(Q,r) model explained

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

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 timeX = Demand during replenishment lead timeg(x) = probability density function of demand during lead timeG(x) = cumulative distribution function of demand during lead time\theta = mean demand during lead timeA = setup or purchase order cost per replenishmentc = unit production costh = annual unit holding costk = cost per stockoutb = annual unit backorder costQ = replenishment quantityr = reorder pointSS = r - \theta , safety stock levelF(Q,r) = order frequencyS(Q,r) = fill rateB(Q,r) = average number of outstanding back-ordersI(Q,r) = average on-hand inventory level

Costs

The number of orders per year can be computed as

F(Q,r) = (D)/(Q) , the annual fixed order cost is F(Q,r)A. The fill rate is given by:S(Q,r) = (1)/(Q) \int ^{r + Q} _{r} G(x)dx

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

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

Inventory holding cost is

hI(Q,r) , average inventory being:I(Q,r) = (Q +1)/(2) + r - \theta + B(Q,r)

Backorder cost approach

The annual backorder cost is proportional to backorder level:

B(Q,r) = (1)/(Q) \int ^{r + Q} _{r} 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) = (D)/(Q) 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) = (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 ^{L} _{t =1} 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} }

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

Notes and References

1. [Thomson M. Whitin|T. Whitin]
2. W.H. Hopp, M. L. Spearman, Factory Physics, Waveland Press 2008