The (Q,r) model is a class of models in inventory theory.[1] A general (Q,r) model can be extended from both the EOQ model and the base stock model[2]

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
  7. There is a fixed cost associated with a replenishment order
  8. There is a constraint on the number of replenishment orders per year

Variables

  • = Expected demand per year
  • = 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
  • = setup or purchase order cost per replenishment
  • = unit production cost
  • = annual unit holding cost
  • = cost per stockout
  • = annual unit backorder cost
  • = replenishment quantity
  • = reorder point
  • , safety stock level
  • = order frequency
  • = fill rate
  • = average number of outstanding back-orders
  • = average on-hand inventory level

Costs

The number of orders per year can be computed as , the annual fixed order cost is F(Q,r)A. The fill rate is given by:

The annual stockout cost is proportional to D[1 - S(Q,r)], with the fill rate beying:

Inventory holding cost is , average inventory being:

Backorder cost approach

The annual backorder cost is proportional to backorder level:

Total cost function and optimal reorder point

The total cost is given by the sum of setup costs, purchase order cost, backorders cost and inventory carrying cost:

The optimal reorder quantity and optimal reorder point are given by:


Normal distribution

In the case lead-time demand is normally distributed:

Stockout cost approach

The total cost is given by the sum of setup costs, purchase order cost, stockout cost and inventory carrying cost:

What changes with this approach is the computation of the optimal reorder point:

Lead-Time Variability

X is the random demand during replenishment lead time:

In expectation:

Variance of demand is given by:

Hence standard deviation is:

Poisson distribution

if demand is Poisson distributed:

See also

References

  1. T. Whitin, G. Hadley, Analysis of Inventory Systems, Prentice Hall 1963
  2. W.H. Hopp, M. L. Spearman, Factory Physics, Waveland Press 2008
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