Chapter 18: Statistical Quality Control 1

Chapter 18: Statistical Quality Control 1

Chapter 18

Statistical Quality Control

LEARNING OBJECTIVES

Chapter 18 presents basic concepts in quality control, with a particular emphasis on statistical quality control techniques, thereby enabling you to:

1.Understand the concepts of quality, quality control, and total quality management.

2.Understand the importance of statistical quality control in total quality management.

3.Learn about process analysis and some process analysis tools, including Pareto charts, fishbone diagrams, and control charts.

4.Learn how to construct charts, R charts, p charts, and c charts.

5.Understand the theory and application of acceptance sampling.

CHAPTER OUTLINE

18.1Introduction to Quality Control

What Is Quality Control?

Total Quality Management

Some Important Quality Concepts

Benchmarking

Just-in-Time Inventory Systems

Reengineering

Six Sigma

Team Building

18.2 Process Analysis

Flowcharts

Pareto Analysis

Cause-and-Effect (Fishbone) Diagrams

Control Charts

18.3Control Charts

Variation

Types of Control Charts

Chart

R Charts

p Charts

c Charts

Interpreting Control Charts

18.4Acceptance Sampling

Single Sample Plan

Double-Sample Plan

Multiple-Sample Plan

Determining Error and OC Curves

KEY TERMS

acceptance samplingPareto analysis

after-process quality controlPareto chart

benchmarkingprocess

c chartproducer’s risk

cause-and-effect diagramproduct quality

centerlinequality

consumer's riskquality circle

control chartquality control

double-sample planR chart

fishbone diagramreengineering

flowchartsingle-sample plan

in-process quality controlSix Sigma

Ishikawa diagramteam building

just-in-time inventory systemstotal quality management (TQM)

lower control limit (LCL)transcendent quality

manufacturing qualityupper control limit (UCL)

multiple-sample planuser quality

operating characteristic (OC) curvevalue quality

p chart chart

STUDY QUESTIONS

1.The collection of strategies, techniques, and actions taken by an organization to assure themselves that they are producing a quality product is ______.

2.Measuring product attributes at various intervals throughout the manufacturing process in an effort to pinpoint problem areas is referred to as ______quality control.

3.Inspecting the attributes of a finished product to determine whether the product is acceptable, is in need of rework, or is to be rejected and scrapped is ______quality control.

4.An inventory system in which no extra raw materials or parts are stored for production is called a ______system.

5.When a group of employees are organized as an entity to undertake management tasks and perform other functions such as organizing, developing, and overseeing projects, it is referred to as ______.

6.A ______is a small group of workers, usually from the same department or work area, and their supervisor, who meet regularly to consider quality issues.

7.The complete redesigning of a company's core business process is called ______. This usually involves innovation and is often a complete departure from the company's normal way of doing business.

8.A total quality management approach that measures the capability of a process to perform defect-free work is called ______.

9.A methodology in which a company attempts to develop and establish total quality management from product to process by examining and emulating the best practices and techniques used in their industry is called ______.

10.A graphical method for evaluating whether a process is or is not in a state of statistical control is called a ______.

11.A diagram that is shaped like a fish and displays potential causes of one problem is called a ______or ______diagram.

12.A bar chart that displays a quantitative tallying of the numbers and types of defects that occur with a product is called a ______.

13.Two types of control charts for measurements are the ______chart and the ______chart. Two types of control charts for attribute compliance are the ______chart and the ______chart.

14.An x bar chart is constructed by graphing the ______of a given measurement computed for a series of small samples on a product over a period of time.

15.An R chart plots the sample ______. The centerline of an R chart is equal to the value of ______.

16.A p chart graphs the proportion of sample items in ______for multiple samples. The centerline of a p chart is equal to ______.

17.A c chart displays the number of ______per item or unit.

18.Normally, an x bar chart is constructed from 20 to 30 samples. However, assume that an x bar chart can be constructed using the four samples of five items shown below:

Sample 1Sample 2 Sample 3 Sample 4

23 21 19 22

22 18 20 24

21 22 20 18

23 19 21 16

22 19 20 17

The value of A2 for this control chart is ______.

The centerline value is ______.

The value of is ______.

The value of UCL is ______.

The value of LCL is ______.

The following samples have means that fall outside the outer control limits ______. In constructing an R chart from these data, the value of the centerline is ______. The value of D3 is ______and the value of D4 is ______. The UCL of the R chart is ______and the value of LCL is ______.

The following samples have ranges that fall outside the outer control limits ______.

19.p charts should be constructed from data gathered from 20 to 30 samples. Suppose, however, that a p chart could be constructed from the data shown below:

Sample n Number out of Compliance

1 70 3

2 70 5

3 70 0

4 70 4

5 70 3

6 70 6

The value of the centerline is ______.

The UCL for this p chart is ______.

The LCL for this p chart is ______.

The samples with sample proportions falling outside the outer control limits are ______.

20.c charts should be constructed using at least 25 items or units. Suppose, however, that a c chart could be constructed from the data shown below:

Item Number of

Number Nonconformities

1 3

2 2

3 2

4 4

5 0

6 3

7 1

The value of the centerline for this c chart is ______.

The value of UCL is ______and the value of LCL is ______.

21.A process is considered to be out of control if ______or more consecutive points occur on one side of the centerline of the control chart.

22.Four possible causes of control chart abnormalities are (at least eight are mentioned in the text) ______, ______, ______, and ______.

23.Suppose a single sample acceptance sampling plan has a c value of 1, a sample size of 10, a p0 of .03, and a p1 of .12. If the supplier really is producing 3% defects, the probability of accepting the lot is ______and the probability of rejecting the lot is ______. Suppose, on the other hand, the supplier is producing 12% defects. The probability of accepting the lot is ______and the probability of rejecting the lot is ______.

24.The Type II error in acceptance sampling is sometimes referred to as the ______risk. The Type I error in acceptance sampling is sometimes referred to as the ______risk.

25.Using the data from question 22, the producer's risk is ______and the consumer's risk is ______. Assume that 3% defects is acceptable and 12% defects is not acceptable.

26.Suppose a two-stage acceptance sampling plan is undertaken with c1 = 2, r1 = 6, and c2 = 7. A sample is taken resulting in 4 rejects. A second sample is taken resulting in 2 rejects. The final decision is to ______the lot.

ANSWERS TO STUDY QUESTIONS

1. Quality Control16. Noncompliance, p (average

proportion)

2. In-Process

17. Nonconformances

3. After-Process

18. 0.577, 20.35, 4.0, 22.658, 18.042,

4. Just-in-Time None, 4.0, 0, 2.115, 8.46, 0.00, None

5. Team Building19. .05, .128, .000, None

6. Quality Circle20. 2.143, 6.535, 0.00

7. Reengineering21. 8

8. Six Sigma22. Changes in the Physical Environment,

Worker Fatigue, Worn Tools, Changes

9. Benchmarking in Operators or Machines,

Maintenance, Changes in Worker

10. Control Chart Skills, Changes in Materials, Process

Modification

11. Fishbone, Ishikawa

23. .9655, .0345, .6583, .3417

12. Pareto Chart

24. Consumer’s, Producer’s

13. , R, p, c

25. .0345, .6583

14. Means

26. Accept

15. Ranges,

SOLUTIONS TO ODD-NUMBERED PROBLEMS IN CHAPTER 18

18.5 = 4.55, = 4.10, = 4.80, = 4.70,

= 4.30, = 4.73, = 4.38

R1 = 1.3, R2 = 1.0, R3 = 1.3, R4 = 0.2, R5 = 1.1, R6 = 0.8, R7 = 0.6

= 4.51 = 0.90

For Chart: Since n = 4, A2 = 0.729

Centerline: = 4.51

UCL: + A2 = 4.51 + (0.729)(0.90) = 5.17

LCL: – A2 = 4.51 – (0.729)(0.90) = 3.85

For R Chart: Since n = 4, D3 = 0 D4 = 2.282

Centerline: = 0.90

UCL: D4 = (2.282)(0.90) = 2.05

LCL: D3 = 0

Chart:

R Chart:

18.7 = .025, = .000, = .025, = .075,

= .05, = .125, = .05

p = .050

Centerline: p = .050

UCL: .05 + 3 = .05 + .1034 = .1534

LCL: .05 – 3 = .05 – .1034 = .000

p Chart:

18.9 = = 1.34375

Centerline: = 1.34375

UCL: = 1.34375 + 3 =

1.34375 + 3.47761 = 4.82136

LCL: = 1.34375 – 3 =

1.34375 – 3.47761 = 0.000

c Chart:

18.11While there are no points outside the limits, the first chart exhibits some problems. The chart ends with 9 consecutive points below the centerline. Of these 9 consecutive points, there are at least 4 out of 5 in the outer 2/3 of the lower region. The second control chart contains no points outside the control limit. However, near the end, there are 8 consecutive points above the centerline. The p chart contains no points outside the upper control limit. Three times, the chart contains two out of three points in the outer third. However, this occurs in the lower third where the proportion of noncompliance items approaches zero and is probably not a problem to be concerned about. Overall, this seems to display a process that is in control. One concern might be the wide swings in the proportions at samples 15, 16 and 22 and 23.

18.13n = 10 c = 0 p0 = .05

P(x = 0) = 10C0(.05)0(.95)10 = .5987

1 – P(x = 0) = 1 – .5987 = .4013

The producer's risk is .4013

p1 = .14 P(x = 0) = 15C0(.14)0(.86)10 =.2213

The consumer's risk is .2213

18.15n = 8 c = 0 p0 = .03 p1 = .1

p Probability

.01 .9227

.02 .8506

.03 .7837

.04 .7214 Producer's Risk for (p0 = .03) =

.05 .6634 1 – .7837 = .2163

.06 .6096

.07 .5596

.08 .5132 Consumer's Risk for (p1 = .10) = .4305

.09 .4703

.10 .4305

.11 .3937

.12 .3596

.13 .3282

.14 .2992

.15 .2725

OC Chart:

18.17

Stop



N



(no)

D  K  L  M (yes)  Stop



 Stop

 

(no) (no)

Start  A  B (yes) C  E  F  G

(yes)



H(no) J  Stop

(yes) 

 

I     

18.19Fishbone Diagram:

18.21 = .06, = .22, = .14, = .04, = .10,

= .16, = .00, = .18, = .02, = .12

p = = .104

Centerline: p = .104

UCL: .104 + 3 = .104 + .130 = .234

LCL: .104 – 3 = .104 – .130 = .000

p Chart:

18.23 n = 15, c = 0, p0 = .02, p1 = .10

p Probability

.01 .8601

.02.7386

.04 .5421

.06 .3953

.08 .2863

.10 .2059

.12 .1470

.14 .1041

Producer's Risk for (p0 = .02) = 1 – .7386 = .2614

Consumer's Risk for (p1 = .10) = .2059

OC Curve:

18.25 = 1.2100, = 1.2050, = 1.1900, = 1.1725,

= 1.2075, = 1.2025, = 1.1950, = 1.1950,

= 1.1850

R1 = .04, R2 = .02, R3 = .04, R4 = .04, R5 = .06, R6 = .02,

R7 = .07, R8 = .07, R9 = .06,

= 1.19583 = 0.04667

For Chart: Since n = 9, A2 = .337

Centerline: = 1.19583

UCL: + A2 = 1.19583 + .337(.04667) =

1.19583 + .01573 = 1.21156

LCL: – A2 = 1.19583 – .337(.04667) =

1.19583 – .01573 = 1.18010

For R Chart: Since n = 9, D3 = .184 D4 = 1.816

Centerline: = .04667

UCL: D4 = (1.816)(.04667) = .08475

LCL: D3 = (.184)(.04667) = .00859

Chart:

R chart:

18.27

= .12, = .04, = .00, = .02667,

= .09333, = .18667, = .14667, = .10667,

= .06667, = .05333, = .0000, = .09333

p = = .07778

Centerline: p = .07778

UCL: .07778 + 3 = .07778 + .09278 = .17056

LCL: .07778 – 3 = .07778 – .09278 = .00000

p Chart:

18.29 n = 10 c = 2 p0 = .10 p1 = .30

p Probability

.05 .9885

.10 .9298

.15 .8202

.20 .6778

.25 .5256

.30 .3828

.35 .2616

.40 .1673

.45 .0996

.50 .0547

Producer's Risk for (p0 = .10) = 1 – .9298 = .0702

Consumer's Risk for (p1 = .30) = .3828

18.31

= .05, = .00, = .15, = .075,

= .025, = .025, = .125, = .00,

= .10, = .075, = .05, = .05,

= .15, = .025, = .000

p = = .06

Centerline: p = .06

UCL: .06 + 3 = .06 + .11265 = .17265

LCL: .06 – 3 = .06 – .112658 = .00000

p Chart:

18.33There are some items to be concerned about with this chart. Only one sample range is above the upper control limit. However, near the beginning of the chart there are eight sample ranges in a row below the centerline. Later in the run, there are nine sample ranges in a row above the centerline. The quality manager or operator might want to determine if there is some systematic reason why there is a string of ranges below the centerline and, perhaps more importantly, why there are a string of ranges above the centerline.

18.35The centerline of the c chart indicates that the process is averaging 0.74 nonconformances per part. Twenty-five of the fifty sampled items have zero nonconformances. None of the samples exceed the upper control limit for nonconformances. However, the upper control limit is 3.321 nonconformances which, in and of itself, may be too many. Indeed, three of the fifty (6%) samples actually had three nonconformances. An additional six samples (12%) had two nonconformances. One matter of concern may be that there is a run of ten samples in which nine of the samples exceed the centerline (samples 12 through 21). The question raised by this phenomenon is whether or not there is a systematic flaw in the process that produces strings of nonconforming items.