# 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

1. Products can be analyzed individually
2. Demands occur one at a time (no batch orders)
3. Unfilled demand is back-ordered (no lost sales)
4. Replenishment lead times are fixed and known
5. Replenishments are ordered one at a time
6. Demand is modeled by a continuous probability distribution

### Variables

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

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

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

${\displaystyle S(r)=G(r+1)}$

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

${\displaystyle S(r)=\phi \left({\frac {r+1-\theta }{\sigma }}\right)}$

Where ${\displaystyle \phi ()}$ 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:

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

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

${\displaystyle Backorders={\begin{cases}0,&x

The expected back order level is therefore given by:

${\displaystyle B(r)=\int _{r}^{+\infty }\left(x-r-1\right)g(x)dx=\int _{r+1}^{+\infty }\left(x-r\right)g(x)dx}$

Again, if demand is normally distributed:[2]

${\displaystyle B(r)=(\theta -r)[1-\phi (z)]+\sigma \phi (z)}$

Where ${\displaystyle z}$ 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:

${\displaystyle TC=hI(r)+bB(r)}$

It can be proven that:[1]

${\displaystyle G(r^{*}+1)={\frac {b}{b+h}}}$

Where r* is the optimal reorder point. If demand is normal then r* can be obtained by:

${\displaystyle r^{*}+1=\theta +z\sigma }$