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The null-space method (NSM) for solving the nonlinear programming (NLP) is used to transform an indefinite system into a symmetric positive definite one of a smaller dimension. In this paper, an NSM optimises the constrained NLP model of the inventories in multi-level supply chains (SCs). The presented NSM can give a direct method with a predictable level of fill without pivoting, decreasing the number of taken iterations to find the optimum solution. We transform the nonlinear equations into an NLP model, which NSM can solved. We also investigate the suitability of using null-space-based factorisations to derive sparse direct methods. Accordingly, an integrated lot-sizing model of the multi-level SC is designed and then optimised using the presented NSM. The paper's objectives are to find the optimum number of stockpiles and the economic period length for inventories. Some numerical examples demonstrate the applicability of the presented NSM to optimise the integrated lot-sizing policy of the multi-level SCs. The presented NSM shows satisfactory performance in optimum solutions, the number of iterations, infeasibility, optimality error, and complementarity.

An integrated lot-sizing policy for the inventory management of constrained multi-level supply chains: null-space method