Base Stock Model

The base stock model is a statistical model in inventory theory.W.H. Hopp, M. L. Spearman, Factory Physics, Waveland Press 2008 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

*$L$ = Replenishment lead time
*$X$ = Demand during replenishment lead time
*$g(x)$ = probability density function of demand during lead time
*$G(x)$ = cumulative distribution function of demand during lead time
*$\theta$ = mean demand during lead time
*$h$ = cost to carry one unit of inventory for 1 year
*$b$ = cost to carry one unit of back-order for 1 year
*$r$ = reorder point
*$SS=r-\theta$, safety stock level
*$S(r)$ = fill rate
*$B(r)$ = average number of outstanding back-orders
*$I(r)$ = 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:

$P(X\leq r+1)=G(r+1)$

Since this holds for all orders, the fill rate is:

$S(r)=G(r+1)$

If demand is normally distributed $\mathcal(\theta,\,\sigma^2)$, the fill rate is given by:

$S(r)=\phi\left( \frac \right)$

Where $\phi()$ 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:

$I(r)=r+1-\theta+B(r)$

In general the number of outstanding orders is X=x and the number of back-orders is:

$Backorders=\begin 0, & x < r+1 \\ x-r-1, & x \ge r+1 \end$

The expected back order level is therefore given by:

$B(r)=\int_^\left( x-r-1 \right)g(x)dx=\int_^\left( x-r \right)g(x)dx$

Again, if demand is normally distributed:Zipkin, Foundations of inventory management, McGraw Hill 2000

B(r)=(\theta-r) -\phi(z)\sigma\phi(z)

Where $z$ 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:

$TC=hI(r)+bB(r)$

It can be proven that:

Where r* is the optimal reorder point.

:

If demand is normal then r* can be obtained by:

$r^+1=\theta+z\sigma$

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

{{reflist

Inventory optimization