The catering manager of La Vista Hotel, Lisa Ferguson, is disturbed

by the amount of silverware she is losing every week. Last Friday

night, when her crew tried to set up for a banquet for $500$ people,

they did not have enough knives. She decides she needs to order

some more silverware, but wants to take advantage of any quantity

discounts her vendor will offer. For a small order or fewer

pieces$),$ her vendor quotes a price of $$\frac{1\mathrm{.}70}{?}$ piece. If she orders

$5,001-8,000$ pieces, the price drops to $$1\mathrm{.}60/$piece$.8,001-12,000$

pieces brings the price to $$1\mathrm{.}50/$piece$,$ and $12,001$ and above

reduces the price to $$\frac{1\mathrm{.}40}{?}$ piece. Lisa's order costs are $$150$ per

order, her annual holding costs are $20\%,$ and the annual demand is

$85,000$ pieces. For the best option:

a$)$ What is the optimal order quantity?

b$)$ What is the total annual cost, including ordering, holding, and

purchasing the silverware $($with the optimal order quantity$)?$

$$121,742\mathrm{.}55$

$$142,586\mathrm{.}91$

$$112,556\mathrm{.}84$

$$182,546\mathrm{.}45$