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
- = Replenishment lead time
- = Demand during replenishment lead time
- = probability density function of demand during lead time
- = cumulative distribution function of demand during lead time
- = mean demand during lead time
- = cost to carry one unit of inventory for 1 year
- = cost to carry one unit of back-order for 1 year
- = reorder point
- , safety stock level
- = fill rate
- = average number of outstanding back-orders
- = 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:
Since this holds for all orders, the fill rate is:
If demand is normally distributed , the fill rate is given by:
Where 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:
In general the number of outstanding orders is X=x and the number of back-orders is:
The expected back order level is therefore given by:
Again, if demand is normally distributed:[2]
Where 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:
It can be proven that:[1]
Where r* is the optimal reorder point.
Proof
|
To minimize TC set the first derivative equal to zero:
And solve for G(r+1).
|
If demand is normal then r* can be obtained by:
See also
References
- ^ a b W.H. Hopp, M. L. Spearman, Factory Physics, Waveland Press 2008
- ^ Zipkin, Foundations of inventory management, McGraw Hill 2000