Consider a deterministic inventory problem which does not allow stock out. Denote d as the demand per unit time, k as the setup cost for ordering one batch of amount o$.$ c as the unit cost of the commodity, h as the holding cost per unit of time held in inventory. $($No points are given without showing the derivation process.$)$

$($a$)$ Formulate the total cost TC per unit time if an amount @ is ordered each period, and find the optimal Q$*.$

$($b$)$ If the cost of the commodity cd is ignored $($TC $=$ TC $-$ cd$),$ show that the optimal

total cost is TC $=$ V$2$hkd$,$ and the ratio of the total cost associated with a non$-$

optimal quantity @ to that associated with the optimal quantity @$*$ iS $76=6+$

$($c$)$ If the order quantity & is twice of the optimal order quantity Q$,$ then how much percent of the total cost will be increased as compared to the optimal cost TC