Problem $4(50$ points$)$

In the batic EOQ motel, when at new shipment atrives, it arrives all int once. Thut is$,$ we order some the following extousion of thasic EOQ model. Instend of arriving oll at once the juventory is replenthed gradially at a rate of $\backslash (\backslash$beta $\backslash )$ items per unt time, until the onder quatity $\backslash ($ Q $\backslash )$ is fullilled. that in renclued whon production is coasud.

$2$

$($b$)$ How long does it tahe to requinthingentory? $($Hint; Write $\backslash ($ T$\_\{1\}\backslash )$ in terms of $\backslash ($ Q $\backslash )$ and $\backslash ($ p $\backslash ).)$

$($c$)$ What is the total amoum of dormand furing the imtarval $7?$

$($e$)$ Write down an expresifon for the cycle time $\backslash (\backslash$left$($T$\_\{1\}+$T$\_\{2\}\backslash$right$)\backslash )$ in tims of $\backslash ($ Q $\backslash ).$ p$,$ dial $\backslash (\backslash$lambda $\backslash )$ and simplify.

$($f$)$ Write down an expmssian for the average inventory in terms of $\backslash ($ Q $\backslash ),\backslash ($ p $\backslash )$ and $\backslash (\backslash$lambda $\backslash ).$

$($B$)$ Assuming thint the fixed ordering $($or sel ap$)$ cost is $\backslash (\backslash$$ K $\backslash ),$ the imventory tholding cost of the item Es Sh per mit time, and the unit purchasing $($or production$)$ const is Ses

$1.$ Whint is the toral coses pur cyclo?

ii$.$ What to the total cost por mit time?

iii. What bo the optimal order quantity for this problem?