An
example on setting target quantities and action limits
SCENARIO
A
packer produces 200 g bags of sweets; the variation due to the
packing process is dependent on the size of sweet.
The
process variation, stated as a standard deviation, is either 4 g, 5 g
or 6 g.
The
packing line produces 4,000 bags an hour and the packer monitors the
average
weight
of the product by taking samples of five bags from the line every 30
minutes.
The
average of each sample is calculated and used to determine when
action is needed.
What
should be the lowest target quantity the packer should aim for, and
what should
the
minimum action limit (1 in 1,000) for the sample mean be, for each of
the three processes?
CALCULATION
1.
Lowest
Target Quantity
1.1. Process
variation
1.1.1. The
average range or standard deviation can be used to consider this
parameter.
The
standard deviation (SD) is more robust, as it is based on all the
data available. In either case, the following assumes that the
distribution of the contents of the packages is ‘normal’. If this
assumption is incorrect the variation from normality also has to be
considered.
1.1.2. In
the scenario the variability of the filling process is given as a
standard deviation. In order to ensure all three packers’ rules are
met, the highest of the following has to be used to address this
variability:
Qn T1+2s T2+3.72s
where
Qn is
the nominal (labelled) quantity
s
is the standard deviation of the filling process,
T1 is
Qn – TNE
T2 is
Qn –2
TNE
TNE is
the tolerable negative error for the nominal quantity
.
WELMEC
document 6.4 at paragraph E2.5 deals with this matter as above. It is
treated differently in the Packers Code at paragraph C15, but the
result is the same.
For
the various process variations the results are:-
Process
variation Qn T1+2s T2+3.72s Largest
4 200 199 196.88 200
5 200 201 200.6 201
6 200 203 204.32 204.32
As
can be seen from this table that which of the packers’ rules is
critical depends on the variability of the filling process. In this
exercise the critical rule is:-
For
s=4 g the first rule, ensuring the average is correct,
For
s=5 g the second rule, ensuring that the number of packages having a
greater negative error than the tolerable negative error
isacceptable, and
For
s=6 g the third rule, ensuring that there are no packages with a
negative error greater than twice the tolerable negative error
produced.
1.1.3. The
above ‘largest’ quantities would be acceptable as a target
quantity if there were no other issues that needed addressing. The
scenario gave information about the sampling plan the packer is using
and so the adequacy of this needs to be considered.
1.2. Sampling
Allowance
1.2.1. Generally
the packer’s sampling plan is considered to be equivalent to the
reference
test if at least 50 items are sample d during the time that 10,000
packs
are
filled (with a minimum time of 1 hour and a maximum time of 1 day or
shift).
See
the Packers Code, paragraph C22, or WELMEC 6.4, paragraph E.3
1.2.2. The
scenario is that the packing line produces 4,000 bags an hour and
the
packer monitors the average weight of the product by taking samples
of five bags
from
the line every 30 minutes. The average of each sample is calculated
and
used
to determine when action is needed.
From
this information, the time taken to fill 10,000 packs, referred to as
the
Production
Period, is equal to 10,000/4,000 hr = 2.5 hr.
The
number of samples (of size 5, i.e. n = 5) taken during this period,
sampling
every
half hour, is k = 2.5/0.5 = 5.
Therefore
the number of items sampled during the
production
period (time taken to produce 10,000 packs) is kn = 5x5 = 25. As this
is less than 50 a sampling
allowance
is needed to ensure that the packers system is as efficient as the
reference
test in detecting non-compliance.
1.2.3. The
appropriate allowance, which is used to enhance the target quantities
established
in 1.1.2 above, is obtained by looking up the tables in the Packers
Code,
Table C1, or WELMEC 6.4, Table E.3.
The
allowances are based on 3 control system:-
-
using
Shewhart Control with Action
Limit
only (1 in 1,000),
-using
Shewhart Control with Warning Limit (1 in 40) & Action Limit (1
in 1,000), and
-using
Cusum control with h=5, f=0.5 (as per BS 5703)
The
exercise indicates that only an Action Limit is used which is
referred to as
Procedure
A in the tables.
1.2.4. Looking
at n=5 and k=5 for procedure A gives a sampling allowance
factor,
z = 0.20. This factor is multiplied with the standard deviation of
the
packing
process to produce an allowance, which is added to the targets
determined
in 1.1.2 above.
1.2.5. The
minimum target quantity taking into account process variability and
sampling
becomes:
Process
SD Qn T1+2s T2+3.72s Largest zs=0.20s Min
Qt
4 200 199 196.88 200 0.8 200.8
5 200 201 200.6 201 1 202
6 200 203 204.32 204.32 1.2 205.52
1.3. Other
Allowances
1.3.1. The
scenario does not give any indications that other allowances are
necessary
but other issues that need to be considered include:-
-a
‘wandering average’,
-storage
allowance, particularly for desiccating products,
-tare
variability, where the quantity determination assumes a constant tare
weight,
-temperature,
if the product is filled hot or cold and the volume changes when
determined
at 200C
Theses
are considered in the Packers Code in paragraphs C21 to C25 and
WELMEC
6.4 in paragraphs E.5.1 to E.5.4.
2. Action
Limits
1.4. The
scenario asks for the ‘minimum action limit (1 in 1,000) for the
sample
mean’. The distribution of the sample mean (of samples of size n)
is
related
to the distribution of the individual items (the process
variability).
1.5. If
the process variability (standard deviation) is s
and
the number of items in a sample is n
then
the standard deviation of the distribution of the means, sometimes
referred to as the standard error of the means, is s/
n
1.6. The
action limit, with a chance of 1 in 1,000 of exceeding, as with the
normal
distribution comes at three times the standard error away from the
target
quantity.
Only the lower action limit is needed for legal metrology, although
upper
limits may be set, for example for safety reasons (aerosols), or
economic
reasons
(duty on alcohol) –but
the corresponding limit must be no nearer the
target
quantity than the lower one.
So
for legal metrology the Action Limit should be no lower than Qt -3
s/n
(If
there was a warning limit of 1 in 40 being used this would be at 2
times the
standard
error.).
An
example is given in the Packers’ Code at paragraph C34, it is also
covered in
WELMEC
6.4 at paragraph E.7.2.
Using
the data from the exercise this gives:
RESULTS
Process
Variation,s Minimum Target Quantity (g) Minimum Action Limit forthe
Mean (g)
4
g 200.8 195.4
5
g 202.0 195.3
6
g 205.5 197.5
So that's all clear now!
