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Statistical Process Control (SPC)
Chapter 6
MGMT 326
Foundations
of Operations
Products &
Processes
Quality
Assurance
Introduction
Managing
Projects
Managing
Quality
Product
Design
Statistical
Process
Control
Strategy
Process
Design
Just-in-Time & Lean Systems
Capacity
and
Facilities
Planning
& Control
Assuring Customer-Based Quality
Product
launch
activities:
Revise
periodically
Customer Requirements
Ongoing
activity
Statistical Process Control:
Measure & monitor quality
Product Specifications
Process Specifications
Statistical Process Control (SPC)
Basic SPC
Concepts
Types of
Measures
Variation
Attributes
Objectives
Variables
First steps
SPC for
Variables
Mean
charts
Range
charts
 and  known
,  unknown
Capable
Processes
 = target
 = target
Variation in a Transformation Process
Inputs
• Facilities
• Equipment
• Materials
• Energy
Transformation
Process
Outputs
Goods &
Services
•Variation in inputs create variation in outputs
• Variations in the transformation process
create variation in outputs
Variation in a Transformation Process
Customer
requirements
are not met
Inputs
• Facilities
• Equipment
• Materials
• Energy
Transformation
Process
Outputs
Goods &
Services
•Variation in inputs create variation in outputs
• Variations in the transformation process
create variation in outputs
Variation



All processes have variation.
Common cause variation is random variation
that is always present in a process.
Assignable cause variation results from changes
in the inputs or the process. The cause can and
should be identified.
 Assignable cause variation shows that the
process or the inputs have changed, at least
temporarily.
Objectives of
Statistical Process Control (SPC)



Find out how much common cause variation the
process has
Find out if there is assignable cause variation.
A process is in control if it has no assignable cause
variation
 Being in control means that the process is
stable and behaving as it usually does.
First Steps in
Statistical Process Control (SPC)


Measure characteristics of goods or services
that are important to customers
Make a control chart for each characteristic

The chart is used to determine whether the
process is in control
Types of Measures (1)
Variable Measures



Continuous random variables
Measure does not have to be a whole
number.
Examples: time, weight, miles per gallon,
length, diameter
Types of Measures (2)
Attribute Measures

Discrete random variables – finite number of
possibilities



Also called categorical variables
The measure may depend on perception or
judgment.
Different types of control charts are used for
variable and attribute measures
Examples of Attribute Measures

Good/bad evaluations



Number of defects per unit


Good or defective
Correct or incorrect
Number of scratches on a table
Opinion surveys of quality


Customer satisfaction surveys
Teacher evaluations
SPC for Variables
The Normal Distribution
 = the population mean
 = the standard deviation
for the population
99.74% of the area under the
normal curve is between
 - 3 and  + 3
SPC for Variables
The Central Limit Theorem


Samples are taken from a distribution with
mean  and standard deviation .
k = the number of samples
n = the number of units in each sample
The sample means are normally distributed

with mean  and standard deviation  x 
n
when k is large.
Control Limits for the Sample Mean x
when  and  are known
x is a variable, and samples of size n are
taken from the population containing x.
Given:  = 10,  = 1, n = 4
Then  x    1  1  0.5

n
4
2
A 99.7% confidence interval for x is

 

(   3 x ,   3 x )     3
, 3

n
n

Control Limits for the Sample Mean x
when  and  are known (2)
The lower control limit for x is

 1 
LCL    3 x    3
 10  3 

n
 4
 10 1.5  8.5
Control Limits for the Sample Mean x
when  and  are known (3)
The upper control limit for x is

 1 
UCL    3 x    3
 10  3 

n
 4
 10 1.5  11.5
Control Limits for the Sample Mean x
when  and  are unknown

If the process is new or has been changed
recently, we do not know  and 
Example 6.1, page 180
Given: 25 samples, 4 units in each sample
 and  are not given

k = 25, n = 4



Control Limits for the Sample Mean x
when  and  are unknown (2)
1.
Compute the mean for each sample. For
example,
15.85  16.02  15.83  15.93
x1 
2.
4
 15.91
Compute
k
x
x
m 1
k
25
m

x
m 1
25
m

15.91  16.00  ...  15.94 398.75

 15.95
25
25
Control Limits for the Sample Mean x
when  and  are unknown (3)
For the ith sample, the sample range is
Ri = (largest value in sample i )
- (smallest value in sample i )
3. Compute Ri for every sample. For example,

R1 = 16.02 – 15.83 = 0.19
Control Limits for the Sample Mean x
when  and  are unknown (4)
4.
Compute R , the average range
k
R

R
i 1
k
i

0.19  0.27  ...  0.30 7.17

 0.29
25
25
We will approximate 3 x by A2 R, where
A2 is a number that depends on the sample
size n. We get A2 from Table 6.1, page 182
Control Limits for the Sample Mean x
when  and  are unknown (5)
Factor for x-Chart
5.
n = the number of
units in each sample
= 4.
From Table 6.1,
A2 = 0.73.
The same A2 is used
for every problem
with n = 4.
Sample Size
(n)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A2
1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.34
0.31
0.29
0.27
0.25
0.24
0.22
Factors for R-Chart
D3
0.00
0.00
0.00
0.00
0.00
0.08
0.14
0.18
0.22
0.26
0.28
0.31
0.33
0.35
D4
3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
1.72
1.69
1.67
1.65
Control Limits for the Sample Mean x
when  and  are unknown (6)
6.
The formula for the lower control limit is
LCL  x  A2 R  15.95  0.73(0.29)  15.74
7.
The formula for the upper control limit is
UCL  x  A2 R  15.95  0.73(0.29)  16.16
Control Chart for
x
The variation between LCL = 15.74 and UCL = 16.16
is the common cause variation.
Common Cause and
Special Cause Variation
 The range between the LCL and UCL, inclusive, is
the common cause variation for the process. When
x is in this range, the process is in control.
 When a process is in control, it is predictable.
Output from the process may or may not meet
customer requirements.
 When x is outside control limits, the process is
out of control and has special cause variation. The
cause of the variation should be identified and
eliminated.
Control Limits for R
Factor for x-Chart
1.
From the table,
get D3 and D4
for n = 4.
D3 = 0
D4 = 2.28
Sample Size
(n)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A2
1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.34
0.31
0.29
0.27
0.25
0.24
0.22
Factors for R-Chart
D3
0.00
0.00
0.00
0.00
0.00
0.08
0.14
0.18
0.22
0.26
0.28
0.31
0.33
0.35
D4
3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
1.72
1.69
1.67
1.65
Control Limits for R (2)
2.
The formula for the lower control limit is
LCL  D3 R  0(0.29)  0
2.
The formula for the upper control limit is
UCL  D4 R  2.28(0.29)  0.66
fig_ex06_03
Statistical Process Control (SPC)
Basic SPC
Concepts
Types of
Measures
Variation
Attributes
Objectives
Variables
First steps
SPC for
Variables
Mean
charts
Range
charts
 and  known
,  unknown
Capable
Processes
 = target
 = target
Capable Transformation Process
Inputs
• Facilities
• Equipment
• Materials
• Energy
Capable
Transformation
Process
Outputs
Goods &
Services
that meet
specifications
a specification that meets customer requirements
+ a capable process (meets specifications)
= Satisfied customers and repeat business
Review of Specification Limits



The target for a process is the ideal value
 Example: if the amount of beverage in a bottle
should be 16 ounces, the target is 16 ounces
Specification limits are the acceptable range of values
for a variable
Example: the amount of beverage in a bottle must be
at least 15.8 ounces and no more than 16.2 ounces.



The allowable range is 15.8 – 16.2 ounces.
Lower specification limit = 15.8 ounces or LSL = 15.8
ounces
Upper specification limit = 16.2 ounces or USL = 16.2
ounces
Control Limits vs. Specification Limits


Control limits show the actual range of
variation within a process
 What the process is doing
Specification limits show the acceptable
common cause variation that will meet
customer requirements.

Specification limits show what the process
should do to meet customer requirements
Process is Capable: Control Limits are
within or on Specification Limits
Upper specification limit
UCL
X
LCL
Lower specification limit
Process is Not Capable: One or Both
Control Limits are Outside Specification Limits
UCL
Upper specification limit
X
LCL
Lower specification limit
Capability and Conformance Quality


A process is capable if
 It is in control and
 It consistently produces outputs that meet
specifications.
 This means that both control limits for the mean
must be within the specification limits
A capable process produces outputs that have
conformance quality (outputs that meet
specifications).
Process Capability Ratio C p

Use C p to determine whether the
process is capable when  = target.
USL  LSL
Cp 
6


If C p  1, the process is capable,
If C p  1 , the process is not capable.
C p Example

Given: Boffo Beverages produces 16-ounce
bottles of soft drinks. The mean ounces of
beverage in Boffo's bottle is 16. The
allowable range is 15.8 – 16.2. The standard
deviation is 0.06. Find C p and determine
whether the process is capable.
C p Example (2)

Given:  = 16,  = 0.06, target = 16
LSL = 15.8, USL = 16.2
USL  LSL 16.2  15.8
Cp 

 1.11
6
6(0.06)
C p  1 The process is capable.
Process Capability Index Cpk
C pk
USL     LSL 
 smaller 
,

3 
 3

If Cpk > 1, the process is capable.

If Cpk < 1, the process is not capable.

We must use Cpk when  does not equal the
target.
Cpk Example

Given: Boffo Beverages produces 16-ounce
bottles of soft drinks. The mean ounces of
beverage in Boffo's bottle is 15.9. The
allowable range is 15.8 – 16.2. The standard
deviation is 0.06. Find C pk and determine
whether the process is capable.
Cpk Example (2)

Given:  = 15.9,  = 0.06, target = 16
LSL = 15.8, USL = 16.2
16.2  15.9 15.9  15.8 
USL     LSL 
C pk  smaller 
,

smaller
,



3(0.06) 
3 
 3
 3(0.06)
 0.3 0.1 
 smaller 
,
  smaller{1.67, 0.56}  0.56
 0.18 0.18 
Cpk < 1. Process is not capable.
Statistical Process Control (SPC)
Basic SPC
Concepts
Types of
Measures
Variation
Attributes
Objectives
Variables
First steps
SPC for
Variables
Mean
charts
Range
charts
 and  known
,  unknown
Capable
Processes
 = target
 = target
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