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 $$175,00010,000$

Toledo $$300,00020,000$

Denver $$375,00030,000$

Kansas City $$500,00040,000$

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.Problem $11-13$

Denver, or Karsas City. The estimated annual fixed cost and the annual capacity for the four proposed plants are as follows:

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

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

SHIPPING COST PER UNIT FOR THE MARTIN$-$BECK SUPPLY CHAIN

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

THE NETWORK REPRESENTATION OF THE MARTIN$-$BECK COMPANY

SUPPLY CHAIN DESIGN PROBLEM

a$.$ 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

$y_1=1if$ a plant $is$ constructed $in$ Detroit; $0if$ not

$y_2=1if$ a plant $is$ constructed $in$ Toledo; $0if$ not

$y_3=1if$ a plant $is$ constructed $in$ Denver; $0if$ not

$y_4=1if$ a plant $is$ constructed $in$ Kansas City; $0if$ not

$x_{0j}=$ the units shipped $in$ thousands from plant $ito$ distribution center $j$

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

b$.$ 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 Oity, and St$.$ Louis. For those boxes in which you must enter subtractive or negative numbers use a

minus sign. $($Example: $-300)$

Let

$y_1=1$ if a plant is constructed in Detroit; $0$ if not

$v_2=1$ if a plant is constructed in Toledo; $0$ if not

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

$y_4=1$ if a plant is constructed in Kansas City; $0$ if not

$x_i=$ the units shipped in thousands from plant $i$ to distribution center $j$

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