4-15. A cargo plane has three compartments for storing cargo: front, center, and back. These compartments have capacity limits on both weight and space , as summarized below : Weight Space Compartment Capacity Capacity Tons) (Cubic Feet ) Front 12 7,000 Center 18 9,000 Back 10 5,000 Furthermore, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity to maintain the balance of the airplane The following four cargoes have been offered for shipment on an upcoming flight as space available : Weight Volume Profit Cargo (Tons) (Cubic Feet/Ton) ($/Ton) 20 500 320 16 700 400 A W N 25 600 360 13 400 290 Any portion of these cargoes can be accepted. The objective is to determine how much (if any) of each cargo should be accepted and how to distribute each among the compartments to maximize the total profit for the flight . (a) Formulate a complete linear programming model for this problem (b) Solve this model by the simplex method to find one of its multiple optimal solutions Solution : a.) Let X. =number of tons per cargo type I = 1,2,3,4 stowed in the compartment j = F (front ), C(center ), B (back ) X +X +X .B X2 +X,+X 2 B X, +X ,+4 3B Maximize : P = 320% 400 360 & )+ 290 %