Optimal cooperative harvesting patterns of agricultural fresh products in case of multiple farmers and multiple markets under periodical flowering

How to make a coordination policy between suppliers is one of key issues in Supply Chain Management. This paper deals with a case where multiple farmers harvest and deliver agricultural fresh products to multiple markets in proportion to each market size provided that the plants related to the fresh products get flowering periodically like tropical fruits such as papaya. A cooperative model is formulated in a mathematical form to obtain the optimal harvesting patterns for multiple farmers who harvest the fresh products cooperatively to maximize the consumption level of fresh products daily used in multiple markets. Although this model becomes a kind of mixed integer linear programming problem hard to solve in general, this paper reduces it into a simple LP problem easy to solve, exploiting some properties of optimal harvesting patterns analytically obtained in an individual un-cooperative model.Numerical analyses provide optimal harvesting patterns for cooperative multiple farmers, and make it clear that the cooperation effect depends on the delivery lead times from multiple farmers to multiple markets and the shift periods between flowering cycles among farmers.In a two-farm, two-market model, it is also shown that the increment of the consumption level of fresh products in the cooperative model compared with that in the individual un-cooperative model becomes largest when the shift period between flowering cycles among two farmers is just half a flowering cycle.