Hardgrave Machine Company produces computer components at its factories in Cincinnati, Kansas City, and Pittsburgh. These factories have not been able to keep up with demand which is 46000 units. As a result, the firm has decided to build a new factory to expand its productive capacity. The two sites being considered are Seattle in Washington and Birmingham in Alabama. Both cities are attractive in terms of labour supply, municipal services, and ease of factory financing. Table 1 presents the production costs and monthly supplies at each of the three existing factories, monthly demands at each of the four warehouses, and estimated production costs at the two proposed factories. Transportation costs from each factory to each warehouse are summarized in Table 2. In addition to this information, Hardgrave estimates that the monthly fixed cost of operating the proposed facility in Seattle would be $400,000. The Birmingham plant would be somewhat cheaper, due to the lower cost of living at that location. Hardgrave therefore estimates that the monthly fixed cost of operating the proposed facility in Birmingham would be $325,000. Note that the fixed costs at existing plants need not be considered here because they will be incurred regardless of which new plant Hardgrave decides to open - that is, they are sunk costs. The question(s) facing Hardgrave is this: Which of the new locations, in combination with the existing plants and warehouses, will yield the lowest cost? Note that the (total) unit cost of shipping from each plant to each warehouse includes both the shipping costs and the corresponding production costs. In addition, the solution needs to consider the monthly fixed costs of operating the new facility. Formulate a mixed integer linear programming (MILP) that Hardgrave will use to solve this problem. Define your decision variables carefully; write the objective function and all relevant constraints

.Table 1: Hardgrave Machine's Demand and Supply Data WAREHOUSE MONTHLY DEMAND (UNITS) PRODUCTION PLANT MONTHLY SUPPLY COST TO PRODUCE ONE UNIT Detroit 10,000 Cincinnati 15,000 $48 Houston 12,000 Kansas City 6,000 $50 New York 15,000 Pittsburgh 14,000 $52 Los Angeles 9,000 Total 46,000 Total 35,000 Note: Supply needed from new plant = 46,000 - 35,000 = 11,000 units per month.

ESTIMATED PRODUCTION COST PER UNIT AT PROPOSED PLANTS Seattle $53 Birmingham $49

Table 2: Hardgrave Machine's Shipping Costs TO FROM DETROIT HOUSTON NEW YORK LOS ANGELES Cincinnati $25 $55 $40 $60 Kansas City $35 $30 $50 $40 Pittsburgh $36 $45 $26 $66 Seattle $60 $38 $65 $27 Birmingham $35 $30 $41 $50