U.S. patent application number 10/842939 was filed with the patent office on 2005-01-13 for method for improving a manufacturing process.
Invention is credited to Bennett, Carl.
Application Number | 20050010308 10/842939 |
Document ID | / |
Family ID | 25104019 |
Filed Date | 2005-01-13 |
United States Patent
Application |
20050010308 |
Kind Code |
A1 |
Bennett, Carl |
January 13, 2005 |
Method for improving a manufacturing process
Abstract
A factorial experiment is conducted on a manufacturing process
to generate a response matrix. The responses are used to calculate
individual contrasts in a document as well as replicates effects.
The contrast sums are also calculated and displayed in the
document. The largest of the contrast sums are identified, and
effects associated with those contrast sums are tested for
significance using an end count method. The information from the
process transformed into "significant effects" information is used
to adjust process variables to improve the manufacturing process by
avoiding the effect or imparting it to a measurable response of the
process.
Inventors: |
Bennett, Carl; (Bellevue,
WA) |
Correspondence
Address: |
WHITE & CASE LLP
PATENT DEPARTMENT
1155 AVENUE OF THE AMERICAS
NEW YORK
NY
10036
US
|
Family ID: |
25104019 |
Appl. No.: |
10/842939 |
Filed: |
May 10, 2004 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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10842939 |
May 10, 2004 |
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09775313 |
Jan 31, 2001 |
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6748279 |
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Current U.S.
Class: |
700/28 |
Current CPC
Class: |
G05B 19/41885 20130101;
G05B 13/021 20130101; Y02P 90/02 20151101; G05B 2219/31318
20130101; Y02P 90/26 20151101; G05B 2219/32015 20130101 |
Class at
Publication: |
700/028 |
International
Class: |
G05B 013/02 |
Claims
What is claimed is:
1. A method for improving a manufacturing process wherein there are
a plurality of process variables and a value of a measurable
response of the manufacturing process is indicative of an
improvement to the process, the method comprising: conducting a
full factorial experiment by setting a plurality of the process
variables at a plurality of settings in a plurality of combinations
of settings and receiving at least one measurement of the response
of the process for each combination of level settings; calculating
individual contrasts for each process variable and each interaction
among the process variables using the received responses of the
full factorial experiment and displaying the individual contrasts
and the sums of the contrasts for each variable and each
interaction; verifying that both variables of an interaction
contrast must be set at the levels of the interaction to impart an
effect substantially equal to the effect of the interaction by
evaluating the variance of the contrasts displayed; and setting the
process variables as a function of the effect of the interaction on
the measurable response of the process.
2. The method of claim 1, further comprising conducting repeat
tests for the full factorial experiment and calculating and
displaying individual replicate effects, wherein only contrast sums
greater than the sums of any set of replicate effects are
identified.
3. A method of improving a manufacturing process wherein a target
is determined for a measurable response, the target being
indicative of an improvement in the process, the method comprising:
conducting a full factorial experiment with at least two process
variables being adjusted between at least two level settings with
output responses being measurements of the response for which the
target is determined; receiving the responses of the full factorial
experiment and using the responses to calculate individual
contrasts for each process variable and each interaction among the
process variables and displaying each of the contrasts in a
document; adding the individual contrasts of each process variable
and each interaction to generate separate contrast sums; selecting
at least one of the contrast sums when it is greater than at least
one of the other contrast sums by a predefined factor; and
adjusting the level settings of the process variables as a function
of an estimated effect associated with the selected contrast
sum.
4. The method claim 3, wherein the level settings of the process
are adjusted as a function of the estimated effect only if the
estimated effect is determined to be significant.
5. The method of claim 4, wherein when the estimated effect is an
interaction effect, determining the significance of the effect
comprises removing lower order effects from the responses of the
full factorial experiment then testing the significance of the
effect.
6. The method of claim 3, wherein there are a plurality of contrast
sums greater than at least one of the other contrast sums by a
predefined factor and a plurality of contrast sums are selected,
and each of the effects of the contrast sums are tested for
significance with each significant effect being removed before
testing the significance of another effect.
7. A method of manufacturing wherein there is a required target for
a measurable response of the process, the method comprising:
conducting a full factorial experiment by setting a plurality of
process variables of the process at a plurality of settings, in a
plurality of combinations of settings, and receiving at least one
measurement of the response of the process for each combination of
level settings; receiving the response of the full factorial
experiment and using the responses to calculate individual
contrasts to calculate individual contrasts for each process
variable and each interaction among the process variables and
displaying each of the contrasts in a display at a particular
location of the display corresponding to a notation, the notation
indicating the level settings of the other of the process variables
not involved in the particular contrasts; adding the individual
contrasts of each process variable and each interaction to generate
separate contrast sums; calculating effects estimates for each of
the contrast sums; identifying contrast sums that are greater than
at least one of the other contrast sums; verifying that both
variables of an estimated interaction effect must be set at the
levels of the interaction to impart an effect substantially equal
to the effect of the interaction by evaluating the variance of the
contrasts displayed when the sum of contrasts for that interaction
has been identified; and adjusting the level settings of the
process variables as a function of at least one of the estimated
effects.
8. A method for improving a manufacturing process wherein there are
a plurality of process variables and a value of a measurable
response of the manufacturing process is indicative of an
improvement to the process, the method comprising: conducting a
full factorial experiment by setting a plurality of process
variables of the process at a plurality of settings, in a plurality
of combinations of settings, and receiving at least one measurement
of the measurable response of the process for each combination of
level settings; receiving the responses of the full factorial
experiment and using the responses to calculate individual
contrasts for each process variable and each interaction among the
process variables; testing the significance of effects associated
with the contrasts, wherein before each effect is tested, the
previously tested effect is removed from the responses if found
significant, and when the effect to be removed is an interaction
effect it is removed to achieve the smallest remaining estimates
for the lower order effects; and adjusting the level settings of
the process variables as a function of a least one of the
significant effects.
9. The method of claim 8, wherein when the effect to be removed is
a spike interaction effect, it is removed only from a response in
which it was observed.
10. The method of claim 8, wherein a graphical representation of
the responses associated with an interaction effect is used to
assist in removal of the interaction effect to achieve the smallest
remaining estimates for the lower order effects.
11. The method of claim 9, wherein each of the remaining lower
order effects of an interaction are estimated with the other of the
process variables of the interaction effect set to a level not
associated with the spike interaction removed.
12. A computer implemented method for use in improving a
manufacturing process wherein the transformed information from
implementation of the method on the computer is used to adjust the
process, the method comprising: receiving a plurality of level
settings for a plurality of process variables and a plurality of
responses from a full factorial experiment; calculating individual
contrasts for each process variable and each interaction among the
process variables; testing the significance of effects associated
with the contrasts, wherein when an effect is found to be
significant and is an interaction effect, it is removed before
testing the significance of another affect, the removal being done
to achieve the smallest remaining estimates for the lower order
effects; and displaying the significant effects of the process and
the associated process variables as well as the settings of the
variables not associated with the effect.
13. A computer implemented method for use in improving a
manufacturing process comprising: receiving level settings for a
plurality of process variables and a plurality of responses of the
process from a full factorial experiment; calculating individual
contrasts for process variables; and testing the significance of
effects associated with the contrasts, wherein when an effect is
found to be significant and is an interaction effect, it is removed
before testing the significance of another affect, the removal
being done to achieve the smallest remaining estimates for the
lower order effects.
14. The method of claim 12 further comprising: receiving the target
response; and setting the process variables as a function of the
significant effects and the target response.
15. A computer readable medium for instructing a computer to
perform a method for improving a manufacturing process, comprising:
receiving level settings and responses for a factorial experiment;
calculating individual contrasts for each process variable and each
interaction among the process variables; and testing the
significance of effects associated with the contrasts, wherein when
an effect is found to be significant and is an interaction effect,
it is removed before testing the significance of another affect,
the removal being done to achieve the smallest remaining estimates
for the lower order effects.
16. The computer readable medium of claim 15 wherein the
significant interaction effect removed is removed from only the
response in which it was observed.
Description
[0001] This continuation application claims the benefit of U.S.
patent application Ser. No. 09/775,313, filed Jan. 31, 2001, the
contents of which are hereby incorporated by-reference in their
entirety.
BACKGROUND OF THE INVENTION
[0002] In the improvement of manufacturing processes and products
it is often necessary to employ empirical methods or techniques. In
most basic terms, this typically involves observing the effects of
variables in a product or process and using the information
observed from those effects to adjust or manipulate the variables,
resulting in an improved or satisfactory product or process.
However, where there are many variables with a multitude of
possible effects on the process or product, arriving at
improvements is more difficult
[0003] Industrial methods of design and analysis of experiments
have been developed to assist in transforming data and improving
manufacturing processes. However, in practical applications, field
experience has shown that existing methods do not yield adequate
solutions. There is a need for a simple and easy to use method that
transforms experimental field data into more revealing and
practical information that can be used to improve processes and
products.
SUMMARY OF THE INVENTION
[0004] The present invention provides a method of manufacturing or
improving a manufacturing process. In addition, the method can be
applied in the design of a manufacturing process or product.
[0005] In one embodiment described herein, a full factorial
experiment is conducted with a plurality of process variables with
each of the variables being tested at a plurality of settings, in a
plurality of combinations of settings. Measurements of the response
of the process for each combination of level settings are
recorded.
[0006] The responses of the full factorial experiment are used to
calculate individual contrasts for each process variable and each
interaction among the process variables. The individual contrasts
are each displayed at a particular location in a document, or other
form of display, corresponding to a particular notation. The
notations indicate the level settings of the other of the process
variables not involved in the particular contrasts.
[0007] The individual contrasts of each process variable and each
interaction are added to generate separate contrast sums which are
also displayed in the document. In addition, effects estimates for
each of the contrast sums are displayed.
[0008] Contrast sums are identified that are greater than at least
one of the other contrast sums by a factor of about 2. If the
contrast sum is that of an interaction effect between a plurality
of process variables, the interaction is verified by referring to
the document. The document provides information as to whether both
variables of the interaction must be set at the levels of the
interaction to impart an effect substantially equal to the effect
of the interaction.
[0009] Furthermore, when at least two trials for the full factorial
experiment are conducted, replicate effects can be generated. The
document can be used to generate replicate effects wherein at least
one hypothetical additional process variable is assumed and one set
of the trail responses are substituted as responses for the
hypothetical variable at one of two levels. Individual contrasts
for the hypothetical variable are calculated, including the
interaction contrasts thereof, to generate replicate effects.
[0010] Contrast sums are identified that are both greater than the
next largest contrast sum by a factor of 2, as well as greater than
all replicate effects calculated. Of the identified contrast sums,
the significance of the contrasts, or associated effects, are
tested using an end count method. Higher order effects are tested
first.
[0011] In order to test the higher order effects, the lower order
effects are temporarily removed. If an effect is found to be
significant, it is permanently removed before testing the
significance of remaining effects associated with identified
contrast sums.
[0012] The raw information from the process is thus transformed
into information regarding the "significant effects" of level
settings of the process variables. The level settings of the
process can be adjusted to impart the "significant effects" to the
process, or to avoid them, depending on whether the effects shift
the process in the direction of an improvement.
BRIEF DESCRIPTION OF THE DRAWINGS
[0013] FIG. 1 is a flow sheet showing the steps of an embodiment of
the method.
[0014] FIG. 2 is the response matrix for Example #1.
[0015] FIG. 3 is the worksheet used to calculate and display
individual contrasts as well as contrast sums.
[0016] FIG. 4 is the worksheet of FIG. 3 completed for Example
#1.
[0017] FIG. 5 is a Pareto chart of the contrast sums calculated in
FIG. 4.
[0018] FIG. 6 is a graph of the responses of cells (1), a, c, and
ac of the response matrix of FIG. 2, for Example #1.
[0019] FIG. 7 is the graph of FIG. 6 with the AC interaction effect
removed from the ac response.
[0020] FIG. 8 is the worksheet of FIG. 4 recalculated after the AC
interaction has been removed for Example #1.
[0021] FIG. 9 is a Pareto chart showing the contrast sums of FIG.
8.
[0022] FIG. 10 is a response matrix for Example #2.
[0023] FIG. 11 is a the worksheet of FIG. 3 completed for Example
#2.
[0024] FIG. 12 shows how the variables in the Yates method table of
FIG. 13 are calculated for Example #2.
[0025] FIG. 13 is a table showing the results of the Yates method
for Example #2.
[0026] FIG. 14 is a block diagram of a general purpose computer for
use with the method.
[0027] FIG. 15 is a representation of the "plane" discussed in
Example #3.
[0028] FIG. 16 is a graph of the responses discussed in Example
#3.
[0029] FIG. 17 is FIG. 16 with the AB interaction removed from the
ab cell.
DETAILED DESCRIPTION OF THE INVENTION
[0030] The present invention relates to a method of manufacturing
or improving a manufacturing or fabrication process, or a product
or article. The various embodiments of the method provide a way of
transforming raw information regarding key variables and the
impacts thereof on the product/process, into focused estimates of
"significant effects" that the input variables have on the key
parameters of the process/product. Once the transformation of
information takes place, the new information is used to adjust the
input variables, resulting in an improved or satisfactory process
or product.
[0031] As illustrated in FIG. 1, one embodiment of the method
comprises the following steps: 1) determine the input variables of
the process that may effect the process or product parameters of
interest; 2) design the experiment and determine the passing end
count required; 3) set levels of the variables in the process
according to the design of the experiment and measure the process
or product parameter; 4) calculate the estimated effects as
individual contrasts and display the effects in a worksheet; 5)
determine which group of effects to test for significance as well
as the order in which the effects will be tested; 6) if the effect
to be tested for significance is an interaction, temporarily remove
the estimates of any lower order effects from the responses; 7)
test the effect for significance; 8) permanently remove the
estimated effect if significant; 9) if the effect removed is an
interaction then recalculate the worksheet; 10) determine if the
largest remaining contrast sum should be tested for significance;
11) iterate steps 6 through 10 above; 12) use the information
transformed to adjust the input variables to impart an improvement
in the process/product.
[0032] The first embodiment of the method is best illustrated by
describing it in conjunction with a simplified example application.
This is done in Example #1 below.
EXAMPLE #1
[0033] The following first example description is directed toward
improving a manufacturing or fabrication process, specifically,
improving quality of an article made by the manufacturing process.
Improving product quality may typically entail meeting product
specifications, exceeding product specifications, or increasing the
amount or percent of units of product that meet specifications. The
steps of the method recited above are described in detail below and
applied to the example.
[0034] For Step 1, it is determined that there are 3 manufacturing
process input variables that are likely to have effects on product
quality. The product quality is measured by an output response, or
a product characteristic, with the measurement being a gage of the
product quality improvement sought. It may be desired to target a
range of values for the product characteristic, or a single value.
The product characteristic measured could be, for example, a
measured tensile strength of the product or component of the
product. Again, the product characteristic can be any parameter
identified as important to the product. The input variables, or
process variables, are physical or operating conditions of the
manufacturing process, process steps, or specifications of parts
/materials used in the process such as equipment or raw
materials.
[0035] In this simplified example, each of the process variables
will be tested at only 2 levels, conditions, or settings. For
example, if one of the process variables is a temperature
parameter, it may be tested at two temperatures, or if it is, for
example, a specification on a part used in the process, it may be
tested at both extremes of the current specification limit.
[0036] Step 2 is to design the experiment and determine the passing
end count. The experimental design applied in this illustration is
a traditional full factorial. Full factorial experiments, with P
number of factors, or input variables, each tested at X number of
levels, or settings, will require X.sup.P number of measurements of
the output response to complete one full factorial experiment. In
this example, there are P=3 process input variables to be tested at
X=2 levels, or settings, each. Thus, the output response must be
measured 2.sup.3=8 times per experiment, to complete the full
factorial experiment, which results in every combination of factor
and level settings being tested once. To acquire the relevant data,
on line (operational) changes are made to the process variables of
interest during manufacturing. The intent of making the changes is
to estimate the impact of the variables on the output response, or
product characteristic, and to then make adjustments to the process
variables to improve the response, or product quality based on
information transformed into "significant effects" information by
the method. Data is limited as it is desired to minimize
disturbances to the manufacturing process, so that a minimal number
of changes can be made to the variables for testing purposes. The
data is thus generated according to the pre-designed full factorial
experiment structure discussed above to maximize the information
yielded by the data. The experiment in Example #1 is run twice to
gather 16 output responses as to product characteristic. Thus there
will be a first and second set of output responses, or repeat tests
or trials, for each combination of level settings.
[0037] In accord with traditional notation used with analysis of
full factorial experiments to help simplify tracking and
recordation of experimental results, each of the level settings for
each process variable is represented by - or +. In addition, the
process variables themselves are represented by A, B, or C. For
example, A+ corresponds to the first of three process variables,
set at the + level.
[0038] FIG. 2 is a response matrix and illustrates how the full
factorial design of the experiment in Example #1 can be illustrated
in matrix form using the notion described above. The response
matrix of FIG. 2 is for a 2.sup.3 full factorial experiment for
Example #1. The matrix is configured to reflect the design of the
experiment and provide a convenient way to record the output
responses (product characteristics). The cells are each labeled in
a lower right hand corner ((1), a, b, ab, c, ac, bc, and abc) in
accordance with traditional or standard cell notation for ordering
combinations, used with factorial experiments. Each cell represents
a particular and unique combination of level settings for the
process variables in the experiment. This can be seen directly from
the structure of the table, and is reflected in the notation for
the cell. For instance, the ab cell is positioned in the A+ column,
the B+ row, and the C- half of the response matrix. The ab notation
indicates that the A and B variables are set at the + level.
[0039] The passing end count must also be determined in Step 2. For
Example #1, a confidence level of 95% is chosen and this will later
be tested by the end count. The end count is a way to verify the
statistical significance of the effects calculated from the
experimental data. The mechanics of checking end count are
discussed in more detail in Step 8.
[0040] Step 3 requires changing the process variables in accord
with the design of the experiment. During the experimentation in
Example #1, the process variables, or input variables, are each set
according to the design of experiment reflected in FIG. 2. For the
first cell in the upper left hand corner of the response matrix of
FIG. 2, labeled "(1)", all of the process variables are set to the
- level since (1) does not correspond to any of the letters of the
input variables. The tester measures the resulting product
characteristic, and records the result in cell (1). This process is
repeated for each of the cells. For example, for the last cell,
labeled "abc" in the lower left corner of the table, all three of
the process variables are set to the + level. The settings of the
variables are represented by A+, B+, and C+. When all of the cells
have been filled with the appropriate output response, or product
characteristic measurement, a full factorial experiment has been
conducted. Pairs, or repeat tests, or trials, are conducted for
each combination of level settings of the process variables, and
the corresponding responses are recorded in pairs in the cells of
FIG. 2. The product characteristic measured for each combination of
level settings for the process variables for Example #1 are
displayed in FIG. 2.
[0041] Step 4 is to calculate individual contrasts for each of the
changes between levels in the variables, and effects of the
variables on the product characteristic. This can be done in the
form of the worksheet shown in FIG. 3.
[0042] The three leftmost columns of the worksheet are labeled
"2-Factors, "3-Factors," and "4-Factors." Each of the cells in
those columns are labeled to correspond to cells of a related
response matrix. In the 3-Factor column, the cells are labeled with
standard notation to represent the cells of a 3-factor response
matrix, such as in Example #1. The fourth column from the left in
the worksheet, labeled "Y", is for recording the output response of
the process, in this case, the measurement of product
characteristic. For Example #1, the product characteristic
measurements for each cell of the response matrix of FIG. 1 are
recorded in the "Y" column in the order indicated by the cell
notation under the "3-Factors" column.
[0043] The remaining cells of the worksheet display contrasts. The
contrasts are estimates of the effects of changes in the level
settings of factors, or process variables A, B, and C in Example
#1. The contrasts have an equal number of + and - signs and are
combinations of the responses, or product characteristics. Each of
the columns displays contrasts for a particular factor or
combinations of factors, as indicated at the top of each column by
the factors, or process variables shown. For example, the first
column is labeled the "A" column to indicate that the column only
displays single factor contrasts for variable A. Single factor
contrasts are displayed for each factor in the worksheet, and
estimate an effect of a change in the level of the factor with the
other factors are set at either the - or + level during the change.
Two factor interaction contrasts are also displayed that estimate
the effect of changes of a factor on the effect of changes of
another factor. Three factor interaction contrasts are also
displayed that provide estimates of the effect of changes of a
factor on a two factor interaction.
[0044] To better illustrate the physical meaning of contrasts, note
that the contrast in cell B1, in the upper left corner of FIG. 3,
is represented by the notation b-(1), as indicated in the cell.
This is equivalent to the difference between the output response
(product characteristic) with B set to the +, and the output
response with B set to the - level, while the other factors are set
at the - level. In addition, cell B2 in the worksheet, positioned
just below cell B1, is represented by the notation ab-a, which is
equivalent to the difference between the output response with B set
at the +, and the output response with B set at the - level, with A
at the + level and C at the - level. To illustrate the physical
meaning of an interaction, or contrast involving two factors, note
that cell AB1 of the worksheet of FIG. 2 provides an estimate of
the effect of a change in the level of A, on the estimated effects
of changes in the level of B discussed above. Hence, cell AB1 is
represented by B2-B1 which is equivalent to the difference between
cell B2 and cell B1 of the worksheet. Each of the cells of the
worksheet are calculated in this manner according to the notations
in the worksheet cells. Contrasts are displayed for each single
factor change, as well as for each interaction, including higher
order interactions involving 3 factors.
[0045] The worksheet in FIG. 3 is directed toward an experiment
with 16 total output response data points and only 2 to 4 factorial
experiments. However, the worksheet can be expanded as needed.
[0046] The four rows at the bottom of the worksheet display: 1) the
sum of contrasts for cells in that column (Contrast Sum); 2) the
orthogonal estimate, or contrast sum divided by half the number of
output responses; 3) the number of individual effects, or
contrasts, in the column (# of Estimates); and 4) the "effect
estimate," which is the average estimated effect, or contrast for
the column.
[0047] FIG. 4 shows the worksheet completed for Example #1, using
the measured product characteristics from the response matrix in
FIG. 2. Note that since there are only 3 process variables, the
columns for contrasts involving changes in a D variable do not have
physical meaning except for measuring "noise" or variation not
associated with the effects being estimated. The contrasts
calculated in those columns are called replicate effects. D is
treated as a hypothetical process variable, and the "noise"
contrasts, or replicates effects, involving changes in the level of
D are calculated by substituting the second of the repeat set of
output responses, which begins with 10 under the "Y" column of the
worksheet, for the hypothetical responses that would be generated
by the D variable at the + level. This is illustrated for Example
#1 by the D1 cell of the Worksheet, which is notated in FIG. 3 as
d-(1). That cell is calculated as (1)-(1), wherein pairs of
responses (product characteristic measurement), recorded in cell
(1) of the response matrix for the repeat tests, are subtracted
from one another to reveal a measurement of variation not
attributable to the effects being tests.
[0048] For Example #1, as can be seen in FIGS. 3 and 4, there is a
first set, or column, of intra-cell replicate effects under column
"D," that measures variation between the repeat tests, or the
variation between output responses within the cells of the response
matrix of FIG. 1. In addition, there are second set replicate
effect, columns "AD," "BD," and "CD," that measure variation
between the intra-cell replicate effects, the set being represented
by interactions between the hypothetical D process variable and
each of the A, B, and C process variables. Lastly, there are third
sets of replicate effects that measure variation between the
inter-cell replicate effects, represented by hypothetical
interactions between D and two of the other three variables. For
Example #1, FIG. 4 shows all of the cells in the D columns
calculated. All of those numbers represent variation not
attributable to the effects being tested and are replicate effects.
Again, if a process variable D was being tested, these cells would
be individual contrasts and not replicate effects.
[0049] The variation in contrast sums as well as in contrasts is
inspected in the worksheet. A large variation in contrasts within a
particular column of the worksheet can be an indicator of an
interaction. In FIG. 4 for Example #1, under column A, the range of
estimated effects, or contrasts, for level changes in the process
variable A vary from between -8 to 16. This is indicative of an
interaction between A and another variable. By cross referencing,
or comparing, the contrasts in FIG. 4 to the notations in the
worksheet of FIG. 3, it can be seen that the interactions are
between the A and B process variables. For example, in cells A1 and
A2, in FIG. 4, the estimated effect on the product characteristic
is -8 and -7, whereas for cells A3 and A4, the estimated effects
are 16 and 16. In FIG. 4 it can be seen from the contrast notations
for cells A1 and A2, a-(1) and ab-b, that the C variable is set at
the - level for both contrasts. However, for cells A3 and A4, the
contrast notations indicate that the C process variable is set at
the + level. This is indicative that C should be set at the + level
while the setting of A is forced from the minus to the plus
level.
[0050] Step 5 is to identify or determine which estimated effects
of the process variables should be tested for significance. The sum
of contrasts displayed for Example #1 in FIG. 4 are plotted on a
Pareto Chart in FIG. 5 to assist in deciding which estimated
effects to test, and which to exclude. Adjacent contrast sums that
drop by about a factor of two or more on the Pareto Chart are
noted. Contrast sums on the high side of the drop are identified
and tested for significance. Those on the low side of the drop may
be excluded. Furthermore, for 2 and 3 factor experiments, the
contrast sums of the replicate effects are compared with the
contrast sums determined to be on the high side of the drop. If the
replicate effects are approximately equal to the contrast sums
identified, those identified contrast sums may be excluded from
testing, since the background noise would appear to be as large as
the effects.
[0051] As seen in FIG. 4, for Example #1, the first drop off, or
break, of about a factor of 2, is between the A contrast sum and
the BC contrast sum. Thus the AC, C, and A contrasts will be tested
for significance. In addition, the highest order contrasts are
tested first. AC is the highest order in Example #1 and should thus
be tested for significance first.
[0052] Step 6 requires that before an interaction effect is tested
for significance, the estimates of all lower order effects involved
in the interaction are temporarily removed from the response matrix
to isolate the effect of the interaction. For Example #1, the
effects of process variables A and C must be removed to test for
the significance of the contrasts for the AC interaction. When
removing lower order effects, such as those of process variables A
and C, the orthogonal estimates, calculated and displayed at the
bottom of the worksheet in FIG. 4, are used. The orthogonal effects
are removed by subtracting the orthogonal estimate from product
characteristics (output responses) that correspond to the + level
settings for the process variables A and B. This is illustrated in
Table 1 below. The Y column of Table 1 displays the product
characteristic measurements that correspond to the indicated cell
of the response matrix of FIG. 2. The orthogonal estimates for
process variables A and C are 4.375 and 6.375, as shown in FIG. 4
for Example #1. These are subtracted from the product
characteristic measurements as shown in Table 1, and as explained
above, to arrive at the results in the last column, which shows the
product characteristics with the orthogonal estimates removed.
1TABLE 1 SUBTRACTING ORTHOGONAL ESTIMATES FOR A AND B FROM THE
PRODUCT CHARACTERISTIC Y. Y with Orthogonal Orth. Est. Orth. Est.
Estimates Cell Y A for A C for B Removed (1) 12 - - 12 a 4 + 4.38 -
-0.38 b 12 - - 12 ab 5 + 4.38 - 0.62 c 6 - + 6.38 -0.38 ac 22 +
4.38 + 6.38 11.24 bc 6 - + 6.38 -0.38 abc 22 + 4.38 + 6.38 11.24
(1) 10 - - 10 a 3 + 4.38 - -1.38 b 11 - - 11 ab 4 + 4.38 - -0.38 c
7 - + 6.38 0.62 ac 23 + 4.38 + 6.38 12.24 bc 5 - + 6.38 -1.38 abc
21 + 4.38 + 6.38 10.24
[0053] Step 7 is to test the estimated effect for significance, in
this case, the interaction effect. The method used is an end count.
To do this, the responses, or product characteristic measurements,
are sorted in rank order (ascending order) and all associated cells
in the Table 1 that are in the same row as the sorted response
cell, are also shifted with the associated response cell. This is
shown in Table 2 below. Table 2 has one more column than Table 1.
The additional column is the rightmost column in the Table 2 and
displays the product of the level settings for process variables A
and C. AC is thus only positive when either both process variables
A and C are positive, or both are negative. The significance of
this is that it is indicative of whether the levels of the
variables are set to permit an interaction. The separation between
+ and - signs in the AC column in Table 2 is indicative of the
amount of overlap between the responses with potential AC
interaction and those without potential AC interaction. As such, an
end count is used to quickly gage the significance of the AC
interaction. The end count is done by first counting - signs from
the top of the AC column until a + sign is encountered. Next, +
signs are counted starting from the bottom of the column until a -
sign is encountered. The two counts are added together to get an
end count. Table 2 shows that the end count for AC for Example #1
is 16. Table 3 shows that an end count of 10 is required for a
confidence level of 95%. The AC interaction is thus identified as
significant.
2TABLE 2 TABLE 1 SORTED IN RESPONSE RANK ORDER Y with Orthogonal
Orth. Est. Orth. Est. Estimates Cell Y A for A C for C Removed AC a
3 + 4.38 - -1.38 - bc 5 - + 6.38 -1.38 - a 4 + 4.38 - -0.38 - c 6 -
+ 6.38 -0.38 - bc 6 - + 6.38 -0.38 - ab 4 + 4.38 - -0.38 - ab 5 +
4.38 - 0.62 - c 7 - + 6.38 0.62 - (1) 10 - - 10 + abc 21 + 4.38 +
6.38 10.24 + b 11 - - 11 + ac 22 + 4.38 + 6.38 11.24 + abc 22 +
4.38 + 6.38 11.24 + (1) 12 - - 12 + b 12 - - 12 + ac 23 + 4.38 +
6.38 12.24 + EC = 16
[0054]
3TABLE 3 REQUIRED ENDCOUNT Required Endcount given the confidence
listed below: # Factors 90% 95% 99% 99.9% 2 8 9 11 14 3 9 10 12 16
4 10 11 13 16
[0055] Step 8 is to permanently removed the estimated effect if
significant. The estimated effect of the interaction of AC must be
removed to test for significance of the remaining identified
effects, process variables A and C. The original product
characteristic measurements are used for this, from the response
matrix in FIG. 2, that is, the lower order effects that were
removed earlier must be replaced.
[0056] The estimated effects of -AC can be mathematically removed
by directly subtracting or adding it to any cells in the ac matrix.
However, the estimate should be removed to achieve the smallest
remaining estimates for the lower order effects. For Example #1, a
graph is created to aid in removing the estimate of the AC
interaction to achieve the smallest remaining estimates. This graph
is illustrated in FIG. 6. The graph indicates that removing the AC
interaction effect from cell ac will leave the smallest A and C
effects.
[0057] FIG. 6 shows that there is not a perfect spike interaction
between the A and C process variables in Example #1. A perfect
spike interaction would have an estimated effect close to zero at
one of the levels of A and a large estimate at the other level of
A. In this case, the C effect is -5 at the A- level and 20 at the
A+ level.
[0058] FIG. 7 is the graph of FIG. 6 with the estimated effect for
AC removed from the ac response. By doing so, the unequal
sensitivity of the A factor when C is set at the + level rather
than the - level, has been set to one of two levels. Now when the
effect of A is estimated, it is estimated when C is set at the -
level. Also, when the effect of C is estimated, it is done with A
set to the - level.
[0059] Factors involved in a removed interaction are set to either
the + or - level. Examples of the possible settings are summarized
in Table 4 below.
4TABLE 4 MAIN EFFECT SETTINGS IN THE MATRIX AFTER AN INTERACTION
HAS BEEN REMOVED Remove AC A effect is C effect is average effect
estimated estimated with estimate from: with C set to: A set to:
(1) C+ A+ a C+ A- c C- A+ ac C- A-
[0060]
5TABLE 5 RESPONSE TABLE FOR Y WITH THE ESTIMATE OF AC REMOVED FROM
CELL AC (AND ABC) Y without Estimate of the estimate Cell Original
Y AC of AC -1 12 12 a 4 4 b 12 12 ab 5 5 c 6 6 ac 22 23.25 -1.25 bc
6 6 abc 22 23.25 -1.25 -1 10 10 a 3 3 b 11 11 ab 4 4 c 7 7 ac 23
23.25 -0.25 bc 5 5 abc 21 23.25 -2.25
[0061] For Example #1, the AC interaction effect is now removed
from the responses using the average estimated effect (not the
orthogonal estimate), as shown in Table 5 above.
[0062] Step 9 is to recalculate the worksheet if the effect removed
is an interaction. Because the last estimate removed was for an
interaction between process variables A and C, the worksheet is
recalculated before proceeding to Step 11. When the removed
estimate is a main effect, the worksheet is not recalculated. FIG.
8 is the worksheet, recalculated with the AC estimated effects
removed from the product characteristics measurements of the ac and
abc cells of the response matrix of FIG. 1.
[0063] Step 10 is to determine if the largest remaining contrast
sum should be tested for significance. The contrast sums from FIG.
8 are again plotted on a Pareto Chart as shown in FIG. 9, and again
checked for an adjacent drop between contrast sums by a factor of 2
or more, as was previously done before the AC interaction was
removed. The contrast sums on the high side represent effects that
should be identified and tested for significance. At this point it
is noted that if enough leverage has been identified to control the
product characteristic, the analysis may be discontinued. Also, if
the factor to be checked for significance is a component of an
interaction where the effect of the interaction will determine the
setting of the factor to be checked, then the analysis may be
discontinued, since no degrees of freedom for the variable remains.
For Example #1, analysis may be discontinued since the remaining
variables that appear on the high side of the break on the Pareto
Chart in FIG. 9 are variables in the interaction AC. Assuming
enough leverage has been identified with the interaction, then the
results of this experiment may cause A and C to be both set at
minus levels if it were desired to keep the process characteristic
low. Nonetheless, for purposes of illustration, the analysis will
continue.
[0064] Table 5 is reorganized in rank order response, shown in
Table 6. This is done in the same manner as was previously done
when the lower order effects of the A and C variable were removed,
in Table 2.
6TABLE 6 TABLE 5 IN RANK ORDER, SHOWING A SETTING LEVELS Y w/o the
Estimate Cell of AC A Level abc -2.25 + ac -1.25 + abc -1.25 + ac
-0.25 + a 3 + a 4 + ab 4 + ab 5 + bc 5 - c 6 - bc 6 - c 7 - -1 10 -
b 11 - -1 12 - b 12 - EC = 16
[0065] The end count is taken using Table 6. The end count is 16
since there is no overlap between the + and - signs of the A level
column. This exceeds a required endcount of 10. A is thus found to
be significant with 95% confidence.
[0066] The A process variable effect is then permanently removed by
subtracting the orthogonal estimates from the responses in FIG. 7.
This is shown in Table 7 below.
[0067] Step 11 is to begin again at step 6. However, the worksheet
does not need to be recalculated at this stage because the effect
of the A process variable is a main effect and has been removed
from the array orthogonally. This means that C effect is still the
third largest contrast sum (-42) and should be the next one checked
for significance. The endcount check for C is shown in Table 7 and
Table 8 below.
7TABLE 7 REMOVE THE EFFECT OF A TO CHECK END COUNT FOR C: Y w/o AC
Y w/o AC Cell effect A Effect of A effect or A effect -1 12 - 12 a
4 + -7.25 11.25 b 12 - 12 ab 5 + -7.25 12.25 c 6 - 6 ac -1.25 +
-7.25 6 bc 6 - 6 abc -1.25 + -7.25 6 -1 10 - 10 A 3 + -7.25 10.25 B
11 - 11 Ab 4 + -7.25 11.25 C 7 - 7 Ac -0.25 + -7.25 7 Bc 5 - 5 Abc
-2.25 + -7.25 5
[0068]
8TABLE 8 TABLE 7 SORTED IN RANK ORDER WITH THE C LEVEL ADDED: Y w/o
AC Cell and A Effects C Level bc 5 + abc 5 + c 6 + ac 6 + bc 6 +
abc 6 + c 7 + ac 7 + -1 10 - a 10.25 - b 11 - a 11.25 - ab 11.25 -
-1 12 - b 12 - ab 12.25 - EC = 16
[0069] The endcount of 16 exceeds the required endcount of 10. C
has been found to be significant with 95% confidence.
[0070] The new and transformed information yielded is that the
largest effect is AC with an estimated effect of 23.25. Setting
both process variables A and C to the + levels causes an increase
of about 23 in the product characteristic. Furthermore, when
process variable C is set to the - level, the A effect is
significant with an estimated effect of -7.25. Also, when the
process variable A is set to the minus level the C effect is
significant with an estimated effect of -5.25. Thus, in order to
maximize the product characteristic, or output response, both A and
C must be set to their plus levels. To minimize the product
characteristic, either and or both A and C should be set at the
minus level.
[0071] Step 12 is to use the information that has been transformed
from process data into information that can be used to directly
control the process, to improve the product/article of manufacture,
by setting the variables as a function of the "significant
effects." It should be determined whether any of the significant
effects, estimated by the contrasts, will impart a shift in the
product characteristic in the direction desired, or whether the
effect is to be avoided. Also, it is noted that the desired product
characteristic may be a range of values. If the estimated effects
are indicative of level settings of the process variables that will
improve the process as whole, taking into consideration costs and
other factors associated with maintaining the level settings, then
the factors may be set at the appropriate level settings to impart
the estimated effects. For Example #1, if the product
characteristic is, for example, percent impurity of some component,
and it is desired to derive a more pure product, both A and C will
be set at minus levels if not cost prohibitive. In that way, even
if one variable goes out of control, the other variable may remain
at the minus level, preventing the interaction effect from
occurring between the variables, causing a high level of impurity.
On the other hand, if the product characteristic is, for example,
tensile strength, and it is desired to have a strong product with
high tensile strength, both A and C may be set at their plus levels
if it is not cost prohibitive.
EXAMPLE #2
[0072] Example #2 compares an embodiment of the method to the Yates
analysis. Example #2 is also directed toward improvement of a
fabrication process where a spike interaction is present between
variables. Example #2, like Example #1, is an alternative
embodiment of the method and is also merely one example application
of the method.
[0073] In Example #2, for Step 1, two input variables are selected
for testing at 2 levels each. Again, a product characteristic is
the measured response or output.
[0074] Step 2 is to design the experiment using a factorial design.
In Example #2, there are 2 factors in the experiment with 2 levels
each. A 2.sup.2 response matrix is thus required. Each combination
of level settings for the variables is to be tested four times, to
produce four repeat responses in each cell of the response
matrix.
[0075] A passing end count is determined in accordance with Step 3
depending on the confidence required.
[0076] Step 4 is to set the levels of the variables and record the
responses to complete the full factorial experiment with four
repeat runs. The results of the experiment are shown in FIG.
10.
[0077] Example #2 is a simplified example and FIG. 10 of the
example shows that an interaction is occurring between process
variables A and B, since the responses in the ab cell are larger
than the responses in the other cells.
[0078] Step 5 is to calculate the estimated effects as individual
contrasts and display the effects in the worksheet. This is shown
in FIG. 11. Evaluating the contrasts in the worksheet reveals that
there is an interaction between A and B when both variable are set
at + levels. First, the set of individual contrasts shown in the
worksheet for A range from 1 to 10. The set of contrasts for B
range from -1 to 7. Finally, the set of contrasts for the
interaction between A and B is 6 to 8. The range of contrasts in
the A and B process variables combined with steady AB contrasts of
limited variance is one indication of an AB interaction effect.
This can be verified by examining the leftmost column of the
worksheet in FIG. 11 for the particular row in which each contrast
is displayed, which provides a notation for the row that indicates
the level setting of variables not involved in the particular
contrast. This is best seen in the worksheet for Example #2 in FIG.
11, showing that the value of each contrast for process variable A
is at the high end of the range in each "ab" row, the "ab" notation
indicating that B is set at the + level. Furthermore, each value
for the contrasts for process variable A is at a low end of the
range in each "a" row, the "a" notation indicating that B is at the
- level. The AB interaction can be verified by examining the B
level settings in the same manner.
[0079] An interaction can thus be predicted based only on the
worksheet, and the level settings of process variables A and B may
be set to impart the AB interaction effect to the response, or to
avoid it, depending on the target value of the response.
[0080] An application of the well known Yates analysis to Example
#2 is shown in FIGS. 12 and 13. The column labeled "Effects" in
these figures show the calculated effects for A, B, and the AB
interaction, in that order from the top of the column. The A effect
and B effect are calculated to be 5 and 4, with the AB effect
calculated to be only 3.5. As can be seen in this simple example in
FIG. 9, it is clear that the response for AB is the largest with
all other responses approximately equivalent. It is thus clear that
the Yates method is yielding the wrong result and the A effect is
not the largest effect.
EXAMPLE #3
[0081] Example #3 provides further explanation of an embodiment of
the method as applied to a spike interaction. Example #3 is again
directed toward improvement of a manufacturing process, having
process variable A, and B, with two level settings, and a
measurable response indicative of improvement to the process.
[0082] As has been shown in the description of the embodiment of
the method in Example #2, a method is provided to analyze full
factorial experiments to identify and quantify spike interactions.
Spike interactions can be explained by viewing a 2{circumflex over
( )}2 experimental matrix plotted as a plane.
[0083] To explain a spike interaction it is helpful to picture a
plane created in space having 4 corners, as illustrated in FIG. 15.
For Example #3, the x, y component of each corner are determined by
the settings of process variables A, B. The z component of each
corner is set by the measurable response, which is equivalent to
the height of each corner of the plane.
[0084] If the responses of all cells are approximately equal and
are, for example, 2 units, the plane will float 2 units above the
zero plane and will be parallel to the zero plane. For Example #3,
there is an A main effect of 0 units, so corners (1) and a will be
the same, in this case 4 units off the zero plane. There is also a
B main effect of 2 units, so corner b will be 2 units higher than
corner (1). If there is no interaction corner ab will be equal to
corner (1) plus both the A and B effects. In this case that would
yield a corner ab at 6 (4+0+2). If there is no interaction the main
effects are superimposed upon each other, and the plane remains
flat, but no longer parallel to the zero plane. However, for
Example #3, there is a spike interaction. This is shown in FIG. 16,
wherein the response of the ab cell is 22. Thus, A and B are
interacting, at one level, to display a higher response than simply
superimposing the A and B effects. The responses are (1)=4, a=4,
b=6, and ab=22.
[0085] Interactions impart a twist on the plane. Traditional
interactions cause opposite cells to move as a pair. For example a
traditional AB interaction will cause cells (1) and ab to both move
in the same direction. Traditional interactions cause the plane to
look like a saddle. Main effects superimposed over traditional
interactions will cause the plane to look like a tilted saddle. The
Yates analysis is based on the analysis of traditional
interactions.
[0086] Field experience has proven the existence of spike
interactions. Spike interactions do not effect the response plane
in the same manner as traditional interactions. Spike interactions
cause one cell of the matrix (not two) to move independent of the
other cells. For example, a positive ab spike interaction will
cause the ab corner of the plane to "spike up" making it
significantly higher than the other 3 corners of the matrix. The
(1) corner which is traditionally paired with the ab corner is
unmoved by the effect of the spike.
[0087] A perfect spike interaction yields contrast sums for both
the interaction and the two associated main effects which are equal
within measurement error. For example, a perfect AB spike will
result in contrast sums of AB, B, and A all being approximately
equal. This is why higher order interactions are tested first. For
the purposes of the embodiment of the method in Example #3, spike
interactions include both perfect spike interactions and
approximated spike interactions.
[0088] Recognizing a spike interaction is one reason why, in step 8
of the embodiment of the method shown in Example #1, the AC
interaction effect was removed from only one response, the ac
response. While the effect of the interaction can be mathematically
subtracted from both cells (1) and ac using the orthogonal estimate
instead of the effect estimate this does not accurately represent
what is physically happening. When a spike interaction is
subtracted from more cells than is physically warranted the
remaining contrasts are artificially large.
[0089] For the present Example #3, FIG. 16 should be used to remove
the AB interaction when permanently removing its effect. The AB
effect is 16, and removing it from the ab cell will achieve the
smallest remaining effects, which is in fact, the location of the
where the spike interaction occurs. FIG. 17 shows Example #3 with
the AB effect removed from the ab cell.
[0090] By removing the AB effect from the ab cell, the effect
estimate of A is now made with B set to the minus level, and the
effect estimate of B is now made with A set to the minus level.
This yields useful information in that, since AB has been shown to
be the interaction of interest, it will be most desirable to also
know the effect of either variable alone with the other set so as
not to interact in the spike interaction. Thus, by graphically
representing the responses, and removing the interaction effect to
achieve the smallest remaining effects, useful information is
obtained that can be directly used to determine settings for
process variables. The same considerations may be given to where to
set the variables as was discussed in Step 12 of Example #1.
EXAMPLE #4
[0091] A full factorial experiment was run for a manufacturing
process wherein electrical components were being manufactured.
Finished parts were failing dielectric testing. Three variables
were identified as possible contributors to the problem. The
variables were tested using a three-factor full factorial
experiment. The response was, arc-volts, the voltage at which the
part failed.
[0092] The present method identified an AB spike interaction when
variables A and B were set at a low value (-). The spike
interaction provided the needed response level. Given the
consequences of building a weak part, and the cost of setting both
A and B to the low level, it was decided to set both A and B to the
low level.
[0093] Examples #1, #2, #3, and #4, have been directed toward the
improvement of a manufacturing process to yield improved product
characteristic. Manufacturing processes can include but are
certainly not limited to, manufacturing of vehicles parts,
vehicles, general electronic apparatus and devices, computers,
computer components, scientific apparatus, medical apparatus,
chemicals, machinery, foods, construction materials, tools,
pharmaceuticals, paper goods and printed matter, paint, rubber
goods, leather goods, furniture, housewares, cordage and fibers,
fabrics, clothing, fancy goods, toys and sporting goods, and
beverages, cosmetics and cleaning preparations, lubricants and
fuels/oil, general metal goods, jewelry, firearms, musical
instruments, and even the processing of natural goods. However, as
will be appreciated, the embodiments of the method have broad
applicability. The output responses monitored can be any form of
product or article characteristic as well as a characteristic of
the fabrication or manufacturing process itself. Thus the
improvement sought and achieved through application of any of the
various embodiments of the method can include improvements not only
to the product or article, but also to the manufacturing or
fabrication process. Examples of measurable responses monitored to
gage improvements to the process include production rate of the
process and any efficiency in the process.
[0094] In addition, embodiments of the method can also be used in
the operation of a manufacturing process, such as, for example,
when a process has temporarily deviated from a target value
required for an operating parameter of the process, and it is
desired to return the process to normal operation. The previous
settings of variables may be unknown, and hence, an embodiment of
the present method can be used to return the variables to the
previous settings to attain the range sought for the operating
parameter. The operating parameter may be related to, but are not
limited to, production rates, manufacturing efficiency parameters,
and product characteristics of the products generated by the
process.
[0095] It will also be appreciated that embodiments of the method
can be applied to the design of processes and products. Such
applications of embodiments of the method may typically be in
connection with bench scale models of a manufacturing or
fabrication process or prototypes of a product or article.
Experimentation can be done on the bench scale, or on the
prototypes, and an embodiment of the method can be used to select
the correct level settings for the variables.
[0096] One skilled in the art will also recognize that the present
invention may be implemented through the use of a general purpose
computer system. For example, the contents of the worksheet of FIG.
3 may be calculated and stored in the computer in a variety of
forms including a spreadsheet, or the iterative steps of the method
as well as the graphical interpretations may be done with the
computer. An embodiment of the method may be implemented by a
computer system, including receiving and adjusting variables
through the input/output devices 4, based on the information
yielded by the method. In one alternative embodiment, an embodiment
of the method is implemented in the computer and a signal is sent
to a controller to adjust the level settings of the process
variables based on information transformed by the embodiment of the
method. Any one of the embodiments of the method may also be stored
on a computer readable medium, such as a memory, which can then be
used with a computer to perform the method.
[0097] FIG. 14 is a block diagram of a general purpose computer for
practicing preferred embodiments of the present invention. The
computer system 1 contains a central processing unit (CPU) 2, a
display screen 3, input/output devices 4, and a computer memory
(memory) 5.
[0098] As the embodiments of the method can be implemented through
the use of a general purpose computer system, wherein the
particular documents described previously are not necessary, so can
the documents be modified and embodied in various forms of display.
For example, the worksheet of FIG. 3 may be implemented in a
variety of forms. A display could be generated in more tabular form
with the fields of the table corresponding to similar notation, or
perhaps in graphical form.
[0099] From the foregoing it will be appreciated that, although
specific embodiments of the invention have been described herein
for purposes of illustration, various modifications may be made
without deviating from the spirit and scope of the invention.
Accordingly, the invention is not limited except as by the appended
claims.
* * * * *