Finance:Base stock model
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.
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
- [math]\displaystyle{ L }[/math] = Replenishment lead time
- [math]\displaystyle{ X }[/math] = Demand during replenishment lead time
- [math]\displaystyle{ g(x) }[/math] = probability density function of demand during lead time
- [math]\displaystyle{ G(x) }[/math] = cumulative distribution function of demand during lead time
- [math]\displaystyle{ \theta }[/math] = mean demand during lead time
- [math]\displaystyle{ h }[/math] = cost to carry one unit of inventory for 1 year
- [math]\displaystyle{ b }[/math] = cost to carry one unit of back-order for 1 year
- [math]\displaystyle{ r }[/math] = reorder point
- [math]\displaystyle{ SS=r-\theta }[/math], safety stock level
- [math]\displaystyle{ S(r) }[/math] = fill rate
- [math]\displaystyle{ B(r) }[/math] = average number of outstanding back-orders
- [math]\displaystyle{ I(r) }[/math] = 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:
[math]\displaystyle{ P(X\leq r+1)=G(r+1) }[/math]
Since this holds for all orders, the fill rate is:
[math]\displaystyle{ S(r)=G(r+1) }[/math]
If demand is normally distributed [math]\displaystyle{ \mathcal{N}(\theta,\,\sigma^2) }[/math], the fill rate is given by:
[math]\displaystyle{ S(r)=\phi\left( \frac{r+1-\theta}{\sigma} \right) }[/math]
Where [math]\displaystyle{ \phi() }[/math] 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:
[math]\displaystyle{ I(r)=r+1-\theta+B(r) }[/math]
In general the number of outstanding orders is X=x and the number of back-orders is:
[math]\displaystyle{ Backorders=\begin{cases} 0, & x \lt r+1 \\ x-r-1, & x \ge r+1 \end{cases} }[/math]
The expected back order level is therefore given by:
[math]\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 }[/math]
Again, if demand is normally distributed:[2]
[math]\displaystyle{ B(r)=(\theta-r)[1-\phi(z)]+\sigma\phi(z) }[/math]
Where [math]\displaystyle{ z }[/math] 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:
[math]\displaystyle{ TC=hI(r)+bB(r) }[/math]
It can be proven that:[1]
[math]\displaystyle{ G(r^{*}+1)=\frac{b}{b+h} }[/math]
Where r* is the optimal reorder point.
Proof [math]\displaystyle{ \frac{dTC}{dr}=h+(b+h)\frac{dB}{dr} }[/math]
[math]\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)] }[/math]
To minimize TC set the first derivative equal to zero:
[math]\displaystyle{ \frac{dTC}{dr} = h - (b+h) [1-G(r+1)]=0 }[/math]
And solve for G(r+1).
If demand is normal then r* can be obtained by:
[math]\displaystyle{ r^{*}+1=\theta+z\sigma }[/math]
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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