Using a computerized Inventory Management System, a Paint Supply Store franchise continuously monitors the inventory of all the paint located at each of their $15$ stores and their distribution warehouse. The Paint Supply Store franchise sells an average of $32$ gallons of Red Paint every week $($for $52$ weeks per year$).$ They purchase Red Paint from their supplier at a price of $$2\mathrm{.}00$ per gallon. $[$The company does not hold Safety Stock$]$

It takes $1\mathrm{.}25$ weeks to receive an order from the supplier.

Administrative costs for Ordering paint have been estimated to be $$15$ per order.

Holding Costs $=20\%$ of the purchase price per gallon per year.

At what inventory level should the company place an order?

Reorder Point, ROP $=dL+SS$

d $=$ Average Demand Rate per Time Period

$D=$ Demand Rate per Year

$L=$ Average Lead time $($time periods$)$

SS $=$ Safety Stock $=z{}_{dL}$

${}_{dL}=\sqrt[\phantom{2}]{L{}_d^2}$

$\backslash$table$[[$Service Level,$z$ Value$],[99\mathrm{.}99\%,3\mathrm{.}719],[99\mathrm{.}90\%,3\mathrm{.}090],[99\mathrm{.}00\%,2\mathrm{.}326],[95\mathrm{.}00\%,1\mathrm{.}645],[90\mathrm{.}00\%,1\mathrm{.}282],[85\mathrm{.}00\%,1\mathrm{.}036],[80\mathrm{.}00\%,0\mathrm{.}842]]$

$($Assuming Lead Time is constant as in Periodic review$)$

${}_d=$ Standard Deviation of the demand

$z=$ Number of Standard deviations corresponding to a service level

$Q=$ Order Quantity

Average Inventory Level $=\frac{0}{2}+SS$

$H=$ Holding Cost Per Year Per Unit

$S=$ Ordering$/$Setup Cost Per Order

Total Annual Inventory Cost $=(\frac{Q}{2})H+(\frac{D}{Q})S+H(SS)$

Economic Order Quantity, EOQ $=\sqrt[\phantom{2}]{\frac{2DS}{H}}$ Time Between

Orders $=$TBO$=\frac{\text{E}\text{O}\text{Q}}{D}$