introduction to dmaic - english (139 pages)

7/31/2019 Introduction to DMAIC - English (139 Pages)

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What is Six Sigma?

It is a business process that allows companies to

drastically improve their bottom line by designing and

monitoring everyday business activities in ways that

minimize waste and resources while increasing customer

satisfaction.

Mikel Harry, Richard Schroeder


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What Six Sigma Can Do For Your Company?

5.154.7

3

2

3

4

5

6

0 1 2 3

years of implementation

Sigma level4.8

D

F

S

S

MAIC

Average company


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SIGMA LEVEL DEFECTS PER MILLION OPPORTUNITIES COST OF QUALITY

2 308,537 ( Noncompetitive companies ) Not applicable

3 66,807 25-40% of sales4 6,210 ( Industry average ) 15-25% of sales

5 233 5-15 of sales

6 3.4 ( World class ) < 1% of sales

Each sigma shift provides a 10 percent net income improvement

THE COST OF QUALITY

What Six Sigma Can Do For Your Company?


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Traditional Cost ofPoor Quality (COQ)

Rework

InspectionWarrantyRejects

5-8%

Lost Opportunity

15-20%Less Obvious Cost of

Quality (COQ)

Set up

The Cost of Quality (COQ)

Note: % of sales


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C

O

R

E

P

H

AS

E

DMAIC : The Yellow Brick Road


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Definition ofOpportunity

Assess the Current Process Confirm f(x)for Y Optimize f(x) for Y Maintain Improvements Sustain the Benefit

1. Project Definition2. Determine

Impact & Priority3. Collect Baseliine

Metric Data4. Savings/Cost

Assessment5. Establish

PlannedTimeline

6. Search Library7. Identify Project

Authority

1. Map the Process2. Determine the Baseline3. Prioritize the Inputs to

Assess4. Assess the

Measurement System5. Capabili ty Assessment6. Short Term7. Long Term8. Determine Entitlement9. Process Improvement10. Financial Savings

1. Determine the VitalVariables Affectingthe Responsef(x) = Y

2. ConfirmRelationships andEstablish the KPIV

1. Determine the BestCombination of Xsfor Producing theBest Y

1. Establish Controls for2. KPIVs and their

settings3. Establish Reaction

Plans

1. FinancialAssessment andInput ActualSavings

2. FunctionalManager/ProcessOwner Monitor

3. Control/Implementation

Breakthrough & People

DEFINE ->>>>>>>>>>>

MEASURE ->>>>>>>>>>>>>>>

ANALYZE ->>>>>>>>>>>>>

IMPROVE ->>>>>>>>>>>>

CONTROL->>>>>>>>>>>>>>

REALIZATION -

>>>>>>>>>>>>>

Champion BlackbeltsFinance Rep.&

Process Owner


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Define

What is my biggest problem? Customer complaints

Low performance metrics

Too much time consumed

What needs to improve?

Big budget items

Poor performance

Where are there opportunities to improve? How do I affect corporate and business group objectives?

Whats in my budget?


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Projects DIRECTLY tie to department and/or business unitobjectives

Projects are suitable in scope

BBs are fit to the project

Champions own and support project selection

Define : The Project


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High Defect Rates

Low Yields

Excessive Cycle Time

Excessive Machine Down Time

High Maintenance Costs

High Consumables Usage

Rework

Customer Complaints

Excessive Test and Inspection

Constrained Capacity with High

anticipated Capital Expenditures

Bottlenecks

Define : The Defect


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Time

RejectRate

Special Cause ( )

Optimum Level

(Chronic)

Define : The Chronic Problem


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0

2

4

6

8

10

12

14

WW01 WW02 WW03 WW04 WW05 WW06 WW07 WW08 WW09 WW10 WW11 WW12

0

5

10

15

20

25


0

2

4

6

8

10

12

14


0

5

10

15

20

25

30

35

40


Is process in control?

Define : The Persistent Problem


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Define : Refine The Defect

Assembly Yield Loss

PSA RSA GramLoad

BentGimbal

SolderDefect

Contam DamperDefect

KPOV

%Y

ieldLoss

a2 a3 a4 a5 a6 a7a1

Refined Defect = a1


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30 - 50

10 - 15

4-8

Potential Key Process

Input Variables (KPIVs)

8 - 10KPIVs

Optimized KPIVs

3-6Key LeverageKPIVs

Inputs Variables

Process Map

Multi-VariStudies,

Correlations

ScreeningDOEs

DOEs, RSM

C&E Matrix and FMEA

Gage R&R, Capability

T-Test, ANOM, ANOVA

Quality Systems

SPC, Control Plans

Measure

Analyze

Improve

Control

MAIC --> Identify Leveraged KPIVs

Tools Outputs


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Measure

TheMeasure phase serves to validate the problem, translate the

practical to statistical problem and to begin the search for root causes


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Measure : Tools

To validate the problem

Measurement System Analysis

To translate practical to statistical problem

Process Capability Analysis

To search for the root cause

Process Map Cause and Effect Analysis

Failure Mode and Effect Analysis


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Work shop #1:

Our products are the distance resulting from the Catapult.

Product spec are +/- 4 Cm. for both X and Y axis

Shoot the ball for at least 30 trials , then collect yield

Prepare to report your result.


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Objectives:

Validate the Measurement / Inspection System

Quantify the effect of the Measurement System variability onthe process variability

Measure : Measurement System Analysis


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To determine if inspectors across all shifts, machines, lines,

etc use the same criteria to discriminategood from bad

To quantify the ability of inspectors or gages to accurately

repeattheir inspection decisions

To identify how well inspectors/gages conform to a known

master (possibly defined by the customer) which includes:

How often operators decide to over reject

How often operators decide to over accept

Attribute GR&R : Purpose


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% REPEATIBILITY OF OPERATOR # 1 = 16/20 = 80%



% Appraiser Score


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% UNBIAS OF OPERATOR # 1 = 12/20 = 60%

% Attribute Score



% Screen Effective Score

% REPEATABILITY OF INSPECTION = 11/20 = 55 %

% Attribute Screen Effective Score

% UNBIAS OF INSPECTION 50 % = 10/20 = 50%



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Study of your measurement system will reveal the relative amount of

variation in your data that results from measurement system error.

It is also a great tool for comparing two or more measurement devicesor two or more operators.

MSA should be used as part of the criteria for accepting a new piece of

measurement equipment to manufacturing.

It should be the basis for evaluating a measurement system which is

suspect of being deficient.


Variable GR&R : Purpose


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Long-Term

Process

Varaition

Short-Term

Process

Variation

Variation

Within

Sample

Actual Variation

Repeatability Reproducibility

Precision Stability Linearity Accuracy

Variation

due to

Gage

Measurement Variation

Observed Variation



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Resolution?

Precision (R&R) Calibration? Stability?

Linearity?Bias?


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Measurement System Variance:

s2meas = s2

repeat + s2

reprod

To determine whether the measurement system is good or bad for a certain

application, you need to compare the measurement variation to the product spec

or the process variation

Comparings2measwith Tolerance: Precision-to-Tolerance Ratio (P/T)

Comparings2measwith Total Observed Process Variation (P/TV): % Repeatability and Reproducibility (%R&R)

Discrimination Index

Measurement System Metrics


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Uses of P/T and P/TV (%R&R)

The P/T ratio is the most common estimate of measurement

system precision Evaluates how well the measurement system can perform

with respect to the specifications

The appropriate P/T ratio is strongly dependent on the

process capability. If Cpk is not adequate, the P/T ratio

may give a false sense of security.

The P/TV (%R&R) is the best measure for Analysis

Estimates how well the measurement system performs withrespect to the overall process variation

%R&R is the best estimate when performing process

improvement studies. Care must be taken to use samples

representing full process range.


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Number of Distinct Categories

Automobile Industry Action Group (AIAG) recommendations:Categories Remarks

< 2 System cannot discern one part from another

= 2 System can only divide data in two groups

e.g. high and low

= 3 System can only divide data in three groups

e.g. low, middle and high

4 System is acceptable


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Variable GR&R : Decision Criterion

% Bias % Linearity DR %P/T %Contribution

BEST < 5 < 5 > 10 < 10 < 2

ACCEPTABLE 5 - 10 5 - 10 5 - 10 10-30 2-7.7

REJECT > 10 > 10 < 5 > 30 > 7.7

Note : Stability is analyzed by control chart


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ANOVA method is preferred.

Enter the data and tolerance information into Minitab.

Stat > Quality Tools > Gage R&R Study (Crossed )

FN: Gageaiag.mtw

Enter Gage Infoand Options.

(see next page)

Example: Minitab


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Enter the data and tolerance information into Minitab.

Stat > Quality Tools > Gage R&R Study

Gage Info (see below) & Options


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Gage name:

Date of study:

Reported by:

Tolerance:

Misc:

0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1 1 2 3

Xbar Chart by Operator

SampleMean

Mean=0.8075UCL=0. 8796

LCL=0.7354

0

0.00

0.05

0.10

0.15 1 2 3

R Chart by Operator

SampleRange

R= 0.03833

UCL=0. 1252

LCL=0

1 2 3 4 5 6 7 8 9 10

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Part

OperatorOperator* Part I nteraction

Average

1

2

3

1 2 3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Operator

By Operator

1 2 3 4 5 6 7 8 9 10

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Part

By Part

%Contribution

%Study Var

%Tolerance

Gage R&R Repeat Reprod Part -t o-Part

0

50

100

Components of Variation

Percent

Gage R&R (ANOVA) for Response

Gage R&R Output


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Gage R&R, Variation Components

%Contribution

Source VarComp (of VarComp)

Total Gage R&R 0.004437 10.67

Repeatability 0.001292 3.10

Reproducibility 0.003146 7.56

Operator 0.000912 2.19

Operator*PartID 0.002234 5.37

Part-To-Part 0.037164 89.33

Total Variation 0.041602 100.00

StdDev Study Var %Study Var %Tolerance

Source (SD) (5.15*SD) (%SV) (SV/Toler)

Total Gage R&R 0.066615 0.34306 32.66 22.87

Repeatability 0.035940 0.18509 17.62 12.34

Reproducibility 0.056088 0.28885 27.50 19.26

Operator 0.030200 0.15553 14.81 10.37

Operator*PartID 0.047263 0.24340 23.17 16.23

Part-To-Part 0.192781 0.99282 94.52 66.19

Total Variation 0.203965 1.05042 100.00 70.03

Variance due to the measurement system (broken down into

repeatability and reproducibility)

Total variance

Variance due

to the parts

Standard deviation foreach variance component

Gage R&R Results


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Gage R&R, Results

%Contribution

Source VarComp (of VarComp)

Total Gage R&R 0.004437 10.67

Repeatability 0.001292 3.10

Reproducibility 0.003146 7.56

Operator 0.000912 2.19

Operator*PartID 0.002234 5.37

Part-To-Part 0.037164 89.33

Total Variation 0.041602 100.00

StdDev Study Var %Study Var %Tolerance

Source (SD) (5.15*SD) (%SV) (SV/Toler)

Total Gage R&R 0.066615 0.34306 32.66 22.87

Repeatability 0.035940 0.18509 17.62 12.34

Reproducibility 0.056088 0.28885 27.50 19.26

Operator 0.030200 0.15553 14.81 10.37

Operator*PartID 0.047263 0.24340 23.17 16.23

Part-To-Part 0.192781 0.99282 94.52 66.19

Total Variation 0.203965 1.05042 100.00 70.03

326600504134300 .

.

.

StudyVarTV/Ptotal

meas

s

s

2287.05.1

3430.0

*15.5/

TolLSLUSL

TP MSs

1067.0041602.0

004437.0

2

2

total

MSonContributi

s

s

Question: What is our conclusion about the measurement system?


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Process capability is a measure of how well the process is

currently behaving with respect to the output specification.

Process capability is determined by the total variation that

comes from common causes -the minimum variation that can be

achieved after all special causes have been eliminated.

Thus, capability represents the performance of the process

itself,as demonstrated when the process is being operated in a

state of statistical control

Measure : Process Capability Analysis


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USLLSLLSL USL

Off-TargetVariationLarge

Characterization


Translate practical problem to statistical problem

LSL USL

Outliers


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Two measures of process capability

Process Potential

Cp

Process Performance

Cpu

Cpl

Cpk

Cpm



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s

6

LSLUSL

ToleranceNatural

TolerancegEngineerinC

p


Process Potential


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The Cp index compares the allowable spread (USL-LSL)against the process spread (6s).

It fails to take into account if the process is centered between

the specification limits.

Process is centered Process is not centered



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Rev. 1 12/98

CapabilityCapability

StudiesStudiesEntitlement

(Short Term)

Performance(Long Term)

Type of Variability Only common cause All causes

# of Data Points 25-50 subgroups 200 points

ProductionExample

(Lumen Output):

-1 lot of raw matl-1 shift; 1 set of people

-Single set-up

-3 to 4 lots of raw matl-All shifts; All people

-Over Several set-ups

CommercialExample

(Response Time):

-Best Cust. Serv. Rep.

-1 Customer (i.e., Grainger)-1 month in the summer

-All Cust. Serv. Reps

-All Customers-Several months

including Dec/Jan

Rule of Thumb:Poor Mans --

Best 2 weeksHistorical data

Process:Running like it was designed

or intended!

Running like it

actually does!

There are 2 kind of variation : Short term Variation and Long term Variation


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Short Term VS LongTerm ( Cp Vs Pp or Cpk vs Ppk )


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Process Potential VS. Process Performance ( Cp Vs Cpk )

1.If Cp > 1.5 , it means the standard deviation is suitable

2.Cp is not equal to Cpk, it means that the process mean is off-centered


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Workshop#3

1. Design the appropriate check sheet

2. Define the subgroup

3. Shoot the ball for at least 30 trials per subgroup

4. Perform process capability analysis, translate Cp, Cpk , Pp

and Ppk into statistical problem

5. Report your results.


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Measure : Process Map

Process Mapis a graphical representation of the flow of a as-is

process. It contains all the major steps and decision points in a

process.

It helps us understand the process better, identify

the critical or problems area, and identify where improvement

can be made.


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OPERATION

All steps in the process where the objectundergoes a change in form or condition.

TRANSPORTATION

All steps in a process where the object moves fromone location to another, outside of the Operation

STORAGE

All steps in the process where the object remainsat rest, in a semi-permanent or storage condition

DELAY

All incidences where the object stops or waits on a

an operation, transportation, or inspection

INSPECTION

All steps in the process where the objects arechecked for completeness, quality, outside of theOperation.

DECISION



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Good

BadBad

Scrap

Warehouse

How many Operational Steps are there?

How many Decision Points? How many Measurement/Inspection Points?

How many Re-work Loops?

How many Control Points?

Good



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Major Step Major StepMajor Step

KPIVsKPIVs KPIVs

KPOVs KPOVs KPOVs

These KPIVs and KPOVs can then be used as inputs to

Cause and Effect Matrix


High Level Process Map


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Workshop #2 : Do the process map and report the

process steps and KPIVs that may be the cause


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Measure : Cause and Effect Analysis

A visual tool used to identify, explore and graphically display, in increasing

detail, all the possible causes related to a problem

or condition to discover root causes

To discover the most probable causes for further analysis

To visualize possible relationships between causes for any problem current or

future

To pinpoint conditions causing customer complaints, process errors or non-conforming products

To provide focus for discussion

To aid in development of technical or other standards or process improvements


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1. Fishbone Diagram - traditional approach to brainstorming and

diagramming cause-effect relationships. Good tool when

there is one primary effect being analyzed.

2. Cause-Effect Matrix - a diagram in table form showing the

direct relationships between outputs (Ys) and inputs (Xs).

Measure : Cause and Effect Matrix

There are two types of Cause and Effect Matrix


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C = Control Factor

N = Noise Factor

X = Factor for DOE (chosen later)

MethodsMaterials

Machinery Manpower

Problem/Desired

Improvement

C/N/X

C

C

C

N N

NNN

C

C


Fishbone Diagram


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Rating of

Importance to

Customer

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Requ

irement

Total

Process Step Process Input

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

10 0

11 0

12 0

13 014 0

15 0

16 0

17 0

18 0

19 0

20 0

0

Total 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Lower Spec

TargetUpper Spec

Cause and Effect

Matrix



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Workshop #4:

Team brainstorming to create the fishbone diagram


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FMEA is a systematic approach used to examine potential

failures and prevent their occurrence. It enhances an

engineers ability to predict problems and provides a system

of ranking, or prioritization, so the most likely failure modes

can be addressed.

Measure : Failure Mode and Effect Analysis


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RPN = S x O x D

Severity ( ) X

Occurrence () X

Detection ()



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(Vital Few)

(Trivial Many)



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Workshop # 5 :

Team Brainstorming to create FMEA


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Check and fix the measurement system

Determine where you are

Rolled throughput yield, DPPM

Process Capability

Entitlement

Identify potential KPIVs

Process Mapping / Cause & Effect / FMEA

Determine their likely impact

Measure : Measure Phases Output


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Analyze

TheAnalyze phase serves to validate the KPIVs, and to study the

statistical relationship between KPIVs and KPOVs

l l


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Analyze : Tools

To validate the KPIVs

Hypothesis Test

2 samples t test

Analysis Of Variances

etc.

To reveal the relationship between KPIVs and KPOVs

Regression analysis

Correlation

A l H h i T i


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The Null Hypothesis

Statement generally assumed to be true unless sufficientevidence is found to the contrary

Often assumed to be the status quo, or the preferred outcome.However, it sometimes represents a state you strongly want to

disprove.

Designated as H0

Analyze : Hypothesis Testing

A l H h i T i


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The Alternative Hypothesis

Statement generally held to be true if the null hypothesis is

rejected

Can be based on a specific engineering difference in acharacteristic value that one desires to detect

Designated as HA

A l H th i T ti


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NULL HYPOTHESIS: Nothing has changed:

For Tests Of Process Mean: H0: =

0

For Tests Of Process Variance: H0:s2 = s2

0

ALTERNATE HYPOTHESIS: Change has occurred:


MEAN VARIANCE

INEQUALITY Ha: 0 Ha:2 20

NEW OLD Ha: 0 Ha:2 20

NEW OLD Ha: 0 Ha:2 20

A l H th i T ti


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Collect and Analyze Data (in Minitab)

Result P-Value 0.05 Do not reject Ho

P-Value < 0.05 Reject Ho

State the practical problem

Common Language Statistical Language

Ho A is the same as B A=B

Ha A is not same as B A>B (or) A = B (or) A


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See Hypothesis Testing Roadmap

Example: Single Mean Compared to Target


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The example will include 10

measurements of a random sample:55 57 58 54 53

56 55 54 54 53

The question is: Is the mean of the samplerepresentative of a target value of 54?

The Hypotheses:

Ho: = 54Ha: 54

Ho can be rejected if p < .05

Single Mean to a Target - Using Minitab


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One-Sample T: C1

Test of mu = 54 vs mu not = 54

Variable N Mean StDev SE Mean

C1 10 54.900 1.663 0.526

Variable 95.0% CI T P

C1 ( 53.710, 56.090) 1.71 0.121

Stat > Basic Statistics > 1-Sample t

Our Conclusion Statement


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Because the p value was greater than our critical confidence level

(.05 in this case), or similarly, because the confidence interval on

the mean contained our target value, we can make the following

statement:

We have insufficient evidence to reject the null hypothesis.

Does this say that the null hypothesis is true (that the true

population mean = 54)? No!

However, we usually then choose to operate under the assumption

that Ho is true.

Single Std Dev Compared to Standard


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A study was performed in order to evaluate the effectiveness of two

devices for improving the efficiency of gas home-heating systems.

Energy consumption in houses was measured after 2 device

(damper=1& damper =2) were installed. The energy consumption

data (BTU.In) are stacked in one column with a grouping column

(Damper) containing identifiers or subscripts to denote the

population. You are interested in comparing the variances of the two

populations to the current (s=2.4).

All Rights Reserved. 2000 Minitab, Inc.

Example: Single Std Dev Compared to Standard


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Example: Single Std Dev Compared to Standard

(Data: Furnace.mtw, Use BTU_in)

Note: Minitab does not provide anindividual c2 test for standarddeviations. Instead, it is necessary tolook at the confidence interval on the

standard deviation and determine ifthe CI contains the claimed value.

Example: Single Standard Deviation


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Stat > Basic Statistics > Display Descriptive Statistics


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4 7 10 13 16

95% Confidence Interval for Mu

9 10 11

95% Confidence Interval for Median

Variable: BTU.In

A-Squared:P-Value:

MeanStDevVariance

SkewnessKurtosisN

Minimum1st QuartileMedian3rd Quartile

Maximum

8.9419

2.4738

8.6170

0.4750.228

9.907753.019879.11960

0.7075240.783953

40

4.00007.88509.5900

11.5550

18.2600

10.8736

3.8776

10.3212

Damper: 1

Anderson-Darling Normality Test


95% Confidence Interval for Sigma


Descriptive StatisticsRunning the Statistics.

Running the Statistics.


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4 7 10 13 16


9 10 11


Variable: BTU.In

A-Squared:P-Value:

MeanStDevVariance

SkewnessKurtosisN

Minimum1st QuartileMedian3rd QuartileMaximum

9.3566

2.3114

8.7706

0.1900.895

10.14302.7670

7.65640

-9.9E-02-2.7E-01

50

2.97008.1275

10.290012.212516.0600

10.9294

3.4481

11.2363

Damper: 2

Anderson-Darling Normality Test


95% Confidence Interval for Sigma


Descriptive Statistics

Two Parameter Testing


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Step 1: State the Practical Problem

Step 2: Are the data normally distributed?

Step 3: State the Null Hypothesis:

For For

Ho: pop1= pop2 Ho: pop1 = pop2(normal data)

Ho: M1 = M2 (non-normal data)

State the Alternative Hypothesis:

For For

Ha: pop1 pop2 Ha: pop1 pop2

Ha: M1 M2 (non-normal data)

Means: 2 Sample t-test

Sigmas: Homog. Of Variance

Medians: Nonparametrics

Failure Rates: 2 Proportions

Two Parameter Testing (Cont.)


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Step 4: Determine the appropriate test statistic

F (calc) to test Ho: pop1 = pop2 T (calc) to test Ho: pop1 = pop2 (normal data)

Step 5: Find the critical value from the appropriate distributionand alpha

Step 6: If calculated statistic > critical statistic, then REJECT Ho.

Or

If P-Value < 0.05 (P-Value < Alpha), then REJECT Ho.

Step 7: Translate the statistical conclusion into process terms.

Comparing Two Independent Sample Means


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The example will make a comparison betweentwo group means

Data in Furnace.mtw ( BTU_in)

Are the mean the two groups the same?

The Hypothesis is:

Ho: 12

Ha : 1

2

Reject Ho if t > t a/2 or t < -t a/2 for n1 +n2 - 2 degrees of freedom


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t-test Using Stacked DataStat >Basic Statistics > 2-Sample t

t-test Using Stacked Data


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Descriptive Statistics Graph: BTU.In by Damper

Two-Sample T-Test and CI: BTU.In, Damper

Two-sample T for BTU.In

Damper N Mean StDev SE Mean

1 40 9.91 3.02 0.48

2 50 10.14 2.77 0.39

Difference = mu (1) - mu (2)

Estimate for difference: -0.235

95% CI for difference: (-1.464, 0.993)

T-Test of difference = 0 (vs not =): T-Value = -0.38 P-Value = 0.704 DF = 80


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2 variances testStat >Basic Statistics > 2 variances

2 variances test


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2 3 4

95% Confidence Intervals for Sigmas

2

1

4 9 14 19

Boxplots of Raw Data

BTU.In

F-Test

Test Statistic: 1.191

P-Value : 0.558

Levene's Test


P-Value : 0.996

Factor Levels

1

2

Test for Equal Variances for BTU.In

Characteristics About Multiple Parameter Testing


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One type of analysis is called Analysis of Variance (ANOVA).

Allows comparison of two or more process means.

We can test statistically whether these samples represent a single population,

or if the means are different.

The OUTPUT variable (KPOV) is generally measured on a continuous scale(Yield, Temperature, Volts, % Impurities, etc...)

The INPUT variables (KPIVs) are known as FACTORS. In ANOVA, the

levels of the FACTORS are treated as categorical in nature even though they

may not be.

When there is only one factor, the type of analysis used is called One-Way

ANOVA. For 2 factors, the analysis is called Two-Way ANOVA. And n

factors entail n-Way ANOVA.

General Method


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Step 1: State the Practical Problem

Step 2: Do the assumptions for the model hold? Response means are independent and normally distributed

Population variances are equal across all levels of the factor

Run a homogeneity of variance analysis--by factor levelfirst

Step 3: State the hypothesisStep 4: Construct the ANOVA TableStep 5: Do the assumptions for the errors hold (residual analysis)?

Errors of the model are independent and normally distributed

Step 6: Interpret the P-Value (or the F-statistic) for the factor effect P-Value < 0.05, then REJECT Ho

Otherwise, operate as if the null hypothesis is true

Step 7: Translate the statistical conclusion into process terms

Step 2: Do the Assumptions for the Model Hold?


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Are the means independent and normally

distributed

Randomize runs during the experiment

Ensure adequate sample sizes

Run a normality test on the data by level

Minitab: Stat > Basic Stats > Normality Test

Population variances are equal for each factor level(run a homogeneity of variance analysis first)

Fors Ho: spop1 = spop2 = spop3 = spop4 = ..

Ha: at least two are different

Step 3: State the Hypotheses


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Ho: s = 0

Ha: k 0

Ho: 1 = 2 = 3 = 4

Ha: At least one k is different

Mathematical Hypotheses:

Conventional Hypotheses:

Step 4: Construct the ANOVA Table


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SOURCE SS df MS Test Statistic

Between SStreatment g - 1 MStreatment = SStreatment / (g-1) F = MStreatment / MSerror

Within SSerror N - g MSerror = SSerror / (N-g)

Total SStotal N - 1

Where:

g = number of subgroups

n = number of readings per subgroup

One-Way Analysis of Variance

Analysis of Variance for TimeSource DF SS MS F POperator 3 149.5 49.8 4.35 0.016Error 20 229.2 11.5

Total 23 378.6

Whats important the probability

that the Operator variation in means

could have happened by chance.

Steps 5 - 7


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Step 5:Do the assumptions for the errors hold (residual analysis) ?

Errors of the model are independent and normally distributed

Randomize runs during the experiment

Ensure adequate sample size

Plot histogram of error terms Run a normality check on error terms

Plot error against run order (I-Chart)

Plot error against model fit

Step 6:Interpret the P-Value (or the F-statistic) for the factor effect

P-Value < 0.05, then REJECT Ho.

Otherwise, operate as if the null hypothesis is true.

Step 7:Translate the statistical conclusion into process terms

Residual

Analysis

Example, Experimental Setup


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Twenty-four animals receive one of four diets.

The type of diet is the KPIV (factor of

interest). Blood coagulation time is the KPOV

During the experiment, diets were assignedrandomly to animals. Blood samples takenand tested in random order. Why ?

DIET A DIET B DIET C DIET D

62 63 68 56

60 67 66 62

63 71 71 60

59 64 67 61

65 68 63

66 68 64

63

59

Example, Step 2


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Do the assumptions for the model hold?

Population by level are normally distributed Wont show significance for small # of samples

Variances are equal across all levels of the factor

Stat > ANOVA > Test for Equal Variances

Ho: _____________

Ha :_____________

1050

95% Confidence Intervals for Sigmas

P-Value : 0.593


Levene's Test

P-Value : 0.644


Bartlett's Test

Factor Levels

4

3

2

1

Test for Equal Variances for Coag_Time

Example, Step 3


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State the Null and Alternate HypothesesHo: diet1= diet2= diet3= diet4 (or) Ho: s = 0

Ha: at least two diets differ from each other(or) Ha:s0

Interpretation of the null hypothesis: the average bloodcoagulation time of each diet is the same (or) what you

eat will NOT affect your blood coagulation time.

Interpretation of the alternate hypothesis: at least one

diet will affect the average blood coagulation time

differently than another (or) what type of diet you keep

does affect blood coagulation time.

Example, Step 4


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Construct the ANOVA Table (using Minitab):

Stat > ANOVA > One-way ...

Hint: Store

Residuals &

Fits for later

use

Example, Step 4


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One-way Analysis of Variance

Analysis of Variance for Coag_Tim

Source DF SS MS F P

Diet_Num 3 228.00 76.00 13.57 0.000

Error 20 112.00 5.60

Total 23 340.00

Individual 95% CIs For Mean

Based on Pooled StDev

Level N Mean StDev ---+---------+---------+---------+---

1 4 61.00 1.826 (------*------)

2 6 66.00 2.828 (-----*----)

3 6 68.00 1.673 (----*-----)

4 8 61.00 2.619 (----*----)

---+---------+---------+---------+---

Pooled StDev = 2.366 59.5 63.0 66.5 70.0

Example, Step 5


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Do the assumptions for the errors hold?

Best way to check is through a residual analysis

Stat > Regression > Residual Plots ...

Determine if residuals are normally distributed

Ascertain that the histogram of the residuals looksnormal

Make sure there are no trends in the residuals

(its often best to graph these as a function of thetime order in which the data was taken)

The residuals should be evenly distributed abouttheir expected (fitted) values

Example, Step 5


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How normal arethe residuals ?

Histogram - bell curve ?Ignore for small data

sets (


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Analysis of Variance for Coag_Tim

Source DF SS MS F P

Diet_Num 3 228.00 76.00 13.57 0.000

Error 20 112.00 5.60

Total 23 340.00

Interpret the P-Value (or the F-statistic) for the factor effect

Assuming the residual assumptions are satisfied:

If P-Value < 0.05, then REJECT Ho

Otherwise, operate as if null hypothesis

is true

4

24

23

22

212 ssss

s

Pooled

When group sizes are equal

If P is less than 5% then

at least one group mean

is different. In this case,

we reject the hypothesis

that all the group means

are equal. At least oneDiet mean is different.

An F-test this large could

happen by chance, but in

less than one time out of

2000 chances. Thiswould be like getting 11

heads in a row from a

fair coin.

F-test is close to 1.00

when group meansare similar. In this

case, The F-test is

much greater.


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Work shop#6:

Run Hypothesis to validate your KPIVs from Measure phase

Analyze : Analyze Phases output


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Refine: KPOV = F(KPIVs)

Which KPIVs cause mean shifts?

Which KPIVs affect the standard deviation?

Which KPIVs affect yield or proportion?

How did KPIVs relate to KPOVs?

Improve


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TheImprovephase serves to optimize the KPIVs and study the

possible actions or ideas to achieve the goal


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Improve : Design Of Experiment


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The GOAL is to obtain a mathematical relationship which characterizes:

Y = F (X1, X2, X3, ...).

Mathematical relationships allow us to identify the most important orcritical factors in any experiment by calculating the effect of each.

Factorial Experiments allow investigation of multiple factors at multiplelevels.

Factorial Experiments provide insight into potential interactionsbetween factors. This is referred to as factorial efficiency.

Factorial Experiments



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Factors: A factor (or input) is one of the controlled or uncontrolled

variables whose influence on a response (output) is being studied in

the experiment. A factor may be quantitative, e.g., temperature in

degrees, time in seconds. A factor may also be qualitative, e.g.,

different machines, different operator, clean or not clean.



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Level: The levels of a factor are the values of the factor being

studied in the experiment. For quantitative factors, each chosen value

becomes a level, e.g., if the experiment is to be conducted at two

different temperatures, then the factor of temperature has two levels.Qualitative factors can have levels as well, e.g for cleanliness , clean

vs not clean; for a group of machines, machine identity.

Coded levels are often used,e.g. +1 to indicate the high level and

-1 to indicate the low level . Coding can be useful in both

preparation & analysis of the experiment



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k1 x k2 x k3 . Factorial : Description of the basic design.The number of ks is the number of factors. The value of each

k is the number of levels of interest for that factor.

Example : A2 x 3 x 3 design indicates three input variables.

One input has two levels and the other two, each have three levels.

Test Run (Experimental Run ) : A single combination of factor

levels that yields one or more observations of the output variable.

Center Point

Method to check linearity of model called Center Point


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Method to check linearity of model called Center Point.

Center Point is treatment that set all factor as center for

quantitative.

Result will be interpreted through curvature in ANOVAtable.

If center points P-value show greater than a level, we cando analysis byexclude center point from model. ( linearmodel )

If center points P-value show less than alevel, thats mean

we can not use equation from software result to be model.( non - linear )

There are no rule to specify how many Center point perreplicate will be take, decision based on how difficult tosetting and control.

Sample Size by Minitab


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Refer to Minitab, sample size will be in menu of

Stat->Power and Sample Size.


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Center Point case


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0 indicated that thesetreatments are center pointtreatment.

Exercise : DOECPT.mtw

Center Point Case


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Estimated Effects and Coefficients for Weight (coded units)

Term Effect Coef StDev Coef T P

Constant 2506.25 12.77 196.29 0.000A 123.75 61.87 12.77 4.85 0.017

B -11.25 -5.62 12.77 -0.44 0.689

C 201.25 100.62 12.77 7.88 0.004

D 6.25 3.12 12.77 0.24 0.822

A*B 120.00 60.00 12.77 4.70 0.018

A*C 20.00 10.00 12.77 0.78 0.491

A*D -17.50 -8.75 12.77 -0.69 0.542

B*C -22.50 -11.25 12.77 -0.88 0.443

B*D 7.50 3.75 12.77 0.29 0.788

C*D 12.50 6.25 12.77 0.49 0.658

A*B*C 16.25 8.13 12.77 0.64 0.570

A*B*D -11.25 -5.63 12.77 -0.44 0.689

A*C*D -18.75 -9.38 12.77 -0.73 0.516

B*C*D 3.75 1.88 12.77 0.15 0.893

A*B*C*D -22.50 -11.25 12.77 -0.88 0.443

Ct Pt -33.75 28.55 -1.18 0.322

P-Value of Ct Pt(center point)show greater thana level, we canexclude CenterPoint from model.

H0 : Model is linear

Ha : Model is non linear

Reduced Model


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Refer to effect table, we can excluded factor thatshow no statistic significance by remove term

from analysis.

For last page, we can exclude 3-Way interactionand 4-Way interaction due to no any term that

have P-Value greater than a level.

We can exclude 2 way interaction except termA*B due to P-value of this term less than a level.

For main effect, we can not remove B whether P-Value of B is greater than a level, due to we needto keep term A*B in analysis.

Center Point Case


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Final equation that we get for model is

Weight = 2499.5 + 61.87A 5.62B + 100.62C + 60AB

Fractional Factorial Fit: Weight versus A, B, C

Estimated Effects and Coefficients for Weight (coded units)

Term Effect Coef SE Coef T P

Constant 2499.50 8.636 289.41 0.000

A 123.75 61.87 9.656 6.41 0.000

B -11.25 -5.62 9.656 -0.58 0.569

C 201.25 100.62 9.656 10.42 0.000

A*B 120.00 60.00 9.656 6.21 0.000

DOE f St d d D i ti


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DOE for Standard Deviations

The basic approach involves taking n

replicates at each trial setting

The response of interest is the standarddeviation (or the variance) of those n values,

rather than the mean of those values There are then three analysis approaches:

Normal Probability Plot of log(s2) or log(s)*

Balanced ANOVA of log(s2

) or log(s)* F tests of the s2 (not shown in this package)

*log transformation permits normal distribution analysis approach

Standard Deviation Experiment

f f


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The following represents the results from 2

different 23 experiments, where 24 replicates

were run at each trial combination

File: Sigma DOE.mtw

*

A B C Expt1 s 2

-1 -1 -1 0.823

1 -1 -1 1.187-1 1 -1 3.186

1 1 -1 2.34

-1 -1 1 0.651

1 -1 1 1.477

-1 1 1 2.048

1 1 1 1.516

Std Dev Experiment Analysis Set Up

After putting this into the proper format as a


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After putting this into the proper format as a

designed experiment:

Stat > DOE > Factorial > Analyze FactorialDesign

Under the Graph option / Effects Plots Normal

ln(s2

)

A B C Expt1 s 2

-1 -1 -1 0.823

1 -1 -1 1.187

-1 1 -1 3.186

1 1 -1 2.34

-1 -1 1 0.6511 -1 1 1.477

-1 1 1 2.048

1 1 1 1.516

Expt1 ln(s^2)

-0.1942

0.17162

1.15888

0.84997

-0.429210.38995

0.71679

0.41602

N l P b bili Pl


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Normal Probability Plots

Plot all the effects of a 23 on a normal

probability plot

Three main effects: A, B and C

Three 2-factor interactions: AB, AC and BC

One 3-factor interaction: ABC If no effects are important, all the points should

lie approximately on a straight line

Significant effects will lie off the line

Single significant effects should be easilydetectable

Multiple significant effects may make it hard

to discern the line.

Probability Plot: Experiment 1


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-0.5 0.0 0.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Effect

Normal Probability Plot of the Effects(response is Expt 1, Alpha = .10)

A: AB: BC: C

Results from Experiment 1 Using

ln(s2

)

B

The plot shows one of the points--corresponding to

the B main effect--outside of the rest of the effects

Minitab does not identify thesepoints unless they are verysignificant. You need to lookat Minitabs Session Window

to identify.

ANOVA Table: Experiment 1


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ANOVA Table: Experiment 1

Results from Experiment 1 Using ln(s

2

)

Analysis of Variance for Expt 1

Source DF SS MS F P

A 1 0.0414 0.0414 0.30 0.611

B 1 1.2828 1.2828 9.39 0.037

C 1 0.0996 0.0996 0.73 0.441

Error 4 0.5463 0.1366

Total 7 1.9701

Sample Size Considerations


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The sample size computed for experiments involving

standard deviations should be based on a and b, aswell as the critical ratio that you want to detect--just

as it is for hypothesis testing

The Excel program Sample Sizes.xls can be used

for this purpose

If m is the sample size for each level (computed

by the program), and the experiment has k

treatment combinations, then the number ofreplicates, n, per treatment combination

= 1 + 2(m-1)*k


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Workshop # 7 : Run DOE to optimize the validate KPIV to

get the desired KPOV

Improve : Improve Phases output


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Which KPIVs cause mean shifts?

Which KPIVs affect the standard deviation?

Levels of the KPIVs that optimize processperformance

Control


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The Controlphase serves to establish the action to ensure

that the process is monitored continuously for consistency

in quality of the product or service.

Control: Tools


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To monitor and control the KPIVs

Error Proofing (Poka-Yoke)

SPC Control Plan

Control: Poka-Yoke


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Strives for zero defects

Leads to Quality Inspection Elimination

Respects the intelligence of workers

Takes over repetitive tasks/actions that depend on

ones memory

Frees an operators time and mind to pursue more

creative and value added activities

Why Poka-Yoke?

Control: Poka-Yoke


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Enforces operational procedures or sequences

Signals or stops a process if an error occurs or a defect is created

Eliminates choices leading to incorrect actions

Prevents product damage

Prevents machine damage

Prevents personal injury

Eliminates inadvertent mistakes

Benefit of Poka-Yoke?

Control: SPC


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SPC is the basic tool for observing variation and using statistical

signals to monitor and/or improve performance. This tool can beapplied to nearly any area.

Performance characteristics of equipment

Error rates of bookkeeping tasks

Dollar figures of gross sales

Scrap rates from waste analysis

Transit times in material management systems

SPC stands for Statistical Process Control. Unfortunately, most

companies apply it to finished goods (Ys) rather than processcharacteristics (Xs).

Until the process inputs become the focus of our effort, the full

power of SPC methods to improve quality, increase productivity,

and reduce cost cannot be realized.

Types of Control Charts


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The quality of a product or process may be assessed by

means of Variables :actual values measured on a continuous scale

e.g. length, weight, strength, resistance, etc Attributes :discrete data that come from classifying units

(accept/reject) or from counting the number

of defects on a unit

If the quality characteristic ismeasurable monitor its mean value and variability

(range or standard deviation)

If the quality characteristic is not measurable monitor the fraction (or number) of defectives monitor the number of defects

Defectives vs Defects


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Defective or Nonconforming Unit

a unit of product that does not satisfy one ormore of the specifications for the product

e.g. a scratched media, a cracked casing, afailed PCBA

Defect or Nonconformity

a specific point at which a specification is notsatisfied

e.g. a scratch, a crack, a defective IC

Shewhart Control Charts - Overview


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Shewhart Control Charts Overview

Walter A Shewhart

Shewhart Control Charts for Variables


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Control: SPC


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ChoosingThe Correct Control Chart Type

Type ofdata

Individualmeasurements or

sub-groups?

NormallyDistributed

data?

Interestedprimarily in

sudden shifts inmean?

Constantsub-group size?

Area of opportunity

constant from sample tosample?

Counting defectsor defectives?

u

c

p, np

p

X, mR

MA, EWMA,

or CUSUM

X-bar, RX-bar, s

Use of modified controlchart rules okay on

x-bar chart

Data tends to be normallydistributed because of central

limit theorem

More effective in

detecting graduallong-term changes

Attributes Variables

Defectives

Yes

No

Defects

No

Measurement

Sub-groups

NoNo

Yes

Yes

Individuals

Yes

Control: Control Phases output


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Y is monitored with suitable tools

X is controlled by suitable tools

Manage the INPUTS and good OUTPUTS will follow

Breakthrough Summary


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Definition ofOpportunity

Assess the Current Process Confirm f(x)for Y Optimize f(x) for Y Maintain Improvements Sustain the Benefit

1. Project Definition2. Determine

Impact & Priority3. Collect Baseliine

Metric Data4. Savings/Cost

Assessment

5. EstablishPlannedTimeline

6. Search Library7. Identify Project

Authority

1. Map the Process2. Determine the Baseline3. Prioritize the Inputs to

Assess4. Assess the

Measurement System5. Capability Assessment

6. Short Term7. Long Term8. Determine Entitlement9. Process Improvement10. Financial Savings

1. Determine the VitalVariables Affectingthe Responsef(x) = Y

2. ConfirmRelationships and

Establish the KPIV

1. Determine the BestCombination of Xs

for Producing theBest Y

1. Establish Controls for2. KPIVs and their

settings

3. Establish ReactionPlans

1. FinancialAssessment andInput ActualSavings

2. FunctionalManager/ProcessOwner Monitor

3. Control/Implementation

DEFINE ->>>>>>>>>>> MEASURE ->>>>>>>>>>>>>>> ANALYZE ->>>>>>>>>>>>> IMPROVE ->>>>>>>>>>>>CONTROL->>>>>>>>>>>>>>

REALIZATION ->>>>>>>>>>>>>

Champion BlackbeltsFinance Rep.&

Process Owner

Hard Savings


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Savings which flow to Net Profit

Before Income Tax (NPBIT)Can be tracked and reported by the

Finance organization

Is usually a reduction in labor,material usage, material cost, oroverhead

Can also be cost of money forreduction in inventory or assets

Finance Guidelines - Savings Definitions


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Hard Savings

Direct Improvement to Company Earnings

Baseline is Current Spending Experience Directly Traceable to Project

Can be Audited

Hard Savings Example

Process is Improved, resulting in lower scrap

Scrap reduction can be linked directly to thesuccessful completion of the project

S i t iti hi h h b

Potential Savings


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Savings opportunities which have been

documented and validated, but requireaction before actual savings could berealized

an example is capital equipment which has

been exceeded due to increased efficiencies inthe process. Savings can not be realizedbecause we are still paying for the equipment.It has the potential for generating savings ifwe could sell or put back into use because ofincreases in schedules.

Some form of a management decision oraction is generally required to realize the

savings


P i l S i


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Potential Savings

Improve Capability of company Resource

Potential Savings Example

Process is Improved, resulting in reducedmanpower requirement

Headcount is not reduced or reduction cannotbe traced to the project

Potential Savings might turn into hard savings if theresource is productively utilized in the future

Identifying Soft Savings


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Dollars or other benefits exist but they

are not directly traceable Projected benefits have a reasonable

probability (TBD) that they will occur

Some or all of the benefits may occuroutside of the normal 12 month trackingwindow

Assessment of the benefit could/should

be viewed in terms of strategic value tothe company and the amount of baselineshift accomplished



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Soft Savings

Benefit Expected from Process Improvement

Benefit cannot be directly traced to SuccessfulCompletion of Project

Benefit cannot be quantified

Soft Savings Example Process is Improved; decreasing cycle time Benefit cannot be quantified

introduction to dmaic - english (139 pages)

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