BITS PilaniPilani Campus

<METHODS IN SYSTEM

ENGINEERING>PROF V.MURALIDHAR

MS(SOFTWARE ENGINEERING)

(TECHNIP)-(2013-2014) BATCH

BITS Pilani, Pilani Campus

Control charts for Mean :

Upper control Limit

Central Line

Lower Control Limit

Data Analysis and Quality Assurance

.

/3

:),(

.

/3

:),(

22

2

2

21

1

1

nofvaluesdifferent

fortablesintabulatedisndAwhere

RAxUCL

XCL

RAxLCLchartRx

nofvaluesdifferent

fortablesintabulatedisncAwhere

sAxUCL

XCL

sAxLCLchartsx

BITS Pilani, Pilani Campus

Control charts for Range (R-chart)

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)/(31

)/(31

:tan

.

33

'

'

'

:tan

24

23

4

3

22221

2

2

1

tablesfromobtainedare

dDD

anddDDwhere

RDUCL

RCL

RDLCL

specifiednotdardS

tablesfromobtainedare

dandDdDandDdDwhere

DUCL

dCL

DLCL

specifieddardS

BITS Pilani, Pilani Campus

The choice between the mean and R-chart is a managerial problem. Mean chart is used to control the quality averages of the samples drawn from a given process, where as R-Chart is used to control the quality variability's of the samples.

Also it is better to construct R-Chart first. If it indicates that the dispersion of the quality by the process is out of control, it is better not to construct Mean chart, until the quality dispersion is brought under the control.

Mean chart reveals undesirable variations between samples while R-Chart reveals any undesirable variations within the samples

INTERPRETATION OF MEAN CHART AND R-CHART

BITS Pilani, Pilani Campus

1. A drilling machine bores holes with a mean diameter of 0.5230cm and a standard deviation of 0.0032cm. Calculate the 2-sigma and 3-sigma upper and lower control limits for means of samples of 4 and prepare a control charts. [Ans: LCL=0.5198cm,UCL=0.5262cm

LCL=0.5182cm,UCL=0.5278CM]

problems

BITS Pilani, Pilani Campus

Suppose random samples of size n are taken from the product output at some time intervals. If d be the number of defectives in a sample, then the fraction defective in that sample is given by p=d/n.

We may suppose that the number of defectives d=np is binomial with E(np)=nP and V(np)=

Where P is the population fraction defective.

The process will be in statistical control if P is the same for all the samples.This will be so if the statistic np lies within the 3-sigma limits nP±3

Control charts for number of defectives (or np-Chart)

)1( PnP

)1( PnP

BITS Pilani, Pilani Campus

np-Chart

)1(3

)1(3

)(

:tan

)'1('3'

'

)'1('3'

:tan

pnpnpUCL

npCL

pnpnpLCL

pdefectivefractionmeanthecalculate

specifiednotdardS

PnPnPUCL

nPCL

PnPnPLCL

specifieddardS

BITS Pilani, Pilani Campus

Control chart for Fraction Defective ( or p-chart)

tablesintabulatedisnAwhere

ppApUCL

pCL

ppApLCL

pdefectivefractionmeanthecalculate

specifiednotdardS

tablesintabulatedisnAwhere

pnpAnpUCL

pCL

ppApLCL

specifieddardS

/3

)1(

)1(

)(

:tan

/3

)'1(''

'

)'1(''

:tan

BITS Pilani, Pilani Campus

When the sample size is constant, it is immaterial whether one uses np-chart or p-chart. However, when the sample size varies, all the three control lines will vary with n, in the case of an np-chart. The resulting chart will be highly confusing.

On the other hand in a p-chart the central line remains invariant. Therefore in case the sample size varies, it is simpler and preferable to use the p-chart.

Choice between p-chart and np-chart

BITS Pilani, Pilani Campus

Defective : It is an item that fails to satisfy some given specification(s)

Defect : It is an event of the item’s lack of conformity to given specifications.

Every defectve item contains one or more defects.

Examples : defective rivets in an aircraft , exposed wires in a refrigerator, surface defects in a roll of a paper, Crumpled pages in a book, loose screws in a bicycle etc.

Control charts for the number of defects (or c-chart)

BITS Pilani, Pilani Campus

C-chart control limits

ccUCL

cCL

ccLCL

cdefectsofnumbermeanthecalculate

specifiednotdardS

ccUCL

cCL

ccLCL

specifieddardS

3

3

)(

:tan

'3'

'

'3'

:tan

BITS Pilani, Pilani Campus

The c-chart relates to samples of constant size. In case of varying sample sizes, it is more convenient to study the control charts for the number of defects per unit, u=c/n.

Control charts for per unit Defects ( or u-chart)

BITS Pilani, Pilani Campus

Control limits for u-chart

nuuUCL

uCL

nuuLCL

cdefectsofnumbermeanthecalculate

specifiednotdardS

defectsunitpermeantheisuwherenuuUCL

uCL

nuuLCL

specifieddardS

/3

/3

)(

:tan

/'3'

'

/'3'

:tan

BITS Pilani, Pilani Campus

An u-chart is applied to control the number of defects per unit in the case of fairly complex assembled units as T.V.sets, aircraft engines, refregirators etc.

Applications of u-chart

BITS Pilani, Pilani Campus

PROBLEMS