6Sigma methods

7/29/2019 6Sigma methods

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2006 Prentice Hall, Inc. S6 1

OperationsManagement

Supplement 6

Statistical Process Control

2006 Prentice Hall, Inc.

PowerPoint presentation to accompanyHeizer/RenderPrinciples of Operations Management, 6eOperations Management, 8e


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Variability is inherent in every process

Natural or common causesSpecial or assignable causes

Provides a statistical signal when

assignable causes are present Detect and eliminate assignable

causes of variation

Statistical Process Control

(SPC)


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Natural Variations

Natural variations in the productionprocess

These are to be expected

Output measures follow a probabilitydistribution

For any distribution there is a measure

of central tendency and dispersion


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Assignable Variations

Variations that can be traced to a specificreason (machine wear, misadjustedequipment, fatigued or untrained workers)

The objective is to discover whenassignable causes are present andeliminate them


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Samples

To measure the process, we take samplesand analyze the sample statistics followingthese steps

(a) Samples of theproduct, say fiveboxes of cerealtaken off the filling

machine line, varyfrom each other inweight

Freque

ncy

Weight

#

#

# #

##

#

#

#

# # ## #

##

# # ## #

## # ##

Each of theserepresents onesample of five

boxes of cereal

Figure S6.1


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Samples

(b) After enoughsamples aretaken from astable process,

they form apattern called adistribution

The solid linerepresents the

distribution

Freque

ncy

WeightFigure S6.1


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Samples

(c) There are many types of distributions, includingthe normal (bell-shaped) distribution, butdistributions do differ in terms of central

tendency (mean), standard deviation orvariance, and shape

Weight

Central tendency

Weight

Variation

Weight

Shape

Frequ

ency

Figure S6.1



Samples

(d) If only naturalcauses ofvariation arepresent, theoutput of aprocess forms adistribution that

is stable overtime and ispredictable

Weight

Frequency

Prediction

Figure S6.1



Samples

(e) If assignablecauses arepresent, theprocess output isnot stable overtime and is notpredicable

Weight

Freq

uency

Prediction

????

??

????

?????????

Figure S6.1



Control Charts

Constructed from historical data, thepurpose of control charts is to helpdistinguish between natural variationsand variations due to assignablecauses



Types of Data

Characteristics thatcan take any real

value

May be in whole orin fractional

numbers Continuous random

variables

Variables Attributes

Defect-relatedcharacteristics

Classify productsas either good orbad or count

defects Categorical or

discrete randomvariables



Control Charts for Variables

For variables that have continuousdimensions

Weight, speed, length, strength, etc.

x-charts are to control the centraltendency of the process

R-charts are to control the dispersion ofthe process



Setting Chart Limits

For x-Charts when we know

Upper control limit(UCL) = x + z x

Lower control limit(LCL) = x - z x

where x = mean of the sample means or a targetvalue set for the process

z = number of normal standard deviationsx = standard deviation of the sample means

= / n

= population standard deviation

n = sample size



Setting Control Limits

Hour 1

Sample Weight ofNumber Oat Flakes

1 17

2 133 16

4 18

5 17

6 16

7 15

8 17

9 16

Mean 16.1

= 1

Hour Mean Hour Mean

1 16.1 7 15.2

2 16.8 8 16.4

3 15.5 9 16.3

4 16.5 10 14.8

5 16.5 11 14.2

6 16.4 12 17.3n = 9

LCLx= x - z x=16 - 3(1/3) = 15 ozs

For99.73% control limits, z= 3

UCLx= x + z x= 16 + 3(1/3) = 17 ozs



17 = UCL

15 = LCL

16 = Mean


Control Chartfor sample of9 boxes

Sample number

| | | | | | | | | | | |1 2 3 4 5 6 7 8 9 10 11 12

Variation dueto assignable

causes

Variation dueto assignable

causes

Variation due tonatural causes

Out ofcontrol

Out ofcontrol




For x-Charts when we dont know

Lower control limit(LCL) = x - A2R

Upper control limit(UCL) = x + A2R

where R = average range of the samples

A2 = control chart factor found in Table S6.1

x = mean of the sample means



Control Chart Factors

Table S6.1

Sample Size Mean Factor Upper Range Lower Rangen A2 D4 D3

2 1.880 3.268 0

3 1.023 2.574 0

4 .729 2.282 0

5 .577 2.115 0

6 .483 2.004 0

7 .419 1.924 0.076

8 .373 1.864 0.1369 .337 1.816 0.184

10 .308 1.777 0.223

12 .266 1.716 0.284


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Process average x= 16.01 ouncesAverage range R= .25Sample size n= 5


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UCLx = x + A2R= 16.01 + (.577)(.25)= 16.01 + .144= 16.154 ounces


FromTable S6.1


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UCLx = x + A2R= 16.01 + (.577)(.25)= 16.01 + .144= 16.154 ounces

LCLx = x - A2R= 16.01 - .144= 15.866 ounces


UCL = 16.154

Mean = 16.01

LCL = 15.866


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R Chart

Type of variables control chart

Shows sample ranges over time

Difference between smallest andlargest values in sample

Monitors process variability

Independent from process mean


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For R-Charts

Lower control limit(LCLR) = D3R

Upper control limit(UCLR) = D4R

where

R = average range of the samples

D3 and D4 = control chart factors from Table S6.1


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UCLR = D4R= (2.115)(5.3)= 11.2 pounds

LCLR = D3R= (0)(5.3)= 0 pounds

Average range R= 5.3 poundsSample size n= 5FromTable S6.1 D4= 2.115, D3 = 0

UCL = 11.2

Mean = 5.3

LCL = 0


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Mean and Range Charts

(a)

Thesesamplingdistributionsresult in the

charts below

(Sampling mean isshifting upward butrange is consistent)

R-chart(R-chart does notdetect change inmean)

UCL

LCL

Figure S6.5

x-chart(x-chart detectsshift in centraltendency)

UCL

LCL


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Mean and Range Charts

R-chart(R-chart detectsincrease indispersion)

UCL

LCL

Figure S6.5

(b)

Thesesamplingdistributionsresult in the

charts below

(Sampling meanis constant butdispersion isincreasing)

x-chart(x-chart does notdetect the increasein dispersion)

UCL

LCL


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Automated Control Charts


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Control Charts for Attributes

For variables that are categorical

Good/bad, yes/no,

acceptable/unacceptable Measurement is typically counting

defectives

Charts may measurePercent defective (p-chart)

Number of defects (c-chart)


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Control Limits for p-Charts

Population will be a binomial distribution,but applying the Central Limit Theorem

allows us to assume a normal distribution

for the sample statistics

UCLp= p + z p^

LCLp= p - z p^

where p = mean fraction defective in the sample

z = number of standard deviations

p = standard deviation of the sampling distribution

n = sample size

^

p(1 - p)np

=^


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p-Chart for Data EntrySample Number Fraction Sample Number FractionNumber of Errors Defective Number of Errors Defective

1 6 .06 11 6 .062 5 .05 12 1 .013 0 .00 13 8 .08

4 1 .01 14 7 .075 4 .04 15 5 .056 2 .02 16 4 .047 5 .05 17 11 .118 3 .03 18 3 .03

9 3 .03 19 0 .0010 2 .02 20 4 .04

Total= 80

(.04)(1 - .04)

100p= = .02^

p= = .0480

(100)(20)


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.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

.01

.00

Sample number

Fractiondefective

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

p-Chart for Data Entry

UCLp= p + z p= .04 + 3(.02) = .10^

LCLp= p - z p= .04 - 3(.02) = 0^

UCLp= 0.10

LCLp= 0.00

p= 0.04


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.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

.01

.00

Sample number

Fractiondefective

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

UCLp= p + z p= .04 + 3(.02) = .10^

LCLp= p - z p= .04 - 3(.02) = 0^

UCLp= 0.10

LCLp= 0.00

p= 0.04

p-Chart for Data Entry

Possibleassignable

causes present


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Control Limits for c-Charts

Population will be a Poisson distribution,but applying the Central Limit Theorem

allows us to assume a normal distribution

for the sample statistics

where c = mean number defective in the sample

UCLc= c +3 c LCLc= c -3 c


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c-Chart for Cab Company

c= 54 complaints/9 days= 6 complaints/day

|1

|2

|3

|4

|5

|6

|7

|8

|9

Day

Numberdefective14

12

10

8

6

42

0

UCLc = c +3 c

= 6 + 3 6= 13.35

LCLc = c -3 c= 3 - 3 6= 0

UCLc= 13.35

LCLc= 0

c= 6


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Patterns in Control Charts

Normal behavior.Process is in control.

Upper control limit

Target

Lower control limit

Figure S6.7


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Upper control limit

Target

Lower control limit


One plot out above (orbelow). Investigate forcause. Process is out

of control.

Figure S6.7


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Upper control limit

Target

Lower control limit


Trends in eitherdirection, 5 plots.Investigate for cause of

progressive change.

Figure S6.7


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Upper control limit

Target

Lower control limit


Two plots very nearlower (or upper)control. Investigate for

cause.

Figure S6.7


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Upper control limit

Target

Lower control limit


Run of 5 above (orbelow) central line.Investigate for cause.Figure S6.7


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Upper control limit

Target

Lower control limit


Erratic behavior.Investigate.

Figure S6.7


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Which Control Chart to Use

Using an x-chart and R-chart:

Observations are variables

Collect20 - 25 samples of n= 4, or n=5, or more, each from a stable processand compute the mean for the x-chart

and range for the R-chartTrack samples of n observations each

Variables Data


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Using the p-chart:

Observations are attributes that canbe categorized in two states

We deal with fraction, proportion, orpercent defectives

Have several samples, each withmany observations

Attribute Data


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Using a c-Chart:

Observations are attributes whosedefects per unit of output can becounted

The number counted is often a smallpart of the possible occurrences

Defects such as number of blemisheson a desk, number of typos in a pageof text, flaws in a bolt of cloth

Attribute Data

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