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