Abstract: With the development of cold chain and the rise of fresh food e-commerce, inventory management for fresh products gets a great deal of attention both from academic and industry areas. Different from managing durable products, the wholesaler faces a difficult challenge of managing fresh products because of their highly perishable nature. This paper focuses on a specific issue that fresh products are sold across several seasons when the wholesaler meets complicated operational risks. On one hand, the product's quality decreases all the time, resulting in a higher demand uncertainty; on the other hand, there's only one chance to purchase in the harvest period while there are several sales periods afterwards. Unlike the traditional inventory problems solving the optimal ordering quantities and times, this paper aims to figure out when and how many to sell across different periods.
Based on the assumption that the product's quality declines in a given stochastic process, a two-stage model is developed to manage the inventory in a dynamic manner. Specifically, in the first stage the wholesaler determines the total procurement quantity for the whole horizon, and in the second stage he determines the sales quantity for each selling period. The dynamic programming approach is used to solve those rationing decisions backwards. By analyzing the structural properties of the profit functions, it is proved that the optimal inventory rationing decision of each period is only determined and can be solved by the KKT condition. Furthermore, the parameter influences on the seller's different rationing decisions are analyzed, for example, with other parameters unchanged, the larger the current inventory amount is, the lower the current price is, the higher the current freshness level is or the less the perishable risk is, the larger the inventory amount rationed to the future selling periods will be. In addition, our numerical experiments show that comparing with the static policy which analyzes the multi-periods as independent markets, the dynamic policy can take the inventory pooling effect on reducing risks and increasing profits, especially when the perishability risk is higher, when the market size is bigger, or when the demand is less uncertain.
A dynamic method is presented in this paper to help the wholesaler match limited supply with multi-period demand effectively, and some meaningful insights into the inventory management of fresh products are provided. Further work is required to extend the model in more realistic conditions, such as adding the freshness-keeping effort decisions and so on.

Key words: fresh products, inventory management, cross-season sales, inventory pooling effect, rationing decisions