A membership function approach to lot size re-order point inventory problems in fuzzy environments

Inventory models play an important role in logistics and supply chain management for reducing cost and increasing customer satisfaction. This paper develops an approach to derive the fuzzy objective value and decision variables of the fuzzy lot size re-order point inventory problem with parameters being fuzzy numbers and the shortages are backordered with extra cost incurred. Different from the existing studies, the idea is based on Zadeh's extension principle. A pair of mixed integer nonlinear programs (MINLP) parameterised by the possibility level α is formulated to calculate the lower and upper bounds of the minimal total cost per unit time at α, through which the membership function of the minimal total cost per unit time is constructed. At the same time the membership functions of the optimal order quantity and the optimal re-order point are also provided. A numerical example studied by previous studies is solved successfully to demonstrate the validity of the proposed method. Compared with previous studies, the obtained results which precisely and completely conserve the fuzziness of the input information are more informative for finding the best inventory policy since they are expressed by membership functions rather than by crisp ones. Moreover, to provide representative crisp solutions for designing inventory systems, the Yager's ranking index method is adopted to defuzzify the obtained membership functions. The successful extension of inventory models to fuzzy environments permits inventory models to have wider applications in practice.