(Q,r) Model

The (Q,r) model is a class of models in inventory theory. T. Whitin, G. Hadley, Analysis of Inventory Systems, Prentice Hall 1963 A general (Q,r) model can be extended from both the EOQ model and the base stock modelW.H. Hopp, M. L. Spearman, Factory Physics, Waveland Press 2008

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 $F(Q,r) = \frac$, the annual fixed order cost is F(Q,r)A. The fill rate is given by:

$S(Q,r)=\frac \int_^ G(x)dx$

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

S(Q,r)=\frac \int_^ G(x) dx = 1 - \frac (r))-B(r+Q)/math>

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

$I(Q,r)=\frac+r-\theta+B(Q,r)$

Backorder cost approach

The annual backorder cost is proportional to backorder level:

$B(Q,r) = \frac \int_^ B(x+1)dx$

=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:

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

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

:

=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 A + kD -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:

$X = \sum_^ D_$

In expectation:

\operatorname = \operatorname \operatorname _=\ell d = \theta

Variance of demand is given by:

\operatorname(x) = \operatorname \operatorname(D_) + \operatorname _\operatorname(L) = \ell \sigma^_ + d^ \sigma^_

Hence standard deviation is:

$\sigma = \sqrt =\sqrt$

Poisson distribution

if demand is Poisson distributed:

$\sigma = \sqrt= \sqrt$

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

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