Combination Sale Problem for a Perishable Product with Fuzzy shortage cost

Abstract

Several researchers have been interested in inventory control for a perishable product. Here we introduce a combinatorial sale in order to reduce the outdating quantities. Here we consider the following problem: (1) Single perishable product with two life time period. That is, for a period, an amount of old one (remaining life time period is 1) in the stock is given as 𝑥_〓 and under this condition we should determine an ordering quantity 𝑥_〓of the fresh one (remaining life time period is 2 (2) Ordering takes a place at the start of the period. The unit purchasing cost of the product is 𝑐. (3) Issuing policy is LIFO, that is, customer buys fresh one first and if fresh one is sold out, only some percent of customers overflowed from purchase of the fresh one buy the old one. This percentage is at most 100𝑞. Unit selling price of the fresh one is 𝑟_〓 and that of old one is 𝑟_〓 satisfying 𝑟_〓〓𝑟_〓〓𝑐 (4)Prominent feature of our model is a combination sale, that is, we sell a set of products consisting of the fresh and old ones at the discount unit price 𝑟_〓 satisfying 𝑟_〓〓𝑐〓𝑟_〓〓𝑟_〓〓𝑟_〓, At most 100𝑝 percent of the customers purchasing the fresh one accept the promotion set, that is, buy the old one with the fresh one at the same time but 𝑝〓𝑞. (5)The old one that is not purchased by the customer and remained outdates and is discarded at the unit cost 𝜃. The fresh one not purchased by the customer is stocked at the unit cost ℎ and we assume that unit shortage costs is 𝐿 fuzzy number. (6)The demand 𝐷 of the customer: nonnegative random variable. Its cumulative distribution function and density function: 𝐹〓𝐷〓,𝑓〓𝐷〓〓. We introduce 𝐿 fuzzy order and we seek some non-dominated ordering quantities considering pessimistic case, most possible case and optimistic case of shortage cost.