Under the prevalent assumption of linear inventory holding and backorder costs, important results have been established in many basic inventory models. However, the assumption is often too restrictive for many applications. In this paper, we study stochastic inventory models under more general backorder costs, namely, the penalty costs for each unit of backorder consisting of fixed and proportional components. Under this backorder cost structure, we derive a necessary and sufficient condition for the quasiconvexity of the one-step loss function. This condition is satisfied by a wide spectrum of demand distributions, but it is not distribution-free. Consequently, existing results based on quasiconvex loss functions are applicable in many, but not all, models with the above cost structure.
Chen, Fangruo, and Yu-Sheng Zheng. "Inventory models with general backorder costs." European Journal of Operations Research 65, no. 2 (March 1993): 175-186.
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