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Operations
Management
Statistical Process Control
Supplement 6
S6-1
Outline
Statistical Process Control (SPC).
Mean charts or X-Charts.
Range chart or R-Charts.
Control charts for attributes.
Managerial issues and control charts.
Acceptance Sampling.
S6-2
Statistical Process Control (SPC)
Statistical technique to identify when nonrandom variation is present in a process.
All processes are subject to variability.
Natural causes: Random variations.
Assignable causes: Correctable problems.
Machine wear, unskilled workers, poor materials.
Uses process control charts.
S6-3
Statistical Process Control Steps
Start
Take Sample
Produce Good
Provide Service
Take Samples
Create
Control Chart
Inspect Sample
Is process
in control?
Yes
No
Stop Process
Find Out Why
S6-4
Process Control Charts
Plot of Sample Data Over Time
Sample Value
80
Upper control limit
60
40
20
0
Lower control limit
1
5
9
13
17
Time
S6-5
21
Control Charts
Process is not in control if:
Sample is not between upper and lower control
limits.
A non-random pattern is present, even when
between upper and lower control limits.
Based on sample being normally distributed.
S6-6
Distribution of Sample Means
Mean of sample means x
x
Standard deviation of
x
the sample means
n
3 x 2 x 1 x
x
x 2 x 3 x
(mean)
95.5% of all x fall within 2 x
99.7% of all x fall within 3 x
S6-7
Central Limit Theorem
Central Limit Theorem
As sample size
gets
large
enough,
distribution of mean
values becomes
approximately normal
for any population
distribution.
X
X
S6-8
Control Chart Types
Control
Charts
Continuous
Numerical Data
Categorical or
Discrete Numerical
Data
Variables
Charts
R
Chart
Attributes
Charts
P
Chart
X
Chart
S6-9
C
Chart
Quality Characteristics
Variables
Attributes
Characteristics that you
measure, e.g., weight,
length.
Characteristics for which
you focus on defects.
Continuous values.
Categorical or discrete
values.
S6-10
‘Good’ or ‘Bad’.
# of defects.
X Chart
Shows sample means over time.
Monitors process average.
Example: Weigh samples of coffee.
Collect many samples, each of n bags.
Sample size = n.
Compute mean and range for each sample.
Compute upper and lower control limits (UCL, LCL).
Plot sample means and control limits.
S6-11
X Chart Control Limits
UCLx x A2 R
LCLx x A2 R
A2 is from Table S6.1
k
k
Ri
R i1
xi
x i1
k
k
sample range
at time i
sample mean
at time i
S6-12
Factors for Computing Control
Chart Limits
Sample
Size, n
2
Mean
Upper
Lower
Factor, A 2 Range, D4 Range, D3
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
S6-13
X Chart - Example 2
Each sample is 4 measurements.
Determine 3 control limits.
sample
1
2
3
4
5
mean
5.02
4.99
4.97
5.03
4.99
x 5.0
range.
.12
.08
.13
.18
.14
4.96, 5.03, 5.01, 5.08
R 0.13
UCLx 5 0.729 0.13 5.095
LCL x 5 0.729 0.13 4.905
S6-14
X Chart - Example 2
Sample Mean
5.1
Upper control limit=5.095
5.0
Lower control limit=4.905
4.9
Time
S6-15
Example 2 – New Samples
sample
6
7
8
values
5.05, 5.00, 4.80, 4.95
5.00, 5.10, 5.10, 5.00
4.80, 5.20, 5.10, 5.00
Sample Mean
5.1
mean
4.95
5.05
5.025
range
0.25
0.10
0.40
Upper control limit=5.095
5.0
Lower control limit=4.905
4.9
Time
S6-16
R Chart
Shows sample ranges over time.
Sample range = largest - smallest value in sample.
Monitors process variability.
Example: Weigh samples of coffee.
Collect many samples, each of n bags.
Sample size = n.
Compute range for each sample & average range.
Compute upper and lower control limits (UCL, LCL).
Plot sample ranges and control limits.
S6-17
R Chart Control Limits
UCL R D 4 R
From Table S6.1
LCL R D 3 R
k
R
Ri
sample range at
time i
i 1
k
S6-18
R Chart - Example 2
Each sample is 4 measurements.
Determine 3 control limits.
sample
1
2
3
4
5
mean
5.02
4.99
4.97
5.03
4.99
x 5.0
range
.12
.08
.13
.18
.14
4.96, 5.03, 5.01, 5.08
R 0.13
UCLR 2.282 0.13 0.297
LCLR 0 0.13 0
S6-19
Sample Range
R Chart - Example 2
0.3
Upper control limit=0.297
0.2
0.1
Lower control limit=0
0
Time
S6-20
Example 2 – New Samples
Sample Range
sample
6
7
8
values
5.05, 5.00, 4.80, 4.95
5.00, 5.10, 5.10, 5.00
4.80, 5.20, 5.10, 5.00
0.3
mean
4.95
5.05
5.025
range
0.25
0.10
0.40
Upper control limit=0.297
0.2
0.1
Lower control limit=0
0
Time
S6-21
Control Chart Steps
Collect 20 to 25 samples of n=4 or n=5 from a
stable process & compute the mean and range.
Compute the overall mean and average range.
Calculate upper and lower control limits.
Collect new samples, and plot the means and
ranges on their respective control charts.
S6-22
Control Chart Steps - Continued
Investigate points or patterns that indicate the
process is out of control. Assign causes for
the variations.
Collect additional samples and revalidate the
control limits.
S6-23
Use of Control Charts
S6-24
Example 3
sample
1
2
3
4
values
4.9, 5.0, 5.1
5.2, 5.3, 5.4
5.5, 5.6, 5.7
5.8, 5.9, 6.0
x 5.45
mean
5.0
5.3
5.6
5.9
R 0.2
UCL 5.45 1.0230.2 5.6546
x
LCL 5.45 1.0230.2 5.2454
x
UCL 2.5740.2 0.5148
R
LCL 00.2 0
R
S6-25
range
0.2
0.2
0.2
0.2
Sample Mean
Example 3 – Control Charts
X Chart
6.0
Upper control limit = 5.6546
5.5
Lower control limit = 5.2454
5.0
Sample Range
Time
R Chart
1.0
0.5
Upper control limit = 0.5148
0.0
Lower control limit = 0
Time
S6-26
Example 4
sample
1
2
3
4
values
5.0, 5.0, 5.0
4.5, 5.0, 5.5
4.0, 5.0, 6.0
3.0, 5.0, 7.0
x 5.0
mean
5.0
5.0
5.0
5.0
range
0.0
1.0
2.0
4.0
R 1.75
UCL 5.0 1.0231.75 6.79025
x
LCL 5.0 1.0231.75 3.20975
x
UCL 2.5741.75 4.5045
R
LCL 01.75 0
R
S6-27
Sample Mean
Example 4 – Control Charts
7.0
Upper control limit = 6.79025
X Chart
5.0
Lower control limit = 3.20975
3.0
Sample Range
Time
6.0
Upper control limit = 4.5045
3.0
0.0
R Chart
Lower control limit = 0
Time
S6-28
p Chart
Attributes control chart.
Shows % of nonconforming items.
Example: Count # defective chairs & divide by
total chairs inspected.
Chair is either defective or not defective.
S6-29
c Chart
Attributes control chart.
Shows number of defects in a unit.
Unit may be chair, steel sheet, car, etc.
Size of unit must be constant.
Example: Count # defects (scratches, chips
etc.) in each chair of a sample of 100 chairs.
S6-30
Acceptance Sampling
Quality testing for incoming materials or
finished goods.
Procedure:
Take one or more samples at random from a lot
(shipment) of items.
Inspect each of the items in the sample.
Decide whether to reject the whole lot based on
the inspection results.
S6-31
Acceptance Sampling
Inspecting all items is too expensive.
The larger the sample inspected:
The greater the cost for inspection.
The less likely you are to accept a “bad” lot or to
reject a “good” lot.
Key questions:
How many should be inspected in each lot?
How confident are you in the accept/reject
decision?
S6-32