Transcript Document 7893980

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  i1
 xi
x  i1
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 1.0230.2  5.6546
x
LCL  5.45 1.0230.2  5.2454
x
UCL  2.5740.2  0.5148
R
LCL  00.2  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 1.0231.75  6.79025
x
LCL  5.0 1.0231.75  3.20975
x
UCL  2.5741.75  4.5045
R
LCL  01.75  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