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

• [math]\displaystyle{ D }[/math] = Expected demand per year
• [math]\displaystyle{ \ell }[/math] = Replenishment lead time
• [math]\displaystyle{ X }[/math] = Demand during replenishment lead time
• [math]\displaystyle{ g(x) }[/math] = probability density function of demand during lead time
• [math]\displaystyle{ G(x) }[/math] = cumulative distribution function of demand during lead time
• [math]\displaystyle{ \theta }[/math] = mean demand during lead time
• [math]\displaystyle{ A }[/math] = setup or purchase order cost per replenishment
• [math]\displaystyle{ c }[/math] = unit production cost
• [math]\displaystyle{ h }[/math] = annual unit holding cost
• [math]\displaystyle{ k }[/math] = cost per stockout
• [math]\displaystyle{ b }[/math] = annual unit backorder cost
• [math]\displaystyle{ Q }[/math] = replenishment quantity
• [math]\displaystyle{ r }[/math] = reorder point
• [math]\displaystyle{ SS=r-\theta }[/math], safety stock level
• [math]\displaystyle{ F(Q,r) }[/math] = order frequency
• [math]\displaystyle{ S(Q,r) }[/math] = fill rate
• [math]\displaystyle{ B(Q,r) }[/math] = average number of outstanding back-orders
• [math]\displaystyle{ I(Q,r) }[/math] = average on-hand inventory level

Costs

The number of orders per year can be computed as [math]\displaystyle{ F(Q,r) = \frac {D}{Q} }[/math], the annual fixed order cost is F(Q,r)A. The fill rate is given by:

[math]\displaystyle{ S(Q,r)=\frac{1}{Q} \int_{r}^{r+Q} G(x)dx }[/math]

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

[math]\displaystyle{ S(Q,r)=\frac{1}{Q} \int_{r}^{r+Q} G(x) dx = 1 - \frac{1}{Q} [B(r))-B(r+Q)] }[/math]

Inventory holding cost is [math]\displaystyle{ hI(Q,r) }[/math], average inventory being:

[math]\displaystyle{ I(Q,r)=\frac{Q+1}{2}+r-\theta+B(Q,r) }[/math]

Backorder cost approach

The annual backorder cost is proportional to backorder level:

[math]\displaystyle{ B(Q,r) = \frac{1}{Q} \int_{r}^{r+Q} B(x+1)dx }[/math]

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:

[math]\displaystyle{ Y(Q,r) = \frac{D}{Q} A + b B(Q,r) +h I(Q,r) }[/math]

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

[math]\displaystyle{ Q^*=\sqrt{\frac{2AD}{h}} }[/math]

[math]\displaystyle{ G(r^* + 1) = \frac{b} {b+h} }[/math]

Proof

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

Normal distribution

In the case lead-time demand is normally distributed:

[math]\displaystyle{ r^* = \theta + z \sigma }[/math]

Stockout cost approach

The total cost is given by the sum of setup costs, purchase order cost, stockout cost and inventory carrying cost:

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

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

[math]\displaystyle{ G(r^*)=\frac{kD}{kD+hQ} }[/math]

Lead-Time Variability

X is the random demand during replenishment lead time:

[math]\displaystyle{ X = \sum_{t=1}^{L} D_{t} }[/math]

In expectation:

[math]\displaystyle{ \operatorname{E}[X] = \operatorname{E}[L] \operatorname{E}[D_{t}] =\ell d = \theta }[/math]

Variance of demand is given by:

[math]\displaystyle{ \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} }[/math]

Hence standard deviation is:

[math]\displaystyle{ \sigma = \sqrt{\operatorname{Var}(X)} =\sqrt{ \ell \sigma^{2}_{D} + d^{2} \sigma^{2}_{L} } }[/math]

Poisson distribution

if demand is Poisson distributed:

[math]\displaystyle{ \sigma = \sqrt{ \ell \sigma^{2}_{D} + d^{2} \sigma^{2}_{L} }= \sqrt{\theta + d^{2} \sigma^{2}_{L}} }[/math]

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