A bank of ten machines requires regular periodic service.Machine running time and service time are both exponential.Machines run for an average of 44 minutes between service requirements,and service time averages 6 minutes per machine. If operators cost $15 per hour in wages and fringe benefits and machine downtime costs $75 per hour in lost production,what is the optimal number of operators for this bank of machines?
A)1
B)2
C)3
D)4
E)5

Question

Quizplus · Accepted Answer

To find the optimal number of operators, we need to minimize the total cost of the system. The total cost includes the operator cost and machine downtime cost.

Let N be the number of operators. The total cost (C) is given by:

C = N * $15 + $75 * (1/N) * (60/44)

The first part of the equation (N * $15) represents the operator cost, and the second part represents the machine downtime cost. In the second part, we divide 60 by 44 to convert the service time and machine running time from minutes to hours.

To minimize the cost, we can take the derivative of the equation with respect to N and equate it to zero:

dC/dN = 15 - (1650/44N^2) = 0

Solving this equation, we get N = 3.

Therefore, the optimal number of operators for this bank of machines is 3.

A Bank of Ten Machines Requires Regular Periodic Service