Base stock model

Fill rate, back-order level and inventory level

In a base-stock system inventory position is given by on-hand inventory-backorders+orders and since inventory never goes negative, inventory position=r+1. Once an order is placed the base stock level is r+1 and if X≤r+1 there won't be a backorder. The probability that an order does not result in back-order is therefore:

${\displaystyle P(X\le r+1)=G(r+1)}$

Since this holds for all orders, the fill rate is:

${\displaystyle S(r)=G(r+1)}$

If demand is normally distributed ${\displaystyle \mathcal{N}(\theta ,{\sigma }^2)}$, the fill rate is given by:

${\displaystyle S(r)=\varphi \left(\frac{r+1-\theta }{\sigma }\right)}$

Where ${\displaystyle \varphi ()}$ is cumulative distribution function for the standard normal. At any point in time, there are orders placed that are equal to the demand X that has occurred, therefore on-hand inventory-backorders=inventory position-orders=r+1-X. In expectation this means:

${\displaystyle I(r)=r+1-\theta +B(r)}$

In general the number of outstanding orders is X=x and the number of back-orders is:

${\displaystyle Backorders=\{\begin{array}{ll}0,& x<r+1\\ x-r-1,& x\ge r+1\end{array}}$

The expected back order level is therefore given by:

${\displaystyle B(r)={\int }_r^{+\mathrm{\infty }}\left(x-r-1\right)g(x)dx={\int }_{r+1}^{+\mathrm{\infty }}\left(x-r\right)g(x)dx}$

Again, if demand is normally distributed:

${\displaystyle B(r)=(\theta -r)[1-\varphi (z)]+\sigma \varphi (z)}$

Where ${\displaystyle z}$ is the inverse distribution function of a standard normal distribution.

Total cost function and optimal reorder point

The total cost is given by the sum of holdings costs and backorders costs:

${\displaystyle TC=hI(r)+bB(r)}$

It can be proven that:

${\displaystyle G(r^{\ast }+1)=\frac{b}{b+h}}$

Where r* is the optimal reorder point.

If demand is normal then r* can be obtained by:

${\displaystyle r^{\ast }+1=\theta +z\sigma }$