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. Common Causes. Assignable Causes. Average. Grams. (a) Mean. Assignable Causes. Average. Grams.

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Statistical Process Control

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Statistical process control

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


Statistical process control

Common Causes


Statistical process control

Assignable Causes

Average

Grams

(a) Mean


Statistical process control

Assignable Causes

Average

Grams

(b) Spread


Statistical process control

Assignable Causes

Average

Grams

(c) Shape


Statistical process control

Control Chart Examples

UCL

Nominal

Variations

LCL

Sample number


Statistical process control

Control Charts

Assignable causes likely

UCL

Nominal

LCL

1 2 3

Samples


Statistical process control

Mean

-3 -2 -1 +1 +2 +3

68.26%

95.44%

99.74%

The Normal

Distribution

 = Standard deviation


Control limits and errors

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 errors1

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 errors2

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 errors3

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


Statistical process control

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


Statistical process control

Types of Data

  • 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


  • Statistical process control

    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


    P charts

    p - Charts

    Based on the binomial distribution


    P chart calculations

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

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

    = 0.190

    p-Chart Calculations

    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


    Example p chart

    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

    c - Charts

    Based on the Poisson distribution


    C chart calculations

    c - Chart Calculations

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

    SampleDefects

    1 12

    2 8

    3 16

    . .

    . .

    15 15

    190


    Example c chart

    Example c - Chart

    24

    21

    18

    15

    .

    12

    Number of defects

    9

    6

    3

    0

    0

    2

    4

    6

    8

    10

    12

    14

    Sample number


    Control charts for variables

    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

    Example: Control Charts for Variable Data

    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


    Constructing a range chart

    Constructing a Range Chart


    Constructing a mean chart

    Constructing A Mean Chart


    Statistical process control

    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


    Statistical process control

    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


    Statistical process control

    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


    Statistical process control

    Acceptance Sampling

    Outline

    • Sampling

    • Some sampling plans

    • A single sampling plan

    • Some definitions

    • Operating characteristic curve

    • Developing a single sampling plan


    Necessity of sampling

    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

    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.


    Statistical process control

    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


    Statistical process control

    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 _______________


    Statistical process control

    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 _____________


    Some definitions

    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)


    Statistical process control

    {

    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


    Example

    Example

    • Develop a sampling plan with

      AQL = 0.1

      LTPD = 0.3

       = 0.05

       = 0.10


    Average outgoing quality

    Average Outgoing Quality


    Statistical process control

    Average Outgoing Quality


    Statistical process control

    Average Outgoing Quality


    Statistical process control

    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


    Statistical process control

    Reading and Exercises

    • 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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