Base stock model

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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[edit]

Assumptions[edit]

  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[edit]

  • = 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[edit]

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[edit]

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

See also[edit]

References[edit]

  1. ^ a b W.H. Hopp, M. L. Spearman, Factory Physics, Waveland Press 2008
  2. ^ Zipkin, Foundations of inventory management, McGrawHill 2000