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DEPARTMENT OF INDUSTRIAL ENGINEERING &MANAGEMENTLaboratory Manual

Quality Assurance &Reliability Engineering

Laboratory

Edition

2011



L A B O R AT O RY M A N U A L

Quality Assurance & Reliability EngineeringLaboratory (07IM61)

Dept. of IEM, R.V.College of EngineeringMysore Road, Bangalore – 560059.

Phone 91-080- 8601700 Extn. 232 • Fax 91 -080-2915050

SEE: 3 hrs SEE Marks: 50 MarksClass Work: 3 Hrs/Week/Batch CIE Marks: 50 Marks

Lab In-charge:

C K Nagendra GupthaKiran K

Approved By

Dr. A.V.Suresh



R.V. College of EngineeringBangalore

Laboratory CertificateThis is to certify that Mr. / Ms. ___________________________________

has satisfactorily completed the course of experiments in Practical

__________________________________________________ prescribed by

University __________________________________________________ course

in the Laboratory of this College in the year _________________

Name of the Candidate _____________________________________________________

USN No. ____________________________ Examination Centre ___________________

Date of Practical Examination: _______________________________________________

Signature of the Teacher in-charge of the batch

Date: Head of the Department

MarksMaximum Obtained



Quality Assurance & Reliability Engineering

Dept. of I.E.M., R.V.C.E. Quality Assurance & Reliability Engineering Laboratory 4

DEPARTMENT OF INDUSTRIAL ENGINEERING & MANAGEMENT.Quality Assurance & Reliability Engineering Laboratory – 07IM61

SCHEME OF CONDUCT AND EVALUATION

CLASS: VI SEMESTER SEE Marks: 50YEAR: 2011 CIE Marks: 50

SL.No.

Expt.No. TITLE

MarksClassWork

Test

CYCLE – I1. QE 01 To test the Goodness of fit for the given quality characteristic using

Normal distribution 15

2. QE 02 To test the Goodness of fit for the given quality characteristic usingPoisson distribution 15

3. QE 03QE04

To test the Goodness of fit for the given quality characteristic usingBinomial & Uniform distribution 30

4. QE 05 Experiments on correlation and Simple regression15

5. QE 06 Conduction of Repeatability and Reproducibility studies for the givenmeasurement system 15

6. QE 07 Estimation of process variability using Deming‟s funnel Experiment /Quincunx Apparatus 15

7. QE 08 Developing Quality Function Deployment Matrix for a Product / Service 158. QE 09 Performing Quality Audit of a System 15

CYCLE – II9. QE 10

QE 11Construction of control chart for attribute quality characteristics &construction of control charts using SYSTAT / SQC PC IV software 30

10. QE 12 Assessing Process Capability of the given manufacturing process using

Normal probability paper method and process capability indices15

11. QE 13 Exercises on Attribute sampling Plans-Single, Double and Multiplesampling plans 15

12. QE 14 Conduction of Design of Experiments-Full Fractional approach for thegiven quality characteristics for machining operation. 15

13. QE 15 Exercises to demonstrate Taguchi‟s orthogonal Array technique throughDOE software 15

14. QE 16 Performing Failure Modes and Effects Analysis for a system 1515. QE 17 Estimation of System Reliability using Reliability Software Package 15

TEST -- 50TOTAL 180

(Min)50

EVALUATION SCHEME: CIE MARKS (Reduced to 50)

Proposed by Prepared by Approved by1. C. K. Nagendra Guptha R. Shekar Dr.A.V.Suresh2. Kiran. K





General Guidelines The Students are required to strictly follow the scheme of conduction of the experiments. The Students are hereby advised to prepare for the Experiments well in advance and the

timing of the lab slots are to be effectively utilized for the computational and analysisaspects of the experiments.

The Students are required to bring in their Blue Books duly complete in all respects,without which the students will not be allowed to do the experiments in the lab.

Submission of Lab Records complete in all respects is to be done in the next subsequentLab Class.

The Outline of the record is as follows: Aim/Objective of the Experiment. Procedure. Inference. Learning Experience.

Students must maintain strict academic discipline in the Laboratory. Use of external hard disks (thumb drives) in the Lab without prior permission of the Lab-in-charge will be viewed very seriously.

CIE Marks will be awarded on the overall assessment based on the discipline and thefulfilment of the Lab requirements.





TABLE OF CONTENTS

SL.No.

Expt.No.

Date TITLE Date of Submission Marks

TEST

TOTAL





Goodness of Fit – Normal Distribution

Aim: To test the goodness of fit for the quality characteristic, width across the flats for theHexagonal Head Bolts manufactured with respect to normal distribution.

Apparatus: Inspection Gauge, Sample bolts and Measuring Guide.

Form No. 01

Theory:A normal distribution is a continuous probability distribution. It is characterized by a bellshaped, symmetric distribution curve. It extends from - to + , the area under the curve is 1.

2

2

2 22

1

x x f exp

= Average = Shift parameter,= Standard Deviation = Scale parameter.

Procedure:

The experiment is conducted using Bolt heads produced from a production process. The qualitycharacteristic measured is width across the flats.

1. The given test sample of bolts, is taken from a group of bolts placed in different boxes

2. Each bolt is taken individually and with the marked face adjacent to the inner verticalface of bar, it is traversed along till it comes to its maximum traverse position. It shouldbe ensured that the bolt head always touch the bottom peak of the gauge.

3. The measuring guide is then traversed on the gauge, keeping the shorter arm on theoutside vertical face of the bar till it touches the bolt body. (The traverse direction isopposite to that of the bolt.)

4. Using this as a reference, the width across flats is gauged by reading the class in whichit falls.

Experiment

QE01





5. The value is tabulated in its appropriate class as shown in the Observation sheet. Chartof scale calibration is used to determine the upper specification and lower specificationof the class interval.

6. The steps repeated for the remaining bolts.

7. The mean and standard deviation of the sample data are calculated using the formula.The calculations are shown below.

8. The standard normal ordinate Z = x- / is calculated, where x is the upperspecification limit of the class interval, is the mean and is the standard deviation of the sample. From the normal tables the probability corresponding to the Z is obtained.This is the cumulative probability. Individual probability for each class interval aredetermined. Expected frequency is given by the formula Pi X Oi.

9. The

i

i i

E

E O 2

gives the Chi Square calculated value.

10. From the table „Chi square table value ‟ is noted for the specified degrees of freedomand confidence level. (DOF = n-k-1, where n = No. of class intervals of pooling, k =No. of parameters estimated, 6-2-1=3) confidence level = 0.5.

11. The data fits the normal distribution provided ² cal less than ²tab,

Experimental Setup – Inspection Gauge

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15





GRIFFIN MADE IN ENGLAND

Scale

1 2 3 4 5 6 7 8 9 10 11 12 13 14

ClassName

A

B

C D E F G H I J K L M

N

W I D T H A C R O S S F L A T

( m m

)

19.8700

19.9496 19.9496

20.0292 20.0292

20.1088 20.1088

20.1884 20.1884

20.2680 20.2680

20.3476 20.3476

20.4272 20.4272

20.5068 20.5068

20.5864 20.5864

20.6660 20.6660

20.7456 20.7456

20.8252

20.8252

20.9048 20.9048

20.9844 20.98

21.0

CHART OF SCALE CALIBRATION FOR INSPECTION GAUGE





Form No. 01DEPARTMENT OF INDUSTRIAL ENGINEERING AND MANAGEMENT.

R. V. COLLEGE OF ENGINEERING, BANGALORE – 560 059.OBSERVATION SHEET FOR

GOODNESS OF FIT TEST OF INDUSTRIAL PRODUCTION(Normal Distribution)

Expt. No: Date :Source of production: Straddle Milling Product Name : Hexagonal BoltVariable Checked:

Sl. No. ClassRange Class

Mid PointObserved Frequency

TotalLowerLimit

UpperLimit





TABLE FOR COMPUTATION OF SAMPLE STATISTICS

Class Name

Range on Scale

Width across flats range for scale class

Class mid point X mi

(mm)

h

AX U mi

i

O i O i x U i O i x U i 2

(XL) Lower(mm)

(XU) Upper(mm)

D 3 – 4

E 4 – 5

F 5 – 6

G 6 – 7

H 7 – 8

I 8 – 9

J 9 – 10

K 10 – 11

L 11 - 12

Total





h = Class width = 0.0796 A = Assumed mean = 20.3874

Mean = AO

U Oh

i

ii

Std. Dev. =22

i

i i

i

i i

O U O

O U O

h





CALCULATION OF ² STATISTIC FOR GOODNESS OF FIT µ ==

Class RANGE Xu ZCUM P i

( P i)P i

O i Pooled Oi E i = P i O i Pooled Ei(O i – E i)

2 E i

D 3 – 4

E 4 – 5

F 5 – 6

G 6 – 7

H 7 – 8

I 8 – 9

J 9 –

10

K 10 – 11

L 11 - 12





Cumulative Pi = P ( X Xu)

=

u X Z P





Note : On Pooling of frequencies: The accuracy of ² test will be better whenever it is ensuredthat the expected frequency is larger than 5. In order to guarantee this, O i, and E i are pooledsuitably.

Conclusion:





Goodness of Fit - Poisson DistributionAim: To determine the goodness of fit for Poisson distribution for the defectives in a sampledrawn from a lot.

Apparatus : Sampling Gadget, Beads - Red (920 Nos.) & Yellow.(200 Nos.)

Form No. 02

Theory:

The Poisson distribution is a discrete distribution bounded at 0 on the low side and unboundedon the high side. The Poisson distribution is a limiting form of the Hyper-geometricdistribution.

The Poisson distribution finds frequent use because it represents the infrequent occurrence of events whose rate is constant. This includes many types of events in time or space such asarrivals of telephone calls, defects in semiconductor manufacturing, defects in all aspects of quality control, molecular distributions, stellar distributions, geographical distributions of plants etc. It is an important starting point in queuing theory and reliability theory. Note that thetime between arrivals (defects) is exponentially distributed, which makes this distribution aparticularly convenient starting point even when the process is more complex.

!

X

e X P

x

Where is the average number of outcomes occurring in the given time interval or specifiedregion.

The Poisson distribution is applicabl e when the conditions such as N= 10n, where „N‟ is the lotsize and „n‟ is the sample size and „P‟ proportion defective should be less than 0.1 are satisfied.

Procedure:

Calculate the No. of red and yellow beads to suit the specified proportion. Mix the beads thoroughly in the top portion of the sampling gadget. Mix thoroughly and draw a sample of beads as per the sample size given. Observe the number of defectives (Yellow beads) in the sample. Use form No: 2 to record the frequency of occurrence of defectives.

Experiment

QE02





Repeat the procedure of drawing the sample and observing the number of defectives in thesample as many times as specified in the class.

The observed frequencies w.r.t the No. of defectives are tabulated in form 2. The calculations for X 2cal are done using a suitably designed spread sheet and the

calculations are shown in the tabulation.

i

ii

e

eOcal

22 )(

X2 tab can be obtained by referring to Chi square distribution and the degrees of freedom. Table value for X 2 tab read off. The degrees of freedom can be determined as Dof = number of classes after grouping –

number of parameters estimates -1. If X 2 cal < X 2 tab then conclude that the sample data provides evidence that the random

variable (no. of defectives in a sample of n items) follow a Poisson distribution. Otherwiseit does not fit the Poisson distribution.

It is desirable to pool the classes on the basis of the expected frequency (If the expectedfrequency is less than 5 pool the observed frequency and calculate the Chi Square value).

Experimental Setup:

BoxTray

Red beads Yellow beads





DEPARTMENT OF INDUSTRIAL ENGINEERING AND MANAGEMENTR. V. COLLEGE OF ENGINEERING, BANGALORE – 560 059.

OBSERVATION SHEET FORGOODNESS OF FIT TEST OF POISSON DISTRIBUTION

Expt. No. : Date :

Sample Size : Quantity in lot :Proportion Defective : Quantity Defective:

Sl.No.

No. ofDefectives Observed Frequency Total

Signature Staff In-Charge





Tabulation:

Sl.No.

No. ofDefects

ObservedFrequency

O i

PooledObservedFrequency

Cum.Probability

IndividualProbability

ExpectedFrequency

e i

PooledExpected

Frequency

i

i i

e e O 2

Specimen Calculation:

For Sl.No. 1





Inference:





Goodness of Fit - Binomial DistributionAim : To determine the goodness of fit for Binomial distribution.

Apparatus : Dies

Form No. 02

Theory:

x n x p p x n x p

1

!!!

x n x n

x n

The Binominal distribution is a discrete distribution bounded by [0,n]. Typically, it is used wherea single trial is repeated over and over, such as the tossing of a coin. The parameter, p, is theprobability of the event, either heads or tails, either occurring or not occurring. Each single trial isassumed to be independent of all others. For large n, the Binomial distribution may beapproximated by the Normal distribution.

The Binomial distribution has had extensive use in games, but is also useful in genetics, samplingof defective parts in a stable process, and other event sampling tests where the probability of theevent is known to be constant or nearly so.

Procedure:

Toss the 6 dies for 70 times.Count the number of times the occurrence of 1 (You may choose occurrence of any number) andrecord the observed frequency in the form No. 2

Determine the Binomial probability using the density function x n x p p

x n

x p

1

Calculate the expected frequency.Determine the Chi square table value with Degrees of Freedom = (No. of Class intervals – No. of parameters estimated – 1) and confidence level = 0.05.If Chi Square calculated value is less than the Chi Square table value data fits binomialdistribution.

Experiment

QE03

x = 0, 1, …, n,p = probability of the event occurringn = number of trials





Experimental Setup:






OBSERVATION SHEET FORGOODNESS OF FIT TEST OF BINOMIAL DISTRIBUTION

Expt. No. : Date :

Sl.No.

No. ofDefectives Observed Frequency Total

Signature of Staff in-charge





Tabulation:

Sl.No.

No. of Defects

ObservedFrequency

O i

PooledObserved

Frequency O i

IndividualProbability

ExpectedFrequency

ei

PooledExpected

Frequency

i

i i

e e O 2


For Sl.No. 1

Inference:





Goodness of Fit - Uniform DistributionAim : To determine the goodness of fit for Uniform distribution .

Apparatus: Dies

Form No. 18

Theory:

1

1minmax

x p

The Discrete Uniform distribution is a discrete distribution bounded on [min, max] with constantprobability at every value on or between the bounds. Sometimes called the discrete rectangulardistribution, it arises when an event can have a finite and equally probable number of outcomes.Note that the probabilities are actually weights at each integer, but are represented by broader barsfor visibility.

Procedure:

Toss the dies for 120 times. Count the number of times the occurrence of 1, 2, 3, etc., and record the observed

frequency in the form No. 18 Determine the probability of uniform distribution (Since there are six faces, the probability

of occurrence of any one face is equal to 1/6) Calculate the expected frequency (Probability of occurrence X Total No. of Observations,

1/6 X 120 = 20) Determine the Chi square table value with Degrees of Freedom = No. of Class intervals –

No. of parameters estimated – 1 and confidence level = 0.05.. If Chi Square calculated value is less than the Chi Square table value data fits uniform

distribution.Experimental Setup

Experiment

QE04

X = min, min+1, … max Min = minimum xMax = maximum x





Rolling Dies


OBSERVATION SHEET FORGOODNESS OF FIT TEST OF UNIFORM DISTRIBUTION

Expt. No. : Date :

Signature Staff in-Charge

FaceNo. Probability P i Observed Frequency Total





Tabulation:

Sl.No.

Face No. ObservedFrequency

O i

P i ExpectedFrequencye i = Pi x O i

i

i i

e e O 2


Inference:





Simple Linear Regression

Aim: To build Simple Linear Regression model and find the dependent variable with help of independent variable.

Apparatus : LVDT, slip gauge set.

Theory:Regression analysis is a technique for the modeling and analysis of numerical data consisting of values of a dependent variable (also called response variable or measurement) and of one or moreindependent variables (also known as explanatory variables or predictors). The dependent variablein the regression equation is modeled as a function of the independent variables, correspondingparameters ("constants"), the parameters are estimated so as to give a "best fit" of the data. Mostcommonly the best fit is evaluated by using the least squares method.

Regression can be used for prediction (including forecasting of time-series data), inference, andhypothesis testing, and modeling of causal relationships.

Procedure:

1. Slip gauges are cleaned.

2. Slip gauge dimension is measured the with the help of LVDT.

3. LVDT readings are tabulated for 20 slip gauges.

4. Simple linear regression model is built.

a. Calculations are done as shown in the table.

5. The graph is plotted.

Experiment

QE05





Department of Industrial Engineering and ManagementR.V. College of Engineering, Bangalore - 59

OBSERVATION SHEETDate: Name: Reg. No.Lab: Class: Expt No.

Title: Simple Linear Regression model

Observation: -Sl no Slip gauge LVDT

reading1

23456789

1011121314151617181920

Signature of the Staff in-charge





CalculationSl no Xi Yi (Xi-x) (yi-y) ((Xi-x)- (yi-y)) (Xi-x) 2

X= y=

n =no of observations

Regression Equation

Equation for st line: y=a+bxb=

a=y-bx Excepted graph:

X Y slope





Inference:





Repeatability and Reproducibility Studies Aim : To determine the repeatability and reproducibility for a given measurement system.

Apparatus : Micrometer

Form No. 16

Theory:Measurement system analysis, QS 9000, Measurement variation – stability, Bias, Linearity,Repeatability, Reproducibility.

Repeatability:

Standard deviation for repeatability

2

[R(Ave)]rangeAveraged e d2 is a control chart Constant

Equipment variation (EV) = Average range[R(Ave)] X k 1 (k1 4.56 for 2 trials)

Reproducibility:

Standard deviation for reproducibility

2

[Xdiff]differenceAveraged o

Appraiser variation (AV) =

nr EV

Average22

2k [Xdiff]difference

k 2 = 3.65 for 2 appraisalsn = number of partsr = number of trials.

Repeatability & Reproducibility = R & R = 22 AV EV

Part-to-part variation:

PV = Range of sample average X k 3 (k 3 – 1.62 for 10 parts)

Experiment

QE06





Total variation:

22m p TV

p = part to part standard deviation

2

[Rp(Ave)]averagepartof Ranged

(d2 – constant)

m = measurement system standards deviation

= 22oe

% Repeatability and Reproducibility = % R & R =100

&

TV

R R

Procedure:

Obtain a sample of 10 parts that represents the expected range of process variation. Number the parts 1 to 10 so that they are not visible to the appraisers, who are referred to

as say A & B. Appraiser A: Measure 10 parts in a random order and enter the results. Appraiser B:

Measure the same 10 parts in another random order without seeing the other‟s readings. Repeat the cycle using a different random order for measurement. Input the values into the Repeatability and Reproducibility software Obtain the result and draw the inference

Analysis R & RError

<10 % The measurement is acceptable

10 – 30% May be acceptable based upon importance of application cost of gauge, costrepairs.

> 30% The measurement system needs improvement.

Repeatability > Reproducibility GaugesReproducibility > Repeatability Appraiser .





Department of Industrial Engineering and ManagementR.V. College of Engineering, Bangalore – 560 059.

Gauges Repeatability and Reproducibility Data Sheet

EquipmentDescription

Part Number

EquipmentNumber

Characteristics

Specification DateLeast Count LocationSl.No. Appraiser A Appraiser B Part Average

Xp I Trial II Trial Range I Trial II Trial Range

1.

2. 3. 4. 5. 6. 7. 8. 9. 10.

Average X

Ra (Ave) Rb(Ave)

(Ave) Xa (Ave) Xb Rp(Ave)

X diff R(Ave)Equipment variation (EV) % EVAppraiser variation (AV) % AVRepeatability & Reproducibility(R&R)

% R&R

Part variation (PV) % PVTotal variation (TV)Remarks





Inference:





Deming’s funnel Experiment

Aim: to demonstrate the losses caused by tampering with real world processes at our places of work.

Theory:The funnel experiment is a mechanical representation of many real world processes at our placesof work. The aim of the experiment is to demonstrate the losses caused by tampering with thesevery same processes. The primary source of this tampering is the use of Management by Results,reactions to every individual result.In the experiment, a marble is dropped through a funnel, and allowed to drop on a sheet of paper,which contains a target. The objective of the process is to get the marble to come to a stop as closeto the target as possible. The experiment uses several methods to attempt to manipulate thefunnel‟s location such that the spread about the target is minimized. These methods are referred toas “rules”. Funnel Experiment: Rule 1During the first setup, the funnel is aligned above the target, and marbles dropped from thislocation. No action is taken to move the funnel to improve performance. This “rule” serves as our initial baseline for comparison with improved rules.The results of rule 1 are a disappointment. The marble does not appear to behave consistently. Themarble rolls off in various directions for various distances. Certainly there must be a better (smart)way to position the funnel to improve the pattern.

Experiment

QE07





Funnel Experiment: Rule 2During rule 2, we examine the previous result and take action to counteract the motion of themarble. We correct for the error of the previous drop. If the marble rolled 2 inches northeast, weposition the funnel 2 inches to the southwest of where it last was.A common example is worker adjustments to machinery. A worker may be working to make a

unit of uniform weight. If the last item was 2 pounds underweight, increase the setting for theamount of material in the next item by 2 pounds.Other real examples of Rule 2 include periodic calibrati ons. One checks a meter‟s measurementagainst a known standard, and adjusts the meter to compensate for the error against the standard.Many automated feedback mechanisms perform this adjustment continuously. Other examplesinclude taking action to change policies and production levels based upon on last month‟s budgetvariances, profit margins, and output.We also see this when setting next year‟s goals and targets based upon last year‟s levels.

Funnel Experiment: Rule 3A possible flaw in rule 2 was that it adjusted the funnel from its last position, rather than relativeto the target. If the marble rolled 2 inches northeast last time, we should set the funnel 2 inchessouthwest of the target. Then when the marble again rolls 2 inches northeast, it will stop on thetarget. The funnel is set at an equal and opposite direction from the target to compensate for thelast error.We see rule 3 at work in systems where two parties react to each other‟s actions. Their goal is tomaintain parity. If one country increases its nuclear arsenal, the rival country increases their

arsenal to maintain the perceived balance.If drug enforcement increases, prices rise due to increased demand, and drug runners haveincentive to go to further lengths due to increased price.A common example provided in economics courses is agriculture.A drought occurs one year causing a drop in crop output. Prices rise, causing farmers to plantmore crops next year. In the next year, there are surpluses, causing the price to drop. Farmers plantless next year. The cycle continues…





Funnel Experiment: Rule 4In an attempt to reduce the variability of the marble drops, we decide to allow the marble to fallwhere it wants to. We position the funnel over the last location of the marble, as that appears to bethe tendency of where the marble tends to stop.A common example of Rule 4 is when we want to cut lumber to a uniform length. We use thepiece we just cut in order to measure the location of the next cut.Other examples of Rule 4 include:

Brainstorming (without outside help) Adjusting starting time of the next meeting based upon actual starting time of the last

meeting Benchmarking, in order to find examples to follow A message is passed from one person to the next, who repeats it to another person, and so

forth. The junior worker trains the next new worker, who then trains the next, and so forth.





TamperingRules 2, 3, and 4 are all examples of process “tampering”. We take action (don‟t just stand there -do something!) as a result of the most recent result.Rule 2 leads to a uniform circular pattern, whose size is 40% bigger than the Rule 1 circle. This isbecause the error in distance from the funnel is independent from one marble drop to the next. Inpositioning the funnel relative to the previous marble drop, we add the error from the first drop (byrepositioning the funnel) to the second drop (the error in the marble).The standard deviation of adding n independent random variables is the square root of n times thestandard deviation of the individual. So the combined standard deviation is 1.4 times the originalstandard deviation. Note, this statistical principle is a standard question that appears on everyCertified Quality Engineer exam in some form or another.

The problems of Rule 2 a re corrected with “dead bands” in automated feedback mechanisms andbetter calibration programs. We wait for a certain error to build up before taking action. But howis the dead band determined? A control chart provides the answer. Plot the results on a controlchart, and recalibrate (or give a feedback signal) when a statistically significant change is detected.Program “dead bands” approximate the control chart action. Rules 3 and 4 tend to “blow up”. In rule 3, results swing back and forth with greater and greateroscillations from the target. In rule 4, the funnel follows a drunken walk off the edge of the table.In both cases, errors accumulate from one “correction” to the next, and the marble (or system)heads off to infinity. Rules 3 and 4 represent unstable systems, with over-corrections tending tooccur.

ConclusionSchemes to control the location of the funnel should be control chart based. In addition, we mayhave to think “outside of the box” to fix this system. If we lowered the height of the fun nel, wewould fundamentally reduce the variation in the process. If we added more layers of cloth or paperto cushion the marble‟s landing, then the marble would roll less. The impact of these changeswould be detected by the control chart, and would prove whether or not an improvement didoccur.





Galton ’s Quincunx Apparatus

Aim: to illustrate the “Law of dispersion”, or Central Limit Theorem .

Theory:

Sir Francis Galton (1822 – 1911) invented the Quincunx in the 1870s to demonstrate the law of error and the normal distribution. He was a prolific inventor, scientist, and mathematician and wasknighted in 1909. Beyond the Quincunx, Galton conceived the standard deviation, invented theuse of the regression line, and was the first to describe the phenomenon of the regression towardthe mean.

The quincunx is such a mechanical device and was first publicly demonstrated at the RoyalInstitute in February 1874 (Stigler, 1986). The device consists of two parallel, vertical planes,between which are many horizontal rows of equally spaced pins, with alternate rows offset by half the pin spacing; see figure 1. At the top is a funnel into which lead shot (small spherical balls of lead) is poured, and at the bottom a row of compartments to collect the shot. In Galton‟s own words (the following quotes are from Natural Inheritance), when lead shot is dropped into thedevice it “scampers deviously down through the pins in a curious and interesting way; each of them darting a step to the right or left, as the case may be, every time it strikes a pin.”

One purpose of this device is to illustrate the “Law of dispersion”, or central limit theorem, in thatthe “cascade [of shot] issuing from the funnel broadens as it descends,” and, at length, when collected at the compartments at the base, approximates the Binomial or Normal distribution.

Galton explains this as follows:

The principle on which the action of the apparatus depends is that a number of small andindependent accidents befall each shot in its career. In rare cases, a long run of luck continues tofavor the course of a particular shot towards either outside place, but in the large majority of instances the number of accidents that cause Deviation to the right, balance in a greater or lessdegree those that cause Deviation to the left. Therefore most of the shot finds its way into thecompartments that are situated near a perpendicular line drawn from the outlet of the funnel, andthe Frequency with which shots stray to different distances to the right or the left of that linediminishes in a much faster ratio than those distances increase.





Figure 1: Schematic of Galton‟s Quincunx Device

The Quincunx Model WD-7 (see figure 1) is an example of an off-the-shelf Quincunx board and iscomprised of a vertical board with 10 rows of pins. Beads are dropped via a funnel into the top of the board. As they descend through the board the beads will bounce either to the left or right asthey encounter each row of pins. The pins may be arranged such that a bead will come in contactwith 4, 6, 8, or 10 rows. As each bead leaves the final row of pins, it is captured in one of severalbead-wide bins which may be numbered for reference by the user. After a sufficient number of beads have been dropped, the height of the beads in the bins begins to resemble the classic bell-shaped curve. In reality, the distribution of beads is binomial; however, the normal distribution isapproximated when n, the number of rows of pins, is large (n = 4, 6, 8, or 10).





Quality Function Deployment

Aim: to conduct quality function deployment study on given product

Theory:Quality Function Deployment, or QFD, is a method used to identify critical customer attributesand to create a specific link between customer attributes and design parameters. Matrices are usedto organize information to help marketers and design engineers answer three primary questions:

What attributes are critical to our customers? What design parameters are important in driving those customer attributes? What should the design parameter targets be for the new design?

The organizing framework for the QFD process is a planning tool called the "house of quality".Working as a team, design engineers and marketers first establish critical customer attributes forthe product. These attributes become the rows of the central matrix of the house of quality. Theteam may group attributes into broader categories in order to simplify planning and analysis.

Steps in building house of quality

Example problem

Conduct a QFD study for handlebar stem for mountain bikes.

Step 1. List the customer requirements

QFD starts with list of goals / objectives. This list is oftenreferred as WHATs that the customer needs or expects inparticular Product.

Experiment

QE08





P r i m a r y

S e c o n d a r y

T e r t i a r y

C u s t o m e r r e q u i r e m e n t s

( W H A T s )

A e s t h e t i c s Reasonable cost

Aerodynamic look Nice finishCorrosion resistant

P e r f o r m a n c e

Light weight

Strength

Durable





Step 2. List Technical Descriptors

The next step in QFD is to come up with engineering characteristics or technical descriptors(HOWs) that will affect one or more customer requirements. The technical descriptors make upthe ceiling or second floor of the house of quality

P r i m a r y

S e c o n d a r y

T e r t i a r y

T e c h n i c a l d e s c r i p t o r s

( H O W s )

M a t e r i a l

s e l e c t i o n

Steel

Aluminum

Titanium

M a n u f a c t u r i n g

p r o c e s s

Welding

Die casting

Sand casting

Forging

Powder metallurgy

Step 3 develop a relation matrix between WHATs & HOWs

The next step in building a house of quality is to compare the customer requirement & technicaldescriptors and determine their respective relationship.

Then fill the relationship matrix with graphical symbols indicating the degree of influencebetween each technical descriptor and each customer requirement like

A solid circle for strong relationship ( )

A single circle for medium relationship( )

A triangle represents a week relationship ( )





Step 4. Develop an inter relationship matrix between HOWs

The roof of the house of quality called the correlation matrix, is used to identify any interrelationship between each of the technical descriptors. Symbols are used to describe the strengthof the interrelationship

A solid circle for strong positive relationship ( ) A single circle for positive relationship( ) A x represents negative relation ship (x) An asterisk representing a strong negative relationship

The symbols describe the direction of correlation. In other words a strong positiveinterrelationship would be a nearly perfectly positive correlation. A strong negativeinterrelationship would be a nearly perfectly negative correlation.





Step 5-Competitive Assessments

The competitive assessments are a pair of weighted tables (or graphs) that depict item for itemhow competitive products compare with current organization products. The competitiveassessment tables are separated into two categories, customer assessment and technical

assessment.

CUSTOMER COMPETITIVE ASSESSMENT

The customer competitive assessment is the block of columns corresponding to each customerrequirement in the house of quality on the right side of the relationship matrix, The numbers 1through 5 are listed in the competitive evaluation column to indicate a rating of 1 for worst and 5for best.

Technical competitive assessment

The technical competitive assessment is often useful in uncovering gaps in engineering judgment.When a technical descriptor directly relates to a customer requirement, a comparison is madebetween the customer's competitive evaluation and the objective measure ranking.





Step 6 - Develop Prioritized Customer Requirements

The prioritized customer requirements make up a block of columns corresponding to eachcustomer requirement in the house of quality on the right side of the customer competitiveassessment. These prioritized customer requirements contain columns for importance to customer,target value, scale-up factor, sales point, and an absolute weight.

IMPORTANCE TO CUSTOMER

Customer requirement by assigned rating. Numbers 1 through 10 & are listed in the importance tocustomer column to indicate a rating of 1 for least important and 10 for very important. In otherwords, the more important the customer requirement, the higher the rating.

TARGET VALUE

The target-value column is on the same scale as the customer competitive assessment (l for worst,5 for best). This column is where the QFD team decides whether they want to keep their productunchanged, improve the product, or make the product better than the competition.

The target value is determined by evaluating the assessment of each customer requirement andsetting a new assessment value that either keeps the product as is, improves the product, orexceeds the competition. For instance, if lightweight has a product rating of 3 and the QFD teamwishes to improve their product, and then the target value could be assigned a value of 4.

SCALE-UP FACTOR





The scale-up factor is the ratio of the target value to the product rating given in the customercompetitive assessment. The higher the number, the more effort is needed. The scale-up factor isdetermined by dividing the target value by the product rating given in the customer competitiveassessment. For instance, if lightweight has a product rating of 3 and the target value is 4, then thescale-up factor is 1.3. The scale-up factor for designing a handlebar stem for a mountain bike isshown in Figure. Note that the numbers for scale-up factor are rounded off in Figure.

SALES POINT

The sales point tells the QFD team how well a customer requirement will sell. The objective hereis to promote the best customer requirement. The sales point is determined by identifying thecustomer requirements that will help the sale of the product. For instance, an aerodynamic look could help the sale of the handlebar stem, so the sales point is given a value of 1.5. If a customerrequirement will not help the sale of the product, the sales point is given a value of 1.

ABSOLUTE WEIGHT

Finally, the absolute weight is calculated by multiplying the importance to customer, scale-upfactor, and sales point:

Absolute Weight = (Importance to Customer) (Scale-up Factor)(Sales Point)

The absolute weight is determined by multiplying the importance to customer, scale-up factor, andsales point for each customer requirement. For instance, for reasonable cost the absolute weight is8 x 1.3 x 1.5 = 16.





Step 7-Develop Prioritized Technical Descriptors

The prioritized technical descriptors make up a block of rows corresponding to each technicaldescriptor in the house of quality below the technical competitive assessment; these prioritizedtechnical descriptors contain degree of technical difficulty, target value, and absolute and relativeweights. The QFD team identifies technical descriptors that are most needed to fulfill customerrequirements and need improvement

DEGREE OF DIFFICULTYDegree of technical difficulty for implementing each technical descriptor, which is expressed inthe first row of the prioritized technical descriptors. The degree of technical difficulty, when used,helps to evaluate the ability to implement certain quality improvements.

The degree of difficulty is determined by rating each technical descriptor from 1 (least difficult) to10 (very difficult). For instance, the degree of difficulty for die-casting is 7, whereas the degree of difficulty for sand casting is 3 because it is a much easier manufacturing process.

TARGET VALUEA target value for each technical descriptor is also included below the degree of technicaldifficulty. This is an objective measure that defines values that must be obtained to achieve thetechnical descriptor. The target value for each technical descriptor is determined in the same waythat the target value was determined for each customer requirement.

ABSOLUTE WEIGHT





The last two rows of the prioritized technical descriptors are the absolute weight-and relativeweight. A popular and easy method for determining the weights is to assign numerical values tosymbols in the relationship matrix symbols.

n

i j RijCia 1

aj= row vector of absolute weight for the technical descriptor.(I=1……,m)

Rij =weight assigne d to the relationship matrix (I=1….,n,j=1….,m)

ci= column vector of importance to customer requirements(I=1…..,n)

m = no of technical descriptor.

m = no of customer requirements.

For example the absolute weight for aluminum is9*8+1*5+9*5+9*2+9*7+3*5+3*3=227

Relative weight

The relative weight for j th technical descriptor is given by replacing the degree of importance forcustomer requirement with absolute weight for customer requirement it is.

n

i

Rijdibj1

b= row vector of relative weight for technical descriptor (j=1….,m)

di =Column vector of absolute weight for customer requirements (I=…..,n)

For example the relative weight for die-casting is

3*16+9*8+9*5+3*2+0*18+3*5+9*3=213




Department of Industrial Engineering and ManagementR.V. College of Engineering, Bangalore - 59

OBSERVATION SHEETDate: Name: Reg. No.Lab: Class: Expt No.

Title quality function deployment study

qare student cycle 1

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