Finance:(Q,r) model

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

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
• $\ell$ = 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:

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

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

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

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:

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

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: $\frac{\partial Y}{\partial Q}=-\frac{DA}{Q^2}+\frac{h}{2}=0$ $\frac{\partial Y}{\partial r}=h+(b+h)\frac{dB}{dr}=0$ $\frac{dB}{dr}=\frac{d}{dr}\underset{r}{\overset{+\mathrm{\infty }}{\int }}(x-r)g(x)dx=-\underset{r}{\overset{+\mathrm{\infty }}{\int }}g(x)dx=-[1-G(r)]$ $\frac{\partial Y}{\partial r}=h-(b+h)[1-G(r)]=0$ 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:

$\mathrm{E}[X]=\mathrm{E}[L]\mathrm{E}[D_t]=\ell d=\theta$

Variance of demand is given by:

$\mathrm{Var}(x)=\mathrm{E}[L]\mathrm{Var}(D_t)+\mathrm{E}[D_t{]}^2\mathrm{Var}(L)=\ell {\sigma }_D^2+d^2{\sigma }_L^2$

Hence standard deviation is:

$\sigma =\sqrt{\mathrm{Var}(X)}=\sqrt{\ell {\sigma }_D^2+d^2{\sigma }_L^2}$

Poisson distribution

if demand is Poisson distributed:

$\sigma =\sqrt{\ell {\sigma }_D^2+d^2{\sigma }_L^2}=\sqrt{\theta +d^2{\sigma }_L^2}$

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