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

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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

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