Using the complete squares method to solve and analyze "Optimal inventory policies when the quantity backordered is uncertain"

Several researchers have recently derived formulae for economic order quantities (EOQs) with some variants without reference to the use of derivatives, neither for first-order necessary conditions nor for second-order sufficient conditions. In addition, this algebraic derivation immediately produces an individual formula for evaluating the minimum expected total annual cost. The purpose of this study is twofold. Solving Mak's (1986) model, i.e. the EOQ model taking into account the case of uncertain partial backordering, using the complete squares method, first we can readily derive global optimal expressions from a non-convex objective function in an algebraic manner, second we can straightforwardly identify some analytic cases in a way that is not as easy to do this using decomposition by projection. A numerical example has been solved to illustrate the solution procedure. Finally, concluding remarks are drawn.