chap 17 statistical quality control

8/3/2019 Chap 17 Statistical Quality Control

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© 2008 Prentice-Hall, Inc.

Chapter 17

To accompanyQuantitative Analysis for Management , Tenth Edition ,

by Render, Stair, and Hanna Power Point slides created by Jeff Heyl

Statistical Quality Control

© 2009 Prentice-Hall, Inc.



© 2009 Prentice-Hall, Inc. 17 – 2

Learning Objectives

After completing this chapter, students will be able to:

Define the quality of a product or service

Develop four types of control charts: x, R, p, and c

Understand the basic theoretical underpinnings ofstatistical quality control, including the centrallimit theorem

Know whether a process is in control




Chapter Outline

17.1 Introduction

17.2 Defining Quality and TQM

17.3 Statistical Process Control

17.4 Control Charts for Variables

17.5 Control Charts for Attributes




Introduction

Quality is often the major issue in a purchasedecision

Poor quality can be expensive for both the

producing firm and the customer Quality management, or quality control (QC ),

is critical throughout the organization

Quality is important for manufacturing andservices

We will be dealing with the most importantstatistical methodology, statistical process control (SPC )




Quality of a product or service is the degree towhich the product or service meets specifications

Increasingly, definitions of quality include anadded emphasis on meeting the customer‟s

needs Total quality management (TQM ) refers to a

quality emphasis that encompasses the entireorganization from supplier to customer

Meeting the customer‟s expectations requires anemphasis on TQM if the firm is to complete as aleader in world markets

Defining Quality and TQM




Several definitions of quality “Quality is the degree to which a specific product

conforms to a design or specification.” (Gilmore, 1974)

“Quality is the totality of features and characteristics of

a product or service that bears on its ability to satisfystated or implied needs.” (Johnson and Winchell, 1989)

“Quality is fitness for use.” (Juran, 1974)

“Quality is defined by the customer; customers wantproducts and services that, throughout their lives, meetcustomers‟ needs and expectations at a cost thatrepresents value.” (Ford, 1991)

“Even though quality cannot be defined, you know whatit is.” (Pirsig, 1974)

Defining Quality and TQM

Table 17.1


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Statistical Process Control

Statistical process control involves establishingand monitoring standards, makingmeasurements, and taking corrective action as aproduct or service is being produced

Samples of process output are examined If they fall outside certain specific ranges, the

process is stopped and the assignable cause islocated and removed

A control chart is a graphical presentation of dataover time and shows upper and lower limits of theprocess we want to control




Patterns to look for in control charts

Normal behavior

Uppercontrol

limit

Target

Lowercontrollimit

One plot out above.Investigate for cause.

One plot out below.

Investigate for cause.

Figure 17.1



Uppercontrollimit

Target

Lowercontrollimit

Uppercontrollimit

Target

Lowercontrollimit



Two plots near lower control.Investigate for cause.

Run of 5 below central

line. Investigate for cause.

Two plots near upper controlInvestigate for cause.

Run of 5 above central line.

Investigate for cause.

Figure 17.1





Trends in either direction 5plots. Investigate for causeof progressive change.

Uppercontrol

limit

Target

Lowercontrollimit Erratic behavior.

Investigate.

Figure 17.1




Building control charts Control charts are built using averages of

small samples

The purpose of control charts is to distinguishbetween natural variations and variations due to assignable causes




Natural variations Natural variations affect almost every

production process and are to be expected,even when the process is n statistical control

They are random and uncontrollable When the distribution of this variation is

normal it will have two parameters

Mean, (the measure of central tendency of

the average) Standard deviation, (the amount by which

smaller values differ from the larger ones)

As long as the distribution remains withinspecified limits it is said to be “in control”




Assignable variations When a process is not in control, we must

detect and eliminate special (assignable )causes of variation

The variations are not random and can becontrolled

Control charts help pinpoint where a problemmay lie

The objective of a process control system is to provide a statistical signal when assignable causes of variation are present



Control Charts for Variables

The x-chart (mean) and R-chart (range) arethe control charts used for processes thatare measured in continuous units

The x-chart tells us when changes haveoccurred in the central tendency of theprocess

The R-chart tells us when there has been a

change in the uniformity of the process Both charts must be used when monitoring

variables




The Central Limit Theorem

The central limit theorem is the foundationfor x-charts

The central limit theorem says that the

distribution of sample means will follow anormal distribution as the sample sizegrows large

Even with small sample sizes the

distribution is nearly normal





The central limit theorem says

1. The mean of the distribution will equal thepopulation mean

2. The standard deviation of the samplingdistribution will equal the population standarddeviation divided by the square root of thesample size

n μx

x x

and

We often estimate and with the average of allsample means ( )

x

x

μ





x

Figure 17.2 shows three possible populationdistributions, each with their own mean () andstandard deviation ( )

If a series of random samples ( and

so on) each of size n is drawn from any of these,the resulting distribution of the „s will appear asin the bottom graph in the figure

Because this is a normal distribution1. 99.7% of the time the sample averages will fall between

±3 if the process has only random variations2. 95.5% of the time the sample averages will fall between±2 if the process has only random variations

If a point falls outside the±3 control limit, weare 99.7% sure the process has changed

,,,,4321

x x x x

i x




|

–3 x

|

–1 x

|

+1 x

|

+2 x

|

–2 x

|

+3 x

|

x =

(mean) n

x

x

errorStandard


Population and sampling distributions

Sampling Distribution of Sample Means (Always Normal)

Normal Beta Uniform

= (mean)

x = S.D.

= (mean)

x = S.D.

= (mean)

x = S.D.

99.7% of all xfall within±3 x

95.5% of all x fall within±2 x

Figure 17.2




Setting the x-Chart Limits

If we know the standard deviation of the process,we can set the control limits using

x z xUCL )(limitcontrolUpper

x z xUCL )(limitcontrolLower

where

x

x

= mean of the sample means z = number of normal standard deviations

= standard deviation of the samplingdistribution of the sample means =

n

x




Box Filling Example

ounces1711636

2316

x xz xUCL

ounces15116

36

2316

x xz x LCL

336216 z n x x ,,,

A large production lot of boxes of cornflakes issampled every hour

To set control limits that include 99.7% of thesample, 36 boxes are randomly selected andweighed

The standard deviation is estimated to be 2ounces and the average mean of all the samplestaken is 16 ounces

So and the control limits

are




Box Filling Example

If the process standard deviation is not availableor difficult to compute (a common situation) theprevious equations are impractical

In practice the calculation of the control limits isbased on the average range rather than thestandard deviation

R A xUCL x 2

R A x LCL x 2

where

= average of the samples A2 = value found in Table 17.2

= mean of the sample means

R

x




Factors for Computing Control Chart Limits

SAMPLE SIZE, n MEAN FACTOR, A2 UPPER RANGE, D4 LOWER RANGE, D3

2 1.880 3.268 0

3 1.023 2.574 0

4 0.729 2.282 0

5 0.577 2.115 0

6 0.483 2.004 07 0.419 1.924 0.076

8 0.373 1.864 0.136

9 0.337 1.816 0.184

10 0.308 1.777 0.223

12 0.266 1.716 0.284

14 0.235 1.671 0.329

16 0.212 1.636 0.364

18 0.194 1.608 0.392

20 0.180 1.586 0.414

25 0.153 1.541 0.459

Table 17.2




Super Cola Example

Super Cola bottles are labeled “net weight 16ounces”

The overall process mean is 16.01 ounces andthe average range is 0.25 ounces

What are the upper and lower control limits forthis process?

R A xUCL x 2

16.01 + (0.577)(0.25) 16.01 + 0.144 16.154

R A x LCL x 2

16.01 – (0.577)(0.25) 16.01 – 0.144 15.866




Setting Range Chart Limits

R

We have determined upper and lower controllimits for the process average

We are also interested in the dispersion orvariability of the process

Averages can remain the same even if variabilitychanges

A control chart for ranges is commonly used tomonitor process variability

Limits are set at±3 for the average range




Setting Range Chart Limits

We can set the upper and lower controls using

where

UCL R = upper control chart limit for the range LCL R = lower control chart limit for the range D4 and D3 = values from Table 17.2

R DUCL R 4

R D LCL R 3




A process has an average range of 53 pounds

If the sample size is 5, what are the upper andlower control limits?

From Table 17.2, D4

= 2.114 and D3

= 0

Range Example

R DUCL R 4

(2.114)(53 pounds)

112.042 pounds

R D LCL R 3

(0)(53 pounds)

0




Five Steps to Follow in Using x and R-Charts

1. Collect 20 to 25 samples of n = 4 or n = 5 from astable process and compute the mean and rangeof each

2. Compute the overall means ( and ), set

appropriate control limits, usually at 99.7% leveland calculate the preliminary upper and lowercontrol limits. If process not currently stable, usethe desired mean, m, instead of to calculatelimits.

3. Graph the sample means and ranges on theirrespective control charts and determine whetherthey fall outside the acceptable limits

x

x

R




Five Steps to Follow in Using x and R-Charts

4. Investigate points or patterns that indicate theprocess is out of control. Try to assign causes forthe variation and then resume the process.

5. Collect additional samples and, if necessary,

revalidate the control limits using the new data

chart x R-chart




Control Charts for Attributes

We need a different type of chart tomeasure attributes

These attributes are often classified as

defective or nondefective There are two kinds of attribute control

charts1. Charts that measure the percent defective in

a sample are called p-charts 2. Charts that count the number of defects in a

sample are called c-charts




p-Charts

Attributes that are good or bad typically followthe binomial distribution

If the sample size is large enough a normaldistribution can be used to calculate the controllimits

p p z pUCL

p p z p LCL

where

= mean proportion or fraction defective in the sample

z = number of standard deviations

= standard deviation of the sampling distribution whichis estimated by where n is the size of each sample

p

p

pˆ

n

p p p

)1(ˆ




ARCO p-Chart Example

Performance of data-entry clerks at ARCO ( n = 100)

SAMPLENUMBER

NUMBER OFERRORS

FRACTIONDEFECTIVE

SAMPLENUMBER

NUMBER OFERRORS

FRACTIONDEFECTIVE

1 6 0.06 11 6 0.06

2 5 0.05 12 1 0.013 0 0.00 13 8 0.08

4 1 0.01 14 7 0.07

5 4 0.04 15 5 0.05

6 2 0.02 16 4 0.04

7 5 0.05 17 11 0.11

8 3 0.03 18 3 0.03

9 3 0.03 19 0 0.00

10 2 0.02 20 4 0.04

80





We want to set the control limits at 99.7% of therandom variation present when the process is incontrol so z = 3

04020100

80

examinedrecordsofnumberTotal

errorsofnumberTotal

.))((

p

020100

0401040.

).)(.(ˆ p

1000203040 .).(.ˆ

p p z pUCL

00203040 ).(.ˆ p p z p LCL

Percentage can‟t be negative





p-chart for data entry

UCL p = 0.10

LCL p = 0.00

040. p

0.12 –

0.11 –

0.10 – 0.09 –

0.08 –

0.07 –

0.06 –

0.05 –

0.04 –

0.03 – 0.02 –

0.01 –

0.00 –

F r a c t i o n

D e f e c t i v e

|

1

|

2

|

3

|

4

|

5

|

6

|

7

|

8

|

9

|

10

|

11

|

12

|

13

|

14

|

15

|

16

|

17

|

18

|

19

|

20

Sample NumberFigure 17.3

Out of Control





Excel QM‟s p-chart program applied to the ARCOdata showing input data and formulas

Program 17.1A





Output from Excel QM‟s p-chart analysis of theARCO data

Program 17.1B




c-Charts

In the previous example we counted the number ofdefective records entered in the database

But records may contain more than one defect

We use c-charts to control the number of defects

per unit of output c-charts are based on the Poisson distribution

which has its variance equal to its mean

The mean is and the standard deviation is equalto

To compute the control limits we use

c

c

c c 3



Red Top Cab Company c-Chart Example

The company receives several complaints eachday about the behavior of its drivers

Over a nine-day period the owner received 3, 0, 8,9, 6, 7, 4, 9, 8 calls from irate passengers for atotal of 54 complaints

To compute the control limits

daypercomplaints69

54 c

Thus

3513452366363 .).( c cUCL c

0452366363 ).( c c LCL c

chap 17 statistical quality control

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