Suppose you are the retailer in the bear game. Your client is the customer, and your supplier is the wholesaler. At each week, you need to place an order to your supplier, and fulfill the customer demand. The lead time is $2$ weeks.

You fully trust your demand forecasting model and use the forecasted demand to manage the inventory. For each unit of over$-$stocked product, the hold cost is $$0\mathrm{.}5$; while the unit cost of the backlogging order is $$1.$ Suppose the inventory maintained at the wholesaler side is always enough to fulfill your order.

The demand forecasting model is given below:

Week $12345678910$ Forecasted$44668810101212$ Demand

$($i$)$ What will be the optimal order quantity placed at week $5,$ if you $100\%$ trust the forecasted demand?

$(10$ points$)$

$($a$)4($b$)8($c$)6($d$)10($e$)12$

$($ii$)$ Suppose you received $4$ units of product at week $1$ and you also received $4$ units of product at week $2.$ Your starting cost is $0.$ Evaluate the total cost $($$$)$ at the end of Week $10$ incurred by your inventory control solution in part $($i$),$ given the below true demand table. $(10$ points$)$

Week $12345678910$ True $4444101010101414$ Demand

$($a$)8($b$)9($c$)10($d$)11($e$)11\mathrm{.}5$