(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]

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

• ${\displaystyle D}$ = 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

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:

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

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

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

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

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

The annual backorder cost is proportional to backorder level:

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

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:

Proof

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

In the case lead-time demand is normally distributed:

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

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:

X is the random demand during replenishment lead time:

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

In expectation:

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

Variance of demand is given by:

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

Hence standard deviation is:

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

if demand is Poisson distributed:

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

• 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

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