You own a factory that make metal patio sets using 2 process

You own a factory that make metal patio sets using 2 processes. The hours of unskilled labor, machine time and skilled labor are Process A - 10 hrs unskilled labor, 1 hr machine time and 5 hrs skilled labor. Process B - 1 hr unskilled labor, 3 hrs machine time and 2 hrs skilled labor. You use up to 4000 hrs of unskilled labor, up to 1500 hrs of machine time and up to 2300 hrs of skilled labor. How many patios sets can you make by each process? Which solutions in the table below are viable or not viable? Please show formula used to determine viability for each.

Solution

Let the number of metal patio sets made using the processes A and B be x and y respectively. Then the number of hours of unskilled labor, machine time and skilled labor used in making x number of metal patio sets by process A are 10x, x and 5x respectively. Similarly, the number of hours of unskilled labor, machine time and skilled labor used in making y number of metal patio sets by process B are y, 3y and 2y respectively. up to 4000 hours of unskilled labor, up to 1500 hours of machine time and up to 2300 hours of skilled labor are available, hence 10x 4000, x 1500 and 5x 2300. Now, x must be the least of the 3 values given by these 3 inequalities. Therefore, x 400. (5x 2300 means x 460). Thus, upto 400 metal patio sets can be made using the process A if only this process is used. Further, for a similar reason, y 4000, 3y 1500 and 2y 2300 so that y 500. This means that upto 500 metal patio sets made using the process B if only this process is used.

If both the processes A and B are used simultaneously, then we have 10x +y 4000, x +3y 1500 and 5x+2y 2300.

Process A

Process B

Valid or not

400

0

Yes

380

200

Yes

400

200

No

350

300

No

300

400

Yes

0

500

Yes

150

450

Yes

250

400

Yes

200

450

No

350

250

Yes

Note:

We have to substitute the value of x in the inequalities 10x +y 4000, x +3y 1500 and 5x+2y 2300 simultaneously and then compute the least value of y.