The base stock model is a statistical model in inventory theory.[1] In this model inventory is refilled one unit at a time and demand is random. If there is only one replenishment, then the problem can be solved with the newsvendor model.
L
X
g(x)
G(x)
\theta
h
b
r
SS=r-\theta
S(r)
B(r)
I(r)
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:
P(X\leqr+1)=G(r+1)
Since this holds for all orders, the fill rate is:
S(r)=G(r+1)
If demand is normally distributed
l{N}(\theta,\sigma2)
S(r)=\phi\left(
| r+1-\theta | |
| \sigma |
\right)
Where
\phi
I(r)=r+1-\theta+B(r)
In general the number of outstanding orders is X=x and the number of back-orders is:
Backorders=\begin{cases}0,&x<r+1\ x-r-1,&x\ger+1\end{cases}
The expected back order level is therefore given by:
| +infty | |
| B(r)=\int | |
| r |
\left(x-r-1
| +infty | |
| \right)g(x)dx=\int | |
| r+1 |
\left(x-r\right)g(x)dx
Again, if demand is normally distributed:[2]
B(r)=(\theta-r)[1-\phi(z)]+\sigma\phi(z)
Where
z
The total cost is given by the sum of holdings costs and backorders costs:
TC=hI(r)+bB(r)
It can be proven that:[1]
Where r* is the optimal reorder point.
If demand is normal then r* can be obtained by:
r*+1=\theta+z\sigma