Problem $4-(20$ marks$)$

A public utility intends to buy a turbine as part of an expansion plan and must now decide on the number of spare

parts to order. One part, X$135,$ can be purchased for $$100$ each. Holding and disposal costs are estimated to be $145$

percent of the purchase price over the life of the turbine. A stock$-$out would cost roughly $$8,000$ due to downtime,

ordering, and "special purchase" factors. Historical records based on the performance of similar equipment

operating under similar conditions suggest that demand for the spare part will tend to approximate a Normal

distribution with a mean of $3\mathrm{.}2$ units and variance of $3\mathrm{.}2$ units for the useful life of the turbine.

a$)$ What is the optimal number of units of this spare part to order? $(10$ marks$)$

b$)$ Carrying six spare parts would be the best strategy for what $[$range of$]$ shortage cost per unit? $(10$ marks$)$