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.
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:
Since this holds for all orders, the fill rate is:
If demand is normally distributed
, the fill rate is given by:
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:
In general the number of outstanding orders is X=x and the number of back-orders is:
The expected back order level is therefore given by:
Again, if demand is normally distributed:[2]
( r ) = ( θ − r ) [ 1 − ϕ ( z ) ] + σ ϕ ( z )
is the inverse distribution function of a standard normal distribution.
The total cost is given by the sum of holdings costs and backorders costs:
Where r* is the optimal reorder point.
To minimize TC set the first derivative equal to zero:
{\displaystyle {\frac {dTC}{dr}}=h-(b+h)[1-G(r+1)]=0}
If demand is normal then r* can be obtained by:
+ 1 = θ + z σ