(Q,r) model
![]() | This article may be too technical for most readers to understand.(January 2022) |
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
[edit]Assumptions
[edit]- 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
- There is a fixed cost associated with a replenishment order
- There is a constraint on the number of replenishment orders per year
Variables
[edit]- = 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
[edit]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
[edit]The annual backorder cost is proportional to backorder level:
Total cost function and optimal reorder point
[edit]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:
Proof To minimize set the partial derivatives of Y equal to zero: And solve for G(r) and Q.
Normal distribution
[edit]In the case lead-time demand is normally distributed:
Stockout cost approach
[edit]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
[edit]X is the random demand during replenishment lead time:
In expectation:
Variance of demand is given by:
Hence standard deviation is:
Poisson distribution
[edit]if demand is Poisson distributed:
See also
[edit]- Infinite fill rate for the part being produced: Economic order quantity
- Constant fill rate for the part being produced: Economic production quantity
- Demand is random: classical Newsvendor model
- Demand is random, continuous replenishment: Base stock model
- Demand varies deterministically over time: Dynamic lot size model
- Several products produced on the same machine: Economic lot scheduling problem