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 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � Table of Contents � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 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 � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� - 2 - 旼컴컴컴컴컴컴컴컴컴컴컴컴컴� � Input Data Restrictions � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴� 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � Naming Conventions and Contents of QMP Output Files � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � Resetting QMP Internal Parameters � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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: 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 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] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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 - 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � Background Information on QMP � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � Equivalent Defects & Defect Expectancy � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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 - 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � QMP Trend Chart Graphical Conventions � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � Trend Chart AXES... � � � � Q 0旼컴컴컴컴컴컴컴컴컴컴컴� <컴훋ero Defect Limit � � u I � � � � a n 1쳐컴컴컴컴컴컴컴컴컴컴컴� <컴횾rocess Capability � � l d � � Threshold � � i e � � (Quality Standard) � � t x � � � � y 5읕컴컴컴컴컴컴컴컴컴컴컴� <컴횭ad Quality Limit � � Time 컴�> (Off-the-Chart Threshold) � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 The Time Scale 컴컴컴컴컴컴컴 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 컴컴컴컴컴컴컴� 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 旼컴컴� ...High Quality 1 쳐 2 � 3 � 4 � 5 읕컴컴� 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 컴컴컴컴컴컴컴컴컴컴컴컴컴� 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: 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � � � 컫� <컴� 5% Quality Percentile � � � � � _ � � � -O_ <컴� Long Run (Process) Average Quality Index � � � - � � 컵� <컴� Best Measure of Current Quality � � � � � � Observed � � X <컴� Current Sample Index = 컴컴컴컴 Ratio � � � Expected � � � � � 컨� <컴�95% Quality Percentile � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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 = 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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: 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � Process Variation, PV - a measure of the period-to-period � � variation in the product's true quality. � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � Sampling Variation, SV - a measure of the deviations � � between the quality in the sample and the product's true � � quality in a rating period. � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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 = 컴컴컴컴� . 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... 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � 컫� 컫� 컫� � � X � � 컫� � � 컫� � X X � � � � 컵� 컵� 컵� - - - - - -O- � � � � - - - - - - O - - - - - - O 컵� � � -O- - - - - O � � X � � � � � � 컨� � � 컵� � � 컨� � � � � 컨� � � X 컨� � � � � � 컨� � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � QMP Trend Charts consist of a time series of "I-plot" � � confidence intervals for the current quality level... � � � � Q 0旼컴컴컴컴컴컴컴컴컴컴커 � � u I � � � � � � � � a n 1� � � � � � � � � � l d � � � � � � � � � � i e � � � � � � � � � t x � � � � � � � y 5읕컴컴컴컴컴컴컴컴컴컴켸 � � Time 컴�> � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 I-Plot = 90 Percent Confidence Interval for the Unknown True the Current Period X and the Process Average 0 - 10 - Percentile Indices 컴컴컴컴컴컴컴컴컴 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � 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旼컴컴컴컴컴컴컴컴컴컴 � � u I � � � � <컴� 5% Percentile � � a n 1� � � � � � and � � l d � � � � � � � <컴�95% Percentile � � i e � � � � � � for last � � t x � � � � Reporting � � y 5읕컴컴컴컴컴컴컴컴컴컴 Period... � � Time 컴�> � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 Location of the I-Plot 컴컴컴컴컴컴컴컴컴컴컴 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 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � 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... � � � � 旼� <컴� 5% Forecast Percentile � � � � � � � � 컴엿 <컴� 5% Present Percentile � � 납 � � \납 � � \� <컴횼ext Period Ahead Forecast coincides with � � Best �\ Average Quality across the History Window � � Present-> 캑�\ � � Measure 납 Observed � � X� <컴� Present Period Index = 컴컴컴컴 Ratio � � 납 Expected � � 냅컴 <컴�95% Forecast Percentile � � 컴� <컴�95% Present Percentile � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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 - 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � 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... � � � � 컴쩡� <컴� 5% Present Percentile � � \ 民� <컴� 5% Process Average Percentile � � \납� � � \납 � � �\� <컴� Process Average BOX is "centered" at � � Best 납\ Average Quality across the History Window � � Present-> 킴�\ � � Measure 납� � � 쿦� <컴� Present Period Quality Index � � 읖� <컴� 95% Process Average Percentile � � � � � 컴좔� <컴�95% Present Percentile � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � SUMMARY of Advantages of Attributes QMP Methodology � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 * 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 - 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � Normal-Theory Quality Measurement Plan � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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.) 旼커 旼컴� � 쳐컴� 旼� 쳐� � � 쳐컴커 旼캑 � � 쳐커 � � � 쳐컴컴� 旼컴� � � � � 쳐컴� � � � � 쳐컴컴컴� 쩡컴컴좔컴좔컨컨컴컨컨컴좔컴좔 탠컨컴컨컴컴좔컴컴좔컴컴컴좔컴� 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � Exponentially Weighted Moving Index Plots � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � Shewhart Z-Score Control Charts � � (like Quesenberry Q-Charts) � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � QMP Trend Chart BENCHMARK Examples � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 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. � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � Advanced Topics in the Interpretation of Trend Charts � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 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. � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� Movement Over Time 컴컴컴컴컴컴컴컴컴 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � 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.) � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 Dynamics of Sudden Change 컴컴컴컴컴컴컴컴컴컴컴컴� 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 컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 컫� � � X � � 컵� 컫� � � O - _ 컫� X � � 컨� - O � S � � 1 ----------�-_----------------------------컵�------- t � � 컵� - 컫� � d � � � - � 컫� 컫� _ - O � � X O - _ � _ - O - 컨� � � 컨� � - O - 컵� � � 컵� 컵� X � � � X 컨� � � � 컨� � � X � � 컨� � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� Sudden Improvement in Quality 컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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 컴컴컴컴컴컴컴컴컴컴컴컴� 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 쿍EST MEASURE OF � EXPECTANCY I-PLOT LENGTH 쿞TABILITY 쿎URRENT QUALITY �(SAMPLE SIZE) 컴컴컴컴컴컴컴컴탠컴컴컴컴컵컴컴컴컴컴컴컴컴탠컴컴컴컴컴컴 Shorter � Steady � Improved � Large 컴컴컴컴컴컴컴컴탠컴컴컴컴컵컴컴컴컴컴컴컴컴탠컴컴컴컴컴컴 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 旼 커 旼 커 � PV � � Best Measure � � 컴컴컴컴컴� � times � 컴컴컴컴컴컴컴 � . � SV + PV � � Expectancy � 읕 켸 읕 켸 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 - 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � APPENDIX 3: C-language source code fragments for Exponentially � � Weighted Moving Average Quality Indices � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � REMINDER: equivalent expectancy = EE � � � � is both the mean and the variance of � � � � equivalent defects = ED � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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 ); }