08 Probabilistic Design_final

8/22/2019 08 Probabilistic Design_final

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© 2001 ConceptFlow 1

Probabilistic Design (BP)




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




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






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






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




Tolerancing Steps

5. Determine ideal short term standard deviation, si, for each

element, Xi, of process



2. Determine ideal short term standard deviation, s 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



1. Determine Ideal Mean

Determining ideal mean is important for:

• Understanding client’s target value for the design output variable

• Determining process element means




• 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



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




• 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.




• 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.



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





Transfer functions can be determined by using:

• Basic principles

• Fundamental business equations

• Fundamental scientific equations

• Design of experiments




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





• 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.





• 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




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





• 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.





• 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




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




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




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.




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





• 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





• 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




Tolerancing Steps




2. Determine ideal short term standard deviation, s Y

, for design output variable, Y







6. Are ideal


achievable?No

Yes




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




Flow Chart for Class Example

Order

Entry

Pick

site A

Trans

site B

Ship to

client

Pack

site B




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





•Since client specified a target value, ideal mean is target value:

mYideal= target value

mYideal= 10.0 days





• 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




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





• 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




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




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




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




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).


2 + s32 + s4

2 + s52 )

= Sqrt (.082 + .172 + .172 + .172 + .332 )= Sqrt ( .0064 + .0289 + .0289 + .0289 + .1089)

= Sqrt (.2020)

= .4494 days




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




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


2 + s32 + s4

2 + s52 )




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


2 + s32 + s4

2 + s52 )

• = Sqrt (.082 + .172 + .172 + .172 + .332 )

• = .4494 days





• 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





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





• 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




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




Flow Chart for Class Exercise

CollectItems

Ship toClient

Order Entry




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




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




Tolerancing Steps




2. Determine ideal short term standard deviation, s Y, for design output variable, Y







6. Are ideal


achievable?No

Yes




Key Learning Points




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



T d k d S i M k



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08 Probabilistic Design_final