Quick Recap
Monitoring and Controlling
Performance Management Objectives
In this training you will learn the most effective methods
to create constructive performance evaluations and how
to communicate with employees during the performance
process.
• To learn the basics of Performance Management
• To understand the purpose and strategies behind
Performance Appraisals
• To gain knowledge of the performance management
forms and tools
• To gain an understanding of the merit/awards process
2
Lesson 13: Monitoring and Controlling Project Performance and Quality
Topic 13A: Perform Quality Control
Topic 13B: Report on Project Performance
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.
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
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
Inspection
• Inspection before and after production involves
acceptance sampling procedure.
• Monitoring during the production process is referred
as process control
Inputs
Acceptance
sampling
Transformation
Process
control
Outputs
Acceptance
sampling
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)
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.
Inspection Costs
Cost
Figure 10.3
Total Cost
Cost of
inspection
Cost of
passing
defectives
Optimal
Amount of Inspection
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
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
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.
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
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.
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
Statistical Process Control
• The essence of statistical process control is to
assure that the output of a process is random so
that future output will be random.
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.
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)
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?
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.
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.
Sampling Distribution
Figure 10.5
Sampling
distribution
Process
distribution
Mean
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).
Process and sampling distribution
Process distribution
Mean = 
Variance = 2
Standard deviation = 
Where:
n = sample size
Sampling distribution
Mean = 
Variance =

2
n

Standard deviation =
n
Normal Distribution
Figure 10.6
Standard deviation


Mean
95.44%
99.74%


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
Control Limits
Figure 10.7
Sampling
distribution
Process
distribution
Mean
Lower
control
limit
Upper
control
limit
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
SPC Errors
• Type I error
– 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.
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
Observations from Sample Distribution
Figure 10.9
UCL
LCL
1
2
Sample number
3
4
Types of control charts
• There 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).
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
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
•
Lower Control Limit (LCL) =
x  z 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 = average of sample means  x


n
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.
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
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 
Control chart
UCL
12.14
x
12.11
LCL
12.08
1
2
3
Sample
4
5
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
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 threesigma 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
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
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
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
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
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.
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
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
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-of-control
signals.
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.
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
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(1  p)
p 
n
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
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.
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
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 
P-Chart
• Solution (cont.)
Fraction
defective
0.20
UCL
0.11
p
0.02
LCL
1
10
Sample number
20
Use of c-Charts
Table 10.3
• Use only when the number of occurrences per unit of
measure can be counted; non-occurrences 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
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
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
C-Chart
• Example
Rolls of coiled wire are
monitored using cchart. 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
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
Managerial consideration concerning
control charts
• 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
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.
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
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.
Counting Runs
Figure 10.12
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
Run test procedure
•
1.
2.
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
E ( r ) med 
E (r ) u / d
1
2
2N  1

3
Where: N is the number of observations or data
points, and E(r) is the expected number of runs
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
Run test
• Example
sample mean sample Mean
Twenty sample means have
1
10
11
10.7
been taken from a process.
The means are shown in
2
10.4
12
11.3
the following table. Use
3
10.2
13
10.8
median and up/down run
test with
4
11.5
14
11.8
z = 2 to determine if
5
10.8
15
11.2
assignable causes of
variation are present.
6
11.6
16
11.6
Assume the median is 11.
7
8
9
10
11.1
11.2
10.6
10.9
17
18
19
20
11.2
10.6
10.7
11.9
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
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
Process Capability
• Tolerances or specifications
– Range of acceptable values established by
engineering design or customer requirements
• Process variability
– Natural variability in a process
• Process capability
– Process variability relative to specification
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.
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
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.
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
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
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)
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
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
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
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
Improving Process Capability
•
•
•
•
•
Simplify the process
Standardize the process
Mistake-proof
Upgrade equipment
Automate
Method
Examples Process Capability
Improving
Simplify
Eliminate steps, reduce number of parts
Standardize
use standard parts, standard procedure
Make mistake-proof Design parts that can only be assembled the correct way;
have simple checks to verify a procedure has been
performed correctly
Upgrade equipment Replace worn-out equipment; take advantage of
technological improvements
Automate
Substitute processing for manual processing
Taguchi Loss Function
Figure 10.17
Traditional
cost function
Cost
Taguchi
cost function
Lower
spec
Target
Upper
spec
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