Advantages

Certain quality characteristics are best measured as attributes. For instance, the taste of a food item is specified as acceptable or not. There are circumstances in which a quality characteristic can be measured as a variable but is instead measured as an attribute because of limited time, money, worker availability, or other resources. Let's consider the inside diameter of a hole. This characteristic could be measured with an inside micrometer, but it may be more convenient and cost-effective to use a go/no-go gage. Of course, the assumption is that attribute information is sufficient; otherwise, the quality characteristic may have to be dealt with as a variable.

In most manufacturing and service operations there are numerous quality characteristics that can be analyzed. If a variable chart (such as an - or R-chart) is selected, one variable chart is needed for each characteristic. The total number of control charts being constructed and maintained can be overwhelming. A control chart for attributes can provide overall quality information at a fraction of the cost.

Let's consider a simple component for which the three quality characteristics of length L, width W, and height H are important. If variable charts are constructed, we will need three charts. However, we can get by with one attribute chart if an item is simply classified as nonconforming when the length, width, or height does not conform to specifications. The attribute chart thus summarizes the information for all three characteristics. An attribute chart can also be used to summarize information about several components that make up a product.

Attributes are encountered at all levels of an organization: the company, plant, department, work center, and machine (or operator) level. Variable charts are typically used at the lowest level, the machine level. When we do not know what is causing a problem, it is sensible to start at a general level and work to the specific. For example, we know that a high proportion of nonconforming items is being detected at the company level, so we keep an attribute chart at the plant level to determine which plants have high proportions of nonconforming items. Once we have identified these plants, we might use an attribute chart for output at the departmental level to pinpoint problem areas. When the particular work center thought to be responsible for the increase in the production of nonconforming items is identified, we could then focus on the machine or operator level and try to further pinpoint the source of the problem. Attribute charts assist in going from the general to a more focused level. Once the lowest problem level has been identified, a variable chart is then used to determine specific causes for an out-of-control situation.

Disadvantages

Attribute information indicates whether a certain quality characteristic is within specification limits. It does not state the degree to which specifications are met or not met. For example, the specification limits for the diameter of a part are 20 ± 0.1 mm. Two parts, one with diameter 20.2 mm and the other with diameter 22.3 mm, are both classified as nonconforming, but their relative conformance to the specifications is not noted on attribute information.

Variable information, on the other hand, indicates the level of the data values. Variable control charts thus provide more information on the performance of a process. Specific information about the process mean and variability can be obtained. Furthermore, for out-of-control situations, variable plots generally provide more information as to the potential causes and hence make identification of remedial actions easier.

If we can assume that the process is very capable (i.e., its inherent variability is much less than the spread between the specification limits), variable charts can forewarn us when the process is about to go out of control. This, of course, allows us to take corrective action before any nonconforming items are produced. A variable chart can indicate an upcoming out-of-control condition even though items are not yet nonconforming.

Figure 8-1 depicts this situation. Suppose that the target value of the process mean is at A, the process is very capable (note the wide spread between the distribution and the specification limits), and the process is in control. The process mean now shifts to B. The variable chart indicates an out-of-control condition. Because of the wide spread of the specification limits, no nonconforming items are produced when the process mean is at B, even though the variable chart is indicating an out-of-control process. The attribute chart, say, a chart for the proportion of nonconforming items, does not detect a lack of control until the process parameters are sufficiently changed such that some nonconforming items are produced. Only when the process mean shifts to C does the attribute chart detect an out-of-control situation. However, were the specifications equal to or tighter than the inherent variability of the process, attribute charts would indicate an out-of-control process in a time frame similar to that for the variable chart.

Attribute charts require larger sample sizes than variable charts to ensure adequate protection against a certain level of process changes. Larger sample sizes can be problematic if the measurements are expensive to obtain or the testing is destructive.

Construction and Interpretation

The general procedures from Chapters 6 and 7 for the construction and interpretation of control charts apply to control charts for attributes as well. They are summarized here.

1. Step 1: Select the objective. Decide on the level at which the p-chart will be used (i.e., the plant, the department, or the operator level). Decide to control either a single quality characteristic, multiple characteristics, a single product, or a number of products.
2. Step 2: Determine the sample size and the sampling interval. An appropriate sample size is related to the existing quality level of the process. The sample size must be large enough to allow the opportunity for some nonconforming items to be present on average. The sampling interval (i.e., the time between successive samples) is a function of the production rate and the cost of sampling, among other factors.

   A bound for the sample size, n, can be obtained using these guidelines. Let represent the process average nonconforming rate. Then, based on the number of nonconforming items that you wish to be represented in a sample—say five, on average—the bound is expressed as

   (8-5)
3. Step 3:Obtain the data and record on an appropriate form. Decide on the measuring instruments in advance.

   A typical data sheet for a p-chart is shown in Table 8-1. The data and time at which the sample is taken, along with the number of items inspected and the number of nonconforming items, are recorded. The proportion nonconforming is found by dividing the number of nonconforming items by the sample size. Usually, 25–30 samples should be taken prior to performing an analysis.

   Sample	Date	Time	Number of Items Inspected, n	Number of Nonconforming Items, x	Proportion Nonconforming,	Comments
   1	10/15	9:00 A.M.	400	12	0.030
   2	10/15	9:30 A.M.	400	10	0.025
   3	10/15	10:00 A.M.	400	14	0.035	New vendor
   .	.	.	.	.	.	.
   .	.	.	.	.	.	.
   .	.	.	.	.	.	.

4. Step 4:Calculate the centerline and the trial control limits. Once they are determined, draw them on the p-chart. Plot the values of the proportion nonconforming for each sample on the chart. Examine the chart to determine whether the process is in control.

The rules for determining out-of-control conditions are the same as those we discussed in Chapters 6 and 7, so are not repeated in this chapter. As usual, the most common criterion for an out-of-control condition is the presence of a point plotting outside the control limits. The means of calculating the centerline and control limits are as follows.

No Standard Specified

When no standard or target value of the proportion nonconforming is specified, it must be estimated from the sample information. Recall that for each sample, the sample proportion nonconforming is given by eq. (8-1). The average of the individual sample proportion nonconforming is used as the centerline (CL_p). That is,

(8-6)

where g represents the number of samples. The variance of is given by eq. (8-4). Since the true value of p is not known, the value of given by eq. (8-6) is used as an estimate. In accordance with the concept of 3σ limits discussed in previous chapters, the control limits are given by

(8-7)

Standard Specified

If the target value of the proportion of nonconforming items is known or specified, the centerline is selected as that target value. In other words, the centerline is given by

(8-8)

where p₀ represents the standard or target value. The control limits in this case are also based on the target value. Thus,

(8-9)

If the lower control limit for p turns out to be negative for eq. (8-7) or (8-9), the lower control limit is simply counted as zero because the smallest possible value of the proportion nonconforming is zero.

1. Step 5:Calculate the revised control limits. Analyze the plotted values of and the pattern of the plot for out-of-control conditions. Typically, one or a few of the rules are used concurrently. On detection of an out-of-control condition, identify the special cause and propose remedial actions. The out-of-control point or points for which remedial actions have been taken are then deleted, and the revised process average is calculated from the remaining number of samples.

   The investigation of special causes must be conducted in an objective manner. If a special cause cannot be identified or a remedial action not implemented for an out-of-control sample point, that sample is not deleted in the calculation of the new average proportion nonconforming.

   The revised centerline and control limits are given by

   (8-10)

   For p-charts based on a standard p₀, the revised limits do not change from those given by eq. (8-9).

2. Step 6: Implement the chart. Use the revised centerline and control limits of the p-chart for future observations as they become available. Periodically revise the chart using guidelines similar to those discussed for variable charts.

A few unique features are associated with implementing a p-chart. First, if the p-chart continually indicates an increase in the value of the average proportion nonconforming, management should investigate the reasons behind this increase rather than constantly revising upward the centerline and control limits. If such an upward movement of the centerline is allowed to persist, it will become more difficult to bring down the level of nonconformance to the previously desirable values. Only if you are sure that the process cannot be maintained at the present average level of nonconformance given the resource constraints should you consider moving the control limits upward.

Possible reasons for an increased level of nonconforming items include a lower incoming quality from vendors or a tightening of specification limits. An increased level of nonconformance can also occur because existing limits are enforced more stringently.

Since the goal is to seek quality improvement continuously, a sustained downward trend in the proportion nonconforming is desirable. Management should revise the average proportion-nonconforming level downward when it is convinced that the proportion nonconforming at the better level can be maintained. Revising these limits provides an incentive not only to maintain this better level but also to seek further improvements.

Variable Sample Size

There are many reasons why samples vary in size. In processes for which 100% inspection is conducted to estimate the proportion nonconforming, a change in the rate of production may cause the sample size to change. A lack of available inspection personnel and a change in the unit cost of inspection are other factors that can influence the sample size.

A change in sample size causes the control limits to change, although the centerline remains fixed. As the sample size increases, the control limits become narrower. As stated previously, the sample size is also influenced by the existing average process quality level. For a given process proportion nonconforming, the sample size should be chosen carefully so that there is ample opportunity for nonconforming items to be represented. Thus, changes in the quality level of the process may require a change in the sample size.

Control Limits for Individual Samples

Control limits can be constructed for individual samples. If no standard is given and the sample average proportion nonconforming is , the control limits for sample i with size n_i are

(8-11)

Standardized Control Chart

Another approach to varying sample size is to construct a chart of normalized or standardized values of the proportion nonconforming. The value of the proportion nonconforming for a sample is expressed as the sample's deviation from the average proportion nonconforming in units of standard deviations. In Chapter 4 we discussed the sampling distribution of a sample proportion nonconforming . It showed that the mean and standard deviation of are given by

(8-12)

(8-13)

respectively, where p represents the true process nonconformance rate and n is the sample size. In practice, p is usually estimated by , the sample average proportion nonconforming. So, in the working versions of eqs. (8-12) and (8-13), p is replaced by .

The standardized value of the proportion nonconforming for the ith sample may be expressed as

(8-14)

where n_i is the size of ith sample. One of the advantages of a standardized control chart for the proportion nonconforming is that only one set of control limits needs to be constructed. These limits are placed ±3 standard deviations from the centerline. Additionally, tests for runs and pattern recognition are difficult to apply to circumstances in which individual control limits change with each sample. They can, however, be applied in the same manner as with other variable charts when a standardized p-chart is constructed. The centerline on a standardized p-chart is at 0, the UCL is at 3, and the LCL is at –3.

Risk-Adjusted p-Charts in Health Care

In health care applications, situations arise where the proportion of adverse outcomes is to be monitored. Examples include the proportion of mortality in patients who undergo surgery for heart disease, proportion of morbidity in patients undergoing cardiac surgery, and proportion of Caesarian sections for newborn deliveries in a hospital. Uniquely different from the manufacturing or service sector, the product units studied (in this case patients) are not homogeneous. Patients differ in their severity of illness on admission. This inherent difference between patients could impact the observed adverse outcomes, which are above and beyond process-related issues. Hence, an adjustment needs to be made in the construction of p-charts that incorporates such differences between patients in impacting the risk of adverse outcomes.

Several measures could be used for the development of risk-adjusted p-charts. One method of stratification of risk for open-heart surgery results in adults is through the use of a Parsonnet score (Parsonnet et al. 1987). For patients on admission to a hospital, the Parsonnet score is calculated based on numerous patient characteristics such as age, gender, level of obesity, degree of hypertension, presence of diabetes, level of reoperation, dialysis dependency, and other rare circumstances such as paraplegia, pacemaker dependency, and severe asthma, among others. The higher the Parsonnet score, the more the severity of risk associated with the patient. A typical range of the Parsonnet scores of patients could be 0–80. Using such patient-related factors, a logistics regression model (see Chapter 13) is used to predict postoperative mortality or morbidity. Such a prediction will vary from patient to patient based on the severity of risk related to the particular patient.

As a simple model, the predicted mortality () for patient j in subgroup i, based on severity of illness, could be given by

(8-15)

where a and b are estimated regression coefficients and PS_ij represents the Parsonnet score of the particular patient. Hence, the centerline and the control limits on the risk-adjusted p-chart will be influenced by the severity of risk of the patients. The observed mortality or morbidity rate will be compared to such risk-adjusted control limits (expected rate) to make an inference on the performance of the process; in this case it could be for a selected health care facility or a chosen surgeon.

Suppose that there are i subgroups, across time, each of size denoted by n_i patients, where i = 1, 2, …, g. As previously stated, the subgroup sizes should be large enough to observe adverse outcomes. The expected number of adverse outcomes, say mortality, should be at least five for each subgroup. Based on the Parsonnet score for patient j in subgroup i, let the predicted (or expected) mortality, based on the patient's identified risk prior to the operation, be denoted by . This is usually found from a logistic regression model that utilizes the Parsonnet score of the patient. For an individual patient j in subgroup i, the variance in the predicted mortality is given by

(8-16)

The predicted (or expected) mortality for all patients in subgroup i, based on the severity of illness, which also represents the centerline for the ith subgroup, is

(8-17)

The variance in predicted mortality for all patients in subgroup i, assuming that the patients are independent of each other, is given by

(8-18)

Hence, for large sample sizes, the control limits for the predicted proportion of mortality, for subgroup i, are given by

(8-19)

where z_α/2 represents the standard normal variate based on a chosen type I error (or false-alarm rate) of α.

Note that for each patient j, in subgroup i, the observed value of the adverse event (say, mortality) is

(8-20)

For subgroup i, the observed mortality is

(8-21)

and the corresponding proportion of mortality is

(8-22)

On a risk-adjusted p-chart, for each subgroup, the observed proportion of mortality, as given by eq. (8-22), is compared to the control limits, based on the predicted risk of individual patients in that subgroup, given by eq. (8-19). If the observed proportion of mortality falls above the upper control limit, one would look for special causes in the process (i.e., the health care facility, its staff, or the surgeon) to determine possible remedial actions that might reduce the mortality proportion. On the other hand, for a surgeon who handles high-risk patients, if the observed mortality proportion is below the lower control limit, this indicates better performance than expected when adjustment is made for the risk of severity in illness associated with the patients.

It is easy to ascertain the importance of risk adjustment in the health care environment. If an ordinary p-chart were constructed to monitor proportion of mortality, a high observed proportion of mortality may indeed be a reflection of the criticality of illness of the patients and not necessarily a measure of the goodness of the facility and its processes or the surgeon and the technical staff. A wrong inference could be made on the process. A risk-adjusted p-chart, on the other hand, makes an equitable comparison between the observed proportion of mortality and the predicted proportion, wherein an adjustment is made to incorporate the severity of illness of the patients.

Special Considerations for p-Charts

No Standard Given

The centerline for an np-chart is given by

(8-23)

where x_i represents the number nonconforming for the ith sample, g is the number of samples, n is the sample size, and is the sample average proportion nonconforming. Since the number of nonconforming items is n times the proportion nonconforming, the average and standard deviation of the number nonconforming are n times the corresponding value for the proportion nonconforming. Thus, the standard deviation of the number nonconforming is

(8-24)

The control limits for an np-chart are

(8-25)

If the lower control limit calculation yields a negative value, it is converted to zero.

Standard Given

Let's suppose that a specified standard for the number of nonconforming items is np₀. The centerline and control limits are given by

(8-26)

No Standard Given

The average number of nonconformities per sample unit is found from the sample observations and is denoted by . The centerline and control limits are

(8-28)

If the lower control limit is found to be less than zero, it is converted to zero.

Standard Given

Let the specified goal for the number of nonconformities per sample unit be c₀. The centerline and control limits are then calculated from

(8-29)

Many of the special considerations discussed for p-charts, such as observations below the lower control limit, comparison with a specified standard, impact of design specifications, and information about overall quality level, apply to c-charts as well.

Probability Limits

The c-chart is based on the Poisson distribution. Thus, for a chosen level of type I error [i.e., the probability of concluding that a process is out of control when it is in control (a false alarm)], the control limits should be selected using this distribution.

The 3σ limits, shown previously, are not necessarily symmetrical. This means that the probability of an observation falling outside either control limit may not be equal. Appendix A-2 lists cumulative probabilities for the Poisson distribution for a given mean. Suppose that the process mean is c₀ and symmetrical control limits are desired for type I error of α. The control limits should then be selected such that

(8-30)

Since the Poisson distribution is discrete, the upper control limit is found from the smallest integer x⁺ such that

(8-31)

Similarly, the lower control limit is found from the largest integer x⁻ such that

(8-32)

Applications in Health Care When Nonoccurrence of Nonconformities Are Not Observable

In certain health care applications, if no nonconformities or defects occur, the outcome is not observable. Examples could be the number of needle sticks, the number of medical errors, or the number of falls, where it is assumed that the sample size or area of opportunity remains constant from sample to sample.

Assuming that the number of occurrences of nonconformities still follows the Poisson distribution, in this situation, the value X = 0 is not observable. If λ represents the constant rate of occurrence of nonconformities, we have, for the positive or zero-truncated Poisson distribution, the probability mass function given by

(8-33)

It can be shown that the mean and variance of X are given by

(8-34)

(8-35)

respectively. Hence, if a sample of n independent and identically distributed observations X₁, X₂, … X_n is observed, the maximum-likelihood estimator () of λ is obtained from

(8-36)

Equation 8-36 may be solved numerically to obtain an estimate , given by

(8-37)

An approximate bound for , given by Kemp and Kemp (2012), is

(8-38)

Once an estimate has been obtained, the center line on a c-chart is

(8-39)

Probability limits, using previously described concepts and eq. (8-33) with found from eq. (8-37), may be found for a chosen false-alarm rate α. The upper control limit is found as the smallest integer x⁺ such that

(8-40)

The lower control limit is found as the largest integer x⁻, such that

(8-41)

Variable Sample Size and No Specified Standard

When the sample size varies, the number of nonconformities per unit for the ith sample is given by

(8-42)

where c_i is the number of nonconformities in the ith sample and n_i is the size of the ith sample. Note that the sample size n_i need not always be an integer. Let's suppose the number of weaving nonconformities is being counted from finished pieces of cloth such that 100 m² represents 1 unit. If three samples of 250, 100, and 350 m² are inspected, the corresponding values of the sample size are 2.5, 1, and 3.5 units, respectively.

The average number of nonconformities per unit , which is also the centerline of a u-chart, is given by

(8-43)

The control limits are given by

(8-44)

It can been seen from eq. (8-44) that the control limits draw closer as the sample size increases. The same behavior is observed for p-charts for variable sample sizes. Thus, the options discussed previously for p-charts with variable sample sizes apply to u-charts as well. Equation (8-44) provides control limits that vary based on each sample size. Note that if n_i = 1, all formulas for u-charts equal those for c-charts.

Risk-Adjusted u-Charts in Health Care

As discussed in the context of p-charts, risk adjustments may be made in creating c-charts or u-charts since patients differ in their severity of illness. An example could be the occurrence of pressure ulcers in hospitalized patients. When the area of opportunity for occurrence of nonconformities, say, for example, patient-days, remains constant from one subgroup to another, the c-chart may be utilized. However, if the area of opportunity varies from subgroup to subgroup, the u-chart is appropriate. The procedure followed could be similar to the risk-adjusted p-chart, where for each patient the expected number of nonconformities (say pressure ulcers) is computed based on the risk level for that particular patient.

Based on patient characteristics such as mobility, the ability to change and control body position, the degree of physical activity, sensory perception indicating the ability to respond to pressure-related discomfort, tissue tolerance consisting of intrinsic factors such as nutrition and arterial pressure, and extrinsic factors such as moisture, friction, and shear, a pressure score such as the Braden score (Bergstrom et al. 1994; Braden and Bergstrom 1987) could be developed. The Braden score ranges from 6 to 23, with lower values implying a higher risk for development of pressure ulcers. Typically, values above 18 represent low risk, values between 15 and 18 represent mild risk, values between 13 and 14 represent moderate risk, values between 10 and 12 represent high risk, and values ≤9 represent very high risk. Utilizing a logistic regression model (see Chapter 13), the predicted probability or rate of nonconformance () for patient j in subgroup i could be given by

(8-45)

where a and b are estimated regression coefficients and BS_ij represents the Braden pressure score for that particular patient. The value , estimated from eq. (8-45), is therefore the risk-adjusted probability for the particular patient.

The centerline and the control limits for each subgroup i will be influenced by the severity of risk of the patients in that subgroup. Suppose that there are i subgroups, across time, each consisting of n_i patients, where i = 1, 2, …, g. To demonstrate the computations, suppose the number of days spent in the health care facility by patient j in subgroup i is n_ij. The expected number of nonconformities, for example, pressure ulcers, is given by

(8-46)

assuming that the occurrence of nonconformities for that patient is influenced by the number of days spent in the facility by that patient. Hence, for subgroup i, the centerline representing the expected number of nonconformities per unit, accounting for the severity of risk for each patient, is given by

(8-47)

where

(8-48)

represents the area of opportunity, for example, total number of days spent by all the patients in subgroup i, that is, the size of the subgroup. The variance of that incorporates an individual patient's risk is given by

(8-49)

For large subgroup sizes, approximate control limits for subgroup i are given by

(8-50)

where z_α/2 represents the standard normal variate for a tail area of α/2, where α is the chosen level of type I error. The observed number of nonconformities per unit for subgroup i is obtained from

(8-51)

where c_ij represents the observed number of nonconformities for patient j in subgroup i. For subgroup i, u_i is compared to the risk-adjusted control limits given by eq. (8-50) and an inference is made on that subgroup.

Classification of Nonconformities

Several systems classify nonconformities according to their degree of seriousness. A defect that causes severe injury compared to that which may lead to minor problems in the functioning of a product is obviously of different degrees of importance. Using this analogy, defects may be classified into the categories of critical, serious, major, or minor as an example. The definition for each of these categories will be influenced by the product/service and the user who makes a determination of the severity of each type of defect.

Once a classification of defects or nonconformities has been established, demerits per unit are assigned to each class. Control charts are then constructed for demerits per unit. The definitions of the classes are not rigid; users adapt them as they see fit. The classification system mentioned here is but one example. The number of categories and the definitions of each should relate specifically to the problem environment. One organization may have three categories of nonconformities—critical, major, and minor—each with its own definitions. Another organization may define nonconformities as either serious or not serious. The assigned weights for defects from each category is user dependent. For example, a weight system of 100, 50, 10, and 1 could be chosen for the categories of very serious, serious, major, and minor, respectively.

Construction of a U-Chart a

Suppose that we have four categories of nonconformities. In general, the procedure described in this section can be applied to any given number of categories. Let the sample size be n, and let c₁, c₂, c₃, and c₄ denote the total number of nonconformities in a sample for the four categories. Let w₁, w₂, w₃, and w₄ denote the weights assigned to each category. We assume that nonconformities in each category are independent of defects in the other categories. We'll also assume that the occurrence of nonconformities in any category is represented by a Poisson distribution. The applicability of a Poisson distribution to such circumstances is discussed in Section 8.7 (c-charts).

For a sample of size n, the total number of demerits is given by

(8-52)

The demerits per unit for the sample are given by

(8-53)

where the quantity U is a linear combination of independent Poisson random variables. The centerline of the U-chart is given by

(8-54)

where represent the average number of nonconformities per unit in their respective classes. Computation of is similar to the calculation of discussed in Section 8.7. If the control limits are based on standard values, those values (say ) should be substituted for in each category.

The estimated standard deviation of U is given by

(8-55)

The control limits for the U-chart are given by

(8-56)

If the lower control limit is calculated to be less than zero, it is converted to zero.

Transformation to Normality

Suppose we assume that the occurrence of nonconforming items or nonconformities is modeled by a Poisson distribution with a constant rate of occurrence. It is known that the times between the occurrences of events (in this case, between the occurrence of nonconforming items or nonconformities) are independent and exponentially distributed. The number of conforming items produced between occurrences of nonconforming items will be the variable to be monitored. This variable, which has an exponential distribution, can be transformed into a Weibull distribution, which resembles a normal distribution. Denoting X_i as the number of conforming items produced between successive nonconforming items, it has been shown (Nelson 1994a,b) that the power transformation

(8-57)

yields values of Y that are approximately normally distributed. Hence, an individuals and moving-range chart for the Y-values could be monitored. If we denote the mean of Y by and the mean of the moving ranges of the Y-values with a window of 2 by , the centerline and control limits on an individuals chart for Y are (for n = 2, d₂ = 1.128)

(8-58)

Note that values of Y above the upper control limit indicate an improvement in quality, while values of Y below the lower control limit indicate a deterioration in quality.

Use of Exponential Distribution for Continuous Variables

When the process under consideration is continuous, an alternative under the assumption that the occurrence of nonconformities conforms to a Poisson process with mean λ is to monitor the time or number of items required (Q) to observe exactly one nonconformity. It is known that the distribution of Q is exponential with parameter λ (mean = 1/λ). Hence, the probability of observing Q units in order to observe a defect is

(8-59)

Probability limits may be calculated based on a chosen type I error rate, α. Here, because the exponential distribution is not symmetrical, probability limits rather than 3σ limits for Q are preferred. The centerline and control limits are given as

(8-60)

(8-61)

(8-62)

In monitoring Q, which is either the time or number of items to observing a nonconformity, we can detect process improvement as well as deterioration. When Q > UCL, a likely improvement has taken place. Any time that a nonconformity is detected, the quantity Q is reset to zero, to keep track of the subsequent conforming items prior to a nonconformity being observed. When the process mean is not known, it may be estimated from prior samples as the average of observed Q values.

Use of Geometric Distribution for Discrete Variables

For a highly conforming process, nonconforming items are few and far between. This implies that monitoring the number of nonconforming items will yield most observations with values of zero, which prevents the detection of a change in the nonconformance rate. An alternative is to monitor the number of items until a nonconforming item is found, sometimes referred to as the number of trials up to the first success (X). When a nonconforming item is found, the count starts anew. The discrete variable X, as defined, has a geometric distribution given by

(8-63)

where p represents the probability of success (nonconforming item) on each trial. The mean and variance of X, which counts the number of trials to the first success, including the trial when the success occurs, are given by

(8-64)

Probability Limits

Let us assume that the probability of a type I error (α) will be divided equally on both sides of the control limits. The centerline and control limits for the count to a nonconforming item are given by (Xie et al. 2002)

(8-65)

These control limits are highly asymmetric, and a logarithmic scale for the vertical axis for monitoring X is recommended. Detection of improvement in the process is usually associated with a value of X falling above the UCL.

A bound can be established for the minimum sample size (n) necessary to detect improvement if the current level of proportion nonconforming is p. The probability of no nonconforming items in a sample of size n is

(8-66)

Assuming that a one-sided limit is being used to detect improvement, with a type I error of α, we have a lower bound for n given by

(8-67)

Applications in Health Care of Low-Occurrence Nonconformities

In certain health care applications, the rate of occurrence of nonconformities such as surgical wound infections, pneumonia, catheter infections, or gastrointestinal infections may be quite low. In such situations, the variable to be monitored is the number of events or surgeries or days between infections, not counting the day on which the next infection occurs. Such a random variable is modeled by the geometric distribution, whose probability mass function is given by

(8-68)

where p represents the probability of “success” (or a nonconforming item) on each trial (such as surgeries). The mean and variance of the random variable X are given by

(8-69)

Since the distribution of the random variable X for small values of p is quite asymmetric, rather than create control limits that are equidistant from the centerline, such limits are calculated using the concept of probability limits, discussed earlier, based on a chosen level of false alarm, α. The cumulative distribution function of X is given by

(8-70)

The upper and lower control limits, using eq. (8-70), are obtained as

(8-71)

(8-72)

In the event that a target occurrence rate (p₀) of a nonconforming item is specified, the centerline is

(8-73)

and the control limits are found from eqs. 8-71 and 8-72 by using the specified value p₀. When such a target value is not specified, it is usually estimated from sample values of the number of cases or time between successive occurrences of the nonconforming item, for example, infection. Here, an estimate of p is obtained, based on , the average time or cases between successive infections, say, and is given by

(8-74)

Control limits may be found using eqs. 8-71 and 8-72, where the estimate is used. A control chart for the time or cases between successive occurrence of a nonconforming item is often referred to as a g -chart. A point above the upper control limit may signal an improvement in the process.

Discussion Questions

1. 8-1 Distinguish between a nonconformity and a nonconforming item. Give examples of each in the following contexts:
   1. Financial institution
   2. Hospital
   3. Microelectronics manufacturing
   4. Law firm
   5. Nonprofit organization
2. 8-2 What are the advantages and disadvantages of control charts for attributes over those for variables?
3. 8-3 Discuss the significance of an appropriate sample size for a proportion-nonconforming chart.
4. 8-4 The CEO of a company has been charged with reducing the proportion nonconforming of the product output. Discuss which control charts should be used and where they should be placed.
5. 8-5 How does changing the sample size affect the centerline and the control limits of a p-chart?
6. 8-6 What are the advantages and disadvantages of the standardized p-chart as compared to the regular proportion-nonconforming chart?
7. 8-7 Discuss the assumptions that must be satisfied to justify using a p-chart. How are they different from the assumptions required for a c-chart?
8. 8-8 Is it possible for a process to be in control and still not meet some desirable standards for the proportion nonconforming? How would one detect such a condition, and what remedial actions would one take?
9. 8-9 Discuss the role of the customer in influencing the proportion-nonconforming chart. How would the customer be integrated into a total quality systems approach?
10. 8-10 Discuss the impact of the control limits on the average run length and the operating characteristic curve.
11. 8-11 Explain the conditions under which a u-chart would be used instead of a c-chart.
12. 8-12 Explain why a p- or c-chart is not appropriate for highly conforming processes.
13. 8-13 Distinguish between 3σ limits and probability limits. When would you consider constructing probability limits?
14. 8-14 Meeting customer due dates is an important goal. What attribute or variables control charts would you select to monitor? Discuss the underlying assumptions in each case.
15. 8-15 Explain the setting under which a U-chart would be used. How does the U-chart incorporate the user's perception of the relative degree of severity of the different categories of defects?
16. 8-16 Which type of control chart (p, np, c, u, U, or charts for highly conforming processes) is most appropriate to monitor the following situations?
    1. Number of potholes in highways
    2. Proportion of customers who are satisfied with the operation of the local housing authority
    3. Satisfaction of customers at a restaurant where customers consider the quality of food and attitude of the server to be more important than the décor
    4. Number of errors in account transactions at a bank where the number of accounts varies from month to month
    5. Number of automobile accident claims filed per month at an insurance dealer, assuming a stable number of insured persons
    6. Responsiveness to customer needs of a local library where customers value the longer hours over short lines to check out books
    7. Needlestick in patients in a hospital
    8. Loan defaults in a financial institution in a year
    9. Number of surgical errors in a health care facility per month
    10. Number of thefts in a city over a period of time
    11. Proportion of people that favor the construction of condominiums in a certain neighborhood
    12. Number of weld defects in the construction of aircraft
    13. Number of seeds that germinate from a large pack of seeds sold by a nursery when the number of seeds in a packet may vary
    14. Performance of the braking mechanism of a certain model of automobile (the car is expected to stop within a certain distance)
    15. Number of firemen to have on duty to provide an acceptable level of service (each fire usually requires four firemen; the department has data on the number of calls they receive during the specified period of time)
    16. Performance of data-entry operators (if one or more input errors are made, the file is rendered useless and a revision is necessary)
    17. Control of the number of typographical errors per page at a document preparation center (data are chosen randomly and obtained daily on the number of errors in 30 pages)
    18. Control of the wiring and transistor defects in an electronic component (wiring defects are considered more serious)
    19. Number of traffic accidents in a city per month
    20. Proportion of successful patent applications by a drug manufacturer
    21. Time between surgical errors in a health care facility
    22. Number of in-patients who develop catheterization infections
    23. Proportion of orders not shipped on time

Problems

1. 8-17 Every employee in a check-processing department goes through a four-month training period, after which the employee is responsible for their operation. The work of one employee who has been on the job for eight months is being studied. Table 8-20 shows the number of errors and the number of items sampled over a period of two months. The first 16 samples were each chosen from 400 items, and the remaining 9 samples were each chosen from 300 items. Determine whether the employee's performance can be judged stable. Comment on the capability of the employee.

   Sample	Errors	Items Sampled	Sample	Errors	Items Sampled
   1	12	400	14	18	400
   2	9	400	15	8	400
   3	13	400	16	6	400
   4	7	400	17	4	300
   5	6	400	18	6	300
   6	10	400	19	5	300
   7	14	400	20	8	300
   8	7	400	21	10	300
   9	5	400	22	7	300
   10	6	400	23	4	300
   11	4	400	24	5	300
   12	9	400	25	3	300
   13	11	400

2. 8-18 The number of customers who are not satisfied with the service provided in a retail store is found for 20 samples of size 100 and is shown in Table 8-21. Construct a control chart for the proportion of dissatisfied customers. Revise the control limits, assuming special causes for points outside the control limits.

   Sample	Number of Dissatisfied Customers	Sample	Number of Dissatisfied Customers
   1	2	11	5
   2	5	12	4
   3	4	13	2
   4	3	14	5
   5	4	15	3
   6	2	16	12
   7	3	17	3
   8	2	18	2
   9	4	19	5
   10	11	20	2

3. 8-19 Refer to Exercise 8-18. Management believes that the dissatisfaction rate is 3%, so establish control limits based on this value. Comment on the ability of the store to meet this standard. If management were to set the standard at 2%, can the store meet this goal? What actions would you recommend?
4. 8-20 Health care facilities must conform to certain standards in submitting bills to Medicare/Medicaid for processing. The number of bills with errors and the number sampled are shown in Table 8-22. Construct an appropriate control chart and comment on the performance of the billing department. Revise the control limits, if necessary, assuming special causes for out-of-control points. Comment on the capability of the department.

   Observation	Bills with Errors	Number Sampled	Observation	Bills with Errors	Number Sampled
   1	8	400	14	3	300
   2	6	400	15	5	300
   3	4	400	16	8	300
   4	9	400	17	11	500
   5	7	400	18	13	500
   6	5	400	19	8	500
   7	5	300	20	7	500
   8	7	300	21	8	500
   9	4	300	22	4	500
   10	15	300	23	3	500
   11	6	300	24	7	500
   12	7	300	25	6	500
   13	4	300

5. 8-21 Observations are taken from the output of a company making semiconductors. Table 8-23 shows the sample size and the number of nonconforming semiconductors for each sample. Construct a p-chart by setting up the exact control limits for each sample. Are any samples out of control? If so, assuming special causes, revise the centerline and control limits.

   Observation	ItemsInspected	Nonconforming Items	Observation	ItemsInspected	Nonconforming Items
   1	80	3	14	90	4
   2	120	6	15	160	5
   3	60	4	16	230	3
   4	150	5	17	200	12
   5	140	8	18	150	8
   6	150	10	19	210	6
   7	160	7	20	190	4
   8	90	6	21	160	9
   9	100	5	22	100	8
   10	160	12	23	100	12
   11	110	8	24	90	7
   12	100	5	25	160	10
   13	200	14

6. 8-22 Refer to Exercise 8-21 and the data shown in Table 8-23. Construct a standardized p-chart and discuss your conclusions.
7. 8-23 The quality of service in a hospital is tracked by determining the proportion of medication errors; this is done by dividing the number of medication errors by 1000 patient-days for each observation. The results of 25 such samples (in percentage of medication errors) are shown in Table 8-24. Construct an appropriate control chart and comment on the quality level. Is a goal of error-free performance reasonable to expect from this system?

   Sample	Medication Errors (%)	Sample	Medication Errors (%)
   1	2.6	14	2.0
   2	1.9	15	4.2
   3	2.8	16	2.2
   4	2.9	17	1.8
   5	2.4	18	2.4
   6	1.8	19	2.3
   7	2.3	20	1.6
   8	2.1	21	1.9
   9	1.4	22	2.0
   10	1.7	23	2.2
   11	2.2	24	2.1
   12	2.0	25	2.3
   13	1.2

8. 8-24 A health care facility is interested in monitoring the primary C-section rate. Monthly data on the number of primary C-sections collected over the last two and a half years is shown in Table 8-25.

   Month	C-Sections	Deliveries	Month	C-Sections	Deliveries
   1	54	350	16	70	449
   2	50	384	17	62	366
   3	63	415	18	65	405
   4	69	422	19	52	386
   5	55	395	20	55	392
   6	63	412	21	61	408
   7	67	407	22	66	442
   8	67	415	23	53	426
   9	51	377	24	47	385
   10	62	404	25	79	413
   11	64	377	26	60	388
   12	62	382	27	69	411
   13	62	425	28	61	378
   14	55	410	29	70	392
   15	68	426	30	65	420

   1. Is the process in control?
   2. There is pressure to make these data public. Can we conclude that the C-section rates had shifted to a higher level in the last six months relative to the previous six months?
   3. What is your prediction on the C-section rate if no changes are made in current obstetrics practices?
   4. Based on benchmarking with comparable facilities in similar metropolitan areas, is it feasible currently to achieve a C-section rate of 10%?
9. 8-25 Refer to Exercise 8-18 and the data shown in Table 8-21. Construct a control chart for the number of dissatisfied customers. Revise the chart, assuming special causes for points outside the control limits.
10. 8-26 The number of processing errors per 100 purchase orders is monitored by a company with the objective of eliminating such errors totally. Table 8-26 shows samples that were selected randomly from all purchase orders. The company is in the process of testing the effects of a new purchase order form that it has designed. The last five samples were made using the new form. Construct a control chart that the company can use for monitoring the quality characteristic selected. What is the effect of the newly designed purchase order form? Is the company capable of achieving the desired goal?

    Sample	Processing Errors	Sample	Processing Errors
    1	6	14	3
    2	4	15	6
    3	2	16	1
    4	3	17	5
    5	4	18	2
    6	7	19	6
    7	5	20	4
    8	7	21	2
    9	11	22	3
    10	4	23	2
    11	2	24	1
    12	5	25	2
    13	4

11. 8-27 The number of dietary errors is found from a random sample of 100 trays chosen on a daily basis in a health care facility. The data for 25 such samples are shown in Table 8-27.

    Sample Number	Number of Dietary Errors	Sample Number	Number of Dietary Errors
    1	9	14	8
    2	6	15	8
    3	4	16	7
    4	7	17	6
    5	5	18	4
    6	6	19	12
    7	16	20	7
    8	8	21	6
    9	7	22	8
    10	9	23	6
    11	3	24	8
    12	6	25	5
    13	10

    1. Construct an appropriate control chart and comment on the process.
    2. How many dietary errors do you predict if no changes are made in the process?
    3. Is the system capable of reducing dietary errors to 2, on average, per 100 trays, if no changes are made in the process?
12. 8-28 A building contractor subcontracts to a local merchant a job involving hanging wallpaper. To have an idea of the quality level of the merchant's work, the contractor randomly selects 300 m² and counts the number of blemishes. The total number of blemishes for 30 samples is 80. Construct the centerline and control limits for an appropriate chart. Is it reasonable for the contractor to set a goal of an average of 0.5 blemish per 100 m²?
13. 8-29 The number of imperfections in bond paper produced by a paper mill is observed over a period of several days. Table 8-28 shows the area inspected and the number of imperfections for 25 samples. Construct a control chart for the number of imperfections per square meter. Revise the limits, if necessary, assuming special causes for the out-of-control points.

    Sample	Area Inspected (m2)	Imperfections	Sample	Area Inspected (m2)	Imperfections
    1	150	6	14	300	8
    2	100	8	15	300	12
    3	200	5	16	200	6
    4	150	4	17	150	4
    5	250	10	18	200	7
    6	100	11	19	150	14
    7	150	3	20	100	4
    8	200	5	21	100	8
    9	300	10	22	200	9
    10	250	10	23	300	12
    11	100	5	24	250	7
    12	200	4	25	200	5
    13	250	12

14. 8-30 Refer to Exercise 8-29. If we want to control the number of imperfections per 100 m², how would this affect the control chart? What would the control limits be? In terms of decision making, would there be a difference between this problem and Exercise 8-29, depending on which chart is constructed? What conclusions can you draw from this?
15. 8-31 The director of a pharmacy department is interested in benchmarking the level of operations in the unit. The director has defined medication errors as being any one of the following: wrong medication; wrong dose; administered to the wrong patient; administered at the wrong time; incorrectly repeating the medication; or omitting the medication. The number of orders filled per day by the pharmacy varies. Table 8-29 shows the number of orders filled and the number of medication errors for 25 days. Construct an appropriate control chart and comment on the stability of the process.

    Sample Number	Number of Orders Filled	Number of Medication Errors	Sample Number	Number of Orders Filled	Number of Medication Errors
    1	1200	11	14	1200	16
    2	1160	10	15	1150	14
    3	1210	12	16	1100	23
    4	1300	9	17	1160	14
    5	1120	10	18	1300	16
    6	1150	12	19	1100	10
    7	1100	14	20	1180	12
    8	1320	12	21	1220	14
    9	1240	10	22	1240	13
    10	1180	15	23	1120	16
    11	1140	4	24	1150	13
    12	1120	13	25	1180	12
    13	1220	7

16. 8-32 Nonconformities in automobiles fall into three categories: serious, major, and minor. Twenty-five samples of five automobiles are chosen, and the total number of nonconformities in each category is reported. Table 8-30 shows the results. Assuming a weighting system of 50, 10, and 1 for serious, major, and minor nonconformities, respectively, construct a demerits per unit control chart. Revise the control limits if necessary, assuming special causes for points that are out of control.

    Sample	Serious Defects	Major Defects	Minor Defects	Sample	Serious Defects	Major Defects	Minor Defects
    1	0	5	8	14	0	7	12
    2	0	3	2	15	0	2	8
    3	1	0	6	16	0	4	3
    4	1	2	1	17	1	0	5
    5	0	6	8	18	0	3	2
    6	0	3	3	19	0	5	8
    7	0	1	10	20	0	2	6
    8	1	2	5	21	1	1	4
    9	0	4	9	22	0	3	10
    10	2	6	6	23	0	2	12
    11	1	3	2	24	0	4	7
    12	0	5	8	25	0	2	4
    13	0	0	9

17. 8-33 Refer to Exercise 8-32. If the weighting systems were different (i.e., 10, 5, and 1), how would the centerline and the control limits change for the U-chart? Discuss changes, if any, in the inferences made about the process.
18. 8-34 The Joint Commission on Accreditation of Healthcare Organizations (JCAHO) requires an accounting of significant medication errors. Data collected over the last 25 months, shown in Table 8-31, indicate the number of orders filled and the number of significant medication errors. Each order is classified either as having or as not having significant medication errors.

    Sample	Orders Filled	Significant Errors	Sample	Orders Filled	Significant Errors
    1	80	3	14	90	4
    2	120	6	15	160	5
    3	60	4	16	230	3
    4	150	5	17	200	12
    5	140	8	18	150	8
    6	150	10	19	210	6
    7	160	7	20	190	4
    8	90	6	21	160	9
    9	100	5	22	100	8
    10	160	12	23	100	12
    11	110	8	24	90	7
    12	100	5	25	160	10
    13	200	14

    1. Construct an appropriate control chart and comment on the process.
    2. If the process is not in control, assuming special causes and appropriate remedial actions, revise the centerline and control limits.
    3. What is your level of expectation from this process?
    4. What would you do if the goal is to reduce the proportion of significant medication errors to 1%. Is this currently achievable?
19. 8-35 Refer to Exercise 8-18. Construct an OC curve for the p-chart. If the process proportion of dissatisfied customers were to rise to 7%, what is the probability of not detecting this shift on the first sample drawn after the change has taken place? What is the probability of detecting the shift by the third sample?
20. 8-36 Refer to Exercise 8-27. Construct an OC curve for the c-chart. If the process average number of dietary errors per 100 trays increases to 10, what is the probability of detecting this on the first sample drawn after the change?
21. 8-37 Refer to Exercise 8-36. Set up 2σ control limits. What is the probability of detecting a change in the process average number of dietary errors per 100 trays to 8 on the first sample drawn after the change? Explain under what conditions you would prefer to have these 2σ control limits over the traditional 3σ limits.
22. 8-38 The number of heart surgery complications is rare. To monitor the effectiveness of such surgeries, data are recorded on the number of such procedures until a complication occurs. These complications occur independently with a constant probability of occurrence and follow a geometric distribution. Table 8-32 shows such data for a sequence of 25 complications. It is estimated from past data that the complication rate is 0.1%. Construct an appropriate control chart and comment on the process assuming a type I error rate of 0.005. What would the control limits be for a type I error rate of 0.05? What factors would influence your selection of the type I error rate?

    Complication Sequence	Number of Procedures	Complication Sequence	Number of Procedures	Complication Sequence	Number of Procedures
    1	654	10	1654	18	1794
    2	981	11	892	19	1112
    3	1508	12	750	20	652
    4	436	13	1333	21	1050
    5	1202	14	1404	22	1085
    6	889	15	909	23	1422
    7	1854	16	822	24	688
    8	3068	17	1609	25	1095
    9	704

23. 8-39 Consider Exercise 8-38. If you were interested in detecting an improvement in the process using a one-sided limit, what should the minimum sample size be for an α of 0.005? What should it be for an α of 0.05? What conclusions can you draw from these results?
24. 8-40 Consider Exercise 8-38 under the assumption that the complication rate is 0.1%. If you were to construct a p-chart using two-sided 3σ limits, what would the minimum sample size be to detect an improvement in the process?
25. 8-41 Consider Exercise 8-38. Determine the sensitivity of the control limits on the complication rate using values of 0.2% and 0.5%.
26. 8-42 Consider Exercise 8-38 under the assumption that the complication rate is 0.1% and a type I error of 0.005. If you reduced the upper control limit to half of its previous value, what type of process complication rates, on average, will this new limit be able to detect improvements?
27. 8-43 Consider Exercise 8-38. However, now assume that the interval between complications follows an exponential distribution. Construct an appropriate control chart and comment on the process assuming a type I error rate of 0.005.
28. 8-44 The JCAHO has standards pertaining to patient restraint use. A checklist has been developed that is to be used each time a restraint is used. The checklist contains five items, all of which should be checked. Table 8-33 shows data collected for 25 months that indicate the number of patients restrained and the number of items that were not checked prior to restraint use.

    Sample	Number Restrained	Number of Items Not Checked	Sample	Number Restrained	Number of Items Not Checked
    1	100	4	14	40	3
    2	50	3	15	80	2
    3	80	8	16	80	4
    4	60	4	17	120	6
    5	120	14	18	100	8
    6	100	8	19	120	10
    7	80	4	20	80	5
    8	100	5	21	60	8
    9	60	2	22	100	4
    10	80	12	23	100	5
    11	100	7	24	80	4
    12	120	5	25	120	5
    13	60	15

    1. Is the process stable?
    2. If not, assuming identifiable remedial actions with the special causes, revise the centerline and control limits.
    3. What is your expectation from this process?
29. 8-45 Mortality data for patients in the intensive care unit (ICU) in a health care facility, collected over 8-hour intervals, are shown in Table 8-34. Since patients have varying risks, the predicted mortality from a logistic regression model that uses the patient's APACHE score (severity of disease) is also shown. Construct a risk-adjusted chart for the mortality proportion and comment on the ICU performance.

    Summary	Subgroup Number
    	1	2	3	4	5	6
    Number of deaths	2	1	2	2	1	1
    Number of patients	8	6	8	7	8	8
    Patient	Predicted Mortality Based on Severity of Disease
    1	0.5	0.6	0.3	0.4	0.7	0.6
    2	0.8	0.5	0.7	0.6	0.3	0.4
    3	0.4	0.8	0.5	0.5	0.6	0.8
    4	0.6	0.4	0.5	0.8	0.5	0.6
    5	0.8	0.6	0.5	0.4	0.7	0.7
    6	0.3	0.8	0.6	0.6	0.4	0.6
    7	0.5		0.8	0.3	0.5	0.8
    8	0.4		0.8		0.4	0.7

30. 8-46 In a health care facility, it is known that nosocomial infections are rare. Table 8-35 shows the number of admissions between occurrences of nosocomial infections. Construct an appropriate control chart and comment on the performance of the health care team.

    Infection Sequence	Number of Admissions	Infection Sequence	Number of Admissions	Infection Sequence	Number of Admissions
    1	120	8	95	15	145
    2	100	9	115	16	160
    3	150	10	135	17	170
    4	140	11	120	18	185
    5	90	12	100	19	190
    6	100	13	90	20	200
    7	110	14	140

31. 8-47 Data from a city bank showing monthly loan applications for home mortgage and the number rejected are given in Table 8-36. Construct an appropriate control chart and comment on the loan approval process.

    Month	Number of Applications	Number Rejected	Month	Number of Applications	Number Rejected
    1	52	3	11	76	5
    2	45	4	12	68	5
    3	68	5	13	70	4
    4	74	4	14	58	4
    5	50	3	15	62	5
    6	66	5	16	65	4
    7	75	4	17	72	3
    8	60	12	18	78	3
    9	55	3	19	70	4
    10	48	4	20	67	4

32. 8-48 An organization that accepts orders via the Web, processes it, and has delivery scheduled based on its inventory levels is known to be proficient in its service. Errors in orders received could be due to wrong shipment of product, incorrect quantity, or not meeting promised delivery date. Given the track record of the organization, Table 8-37 shows the number of error-free orders filled between successive orders that have at least one error. Construct an appropriate control chart and comment on the performance of the organization. Use a level of significance of 0.05.

    Error Sequence	Orders Filled	Error Sequence	Orders Filled	Error Sequence	Orders Filled
    1	150	8	200	15	210
    2	120	9	140	16	240
    3	90	10	165	17	285
    4	140	11	180	18	290
    5	110	12	150	19	320
    6	85	13	155	20	340
    7	160	14	170

33. 8-49 A major airline surveys passengers to monitor the satisfaction level of its customers. Even though the airline uses a five-point Likert scale (5: outstanding service; 4: good; 3: average; 2: below average; 1: poor), it wants to monitor the proportion that consider the service level as either outstanding or good. Table 8-38 shows the number of passengers responding to the survey and the number that consider the service as either outstanding or good. Construct an appropriate control chart and comment on the service level of the airline.

    Week	Number Responded	Number Checking “Outstanding” or “Good”	Week	Number Responded	Number Checking “Outstanding” or “Good”
    1	80	65	14	135	105
    2	110	80	15	125	90
    3	250	210	16	150	118
    4	150	130	17	100	82
    5	90	70	18	110	75
    6	100	85	19	85	64
    7	140	108	20	140	114
    8	220	202	21	165	122
    9	200	184	22	185	148
    10	180	140	23	130	105
    11	70	55	24	125	114
    12	95	70	25	120	105
    13	115	82

34. 8-50 The marketing department of a large organization is interested in being responsive to its customers' needs. Accordingly, each week it monitors the number of complaints received. It is possible for a customer to have more than one complaint. Table 8-39 shows the feedback from its customers. Construct an appropriate control chart and comment on the level of customer service by the organization.

    Sample	Number Responded	Number of Complaints	Sample	Number Responded	Number of Complaints
    1	75	8	14	90	10
    2	120	12	15	145	18
    3	110	6	16	150	20
    4	135	15	17	124	16
    5	150	12	18	108	12
    6	175	28	19	115	15
    7	80	11	20	88	12
    8	70	14	21	92	14
    9	130	20	22	95	15
    10	105	10	23	120	18
    11	118	26	24	135	6
    12	130	18	25	130	5
    13	125	16