(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

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:

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

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:

$$Y(Q,r) = \frac{D}{Q} A + b B(Q,r) +h I(Q,r)$$

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


 * {| class="toccolours collapsible collapsed" width="90%" style="text-align:left"

!Proof $$\frac{\partial Y}{\partial Q} =-\frac{DA}{Q^2}+\frac{h}{2}=0$$
 * To minimize set the partial derivatives of Y equal to zero:
 * To minimize set the partial derivatives of Y equal to zero:

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

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

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

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:

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

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

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:

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

Hence standard deviation is:

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

Poisson distribution
if demand is Poisson distributed:

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