what is six sigma? it is a business process that allows companies to drastically improve their...

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

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

SIGMA LEVEL DEFECTS PER MILLION OPPORTUNITIES COST OF QUALITY

2 308,537 ( Noncompetitive companies ) Not applicable

3 66,807 25- 40% of sales

4 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?

Traditional Cost of Traditional Cost of Poor Quality (COQ)Poor Quality (COQ)

Rework

InspectionWarranty

Rejects

5-8%

Lost OpportunityLost Opportunity

15-20%Less Obvious Cost of Less Obvious Cost of

Quality (COQ)Quality (COQ)

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The Cost of Quality (COQ)

Note: % of sales

C

O

R

E

P

H

A

S

E

DMAIC : The Yellow Brick Road

Breakthrough & People

Champion

Blackbelts

Finance Rep.&Process Owner

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? What’s in my budget?

Projects DIRECTLY tie to department and/or business unit objectives

Projects are suitable in scope

BBs are “fit” to the project

Champions own and support project selection

Define : The Project

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

Time

Rej

ect

Rat

e

Special Cause ( ปั�ญหานาน ๆ ครั้��ง )

Optimum Level

ป/ญหาฝั/งแน�น (Chronic)

Define : The Chronic Problem

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

Define : Refine The Defect

Assembly Yield Loss

PSA RSA GramLoad

BentGimbal

SolderDefect

Contam DamperDefect

KPOV

% Y

ield

Lo

ss

a2 a3 a4 a5 a6 a7a1

Refined Defect = a1

30 - 50

10 - 15

4-8

Potential Key Process

Input Variables (KPIVs)

8 - 10 KPIVs

Optimized KPIVs

3-6 Key Leverage KPIVs

Inputs VariablesProcess Map

Multi-Vari Studies,Correlations

Screening DOE’s

DOE’s, 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 KPIV’s

Tools Outputs

Measure

The Measure phase serves to validate the problem, translate the practical to statistical problem and to begin the search for root causes

Measure : Tools

To validate the problem Measurement System Analysis

To translate practical to statistical problem

Process Capability AnalysisTo search for the root cause

Process Map Cause and Effect Analysis Failure Mode and Effect Analysis

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.

Objectives:

Validate the Measurement / Inspection System

Quantify the effect of the Measurement System variability on the process variability

Measure : Measurement System Analysis


To determine if inspectors across all shifts, machines, lines, etc… use the same criteria to discriminate “good” from “bad”

To quantify the ability of inspectors or gages to accurately repeat their 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


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



% Appraiser Score

•% 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%


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 devices or 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.

Measure : Measurement System AnalysisVariable GR&R :

Purpose

Long-T ermProcessVa ra ition

Short-T ermProcessVa ria tion

Va ria tionW ithin

Sa m ple

Actual Variation

R epeatability R eproducib ility

Precis ion S tability L inearity Accuracy

Va ria tiondue toG a ge

M easurem ent Variation

Observed Variation



Resolution?

“Precision” (R&R) Calibration? Stability?

Linearity?Bias?

Measurement System Variance:

2meas = 2

repeat + 2reprod

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

• Comparing 2meas with Tolerance:

– Precision-to-Tolerance Ratio (P/T)

• Comparing 2meas with Total Observed Process

Variation (P/TV):– % Repeatability and Reproducibility (%R&R)– Discrimination Index

Measurement System Measurement System MetricsMetrics

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

Number of Distinct Number of Distinct CategoriesCategories

•Automobile Industry Action Group (AIAG) recommendations:•CategoriesRemarks

< 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


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

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: MinitabExample: Minitab

Enter the data and tolerance information into Minitab. –Stat > Quality Tools > Gage R&R Study–Gage Info (see below) & Options

Gage name:Date of study:Reported by:Tolerance:Misc:

00.30.40.50.60.70.80.91.01.1 1 2 3

Xbar Chart by Operator

Sam

ple

Mea

n

Mean= 0.8075UCL= 0.8796

LCL= 0.7354

0

0.00

0.05

0.10

0.15 1 2 3

R Chart by Operator

Sam

ple

Ran

ge

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 Interaction

Ave

rage

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

%Cont ribut ion

%Study Var %Tolerance

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

0

50

100

Components of Variation

Perc

ent

Gage R&R (ANOVA) for Response

Gage R&R Output

Gage R&R Output

Gage R&R, Variation Gage R&R, Variation ComponentsComponents

%ContributionSource 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 %ToleranceSource (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 for each variance component

Gage R&R, Gage R&R, ResultsResults

%ContributionSource 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 %ToleranceSource (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

3266005041

34300.

.

.

StudyVarTV/Ptotal

meas

2287.05.1

3430.0

*15.5/

TolLSLUSL

TP MS

1067.0041602.0

004437.0

2

2

total

MSonContributi

Question: What is our conclusion about the measurement system?

•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

USLLSLLSL USL

Off-TargetVariationLarge

Characterization

Measure : Process Capability AnalysisTranslate practical problem to statistical problem

LSL USL

Outliers

Two measures of process capability

Process PotentialCp

Process PerformanceCpu

Cpl

Cpk

Cpm


6

LSLUSL

ToleranceNatural

TolerancegEngineerinCp


Process Potential

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

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

Process is centered Process is not centered



The Cpk index relates the scaled distance between the process mean and the nearest specification limit.

3

USLCpu

3

LSLCpl

3

NSLC,CMinimumC plpupk

Process Performance


Rev. 1 12/98

Capability Capability

StudiesStudiesEntitlement

(Short Term)Performance

(Long Term)

Type of Variability Only common cause All causes

# of Data Points 25-50 subgroups 200 points

Production Example

(Lumen Output):

-1 lot of raw mat’l-1 shift; 1 set of people-Single “set-up”

-3 to 4 lots of raw mat’l-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 Man’s --

“Best 2 weeks”Historical data

Process:Running like it was designed

or intended!Running like itactually does!

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


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


Process Potential VS. Process Performance ( Cp Vs Cpk )

1.If Cp > 1.5 , it means the standard deviation is suitable2.Cp is not equal to Cpk, it means that the process mean is off-centered

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.

Measure : Process Map

Process Map is 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.

OPERATION All steps in the process where the object undergoes

a change in form or condition.TRANSPORTATION All steps in a process where the object moves from

one location to another, outside of the OperationSTORAGE All steps in the process where the object remains at

rest, in a semi-permanent or storage conditionDELAY All incidences where the object stops or waits on a

an operation, transportation, or inspectionINSPECTION All steps in the process where the objects are

checked for completeness, quality, outside of the Operation.

DECISION


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


Major StepMajor Step Major StepMajor StepMajor 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

Workshop #2 : Do the process map and report the process steps and KPIVs that may be the cause

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

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 (Y’s) and inputs (X’s).

Measure : Cause and Effect Matrix

There are two types of Cause and Effect Matrix

C = Control FactorN = Noise FactorX = Factor for DOE (chosen later)

MethodsMaterials

Machinery Manpower

Problem/Desired

Improvement

C/N/X

C

C

C

N N

NNN

C

C


Fishbone Diagram

Rating of Importance to

Customer

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

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Requirem

ent

Total

Process Step Process Input

1 02 03 04 05 06 07 08 09 010 011 012 013 014 015 016 017 018 019 020 0

0

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

Lower Spec

Target

Upper Spec

Cause and Effect Matrix


Workshop #4:

Team brainstorming to create the fishbone diagram

FMEA is a systematic approach used to examine potential failures and prevent their occurrence. It enhances an engineer’s 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

RPN = S x O x D

Severity (ค่วามร"นแรง ) X

Occurrence (โอกาส่การเก�ดข3�น) X

Detection (การติรวจำจำ�บ )


ส่��งส่%าค่�ญม�น�อย (Vital Few)

ส่��งจำ�4บจำ5อยม�มาก(Trivial Many)


Workshop # 5 :

Team Brainstorming to create FMEA

Check and fix the measurement system

Determine “where” you are Rolled throughput yield, DPPM Process Capability Entitlement

Identify potential KPIV’s Process Mapping / Cause & Effect /

FMEA Determine their likely impact

Measure : Measure Phase’s Output

Analyze

The Analyze phase serves to validate the KPIVs, and to study the statistical relationship between KPIVs and KPOVs

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

The Null Hypothesis

Statement generally assumed to be true unless sufficient evidence 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


The Alternative Hypothesis

Statement generally held to be true if the null hypothesis is rejected

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

Designated as HA

NULL HYPOTHESIS: Nothing has changed: For Tests Of Process Mean: H0: = 0

For Tests Of Process Variance:H0: 2 = 2

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

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<B

Conclusion about the claim:

If A is the same B Then

If A is NOT better than B Then

Actions to be taken:



See Hypothesis Testing Roadmap

Example: Single Mean Example: Single Mean Compared to TargetCompared to Target

• The example will include 10 measurements of a random sample:

– 55 57 58 54 5356 55 54 54 53The question is: Is the mean of the sample representative of a target value of 54?

• The Hypotheses:Ho: = 54Ha: 54

Ho can be rejected if p < .05

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

Single Mean to a Single Mean to a Target - Using MinitabTarget - Using Minitab

Stat > Basic Statistics > 1-Sample t

P-value is greater than 5%, so we say the sample mean

is representative of 54

Our Conclusion Our Conclusion StatementStatement

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.

•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 (=2.4).

• ฉ All Rights Reserved. 2000 Minitab, Inc.

Single Std Dev Compared to Standard

Example: Single Std Dev Example: Single Std Dev Compared to StandardCompared to Standard

(Data: Furnace.mtw, Use “BTU_in”)

Note: Minitab does not provide an individual 2 test for standard deviations. Instead, it is necessary to look at the confidence interval on the standard deviation and determine if the CI contains the claimed value.

Example: Single Example: Single Standard DeviationStandard Deviation

Stat > Basic Statistics > Display Descriptive Statistics

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:

MeanStDevVarianceSkewnessKurtosisN

Minimum1st QuartileMedian3rd QuartileMaximum

8.9419

2.4738

8.6170

0.4750.228

9.907753.019879.11960

0.7075240.783953

40

4.0000 7.8850 9.590011.555018.2600

10.8736

3.8776

10.3212

Damper: 1

Anderson-Darling Normality Test


95% Confidence Interval for Sigma


Descriptive StatisticsRunning the Statistics….Running the Statistics….

4 7 10 13 16


9 10 11


Variable: BTU.In

A-Squared:P-Value:

MeanStDevVarianceSkewnessKurtosisN

Minimum1st QuartileMedian3rd QuartileMaximum

9.3566

2.3114

8.7706

0.1900.895

10.1430 2.76707.65640-9.9E-02-2.7E-01

50

2.9700 8.127510.290012.212516.0600

10.9294

3.4481

11.2363

Damper: 2

Anderson-Darling Normality Test


95% Confidence Interval for Sigma


Descriptive Statistics

Running the Statistics….

Two Parameter Two Parameter TestingTesting

Step 1: State the Practical Problem

Step 2: Are the data normally distributed?

Step 3: State the Null Hypothesis:

ForFor

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

Ho: M1 = M2 (non-normal data)

State the Alternative Hypothesis:

ForFor

Ha: pop1 pop2 Ha: pop1 pop2

Ha: M1 M2 (non-normal data)

Means: 2 Sample t-testSigmas: Homog. Of VarianceMedians: NonparametricsFailure Rates: 2 Proportions

Two Parameter Testing (Cont.)Two Parameter Testing (Cont.)

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 distribution and 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 Comparing Two Independent Sample Independent Sample

MeansMeans

• The example will make a comparison between two group means

• Data in Furnace.mtw ( BTU_in)

• Are the mean the two groups the same?

• The Hypothesis is:

– Ho:

– Ha :

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

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

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

t-test Using Stacked Data

2 variances test2 variances testStat >Basic Statistics > 2 variances

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

2 variances test

Characteristics About Multiple Characteristics About Multiple Parameter TestingParameter Testing

• 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 (KPIV’s) 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 MethodGeneral Method

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 level—first

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 Step 2: Do the Assumptions for the Model Hold?the Model Hold?

• 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)

• For Ho: pop1 = pop2 = pop3 = pop4 = ..

Ha: at least two are different

Step 3: State the HypothesesStep 3: State the Hypotheses

Ho: ’s = 0Ha: k 0

Ho: 1 = 2 = 3 = 4

Ha: At least one k is different

Mathematical Hypotheses:

Conventional Hypotheses:

Step 4: Construct the ANOVA Step 4: Construct the ANOVA TableTable

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 VarianceAnalysis of Variance for TimeSource DF SS MS F POperator 3 149.5 49.8 4.35 0.016Error 20 229.2 11.5Total 23 378.6

What’s important the probability

that the Operator variation in means

could have happened by chance.

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

Steps 5 - 7Steps 5 - 7

ResidualAnalysis

Example, Experimental Example, Experimental SetupSetup

• 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 assigned

randomly to animals. Blood samples taken and tested in random order. Why ?

DIET A DIET B DIET C DIET D62 63 68 5660 67 66 6263 71 71 6059 64 67 61

65 68 6366 68 64

6359

Example, Step 2Example, Step 2

• Do the assumptions for the model hold?• Population by level are normally distributed

– Won’t 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


• State the Null and Alternate HypothesesHo: µ diet1= µ diet2= µ diet3= µ 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 onediet will affect the average blood coagulation timedifferently than another (or) what type of diet you keepdoes affect blood coagulation time.

Example, Step 4Example, Step 4• Construct the ANOVA Table (using Minitab):

Stat > ANOVA > One-way ...

Hint: Store Residuals & Fits for later use


One-way Analysis of Variance

Analysis of Variance for Coag_TimSource DF SS MS F PDiet_Num 3 228.00 76.00 13.57 0.000Error 20 112.00 5.60Total 23 340.00 Individual 95% CIs For Mean Based on Pooled StDevLevel 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

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 looks normal

• Make sure there are no trends in the residuals (it’s often best to graph these as a function of the time order in which the data was taken)

• The residuals should be evenly distributed about their expected (fitted) values



How normal arethe residuals ?

Histogram - bell curve ?Ignore for small data

sets (<30)

Individual residuals -trends? Or outliers?

This graph investigateshow the Residuals behave across the experiment. This is probably the mostimportant graph, since itwill signal that somethingoutside the experimentmay be operating.Nonrandom patternsare warnings.

This graph investigateswhether the mathematicalmodel fits equally for lowto high values of the Fits

Random about zerowithout trends?

Analysis of Variance for Coag_Tim

Source DF SS MS F PDiet_Num 3 228.00 76.00 13.57 0.000Error 20 112.00 5.60Total 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

Pooled

When group sizes are equal

If P is less than 5% thenat least one group meanis different. In this case,we reject the hypothesisthat all the group meansare equal. At least oneDiet mean is different.

An F-test this large couldhappen by chance, but inless than one time out of2000 chances. Thiswould be like getting 11heads in a row from afair coin.

F-test is close to 1.00when group meansare similar. In thiscase, The F-test ismuch greater.

Work shop#6:

Run Hypothesis to validate your KPIVs from Measure phase

Analyze : Analyze Phase’s output

Refine: KPOV = F(KPIV’s)

Which KPIV’s cause mean shifts?

Which KPIV’s affect the standard deviation?

Which KPIV’s affect yield or proportion?

How did KPIV’s relate to KPOV’s?

Improve

The Improve phase serves to optimize the KPIV’s and study the possible actions or ideas to achieve the goal

Improve : Tools

To optimize KPIV’s in order to achieve the goal

Design of Experiment

Evolutionary Operation

Response Surface Methodology

The GOAL is to obtain a mathematical relationship which characterizes:

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

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

Factorial Experiments allow investigation of multiple factors at multiple levels.

Factorial Experiments provide insight into potential “interactions” between factors. This is referred to as factorial efficiency.

Improve : Design Of Experiment

Factorial Experiments

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.


• 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


k1 x k2 x k3 …. Factorial : Description of the basic design.

The number of “ k’s ” 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 PointCenter Point• 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 ANOVA table.

• If center point’s P-value show greater than level, we can do analysis by exclude center point from model. ( linear model )

• If center point’s P-value show less than level, that’s mean we can not use equation from software result to be model. ( non - linear )

• There are no rule to specify how many Center point per replicate will be take, decision based on how difficult to setting and control.

Sample Size by Sample Size by MinitabMinitab

• Refer to Minitab, sample size will be in menu of Stat->Power and Sample Size.

Sample Size By Sample Size By MinitabMinitab

Specify number of factor in experiment design.

Specify number of run per replicated.

Enter power value, 1-, which can enter more than one. And effect is critical difference that would like to detect ().

Process sigma

Center Point Center Point casecase

“0” indicated that these treatments are center point treatment.

Exercise : DOECPT.mtw

Center Point Center Point CaseCase

Estimated Effects and Coefficients for Weight (coded units)

Term Effect Coef StDev Coef T P

Constant 2506.25 12.77 196.29 0.000

A 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 than a level, we can exclude Center Point from model.

H0 : Model is linear

Ha : Model is non linear

Reduced ModelReduced Model• Refer to effect table, we can excluded factor that

show no statistic significance by remove term from analysis.

• For last page, we can exclude 3-Way interaction and 4-Way interaction due to no any term that have P-Value greater than level.

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

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

Center Point Center Point CaseCase

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 for Standard DeviationsDOE for Standard Deviations• The basic approach involves taking “n”

replicates at each trial setting

• The response of interest is the standard deviation (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 ExperimentStandard Deviation ExperimentThe following represents the results from 2 different 23 experiments, where 24 replicates were run at each trial combination

File: Sigma DOE.mtw*

Std Dev Experiment Analysis Set Std Dev Experiment Analysis Set UpUp

After putting this into the proper format as a designed experiment:

Stat > DOE > Factorial > Analyze Factorial Design

Under the Graph option / Effects Plots Normal

ln(s2)

Normal Probability PlotsNormal 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 easily detectable

– Multiple significant effects may make it hard to discern the line.

-0.5 0.0 0.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Effect

Nor

mal

Sco

reNormal Probability Plot of the Effects

(response is Expt 1, Alpha = .10)

A: AB: BC: C

Probability Plot: Experiment 1Probability Plot: Experiment 1

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 these points unless they are very significant. You need to look at Minitab’s Session Window to identify.

ANOVA Table: Experiment 1ANOVA Table: Experiment 1Results from Experiment 1 Using ln(s2)

Analysis of Variance for Expt 1

Source DF SS MS F PA 1 0.0414 0.0414 0.30 0.611B 1 1.2828 1.2828 9.39 0.037C 1 0.0996 0.0996 0.73 0.441Error 4 0.5463 0.1366Total 7 1.9701

Sample Size ConsiderationsSample Size Considerations• The sample size computed for experiments involving

standard deviations should be based on and , as well 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 of replicates, n, per treatment combination

= 1 + 2(m-1)*k

Workshop # 7 : Run DOE to optimize the validate KPIV to get the desired KPOV

Improve : Improve Phase’s output

Which KPIV’s cause mean shifts?

Which KPIV’s affect the standard deviation?

Levels of the KPIV’s that optimize process performance

Control

The Control phase serves to establish the action to ensure that the process is monitored continuously for consistency in quality of the product or service.

Control: Tools

To monitor and control the KPIV’s Error Proofing (Poka-Yoke) SPC Control Plan

Strives for zero defects

Leads to Quality Inspection Elimination

Respects the intelligence of workers

Takes over repetitive tasks/actions that depend on one’s memory

Frees an operator’s time and mind to pursue more creative and value added activities

Control: Poka-Yoke

Why Poka-Yoke?

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: Poka-Yoke

SPC is the basic tool for observing variation and using statistical signals to monitor and/or improve performance. This tool can be applied 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 (Y’s) rather than process characteristics (X’s).

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.

Control: SPC

Types of Control ChartsTypes of Control Charts

The quality of a product or process may be assessed by means of

• Variables :actual values measured on a continuous scalee.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 is measurable • 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 DefectsDefectives vs Defects

• Defective or Nonconforming Unit• a unit of product that does not satisfy one or

more of the specifications for the product– e.g. a scratched media, a cracked casing, a

failed PCBA

• Defect or Nonconformity• a specific point at which a specification is not

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

Shewhart Control Charts - Shewhart Control Charts - OverviewOverview

Walter A Shewhart

Shewhart Control Charts Shewhart Control Charts for Variablesfor Variables

Choosing The Correct Control Chart Type

Type of data

Individual measurements or

sub-groups?

Normally Distributed data?

Interested primarily in

sudden shifts in mean?

Constant sub-group size?

Area of opportunity constant from sample to

sample?

Counting defects or defectives?

u

c

p, np

p

X, mR

MA, EWMA,

or CUSUM X-bar, RX-bar, s

Use of modified control chart rules okay on

x-bar chart

Data tends to be normally distributed because of central

limit theorem

More effective in

detecting gradual long-term changes

Attributes Variables

Defectives

Yes

No

Defects

No

Measurement

Sub-groups

NoNo

Yes

Yes

Individuals

Yes

Control: SPC

Control: Control Phase’s output

Y is monitored with suitable tools

X is controlled by suitable tools

Manage the INPUTS and good OUTPUTS will follow

Breakthrough Summary

Champion

Blackbelts

Finance Rep.&Process Owner

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, or overhead

Can also be cost of money for reduction in inventory or assets

Hard SavingsHard Savings

Finance Guidelines - Finance Guidelines - Savings DefinitionsSavings Definitions

• 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 the successful completion of the project

Savings opportunities which have been documented and validated, but require action before actual savings could be realizedan example is capital equipment which has been

exceeded due to increased efficiencies in the process. Savings can not be realized because we are still paying for the equipment. It has the potential for generating savings if we could sell or put back into use because of increases in schedules.

Some form of a management decision or action is generally required to realize the savings

Potential SavingsPotential Savings

Finance Guidelines - Savings Finance Guidelines - Savings DefinitionsDefinitions

Potential Savings

• Improve Capability of company Resource

Potential Savings Example• Process is Improved, resulting in reduced manpower requirement• Headcount is not reduced or reduction cannot be traced to the project

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

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 occur outside of the normal 12 month tracking window

Assessment of the benefit could/should be viewed in terms of strategic value to the company and the amount of baseline shift accomplished

Identifying Soft Identifying Soft SavingsSavings

Finance Guidelines - Savings Finance Guidelines - Savings DefinitionsDefinitions

Soft Savings

• Benefit Expected from Process Improvement• Benefit cannot be directly traced to Successful Completion of Project• Benefit cannot be quantified

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

what is six sigma? it is a business process that allows companies to drastically improve their...

Documents

defect slide

project slide

purpose slide

sales slide

a1 slide

effect analysis slide

measurement inspection

process owner slide