Inventory Control Guide Page 29
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404.4 Objective 2: Fill Rate (service level)
In the previous examples, the reorder point R was computed to satisfy some stockout
probability. Here we set R in order to satisfy an objective in terms of fill rate.
Decision
Select Q and R = Lt × D + SS
Objective 2
Minimize the total costs and
Guarantee a minimum fill rate ( β )
Remember that the fill rate is the percentage of units delivered immediately from the shelves,
that is without delay.
β = ( Q - n( R ) ) / Q
if backorder :
β = Q / ( Q + n( R ) )
if lost sales :
where n(R) is the average number of units in a cycle that are not served from the shelves. To
derive the fill rate from n(R), we need to distinguish the backorder model from the lost sales
model.
In a backorder model, the total demand during a cycle is Q. When we receive a delivery of Q
units, n(R) units out of Q will be used for the backlogged demand. The remaining Q-n(r) units
will be used to serve customers directly from the shelves. Then the fill rate is β = (Q-n(R))/Q
In a lost model, the total demand during a cycle is (Q+n(R)) from which Q only are served
from the shelves and n(R) are lost. Then the fill rate is β = Q/(Q+n(R)). Note that when n(R) is
small, both definitions of β are close. Try for example with Q=100 and n(R)=1.
The cost function is still the same and leads to the same conclusions.
Minimize C( Q, R ) ≈ O(D /Q ) + H( Q /2+ R- LtD)+ ID
Q = 2OD / H ;
minimize R
We should thus take R as small as possible. However, we want to guarantee a fill rate.
β
! R
n( R )
update R
Starting with a R value, we compute n(R) and derive β. If β is too small, we increase R and if β
is too large, we decrease R and so on. We will see that in some cases we can immediately
derive R from the β value.
Here is the method for the backorder model. A similar method can be used for the lost sales
model.
Method 2
If a fill rate β is required, then
Set Q = EOQ
Compute n( R ) = Q (1- β )
Compute R to reach this value for n(R).
Note that R implicitly depends on the selected lot size Q. This will be made more clear in the
numerical examples below.
Prod 2100-2110
Inventory Control
28
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