Strong formulations for a capacitated multi-vehicle production-distribution problem

We consider a supply chain management problem which integrates both production and distribution decisions. The supply chain is composed of one supplier production facility and several customers. The supplier is responsible for the inventory of customers and has to decide about the quantities to be delivered to these customers and which vehicles to be used. The demand of customers is deterministic over a limited planning horizon. The objective is to minimize a total cost composed of production, transportation and inventory holding costs. Tow mixed integer linear programming formulations are proposed and valid inequalities are added to strengthen these formulations. The numerical results show that the valid inequalities considerably improve the quality of the formulations.