Statistical Process Control & Operations Management

©The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin

Statistical Process Control and Quality Management

Chapter 19

2

GOALS

Discuss the role of quality control in production and service operations.

Define and understand the terms chance cause, assignable cause, in control, out of control, attribute, and variable.

Construct and interpret a Pareto chart. Construct and interpret a fishbone diagram. Construct and interpret mean and range charts. Construct and interpret percent defective and a c-bar

charts. Discuss acceptance sampling. Construct an operating characteristic curve for various

sampling plans.

3

Control Charts

Statistical Quality Control emphasizes in-process control with the objective of controlling the quality of a manufacturing process or service operation using sampling techniques.

Statistical sampling techniques are used to aid in the manufacturing of a product to specifications rather than attempt to inspect quality into the product after it is manufactured.

Control Charts are useful for monitoring a process.

4

Causes of Variation

There is variation in all parts produced by a manufacturing process. There are two sources of variation:

Chance Variation is random in nature and cannot be entirely eliminated. Assignable Variation is nonrandom in nature and can be reduced or

eliminated.

5

Diagnostic Charts

There are a variety of diagnostic techniques available to investigate quality problems. Two of the more prominent of these techniques are Pareto charts and fishbone diagrams.

6

Pareto Charts

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

The chart is named after a nineteenth-century Italian scientist, Vilfredo Pareto. He noted that most of the “activity” in a process is caused by relatively few of the “factors.”

Pareto’s concept, often called the 80–20 rule, is that 80 percent of the activity is caused by 20 percent of the factors. By concentrating on 20 percent of the factors, managers can attack 80 percent of the problem.

7

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 in Grove City. To investigate, she selects a sample of 100 homes and determines the typical daily water usage for various purposes. These sample results are as follows.

8

Pareto Chart - Minitab

9

Fishbone Diagrams

Another diagnostic chart is a cause-and-effect diagram or a fishbone diagram. It is called a cause-and-effect diagram to emphasize the relationship between an effect and a set of possible causes that produce the particular effect.

This diagram is useful to help organize ideas and to identify relationships. It is a tool that encourages open brainstorming for ideas. By identifying these relationships we can determine factors that are the cause of variability in our process.

The name fishbone comes from the manner in which the various causes and effects are organized on the diagram. The effect is usually a particular problem, or perhaps a goal, and it is shown on the right-hand side of the diagram. The major causes are listed on the left-hand side of the diagram.

10

Purpose of Quality Control Charts

The purpose of quality-control charts is to portray graphically when an assignable cause enters the production system so that it can be identified and corrected.

This is accomplished by periodically selecting a random sample from the current production.

11

Mean Chart for Variables

The mean or the x-bar chart is designed to control variables such as weight, length, etc. The upper control limit (UCL) and the lower control limit (LCL) are obtained from the equation:

where is the mean of the sample means and is the mean of the sample ranges

RAXLCLandRAXUCL 22

X R

12

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 11 P.M. daily. It is impossible to have every call answered immediately by a technical representative, but it is important customers do not wait too long for a person to come on the line. Customers become upset when they hear the message “Your call is important to us. The next available representative will be with you shortly” too many times. To understand its process, Statistical Software decides to develop a control chart describing the total time from when a call is received until the representative answers the call and resolves the issue raised by the caller. Yesterday, for the 16 hours of operation, five calls were sampled each hour. This information is on the table, in minutes, until the issue was resolved.

Based on this information, develop a control chart for the mean duration of the call. Does there appear to be a trend in the calling times? Is there any period in which it appears that customers wait longer than others?

Mean Chart for Variables - Example

13

Appendix B.8 (portion)

14

Constructing a Mean Chart

413.916

60.150X

375.616

102R

15

13.091

)375.6)(577(.413.9

2

andRAXUCL

5.735

)375.6)(577(.413.9 2

RAXLCL

Mean Chart for Variables - Example

16

Range Charts for Variables

A range chart shows the variation in the sampleranges.

17

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 Table 19–1.

Develop a control chart for the range. Does it appear that there is any time when there is too much variation in the operation?

18

375.616

102R

Range Chart - Example

19

Range Chart - Example

13.483

)375.6(115.2

4

RDUCL

0

)375.6(0 3

RDLCL

20

Mean and Range Charts - Minitab

21

In-Control Situation

22

Mean In-control, Range Out-of-control

23

Mean Out-of-control, Range In-control

24

Attribute Control Chart: The p-Chart

The percent defective chart is also called a p-chart or the p-bar chart. It graphically shows the proportion of the production that is not acceptable. The proportion of defectives is found by:

samples ofNumber

Defectives ofNumber Totalp

25

Attribute Control Chart – The p-Chart

The UCL and LCL are computed as the mean percent defective plus or minus 3 times the standard error of the percents:

n

pppLCLandUCL

)1(3

26

p-Chart Example

Jersey Glass Company, Inc., produces small hand mirrors. Jersey Glass runs a day and evening shift each weekday. Each day, the quality assurance department (QA) monitors the quality of the mirrors twice during the day shift and twice during the evening shift. After each four-hour period, QA selects and carefully inspects a random sample of 50 mirrors. 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. List below is the result of these checks over the last 10 business days.

Construct a percent defective chart for this process. What are the upper and lower control limits? Interpret the results. Does it appear the process is out of control during the period?

27

Computing the Control Limits

28

p-Chart using Minitab

29

The c-chart or the c-bar chart is designed to control the number of defects per unit. The UCL and LCL are found by:

UCL and LCL c c 3

Attribute Control Chart : The c-Chart

30

The publisher of the Oak Harbor Daily Telegraph is concerned about the number of misspelled words in the daily newspaper. It does not print a paper on Saturday or Sunday. 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 found in the final edition of the paper for the last 10 days is: 5, 6, 3, 0, 4, 5, 1, 2, 7, and 4.

Determine the appropriate control limits and interpret the chart. Were there any days during the period that the number of misspelled words was out of control?

0 07.277.57.3

47.977.57.3

7.337.3

7.310

37

LCL

UCL

LCLandUCL

c

c-Chart Example

31

c-Chart in Minitab

32

Acceptance Sampling

Acceptance sampling is a method of determining whether an incoming lot of a product meets specified standards. – It is based on random sampling

techniques.– A random sample of n units is

obtained from the entire lot.– c is the maximum number of

defective units that may be found in the sample for the lot to still be considered acceptable.

33

Acceptance Sampling Procedure

Accept shipment or reject shipment? The usual procedure is to screen the quality of incoming parts by using a statistical sampling plan.

According to this plan, a sample of n units is randomly selected from the lots of N units (the population). This is called acceptance sampling. The inspection will determine the number of defects in the sample. This number is compared with a predetermined number called the critical number or the acceptance number. The acceptance number is usually designated c.

– If the number of defects in the sample of size n is less than or equal to c, the lot is accepted.

– If the number of defects exceeds c, the lot is rejected and returned to the supplier, or perhaps submitted to 100 percent inspection.

34

Consumer’s Risk vs. Producer’s Risk in Acceptance Sampling

Type I Error

Type II Error

35

Operating Characteristic Curve

An OC curve, or operating characteristic curve, is developed using the binomial probability distribution in order to determine the probabilities of accepting lots of various quality level.

36

OC Curve - Computation Example

Sims Software purchases DVDs from DVD International. The DVDs are packaged in lots of 1,000 each. Todd Sims, president of Sims Software, has agreed to accept lots with 10 percent or fewer defective DVDs. Todd has directed his 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 percent defective?

37

OC Curve - Computation Example

This type of sampling is called attribute sampling because the sampled item, a DVD in this case, is classified as 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.

38

OC Curve - Computation Example

The binomial distribution is used to compute the various values on the OC curve. Recall that the binomial has four requirements:

1. There are only two possible outcomes. Here the DVD is either acceptable or unacceptable.

2. There is a fixed number of trials. In this instance the number of trials is the sample size of 20.

3. There is a constant probability of success. A success is finding a defective DVD. The probability of success is assumed to be .10.

4. The trials are independent. The probability of obtaining a defective DVD on the third one selected is not related to the likelihood of finding a defect on the fourth DVD selected.

39

OC Curve - Computation Example

40

OC Curve - Computation Example

To begin we determine the probability of accepting a lot that is 5 percent defective. This means that = .05, c = 2, and n = 20. From the Excel output, the likelihood of selecting a sample of 20 items from a shipment that contained 5 percent defective and finding exactly 0 defects is .358. The likelihood of finding exactly 1 defect is .377, and finding 2 is .189. Hence the likelihood of 2 or fewer defects is .924, found by .358 +.377 + .189. This result is usually written in shorthand notation

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

The likelihood of accepting a lot that is actually 10 percent defective is .677. P(x≤ 2 | = .10 and n = 20) = .122 + .270 + .285 = .677

The complete OC curve in the next slide (Chart 19–8) shows the smoothed curve for all values of between 0 and about 30 percent.

41

OC Curve - Computation Example

42

End of Chapter 19