Optimal inventory policy for the fuzzy newsboy problem with quantity discounts

Newsboy models have wide applications in solving real-world inventory problems. This paper analyzes the optimal inventory policy for the single-order newsboy problem with fuzzy demand and quantity discounts. The availability of the quantity discount causes the analysis of the associated model to be more complex, and the proposed solution is based on the ranking of fuzzy numbers and optimization theory. By applying the Yager ranking method, the fuzzy total cost functions with different unit purchasing costs are transformed into convex, piecewise nonlinear functions. By proving certain properties of the ranking index of the fuzzy total cost, several possible cases are identified for investigation. After analyzing the relative positions between the price break and the minimums of these nonlinear functions, the optimal inventory policies are provided and closed-form solutions to the optimal order quantities are derived. Several cases of a numerical example are solved to demonstrate the validity of the proposed analysis method. The advantage of using the proposed approach is also demonstrated by comparing it to the classic stochastic approach. It is clear that the proposed methodology is applicable to other cases with different types of quantity discounts and more complicated cases.