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10-1
Quality Control
Production planning and
control
Chapter 1
Quality Control
William J. Stevenson
9th edition
10-2

Quality Control
Quality Control
Quality control is a process that measures
output relative to standard, and acts when
output doesn't meet standards.
 The purpose of quality control is to assure that
processes are performing in an acceptable
manner.
 Companies accomplish quality control by
monitoring process output using statistical
techniques.
10-3
Quality Control
Phases of Quality Assurance
Figure 10.1
Inspection
before/after
production
Acceptance
sampling
The least
progressive
Inspection and
corrective
action during
production
Process
control
Quality built
into the
process
Continuous
improvement
The most
progressive
10-4

Quality Control
Inspection
Inspection is an appraisal activity that compares
goods or services to a standard.
 Inspection can occur at three points:
- before production: is to make sure that inputs
are acceptable.
- during production: to make sure that the
conversion of inputs into outputs is proceeding
in an acceptable manner.
- after production: to make a final verification
of conformance before passing goods to
customers
10-5
Quality Control
Inspection

Inspection before and after production
involves acceptance sampling procedure.
 Monitoring during the production process is
referred as process control
Inputs
Acceptance
sampling
Figure 10.2
Transformation
Process
control
Outputs
Acceptance
sampling
10-6

Quality Control
Inspection
The purpose of inspection is to provide
information on the degree to which items
conform to a standard.
 The basic issues of inspection are:
1 - how much to inspect and how often
2- At what points in the process inspection
should occur.
3 - whether to inspect in a centralized or on-site
location.
4- whether to inspect attributes (counts) or
variables (measures)
10-7
Quality Control
How much to inspect and how often





The amount of inspection can range from no inspection to
inspection of each item many times.
Low-cost, high volume items such as paper clips and pencils
often require little inspection because:
1. the cost associated with passing defective items is quite
low.
2. the process that produce these items are usually highly
reliable, so that defects are rare.
High-cost, low volume items that have large cost associated
with passing defective items often require more intensive
inspection such as airplanes and spaceships.
The majority of quality control applications ranges between
these two extremes.
The amount of inspection needed is governed by the cost of
inspection and the expected cost of passing defective items.
10-8
Quality Control
Inspection Costs
Cost
Figure 10.3
Total Cost
Cost of
inspection
Cost of
passing
defectives
Optimal
Amount of Inspection
10-9
Quality Control
Where to Inspect in the Process
Inspection always adds to the cost of the product;
therefore, it is important to restrict inspection efforts to
the points where they can do the most good. In
manufacturing, some of the typical inspection points
are:

Raw materials and purchased parts

Finished products

Before a costly operation

Before an irreversible process

Before a covering process
10-10
Quality Control
Examples of Inspection Points
Table 10.1
Type of
business
Fast Food
Inspection
points
Cashier
Counter area
Eating area
Building
Kitchen
Hotel/motel Parking lot
Accounting
Building
Main desk
Supermarket Cashiers
Deliveries
Characteristics
Accuracy
Appearance, productivity
Cleanliness
Appearance
Health regulations
Safe, well lighted
Accuracy, timeliness
Appearance, safety
Waiting times
Accuracy, courtesy
Quality, quantity
10-11
Quality Control
Centralized versus on-site inspection

Some situations require that inspections be
performed on site such as inspecting the hull of
a ship for cracks.

Some situations require specialized tests to be
performed in a lab such as medical tests,
analyzing food samples, testing metals for
hardness, running viscosity tests on lubricants.
10-12

Quality Control
Statistical process control
Quality control is concerned with the quality of
conformance of a process: Does the output of a
process conform to the intent of design?
 Managers use Statistical Process Control (SPC)
to evaluate the output of a process to determine if
it is statistically acceptable.
 Statistical Process Control:
Statistical evaluation of the output of a process
during production
 Quality of Conformance:
A product or service conforms to specifications
10-13

Quality Control
Control Chart
Control Chart: an important tool in SPC

Purpose: to monitor process output to see if it is
random (in control) or not (out of control).

A time ordered plot representative sample
statistics obtained from an on going process (e.g.
sample means).

Upper and lower control limits define the range of
acceptable variation.
10-14
Quality Control
Control Chart
Figure 10.4
Abnormal variation
due to assignable sources
Out of
control
UCL
Mean
Normal variation
due to chance
LCL
Abnormal variation
due to assignable sources
0
1
2
3
4
5
6
7
8
9
Sample number
10 11 12 13 14 15
10-15
Quality Control
 The
Statistical Process Control
essence of statistical process
control is to assure that the output of a
process is random so that future output
will be random.
10-16

Quality Control
Statistical Process Control
The Control Process include

Define what is to be controlled.
 Measure the attribute or the variable to be
controlled
 Compare with the standard
 Evaluate if the process in control or out of control
 Correct when a process is judged out of control
 Monitor results to ensure that corrective action is
effective.
10-17

Quality Control
Statistical Process Control
Variations and Control

Random variation: Common natural variations
in the output of a process, created by countless
minor factors. It would be negligible.

Assignable variation: A special variation whose
source can be identified (it can be assigned to a
specific cause)
10-18

Quality Control
Sampling Distribution
The variability of a sample statistic can be
described by its sampling distribution.
 The sampling distribution is a theoretical
distribution that describe the random variability of
a sample statistic.
 The goal of the sampling distribution is to
determine whether nonrandom-and thus,
correctable-source of variation are present in the
output of a process. How?
10-19

Quality Control
Sampling distribution
Suppose there is a process for filling bottles with
soft drink. If the amount of soft drink in a large
number of bottles (e.g., 100) is measured
accurately, we would discover slight differences
among the bottles.
 If these amounts were arranged in a graph, the
frequency distribution would reflect the process
variability.
 The values would be clustered close to the process
average, but some values would vary somewhat
from the mean.
10-20

Quality Control
Sampling distribution (cont.)
If we return back to the process and take samples
of 10 bottles each and compute the mean amount
of soft drink in each sample, we would discover
that these values also vary, just as the individual
values varied. They, too, would have a distribution
of values.
 The following figure shows the process and the
sampling distribution.
10-21
Quality Control
Sampling Distribution
Figure 10.5
Sampling
distribution
Process
distribution
Mean
10-22
Quality Control
Sampling distribution
Properties
 The sampling distribution exhibits much less
variability than the process distribution.
 The sampling distribution has the same mean as
the process distribution.
 The sampling distribution is a normal distribution
regardless of the shape of the process distribution.
(central limit theorem).
10-23
Quality Control
Process and sampling distribution
Process distribution
Mean = 
Variance = 2
Standard deviation = 
Where:
n = sample size
Sampling distribution
Mean = 
Variance =

2
n
Standard deviation =

n
10-24
Quality Control
Normal Distribution
Figure 10.6
Standard deviation


Mean
95.44%
99.74%


10-25





Quality Control
Control limits
Control charts have two limits that separate random
variation and nonrandom variation.
Control limits are based on sampling distribution
Theoretically, the normal distribution extends in either
direction to infinity. Therefore, any value is theoretically
possible.
As a practical matter, we know that 99.7% of the values
will be within ±3 standard deviation of the mean of the
distribution.
Therefore, we could decide to set the control limit at the
values that represent ±3 standard deviation from the mean
10-26
Quality Control
Control Limits
Figure 10.7
Sampling
distribution
Process
distribution
Mean
Lower
control
limit
Upper
control
limit
10-27
Quality Control
SPC hypotheses
Null hypothesis
H0: the process is in control
Alternative hypothesis
H1: the process is out of control
Actual situation
Decision
Reject H0
Don’t reject H0
H0 is true
Type I error
H0 is false
Correct
Correct
Type II error
10-28

Quality Control
Type I error


SPC Errors
Concluding a process is not in control when it
actually is. The probability of rejecting H0 when
it is actually true.
Type II error

Concluding a process is in control when it is not.
The probability of accepting H0 when it is
actually not true.
10-29
Quality Control
Type I Error
Figure 10.8
/2
/2
Mean
Probability
of Type I error
LCL
UCL
Using wider limits (e.g., ± 3 sigma limits) reduces
the probability of Type I error
10-30
Quality Control
Observations from Sample Distribution
Figure 10.9
UCL
LCL
1
2
Sample number
3
4
10-31
Quality Control
 There
Types of control charts
are four types of control charts; two
for variables, and two for attributes
 Attribute: counted data (e.g., number of
defective items in a sample, the number of
calls per day)
 Variable: measured data, usually on a
continuous scale (e.g., amount of time
needed to complete a task, length, width,
weight, diameter of a part).
10-32
Quality Control
Variables Control Charts


Mean control charts

Used to monitor the central tendency of a
process.

X-bar charts
Range control charts

Used to monitor the process dispersion

R charts
10-33
Quality Control
Mean Chart (X-bar chart)

The control limits of the mean chart is calculated as follows: (first approach)

Upper Control Limit (UCL) =
x  z x
x  z

Lower Control Limit (LCL) =
x
Where:
n = sample size
z = standard normal deviation (1,2 and 3; 3 is recommended)
 = process standard deviation
x
= standard deviation of the sampling distribution of the means
x 
x= average of sample means

n
10-34

Quality Control
Mean Chart (X-bar chart)
Example
A quality inspector took five samples, each with
four observations, of the length of time for glue to
dry. The analyst computed the mean of each
sample and then computed the grand mean. All
values are in minutes. Use this information to
obtain three-sigma (i.e., z = 3) control limits for
the means of future time. It is known from
previous experience that the standard deviation of
the process is 0.02 minute.
10-35
Quality Control
Mean chart
Sample
1
Observation
2
3
4
5
1
12.11 12.15 12.09 12.12 12.09
2
12.10 12.12 12.09 12.10 12.14
3
12.11 12.10 12.11 12.08 12.13
4
12.08 12.11 12.15 12.10 12.12
x
12.10 12.12 12.11 12.10 12.12
10-36



Quality Control
Solution
n=4
z=3
 = 0.02
12.10  12.12  12.11  12.10  12.12
x
 12.11
5
 0.02 
UCL : 12.11  3
  12.14
 4 
 0.02 
LCL : 12.11  3
  12.08
 4 
10-37
Quality Control
Control chart
UCL
12.14
x
12.11
LCL
12.08
1
2
3
Sample
4
5
10-38


Quality Control
Mean chart
A second approach to calculate the control limits:
This approach assumes that the range is in control
UCL  x  A2 R
LCL  x  A2 R
Where:
A2 = A factor from table 10.2 Page 441
R
= Average of sample ranges
This approach is
recommended when
the process standard
deviation is not
known
10-39


Quality Control
Example
Twenty samples of n = 8 have been taken from a
cleaning operations. The average sample range for the
20 samples was 0.016 minute, and the average mean
was 3 minutes. Determine three-sigma control limits
for this process.
Solution
x = 3 min. R,
= 0.016, A2 = 0.37 for n = 8 (table
10.2)
UCL  x  A2 R  3  0.37(0.016)  3.006
LCL  x  A2 R  3  0.37(0.016)  2.994
10-40


Quality Control
Range Control Chart (R-chart)
The R-charts are used to monitor process dispersion;
they are sensitive to changes in process dispersion.
Although the underlying sampling distribution of the
range is not normal, the concept for use of range charts
are much the same as those for use of mean chart.
Control limits:
UCL  D4 R
LCL  D3 R
Where values of D3 and D4 are obtained from table
10.2 page 441
10-41

Quality Control
R-chart
Example
Twenty-five samples of n = 10 observations have been
taken from a milling process. The average sample range
was 0.01 centimeter. Determine upper and lower control
limits for sample ranges.
 Solution
R = 0.01 cm, n = 10
From table 10.2, for n = 10, D4 = 1.78 and D3 = 0.22
UCL = 1.78(0.01) = 0.0178 or 0.018
LCL = 0.22(0.01) = 0.0022 or 0.002
10-42

Quality Control
R-Chart
Example
Small boxes of cereal are labeled “net weight 10
ounces.” Each hour, a random sample of size n = 4
boxes are weighted to check process control. Five hours
of observation yielded the following:
Time
9 A.M.
10 A.M
11 A.M
Noon
1 P.M
Box 1
9.8
10.1
9.9
9.7
9.7
Box 2
10.4
10.2
10.5
9.8
10.1
Box 3
9.9
9.9
10.3
10.3
9.9
Box 4
10.3
9.8
10.1
10.2
9.9
Range
0.6
0.4
0.6
0.6
0.4
10-43

Quality Control
R-Chart
Solution
n=4
For n = 4 , D3 = 0 and D4 = 2.28
0.6  0.4  0.6  0.6  0.4
 0.52
5
UCL  D4 R  2.28(0.52)  1.1865
R 
LCL  D3 R  0(0.52)  0
Since all ranges are between the upper and lower
limits, we conclude that the process is in control
10-44
Quality Control
Using Mean and Range Charts

Mean control charts and range control charts
provide different perspectives on a process.

The mean charts are sensitive to shifts in process
mean, whereas range charts are sensitive to
changes in process dispersion.

Because of this difference in perspective, both
types of charts might be used to monitor the same
process.
10-45
Quality Control
Mean and Range Charts
Figure 10.10A
(process mean is
shifting upward)
Sampling
Distribution
UCL
Detects shift
x-Chart
LCL
UCL
R-chart
LCL
Does not
detect shift
10-46
Quality Control
Mean and Range Charts
Figure 10.10B
Sampling
Distribution
(process variability is increasing)
UCL
x-Chart
LCL
Does not
reveal increase
UCL
R-chart
Reveals increase
LCL
10-47
Quality Control
Using the Mean and Range Chart
To use the Mean and Range control chart, apply the
following procedure:
1.
2.
3.
4.
5.
Obtain 20 to 25 samples. Compute the appropriate
sample statistics (mean and range) for each sample.
Establish preliminary control limits using the formulas.
Determine if any points fall outside the control limits.
If you find no out-of-control signals, assume that the
process is in control. If not, investigate and correct
assignable cause of variation. Then resume the process
and collect another set of observations upon which
control limits can be based.
Plot the data on a control chart and check for out-ofcontrol signals.
10-48
Quality Control
Control Chart for Attributes

Control charts for attributes are used when
the process characteristic is counted rather
than measured. Two types are available:

P-Chart - Control chart used to monitor the
proportion of defectives in a process

C-Chart - Control chart used to monitor the
number of defects per unit
Attributes generate data that are counted.
10-49
Quality Control
Use of p-Charts
Table 10.3


When observations can be placed into two
categories.

Good or bad

Pass or fail

Operate or don’t operate
When the data consists of multiple samples
of several observations each
10-50




Quality Control
P-Charts
The theoretical basis for the P-chart is the binomial
distribution, although for large sample sizes, the normal
distribution provides a good approximation to it.
A P-chart is constructed and used in much the same way
as a mean chart.
The center line on a P-chart is the average fraction
defective in the population, P.
The standard deviation of the sampling distribution when
P is known is:
p 
p(1  p)
n
10-51
Quality Control

P-Chart
The Control limits
UCL  p  z p
LCL  p  z p
If p is unknown, it can be estimated from the samples. That
estimates p, replaces p in the preceding formulas, and
^
 p replaces p.
p
Total number of defectives
Total number of observations
10-52

Quality Control
P-Chart
Example
An inspector counted the number of defective
monthly billing statements of a company
telephone in each of 20 samples. Using the
following information, construct a control chart
that will describe 99.74 percent of the chance
variation in the process when the process is in
control. Each sample counted 100 statements.
10-53

Quality Control
P-Chart
Example (cont.)
Sample
# of defective
Sample
# of defective
1
4
11
8
2
10
12
12
3
12
13
9
4
3
14
10
5
9
15
21
6
11
16
10
7
10
17
8
8
22
18
12
9
13
19
10
10
10
20
16
Total
220
10-54
Quality Control
P-Chart

Solution
Z for 99.74 percent is 3
p
^

p
220
 0.11
20(100)

p (1  p )

n
0.11(1  0.11)
 0.03
100
Control limits are
^
UCL  p  z 
p
 0.11  3(0.03)  0.20
p
 0.11  3(0.03)  0.02
^
LCL  p  z 
10-55

Quality Control
P-Chart
Solution (cont.)
Fraction
defective
0.20
UCL
0.11
p
0.02
LCL
1
10
Sample number
20
10-56
Quality Control
Use of c-Charts
Table 10.3

Use only when the number of occurrences per
unit of measure can be counted; nonoccurrences cannot be counted.

Scratches, chips, dents, or errors per item
 Cracks or faults per unit of distance
 Breaks or Tears per unit of area
 Bacteria or pollutants per unit of volume
 Calls, complaints, failures per unit of time
10-57





Quality Control
C-Chart
When the goal is to control the number of occurrences (e.g.,
defects) per unit, a C-chart is used.
Units might be automobiles, hotel rooms, typed papers, or rolls of
carpet.
The underlying sampling distribution is the Poisson distribution.
Use of Poisson distribution assumes that defects occur over some
continuous region and that the probability of more than one
defect at any particular point is negligible.
The mean number of defects per unit is c and the standard
deviation is:
c
10-58

Quality Control
C-Chart
Control Limits
UCL  c  z c
LCL  c  z c
If the value of c is unknown, as is generally the
case, the sample estimate, c , is used in place
of c. where:
c = Number of defects ÷ Number of samples
10-59

Quality Control
C-Chart
Example
Rolls of coiled wire are
monitored using c-chart.
Eighteen rolls have been
examined, and the number
of defects per roll has
been recorded in the
following table. Is the
process in control? Plot
the values on a control
chart using three standard
deviation control limit.
sample # of
Sample # of
defects
defects
1
2
3
4
5
6
7
8
9
3
2
4
5
1
2
4
1
2
10
11
12
13
14
15
16
17
18
1
3
4
2
4
2
1
3
1
45
10-60

Quality Control
C-Chart
Solution
Average number of defects per coil = c = 45/18 =2.5
UCL  c  3 c  2.5  3 2.5  7.24
LCL  c  3 c  2.5  3 2.5  2.24  0
When the computed lower control limit is negative, the
effective lower limit is zero. The calculation sometimes
produces a negative lower limit due to the use of normal
distribution as an approximation to the Poisson
distribution.
The control chart is left for the student as a homework
10-61
Managerial consideration concerning
control charts
Quality Control

At what point in the process to use control charts: at the
part of the process that (1) have tendency to go out of
control, (2) are critical to the successful operation of
the product or service.

What size samples to take: there is a positive relation
between sample size and the cost of sampling.

What type of control chart to use:

Variables: gives more information than attributes

Attributes: less cost and time than variables
10-62
Quality Control
Run Tests

Run test – a test for randomness

Control charts test for points that are too extreme to be
considered random.

However, even if all points are within the control
limits, the data may still not reflect a random process.

Any sort of pattern in the data would suggest a nonrandom process.

The presence of patterns, such as trends, cycles, or bias
in the output indicates that assignable, or nonrandom,
cause of variation exist.

Analyst often supplement control charts with a run
test, which is another kind of test for randomness.
10-63
Quality Control
Nonrandom Patterns in Control charts
Figure 10.11

Trend: sustained upward or downward
movement.
 Cycles: a wave pattern
 Bias: too many observations on one side of
the center line
 Mean shift: A shift in the average
 Too much dispersion: the values are too
spread out
10-64

Quality Control
Run Test
A run is defined as a sequence of observations with a
certain characteristic, followed by one or more
observations with a different characteristic.
 The characteristic can be anything that is observable.
 For example, in a series AAAB, there are two runs; a
run of three A’s followed by a run of one B.
 The series AABBBA , indicates three runs; a run of
two A’s followed by a run of three B’s, followed by
a run of one A.
10-65
Quality Control

Run test
There are two types of run test:
1.
2.
Runs up and down
Runs above and below the median

In order to count these runs, the data are transformed
into a series of U’s and D’s (for up and down) and into
a series of A’s and B’s (for above and below the
median).

There are three U/D and four A/B runs for the data:
25
29
42
40
35 38
U
U
D
D U
B
B
A
A
B
A
Where the median is 36.5
10-66
Quality Control
Figure 10.12
Counting Runs
Counting Above/Below Median Runs
B A
Figure 10.13
A
B
A
B
B
B A
Counting Up/Down Runs
U
U
D
U
(7 runs)
A
B
(8 runs)
D
U
D U
U D
10-67

1.
2.
Quality Control
Run test procedure
To determine whether any patterns are present in
control charts, one must do the following:
Transform the data into both A’s and B’s and U’s
and D’s, and then count the number of runs in each
case.
Compare the number of runs with the expected
number of runs in a completely random series, which
is calculated as follows:
N
1
2
2N  1

3
E ( r ) med 
E (r ) u / d
Where: N is the number of observations or data
points, and E(r) is the expected number of runs
10-68
Quality Control
Run test procedure (cont.)
3. Calculate the standard deviations of the runs as:
 med 
N 1
4
u/d 
16 N  29
90
4. Calculate the test statistic (Ztest) as following:
Z test
observed number of runs – expected number of runs
standard deviation of number of runs
N
 1)
2

N 1
4
2N  1
r  (
)
3

16 N  29
90
r  (
Z t est
Z t est
For the median
Up and down
If the Ztest is
within ± 2 or ± 3;
then the process
is random;
otherwise, it is
not random
10-69

Quality Control
Run test
Example
Twenty sample means
have been taken from a
process. The means are
shown in the following
table. Use median and
up/down run test with
z = 2 to determine if
assignable causes of
variation are present.
Assume the median is 11.
sample
1
2
3
4
5
6
7
8
9
10
mean sample Mean
10
11
10.7
10.4
12
11.3
10.2
13
10.8
11.5
14
11.8
10.8
15
11.2
11.6
16
11.6
11.1
17
11.2
11.2
18
10.6
10.6
19
10.7
10.9
20
11.9
10-70

Quality Control
Run test
Solution
sample
mean
A/B
U/D
Sample
Mean
A/B
U/D
1
10
B
-
11
10.7
B
D
2
10.4
B
U
12
11.3
A
U
3
10.2
B
D
13
10.8
B
D
4
11.5
A
U
14
11.8
A
U
5
10.8
B
D
15
11.2
A
D
6
11.6
A
U
16
11.6
A
U
7
11.1
A
D
17
11.2
A
D
8
11.2
A
U
18
10.6
B
D
9
10.6
B
D
19
10.7
B
U
10
10.9
B
U
20
11.9
A
U
10-71
Quality Control
Run test
Solution (cont.)
1. A/B: 10 runs
and
U/D: 17 runs
2. Expected number of runs for each test is:
N
20
1 
 1  11
2
2
2 N  1 2(20)  1


 13
3
3
E (r ) med 
E (r ) u / d
3. The standard deviations are:
 med 
N 1

4
20  1
 2.18
4
u/d 
16 N  29

90
16( 20)  29
 1.8
90
4. The ztest values are:
10  11
 0.46
2.18
17  13

 2.22
1.8
Z med 
Zu / d
Although the median
test doesn’t reveal any
pattern, because its Ztest
value is within ±2, the
up/down test does; its
value exceed +2.
consequently,
nonrandom variations
are probably present in
the data and, hence, the
process is not in control
10-72

Quality Control
Tolerances or specifications


Range of acceptable values established by
engineering design or customer requirements
Process variability


Process Capability
Natural variability in a process
Process capability

Process variability relative to specification
10-73




Quality Control
Capability analysis
Capability analysis is the determination of whether the
variability inherent in the output of a process falls within
the acceptable range of variability allowed by the design
specification for the process output.
If it is within the specifications, the process is said to be
“capable.” if it is not, the manager must decide how to
correct the situation.
We cannot automatically assume that a process that is in
control will provide desired output. Instead, we must
specifically check whether a process is capable of meeting
specifications and not simply set up a control chart to
monitor it.
A process should be both in control and within
specifications before production begins.
10-74
Quality Control
Process Capability
Figure 10.15
Lower
Specification
Upper
Specification
A. Process variability
matches specifications
Lower
Specification
Upper
Specification
B. Process variability
Lower
Upper
well within specifications Specification Specification
C. Process variability
exceeds specifications
10-75

Quality Control
Capability analysis
If the product doesn’t meet specifications (not capable) a
manager might consider a range of possible solutions such
as:
1. Redesign the process.
2. Use an alternative process.
3. Retain the current process but attempt to eliminate
unacceptable output using 100% inspection.
4. Examine the specifications to see whether they are
necessary or could be relaxed without adversely affecting
customer satisfaction.
10-76
Quality Control
Process Capability Ratio
Calculate the capability and compare it to
specification width. If the capability is less than the
specification width, the process is capable.
Where: Capability = 6; where  is the process SD
Or calculate
Process capability ratio, Cp =
Cp =
specification width
process width
Upper specification – lower specification
6
The process is capable if Cp is at least 1.33, this ratio
implies only about 30 parts per million can be expected
to not be within the specification
10-77

Quality Control
Capability analysis
Example
A manager has the option of using any one of three
machines for a job. The machines and their standard
deviations are listed below. Determine which machines
are capable if the specifications are 10 mm and 10.8
mm.
Machine
A
Standard deviation
(mm)
0.13
B
0.08
C
0.16
10-78
Quality Control
Capability analysis

Solution
Capability = 6
Machine
A
B
C
Standard
deviation (mm)
0.13
0.08
0.16
Machine
capability
0.78
0.48
0.96
Capable
Yes
Yes
No
It is clear that machine A and machine B are
capable, since the capability is less than the
specification width (10.8 – 10 = 0.8)
10-79
Quality Control
Capability ratio
Example
Compute the process capability ratio for each machine
in the previous example
Solution
Machine Standard Machine
deviation capability
(mm)
6
A
0.13
0.78
B
C
0.08
0.16
0.48
0.96
Cp
Capable
0.8/0.78= 1.03
No
0.8/0.48 = 1.67
0.8/0.96 = 0.83
Yes
No
Only machine B is capable because its ratio exceed 1.33
10-80
Quality Control
3 Sigma and 6 Sigma Quality
Upper
specification
Lower
specification
1.350 ppm
1.350 ppm
1.7 ppm
1.7 ppm
Process
mean
+/- 3 Sigma
+/- 6 Sigma
10-81

Quality Control
Cpk ratio
If a process is not centered (the mean of the
process is not in the center of the specification), a
more appropriate measure of process capability is
the Cpk ratio, because it does take the process
mean into account.
 The Cpk is equal the smaller of
Upper specification – process mean
3
And
Process mean – lower specification
3
10-82
Quality Control
Cpk Ratio

Example
A process has a mean of 9.2 grams and a standard
deviation 0f 0.3 grams. The lower specification limit
is 7.5 grams and upper specification limit is 10.5
grams. Compute Cpk
Solution
1. Compute the ratio for the lower specification:
9.2  7.5 1.7

 1.89
3(.3)
0.9
2. Compute the ratio for the upper specification:
10.5  9.2 1.3

 1.44
3(0.3)
.9
The smaller of the two ratios is 1.44
(greater than 1.33), so this is the Cpk .
Therefore, the process is capable
10-83
Quality Control
Improving Process Capability

Simplify the process
 Standardize the process
 Mistake-proof
 Upgrade equipment
 Automate
10-84
Quality Control
Improving Process Capability
Method
Examples
Simplify
Eliminate steps, reduce number of parts
Standardize
use standard parts, standard procedure
Make
Design parts that can only be assembled
mistake-proof the correct way; have simple checks to
verify a procedure has been performed
correctly
Upgrade
Replace worn-out equipment; take
equipment
advantage of technological improvements
Automate
Substitute processing for manual
processing
10-85
Quality Control
Taguchi Loss Function
Figure 10.17
Traditional
cost function
Cost
Taguchi
cost function
Lower
spec
Target
Upper
spec
10-86
Quality Control
Limitations of Capability Indexes
1.
Process may not be stable
2.
Process output may not be normally
distributed
3.
Process not centered but Cp is used