$TITLE ECON 480 FINAL EXAM MATHEMATICAL PROGRAMMING QUESTION T.docx

$TITLE ECON 480 FINAL EXAM MATHEMATICAL
PROGRAMMING QUESTION
* This problem is from "Computational Economics" CH 1,
Problem 7
* The product is Bar Stool.
$OFFSYMLIST OFFSYMXREF
*OPTION LIMROW = 0
*OPTION LIMCOL = 0
SETS
I PLANTS Production Sites
/P1,P2,P3,P4/
J MARKETS Market Sites
/M1,M2,M3,M4,M5,M6/;

*Data
TABLE NETREV(I,J) Net Revenue in Dollars
M1 M2 M3 M4 M5 M6
P1 29 51 59 54 56 27
P2 44 43 29 63 53 46
P3 63 37 63 62 46 47
P4 51 53 54 51 43 38;
* The Net Revenues are different because the product is
* sold at different prices in the six markets and the
* transportation cost from each plant to each market is also
different
PARAMETERS
CAPACITY(I) Capacity at each plant

/P1 1200
P2 1800
P3 1100
P4 1900/
DEMAND(J) Demand at each market
/M1 1000
M2 1200
M3 700
M4 1500
M5 500
M6 1000/;
VARIABLES
X(I,J) Montly Shippments from plant i to market j
TOTPROF Objective Function Value;

POSITIVE VARIABLE X;
EQUATIONS
OBJFUNC OBJECTIVE FUNCTION
PLANCONS(I) PLANT CONSTRAINT
DEMCONS(J) DEMAND CONSTRAINT;
OBJFUNC .. TOTPROF =E= SUM((I,J), NETREV(I,J) *
X(I,J));
PLANCONS(I) .. SUM(J, X(I,J)) =L= CAPACITY(I);
DEMCONS(J) .. SUM(I, X(I,J)) =G= DEMAND(J);
MODEL MAXPROF /OBJFUNC,PLANCONS,DEMCONS/;
SOLVE MAXPROF USING LP MAXIMIZING TOTPROF;

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NOT Transforming the Data Can Be Fatal to Your Analysis
A case study, with real data, describes the need for data
transformation.

[Type here]
Donald Wheeler stated in his second article "Transforming the
Data Can Be Fatal to Your Analysis,"
"out of respect for those who are interested in learning how to
better analyze data, I feel the need to
further explain why the transformation of data can be fatal to
your analysis."
My motivation for writing this article is not only improved data
analysis but formulating analyses so
that a more significant business performance reporting question
is also addressed within the
assessment; i.e., a paradigm shift from the objectives of the
traditional four-step Shewhart system,
which Wheeler referenced in his article.
The described statistical business performance charting
methodology can, for example, reduce
firefighting when the approach replaces organizational goal-
setting red-yellow-green scorecards,
which often have no structured plan for making improvements.
This article will show, using real
data, why an appropriate data transformation can be essential to

determine the best action or non-
action to take when applying this overall system in both
manufacturing and transactional processes.
Should Traditional Control Charting Procedures be Enhanced?
I appreciate the work of the many statistical-analysis icons. For
example, more than 70 years ago
Walter Shewhart introduced statistical control charting. W.
Edwards Deming extended this work to
the business system with his profound knowledge philosophy
and introduced the terminology
common and special cause variability. I also respect and
appreciate the work that Wheeler has done
over the years. For example, his book Understanding Variation
has provided insight to many.
However, Wheeler and I have a difference of opinion about the
need to transform data when a
transformation makes physical sense. The reason for writing
this article is to provide additional
information on the reasoning for my position. I hope that this
supplemental explanation will provide
readers with enough insight so that they can make the best
logical decision relative to considering
data transformations or not.

In his last article, Wheeler commented: "However, rather than
offering a critique of the points raised
in my original article, he [Breyfogle] chose to ignore the
arguments against transforming the data
and to simply repeat his mantra of ‘transform, transform,
transform.' "
http://www.qualitydigest.com/inside/quality-insider-
column/transforming-data-can-be-fatal-your-analysis.html
transformation.
[Type here]
In fact, I had agreed with Wheeler’s stated position that, "If a
transformation makes sense both in
terms of the original data and the objectives of the analysis,
then it will be okay to use that
transformation."
What I did take issue with was his statement: "… Therefore, we
do not have to pre-qualify our data
before we place them on a process behavior chart. We do not
need to check the data for normality,

nor do we need to define a reference distribution prior to
computing limits. Anyone who tells you
anything to the contrary is simply trying to complicate your life
unnecessarily."
In my article, I stated "I too do not want to complicate people’s
lives unnecessarily; however, it is
important that someone’s over-simplification does not cause
inappropriate behavior."
Wheeler kept criticizing my random data set parameter selection
and the fact that I did not use real
data. However, Wheeler failed to comment on the additional
points I made relative to addressing a
fundamental issue that was lacking in his original article, one
that goes beyond the transformation
question. This important issue is the reporting of process
performance relative to customer
requirement needs, i.e., a goal or specification limits.
This article will elaborate more on the topic using real data
which will lead to the same conclusion as
my previous article: For some processes an appropriate
transformation is a very important step in
leading to the most suitable action or non-action. In addition,
this article will describe how this
metric reporting system can create performance statements,

offering a prediction statement of how
the process is performing relative to customer needs. It is
important to reiterate that appropriate
transformation selection, when necessary, needs to be part of
this overall system.
In his second article, Wheeler quoted analysis steps from
Shewhart’s Economic Control of Quality of
Manufactured Product. These steps, which were initiated over
seven decades ago, focus on an
approach to identify out-of-control conditions. However, the
step sequence did not address whether
the process output was capable of achieving a desired
performance output level, i.e., expected
process non-conformance rate from a stable process.
Wheeler referenced a study "… encompassing more than 1,100
probability models where 97.3
percent of these models had better than 97.5-percent coverage at
three-sigma limits." From this
statement, we could infer that Wheeler believes that industry
should be satisfied with a greater than
2-percent false-signal rate. For processes that do not follow a
normal distribution, this could

transformation.
[Type here]
translate into huge business costs and equipment downtime
searching for special-cause conditions
that are in fact common-cause. After an organization chases
phantom special-cause-occurrences over
some period of time, it is not surprising that they would
abandon control charting all together.
Wheeler also points out how traditional control charting has
been around for more than 70 years and
how the limits have been thoroughly proven. I don't disagree
with the three sampling standard
deviation limits; however, how often is control charting really
used throughout businesses? In the
1980s, there was a proliferation of control charts; however, look
at us now. Has the usage of these
charts continued and, if they are used, are they really applied
with the intent originally envisioned by
Shewhart? I suggest there is not the widespread usage of control
charts that quality-tools training

classes would lead students to believe. But why is there not
more frequent usage of control charts in
both manufacturing and transactional processes?
To address this underutilization, I am suggesting that while the
four-step Shewhart methodology has
its applicability, we now need to revisit how these concepts can
better be used to address the needs of
today's businesses, not only in manufacturing but transactional
processes as well.
To assess this tool-enhancement need, consider that the output
of a process (Y) is a function of its
inputs (Xs) and its step sequence, which can be expressed as
Y=f(X). The primary purpose of
Wheeler's described Shewhart four-step sequence is to identify
when a special cause condition occurs
so that an appropriate action can be immediately taken. This
procedure is most applicable to the
tracking of key process input variables (Xs) that are required to
maintain a desired output process
response for situations where the process has demonstrated that
it provides a satisfactory level of
performance for the customer-driven output, i.e., Y.
However, from a business point of view, we need to go beyond
what is currently academically

provided and determine what the enterprise needs most as a
whole. One business-management need
is an improved performance reporting system for both
manufacturing and transactional processes.
This enhanced reporting system needs to structurally evaluate
and report the Y output of a process
for the purpose of leading to the most appropriate action or non-
action.
An example application transactional process need for such a
charting system is telephone hold time
in a call center. For situations like this, there is a natural
boundary: hold time cannot get below zero.
For this type of situation, a log-normal transformation can often
be used to describe adequately the
distribution of call-center hold time. Is this distribution a
perfect fit? No, but it makes physical sense
transformation.
[Type here]

and is often an adequate representation of what physically
happens; that is, a general common-cause
distribution consolidation bounded by zero with a tail of hold
times that could get long.
From a business view point, what is desired at a high level is a
reporting methodology that describes
what the customer experiences relative to hold time, the Y
output for this process. This is an
important business requirement need that goes beyond the four-
step Shewhart process.
I will now describe how to address this need through an
enhancement to the Shewhart four-step
control charting system. This statistical business performance
charting (SBPC) methodology
provides a high-level view of how the process is performing.
With SBPC, we are not attempting to
manage the process in real time. With this performance
measurement system, we consider
assignable-cause differences between sites, working shifts,
hours of the day, and days of the week to
be a source of common-cause input variability to the overall
process—in other words, Deming's
responsibility of management variability.
With the SBPC system, we first evaluate the process for

stability. This is accomplished using an
individuals chart where there is an infrequent subgrouping time
interval so that input variability
occurs between subgroups. For example, if we think that
Monday's hold time could be larger than the
other days of the week because of increased demand, we should
consider selecting a weekly
subgrouping frequency.
With SBPC, we are not attempting to adjust the number of
operators available to respond to phone
calls in real time since the company would have other systems
to do that. What SBPC does is assess
how well these process-management systems are addressing the
overall needs of its customers and
the business as a whole. The reason for doing this is to
determine which of the following actions or
non-actions are most appropriate, as described in Table 1.
Table 1: Statistical Business Performance Charting (SBPC)
Action Options
1. Is the process unstable or did something out of the ordinary
occur,
which requires action or no action?
2. Is the process stable and meeting internal and external
customer

needs? If so, no action is required.
3. Is the process stable but does not meet internal and external
customer needs? If so, process improvement efforts are needed.
The four-step Shewhart model that Wheeler referenced focuses
only on step number one.
transformation.
[Type here]
In my previous article, I used randomly generated data to
describe the importance of considering a
data transformation when there is a physical reason for such a
consideration. I thought that it would
be best to use random data since we knew the answer; however,
Wheeler repeatedly criticized me for
selecting a too-skewed distribution and not using real data.
I will now use real data, which will lead to the same conclusion
as my previous article.
Real-data Example
A process needs to periodically change from producing one

product to producing another. It is
important for the changeover time to be as small as possible
since the production line will be idle
during changeover. The example data used in this discussion is
a true enterprise view of a business
process. This reports the time to change from one product to
another on a process line. It includes
six months of data from 14 process lines that involved three
different types of changeouts, all from a
single factory. The factory is consistently managed to rotate
through the complete product line as
needed to replenish stock as it is purchased by the customers.
The corporate leadership considers
this process to be predictable enough, as it is run today, to
manage a relatively small finished goods
inventory. With this process knowledge, what is the optimal
method to report the process behavior
with a chart?
Figure 1 is an individuals chart of changeover time. From this
control chart, which has no
transformation as Wheeler suggests, nine incidents are noted
that should have been investigated in
real time. In addition, one would conclude that this process is
not stable or is out of control. But, is

it?
transformation.
[Type here]
Figure 1: Individuals Chart of Changeover Time
(Untransformed Data)
One should note that in Figure 1 the lower control limit is a
negative number, which makes no
physical sense since changeover time cannot be less than zero.
Wheeler makes the statement, "Whenever we have skewed data
there will be a boundary value on
one side that will fall inside the computed three-sigma limits.
When this happens, the boundary
value takes precedence over the computed limit and we end up
with a one-sided chart." He also says,
"The important fact about nonlinear transformations is not that
they reduce the false-alarm rate, but

rather that they obscure the signals of process change."
It seems to me that these statements can be contradictory.
Consider that the response that we are
monitoring is time, which has a zero boundary, and where a
lower value is better. For many
situations, our lower-control limit will be zero with Wheeler’s
guidelines (e.g., Figure 1). Consider
that the purpose of an improvement effort is to reduce
changeover time. An improved reduction in
changeover time can be difficult to detect using this “one-sided”
control chart, when the lower-
control limit is at the boundary condition—zero in this case.
Wheeler's article makes no mention of what the process
customer requirements are or the reporting
of its capability relative to specifications, which is an important
aspect of lean Six Sigma programs.
Let's address that point now.
transformation.
[Type here]

The current process has engineering evaluating any change that
takes longer than twelve hours. This
extra step is expensive and distracts engineering from its core
responsibility. The organization's
current reporting system does not address how frequently this
engineering intervention occurs.
In his article, Wheeler made no mention of making such a
computation for either normal or non-
normal distributed situations. This need occurs frequently in
industry when processes are to be
assessed on how they are performing relative to specification
requirements.
Let's consider making this estimate from a normal probability
plot of the data, as shown in Figure 2.
This estimate would be similar to a practitioner manually
calculating the value using a tabular z-
value with a calculated sample mean and standard deviation.
Figure 2: Probability Plot of the Untransformed Data
We note from this plot how the data do not follow a straight
line; hence, the normal distribution does

not appear to be a good model for this data set. Because of this
lack-of-fit, the percentage-of-time
estimate for exceeding 12 hours is not accurate; i.e., 46% (100-
54 = 46).
transformation.
[Type here]
We need to highlight that technically we should not be making
an assessment such as this because
the process is not considered to be in control when plotting
untransformed data. For some, if not
most, processes that have a long tail, we will probably never
appear to have an in-control process, no
matter what improvements are made; however, does that make
sense?
The output of a process is a function of its steps and input
variables. Doesn’t it seem logical to expect
some level of natural variability from input variables and the
execution of process steps? If we agree
to this assumption, shouldn’t we expect a large percentage of

process output variability to have a
natural state of fluctuation; i.e., be stable?
To me this statement is true for most transactional and
manufacturing processes, with the exception
of things like naturally auto-correlated data situations such as
the stock market. However, with
traditional control charting methods, it is often concluded that
the process is not stable even when
logic tells us that we should expect stability.
Why is there this disconnection between our belief and what
traditional control charts tell us? The
reason is that underlying control-chart-creation assumptions and
practices are often not consistent
with what occurs naturally in the real world. One of these
practices is not using suitable
transformations when they are needed to improve the
description of process performance, for
instance, when a boundary condition exists.
It is important to keep in mind that the reason for process
tracking is to determine which actions or
non-actions are most appropriate, as described in Table 1. Let’s
now return to our real-data example
analysis.

For this type of bounded situation, often a log-normal
distribution will fit the data well, since
changeover time cannot physically go below a lower limit of
zero, such as the previously described
call-center situation. With the SBPC approach, we want first to
assess process stability. If a process
has a current stable region, we can consider that this process is
predictable. Data from the latest
region of stability can be considered a random sample of the
future, given that the process will
continue to operate as it has in the recent past's region of
stability.
When these continuous data are plotted on an appropriate
probability plot coordinate system, a
prediction statement can be made: What percentage of time will
the changeover take longer than 12
transformation.
[Type here]

hours? Figure 3 shows a control chart in conjunction with a
probability plot. A netting-out of the
process analysis results is described below the graphics: The
process is predictable where about 38
percent of the time it takes longer than 12 hours.
Unlike the previous non-transformed analysis, we would now
conclude that the process is stable, i.e.,
in control. Unlike the non-transformed analysis, this analysis
considers that the skewed tails, which
we expect from this process, to be the result of common-cause
process variability and not a source
for special cause investigation. Because of this, we conclude,
for this situation, that the
transformation provides an improved process discovery
foundation model to build upon, when
compared to a non-transformed analysis approach.
transformation.
[Type here]

The process is predictable where about 38% percent of the time
it takes
longer than 12 hours.
Figure 3: Report-out.
Let’s now compare the report-outs of both the untransformed
(Figure 1) and transformed data
(Figure 3). Consider what actions your organization might take
if presented each of these report-outs
separately.
transformation.
[Type here]
The Figure 1 report can be attempting to explain common cause
events as though each occurrence
has an assignable cause that needs to be addressed. Actions
resulting from this line of thinking can
lead to much frustration and unnecessary process-procedural

tampering that result in increased
process-response variability, as Deming illustrated in his funnel
experiment.
When Figure 3's report-out format is used in our decision
making process, we would assess the
options in Table 1 to determine which action is most
appropriate. With this data-transformed
analysis, number three in Table 1 would be the most appropriate
action, assuming that we consider
the 38 percent frequency of occurrence estimate above 12 hours
excessive.
Wheeler stated, "If you are interested in looking for assignable
causes you need to use the process
behavior chart (control chart) in real time. In a retrospective use
of the chart you are unlikely to ever
look for any assignable causes …"
This statement is contrary to what is taught and applied in the
analyze phase of lean Six Sigma’s
define-measure-analyze-improve-control (DMAIC) process
improvement project execution
roadmap. Within the DMAIC analyze phase, the practitioner
evaluates historical data statistically for
the purpose of gaining insight into what might be done to
improve his process improvement project’s

process.
It can often be very difficult to determine assignable causes
with a real-time-search-for-signals
approach that Wheeler suggests, especially when the false
signal rate can be amplified in situations
where an appropriate transformation was not made. Also, with
this approach, we often do not have
enough data to test that the hypothesis of a particular assignable
cause is true; hence, we might think
that we have identified an assignable cause from a signal, but
this could have been a chance
occurrence that did not, in fact, negatively impact our process.
In addition, when we consider how
organizations can have thousands of processes, the amount of
resources to support this search-for-
signal effort can be huge.
Deming in Out of the Crisis stated, "I should estimate that in
my experience most troubles and most
possibilities for improvement add up to proportions something
like this: 94 percent belong to the
system (responsibility of management), 6 percent [are] special."

transformation.
[Type here]
With a search-for-signal strategy it seems as if we are trying to
resolve the 6 percent of issues that
Deming estimates. It would seem to me that it would be better
to focus our efforts on how we can
better address the 94 percent common-cause issues that Deming
describes.
To address this matter from a different point of view, consider
extending a classical assignable cause
investigation from special cause occurrences to determining
what needs to be done to improve a
process’ common-cause variability response if the process does
not meet the needs of the customer
or the business.
I have found with the SBPC reporting approach that assignable
causes that negatively impact process
performance from a common-cause point of view can best be
determined by collecting data over
some period of time to test hypotheses that assess differences

between such factors as machines,
operators, day of the week, raw material lots, and so forth.
When undergoing process improvement
efforts, the team can use collected data within the most recent
region of stability to test out compiled
hypothesis statements that it thinks could affect the process
output level. These analyses can provide
guiding light insight to process-improvement opportunities.
This is a more efficient analytical
discovery approach than a search-for-signals strategy where the
customer needs are not defined or
addressed relative to current process performance.
For the example SBPC plot in Figure 3, the team discovered
through hypotheses tests that there was
a significant difference in the output as a function of the type of
change, the shift that made the
change, and the number of performed tests made during the
change.
This type of information helps the team determine where to
focus its efforts in determining what
should be done differently to improve the process. Improvement
to the system would be
demonstrated by a statistically significant shift of the SPBC
report-out to a new-improved level of

stability. This system of analysis and discovery would apply for
both processes that need data
transformation and those that don't, that is, the SPBC system
highlights and does not obscure the
signals of process change.
Detection of an Enterprise Process Change
Wheeler stated, "However, in practice, it is not the false-alarm
rate that we are concerned with, but
rather the ability to detect signals of process changes. And that
is why I used real data in my article.
There we saw that a nonlinear transformation may make the
histogram look more bell-shaped, but in
transformation.
[Type here]
addition to distorting the original data, it also tends to hide all
of the signals contained within those
data."

Let's now examine how well a shift can be detected with our
real-data example, using both non-
transformed and transformed data control charts. A traditional
test for a process control chart is the
average run length until an out-of-control indication is detected,
typically for a one standard
deviation shift in the mean. This does not translate well to
cycle-time-based data where there are
already values near the natural limit of zero. For our analysis,
we will assume a 30-percent reduction
in the average cycle time to be somewhat analogous to a shift of
one standard deviation in the mean
of a normally distributed process.
In our sample data, there were 590 changeovers in the six-
month period. The 30-percent reduction
in cycle time was introduced after point 300. A comparison of
the transformed and untransformed
process behavior charting provides a clear example of the
benefits of the transformation, noting that,
for the non-transformed report-out, a lower control limit of
zero, per Wheeler's suggestion, was
included as a lower bound reference line.
In Figure 4a, the two charts show the untransformed data
analysis, the upper chart appearing to

have become stable after the simulated process change. When a
staging has been created (lower
chart in Figure 4a), the new chart stage identifies four special
cause incidents that were considered
common-cause events in the transformed data set, as shown in
the lower chart in Figure 4b.
transformation.
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Figure 4a: Non-transformed Analysis of a Process Shift
transformation.
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Figure 4b: Transformed Analysis of a Process Shift
The two transformed charts in Figure 4b show a change in the
process with the introduction of the
special-cause indications after the simulated change was
implemented.
When staging is introduced into this charting, the special-cause
indications are eliminated; i.e., the
process is considered stable after the change.
transformation.
[Type here]
Using Wheeler’s guidelines, a simple reduction in process cycle
time would appear to be the removal
of special causes. This is fine in the short term, but as we
collect more data on the new process and
introduce staging into the data charting we would surely, for
skewed process performance, return to
a situation were special-cause signals would reoccur, as shown
in the lower chart in Figure 4a. We

should highlight that these special cause events appear as
common-cause variability in the
transformed-data-set analysis shown in the lower chart in Figure
4b.
If you follow a guideline of examining a behavioral chart that
has an appropriate transformation, you
would have noted a special cause occurring when the simulated
process change was interjected.
From this analysis, the success of the process improvement
effort would have been recognized and
the process behavior chart limits would be re-set to new limits
that reflect the new process response
level.
Since the transformed data set process is stable, we can report-
out a best estimate for how the
process is performing relative to the 12 hour cycle time criteria.
In Figure 5, the SBPC description
provides an estimate of how the process performed before and
after the change, where 100-63 = 37%
and 100-87 = 13%. Since the process has a recent region of
stability, the data from this recent region
can provide a predictive estimate of future performance unless
something changes within the process

and/or its inputs; i.e., our estimate is that 13 percent of the
cycle times will be above 12 hours.
transformation.
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The process has been predictable since observation 300 where
now about
13% of the time it takes longer than 12 hours for a changeover.
This 13%
value is a reduction from about 37% before the new process
change was
made.
Figure 5: SBPC Report-out Describing Impact of Process
Change.
I wonder if much of the dissatisfaction and lack of use of

business control charting derive from the
use of non-transformed data. Immediately after the change, the
process looks to be good, and the
improvement effort is recognized as a success. However, in a
few weeks or months after the control
chart has been staged, the process will show to be out of control
again because the original data is no
longer the primary driver for the control limits. The
organization will assume the problem has
returned and possibly consider the earlier effort to now be a
failure.
In addition, Wheeler’s four-step Shewhart process made no
mention of how to assess how well a
process is performing relative to customer needs, a very
important aspect in the real business world.
Conclusions
The purpose of traditional individuals charting is to identify in
a timely fashion when special-cause
conditions occur so that corrective actions can be taken. The
application of this technique is most

transformation.
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beneficial when tracking the inputs to a process that has an
output level which is capable of meeting
customer needs.
False signals can occur if the process measurement by nature is
not normally distributed, for
example, in processes that cannot be below zero. Investigation
into these false signals can be
expensive and lead to much frustration when no reason is found
for out-of-control conditions.
The Wheeler suggested four-step Shewhart process has its
application; however, a more pressing
issue for businesses is in the area of high-level predictive
performance measurements. SBPC
provides an enhanced control charting system that addresses
these needs; e.g., an individuals chart
in conjunction with an appropriate probability plot for
continuous data. Appropriate data
transformation considerations need to be part of the overall
SBPC implementation process.
With SBPC, we are not limited to identifying out-of-control

conditions but also are able to report the
capability of the process in regions of stability in terms that
everyone can understand. With this form
of reporting, when there is a recent region of stability, we can
consider data from this region to be a
random sample of the future. With statistical business
performing charting approach, we might be
able to report that our process has been stable for the last 14
weeks with a prediction that 10 percent
of our incoming calls will take longer than a goal of one minute.
I expect that the one real issue behind this entire discussion is
the idea of "what is good enough?"
Wheeler shared a belief that a control charting method that
allows up to 2.5 percent of process
measures to trigger a cause and corrective action effort that will
not find a true cause in a business as
"good enough." Wheeler goes so far as to relate the 95 percent
confidence concept from hypothesis
testing to imply that up to 5 percent false special cause
detections are acceptable. Using the above-
described concept of transforming process data where the
transformation is appropriate for the
process-data type will lead to process behavior charting that
matches the sensitivity and false-cause

detection that we have all learned to expect when tracking
normally distributed data in a typical
manufacturing environment.
Why would anyone want to have a process behavior chart that
will be interpreted differently for each
use in an organization? The answer should be clear: use
transformations when they are appropriate
and then your organization can interpret all control charts in the
same manner. Why be "good
enough" when you have the ability to be correct?
transformation.
[Type here]
The "to transform or not transform" issue addressed in this
paper led to SBPC reporting and its
advantages over the classical control-charting approach
described by Wheeler. However, the
potential for SBPC predictive reporting has much larger
implications than reporting a widget

manufacturing process output.
Traditional organizational performance measurement reporting
systems have a table of numbers,
stacked bar charts, pie charts, and red-yellow-green goal-based
scorecards that provide only
historical data and make no predictive statements. Using this
form of metric reporting to run a
business is not unlike driving a car by only looking at the rear
view mirror, a dangerous practice.
When predictive SBPC system reporting is used to track
interconnected business process map
functions, an alternative forward-looking dashboard
performance reporting system becomes
available. With this metric system, organizations can
systematically evaluate future expected
performance and make appropriate adjustments if they don't like
what they see, not unlike looking
out a car's windshield and turning the steering wheel or
applying the brake if they don't like where
they are headed.
How SBPC can be integrated within a business system that
analytically/innovatively determines
strategies with the alignment of improvement projects that

positively impact the overall business will
be described in a later article.
DISCUSS
REGISTER )
ABOUT THE AUTHOR
Forrest Breyfogle—New Paradigms
CEO and president of Smarter
Solution
s Inc., Forrest W. Breyfogle III is the creator of the integrated
enterprise excellence (IEE) management system, which takes
lean Six Sigma and the balanced
scorecard to the next level. A professional engineer, he’s an
ASQ fellow who serves on the board of
advisors for the University of Texas Center for Performing
Excellence. He received the 2004 Crosby

Medal for his book, Implementing Six Sigma. E-mail him at
[email protected]
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transformation.
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Comments

Non-normal data: To Transform or Not to Transform
Sometimes you need to transform non-normal data.
1
In “Do You Have Leptokurtophobia?” Don Wheeler stated,
“‘But the software suggests transforming
the data!’ Such advice is simply another piece of confusion. The
fallacy of transforming the data is as
follows:
“The first principle for understanding data is that no data have
meaning apart from their context.
Analysis begins with context, is driven by context, and ends
with the results being interpreted in the
context of the original data. This principle requires that there

must always be a link between what
you do with the data and the original context for the data. Any
transformation of the data risks
breaking this linkage. If a transformation makes sense both in
terms of the original data and the
objectives of the analysis, then it will be okay to use that
transformation. Only you as the user can
determine when a transformation will make sense in the context
of the data. (The software cannot do
this because it will never know the context.) Moreover, since
these sensible transformations will tend
to be fairly simple in nature, they do not tend to distort the
data.”
I agree with Wheeler in that data transformations that make no
physical sense can lead to the wrong
action or nonaction; however, his following statement concerns
me. “Therefore, we do not have to

pre-qualify our data before we place them on a process behavior
chart. We do not need to check the
data for normality, nor do we need to define a reference
distribution prior to computing limits.
Anyone who tells you anything to the contrary is simply trying
to complicate your life unnecessarily.”
I, too, do not want to complicate people’s lives unnecessarily;
however, it is important that
someone’s oversimplification does not cause inappropriate
behavior.
The following illustrates, from a high level, or what I call a
30,000-foot-level, when and how to apply
transformations and present results to others so that the data
analysis leads to the most appropriate
action or nonaction. Statistical software makes the application
of transformations simple.

Why track a process?
There are three reasons for statistical tracking and reporting of
transactional and manufacturing
process outputs:
occur, which requires action or no
action?
customer needs? If so, no action is
required.
http://www.qualitydigest.com/inside/quality-insider-column/do-
you-have-leptokurtophobia.html

2
customer needs? If so, process
improvement efforts are needed.
W. Edwards Deming, in his book, Out of the Crisis
(Massachusetts Institute of Technology, 1982)
stated, “We shall speak of faults of the system as common
causes of trouble, and faults from fleeting
events as special causes…. Confusion between common causes
and special causes leads to frustration
of everyone, and leads to greater variability and to higher costs,
exactly contrary to what is needed. I
should estimate that in my experience, most troubles and most
possibilities for improvement add up
to proportions something like this: 94 percent belong to the

system (responsibility of management),
6 percent special.”
With this perspective, the second portion of item No. 1 could be
considered a special-cause
occurrence, while items No. 2 and 3 could be considered
common-cause occurrence.
A tracking system is needed for determining which of the three
above categories best describes a
given situation.
Is the individuals control chart robust to non-normality?
The following will demonstrate how an individuals control chart
is not robust to non-normally
distributed data. The implication of this is that an erroneous
decision could be made relative to the
three listed reasons, if an appropriate transformation is not
made.

To enhance the process of selecting the most appropriate action
or nonaction from the three listed
reasons, an alternate control charting approach will be
presented, accompanied by a procedure to
describe process capability/performance reporting in terms that
are easy to understand and
visualize.
Let’s consider a hypothetical application. A panel’s flatness,
which historically had a 0.100 in. upper
specification limit, was reduced by the customer to 0.035 in.
Consider, for purpose of illustration,
that the customer considered a manufacturing nonconformance
rate above 1 percent to be
unsatisfactory.
Physical limitations are that flatness measurements cannot go
below zero, and experience has shown

that common-cause variability for this type of situation often
follows a log-normal distribution.
3
The person who was analyzing the data wanted to examine the
process at a 30,000-foot-level view to
determine how well the shipped parts met customers’ needs. She
thought that there might be
differences between production machines, shifts of the day,
material lot-to-lot thickness, and several
other input variables. Because she wanted typical variability of
these inputs as a source of common-

cause variability relative to the overall dimensional
requirement, she chose to use an individuals
control chart that had a daily subgrouping interval. She chose to
track the flatness of one randomly-
selected, daily-shipped product during the last several years that
the product had been produced.
She understood that a log-normal distribution might not be a
perfect fit for a 30,000-foot-level
assessment, since a multimodal distribution could be present if
there were a significant difference
between machines, etc. However, these issues could be checked
out later since the log-normal
distribution might be close enough for this customer-product-
receipt point of view.
To model this situation, consider that 1,000 points were
randomly generated from a log-normal

distribution with a location parameter of two, a scale parameter
of one, and a threshold of zero (i.e.,
log normal 2.0, 1.0, 0). The distribution from which these
samples were drawn is shown in figure 1. A
normal probability plot of the 1,000 sample data points is shown
in figure 2.
Figure 1: Distribution From Which Samples Were Selected
4
Figure 2: Normal Probability Plot of the Data

From figure 2, we statistically reject the null hypothesis of
normality technically, because of the low
p-value, and physically, since the normal probability plotted
data does not follow a straight line. This
is also logically consistent with the problem setting, where we
do not expect a normal distribution for
the output of such a process having a lower boundary of zero. A
log-normal probability plot of the
data is shown in figure 3.
5

Figure 3: Log-Normal Probability Plot of the Data
From figure 3, we fail to statistically reject the null hypothesis
of the data being from a log-normal
distribution, since the p-value is not below our criteria of 0.05,
and physically, since the log-normal
probability plotted data tends to follow a straight line. Hence, it
is reasonable to model the
distribution of this variable as log normal.
If the individuals control chart is robust to data non-normality,
an individuals control chart of the
randomly generated log-normal data should be in statistical
control. In the most basic sense, using
the simplest run rule (a point is “out of control” when it is
beyond the control limits), we would
expect such data to give a false alarm on the average three or

four times out of 1,000 points. Further,
we would expect false alarms below the lower control limit to
be equally likely to occur, as would
false alarms above the upper control limit.
Figure 4 shows an individuals control chart of the randomly
generated data.
6
Figure 4: Individuals Control Chart of the Random Sample Data
The individuals control chart in figure 4 shows many out-of-
control points beyond the upper control

limit. In addition, the individuals control chart shows a physical
lower boundary of zero for the data,
which is well within the lower control limit of 22.9. If no
transformation is needed when plotting
non-normal data in a control chart, then we would expect to see
a random scatter pattern within the
control limits, which is not prevalent in the individuals control
chart.
Figure 5 shows a control chart using a Box-Cox transformation
with a lambda value of zero, the
appropriate transformation for log-normally distributed data.
This control chart is much better
behaved than the control chart in figure 4. Almost all 1,000
points in this individuals control chart
are in statistical control. The number of false alarms is
consistent with the design and definition of

the individuals control chart control limits.
7
Figure 5: Individuals Control Chart With a Box-Cox
Transformation Lambda Value of Zero
Determining actions to take
Previously three decision-making action options were described,
where the first option was:
action?

For organizations that did not consider transforming data to
address this question, as illustrated in
figure 4, many investigations would need to be made where
common-cause variability was being
reacted to as though it were special cause. This can lead to
much organizational firefighting and
frustration, especially when considered on a plantwide or
corporate basis with other control chart
metrics. If data are not from a normal distribution, an
individuals control chart can generate false
signals, leading to unnecessary tampering with the process.
For organizations that did consider transforming data to address
this question, as illustrated in
figure 5, there is no over reaction to common-cause variability
as though it were special cause.
For the transformed data analysis, let’s next address the other

questions:
required.
8
When a process has a recent region of stability, we can make a
statement not only about how the

process has performed in the stable region but also about the
future, assuming nothing will change in
the future either positively or negatively relative to the process
inputs or the process itself. However,
to do this, we need to have a distribution that adequately fits the
data from which this estimate is to
be made.
For the previous specification limit of 0.100 in., figure 6 shows
a good distribution fit and best-
estimate process capability/performance nonconformance
estimate of 0.5 percent (100.0 - 99.5). For
this situation, we would respond positively to item number two
since the percent nonconformance is
below 1 percent; i.e., we determined that the process is stable
and meeting internal and external
customer needs of a less than 1-percent nonconformance rate;
hence, no action is required.

However, from figure 6 we also note that the nonconformance
rate we expect to increase to about 6.3
percent (100–93.7) with the new specification limit of 0.35 in.
Because of this, we would now
respond positively to item number three, since the
nonconformance percentage is above the 1-
percent criterion. That is, we determined that the process is
stable but does not meet internal and
external customer needs; hence, process improvement efforts
are needed. This metric improvement
need would be “pulling” for the creation of an improvement
project.

9
Figure 6: Log-Normal Plot of Data and Nonconformance Rate
Determination for Specifications of
0.100 in. and 0.35 in.
10

Figure 7: Predictability Assessment Relative to a Specification
of 0.35 in.
It is important to present the results from this analysis in a
format that is easy to understand, such as
described in figure 7. With this approach, we demonstrate
process predictability using a control chart
in the left corner of the report-out and then use, when
appropriate, a probability plot to describe
graphically the variability of the continuous-response process
with its demonstrated predictability
statement. With this form of reporting, I suggest including a
box at the bottom of the plots that nets
out how the process is performing; e.g., with the new
specification requirement of 0.035, our process
is predictable with an approximate nonconformance rate of 6.3
percent.

A lean Six Sigma improvement project that follows a project
define-measure-analyze-improve-
control execution roadmap could be used to determine what
should be done differently in the
process so that the new customer requirements are met. Within
this project it might be determined
in the analyze phase that there is a statistically significant
difference in production machines that
now needs to be addressed because of the tightened 0.035
tolerance. This statistical difference
between machines was probably also prevalent before the new
specification requirement; however,
this difference was not of practical importance since the
customer requirement of 0.100 was being
met at the specified customer frequency level of a less than 1-
percent nonconformance rate.

11
Upon satisfactory completion of an improvement project, the
30,000-foot-level control chart would
need to shift to a new level of stability that had a process
capability/performance metric that is
satisfactory relative to a customer 1 percent maximum
nonconformance criterion.
Generalized statistical assessment
The specific distribution used in the prior example, log normal
(2.0, 1.0, 0), has an average run
length (ARL) for false rule-one errors of 28 points. The single
sample used showed 33 out-of-control

points, close to the estimated value of 28. If we consider a less
skewed log-normal distribution, log
normal (4, 0.25, 0), the ARL for false rule-one errors drops to
101. Note that a normal distribution
will have a false rule-one error ARL of around 250.
The log-normal (4, 0.25, 0) distribution passes a normality test
over half the time with samples of 50
points. In one simulation, a majority, 75 percent, of the false
rule-one errors occurred on the samples
that tested as non-normal. This result reinforces the conclusion
that normality or a near-normal
distribution is required for a reasonable use of an individuals
chart or a significantly higher false
rule-one error rate will occur.
Conclusions

The output of a process is a function of its steps and inputs
variables. Doesn’t it seem logical to
expect some level of natural variability from input variables and
the execution of process steps? If we
agree to this presumption, shouldn’t we expect a large
percentage of process output variability to
have a natural state of fluctuation, that is, to be stable?
To me this statement is true for many transactional and
manufacturing processes, with the exception
of things like naturally auto-correlated data situations such as
the stock market. However, with
traditional control charting methods, it is often concluded that
the process is not stable even when
logic tells us that we should expect stability.
Why is there this disconnection between our belief and what
traditional control charts tell us? The

reason is that often underlying control-chart-creation
assumptions are not valid in the real world.
Figures 4 and 5 illustrate one of these points where an
appropriate data transformation is not made.
The reason for tracking a process can be expressed as
determining which actions or nonactions are
most appropriate.
12
action?

required.
This article described why appropriate transformations from a
physical point of view need to be a
part of this decision-making process.
The box at the bottom of figure 7 describes the state of the
examined process in terms that everyone
can understand; i.e., the process is predictable with an estimate
6.7-percent nonconformance rate.
An organization gains much when this form of scorecard-value-
chain reporting is used throughout

its enterprise and is part of its decision-making process and
improvement project selection.
DISCUSS
ABOUT THE AUTHOR
Forrest Breyfogle—New Paradigms
CEO and president of Smarter

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