AND QUALITY MANAGEMENT TOPICS
Chapter 18. Minitab offers a variety of control charts like those just described,
8. Eight points in a row are more than 1 sigma from the centerline
Process Capability Index
Earlier, in our discussion of Taguchi methods, we briefly introduced the concept of upper and lower specification limits. Such specifications may be necessary so an individual item will fit or function properly with other components in a fin- ished product. For example, it may be required that the lengths of aluminum rods in an industrial application be no greater than 1.575 inches and no less than
( 20.9 )
1.525 inches. In this case, the upper specification limit (USL) would be 1.575 inches and the lower specification limit (LSL) would be 1.525. Thus, any rod resulting from the production process would be deemed acceptable if it fell within these limits. Think of these limits as the goalposts on an American football field; a kicked ball must go between the vertical posts in order to be successful and earn points.
We can express the potential ability of a process to generate products fall- ing within the required specification limits with a measure known as the Process Potential Index, or Cp. It can be calculated as shown below.
Process Potential Index, Cp: Cp5 USL 2 LSL
wwwww6 , where Cp5 Process Potential Index USL 5 upper specification limit
LSL 5 lower specification limit
5 process standard deviation (If is unknown, use s.)
Assumptions: The process is in control, the process mean is midway between the upper and lower specification limits, and the output is at least ap- proximately normally distributed.
Process Capability Index, Cpk:
Cpk5 the lower of ( wwww}}x 23sLSL ) or ( wwwwUSL 3s2}}x )
where Cpk5 Process Capability Index }}
x 5 overall mean for all units sampled s 5 standard deviation for all units sampled USL 5 upper specification limit
LSL 5 lower specification limit
Assumptions: The process is in control and the output is at least approximately normally distributed.
This index is essentially the allowable spread in the output divided by the actual spread in the output. Higher numerical values for Cp are preferable to lower ones, a value of at least 1.0 is necessary for a process to be judged as po- tentially capable, and the ideal Cp will be as high as possible.9
Note that this measure assumes that the process mean is exactly midway between the upper and lower specification limits. In actuality, the process mean might be closer to the upper limit, closer to the lower limit, or even lie outside the range designated by the limits. Because of these possibilities, we will focus on the more realistic measure discussed below.
The Process Capability Index (Cpk) takes into consideration that the process mean might be closer to one of the specification limits than to the other. In addi- tion, we generally do not know the actual process mean and standard deviation, so we must typically estimate them based on the mean and standard deviation of a sample of output, such as the entire group of measurements we may have used in constructing a statistical process control chart for variables. In this case, we can calculate Cpk as shown below.
9“Process Capability, Minding Your Cpk’s,” qualitydigest.com, September 1, 2009.
EXAMPLE
Determination of Process Capability Index
In Computer Solutions 20.1, we were provided with the lengths (inches) of 80 rivets from a manufacturing process. These lengths are in file CX20RIVS, and we can use Minitab or Excel to calculate the sample mean and sample standard deviation for all 80 rivets. In either case, the results are as shown below.
Overall mean for all units sampled: }}x 5 1.3983 inches Sample standard deviation for all units sampled: s 5 0.0103 inches We have already concluded that the process is in control, and we have rea- sonably assumed that the distribution of process output measurements is at least approximately normal. The preceding items describe our knowledge of this pro- duction process. (Note that we are using the actual sample standard deviation for the sample of 80, not the estimated value that was used in constructing the statistical process control chart in Computer Solutions 20.1.)
Suppose we are the producer involved, and that the contract with our rivet cus- tomer requires that acceptable rivets have a length no greater than 1.4900 inches (USL) and no less than 1.3300 inches (LSL). Given this contractual requirement and the sample information from our production process, what is the Process Capability Index (Cpk) of our rivet production line?
SOLUTION
In determining our Process Capability Index (Cpk), we must first compute how well we’re doing with regard to the lower (LSL) and the upper (USL) specifica- tion limits, respectively, then identify which value is the lower of the two.
Pertaining to the lower specification limit (LSL):
( }}x 2wwww3sLSL ) 5 ( 1.3983 2wwwwwwww3(0.0103) 1.3300 ) 5 2.21
Pertaining to the upper specification limit (USL):
( wwwwwUSL 3s 2}}x ) 5 ( wwwwwwww1.4900 3(0.0103)21.3983 ) 5 2.97
The lower of these two values is 2.21, so our Process Capability Index is Cpk 5 2.21. Both we and our customer should be impressed with a Cpk this high.
In order to officially qualify as a Six-Sigma process, the Cpk performance must be at least 2.0.
The Process Capability Index should be Cpk $1.0 to demonstrate that a pro- cess is capable, and higher values are even better. However, diminishing returns may set in—for example, a Cpk greater than 2.5 or 3.0 might suggest that we are paying for levels of precision we don’t really need. For processes involving both upper and lower specification limits, as we’ve discussed here, one expert has recommended minimum Cpk values of 1.33 for existing processes, 1.50 for new processes or for existing processes involving a safety or critical parameter,
EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAM
1.67 for new processes involving a safety or critical parameter, and 2.00 for a Six-Sigma quality process.10
Because the Cpk evaluates how effectively a process meets the intended speci- fications for its output, it helps managers design and refine the operations gen- erating the products on which their company’s profits and reputations rely. Of necessity, our coverage of this key descriptor has been relatively brief, and the interested reader should refer to more advanced works for additional insights into the Process Capability Index and its managerial applications.
10Douglas Montgomery, Introduction to Statistical Quality Control, New York: John Wiley & Sons, 2004, p. 776.
20.54 Differentiate between the Process Potential Index (Cp) and the Process Capability Index (Cpk). In what way(s) are they similar? In what way(s) do they differ?
20.55 Can the Process Capability Index (Cpk) ever be greater than the Process Potential Index (Cp)? If so, how?
If not, why not?
20.56 A process is in control and the results from a series of samples show the overall mean and standard deviation of all the units sampled to be }}x 5 23.50 grams and s 5 0.21 grams. The upper and lower specification limits for the output are USL 5 24.40 grams and LSL 5 22.80 grams. Based on this information, determine and interpret the value of the Process Capability Index for this process.
20.57 A process is in control and the results from a series of samples show the overall mean and standard deviation of all the units sampled to be }}x 5 2.320 inches and s 5 0.003 inches. The upper and lower specification limits for the output are USL 5 2.328 inches and LSL 5 2.272 inches. Based on this information, deter- mine and interpret the value of the Process Capability Index for this process.
( DATA SET ) Note: Exercises 20.58–20.64 require a computer and statistical software.
20.58 Repeat Exercise 20.42 and apply all of the diag- nostic tests available with your statistical software. Has applying these diagnostics to the mean and range charts raised any concerns regarding the state of control of the process? If so, in what way(s)?
20.59 Repeat Exercise 20.43 and apply all of the diag- nostic tests available with your statistical software. Has applying these diagnostics to the mean and range charts raised any concerns regarding the state of control of the process? If so, in what way(s)?
20.60 Repeat Exercise 20.52 and apply all of the diag- nostic tests available with your statistical software. Has applying these diagnostics to the c-chart raised any con- cerns regarding the state of control of the process? If so, in what way(s)?
20.61 The data in file XR20061 represent 30 samples, each with n 5 3 items measured in inches. For the associated mean and range charts, have your statistical software apply all of the diagnostic tests of which it is capable.
Has applying these diagnostics to the mean and range charts raised any concerns regarding the state of control of the process? If so, in what way(s)?
20.62 The data in file XR20062 represent the number of anonymous tips phoned in during each of 50 consecutive days to a local law enforcement hotline. Use your statisti- cal software in generating an appropriate control chart for these data and diagnosing unusual conditions and pat- terns. Have the diagnostics cast any doubt on whether the process is in control? If so, in what way(s)?
20.63 A process has generated the output (in centimeters) listed in file XR20063.
a. For a series of samples, each with n 5 4, construct and interpret the }x and range charts for this process.
Does it appear to be in control?
b. Given your response to part (a), is it appropriate to compute the Process Capability Index (Cpk) for this process? If so, determine the overall mean and stan- dard deviation for all the units measured, then com- pute and interpret the Process Capability Index if the specification limits are USL 5 5.700 centimeters and LSL 5 5.100 centimeters.
20.64 A process has generated the output (in ounces) listed in file XR20064.
a. For a series of samples, each with n 5 5, construct and interpret the }x and range charts for this process.
Does it appear to be in control?
E X E R C I S E S
b. Given your response to part (a), is it appropriate to compute the Process Capability Index (Cpk) for this process? If so, determine the overall mean and standard deviation for all the units measured, then
compute and interpret the Process Capability Index if the specification limits are USL 5 45.400 ounces and LSL 5 43.800 ounces.
SUMMARY
• TQM and the process orientation
Organizations seeking to improve the quality of their goods and services are increasingly adopting the principles and practices of total quality management (TQM). Companies are finding a close parallel between the adage, “a stitch in time saves nine” and the TQM concept of dealing with defects through a pro- cess orientation that concentrates on preventing them from occurring in the first place.
• Process variation and defect prevention
Much of the emphasis in TQM is on understanding and reducing variation within the process through which the product is produced, and samples of output are used in reaching inferences about the stability of the process. The two sources of process variation are random (common-cause) variation and assignable (special- cause) variation.
The traditional philosophy of defect detection generally involved a corporate atmosphere where quality was a separate entity within the firm. The TQM phi- losophy of defect prevention has quality integrated throughout the organizational structure and central to corporate thinking in all matters. The emergence of TQM saw pioneers like W. Edwards Deming spreading the teachings of TQM and its process orientation.
• Kaizen and TQM tools
The practices of TQM are closely related to the Kaizen philosophy of continuing improvement and include guidelines such as Deming’s 14 points, quality audits, competitive benchmarking, just-in-time manufacturing, and worker empower- ment. The chapter discusses several of the statistical tools used in applying TQM, including the process flow chart, the cause-and-effect diagram, the Pareto dia- gram, the check sheet, and the Taguchi view of quality loss.
• Statistical process control charts
A popular and important tool for applying TQM is statistical process control, where sampled products are used in making inferences about the process from which they came. The control charts used in monitoring the process can provide a warning signal that the process may have become unstable due to the presence of assignable variation. In this event, the process should receive attention and corrective action. A process that is unstable will tend to generate products that are of unpredictable and/or unsatisfactory quality.
• Control charts for variables versus attributes
Control charts may be constructed for either variables (measurements) or at- tributes (counts). The mean chart and the range chart apply to variables and monitor central tendency and variability, respectively. Applicable to attributes, the p-chart shows the proportion of defects in successive samples, while the c- chart reflects a count of the number of defects found in units of the output. In each of these charts, the upper and lower control limits are the boundaries for the amount of variation that can be attributed to chance alone. These and other control charts can be generated with the use of a computer statistical package.
( 20.10 )
• Process Capability Index
The process capability index measures the capability of a process to generate output falling within specified upper and lower specification limits. It is an essential component of the Six-Sigma quality system.
Mean Chart When the Process Standard Deviation Is Known
• Centerline 5x , the mean of the sample means. }}
• Upper and lower control limits (UCL and LCL):
}}
x 6 3 ( ww ẽwn ) where 5 the process standard deviation n 5 the sample size
y ẽwn 5 the standard error of the sample means Mean Chart When the Process Standard Deviation Is Not Known
• Centerline 5}}x , the mean of the sample means.
• Upper and lower control limits (UCL and LCL):
}}x 6 A2}R where }R5 the average range for the samples
A25 value from the 3-sigma control chart factors table Range Chart
• Centerline 5 }R , the average of the sample ranges.
• Upper and lower control limits:
UCL 5 D4R where } R 5 the average range for the samples}
LCL 5 D3}R D3 and D45 values from 3-sigma control chart factors table p-Chart
• Centerline: }p , the average of the sample proportions, or
op
wwwwwwwww Number of samples
• Upper and lower control limits (UCL and LCL):
}p 6 zẽwwww }wwwwp(1n2}p) where }p 5 the average of the sample proportions n 5 the size of each sample
z 5 standard error units, normal distribution c-Chart
• Centerline: }c , the average number of defects per sample, or
oc
wwwwwwwww Number of samples
• Upper and lower control limits (UCL and LCL):
}c z ẽw}c where c }5 the average number of defects per sample z 5 standard error units, normal distribution
E Q U A T I O N S
Process Potential Index, Cp Cp5 USL 2LSL
wwwww6 where Cp5 Process Potential Index USL 5 upper specification limit
LSL 5 lower specification limit
5 process standard deviation (If is unknown, use s.)
Process Capability Index, Cpk
Cpk5 the lower of ( }}x wwww23sLSL ) or ( wwwwwUSL 3s 2}}x )
where Cpk5 Process Capability Index
x
5 overall mean for all units sampled s 5 standard deviation for all units sampled USL 5 upper specification limit
LSL 5 lower specification limit
20.65 For a person who eats hamburgers with ketchup, how might the traditional and the Taguchi approaches to quality loss apply with regard to the amount of ketchup that is put on the hamburger?
20.66 For each of the following, identify which kind of variation is at work—random or assignable—and explain your reasoning.
a. During the noon hour, there is an electrical power disruption and nuts machine-tightened during the five minutes that emergency power is on have 15%
less torque than normal.
b. Although they are all labeled “9–D,” the shoes on a retailer’s shelf are not exactly the same size. Any given shoe is just as likely to be slightly larger than 9–D as it is to be slightly smaller.
c. The number of cars leaving downtown Cleveland during a Friday rush hour is greater than during other days of the week.
d. In a production run of curtain rods, the lengths are normally distributed, with a mean of 24.135 inches and a standard deviation of 0.142 inches.
20.67 A university is starting a new college of engi- neering, has extensive funding, and wants to use com- petitive benchmarking. What U.S. universities might be useful as objects of the competitive benchmarking effort?
20.68 As a machine produces steel rods, accumulated deposits and mechanical wear cause the rod diameters to increase gradually over time. What type of control chart should be used in determining when the process has gone out of control? Why?
20.69 In producing bulk rolls of paper for sale to newspapers, a firm wishes to control the quality of the paper by keeping track of how many discolored spots or other flaws are present in continuous strips of a given length. What type of control chart should be used and why?
20.70 A process is in control and the results from a series of samples show the overall mean and standard deviation of all the units sampled to be }}
x 5 21.4500 inches and s 5 0.0969 inches. The upper and lower specification limits for the output are USL 5 21.8600 inches and LSL 5 21.0600 inches.
Based on this information, determine and interpret the value of the Process Capability Index for this process.
20.71 A process is in control and the results from a series of samples show the overall mean and standard deviation of all the units sampled to be }}x 5 17.35 kilo- grams and s 5 0.95 kilograms. The upper and lower specification limits for the output are USL 5 20.50 kilo grams and LSL 5 14.30 kilograms. Based on this information, determine and interpret the value of the Process Capability Index for this process.
( DATA SET ) Note: For Exercises 20.72–20.80, a com- puter and statistical software are recommended.
20.72 In a production process for canvas, a sample con- sisting of 25 square feet contains 4 discolored spots that must be touched up before shipping. Data for this and subsequent samples, each consisting of 25 square feet of
CHAPTE R EXERCISES
canvas, are shown here. Use an appropriate 3-sigma control chart in determining whether the process is in control.
Sample Number Sample Number Number of Spots Number of Spots
1 4 16 13
2 6 17 10
3 3 18 9
4 9 19 8
5 9 20 12
6 8 21 10
7 3 22 12
8 5 23 9
9 8 24 12
10 12 25 10
11 15 26 12
12 8 27 12
13 10 28 10
14 8 29 11
15 10 30 16 20.73 The number of defective products in each of 40 sam- ples taken from a production run are shown here. In each sample, 200 units were examined and classified as either good or defective, depending on whether dimensional toler- ances had been met. Use an appropriate 3-sigma control chart in commenting on the state of control of the process.
Sample Number of Sample Number of Number Defectives Number Defectives
1 12 21 10
2 9 22 10
3 15 23 7
4 11 24 8
5 15 25 12
6 8 26 9
7 14 27 12
8 13 28 16
9 10 29 12
10 12 30 11
11 7 31 11
12 12 32 9
13 12 33 7
14 14 34 12
15 8 35 8
16 9 36 8
17 7 37 12
18 8 38 9
19 10 39 6
20 12 40 11
20.74 Following their manufacture, plastic shims are sampled at periodic intervals and their thickness (in millimeters) measured. Each sample consists of n 5 3 shims, and the following data have been collected for the past 20 samples. Given these measurements, use one or more appropriate 3-sigma control charts in evaluating whether the process is in control.
Sample Measurement (mm) Number 1 2 3
1 47.7 51.8 42.7
2 55.4 47.9 53.9
3 51.6 47.9 47.6
4 45.3 45.5 57.4
5 48.9 49.4 47.0
6 46.5 46.3 47.5
7 55.0 44.3 51.1
8 49.5 50.3 43.9
9 53.7 47.6 49.2
10 48.4 49.2 41.8
11 46.7 51.1 43.9
12 48.8 49.8 48.7
13 48.1 48.8 50.9
14 55.6 52.5 42.8
15 54.3 54.6 44.4
16 48.4 48.3 47.6
17 46.2 46.3 50.3
18 52.4 51.8 49.1
19 48.1 51.7 53.5
20 52.7 46.5 49.2
20.75 An adhesives manufacturer periodically selects a sample of its product and glues 6 toothpicks together, in pairs. A measurement is then made of the number of pounds required to separate the toothpicks. The follow- ing data represent 15 samples, each consisting of 3 pairs of toothpicks, and the number of pounds required to separate each pair. Given these measurements, use one or more appropriate 3-sigma control charts in evaluating whether the process is in control.
Pounds to Separate Pair Sample
Number 1 2 3
1 28 24 27
2 29 27 27
3 27 26 28
4 28 31 30
5 31 25 25
6 22 27 29
7 30 27 26
8 29 25 28
9 28 29 34
10 32 28 32
11 26 25 28
12 22 27 29
13 26 28 34
14 27 30 24
15 26 24 28
20.76 During the production of 15-ampere circuit break- ers, samples are taken at periodic intervals and tested for the exact amperage at which they break a circuit.
Each sample consists of n 5 4 circuit breakers. For the following data, representing the results from the past
20 samples, construct one or more appropriate control charts and comment on the state of control of the process.
Measurement Number Sample
Number 1 2 3 4
1 14.8 15.1 15.2 14.7
2 15.1 14.8 15.1 15.2
3 15.0 15.0 15.0 15.2
4 15.1 15.2 14.7 15.1
5 15.2 15.2 15.2 14.7
6 15.1 15.0 15.2 15.2
7 15.6 14.7 14.8 15.2
8 15.2 14.9 14.7 15.0
9 14.9 14.9 15.4 14.9
10 15.1 15.0 14.6 15.2 1 1 15.1 14.7 14.7 14.8 12 15.3 15.1 15.0 14.8 13 15.3 14.6 14.5 14.9 14 15.2 15.1 15.1 15.0 15 15.2 14.1 14.9 15.4 16 15.3 15.9 15.4 15.6 17 16.2 15.9 15.5 15.1 18 15.9 15.0 14.9 14.9 19 14.7 16.6 15.8 14.7 20 15.1 15.2 16.9 15.6 20.77 Telemarketers for a mail-order firm are assigned to make a total of 200 calls each day, then record the number of potential customers who are interested in receiving a catalog and first-order discount coupon in the mail. For each of the past 50 days, the number of catalog/coupon packages sent is as shown here.
Construct one or more appropriate control charts and comment on the state of control of the process.
Days Successes
1–15 21 16 23 16 22 19 23 25 15 22 17 21 21 23 15 16–30 18 20 14 24 18 18 22 27
15 13 29 18 21 21 25 31–45 20 24 15 20 21 11 19 17
22 16 17 28 23 22 16 46–50 22 18 19 15 21
20.78 A cabinet manufacturer randomly selects cabinets from the production line and examines them for defects.
For each of the 30 cabinets sampled during today’s pro- duction, the number of defects is shown below. Given these data, construct one or more appropriate control charts and comment on the state of control of the process.
Cabinet Number Cabinet Number Number of Defects Number of Defects
1 4 16 3 2 3 17 5 3 2 18 4 4 3 19 3 5 3 20 3 6 2 21 1 7 2 22 3 8 3 23 9 9 3 24 2 10 4 25 2 11 6 26 5 12 1 27 4 13 4 28 4 14 2 29 4 15 1 30 3
20.79 Given the conclusion reached in the solution for Exercise 20.75, with the data in file XR20075, is it appropriate to compute the Process Capability Index (Cpk) for this process? If so, determine the overall mean and standard deviation for all the units measured, then compute and interpret the Process Capability Index if the specification limits are USL 5 33.700 pounds and LSL 5 21.700 pounds.
20.80 Given the conclusion reached in the solution for Exercise 20.74, with the data in file XR20074, is it appro- priate to compute the Process Capability Index (Cpk) for this process? If so, determine the overall mean and stan- dard deviation for all the units measured, then compute and interpret the Process Capability Index if the specifi- cation limits are USL 5 61.000 millimeters and LSL 5 37.000 millimeters.
Like any other large firm, Thorndike Sports Equipment receives daily telephone calls in which customers either praise or complain about a Thorndike product. Luke
Thorndike is especially sensitive about the number of complaint calls received daily. Twelve weeks ago, he asked the customer service department to keep a daily log of the
Thorndike Sports Equipment