Question

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

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 $100 per hour worked and craftsman time is costed at $20 per hour worked. Both machine and craftsman idle times incur no costs. The revenue received for each item produced (all production is sold) is $200 for X and $300 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.
• Solve this linear program graphically.

Please solve step by step and explain

Thank you!

Answer 1

Number of Product X=X1

Number of Product Y=X2

As, we will maximize revenue, so Objective function Z=200*X1+300*X2

Constraints:

13*X1+19*X2=<40*60 or 2400

20*X1+29*X2=<35*60 or 2100

X1>=10

Here, X1 and X2 >=0

13*X1+19*X2=<40*60 or 2400

X1	X2
0.00	126.32
184.62	0.00

20*X1+29*X2=<35*60 or 2100

X1	X2
0.00	72.41
105.00	0.00

optimal solution:

Number of Product X=X1=10

Number of Product Y=X2=65.52

Revenue=21655.17

Machine Time

13*X1+19*X2=<40*60 or 2400

13*10+19*65.52=<40*60 or 2400

Machine time is 1374.88 =1375 Minutes=22.92 hours

Crafts man time

20*X1+29*X2=<35*60 or 2100

20*10+29*65.52=<35*60 or 2100

Crafts man time=2100 Minutes=35 Hours

Machine cost =22.92*100=$2292

Crafts man =20*35=$700