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Statistical Process Control

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- Take periodic samples from a process
- Plot the sample points on a control chart
- Determine if the process is within limits
- Correct the process before defects occur

Common Causes

Assignable Causes

Average

Grams

(a) Mean

Assignable Causes

Average

Grams

(b) Spread

Assignable Causes

Average

Grams

(c) Shape

Control Chart Examples

UCL

Nominal

Variations

LCL

Sample number

Control Charts

Assignable causes likely

UCL

Nominal

LCL

1 2 3

Samples

Mean

-3 -2 -1 +1 +2 +3

68.26%

95.44%

99.74%

The Normal

Distribution

= Standard deviation

Type I error:

Probability of searching for

a cause when none exists

UCL

Process

average

LCL

(a) Three-sigma limits

Type I error:

Probability of searching for

a cause when none exists

UCL

Process

average

LCL

(b) Two-sigma limits

Type II error:

Probability of concluding

that nothing has changed

UCL

Shift in process

average

Process

average

LCL

(a) Three-sigma limits

Type II error:

Probability of concluding

that nothing has changed

UCL

Shift in process

average

Process

average

LCL

(b) Two-sigma limits

Constructing a Control Chart

- Decide what to measure or count
- Collect the sample data
- Plot the samples on a control chart
- Calculate and plot the control limits on the control chart
- Determine if the data is in-control
- If non-random variation is present, discard the data (fix the problem) and recalculate the control limits

Types of Data

- Attribute data
- Product characteristic evaluated with a discrete choice
- Good/bad, yes/no

- Product characteristic evaluated with a discrete choice

- Product characteristic that can be measured
- Length, size, weight, height, time, velocity

Control Charts For Attributes

- p Charts
- Calculate percent defectives in a sample;
- an item is either good or bad

- c Charts
- Count number of defects in an item

Based on the binomial distribution

UCL = p + 3 p(1-p) /n

= 0.10 + 3 0.10 (1-0.10) /100

= 0.190

Proportion

SampleDefect Defective

1 6 .06

2 0 .00

3 4 .04

. . .

. . .

20 18 .18

200

- LCL= p - 3 p(1-p) /n
- = 0.10 + 3 0.10 (1-0.10) /100
- = 0.010

100 jeans in each sample

total defectives

total sample observations

200

20 (100)

p =

=

= 0.10

0.2

0.18

0.16

0.14

0.12

0.1

Proportion defective

0.08

0.06

0.04

0.02

0

Sample number

0

2

4

6

8

10

12

14

16

18

20

.

.

Based on the Poisson distribution

Count # of defects per roll in 15 rolls of denim fabric.

SampleDefects

1 12

2 8

3 16

. .

. .

15 15

190

24

21

18

15

.

12

Number of defects

9

6

3

0

0

2

4

6

8

10

12

14

Sample number

- Mean chart (X-Bar Chart)
- Measures central tendency of a sample

- Range chart (R-Chart)
- Measures amount of dispersion in a sample

- Each chart measures the process differently. Both the process average and process variability must be in control for the process to be in control.

Slip Ring Diameter (cm)

Sample 12345 X R

15.025.014.944.994.96 4.98 0.08

25.015.035.074.954.96 5.00 0.12

34.995.004.934.924.99 4.97 0.08

45.034.915.014.984.89 4.96 0.14

54.954.925.035.055.01 4.99 0.13

64.975.065.064.965.03 5.01 0.10

75.055.015.104.964.99 5.02 0.14

85.095.105.004.995.08 5.05 0.11

95.145.104.995.085.09 5.08 0.15

105.014.985.085.074.99 5.030.10

50.09 1.15

3s Control Chart Factors

Sample size X-chartR-chart

nA2D3D4

21.8803.27

31.0202.57

40.7302.28

50.5802.11

60.4802.00

70.420.081.92

80.370.141.86

Process Capability

- Range of natural variability in process
- Measured with control charts

- Process cannot meet specifications if natural variability exceeds tolerances
- 3-sigma quality
- specifications equal the process control limits.

- 6-sigma quality
- specifications twice as large as control limits

Process Capability

Natural

control

limits

Natural

control

limits

Design

specs

PROCESS

PROCESS

Process can meet specifications

Process cannot meet specifications

Natural

control

limits

Design

specs

PROCESS

Process capability exceeds specifications

Acceptance Sampling

Outline

- Sampling
- Some sampling plans
- A single sampling plan
- Some definitions
- Operating characteristic curve
- Developing a single sampling plan

- In most cases 100% inspection is too costly.
- In some cases 100% inspection may be impossible.
- If only the defective items are returned, repair or replacement may be cheaper than improving quality. But, if the entire lot is returned on the basis of sample quality, then the producer has a much greater motivation to improve quality.

- Single sampling plans:
- Most popular and easiest to use
- Two numbers n and c
- If there are more than c defectives in a sample of size n the lot is rejected; otherwise it is accepted

- Double sampling plans:
- A sample of size n1 is selected.
- If the number of defectives is less than or equal to c1 than the lot is accepted.
- Else, another sample of size n2 is drawn.
- If the cumulative number of defectives in both samples is more than c2 the lot is rejected; otherwise it is accepted.

Some Sampling Plans

- Sequential sampling
- An extension of the double sampling plan
- Items are sampled one at a time and the cumulative number of defectives is recorded at each stage of the process.
- Based on the value of the cumulative number of defectives there are three possible decisions at each stage:
- Reject the lot
- Accept the lot
- Continue sampling

A Single Sampling Plan

Consider a single sampling plan with n = 10 and c = 2

- Compute the probability that a lot will be accepted with a proportion of defectives, p = 0.10
- If a producer wants a lot with p = 0.10 to be accepted, the sampling plan has a risk of _______________

A Single Sampling Plan

- Compute the probability that a lot will be accepted with a proportion of defectives, p = 0.30
- If a consumer wants to reject a lot with p = 0.30, the sampling plan has a risk of _____________

- Acceptable quality level (AQL)
Acceptable fraction defective in a lot

- Lot tolerance percent defective (LTPD)
Maximum fraction defective accepted in a lot

- Producer’s risk,
Type I error = P(reject a good lot)

- Consumer’s risk,
Type II error = P(accept a bad lot)

{

0.80

0.60

0.40

0.20

{

0.20

0.18

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Operating Characteristic Curve

1.00

1-a = 0.05

OC curve for n and c

Probability of acceptance, Pa

b = 0.10

Percent defective

AQL

LTPD

- Develop a sampling plan with
AQL = 0.1

LTPD = 0.3

= 0.05

= 0.10

Average Outgoing Quality

Average Outgoing Quality

AOQ Curve

0.015

AOQL

0.010

Average

Outgoing

Quality

0.005

0.10

0.09

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

AQL

LTPD

(Incoming) Percent Defective

Reading and Exercises

- Reading
- Chapter 8 pp. 323-36 (self-study)
- Technical Note 8 pp. 346-60

- Exercises:
- Technical Note Problems 3,12,13