- Source: Base stock model
The base stock model is a statistical model in inventory theory. 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.
Overview
= Assumptions
=Products can be analyzed individually
Demands occur one at a time (no batch orders)
Unfilled demand is back-ordered (no lost sales)
Replenishment lead times are fixed and known
Replenishments are ordered one at a time
Demand is modeled by a continuous probability distribution
= Variables
=L
{\displaystyle L}
= Replenishment lead time
X
{\displaystyle X}
= Demand during replenishment lead time
g
(
x
)
{\displaystyle g(x)}
= probability density function of demand during lead time
G
(
x
)
{\displaystyle G(x)}
= cumulative distribution function of demand during lead time
θ
{\displaystyle \theta }
= mean demand during lead time
h
{\displaystyle h}
= cost to carry one unit of inventory for 1 year
b
{\displaystyle b}
= cost to carry one unit of back-order for 1 year
r
{\displaystyle r}
= reorder point
S
S
=
r
−
θ
{\displaystyle SS=r-\theta }
, safety stock level
S
(
r
)
{\displaystyle S(r)}
= fill rate
B
(
r
)
{\displaystyle B(r)}
= average number of outstanding back-orders
I
(
r
)
{\displaystyle I(r)}
= average on-hand inventory level
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:
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
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:
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.
Total cost function and optimal reorder point
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:
Where r* is the optimal reorder point.
If demand is normal then r* can be obtained by:
r
∗
+
1
=
θ
+
z
σ
{\displaystyle r^{*}+1=\theta +z\sigma }
See also
Infinite fill rate for the part being produced: Economic order quantity
Constant fill rate for the part being produced: Economic production quantity
Demand is random: classical Newsvendor model
Continuous replenishment with backorders: (Q,r) model
Demand varies deterministically over time: Dynamic lot size model
Several products produced on the same machine: Economic lot scheduling problem
References
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