Question $1(10$ points$)$: EOQ Model

A tire store expects to sell approximately $100$ all$-$season tires of a certain make, model, and size next year. Annual holding cost is $$16$ per tire and ordering cost is $$20$ per order. The store operates $360$ days a year and has enough storage space in the back.

a$)$ What is the EOQ?

b$)$ How many times per year would the store reorder if EOQ units are ordered each time?

c$)$ What is the length of time between each two shipments?

Question $2(10$ points$)$: EPQ Model

HAL Ltd$.$ produces a line of high$-$capacity disk drivers for mainframe computers. It can produce the disk drive housings at the rate of $150$ housings per month. The housing cost $85$$ each to produce and the setup cost for beginning of a production run is $700$$$.$ Assume the annual interest rate of $28\%$ for determining the holding cost and the demand is $720$ units per year.

a$)$ What is the optimal number of housing for HAL Ltd$.$ to produce in each production run?

b$)$ Find the time between initiation of production runs, the time devoted to production and the downtime each production cycle.

c$)$ Calculate the maximum level of inventory in the warehouse in this case.

Question $3(15$ points$)$: Safety Inventory

Weekly demand for private label washing machines at Karstadt, a German department store chain, is normally distributed with a mean of $700$ and a standard deviation of $320.$ Karstadt currently has a supply source in China that delivers machines at a cost of $200$ euros. The lead time required by the supplier is normally distributed with a mean of $5$ weeks and a standard deviation of $2$ weeks. A European supplier has offered to deliver washing machines with a guaranteed lead time of $1\mathrm{.}5$ weeks at a cost of $250$ euros.

$1$

Karstadt has a holding cost of $20$ percent and targets a cycle service level of $94$ percent. Should Karstadt accept the local supplier's offer in terms of annual total cost?

$($Hint: calculate the holding cost related to the safety inventory for each suppler separately, then calculate the annual purchasing cost for each supplier separately; finally compare the total cost for each supplier and choose the best option$).$

Question $4(15$ points$)$: Newsvendor Model

A retail outlet sells a perishable product for $$13$ per unit. The cost of the product is $$6$ per unit. The product not sold has the salvage value of $$2.$

a$)$ Compute the overage cost and underage cost per unit.

b$)$ If the demand follows the normal distribution with an average of $350,$ and the standard deviation of $110.$ What is the optimal order quantity to maximize the expected profit?

c$)$ What is the optimal ordering quantity if the retailer uses the following discrete distribution for demand?

$\backslash$table$[[$quantity$,100,150,200,250,300,350,400,450,500],[$probability$,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}15,0\mathrm{.}15,0\mathrm{.}20,0\mathrm{.}15,0\mathrm{.}15,0\mathrm{.}05,0\mathrm{.}05]]$

Question $5(10$ points$)$: Lot$-$Sizing Problem

The time$-$phased net requirements for the base assembly in a table lamp over the next six weeks are

$\backslash$table$[[$Week$,1,2,3,4,5,6],[$Requirements$,335,200,140,440,300,200]]$

The setup cost for the construction of the base assembly is $$300,$ and the holding cost is $$0\mathrm{.}50$ per assembly per week.

a$)$ Determine the lot sizes using the Silver$-$Meal heuristic.

b$)$ Determine the lot sizes using the least unit cost heuristic.