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© 2001 ConceptFlow 1
Probabilistic Design (BP)
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© 2001 ConceptFlow 2
By the end of this module, participant will beable to:
• Apply probabilistic design to business processes to achieve six sigma
capability
• Establish ideal mean and short term standard deviation for business
process
• Establish ideal means and short term standard deviations for elementsof process
• Apply capability flow up to business processes to predict capability of
newly designed process
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© 2001 ConceptFlow 3
Why Use Probabilistic Design?
Probabilistic design can be used to:
• Determine ideal mean and short term standard deviation of design (Y)
to satisfy client requirements at a six sigma level
• Determine ideal means and short term standard deviations of process
elements (Xs) to satisfy client requirements at a six sigma level• Establish operational limits for elements (Xs) of design
• Apply capability flow up to business processes to predict capability of
newly designed process
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© 2001 ConceptFlow 4
What Is Probabilistic Design?
Probabilistic design is a statistical methodology used to:
• Determine ideal mean and short term standard deviation of design (Y)
to satisfy client requirements at a six sigma level
• Determine ideal means and short term standard deviations of process
elements (Xs) to satisfy client requirements at a six sigma level• Establish operational limits for elements (Xs) of design
• Apply capability flow up to business processes to predict capability of
newly designed process
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© 2001 ConceptFlow 5
Tolerancing Steps
5. Determine ideal short term standard deviation, si, for each element, Xi, of process
4. Determine ideal mean, mi, for each element, Xi, of process
3. Determine transfer function, Y = f(X), for process
2. Determine ideal short term standard deviation, s Y, for design output variable, Y
1. Determine ideal mean, m Y, for design output variable, Y
7. Trade-off element means and short term standarddeviations to combination that is achievable
8. Flow up element means and short term standard deviations to assure
design output ideal mean and short term standard deviation are achieved.
9. Use element ideal means and short term standard deviations to
determine upper and lower operational limits for each element
6. Are ideal
means and short term standard deviations
achievable?No
Yes
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Tolerancing Steps
5. Determine ideal short term standard deviation, si, for each
element, Xi, of process
4. Determine ideal mean, mi, for each element, Xi, of process
3. Determine transfer function, Y = f(X), for process
2. Determine ideal short term standard deviation, s Y, for
design output variable, Y
1. Determine ideal mean, m Y, for design output variable, Y
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Tolerancing Steps - Continued
6. Are idealmeans and short term standard deviations
achievable?
7. Trade-off element means and
short term standard deviations to
combination that is achievable
8. Flow up element means and short term
standard deviations to assure design output ideal
mean and short term standard deviation are
achieved.
9. Use element ideal means and short term
standard deviations to determine upper and
lower operational limits for each element
Yes
No
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1. Determine Ideal Mean
Determining ideal mean is important for:
• Understanding client’s target value for the design output variable
• Determining process element means
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1. Determine Ideal Mean
• If client specifies a target value for design output variable (Y), thistarget becomes ideal mean.
mYideal= target value
• If a target value is not specified, but only upper and lower operational
limits, convention will be to use mid-point of upper and lower limits.
mYideal= (upper limit + lower limit) / 2
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2. Determine Ideal Standard Deviation
Determining ideal short term standard deviation is important for:
• Understanding client’s acceptable level of variability in design output
variable
• Determining process element short term standard deviations
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2. Determine Ideal Standard Deviation
• Clients typically can specify a target value for design output variable.However, it is not typical that clients can specify acceptable variability
or standard deviation of output design variable.
• To determine ideal standard deviation of design output variable, ideal
short term standard deviation will be used.
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2. Determine Ideal Standard Deviation
• To calculate ideal short term standard deviation for Y, use followingformulas:
sYideal = (upper limit – mYideal
) / 6
or,
sYideal = (mYideal
- lower limit ) / 6
• Divisor is 6 in order to have a six sigma process, using short term
standard deviation.
• If specification is asymmetric, use smaller short term standarddeviation.
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3. Determine Transfer Function
Determining transfer function is important for:
• Understanding relationship between design output variable and
process elements
• Flowing down requirements for process elements
• Determining ideal means for process elements
• Determining ideal short term standard deviations for process elements
• Flowing up capability of current process
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3. Determine Transfer Function
Transfer functions can be determined by using:
• Basic principles
• Fundamental business equations
• Fundamental scientific equations
• Design of experiments
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4. Determine Ideal Element Means
Determining ideal element means is important for:
• Knowing where to center process elements in order to achieve
proper centering of design output variable
• Establishing operational limits for process elements
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4. Determine Ideal Element Means
• To determine ideal element means, use transfer function as follows:
mYideal= f(m1, m2, …, mk).
• Establish m1, m2, …, mk such that through transfer function, mYidealis
achieved.
• This is sometimes known as Flowing Down mean requirements.
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4. Determine Ideal Element Means
• Determining ideal element means may not be simple. There are manydifferent ways to configure element means to achieve proper centering
of output. Some considerations are
• Process knowledge
• Current process capability• Benchmark levels
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5. Determine Ideal Element Short TermStandard Deviations
Determining ideal element short term standard deviations isimportant for:
• Achieving acceptable level of variability in design output variable
• Establishing operational limits for process elements
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5. Determine Ideal Element Short TermStandard Deviations
• To determine ideal element short term standard deviations, usemethod of Root Sum of Squares.
sY = Sqrt ( s12 + s2
2 + … + sk2 )
• This is sometimes known as Flowing Down standard deviationrequirements.
Technical note: Root sum of squares method is valid only for
linear transfer functions. For non-linear transfer functions, use
Monte Carlo simulation analysis.
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5. Determine Ideal Element Short TermStandard Deviations
• Determining ideal element short term standard deviations may not besimple. There are many different ways to configure element short term
standard deviations to achieve proper variability of output. Some
considerations are
• Process knowledge• Current process capability
• Benchmark levels
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6. Are Ideal Means and Standard Deviations Achievable?
Once ideal element means and short term standard deviations havebeen determined, it is crucial that they are achievable. Items to
consider are
• Process knowledge
• Current process capability• Benchmark levels
• Process entitlement
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7. Trade-Off Element Means and Short TermStandard Deviations
If ideal element means and short term standard deviations are notachievable, then trade-offs, or changes, must be made in ideal values
to achieve six sigma capability in design output variable.
These trade-offs can be changes in:
• Means• Short term standard deviations
• Both
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8. Flow Up Element Means and Short TermStandard Deviations
• As changes are made to element means, short term standarddeviations, or both, it is crucial to use these new combinations in
transfer function to assure design output ideal mean and standard
deviation are achieved.
• Using trade-off means and standard deviations in transfer functions issometimes called Flow Up.
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9. Determine Upper and Lower OperationalLimits
Determining upper and lower operational limits for process elements isimportant for:
• Establishing operational range for process elements
• Assuring six sigma capability of design output variable
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9. Determine Upper and Lower OperationalLimits
• Once element ideal means and ideal short term standard deviationsare determined (perhaps, after trade-off analysis), operational limits for
each process element can be established as:
• Ideal Element Mean +/- Tolerance
or, mi +/- Toli
where, Toli = 6 * si
si
= short term standard deviation
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9. Determine Upper and Lower OperationalLimits
• With element operational limits set as:
mi +/- 6 * si
• design output variable is now assured of achieving six sigma
capability, even if shifting of element means occurs over time.
T l i St
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Tolerancing Steps
5. Determine ideal short term standard deviation, si, for each element, Xi, of process
4. Determine ideal mean, mi, for each element, Xi, of process
3. Determine transfer function, Y = f(X), for process
2. Determine ideal short term standard deviation, s Y
, for design output variable, Y
1. Determine ideal mean, m Y, for design output variable, Y
7. Trade-off element means and short term standarddeviations to combination that is achievable
8. Flow up element means and short term standard deviations to assure
design output ideal mean and short term standard deviation are achieved.
9. Use element ideal means and short term standard deviations to
determine upper and lower operational limits for each element
6. Are ideal
means and short term standard deviations
achievable?No
Yes
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Class Example
• A new order entry process is to be designed. client requirement is 10days +/- 3 days.
• There are five steps to process:
• Order entry
• Picking at site A
• Transfer to site B
• Packing at site B
• Ship to client
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Flow Chart for Class Example
Order
Entry
Pick
site A
Trans
site B
Ship to
client
Pack
site B
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Class Example
• From client requirement of 10 days +/- 3 days total time, operationallimits can be determined.
• Thus:
Upper operational limit: 13 daysTarget: 10 days
Lower operational limit: 7 days
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1. Determine Ideal Mean
•Since client specified a target value, ideal mean is target value:
mYideal= target value
mYideal= 10.0 days
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2. Determine Ideal Standard Deviation
• To calculate ideal short term standard deviation, use either upper or lower limit formulas:
sYideal = (upper limit – mYideal
) / 6
= (13 days – 10 days) / 6
= .5 days
or,
sYideal = (mYideal
- lower limit ) / 6
= (10 days – 7 days ) / 6
= .5 days
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Ideal Mean and Standard Deviation
• For our new design, ideal mean and ideal short term standarddeviation are established, such that process will satisfy client at a six
sigma level:
mYideal= 10.0 days
sYideal = .5 days
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3. Determine Transfer Function
• From flow chart of class example, total time is sum of five elementtimes. The transfer function can be determined from basic principles.
• Y = tOE + tPA + tTB + tPB + tSC
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4. Determine Element Means
• To calculate ideal means for five element steps, use transfer function
with following mean times. These times were taken from
benchmarking best-in-class.
Step Mean
Order Entry 0.5Pick at Site A 3.0
Transfer to Site B 1.5
Pack at Site B 2.0
Ship to Client 3.0
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4. Determine Element Means
• Entering ideal mean times in transfer function yields client target of 10
days.
Y = tOE + tPA + tTB + tPB + tSC
mYideal = m1 + m2 + m3 + m4 + m5
= 0.5 + 3.0 + 1.5 + 2.0 + 3.0
= 10.0 days
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5. Determine Element Short Term StandardDeviations
• To calculate ideal short term standard deviations for five element
steps, use root sum of squares method with following short term
standard deviations. These short term standard deviations were
determined from benchmarking best-in-class.
Standard Deviation
Step Hours DaysOrder Entry 2 .08
Pick at Site A 4 .17
Transfer to Site B 4 .17
Pack at Site B 4 .17
Ship to Client 8 .33
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5. Determine Element Short Term StandardDeviations
• Entering ideal short term standard deviations in root sum of squares
formula yields required ideal total short term standard deviation of .5
days (or slightly less).
sY = Sqrt ( s12 + s2
2 + s32 + s4
2 + s52 )
= Sqrt (.082 + .172 + .172 + .172 + .332 )= Sqrt ( .0064 + .0289 + .0289 + .0289 + .1089)
= Sqrt (.2020)
= .4494 days
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© 2001 ConceptFlow 39
6. Are Ideal Means and Short Term StandardDeviations Achievable
• Based on previous process knowledge it is determined that means and
short term standard deviations are achievable. The following table is
a summary of element requirements:
Ideal
Step Mean SDOrder Entry 0.5 0.08
Pick at Site A 3.0 0.17
Transfer to Site B 1.5 0.17
Pack at Site B 2.0 0.17
Ship to Client 3.0 0.33
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7. Trade-Off Element Means and Short TermStandard Deviations
• Had it been determined that element means and short term standard
deviations were not achievable, trade-offs must be done. Trade-off
means and standard deviation would be entered into the following
formulas and flowed up to assure that ideal design mean (10.0) and
standard deviation (0.5) are met.
mYideal= m1 + m2 + m3 + m4 + m5
sY = Sqrt ( s12 + s2
2 + s32 + s4
2 + s52 )
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8. Flow Up Element Means and Short TermStandard Deviations
• Achievable means and short term standard deviations have been
determined. These must be flowed up to assure ideal design mean
and short term standard deviation are met.
• mYideal= m1 + m2 + m3 + m4 + m5
• = 0.5 + 3.0 + 1.5 + 2.0 + 3.0• = 10.0 days
sY = Sqrt ( s12 + s2
2 + s32 + s4
2 + s52 )
• = Sqrt (.082 + .172 + .172 + .172 + .332 )
• = .4494 days
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9. Determine Upper and Lower OperationalLimits
• With element ideal means and ideal short term standard deviations
determined, tolerances for each element can be established as:
Ideal Mean +/- Tolerance
mi +/- 6 * si
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9. Determine Upper and Lower OperationalLimits
Summary table of tolerance calculations:
Ideal
Step SD 6 * SD
Order Entry 0.08 0.48
Pick at Site A 0.17 1.02
Transfer to Site B 0.17 1.02
Pack at Site B 0.17 1.02
Ship to Client 0.33 2.00
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9. Determine Upper and Lower OperationalLimits
• The following table is a summary of element operational requirements,
with ideal mean, ideal short term standard deviation, and tolerance:
Ideal
Step Mean SD Tol
Order Entry 0.5 0.08 0.48Pick at Site A 3.0 0.17 1.02
Transfer to Site B 1.5 0.17 1.02
Pack at Site B 2.0 0.17 1.02
Ship to Client 3.0 0.33 2.00
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Class Exercise
• A new order fulfillment system is to be designed.
• Client requirement is to fill order in 18 +/- 6 hours.
• There are three steps to process:
• Order entry
• Item collection
• Ship to client
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Flow Chart for Class Exercise
CollectItems
Ship toClient
Order Entry
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© 2001 ConceptFlow 47
Data for Process Elements
Data values of current mean times (in hours) and short term standard
deviations for each element are recorded below:
Estimates
Step Mean SD
Order Entry 2.0 0.2
Item Collection 8.0 0.8Ship to Client 10.0 1.0
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Class Exercise
• Use nine steps for statistical tolerancing to determine element
operational limits so that total time will meet client requirements at a six
sigma level.
Tolerancing Steps
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Tolerancing Steps
5. Determine ideal short term standard deviation, si, for each element, Xi, of process
4. Determine ideal mean, mi, for each element, Xi, of process
3. Determine transfer function, Y = f(X), for process
2. Determine ideal short term standard deviation, s Y, for design output variable, Y
1. Determine ideal mean, m Y, for design output variable, Y
7. Trade-off element means and short term standarddeviations to combination that is achievable
8. Flow up element means and short term standard deviations to assure
design output ideal mean and short term standard deviation are achieved.
9. Use element ideal means and short term standard deviations to
determine upper and lower operational limits for each element
6. Are ideal
means and short term standard deviations
achievable?No
Yes
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Key Learning Points
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By end of this module, participant will be able to:
• Apply statistical tolerancing to business processes to achieve six
sigma capability
• Establish ideal mean and short term standard deviation for business
process
• Establish ideal means and short term standard deviations for elementsof process
• Apply capability flow up to business processes to predict capability of
newly designed process
T d k d S i M k
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Trademarks and Service Marks
Six Sigma is a federally registered trademark of Motorola, Inc.
Breakthrough Strategy is a federally registered trademark of Six Sigma Academy.
VISION. FOR A MORE PERFECT WORLD is a federally registered trademark of Six Sigma Academy.
ESSENTEQ is a trademark of Six Sigma Academy.
FASTART is a trademark of Six Sigma Academy.
Breakthrough Design is a trademark of Six Sigma Academy.
Breakthrough Lean is a trademark of Six Sigma Academy.
Design with the Power of Six Sigma is a trademark of Six Sigma Academy.
Legal Lean is a trademark of Six Sigma Academy.
SSA Navigator is a trademark of Six Sigma Academy.
SigmaCALC is a trademark of ix Sigma Academy.
SigmaFlow is a trademark of Compass Partners, Inc.
SigmaTRAC is a trademark of DuPont.
MINITAB is a trademark of Minitab, Inc.