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 PlantAnnual Fixed CostAnnual CapacityDetroit$$175,00040,000$Toledo$$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 CenterAnnual DemandBoston$20,000$Atlanta$30,000$Houston$20,000$

The shipping cost per unit $($in dollars$)$ from each plant to each distribution center is as follows:

Distribution CentersPlant SiteBostonAtlantaHoustonDetroit$523$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 $-$Kansas CityDetroit & ToledoToledoDetroitItem $2($b$)$Using equation $15\mathrm{.}1,$ find a second$-$best solution. What is the optimal set of plants to open?$-$ Select your answer $-$DetroitDetroit & DenverDenverDetroit & ToledoItem $3$ What is the increase in cost versus the best solution from part $($a$)?$$