A Convex Optimization Model for Business Management

In this paper we first present a supply chain network model with four different tiers of decision makers (suppliers of raw materials, manufacturers, retailers, demand markets), we derive the optimality conditions and the associated variational inequality problem for the representatives of each level and for the total supernetwork and focusing our attention on the behavior of the manufacturers. Then, in a more detailed model, we introduce the distinction by brand of the products of manufacturers and we add the e-commerce to the traditional physical links for the shipments from manufacturers to demand markets. Moreover, to the forward chain we add a reverse chain model where manufacturers, using the unsold product given back from retailers, after reworking, produce a new commodity which will be sold to new retailers. Also in this case we study the behavior of manufacturers obtaining their optimality conditions and the governing variational formulation. Finally, we apply our model to a concrete company (Valle del Dittaino, Italy), obtaining, after introducing additional constraints, the optimal amount of raw material, the optimal shipment of new product as well as the optimal production periods.