The RedBrand Company produces a tomato product at three plants. This product can be shipped directly to the company’s two customers, or it can first be shipped to the company’s two warehouses and then to the customers. Figure 5.17 is a network representing RedBrand’s problem. Nodes 1, 2, and 3 represent the plants (suppliers denoted by S), nodes 4 and 5 represent the warehouses (transshipments nodes denoted by T), and nodes 6 and 7 represent the customers (demanders denoted by D). Note that we allow the possibility of some shipments among plants, among warehouses, and among customers. Also, some arcs have arrows on both ends, which means that flow is allowed in either direction. The cost of producing is the same at each plant, so RedBrand is concerned with minimizing the total shipping cost incurred in meeting customer demands. The production capacity of each plant (in tons per year) and the demand of each customer are shown in Figure 5.17. In addition, the cost (in thousands of dollars) of shipping a ton of the product between each pair of locations is listed in Table 5.7, where a blank indicates that RedBrand cannot ship along that arc. We also assume that at most 200 tons of the product can be shipped between any two nodes. This is the common arc capacity. RedBrand wants to determine a minimum-cost shipping schedule.
Example 5.7: Crew Scheduling at Braneast Airlines
Braneast Airlines must staff the daily flights between New York and Chicago shown in the Table 5.11 below. Braneast’s crews live in either New York or Chicago. Each day, a crew must fly one New York-Chicago flight and one Chicago-New York flight with at least one hour of downtime between flights. Braneast wants to schedule crews to cover all flights and minimize the total downtime.