Q$1($Quality Management$).(10$ points$)$ C$-$Spec, Inc., is attempting to determine whether an existing machine is capable of milling an engine part that has a key specification of $\backslash (4\backslash$pm $0\mathrm{.}003\backslash )$ inches. After trial runs on this machine, C$-$Spec has determined that the machine has a sample mean of $4\mathrm{.}001$ inches with a standard devition of $0\mathrm{.}002$ inch. Calcuate the $\backslash ($ C$\_\{$p k$\}\backslash )$ for this machine. Should C$-$Spec use this machine to produce this part?

Q$2.(15$ points$)$ Betty bakes and sells bagels $7$ days a week all year around. Betty has to plan and manage inventories of paper take$-$out bags with her logo printed on them to satisfy customer demand. Average daily demand for take$-$out bags is $90$ bags. Betty's printer charges her $\backslash (\backslash$$ $10\backslash )$ per order for print setup independent of order size. Bags are printed at $5$ cents $(\backslash (\backslash$$ $0\mathrm{.}05\backslash ))$ each bag. It takes $4$ days for an order to be printed and delivered. Betty has a storage room big enough to hold all reasonable quantities of bags; its operating expenses may be regarded as fixed. The only holding cost is the opportunity cost of capital, which is estimated to be $\backslash (\backslash$$ $0\mathrm{.}0125\backslash )$ per unit per year. Assume $360$ days per year.

a$.$ What is the optimal order quantity per order for Betty? $(4$ points$)$

b$.$ How many times per year does Betty need to order? $(2$ point$)$

c$.$ How many days will elapse between two consecutive orders? $(2$ point$)$

d$.$ What is Betty's minimum total annual cost of placing orders $\backslash$& carrying inventory? $(3$ points$)$

e$.$ If Betty wants to carry a safety stock of $140$ bags during the order lead time, what is her recorder point? $(4$ points$)$