Essentials of Lean Six Sigma

104

105

106

106

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x

Contents

Chapter 8

Road Map to Lean Six Sigma Continuous Improvement

Engineering Strategies

8.1 Six-Sigma Continuous Improvement Engineering

8.2 Defi nition and Measurement

108

8.2.1 Phase 0: Process Defi nition/Project Selection

8.2.2 Phase I: Process Measurement

116

107

108

8.3 Evaluation of Existing Process Sigma/Baseline Sigma

8.4 Data Analysis

143

143

8.4.1 Phase II: Process Analysis

8.5 Optimization and Improvement

143

150

8.5.1 Phase III: Process Improvement

150

8.6 Evaluation of New Sigma

160

8.7 Process Control

8.7.1 Phase IV: Process Control and Maintain

159

160

PART III

Case Studies

Chapter 9

Six Sigma Green and Black Belt Level Case Studies

9.1

9.2

175

175

175

179

Phase 3: Improve and Verify Analyzed Data

Case Study: Methodology for Machine Downtime Reduction—

A Green Belt Methodology

9.1.1 Phase 0: Problem Statement

9.1.2 Phase 1: Data Collection and Measurement

9.1.3 Phase 2: Analysis of Measurement

9.1.4

9.1.5 Phase 4: Control and Maintain

Case Study: Methodology for Defect Reduction in Injection

Molding Tools—A Black Belt Methodology

9.2.1 Phase 0: Defi nition and Statement of Issues

9.2.2 Phase 1: Data Collection and Measurement

9.2.3 Phase 2: Analysis of Collected Data

9.2.4 Phase 3: The Process of Improvement

9.2.5

184

184

184

182

179

191

191

Phase 4: The Process of Control and

Maintenance

197

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Contents

xi

Chapter 10

Six Sigma Master Black Belt Level Case Study

10.1 Case Study: Defect Reduction in Injection Molding a Multifactor

Lean Central Composite Design Approach

10.1.1 Scope of Injection Molded Parts

10.1.2 Scope of Study

205

205

206

205

10.2 Composite Design Methodology

10.3 Modeling

10.4 Simulation

10.5 Conclusion

Bibliography

215

222

223

212

Appendix: Statistical Tables Used for Lean Six Sigma

AI: Highlights of Symbols and Abbreviations

AII: Chapter 10 Case Study Extended Equations

AIII: Values of y = exp(−h)

AIV: DPMO to Sigma to Yield % Conversion Table

233

227

229

235

AV: Standard Normal Distribution

AVI: Critical Values of t-Distribution

AVII: Critical Values of Chi-Square Distribution with Degrees

237

241

of Freedom

245

AVIII: Upper Critical Values of the F-Distribution for df1

Numerator Degrees of Freedom and df2 Denominator

251

Degrees of Freedom

AIX: Deming’s Condensation of the 14 Points for

Management

265

AX: Scorecard for Performance Reporting

AXI: Scorecard for Performance Reporting (Partly Completed

267

Index

271

Example)

269

www.EngineeringEbooksPdf.com

www.EngineeringEbooksPdf.comPreface

ABOUT THIS BOOK

Before the 1970s, the industry standards were based on ±3s and percent (%)

defect. Now, as population grows and industrial volume due to global economy

becomes mass production, the ±3s and percent defect evaluations are no longer

valid. The ±6s and defect per million are today’s standard for ultimate customer

satisfaction and maximum profi tability. Knowing that customer satisfaction is the

number one priority on any organization’s list, the success of any company depends

on quality and competitive product pricing. Today the globalized market allows no

space for error. Thus, Six Sigma is necessary for all organizations. The theory of

Six Sigma demonstrates the bottom line and customer satisfaction improvement.

Unlike other programs that concentrate on quality only, Six Sigma focuses on

customer satisfaction and the bottom line. This also means the highest quality:

As defects drop to 3.4 per million, quality improves dramatically.

This book explains the Lean Six Sigma concepts, the essential theory and analysis

from the engineering point of view in three different parts. Part I: Statistical Theory

and Concepts; Part II: Six Sigma Engineering and Implementation; and Part III: Case

Studies. Throughout this book numerous examples have been cited, particularly in

the plastics industry of injection molding. All other manufacturers may also benefi t

to a great extent. Consequently, any other organization may engineer their Six Sigma

program using this book, as well. A brief description of each chapter follows:

Part I. Statistical Theory and Concepts

Chapter 1. Reviews Lean Six Sigma concepts and background (history).

Chapter 2.

Demonstrates normal distribution, process capability estimation of

1 sigma through Six Sigma.

Explains essentials of mathematical concepts in Lean Six Sigma

engineering strategies, as well as a review of standard normal dis-

tribution and normality tests.

Chapter 3.

xiii

www.EngineeringEbooksPdf.comxiv

Preface

Part II. Six Sigma Engineering and Implementation

Chapter 4.

The essentials of Six Sigma continuous improvement principles and

training models.

The essentials of design for Six Sigma principles, tools, and

techniques.

Chapter 5.

Chapter 6. The essentials of design for Lean Six Sigma and training models.

Chapter 7.

The roles and responsibilities to Six Sigma philosophy and Six

Sigma infrastructure.

The road map to Lean Six Sigma continuous improvement engineer-

ing strategies.

Chapter 8.

Part III. Case Studies

Chapter 9.

Case studies with complete Six Sigma applications in injection

molding plastics manufacturing for Green and Black Belts.

Chapter 10. An expanded version of the case study published by the author for

the Black Belt level in reduction or minimization of variation in an

injection molding plastic industry.

Salman Taghizadegan

www.EngineeringEbooksPdf.comAcknowledgments

The author wishes to thank Keith Boyle (Quality & Productivity Resources)

for his inspiration, assistance, and contributions throughout this book. He also

provided my Black Belt training at the University of California at San Diego.

Much appreciation goes to my family. To my wife, Leila, and my daughter, Sara,

thank you for your long-lasting patience and continuous support. I would like to

acknowledge Hunter Industries for supporting and giving me the opportunity to

implement Lean Six Sigma concepts in practice. I thank Diane Clark for her

support of the publication of this book (and for putting in long hours and making

this book easier to read). Finally, my thanks to Joel Stein at Elsevier Science for

his support and patience throughout the publishing process and other members

of the Elsevier Science team for their support and assistance in making this work

a reality. My thanks to Shelley Burke for her assistance throughout the publishing

process. Furthermore, thanks goes to Carl M. Soares for managing the Publica-

tion process.

xv

www.EngineeringEbooksPdf.comwww.EngineeringEbooksPdf.comAbout the Author

Dr. Salman Taghizadegan has substantial experience in chemical and plas-

tics processing, design, control, and analysis. He received his B.S. in chemistry

from Western Illinois University, his B.S. in chemical engineering from the

University of Arkansas, his M.S. in chemical engineering from the Texas A&M

University, and his Ph.D. in chemical engineering with emphasis in plastics from

the University of Louisville.

He has over 20 years of academic and full-time industrial experience in plas-

tics and chemical processing, design, and control engineering, primarily in injec-

tion molding industries. He has authored numerous technical publications and

has spent most of his professional career as an adjunct professor in engineering,

as a highly technical specialist in the plastics industry, as a leader in quality and

process improvement, and as the manager of waste reduction in the manufactur-

ing environment. Dr. Taghizadegan is certifi ed Six-Sigma Black Belt and Master

Black Belt through the University of California at San Diego and the university

of San Diego. He is a member of the Society of Plastics Engineers.

xvii

www.EngineeringEbooksPdf.comwww.EngineeringEbooksPdf.comChapter 1

Introduction to Essentials of

Lean Six Sigma (6s) Strategies

Lean Six Sigma: Six Sigma Quality with

Lean Speed

1.1 LEAN SIX SIGMA (6s ) CONCEPT REVIEW

1.1.1 THE PHILOSOPHY

In any organization customer satisfaction is the number one priority. Customer

satisfaction also means profi tability. The success of any company depends on the

ability to ensure the highest quality at the lowest cost. In the 1980s when most

companies believed that producing quality products was too costly, Motorola

believed the opposite: “the better, the cheaper.” It realized that by producing a

higher-quality product, the cost of producing goes down. Motorola knew that

greater customer satisfaction generates higher profi tability.

Today the competitive market leaves no space for error. It is now necessary

to implement the concepts of Lean Six Sigma. Lean Six Sigma is a business

strategy in which the focus is to improve the bottom line and increase customer

satisfaction.

Six Sigma philosophies are related to statistical process control, stochastic

control (relating to probability), and engineering process control. In addition, it

requires process and data analysis, optimization methods, lean manufacturing,

design of experiment, analysis of variance, statistical methods, mistake-proofi ng,

on-time and or on-schedule shipping, waste reduction, and consistency as-

surance. It is a process capability that continuously improves the quality of the

product and maximizes productivity. In simpler terms, Lean Six Sigma is the

following:

Essentials of Lean Six Sigma

Copyright © 2006 by Academic Press, Inc. All rights of reproduction in any form reserved.

1

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Introduction to Essentials of Lean Six Sigma (6s) Strategies

1. It is a data-driven approach and methodology to analyze the root causes

of manufacturing and business problems/processes by eliminating

defects (driving toward six standard deviations between the mean and

the nearest specifi cation limit), and dramatically improving the product.

2. It improves the employee’s knowledge of business management to distinguish

the business from the bottom line, customer satisfaction, and on-time delivery.

Thus, Six Sigma is not just process-improvement techniques but a

management strategy to manage the projects to fi nancial goals.

3. It combines robust design engineering philosophy and techniques with

low risks (Lean Six Sigma tools: measure, analyze, develop, and verify).

It would be very diffi cult to achieve this goal without teamwork and proper

training of the entire organization to a higher level of competency. During the

1980s Six Sigma grew into a distinct manufacturing discipline. It now encompasses

a wide range of disciplines, including transportation, administration, manufactur-

ing, medical, and a variety of other operating organizations and processes (by defi -

nition a process is any operation that has an input and produces an output).

1.1.2 LEAN/KAIZEN SIX SIGMA ENGINEERING

Lean speed is a technique as well as a continuous effort that is used to acceler-

ate and minimize the cost of any process by eliminating the waste in either

manufacturing or service. Basically, Lean philosophy identifi es and removes

ineffi ciencies like the nonvalue-added (waste) cost or unneeded wait time within

the process caused by defects, excess production, and other processes to expand

any organization. For example, in most cases 95% of the lead time (from the

beginning to the end of a process) is the wait time. Further, 80% of process delays

are caused by a 20% time trap (activities in the workstation). By improving 20%

time trap, it can eliminate 80% of process delays. Hence, Lean is associated with

speed, effi ciency, and acceleration of the process. Therefore, by integrating ele-

ments of Lean enterprise methodology with Six Sigma, which lacks tools that

control and reduce lead time, the feedback will be faster than planned.

The combination of these two powerful tools, Lean manufacturing and Six

Sigma strategy, will result in process variation reduction and dramatic bottom-

line (language of CEO) improvement. Since all companies are in the business of

achieving faster return on investments, particularly for their shareholders, using

Lean principles in Six Sigma is extremely important. For the company architect-

ing Six Sigma philosophy in its infrastructure, Lean manufacturing speed can

accelerate the implementation and benefi ts of the manufacturing process.

Here are some of the basic Lean manufacturing techniques and principles that

are used in Lean Six Sigma:

www.EngineeringEbooksPdf.comIntroduction to Essentials of Lean Six Sigma (6s) Strategies

3

1. 5S

• Sort (keep things that are essential), Shine (keep everything clean),

Straighten (make everything visible and accessible), Standardize

(implement the fi rst 3S and maintain them), and Sustain.

• The fi rst 3S are actions, and the last two are sustaining and

progressive.

2. Value-stream mapping

• A method of mapping a product’s production path from manufacturing

facility to customer’s door.

• A visual tool for identifying all steps of operations in the

manufacturing process with cost-effective results.

3. Kaizen event

• Continuous improvement.

4. Mistake-proofi ng

• Process analysis and implementation of robust engineering to build

quality into an assembly or manufacturing process with cost-effective

results.

5. Cycle time reduction

6. Inventory reduction

7. Setup time reduction

8. Waste identifi cation and elimination

In other words, the Lean speed is merged with or is embedded within the Six

Sigma principles. The integration of these two concepts will both deliver faster

results and achieve the best competitive position by concentrating on the use of tools

that have the highest impact on the already established performance levels. Another

example is the design of experiment that may require about 16 runs to determine

optimum factors and reduce variation. Minimizing the lead time by 80% will allow

the experiment to be completed fi ve times faster using fractional factorial design.

Basically Lean contributes to Six Sigma in the following manner:

1. Eliminates all the waste time that slows the project.

2. Maintains customer satisfaction with speed in delivery.

3. Gets the project done under the deadline and possibly under budget.

4. Continuously improves the profi tability (e.g., in a shorter period of time

than planned).

1.2 SIX SIGMA BACKGROUND

Motorola engineering scientist William Smith, known as the father of Six Sigma,

developed the concept in the 1980s. For many years, he and other pioneering engineers

and scientists worked on this or similar concepts to reduce variation, improve quality,

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Introduction to Essentials of Lean Six Sigma (6s) Strategies

and maximize productivity, including Walter A. Shewhart, W. Edwards Deming (see

Appendix for Demingr’s 14 points for management), Philip R. Crosby, Shiego Shingo,

Taiichi Ohno, and Joseph Juran. Each one studied quality from a different angle.

The methodology of Six Sigma uses the statistical theory and thus assumes

that every process factor can be characterized by a statistical distribution curve.

The objective is to free all the defects from every process, product, and transac-

tion. It is a process that provides tools to achieve nearly error-free products and

services with maximum profi tability. In the 1960s and 1970s, statistical process

control limits were based on plus or minus three sigma (±3 standard deviation)

from the mean. However, in this concept the process limits are plus or minus Six

Sigma from the mean.

Just like three sigma, Six Sigma is applicable to batch-to-batch process, dis-

crete, and continuous applications. The goal is to produce less than four defects

per one million operations. Six Sigma will enable a company to capture substan-

tial market share in the competitive global markets. Global competitiveness

almost becomes impossible without Six Sigma. Every company would benefi t by

adopting Six Sigma concepts and philosophy. Profi tability improves tremen-

dously if it is applied to all workforces in every department of the corporation.

1.3 SIX SIGMA SUCCESSES

An example of a Six Sigma successe is Motorola Corporation, which increased

net income from $2.3 billion in 1978 to $8.3 billion in 1988, using the Six Sigma

program. As a result, Motorola received the Malcolm Baldrige National Quality

Award by President Reagan in 1988. The award is presented to the industries that

become quality role models for others. GE also implemented Six Sigma in the

mid-1990s in a fi ve-year program and boosted its profi ts by a substantial amount.

By the year 2002 GE had achieved $4 billion in savings per year. Other compa-

nies that benefi t from Six Sigma are Allied Signal, Inc.; Polaroid Corporation;

Asea Brown Boveri Power Transformer Company; and DuPont.

At three sigma the cost of quality is 25 to 40% of sales revenue. At Six Sigma

it reduces cost of quality to less than 1% of sales revenue. In fact, Lean Six Sigma

is the epitome of quality and should be adopted by all manufacturing companies

to remain in business. Therefore, one must change measurement of quality in

parts per hundred (percentages) to parts per million. This has changed the

makeup and culture of industries that adopted Lean Six Sigma.

Sigma Variation

Mathematically variation and reproducibility are inversely related to each

other—for example, as variation increases, producibility decreases due to increase

www.EngineeringEbooksPdf.comIntroduction to Essentials of Lean Six Sigma (6s) Strategies

5

Comparisons of 3.8 Sigma and Six Sigma Defect Examples

Table 1.1

3.8 Sigma (99% Good)

Six Sigma (99.99966% 6s)

• 200,000 wrong drug prescriptions per year

• 5,000 incorrect surgical operations per week

• More than 15,000 newborn babies accidentally

dropped per year

• 2 short or long landings at major airports per day

• 20,000 articles of mail lost per hour

• 680 wrong prescriptions per year

• 88 incorrect operations per week

• 5 newborn babies dropped per year

• Less than 1 short or long landing

every 8 years

• 7 articles lost per hour

Comparisons of Old (Traditional) and New (Lean Six Sigma) Methods

Table 1.2

Problem

Design

Analysis

Issue

Manufacturing/

Molding

Inventory level

People

Management

Employee goal

Old methods

New methods

Product performance

Product producibility

Experience based

Fixing problems

Data based

Preventing problems

Trial & error process

Robust design process

High production quantity

Low production quantity as needed

Cost to company

Cost & time

Company

Asset to company

Quality & time

Customer

Product engineering

Little input from customer

High input from customer

Quality focus

Product

Process

Dominant process

factors—selection

Apply one factor at a time

Apply design of experiment

Process improvement

Robotic technique

Optimization technique

Proving

Experience based

Company outlook

Short-term plan

Statistically based

Long-term plan

Customer satisfaction

Production at statistical

acceptance quality level

Fewer defects, when and what

quantity customer wants

External relationship

Price relationship

Long-term relationship

Layout

Functional

Production schedules

Forecast

Cell type

Customer order

Manufacturing cost

Continuously rising

Stable and decreasing

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6

Introduction to Essentials of Lean Six Sigma (6s) Strategies

of nonconformance (in the technical sense called a rejection or defect) probabil-

ity. Additional workforce, cost, scrap, and cycle time reduce the sigma level

where such variation comes from design, process, and material of the fi nished

products. Consequently, sigma variation reduces customer satisfaction and has

negative impact in the profi tability, which is one of the main focus areas of Six

Sigma.

It is too diffi cult to convert any operation from three sigma (3.0s) to Six Sigma

(6.0s) in one step. It will require several steps of improvements from 3.0s to

4.0s, 4.5s, 5.0s, 5.5s, and fi nally 6s (Tables 2.2 through 2.4 illustrate how as

sigma increases, product quality and profi tability also increase). This also means

that cycle time is reduced, quality checks are minimized, operating cost goes

down, variable costs shrink, and customer satisfaction goes up. At Six Sigma, all

products conform to a worldwide standard and are nearly defect free. In other

words, Six Sigma determines the capability of the process to accomplish a defect-

free work environment. So sigma range dictates how often defects are likely to

happen in the system. Six Sigma is not twice as good as three sigma but almost

20,000 times better.

Some examples of Six Sigma quality for long-term processes are shown in

Table 1.1.

Highlights of some of the Six Sigma cultural changes are listed in Table 1.2.

www.EngineeringEbooksPdf.comChapter 2

Statistical Theory of Lean Six

Sigma (6s) Strategies

2.1 NORMAL DISTRIBUTION CURVE

The concept of the normal distribution curve is the most important continuous

distribution in statistics. The normal distribution curve plays a key role in statisti-

cal methodology and applications. For instance, suppose for each of six days

samples of 11 parts were collected and measured for a critical dimension con-

cerning a shrinkage issue. The number of parts with dimensions is listed in Table

2.1.

Figure 2.1 illustrates the graphical representation of frequency distribution for

the data in Table 2.1, having an upper specifi cation limit (USL) 0.629, mean

0.625, and lower specifi cation limit (LSL) 0.621 (tolerance = ±0.004). This means

that any data above 0.629 and below 0.621 are assumed defects (out of specifi ca-

tion). Figure 2.1 indicates that data (population) are symmetrically distributed.

By locating bullets on the middle top of each column (as shown in Figure 2.2)

and connecting them, we set a bell-shaped curve, as in Figure 2.3, which is also

called a normal distribution curve (Figure 2.4). (This is discussed in detail in

Section 3.2.) The area under the distribution curve is the probability of variations

from the mean of any process.

2.2 SIX SIGMA PROCESS CAPABILITY CONCEPTS

Any manufactured part (or fi nished product) is considered scrap if it does not

meet the required measurement (e.g., dimensional, physical, mechanical,

or chemical properties). In other words, anything that results in customer dis-

satisfaction is called a defect. What this tells us is that the value is outside the

Essentials of Lean Six Sigma

Copyright © 2006 by Academic Press, Inc. All rights of reproduction in any form reserved.

7

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Statistical Theory of Lean Six Sigma (6s) Strategies

Table 2.1

Shrinkage Study Samples

Row

Number of parts

Dimensions

1

2

3

4

5

6

7

8

9

10

11

1

3

5

8

10

12

10

8

5

3

1

0.620

0.621 = LSL

0.622

0.623

0.624

0.625 = Mean

0.626

0.627

0.628

0.629 = USL

0.630

LSL = 0.621, Mean = 0.625, USL = 0.629

12

10

10

8

8

5

5

3

1

3

1

Dimension

LSL

USL

0.620 0.621 0.622 0.623 0.624 0.625 0.626 0.627 0.628

Figure 2.1 Histogram of sample dimensions.

www.EngineeringEbooksPdf.comStatistical Theory of Lean Six Sigma (6s) Strategies

9

LSL

LSU

Figure 2.2 Histogram of sample dimensions with bullets on the frequency.

LSL

Mean

USL

Figure 2.3 Histogram and distribution curve of sample dimensions.

www.EngineeringEbooksPdf.com

10

Statistical Theory of Lean Six Sigma (6s) Strategies

Probability of

defects

Probability of

defects

LSL

Mean

LSU

Figure 2.4 Distribution curve of sample dimensions.

customer specifi cation limits—that is, the lower specifi cation limit (LSL) and

upper specifi cation limit (USL). The LSL and USL are determined by customer

requirements. The center in between LSL and USL is known as the mean or

target value.

Once a production process is in control (using statistical process control

methods), the question is, can the process maintain its capability for a longer period

of time? To determine this, one may investigate the short- and long-term capability

of a desired process. The difference between short and long capability is shown in

the following sections. (See Chapter 8 for more on process capability.)

2.2.1 SIX SIGMA SHORT-TERM CAPABILITY

Six Sigma short-term capability occurs when the process is centered on the

target and there is no distribution shift. It also assumes continuous uniform

process with no changes. The process capability for the short term is shown in

Tables 2.2 and 2.3.

For simplicity, Figure 2.5 illustrates Table 2.2 in x and y coordinates

(magnitude).

Table 2.3 illustrates the values of Table 2.2 in percentages.

The theoretical value of 100% cannot be reached in practice because the curve

meets the x-axis in infi nity. The statistical representation (distribution curves) of

Table 2.2 for three sigma capability and Six Sigma capability when the process

is centered on the target is shown in Figure 2.6.

Figure 2.7 illustrates the distribution curve/ process and design width for Six

Sigma capability when the process is centered at the target.

www.EngineeringEbooksPdf.comStatistical Theory of Lean Six Sigma (6s) Strategies

11

Impact of Process Capability of One Sigma through Six Sigma for Short Term

When the Process Is Centered on the Target

Table 2.2

Sigma capability

Defect free/million

1.0 Sigma

2.0 Sigma

3.0 Sigma

3.5 Sigma

4.0 Sigma

4.5 Sigma

5.0 Sigma

6.0 Sigma

682,690

954,500

997,300

999,535

999,937

999,993.2

999,999.4

999,999.998

Table 2.3

Defect/million

(expected nonconformances)

317,310

45,500

2,700 (Traditional quality)

465

63

6.8

0.6

0.002 (2 parts per billion)

Mathematical Comparison of Sigma Capability Concepts for Short Term

Defect free

per million

68.269000%

95.450000%

99.730000%

99.953500%

99.993700%

99.999320%

99.999940%

99.9999998%

Defect

per million

31.731000%

04.550000%

00.270000%

00.465000%

00.006300%

00.000680%

00.000060%

00.0000002%

Quality/

profi tability

loss

an industry average

above average

Sigma

capability

1.0 Sigma

2.0 Sigma

3.0 Sigma*

3.5 Sigma

4.0 Sigma

4.5 Sigma

5.0 Sigma

6.0 Sigma

*Traditional quality

s

e

i

t

i

n

u

t

r

o

p

p

O

n

o

i

l

l

i

M

r

e

p

s

t

c

e

f

e

D

1,000,000.00

100,000.00

10,000.00

1,000.00

100.00

10.00

1.00

0.10

0.01

0.00

1

2

3

3.5

4

4.5

5

6

Sigma Level

Figure 2.5 Capability of one sigma through Six Sigma for short term.

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12

Statistical Theory of Lean Six Sigma (6s) Strategies

1350 dpm

LSL

Target

(A)

3 Sigma Process Centered

1350 dpm

USL

6 Sigma Process Centered

0.001 dpm

0.001 dpm

LSL

(B)

LSU

Figure 2.6 A. Three Sigma capability. B. Six Sigma capability. When the process is centered on

the target.

-6σ

+6σ

-6σ

-3σ

+3σ

+6σ

LSL

Process Range

Design Range

USL

Figure 2.7 Distribution curve centered at the target.

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Statistical Theory of Lean Six Sigma (6s) Strategies

13

2.2.2 ESTIMATION OF SIX SIGMA LONG-TERM CAPABILITY

Six Sigma assumes that the process mean changes from lot to lot, especially

within a large quantity of lots. In addition, this change could result in an average

of 1.5 sigma distribution shift in either direction for long-term performance. This

is due to the fact that operator error and machine wear and tear contribute to the

offset of sigma from the target line. This is demonstrated in Table 2.4 for long-

term operation. Using this concept one can use an Excel spreadsheet and Equa-

tion 2.1 to calculate the sigma capability for various nonconformance probabilities

of product yield.

Sigma Capability NORMSINV Probability

=

(





)

Defects

6

10





Sigm

aa Capability NORMSINV 1

=

−

+

1 5

.

(2.1)

where NORMSINV stands for the inverse of the standard normal cumulative

distribution. Basically, NORMSINV uses an iterative method for evaluating the

function. By giving a probability value [1 − (Total defect/total op...