Question

Problem 11-13

The Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas City. The estimated annual fixed cost and the annual capacity for the four proposed plants are as follows:

Proposed Plant	Annual Fixed Cost	Annual Capacity
Detroit	$175,000	10,000
Toledo	$300,000	20,000
Denver	$375,000	30,000
Kansas City	$500,000	40,000

The company's long-range planning group developed forecasts of the anticipated annual demand at the distribution centers as follows:

Distribution Center	Annual Demand
Boston	30,000
Atlanta	20,000
Houston	20,000

The shipping cost per unit from each plant to each distribution center is shown in table below.

A network representation of the potential Martin-Beck supply chain is shown in figure below.

Each potential plant location is shown; capacities and demands are shown in thousands of units. This network representation is for a transportation problem with a plant at St. Louis and at all four proposed sites. However, the decision has not yet been made as to which new plant or plants will be constructed.

1. Formulate a model that could be used for choosing the best plant locations and for determining how much to ship from each plant to each distribution center. There is a policy restriction that a plant must be located either in Detroit or in Toledo, but not both. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300)

Let
	y1 = 1 if a plant is constructed in Detroit; 0 if not
	y2 = 1 if a plant is constructed in Toledo; 0 if not
	y3 = 1 if a plant is constructed in Denver; 0 if not
	y4 = 1 if a plant is constructed in Kansas City; 0 if not
	xij = the units shipped in thousands from plant i to distribution center j
	i= 1,2,3,4,5, and j = 1,2,3

Min

x11

+

x12

+

x13

+

x21

+

x22

+

x23

+

x31

+

x32

+

x33

+

x41

+

x42

+

x43

+

x51

+

x52

+

x53

+

y1

+

y2

+

y3

+

y4

1. Formulate a model that could be used for choosing the best plant locations and for determining how much to ship from each plant to each distribution center. There is a policy restriction that no more than two plants can be located in Denver, Kansas City, and St. Louis. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300)

Let
	y1 = 1 if a plant is constructed in Detroit; 0 if not
	y2 = 1 if a plant is constructed in Toledo; 0 if not
	y3 = 1 if a plant is constructed in Denver; 0 if not
	y4 = 1 if a plant is constructed in Kansas City; 0 if not
	xij = the units shipped in thousands from plant i to distribution center j
	i= 1,2,3,4,5, and j = 1,2,3

Min

x11

+

x12

+

x13

+

x21

+

x22

+

x23

+

x31

+

x32

+

x33

+

x41

+

x42

+

x43

+

x51

+

x52

+

x53

+

y1

+

y2

+

y3

+

y4

Please show how to solve parts a and b using Excel

Answer 1

The shipping cost per unit from each plant to each distribution center is as follows:

Distribution Center

Plant Site Boston Atlanta Houston Fixed Cost Capacity

Detroit 5 2 3 $ 175,000 1000

Toledo 4 3 4 $ 300,000 2000

Denver 9 7 5 $ 375,000 3000

Kansas City   10 4 2 $ 500,000 4000

St.Louis 8 4 3   

Demand 30000 20000 20000

The formulation of a mixed-integer programming model that would optimize the plant location selection for the company as shown below:

Consider that,

y1 = 1 if a plant constructed in Detroit; 0 if not

y2 = 1 if a plant is constructed in Toledo; 0 if not

y3 = 1 if a plant is constructed in Denver; 0 if not

y4 = 1 if a plant is contructed in Kansas City; 0 if not

x_ij = The units shipped in thousands from plant i to distribution center j

i = 1, 2, 3, 4, 5 and j = 1, 2, 3

The formulation is as written below:

Minimize

Subject to,

Detroit Capacity

Toledo Capacity

Denver Capacity

Kansas City Capacity

St. Louis Capacity

Boston demand

Atlanta demand

Houston demand

for all i and j, binary.

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