1

Chapter 9A

Process Capability & SPC

2

Process Variation

Process Capability

Process Control Procedures

Variable data

Attribute data

Acceptance Sampling

Operating Characteristic Curve

OBJECTIVES

3

Basic Forms of Variation

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 molding process that

always leaves “burrs” or flaws on a

molded item.

4

Taguchi’s View of Variation

Incremental

Cost of

Variability

High

Zero

Lower

Spec

Target

Spec

Upper

Spec

Traditional View

Incremental

Cost of

Variability

High

Zero

Lower

Spec

Target

Spec

Upper

Spec

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, where Taguchi views costs as increasing as variability

increases, so seek to achieve zero defects and that will truly minimize quality costs.

5

Process Capability

Process limits

Specification limits

How do the limits relate to one another?

6

Process Capability Index, Cpk

3

X-UTLor

3

LTLXmin=Cpk

Shifts in Process Mean

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.

7

A simple ratio: Specification Width

_________________________________________________________

Actual “Process Width”

Generally, the bigger the better.

Process Capability –

A Standard Measure of How Good a Process Is.

8

This is a “one-sided” Capability Index

Concentration on the side which is closest to

the specification - closest to being “bad”

3

;3

XUTLLTLXMinCpk

Process Capability

9

Coffee Bean Bag Example

Ron Valdez Café roasts and nationally distributes 16 ounce bags of coffee beans. The Omaha World Herald has just published an article that claims that the bags frequently have less than 15 ounces of coffee beans in out 16 ounce bags.

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

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 bags of coffee beans and find that they weight an average of 16.102 ounces with a standard deviation of .629 ounces.

10

Cereal Box Process Capability

Specification or Tolerance Limits

Upper Spec = 16.8 oz

Lower Spec = 15.2 oz

Observed Weight

Mean = 16.102 oz

Std Dev = .629 oz

3

;3

XUTLLTLXMinCpk

)629(.3

102.168.16;

)629(.3

2.15102.16MinCpk

370.;478.MinCpk

370.pkC

11

What does a Cpk of .370 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!

Cpk < 1 The process is not capable

Cpk = 1 The process is just capable

Cpk > 1 The process is capable

12

Types of Statistical Sampling

Attribute (Go or no-go information)

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

13

Types of Statistical Sampling

Variable (Continuous)

Usually measured by the mean and the

standard deviation.

X-bar and R chart applications

14 Statistical

Process

Control

(SPC) Charts

UCL

LCL

Samples

over time

1 2 3 4 5 6

UCL

LCL

Samples

over time

1 2 3 4 5 6

UCL

LCL

Samples

over time

1 2 3 4 5 6

Normal Behavior

Possible problem, investigate

Possible problem, investigate

15

Control Limits are based on the Normal Curve

x

0 1 2 3 -3 -2 -1 z

m

Standard deviation units or “z”

units.

16

Control Limits

We establish the Upper Control Limits (UCL) and the Lower Control Limits (LCL) with plus or minus 3 standard deviations from some x-bar or mean value. Based on this we can expect 99.7% of our sample observations to fall within these limits.

x

LCL UCL

99.7%

17

Example of Constructing a p-Chart:

Required Data

1 100 4

2 100 2

3 100 5

4 100 3

5 100 6

6 100 4

7 100 3

8 100 7

9 100 1

10 100 2

11 100 3

12 100 2

13 100 2

14 100 8

15 100 3

Sample

Number

Number of

Samples

Number of defects found

in each sample

18

Given:

Statistical Process Control Formulas:

Attribute Measurements (p-Chart)

Compute control limits:

nsObservatio ofNumber Total

Defectives ofNumber Total=p

ns

)p-(1 p = p

p

p

z - p = LCL

z + p = UCL

s

s

19

1. Calculate the sample proportions, p

(these are what can be plotted on the

p-chart) for each sample

Sample n Defectives 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

Example of Constructing a p-chart: Step 1

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2. Calculate the average of the sample proportions

3. Calculate the standard deviation of the sample proportion

Example of Constructing a p-chart:

Steps 2&3

0.036=1500

55 = p

.0188= 100

.036)-.036(1=

)p-(1 p = p

ns

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4. Calculate the control limits

UCL = 0.0924

LCL = -0.0204 (or 0)

Example of Constructing a p-chart: Step 4

p

p

z - p = LCL

z + p = UCL

s

s

3(.0188) .036

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Example of Constructing a p-Chart: Step 5

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

5. Plot the individual sample proportions, the average

of the proportions, and the control limits

23

Example of x-bar and R Charts:

Required Data

Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5

1 10.68 10.689 10.776 10.798 10.714

2 10.79 10.86 10.601 10.746 10.779

3 10.78 10.667 10.838 10.785 10.723

4 10.59 10.727 10.812 10.775 10.73

5 10.69 10.708 10.79 10.758 10.671

6 10.75 10.714 10.738 10.719 10.606

7 10.79 10.713 10.689 10.877 10.603

8 10.74 10.779 10.11 10.737 10.75

9 10.77 10.773 10.641 10.644 10.725

10 10.72 10.671 10.708 10.85 10.712

11 10.79 10.821 10.764 10.658 10.708

12 10.62 10.802 10.818 10.872 10.727

13 10.66 10.822 10.893 10.544 10.75

14 10.81 10.749 10.859 10.801 10.701

15 10.66 10.681 10.644 10.747 10.728

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Example of x-bar and R charts: Step 1. Calculate sample

means, sample ranges, mean of means, and mean of ranges.

Sample 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 0.116

2 10.787 10.86 10.601 10.746 10.779 10.755 0.259

3 10.78 10.667 10.838 10.785 10.723 10.759 0.171

4 10.591 10.727 10.812 10.775 10.73 10.727 0.221

5 10.693 10.708 10.79 10.758 10.671 10.724 0.119

6 10.749 10.714 10.738 10.719 10.606 10.705 0.143

7 10.791 10.713 10.689 10.877 10.603 10.735 0.274

8 10.744 10.779 10.11 10.737 10.75 10.624 0.669

9 10.769 10.773 10.641 10.644 10.725 10.710 0.132

10 10.718 10.671 10.708 10.85 10.712 10.732 0.179

11 10.787 10.821 10.764 10.658 10.708 10.748 0.163

12 10.622 10.802 10.818 10.872 10.727 10.768 0.250

13 10.657 10.822 10.893 10.544 10.75 10.733 0.349

14 10.806 10.749 10.859 10.801 10.701 10.783 0.158

15 10.66 10.681 10.644 10.747 10.728 10.692 0.103

Averages 10.728 0.220400

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Example of x-bar and R charts: Step 2. Determine Control

Limit Formulas and Necessary Tabled Values

x Chart Control Limits

UCL = x + A R

LCL = x - A R

2

2

R Chart Control Limits

UCL = D R

LCL = D R

4

3

From Exhibit TN8.7

n A2 D3 D4

2 1.88 0 3.27

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 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

26 Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart

and Plot Values

10.601

10.856

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

=).58(0.220410.728RA + x = UCL

2

2

10.55

10.60

10.65

10.70

10.75

10.80

10.85

10.90

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

Sample

Mea

ns

UCL

LCL

27

Example of x-bar and R charts:

Steps 5&6. Calculate R-chart and Plot Values

0

0.46504

)2204.0)(0(R D= LCL

)2204.0)(11.2(R D= UCL

3

4

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

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

Sample

RUCL

LCL

28

Basic Forms of Statistical Sampling for Quality Control

Acceptance Sampling is sampling to accept or reject

the immediate lot of product at hand

Statistical Process Control is sampling to determine

if the process is within acceptable limits

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Acceptance Sampling

Purposes

Determine quality level

Ensure quality is within predetermined level

Advantages

Economy

Less handling damage

Fewer inspectors

Upgrading of the inspection job

Applicability to destructive testing

Entire lot rejection (motivation for improvement)

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Acceptance Sampling (Continued)

Disadvantages

Risks of accepting “bad” lots and rejecting “good”

lots

Added planning and documentation

Sample provides less information than 100-percent

inspection

31

Acceptance Sampling:

Single Sampling Plan

A simple goal

Determine (1) how many units, n, to sample from a lot, and (2) the maximum number of defective items, c, that can be found in the sample before the lot is rejected

32

Risk

Acceptable Quality Level (AQL)

Maximum acceptable percentage of defectives defined by

producer.

The a (Producer’s risk) is the probability of rejecting a good lot

Lot Tolerance Percent Defective (LTPD)

Percentage of defectives that defines consumer’s rejection

point.

The (Consumer’s risk) is the probability of accepting a bad lot

33

Operating Characteristic Curve

n = 99

c = 4

AQL LTPD

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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

Lot Tolerance Percent defective

Pro

ba

bil

ity o

f a

ccep

tan

ce

=.10

(consumer’s risk)

a = .05 (producer’s risk)

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

Acceptable Quality Level

34

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.

35

Example: Step 1. What is given and what is not?

In this problem,

Acceptable Quality Level, AQL = 0.01

Tolerance Percent Defective, LTPD = 0.03

Producer’s risk, a = 0.05

Consumer’s risk, = 0.10

You need c and n to determine your sampling plan

36

Example: Step 2. Determine “c”

First divide LTPD by AQL.

Then find the value for “c” by selecting the value in the approprate table

“n AQL” column that is equal to or just greater than the ratio above.

c LTPD/AQL n AQL c LTPD/AQL n AQL

0 44.890 0.052 5 3.549 2.613

1 10.946 0.355 6 3.206 3.286

2 6.509 0.818 7 2.957 3.981

3 4.890 1.366 8 2.768 4.695

4 4.057 1.970 9 2.618 5.426

0.10 0.05,for Plan table Sample a

301.

03.

AQL

LTPD

37

Example: Step 3. Determine Sample Size

Sampling Plan:

Take a random sample of 329 units from a lot.

Reject the lot if more than 6 units are defective.

Now given the information below, compute the sample size in units to generate your sampling

plan

up) round (always 329

6.32801./286.3/

AQLAQLn

problemin given ,01.

Table from ,286.3

Table from ,6

AQL

AQLn

c