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## Statistical Process Control

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Statistical Process Control

- 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

Control Limits and Errors

Type I error:

Probability of searching for

a cause when none exists

UCL

Process

average

LCL

(a) Three-sigma limits

Control Limits and Errors

Type I error:

Probability of searching for

a cause when none exists

UCL

Process

average

LCL

(b) Two-sigma limits

Control Limits and Errors

Type II error:

Probability of concluding

that nothing has changed

UCL

Shift in process

average

Process

average

LCL

(a) Three-sigma limits

Control Limits and Errors

Type II error:

Probability of concluding

that nothing has changed

UCL

Shift in process

average

Process

average

LCL

(b) Two-sigma limits

- 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

- Attribute data
- Product characteristic evaluated with a discrete choice
- Good/bad, yes/no
- Variable data
- Product characteristic that can be measured
- Length, size, weight, height, time, velocity

- p Charts
- Calculate percent defectives in a sample;
- an item is either good or bad
- c Charts
- Count number of defects in an item

p - Charts

Based on the binomial distribution

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

= 0.190

p-Chart CalculationsProportion

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

Example p - Chart

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

.

.

c - Charts

Based on the Poisson distribution

c - Chart Calculations

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

SampleDefects

1 12

2 8

3 16

. .

. .

15 15

190

Control Charts For Variables

- 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.

Example: Control Charts for Variable Data

Slip Ring Diameter (cm)

Sample 1 2 3 4 5 X R

1 5.02 5.01 4.94 4.99 4.96 4.98 0.08

2 5.01 5.03 5.07 4.95 4.96 5.00 0.12

3 4.99 5.00 4.93 4.92 4.99 4.97 0.08

4 5.03 4.91 5.01 4.98 4.89 4.96 0.14

5 4.95 4.92 5.03 5.05 5.01 4.99 0.13

6 4.97 5.06 5.06 4.96 5.03 5.01 0.10

7 5.05 5.01 5.10 4.96 4.99 5.02 0.14

8 5.09 5.10 5.00 4.99 5.08 5.05 0.11

9 5.14 5.10 4.99 5.08 5.09 5.08 0.15

10 5.01 4.98 5.08 5.07 4.99 5.03 0.10

50.09 1.15

Sample size X-chart R-chart

nA2D3D4

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

- 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

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

Outline

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

Necessity of Sampling

- 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.

Some Sampling Plans

- 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.

- 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

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 _______________

- 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 _____________

Some Definitions

- 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

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
- Chapter 8 pp. 323-36 (self-study)
- Technical Note 8 pp. 346-60
- Exercises:
- Technical Note Problems 3,12,13

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