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qmp9210.txt
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QMP makes CGA-graphics displays of six types of QMP and/or Alternative
Charts for Quality-Trend-Monitoring. QMP accepts Keyboard or Batch
data file input, writes detailed results to an output file, and makes
optional dumps of its graphics screen to your PC's slave printer.
Trend Chart Type: I ==> QMP I-plots [Bayesian Confidence Intervals]
----------------- F ==> QMP Forecast Ahead Intervals [and I-s]
B ==> QMP Process Standard Error Boxes [and I-s]
N ==> Normal Theory QMP I-plots
E ==> Exp.Weighted Moving Indices and Forecasts
S ==> Shewhart t-Rate Control Charts
┌─────────────────────────────────────────────────────────────────────┐
│ Table of Contents │
├─────────────────────────────────────────────────────────────────────┤
│ Input Data Restrictions. . . . . . . . . . . . . . . . . . . . . 1 │
│ Naming Conventions and Contents of QMP Output Files. . . . . . . 2 │
│ Resetting QMP Internal Parameters. . . . . . . . . . . . . . . . 3 │
│ Audible Signaling Options. . . . . . . . . . . . . . . . . . . . 3 │
│ Background Information on QMP. . . . . . . . . . . . . . . . . . 4 │
│ Equivalent Defects & Defect Expectancy . . . . . . . . . . . . . 5 │
│ QMP Trend Chart Graphical Conventions. . . . . . . . . . . . . . 5 │
│ SUMMARY of Advantages of Attributes QMP Methodology. . . . . . .11 │
│ Normal Theory QMP. . . . . . . . . . . . . . . . . . . . . . . .12 │
│ Exponentially Weighted Moving Indices. . . . . . . . . . . . . .12 │
│ Shewhart t-Rate Control Charts . . . . . . . . . . . . . . . . .13 │
│ REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . .14 │
│ Appendix 1: The EIGHT Trend Chart BENCHMARK Examples . . . . . .14 │
│ Appendix 2: Advanced Topics in Interpretation. . . . . . . . . .16 │
│ Appendix 3: Exp.Weighted Moving Average Algorithm. . . . . . . .20 │
└─────────────────────────────────────────────────────────────────────┘
- 2 -
┌───────────────────────────┐
│ Input Data Restrictions │
└───────────────────────────┘
Data must be input to QMP in the form of Equivalent Defects (ED) and
Equivalent Expectancies (EE) for a series of consecutive reporting
periods (at most 50 history periods and 12 current, DISPLAY periods.)
For details on ED's and EE's, see the material starting on page 4.
Here is a simple example of "expectancy" at standard quality: If you
inspect 250 units where the quality standard is 0.004 defects/unit,
then you "expect" to see one defect.
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 this "Poissonization",
Obenchain(1991), each EE is (at least, approximately,) both the mean
and the variance if its corresponding ED.
When forming composite quality indicies, sum together all ED's and
EE's for a reporting period. Enter only these sums into QMP.
┌─────────────────────────────────────────────────────────┐
│ Naming Conventions and Contents of QMP Output Files │
└─────────────────────────────────────────────────────────┘
The 3-character filename EXTension of the QMP output file indicates
which of the three types of calculations were performed...
OTI ==> The following 11 measures for each Attributes QMP I-plot:
defct = equivalent defects
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
OTF ==> Basic 11 Measures of OTI plus two Forecast Ahead Bounds:
f95 = lower 95% forecast index for next period
f5 = upper 5% forecast index for next period
OTB ==> Basic 11 Measures of OTI plus two Process Average Bounds:
p95 = lower 95% quality index for process average
p5 = upper 5% quality index for process average
- 3 -
OTN ==> Basic 11 Measures of OTI except uses Normal Theory QMP
OTE ==> The following 9 measures for each reporting period
defct = equivalent defects
exptc = equivalent expectancy
q95 = lower 95% EWMA Forecast Limit
q5 = upper 5% EWMA Forecast Limit
ewmdef = Exp.Weighted Moving Average Defects
ewmexp = Exp.Weighted Moving Average Expectancy
ewavg = Exp.Weighted Moving Average INDEX = ewmdef/ewmexp
indx = raw quality index for current period = defct/exptc
sigma2 = Variance of ewmdef
OTS ==> The following 3 measures for each reporting period
defct = equivalent defects
exptc = equivalent expectancy
t-rate = Shewhart t-Rate = ( exptc - defct ) / sqrt( exptc )
...i.e. upward (+) values represent good quality,
while downward (-) values represent poor quality.
┌──────────────────────────────────────────┐
│ Resetting QMP Internal Parameters │
└──────────────────────────────────────────┘
If a file named QMP-PARM.SET is within the "current working directory"
from which QMP.EXE is invoked, QMP will read that file to initialize
its parameters. Otherwise the following defaults will be used:
┌─────────────────────────────────────────────────────────────────────┐
│ 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 = QMPtunes, audible signaling time factor [0.0,2.0] │
└─────────────────────────────────────────────────────────────────────┘
NOTE: Setting the QMPtunes time factor to 0.0 in a QMP-PARM.SET file
elimates "audible signaling." But, if either no QMP-PARM.SET file is
present in the current working directory or else the QMPtunes time
factor is set to a strictly positive value, audible signaling can
occur. Under these latter circumstances, the 'Y'es or 'N'o signaling
options specified either from your Keyboard or else within your Batch
Input (.IN) file will then either retain or eliminate audible
signaling. The duration of audible signaling can vary from quick
[small time factor = 0.1] to slow [max time factor = 2.0].
QMPtune's audible tones provide some not-particularly-subtle "aids-to-
interpretation" of Quality Index Intervals on Trend Charts...
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.
- 4 -
┌──────────────────────────────────────────┐
│ Background Information on QMP │
└──────────────────────────────────────────┘
QMP is a totally-integrated, graphical approach to quality monitoring
that offers both theoretical and practical advantages over
conventional control charts for revealing quality trends. An
individual quality manager may wish to view different processes in
different ways using his or her favorite methodology (Shewart, X-bar &
R or S, Moving Average, Cusum, P-Charts, Summary Tables, Verbal
Commentaries) for each different situation. But the tasks of making
comparisons across processes and developing consensus views are
greatly facilitated by having a single, common way to monitor all
quality processes. QMP makes "common ground" visible by incorporating
state-of-the-art statistical methodology that is applicable to all
types of quality trend monitoring activities.
QMP graphical displays (called Trend Charts) have a "look" that is
different from the control charts produced by other methodologies.
New users do need to be trained how to read these Trend Charts. In
fact, the interpretation of Trend Charts is one of the main topics of
this introduction. Study of the material discussed here will convince
you that the "rules" for reading Trend Charts are simple, straight-
forward, and objective. You will find that QMP results are actually
rather difficult to "misinterpret" because no subjective opinions or
visual extrapolations/interpolations are needed to read them properly.
On the other hand, QMP displays might be even easier to read if they
were not as fully informative as they are. Consider the following
list of features of QMP Trend Charts:
* Definitive "Zero-Defect" Limit...attained only when all
observed values fall at their Process Target Value(s),
* Process Capability "Standard" for Deviations from Target,
* Quality INDEX Scale...showing the ratio of the Observed
Cost-of-Poor-Quality to the Expected Cost at the Process
Capability Threshold,
* Purely Descriptive Measure of Quality within the Current
Reporting Period (Month or Week),
* Powerful Statistical Inferences (Confidence Intervals) that
utilize the Process "Track Record" (History) over a Window
of Recent Reporting Periods, and
* Special Signals to indicate Missing Data (Skipped Periods) or
Poor Performance demanding immediate Management Action.
Every "conventional" control charting method (Shewart, X-bar & R or S,
Moving Average, Cusum, P-Charts) fails to inform its reader about at
least one of the points that are prominently displayed on every QMP
Trend Chart. In fact, nothing less than QMP can provide sound
- 5 -
quantitative underpinnings for the full spectrum of modern "Total
Quality Control" and "Statistical Process Control" rhetoric.
┌──────────────────────────────────────────┐
│ Equivalent Defects & Defect Expectancy │
└──────────────────────────────────────────┘
EXPECTANCY is a better measure than sample size of the amount of
information in a sample. EXPECTANCY is the number of defects
expected at Standard Quality. It is defined as the sample size, n,
times the Standard Quality per unit, s. It allows a more equitable
comparison of Trend Charts for different products, where the sample
sizes may be the same, but the Standard Quality per unit is
different, as shown by the following example. A sample of 100
cartons of aerosol cans has a relatively large defect expectancy as
compared to a sample of 100 tablets from one bottle, which has a
much smaller defect expectancy. We obtain a lot more information
from inspecting 100 cartons than from inspecting 100 tablets even
though the sample sizes are the same.
Attributes Data: Proportion of Non-Conforming (Defective) Units
---------------------------------------------------------------
n = sample size inspected
s = quality standard in proportion defective
ED = observed number of defects
EE = defect expectancy = n times s
Variables Data: Quantitative Measurements
-----------------------------------------
Q = Cost-of-Poor-Quality Loss Function that Measures the $$$
Impact of Deviations of Measurements from Target Value(s).
E(Q) = Expected Value of Q at Present Capability
V(Q) = Variance of Q at Present Capability
ED = Equivalent Defects = Q times [ E(Q) divided by V(Q) ]
EE = Equivalent Expectancy = [ square of E(Q) ] divided by V(Q)
The process of converting observed LOSS into Equivalent Defects and
Equivalent Expectancy is called "Poissonization", Obenchain(1991.)
Nothing more is involved in this process than a simple "Change-of-
Scale"...ED equals Q times the known constant value that assures that
the Mean and Variance of ED will be EQUAL (to EE = Equivalent
Expectancy.) This equality of mean and variance is characteristic of
Poisson Distributions.
- 6 -
┌───────────────────────────────────────────┐
│ ┌───────────────────────────────────────┐ │
│ │ QMP Trend Chart Graphical Conventions │ │
│ └───────────────────────────────────────┘ │
└───────────────────────────────────────────┘
Trend Charts provide both a series of I-Plots and a process average
quality line that show, respectively, product quality within the
current period and long-range quality trends over time.
QMP utilizes not only data from the current rating period, but also
makes an assessment of how well all of the data within a WINDOW of
recent rating periods are related to CURRENT QUALITY. (This Window is
usually either 6 months or 13 weeks wide.) Using that assessment, QMP
then combines current data with recent process history to make an
optimal statistical evaluation of the CURRENT QUALITY of the process.
How much better the evaluation turns out to be depends on the relevance
of the historical data.
┌──────────────────────────────────────────────────────────────┐
│ Trend Chart AXES... │
│ │
│ Q 0┌───────────────────────┐ <───Zero Defect Limit │
│ u I │ │ │
│ a n 1├───────────────────────┤ <───Process Capability │
│ l d │ │ Threshold │
│ i e │ │ (Quality Standard) │
│ t x │ │ │
│ y 5└───────────────────────┘ <───Bad Quality Limit │
│ Time ───> (Off-the-Chart Threshold) │
└──────────────────────────────────────────────────────────────┘
The Time Scale
──────────────
The symbols running horizontally across the top of the Trend Chart
identify monthly or weekly "rating periods." Months are identified by
the initial letter of their name: J F M A M J J A S O N D. The 52 weeks
of a year are divided into 4 quarters; the 13 weeks of each quarter are
identified as 1 2 3 4 5 6 7 8 9 0 A B C. Quality in each rating period
is shown by the I-Plot below the period symbol.
The Index Scale
───────────────
On the left side of a Trend Chart is an "Index Scale" used for measuring
quality. This scale runs from 0 to 5, with 1 representing Standard
Quality. When Observed Quality is equal to what would be Expected at
Standard, the index (or ratio) equals 1. Values of the index less that
1 represent better-than-standard quality; values greater than 1 show
worse-than-standard quality. For example, the value 2 would mean that
there are twice as many defects observed as expected under the quality
standard.
- 7 -
UP is GOOD
0 ┌────── ...High Quality
1 ├─
2 │
3 │
4 │
5 └────── DOWN is BAD
...Low Quality
Standard Quality is defined to be the Quality Level corresponding to the
Threshold of the Present Capability of a Process. Using Statistical
Process Control, Processes can be maintained either right at their
Standard Level or even BETTER...i.e. fewer defects or lower Cost-of-Poor-
Quality. The Standard is usually specified at the bottom of each Trend
Chart in terms of a (Per Measurement) Expected Value and Variance for
LOSS (Cost-of-Poor-Quality.)
The Symbols on Trend Charts
───────────────────────────
For each reporting period (month or week) in a Trend Chart there is an
I-Plot consisting of four elements: a vertical line segment with two end
points, a horizontal dash, an X, and a circle. The circles in the
figures are connected from period to period by lines which run generally
horizontally across the Trend Chart. Let's take a closer look at one of
the I-Plots:
┌──────────────────────────────────────────────────────────────┐
│ │
│ ─┬─ <─── 5% Quality Percentile │
│ │ │
│ _ │ │
│ -O_ <─── Long Run (Process) Average Quality Index │
│ │ - │
│ ─┼─ <─── Best Measure of Current Quality │
│ │ │
│ │ Observed │
│ X <─── Current Sample Index = ──────── Ratio │
│ │ Expected │
│ │ │
│ ─┴─ <───95% Quality Percentile │
│ │
└──────────────────────────────────────────────────────────────┘
CURRENT SAMPLE INDEX (the X): The quality index for the sample
taken in the current rating period (month or week) is called the
CURRENT SAMPLE INDEX and is simply the ratio of the number of
defects actually found in the sample to the number of defects
expected at Standard Quality.
- 8 -
Observed Cost-of-Poor-Quality
X Symbol = Current Quality INDEX = ─────────────────────────────────
Expected Cost at Standard Quality
LONG RUN AVERAGE (the circle): The data inside the WINDOW (usually
the 5 prior months plus the current month or the 12 prior weeks and
the current week) are combined into a summary index called the LONG
RUN AVERAGE. This number is a weighted average of individual
indices, where the weighting merely accounts for period-to-period
variation in the sample size.
0 Symbol = Process Average Quality Level over the Window
of Recent Reporting Periods.
Note: The 0 symbols for adjacent reporting periods
are connected by a straight line segment.
BEST MEASURE OF CURRENT QUALITY (the horizontal dash) is a weighted
average of the LONG RUN AVERAGE and the CURRENT SAMPLE INDEX. It is the
"best" estimate of the true (but unknown) quality of current production.
The weights used in the average depend on the relative magnitudes of two
sources of variation:
┌────────────────────────────────────────────────────────────┐
│ Process Variation, PV - a measure of the period-to-period │
│ variation in the product's true quality. │
├────────────────────────────────────────────────────────────┤
│ Sampling Variation, SV - a measure of the deviations │
│ between the quality in the sample and the product's true │
│ quality in a rating period. │
└────────────────────────────────────────────────────────────┘
These two sources of variation account for the Total Variation in
CURRENT QUALITY: TV = PV + SV.
The BEST MEASURE OF CURRENT QUALITY is of the form
BM = Weight [LONG RUN AVERAGE] + (1-Weight) [CURRENT SAMPLE INDEX]
= Empirical Bayes Weighted Average of the "O" and the "X",
where
SV
Weight = ───────── .
SV + PV
The larger is the sampling variation, SV, relative to the process
variation, PV, the more weight is put on the LONG RUN AVERAGE. The
observed percent shrinkage, or movement, of the BEST MEASURE OF CURRENT
QUALITY toward the LONG RUN AVERAGE (and away from the CURRENT SAMPLE
INDEX) is defined by the Weight times 100 %.
- 9 -
When the sample is small, producing a large sampling variation, the BEST
MEASURE gets most of its information from the LONG RUN AVERAGE and
shrinks towards the LONG RUN AVERAGE. On the Trend Chart, the BEST
MEASURE will appear close to the LONG RUN AVERAGE. When the sample gets
larger, the BEST MEASURE gets more information from the sample and
shrinks less toward the LONG RUN AVERAGE. As sampling variation gets
small enough to be in the same numerical range as the process variation,
W approaches 1/2 and the BEST MEASURE appears to split the distance
between the X, the CURRENT SAMPLE INDEX, and the circle, the LONG RUN
AVERAGE...
┌──────────────────────────────────────────────────────────────┐
│ ─┬─ ─┬─ ─┬─ │
│ X │ │ ─┬─ │
│ ─┬─ │ X X │ │
│ │ ─┼─ ─┼─ ─┼─ - - - - - -O- │
│ │ │ - - - - - - O - - - - - - O ─┼─ │
│ -O- - - - - O │ │ X │
│ │ │ │ │ ─┴─ │
│ ─┼─ │ │ ─┴─ │
│ │ │ ─┴─ │
│ X ─┴─ │
│ │ │
│ ─┴─ │
└──────────────────────────────────────────────────────────────┘
Remember that the BEST MEASURE OF CURRENT QUALITY will always fall
between the O, representing the LONG RUN AVERAGE, and the X
representing the CURRENT SAMPLE INDEX. This is always the case
because the BEST MEASURE OF CURRENT QUALITY is a weighted average of
the two. The distance of the X and the circle on the index scale from
the BEST MEASURE OF CURRENT QUALITY is an indication of which of the
two carried more weight in the calculation of the BEST MEASURE. The
closer one carried more weight for the period.
┌──────────────────────────────────────────────────────────────┐
│ QMP Trend Charts consist of a time series of "I-plot" │
│ confidence intervals for the current quality level... │
│ │
│ Q 0┌──────────────────────┐ │
│ u I │ ┬ ┬ ┬ ┬ │ │
│ a n 1├ │ ┬ ┬ │ │ ┴ ┤ │
│ l d │ │ │ ┬ │ │ ┴ │ │
│ i e │ │ │ │ ┴ ┴ │ │
│ t x │ ┴ ┴ ┴ │ │
│ y 5└──────────────────────┘ │
│ Time ───> │
└──────────────────────────────────────────────────────────────┘
I-Plot = 90 Percent Confidence Interval for the Unknown True
the Current Period X and the Process Average 0
- 10 -
Percentile Indices
──────────────────
The two end points of the I-Plot are called the 5th and 95th
percentile indices, respectively. There is a 95% chance that the true
quality level is worse than the 5th percentile index at the TOP of the
I-Plot; similarly, there is a 5% chance that the true quality is even
worse than the 95th percentile index at the BOTTOM of the I-Plot.
Therefore, there is a 90% chance that the unknown, TRUE quality level
is COVERED by the I-Plot interval estimate for CURRENT QUALITY.
┌──────────────────────────────────────────────────────────────┐
│ If the quality characteristics of a manufacturing │
│ process are changing over time, its confidence limits │
│ will move up, move down, or change height... │
│ │
│ Q 0┌───────────────────── │
│ u I │ ┬ ┬ ┬ <─── 5% Percentile │
│ a n 1├ │ ┬ ┬ │ │ and │
│ l d │ │ │ ┬ │ │ ┴ <───95% Percentile │
│ i e │ │ │ │ ┴ ┴ for last │
│ t x │ ┴ ┴ ┴ Reporting │
│ y 5└───────────────────── Period... │
│ Time ───> │
└──────────────────────────────────────────────────────────────┘
Location of the I-Plot
──────────────────────
The location of the I-Plot on the index scale for the Current Period
is very important in the interpretation of a Trend Chart.
Nominal Case: The I-Plot INCLUDES or is entirely ABOVE the
Process Capability Standard Value of One.
In this case, one does not have strong evidence (there is less
than a 95% chance) that the true quality is worse than Standard.
Below Normal: The I-Plot has fallen BELOW the Index = 1 line that
represents Present Process Capability.
In such a case, where the 5th percentile index falls below
Standard, there is more than a 95% chance that the product or
service being tested/inspected is worse than Standard. This is
strong evidence of a quality problem.
Occasionally, an entire I-Plot will fall Off-the-Chart. When
this occurs, the numerical value of the 5th percentile index is
printed on the Trend Chart as a ** Special Signal ** to warn
management that CURRENT QUALITY is totally unacceptable.
N Symbol = ** Special Signal ** that no sample data were taken that
reporting period (month or week.) In this case, the LONG
RUN AVERAGE remains the same as in the previous period.
- 11 -
Next Period Ahead FORECASTS and Process Average BOXES
─────────────────────────────────────────────────────
QMP's measure of period-to-period Process Variation (PV) can be used
to make (i) a FORECAST confidence interval for the next reporting
period or (ii) a BOX confidence interval for average quality across the
entire window of recent reporting periods.
In forecast mode, the usual QMP I-plot is replaced by a PAIR of
back-to-back Confidence Intervals; the present-period interval is on
the left and the next-period-ahead forecast is on the right. The
forecast interval is centered at the process average over the window
of recent quality history and its width is based upon the SUM of the
variances within the current period and across the history window.
┌──────────────────────────────────────────────────────────────┐
│ All types of Confidence Intervals on QMP charts are │
│ designed to cover the unknown true Quality Index (present- │
│ period or next-period-ahead) with probability 0.9... │
│ │
│ ┌── <─── 5% Forecast Percentile │
│ │ │
│ │ │
│ ──┐│ <─── 5% Present Percentile │
│ ││ │
│ \││ │
│ \│ <───Next Period Ahead Forecast coincides with │
│ Best │\ Average Quality across the History Window │
│ Present-> ─┤│\ │
│ Measure ││ Observed │
│ X│ <─── Present Period Index = ──────── Ratio │
│ ││ Expected │
│ │└── <───95% Forecast Percentile │
│ ──┘ <───95% Present Percentile │
└──────────────────────────────────────────────────────────────┘
In process average mode, the usual QMP I-plot is augmented with a BOX
that defines a Confidence Interval for average quality across the
current history window. Like next-period-ahead forecasts, the Process
Average BOX is "centered" at the process average across that window.
But the height of the BOX depends only upon the current best measure
of Process Variation (PV) across the current history window.
- 12 -
┌──────────────────────────────────────────────────────────────┐
│ All types of Confidence Intervals on QMP charts are │
│ designed to cover the unknown true Quality Index (present- │
│ period or process-average) with probability 0.9... │
│ │
│ ──┬── <─── 5% Present Percentile │
│ \ ┌┼┐ <─── 5% Process Average Percentile │
│ \│││ │
│ \││ │
│ │\│ <─── Process Average BOX is "centered" at │
│ Best ││\ Average Quality across the History Window │
│ Present-> ┼┤│\ │
│ Measure │││ │
│ │X│ <─── Present Period Quality Index │
│ └┼┘ <─── 95% Process Average Percentile │
│ │ │
│ ──┴── <───95% Present Percentile │
└──────────────────────────────────────────────────────────────┘
┌──────────────────────────────────────────────────────┐
│ SUMMARY of Advantages of Attributes QMP Methodology │
└──────────────────────────────────────────────────────┘
* QMP Trend Charts can display quality monitoring results derived
from either attributes data (observed defects) or variables data
(economic loss due to deviations from an intended target value.)
* QMP Trend Charts utilize the quality "track record" of a process
to form statistically optimal estimates and confidence intervals for
present quality...moving the index computed from data collected only
within the present reporting period toward the process average level
across a window of recent "history."
* QMP Trend Charts utilize an "index scale" defined by the ratio of
the observed defect or loss level to the level that would be
expected to occur, on long-run-average, when quality is at the level
that defines the limit of the present capability of the process.
Thus, unlike conventional control charts, trend charts have not only
a clearly defined "Zero Defect" limit (ultimate quality objective)
at INDEX = 0 but also a well defined process capability "standard"
for quality operations at INDEX = 1.
* QMP Trend Charts always display results with the quality "index"
(observed to expected ratio) on the vertical axis restricted to the
range from 0 to 5. This convention makes trend charts easier to
draw and to understand. After all, if any results are
"Off-the-Chart" (INDEX > 5), they represent quality levels which are
intolerable and which demand immediate management action. (Quality
levels this bad do not need to be accurately reported; they do need
to be immediately eliminated!)
- 13 -
┌────────────────────────────────────────────────┐
│ Normal-Theory Quality Measurement Plan │
└────────────────────────────────────────────────┘
This option provides QMP I-plots that are based upon Normal
distribution theory ( the familiar "bell shaped," symmetric curve.) As
a result, these plots are rather more sensitive to unusual
observations (outliers) than are the Poisson/Gamma theory QMP methods.
And the Poisson/Gamma QMP are also much more appropriate than Normal
theory when the cost-of-poor-quality distribution is highly skewed.
Normal (or Poisson-Gamma Poisson-Gamma (skewed,
with large expected value.) small expected value.)
┌──┐
┌───┐ │ ├───┐
┌─┤ ├─┐ │ │ ├────┐
┌──┤ │ │ ├──┐ │ │ │ ├─────┐
┌───┤ │ │ │ │ ├───┐ │ │ │ │ ├───────┐
┬─────┴───┴──┴─┴───┴─┴──┴───┴─ ┼──┴───┴────┴─────┴───────┴────
0 Cost-of-Poor-Quality --> 0 Cost-of-Poor-Quality -->
Rather than use equivalent defects (ED) and equivalent expectancy (EE)
as its two inputs, the Normal-theory QMP algorithm is usually
described as requiring input of the quality index and its variance.
There is, however, a simple relationship between these four
quantities:
ED Q V(Q) 1
Quality Index = ---- = ------ & Index Variance = ------ = ----
EE E(Q) E(Q)^2 EE
QMPAL uses these last two equations to convert the "standard" inputs,
ED and EE, into a quality index and its variance for analysis by the
Normal-theory QMP algorithm.
┌────────────────────────────────────────────────────┐
│ Exponentially Weighted Moving Index Plots │
└────────────────────────────────────────────────────┘
This option provides Exponentially Weighted Moving "Indices." As in
QMP methodology, the index is an observed to expected "equivalent
defects" ratio, but both numerator and denominator are exponentially
weighted averages in this case.
Use of Exponentially Weighted Moving Averages (EWMAs) in quality
monitoring was pioneered (at AT&T/Bell Laboratories/Western Electric)
by Spence Roberts (1959), Technometrics 1, 239-250, who called them
"Geometric" weighting schemes. Numerical EWMA computations are
extremely simple when successive observations have equal variance, the
"homoscedastic" case. Recent advocates of EWMA schemes include Stuart
- 14 -
Hunter (1986), Journal of Quality Technology 18, 203-210, who stresses
the interpretation of the EWMA as a (next-period-ahead) forecast.
Generalization of the EWMA concept for application to series of
observations with unequal variances, the "heteroscedastic" case, is
neither "obvious" nor straight-forward. Here we use one of many
possible such formulations. Rather than average a series of quality
indices (ratios) as if they had constant precision, our exponentially
weighted index is the ratio of the two EWMAs for the numerator
(defect) and denominator (expectancy) series, respectively. In other
words, we treat the denominator series like known constants containing
no uncertainty. This approach is consistent with the "tradition" of
QMP, usually motivated by the "additivity" of Poisson distributions,
in which composite indices are always formed by summing (over time
and/or across products) the numerators and denominators before forming
the defect/expectancy ratio.
This option calls analysis routines for exponentially weighted
averages, and displays approximate central 90% (+/- 1.65 sigma)
forecast intervals. See Appendix 3 for calculation details.
┌──────────────────────────────────────────┐
│ Shewhart Z-Score Control Charts │
│ (like Quesenberry Q-Charts) │
└──────────────────────────────────────────┘
This option displays Shewhart "standardized" (mean=0, variance=1)
normal-distribution-theory control charts. Walter Shewhart developed
this methodology in the 1920's (at Bell Telephone Laboratories for
AT&T/Western Electric) for "Quality Control Charting" that adjusts for
potential differences in sample sizes between successive reporting
periods. The statistics that he called "t-Rates" are, today, commonly
referred to as (standardized) normal "Z-Scores"...the notation "t"
being reserved for "Studentized" statistics, Gosset(1908). The
empirical distributions of Shewhart "t-Rates" are frequently well
approximated by Studentized-t distributions with very few (2 or 3)
degrees-of-freedom, but Shewhart "t-Rates" are still commonly plotted
as if they were Normal(0,1), which is a Student-t distribution with
"infinite" degrees-of-freedom for error.
REFERENCES
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.)
- 15 -
OBENCHAIN, Robert. (1991). "Regret Indices and Quantification of
Process Capability." Submitted to Journal of Quality
Technology.
QUALITY MEASUREMENT PLAN (QMP). Bell Communications Research Technical
Reference, TR-TSY-000438. Issue 1, April 1987.
QUESENBERRY, Charles. (1991). "SPC Q Charts for Start-Up
Processes and Short or Long Runs." Journal of Quality
Technology, 23, 213-224.
APPENDIX ONE
┌───────────────────────────────────────────────────────────────────────┐
│ QMP Trend Chart BENCHMARK Examples │
├───────────────────────────────────────────────────────────────────────┤
│ This Appendix presents eight canonical examples that cover a variety │
│ of different types of quality process behavior. In addition to the │
│ graphical displays and verbal commentaries given here, the BENCHMARKS │
│ include ASCII files of input data and output results that are used to │
│ validate alternative QMP software systems. We recommend that readers │
│ first examine each Trend Chart and try to decide what information it │
│ conveys. The reader can then confirm/guide his or her interpretation │
│ by examining the verbal commentary associated with that Trend Chart. │
└───────────────────────────────────────────────────────────────────────┘
Example One...Stable Compound
This is an example of a well controlled process where quality is
beinging maintained right at the Standard quality level. There is
very little process variation.
Example Two...High Quality
This stable, well controlled process is running much better than
Standard; i.e. well within the threshold of present process
capability. From May through November, there were no defects
observed, as shown by the X's right at the Zero-Defect Limit. There
is very little variation (either process or sampling) in the process.
Example Three...Process Jitter
This is an example of an unstable process...a process with
considerable sampling variability as well as process variability.
Each period the interval for CURRENT QUALITY is moves abruptly. This
interval bounces from better than Standard to Standard to worse than
Standard, etc. Process variability is more likely to cause this sort
of fluctuation than is sampling variation. The trend over the last
five months hovers around 1.5 times Standard...even though October
does show a BEST MEASURE better than Standard. This process has
severe quality problems.
- 16 -
Example Four...Out of Control
This process is literally Off-the-Chart. The numbers at the bottom
indicate that the entire 90% confidence interval estimate for CURRENT
QUALITY is worse than 5 times Standard in June, August, and September.
This represents a severe quality problem that demands immediate
management action.
Example Five...Dramatic Decay
During February, March and April, this process was running above
Standard. But quality started deteriorating toward Standard in May
and June. Then quality plummeted in July, and the CURRENT INDEX
remained worse than two times Standard for the next four months. From
May onward, there is increased process and sampling variation. Even
though there is insufficient statistical evidence to say with 95%
confidence that quality is worse than Standard during these months,
there is a degradation, or downward trend, in quality. As of
September, on the other hand, one can say with 95% confidence that
quality is running significantly worse than Standard.
Example Six...Questionable Quality
This process quickly drifts to between 1.5 and 2 times Standard,
showing considerable sampling variation. Note that the interval
estimate for CURRENT QUALITY is entirely below Standard in April and
October indicating a quality problem. During May through September,
the CURRENT QUALITY is probably below Standard at least part of the
time. This is a fair interpretation even though the strenght of
evidence does not exceed 95% confidence for any single month between
May and September...all interval estimates cover the Standard value of
1.
Example Seven...Skip Lot Sampling
This is a well controlled process. Very few defects have ever been
observed. No samples were collected for this process in September or
October; inspection resources are apparently concentrated on other,
more troublesome products during those months. However, an occasional
check is needed to verify that the process has not changed
significantly. Samples were again collected in November.
Example Eight...Subtle Window Effects
This example shows considerable sampling variation; the I-Plots tend
to be wide because defect expectancy is low most months. Between
February and November, there is insufficient evidence to confirm that
quality for any particular month is worse than Standard. But the LONG
RUN AVERAGE shows quality between 1 and 2 times Standard most of the
year. The downward trend in the LONG RUN AVERAGE in September-October
combined with the below-standard signal for November should cause the
- 17 -
reader to be seriously concerned about the quality of product to be
produced in December.
ADVANCED COMPREHENSION QUESTIONS:
(1) Why did the LONG RUN AVERAGE move up in August when the CURRENT
SAMPLE INDEX was down from July?
(2) Why did the LONG RUN AVERAGE move downward in October when the
CURRENT SAMPLE INDEX was up from September?
These sorts of effects are explained by realizing that the LONG RUN
AVERAGE is calculated over the WINDOW of the 6 most recent months.
ANSWERS:
(1) Note that when August (with twice the expected number of defects)
entered the window, February (with more than FIVE times the
expected number of defects) exited out the back of the window.
Thus the LONG RUN AVERAGE moved upward in August primarily because
the very poor perfromance from Feburary became outdated as of
August.
(2) When October (with slightly more than the expected number of
defects) entered the window, April (with NO observed defects)
exited out the back of the window. Note also that the I-Plot for
October is more narrow than those for most previous
months...meaning that a larger sample was taken in October (than
in April, May, or July when NO defects were observed). Thus more
weight is assigned to October results in the current LONG RUN
AVERAGE than was given to April data in the previous LONG RUN
AVERAGE. Both of these factors combine to move the LONG RUN
AVERAGE downward in October.
┌───────────────────────────────────────────────────────────────────┐
│ Advanced Topics in the Interpretation of Trend Charts │
├───────────────────────────────────────────────────────────────────┤
│ A number of factors affect the interpretation of Trend Chart │
│ results. In this Appendix, we outline both major factors and │
│ also subtle elements you can look for in interpreting a chart. │
└───────────────────────────────────────────────────────────────────┘
Movement Over Time
──────────────────
Trend Charts are "time series" graphs which frequently show movement
over consecutive rating periods (months or weeks.) The three types of
symbols on Trend Charts tend to move differently.
The Os, which represent LONG RUN AVERAGEs, exhibit relatively small
fluctuations from period to period. After all, these PROCESS
averages are MOVING averages over a WINDOW of recent periods. The
Os are connected to give an indication of any sort of trend.
The Xs, which are CURRENT SAMPLE INDICES, tend to fluctuate much
more than the Os. Each X depicts results from only the present
period.
- 18 -
The BEST MEASURES of Current Quality (Cross Bars on the I-Plots)
tend to move up and down more than the Os but less than the Xs.
(The last section of this Appendix discusses how the HEIGHT of
I-Plots can also change from period-to-period.)
Trend Charts are displayed as a "time series" so that the "psychology
of graphical perception" shared by MOST (if not by all) readers will
help them visualize WHY I-Plots move up and down or change height. On
the other hand, it is NOT necessary to look at EARLIER I-Plots in
order to understand the present I-Plot. In fact, it is actually
INAPPROPRIATE to look backward toward the past (let alone forward into
the future) when reading the current I-Plot. The reader should simply
TRUST the QMP algorithm to appropriately position both the Best
Measure and the 90% Confidence Interval.
┌──────────────────────────────────────────────────────────────────┐
│ QMP methodology may well seem to be technically complex to │
│ novices, but users soon find that the resulting Trend Charts │
│ are EASY to read because they are DIFFICULT to misinterpret. │
│ Either the Current I-Plot is "above" or "at" Standard (above or │
│ intersects Std) or else it is "below" Standard (indicating at │
│ least 95% Confidence that the unknown true level is below Std.) │
└──────────────────────────────────────────────────────────────────┘
Dynamics of Sudden Change
─────────────────────────
Since QMP results are calculated using a multi-period WINDOW of recent
history, it is important to appreciate the degree of responsiveness of
I-Plots to sudden changes in quality. A prior history of standard or
substandard quality should not prevent the recognition of a sudden
change in quality.
If a chronic quality problem is finally solved, then the I-Plots
should quickly signal this improvement. Conversely, if there is a
sudden degradation in quality, a good quality trend monitoring system
should quickly detect that too. However, a good system should also be
able to distinguish between a REAL CHANGE in the quality level and
temporary "blips" due to SAMPLING VARIATION.
Sudden Degradation of Quality
─────────────────────────────
The illustration below portrays both a sudden degradation and a sudden
improvement in quality. In period 2, the CURRENT SAMPLE INDEX drops
below Standard, providing some evidence for degradation in Quality
from period 1. But QMP detects that this change in the sample result
is explainable by sampling variation, and the I-Plot for period 2
remains at Standard. In other words, the downward shift is not
significant enough to indicate a change in process quality.
- 19 -
In rating period 3, we see a second downward shift in the CURRENT
SAMPLE INDEX. This time QMP detects a downward shift too large to be
due to sampling variation. The entire I-Plot has fallen below
Standard. Thus we say we have 95% confidence that the process is
BELOW Standard as of period 3.
┌───────────────────────────────────────────────────────────┐
│ ─┬─ │
│ X │
│ ─┼─ ─┬─ │
│ O - _ ─┬─ X │
│ ─┴─ - O │ S │
│ 1 ----------│-_----------------------------─┼─------- t │
│ ─┼─ - ─┬─ │ d │
│ │ - │ ─┬─ ─┬─ _ - O │
│ X O - _ │ _ - O - ─┴─ │
│ ─┴─ │ - O - ─┼─ │
│ ─┼─ ─┼─ X │
│ │ X ─┴─ │
│ │ ─┴─ │
│ X │
│ ─┴─ │
└───────────────────────────────────────────────────────────┘
Sudden Improvement in Quality
─────────────────────────────
Continuing with the above illustration, there is a significant
improvement in quality from period 5 to period 6. Quality had
remained below Standard for three periods (3, 4 & 5.) But the CURRENT
SAMPLE INDEX (X) showed a dramatic rise in period 6, moving to better
than Standard. There is a similar improvement in the BEST MEASURE OF
CURRENT QUALITY (the dash). In fact, QMP reveals that there no longer
is much evidence that true quality might be worse than Standard. If
the CURRENT SAMPLE INDEX stays at or better than Standard for the next
few periods, both the LONG RUN AVERAGE and the BEST MEASURE will
gradually move up to Standard or better.
The Length of Each I-Plot
─────────────────────────
The length of each I-Plot is a measure of the precision of the BEST
MEASURE. A very short I-Plot indicates that there is little
uncertainty about the quality implied by that rating; the range from
the 5th to the 95th percentile indices is very small. On the other
hand, a tall I-Plot indicates a wide range of likely values for the
true quality from the 5th to the 95th percentile indices. Thus, there
is more uncertainty.
Precision depends on several factors, one of which is the size of the
sample taken. In any sample, complete accuracy can never be totally
assured, unless the entire population is tested/inspected. Since one
- 20 -
usually can't examine the entire population, there is "uncertainty" in
our result. However, the more items we sample, the better our
evaluation.
The sample size determines the EQUIVALENT EXPECTANCY; a certain
expectancy is derived from the conversion of each measurement into a
Cost-of-Poor-Quality The smaller the EXPECTANCY, the taller the
I-Plot, and the greater the resulting uncertainty about quality. The
larger the EXPECTANCY, the shorter the I-Plot, and thus the greater
the precision in our estimate of quality.
The length of the I-Plot actually depends on an intricate relationship
between the size of the expectancy (or sample size) and two other
factors: process stability and the BEST MEASURE OF CURRENT QUALITY.
Although the relationship between these factors is quite complex, a
change in the length of the I-Plot is generally attributable to one,
or a combination, of these three conditions:
CHANGE IN │ PROCESS │BEST MEASURE OF │ EXPECTANCY
I-PLOT LENGTH │STABILITY │CURRENT QUALITY │(SAMPLE SIZE)
────────────────┼──────────┼────────────────┼─────────────
Shorter │ Steady │ Improved │ Large
────────────────┼──────────┼────────────────┼─────────────
Longer │ Erratic │ Deteriorated │ Small
A more concise way to view this relationship is as follows. The
length of the I-Plot is approximately proportional to the square
root of
┌─ ─┐ ┌─ ─┐
│ PV │ │ Best Measure │
│ ─────────── │ times │ ────────────── │ .
│ SV + PV │ │ Expectancy │
└─ ─┘ └─ ─┘
In the illustration above, notice the reduction in height of the
I-Plots for periods 4 and 5 relative to period 3. Those I-Plots
undoubtedly got shorter because management requested larger samples
when the quality was looking bad.
Notice also that the I-Plot got longer again in period 6. This
probably didn't happen because a smaller sample was taken, and it
certainly didn't happen because the Current Qquality was judged to be
especially good. (Besides, a small Best Measure would tend to make the
I-Plot shorter, not longer.) The logical explanation here is that the
Process Variation has greatly increased; the Process Quality level is
indeed changing (and in the good, upward direction too!)
- 21 -
┌─────────────────────────────────────────────────────────────────┐
│ APPENDIX 3: C-language source code fragments for Exponentially │
│ Weighted Moving Average Quality Indices │
└─────────────────────────────────────────────────────────────────┘
Using the symbols...
weight => short-term memory weight
defects[] => numerator array
expect[] => denominator array
qindx => pointer to quality index series
ewmi => pointer to exponentially weighted
moving index series
Start-Up Defaults, Minimums, and Maximums...
--------------------------------------------
weight = 0.2 ...0.1 <= weight <= 0.9
ewmexp = 2.0 ...0.1 <= ewmexp <= 10.0
ewmdef = ewmexp ...0.0 <= ewmdef <= 5.0 * ewmexp
ewmi = ewmdef / ewmexp
sigma2 = 0.0 ...for start-up at given CONTSANT ED numerator
or
sigma2 = ewmexp * weight / (2.0 - weight )
...for start-up at asymptotic ED numerator var.
...0.0 <= sigma2
confact = 1.65
lowexp = 0.002
┌─────────────────────────────────────────────────────────────┐
│ REMINDER: equivalent expectancy = EE │
│ │
│ is both the mean and the variance of │
│ │
│ equivalent defects = ED │
└─────────────────────────────────────────────────────────────┘
The following C-language function documents exactly how exponential
weighting is implemented:
/*
* Calculate EWMA Results...
*/
int ewmidx( weight, defects, expect, qindex,
ewmdef, ewmexp, ewmi, sigma2, quantiles )
double weight, defects, expect, *qindex,
*ewmdef, *ewmexp, *ewmi, *sigma2, quantiles[]; {
double onemwgt, forcserr;
extern double confact, lowexp;
- 22 -
if( defects < 0.0 )
defects = 0.0;
if( expect < 0.0 )
expect = 0.0;
if( weight < 0.1 )
weight = 0.1;
else if( weight > 0.9 )
weight = 0.9;
onemwgt = 1.0 - weight;
*qindex = -1.0;
if( expect > lowexp )
*qindex = defects / expect;
*ewmexp = weight * expect + onemwgt * (*ewmexp);
*ewmdef = weight * defects + onemwgt * (*ewmdef);
*sigma2 = weight * weight * expect + onemwgt * onemwgt * (*sigma2);
*ewmi = -1.0;
quantiles[1] = quantiles[2] = -1.0;
if( *ewmexp > lowexp ) {
*ewmi = *ewmdef / *ewmexp;
/* forecast variance = EWMA var. + var. of single obs. */
forcserr = sqrt( ( (*sigma2) / (*ewmexp) + 1.0 ) / (*ewmexp) );
quantiles[1] = *ewmi - confact * forcserr;
if( quantiles[1] < 0.0 )
quantiles[1] = 0.0;
quantiles[2] = *ewmi + confact * forcserr;
}
return( 0 );
}
