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┌───────────────────────────────────────────────────────────────┐
│ QTMSIM.EXE...Version 9403 │
├───────────────────────────────────────────────────────────────┤
│ A Quality Assurance Training Tool: │
│ Statistics Committee of the QA Section of the PMA │
├───────────────────────────────────────────────────────────────┤
│ Bob Obenchain, CompuServe User ID=72007,467 │
│ 5261 Woodfield Drive North, Carmel,IN 46033 │
└───────────────────────────────────────────────────────────────┘
Note: PMA = Pharmaceutical Manufacturers Association
Table-of-Contents
Introduction to QTMSIM Simulation Scenarios. . . . . . . . . . . . . 1
Input Data Restrictions. . . . . . . . . . . . . . . . . . . . . . . 4
Naming Conventions for QTMSIM Input/Output Files . . . . . . . . . . 5
Resetting QMP/WAMOC Internal Parameters. . . . . . . . . . . . . . . 8
Example .SIM Scenario Files. . . . . . . . . . . . . . . . . . . . . 8
Results from QTMSIM Simulation Scenarios . . . . . . . . . . . . . . 9
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
QTMSIM Software Update History . . . . . . . . . . . . . . . . . . .22
┌────────────────┐
│ Introduction │
└────────────────┘
QTMSIM.EXE allows the user to define "Simulation Scenarios" in which
true quality level may vary over a sequence of 12 consecutive
reporting periods (say, 12 months.) For each period in the sequence,
the user must specify:
(i) the "volume" of information that will be available from quality
inspections that period, expressed as an Equivalent Expectancy
(EE) at standard quality,
and
(ii) the "true" quality level attained that period, expressed as a
ratio of true expected nonconformance rate to the standard rate.
QTMSIM.EXE then generates pseudo-random Poisson or standardized Gamma
deviates representing "observed" nonconformances for each period. In
other words, period-to-period changes in nonconformance expectancy
follow the scenario input by the user, but state-of-the-art computer
algorithms assure that the observed noncomformances for successive
reporting periods are statistically independent (although usually not
identically distributed.)
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 2 of 22
Next, QTMSIM analyses the generated nonconformance data using either
of two powerful Moving Average methodologies...
QMP = Quality Measurement Plan
or
WAMOC = Windowed At-Most-One Change.
In other words, these Quality-Trend-Monitoring methodologies attempt
to recover the pattern of period-to-period quality variation specified
in the scenario. We illustrate these concepts with an example.
*********************************************************************
EXAMPLE: Suppose a simulation scenario is specified to QTMSIM as...
Period Number: 01 02 03 04 05 06 07 08 09 10 11 12
True Quality Level: 1 1 1 1 1 1 3 3 3 3 3 3
Expectancy at Standard Quality: 2 2 2 2 1 1 1 1 2 2 2 2
This is a scenario with only one shift in level over 12 periods;
quality is at Standard for the 1st 6 periods, then shifts to 3 times
Standard. Meanwhile, during the middle 4 periods, sampling intensity
is half of what it is in the 1st and 3rd 4 periods.
On the type of chart used by QTMSIM for both QMP and WAMOC, this
scenario would be graphed as follows...
Zero Defects ┌──────────────────────────────────────┐
│ │
At Standard ├──T──T──T──T──T──T────────────────────┤
TRUE │ │
QUALITY 2 x Standard │ │
LEVEL │ │
3 x Standard ├────────────────────T──T──T──T──T──T──┤
│ │
4 x Standard │ │
│ │
5 x Standard └──────────────────────────────────────┘
01 02 03 04 05 06 07 08 09 10 11 12
PERIOD -->
The True Nonconformance Expectancy for each period is the product of
its Expectancy at Standard Quality times its True Quality Level...
Period Number: 01 02 03 04 05 06 07 08 09 10 11 12
True Expectancy: 2 2 2 2 1 1 3 3 6 6 6 6
Thus there are 3 changes in per-period Nonconformance Expectancy even
though there is only 1 shift in Quality Level over the 12 periods.
*********************************************************************
Results from the very first "cycle" of at most 12 periods are
displayed in CGA-graphics and also "smoothed" using the QMP or WAMOC
algorithms. Generated quality level estimates (plotted below with the
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 3 of 22
X symbol) will deviate from their true values (T symbol) due to random
sampling error. But QMP or WAMOC I-plots provide nominal 90%
confidence intervals for the (unknown) true level...
Zero Defects ┌──────────────────────────────────────┐
│ X ┬ X ┬ X │
At Standard ├──T┼─T┼─T┬─T┼─TX─T┼───────────────────┤
ESTIMATED │ ┴ X │ ┴ ┴ ┴ ┬ ┬ │
QUALITY 2 x Standard │ ┴ X X ┬ │ ┬ ┬ │
LEVEL │ ┴ ┴ X │ X ┬ X │
3 x Standard ├────────────────────T──T┴─TX─T┼─T┼─T┼─┤
│ ┴ │ X ┴ │
4 x Standard │ ┴ │ │
│ ┴ │
5 x Standard └──────────────────────────────────────┘
01 02 03 04 05 06 07 08 09 10 11 12
PERIOD -->
The numbers of times that the true-Quality-Level was ABOVE, WITHIN, or
BELOW the QMP or WAMOC I-plot limits are then counted and displayed on
your graphics screen. QMP and WAMOC I-plots are designed to cover the
true quality level about 90% of the time, on long range average, at
least when per period expectancy is large. But QTMSIM lets the user
investigate how good those coverage claims are in actual practice!
Of course, any single quality scenario needs to be evaluated hundreds
or even thousands of times in order to develop good estimates of
"coverage probabilities," especially at any points within the scenario
where the true quality level is changing rapidly between periods.
Thus, after that first cycle, QTMSIM uses a "non-graphical mode" so
that many cycles may be evaluated rapidly.
QTMSIM is designed, primarily, to help users new to QMP and/or WAMOC
methodology to "visualize" what can happen. For example, QTMSIM can
be used to show "trade-offs" between conflicting objectives, such as:
...making good use of recent process history to predict
performance during the next reporting period, and
...responding quickly to abrupt shifts in quality level.
In summary, QTMSIM can help new users develop QMP and WAMOC
"intuition." See the QMPCHART.DOC and WAMOC.DOC documentation files
that are distributed with QMPCHART.EXE and WAMOC.EXE, respectively,
for detailed information on these powerful new Moving Average
methodologies.
QTMSIM.EXE...
- accepts either Keyboard or Batch input (from a .SIM file),
- writes detailed simulation results to a .OUT file,
- can make optional dumps of its graphics screen if your PC's has a
slave printer connected to LPT1, and
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 4 of 22
- creates a batch data file (.WIN) of results from only its very
first cycle that is suitable for input to QMPCHART or WAMOC.
┌───────────────────────────┐
│ Input Data Restrictions │
└───────────────────────────┘
Variable Minimum Default Maximum
Name Brief Explanation Value Value Value
------- ------------------------ ------- ------- -------
wind Moving Avg. Window Width 1 6 12
histp number of history periods 0 0 wind-1
preceeding the first period
actually plotted by QTMSIM
E-factor defect expectancy factor .001 1 1
expect Equivalent Expectancy at 0 1 999
standard quality for the
current reporting period
trueindx true quality level for .1 1 6
current reporting period
idum Random Number SEED Value -32767 0* 0
For pseudo-random number generation, QTMSIM uses the poidev(),
gamdev() and gammln() functions of Press, Flannery, Teukolsky, and
Vetterling(1988). The Poisson defect "intensity" parameter used for
the current reporting period is always of the form:
Poisson Intensity = trueindx * expect
NOTE 1: Specifying an E-factor < 1.0 implies that you wish to specify
inspection intensity in terms of SAMPLE SIZES inspected. In this
case, QTMSIM will compute expectancies for you as follows:
EE = Equivalent Expectancy = Sample Size * E-factor.
For example, if you inspect 250 units and the E-factor (quality
standard) is 0.004 defects/unit, then you "expect" to see one
defect, EE = 1, when quality is indeed at its "standard" level.
NOTE 2: idum = 0 causes the start-up seed value to be generated using
your personal computer's System Clock. When QTMSIM.EXE writes a
.SIM file in response to keyboard input, QTMSIM will always set
idum=0 so that successive QTMSIM runs can start at different points
in the pseudo-random sequence and, therefore, give different
results. The seed value that actually gets used is recorded,
however, in the QTMSIM .OUT file. If you use an editor to modify a
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 5 of 22
.SIM file and specify a non-zero seed, the exact same results will
be generated each time that scenario is executed by QTMSIM.
NOTE 3: CAPQUANT software (a separate module in this "QA Training Tool"
series) can be used to convert almost any single (or pair) of
quantitative variable(s) into equivalent defects (nonconformances)
and expectancies. The user can specify any one of ten types of
cost-of-poor-quality loss functions. Due to "Poissonization",
Obenchain(1991), each EE is (at least, approximately,) both the
mean and the variance if its corresponding ED. Thus, once you
know the range of EE values generated by ANY QUALITY PROCESS, you
have all the information needed to use QTMSIM to simulate behavior
of the QMP or WAMOC algorithms in trending that process!
┌────────────────────────────────────────────────────┐
│ Naming Conventions for QTMSIM Input/Output Files │
└────────────────────────────────────────────────────┘
QTMSIM uses three different sorts of input/output files, with 4-
character filename EXTensions of .WIN, .OUT, and .SIM. The contents
of these three types of QTMSIM input/output files are as follows...
=======================================================================
.WIN ==> The QTMSIM output file suitable for input to either the
QMPCHART.EXE or WAMOC.EXE trend charting modules (VGA.)
=======================================================================
I ...I, F, or B
Year-to-Date ...Year Identifier
12 ...Number of Periods
1 2 3 4 5 6 7 8 9 0 A B ...Period Labels
QTM Simulation Scenario ...1st Title
Batch Input File = FILENAME.WIN ...2nd Title
Equivalent Nonconformances ...1st Footer
and Equivalent Expectancies ...2nd Footer
QMP Window Width ...1 to 12
Expectancy Factor ...0 to 999
Audible Signaling Speed ...0.0 to 2.0
Number of History Periods ...0 to WinWidth-1
expectancy defects
expectancy defects [2 numerical values on each row]
expectancy defects
...
NOTE: You may use your favorite text file editor or word processor to
modify/customize the first 8 to 12 lines of a .WIN file. In
fact, it will be NECESSARY to modify the 3rd and 4th lines
whenever your simulation scenario does not include all 12 of the
plotting periods supported by QTMSIM.
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 6 of 22
=======================================================================
.OUT ==> The QTMSIM detailed output file, including...
=======================================================================
Wed Feb 23 22:33:52 1994 ...Date/Time Stamp
Q ...Moving Average Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
FILENAME.SIM ...Simulation Save-Input
FILENAME.WIN ...Moving Average Analysis File
-19023 ...Actual Startup Seed
6 ...Window Width
1.000 ...Expectancy Factor
5 ...History Periods
NEXT: 3 measures for each history period in FIRST CYCLE...
truei = true quality level index for that period
defct = Poisson defectes generated (intensity = truei*exptc)
exptc = equivalent expectancy
truei defct exptc
0.65 64.00 100.00
2.71 279.00 100.00
0.35 37.00 100.00
1.32 151.00 100.00
1.60 164.00 100.00
NEXT: 12 measures for each plotting period in FIRST CYCLE...
truei = true quality level index for that reporting period
defct = Poisson defectes generated (intensity = truei*exptc)
exptc = equivalent expectancy
q95 = lower 95% quality index for current period
q5 = upper 5% quality index for current period
best = best measure of current quality
pavg = process average quality over current window
indx = raw quality index for current period
symb = plotting symbol for current index
pvar = variance of current process average
bvar = variance of current best measure
pfac = P - factor
truei defct exptc q95 q5 best pavg indx symb pvar bvar pfac
1.07 94.00 100.00 1.113 0.796 0.949 1.306 0.940 X 0.518 0.009 0.071
1.06 108.00 100.00 1.264 0.925 1.089 1.377 1.080 X 0.456 0.011 0.064
0.38 44.00 100.00 0.587 0.361 0.468 0.997 0.440 X 0.185 0.005 0.023
0.23 31.00 100.00 0.447 0.250 0.342 0.987 0.310 X 0.197 0.004 0.024
2.36 227.00 100.00 2.487 1.998 2.237 1.110 2.270 X 0.374 0.022 0.048
0.66 69.00 100.00 0.839 0.567 0.697 0.956 0.690 X 0.342 0.007 0.042
0.77 73.00 100.00 0.880 0.601 0.735 0.922 0.730 X 0.349 0.007 0.043
1.21 112.00 100.00 1.292 0.949 1.115 0.929 1.120 X 0.350 0.011 0.043
1.82 194.00 100.00 2.149 1.698 1.918 1.171 1.940 X 0.404 0.019 0.053
0.14 8.00 100.00 0.169 0.054 0.105 1.134 0.080 X 0.466 0.001 0.060
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 7 of 22
0.51 57.00 100.00 0.708 0.461 0.579 0.859 0.570 X 0.270 0.006 0.033
1.11 93.00 100.00 1.091 0.778 0.929 0.898 0.930 X 0.265 0.009 0.032
NEXT: I-plot coverage summary for FIRST CYCLE...
True Index Above I ... 1 times
True Index Within I ... 10 times
True Index Below I ... 1 times
FINALLY: I-plot coverage summary for all non-display cycles...
Elapsed Simulation Time (seconds) = 509
Summary of QTMSIM Results in 5002 Cycles...
Expect TruIdx Below Within Above
100.00 1.07 4.82 90.30 4.88
100.00 1.06 4.56 90.14 5.30
100.00 0.38 1.44 88.76 9.80
100.00 0.23 0.56 87.23 12.22
100.00 2.36 8.02 88.74 3.24
100.00 0.66 4.56 90.40 5.04
100.00 0.77 4.80 90.14 5.06
100.00 1.21 6.06 90.02 3.92
100.00 1.82 6.96 89.36 3.68
100.00 0.14 0.64 88.42 10.94
100.00 0.51 3.64 90.44 5.92
100.00 1.11 6.40 89.96 3.64
Overall % : 4.37 89.50 6.13
Precision +/- 0.289 0.434 0.339
=======================================================================
.SIM ==> The QTMSIM simulation scenario batch input file, including...
=======================================================================
0 ...Seed (0 => Use Clock to get Startup Seed.)
6 ...Moving Average Window Width
1.000 ...Expectancy Factor
0.0 ...Audible Signaling Speed (0.0 => NONE.)
5 ...Number of History Periods
2.00 ...expectancy
1.00 ...trueindx
...
quit ...an end of data signal is needed if fewer than
twelve simulation/display periods are defined.
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 8 of 22
┌───────────────────────────────────────────┐
│ Resetting QMP/WAMOC Internal Parameters │
└───────────────────────────────────────────┘
If files named QMP-PARM.SET and WAMOPARM.SET are located within the
"current working directory" from which QTMSIM.EXE is invoked, QTMSIM
will read in one of these files to initialize its parameters.
Otherwise the following defaults will be used:
┌────────────────┐
│ QMP-PARM.SET │
├────────────────┴────────────────────────────────────────────────────┐
│ 0.4 = e0, start-up equivalent expectancy │
│ 0.4 = x0, start-up equivalent defects = theta0 * e0 │
│ 1.0 = I0, start-up quality index │
│ 0.55 = gamma0, prior mean of gammasq, the process variance │
│ 2.2 = maxgammasq, 95% point of the prior distribution of gammasq │
│ 1.0 = sigma0, variables QMP prior sampling variance │
│ 1.0 = QTMtunes, audible signaling speed factor [0.0,2.0] │
└─────────────────────────────────────────────────────────────────────┘
┌────────────────┐
│ WAMOPARM.SET │
├────────────────┴────────────────────────────────────────────────────┐
│ 0.1 = e0, equivalent expectancy for diffuse prior │
│ 0.1 = d0, equivalent defects for diffuse prior │
│ 1.0 = theta0, default quality index value │
│ 0.35 = gamma0, prior mean of gammasq, the process variance │
│ 1.0 = QTMtunes, audible signaling time factor [0.0,2.0] │
└─────────────────────────────────────────────────────────────────────┘
NOTE: Setting the QTMtunes parameter value to 0.0 in QMP-PARM.SET and
WAMOPARM.SET would make it impossible for QTMSIM.EXE to use "audible
signaling." But, as long as either no ".SET" file is present in the
current working directory or else the QMPtunes parameter setting is
greater than 0.0 in the .SET file, audible signaling can occur. Under
these latter circumstances, the QMPtunes setting either entered from
your Keyboard or read from a Scenario (.SIM) input file will govern
audible signaling.
QMPtunes = 1.0 gives acceptable signaling speed on 20 to 40 Mhz
386/486 machines.
A much faster machine might need the QTMtunes = 2.0 setting to
make the tunes understandable, while a very slow machine might
need the QTMtunes = 0.5 setting to speed the tunes up.
QMPtune's audible tones provide some not-particularly-subtle "aids-to-
interpretation" of each I-plot on a QTM Trend Chart...
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 9 of 22
Case 1: q95 <= 1 High Quality...CHARGE Ahead!!!
Case 2: q5<1 & q95>1 At Standard....no sound
Case 3: q5 > 1 Below Normal...a DEATH in your
immediate family.
┌────────────────────────────────────┐
│ Example .SIM Scenario Files │
└────────────────────────────────────┘
The five simulation scenario files distributed with QTMSIM.EXE
illustrate a variety of situations:
STABLE.SIM ...quality stays fixed at the standard level
LARGESAM.SIM ...very high per period sampling information
ONESHIFT.SIM ...sudden shift in quality level
TRENDOWN.SIM ...steady downward trend in quality
ZIGZAG.SIM ...quality level jumps between just 2 levels
Because the type of Moving Average smoothing (QMP or WAMOC) and the
parametric form of simulation distribution (Poisson or standardized
Gamma) is not specified within the .SIM file, each of the above five
files can actually be used to simulate four different QTMSIM parameter
combinations.
┌────────────────────────────────────────────┐
│ Results from QTMSIM Simulation Scenarios │
└────────────────────────────────────────────┘
[1] The "Large Sample" Scenario...
==================================
Five History Periods ( Window Width = 6 ) followed by Twelve Moving
Average Analysis Periods in which...
Per Period Nonconformance Expectancy is VERY large (100.0).
...As a direct result, the variability in sample quality
indices (X) about their true levels (T) is SMALL.
The true quality levels for the 17 periods is a pseudo-random
sample from a Gamma distribution with mean = variance = 1.0.
NOTE: Almost regardless of the pattern of variability in the true
quality levels in this sort of very-low-sampling-variability scenario,
the confidence intervals resulting from moving average analyses
"should" come very close to achieving their nominal 90% confidence
levels...even though these intervals will tend to be quite narrow in
these "high information" cases!
***************************************************
Q ...Moving Average Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
-11336 ...Actual Startup Seed
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 10 of 22
Elapsed Simulation Time (seconds) = 509
Summary of QTMSIM Results in 5002 Cycles...
Expect TruIdx Below Within Above
100.00 1.07 4.82 90.30 4.88
100.00 1.06 4.56 90.14 5.30
100.00 0.38 1.44 88.76 9.80
100.00 0.23 0.56 87.23 12.22
100.00 2.36 8.02 88.74 3.24
100.00 0.66 4.56 90.40 5.04
100.00 0.77 4.80 90.14 5.06
100.00 1.21 6.06 90.02 3.92
100.00 1.82 6.96 89.36 3.68
100.00 0.14 0.64 88.42 10.94
100.00 0.51 3.64 90.44 5.92
100.00 1.11 6.40 89.96 3.64
Overall % : 4.37 89.50 6.13
Precision +/- 0.289 0.434 0.339
***************************************************
Q ...Moving Average Type, QMP or WAMOC
G ...Distribution, Gamma or Poisson
-13182 ...Actual Startup Seed
Elapsed Simulation Time (seconds) = 468
Summary of QTMSIM Results in 5000 Cycles...
Expect TruIdx Below Within Above
100.00 1.07 4.78 90.00 5.22
100.00 1.06 4.12 90.08 5.80
100.00 0.38 1.02 89.36 9.62
100.00 0.23 0.20 87.14 12.66
100.00 2.36 7.68 88.42 3.90
100.00 0.66 3.88 91.28 4.84
100.00 0.77 4.78 90.16 5.06
100.00 1.21 5.54 90.04 4.42
100.00 1.82 6.64 89.38 3.98
100.00 0.14 0.20 89.48 10.32
100.00 0.51 3.44 90.90 5.66
100.00 1.11 5.04 91.40 3.56
Overall % : 3.94 89.80 6.25
Precision +/- 0.275 0.428 0.342
***************************************************
W ...Moving Average Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
-12516 ...Actual Startup Seed
Elapsed Simulation Time (seconds) = 613
Summary of QTMSIM Results in 5000 Cycles...
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 11 of 22
Expect TruIdx Below Within Above
100.00 1.07 2.30 91.06 6.64
100.00 1.06 4.46 90.94 4.60
100.00 0.38 1.02 69.02 29.96
100.00 0.23 3.34 87.96 8.70
100.00 2.36 10.14 86.98 2.88
100.00 0.66 1.28 91.00 7.72
100.00 0.77 1.92 91.06 7.02
100.00 1.21 4.64 91.08 4.28
100.00 1.82 7.52 90.50 1.98
100.00 0.14 0.52 77.54 21.94
100.00 0.51 4.34 91.02 4.64
100.00 1.11 6.98 91.08 1.94
Overall % : 4.04 87.44 8.53
Precision +/- 0.278 0.469 0.395
***************************************************
W ...Moving Average Type, QMP or WAMOC
G ...Distribution, Gamma or Poisson
-11909 ...Actual Startup Seed
Elapsed Simulation Time (seconds) = 574
Summary of QTMSIM Results in 5000 Cycles...
Expect TruIdx Below Within Above
100.00 1.07 2.26 91.22 6.52
100.00 1.06 5.28 89.80 4.92
100.00 0.38 0.42 71.44 28.14
100.00 0.23 1.94 89.66 8.40
100.00 2.36 9.54 87.42 3.04
100.00 0.66 0.90 91.22 7.88
100.00 0.77 1.40 91.76 6.84
100.00 1.21 5.18 90.62 4.20
100.00 1.82 8.16 89.42 2.42
100.00 0.14 0.16 81.38 18.46
100.00 0.51 3.64 91.62 4.74
100.00 1.11 6.90 91.22 1.88
Overall % : 3.82 88.07 8.12
Precision +/- 0.271 0.458 0.386
***************************************************
Conclusions:
(a) In this "Large Sample" scenario, both moving average methods (QMP
and WAMOC) performed close to expectations.
The coverage probabilities of QMP I-plots (89.50% and
89.80%) were somewhat closer to their nominal 90% level than
were those of WAMOC I-plots (87.44% and 88.07%.)
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 12 of 22
Both QMP and WAMOC I-plots are biased very slightly downward
(toward undesirable, high quality indices.) When the true
level is not within the nominal 90% I-plot interval, it is
somewhat more likely to be above the I than below it.
(b) Results using Gamma and Poisson deviates were quite similar.
[2] The "Perfectly Stable Process" Scenario...
==============================================
Five History Periods ( Window Width = 6 )...
Followed by Twelve Moving Average Analysis Periods...
For EVERY Period, True Index = 1.00 and Expectancy = 2.00
***************************************************
Q ...Moving Average Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
-19536 ...Actual Startup Seed
Elapsed Simulation Time (seconds) = 55
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 2.24 97.00 0.76
2.00 1.00 1.96 97.04 1.00
2.00 1.00 2.04 97.08 0.88
2.00 1.00 1.72 97.28 1.00
2.00 1.00 1.64 97.52 0.84
2.00 1.00 2.04 97.00 0.96
2.00 1.00 1.88 97.48 0.64
2.00 1.00 1.32 97.88 0.80
2.00 1.00 1.60 97.64 0.76
2.00 1.00 1.88 97.40 0.72
2.00 1.00 1.76 97.20 1.04
2.00 1.00 1.88 97.52 0.60
Overall % : 1.83 97.34 0.83
***************************************************
Q ...Moving Average Type, QMP or WAMOC
G ...Distribution, Gamma or Poisson
-24842 ...Actual Startup Seed
Elapsed Simulation Time (seconds) = 61
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 0.44 99.04 0.52
2.00 1.00 0.40 98.72 0.88
2.00 1.00 0.08 99.24 0.68
2.00 1.00 0.24 99.04 0.72
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 13 of 22
2.00 1.00 0.16 98.76 1.08
2.00 1.00 0.32 98.64 1.04
2.00 1.00 0.48 98.60 0.92
2.00 1.00 0.24 98.84 0.92
2.00 1.00 0.20 98.80 1.00
2.00 1.00 0.00 99.08 0.92
2.00 1.00 0.16 98.56 1.28
2.00 1.00 0.32 98.80 0.88
Overall % : 0.25 98.84 0.90
***************************************************
W ...Moving Average Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
-24687 ...Actual Startup Seed
Elapsed Simulation Time (seconds) = 63
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 2.68 96.64 0.68
2.00 1.00 2.96 96.48 0.56
2.00 1.00 2.48 97.28 0.24
2.00 1.00 2.36 96.96 0.68
2.00 1.00 2.68 96.88 0.44
2.00 1.00 3.04 96.72 0.24
2.00 1.00 3.20 96.08 0.72
2.00 1.00 3.28 96.32 0.40
2.00 1.00 3.44 96.16 0.40
2.00 1.00 3.00 96.64 0.36
2.00 1.00 2.96 96.52 0.52
2.00 1.00 2.96 96.28 0.76
Overall % : 2.92 96.58 0.50
***************************************************
W ...Moving Average Type, QMP or WAMOC
G ...Distribution, Gamma or Poisson
-19630 ...Actual Startup Seed
Elapsed Simulation Time (seconds) = 62
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 0.16 98.52 1.32
2.00 1.00 0.28 98.68 1.04
2.00 1.00 0.20 99.32 0.48
2.00 1.00 0.28 99.12 0.60
2.00 1.00 0.32 98.84 0.84
2.00 1.00 0.32 98.88 0.80
2.00 1.00 0.24 98.92 0.84
2.00 1.00 0.20 98.76 1.04
2.00 1.00 0.04 98.72 1.24
2.00 1.00 0.24 99.08 0.68
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 14 of 22
2.00 1.00 0.12 99.20 0.68
2.00 1.00 0.24 99.04 0.72
Overall % : 0.22 98.92 0.86
***************************************************
Conclusions:
(a) In this "Perfectly Stable Process" scenario, the two moving
average methodologies (QMP and WAMOC) performed quite similarly.
In both cases, the confidence levels of I-plots were
consistently much higher (averaging 97% to 98%) than their
intended 90% nominal level.
(b) Results using Gamma and Poisson deviates are quite similar.
┌──────────────────────┬───────────────────────┐
│ Poisson │ Std. Gamma │
┌────────────┼──────────────────────┼───────────────────────┤
│ │ below within above │ below within above │
│ QMP │ 1.83 97.34 0.83 │ 0.25 98.84 0.90 │
├────────────┼──────────────────────┼───────────────────────┤
│ │ below within above │ below within above │
│ WAMOC │ 2.92 96.58 0.50 │ 0.22 98.92 0.86 │
└────────────┴──────────────────────┴───────────────────────┘
[3] The "Sudden Downward Shift" Scenario...
===========================================
Five Stable History Periods ( Window Width = 6 )...
Expectancy = 2; True Quality Level = 1.00.
Followed by Twelve Moving Average Alanysis Periods...
Period Number: 01 02 03 04 05 06 07 08 09 10 11 12
True Quality Level: 1 1 1 1 1 1 3 3 3 3 3 3
Expectancy: 2 2 2 2 1 1 1 1 2 2 2 2
***************************************************
Q ...MA Type, QMP or WAMOC
G ...Distribution, Gamma or Poisson
-145 ...Actual Startup Seed
6 ...Window Size
5 ...History Periods
Elapsed Simulation Time (seconds) = 209
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 0.84 97.68 1.48
2.00 1.00 1.04 97.64 1.32
2.00 1.00 0.76 97.92 1.32
2.00 1.00 1.12 97.88 1.00
1.00 1.00 0.52 98.60 0.88
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 15 of 22
1.00 1.00 0.44 98.60 0.96
1.00 3.00 62.00 37.64 0.36
1.00 3.00 46.68 53.12 0.20
2.00 3.00 24.88 75.04 0.08
2.00 3.00 12.20 87.52 0.28
2.00 3.00 6.24 93.44 0.32
2.00 3.00 2.80 96.08 1.12
Overall % : 13.29 85.93 0.78
***************************************************
Q ...MA Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
-171 ...Actual Startup Seed
6 ...Window Size
5 ...History Periods
Elapsed Simulation Time (seconds) = 212
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 2.40 96.36 1.24
2.00 1.00 1.84 97.28 0.88
2.00 1.00 1.92 97.48 0.60
2.00 1.00 2.20 96.96 0.84
1.00 1.00 1.52 97.96 0.52
1.00 1.00 1.56 97.72 0.72
1.00 3.00 59.40 40.56 0.04
1.00 3.00 41.92 58.08 0.00
2.00 3.00 23.32 76.68 0.00
2.00 3.00 11.16 88.76 0.08
2.00 3.00 5.20 94.48 0.32
2.00 3.00 3.36 95.84 0.80
Overall % : 12.98 86.51 0.50
***************************************************
W ...MA Type, QMP or WAMOC
G ...Distribution, Gamma or Poisson
-9124 ...Actual Startup Seed
6 ...Window Width
5 ...History Periods
Elapsed Simulation Time (seconds) = 244
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 0.52 98.72 0.76
2.00 1.00 0.40 98.80 0.80
2.00 1.00 0.28 98.68 1.04
2.00 1.00 0.28 98.72 1.00
1.00 1.00 0.00 99.40 0.60
1.00 1.00 0.08 99.52 0.40
1.00 3.00 14.00 85.20 0.80
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 16 of 22
1.00 3.00 3.68 95.40 0.92
2.00 3.00 2.64 95.72 1.64
2.00 3.00 1.00 97.24 1.76
2.00 3.00 0.52 97.68 1.80
2.00 3.00 0.40 98.60 1.00
Overall % : 1.98 96.97 1.04
***************************************************
W ...MA Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
-8696 ...Actual Startup Seed
6 ...Window Width
5 ...History Periods
Elapsed Simulation Time (seconds) = 245
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 2.64 96.88 0.48
2.00 1.00 2.28 97.36 0.36
2.00 1.00 2.28 97.36 0.36
2.00 1.00 2.44 96.96 0.60
1.00 1.00 0.96 99.00 0.04
1.00 1.00 1.40 98.48 0.12
1.00 3.00 12.24 87.24 0.52
1.00 3.00 5.88 93.60 0.52
2.00 3.00 4.44 94.20 1.36
2.00 3.00 2.08 96.84 1.08
2.00 3.00 1.64 97.68 0.68
2.00 3.00 2.00 97.60 0.40
Overall % : 3.36 96.10 0.54
***************************************************
Conclusions:
(a) WAMOC seems to detect a sudden (downward) shift better than QMP.
The confidence level of QMP I-plots quickly drops below 50% and
remains low for 3 or 4 periods; WAMOC I-plots maintain a
confidence level of at least 85% and also recover within 1 or 2
periods.
(b) Results using Gamma and Poisson deviates are quite similar.
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 17 of 22
[4] The "Steady Downward Trend" Scenario...
===========================================
Five Stable History Periods ( Window Width = 6 )...
Expectancy = 2; True Quality Level = 1.00.
Followed by Twelve Moving Average Alanysis Periods...
Expectancy = 2; True Quality Level = 1.00 to 4.67 by 0.33
***************************************************
Q ...MA Type, QMP or WAMOC
G ...Distribution, Gamma or Poisson
-13704 ...Actual Startup Seed
6 ...Window Width
5 ...History Periods
Elapsed Simulation Time (seconds) = 234
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 0.96 97.60 1.44
2.00 1.33 6.28 93.16 0.56
2.00 1.67 15.92 83.72 0.36
2.00 2.00 22.40 77.08 0.52
2.00 2.33 26.24 73.48 0.28
2.00 2.67 25.84 73.72 0.44
2.00 3.00 23.52 76.36 0.12
2.00 3.33 22.56 77.24 0.20
2.00 3.67 22.00 77.92 0.08
2.00 4.00 21.68 78.28 0.04
2.00 4.33 20.96 79.04 0.00
2.00 4.67 21.76 78.12 0.12
Overall % : 19.18 80.48 0.35
***************************************************
Q ...MA Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
-14076 ...Actual Startup Seed
6 ...Window Width
5 ...History Periods
Elapsed Simulation Time (seconds) = 224
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 1.96 97.00 1.04
2.00 1.33 7.24 92.60 0.16
2.00 1.67 15.36 84.56 0.08
2.00 2.00 20.44 79.36 0.20
2.00 2.33 21.84 78.12 0.04
2.00 2.67 25.04 74.96 0.00
2.00 3.00 24.00 75.88 0.12
2.00 3.33 21.60 78.36 0.04
2.00 3.67 22.48 77.52 0.00
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 18 of 22
2.00 4.00 20.04 79.96 0.00
2.00 4.33 20.80 79.20 0.00
2.00 4.67 19.96 80.04 0.00
Overall % : 18.40 81.46 0.14
***************************************************
W ...MA Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
-11308 ...Actual Startup Seed
6 ...Window Width
5 ...History Periods
Elapsed Simulation Time (seconds) = 262
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 2.56 97.04 0.40
2.00 1.33 2.76 96.88 0.36
2.00 1.67 3.52 96.12 0.36
2.00 2.00 3.48 96.12 0.40
2.00 2.33 3.16 96.28 0.56
2.00 2.67 2.64 96.84 0.52
2.00 3.00 2.64 96.76 0.60
2.00 3.33 2.48 96.92 0.60
2.00 3.67 2.52 96.96 0.52
2.00 4.00 1.76 97.84 0.40
2.00 4.33 1.68 97.76 0.56
2.00 4.67 1.40 97.84 0.76
Overall % : 2.55 96.95 0.50
***************************************************
W ...MA Type, QMP or WAMOC
G ...Distribution, Gamma or Poisson
-10988 ...Actual Startup Seed
6 ...Window Width
5 ...History Periods
Elapsed Simulation Time (seconds) = 272
Summary of QTMSIM Results in 2500 Cycles...
Expect TruIdx Below Within Above
2.00 1.00 0.12 98.92 0.96
2.00 1.33 1.28 98.04 0.68
2.00 1.67 2.08 96.92 1.00
2.00 2.00 2.20 96.68 1.12
2.00 2.33 1.48 97.44 1.08
2.00 2.67 1.12 98.08 0.80
2.00 3.00 1.56 97.60 0.84
2.00 3.33 1.20 97.84 0.96
2.00 3.67 0.64 98.60 0.76
2.00 4.00 1.04 98.28 0.68
2.00 4.33 0.72 98.80 0.48
2.00 4.67 1.12 97.72 1.16
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 19 of 22
Overall % : 1.21 97.91 0.88
***************************************************
Conclusions:
(a) WAMOC seems to detect steady (downward) trends better than QMP.
The confidence level of QMP I-plots quickly drops from the high
90 percentiles down below 80%, while WAMOC I-plots maintain
confidence levels in the high 90 percentiles.
(b) Results using Gamma and Poisson deviates are quite similar.
[5] The "ZigZag" Scenario...
============================
Five History Periods ( Window Width = 6 )...
Followed by Twelve Moving Average Alanysis Periods...
True Quality level jumps back an forth between 0.50 and 1.50.
Note that true quality thus averages 1.00 and has
variance 0.25 within each window of 6 periods.
Every Period has Nonconformance Expectancy = 2.00.
Note that the per period sampling variance at standard
quality (level 1.00) is thus 0.50.
***************************************************
Q ...Moving Average Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
-10468 ...Actual Startup Seed
At Cycles = 5000, Overall % : 7.98 85.87 6.15
At Cycles = 10000, Overall % : 8.13 85.79 6.09
At Cycles = 15000, Overall % : 8.21 85.76 6.03
At Cycles = 20000, Overall % : 8.18 85.71 6.11
At Cycles = 25000, Overall % : 8.15 85.69 6.16
At Cycles = 30000, Overall % : 8.16 85.69 6.15
At Cycles = 35000, Overall % : 8.14 85.70 6.16
Elapsed Simulation Time (seconds) = 2930
Summary of QTMSIM Results in 36004 Cycles...
Expect TruIdx Below Within Above
2.00 1.50 16.28 83.57 0.16
2.00 0.50 0.01 87.91 12.08
2.00 1.50 16.45 83.40 0.16
2.00 0.50 0.00 87.88 12.12
2.00 1.50 16.27 83.56 0.17
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 20 of 22
2.00 0.50 0.01 87.86 12.13
2.00 1.50 16.46 83.39 0.14
2.00 0.50 0.00 87.76 12.24
2.00 1.50 16.10 83.76 0.14
2.00 0.50 0.01 87.65 12.34
2.00 1.50 16.07 83.77 0.16
2.00 0.50 0.01 87.83 12.16
Overall % : 8.14 85.69 6.17
***************************************************
Q ...Moving Average Type, QMP or WAMOC
G ...Distribution, Gamma or Poisson
-10468 ...Actual Startup Seed
At Cycles = 2000, Overall % : 7.66 86.78 5.57
At Cycles = 4000, Overall % : 7.72 86.81 5.46
At Cycles = 6000, Overall % : 7.66 86.84 5.50
At Cycles = 8000, Overall % : 7.65 86.83 5.52
Elapsed Simulation Time (seconds) = 768
Summary of QTMSIM Results in 10002 Cycles...
Expect TruIdx Below Within Above
2.00 1.50 15.93 83.61 0.46
2.00 0.50 0.00 89.57 10.43
2.00 1.50 15.45 84.01 0.54
2.00 0.50 0.00 89.60 10.40
2.00 1.50 15.21 84.20 0.59
2.00 0.50 0.00 89.12 10.88
2.00 1.50 15.31 84.16 0.53
2.00 0.50 0.00 89.41 10.59
2.00 1.50 14.91 84.61 0.48
2.00 0.50 0.00 89.50 10.50
2.00 1.50 15.67 83.77 0.56
2.00 0.50 0.00 89.64 10.36
Overall % : 7.71 86.77 5.53
***************************************************
W ...Moving Average Type, QMP or WAMOC
P ...Distribution, Gamma or Poisson
-10468 ...Actual Startup Seed
At Cycles = 2000, Overall % : 3.05 96.15 0.80
At Cycles = 4000, Overall % : 3.03 96.16 0.81
At Cycles = 6000, Overall % : 3.01 96.19 0.80
At Cycles = 8000, Overall % : 3.05 96.14 0.81
Elapsed Simulation Time (seconds) = 952
Summary of QTMSIM Results in 10000 Cycles...
Expect TruIdx Below Within Above
2.00 1.50 5.58 94.03 0.39
2.00 0.50 0.98 97.87 1.15
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 21 of 22
2.00 1.50 5.49 94.07 0.44
2.00 0.50 0.98 97.78 1.24
2.00 1.50 4.93 94.68 0.39
2.00 0.50 0.77 98.02 1.21
2.00 1.50 5.11 94.49 0.40
2.00 0.50 0.90 97.89 1.21
2.00 1.50 5.51 94.15 0.34
2.00 0.50 1.07 97.58 1.35
2.00 1.50 5.12 94.47 0.41
2.00 0.50 0.83 97.87 1.30
Overall % : 3.11 96.08 0.82
***************************************************
W ...Moving Average Type, QMP or WAMOC
G ...Distribution, Gamma or Poisson
-10468 ...Actual Startup Seed
At Cycles = 2000, Overall % : 1.43 97.11 1.46
At Cycles = 4000, Overall % : 1.42 97.13 1.45
At Cycles = 6000, Overall % : 1.42 97.12 1.47
At Cycles = 8000, Overall % : 1.42 97.13 1.45
Elapsed Simulation Time (seconds) = 916
Summary of QTMSIM Results in 10000 Cycles...
Expect TruIdx Below Within Above
2.00 1.50 3.04 96.07 0.89
2.00 0.50 0.06 98.02 1.92
2.00 1.50 2.95 96.07 0.98
2.00 0.50 0.06 98.08 1.86
2.00 1.50 2.64 96.28 1.08
2.00 0.50 0.02 98.02 1.96
2.00 1.50 2.76 96.20 1.04
2.00 0.50 0.08 98.05 1.87
2.00 1.50 2.74 96.32 0.94
2.00 0.50 0.03 98.28 1.69
2.00 1.50 2.91 96.00 1.09
2.00 0.50 0.05 98.09 1.86
Overall % : 1.45 97.12 1.43
***************************************************
Conclusions:
(1) In this "ZigZag" scenario, the two moving average methodologies
displayed very different confidence levels. Coverage of QMP
I-plots dropped well below their nominal 90% level to only 86%
to 87%, while WAMOC I-plot coverages remained well above their
nominal 90% level at 96% to 97%.
QTMsim, version 9403 . . . . . . . . . . . . . . . Page 22 of 22
(2) Results using Gamma and Poisson deviates are quite similar.
┌──────────────────────┐
│ REFERENCES │
└──────────────────────┘
AHRENS and DIETER(1974). "Computing methods for sampling from gamma,
beta, poisson and binomial distributions." Computing 12,
223-246.
HOADLEY, Bruce. (1981). "The Quality Measurement Plan, QMP." Bell
System Technical Journal, 60, 215-273.
HOADLEY, Bruce. (1986). "QUALITY MEASUREMENT PLAN(QMP)." Encyclopedia
of Statistical Sciences. (Kotz, Johnson and Read, Editors)
Volume 7, pages 393-398. New York, John Wiley.
INTRODUCTION TO QUALITY TREND CHARTS. Bell Communications Research,
Inc., January 1986 (18 page booklet.)
L'ECUYER, Pierre (1988). "Efficient and Portable Combined Random
Number Generators." Communications of the ACM 31, 742-749,774.
OBENCHAIN, Robert. (1994). "Cumulative Capability Curves." Submitted
to Technometrics.
PRESS, FLANNERY, TEUKOLSKY, and VETTERLING (1988), "Numerical Recipes
in C: The Art of Scientific Computing." [especially Chapter 7:
Random Number Generation.] Cambridge University Press. {Source
Code: Copyright 1985, 1987 by Numerical Recipes Software P.O.Box
243, Cambridge, MA 02238.}
QUALITY MEASUREMENT PLAN (QMP). Bell Communications Research Technical
Reference, TR-TSY-000438. Issue 1, April 1987.
Software Update History:
========================
9403 ...Original Version
