(Q,r) model

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]

## Contents

* 1 Overview
* 1.1 Assumptions
* 1.2 Variables
* 1.3 Costs
* 1.3.1 Backorder cost approach
* 1.3.1.1 Total cost function and optimal reorder point
* 1.3.1.2 Normal distribution
* 1.3.2 Stockout cost approach
* 1.4 Lead-Time Variability
* 1.4.1 Poisson distribution
* 2 See also
* 3 References

## 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

* D = Expected demand per year
* 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
* θ = mean demand during lead time
* A = setup or purchase order cost per replenishment
* c = unit production cost
* h = annual unit holding cost
* k = cost per stockout
* b = annual unit backorder cost
* Q = replenishment quantity
* r = reorder point
* SS=r−θ, safety stock level
* F(Q,r) = order frequency
* S(Q,r) = fill rate
* B(Q,r) = average number of outstanding back-orders
* I(Q,r) = average on-hand inventory level

### Costs

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

S(Q,r)=1Q∫rr+QG(x)dx

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

S(Q,r)=1Q∫rr+QG(x)dx=1−1Q[B(r))−B(r+Q)]

Inventory holding cost is hI(Q,r), average inventory being:

I(Q,r)=Q+12+r−θ+B(Q,r)

#### Backorder cost approach

The annual backorder cost is proportional to backorder level:

B(Q,r)=1Q∫rr+QB(x+1)dx

##### 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:

Y(Q,r)=DQA+bB(Q,r)+hI(Q,r)

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

Q*=2ADh

G(r*+1)=bb+h

Proof

To minimize set the partial derivatives of Y equal to zero:

∂Y∂Q=−DAQ2+h2=0

∂Y∂r=h+(b+h)dBdr=0

dBdr=ddr∫r+∞(x−r)g(x)dx=−∫r+∞g(x)dx=−[1−G(r)]

∂Y∂r=h−(b+h)[1−G(r)]=0

And solve for G(r) and Q.

##### Normal distribution

In the case lead-time demand is normally distributed:

r*=θ+zσ

#### Stockout cost approach

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

Y(Q,r)=DQA+kD[1−S(Q,r)]+hI(Q,r)

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

G(r*)=kDkD+hQ

### Lead-Time Variability

X is the random demand during replenishment lead time:

X=∑t=1LDt

In expectation:

E⁡[X]=E⁡[L]E⁡[Dt]=ℓd=θ

Variance of demand is given by:

Var⁡(x)=E⁡[L]Var⁡(Dt)+E⁡[Dt]2Var⁡(L)=ℓσD2+d2σL2

Hence standard deviation is:

σ=Var⁡(X)=ℓσD2+d2σL2

#### Poisson distribution

if demand is Poisson distributed:

σ=ℓσD2+d2σL2=θ+d2σL2

## See also

* 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

## 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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