Transcript UCL

Koç University
OPSM 301 Operations Management
Class 17:
Quality: Statistical process control
Zeynep Aksin
[email protected]
Announcement
 Group Case Assignment 2
 See Web page to download a copy
 Due Thursday in class or Friday in my or
Buse’s office
Statistical Quality Control Objectives
 1.Reduce normal variation (process
capability)
– If normal variation is as small as desired, Process
is capable
– We use capability index to check for this
 2.Detect and eliminate assignable variation
(statistical process control)
– If there is no assignable variation, Process is in
control
– We use Process Control charts to maintain this
Natural Variations

Also called common causes

Affect virtually all production processes

Expected amount of variation, inherent due to:
- the nature of the system
- the way the system is managed
- the way the process is organised and operated

can only be removed by
- making modifications to the process
- changing the process

Output measures follow a probability distribution

For any distribution there is a measure of central tendency
and dispersion
Assignable Variations

Also called special causes of variation

Exceptions to the system
 Generally this is some change in the process

Variations that can be traced to a specific reason
 considered abnormalities
 often specific to a
certain operator
certain machine
certain batch of material, etc.

The objective is to discover when assignable causes are
present
 Eliminate the bad causes
 Incorporate the good causes
1. Process Capability
Example:Producing bearings for a rotating shaft
Specification Limits
Design requirements:
Diameter: 1.25 inch ±0.005 inch
Lower specification Limit:LSL=1.25-0.005=1.245
Upper Specification Limit:USL=1.25+0.005=1.255
Relating Specs to Process Limits
Process performance (Diameter of the products produced=D):
Average 1.25 inch
Std. Dev: 0.002 inch
Frequency
Question:What is the probability
That a bearing does not meet specifications?
(i.e. diameter is outside (1.245,1.255) )
1.25
Diameter
1.245  1.25
)  P( z  2.5)  NORMSDIST (2.5)  0.006
0.002
1.255  1.25
P( D  1.255)  P ( z 
)  P( z  2.5)  1  NORMSDIST (2.5)  0.006
0.002
P( D  1.245)  P ( z 
P(defect)=0.006+0.006=0.012
or 1.2%
This is not good enough!!
Process capability
•If P(defect)>0.0027 then the process is not capable of producing
according to specifications.
•To have this quality level (3 sigma quality), we need to have:
•Lower Spec: mean-3
•Upper Spec:mean+3
If we want to have P(defect) 0, we aim for 6 sigma quality, then, we need:
Lower Spec: mean-6
Upper Spec:mean+6
 What can we do to improve capability of our process? What should be to
have Six-Sigma quality?
 We want to have: (1.245-1.25)/ = 6  =0.00083 inch
 We need to reduce variability of the process. We cannot change specifications
easily, since they are given by customers or design requirements.
Six Sigma Quality
Process Capability Index Cpk
 Shows how well the parts being produced fit into the range specified
by the design specifications
 Want Cpk larger than one
 X  LSL USL - X 

Cpk = min 
or

3

3



For our example:
C pk  min(
1.25  1.245 1.255  1.25
,
)  0.83  1
3x0.002
3x0.002
Cpk tells how many standard deviations can fit between the mean and
the specification limits.
Ideally we want to fit more, so that probability of defect is smaller
Process Capability Index Cp
LS
US
Process Interval = 6
Specification interval = US –LS
99.73%
Cp= (US-LS) / 6
Process Interval = 60
Specification Interval = US – LS = 60
100
m  130
 = 10
Process Interval
Specification Interval
160
Cp= (US-LS) / 6 = 60 / 60 = 1
Process Capability Index Cp
LS
US
Process Interval = 6 = 30
Specification Interval = US – LS =60
99.99998%
Cp= (US-LS) / 6 =2
99.73%
100
m  130
=5
3 Process Interval
Specification Interval
6 Process Interval
160
Process Mean Shifted
LS
US
Cpk = min{ (US - m)/3, (m - LS)/3 }
Cpk = min(2,0)=0
Specification
3 Process
70
100
m  100
 = 10
130
160
2. Statistical Process Control: Control
Charts
Can be used to monitor ongoing production process
quality
1020
UCL
1010
1000
990
LCL
980
970
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Setting Chart Limits
For x-Charts when we know 
Upper control limit (UCL) = x + zx
Lower control limit (LCL) = x - zx
where
x = mean of the sample means or a target value set for
the process
z = number of normal standard deviations (=3)
x = standard deviation of the sample means
= / n
 = population standard deviation
n = sample size
Setting Control Limits
Hour 1
sample item
Weight of
Number
Oat Flakes
1
17
Sample
2
13
size
3
16
4
18
n=9
5
17
6
16
7
15
8
17
9
16
Mean
16.1
=
1
Hour
1
2
3
4
5
6
Mean
16.1
16.8
15.5
16.5
16.5
16.4
Hour
7
8
9
10
11
12
Mean
15.2
16.4
16.3
14.8
14.2
17.3
For 99.73% control limits, z = 3
Setting Control Limits
Hour 1
Hour Mean Hour Mean
Sample
Weight of
1
16.1
7
15.2
Number
Oat Flakes
2
16.8
8
16.4
1
17
3
15.5
9
16.3
2
13
4
16.5
10 14.8
3
16
5
16.5
11 14.2
4
18
6
16.4
12 17.3
n=9
5
17
6
16
For 99.73% control limits, z = 3
7
15
8
17
UCLx = x + zx = 16 + 3(1/3) = 17
9
16
LCLx = x - zx = 16 - 3(1/3) = 15
Mean 16.1
=
1
Setting Control Limits
Control Chart
for sample of
9 boxes
Variation due
to assignable
causes
Out of
control
17 = UCL
Variation due to
natural causes
16 = Mean
15 = LCL
| | | | | | | | | | | |
1 2 3 4 5 6 7 8 9 10 11 12
Sample number
Out of
control
Variation due
to assignable
causes
R – Chart
 Type of variables control chart
 Shows sample ranges over time
 Difference between smallest and
largest values in sample
 Monitors process variability
 Independent from process mean
Setting Chart Limits
For R-Charts
Upper control limit (UCLR) = D4R
Lower control limit (LCLR) = D3R
where
R = average range of the samples
D3 and D4 = control chart factors
Setting Control Limits
Average range R = 5.3 pounds
Sample size n = 5
D4 = 2.115, D3 = 0
UCLR
= D4R
= (2.115)(5.3)
= 11.2 pounds
UCL = 11.2
Mean = 5.3
LCLR
= D3R
= (0)(5.3)
= 0 pounds
LCL = 0
Mean and Range Charts
(a)
(Sampling mean is
shifting upward but
range is consistent)
These
sampling
distributions
result in the
charts below
UCL
(x-chart detects
shift in central
tendency)
x-chart
LCL
UCL
(R-chart does not
detect change in
mean)
R-chart
LCL
Mean and Range Charts
(b)
These
sampling
distributions
result in the
charts below
(Sampling mean
is constant but
dispersion is
increasing)
UCL
(x-chart does not
detect the increase
in dispersion)
x-chart
LCL
UCL
(R-chart detects
increase in
dispersion)
R-chart
LCL
Process Control and Improvement
Out of Control
UCL
m
LCL
In Control
Improved
Six Sigma Quality
Six Sigma







a vision;
a philosophy;
a symbol;
a metric;
a goal;
a methodology
All of the
Above
Six Sigma : Organizational Structure
 Champion
– Executive Sponsor
 Master Black Belts
– Process Improvement Specialist
– Promotes Org / Culture Change
 Black Belts
– Full Time
– Detect and Eliminate Defects
– Project Leader
 Green Belts
– Part-time involvement
Six Sigma Quality: DMAIC Cycle (Continued)
1. Define (D)
Customers and their priorities
2. Measure (M)
Process and its performance
3. Analyze (A)
Causes of defects
4. Improve (I)
Remove causes of defects
5. Control (C)
Maintain quality
Process Control and Capability: Review
 Every process displays variability: normal or abnormal
 Do not tamper with process “in control” with normal variability Correct “out of
control” process with abnormal variability
 Control charts monitor process to identify abnormal variability
 Control charts may cause false alarms (or missed signals) by mistaking
normal (abnormal) for abnormal (normal) variability
 Local control yields early detection and correction of abnormal
 Process “in control” indicates only its internal stability
 Process capability is its ability to meet external customer needs
 Improving process capability involves changing the mean and reducing
normal variability, requiring a long term investment
 Robust, simple, standard, and mistake - proof design improves process
capability
 Joint, early involvement in design improves quality, speed, cost