A company is making plans for a new factory. One department has been allocated 12 semiautomatic machines. A small number (yet to be determined) of operators will be hired to provide the machines the needed occasional servicing (loading, unloading, adjusting, setup, and so no). A decision now needs to be made on how to organize the operators to do this. Three alternatives are being considered:

1. Assign each operator to his or her own machines.

2. Pool the operators so that any idle operator can take the next machine that needs servicing.

3. Combine the operators into a single crew that will work together on any machine needing servicing.

The running time (time between completing service on a machine and its requiring service again) of each machine is expected to have an exponential distribution, with a mean of 150 minutes. The service time is expected to have an exponential distribution, with a mean of 15 minutes for Alternatives 1 and 2, and 15 minutes divided by the number of operators for Alternative 3.

For the department to achieve the required production rate, there must be an average of at least 10 machines running at any time.

a. Suppose that Alternative 1 is used with a total of 3 operators (4 machines per operator), on average how many machines are down (either being serviced or waiting for service)? On average how many machines are running?

b. For Alternative 1, what is the maximum number of machines that can be assigned to a single operator while still achieving the required production rate?

c. What is the minimum number of operators needed to achieve the required production rate with Alternative 2?

d. For Alternative 3, what is the minimum size of the crew needed to achieve the required production rate?

e. Which Alternative maximizes the amount of time that all 12 machines are running?