Page 1/Lect. 10
Shewharts’ Model Buildup and Interpretation
Standard implementation of the Shewharts’ model:
Product
Process
Action execution
Decision-making
(action choice for
correction)
Observation
(data gathering)
Diagnostics
(failure
determinatio
n)
Data evaluation
(data
preprocessing)
Key elements:
1. Observation
- Process sampling
- Data stratification (value discretization)
2. Data evaluation (!)
- probabilistic descriptors
o mean(X) – mean in a set of X values
o R – range of a set of X values
3. Diagnostics (!)
- Tools for error detection in gathered data (nonlinear
analysis → heuristics → tests)
Page 2/Lect. 10
Build up of the mean (X), R charts (or simply X, R charts):
Main steps:
Sampling
- Each sample consists of a set of values (and corresponds
to the set of measurements for the mean(X) and R
calculation).
- Sets of measurements (samples) should be chosen in
order to enable elimination of special effects in the
observed process (by the control process later on) →
adjusting the best possibility to detect a special effect.
On the other hand, some specific situations should be possibly avoided
(why?) as:
- Sampling from different sources (e.g. production lines)
- Sparse sampling (long time intervals between samples)
- Sampling of products being combined from different
sources, etc.
Remark:
Empirical experiences say (for standard average production process):
- 25 to 50 samples are sufficient to calculate the mean
value and the dispersion of a process
- Standard “sample” size is about 3 to 6 values
(measurements)
For each particular sample are calculated the following values of:
- Sample mean:
n
mean(Xi) = SUM Xij/n
j=1
where Xij stands for the ith sample
containing j=1..n measurements
Page 3/Lect. 10
- Dispersion within a sample, expressed by the “range” of X:
Ri=R(Xi) = max (Xij – Xik), for all j, k = 1..n
- Total average (as mean value of all samples mean values):
k
avg(X) = 1/k SUM mean (Xi),
i=1
where k is total no. of samples
- Mean of the range of X as:
k
mean(R) = 1/k SUM Ri
i=1
Real range of n-sized samples is related to standard deviation of the
population (process), so that:
mean(R)/σx =d2,
where: d2 =d2(n) are given values on condition of normal
distribution of X
The previous enables to define control limit for the mean(X) as:
avg(X) ± 3* σmean(x)
(standard deviation of mean(X) )
The Central Limit Theorem gives more relations as follows:
as σmean(X) = σx/SQRT(n) and σx = mean(R)/d2
so that standard deviation of a sample mean value is also:
σmean(X) = mean(R)/(d2*SQRT(n))
□
Page 4/Lect. 10
The previous then defines the upper and lower control limits (decision
levels) as:
upper limit:
avg(X) + mean(R) * 3/(d2* SQRT(n))
lower limit:
avg(X) - mean(R) * 3/(d2* SQRT(n)),
where expression 3/(d2* SQRT(n)) is exclusively function of n, so that
constant for fixed n.
Final remarks:
- Probabilistic distribution of the variable R is typically hard to be
determined
- It is not symmetric (!) with respect to mean(R) – to simplify the
situation it is often approximated by a symmetric substitute.
Page 5/Lect. 10
Shewharts’ Model Interpretation
The goal:
To describe general situations indicating dysfunction of the
process control.....
Key idea (Shewharts’ Control Chart Patterns):
(a)
Process monitoring through periodic sampling
(of the observed feature)
(b)
Estimation of the feature mean and standard deviation
(scatter)
(c)
Building the Shewharts’ X, R charts
Necessary condition for X,R charts use:
The system needs to be purely under influence of “standard disturbances”
(so it has to operate in a locked-loop mode of the statistic control)
The X, R charts allow monitoring and prediction of future behavior of the
system (estimation or possible improvements in process quality, etc).
Features of inconsistencies in the Shewharts’ model:
- Normal distribution of observed values is o.k.
- Deviations from normal distribution indicate significant cases for
optimization (even all the values are within the given bounds)
Deviations from the normal distribution can be classified as:
(1) Too many values near the control level indicate “overshooting”of
the process to control the mean(X) →
a. Too high feedback loop gain
b. Too large variance of the process inputs (e.g. inputs quality)
Page 6/Lect. 10
(2) Steady trend or oscillations of mean(X) show:
a. Operator failure
b. Usage of tools after their lifetime (exhausted)
c. Systematic adjustments in the process (!)
d. Collecting of low quality materials or intermediate products,
etc.
(3) Sudden changes of values may be caused by:
a. New machine startup, new worker
b. Malfunction of technology
c. New material delivery, new quality (changes in process
inputs)
d. Changes of measuring methodology
e. Technology switching
Evaluation of the process quality:
Basic criteria for sample evaluation (that describe a process having a
“good quality”):
(1) No values are out the upper and lower control limit
(all within avg(X) ± 3* σmean(x) for X, the same for R)
(2) Value distribution in the range of the control interval is
approximately normal
(3) No steadily raising or sinking trends or oscillations in the
measured values
(4) The samples are distributed more or less randomly in time
Page 7/Lect. 10
Illustration of a statistically “good” process:
mean (X)
range(X)
UCL
avg(X)
mean(R)
time
L
LCL
L
Zone classification
- Denotes rules for understanding X, R charts
o Divides the admissible interval (in between the UCL and
LCL levels in X or R charts) into zones around avg(X) or
mean(R) defined by ±σ, ±2σ and ±3σ, the zones ale entitled
C, B and A
o Applies 8 tests (heuristics) targeted on indications that the
observed process has substantially changed its’ mean value
or variance (or the both)
mean (X)
range(X)
UCL
avg(X)
mean(R)
A
B
C
C
B
A
LCL
L
+3σ
+2σ
+σ
-σ
- 2σ
- 3σ
Page 8/Lect. 10
TEST 1: Extreme values - the values are out of admissible interval
in-between of UCL and LCL
 Suitable for use with mean(X) and range(X)
 Indicates situations totally our of process control
 Very rough (insensitive) test
mean (X)
range(X)
UCL
avg(X)
mean(R)
LCL
L
TEST 2: 2 of 3 in zone A – existence of at least 2 values from 3
strictly succeeding triples in the zone A (or overshooting the zone A)
 Suitable for use with mean(X) only
 Indicates loosing process control
 More sensitive criterion than the test 1.
range(X)
UCL
avg(X)
A
B
C
C
B
A
LCL
L
Page 9/Lect. 10
TEST 3: 4 of 5 in zone B or outside – existence of at least 4 values
from 5 strictly succeeding sequences in the zone B (or overshooting the
zone B), similar to test 2.
 Suitable for use with mean(X) only
 Similar to test 2, indicates loosing process control
 More sensitive criterion than test 2 (and test 1 as well).
 Recommended for prediction of rapid changes.
range(X)
UCL
avg(X)
A
B
C
C
B
A
LCL
L
TEST 4: Long runs strictly above or below the reference value - the
values keep steadily (for ≥8 succeeding values) above or below the
avg(X) or the mean(R)
 Suitable for use with mean(X) and range(X)
 Indicates changes of the average (mean) of the process
mean (X)
range(X)
UCL
avg(X)
mean(R)
LCL
L
Page 10/Lect. 10
TEST 5: Linear trend indication - the values keep steadily raising or
dropping for ≥6 succeeding values.
 Suitable for use with mean(X) and range(X)
 Neither zones nor mean values influence linear trend
recognition.
mean (X)
range(X)
UCL
avg(X)
mean(R)
LCL
L
TEST 6: Identification of oscillations - indication of periodic
changes, similar to the test 5, interlaces 7 raising and 7 dropping values
(7 periods).
 Suitable for use with mean(X) and range(X)
 Neither zones nor mean values influence recognition
of oscillations.
mean (X)
range(X)
UCL
avg(X)
mean(R)
LCL
L
Page 11/Lect. 10
TEST 7: Overshooting zone C – at least 8 succeeding values (either
above or below the reference value (avg (X)) do not fit into the zone C.
 Suitable for use with mean(X) only
 Indicates:
 Overshooting of the process control (high gain
in the feedback loop),
 Attempt to process more than one process in a
single chart.
range(X)
UCL
avg(X)
A
B
C
C
B
A
LCL
L
TEST 8: Exclusive placement in zone C – at least 15 succeeding
values is located exclusively in the zone C, either above or bellow the
reference value.
 Suitable for use with mean(X) only
 Indicates:
 Loosing process control (due to low gain in the
feedback loop),
 Improper sampling of the process (see the
sample stratification)
 Sinking dispersion of the observed process
which is not proper to the UCL and LCL
control levels setup (set too high for the case).
Page 12/Lect. 10
range(X)
UCL
avg(X)
A
B
C
C
B
A
LCL
L
________________________________________________________□
It is expected simultaneous application of the test1-test8 heuristics in
single chart → multiple marking of end-points of sequences satisfying
particular tests.
Ex.:
A
UCL
A
B
C
C
B
A
LCL
L
B
Value A satisfies: test 7 and test 3
(8pts., 4 of 5pts.)
Value B satisfies: test 7, test 2 and test 3
(8pts., 2 of 3pts., 4 of 5pts.)
Page 13/Lect. 10
Application remarks:
 Simple but efficient!
 The tests are easy to be programmed → computer controlled
manufacturing
 Requires careful adjustment of the sampling, UCL and LCL to
provide expected performance
 The method is adaptive (on-line) via evaluation of the test 8 results.