The Becker Company factory has been experiencing long delays in jobs going through the turret lathe department because of inadequate capacity. The head of this department contends that five machines are required, as opposed to the three machines now in place. However, because of pressure from management to hold down capital expenditures, only one additional machine will be authorized unless there is solid evidence that a second one is necessary This shop does three kinds of jobs, namely, government jobs, commercial jobs, and standard products. Whenever a turret lathe operator finishes a job, he starts a government job if one is waiting; if not, he starts a commercial job if any are waiting; if not, he starts on a standard product if any are waiting. Jobs of the same type are taken on a first-come-first-served basis.

Although much overtime work is required currently, management wants the turret lathe department to operate on an 8-hour, 5-day-per-week basis. The probability distribution of the time required by a turret lathe operator for a job appears to be approximately exponential, with a mean of 10 hours. Jobs come into the shop randomly (a Poisson process) at a mean rate of 6 per week for governments jobs, 4 per week for commercial jobs, and 2 per week for standard products. (These figures are expected to remain the same for the indefinite future.)

Management feels that the average waiting time before work begins in the turret lathe department should not exceed 0.25 (working) day for government jobs, 0.5 day for commercial jobs, and 2 days for standard products.

(a) Determine how many additional turret lathes need to be obtained to satisfy these management guidelines.

(b) It is worth about $750, $450, and $150 to avoid a delay of 1 additional (working) day in a government, commercial, and standard job, respectively. The incremental capitalized cost of providing each turret lathe (including the operator and so on)

is estimated to be $250 per working day. Determine the number of additional turret lathes that should be obtained to minimize the expected total cost.