Process control is achieved by taking periodic samples from the process and plotting these sample points on a chart, to see if the process is within statistical control limits. A sample can be a single item or a group of items. If a sample point is outside the limits, the process may be out of control, and the cause is sought so that the problem can be corrected. If the sample is within the control limits, the process continues without interference but with continued monitoring. In this way, 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 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. These causes tend 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 or materials, broken machinery or equipment, operator fatigue or poor work methods, or errors due to lack of training.

•Sample: a subset of the items produced to use for inspection.

All processes have variability—random and nonrandom (identifiable, correctable).

A statistical control chart like this one is a graph to monitor a production process. Samples are taken from the process periodically, and the observations are plotted on the graph. If an observation is outside the upper or lower limits on the graph, it may indicate that something is wrong in the process; that is, it is not in control, which may cause defective or poor-quality items. By monitoring a process with a control chart, the employee and management can detect problems quickly and prevent poor-quality items from passing on through the remainder of the process and ending up as defective products that must be thrown away or reworked, thus wasting time and resources.

QUALITY MEASURES: ATTRIBUTES AND VARIABLES

The quality of a product or service can be evaluated using either An attribute of the product or service or a variable measure. An attribute is a product characteristic such as color, surface texture, cleanliness, or perhaps smell or taste. Attributes 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 whether or not a product or service is 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.

•Attribute: a product characteristic that can be evaluated with a discrete response (good/bad, yes/no).

Housekeepers at luxury hotels like the Ritz-Carlton strive to achieve the hotel's goal of a totally defect-free guest experience.

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 specifications. Or the time it takes to serve a customer 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 a 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 a product is more informative than simply saying the product is good or bad.

•Variable measure: a product characteristic that is continuous and can be measured (weight, length).

Control charts are graphs that visually show if a sample is within statistical control limits. They have two basic 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 (

•Control chart: a graph that establishes the control limits of a process.

•Control limits: the upper and lower bands of a control chart.

The formulas for conducting upper and lower limits in control charts are based on a number of standard deviations, "z," from the process average (e.g., center line) according to a normal distribution. Occasionally, z is equal to 2.00 but most frequently is 3.00. A z value of 2.00 corresponds to an overall normal probability of 95%, and z = 3.00 corresponds to a normal probability of 99.73%.

The normal distribution in Figure 3.2 on page 114 shows the probabilities corresponding to z values equal to 2.00 and 3.00 standard deviations (σ).

Types of charts: attributes, p, and c; variables,

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 deviations), whereas z = 3.00 means 3-sigma (3σ) limits.

Sigma limits are the number of standard deviations.

Figure 3.1. Process Control Chart

Figure 3.2. The Normal Distribution

Management usually selects 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 normal, random variations. Alternatively, wider limits make it harder to detect changes in the process that are not random and have an assignable cause. A process might change because of 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.

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 3.1. A process is generally considered to be in control if, for example,

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.

Sample 9 in Figure 3.1 is above the upper control limit, suggesting the process is out of control (i.e., something unusual has happened). The cause is not likely to be random since the sample points have been moving toward the upper limit, so management should attempt to find out what is wrong with the process and bring it back in control. Perhaps the employee was simply interrupted. Although the other samples display some degree of variation from the process average, they are usually considered to be caused by normal, random variability in the process and are thus in control. However, it is possible for sample observations to be within the control limits and the process to be out of control anyway, if the observations display a discernible, abnormal pattern of movement. We discuss such patterns in a later section.

A sample point can be within the control limits and the process still be out of control.

After a control chart is established, it is used to determine when a process goes out of control and corrections need to be made. As such, a process control chart should be based only on sample observations from when the process is in control so that the control chart reflects a true benchmark for an in-control process. However, it is not known whether or not the process is in control until the control chart is first constructed. Therefore, when a control chart is first developed if the process is found to be out of control, the process should be examined and corrections made. A new center line and control limits should then be determined from a new set of sample observations. This "corrected" control chart is then used to monitor the process. It may not be possible to discover the cause(s) for the out-of-control sample observations. In this case a new set of samples is taken, and a completely new control chart constructed. Or it may be decided to simply use the initial control chart, assuming that it accurately reflects the process variation.

The development of a control chart.

The quality measures used in attribute control charts are discrete values reflecting a simple decision criterion such as good or bad. A p-chart uses the proportion of defective items in a sample as the sample statistic; a c-chart uses the actual number of defects per item in a sample. A p-chart can be used when it is possible to distinguish between defective and nondefective items and to state the number of defectives as a percentage of the whole. In some processes, 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.

•p-chart: uses the proportion defective in a sample.

•c-chart: uses the number of defective items in a sample.

p-CHART

With a p-chart a sample of n items is taken periodically from the production or service 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 defective items) and thus is not continuous, it is assumed that as the sample size (n) 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 deviations from the process average

σ_p = the standard deviation of the sample proportion

The sample standard deviation is computed as

where n is the sample size.

Example 3.1 demonstrates how a p-chart is constructed.

Construction of a p-Chart

The Western Jeans Company produces denim jeans. The company wants to establish a p-chart to monitor the production process and maintain high quality. Western believes that approximately 99.74% of the variability in the production process (corresponding to 3-sigma limits, or z = 3.00) is random and thus should be within control limits, whereas 0.26% 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 (n = 100), and inspected them for defects, the results of which are as follows.

Sample	Number of Defectives	Proportion Defectives
1	6	.06
2	0	.00
3	4	.04
4	10	.10
5	6	.06
6	4	.04
7	12	.12
8	10	.10
9	8	.08
10	10	.10
11	12	.12
12	10	.10
13	14	.14
14	8	.08
15	6	.06
16	16	.16
17	12	.12
18	14	.14
19	20	.20
20	18	.18
	200

The proportion defective for the population is not known. The company wants to construct a p-chart to determine when the production process might be out of control.

Solution

Since p is not known, it can be estimated from the total sample:

The control limits are computed as follows:

The p-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 problem with the inspection process, then management would want to know what caused the quality of the process to improve. Perhaps "better" denim material from a new supplier that week or a different operator was working.

The process was above the upper limit during day 19. This suggests that the process may not be in control and the cause should be investigated. The cause could be defective or maladjusted machinery, 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 suggests 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 indicated an out-of-control situation before day 19, which would have alerted the operator to make corrections. Patterns are discussed in a separate section later in this chapter (page 124).

This initial control chart shows two out-of-control observations 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 the remaining sample values.

ALONG THE SUPPLY CHAIN

Using Control Charts for Improving Health-Care Quality

The National Health Service (NHS) in the United Kingdom makes extensive use of SPC charts to improve service delivery to patients. SPC charts are frequently developed using Excel spreadsheets. Employees throughout NHS attend a two-day training course where they learn about X- and R-charts and their application. They also learn about Deming's 14 points and principles of quality management and how to improve processes in their own operation.

The number of possible applications of SPC charts for a health-care facility is very high, with many applications relating to waiting times (as is the case with many service organizations); for example, the time to see a doctor, nurse, or other medical staff member, or to get medical results. One specific example in the NHS is "door to needle" times, which is the time it takes after a heart attack patient is registered into the hospital to receive an appropriate injection. The system-wide goverment-mandated target value for this process is that 75% of heart patients receive treatment within 20 minutes of being received into the hospital. Control charts are also used to monitor administrative errors.

Bellin Health System is as an integrated health-care organization serving Green Bay, Wisconsin, and the surrounding region. It specializes in cardiac care and its 167-bed hospital is known as the regon's heart center. In addition it operates a psychiatric center, 20 regional clinics, and a college of nursing. Bellin's quality management program is based on improving processes by identifying measures of success, setting goals for improvement that can be measured, applying Deming's PDCA cycle (see Chapter 2) for improving processes, and using statistical process control charts to monitor processes for stability and to see if improvement efforts are successful. Bellin has over 1,250 quality indicators reported monthly, quarterly, or annually, and over 90% are monitored with SPC charts. For example, one such quality indicator is for compliance with the Centers for Disease Control guidelines on health-care hand hygiene, which Bellin measures across its entire system. The lack of hand hygiene is a leading cause of fatal hospital-borne infections, and thus a very important quality indicator. The quality resource department (QRD) at Bellin uses a p-chart to monitor hand hygiene for system nursing care, where the control measure is the portion of hand opportunities that meet the criteria divided by the total number of opportunities available to meet the criteria. Using SPC charts Bellin was able to improve the hand hygiene process such that a 90% target level was acheived. Throghout the Bellin Healthcare System control charts are employed to identify special cause variations outside the norm in processes that require improvement. SPC has become an essential part of the continuous quality improvement program at Bellin used by all members of the organization to improve service and health care, and reduce costs.

Identify and discuss other applications of SPC charts in a healthcare facility that you think might improve processes.

Sources: M. Owen, "From Sickness to Health," Qualityworld 29 (8: August 2003), pp. 18–22; and C. O'Brien and S. Jennings, Quality Progress 41 (3; March 2000), pp. 36–43.

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 the observations are checked on the control chart to see if the process is in control.

Checking cracker texture at a Nabisco plant as part of the quality-control process.

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 proportion cannot be computed, since the total number of possible blemishes is not known. In this case a single car is the sample. Since the number of defects per sample is assumed to derive from some extremely large population, the probability of a single defect is very small. As with the p-chart, the normal distribution can be used to approximate the distribution of defects. The process average for the c-chart is the mean number of defects per item,

Construction of a c-Chart

The Ritz Hotel has 240 rooms. The hotel's housekeeping department is responsible for maintaining the quality of the rooms' appearance and cleanliness. Each individual housekeeper is responsible for an area encompassing 20 rooms. Every room in use is thoroughly cleaned and its supplies, toiletries, and so on are restocked each day. Any defects that the housekeeping staff notice that are not part of the normal housekeeping service are supposed to be reported to hotel maintenance. Every room is briefly inspected each day by a housekeeping supervisor. However, hotel management also conducts inspection tours at random for a detailed, thorough inspection for quality-control purposes. The management inspectors not only check for normal housekeeping service defects like clean sheets, dust, room supplies, room literature, or towels, but also for defects like an inoperative or missing TV remote, poor TV picture quality or reception, defective lamps, a malfunctioning clock, tears or stains in the bedcovers or curtains, or a malfunctioning curtain pull. An inspection sample includes 12 rooms, that is, one room selected at random from each of the twelve 20-room blocks serviced by a housekeeper. Following are the results from 15 inspection samples conducted at random during a one-month period:

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

The hotel believes that approximately 99% of the defects (corresponding to 3-sigma limits) are caused by natural, random variations in the housekeeping and room maintenance service, with 1% caused by nonrandom variability. They want to construct a c-chart to monitor the housekeeping service.

Solution

Because c, the population process average, is not known, the sample estimate,

The control limits are computed using z = 3.00, as follows:

The resulting c-chart, with the sample points, is shown in the following figure:

All the sample observations are within the control limits, suggesting that the room quality is in control. This chart would be considered reliable to monitor the room quality in the future.

Variable control charts are used for continuous variables that can be measured, such as weight or volume. Two commonly used variable control charts are the range chart, or R-chart, and the mean chart, or

Range (R-) chart: uses the amount of dispersion in a sample.

Mean (

MEAN (

-) CHART

In a mean (or

The

The formulas for computing the upper control limit (UCL) and lower control limit (LCL) are

where

Example 3.3 illustrates how to develop an

Constructing an

-Chart

The Goliath Tool Company produces slip-ring bearings, which look like flat doughnuts or washers. They fit around shafts or rods, such as drive shafts in machinery or motors. At an early stage in the production process for a particular slip-ring bearing, the outside diameter of the bearing is measured. Employees have taken 10 samples (during a 10-day period) of 5 slip-ring bearings and measured the diameter of the bearings. The individual observations from each sample (or subgroup) are shown as follows:

Subgroup k	Observations (Slip-Ring Diameter, cm), n
	1	2	3	4	5
1	5.02	5.01	4.94	4.99	4.96	4.98
2	5.01	5.03	5.07	4.95	4.96	5.00
3	4.99	5.00	4.93	4.92	4.99	4.97
4	5.03	4.91	5.01	4.98	4.89	4.96
5	4.95	4.92	5.03	5.05	5.01	4.99
6	4.97	5.06	5.06	4.96	5.03	5.01
7	5.05	5.01	5.10	4.96	4.99	5.02
8	5.09	5.10	5.00	4.99	5.08	5.05
9	5.14	5.10	4.99	5.08	5.09	5.08
10	5.01	4.98	5.08	5.07	4.99	5.03
						50.09

From past historical data it is known that the process standard deviation is .08. The company wants to develop a control chart with 3-sigma limits to monitor this process in the future.

The process average is computed as

The control limits are

None of the sample means (

In the second approach to developing an

where

A Nestle's quality control team tests samples of candies. The sample results can be plotted on a control chart to see if the production process is in control. If not, it will be corrected before a large number of defective candies are produced, thereby preventing costly waste.

Table 3.1. Factors for Determining Control Limits for

and R-Charts

Sample Size	Factor for	Factors for R-Chart
n	A2	D3	D4
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

An

-Chart

The Goliath Tool Company desires to develop an

Sample k	Observations (Slip-Ring Diameter, cm), n
	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

Solution

Using the value of A₂ = 0.58 for n = 5 from Table 3.1 and

The

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

•Range: the difference between the smallest and largest values in a sample.

where

R = range of each sample

k = number of samples (subgroups)

D₃ and D₄ are table values like A₂ for determining control limits that have been developed based on range values rather than standard deviations. Table 3.1 also includes values for D₃ and D₄ for sample sizes up to 25.

Constructing an R-Chart

The Goliath Tool Company from Examples 3.3 and 3.4 wants to develop an R-chart to control process variability.

From Example 3.4,

These limits define the R-chart shown in the following figure. It indicates that the process appears to be in control; any variability observed is a result of natural random occurrences.

This example illustrates the need to employ the R-chart and the

USING

- AND R-CHARTS TOGETHER

The

Both the process average and variability must be in control.

For example, consider two samples, the first having low and high values of 4.95 and 5.05 centimeters, and the second having low and high values of 5.10 and 5.20 centimeters. The range of both is 0.10 centimeters, but

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

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

In other situations, a company may have studied and collected data for a process for a long time and already know what the mean and standard deviation of the process are; all they want to do is monitor the process average by taking periodic samples. In this case it would be appropriate to use the mean chart where the process standard deviation is already known as shown in Example 3.3.

A pattern can indicate an out-of-control process even if sample values are within control limits.

Even though a control chart may indicate that a process is in control, it is possible the sample variations within the control limits are not random. If the sample values display a consistent pattern, even within the control limits, it suggests that this pattern has a nonrandom cause that might warrant investigation. We expect the sample values to "bounce around" above and below the center line, reflecting the natural, random variation in the process that will be present. However, if the sample values are consistently above (or below) the center line for an extended number of samples or if they move consistently up or down, there is probably a reason for this behavior; that is, it is not random. Examples of nonrandom patterns are shown in Figure 3.3.

Figure 3.3. Control Chart Patterns

Figure 3.4. Zones for Pattern Tests

A pattern in a control chart is characterized by a sequence of sample observations that display the same characteristics—also called a run. One type of pattern is a sequence of observations either above or below the center line. For example, three values above the center line followed by two values below the line represent two runs of a pattern. Another type of pattern is a sequence of sample values that consistently go up or go down within the control limits. Several tests are available to determine if a pattern is nonrandom or random.

• Run: a sequence of sample values that display the same characteristic.

One type of pattern test divides the control chart into three "zones" on each side of the center line, where each zone is one standard deviation wide. These are often referred to as 1-sigma, 2-sigma, and 3-sigma limits. The pattern of sample observations in these zones is then used to determine if any nonrandom patterns exist. Recall that the formula for computing an

•Pattern test: determines if the observations within the limits of a control chart display a nonrandom pattern

There are several general guidelines associated with the zones for identifying patterns in a control chart, where none of the observations are beyond the control limits:

Guidelines for identifying patterns.

If any of these guidelines applied to the sample observations in a control chart, it would imply that a nonrandom pattern exists and the cause should be investigated. In Figure 3.4, rules 1, 4, and are violated. Example 3.6 on the next page demonstrates how a pattern test is performed.

SAMPLE SIZE DETERMINATION

In our examples of control charts, sample sizes varied significantly. For p-charts and c-charts, we used sample sizes in the hundreds and as small as one item for a c-chart, whereas for

Attribute charts require larger sample sizes; variable charts require smaller samples.

Performing a Pattern Test

The Goliath Tool Company

Solution

In order to perform the pattern test, we must identify the runs that exist in the sample data for Example 3.4, as follows. Recall that

Sample		Above/Below	Up/down	Zone
1	4.98	B	—	B
2	5.00	B	U	C
3	4.97	B	D	B
4	4.96	B	D	A
5	4.99	B	U	C
6	5.01	—	U	C
7	5.02	A	U	c
8	5.05	A	U	B
9	5.08	A	U	A
10	5.03	A	D	B

No pattern rules appear to be violated.

Some companies use sample sizes of just two. They inspect only the first and last items in a production lot under the premise that if neither is out of control, then the process is in control. This requires the production of small lots (which are characteristic of some Japanese companies), so that the process will not be out of control for too long before a problem is discovered.

Size may not be the only consideration in sampling. It may also be important that the samples come from a homogeneous source so that if the process is out of control, the cause can be accurately determined. If production takes place on either one of two machines (or two sets of machines), mixing the sample observations between them makes it difficult to ascertain which operator or machine caused the problem. If the production process encompasses more than one shift, mixing the sample observation between shifts may make it more difficult to discover which shift caused the process to move out of control.

A number of computer software and spreadsheet packages are available that perform statistical quality control analysis, including the development of process control charts. We will demonstrate how to develop a statistical process control chart on the computer using Excel and OM Tools. The Excel spreadsheet in Exhibit 3.1 shows the data for Example 3.1 in which we constructed a p-chart to monitor the production process for denim jeans at the Western Jeans Company. The values for

Figure E3.1. Exhibit 3.1

The Excel file for the example problem spreadsheet shown in Exhibit 3.1 is provided on the text Web site, as are all Excel exhibits in the text. These files can be easily accessed and downloaded for use. The exhibit spreadsheets can often be used as templates for solving end-of-chapter homework problems.

The control chart shown in the lower-right-hand corner of Exhibit 3.1 was constructed by clicking on the "Insert" tab on the toolbar at the top of the spreadsheet, then covering the data points B4:E25. The next step was to click on the "Charts" tab and select a line chart, then after clicking "OK," the chart in Exhibit 3.1 appeared.

Exhibit 3.2 shows the

Control limits are occasionally mistaken for tolerances; however, they are quite different things. Control limits provide a means for determining natural variation in a production process. They are statistical results based on sampling. Tolerances are design specifications reflecting customer requirements for a product. They specify a range of values above and below a designed target value (also referred to as the nominal value), within which product units must fall to be acceptable. For example, a bag of potato chips might be designed to have a net weight of 9.0 oz of chips with a tolerance of ±0.5 oz. The design tolerances are thus between 9.5 oz (the upper specification limit) and 8.5 oz (the lower specification limit). The packaging process must be capable of performing within these design tolerances or a certain portion of the bags will be defective, that is, underweight or overweight. Tolerances are not determined from the production process; they are externally imposed by the designers of the product or service. Control limits, on the other hand, are based on the production process, and they reflect process variability. They are a statistical measure determined from the process. It is possible for a process in an instance to be statistically "in control" according to control charts, yet the process may not conform to the design specifications. To avoid such a situation, the process must be evaluated to see if it can meet product specifications before the process is initiated, or the product or service must be redesigned.

•Tolerances: design specifications reflecting product requirements.

Figure E3.2. Exhibit 3.2

Process capability refers to the natural variation of a process relative to the variation allowed by the design specifications. In other words, how capable is the process of producing acceptable units according to the design specifications? Process control charts are used for process capability to determine if an existing process is capable of meeting design specifications.

The three main elements associated with process capability are process variability (the natural range of variation of the process), the process center (mean), and the design specifications. Figure 3.5 shows four possible situations with different configurations of these elements that can occur when we consider process capability

•Process capability: the range of natural variability in a process–what we measure with control charts.

Figure 3.5a depicts the natural variation of a process, which is greater than the design specification limits. The process is not capable of meeting these specification limits. This situation will result in a large proportion of defective parts or products. If the limits of a control chart measuring natural variation exceed the specification limits or designed tolerances of a product, the process cannot produce the product according to specifications. The variation that will occur naturally, at random, is greater than the designed variation.

Parts that are within the control limits but outside the design specification must be scrapped or reworked. This can be very costly and wasteful. Alternatives include improving the process or redesigning the product. However, these solutions can also be costly. As such, it is important that process capability studies be done during product design, and before contracts for new parts or products are entered into.

If the natural variability in a process exceeds tolerances, the process cannot meet design specifications.

Figure 3.5b shows the situation in which the natural control limits and specification limits are the same. This will result in a small number of defective items, the few that will fall outside the natural control limits due to random causes. For many companies, this is a reasonable quality goal. If the process distribution is normally distributed and the natural control limits are three standard deviations from the process mean—that is, they are 3-sigma limits—then the probability between the limits is 0.9973. This is the probability of a good item. This means the area, or probability, outside the limits is 0.0027, which translates to 2.7 defects per thousand or 2700 defects out of one million items. However, according to strict quality philosophy, this is not an appropriate quality goal. As Evans and Lindsay point out in the book The Management and Control of Quality, this level of quality corresponding to 3-sigma limits is comparable to "at least 20,000 wrong drug prescriptions each year, more than 15,000 babies accidentally dropped by nurses and doctors each year, 500 incorrect surgical operations each week, and 2000 lost pieces of mail each hour."^[8]

Figure 3.5. Process Capability

As a result, a number of companies have adopted "6-sigma" quality. This represents product-design specifications that are twice as large as the natural variations reflected in 3-sigma control limits. This type of situation, where the design specifications exceed the natural control limits, is shown graphically in Figure 3.5c. The company would expect that almost all products will conform to design specifications—as long as the process mean is centered on the design target. Statistically, 6-sigma corresponds to only 0.0000002 percent defects or 0.002 defective parts per million (PPM), which is only two defects per billion! However, when Motorola announced in 1989 that it would achieve 6-sigma quality in five years they translated this to be 3.4 defects per million. How did they get 3.4 defects per million from 2 defects per billion? Motorola took into account that the process mean will not always exactly correspond to the design target; it might vary from the nominal design target by as much as 1.5 sigma (the scenario in Figure 3.5d), which translates to a 6-sigma defect rate of 3.4 defects per million. This value has since become the standard for 6-sigma quality in industry and business. Applying this same scenario of a 1.5-sigma deviation from the process mean to the more typical 3-sigma level used by most companies, the defect rate is not 2700 defects per million, but 66,810 defects per million.

In 6-sigma quality, we have 3.4 defective PPM, or zero defects.

As indicated, Figure 3.5d shows the situation in which the design specifications are greater than the process range of variation; however, the process is off center. The process is capable of meeting specifications, but it is not because the process is not in control. In this case a percentage of the output that falls outside the upper design specification limit will be defective. If the process is adjusted so that the process center coincides with the design target (i.e., it is centered), then almost all of the output will meet design specifications.

Determining process capability is important because it helps a company understand process variation. If it can be determined how well a process is meeting design specifications, and thus what the actual level of quality is, then steps can be taken to improve quality. Two measures used to quantify the capability of a process, that is, how well the process is capable of producing according to design specifications, are the capability ratio (C_p) and the capability index (C_pk)

PROCESS CAPABILITY MEASURES

One measure of the capability of a process to meet design specifications is the process capability ratio (C_p). It is defined as the ratio of the range of the design specifications (the tolerance range) to the range of the process variation, which for most firms is typically ±3σ or 6σ.

If C_p is less than 1.0, the process range is greater than the tolerance range, and the process is not capable of producing within the design specifications all the time. This is the situation depicted in Figure 3.5a. If C_p equals 1.0, the tolerance range and the process range are virtually the same—the situation shown in Figure 3.5b. If C_p is greater than 1.0, the tolerance range is greater than the process range—the situation depicted in Figure 3.5c. Thus, companies would logically desire a C_p equal to 1.0 or greater, since this would indicate that the process is capable of meeting specifications.

Computing C_p

The Munchies Snack Food Company packages potato chips in bags. The net weight of the chips in each bag is designed to be 9.0 oz, with a tolerance of ±0.5 oz. The packaging process results in bags with an average net weight of 8.80 oz and a standard deviation of 0.12 oz. The company wants to determine if the process is capable of meeting design specifications.

Solution

Thus, according to this process capability ratio of 1.39, the process is capable of being within design specifications.

ALONG THE SUPPLY CHAIN

Design Tolerances at Harley-Davidson Company

Harley-Davidson, once at the brink of going out of business, is now a successful company known for high quality. It has achieved this comeback by combining the classic styling and traditional features of its motorcycles with advanced engineering technology and a commitment to continuous improvement. Harley-Davidson's manufacturing process incorporates computer-integrated manufacturing (CIM) techniques with state-of-the-art computerized numerical control (CNC) machining stations. These CNC stations are capable of performing dozens of machining operations and provide the operator with computerized data for statistical process control.

Harley-Davidson uses a statistical operator control (SOC) quality-improvement program to reduce parts variability to only a fraction of design tolerances. SOC ensures precise tolerances during each manufacturing step and predicts out-of-control components before they occur. SOC is especially important when dealing with complex components such as transmission gears.

The tolerances for Harley-Davidson cam gears are extremely close, and the machinery is especially complex. CNC machinery allows the manufacturing of gear centers time after time with tolerances as close as 0.0005 inch. SOC ensures the quality necessary to turn the famous Harley- Davidson Evolution engine shift after shift, mile after mile, year after year.

Discuss how reducing "parts variability to only a fraction of design tolerances" is related to a goal of achieving Six Sigma quality.

A second measure of process capability is the process capability index (C_pk). The C_pk differs from the C_p in that it indicates if the process mean has shifted away from the design target, and in which direction it has shifted—that is, if it is off center. This is the situation depicted in Figure 3.5d. The process capability index specifically measures the capability of the process relative to the upper and lower specifications.

If the C_pk index is greater than 1.00, then the process is capable of meeting design specifications. If C_pk is less than 1.00, then the process mean has moved closer to one of the upper or lower design specifications, and it will generate defects. When C_pk equals C_p, this indicates that the process mean is centered on the design (nominal) target.

Computing C_pk

Recall that the Munchies Snack Food Company packaged potato chips in a process designed for 9.0 oz of chips with a tolerance of ±0.5 oz. The packaging process had a process mean (

Although the C_p of 1.39 computed in Example 3.7 indicated that the process is capable (it is within the design specifications), the C_pk value of 0.83 indicates the process mean is off center. It has shifted toward the lower specifications limit; that means underweight packages of chips will be produced. Thus, the company needs to take action to correct the process and bring the process mean back toward the design target.

PROCESS CAPABILITY WITH EXCEL AND OM TOOLS

Exhibit 3.3 shows the Excel solution screen for the computation of the process capability ratio and the process capability index for Examples 3.7 and 3.8. The formula for the process capability index in cell D16 is shown on the formula bar at the top of the screen.

Exhibit 3.4 shows the computation of the process capability ratio and the process capability index for Example 3.7 and 3.8 using the "quality control" module from the OM Tools software.

Figure E3.3. Exhibit 3.3

Figure E3.4. Exhibit 3.4

Statistical process control is one of the main quantitative tools used in most quality management systems. Quality focused companies provide extensive training in SPC methods for all employees at all levels. In this environment employees have more responsibility for their own operation or process. Employees recognize the need for SPC for accomplishing a major part of their job, product quality, and when employees are provided with adequate training and understand what is expected of them, they have little difficulty using statistical process control methods.

Control Limits for p-Charts

Control Limits for c-Charts

Control Limits for R-Charts

Control Limits for

Process Capability Ratio

attribute a product characteristic that can be evaluated with a discrete response such as yes or no, good or bad.

c-chart a control chart based on the number of defects in a sample.

control chart a graph that visually shows if a sample is within statistical limits for defective items.

control limits the upper and lower bands of a control chart.

mean(

p-chart a control chart based on the proportion defective of the samples taken.

pattern test a statistical test to determine if the observations within the limits of a control chart display a nonrandom pattern.

process capability the capability of a process to accommodate design specications of a product.

range the difference between the smallest and largest values in a sample.

range (R-) chart a control chart based on the range (from the highest to the lowest values) of the samples taken.

run a sequence of sample values that display the same tendency in a control chart.

sample a portion of the items produced used for inspection.

statistical process control (SPC) a statistical procedure for monitoring the quality of the production process using control charts.

tolerances product design specications required by the customer.

variable measure a product characteristic that can be measured, such as weight or length.

1. p-CHARTS

Twenty samples of n = 200 were taken by an operator at a workstation in a production process. The number of defective items in each sample were recorded as follows.

Management wants to develop a p-chart using 3-sigma limits. Set up the p-chart and plot the observations to determine if the process was out of control at any point.

SOLUTION

step 1. Compute

2. PATTERN TESTS

In the preceding problem, even though the control chart indicates that the process is in control, management wants to use pattern tests to further determine if the process is in control.

SOLUTION

Step 1. Determine the "up-and-down" and "above-and-below" runs and zone observations. Construct the zone boundaries on the control chart as follows.

Step 2. Determine the control limits:

Step 3. Construct the

SAMPLE	ABOVE/BELOW	UP/DOWN	ZONE
(Note: Ties are broken in favor of A and U.)
1	B	—	B
2	A	U	C
3	B	D	B
4	B	U	C
5	A	U	C
6	A	U	B
7	A	D	C
8	B	D	B
9	B	D	B
10	B	U	C
11	A	U	C
12	B	D	C
13	B	D	C
14	A	U	C
15	A	U	C
16	A	U	C
17	A	D	C
18	A	U	B
19	A	U	B
20	A	U	B

Because four of five consecutive points are in Zone B (i.e., points 16 to 20), it suggests that nonrandom patterns may exist and that the process may not be in control.

3-1. Explain the difference between attribute control charts and variable control charts.

3-2. How are mean (

3-3. What is the purpose of a pattern test?

3-4. What determines the width of the control limits in a process chart?

3-5. Under what circumstances should a c-chart be used instead of a p-chart?

3-6. What is the difference between tolerances and control limits?

3-7. Why have companies traditionally used control charts with 3-sigma limits instead of 2-sigma limits?

3-8. Select three service companies or organizations you are familiar with and indicate how process control charts could be used in each.

3-9. Visit a local fast-food restaurant, retail store, grocery store, or bank, and identify the different processes that control charts could be used to monitor.

3.10. Explain the different information provided by the process capability ratio and the process capability index.

3.11. For the Goliath Tool Company in Example 3.4, if the design tolerances are ±0.07 cm, is the process capable of meeting tolerances of ±0.08 cm for the slip-ring bearings?

3-1. The Great North Woods Clothing Company sells specialty outdoor clothing through its catalogue. A quality problem that generates customer complaints occurs when a warehouse employee fills an order with the wrong items. The company has decided to implement a process control plan by inspecting the ordered items after they have been obtained from the warehouse and before they have been packaged. The company has taken 30 samples (during a 30-day period), each for 100 orders, and recorded the number of "defective" orders in each sample, as follows:

Construct a p-chart for the company that describes 99.74% (3σ) of the random variation in the process, and indicate if the process seems to be out of control at any time.

3-2. The Road King Tire Company in Birmingham wants to monitor the quality of the tires it manufactures. Each day the company quality-control manager takes a sample of 100 tires, tests them, and determines the number of defective tires. The results of 20 samples have been recorded as follows:

Construct a p-chart for this process using 2σ limits and describe the variation in the process.

3-3. 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 30 days and checking their accuracy. The sample results are as follows:

Develop a p-chart for the billing process using 3σ control limits and indicate if the process is out of control.

3-4. In the assembly process for automobile engines, at one stage in the process a gasket is placed between the two sections of the engine block before they are joined. If the gasket is damaged (e.g., bent, crimped), oil can leak from the cylinder chambers and foul the spark plugs, in which case the entire engine has to be disassembled and a new gasket inserted. The company wants to develop a p-chart with 2σ limits to monitor the quality of the gaskets prior to the assembly stage. Historically, 2% of the gaskets have been defective, and management does not want the upper control limit to exceed 3% defective. What sample size will be required to achieve this control chart?

3-5. The Great North Woods 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 classied 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:

3-6. One of the stages in the process of making denim cloth at the Southern Mills Company is to spin cotton yarn onto spindles for subsequent use in the weaving process. Occasionally the yarn breaks during the spinning process, and an operator ties it back together. Some number of breaks is considered normal; however, too many breaks may mean that the yarn is of poor quality. In order to monitor this process, the quality-control manager randomly selects a spinning machine each hour and checks the number of breaks during a 15-minute period. Following is a summary of the observations for the past 20 hours:

Construct a c-chart using 3σ limits for this process and indicate if the process was out of control at any time.

3-7. The Xecko Film Company manufactures color photographic film. The film is produced in large rolls of various lengths before it is cut and packaged as the smaller roles purchased in retail stores. The company wants to monitor the quality of these rolls of film using a c-chart. Twentyfour rolls have been inspected at random, and the numbers of defects per roll are as follows:

Construct a c-chart with 2σ limits for this process and indicate if the process was out of control at any time.

3-8. Telecom manufactures electronic components for computers. One measure it uses to monitor the quality of its distribution process is the number of customer invoice errors. The distribution center manager monitored the company's order processing and distribution by recording the number of invoice errors for 30 days. The sample results are as follows:

Construct a c-chart with 3σ limits for invoice errors and indicate if the process was out of control at any time.

3-9. The National Bread Company daily delivers multiple orders by truck from its regional distribution center to stores in the Wayman's Supermarket chain. One measure of its supply chain performance is the number of late deliveries. The company's goal is to make all deliveries within one day, so a delivery is late if it exceeds one day. The total number of late deliveries for each of the past 20 days are as follows:

Construct a c-chart for late deliveries with 3σ control limits and indicate if the delivery process was out of control at any time.

3-10. BooksCDs.com sells books, videos, DVDs, and CDs through its Internet Web site. The company ships thousands of orders each day from seven national distribution centers. BooksCDs.com wants to establish a p-chart to monitor the quality of its distribution process, specically the number of "problem" orders. A problem order is one that is delivered to a customer late (i.e., after five days), incorrect or incomplete. The company sampled 500 orders every other day for 20 samples and tracked them to final customer delivery, the results of which are as follows:

Construct a p-chart for this process using 3σ limits and indicate if the process was out of control.

3-11. Valtec Electronics fills orders for its electronic components and parts by truck to customers through several distribution centers. A measure of its supply chain responsiveness is order fulfillment lead time, which is the number of days from when a company receives an order to when it is delivered to the customer. A distribution center manager has taken 20 samples of 5 orders each during the month and recorded the lead time for each as follows:

Construct an

3-12. The Stryker Baseball Bat Company manufactures wooden and aluminum baseball bats at its plant in New England. Wooden bats produced for the mass market are turned on a lathe, where a piece of wood is shaped into a bat with a handle and barrel. The bat is cut to its specified length and then finished in subsequent processes. Once bats are cut to length, it is difficult to rework them into a different style, so it is important to catch defects before this step. As such, bats are inspected at this stage of the process. A specific style of wooden bat has a mean barrel circumference of 9 inches at its thickest point with a standard deviation of 0.6 inch. (The process variability is assumed to be normally distributed.)

3-13. A machine at the Pacific Fruit Company fills boxes with raisins. The labeled weight of the boxes is 9 ounces. The company wants to construct an R-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:

Construct an R-chart from these data with 3σ control limits, plot the sample range values, and comment on process control.

3-14. The City Square Grocery and Meat Market has a large meat locker in which a constant temperature of approximately 40° F should be maintained. The market manager has decided to construct an R-chart to monitor the temperature inside the locker. The manager had one of the market employees take sample temperature readings randomly five times each day for 20 days in order to gather data for the control chart. Following are the temperature sample observations:

3-15. The Oceanside Apparel Company manufactures expensive, polo-style men's and women's short-sleeve knit shirts at its plant in Jamaica. The production process requires that material be cut into large patterned squares by operators, which are then sewn together at another stage of the process. If the squares are not of a correct length, the final shirt will be either too large or too small. In order to monitor the cutting process, management takes a sample of four squares of cloth every other hour and measures the length. The length of a cloth square should be 36 inches, and historically, the company has found the length to vary across an acceptable average of 2 inches.

Plot the new sample data on the control chart constructed in part (a) and comment on the process variability.

3-16. For the sample data provided in Problem 3-13, construct an

3-17. For the sample data provided in Problem 3-14, construct an

3-18. Using the process information provided in Problem 3-15, construct an

3-19. Use pattern tests to determine if the sample observations used in the

3-20. Use pattern tests to determine if the sample observations in Problem 3-5 reflect any nonrandom patterns.

3-21. Use pattern tests to determine if the sample observations in Problem 3-14 reflect any nonrandom patterns.

3-22. Use pattern tests to determine if the sample observations used in the

3-23. Use pattern tests to determine if the sample observations used in the p-chart in Problem 3-1 reflect any nonrandom patterns.

3-24. Dave's Restaurant is a chain that employs independent evaluators to visit its restaurants as customers and assess the quality of the service by filling out a questionnaire. The company evaluates restaurants in two categories, products (the food) and service (e.g., promptness, order accuracy, courtesy, friendliness). The evaluator considers not only his or her order experiences but also observations throughout the restaurant. Following are the results of an an evaluator's 20 visits to one particular restaurant during a month showing the number of "defects" noted in the limits. service category:

Construct a control chart for this restaurant using 3σ limits to monitor quality service and indicate if the process is in control.

3-25. The National Bank of Warwick is concerned with complaints from customers about its drive-through window operation. Customers complain that it sometimes takes too long to be served and since there are often cars in front and back of a customer, they cannot leave if the service is taking a long time. To correct this problem the bank installed an intercom system so the drive-through window teller can call for assistance if the line backs up or a customer has an unusually long transaction. The bank's objective is an average custome's waiting and service time of approximately 3 minutes. The bank's operations manager wants to monitor the new drive-through window system with SPC. The manager has timed five customers' waiting and service times at random for 12 days as follows:

Develop an

3-26. The Great Outdoors Clothing Company is a mail-order catalogue operation. Whenever a customer returns an item for a refund, credit, or exchange, he or she is asked to complete a return form. For each item returned the customer is asked to insert a code indicating the reason for the return. The company does not consider the returns related to style, size, or "feel" of the material to be a defect. However, it does consider returns because the item "was not as described in the catalogue," "didn't look like what was in the catalogue," or "the color was different than shown in the catalogue," to be defects in the catalogue. The company has randomly checked 100 customer return forms for 20 days and collected the following data for catalogue defects:

Construct a control chart using 3σ limits to monitor catalogue defects and indicate if the process is in control. Use pattern tests to verify an in-control situation.

3-27. The dean of the College of Business at State University wants to monitor the quality of the work performed by the college's secretarial staff. Each completed assignment is returned to the faculty with a check sheet on which the faculty member is asked to list the errors made on the assignment. The assistant dean has randomly collected the following set of observations from 20 three-day work periods:

Construct a process control chart for secretarial work quality using 3σ limits and determine if the process was out of control at any point. Use pattern tests to determine if any nonrandom patterns exist.

3-28. Martha's Wonderful Cookie Company makes a special super chocolate-chip peanut butter cookie. The company would like the cookies to average approximately eight chocolate chips apiece. Too few or too many chips distort the desired cookie taste. Twenty samples of five cookies each during a week have been taken and the chocolate chips counted. The sample observations are as follows:

Construct an

3-29. Thirty patients who check out of the Rock Creek County Regional Hospital each week are asked to complete a questionnaire about hospital service. Since patients do not feel well when they are in the hospital, they typically are very critical of the service. The number of patients who indicated dissatisfaction of any kind with the service for each 30-patient sample for a 16-week period is as follows:

Construct a control chart to monitor customer satisfaction at the hospital using 3σ limits and determine if the process is in control.

3-30. An important aspect of customer service and satisfaction at the Big Country theme park is the maintenance of the restrooms throughout the park. Customers expect the restrooms to be clean; odorless; well stocked with soap, paper towels, and toilet paper; and to have a comfortable temperature. In order to maintain quality, park quality-control inspectors randomly inspect restrooms daily (during the day and evening) and record the number of defects (incidences of poor maintenance). The goal of park management is approximately 10 defects per inspection period. Following is a summary of the observations taken by these inspectors for 20 consecutive inspection periods:

Construct the appropriate control chart for this maintenance process using 3σ limits and indicate if the process was out of control at any time. If the process is in control, use pattern tests to determine if any nonrandom patterns exist.

3-31. The Great Outdoors Clothing Company, a mail-order catalogue operation, contracts with the Federal Parcel Service to deliver all of its orders to customers. As such, Great Outdoors considers Federal Parcel to be part of its QMS program. Great Outdoors processes orders rapidly and requires Federal Parcel to pick them up and deliver them rapidly. Great Outdoors has tracked the delivery time for fiive randomly selected orders for 12 samples during a two-week period as follows:

Construct an

3-32. The Great Outdoors Clothing Company in Problem 3-31 has designed its packaging and delivery process to deliver orders to a customer within 3 business days ±1 day, which it tells customers. Using the process mean and control limits developed in Problem 3-31 compute the process capability ratio and index, and comment on the capability of the process to meet the company's delivery commitment.

3-33. Martha's Wonderful Cookie Company in Problem 3-28 has designed its special super chocolate-chip peanut butter cookies to have 8 chocolate chips with tolerances of ±2 chips. Using the process mean and control limits developed in Problem 3-28, determine the process capability ratio and index, and comment on the capability of the cookie production process.

3-34. The Pacific Fruit Company in Problem 3-13 has designed its packaging process for boxes to hold a net weight (nominal value) of 9.0 oz of raisins with tolerances of ±0.5 oz. Using the process mean and control limits developed in Problem 3-13, compute the process capability ratio and index, and comment on the capability of the process to meet the company's box weight specifications.

3-35. Sam's Long Life 75-watt light bulbs are designed to have a life of 1125 hours with tolerances of ±210 hours. The process that makes light bulbs has a mean of 1050 hours, with a standard deviation of 55 hours. Compute the process capability ratio and the process capability index, and comment on the overall capability of the process.

3-36. Elon Corporation manufactures parts for an aircraft company. It uses a computerized numerical controlled (CNC) machining center to produce a specific part that has a design (nominal) target of 1.275 inches with tolerances of ±0.024 inch. The CNC process that manufactures these parts has a mean of 1.281 inches and a standard deviation of 0.008 inch. Compute the process capability ratio and process capability index, and comment on the overall capability of the process to meet the design specifications.

3-37. Explain to what extent the process for producing Sam's Long Life bulbs in Problem 3-35 would have to be improved in order to achieve 6-sigma quality.

3-38. The Elon Company manufactures parts for an aircraft company using three computerized numerical controlled (CNC) turning centers. The company wants to decide which machines are capable of producing a specific part with design specifications. of 0.0970 inch ±0.015 inch. The machines have the following process prameters—machine 1 (

3-39. The emergency medical response (EMR) team in Brookville has instituted a quality improvement program and it makes extensive use of SPC charts. It wants to monitor the response times for emergency calls to make sure they stay around nine minutes on average. The EMR administration staff has randomly timed five emergency calls each month during a 12-month period and collected the following data.

Develop a

3-40. The Balston Healthcare System uses SPC charts extensively to monitor various quality indicators and improve processes at its hospitals. One of the hospitals has discovered a potential problem in hand hygiene among its nursing staff. Lack of hand hygiene can be a major cause of fatal hospital-borne infections. The Centers for Disease Control has established criteria for hand hygiene in hospitals and the hospital suspects it is not meeting these criteria. When the CDC published the criteria for hand hygiene the first of January the hospital began collecting data—it sampled 150 opportunities for hand hygiene each week and recorded how many times hand hygiene was actually practiced according to the CDC criteria. The data showed a deficiency in hand hygiene among the nurses in the first six weeks and in Week 7 a program to improve hand hygiene among the nurses was implemented with a goal of consistently reaching a target value of meeting 90% of all hand hygiene opportunities. Following is a table showing the data for the year.

Develop a control chart based on this data (using 3σ control limits) and explain what the chart means and the steps in a quality improvement process the SPC chart results would lead the quality management staff to take.

3-41. The Rollins Sporting Goods Company manufactures baseballs for the professional minor and major leagues at its plants in Costa Rica. According to the rules of major league baseball, a baseball must weigh between 142 and 149 grams. The company has taken 20 samples of five baseballs each and weighed the baseballs as follows:

Construct an

3-42. Explain to what extent the process for manufacturing baseballs in Problem 3-41 must be improved in order to achieve 6σ quality.

3-43. At Samantha's Super Store the customer service area processes customer returns, answers customer questions and provides information, addresses customer complaints, and sells gift certificates. The manager believes that if customers must wait longer than 8 minutes to see a customer service representative they get very irritated. The customer service process has been designed to achieve a customer wait time of between 6 and 12 minutes. The store manager has conducted 10 samples of five observations each of customer waiting time over a two-week period as follows:

Construct an

3-44. Metropolitan General Hospital is a city-owned and operated public hospital. Its emergency room is the largest and most prominent in the city. Approximately 70% of emergency cases in the city come or are sent to Metro General's emergency room. As a result, the emergency room is often crowded and the staff is overworked, causing concern among hospital administrators and city officials about the quality of service and health care the emergency room is able to provide. One of the key quality attributes administrators focus on is patient waiting time—that is, the time between when a patient checks in and registers and when the patient first sees an appropriate medical staff member. Hospital administration wants to monitor patient waiting time using statistical process control charts. At different times of the day over a period of several days, patient waiting times were recorded at random with the following results:

3-45. The three most important quality attributes at Mike's Super Service fast-food restaurant are considered to be good food, fast service, and a clean environment. The restaurant manager uses a combination of customer surveys and statistical measurement tools to monitor these quality attributes. A national marketing and research firm has developed data showing that when customers are in line up to five minutes their perception of that waiting time is only a few minutes; however, after five minutes customer perception of their waiting time increases exponentially. Furthermore, a five-minute average waiting time results in only 2% of customers leaving. The manager wants to monitor speed of service using a statistical process control chart. At different times during the day over a period of several days the manager had an employee time customers' waiting times (from the time they entered an order line to the time they received their order) at random, with the following results:

3-46. The family of a patient at County General Hospital complained when the patient fell in the hospital and broke her hip. The family was threatening a lawsuit, and there had been some negative publicity about the hospital in the local media suggesting that patient falls might be common. The hospital administration decided to investigate this potential problem by developing a control chart based on two years of monthly data for the number of patient falls each month as follows:

Develop a control chart with 3σ limits to monitor patient falls, and discuss if you think there appears to be a quality problem at the hospital.

3-47 The Shuler Motor Mile is a high-volume discount car dealership that stocks several different makes of cars. They also have a large service department. The dealership telephones six randomly selected customers each week and conducts a survey to determine their satisfaction with the service they received. Each survey results in a customer satisfaction score based on a 100-point scale, where 100 is perfect (and what the dealership ultimately aspires to). Following are the survey results for three months:

3-48. The time from when a patient is discharged from North Shore Hospital to the time the discharged patient's bed is ready to be assigned to a new patient is referred to as the bed assignment turnaround time. If the bed turnaround time is excessive, it can cause problems with patient flow and delay medical procedures throughout the hospital. This can cause long waiting times for physicians and patients thus creating customer dissatisfaction. The admissions RN has assigned a patient care associate to measure the bed turnaround time for a randomly selected bed each morning, afternoon, and evening for 30 days. Following are the bed turnaround time sample observations:

Quality Control at Rainwater Brewery

Bob Raines and Megan Waters own and operate the Rainwater Brewery, a micro-brewery that grew out of their shared hobby of making home-brew. The brewery is located in Whitesville, the home of State University where Bob and Megan went to college.

Whitesville has a number of bars and restaurants that are patronized by students at State and the local resident population. In fact, Whitesville has the highest per capita beer consumption in the state. In setting up their small brewery, Bob and Megan decided that they would target their sales toward individuals who would pick up their orders directly from the brewery and toward restaurants and bars, where they would deliver orders on a daily or weekly basis.

The brewery process essentially occurs in three stages. First, the mixture is cooked in a vat according to a recipe; then it is placed in a stainless-steel container, where it is fermented for several weeks. During the fermentation process the specific gravity, temperature, and pH need to be monitored on a daily basis. The specific gravity starts out at about 1.006 to 1.008 and decreases to around 1.002, and the temperature must be between 50 and 60°F. After the brew ferments, it is filtered into another stainless-steel pressurized container, where it is carbonated and the beer ages for about a week (with the temperature monitored), after which it is bottled and is ready for distribution. Megan and Bob brew a batch of beer each day, which will result in about 1000 bottles for distribution after the approximately three-week fermentation and aging process.

In the process of setting up their brewery, Megan and Bob agreed they had already developed a proven product with a taste that was appealing, so the most important factor in the success of their new venture would be maintaining high quality. Thus, they spent a lot of time discussing what kind of quality-control techniques they should employ. They agreed that the chance of brewing a "bad," or "spoiled," batch of beer was extremely remote, plus they could not financially afford to reject a whole batch of 1000 bottles of beer if the taste or color was a little "off" the norm. So they felt as if they needed to focus more on process control methods to identify quality problems that would enable them to adjust their equipment, recipe, or process parameters rather than rejecting the entire batch.

Describe the different quality-control methods that Rainwater Brewery might use to ensure good-quality beer and how these methods might fit into an overall quality management program.

Quality Control at Grass, Unlimited

Mark Sumansky owns and manages the Grass, Unlimited, lawn-care service in Middleton. His customers include individual homeowners and businesses that subscribe to his service for lawn care beginning in the spring and ending in the fall with leaf raking and disposal. Thus, when he begins his service in April he generally has a full list of customers and does not take on additional customers unless he has an opening. However, if he loses a customer any time after the first of June, it is difficult to find new customers, since most people make lawn-service arrangements for the entire summer.

Mark employs five crews, with three to five workers each, to cut grass during the spring and summer months. A crew normally works 10-hour days and can average cutting about 25 normal-size lawns of less than a halfacre each day. A crew will normally have one heavy-duty, wide-cut riding mower, a regular power mower, and trimming equipment. When a crew descends on a lawn, the normal procedure is for one person to mow the main part of the lawn with the riding mower, one or two people to trim, and one person to use the smaller mower to cut areas the riding mower cannot reach. Crews move very fast, and they can often cut a lawn in 15 minutes.

Unfortunately, although speed is an essential component in the profitability of Grass, Unlimited, it can also contribute to quality problems. In his or her haste, a mower might cut flowers, shrubs, or border plants, nick and scrape trees, "skin" spots on the lawn creating bare spots, trim too close, scrape house paint, cut or disfigure house trim, and destroy toys and lawn furniture, among other things. When these problems occur on a too-frequent basis, a customer cancels service, and Mark has a difficult time getting a replacement customer. In addition, he gets most of his subscriptions based on word-of-mouth recommendations and retention of previous customers who are satisfied with his service. As such, quality is a very important factor in his business.

In order to improve the quality of his lawn-care service, Mark has decided to use a process control chart to monitor defects. He has hired Lisa Anderson to follow the teams and check lawns for defects after the mowers have left. A defect is any abnormal or abusive condition created by the crew, including those items just mentioned. It is not possible for Lisa to inspect the more than 100 lawns the service cuts daily, so she randomly picks a sample of 20 lawns each day and counts the number of defects she sees at each lawn. She also makes a note of each defect, so that if there is a problem, the cause can easily be determined. In most cases the defects are caused by haste, but some defects can be caused by faulty equipment or by a crew member using a poor technique or not being attentive.

Over a three-day period Lisa accumulated the following data on defects:

Develop a process control chart for Grass, Unlimited, to monitor the quality of its lawn service using 2-sigma limits. Describe any other quality-control or quality-management procedures you think Grass, Unlimited, might employ to improve the quality of its service.

Improving Service Time at Dave's Burgers

Dave's Burgers is a fast-food restaurant franchise in Georgia, South Carolina, and North Carolina. Recently, Dave's Burgers has followed the lead of larger franchise restaurants like Burger King, McDonald's, and Wendy's and constructed drive-through windows at all its locations. However, instead of making Dave's Burgers more competitive, the drive-through windows have been a source of continual problems, and it has lost market share to its larger competitors in almost all locations. To identify and correct the problems, top management has selected three of its restaurants (one in each state) as test sites and has implemented a quality management program at each of them. A quality team made up of employees, managers, and quality specialists from company headquarters, at the Charlotte, North Carolina, test restaurant using traditional quality tools like Pareto charts, checksheets, fishbone diagrams, and process flowcharts, have determined that the primary problem is slow, erratic service at the drive-through window. Studies show that from the time a customer arrives at the window to the time the order is received averages 2.6 minutes. To be competitive, management believes service time should be reduced to at least 2.0 minutes and ideally 1.5 minutes.

The Charlotte Dave's Burgers franchise implemented a number of production process changes to improve service time at the drive-through window. It provided all employees with more training across all restaurant functions, improved the headset system, improved the equipment layout, developed clearer signs for customers, streamlined the menu, and initiated even-dollar (tax-inclusive) pricing to speed the payment process. Most importantly the restaurant installed large, visible electronic timers that showed how long a customer was at the window. This not only allowed the quality team to measure service speed but also provided employees with a constant reminder that a customer was waiting.

These quality improvements were implemented over several months, and their effect was immediate. Service speed was obviously improved, and market share at the Charlotte restaurant increased by 5%. To maintain quality service, make sure the service time remained fast, and continue to improve service, the quality team decided to use a statistical process control chart on a continuing basis. They collected six service time observations daily over a 15-day period, as follows: