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.

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:

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

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:

Where r* is the optimal reorder point.


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!Proof

$$\frac{dTC}{dr}=h+(b+h)\frac{dB}{dr}$$

$$\frac{dB}{dr}=\frac{d}{dr} \int_{r+1}^{+\infty} (x-r-1) g(x) dx = - \int_{r+1}^{+\infty} g(x) dx = -[1 - G(r+1)]$$

To minimize TC set the first derivative equal to zero:

$$\frac{dTC}{dr} = h - (b+h) [1-G(r+1)]=0$$

And solve for G(r+1).
 * }

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

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