# 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

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

• $L$ = 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 ${\mathcal {N}}(\theta ,\,\sigma ^{2})$ , the fill rate is given by:

$S(r)=\phi \left({\frac {r+1-\theta }{\sigma }}\right)$ 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{cases}0,&x The expected back order level is therefore given by:

$B(r)=\int _{r}^{+\infty }\left(x-r-1\right)g(x)dx=\int _{r+1}^{+\infty }\left(x-r\right)g(x)dx$ Again, if demand is normally distributed:

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

$G(r^{*}+1)={\frac {b}{b+h}}$ Where r* is the optimal reorder point. If demand is normal then r* can be obtained by:

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