(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]

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

• ${\displaystyle D}$ = Expected demand per year
• ${\displaystyle \ell }$ = Replenishment lead time
• ${\displaystyle X}$ = Demand during replenishment lead time
• ${\displaystyle g(x)}$ = probability density function of demand during lead time
• ${\displaystyle G(x)}$ = cumulative distribution function of demand during lead time
• ${\displaystyle \theta }$ = mean demand during lead time
• ${\displaystyle A}$ = setup or purchase order cost per replenishment
• ${\displaystyle c}$ = unit production cost
• ${\displaystyle h}$ = annual unit holding cost
• ${\displaystyle k}$ = cost per stockout
• ${\displaystyle b}$ = annual unit backorder cost
• ${\displaystyle Q}$ = replenishment quantity
• ${\displaystyle r}$ = reorder point
• ${\displaystyle SS=r-\theta }$, safety stock level
• ${\displaystyle F(Q,r)}$ = order frequency
• ${\displaystyle S(Q,r)}$ = fill rate
• ${\displaystyle B(Q,r)}$ = average number of outstanding back-orders
• ${\displaystyle I(Q,r)}$ = average on-hand inventory level

Costs

The number of orders per year can be computed as ${\displaystyle F(Q,r)={\frac {D}{Q}}}$, the annual fixed order cost is F(Q,r)A. The fill rate is given by:

${\displaystyle S(Q,r)={\frac {1}{Q}}\int _{r}^{r+Q}G(x)dx}$

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

${\displaystyle S(Q,r)={\frac {1}{Q}}\int _{r}^{r+Q}G(x)dx=1-{\frac {1}{Q}}[B(r))-B(r+Q)]}$

Inventory holding cost is ${\displaystyle hI(Q,r)}$, average inventory being:

${\displaystyle I(Q,r)={\frac {Q+1}{2}}+r-\theta +B(Q,r)}$

Backorder cost approach

The annual backorder cost is proportional to backorder level:

${\displaystyle B(Q,r)={\frac {1}{Q}}\int _{r}^{r+Q}B(x+1)dx}$

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:

${\displaystyle Y(Q,r)={\frac {D}{Q}}A+bB(Q,r)+hI(Q,r)}$

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

${\displaystyle Q^{*}={\sqrt {\frac {2AD}{h}}}}$

${\displaystyle G(r^{*})={\frac {b}{b+h}}}$

Proof

To minimize set the partial derivatives of Y equal to zero: ${\displaystyle {\frac {\partial Y}{\partial Q}}=-{\frac {DA}{Q^{2}}}+{\frac {h}{2}}=0}$ ${\displaystyle {\frac {\partial Y}{\partial r}}=h+(b+h){\frac {dB}{dr}}=0}$ ${\displaystyle {\frac {dB}{dr}}={\frac {d}{dr}}\int _{r}^{+\infty }(x-r)g(x)dx=-\int _{r}^{+\infty }g(x)dx=-[1-G(r)]}$ ${\displaystyle {\frac {\partial Y}{\partial r}}=h-(b+h)[1-G(r)]=0}$ And solve for G(r) and Q.

Normal distribution

In the case lead-time demand is normally distributed:

${\displaystyle r^{*}=\theta +z\sigma }$

Stockout cost approach

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

${\displaystyle Y(Q,r)={\frac {D}{Q}}A+kD[1-S(Q,r)]+hI(Q,r)}$

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

${\displaystyle G(r^{*})={\frac {kD}{kD+hQ}}}$

Lead-Time Variability

X is the random demand during replenishment lead time:

${\displaystyle X=\sum _{t=1}^{L}D_{t}}$

In expectation:

${\displaystyle \operatorname {E} [X]=\operatorname {E} [L]\operatorname {E} [D_{t}]=\ell d=\theta }$

Variance of demand is given by:

${\displaystyle \operatorname {Var} (x)=\operatorname {E} [L]\operatorname {Var} (D_{t})+\operatorname {E} [D_{t}]^{2}\operatorname {Var} (L)=\ell \sigma _{D}^{2}+d^{2}\sigma _{L}^{2}}$

Hence standard deviation is:

${\displaystyle \sigma ={\sqrt {\operatorname {Var} (X)}}={\sqrt {\ell \sigma _{D}^{2}+d^{2}\sigma _{L}^{2}}}}$

Poisson distribution

if demand is Poisson distributed:

${\displaystyle \sigma ={\sqrt {\ell \sigma _{D}^{2}+d^{2}\sigma _{L}^{2}}}={\sqrt {\theta +d^{2}\sigma _{L}^{2}}}}$

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