Statistical Process Control

Dr. Keong LeongManagement Department

UNLV

(Principles of Operations Management, Heizer & Render, 5th Edition)

Outline

• Statistical Quality Control• Common causes vs. assignable causes• SPC charts

– Control charts for variables– Control charts for attribute

(Principles of Operations Management, Heizer & Render, 5th Edition)

Variations

• Natural variations (due to chance/randomness)– A certain amount of natural variability will always

exist. – A process with only natural variations is in

statistical control• Assignable variations (correctable problem):

can be traced to a specific reason– operators errors– defective raw materials– improperly adjusted machines

(Principles of Operations Management, Heizer & Render, 5th Edition)

Natural & Assignable Variation

(Principles of Operations Management, Heizer & Render, 5th Edition)

Quality Characteristics

• Characteristics for which you focus on defects

• Classify products as either ‘good’ or ‘bad’, or count number of defects– e.g., radio works or not

• Categorical or discrete random variables

AttributesVariables• Characteristics that you

measure, e.g., weight, length

• May be in whole or in fractional numbers

• Continuous random variables

(Principles of Operations Management, Heizer & Render, 5th Edition)

Theoretical Basis of Control Charts

X

As sample size gets large enough,

sampling distribution becomes almost normal regardless of population distribution.

Central Limit Theorem

XX

(Principles of Operations Management, Heizer & Render, 5th Edition)

The Normal Distribution

-3σ -2σ -1σ +1σ +2σ +3σMean

68.26%95.44%99.74%

σ = Standard deviation

(Principles of Operations Management, Heizer & Render, 5th Edition)

X Chart Control Limits

Range for sample i

# Samples

Mean for sample i

From Table S6.1

RAxxLCL

RAxxUCL

2−=

2+=

n

R R

in

1i =∑

=n x

in

ix

1=∑

=

(Principles of Operations Management, Heizer & Render, 5th Edition)

(Principles of Operations Management, Heizer & Render, 5th Edition)

Factors for Computing Control Chart Limits

Sample Size, n

Mean Factor, A2

Upper Range, D4

Lower Range, D3

2 1.880 3.268 0 3 1.023 2.574 0 4 0.729 2.282 0 5 0.577 2.115 0 6 0.483 2.004 0 7 0.419 1.924 0.076 8 0.373 1.864 0.136 9 0.337 1.816 0.184

10 0.308 1.777 0.223 12 0.266 1.716 0.284

0.184

(Principles of Operations Management, Heizer & Render, 5th Edition)

R Chart Control Limits

Range for Sample i

# Samples

From Table S6.1

n

R R

R D LCL

R D UCL

in

1i

3R

4R

=∑

=

=

=

(Principles of Operations Management, Heizer & Render, 5th Edition)

(Principles of Operations Management, Heizer & Render, 5th Edition)

Control Charts for Variables

Sample SampleNumber 1 2 3 4 Range Mean

1 0.5014 0.5022 0.5009 0.50272 0.5021 0.5041 0.5032 0.50203 0.5018 0.5026 0.5035 0.50234 0.5008 0.5034 0.5024 0.50155 0.5041 0.5056 0.5034 0.5039

Special Metal Screw

(Principles of Operations Management, Heizer & Render, 5th Edition)

Control Charts for Variables

Sample SampleNumber 1 2 3 4 Range Mean

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.50182 0.5021 0.5041 0.5024 0.5020 0.0021 0.50273 0.5018 0.5026 0.5035 0.5023 0.0017 0.50264 0.5008 0.5034 0.5024 0.5015 0.0026 0.50205 0.5041 0.5056 0.5034 0.5047 0.0022 0.5045

Average 0.0021 0.5027

Special Metal Screw

(Principles of Operations Management, Heizer & Render, 5th Edition)

Control Charts for Variables

Control Charts - Special Metal ScrewR - Charts R = 0.0021

UCLR = D4R = 2.282(0.0021) = 0.00479LCLR = D3R = 0(0.0021) = 0

(Principles of Operations Management, Heizer & Render, 5th Edition)

Control Charts for Variables

Control Charts - Special Metal ScrewR - Charts R = 0.0020 D4 = 2.2080

Control Chart Factors (Three Sigma Limits)Control Chart Factors (Three Sigma Limits)Factor for UCLFactor for UCL Factor forFactor for FactorFactor

Size ofSize of and LCL forand LCL for LCL forLCL for UCL forUCL forSampleSample xx--ChartsCharts RR--ChartsCharts RR--ChartsCharts

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

22 1.8801.880 00 3.2683.26833 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

(Principles of Operations Management, Heizer & Render, 5th Edition)

Range Chart - Special Metal Screw

0.005

0.004

0.003

0.002

0.001

01 2 3 4 5 6

Ran

ge (i

n.)

Sample number

UCLR = 0.00479

LCLR = 0

R = 0.0021

(Principles of Operations Management, Heizer & Render, 5th Edition)

Control Charts for Variables

Control Charts - Special Metal ScrewR = 0.0021x = 0.5027

x - Charts

UCLx = x + A2R = 0.5027 + 0.729(0.0021)LCLx = x - A2R = 0.5027 - 0.729(0.0021)UCL = 0.5042LCL = 0.5012

(Principles of Operations Management, Heizer & Render, 5th Edition)

Control Charts for Variables

Control Charts - Special Metal ScrewR = 0.0020x = 0.5025

x - Charts

UCLx = x + A2RLCLx = x - A2R

Control Chart Factors (Three Sigma Limits)Control Chart Factors (Three Sigma Limits)Factor for UCLFactor for UCL Factor forFactor for FactorFactor

Size ofSize of and LCL forand LCL for LCL forLCL for UCL forUCL forSampleSample xx--ChartsCharts RR--ChartsCharts RR--ChartsCharts

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

22 1.8801.880 00 3.2683.26833 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

(Principles of Operations Management, Heizer & Render, 5th Edition)

x Chart - Special Metal Screw

0.5050

0.5040

0.5030

0.5020

0.5010

1 2 3 4 5

Ave

rage

(in.

)

Sample number

x = 0.5027

UCLx = 0.5042

LCLx = 0.5012

0.5045

(Principles of Operations Management, Heizer & Render, 5th Edition)

Key Learning Outcome

• Ability to Identify or Define:– Natural and assignable variations– Attribute and variable inspection – Process control – X-bar charts and R charts