Problem $1.(20$ pts$)$ Talisman Games, a local gaming store, stocks the $4$th edition of

the rulebook of a popular pen$-$and$-$paper role$-$playing game $-$ Daggers and Dungeons.

Talisman orders the rulebooks directly from its publisher $-$ Wizards of the Shore

$($WotS$).$ The fixed order cost is $$20$ per order. Demand for the rulebook each week is

$4$ units with no variability. Talisman estimates that the average cost of managing the

inventories is about $$5$ per unit per week.

Q$1.$ What is Talisman's optimal inventory control policy; that is$,$ its optimal order

quantity $Q^{**}?Q^{**}$ :

Q$2.$ What is the average inventory of the rulebooks in Talisman? How long on

average does a rulebook spend in Talisman before it is sold to a customer?

$=\frac{4}{7},CT=\frac{7}{4},$INV$=CT=1$

This morning a sales rep from WotS contacted the owners of Talisman. The owners

learn that the new $5$th edition of Daggers and Dungeons will be released in exactly $12$

weeks. The sales rep told them that today was their last chance to order $4$th edition

books before they go out of print.

Talisman forecasts that starting from today, the weekly demand for the $4$th edition will

follow a normal distribution with a mean of $4$ units and a standard deviation of $2$ units

until the $5$th edition is released. Assume that the demands for each week are i$.$i$.$d$.$

Besides the cost of managing the inventories, Talisman finds that ordering the

rulebook from WotS has a cost of $$9.$ Talisman can sell to the customers at a price of

$$20.$ After $5^{th}$ edition is released, they will have to sell the $4$ th edition book by only $20\%$

of the original market price to sell the remaining units $($assume all are eventually sold

at this reduced price$).$

Q$3.$ What inventory level for $4$th edition books should Talisman target to have in the

last order?