$38.(20$ points$)$ FlexSteal, marketed as a miracle brush$-$on sealant and baldness cure, is

manufactured at three plants that are struggling to keep up with the demand at the

four regional centers. The cost to ship a truckload from each of the existing plants to

the regional centers is shown in the table. Note that the capacity and the total demand

are both measured in truckloads.

Philadelphia Atlanta Dallas Los Angeles Capacity

Plant $1897593115400$

Plant $21208810393400$

Plant $3958211298400$

Total Demand $350350350350$

The supply chain manager has explored plant construction costs in two other cities as

well as the cost to ship a truckload from each of the plants to the regional centers. The

current plan is to build another plant with a capacity of $400$ to allow room for sales

growth. The construction cost for the new plants and cost to ship to each of the

regional centers is shown here.

Philadelphia Atlanta Dallas Los Angeles Cost to Build

Plant A $94789595$ $$23,500$

Plant B $95888894$ $$26,000$

$($a$)(5$ points$)$ What is the lowest cost solution for the existing set of plants and

regional centers?

$($b$)(5$ points$)$ Setup objective function.

$($c$)(10$ points$)$ Setup constraints.