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### Statistical ProcessControl

A. A. Elimam

Two Primary Topics in Statistical Quality Control

- Statistical process control (SPC) is a statistical method using control charts to check a production process - prevent poor quality. In TQM all workers are trained in SPC methods.

Two Primary Topics in Statistical Quality Control

- Acceptance Sampling involves inspecting a sample of product. If sample fails reject the entire product - identifies the products to throw away or rework. Contradicts the philosophy of TQM. Why ?

Inspection

- Traditional Role: at the beginning and end of the production process
- Relieves Operator from the responsibility of detecting defectives & quality problems
- It was the inspection's job
- In TQM, inspection is part of the process & it is the operator’s job
- Customers may require independent inspections

How Much to Inspect?

- Complete or 100 % Inspection.
- Viable for products that can cause safety problems
- Does not guarantee catching all defectives
- Too expensive for most cases
- Inspection by Sampling
- Sample size : representative
- A must in destructive testing (e.g... Tasting food)

Where To Inspect ?

- In TQM , inspection occurs throughout the production process
- IN TQM, the operator is the inspector
- Locate inspection where it has the most effect (e.g.... prior to costly or irreversible operation)
- Early detection avoids waste of more resources

Destructive Testing

Product cannot be used after testing (e.g.. taste or breaking item)

Sample testing

Could be costly

Non-Destructive Testing

Product is usable after testing

100% or sampling

Quality TestingQuality Measures:Attributes

- Attribute is a qualitative measure
- Product characteristics such as color, taste, smell or surface texture
- Simple and can be evaluated with a discrete response (good/bad, yes/no)
- Large sample size (100’s)

Quality Measures:Variables

- A quantitative measure of a product characteristic such as weight, length, etc.
- Small sample size (2-20)
- Requires skilled workers

Variation & Process Control Charts

- Variation always exists
- Two Types of Variation
- Causal: can be attributed to a cause. If we know the cause we can eliminate it.
- Random: Cannot be explained by a cause. An act of nature - need to accept it.
- Process control charts are designed to detect causal variations

Control Charts: Definition & Types

- A control chart is a graph that builds the control limits of a process
- Control limits are the upper and lower bands of a control chart
- Types of Charts:
- Measurement by Variables: X-bar and R charts
- Measurement by Attributes: p and c

Process Control Chart & Control Criteria

1. No sample points outside control limits.

2. Most points near the process average.

3. Approximately equal No. of points above

& below center.

4. Points appear to be randomly distributed

around the center line.

5. No extreme jumps.

6. Cannot detect trend.

Basis of Control Charts

- Specification Control Charts
- Target Specification: Process Average
- Tolerances define the specified upper and lower control limits
- Used for new products (historical measurements are not available)
- Historical Data Control Charts
- Process Average, upper & lower control limits: based on historical measurements
- Often used in well established processes

425 Grams

Using Control Charts for Process Improvement

- Measure the process
- When problems are indicated, find the assignable cause
- Eliminate problems, incorporate improvements
- Repeat the cycle

The Normal Distribution

- Measures of Variability:
- Most accurate measure
- = Standard Deviation
- Approximate Measure - Simpler to compute
- R = Range
- Range is less accurate as the sample size
- gets larger
- Average = Average R when n = 2

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

for Variables

Special Metal Screw

Sample Sample

Number 1 2 3 4 Range Mean

1 0.5014 0.5022 0.5009 0.5027

2 0.5021 0.5041 0.5032 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

for Variables

Special Metal Screw

Sample Sample

Number 1 2 3 4 Range Mean

1 0.5014 0.5022 0.5009 0.5027

2 0.5021 0.5041 0.5032 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

0.5027 - 0.5009 = 0.0018

for Variables

Special Metal Screw

Sample Sample

Number 1 2 3 4 Range Mean

1 0.5014 0.5022 0.5009 0.5027 0.0018

2 0.5021 0.5041 0.5032 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

0.5027 - 0.5009 = 0.0018

for Variables

Special Metal Screw

Sample Sample

Number 1 2 3 4 Range Mean

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5032 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

0.5027 - 0.5009 = 0.0018

(0.5014 + 0.5022 +

0.5009 + 0.5027)/4 = 0.5018

for Variables

Special Metal Screw

Sample Sample

Number 1 2 3 4 Range Mean

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5032 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

0.5027 - 0.5009 = 0.0018

(0.5014 + 0.5022 +

0.5009 + 0.5027)/4 = 0.5018

for Variables

Special Metal Screw

Sample Sample

Number 1 2 3 4 Range Mean

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5032 0.5020 0.0021 0.5029

3 0.5018 0.5026 0.5035 0.5023 0.0017 0.5026

4 0.5008 0.5034 0.5024 0.5015 0.0026 0.5020

5 0.5041 0.5056 0.5034 0.5039 0.0022 0.5043

R = 0.0020

x = 0.5025

for Variables

R = 0.0020

UCLR = D4R

LCLR = D3R

Control Charts - Special Metal Screw

R - Charts

Factor for UCL Factor for Factor

Size of and LCL for LCL for UCL for

Sample x-Charts R-Charts R-Charts

(n) (A2) (D3) (D4)

2 1.880 0 3.267

3 1.023 0 2.575

4 0.729 0 2.282

5 0.577 0 2.115

6 0.483 0 2.004

7 0.419 0.076 1.924

R = 0.0020 D4 = 2.2080

Control Charts for Variables

Control Charts - Special Metal Screw

R - Charts

for Variables

R = 0.0020 D4= 2.282

D3 = 0

UCLR = D4R

LCLR = D3R

Control Charts - Special Metal Screw

R - Charts

UCLR = 2.282 (0.0020) = 0.00456 in.

LCLR = 0 (0.0020) = 0 in.

Range Chart - Special Metal Screw

R = 0.0020

0.005

0.004

0.003

0.002

0.001

0

UCLR = 0.00456

Range (in.)

LCLR = 0

1 2 3 4 5 6

Sample number

Control Chart Factors

Factor for UCL Factor for Factor

Size of and LCL for LCL for UCL for

Sample x-Charts R-Charts R-Charts

(n) (A2) (D3) (D4)

2 1.880 0 3.267

3 1.023 0 2.575

40.729 0 2.282

5 0.577 0 2.115

6 0.483 0 2.004

7 0.419 0.076 1.924

x - Charts

R = 0.0020

x = 0.5025

UCLx = x + A2R

LCLx = x - A2R

Control Charts - Special Metal Screw

for Variables

x - Charts

R = 0.0020 A2 = 0.729

x = 0.5025

UCLx = x + A2R

LCLx = x - A2R

UCLx = 0.5025 + 0.729 (0.0020) = 0.5040 in.

Control Charts - Special Metal Screw

for Variables

x - Charts

R = 0.0020 A2 = 0.729

x = 0.5025

UCLx = x + A2R

LCLx = x - A2R

UCLx = 0.5025 + 0.729 (0.0020) = 0.5040 in.

LCLx = 0.5025 - 0.729 (0.0020) = 0.5010 in.

Control Charts - Special Metal Screw

x = 0.5025

LCLx = 0.5010

Average Chart - Special Metal Screw0.5050

0.5040

0.5030

0.5020

0.5010

Average (in.)

1 2 3 4 5 6

Sample number

x = 0.5025

LCLx = 0.5010

1 2 3 4 5 6

Sample number

Average Chart - Special Metal Screw0.5050

0.5040

0.5030

0.5020

0.5010

- Measure the process
- Find the assignable cause
- Eliminate the problem
- Repeat the cycle

Average (in.)

UCLp = p + zp

p = 0.0049

LCLp = p - zp

p = p(1 - p)/n

Sample Wrong Proportion

Number Account Number Defective

1 15 0.006

2 12 0.0048

3 19 0.0076

4 2 0.0008

5 19 0.0076

6 4 0.0016

7 24 0.0096

8 7 0.0028

9 10 0.004

10 17 0.0068

11 15 0.006

12 3 0.0012

Total 147

MANDARA Bank

n = 2500

for Attributes

n = 2500 p = 0.0049

UCLp = p + zp

LCLp = p - zp

MANDARA Bank

p = 0.0049(1 - 0.0049)/2500

for Attributes

n = 2500 p = 0.0049

MANDARA Bank

UCLp = 0.0049 + 3(0.0014)

LCLp = 0.0049 - 3(0.0014)

p = 0.0014

Wrong Account Numbers

0.011

0.010

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

UCL

p

LCL

Proportion defective in sample

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

Sample number

Wrong Account Numbers

0.011

0.010

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

UCL

p

LCL

Proportion defective in sample

- Measure the process
- Find the assignable cause
- Eliminate the problem
- Repeat the cycle

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

Sample number

value

Process distribution

Lower

specification

Upper

specification

Hours

80

100

120

(a) Process is capable

Process Capabilityvalue

Process distribution

Lower

specification

Upper

specification

Hours

80

100

120

(b) Process is not capable

Process CapabilityProcess Capability

Light-bulb Production

Upper specification = 120 hours

Lower specification = 80 hours

Average life = 90 hours s = 4.8 hours

Upper specification -

Lower specification

6s

Cp =

Process Capability Ratio

6(4.8)

Process CapabilityLight-bulb Production

Upper specification = 120 hours

Lower specification = 80 hours

Average life = 90 hours s = 4.8 hours

Cp =

Process Capability Ratio

Process Capability

Light-bulb Production

Upper specification = 120 hours

Lower specification = 80 hours

Average life = 90 hours s = 4.8 hours

Cp = 1.39

Process Capability Ratio

3s

Upper specification - x

3s

Process CapabilityLight-bulb Production

Upper specification = 120 hours

Lower specification = 80 hours

Average life = 90 hours s = 4.8 hours

Cp = 1.39

Cpk = Minimum of

,

Process

Capability

Index

Process Capability

Light-bulb Production

Upper specification = 120 hours

Lower specification = 80 hours

Average life = 90 hours s = 4.8 hours

Cp = 1.39

Cpk = Minimum of

90 - 80

3(4.8)

,

Process

Capability

Index

120 - 90

3(4.8)

Process Capability

Light-bulb Production

Upper specification = 120 hours

Lower specification = 80 hours

Average life = 90 hours s = 4.8 hours

Cp = 1.39

Cpk = Minimum of [0.69, 2.08 ]

Process

Capability

Index

Process Capability

Light-bulb Production

Upper specification = 120 hours

Lower specification = 80 hours

Average life = 90 hours s = 4.8 hours

Cpk = 0.69

Cp = 1.39

Process

Capability

Index

Process

Capability

Ratio

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