Chapter 13 - Statistical Quality Control PDF

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Statistical Quality Control...

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Statistical Quality Control

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Learning Objectives LO 13–1 Illustrate process variation and explain how to measure it. LO 13–2 Analyze process quality using statistics. LO 13–3 Analyze the quality of batches of items using statistics.

C O N TR O L C H A R TS S H OW YO U VA R I ATI O N TH AT M ATTER S We all like variation, but when it comes to business it can create major problems. Doing business consistently is one key to success, whether it be a service or a manufacturing endeavor. Control charts are an important tool for understanding the variation that we have in business processes. Control charts allow us to differentiate whether the variation we have is “normal” and acceptable or if it has something “special” happening that should be addressed. When you buy a burger from a fast-food restaurant, you want consistency, not unpredictability. Now, the pickle on your burger may be closer to the edge of the bun today than it was last week—but as long as the pickle is there, it’s acceptable. Businesses use statistical process control (SPC) to keep processes stable, consistent, and predictable so they can ensure the quality of products and services. And one of the most common and useful tools in SPC is the control chart. The control chart shows how a process or output attribute varies over time so you can easily distinguish between “common cause” and “special cause” variations. Identifying the different causes of the variation in an attribute prompts action to repair a process when it is needed. One example of common cause variation at our burger joint would be the number of pickles placed on different areas of the buns. Say we expect

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2 to 4 pickles to be placed randomly on each burger and that the pickles should not hang outside the edge of the bun; this is the “common” variation that is expected. Special cause variation would be a sudden rash of burgers that have 10 pickles instead of an average of 3. Clearly, something unusual is causing “special” and unacceptable variation, and it needs to be addressed! A control chart, used with some type of simple periodic sampling of the burgers, can allow us to detect this unacceptable variation when it occurs. This would then trigger a look at the process to find what is causing the variation. Possibly a machine Adapted from Eston Martz, Control Charts Show You Variation That Matters, The Minitab Blog, July 29, 2011.

is not working correctly, the pickles might not be prepared properly, or maybe an employee needs some training.

S TATI STI C A L Q UA LI TY C O N TR O L LO 13–1 Illustrate process variation and explain how to measure it.

Statistical quality control (SQC) A number of different techniques designed to evaluate quality from a conformance view.

This chapter on statistical quality control (SQC) covers the quantitative aspects of quality management. In general, SQC is a number of different techniques designed to evaluate quality from a conformance view; that is, how well are we doing at meeting the specifications that have been set during the design of the parts or services we are providing? Managing quality performance using SQC techniques usually involves periodic sampling of a process and analysis of these data using statistically derived performance criteria. As you will see, SQC can be applied to logistics, manufacturing, and service processes. Here are some examples of situations where SQC can be applied: ∙ ∙

Assignable variation Deviation in the output of a process that can be clearly identified and managed.

Common variation Deviation in the output of a process that is random and inherent in the process itself.

∙ ∙

How many paint defects are there in the finish of a car? Have we improved our painting process by installing a new sprayer? How long does it take to execute market orders in our Web-based trading system? Has the installation of a new server improved the service? Does the performance of the system vary over the trading day? How well are we able to maintain the dimensional tolerance on our three-inch ball bearing assembly? Given the variability of our process for making this ball bearing, how many defects would we expect to produce per million bearings that we make? How long does it take for customers to be served from our drive-thru window during the busy lunch period?

Processes that provide goods and services usually exhibit some variation in their output. This variation can be caused by many factors, some of which we can control and others that are inherent in the process. Variation that is caused by factors that can be clearly identified and possibly even managed is called assignable variation. For example, variation caused by workers not being equally trained or by improper machine adjustment is assignable variation. Variation that is inherent in the process itself is called common variation. Common variation is often referred to as random variation and may be the result of the type of equipment used to complete a process, for example. As the title of this section implies, this material requires an understanding of very basic statistics. Recall the definition of the mean and standard deviation from your study of statisX( ) is just the average value of tics involving numbers that are normally distributed. The mean ¯ a set of numbers. Mathematically, this is n

∑ xi

where xi = Observed value n = Total number of observed values 318

i=1 ¯ X = ____ n

[13.1 ]

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THE ELMO CHICKEN DANCE TOY GETS A SOUND CHECK AT A MATTEL LAB IN SHENZHEN, CHINA. MATTEL LOBBIED TO LET ITS LABS CERTIFY TOY SAFETY. THE CALIFORNIA COMPANY HAS 10 LABS IN 6 COUNTRIES. © Chang W. Lee/The New York Times/Redux

The standard deviation is

__________

√ σ = ____________ n

¯ )2 ∑ (xi − X

[13.2 ] n In monitoring a process using SQC, samples of the process output would be taken and sample statistics calculated. The distribution associated with the samples should exhibit the same kind of variability as the actual distribution of the process, although the actual variance of the sampling distribution would be less. This is good because it allows the quick detection of changes in the actual distribution of the process. The purpose of sampling is to find when the process has changed in some nonrandom way, so that the reason for the change can be quickly determined. In SQC terminology, sigma (σ) is often used to refer to the sample standard deviation. As you will see in the examples, sigma is calculated in a few different ways, depending on the underlying theoretical distribution (i.e., a normal distribution or a Poisson distribution). i=1

Under s t a ndi ng a nd Mea s ur i ng Pr o ces s Va r i a t i o n

It is generally accepted that as variation is reduced, quality is improved. Sometimes that knowledge is intuitive. If a commuter train is always on time, schedules can be planned more precisely. If clothing sizes are consistent, time can be saved by ordering from a catalog. But rarely are such things thought about in terms of the value of low variability. When engineering a mechanical device such as an automobile, the knowledge is better defined. Pistons must fit cylinders, doors must fit openings, electrical components must be compatible, and tires must be able to handle the required load—otherwise, quality will be unacceptable and customers will be dissatisfied. However, engineers also know that it is impossible to have zero variability. For this reason, designers establish specifications that define not only the target value of something but also acceptable limits about the target. For example, if the target value of a dimension is 10 inches, the design specifications might then be 10.00 inches ± 0.02 inch. This would tell the Upper and lower manufacturing department that, while it should aim for exactly 10 inches, anything between specification limits 9.98 and 10.02 inches is OK. These design limits are often referred to as the upper and lower The range of values in a measure associated with specification limits. a process that is allowable A traditional way of interpreting such a specification is that any part that falls within the given the intended use of allowed range is equally good, whereas any part falling outside the range is totally bad. This the product or service.

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ex h i b i t 13. 1

A Traditional View of the Cost of Variability

High Incremental cost to society of variability

Zero

ex h i b i t 13. 2

Lower spec

Aim spec

Upper spec

Taguchi’s View of the Cost of Variability High

Incremental cost to society of variability

Zero Lower spec

Aim spec

Upper spec

is illustrated in Exhibit 13.1. (Note that the cost is zero over the entire specification range, and then there is a quantum leap in cost once the limit is violated.) Genichi Taguchi, a noted quality expert from Japan, has pointed out that the traditional view illustrated in Exhibit 13.1 is nonsense for two reasons: 1.

From the customer’s view, there is often practically no difference between a product just inside specifications and a product just outside. Conversely, there is a far greater difference in the quality of a product that is at the target and the quality of one that is near a limit.

2.

As customers get more demanding, there is pressure to reduce variability. However, Exhibit 13.1 does not reflect this logic.

Taguchi suggests that a more correct picture of the loss is shown in Exhibit 13.2. Notice that in this graph the cost is represented by a smooth curve. There are dozens of illustrations of this notion: the meshing of gears in a transmission, the speed of photographic film, the temperature in a workplace or department store. In nearly anything that can be measured, the customer sees not a sharp line, but a gradation of acceptability away from the “Aim” specification. Customers see the loss function as Exhibit 13.2 rather than Exhibit 13.1. Of course, if products are consistently scrapped when they are outside specifications, the loss curve flattens out in most cases at a value equivalent to scrap cost in the ranges outside specifications. This is because such products, theoretically at least, will never be sold so there is no external cost to society. However, in many practical situations, either the process is capable of producing a very high percentage of product within specifications and 100 percent checking is not done, or if the process is not capable of producing within specifications, 100 percent checking is done and out-of-spec products can be reworked to bring them within specs. In any of these situations, the parabolic loss function is usually a reasonable assumption.

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Taguchi argues that being within specification is not a yes/no decision, but rather a continuous function. Motorola quality experts, on the other hand, argue that the process used to produce a good or deliver a service should be so good that the probability of generating a defect should be very, very low. Motorola made process capability and product design famous by adopting Six Sigma limits. When a part is designed, certain dimensions are specified to be within the upper and lower specification limits. As a simple example, assume engineers are designing a bearing for a rotating shaft—say, an axle for the wheel of a car. There are many variables involved for both the bearing and the axle—for example, the width of the bearing, the size of the rollers, the size of the axle, the length of the axle, how it is supported, and so on. The designer specifies limits for each of these variables to ensure that the parts will fit properly. Suppose that initially a design is selected and the diameter of the bearing is set at 1.250 inches ± 0.005 inch. This means that acceptable parts may have a diameter that varies between 1.245 and 1.255 inches (which are the lower and upper specification limits). Next, consider the process in which the bearing will be made. Consider that many different processes for making the bearing are available. Usually, there are trade-offs that need to be considered when designing a process for making a part. The process, for example, might be fast but not consistent, or alternatively it might be slow but consistent. The consistency of a process for making the bearing can be measured by the standard deviation of the diameter measurement. A test can be run by making, say, 100 bearings and measuring the diameter of each bearing in the sample. After running the test, the average or mean diameter is found to be 1.250 inches. Another way to say this is that the process is “centered” right in the middle of the upper and lower specification limits. In reality, it may be difficult to have a perfectly centered process like this example. Consider that the diameter values have a standard deviation or sigma equal to 0.002 inch. What this means is that the process does not make each bearing exactly the same size. As is discussed later in this chapter, normally a process is monitored using control charts such that if the process starts making bearings that are more than three standard deviations KEY IDEA (± 0.006 inch) above or below 1.250 inches, the process is stopped. This means that the proThe main point of this is cess will produce parts that vary between 1.244 (this is 1.250 – 3 × .002) and 1.256 (this is that the process should 1.250 + 3 × .002) inches. The 1.244 and 1.256 are referred to as the upper and lower process be able to make a limits. Be careful and do not get confused with the terminology here. The process limits relate part well within design to how consistent the process is for making the bearing. The goal in managing the process is specifications. Here, we to keep it within plus or minus three standard deviations of the process mean. The specifica- show how statistics are tion limits are related to the design of the part. Recall that, from a design view, acceptable used to evaluate how good a process is. parts have a diameter between 1.245 and 1.255 inches (which are the lower and upper specification limits). As can be seen, process limits are slightly greater than the specification limits given by the designer. This is not good because the process will produce some parts that do not meet specifications. Companies with Six Sigma processes insist that a process making a part be capable of operating so that the design specification limits are six standard deviations away from the process mean. For the bearing process, how small would the process standard deviation need to be for it to be Six Sigma capable? Recall that the design specification was 1.250 inches plus or minus 0.005 inch. Consider that the 0.005 inch must relate to the variation in the process. Divide 0.005 inch by 6, which equals 0.00083, to determine the process standard deviation for a Six Sigma process. So for the process to be Six Sigma capable, the mean diameter produced by the process would need to be exactly 1.250 inches and the process standard deviation would need to be less than or equal to 0.00083 inch. We can imagine that some of you are really confused at this point with the whole idea of Six Sigma. Why doesn’t the company, for example, just check the diameter of each bearing and throw out the ones with a diameter less than 1.245 or greater than 1.255? This could certainly be done and for many, many parts 100 percent testing is done. The problem is for a company that is making thousands of parts each hour, testing each critical dimension of each

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part made can be very expensive. For the bearing, there could easily be 10 or more additional critical dimensions in addition to the diameter. These would all need to be checked. Using a 100 percent testing approach, the company would spend more time testing than it takes to actually make the part! This is why a company uses small samples to periodically check that the process is in statistical control. We discuss exactly how this statistical sampling works later in the chapter. We say that a process is capable when the mean and standard deviation of the process are operating such that the upper and lower control limits are acceptable relative to the upper and lower specification limits. Consider diagram A in Exhibit 13.3. This represents the distribution of the bearing diameter dimension in our original process. The average or mean value is 1.250 and the lower and upper design specifications are 1.245 and 1.255, respectively. Process control limits are plus and minus three standard deviations (1.244 and 1.256). Notice that there is a probability (the red areas) of producing defective parts. If the process can be improved by reducing the standard deviation associated with the bearing diameter, the probability of producing defective parts can be reduced. Diagram B in Exhibit13.3 shows a new process where the standard deviation has been reduced to 0.00083 (the area outlined in yellow). Even though we cannot see it in the diagram, there is some probability that a defect could be produced by this new process, but that probability is very, very small. Suppose that the central value or mean of the process shifts away from the mean. Exhibit13.4 shows the mean shifted one standard deviation closer to the upper specification limit. This, of course, causes a slightly higher number of expected defects, but we can see that this is still very, very good. The capability index is used to measure how well our process is capable of producing relative to the design specifications. A description of how to calculate this index is in the next section.

ex h i b i t 13. 3 Diagram A

Process Capability

1.244

1.250

1.256

1.245

1.255 Upper spec limit

Lower spec limit

1.244

1.250

Diagram B

1.248

1.250

1.256

1.252 Improved process

1.245

1.255 Original process

1.244

1.250

1.256

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ex h i b i t 13. 4

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Process Capability with a Shift in the Process Mean LTL

1.249

1.251

1.254 UTL

1.245

1.244

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1.255

1.250

1.256

C a p a b i l i t y In d e x ( C p k ) The capability index (Cpk) shows how well the parts being Capability index produced fit into the range specified by the design specification limits. If the specification limits The ratio of the range of are larger than the three sigma allowed in the process, then the mean of the process can be allowed values allowed by the to drift off-center before readjustment, and a high percentage of good parts will still be produced. design specifications divided by the range of Referring to Exhibits 13.3 and 13.4, the capability index (Cpk) is the position of the mean values produced by a and tails of the process relative to design specifications. The more off-center, the greater the process. chance to produce defective parts. Because the process mean can shift in either direction, the direction of shift and its distance from the design specification set the limit on the process capability. The direction of shift is toward the smaller number. Formally stated, the capability index (Cpk) is calculated as the smaller of the two numbers as follows: ¯ X X − LSL USL − ¯ Cpk = min ___...