Integrated production and distribution scheduling problems related with fixed delivery departure dates and number of late orders

We consider an integrated production and distribution scheduling problem faced by a typical make-to-order manufacturer which relies on a third-party logistics provider for finished product delivery to customers. In the beginning of a planning horizon, the manufacturer has received a set of orders to be processed on a single production line. Completed orders are delivered to customers by a finite number of vehicles (e.g. trucks, air freight containers on specific air flights) provided by the 3PL company which follows a fixed daily or weekly shipping schedule such that the vehicles have fixed departure dates which are not part of the decisions. The problem is to find a feasible schedule that minimizes one of the following objective functions: (1) the number of late orders, (2) the number of vehicles used subject to the condition that the number of late orders is minimum. We show that both problems are solvable in polynomial time.