Six-Sigma Quality, Process Capability, and Statistical Process

ControlSelected Slides from Jacobs et al, 9th Edition

Operations and Supply Management Chapter 9 and 9A

Edited, Annotated and Supplemented byPeter Jurkat

Total Quality Management (TQM)• Total quality

management is defined as managing the entire organization so that it excels on all dimensions of products and services that are important to the customer

• Design quality: Inherent value of the product in the marketplace

– Dimensions include: Performance, Features, Reliability/Durability, Serviceability, Aesthetics, and Perceived Quality.

• Conformance quality: Degree to which the product or service design specifications are met

Six Sigma Quality

• A philosophy and set of methods companies use to eliminate defects in their products and processes

• Seeks to reduce variation in the processes that lead to product defects

9-3

McGraw-Hill/Irwin

Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

Gurus and their wisdom

Consensus

• Gurus had considerable differences• After 30 years get some consensus

– Senior level leadership– Customer focus– Work force involvement– Process analysis– Continuous improvement

Costs of Quality

External Failure Costs

Appraisal Costs

Prevention Costs

Internal FailureCosts

Costs ofQuality

9-6

9-7Cost of Quality Example

At 20% of sales, represents about $2M sales, at 2.5% about $73M sales

Six Sigma Quality Measurement

• Six Sigma allows managers to readily describe process performance using a common metric: Defects Per Million Opportunities (DPMO)

• Defect associated with a critical-to-quality characteristic: a measurable quantity used to identify failure

• Statistical six sigma goal is 3.4 failures per 1,000,000 opportunities 1 failure in about 300,000 DPMO (actually 1 in 294,118)

1,000,000 x

units of No. x

unit per error for

iesopportunit ofNumber

defects ofNumber

DPMO 1,000,000 x

units of No. x

unit per error for

iesopportunit ofNumber

defects ofNumber

DPMO

9-8

9A-9Process Capability• Shows to what extent (probability) parts are produced that meet and fall outside of specifications•Achieved when process variation (s.d.) is so small that an acceptable proportion are defects – Six-Sigma goal is 3.4 out of one million

Bearing Diameter

Why 3.4 DPMO?• Six sigma between mean and, say, upper

specification limit results 1 defect in 100,000,000 (see SixSigmaOrigin.xls)

• Experience has shown that in long term processes have a wider variation than in short term studies, which results in defects with probability beyond 4.5 sigma, i.e. 3.4 DPMO (1.5 sigma less than 6 sigma)

• See origin of this at http://en.wikipedia.org/wiki/Six_Sigma ( which bases above on Tennant, Geoff (2001). SIX SIGMA: SPC and TQM in Manufacturing and Services. Gower Publishing, Ltd., p. 25. ISBN 0566083744.)

9A-11

Mean shift during process improvement

Still an improvement but capability is now measured against closest of LTL and UTL

LTL = lower tolerance limit UTL = upper tolerance limit

Measure Example

Example of Defects Per Million Opportunities (DPMO) calculation. Suppose we observe 200 letters delivered incorrectly to the wrong addresses in a small city during a single day when a total of 200,000 letters were delivered. What is the DPMO in this situation?

Example of Defects Per Million Opportunities (DPMO) calculation. Suppose we observe 200 letters delivered incorrectly to the wrong addresses in a small city during a single day when a total of 200,000 letters were delivered. What is the DPMO in this situation?

000,1 1,000,000 x

200,000 x 1

200DPMO

000,1 1,000,000 x

200,000 x 1

200DPMO

So, for every one million letters delivered this city’s postal managers can expect to have 1,000 letters incorrectly sent to the wrong address.

So, for every one million letters delivered this city’s postal managers can expect to have 1,000 letters incorrectly sent to the wrong address.

Cost of Quality: What might that DPMO mean in terms of over-time employment to correct the errors?

Cost of Quality: What might that DPMO mean in terms of over-time employment to correct the errors?

9-12

8-13Service Blueprint, Failure Anticipation, and Poka-Yokes

Complete blueprint (p262-3) identifies 16 failure opportunities

Toyota Dealer Service Example

• Blueprint identified 16 failure opportunities per customer

• Assume 20 customers /day => 80,000 customers/year for 250 working days per year

• At 3.4 failures per 1,000,000 opportunities this would allow .272 failures/year, or 3 2/3 years between failures

• What is the critical-to-quality characteristic of the first identified failure? Second failure?

DMAIC Cycle• GE developed

methodology• Overall focus is to

understand and achieve what the customer wants (Juran)

• Identifies defects and variation in processes as underlying cause of defects (Deming)

• A 6-sigma program seeks to reduce the variation in the processes that lead to these defects

• Define customers and their priorities

• Measure process and its performance

• Analyze causes of defects

• Improve by removing causes

• Control to maintain quality

DMAIC in Action• We are the maker of a

cereal. Consumer Reports has just published an article that shows that we frequently have less than 16 ounces of cereal in a box.

• What should we do?1. Define

a. What is the critical-to-quality characteristic?

b. The CTQ (critical-to-quality) characteristic in this case is the weight of the cereal in the box.

2. Measurea. How would we measure to

evaluate the extent of the problem?

b. What are acceptable limits on this measure?

c. Let’s assume that the government says that we must be within ± 5 percent of the weight advertised on the box.

d. Upper Tolerance Limit = 16 + .05(16) = 16.8 ounces

e. Lower Tolerance Limit = 16 – .05(16) = 15.2 ounces

f. Survey: 1000 boxes have mean weight = 15.875 oz with s.d. = .529

Upper Tolerance = 16.8

Lower Tolerance = 15.2

ProcessMean = 15.875Std. Dev. = .529

What percentage of boxes are defective (i.e. less than 15.2 oz)?

Z = (x – Mean)/Std. Dev. = (15.2 – 15.875)/.529 = -1.276

NORMSDIST(Z) = NORMSDIST(-1.276) = .100978

Approximately, 10 percent of the boxes have less than 15.2 Ounces of cereal in them – way out of six-sigma specs

9-17

DMAIC in Action

3. Analyze - how can we improve the capability of our cereal box filling process?a. Decrease Variationb. Center Processc. Tighten Specifications

4. Improve – How good is good enough?

a. Set center spec at goal (16 oz in this case)

b. Set controls so that a deviation of 6 s.d. occurs only 3.4 times out of a million

DMAIC in Action5. Statistical Process

Controla. Use data from actual

processb. Estimate distributionsc. Calculate capability – do

better if not adequate (actually do better all the time)

d. Statistically monitor the process over time

e. Tools1) Process flow charts (e.g.,

Toyota service blueprint)2) Run charts3) Pareto charts4) Check sheets5) Cause-and-effect

diagrams6) Opportunity flow

diagrams7) Failure mode and effect

analysis (FMEA)8) Statistical Process Control

(SPC) and Control charts9) Design of Experiments

(DOE)

9-20

9-21

9-22

9-23

Severity: cost of damage, rating numberOccurrence: observed relative frequency, predicted probabilityDetection: probability of detectionRPN = Occurrence X Severity X Detection

Failure Mode and Effect Analysis

9-24

Part of Statistical Process Control•Uses statistical theory and practice to follow processes in order to determine if they are within specification/control•Also used to predict if a process might be going out of control while still within specs•General approach is to sample a process at intervals, plot the results and compare these to control limits

Upper Control Limit

Lower Control Limit

Statistical Process Control

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

• Common variation is inherent in the production process

Example: A poorly trained employee that creates variation in finished product output.

Example: A poorly trained employee that creates variation in finished product output.

Example: A molding process that always leaves “burrs” or flaws on a molded item.

Example: A molding process that always leaves “burrs” or flaws on a molded item.

9A-25

•Based on statistical theory of variation (dispersion)•Defines process capability•Establishes process control limits•Controls process bases on periodic sampling (small samples as opposed to inspecting/measuring everything)

Variation

Taguchi’s View of Variation

IncrementalCost of Variability

High

Zero

LowerSpec

TargetSpec

UpperSpec

Traditional View

IncrementalCost of Variability

High

Zero

LowerSpec

TargetSpec

UpperSpec

Taguchi’s View

Traditional view is that quality within the LS and US is good and that the cost of quality outside this range is constant, Taguchi views costs as increasing as variability increases, so seek to achieve zero defects and that will truly minimize quality costs.

Traditional view is that quality within the LS and US is good and that the cost of quality outside this range is constant, Taguchi views costs as increasing as variability increases, so seek to achieve zero defects and that will truly minimize quality costs.

9A-26

Upper and lower specs are also called upper and lower tolerance limits (UTL and LTL)

Process Capability Index, Cpk

3

X-UTLor

3

LTLXmin=C pk

Shifts in Process Mean

Capability Index shows how well parts being produced fit into design limit specifications.

Capability Index shows how well parts being produced fit into design limit specifications.

As a production process produces items small shifts in equipment or systems can cause differences in production performance from differing samples.

As a production process produces items small shifts in equipment or systems can cause differences in production performance from differing samples.

9A-27

LTL/UTL = lower/upper tolerance limit

The Cereal Box Example

• We are the maker of this cereal. Consumer reports has just published an article that shows that we frequently have less than 16 ounces of cereal in a box.

• Let’s assume that the government says that we must be within ± 5 percent of the weight advertised on the box.

• Upper Tolerance Limit = 16 + .05(16) = 16.8 ounces• Lower Tolerance Limit = 16 – .05(16) = 15.2 ounces• We go out and buy 1,000 boxes of cereal and find that

they weight an average of 15.875 ounces with a standard deviation of .529 ounces.

9A-28

Cereal Box Process Capability

• Specification or Tolerance Limits– Upper Spec = 16.8 oz– Lower Spec = 15.2 oz

• Observed Weight– Mean = 15.875 oz– Std Dev = .529 oz

3

;3

XUTLLTLXMinC pk

)529(.3

875.158.16;

)529(.3

2.15875.15MinC pk

5829.;4253.MinC pk

4253.pkC

9A-29

What does a Cpk of .4253 mean?

• An index that shows how well the units being produced fit within the specification limits.

• This is a process that will produce a relatively high number of defects.

• Many companies look for a Cpk of 1.3 or better… 6-Sigma company wants 2.0!

9A-30

Types of Statistical Sampling in SPC

• Attribute (overall acceptable or not)– Defectives refers to the acceptability of product across a

range of characteristics.– Defects refers to the number of defects per unit which

may be higher than the number of defectives.– p-chart application (p for proportion)

• Variable (Continuous)– Usually actual dimensions (length, weight, hardness, …)– Usually measured by the mean and the standard

deviation.– X-bar and R chart applications (x-bar for mean and R for

range – much easier to measure than s.d.)

9A-31

9A-32

Control Charts

Statistical Process Control Formulas:Attribute Measurements (p-Chart)

p =Total Number of Defectives

Total Number of Observationsp =

Total Number of Defectives

Total Number of Observations

ns

)p-(1 p = p n

s)p-(1 p

= p

p

p

z - p = LCL

z + p = UCL

s

s

p

p

z - p = LCL

z + p = UCL

s

s

Given:

Compute control limits:

9A-33

1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sample

1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sample

Sample n Defectives p1 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

Example of Constructing a p-chart

9A-34

2. Calculate the average and s.d. of the sample proportions

2. Calculate the average and s.d. of the sample proportions

0.036=1500

55 = p 0.036=1500

55 = p

.0188= 100

.036)-.036(1=

)p-(1 p = p n

s .0188= 100

.036)-.036(1=

)p-(1 p = p n

s

Example of Constructing a p-Chart

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

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

Observation

p

UCL

LCL

Calculate control limits and plot the individual sample proportions, the average of the proportions, and the control limits

Calculate control limits and plot the individual sample proportions, the average of the proportions, and the control limits

9A-35

3(.0188) .036 3(.0188) .036 UCL = 0.0924LCL = -0.0204 (or 0)

UCL = 0.0924LCL = -0.0204 (or 0)

Example of x-bar and R charts: Steps: Calculate x-bar Chart and Plot Values

10.601

10.856

=).58(0.2204-10.728RA - x = LCL

=).58(0.2204-10.728RA + x = UCL

2

2

10.601

10.856

=).58(0.2204-10.728RA - x = LCL

=).58(0.2204-10.728RA + x = UCL

2

2

10.550

10.600

10.650

10.700

10.750

10.800

10.850

10.900

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

Sample

Mea

ns

UCL

LCL

9A-36

Example of x-bar and R charts: Calculate R-chart and Plot Values

0

0.46504

)2204.0)(0(R D= LCL

)2204.0)(11.2(R D= UCL

3

4

0

0.46504

)2204.0)(0(R D= LCL

)2204.0)(11.2(R D= UCL

3

4

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

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

Sample

RUCL

LCL

9A-37

9A-38

Common criteria for concluding process is out of control or in

danger of being so

• Advantages– Economy– Less handling damage– Fewer inspectors– Upgrading of the

inspection job– Applicability to destructive

testing– Entire lot rejection

(motivation for improvement)

• Disadvantages– Risks of accepting “bad”

lots and rejecting “good” lots

– Added planning and documentation

– Sample provides less information than 100-percent inspection

Acceptance Sampling

•Purposes•Determine quality level of acquired goods or services (“after the fact”) when no sampling of production process is available•Ensure quality is within predetermined level

Risk

• Acceptable Quality Level (AQL)– Max. acceptable percentage of defectives defined

by producer

• The(Producer’s risk)– The probability of rejecting a good lot– Probability of Type I error based on consumer’s

null hypothesis that lot is good

• Lot Tolerance Percent Defective (LTPD)– Percentage of defectives that defines consumer’s

rejection point

• The (Consumer’s risk)– The probability of accepting a bad lot– Probability of Type II error based on consumer’s

null hypothesis that lot is good

9A-40

Operating Characteristic Curve

n = 99c = 4

AQL LTPD

00.10.20.30.40.50.60.70.80.9

1

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

Percent defective

Pro

bab

ilit

y of

acc

epti

ng

lots

w

ith

giv

e %

of

def

ecti

ves

=.10(consumer’s risk = accept bad lot)

= .05 (producer’s risk = reject good lot)

The OCC brings the concepts of producer’s risk, consumer’s risk, sample size, and maximum defects allowed together

The OCC brings the concepts of producer’s risk, consumer’s risk, sample size, and maximum defects allowed together

The shape or slope of the curve is dependent on a particular combination of the four parameters

The shape or slope of the curve is dependent on a particular combination of the four parameters

9A-41

n = sample sizec = acceptance number (max defectives allowed before lot is rejected)

H0: Lot is goodHa: Lot is bad

Example: Acceptance Sampling Problem

Zypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard. Procard has set an acceptable quality level of 1% and accepts a 5% risk of rejecting lots at or below this level. Zypercom considers lots with 3% defectives to be unacceptable and will assume a 10% risk of accepting a defective lot.

Develop a sampling plan for Zypercom and determine a rule to be followed by the receiving inspection personnel.

Zypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard. Procard has set an acceptable quality level of 1% and accepts a 5% risk of rejecting lots at or below this level. Zypercom considers lots with 3% defectives to be unacceptable and will assume a 10% risk of accepting a defective lot.

Develop a sampling plan for Zypercom and determine a rule to be followed by the receiving inspection personnel.

9A-42

Example: What is given and what is not?

In this problem, AQL is given to be 0.01 and LTDP is given to be 0.03. We are also given an alpha of 0.05 and a beta of 0.10.

In this problem, AQL is given to be 0.01 and LTDP is given to be 0.03. We are also given an alpha of 0.05 and a beta of 0.10.

What you need to determine is your sampling plan is “c” and “n.”

What you need to determine is your sampling plan is “c” and “n.”

9A-43

LTPD = Lot tolerant percent defective (buyers)AQL = Acceptable quality level (seller)

For a give allowed sampling error SE and confidence C = 1 - α the sample size is determined by:

2

22/α

SE

pqzn =

where p = probability of a defective and q = 1 - p

Example: Step 2. Determine “c”

First divide LTPD by AQLFirst divide LTPD by AQL

LTPD

AQL =

.03

.01 = 3

LTPD

AQL =

.03

.01 = 3

Then find the value for “c” by selecting the value in the QA-12 (on disk) “n(AQL)”column that is equal to or just greater than the ratio above.

Then find the value for “c” by selecting the value in the QA-12 (on disk) “n(AQL)”column that is equal to or just greater than the ratio above.

Exhibit QA-12Exhibit QA-12

c LTPD/AQL n AQL c LTPD/AQL n AQL0 44.890 0.052 5 3.549 2.6131 10.946 0.355 6 3.206 3.2862 6.509 0.818 7 2.957 3.9813 4.890 1.366 8 2.768 4.6954 4.057 1.970 9 2.618 5.426

So, c = 6.So, c = 6.

9A-44

LTPD = Lot tolerant percent defective (buyers)AQL = Acceptable quality level (seller)