Question#1 (5 marks). MaterialCo desires to blend a new alloy of 30% lead, 20 % zinc, and 50% tin from several available alloys having the following properties:

Alloy	1	2	3	4	5
Percentage lead	20	10	50	10	50
Percentage zinc	60	20	20	10	10
Percentage tin	20	70	30	80	40
Cost ($/kg)	17	12	18.5	11	18

The objective is to determine the proportions of these alloys that should be blended to produce the new alloy at a minimum cost.

Formulate the linear programming model for this problem.

Question#2 (10 marks). A cargo plane has three compartments for storing cargo: front, centre, and rear. These compartments have capacity limits on both weight and space, as summarized below:

Compartment	Weight Capacity (tons)	Space Capacity (cu ft)
Front	8	5,000
Centre	12	7,000
Rear	7	3,000

The following four cargoes have been offered for shipment on an upcoming flight of the cargo plane.

Cargo	Weight (tons)	Volume (cu ft / ton)	Profit ($/ton)
1	15	500	100
2	9	700	140
3	18	600	110
4	10	400	95

Any portion of each 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.

Formulate the linear programming model for this problem.

Bonus Question (2 marks). Furthermore in Question#2, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity in order to maintain the balance of the airplane.

Using the decision variables you have defined in Question#2, write down this additional constraint.