Problem $1.(25$pt$)$

Consider a hardware supply warehouse that is contractually obligated to deliver $2,000$ units of a specialized fastener to a local

manufacturing company each week. Each time the warehouse places an order for these items from its supplier, an ordering

and transportation fee of $$30$ is charged to the warehouse. The warehouse pays $$1\mathrm{.}00$ for each fastener and charges the local

firm $$4\mathrm{.}00$ for each fastener. Annual holding cost is $25$ percent of inventory value, or $o$\mathrm{.}25$ per year. Assume the

manufacturing plant operates for $50$ weeks a year.

$(1)[5$ pt$]$ How much the warehouse manager need to order when inventory gets to zero to minimize the average cost?

$(2)[10$ pt$]$ If the supplier only allows orders in thousands of units, then how much the warehouse manager need to order?

Compared with the optimal average cost in $(1),$ what is the percentage of increase in this case?

$(3)[10$ pt$]$ In an EOQ model, denote K the fixed order cost, h the holding cost per unit per day, D the constant demand per

day. Let Q$*$ be the opitmal order quantity and C$($O$*)$ be the corresponding average cost in terms of time. Prove that for any

b$>0,$

C$($bQ$*)$

$/($b$+)$

C$($Q$+)$

Hint: Average inventory cost function C$($Q$)="+"9$