H&I Industry produces a special machine with different production rates (pieces per hour) to meet customer specifications. A shop owner is considering buying one of these machines and wants to decide on the most economical speed (in pieces per hour) to be ordered. From past experience, the owner estimates that orders from customers arrive at the shop according to a Poisson distribution at the rate of three orders per hour. Each order averages about 500 pieces. Contracts between the owner and the customers specify a penalty of $100 per late order per hour.

(a) Assuming that the actual production time per order is exponential, develop a general cost model as a function of the production rate, m.

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(b) From the cost model in (a), determine an expression for the optimal production rate.

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(c) Using the data given in the problem, determine the optimal production rate the owner should request from H&I.