Sean Skippers is in charge of load planning for cargo ships for Chase Shipping Inc. (CSI). Sean is preparing a load plan for a CSI freighter destined for China. An agricultural commodities dealer would like to transport three commodity products aboard this ship. The shipping characteristics of these commodities are shown below.

Amount Volume per Ton Profit per Ton
Commodity Available (tons) (cubic feet) ($)
1 4800 40 70
2 2500 25 50
3 2900 60 80

Sean 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 (tons) Volume Capacity (cubic feet)
Forward 3000 145,000
Center 6000 180,000
Rear 4000 155,000

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. Structure as a linear programming problem (decision variables, objective function, and constraints). You do not need to solve this problem, just structure it.