1. state your model by clearly defining your decision variables (with appropriate units), the objective function, and all the relevant (including the non-negativity and binary if applicable) constraints.
2. you only need to give the model formulation, no solution required.

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 for orders at Hardgrave's four warehouses in Detroit, Houston, New York, and Los Angeles. As a result, the firm has decided to build a new factory to expand its productive capacity.The two sites being considered are Seattle, Washington, and Birmingham, 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