In inventory theory, the (Q,r) model is used to determine optimal ordering policies.
[1] It is a class of inventory control models that generalize and combine elements of both the Economic Order Quantity (EOQ) model and the base stock model.
[2] The (Q,r) model addresses the question of when and how much to order, aiming to minimize total inventory costs, which typically include ordering costs, holding costs, and shortage costs.
It specifies that an order of size Q should be placed when the inventory level reaches a reorder point r. The (Q,r) model is widely applied in various industries to manage inventory effectively and efficiently.
The number of orders per year can be computed as
, the annual fixed order cost is F(Q,r)A.
The fill rate is given by:
{\displaystyle 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:
Inventory holding cost is
, average inventory being:
+ r − θ +
The annual backorder cost is proportional to backorder level:
The total cost is given by the sum of setup costs, purchase order cost, backorders cost and inventory carrying cost:
The optimal reorder quantity and optimal reorder point are given by:
{\displaystyle {\frac {\partial Y}{\partial r}}=h+(b+h){\frac {dB}{dr}}=0}
{\displaystyle {\frac {dB}{dr}}={\frac {d}{dr}}\int _{r}^{+\infty }(x-r)g(x)dx=-\int _{r}^{+\infty }g(x)dx=-[1-G(r)]}
In the case lead-time demand is normally distributed:
= θ + z σ
The total cost is given by the sum of setup costs, purchase order cost, stockout cost and inventory carrying cost:
What changes with this approach is the computation of the optimal reorder point:
{\displaystyle G(r^{*})={\frac {kD}{kD+hQ}}}
X is the random demand during replenishment lead time:
] = ℓ d = θ
Variance of demand is given by:
Var ( x ) =
] Var (
Hence standard deviation is:
ℓ
if demand is Poisson distributed: