It is about the variance of the reorder point process in an inventory management system.

It is assumed that the demand X during lead time has a normal distribution with mean E(X) and variance Var(X). Then, the re-order point is set equal to:

R = E(X) + Z StdDev(X)

Where

X is a RV representing the demand for one lead time

R is the re-order point i.e. the inventory level at which a replenishment order should be placed

Z is the inverse CDF of the desired service level

The service level is the probability that there will be sufficient stock to meet demand

This makes sense to me because it transposes to

[tex] SL = F(\frac {R - x} {StdDev(X)}) [/tex]

so R is the mean of this process and Z.StdDev(X) is the safety stock

The lead time is the time it takes for a replenishment order to arrive and is also a RV independant of the demand

So,

X = d x LT

Where

d is the demand per unit time process (a RV) and

LT is a RV representing the lead time and

d an LT are independant

So my understanding is that

[tex]E(X) = E(d)E(LT)[/tex]

[tex]Var(X) = Var(d)(E(LT))^2 + Var(LT)(E(d))^2 + Var(d) Var(LT) [/tex]

The result that is normally given is (due to Hadley and Whitin 1963)

[tex] R = E(d)E(LT) + Z \sqrt {E(LT)Var(d) + E(d)^2Var(LT)} [/tex]

But I can not see why.

can anyone help me out?