Paul Bergey is in charge of loading cargo ships for International Cargo Company $($ICC$)$ at the port in Newport News, Virginia. Paul is preparing a loading plan for an ICC freighter destined for Ghana. An agricultural commodities dealer wants to transport the following products aboard this ship.

Commodity Amount Available Volume per ton Profit per ton

$($tons$)($cubic feet$)($$$)$

$148004070$

$225002550$

$312006060$

$417005580$

Paul can elect to load any and$/$or all of the available commodities. However, the ship has three cargo holds with the following capacity restrictions:

Cargo Hold Weight Capacity Volume Capacity

$($tons$)($cubic feet$)$

Forward $3000145,000$

Center $6000180,000$

Rear $4000155,000$

More than one type of commodity can be placed in the same cargo hold. However, because of balance considerations, the weight in the forward cargo hold must be within $10\%$ of the weight in the rear cargo hold and the ce nter cargo hold must be between $40\%$ and $60\%$ of the total weight on board.a$)$ Is the solution degenerate?

b$)$ Is the solution unique?

c$)$ How much can the profit per ton on commodity one decrease before the optimal solution would change?

d$)$ If the shipping company wanted to increase the price of transporting one of the commodities, which one would have the greatest influence on total profits?