freight plane has three large compartments to carry cargo. Weight and volume limitations of these compartments are:
Compartment
Weight (Tons)
Volume (m3 )

Front
10
6800

Center
16
8700

Rear
8
5300


There are four cargos waiting to be loaded in this plane. Properties of these cargos are shown on the table below:
Cargo
Total Weight (Tons)
Total Volume (m3 )
Profit (TL/ton)

K1
18
8640
310

K2
15
9750
380

K3
13
13340
350

K4
12
4680
285


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 plane. Any proportion of these cargoes can be accepted. Formulate a linear programming model to maximize the profit by choosing how many tons of which cargo to load on the plane under these circumstances.
QUESTION TWO
A company is involved in the production of two items (X and Y). The resources need to produce X and Y are twofold, namely machine time for automatic processing and craftsman time for hand finishing. The table below gives the number of minutes required for each item:
Machine time Craftsman time
Item X 13 20
Item Y 19 29
The company has 40 hours of machine time available in the next working week but only 35 hours of craftsman time. Machine time is costed at £10 per hour worked and craftsman time is costed at £2 per hour worked. Both machine and craftsman idle times incur no costs. The revenue received for each item produced (all production is sold) is £20 for X and £30 for Y. The company has a specific contract to produce 10 items of X per week for a particular customer. Formulate the problem of deciding how much to produce per week as a linear program.