in terms of optimality error and solution quality. However, the number of taken iterations by IP is less than SQP algorithm. Finally, the result of sensitivity analyses confirmed the excellent performance of the presented methods for solving the large-scale NLP models.

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