When both demand and lead time are uncertain, we use the following formula to calculate the reorder point $($ROP$)$:

$$

$$

$$

$=$

$$

$$

$+$

$$

$\backslash$times

$($

$$

$$

$2$

$\backslash$times

$$

$)$

$+$

$($

$$

$2$

$\backslash$times

$$

$$

$2$

$)$

ROP$=$dL$+$Z$\backslash$times

$(\backslash$sigma

D

$2$

$$

$\backslash$times L$)+($d

$2$

$\backslash$times $\backslash$sigma

L

$2$

$$

$)$

$$

Where:

$$

$$

dL is the average demand during the lead time,

$$

Z is the z$-$score for the desired service level $(1\mathrm{.}65$ for $95\%),$

$$

$$

$\backslash$sigma

D

$$

is the standard deviation of demand per month $(20$ kits$),$

$$

L is the average lead time $(1\mathrm{.}5$ months$),$

$$

$$

$\backslash$sigma

L

$$

is the standard deviation of lead time $(1$ month$),$

$$

d is the average monthly demand $(400$ units$).$

Step $1$: Calculate the average lead time demand $($

$$

$$

dL$)$

Monthly demand is $400$ units.

The new lead time is $1\mathrm{.}5$ months.

$$

$$

$=$

$400$

$\backslash$times

$1\mathrm{.}5$

$=$

$600$

$$units

dL$=400\backslash$times $1\mathrm{.}5=600$units

Step $2$: Calculate the combined standard deviation $($

$$

$$

$$

$\backslash$sigma

LT

$$

$)$

$$

$$

$$

$=$

$($

$$

$$

$2$

$\backslash$times

$$

$)$

$+$

$($

$$

$2$

$\backslash$times

$$

$$

$2$

$)$

$\backslash$sigma

LT

$$

$=$

$(\backslash$sigma

D

$2$

$$

$\backslash$times L$)+($d

$2$

$\backslash$times $\backslash$sigma

L

$2$

$$

$)$

$$

$$

$$

$$

$=$

$($

$2$

$0$

$2$

$\backslash$times

$1\mathrm{.}5$

$)$

$+$

$($

$40$

$0$

$2$

$\backslash$times

$1$

$2$

$)$

$=$

$($

$400$

$\backslash$times

$1\mathrm{.}5$

$)$

$+$

$160000$

$=$

$600$

$+$

$160000$

$=$

$160600$

$$

$400\mathrm{.}75$

$\backslash$sigma

LT

$$

$=$

$(20$

$2$

$\backslash$times $1\mathrm{.}5)+(400$

$2$

$\backslash$times $1$

$2$

$)$

$$

$=$

$(400\backslash$times $1\mathrm{.}5)+160000$

$$

$=$

$600+160000$

$$

$=$

$160600$

$$

$400\mathrm{.}75$

Step $3$: Calculate the new reorder point $($ROP$)$

$$

$$

$$

$=$

$600$

$+$

$1\mathrm{.}65$

$\backslash$times

$400\mathrm{.}75$

$=$

$600$

$+$

$661\mathrm{.}24$

$=$

$1261\mathrm{.}24$

$$

$1261$

$$units

ROP$=600+1\mathrm{.}65\backslash$times $400\mathrm{.}75=600+661\mathrm{.}24=1261\mathrm{.}241261$units

Final Answer:

The new reorder point is $1261$ units to maintain a $95\%$ service level with the new, unreliable lead time.