GRAPmCAL EVALUATION OF PRODUCT CHARACTERISTICS

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1 GRAPmCAL EVALUATION OF PRODUCT CHARACTERISTICS Melissa A. Durfee, Wyman-Gordon Company Abstract Utilizing SAS/GRAPH, many aspects of product characteristics may be displayed and analyzed. Besid.es using regression plots to examine relationships between variables, contour and 3-D response surface plots may be constructed on two independent variables. In designed experiments, potentially significant factors may be indicated by Bayes posterior plot of effect estimates coupled with a cube plot to examine the response. In a Taguchi experiment, the optimal design may be identified through a main effects plot using the appropriate signal-to-noise ratio. In addition, the nature of product variation may be assessed using multi-vari (multivariate),&iots. By interfacing SAS/GRAPH with SAS/QC procedures, process stability and capability may be evaluated. These graphical applications facilitate and simplify the use of statistics in industry. Introduction Wyman-Gordon Company is a major producer of structural and turbine aerospace forgings. The company utilizes basic techniques of Statistical Process Control to evaluate process stability and capability and advanced methods such as regression analysis and experimental design to determine process parameters with significant effects. Process Stability To assess whether a process IS In a state of statistical control, a Shewhart control chart (Exhibit 1) is used. Two types of data are analyzed: o variableso attributes - quality characteristics measured on a continuous scale (e.g. J machined dimensions, mechanical properties, weights, temperatures); quality characteristics measured by counting the number of nonconformities (defects) in an item, the number of nonconforming (defective) items in a sample, or the number of occurrences of an undesirable aspect in a time period (e.g., scratches, cracks, dents, absenteeism, delays). Depending on. the type of data and the subgroup size, the appropriate chart type is determined. The tests for special causes available in PROC SHEWHART facilitate assessing process stability. Besides indicating control limit violations (Test 1), runs (Test 2), trends (Test 3), and cycles (Test 4) may be identified, and zone analyses (Tests 5-8) may be.performed. When a test is positive, an assignable (special) cause of process variation is present. Additional analysis through techniques such as regression, experimental design, and multi-vari is required. to determine the nature of the problem and eliminate it. Process Capability For an in-control process (no assignable causes of process variation present), a process capability analysis is performed to assess the ability to consistently meet design specifications. PROC CAP ABILITY provides the graphical options of histogram and cumulative distribution function, quantile-quantile, probability-probability and probability plots. The histogram (Exhibit 2) may be superimposed with specification limits and fitted probability density curves from the normal, lognormal, beta, exponential, gamma, and Weibull distribution. The HISTOG RAM statement can create an output data set, the OUTFIT= data set, which indicates parameters of the fitted density curves and results of the chi-square goodness-()f-fit tests. These results are used to evaluate the adequacy of the selected distribution. Depending on the results of the process capability evaluation, additional analyses may be warranted. To examine excessive process variation (spread), multi-vari plots may be constructed. If the process mean is shifted from the design nominal (midpoint of the specification limits), regression, contour, and response surface analyses may be performed using historical data to determine the significant variables affecting the response variable. If no historical data is available or further optimization is needed, then graphical analysis through experimental design should be completed. 741

2 Multi-Vari Plots A multi-vari plot (Exhibit 3), which aids in assessing the nature of product variation, may be constructed utilizing PROC GPLOT. This plot determines whether the largest source of variation is within one piece, piece-to-piece, run-to-run, etc. Determining the source of the largest product variation improves the efficiency of statistical problem-solving methods which are subsequently used to determine the root cause(s) of product vahation. Therefore, the likelihood that a significant effect or interaction would be identified through a designed experiment is increased. Regression Regression analyses and associated plots are useful for examining the relationship between two variables. The GPLOT procedure is utilized to generate regression plots (Exhibit 4) where the independent (input) variable is plotted on the x-axis and the dependent (response) variable is plotted on the y-axis. Using regression and confidence limits, extrapolation beyond existing data may be accomplished. Contour and 3-D Response Surface When regression analysis through procedures such as PROC REG or PROC RSREG indicate two significant independent variables, the effect on the response variable is effectively displayed through a contour plot (Exhibit 5) or 3-D response surface plot (Exhibit 6). The GCONTOUR procedure produces contour plots that represent three-dimensional relationships in two dimensions. A contour plot should be used when the levels - not the shape - of the response are important. The G3D procedure produces three-dimensional graphs that plot one vertical response variable (z) versus two horizontal independent variables (x and y). The G3GRID procedure may be used to create a data set for plotting for subsequent use by the GCONTOUR and G3D procedures. Bayes Posterior Plot Available in the ADX menu system of SAS/QC, the Bayes posterior plot of effect estimates (Exhibit 7) is useful in analyzing saturated two-level fractional factorial designs. Based on the Pareto principle, which assumes that most effects in the model are insignificant, the Bayes plot displays the individual probability for each effect to assess significance. The computed probabilities are affected by the choice of the prior probability and scale factor. The ADX defaults are a prior probability of.2 and a scale factor of 1. Therefore, 2% of the effects will be 1 times larger than the remaining effects. However, constructing posterior probability plots using a range of prior probabilities and scale factors is recommended. After identifying possibly significant factors, a cube plot (Exhibit 7 insert) of the data may be constructed to examine the nature of the response. Main Effects Plots In a Taguchiexperiment with an orthogonal array, a main effects plot (Exhibit 8) may be constructed on factors to pinpoint the optimal parameter settings. The optimal design is determined by plotting the appropriate signal-to-noise ratio and identifying its maximum. If a particular factor does not indicate much difference in signal-to-noise levels, the optimum is selected based on cost. Conclusion Graphical analysis is not only useful in assessing process stability and capability but also in examining relationships between variables, identifying optimal design settings, and determining sources of excessive product variation. This pictorial approach enhances, yet simplifies, the application of statistics, in industry. References Jason J. Brown and Randall D. Tobias. ADX Menu System Examples. Cary, NC: SAS Institute Inc., SASIGRAPH Software: Reference, Volume 2, Version 6, First Edition. Cary, NC: SAS Institute Inc., 199. SASIQC Software: Reference, Version 6, First Edition. Cary, NC: SAS Institute Inc., SAS/GRAPH and SAS/QC are registered trademarks of SAS Institute Inc., Cary, NC, USA. The Author Melissa A. Durfee Wyman-Gordon Company 244 Worcester Street Box 81 North Grafton, MA (58)

3 EXHIBIT 1 IX &; MR CHA.RT ON MACHINED DIMENSIONS AS INSPECTED BY NUMERI-PROBE z WG=9(}628 OPERAllN=774 SEQUENCE=5AXIS=X NOMINA.L= : ~ '.'. :. '.'..'. '.'..'. '.' '.'. '....'. '... ". ',' '.' UCL=15.18J4 '.....'... _ ' '.. " _ ' ' ' ' _ ~ :::; CI LCL= & (}.3 (} ~ I _.'._ '.' ' -. ' _.' "._ '.' '._ " r;...,...,.;..,-,...;,...,...,;~~...,;..,.~~~pr_j!>~h~r:;;f;~"...~ o UCL=.QOJ7 R=.11 LCL=O o PIECE I (ORDER W.CHINED) EXHIBIT 2 6 5!z 4 w ~ 3 w. 2 HISTOGRAM OF MA.CHINED DIMENSIONS AS INSPECTED BY NUMERI-PROBE WG=9(}628 OPERAllN= 774 SEQUENCE=5 AXlS=X NOMINA.L= olb~~~l_~---l--~--l-~--~~~~ MA.CHINED DIMENSION LSL= USL= CPK=8.9 Curve: -- NQrmaI(Mu=15.18 Sigma=.1) 743

4 EXHIBIT 3 CR Multi-Vari Ch(lrt n 6-4 Reg Top, Average. and BQttQm I:>y Ingot MELT=TRIPLE DIAMErER=3J INCH ELECl'RODE T't'PE=COMPACT 14f>..PR93 ct:: (.) O.OJO.25.2 m m OJ ') m ('>1 '<t" III <D 1'('>1 I'l '<t I' ~ N r<"l '<t" ~ C\I r'ii C\I Nt"') t') t"') t').;t < 1 < Ol GOl G Ol m<ll m m Ol mol m <ll m <ll m m m ~ ~... ~ I'- II:! m... I' I{} <D I' It) LO It) """ C\I N C\I N INGOT LEGEND 'i1 II II BOTTOM AVE. JOINED /:J. /:J. /:J. TO P EXHIBIT 4 12 F TENSILE::'I RED OF AREA B.A.R PAIRED COMPARISON OF PRODVCTVS BILLET ACCEPTMlCE (AVE Of CENTER &: SURFACE) WG= X---,... y ct:: 4 Y - II( y X I- _L y- - U ---y<{ ::J ~ a:: a JO ~ _-"'Y Z -)( X --- ACCEPTANCE ~ RIA -- PROD RA = >I\A.CCP RA N = 61 P =.1 R-SQ =

5 EXHIBIT 5 CONTOUR PLOT OF CUT-UP FRACTURE TOUGHNESS VS. SECTION SIZE.AND W-G H (MID-RAD) FROM MATCHING OR CLOSEST BAR Ti 6-4 REG FAN DISKS., / T Cl Cl.45 T ~ R T I o..35.;; :::c.25 _.-' (!) I 3 Cl.15 R R R T....../ SECTION SIZE AT CUT-UP LOCATION (IN) AVE CUT-UP FRACTURE TOUGHNESS: _. 66 AVE CUT-UP F.T. (KSI) = ;hSIZE *H (MR) N=22 P=.OCl2 R-SQ=.48 EXHIBIT 6 3D RESPONSE SURFACE OF CUT-UP FRJl.CTURE TOUGHNESS VS. SECTION SIZE.AND W-G H (MID-RAD) FROM MATCHING OR CLOSEST BAR. Ti 6-4 REG FAN DISKS KSI o. uu <1l ':; W-G H (MID-RAD) SIZE (IN)' O.DOCl9 AVE CUT-UP F.T. (KSI) = *SIZE *H (MR) N=22 P=.OCl2 R-SQ=

6 EXHIBIT 7 Boye~ plot of estimotes for prior =.2, 6<:1e = 1 Cube plot of TIME means by GEAR*DYNAMO*SEAT / /1 / / 1 8 / / 1 on I. 1 1 DYNAMO down I / I /.4 I / I / SEAT Z-axi~ 1/ 1/ off up O 2 low medium X-axis GEAR O~~~~~-T~~~==~~~==~==~~ Bars: ES3 Highest 2 Effect EXHIBIT 8 Meon:s of ADXSNR for TENSMACH main effect (Vertical bars represent 2 std. errors above (I: below the mean) S 55- N R 53- f F U 45- L T I FTENSILE MACHINE 746