Final answer:
The optimal number of operators for the bank of machines can be determined by balancing the cost of operator wages against the cost of machine downtime, using the average running and service times, and considering the costs per hour for both operators and downtime.
Step-by-step explanation:
To determine the optimal number of operators for the bank of machines, we need to balance the cost of operators against the cost of machine downtime. To calculate this, we need to know the average running time and service time for each machine, as well as the cost of operators and the cost of downtime. We can formulate the problem by setting up equations involving the average running and service times to minimize the total cost.
Each machine runs for an average of 44 minutes (0.733 hours) between service requirements, and service time averages 6 minutes (0.1 hours) per machine. Operators cost $15 per hour, and machine downtime costs $75 per hour. By calculating the machine downtime cost per minute, we find that each minute of downtime costs $1.25. Given that an operator can prevent this downtime by servicing a machine in 6 minutes, the cost of having an operator is $0.90 per minute ($15/60 minutes). It's cheaper to employ an operator for service than to incur downtime costs.
The optimal number might not be a whole number because of the continuous nature of the exponential distribution, but in practice, you would need a whole number of operators. To find the exact number, we would use queuing theory or simulation to determine the point at which the additional cost of hiring an extra operator exceeds the savings from reduced downtime.