Wk7 DQ - Data Analysis & Business Intelligence

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Statistical Process Control and Quality Management

Chapter 19

19-1

In this chapter, we present a somewhat different application of hypothesis testing, called statistical process control. SPC is a collection of strategies, techniques, and actions taken by an organization to ensure it is producing a quality product or providing a quality service. To effectively use quality control, measurable attributes and specifications are developed against which the actual attributes are compared. Companies today are recognized for their quality achievements. The Malcolm Baldrige National Quality Award is awarded annually to U.S. companies that demonstrate excellence in quality achievement and management.

Learning Objectives

LO19-1 Explain the purpose of quality control in production and service operations

LO19-2 Define the two sources of process variation and explain how they are used to monitor quality

LO19-3 Explain the use of charts to investigate the sources of process variation

LO19-4 Compute control limits for mean and range control charts for a variable measure of quality

LO19-5 Evaluate control charts to determine if a process is out of control

LO19-6 Compute control limits of control charts for an attribute measure of quality

LO19-7 Explain the process of acceptance sampling

19-2

Statistical Process Control

The objective of statistical quality control is to monitor the quality of the product or device as it is being developed

Interest in quality has become one of the “in” topics in business today

A noted quality consultant reports that 20 to 25% of the cost of production in the U.S. is spent finding and correcting mistakes

The additional cost of repairing or replacing faulty products in the field drives the cost of poor quality to nearly 30%—it’s 3% in Japan

19-3

Product quality wasn’t always considered important; in fact, it wasn’t an important business topic in the U.S. until the late 1980s. Japan adopted quality control practices right after WWII. Dr. Shewhart, of Bell Telephone Laboratories, first developed the concepts of quality control. He emphasized controlling the quality of a product as it was being manufactured rather than just inspecting it after it was made. Charting techniques were developed for controlling in-process manufacturing operations.

Six Sigma

An organization-wide program designed to improve quality and performance throughout an organization

The focus is to reduce variation in any process used to produce and deliver goods and services to customers

A successful Six Sigma program will

Reduce the costs of defects and errors

Increase customer satisfaction

Increase sales and profits

Sigma means standard deviation and plus or minus 3 standard deviations, which gives a range of six

That means that no more than 3.4 defects per million

19-4

Six Sigma applies to accounting and other support processes as well, not just production processes. To achieve a company’s quality goals, a Six Sigma program will train every organization member in processes in order to identify sources of process variation that significantly affects quality.

Sources of Variation

Examples of chance variation: slight machine friction, slight variation in materials, temperature variation

Chance variation usually is not correctable

Examples of assignable variation: an operator who always sets up the machine incorrectly, using a dull drill bit

Assignable variation is usually correctable

CHANCE VARIATION Variation that is random in nature. This type of variation cannot be completely eliminated unless there is a major change in the techniques, technologies, methods, equipment, or materials used in the process.

ASSIGNABLE VARIATION Variation that is not random. It can be eliminated or reduced by investigating the problem and finding the cause.

19-5

No two products are exactly the same, there is always some variation. One problem with variation is that it will change the shape, dispersion, and central location of the distribution of the product characteristic being measured. There are two types of variation; chance and assignable.

Diagnostic Charts

There are a variety of diagnostic techniques available to investigate quality problems, like the Pareto Chart

A Pareto chart is a technique for tallying the number and type of defects that happen within a product or service

Named after an Italian scientist, Vilfredo Pareto

The 80-20 rule is the concept that 80% of the activity is caused by 20% of the factors

Example

Emily’s Family Restaurant is investigating customer complaints, out of the 5 most frequently heard, they find that more than 85% of the complaints are about discourteous service and cold food

19-6

The Pareto rule is that by concentrating on 20% of the factors, managers can attack 80% of the problem. If Emily’s Restaurant focuses on the two issues of discourteous service and cold food first, this will yield the largest reduction in complaints and the largest increase in customer satisfaction.

Pareto Chart Example

The city manager of Grove City, Utah is concerned with water usage, particularly in single-family homes. She would like to develop a plan to reduce the water usage. To investigate, she selects a sample of 100 homes and determines the typical daily water usage for various purposes. The sample results are shown below.

Where should she concentrate her efforts to reduce the water usage?

19-7

To develop a Pareto chart, we begin by tallying the type of defects; or in this case, reasons for water usage. Convert each of these activities to a percent and then order them from largest to smallest.

Pareto Chart Example Continued

19-8

She found watering the lawn, personal bathing, and pools account for 82.1% of water usage!

To draw the Pareto chart, scale the number of gallons used on the left vertical axis and the corresponding percent on the right vertical axis. Then draw vertical bars corresponding to each activity (for instance, watering lawns is drawn up to a height of 143.7; this is referred to as the count). Below the chart, list the activities, their frequency of occurrence, and the percent of occurrence. In the last row, include the cumulative percentage. In this example, watering the lawn, personal bathing, and pools account for 82.1% of the water usage. The city manager can attain the greatest gain by looking to reduce the water usage in these three areas. Minitab was used to create the Pareto chart in this example.

Fishbone Diagram Example

A fishbone diagram emphasizes the relationship between a possible cause that produced the particular effect

The usual approach is to consider four problem areas: methods, materials, equipment, and personnel

19-9

This is also called a cause-and-effect diagram. To illustrate, we investigate the causes of cold food served at Emily’s Family Restaurant. Recall that a Pareto analysis showed that cold food was one of the top two complaints. In the chart, notice that each of the subcauses is listed as an assumption. Each of these subcauses must be investigated to find the real problem regarding the cold food. In the fishbone diagram, there is no weighting of the subcauses.

Purpose and Type of Quality Control Charts

The purpose of the control chart is to monitor graphically the quality of a product or service

There are two types of control charts

A variable control chart is the result of a measurement

An attribute chart shows whether the product or service is acceptable or not acceptable

There are two sources of variation in the quality of a product or service

Chance variation is random in nature and cannot be controlled or eliminated

Assignable variation is not due to random causes and can be eliminated

19-10

Control charts identify when assignable causes of variation or changes have entered the process. It is important to know when changes have entered the process, so that the cause may be identified and corrected before a large number of unacceptable items are produced.

Control Charts for Variables

To develop control charts, we rely on sampling theory

Find the overall or grand mean with formula 19-1

Find the standard error of the distribution of sample means with formula 19-2

These allow us to establish limits for the sample means to show how much variation can be expected

These limits are the upper control limit (UCL) and the lower control limit (LCL)

19-11

To develop control charts for variables, we rely on sampling theory discussed in connection with the central limit theorem in chapter 8.

Control Charts for Variables Example

Statistical Software Inc. offers a toll-free number where customers can call with problems involving the use of their products from 7 a.m. until 11p.m. daily. It is impossible for a technical representative to answer every call immediately. To understand its service-call process, Statistical Software decides to develop a control chart describing the total time for the call to be answered until the problem is resolved. Yesterday, for the 16 hours of operation, five calls were sampled each hour and the total time to resolve a customer’s problem was recorded.

Does there appear to be a trend in the calling times?

19-12

Is there any period in which it appears that customers wait longer than others?

Control Limits for the Mean

19-13

The upper and lower control limits are calculated by:

The number 3 represents the 99.74% confidence limits. The limits are often called the 3-sigma limits.

For fixed-sized samples, there is a constant relationship between the range and the standard deviation, so we can use the following formulas to determine the 99.74% control limits for the mean.

Control Charts for Variables Example Continued

19-14

Notice that in the calculation of the upper and lower control limits with formula 19-3, the number 3 appears. It represents the 99.74% confidence limits. The limits are often called the 3-sigma limits. (Though other confidence limits can be used.) Rather than calculate the standard deviation from each sample, it is easier to use the range (R) and formula 19-4. In this formula, the value for A2 is found in Appendix B.10, a portion of Appendix B.10 is provided here, n is the sample size and is 5, then just move horizontally over to the A2 column; the value is 0.577 and is used in formula 19-4., is the mean of the sample means and the center of the control chart; here 9.413, the UCL is located at 13.091 and the LCL is located at 5.735. Before using control charts in practice, at least 25 samples need to be collected to establish control chart limits.

Range Charts

A range chart shows the variation in the sample ranges

If the points representing the ranges fall between the upper and lower control limits, it is concluded the operation is in control

If the range should fall above the limits, we conclude that an assignable cause affected the operation and an adjustment to the process is needed

Use formula 19-5 to determine the control limits of the range chart

19-15

According to chance, about 997 times out of 1,000 the range of the samples will fall within the limits. For small samples, the lower limit is often zero. In the formula, the values for D4 and D3 are the 3 sigma limits for various sample sizes and can be found in Appendix B.10.

Range Chart Example

The length of time customers of Statistical Software Inc. waited from the time their call was answered until a technical representative answered their question or solved their problem is recorded in the table. Develop a control chart for the range.

Does it appear that there is any time when there is too much variation in the operation?

19-16

We’ll need to find the range for each sample next and then calculate the mean of the sample ranges. This is on the next slide.

Range Chart Example Continued

The first step is to find the mean of the sample ranges. The range for the five calls at 7 a.m. is 11. The longest call that hour was 15 minutes and the shortest call was 4 minutes. In the 8 a.m. hour, the range is 4 and so on. Total the ranges and divide by 16 to get . So is 102 ÷ 16 = 6.375. Refer to Appendix B.10 for D3 (0) and D4 (2.115) to use in Formula 19-5.

19-17

is the average range. The chart shows all the ranges are well within the control limits. We conclude the variation in the time to service the customers calls is within normal limits, that is, “in control.” Statistical software can compute the statistics and draw the control charts.

In-Control Situation

The mean chart and the range chart together indicate the process is in control. The sample means and sample ranges are clustered close to the center lines. Some are above and some are below, indicating the process is quite stable.

19-18

There is no visible tendency for the means and ranges to move toward the out-of-control areas.

In-Control and Out-of-Control Situation

The sample means are in control, but the ranges of the last two samples are out of control. Some sample ranges are large; others are small. An adjustment in the process is probably necessary.

19-19

This looks like there is considerable variation in the samples.

In-Control and Out-of-Control Situation Continued

The sample means are in control for the first few samples, but there is an upward trend toward the UCL. An adjustment in the process is indicated.

19-20

The last two sample means are out of control. Also, notice the direction of the last 5 observations of the mean; they are all above and increasing. The fact that the sample means were increasing for the last six observations is very improbable and another indication the process is out-of-control.

Attribute Control Charts

A p-chart is an attribute chart that shows the proportion of the product or service that does not conform to the standard

The data we collect are the result of counting something rather than measuring

Examples

A screw top on a shampoo bottle either fits and does not leak (acceptable) or does not fit and leaks (unacceptable)

A bank makes a loan to a customer and the loan is either repaid or not repaid

British Airways might count the number of flights arriving late each day at Gatwick Airport

19-21

If the item recorded is the proportion of unacceptable parts made in a larger batch of parts, the appropriate control chart is the p-chart. It is based on the binomial distribution and proportions. The centerline is at p, the mean proportion defective.

p-Charts

The mean proportion defective is found by

The variation is the standard error of the proportion

The control limits for the proportion defective are determined from the equation

19-22

The UCL and the LCL are computed as the mean proportion defective plus or minus three times the standard error of the proportions.

p-Chart Example

Jersey Glass Company Inc. produces small hand mirrors. Jersey Glass runs day and evening shifts each weekday. The quality assurance department (QA) monitors the quality of the mirrors twice during the day shift and twice during the evening shift. QA selects and carefully inspects a random sample of 50 mirrors once every four hours. Each mirror is classified as either acceptable or unacceptable. Finally, QA counts the number of mirrors in the sample that do not conform to quality specifications. Listed below are the results of the checks for the last 10 business days.

Construct a p-chart for this process. Does it appear the process is out of control?

19-23

First determine the overall proportion defective with formula 19-6. This is on the next slide.

p-Chart Example Continued

19-24

First determine the overall proportion defective with formula 19-6. We estimate that .098 of the mirrors do not meet specifications and set the upper control limit at .098 + .1261 = .2241 and the lower control limit at zero. The lower control limit is set at zero because it is not possible to have a negative amount of defects.

p-Chart Example Concluded

After establishing the limits, the process is monitored the next week—five days, two shifts per day, two quality checks per shift.

QA should report this to the production department!

19-25

The process was out of control on 2 occasions, October 24 when the proportion of defects was .26 and again on October 27 when the proportion of defects was .24. QA should report this to the production department for the appropriate action.

c-Bar Charts

A c-bar chart plots the number of defects per unit

It is based on the Poisson distribution

The mean number of defects per unit is

The control limits are determined from the following

Examples

Blue Sky Airlines might monitor the number of bags mishandled on a flight with a c-bar chart

The IRS might count and develop a control chart for the number of errors per tax return

19-26

For Blue Sky Airlines, the unit under consideration is the flight. On most flights, there are no bags mishandled, on another flight, there might be one, on others two and so on. For the IRS most returns will not have any errors, some will have one, or two, and so on.

c-Bar Chart Example

The publisher of the Oak Harbor Daily Telegraph is concerned about the number of misspelled words in the daily newspaper. In an effort to control the problem and promote the need for correct spelling, a control chart will be used. The number of misspelled words in the final edition of the paper for the last 10 days are 5, 6, 3, 0, 4, 5, 1, 2, 7, and 4. Determine the appropriate control limits.

During the 10-day period there were 37 misspelled words. So the mean number is 3.7 words. The number of misspelled words per edition follows the Poisson probability distribution. The standard deviation is the square root of the mean, 1.924

The number of misspelled words

is “in control”!

19-27

To find the UCL, use formula 19-9. It is 9.47 and the lower control limit is zero. All the data points for misspelled words are within the control limits. Control charting offers a means of tracking daily results and determining whether there has been a change. For example, if a new proofreader was hired, her work could be compared with others.

Acceptance Sampling

Acceptance sampling is a method to determine whether an incoming lot of a product meets specified standards

It is based on random sampling techniques

A random sample of n units is selected from a population of N units

c is the maximum number of defective units that may be found in the sample of n and the lot is still considered acceptable

An OC (operating characteristic) curve is developed using the binomial probability distribution to determine the probability of accepting lots of various quality levels

19-28

This provides a method to verify that incoming product meets the stipulated requirements. The best protection against inferior shipments is 100% inspections but that is often just not practical. The usual procedure is to use a statistical sampling plan.

Acceptance Sampling Continued

Acceptance sampling is a decision making process

The acceptance number is predetermined

Accept or reject the lot, but the lot could be good or it could be bad

Consumer’s risk occurs when the lot contains more defects than it should but it is accepted

Producer’s risk occurs when the lot is within the agreed upon limits but is rejected

19-29

Here is a summary of the possible outcomes when making the decision to accept or reject a lot. To evaluate a sampling plan and determine that it is fair to both the producer and the consumer, develop an operating characteristic curve, an OC curve.

Acceptance Sampling Example

Sims Software purchases DVDs from DVD International. The DVDs are packaged in lots of 1,000. Todd Sims, president of Sims Software, has agreed to accept lots with 10% or fewer defective DVDs. He instructed the inspection department to select a random sample of 20 DVDs and examine them carefully. He will accept the lot if it has two or fewer defectives in the sample. Develop an OC curve for this inspection plan. What is the probability of accepting a lot that is 10% defective?

This type of sampling is called attribute sampling because the sampled item, a DVD, is classified as either acceptable or unacceptable. Let represent the actual proportion defective in the population.

The lot is good if ≤ 10

The lot is bad if > 10

Let x be the number of defects in the sample, the decision rule is

Accept the lot if x ≤ 2

Reject the lot if x ≥ 3

This meets the binomial requirements.

19-30

The binomial requirements are that there are only two possible outcomes, there are a fixed number of trials, there is a constant probability of success, and the trials are independent. Appendix B.1 gives various binomials probabilities but only go up to n =15, so we’ll use Excel.

Acceptance Sampling Example Continued

The probability of accepting a lot that is 5% defective and finding 0 defects is .358. The likelihood of finding 1 defect is .377 and finding two defects is .189

P(x ≤ 2 | = .05 and n = 20) = .358+.377+.189 = .924

The probability of accepting a lot that is 10% defective is:

P(x ≤ 2 | = .10 and n = 20) =

.122+.270+.285 = .677

19-31

The Excel output shows the binomial probabilities for n=20 when is equal to .05, .10, .15, .20, .25, and .30.

Acceptance Sampling Example Concluded

With the curve, the management of Sims Software will be able to quickly evaluate the probabilities of various quality levels.

19-32

The complete OC curve shows the smoothed curve for all values of between 0 and about 30%. There is no need to show values greater than 30% because their probability is very close to 0. The likelihood of accepting lots of selected quality levels is shown to the right of the OC curve.

Chapter 19 Practice Problems

19-33

Question 1

19-34

Tom Sharkey is the owner of Sharkey Chevy, Buick, GMC. At the start of the year, Tom instituted a customer opinion program to find ways to improve service. The day after the service is performed, Tom’s administrative assistant calls the customer to find out whether the service was performed satisfactorily and how the service might be improved. A summary of the complaints for the first 6 months follows. Develop a Pareto chart. How should Tom prioritize the complaints to improve the quality of service?

LO19-3

Question 7

19-35

A new industrial oven has just been installed at Piatt Bakery. To develop experience regarding the oven temperature, an inspector reads the temperature at four different places inside the oven each half hour starting at 8:00 a.m. The last reading was at 10:30 a.m., for a total of six samples. The first reading, taken at 8:00 a.m., was 340 degrees Fahrenheit. (Only the last two digits are given in the following table to make the computations easier.)

On the basis of this initial experience, determine the control limits for the mean temperature. Determine the grand mean. Plot the results on a control chart.

Interpret the chart. Does there seem to be a time when the temperature is out of control?

LO19-4

Question 13

19-36

Sam’s Supermarkets monitors the checkout scanners by randomly examining the receipts for scanning errors. On October 27, they recorded the following number of scanner errors on each receipt: 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0. Construct a control chart for this process and comment on whether the process is “in control.”

LO19-6

Question 17

19-37

Warren Electric manufactures fuses for many customers. To ensure the quality of the outgoing product, it tests 10 fuses each hour. If no more than one fuse is defective, it packages the fuses and prepares them for shipment. Develop an OC curve for this sampling plan. Compute the probabilities of accepting lots that are 10%, 20%, 30%, and 40% defective. Draw the OC curve for this sampling plan using the four quality levels.

LO19-7