Six Sigma and Statistical Quality Control. Outline Quality and Six Sigma: Basic ideas and history...

Six Sigma andStatistical Quality

Control

Outline• Quality and Six Sigma: Basic ideas and history

• Juran Trilogy– Control

– Improvement

– Planning

• Quality Strategy

• Focus on Statistical Methods– Process Capability ideas and metrics

– Control charts for attributes and variables

A Brief History• The Craft System• Taylorism (Scientific Management)• Statistical Quality Control

– Pearson, Shewhart, Dodge• Human Relations School

– Mayo, Maslow, Simon, Herzberg, Likert• The Japanese Revolution (1950)

– Ishikawa, Taguchi, Deming, Juran, Feigenbaum• The USA Wakes Up (1980)

– Crosby• 1990s: Six Sigma• The Need for Organizational Change

Operations -- Prof. Juran

4

JIT and TQM

Walter Shewhart1891 - 1967

W. Edwards Deming

1900 - 1993

Joseph M. Juran1904 - 2008

http://upload.wikimedia.org/wikipedia/commons/a/a4/Flag_of_the_United_States.svg

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What is Quality?

• Freedom from Defects– Quality Costs Less– Affects Costs

• Presence of Features– Quality Costs More– Affects Revenue

Juran TrilogyPlanning, Control, Improvement

Juran Trilogy

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Juran TrilogyPlanning, Control, Improvement

Juran Trilogy

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Chronic Waste

Chronic Waste

Sporadic Spike

Planning Control ControlImprovement

Quality Control

• Aimed at preventing unwanted changes

• Works best if deployed at the point of production or service delivery (Empowerment)

• Tools:– Established, measurable standards– Measurement and feedback– Control charts– Statistical inference

Quality Control

Operate

Establish Standard

Corrective Action

Measure Performance

Compare to StandardOK?

Yes

No

Quality Improvement• Aimed at creating a desirable change• Two distinct “journeys”

– Diagnosis– Remedy

• Project team approach• Tools

– Process flow diagram– Pareto analysis– Cause-effect (Ishikawa, fishbone) diagram– Statistical tools

Quality Improvement

• Identify problem• Analyze symptoms• Formulate theories• Test theories - Identify root

cause• Identify remedy• Address cultural resistance• Establish control

Quality Planning• Aimed at creating or redesigning

(re-engineering) a process to satisfy a need

• Project team approach• Tools

– Market research– Failure analysis– Simulation– Quality function deployment– Benchmarking

Quality Planning• Verify goal• Identify customers• Determine customer needs• Develop product• Develop process• Transfer to operations• Establish control

Strategic Quality Planning• Mission• Vision• Long-term objectives• Annual goals• Deployment of goals• Assignment of resources• Systematic measurement• Connection to rewards and

recognition

Strategic Quality Planning• Aimed at establishing long-range

quality objectives and creating an approach to meeting those objectives

• Top management’s job• Integrated with other objectives

– Operations– Finance– Marketing– Human Resources

Six Sigma Defined (High-Level)

• The application of rigorous quantitative tools in the planning, control, and improvement processes in an effort to meet customer requirements 100% of the time.

• Old wine in new bottles?

Process Capability

• The Relationship between a Process and the Requirements of its Customer

• How Well Does the Process Meet Customer Needs?

Process Capability

• Specification Limits reflect what the customer needs

• Natural Tolerance Limits (a.k.a. Control Limits) reflect what the process is capable of actually delivering

• These look similar, but are not the same

Specification Limits

• Determined by the Customer• A Specific Quantitative Definition

of “Fitness for Use”• Not Necessarily Related to a

Particular Production Process• Not Represented on Control Charts

Tolerance (Control) Limits• Determined by the inherent central

tendency and dispersion of the production process

• Represented on Control Charts to help determine whether the process is “under control”

• A process under control may not deliver products that meet specifications

• A process may deliver acceptable products but still be out of control

Measures of Process Capability

• Cp

• Cpk

• Percent Defective• Sigma Level

Example: Cappuccino• Imagine that a franchise food service

organization has determined that a critical quality feature of their world-famous cappuccino is the proportion of milk in the beverage, for which they have established specification limits of 54% and 64%.

• The corporate headquarters has procured a custom-designed, fully-automated cappuccino machine which has been installed in all the franchise locations.

• A sample of one hundred drinks prepared at the company’s Stamford store has a mean milk proportion of 61% and a standard deviation of 3%.

Example: Cappuccino• Assuming that the process is in control and normally

distributed, what proportion of cappuccino drinks at the Stamford store will be nonconforming with respect to milk content?

• Try to calculate the Cp, Cpk, and Parts per Million for this process.

• If you were the quality manager for this company, what would you say to the store manager and/or to the big boss back at headquarters? What possible actions can be taken at the store level, without changing the inherent variability of this process, to reduce the proportion of non-conforming drinks?

Lower Control Limit

52

03361

3

.

..

LNTL

Upper Control Limit

70

03361

3

.

..

UNTL

Nonconformance

001036164

332036154

2

1

../..Z

../..Z

Nonconformance

-4 -3 -2 -1 0 1 2 3 4values of z

Nonconformance• 0.00990 of the drinks will fall

below the lower specification limit.• 0.84134 of the drinks will fall

below the upper limit. • 0.84134 - 0.00990 = 0.83144 of

the drinks will conform.• Nonconforming:

1.0 - 0.83144 = 0.16856 (16.856%)

Cp Ratio

pC6

LSLUSL

0306540640

...

180100

.

.

5550 .

Cpk Ratio

pkC

33

LSL,

USLmin

0303540610

0303610640

...

,.

..min

77703330 .,.min

3330 .

Parts per Million

PPM 16900000001 .,, 000169,

(about 1.38 Sigma)

1234

A B C D E F

Defects per million 169000

Sigma level 1.375424

=1000000*2*(1-NORMSDIST(B4))

Quality Improvement

• Two Approaches:– Center the Process between the

Specification Limits– Reduce Variability

Approach 1: Center the Process


671035964

671035954

../..Z

../..Z

USL

LSL


• 0.04746 of the drinks will fall below the lower specification limit.

• 0.95254 of the drinks will fall below the upper limit.

• 0.95254 - 0.04746 = 0.90508 of the drinks will conform.

• Nonconforming:1.0 - 0.90508 = 0.09492 (9.492%)


• Nonconformance decreased from 16.9% to 9.5%.

• The inherent variability of the process did not change.

• Likely to be within operator’s ability.

Approach 2: Reduce Variability

• The only way to reduce nonconformance below 9.5%.

• Requires managerial intervention.

Quality Control

Operate

Establish Standard

Corrective Action

Measure Performance

Compare to StandardOK?

Yes

No

Quality Control

• Aimed at preventing and detecting unwanted changes

• An important consideration is to distinguish between Assignable Variation and Common Variation

• Assignable Variation is caused by factors that can clearly be identified and possibly managed

• Common Variation is inherent in the production process

• We need tools to help tell the difference

When is Corrective Action Required?

• Operator Must Know How They Are Doing

• Operator Must Be Able to Compare against the Standard

• Operator Must Know What to Do if the Standard Is Not Met


• Use a Chart with the Mean and 3-sigma Limits (Control Limits) Representing the Process Under Control

• Train the Operator to Maintain the Chart

• Train the Operator to Interpret the Chart

Example: Run Chart

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Here are four indications that a process is “out of control”. If any one of these things happens, you should stop the machine and call a quality engineer:

• One point falls outside the control limits.

• Seven points in a row all on one side of the center line.

• A run of seven points in a row going up, or a run of seven points in a row going down.

• Cycles or other non-random patterns.

Example: Run Chart

14.000

14.100

14.200

14.300

14.400

14.500

14.600

14.700

14.800

14.900

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15.200

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15.400

15.500

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Type I and Type II Errors

In Control Out of Control

Assume Process is OK Correct Decision ERROR (Type II)

Take Corrective Action ERROR (Type I) Correct Decision


One point falls outside the control limits.

• 0.27% chance of Type I Error

Seven points in a row all on one side of the center line.


A run of seven points in a row going up, or a run of seven points in a row going down.


Basic Types of Control Charts

Attributes (“Go – No Go” data)• A simple yes-or-no issue, such as “defective or not”

• Data typically are “proportion defective”

• p-chart

Variables (Continuous data)• Physical measurements such as dimensions, weight,

electrical properties, etc.

• Data are typically sample means and standard deviations

• X-bar and R chart

Statistical Symbols (Attributes)

n Number of observations in each sample

p Proportion defective (an unknown population parameter)

p Sample proportion defective (estimate of p from one sample)

p Sample proportion defective (estimate of p from all samples)

sp Standard deviation of sample proportions

p-chart Example

size Samplesamplein defectives of Number

p

samples of Number * size Samplesamples allin defectives of Number

p

p-chart Example

n

ppsp

1

pzspUCL

pzspLCL

Sample Sample Size Number of Defects p

1 100 4 0.04 2 100 2 0.02 3 100 5 0.05 4 100 3 0.03 5 100 6 0.06 6 100 4 0.04 7 100 3 0.03 8 100 7 0.07 9 100 1 0.01 10 100 2 0.02 11 100 3 0.03 12 100 2 0.02 13 100 2 0.02 14 100 8 0.08 15 100 3 0.03

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A B C D E F G HSample Sample Size Number of Defects p-bar

1 100 4 0.042 100 2 0.023 100 5 0.054 100 3 0.035 100 6 0.066 100 4 0.047 100 3 0.038 100 7 0.079 100 1 0.01

10 100 2 0.0211 100 3 0.0312 100 2 0.0213 100 2 0.0214 100 8 0.0815 100 3 0.03

p-bar-bar 0.0367stdev(p-bar) 0.0188

=AVERAGE(D2:D16)=SQRT(((D17)*(1-D17))/(B16))

Note: If the LCL is negative, we round it up to zero.

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A B C D E F GSample Sample Size Number of Defects p-bar LCL UCL

1 100 4 0.04 0 0.09302 100 2 0.02 0 0.09303 100 5 0.05 0 0.09304 100 3 0.03 0 0.09305 100 6 0.06 0 0.09306 100 4 0.04 0 0.09307 100 3 0.03 0 0.09308 100 7 0.07 0 0.09309 100 1 0.01 0 0.0930

10 100 2 0.02 0 0.093011 100 3 0.03 0 0.093012 100 2 0.02 0 0.093013 100 2 0.02 0 0.093014 100 8 0.08 0 0.093015 100 3 0.03 0 0.0930

p-bar-bar 0.0367stdev(p-bar) 0.0188

UCL 0.0930LCL -0.0197

=D17+3*D18=D17-3*D18

=$D$20

0.000

0.025

0.050

0.075

0.100

0.125

0.150

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Statistical Symbols (Variables)

X A Random Variable

xi One Observation of X

n The Number of Observations of X

Mean (A Population Parameter)

X Mean (An Estimate of from Sample Data) Standard Deviation (A Population Parameter)

s Standard Deviation (An Estimate of from Sample Data)

X-bar, R chart Example

For X-bar chart

RAXLCL 2 RAXUCL 2

For R chart

RDLCL 3 RDUCL 4

Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 1 10.682 10.689 10.776 10.798 10.714 2 10.787 10.860 10.601 10.746 10.779 3 10.780 10.667 10.838 10.785 10.723 4 10.591 10.727 10.812 10.775 10.730 5 10.693 10.708 10.790 10.758 10.671 6 10.749 10.714 10.738 10.719 10.606 7 10.791 10.713 10.689 10.877 10.603 8 10.744 10.779 10.110 10.737 10.750 9 10.769 10.773 10.641 10.644 10.725 10 10.718 10.671 10.708 10.850 10.712 11 10.787 10.821 10.764 10.658 10.708 12 10.622 10.802 10.818 10.872 10.727 13 10.657 10.822 10.893 10.544 10.750 14 10.806 10.749 10.859 10.801 10.701 15 10.660 10.681 10.644 10.747 10.728

From Exhibit TN7.7

n A2 D3 D4 2 1.88 0.00 3.27 3 1.02 0.00 2.57 4 0.73 0.00 2.28 5 0.58 0.00 2.11 6 0.48 0.00 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74

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1011121314151617

A B C D E F G H I J KSample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Avg Range

1 10.682 10.689 10.776 10.798 10.714 10.732 0.1162 10.787 10.860 10.601 10.746 10.779 10.755 0.2593 10.780 10.667 10.838 10.785 10.723 10.759 0.1714 10.591 10.727 10.812 10.775 10.730 10.727 0.2215 10.693 10.708 10.790 10.758 10.671 10.724 0.1196 10.749 10.714 10.738 10.719 10.606 10.705 0.1437 10.791 10.713 10.689 10.877 10.603 10.735 0.2748 10.744 10.779 10.110 10.737 10.750 10.624 0.6699 10.769 10.773 10.641 10.644 10.725 10.710 0.132

10 10.718 10.671 10.708 10.850 10.712 10.732 0.17911 10.787 10.821 10.764 10.658 10.708 10.748 0.16312 10.622 10.802 10.818 10.872 10.727 10.768 0.25013 10.657 10.822 10.893 10.544 10.750 10.733 0.34914 10.806 10.749 10.859 10.801 10.701 10.783 0.15815 10.660 10.681 10.644 10.747 10.728 10.692 0.103

Averages 10.728 0.220

=AVERAGE(B10:F10)

=MAX(B12:F12)-MIN(B12:F12)

=AVERAGE(H2:H16)

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1011121314151617

A B C D E F G H I J K L MSample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Avg Range n A2 D3 D4

1 10.682 10.689 10.776 10.798 10.714 10.732 0.116 2 1.88 0.00 3.272 10.787 10.860 10.601 10.746 10.779 10.755 0.259 3 1.02 0.00 2.573 10.780 10.667 10.838 10.785 10.723 10.759 0.171 4 0.73 0.00 2.284 10.591 10.727 10.812 10.775 10.730 10.727 0.221 5 0.58 0.00 2.115 10.693 10.708 10.790 10.758 10.671 10.724 0.119 6 0.48 0.00 2.006 10.749 10.714 10.738 10.719 10.606 10.705 0.143 7 0.42 0.08 1.927 10.791 10.713 10.689 10.877 10.603 10.735 0.274 8 0.37 0.14 1.868 10.744 10.779 10.110 10.737 10.750 10.624 0.669 9 0.34 0.18 1.829 10.769 10.773 10.641 10.644 10.725 10.710 0.132 10 0.31 0.22 1.78

10 10.718 10.671 10.708 10.850 10.712 10.732 0.179 11 0.29 0.26 1.7411 10.787 10.821 10.764 10.658 10.708 10.748 0.16312 10.622 10.802 10.818 10.872 10.727 10.768 0.250 A2 0.5813 10.657 10.822 10.893 10.544 10.750 10.733 0.349 D3 014 10.806 10.749 10.859 10.801 10.701 10.783 0.158 D4 2.1115 10.660 10.681 10.644 10.747 10.728 10.692 0.103 Mean 10.728

Averages 10.728 0.220 Range 0.220

=K5

=G17=H17

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1011121314151617181920212223

A B C D E F G H I J KSample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Avg Range

1 10.682 10.689 10.776 10.798 10.714 10.732 0.1162 10.787 10.860 10.601 10.746 10.779 10.755 0.2593 10.780 10.667 10.838 10.785 10.723 10.759 0.1714 10.591 10.727 10.812 10.775 10.730 10.727 0.2215 10.693 10.708 10.790 10.758 10.671 10.724 0.1196 10.749 10.714 10.738 10.719 10.606 10.705 0.1437 10.791 10.713 10.689 10.877 10.603 10.735 0.2748 10.744 10.779 10.110 10.737 10.750 10.624 0.6699 10.769 10.773 10.641 10.644 10.725 10.710 0.132

10 10.718 10.671 10.708 10.850 10.712 10.732 0.17911 10.787 10.821 10.764 10.658 10.708 10.748 0.16312 10.622 10.802 10.818 10.872 10.727 10.768 0.250 A2 0.5813 10.657 10.822 10.893 10.544 10.750 10.733 0.349 D3 014 10.806 10.749 10.859 10.801 10.701 10.783 0.158 D4 2.1115 10.660 10.681 10.644 10.747 10.728 10.692 0.103 Mean 10.728

Averages 10.728 0.220 Range 0.220

X-Bar UCL 10.85625LCL 10.60058

R-Bar UCL 0.465044LCL 0

=K16+K13*K17=K16-K13*K17

=K15*K17=K14*K17

X-bar Chart

10.45

10.50

10.55

10.60

10.65

10.70

10.75

10.80

10.85

10.90

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R chart

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Interpretation

• Does any point fall outside the control limits?

• Are there seven points in a row all on one side of the center line?

• Is there a run of seven points in a row going up, or a run of seven points in a row going down?

• Are there cycles or other non-random patterns?

A Process in which the Specification Limits are Six Standard Deviations above and below the Process Mean

Two Approaches:• Move the Specification Limits Farther Apart

• Reduce the Standard Deviation

Six Sigma Defined (Low-Level)

Approach #1

Ask the Customer to Move the Specification Limits Farther Apart.

2-sigma ProcessCp = 0.667Cpk = 0.66745,500 ppm

Spec Limits


Spec Limits

4-sigma ProcessCp = 1.333Cpk = 1.333

63 ppm

Spec Limits


0.57 ppm

Spec Limits

6-sigma ProcessCp = 2.0Cpk = 2.00.002 ppm

Spec Limits

Approach #2

Reduce the Standard Deviation.


Spec Limits


63 ppm

Spec Limits


0.57 ppm

Spec Limits

6-sigma ProcessCp = 2.0Cpk = 2.00.002 ppm

Spec Limits

Process Drift

What Happens when the Process Mean Is Not Centered between the Specification Limits?

3-sigma Process(centered)Cp = 1.0Cpk = 1.02,700 ppm

Spec Limits

3-sigma Process(shifted 0.5 std. dev.)

Cp = 1.0Cpk = 0.833ppm = 6,442

(about 2.72-sigma)

Spec Limits


Cp = 1.0Cpk = 0.667

ppm = 22,782(about 2.28-sigma)

Spec Limits


Cp = 1.0Cpk = 0.5

ppm = 66,811(about 1.83-sigma)

Spec Limits

Six Sigma: Many Meanings

• A Symbol • A Measure • A Benchmark or Goal• A Philosophy• A Method

Six Sigma: A Symbol

• is a Statistical Symbol for Standard Deviation

• Standard Deviation is a Measure of Dispersion, Volatility, or Variability

Six Sigma: A Measure

• The “Sigma Level” of a process can be used to express its capability — how well it performs with respect to customer requirements.

• Percent Defects, Cp, Cpk, PPM

Six Sigma: A Benchmark or Goal

• The specific value of 6 Sigma (as opposed to 5 or 4 Sigma) is a benchmark for process excellence.

• Adopted by leading organizations as a goal for process capability.

Six Sigma: A Philosophy

• A vision of process performance• Tantamount to “zero defects”• A “Management Mantra”

Six Sigma: A Method

• Really a Collection of Methods:– Product/Service Design– Quality Control– Quality Improvement– Strategic Planning

Where Does “3.4 PPM” Come From?

• Six Sigma is commonly defined to be equivalent to 3.4 defective parts per million.

• Juran says that a Six Sigma process will produce only 0.002 defective parts per million.

• What gives?

Normal Curve Probabilities

±1 Sigma 68.3% of Data ±2 Sigmas 95.4% ±3 Sigmas 99.73% ±4 Sigmas 99.994% ±5 Sigmas 99.99994% ±6 Sigmas 99.9999998%

12345678

A B CArea under the curve z > 6

0.00000000099

Total area outside of 6 sigma limits0.00000000198

Parts per Million0.00198

=1-NORMSDIST(6)

=2*A2

=A5*1000000

Process Centered between Spec Limits

Sigma Level Cp Cpk PPM1 0.333 0.333 317,3102 0.667 0.667 45,5003 1.000 1.000 2,7004 1.333 1.333 63.45 1.667 1.667 0.576 2.000 2.000 0.002

Process Shifted by 1.5 Standard Deviations

Sigma Level Cp Cpk PPM1 0.333 -0.167 697,6722 0.667 0.167 308,7703 1.000 0.500 66,8104 1.333 0.833 6,2105 1.667 1.167 232.76 2.000 1.500 3.4

Where Does “3.4 PPM” Come From?

• The 3.4 defective parts per million definition of Six Sigma includes a “worst case” scenario of a 1.5 standard deviation shift in the process.

• It is assumed that there is a very high probability that such a shift would be detected by SPC methods (low probability of Type II error).

Six Sigma in Context

• Is Six Sigma dramatically different from old-fashioned quality control?

• Is Six Sigma a departure from 1980’s-style TQM?

Six Sigma in Context• What Is New?

– Focus on Quantitative Methods– Focus On Control– A Higher Standard– A New Metric for Defects (PPM)– Lots of training– Linkage between quality goals and

employee incentives?

Using Six Sigma

• A New Standard; Not Adopted Uniformly across Industries

• Beyond Generalities, Need to Develop Organization-Specific Methods

• Hard Work, Not Magic• “A Direction Not a Place”

Summary• Quality and Six Sigma: Basic ideas and history

• Juran Trilogy– Control

– Improvement

– Planning

• Quality Strategy

• Focus on Statistical Methods– Process Capability ideas and metrics

– Control charts for attributes and variables

Six Sigma and Statistical Quality Control. Outline Quality and Six Sigma: Basic ideas and history...

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