statistical process control a. a. elimam a. a. elimam

Statistical ProcessStatistical ProcessControlControl

A. A. ElimamA. A. Elimam

Two Primary Topics in Two Primary Topics in Statistical Quality ControlStatistical 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 Two Primary Topics in Statistical Quality ControlStatistical 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 ?

InspectionInspection 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?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 ?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

Quality Testing Quality Testing 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 Measures:AttributesQuality 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:VariablesQuality 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 ChartsVariation & 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 & TypesControl 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 Process ControlControl Chart Chart & Control Criteria& 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 ChartsBasis 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

Common CausesCommon Causes

xx

n

ii

n

1

x x

ni

2

1

425 Grams

Assignable CausesAssignable Causes

(a) LocationGrams

Average


(b) SpreadGrams

Average


(c) ShapeGrams

Average

Effects of Assignable Causes on Effects of Assignable Causes on Process ControlProcess Control

Assignable Assignable causes presentcauses present

Effects of Assignable Effects of Assignable Causes on Process ControlCauses on Process Control

No No assignable causesassignable causes

Sample Means and theSample Means and theProcess DistributionProcess Distribution

425 Grams

Mean

Processdistribution

Distribution ofsample means

The NormalThe NormalDistributionDistribution

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

68.26%95.44%99.97%

= Standard deviation

Control Charts

UCL

Nominal

LCL

Assignable causes likely

1 2 3Samples

Using Control Charts for Using Control Charts for Process ImprovementProcess Improvement

Measure the processMeasure the process When problems are indicated, When problems are indicated,

find the assignable causefind the assignable cause Eliminate problems, incorporate Eliminate problems, incorporate

improvementsimprovements Repeat the cycleRepeat the cycle

Control Chart Examples

Nominal

UCL

LCL

Sample number(a)

Var

iati

on

s


Nominal

UCL

LCL

Sample number(b)

Var

iati

on

s


Nominal

UCL

LCL

Sample number(c)

Var

iati

on

s


Nominal

UCL

LCL

Sample number(d)

Var

iati

on

s


Nominal

UCL

LCL

Sample number(e)

Var

iati

on

s

The Normal DistributionThe Normal DistributionMeasures 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

LCL

Processaverage

UCL

(a) Three-sigma limits

Type I error:Probability of searching for a cause when none exists


Type I error:Probability of searching for a cause when none exists

UCL

LCL

Processaverage

(b) Two-sigma limits

Type II error:Probability of concludingthat nothing has changed


UCL

Shift in process average

LCL

Processaverage

(a) Three-sigma limits

Type II error:Probability of concludingthat nothing has changed


UCL

Shift in process average

LCL

Processaverage

(b) Two-sigma limits

Control ChartsControl Chartsfor Variablesfor Variables

Mandara Mandara IndustriesIndustries


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

Special Metal Screw


Sample Sample


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 - 0.50090.5027 - 0.5009 == 0.00180.0018

Special Metal Screw


Sample Sample


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 - 0.50090.5027 - 0.5009 == 0.00180.0018

Special Metal Screw


Sample Sample


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 - 0.50090.5027 - 0.5009 == 0.00180.0018(0.5014 + 0.5022 +(0.5014 + 0.5022 + 0.5009 + 0.5027)/40.5009 + 0.5027)/4 == 0.50180.5018

Special Metal Screw


Sample Sample


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

Special Metal Screw


Control Charts - Special Metal Screw

R - Charts R = 0.0020

UCLR = D4RLCLR = D3R

Control Charts for VariablesControl Charts for Variables


R - Charts R = 0.0020 D4 = 2.2080

Control Chart FactorsControl Chart Factors

Factor for UCLFactor for UCL Factor forFactor for FactorFactorSize ofSize of and LCL forand LCL for LCL forLCL for UCL forUCL forSampleSample xx-Charts-Charts RR-Charts-Charts RR-Charts-Charts

((nn)) ((AA22)) ((DD33)) ((DD44))

22 1.8801.880 00 3.2673.26733 1.0231.023 00 2.5752.57544 0.7290.729 00 2.2822.28255 0.5770.577 00 2.1152.11566 0.4830.483 00 2.0042.00477 0.4190.419 0.0760.076 1.9241.924



R - Charts R = 0.0020 D4 = 2.282D3 = 0

UCLR = 2.282 (0.0020) = 0.00456 in.LCLR = 0 (0.0020) = 0 in.

UCLR = D4RLCLR = D3R

0.005

0.004

0.003

0.002

0.001

0 1 2 3 4 5 6

Ran

ge

(in

.)

Sample number

UCLR = 0.00456

LCLR = 0

R = 0.0020

Range Chart - Special Metal Screw

Control Charts for Variables


R = 0.0020x = 0.5025

x - Charts

UCLx = x + A2RLCLx = x - A2R

Control Chart FactorsControl Chart Factors

Factor for UCLFactor for UCL Factor forFactor for FactorFactorSize ofSize of and LCL forand LCL for LCL forLCL for UCL forUCL forSampleSample xx-Charts-Charts RR-Charts-Charts RR-Charts-Charts

((nn)) ((AA22)) ((DD33)) ((DD44))

22 1.8801.880 00 3.2673.26733 1.0231.023 00 2.5752.57544 0.7290.729 00 2.2822.28255 0.5770.577 00 2.1152.11566 0.4830.483 00 2.0042.00477 0.4190.419 0.0760.076 1.9241.924



R = 0.0020 A2 = 0.729x = 0.5025

x - Charts


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



R = 0.0020 A2 = 0.729x = 0.5025

x - Charts


UCLx = 0.5025 + 0.729 (0.0020) = 0.5040 in.LCLx = 0.5025 - 0.729 (0.0020) = 0.5010 in.

0.5050

0.5040

0.5030

0.5020

0.5010

1 2 3 4 5 6

Ave

rag

e (i

n.)

Sample number

x = 0.5025

UCLx = 0.5040

LCLx = 0.5010

Average Chart - Special Metal Screw

0.5050

0.5040

0.5030

0.5020

0.5010

Ave

rag

e (i

n.)

x = 0.5025

UCLx = 0.5040

LCLx = 0.5010

1 2 3 4 5 6Sample number

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

Average Chart - Special Metal Screw

Control ChartsControl Chartsfor Attributesfor Attributes

MANDARA BankMANDARA Bank

UCLUCLpp = = pp + + zzpp

LCLLCLpp = = pp - - zzpp

pp = = pp(1 - (1 - pp))//nn




pp = = pp(1 - (1 - pp))//nn

Sample Wrong ProportionNumber Account Number Defective

1 15 0.0062 12 0.00483 19 0.00764 2 0.00085 19 0.00766 4 0.00167 24 0.00968 7 0.00289 10 0.004

10 17 0.006811 15 0.00612 3 0.0012

Total 147

p = 0.0049

n = 2500

Control Charts for AttributesControl Charts for Attributes





pp = 0.0049(1 - 0.0049)/2500 = 0.0049(1 - 0.0049)/2500

n = 2500 p = 0.0049





pp = 0.0014 = 0.0014

n = 2500 p = 0.0049



UCLUCLpp = 0.0049 + 3(0.0014) = 0.0049 + 3(0.0014)

LCLLCLpp = 0.0049 - 3(0.0014) = 0.0049 - 3(0.0014)

pp = 0.0014 = 0.0014

n = 2500 p = 0.0049



UCLUCLpp = 0.0091 = 0.0091

LCLLCLpp = 0.0007 = 0.0007

pp = 0.0014 = 0.0014

n = 2500 p = 0.0049

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

Sample number

UCL

p

LCL

0.011

0.010

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0Pro

po

rtio

n d

efec

tive

in s

amp

lep-Chart

Wrong Account Numbers

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

Sample number

UCL

p

LCL

0.011

0.010

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0Pro

po

rtio

n d

efec

tive

in s

amp

lep-Chart

Wrong Account Numbers

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

Process CapabilityProcess Capability

Nominalvalue

80 100 120 Hours

Upperspecification

Lowerspecification

Process distribution

(a) Process is capable


Nominalvalue

80 100 120 Hours

Upperspecification

Lowerspecification

Process distribution

(b) Process is not capable


Lowerspecification

Mean

Upperspecification

Two sigma


Lowerspecification

Mean

Upperspecification

Four sigma

Two sigma


Lowerspecification

Mean

Upperspecification

Six sigma

Four sigma

Two sigma

Upper specification = 120 hoursLower specification = 80 hoursAverage life = 90 hours s = 4.8 hours

Process CapabilityProcess CapabilityLight-bulb Production

Cp =

Upper specification - Lower specification

6s

Process Capability RatioProcess Capability Ratio



Cp = 120 - 80

6(4.8)




Cp = 1.39




Cp = 1.39

Cpk = Minimum of

Upper specification - x

3s

x - Lower specification

3s

ProcessProcessCapabilityCapabilityIndexIndex

,



Cp = 1.39

Cpk = Minimum of

120 - 90

3(4.8)

90 - 80

3(4.8)


,



Cp = 1.39

Cpk = Minimum of [ 0.69, 2.08 ]




Cp = 1.39Cpk = 0.69


ProcessProcessCapabilityCapabilityRatioRatio

statistical process control a. a. elimam a. a. elimam

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