Title : STATISTICAL PROCESS CONTROL (SPC) Period : 4 hours
I. Objectives
At the end of the lesson, students are expected to: 1. define statistical process control
2. describe the fundamentals of SPC and understand its statistical concepts 3. identify and explain the different process charts for monitoring quality services 4. describe the quality characteristics for the different services that can be
measured and monitored using control charts
5. illustrate how to develop/construct and interpret control charts 6. explain SPC as a tool for continuous process improvement
II. Subject Matter
1. Topics
a. Statistical Process Control b. SPC in TQM
c. Control Charts
d. Control Charts for Attributes e. Control Charts for Variables 2. Educational Resource(s)
a. Anderson, D. R. (2004). An Introduction to Management Science:
Quantitative Approaches to Decision Making. Amazon.com: New Jersey
b. Render, B. & Heizer, J. (2006). Principles of Operations Management, 7th Edition. Prentice hall: USA
c. Russell, R.S., & Taylor, W. III B. (2006). Operations Management, 5th Edition.
John Wiley & Sons: New Jersey d. http://www.sandia.gov. e. http://www.margaret.net/spc f. http://www.isixsigma.com 3. Materials a. Lecture notes b. PowerPoint presentation c. PCs/Calculators
III. Learning Procedures and Strategies
1. Preparatory Activity
a. Introduction:
After World War II, W. E. Deming, the quality expert and consultant, was invited to Japan by the Japanese government to give a series of lectures on improving product reliability. This was probably the single most important event that started the Japanese toward a global quality revolution. These lectures were based on statistical quality control, and they became a cornerstone of the Japanese commitment to quality management.
A major topic in statistical quality control is statistical process control (SPC).
Statistical process control (SPC) involves monitoring the production
process to prevent poor quality. Employee training in SPC is a fundamental principle of TQM.
In TQM, operators use SPC to inspect and measure the production process to see if it is varying from what it is supposed to be doing. If there is unusual or undesirable variability, the process is corrected so that defects will not occur. In this way statistical process control is used to prevent poor quality before it occurs. It is such an important part of quality management in Japan that nearly all workers at all levels in Japanese companies are given extensive and continual training in SPC. Conversely, in the United States one reason cited for past failure to achieve high quality is the lack of comprehensive training for employees in SPC methods. U.S. companies that have been successful in adopting TQM train all their employees in SPC methods and make extensive use of SPC for continuous process improvement.
SPC Applied to Services
Control charts have historically been used to monitor the quality of manufacturing processes. However, SPC is just as useful for monitoring quality in service. The difference is the nature of the “defect” being measured and monitored. Following is a list of several services and the quality characteristic for each that can be measured and monitored with control charts.
Hospital: Timelines and quickness of care, staff responses to request,
accuracy of lab test, cleanliness, courtesy, accuracy of paperwork, speed of admittance and check outs.
Grocery store: Waiting time to check out, frequency of out-of-stock items,
quality of food items, cleanliness, costumer complaints, check-out register errors.
Airlines: Flight delays, lost luggage and luggage handling, waiting time at
ticket counters and check-in, agent and flight attendant courtesy, passenger cabin cleanliness and maintenance.
Fast-food restaurants: Waiting time for service, customer complaint,
cleanliness, food quality, order accuracy, employee courtesy.
Catalog-order companies: Order accuracy, operator knowledge and
courtesy, packaging, delivery time, phone order waiting time.
Insurance companies: Billing accuracy, timeless of claims processing,
Another topic in statistical quality control, acceptance sampling, involves inspecting a sample of product input (i.e., raw material or parts) and/or the final product and comparing it to a quality standard. If the sample fails the comparison, or test, then that implies poor quality, and the entire group of items from which the sample was taken is rejected. This additional approach to quality control conflicts with the philosophy of TQM because it assumes that some degree of poor quality will occur and that it is acceptable. TQM
seeks ultimately to have zero defects and 100 percent quality. Acceptance
sampling identifies defective items after the product is already finished, whereas TQM preaches the prevention of defects altogether. To disciples of TQM, acceptance sampling is simply a means of identifying products to throw away or rework. It does nothing to prevent poor quality and to ensure good quality in the future. Nevertheless, acceptance sampling is still used by firms that have not adopted TQM. There may come a time when TQM becomes so pervasive that acceptance sampling will be used not at all or to a very limited extent. However, that time has not yet arrived, and until it does acceptance sampling still is a topic that requires some attention.
Acceptance sampling involves taking a random sample to determine if a
lot is acceptable. 2. Lesson Proper
a. Statistical Process Control
Process control is achieved by taking periodic sample from the process and plotting these sample points on the chart, to see if the process is within statistical control limits. If a sample point is outside the limits, the process may be out of control, and the cause is sought so the problem can be corrected. If the sample is within the control limits, the process continuous without interference but with continued monitoring.
Statistical process control (SPC) is a statistical procedure using control
charts to see if any part of a production process is not functioning properly and could cause poor quality.
SPC prevents quality problems by correcting the process before it starts producing defects. No production process produces exactly identical items, one after the other. All processes contain a certain amount of variability that makes some variation between units inevitable. There are two reasons why a production process might vary. The first is the inherent random variability of the process, which depends on the equipment and machinery, engineering, the operator, and the system used for measurement. This kind of variability is a result of natural occurrences. The other reason for variability is unique or special causes that are identifiable and can be corrected. This cause tends to be nonrandom and, if left unattended, will cause poor quality. These might include equipment that is out of adjustment, defective materials, changes in
parts of materials, broken machinery or equipment, operator fatigue or poor work methods, or errors due to lack of training.
SPC is a tool for identifying problems in order to make improvements.
b. Quality Measures: Attributes and Variables
The quality of a product can be evaluated using either an attribute of the product or a variable measure. An attribute is a product characteristic such as a color, surface texture, or perhaps smell or taste. Attribute can be evaluated quickly with a discrete response such as good or bad, acceptable or not, or yes or no. Even if quality specifications are complex and extensive, a simple attribute test might be used to determine if a product is or is not defective. For example, an operator might test a light bulb by simply turning it on and seeing if it lights. If it does not, it can be examined to find out the exact technical cause for failure, but for SPC purposes, the fact that it is defective has been determined.
A variable measure is a product characteristic that is measured on a continuous scale such as length, weight, temperature, or time. For example, the amount of liquid detergent in a plastic container can be measured to see if it conforms to the company’s product specification. Or the time it takes to serve a costumer at McDonald’s can be measured to see if it is quick enough. Since a variable evaluation is the result of some form of measurement, it is sometimes referred to as a quantitative classification method.
An attribute evaluation is sometimes referred to as qualitative classification, since the response is not measured. Because it is a measurement, a variable classification typically provides more information about the product –the weight of the product is more informative than simply saying the product is good or bad.
c. Control charts
Control charts are graphs that visually show if a sample is within statistical control limits. They have two basics purposes, to establish the control limits for
a process and then to monitor the process to indicate when it is out of control. Control charts exist for attributes and variables; within each category there are several different types of control charts. We will present four commonly used control charts, two in each category: p-charts and c-charts for attributes and mean ( ) and range (R) control charts for variables. Even though these control charts differ in how they measure process control, they all have certain similar characteristics. They all look alike, with a line through the center of a graph that indicates the process average and lines above and below the center line that represent the upper and lower limit of the process, as shown in the figure 10.1.
Figure 10.1 Process Control Chart
Each time a sample is taken, the mathematical average of the sample is plotted as a point on the control chart as shown in figure 10.1. A process is generally considered to be in control if, for example,
1. There are no sample points outside the control limits.
2. Most points are near the process average (i.e., the center line), without too many close to the control limits.
3. Approximately equal numbers of sample points occur above and below the center line.
4. The points appear to be randomly distributed around the center line (i.e., no discernible pattern).
If any of these conditions are violated, the process may be out of control. The reason must be determined, and if the cause is not random, the problem must be corrected.
d. Control charts for attributes
The quality measures used in attribute control charts are discrete values reflecting a simple decision criterion such as good or bad. A þ-chart uses the proportion defective items in a sample as the sample statistic; a c-chart uses the actual number of defective items in a sample. A p-chart can be used when it is possible to distinguish between defective and non-defective items and to state the number of defectives as a percentage of the whole. In some process, the proportion defective cannot be determined. For example, when counting the number of blemishes on a roll of upholstery material (at periodic intervals), it is not possible to compute a proportion. In this case a c-chart is required.
þ-Chart
With a p-chart a sample is taken periodically from the production process, and the proportion of defective items in the sample is determined to see if the proportion falls within the control limits on the chart. Although a p-chart employs a discrete attribute measure (i.e., number of detective items) and thus is not continuous, it is assumed that as the sample size gets larger, the normal distribution can be used to approximate the distribution of the proportion defective. This enables us to use the following formulas based on the normal distribution to compute the upper control limit (UCL) and lower control limit (LCL) of a p-chart.
where
z = the number of standard deviation from the process average
þ = the sample proportion defective; an estimate the process average
σþ = the standard deviation of the sample proportion
The sample standard deviation is computed as
where
is the sample size.
In the control limit formulas for -chart (and other control charts), z is occasionally equal to 2.00 but most frequently 3.00. A z value of 2.00 corresponds to an overall normal probability of 95 percent and z = 3.00 correspond to a normal probability of 99.74 percent.
The normal distribution in the figure 10.2 shows the probabilities correspond to z values 2.00 and 3.00 standard deviation (σ).
The smaller the value of z, the more narrow the control limits are and the more sensitive the chart is to changes in the production process. Control charts using z = 2.00 are often referred to as having “2-sigma” (2σ) limits (referring to two standard deviation), whereas z = 3.00 means “3-sigma” (3σ) limits.
Management usually select z = 3.00 because if the process is in control it wants a high probability that the sample values will fall within the control limits. in other words, with wider limits management is less likely to (erroneously)conclude that the process is out of control when points outside the control limits are due to the normal, random variations. Alternatively, wider limits make it harder to detect the wider changed in the process that are not random and have an assignable cause. A process might change because a nonrandom, assignable cause and be detectable with the narrower limits but not with the wider limits. However, companies traditionally use the wider control limits.
The following example demonstrates how þ-chart is constructed.
Figure 10.2 The Normal Distribution
Construction of p-chart
Example 10.1: The Western Jeans Company produces denim jeans. The
company wants to establish a þ-chart to monitor the production process and maintain the high quality. Western believes that approximately 99.74 percent of the variability in the production process (corresponding to a 3-sigma limits, or z = 3.00 ) is random and thus should be within control limits, whereas .26 percent of the process variability is not random and suggests that the process is out of control. The company has taken 20 samples (one per day for 20 days), each containing 100 pairs of jeans ( = 100), and inspected them for defects, the results of which are as follows.
The proportion defective for the population is not known. The company wants to construct a þ-chart to determine when the production process might be out of control.
Sample Number of defectives Proportion 1 6 0.06 2 0 0.00 3 4 0.04 4 10 0.10 5 6 0.06 6 4 0.04 7 12 0.12 8 10 0.10 9 8 0.08 10 10 0.10 11 12 0.12 12 10 0.10 13 14 0.14 14 8 0.08 15 6 0.06 16 16 0.16 17 12 0.12 18 14 0.14 19 20 0.20 20 18 0.18 200 Solution:
Since þ is not known, it can be estimated from the total sample:
The control limits are computed as follows.
The -chart, including sample points, is shown in the following figure.
The process was below the lower control limits for sample 2 (i.e., during day 2). Although this could be perceived as a ”good” result since it means there were very few defects, it might also suggest that something was wrong with the inspection process during that week that should be checked out. If there is no problems with the inspection process, then would want to know what caused the quality of the process to improve. Perhaps “better” denim materials form a new supplier that week or a different operator was working.
The process was above the upper limit during day 19. This suggested that the process may not be control and the cause should be investigated. The cause could be defective or maladjusted machinery or a problem with an operator, defective materials (i.e., denim cloth), or a number of other correctable problems. In fact, there is an upward trend in the number of defectives throughout the 20-day test period. The process was consistently moving toward an out-of-control situation. This trend represents a pattern in the observations, which suggest a nonrandom cause. If this was the actual control chart used to monitor the process (and not the initial chart), it is likely this pattern would have indicate an out-of-control situation before day 19, which would have alerted the operator to make correction.
This initial control chart shows two out-of-control observation and a distinct pattern of increasing defects. Management would probably want to discard this set of samples and develop a new center line and control limits from a different set of sample values after the process has been corrected. If the pattern had not existed and only the two out-of-control observations were present, these two observations could be discarded, and a control chart could be developed from remaining sample values, if the problem is corrected.
Once a control chart is established based solely on natural, random variation in the process, it is used to monitor the process. Samples are taken periodically, and then observations are checked on the control chart to see if the process is in control.
c-chart
A c-chart is used when it is not possible to compute a proportion defective and the actual number of defects must be used. For example, when automobiles are inspected, the number of blemishes (i.e., defects) in the paint job can be counted for each car, but a proportional cannot be computed, since the total number of possible blemishes is not known.
Since the number of defects per sample is assumed to arrive from some extremely large population, the probability of the single defects is very small. As with the -chart, the normal distribution can be used to approximate the distribution of defects. The process average for the -chart is the sample mean number of defects, , computed by dividing the total number of defects by the number of sample. The sample standard deviation, σc is .
The following formulas for the control limits are used:
Construction of a c-chart
Example 10.2 : Barrett Mills in North Carolina produces denim cloth used by
manufactures (such as the Western jeans company) to make jeans. The denim cloth is woven from thread on the weaving machine, and the thread occasionally breaks and is repaired by the operator, sometimes causing blemishes (called “picks”). The operator inspects rolls of denim on a daily basis as they come off the loom and counts the number of blemishes. Following are the results from inspecting fifteen rolls of denim during a three-week period.
The company believes that approximately 99 percent of the defects (corresponding to 3-sigma limits) are caused by natural, random variations in the weaving process, with 1 percent caused by nonrandom variability. They want to construct a -chart to monitor the weaving process.
Sample Number of defects
1 12 2 8 3 16 4 14 5 10 6 11 7 9 8 14 9 13 10 15 11 12 12 10 13 14 14 17 15 15 190 Solution:
Since , the population process average, is not known, the sample estimate, , can be used instead:
The control limits are computed using , as follows.
The resulting -chart, with the sample points, is shown in the following figure.
All the sample observations are within the control limits, suggesting that the weaving process is in control. This chart would be considered reliable to monitor the weaving process in the future.
e. Control charts for variables
Variable control charts are for continuous variables that can be measured, such as weight or volume. Two commonly used variable control charts are the rage chart, or R-chart, and the mean chart, or -chart. A range
(R-) chart reflects the amount of dispersion present in each example; a mean ( -) chart indicate how sample results relate to the process average or mean.
These chart are normally used together to determine if a process is in the control.
Range (R) chart
In an R-chart, the range is the difference between the smallest and largest values in a sample. This range reflects the process variability instead of the tendency toward a mean value. The formulas for determining control limits are,
the average (and center line) for the samples,
where
= range of each sample = number of samples
D3 and D4 are tables values for determining control limits that have been
developed based on range values rather than standard deviation. Table 10.1 includes values for D3
Table 10.1
Factors for determining control limits for x- and R-charts
Sample Factor of x-chart Factor for R-chart
N 2 1.88 0 3.27 3 1.02 0 2.57 4 0.73 0 2.28 5 0.58 0 2.11 6 0.48 0 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74 12 0.27 0.28 1.72 13 0.25 0.31 1.69 14 0.24 0.33 1.67 15 0.22 0.35 1.65 16 0.21 0.36 1.64 17 0.20 0.38 1.62 18 0.19 0.39 1.61 19 0.19 0.40 1.60 20 0.18 0.41 1.59 21 0.17 0.43 1.58 22 0.17 0.43 1.57 23 0.16 0.44 1.56 24 0.16 0.45 1.55 25 0.15 0.46 1.54
And D4 for sample sizes up to 25. These tables are available in many texts
on operation management and quality control. They provide control limits comparable to three standard deviations for different sample sizes.
Constructing an R-chart
Example 10.3: The Goliath Tool Company produces slip-ring bearing which
look like flats doughnuts or washers. They fit around shafts or round, such as drive shafts in machinery or motors. In the production process for a particular slip-ring bearing the employees have taken 10 samples (during a 10-day
period) of 5 slip-ring bearing (i.e., n = 5). The individual observations from each sample are shown as follows:
Observations (slip-ring diameter, cm)
Sample 1 2 3 4 5 R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15
The company wants to develop an R-chart to monitor the process variability.
Solution:
is computed by first determining the rage for each sample by computing the difference between the highest and lowest values shown in the last column in our table of the sample observation.
These ranges are assumed and then divided by the number of samples, , as follows:
D3 = 0 and D4 = 2.11 from table 4.1 for . Thus, thus the control limits
Mean ( ) chart
For an -chart, the mean of each sample is computed and plotted on the chart; the points are the sample means. Each sample mean is a value of . The samples tend to be small, usually around 4 or 5. The center line of the chart is the overall process average, the sum of the average of samples,
When an -chart is used in conjunction with an R-chart, the following formulas for control limits are used:
Where is the average of the sample means and is the average range value. A2 is a tabular value like D3 and D4 that is used to establish the control
limits. Values of A2 are included in Table 4.1. They were developed specifically
for determining the control limits for -chart and are comparable to
3-standard deviation (3σ) limits.
Since the formulas for the control limits of the -chart use the average range values, , the R-chart must be constructed before the -chart.
Example 10.4: The Goliath Tool Company desires to develop an - chart to be
used in conjunction with the R-chart developed in Case Problem 10.3.
Solution:
The data provide in Example 10.3 for the samples allows to compute as follows:
Using the value of A2 = 0.58 for from Table 4.1 and = 0.115 from
Example 10.3, the control limits are computed as
The -chart defined by these control limits is shown in the following figure. Notice that the process is out of control for sample 9; in fact, samples 4 to 9 show an upward trend. This would suggest the process variability is subjected to nonrandom causes and should be investigated.
This example illustrates the need to employ the R-chart and the -chart together. The R-chart in Example 10.3 suggested that the process was in control, and none of the ranges for the samples were close to the control limits. However, the -chart suggests that the process is not in control. In fact, the ranges for sample 8 and 10 were relatively narrow, whereas the means of both samples were relatively high. The use of both charts together provided a more complete picture of the overall process variability.
The -chart is used with the R-chart under the premise that both the process average and variability must be in control for the process to be in control. This is logical. The two charts measure the process differently. It is possible for samples to have very narrow ranges, suggesting little process variability, but the sample averages might be beyond the control limits.
For example, consider two samples with the first having low and high values of 4.95 and 5.50 centimeters, and the second having low and high values of 5.10 and 5.20 centimeters. The range of both is 0.10cm but for the first is 5.00 centimeters and for the second is 5.15 centimeters. The two sample ranges might indicate the process is in control and = 5.00 might be okay, but = 5.15 could be outside the control limit.
Conversely, it is possible for the sample averages to be in control, but the ranges might be very large. For example, two samples could both have = 5.00 centimeters, but sample 1 could have a range between 4.95 and 5.05 (R = 0.10 centimeters) and Sample 2 could have range between 4.80 and 5.20 (R = 0.40 centimeters). Sample 2 suggests the process is out of control.
It is also possible for an R-chart to exhibit a distinct downward trend in the range values, indicating that the ranges are getting narrower and there is less variation. This would be reflected on the -chart by mean values closer to the center line. Although this occurrence does not indicate that the process is out of control, it does suggest that some nonrandom cause is reducing process variation. This cause needs to be investigated to see if it is sustainable. If so, new control limits would need to be developed.
Sometimes an -chart is used alone to see if a process is improving. For example, in the “competitive edge” box for Kentucky Fried Chicken, an -chart is used to see if average service time at a drive-through window are continuing to decline over time toward a special goal.
In this situation a company may have studied and collected data for process for a long period of time and already know what the mean and standard deviation of the process is and all they want to do is monitor the
process average by taking the periodic samples. In this case it would be appropriate to use just the mean chart and the following formulas for
establish control limits,
The sample standard deviation, is computed as
Constructing a Mean Chart
Example 10.5: The Goliath Tool Company has studied its process for
manufacturing slip-ring bearings and determined that the bearings have a mean diameter of 5 centimeters and a standard deviation of 0.40 centimeters. The company wants to develop a mean chart for this production process that will include 99.74 percent (i.e., three standard deviations) of the process variability using samples of size 20.
Solution:
The control limits are computed as follows:
If the sample is taken and the sample mean falls outside of these control limits, it suggests that the process is out of control, so the cause is probably nonrandom and should be investigated.
3. Ending Activity
a. Summary
This module provided a general overview of Statistical Process Control. Statistical Process Control and acceptance sampling are the main quantitative tools of quality control.
Control charts for SPC help the operations manager distinguish between natural and assignable variations. The -chart and the R-chart are used for variable sampling and the p-chart and the c-chart for attribute sampling. Operating characteristic curves facilitate acceptance sampling and provide the manager with tools to evaluate the quality of a production run or shipment.
IV. Evaluation/Assessment
a. Self-test 10: Test your understanding of this lesson.
1. A c-chart can be used when it is possible to distinguish between defective
and non-defective items and to state the number of defectives as a percentage of the whole.
a. True b. False
2. Control charts for attributes are a. p-chart
b. c-chart c. R-chart d. -chart
3. If a sample parts are measured and the mean of the sample measurement is outside the tolerance limits,
a. The process is out of control and the cause can be established b. The process is in control, but not capable of producing within the
established control limits
c. The process is within the established control limits with only natural causes of variation
d. All of the above are true
4. The type of chart used to control the number of defects per unit of output is
a. - bar chart b. R-chart c. p-chart
d. all of the above e. none of the above
5. If a sample of parts are measured and the mean of the sample measurement is in the middle of the tolerance limits but some parts measure too low and other parts measure too high
a. The process is out of control and the cause can be established b. The process is in control, but not capable of producing within the
established control limits
c. The process is within the established control limits with only natural causes of variation
d. All of the above are true
a. Problem Set 10
1. The Commonwealth Banking Corporation issues a national credit card through its various bank branches in five southeastern states. The bank credit
card business is highly competitive and interest rates do not vary substantially, so the company decided to attempt to retain its customers by improving customer service through a reduction in billing errors. The credit card division monitored its billing department process by taking daily samples of 200 customer bills for 30days and checking their accuracy. The sample results are as follows:
Sample Number of
Defectives Sample Number of Defectives Sample Number of Defectives
1 7 11 9 21 13 2 12 12 6 22 9 3 9 13 3 23 10 4 6 14 2 24 12 5 5 15 8 25 15 6 8 16 10 26 14 7 10 17 12 27 16 8 11 18 14 28 12 9 14 19 16 29 15 10 10 20 15 30 14
Develop a p-chart for the billing process using 3σ control limits and indicate if process is out of control.
2. A machine at the Pacific Fruit Company fills boxes with raisins. The labeled weight of the boxes is 10 ounces. The company wants to construct an R-chart and a mean chart to monitor the filling process and make sure the box weights are in control. The quality control department for the company sampled five boxes every two hours for three consecutive working days. The sample observations are as follows.
Sample Box Weights (oz) Sample Box Weights (oz)
1 9.06 9.13 8.97 8.85 8.46 7 9.00 9.21 9.05 9.23 8.78 2 8.52 8.61 9.09 9.21 8.95 8 9.15 9.20 9.23 9.15 9.06 3 9.35 8.95 9.20 9.03 8.42 9 8.98 8.90 8.81 9.05 9.13 4 9.17 9.21 9.05 9.01 9.53 10 9.03 9.10 9.26 9.46 8.47 5 9.21 8.87 8.71 9.05 9.35 11 9.53 9.02 9.11 8.88 8.92 6 8.74 8.35 8.50 9.06 8.89 12 8.95 9.10 9.00 9.06 8.95
a. Construct a R-chart and a mean chart from these data with 3σ control limits, and plot the sample range and mean values.
b. What does the R-chart suggest about the process variability?
3. The great Northwoods Clothing Company is a mail-order company that processes thousands of mail and telephone orders each week. They have a customer service number to handle customer order problems, inquiries, and complaints. The company wants to monitor the number of customer calls that can be classified as complaints. The total number of complaint calls the
customer service department has received for each of the last 30 weekdays are shown as follows.
Day Complaint Calls Day Complaint Calls Day Complaint Calls
1 27 11 26 21 31 2 15 12 42 22 14 3 38 13 40 23 18 4 41 14 35 24 26 5 19 15 25 25 27 6 23 16 19 26 35 7 21 17 12 27 20 8 16 18 17 28 12 9 33 19 18 29 16 10 35 20 26 30 15
a. Construct a c-chart for this process with 3σ control limits, and indicate if the process was out of control.
b. What nonrandom (i.e. assignable) causes might result in the process being out of control?
