Statistically Based Quality Improvement

Statistically Based Quality Improvement
Chapter 12 Statistically Based Quality Improvement S. Thomas Foster, Jr. Boise State University Slides Prepared by Bruce R. Barringer University of Central Florida ©2001 Prentice-Hall

Chapter Overview Statistical Fundamentals Process Control Charts Some Control Chart Concepts Process Capability Other Statistical Techniques in Quality Management

Introduction Statistics are everywhere.
Statistics are a group of tools that allow us to analyze data, make summaries, draw inferences, and generalize from the data. Statistics are very important in the field of quality. In fact, during the first half century of the quality movement, nearly all the work done in the field of quality related to statistics. It is not enough to learn the different charts and statistical techniques. We also must know how to apply these techniques in a way that will document and motivate continual improvement in organizations.

Statistical Fundamentals Slide 1 of 8
Statistical Thinking Is a decision-making skill demonstrated by the ability to draw to conclusions based on data. Intuitive decisions are sometimes biased and wrong-headed. As a result, decisions are sometimes made that satisfy the few but irritate the many.

Statistical Fundamentals
Statistical thinking is based on three concepts: . All work occurs in a system of interconnected process. . All processes have variation. . Understanding variation and reducing variation are important keys to success.

In business, decisions need to be made based on data. Statistical thinking guides us to make decisions based on the analysis of data ( see Quality Highlight 12.1)

Why Do Statistics Sometimes Fail in the Workplace? Regrettably, many times statistical tools do not create the desired result. Why is this so? Many firms fail to implement quality control in a substantive way. That is they prefer form over substance.

Reasons for Failure of Statistical Tools Lack of knowledge about the tools; therefore, tools are misapplied. General disdain for all things mathematical creates a natural barrier to the use of statistics. Cultural barriers in a company make the use of statistics for continual improvement difficult. Statistical specialists have trouble communicating with managerial generalists.

Reasons for Failure of Statistical Tools (continued) Statistics generally are poorly taught, emphasizing mathematical development rather than application. People have a poor understanding of the scientific method. Organization lack patience in collecting data. All decisions have to be made “yesterday.”

Reasons for Failure of Statistical Tools (continued) Statistics are view as something to buttress an already-held opinion rather than a method for informing and improving decision making. People fear using statistics because they fear they may violate critical statistical assumptions.

Reasons for Failure of Statistical Tools (continued) --Most people don’t understand random variation resulting in too much process tampering. --Statistical tools often are reactive and focus on effect rather than causes.

Reasons for Failure of Statistical Tools (continued) -- Another reason people make mistakes with statistics is founded in the notions of Type Ⅰand Type Ⅱerrors. -- In the study of quality, we call Type Ⅰ error producer’s risk and Type Ⅱ error, consumer’s risk.

Producer’s and Consumer’s risk
Producer’s risk is the probability that a good product will be rejected. Consumer’s risk is the probability that a nonconforming product will be available for sale. Consumer’s risk happens when statistical quality analysis fails to result in the scrapping or reworking of a defective product. When either Type Ⅰ or Type Ⅱ errors occur, erroneous decision are made relative to product which result in high cost or lost future sales.

What do we mean by the term Statistical Quality Control? -- The age of control-oriented management is over. -- Now the focus is on continual improvement – not on process or organization control. -- We prefer to not use the term quality control here. -- We use the term control sparsely in this chapter, such as in control limits.

Understanding Process Variation All process exhibit variation. There is some variation that we can control and other variation that we cannot control. If there is too much variation, process parts will not fit correctly, products will nit function properly, and a firm will gain a reputation for poor quality.

Understanding Process Variation Two types of variation commonly occur. These are random and nonrandom variation. Random variation is uncontrollable and nonrandom variation has a cause that can be identified.

Understanding Process Variation Random variation is centered around a mean and occurs with a consistent amount of dispersion. This type of variation cannot be controlled. Hence, we refer to it as “uncontrolled variation.” When the random variation is large, processes may not meet specifications on a consistent basis.

Understanding Process Variation The statistical tools discussed in this chapter are not designed to detect random variation. Figure 12.1 shows normal distributions resulting from a variety of samples taken from the same population over time. We find a consistency in the amount of dispersion and the mean of the process. The consistency of the variation shows that only random causes of variation are present within the process. This means that in the future when we gather samples from the process, we can expect that the distributions associated with such samples will also take the same form.

Figure 12.1 Random variation

Nonrandom Variation -- Nonrandom or “special cause” variation results from some event. The event may be a shift in a process mean or some unexpected occurrence. -- Figure 12.2 shows distributions resulting from a number of samples taken from the same population over time where nonrandom variation is exhibited. Notice that from one sample to the next, the dispersion, and average of the process are changing. It is clear that nonrandom variation results in a process that is not predictable.

Figure 12.2 Nonrandom Variation

Process Stability Means that the variation we observe in the process is random variation and nor nonrandom variation. To determine process stability we use process charts. Process charts are graphs designed to signal process workers when nonrandom variation is occurring in a process.

Sampling Methods To ensure that processes are stable, data are gathered in samples. For the most part, sampling methods have been preferred to the alternative of 100% inspection.

Sampling Methods -- The reasons for sampling are well established. . Samples are - cheaper, - take less time, - less intrusive, - allow the user to frame the sample, and - destructive testing, when 100% inspection is impossible.

100% inspection -- Recent experience has shown that 100% inspection can be effective in certain instances. -- 100% samples are also known as screening samples, sorting samples, rectifying samples, or detailing samples. -- Another example of 100% inspection is used when performing in-process inspection. -- We should clarify that in-process inspection also can be performed on a sampling basis.

Sampling Methods -- Random samples. Randomization is useful because it ensures independence among observations. To randomize means to sample is such a way that every piece of product has an equal chance of being selected for inspection. -- Random samples are often the preferred form of sampling and yet often the most difficult to achieve. -- This is especially true in process industries were multiple products are made by the same machines, workers, and processes in sequence.

Sampling Methods -- Systematic samples. Systematic samples have some of the benefits of random samples without the difficulty of randomizing. -- Samples can be systematic according to time or according to sequence.

Sampling Methods Sampling by Rational Subgroup. A rational subgroup is a group of data that is logically homogenous; variation within the data can provide a yardstick for setting limits on the standard variation between subgroups. If variation among different subgroups is not accounted for, then an unwanted source of nonrandom variation is being introduced.

Planning for inspection -- Questions must be answered about: - What type of sampling plan will be used? - Who will perform the inspection? - Who use in-process inspection sample size? - What the critical attributes to be inspection are? - Where inspection should be perform? -- There are rules for inspection that help to prioritize where inspection should be performed. Many firms compute the ratio between the cost of inspection and the cost of failure to determine the prioritize.

Process Control Charts Slide 1 of 37
Process Charts SPC charts are tools for monitoring process variation. Figure 12.3 shows a process control chart. It has an upper limit, a center line, and a lower limit. There is a generalized process for implementing all types of process charts.

Figure 12.3 Control Chart

Variables and Attributes To select the proper process chart, we must differentiate between variables and attributes. A variable is a continuous measurement such as weight, height, or volume. An attribute is the result of a binomial process that results in an either-or-situation. Table 12.1 shows the most common types of variable and attribute charts.

Table 12.1 Variables and Attributes Control Chart Variables Attributes X (process population average) P (proportion defective) X-bar (mean for average) np (number defective) R (range) C (number conforming) MR (moving range) U (number nonconforming) S (standard deviation)

There are four central requirements for properly using process charts 1. You must understand the generic process for implementing process charts. 2. You must know how to interpret process charts. 3. You need to know when different process charts are used. 4. You need to know how to compute limits for the different types of process charts.

A Generalized Procedure for Developing Process Charts 1. Identify critical operations in the process where inspection might be needed. These are operations in which, if the operation is performed improperly, the product will be negatively affected. 2. Identify critical product characteristics. These are the attributes of the product that will result in either good or poor function of the product.

A Generalized Procedure for Developing Process Charts (continued) 3. Determine whether the critical product characteristic is a variable or an attribute. 4. Select the appropriate process control chart from among the many types of control charts. This decision process and types of charts available are discussed later. 5. Establish the control limits and use the chart to continually monitor and improve.

A Generalized Procedure for Developing Process Charts (continued) 6. Update the limits when changes have been made to the process.

Understanding Control Charts A process chart is nothing more than an application of hypothesis testing where the null hypothesis is that the product meets requirements. An X-bar chart is a variables chart that monitors average measurement.

Process Control Charts
We want to inspect a sample of sheets to see whether the sheets are indeed 11 inches long. We first use a hypothesis test instead of a control chart to determine whether the paper is really 11 inches long. The null hypothesis is H0 :μ= 11 inches The alternative hypothesis is H1 :μ ≠11 inches We establish a distribution as Figure 12.4 shows the rejection regions.

Figure 12.4 Hypothesis testing

To test this hypothesis we draw a sample of n = 10 sheets of paper and measure the sheets. The measurements are shown in the table shown in page 356 . Because sample mean does not fall within either of the rejection regions shown in Figure 12.4, we fail to reject the null hypothesis and conclude that the sheets do not differ significantly from an average of 11 inches.

With process charts, we place the distribution on its side as shown in Figure 12.5. We draw a central line and upper and lower rejection lines, which we call control limits. We then plot the sample average on the control chart. Because the point falls between the control limits, we conclude that the process is in control This mean that the variation in the process is random.

Figure 12.5 process control chart

The Central Limit Theorem ( CLT) states that when we plot the sample means, the distribution approximates a normal distribution.

X-bar and R Charts When we are interested in monitoring a measurement for a particular product in a process, there are two primary variables of interest: the mean of the process or average and the dispersion of the process. The x-bar chart aids us in monitoring the process mean or average. The R chart is used in monitoring process dispersion.

X-bar and R Charts The X-bar chart is a process chart used to monitor the average of the characteristics being measured. To set up an X-bar chart select samples from the process for the characteristic being measured. Then form the samples into rational subgroups. Next, find the average value of each sample by dividing the sums of the measurements by the sample size and plot the value on the process control X-bar chart.

X-bar and R Charts (continued) The R chart is used to monitor the variability or dispersion of the process. It is used in conjunction with the X-bar chart when the process characteristic is variable. To develop an R chart, collect samples from the process and organize them into subgroups, usually of three to six items. Next, compute the range, R, by taking the difference of the high value in the subgroup minus the low value. Then plot the R values on the R chart. A standard process chart form is shown in Figure 12.6.

Process Control Charts Slide 12 of 37 Figure 12.6 X-bar and R Charts

In the example in Figure 12.7, our control chart form is filled out with measurements from a process ( There are k = 25 samples of size n = 4). The formulas for computing control lines are given in Figure Figure 12.9 shows the completed formulas for the example in Figure 12.7. The lower limit of R is zero for sample sizes less than or equal to six. For sample size greater than six, D3 values must be used from Table A-1 in the appendix (page 449).

Figure 12.7 Completed x-bar and R chart

Figure 12.8 x-bar and R chart calculation work sheet

Figure 12.9 Calculation work sheet for Figure 12.7 data

Interpreting Control Charts Before introducing other types of process charts, we discuss the interpretation of the charts. Figure show different signals for concern that are sent by a control chart, as in the second and third boxes. When a point is found to be outside of the control limits, we call this an “out of control” situation. When a process is out of control, the variation is probably not longer random, this was a nonrandom event and search for an assignment cause of variability.

Figure Control chart evidence for investigation

To determine nonrandom event, base on the chances of this happening at random are very low. There are some cases of nonrandom event: -- Process run: five points in succession ( either all above or below the center line). -- Process drift: seven points that are all either increasing or decreasing. -- erratic behavior: large jumps of more than three or four standard deviations.

Implications of a Process Out of Control If a process loses control and becomes nonrandom, the process should be stopped immediately. In many modern process industries where just-in-time is used, this will result in the stoppage of several work stations. The team of workers who are to address the problem should use a structured problem solving process such as discussed in chapter 10 to identify the root cause of the out-of-control situation.

Once the assignable cause of variation has been discovered, corrective action can be taken to eliminate the cause. The process is than restart and people return to work.

The cause of the problem should be documented and discussed later during the weekly departmental meeting. All worker should know why a problem in the process occurred. They should understand the causes and the corrective action that was taken to solve the problem. As time passes, the process become more stable as causes of errors are detected and eliminated.

As a result, they had increased their master production schedules by 20% to cover up this problem. This is result of less rework, scrap, and other problems because of poor quality. We should know that it is the process and the management problem, it is not the worker problem.

Example 12.1: Using x-bar and R charts -- The data is shown as table that is shown in page 362 in textbook. -- The grand mean is and R-bar is 2. -- Figure shows the values for the process control limits. -- The averages and ranges fall within the control limits and no other signals of nonrandom activity are present, we conclude that the process is random.

Figure Calculation work sheet and x-bar chart

X and Moving Range (MR) Charts for Population Data At times, it may not be possible to draw samples. This may occur because a process is so slow that only one or two units per day are produced. If you have a variable measurement that you want to monitor, the X and MR charts might be the thing for you.

X and Moving Range (MR) Charts for Population Data (continued) X chart. A chart used to monitor the mean of a process for population values. MR chart. A chart for plotting variables when samples are not possible. If data are not normally distributed, other charts are available.

g and h Charts A g chart is used when data are geometrically distributed, and h charts are useful when data are hyper-geometrically distributed. Figure presents pictures of geometric and hyper-geometric distributions. If you develop a histogram of your data, and it appears like either of these distributions, you may want to use either an h or a g chart instead of an X chart.

Figure g and h Distributions

X and Moving Range (MR) Charts for Population Data (continued) -- In statistics, an X is an individual observation from a population. -- The X chart reflects a population distribution. -- We call the three standard deviation limits in an X chart the natural variation in a process.

X and Moving Range (MR) Charts for Population Data (continued) -- This natural variation can be compared with specification limits. -- Strictly speaking, X chart limits are not control limits; they are natural limits.

X and Moving Range (MR) Charts for Population Data (continued) -- The formula for the center line and the natural limits for an X chart is as follows from Table 12.2 ( page 371). -- -- E2 = 2.66 ( n = 2 ), ( see Table A-1 in the Appendix).

X and Moving Range (MR) Charts for Population Data (continued) -- The formula for the MR chart is similar to that for the R chart ( where n = 2). --

Example 12.2: X and MR charts in action -- The data is shown on page 365. -- The results shown in Figure -- A run ( from point 9 to point 15) indicates that trip times may be increasing. -- This does not imply that the girlfriend is the cause. -- Further investigation may be needed.

Figure X chart for trips to Lincoln

Control Charts for Attributes We now shift to charts for attributes. These charts deal with binomial and Poisson processes that are not measurements. We will now be thinking in terms of defects and defectives rather than diameters or widths. A defect is an irregularity or problem with a larger unit. The larger unit may contain many defects. Defects are monitored using c and u charts. A defective is a unit that, as a whole, is not acceptable or does not meet specifications. Defectives are monitored using p and np charts.

p Charts for Proportion Defective The p chart is a process chart that is used to graph the proportion of items in a sample that are defective (nonconforming to specifications). p charts are effectively used to determine when there has been a shift in the proportion defective for a particular product or service. Typical applications of the p chart include things like late deliveries, incomplete orders, calls not getting dial tones, accounting transaction errors, clerical errors on written forms, or parts that don’t mate properly.

p Charts for Proportion Defective -- The formulas for the p chart are as follows: -- -- Example 12.3: p charts in action - The data is shown on page 366. - The results are shown on Figure - It shows that two month have poor performance. - Investigations should be undertaken to identify assignable causes of variation.

Figure p charts for Example 12.3

np Charts The np chart is a graph of the number of defectives (or nonconforming units) in a subgroup. The np chart requires that the sample size of each subgroup be the same each time a sample is drawn. When subgroup sizes are equal, either the p or np chart can be used. They are essentially the same chart.

np Charts (continued) Some people find the np chart easier to use because it reflects integer numbers rather than proportions. The uses for the np chart are essentially the same as the uses for the p chart.

np Charts (continued) -- To compute the control limits on an np chart, the following formula is used: -- -- Example 12.4: np charts in action - The data is shown on page 368. - The result is shown in Figure - The chart shows that rating errors are increasing. Assignable causes should be identified through investigation.

Figure np chart for Example 12.4

c and u Charts The c chart is a graph of the number of defects (nonconformities) per unit. The units must be of the same sample space; this includes size, height, length, volume and so on. This means that the “area of opportunity” for finding defects must be the same for each unit. Several individual unites can comprise the sample but they will be grouped as if they are one unit of a larger size. The control limits for the c chart are computed based on the Poisson distribution.

c and u Charts (continued) Like other process charts, the c chart is used to detect nonrandom events in the life of a production process. Typical applications of the c chart include number of flaws in an auto finish, number of flaws in a standard typed letter, and number of incorrect responses on a standardized test.

c and u Charts (continued) The u chart is a graph of the average number of defects per unit. This is contrasted with the c chart, which shows the actual number of defects per standardized unit. The u chart allows for the units sampled to be different sizes, areas, heights and so on, and allows for different numbers of units in each sample space. The uses for the u chart are the same as the c chart.

The formulas for the c and u charts are -- -- Example 12.5: c and u charts in action - The data is shown on page 369 and page 370. - The results are shown in Figure and Figure - The process for Pokas appears to be in control. - The process for Pokas shows a run of five points below the mean. An assignable cause should be sought.

Figure c chart for Pokas

Figure u chart for Yokas

Other Control Charts s Chart. The s (standard deviation) chart is used in place of the R chart when a more sensitive chart is desired. These charts are commonly used in semiconductor production where process dispersion is watched very closely. The formulas for the s chart are shown in Table 12.2.

Table 12.2 Summary of chart formulas

Other Control Charts (continued) Moving Average Chart. The moving average chart is an interesting chart that is used for monitoring variables and measurement on a continuous scale. The chart uses past information to predict what the next process outcome will be. Using this chart, we can adjust a process in anticipation of its going out of control.

Other Control Charts (continued) Cusum Chart. The cumulative sum, or cusum, chart is used to identify slight but sustained shifts in a universe where there is no independence between observations. A Cusum chart looks very different from a Shewhart process chart as shown in Figure

Figure Cusum chart

Some Control Chart Concepts Choosing the Correct Control Chart Obviously, it is key to choose the correct control chart. Figure shows a decision tree for the basic control charts. This flow chart helps to show when certain charts should be selected for use.

Figure Process for selecting the right chart

Some Control Chart Concepts (continued) Corrective Action. When a process is out of control, corrective action is needed. Correction action steps are similar to continuous improvement processes. They are 1. Carefully identify the problem. 2. Form the correct team to evaluate and solve the problem. 3. Use structured brainstorming along with fishbone diagrams or affinity diagrams to identify causes of the problem.

Some Control Chart Concepts (continued) Corrective Action (continued) 4. Brainstorm to identify potential solutions to problems. 5. Eliminate the cause. 6. Restart the process. 7. Document the problem, root causes, and solutions. 8. Communicate the results of the process to all personnel so that this process becomes reinforced and ingrained in the operations.

Some Control Chart Concepts (continued) How Do We Use Control Charts to Continuously Improve? One of the goals of the control chart user is to reduce variation. Over time, as processes are improved, control limits are recomputed to show improvements in stability. As upper and lower control limits get closer and closer together, the process improving.

Some Control Chart Concepts (continued) How Do We Use Control Charts to Continuously Improve? There are two key concepts here: 1. The focus of control charts should be on continuous improvement and they should be updated only when there is a change in the process. 2. Control chart limits should be updated only when there is a change to the process. Otherwise any changes are unexpected.

Some Control Chart Concepts (continued) -- Tampering with the Process - One of the cardinal rules of process charts is that you should never tamper with the process. - If we make adjustments to a random process, we actually inject nonrandom activity into the process. - Figure shows a random process. - Suppose we had decided to adjust the process after the fourth observation. We would have shifted the process– signaled by out-of-control observations during samples 12 and 19.

Figure The effects of tampering

Process Capability Slide 1 of 4
Process Stability and Capability Once a process is stable, the next emphasis is to ensure that the process is capable. Process capability refers to the ability of a process to produce a product that meets specifications. World-class levels of process capability are measured by parts per million (PPM) defects levels, which means that for every million pieces produced, only a small number (less than 100) are defective.

Process Capability Process Stability and Capability
Six-sigma program such as those pioneered by Motorola Corporation result in highly capable processes. Six sigma is a design program that emphasizes engineering parts so that they are highly capable. As shown in Figure 12.21, these processes are characterized by specifications that +/- 6 standard deviations from the process mean. This means that even large shifts in process mean and dispersion will not result in defective products being built.

Six-Sigma Quality (Figure in the textbook)

Process Capability Six-Sigma Quality
-- If a process average is on the center line, a six-sigma process will result in an average of only 3.4 defect per million units produced. -- The Taguchi method ( see chapter 7S) is a valuable tool for achieving six-sigma quality by helping to develop robust designs that are insensitive to variation.

Population Versus Sampling Distributions To understand process capability we must first understand the differences between population and sampling distributions. Population distributions are distributions with all the items or observations of interest to a decision maker. A population is defined as a collection of all the items or observations of interest to a decision maker.

Process Capability Population Versus Sampling Distributions
A sample is subset of the population. Sampling distributions are distributions that reflect the distributions of sample means. Sampling distributions are distributions that reflect the distribution of sample means.

Population Versus Sampling Distributions
Population distributions have much more dispersion than sampling distribution. As shown in Figure 12.22, student height for this population is normally distribution, with a mean of 5 feet 8 inches and a distribution ranging from 5 feet to 6 feet 4 inches. Notice in Figure that the mean of the sample ( sample size is 5 and replacement) is still 5 feet 8 inches, but the distribution ranges only from 5 feet 4 inches to 6 feet. We see that sampling distributions have much less dispersion than population distributions.

Figure Population and sampling distributions for class heights

In the context of quality, specification and capability are associated with population distributions. Sample-based process charts and stability are computed statistically and reflect sampling distribution. Quality practitioners should never compare process chart limits with product specifications. Specification ( or tolerance) limits are set by design engineers who establish limits based on the design requirements for a product.

Process Capability Capability Studies
-- There are two purposes for performing process capability studies: 1. To determine whether a process consistently results in product that meet specifications. 2. To determine whether a process is in need of monitoring through the use of permanent process charts.

Capability Studies There are five steps in performing process capability studies: 1. Select a critical operation. 2. Take k samples of size n, where x is an individual observation. -- where 19 < k < 26 -- if x is an attribute n > 50 -- or if x is a measurement 1 < n < 11

Capability Studies There are five steps in performing process capability studies ( continued): 3. Use a trial control chart to see whether the process is stable. 4. Compare process natural tolerance limits with specification limits. 5. Compute capability indexes. ( see next slide)

Capability Studies Compute capability indexes -- Cpu = ( USL - μ)/3σ
-- Cpl = ( μ- LSL)/3σ -- Cpk = min{ Cpu, Cpl} -- We will say that processes that achieve capability indexes (Cpk) of 1.25 are capable, 1.33 are highly capable, and 2.0 are world-class ( six sigma).

Capability Studies Example 12.6: Process capability
-- Cpu = ( 40 – 34 ) / (3*3.5) = 0.57 -- Cpl = ( 34 – 30 ) / (3*3.5) = 0.38 -- Cpk = 0.38 -- The process capability in this case is poor.

Capability Studies The proportion of nonconforming product is
The probability is shown as Figure This means that on average, more than 17% of the product produced does not meet specification. This is unacceptable in almost any circumstance.

Capability Studies Figure Proportion of product not conforming for Example 12.6

The Difference Between Capability and Stability? Once again, a process is capable if individual products consistently meet specifications. A process is stable if only common variation is present in the process. It is possible to have a process that is stable but not capable. This would happen where random variation was very high. It is probably not so common that an incapable process would be stable.

Other Statistical Techniques in Quality Management
Correlation and regression also can be useful tools for improving quality, particularly in services. There are other types of data that can be correlated and regressed to understand the customer. Figure shows there conformance rates and quality cost were correlated in one company. Table 12.3 shows that these variables were significantly and positive related. The R2 values show the strength of the relationships between the variables for linear and nonlinear models. Such correlation is called interlinking.

Figure Plot of prevention and appraisal costs with conformance

Table 12.3 Relationship between conformance and PA costs

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