(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

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

$D$ = Expected demand per year
$\ell$ = 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
$\theta$ = 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
$SS=r-\theta$ , 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 $F(Q,r)={\frac {D}{Q}}$ , the annual fixed order cost is F(Q,r)A. The fill rate is given by:

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

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

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

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

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

Backorder cost approach

The annual backorder cost is proportional to backorder level:

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

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:

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

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

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

$G(r^{*}+1)={\frac {b}{b+h}}$

Proof

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

${\frac {\partial Y}{\partial Q}}=-{\frac {DA}{Q^{2}}}+{\frac {h}{2}}=0$

${\frac {\partial Y}{\partial r}}=h+(b+h){\frac {dB}{dr}}=0$

${\frac {dB}{dr}}={\frac {d}{dr}}\int _{r}^{+\infty }(x-r)g(x)dx=-\int _{r}^{+\infty }g(x)dx=-[1-G(r)]$

${\frac {\partial Y}{\partial r}}=h-(b+h)[1-G(r)]=0$

And solve for G(r) and Q.

Normal distribution

In the case lead-time demand is normally distributed:

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

Stockout cost approach

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

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

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

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

Lead-Time Variability

X is the random demand during replenishment lead time:

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

In expectation:

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

Variance of demand is given by:

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

Hence standard deviation is:

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

Poisson distribution

if demand is Poisson distributed:

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

References

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

[1] T. Whitin, G. Hadley, Analysis of Inventory Systems, Prentice Hall 1963

[FacPhy-2] W.H. Hopp, M. L. Spearman, Factory Physics, Waveland Press 2008

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