Question

The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring, versus 2 man-hours to make one SST ring. How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP ring is $40 and on an SST ring is $35?

Answer 1

Answer:

The combination that gives the most profit is 12 VIP rings and 12 SST rings (900 $/day).

Step-by-step explanation:

This is a linear programming problem.

The objective function is profit R, which has to be maximized.

$R=40V+35S$

being V: number of VIP rings produced, and S: number of SST rings produced.

The restrictions are

- Amount of rings (less or equal than 24 a day):

$V+S\leq24$

- Amount of man-hours (up to 60 man-hours per day):

$3V+2S\leq60$

- The number of rings of each type is a positive integer:

$V, \;S\geq 0$

This restrictions can be graphed and then limit the feasible region. The graph is attached.

We get 3 points, in which 2 of the restrictions are saturated. In one of these three points lies the combination of V and S that maximizes profit.

The points and the values for the profit function in that point are:

Point 1: V=0 and S=24.

$R=40V+35S=40\cdot 0+35\cdot 24=0+840\\\\R=840$

Point 2: V=12 and S=12

$R=40V+35S=40\cdot 12+35\cdot 12=480+420\\\\R=900$

Point 3: V=20 and S=0

$R=40V+35S=40\cdot 20+35\cdot 0=800+0\\\\R=800$

The combination that gives the most profit is 12 VIP rings and 12 SST rings (900 $/day).