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:
P ( X ≤ r + 1 ) = G ( r + 1 ) {\displaystyle P(X\leq r+1)=G(r+1)}
Since this holds for all orders, the fill rate is:
S ( r ) = G ( r + 1 ) {\displaystyle S(r)=G(r+1)}
If demand is normally distributed N ( θ , σ 2 ) {\displaystyle {\mathcal {N}}(\theta ,\,\sigma ^{2})} , the fill rate is given by:
S ( r ) = ϕ ( r + 1 − θ σ ) {\displaystyle S(r)=\phi \left({\frac {r+1-\theta }{\sigma }}\right)}
Where ϕ ( ) {\displaystyle \phi ()} 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:
I ( r ) = r + 1 − θ + B ( r ) {\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:
B a c k o r d e r s = { 0 , x < r + 1 x − r − 1 , x ≥ r + 1 {\displaystyle Backorders={\begin{cases}0,&x<r+1\\x-r-1,&x\geq r+1\end{cases}}}
The expected back order level is therefore given by:
B ( r ) = ∫ r + ∞ ( x − r − 1 ) g ( x ) d x = ∫ r + 1 + ∞ ( x − r ) g ( x ) d x {\displaystyle B(r)=\int _{r}^{+\infty }\left(x-r-1\right)g(x)dx=\int _{r+1}^{+\infty }\left(x-r\right)g(x)dx}
Again, if demand is normally distributed:[2]
B ( r ) = ( θ − r ) [ 1 − ϕ ( z ) ] + σ ϕ ( z ) {\displaystyle B(r)=(\theta -r)[1-\phi (z)]+\sigma \phi (z)}
Where z {\displaystyle 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:
T C = h I ( r ) + b B ( r ) {\displaystyle TC=hI(r)+bB(r)}
It can be proven that:[1]
G ( r ∗ + 1 ) = b b + h {\displaystyle G(r^{*}+1)={\frac {b}{b+h}}}
Where r* is the optimal reorder point.
d T C d r = h + ( b + h ) d B d r {\displaystyle {\frac {dTC}{dr}}=h+(b+h){\frac {dB}{dr}}}
d B d r = d d r ∫ r + 1 + ∞ ( x − r − 1 ) g ( x ) d x = − ∫ r + 1 + ∞ g ( x ) d x = − [ 1 − G ( r + 1 ) ] {\displaystyle {\frac {dB}{dr}}={\frac {d}{dr}}\int _{r+1}^{+\infty }(x-r-1)g(x)dx=-\int _{r+1}^{+\infty }g(x)dx=-[1-G(r+1)]}
To minimize TC set the first derivative equal to zero:
d T C d r = h − ( b + h ) [ 1 − G ( r + 1 ) ] = 0 {\displaystyle {\frac {dTC}{dr}}=h-(b+h)[1-G(r+1)]=0}
And solve for G(r+1).
If demand is normal then r* can be obtained by:
r ∗ + 1 = θ + z σ {\displaystyle r^{*}+1=\theta +z\sigma }