Download Graphical Evaluation of Product Characteristics

GRAPmCAL EVALUATION OF PRODUCT CHARACTERISTICS
Melissa A. Durfee, Wyman-Gordon Company
Abstract
Depending on. the type of data and the subgroup
size, the appropriate chart type is determined.
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.
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.
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.
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.
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 variables-
quality characteristics measured on
a continuous scale (e.g. machined
dimensions, mechanical properties,
weights, temperatures);
J
o attributes -
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.
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).
741
Multi-Vari Plots
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 0.2 and a scale factor of 10.
Therefore, 20% of the effects will be 10 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.
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
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.
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.
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.
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.
References
Jason J. Brown and Randall D. Tobias. ADX Menu
System Examples. Cary, NC: SAS Institute Inc.,
1991.
SASIGRAPH Software: Reference, Volume 2,
Version 6, First Edition. Cary, NC: SAS Institute
Inc., 1990.
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.
SASIQC Software: Reference, Version 6, First
Edition. Cary, NC: SAS Institute Inc., 1989.
SAS/GRAPH and SAS/QC are registered
trademarks of SAS Institute Inc., Cary, NC, USA.
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
The Author
Melissa A. Durfee
Wyman-Gordon Company
244 Worcester Street Box 8001
North Grafton, MA 01536-8001
(508) 756-5111
742
EXHIBIT 1
IX &; MR CHA.RT ON MACHINED DIMENSIONS AS INSPECTED BY NUMERI-PROBE
WG=9(}628 OPERAll0N=774 SEQUENCE=5AXIS=X NOMINA.L=15.185
z
~ 15.184
~
:::; 15.182
:
··
·
•
.
. .
. .' .
. .
. .. . . . .
..
--------------------.. .. .. .. .. .. .. .. .. .. ..
.. .. .. ..
:
'.'
••
.
'.
'.'
.
.
...
.'
'.'
.
.
. . . . . ._
.'
••
'.'
'.
'.'
..
' • • •' . .
•
.
0"
'.
.'
••
','
"
'.'
UCL=15.18J4
.....
_ • • •'
••
' • • • '
••
' ••• _
CI
15.178
15.176
. ···. . ... .
.. .. - ..
-- -
··•
•
•••
•
&
•
•••
. .. . ... . . ... . ... . . ...
- .. -.. -...
.. -..
. . .. . . ...
.. ..
..
.... .....
----.. .. ..
. ... . ....
.. - ..
----- -
•
•
•
•••
•
••
•
•••
•••
•
••
•
•
-
•
•••
•
••
•
•••
•••
•
LCL=15. 177 4-
••
0.004
·
. . . . . . . . . . . . . . UCL=.QOJ7
(}.003
· . .·
.
..
. .. ..
. .. . . . . .
.
.
(}.002
0.001 -r;....,....,.;..,-,....;,...,...,;~~...,;..,.~~~Pr_j!>~h~r:;;f;~"...~ R=.0011
•
•
~
••
•••
I
•
_.'
•
._
•
'.'
•
0'
•
-.
•
•
0'
•
_.'
"
._
•
'.'
'._
•
"
o
LCL=O
o
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
PIECE
I
(ORDER W.CHINED)
EXHIBIT 2
HISTOGRAM OF MA.CHINED DIMENSIONS AS INSPECTED BY NUMERI-PROBE
WG=9(}628 OPERAll0N= 774 SEQUENCE=5 AXlS=X NOMINA.L= 15.185
60
50
!z
w
40
~ 30
w
0.
20
10
oLb~~~L_~---L--~--L-~--~~~~
15.1782
15.1794
15.1806
15.1818
MA.CHINED DIMENSION
LSL=15.155
USL=15.215
Curve:
CPK=8.09
- - NQrmaI(Mu=15.18 Sigma=0.001)
743
15.1830
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
O.OJO
0.025
0.020
ct::
(.)
m 0
m 0
m
..... ('>1 '<t" III <D 1'('>1 I'l
0
.....
.....
C\I r'II C\I Nt"') t')
OJ Ol GOl G Ol m<ll m m Ol mOl m
.... ..... .......... ..... ..... ..... ..... .... ..... ..... ..... .....
.....
1')
00 0
'<t
t"')
~
0
00 0
0
0
00 0
0
~
N
<0
t').;t
<ll m <ll m
~
0
I'
~
00 0
0
.....
r<"l
'<t"
m
m
0
0
10
<0
..... .... .....
~
00 0
I'- I ' I{} <D
It) LO
II:!
m 0
0 0
C\I N C\I
....0 """
0
0
I'
It)
0
N
0
0
INGOT
LEGEND
'i1 II II BOTTOM
- - - AVE. JOINED
/:J. /:J. /:J.
TO P
EXHIBIT 4
1200 F TENSILE::'I RED OF AREA
B.A.R PAIRED COMPARISON OF PRODVCTVS BILLET ACCEPTMlCE (AVE Of CENTER &: SURFACE)
WG=15397
50
~
ct::
40
II(
I-
U
::J
0
0
a::
30
----
a.
20
-)(
X
20
--- --
Y
y
¥
.....-X--y
--
_L y- -
- - ---_-"'Y
-- -
-- -
---y<{
Z
.-
~
...-
--- -
40
50
ACCEPTANCE ~ RIA
PROD RA = 10.55
+ 0.6903>I\A.CCP RA
744
N = 61
P = .0001
X
---.----- --- --
JO
,...
R-SQ = .68
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
.,.... 0.0055 /
~
Cl.0045
o
0.00.35
Cl
I
T
T
R
.;;
:::c 0.0025
T
R
_.-'
(!)
I
3
T .. ····
R
Cl.0015
1.0
1.5
R
2.0
2.5
3.0
3.5
......
4.0
4.5
5.0
5.5
./
6.0
6.5
SECTION SIZE AT CUT-UP LOCATION (IN)
AVE CUT-UP FRACTURE TOUGHNESS:
AVE CUT-UP F.T. (KSI) = 55.483
50
........ 62
+ 2.036;hSIZE -
- - - 54
. _ . 66
-----58
1572.54*H (MR) N=22 P=.OCl20 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
67
64
61
58
55
52
49
o.
uu
0.0051
0.0037
5 ..32.<1l ':;4-3.
329
0.0023 W-G H (MID-RAD)
96
SIZE (IN)' 2.611.931.25 O.DOCl9
AVE CUT-UP F.T. (KSI) = 55.483
+ 2.0363*SIZE 745
1572.54*H (MR) N=22 P=.OCl20 R-SQ=.48
EXHIBIT 7
Boye~
plot of estimotes for prior = 0.2, 6<:01e = 10
Cube plot of TIME means
1.0
/
/1
/
08
•
on
06
/
/
59.5
1
DYNAMO
0.4
1
51
I
/
·
1
1
1
I
I /
off
1
1
1
-------1
down
/
I /
1/
O2
1
/
--------
I.
•
GEAR*DYNAMO*SEAT
by
-------- 85.5
-------low
X-axis
1/
70
SEAT
Z-axi~
up
medium
GEAR
O~~~~~-T~~~==~~~==~==~~
Effect
Bars:
ES3
EXHIBIT 8
Highest 2
Meon:s of ADXSNR for TENSMACH main effect
(Vertical bars represent 2 std. errors above (I: below the mean)
S 55N
R 530
f
51
..
1
2 49 0
0 47F
U 45L
T
I
1
1
41200FTENSILE MACHINE
746