Gator Electric Co$.$ manufactures generators for home owners. The company$$s most popular model is GEC$1.$ Gator Electric Co$.$ manufactures these generators at their manufacturing plants located in Fort Lauderdale, Gainesville, Charlotte and Richmond and sells them through their six stores all of which are located in the Eastern United States. Table $3$ gives the unit cost of manufacturing and shipping for each plant$-$store combination along with the capacity available at each plant and the demand at each store location. Also, the company charges a different price for GEC$1$ at different store locations. These prices are also included in Table $3.$

Table $3.$ Gator Electric Co$.$ Cost, Price, Capacity and Demand Data

Store $1$ Store $2$ Store $3$ Store $4$ Store $5$ Store $6$ Supply at plant

Ft$.$ Lauderdale Plant $320350280290280320190$

Gainesville Plant $350330300300310330250$

Charlotte Plant $290290320310330290200$

Richmond Plant $280290300300290340200$

Demand at store $1502008040120160$

Price at Store$($$$)450500400380430480$

a$.$ Find a profit$-$maximizing distribution plan for Gator Electric Co$.$

b$.$ Assume that the Gainesville plant has to manufacture at least $200$ units in accordance with a labor agreement. Will this change the optimal solution you found in $($a$)$ above? If so$,$ what is the new optimal solution?

c$.$ Now go back to the original problem, i$.$e$.$ ignore the condition described in $($b$)$ above. If Charlotte plant had only $100$ units of supply, how would the solution you obtained in $($a$)$ will change?