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 $$175,00040,000$

Toledo $$300,00010,000$

Denver $$375,00020,000$

Kansas City $$500,00030,000$

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 $($in dollars$)$ from each plant to each distribution center is as follows:

Distribution Centers

Plant Site Boston Atlanta Houston

Detroit $523$

Toledo $434$

Denver $975$

Kansas City $1042$

St$.$ Louis $843$

$($a$)$ Develop a mixed$-$integer programming model that could be used to help Martin$-$Beck determine which new plant or plants to open in order to satisfy anticipated demand. Solve the model and answer the following questions. What is the optimal cost?

$

What is the optimal set of plants to open?

$-$ Select your answer $-$

$($b$)$ Using equation $15\mathrm{.}1,$ find a second$-$best solution. What is the optimal set of plants to open?

$-$ Select your answer $-$

What is the increase in cost versus the best solution from part $($a$)?$

$