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है”ह”ह
IS 10427-2 (2006): Designs for industrial experimentation,Part 2: Orthogonal arrays [MSD 3: Statistical Methods forQuality and Reliability]
IS 10427( Part 2 ): 2006
Indian Standard
DESIGN FOR ~DUSTRIAL EXPERIMENTATION
PART 2 ORTHOGONAL ARRAYS
(First Revision )
ICS 03.120.30
0 BIS 2006
B-U-REAU OF .INDIAN STANDARDS
MANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG
NEW DELHI 110002
June 2006 Pdce Group 10
.—. — ..
Statistical Methods for Quality and Reliability Sectional Committee, MSD 3
FOMWORD
This Indian Standard ( Part 2 ) ( First Revision ) was adopted by the Bureau of Indian Standards, afier the drafifinalized by the Statistical Methods for Quality and Reliability Sectional Committee had been approved by theManagement and Systems Division Council.
Industrial organizations constantly face~he problem of decision making regarding product/process design, processspecifications, quality improvement, identification of dominant factors affecting quality, cost reduction, importsubstitution, etc. In all such problems, one is confronted with several alternatives and one has to choose thatalternative which satisfies the requirements at minimum cost. For taking a right decision in all such cases, anexperiment may have to be carried out either to discover something about a particular process or to compare theeffect of several conditions on the phenomenon under study.
The effectiveness of an experiment depends to a large extent on the manner in which the data are collected. Themethod of data collection may adversely affect the conclusion that can be drawn from the experiment. If, therefore,proper designing of an experiment is not made, no inferences maybe drawn or if drawn may not answer the questionsto which the experimenter is seeking an answer. The designing of an.experiment is essenti?.lly the determinationof the pattern of observations to be collected. A good experimental design is one that answer efficiently andunambiguously these questions, which are to be resolved and furnishes the required information with a minimumof experimental effort. For this purpose the experiments maybe statistically designed.
Part 1 of this standard covers the basic designs, namely, completely randomized design ( CRD ), randomized blockdesigns ( RBD ), latin square designs, balanced incomplete block designs ( BIBD ) and factorial designs. Thefactorial designs enable evaluation of main effects and interactions and also provide more efficient estimates,Iiowever, one disadvantage with the factorial designs is that it calls for a large number of experiments. It is possibleto reduce the numkr of experiments and still estimate most of the important effects. This is achieved by fractionalfactorial experiments. By carrying out fractional factorial experiments, some information is lost. But when thereare several factors, higher order interactions are generally not of much importance and in some cases difficult tointerpret. Hence information on these “higher order interactions are deliberately ignored to reduce the number ofexperiments.
Orthogonal array ( OA ) designs, which are discussed in this part, constitute one particulartype of the fractionalfactorial designs. A special feature of these designs is the associated concept of linear graphing, which enablesa scientist or an engineer to design complicated experiments without requiring sophisticated statistical knowledge.The OA designs can meet the needs of various practical situations, such as:
a) studyiwg the effect of various factors having different number of levels,
b) analyzing nested factorial effects when nested factors coexist with some other common factors, and
c) estimating all the main effects along with a few desired lower order interactions.
The following changes have been made in this revision:
a) More commonly used designs like L12(2”) for 2“ series and L18(2 x 37) have been included.
b) Table 2 and Table 7 of pre-revised version have been corrected.
c) Other editorial corrections have been incorporated,
The other part in the series is:
Part 1 Standard designs
The composition of the Committee responsible for the formulation of this standai-d is given in Annex G.
—- -. -.
.1S 10427( Part 2 ): 2006
Indian Standard
DESIGN FOR WDUSTNAL EXPERIMENTATION
PART 2 ORTHOGONAL ARRAYS
(First Revision )
1 SCOPE
This standard ( Part 2 ) provides methods ofplanning and conducting experiments usingorthogonal array ( OA ) tables when all the factorsare either at two levels or at three levels. Theprocedure is also discussed when some of the factorsare at two levels and the remaining at three or fourlevels. It also describes the procedure for the analysisof the data and selection of the optimum level ofeach factor.
2 REFERENCES
The fotlowing standards contain provisions, whichthrough reference in this text constitute provisionsof this standard. At the time of publication, the editionsindicated were valid. All standards are subject torevision and parties to agreements based on thisstandard are encouraged to investigate the possibilityof applying the most recent editions of the standardsindicated below:
IS No. Title
6200 Statistical tests of significance:( Part 1 ): 2003 Part 1 t-, Normal and F-tests
7920 Statistical vocabulary and symbols:( Part 3 ): 1996 Part 3 Design of experiments
4905:1968 Methods for random sampling
3 TERMINOLOGY
For the purpose of this standard, the definitionsgiven in IS 7920 ( Part 3 ) and the following shallapply.
3.1 Orthogonal Array ( OA ) Tables — An N x narray ofs symbols is said to be an Orthogonal Arrayof strength t if every N x t sub-array contains everyt-plet ofs- symbols an equal number of times, say k.Thus N=L. St. An OA of strength t is represented asOA ( N, n, S, f ), where N denotes the number of
experiments, n denoted the number of factors andsis the number of levels of the factors.
It is known that OAS have a close association withfractional factorial experiments. An OA of strengthtwo is an orthogonal main effect plan.
1
4 USE OF ORTHOGONAL ARRAY ( OA )TABLES
4.1 In general, an experiment in which all possiblecombinations of factor levels are realized is calledafull factorial experiment. Therefore, the total numberof experiments ( N ) to be conducted is equal to s“,wheres is the number of levels of each factor and-n isnumber of factors, if there are 15 factors, and eachfactor has 2 levels, then the total number ofexperiments to be conducted is 2*5. The number ofexperiments in a factorial experiment is considerablyhigh and sometimes prohibitive in actual use. In fact,orthogonal arrays ( OA ) tables evolved through theconcept of fractional replication, that is, sacrificinginformation about interactions which are usually notvery important in an industrial.proj ect, could find itselfin sound foating in minimizing the number ofexperiments. It is seen that while investigating theinfluence of 15 factors ( each at two levels ), thenumber of experiments can be reduced to 16 by usingOA tables. The effectiveness of using OA tablesdepends solely Qn the successful selection of thescheme of confounding the interaction effects andon the skilful strategy of choosing the levels ofthe factors and running the experiment. A priorinform~tion on interactions render a great serviceto experimenter in this case.
4.2 An orthogonal array is also representedas L~[ (s)”],
where
n = number of factors;
s = number of levels of each factor; and
N = total number of experiments to be conducted.
4.3 Orthogonal arrays were known as square gamesin former days. Recently these arrays have beeneffectively applied in the layout of experiment. Anexample of orthogonal array of 27 or simply L8( 27)array is given in Table 1, as.per Annex A.
4.3.1 There are 8 experiments in this array. Eachcolumn consists of 1 and 2 each four times. Whentwo columns consist of figures 1 and 2 and alsothey have same number of combinations, that is
(1, 1 ),(1,2),(2, 1 ),and(2,2) arerepeatedsamenumber of times, the two columns are said to bebalanced or orthogonal,
— .-____, -_._.. _
IS 10427( Part 2 ): 2006
Table 1 Orthogonal Array of27
( Clause 4.3)
1 2 3 4 5 6 7
,ment No.(1) (2) (3) (4) (5) (6) (7) (8)
1 I 1 1 1 1 1 1
2 2 1 2 1 2 ] 2
3 1 2 2 1 1 2 2
4 2 2 1 1 2 2 1
5 I 1 1 2 2 2 2
6 2 I 2 2 1 2 I
7 1 2 2 2 2 1 ]
8 2 2 1 2 1 1 2
4,3.2 The necessary condition for an array to beorthogonal is that for all pairs of columns, particularlevels appear together an equal number of times. Forexample, in Table 1, by taking any pair of columns,the level combination ( 1, 1 ); ( 1,2 ); ( 2, 1 ); ( 2,2 )appear equal number of times, that is, twice.Mathematically, this condition may be written asfollows:
‘i. x ‘j For every combination of ( i, j )n,. =lJ
where
nij =
ni, =
n.j =
N=
nll
nl.
and
N level and every pair of columns
number of times the level combination(i, j ) occurs in any two columns,
number of times the level i occurs in onecolumn,
number of times the levelj occurs in othercolumn, and
total number of experiment in Table 1,
——n12 = n21 = n22 =2
=n2. = n.1 = n.2 = 4
N=8
The above condition holds good for everycombination of levels and every pair of columns.The~efore, Table 1 is an orthogonal array.
4.3.3 One factor can be assigned to a column ofTable 1. Let the seven factors assigned to columns1to 7 be called A, B, C, D, E, Fand G. For experimentnumber 1 ( see Table 1 ) all the figures are 1 whichmeans that all the factors in the experiment arein the first level. It is expressed as Al, B,, Cl, D], El,F1 and G,, Lg ( 27) orthogonal arrays is method tocarry out 8 experiments to independently comparethe effects between A, and AI , B] and B2... . . , G1
and G2. Against each experimental trial, let theresponse be recorded as y] , y2 .... yg. In order tocompare the effect caused by factor A, the total ofresponses resulting under conditions A~and AZ arecalculated separately, that is, to sum up the responsesin experiment number 1,3, 5, and 7 which were carriedout under conditions Al and also sum up theresponses in experiment’2, 4, 6, and 8 underconditions A2. Let A, and A2 denote totals of resultsunder conditions A, and A2 respectively.
A] ‘Yl +Y3+Y5+Y7
A2=Y2+Y4+Y6+Y8
Dividing the above results by 4, the average responseof A1 and A2 are calculated as:
i=(Yl+Y3+Y5+Y7)/4
‘2=(y2+y~+y6+Y*)/4
B, and B2 are similarly compared by the averagesresponses of the results under condition BI( experiment number 1, 2, 5 and 6 ) and B2( experiment number 3,4,7 and 8 ). Other factors arecompared in the same way.
4.3.4 So, even by reducing the size of the experimentit is easy to conclude which factor influences theultimate response under consideration and shouldbe controlled. The advantage of OA techniques liesin the high reproducibility of the fictorial effect. InOA experiment the difference between the two levelsA, and A2 is determined as the average effect whilethe conditions of.other.factors vary in equal measure.If the influence of A, and A2 to the experimentalresult is consistent while the conditions of otherfactors vary, the effect obtained from theexperiments using OA tends to be insignificant. Onthe other hand, if the difference between A, and Azvaries significantly, once levels of other factorschange, effect of A tends to be significant. If OAtechnique is used, a factor having consistent effectwith different conditions of other factors will besignificantly estimated. That means a large factorialeffect is obtained from OA experimentation ( or theorder of the preferable level ) that does not varyeven if there is some variation in the levels of otherfactors.
4.4 The orthogonal arrays being discussed in thisstandard are for 2“ and 3“series, that is, all the factorsare either at two levels or three levels. The procedurefor experiments with factors at different levels is alsodiscussed. The orthogonal array tables for 2“ seriesand 3“ series are given in Annexes A and Brespectively. For 2“ series, the orthogonal arraytables are given for Ld( 2S), L8( 27), Llb ( 215) andL32( 231) designs. For 3“ series, the orthogonal array
2
tables are given for Lg( 34) and LZ7(3’3 ) designs.The tables for interactions between two coiumns arealso given.
5 LINEAR GRAPHS.
5.1 Information to be derived from an experiment isnot always limited to the main effects, some timesinteractions are also necessary. It is not very usualto design an experiment with all two-level factors orall three-level factors. If there are four-level factors,co-existing with two-or-three-level factors, it isnecessary to modi~ a two-or-three-level series OAtable so as to meet the requirements. Linear graphsare useful for this purpose. The linear graphs for allthe OA tables are given in Annexes C and D.
5.2 A linear graph associated with an orthogonalarray pictorially presents the information about theinteraction between some specified columns of thatarray. Such a graph consists of a sets of nodes andset of edges, each of which joins certain pair of nodes.A node denotes a column of the array and the edgejoining the two nodes denotes another column ofthe array which is the interaction of the pair ofcolumns under consideration.
5.3 For example, one of the two standard lineargraphs associated for Lg( 27) is as follows:
1
A3 5
02 6 4 7
This linear graph shows that the interactionbetween columns 1 and 2 comes out as column 3, theinteraction between columns 1 and 4 comes out ascolumn 5 and so on. This is in line with the interactiontable given after orthogonal tables in Annex A.Column 7 is shown as independent node which isapart from the triangle. This means that thiscolumn should .be allotted to that factor whereinteraction with the other factor is not required. Itcan be noted that all the columns appear as anode oran edge in the linear graph.
6 ORTHOGONAL ARRAY FOR 2“ SERIES
6.1 The simplest case of factorial experiment is whenall the factors are at ~ levels each. In the experiment,if the degrees of freedom are fully consumed by themain effects and interactions, then the degrees offreedom for error will be zero. In order to generatethe degree of freedom for the error, the experiment
IS 10427( Part 2 ): 2006
may be replicated at least twice.
6.2 In this case, each column of-orthogonal arraytables given in Annex A has one degree of freedom.Therefore, one column will be used for each maineffect. Similarly, as interaction between two maineffects will also have one degree of freedom, onecolumn will be used for each interaction.
6.3 The various steps for the selection of 2“orthogonal array design are given below.
6.3.1 Under the given situation, estimate the totaldegrees of freedom required. The total degrees offreedom are equal to the sum of degrees of freedomfor main effects and interaction effects which arerequired to be estimated.
6.3.2 Decide, depending upon the number of degreesof freedom as to which the orthogonal array tableswill be used, namely, Ld( 23), Lg( 27), L16( 2*5) orL32(23’).
6.3.3 Depending upon the situation, that is, whichof the main effects and interactions are required, drawthe required linear graph.
6.3.4 Select a standard linear graph from Annex Cwhich is closest to the required linear graph.
6.3.5 Make the required changes, if any, by deletingsome lines or joining nodes by lines in the standardlinear graph so that the required linear graph isobtained. Write down column numbers to variousmain effects and interactions and obtain the designmatrix.
6.3.6 Translate the design matrix into physicallayout.
6.3.7 Construct a random sequence of experimentsto be used while carrying out the total experiment.
6.4 Example
In a telephone industry, an experiment was plannedto find the infience of different &omponentdimensions, on the performance of a receiver. Forthis purpose, it was decided to choose the followingfive factors, each at two levels:
S1 Factor First SecondNo. .Level Level
~ Armature thickness Al= 0.73 Az= 0.75(A)
ii) Pole piece height B,= 3.41 Bz= 3.46(top)(B)
iii) Pole piece height C,= 3.41 Cz= 3.46( bottom)(C)
iv) Magnet height ( D ) D,= 7.995 Dz= 8.005
v) Acoustic resistance El =26.27 Ez= 30.31(E)
3
1S 10427-( Part 2 ): 2006
Besides the main effects, it is also required toexamine the interactions AB and BC.
6.4.1 Selection of Design
The various steps fQr selection of the requireddesign are as follows:
a) Total degrees of freedom required=A(l)+B(l)+ C(l)+ D(l)+ E(l)+ AB(l)+BC(1)=7
b) Since there are 7 degrees of freedom, thisexperiment may be tried in L8( 27);
c) The required linear graph is as follows:
B
AAB SC
00A c D E
d) The two standard linear graphs(see Annex C ) for L8 ( 27) are as follows:
1 2 6
A3 5
-o Y3f7
5
2 6 47 4
e) In this case both the standard linear graphscan be used with equal ease. The changesrequired in the standard linear graphs areasfollows:
A.oYa2 6 467 4
0
g)
Allocation of main effects and interactionsto various columns is as follows:
B:l
A:AB:3 BC:5
~6 oA:2 C:4 E:?
As the columns 1,2,4,6, and 7 are allottedto factors B, A, C, D and E, write down thesecolumns from orthogonal tables for L8 (27)from Annex A, and above each column, the
h)
J]
k)
respective factor. The theoretical design soobtained is given in Table 2.
Translate the theoretical design into actualas given in Table 3.
This is a saturated design as no degrees offreedom are available for estimating error.Therefore each experiment will be conductedtwice to generate 8 degrees of freedom forerror.
Select two random sequences of numbersfrom 1 to 8. For this purpose referencemay be made to 1S 4905. Let the randomsequences are:
Replication 1 :4,1, 3,2,8,7,5,6
Replication 11:6,5,2,4,7, 1,3,8
using these sequences, the 16 experimentsare conducted.
Table 2 Theoretical Design
[ Cfause6.4.l ( g) ]
c A D B E AB E
(4) (2) (6) (1) (5) (3) (7)ment No.
(-t ) (2) (3) (4) (5) (6) (7) (8)
I I I 1 1 I 1 I
2 2 1 2 1 2 ] 2
3 1 2 2 1 1 2 2
4 2 2 1 ] 2 2 ]
5 I 1 1 2 2 2 2
6 2 1 2 2 I 2 1
7 I 2 2 2 2 ] 1
8 2 2 I 2 ] 1 2
NOTE — Columns (3) and (5) are used only for thecomputation and not in the actual conduct of theexperiment. So these columns do not appear in thephysical layout.
Table 3 Actual Design
[Clause 6.4.l (h ) ]
ExperimentNo.
(1)
1
2
3
4
5
6
7
8
Level of FactorsAt \
A BC D E
(2) (3) (4) (5) (6)
0.73 3.41 3.4.1 7.995 26.27
0.73 3.41 3.46 8.005 30.31
0.75 3.41 3.41 8.005 30.31
0.75 3.41 3.46 7.995 26.27
0.73 3.46 3.41 7.995 30.31
0.73 3.46 3,46 8.005 26.27
0.75 3.46 3.41 8.005 26.27
0.75 3.46 3.46 7.995 30.31
4
.- ..——, ..,—— ____ — -.-—. -—.- --.. -.,. .... ~. .—x.. .._, ..
6.4.2 Analysis
The various steps in the analysis of above designedexperiment are as follows:
a)
b)
c)
d)
e)
o
g)
h)
Let Y,, Y2, ... ... ... ... ..ygbe thetest responsesin the first replication andyl ‘,y2’, ... ... ... ....y~’ in the second replication for experiments1,2, . .. .. . . . . . . . ... 8 respectively. Denote ~ =(yi +X’) as total response from ith experimentfor both the replications.
Prepare the total and.average response tablefor the main effects and interactions, with thehelp of theoretical design obtained in 6.4.1as given in Tables 4 and 5.
Sums of squares for the main effects areobtained from COI 2 of Table 4. Forexample,
Grand total = G = z (yi +Yi’)
Correction factor (CF)=G2/16
Sum of squares due to factor A, ( SSA ) =
{( T2A1+T2A2)/8}-CF
The sum of squares for other main effectsmay also be obtained in a similar way.
Total sum of squares ( TSS ) =x(y2i+y2i’)-cF
Sum of squares due to interaction effectsAB (~~m)= { [T2A1~1+ T2A1~2+T2A2B,+
T2A2B2]/4 ] - CF - SSA - SS~
The sum of squares due to interaction BCmay be obtained in similar way.
Sum of squared due to error ( SSE ) =TSS – SSA – SS~ – SSC – SS~ – SS~’– SSA~– SSBC
The above sum of squares may be enteredin the analysis of variance table ( Table 6 ).
The mean squares due to main effects orinteractions, as obtained in Table 6, arecompared with mean square due to error andthe significance of the main effects andinteractions are tested.
For the main effects and interactions,which are significant, the optimum level ischosen with the help of average responsetables ( see Tables 4 and 5 ). The level forwhich the response is optimum ( maximumor minimum, as the case maybe) is selected.For other non-significant factors andinteractions, the level for which the costitime/Iabour is minimum, is selected.
IS 10427( Part 2 ): 2006
Table 4 Response Table for Main Effects
[ Clause 6.4.2(b) and(h)]
Factor Total Response Average ResponseLevel
(1)
A,
A2
B,
B2
c,
C*
D,
D2
E,
E2
(2)
T,+ T2+T5+T6(=TA, )
T3+T4+T7+T8(=TA2)
TI+T2+T3+T4(=TB, )
T5+T6+r7+T8(=TB2)
TI+T3+T5+T7(=TC, )
T2+T4+T6+T8(=TC2)
TI+T4+T5+T8(=TD1)
T2+T3+T6+T7(=TD2)
T1+T4+T6+T7(=TE1)
T2+TJ+T5+T8(=TE, )
(3)
(T, +T2+T5+T6 )/8
(T, +T4+T7+T8 )/8
(T, +T2+T3+T4 )/8
(T5+T6+T7+T5 )/8
(T, +T3+T5+T7 )/8
(T2+T4+T6+T5 )18
(T, + Td+ T5+ T5)/8
( T2+ T3+ T6+ T7)18
(T, + Td+ T6+ T7)/8
( T2+ T3+ T5+ T5)/8
Table 5 Table for Interaction Effects
[ Clause 6.4.2(b) and(h)]
Factor Total Response AverageCombination Response
Level
(1) (2) (3)
Al B, Tl+Tz(=TA, ~,) ( T, + T2)/4
A,B2 T5+T6(=TA; ~;) ( T5+ T6)/4
A2B, T3+T4(=TA2~, ) ( T3+ T, )/4
A2Bz T7+T8(=TA2B2)
B, C, TI+TJ(=TB, C,)
B, C2 T2+T4(=TB, C2)
B2C, T5+T7(=TB2C, )
B2C2 Tb+T8(=T~2C2)
T7+ T*)/4
T, + T3)/4
T2+ Td)/4
T5+ T7)/4
T6+ T8)/4
Table 6 Analysis of Variance Table
[Clause 6.4.2 (f)and (g)]
Source of Sum of Degree of Mean FVariation Square Freedom Square Ratio
(1) (2) (3) (4) 1(5)[=(4)/(3)1Main effects
Factor A SSA I MSA MsAmsE
Factor B SSB 1 MS~ M3BIMSEFactor C Ssc 1 MSC MS~ MSE
Factor D SSD I MS~ MS~l MSEFactor E SSE I MS~ MS~/MSE
I InteractionsI I
II AB SSAB I MSAB MSm/MSEBC “Ssnn I I MS~C MSBC/MSE
I Error I 1ss.18 I M.V-k 1 I . . I I
I I I I. ..- E
Total I TS;I
15I 1 1 I I 1 I
5
..—-, .,—.— —— ___ ——... -.—-—__, . . .
IS 1-0427( Part 2 ): 2006
6.5 Example
For minimizing the value of the tan 5 of HV insulationsystem, the following four factors each at two levels,were studied:
kind of conducting tapes — adhesive ornon-adhesive,
thickness of conducting tapes,
curing temperature, and
curing pressure.
From tetihnical considerations, it was felt that theinteractions AC and CD may exist. For this purpose,Lg( 27) experiment was conducted and the responses( coded) are given in Table 7. Set up an analysis ofvariance table, examine the significance of main effectsand interactions AC and CD; and find the optimumlevel for each factor.
6.5.1 For obtaining the sum of squares due to maineffects and interactions, the total and averageresponses are calculated as given in Table 8.
From the response table,
a) Total sum ofsquares ( TSS ) = Z Y*– CF =281.96
b) Correction factor ( CF ) = ( 1339 )2 /24=74 705.04
c) Sum of squares due to factor A ( SSA ) =(6442 +6952 )/12-CF= 108.38
d) SS~= [ (6652+6742)/12 ]-CF ‘3.38
e) SSC= [(6792 +660z)/12]– CF= 15.04
0 SS~=[(6722+ 6672)/12 ]-CF= 1.00
g)
h)
j)
SSAc=[( 3442+ 3092+ 3512+ 3352 )/6]–SSA-SSC - CF=45.38
SSc~ = [ ( 3322+ 3392+ 3282+ 3402)/6 ]-SSC-SS~-CF=0.3-8
Sum of squares due to error ( SSE ) = TSS -SS~-SS~ -SSc-SS~ -SSAc–SSc~ = 108.40
6.5.2 The above sums of squares maybe entered inthe analysis of variance table ( see Table 9 ).
=6.5.3 The tabulated value of F for ( 1, 17) degrees offreedom at 5 and 1percent level of significance is 4.45and 8.40 respectively [ see IS 6200 ( Part 1 ) ]. Sincethe calculated value of F for the main effect A k morethan the tabulated value at 1 percent level ofsignificance, this factor is highly significant. Similarlyas the calculated value of F far the interaction AC ismore than the tabulated value at 5 percent level ofsignificance, this interaction is significant.
6.5.4 As the aim is to minimize the value of tan 6, thesecond level of factor, A which gives the lower response,is selected as optimum level. Similarly for interactionsAC, which is significant, the combination A2 Cl hasminimum value. Hence first level of factor C is selected.For the other two factors, namely, B and D, the costconsiderations may be taken into account for selectionof level.
7 ORTHOGONAL ARRAY FOR 3“ SERIES
7.1 In this case, each column of orthogonal tables( see Annex B ) has 2 degrees of freedom. Thereforefor each main effect ( as it has two degrees offreedom ) one column will be used, whereas twocolumns will be used by an interaction, as it has fourdegrees of freedom.
Table 7 Test Responses
( Clause 6.5)
Experiment Column Number Response ( Hardness ) Total
No. \ .2 1 4 7 Replicate Replicate Replicate
Factor 1 2 3
A B c D
(1) (2) (3) (4) (5) (6) (7) (8) (9)
I A, B, c, D, 61 60 57 178
2 A, B2 c, Dz 57 60 ’56 173
3 Az B, c, Dz 51 58 50 159
4 A2 Bz c, D, 49 49 52 150
5 A, B, C2 D2 57 57 57 171
6 A, B2 C* D, 57 58 58 173
7 AZ B, c1 D, ~7 51 58 166
8 AZ B2 C* Dz 57 55 57 169
Total 1339
6
Table 8 Total and Average Response
( Clause 6.5.1 )
Factor Total Response AverageLevel Response
.(1) (2) (3)
Al
A2
B,
4c,
C2
D,
Dz
A,C,
AIC2
A2C,
A,C2
C, D,
C, Dz
C2DI
C2DZ
178+173+ 171+ 173=695
159+ 150+ 166+ 169=644
178+ 159+ 171 + 166=674
173+ 150+ 173+ 169=665
178+ 173+ 159+ 150=660
171 + 173 + 166+ 169=679
178+ 150+ 173+ 166=667
173+ 159+ 171 + 169=672
178+ 173=351
171 + 173=344
159+ 150=309
166 + 169=335
178+ 150=328
!73 +159=332
173+ 166=339
171 + 169=340
695112= 57.9
644/12= 53.7
674/12= 56.2
665/12= 55.4
660/12= 55.0
679112= 56.6
667/12= 55.6
672/12= 56.0
351/6 = 58.5
344/6 = 57.3
309/6 = 51.5
335/6 = 55.8
32816 = 54.7
33216 = 55.3
339/6 = 56.5
340/6 = 56.7
Table 9 Analysis of Variance Table
( Clause 6.5.2)
Source of Degrees of Sum of Mean FVariation Freedom Squares Square
(1) (2) (3) (4) (5)
Main Effects
Factor A 1 108.38 I 108.38 ] 17.00
Factor B 11 I 3.38 I 3.38 I 0.53 I
Factor C 1 15.04 15.04 2.36
Factor D 1 1.00. 1.00 0.16
Interactions
IAC I 1 I 45.38 I 45.38 I 7. I I I
CD 1 0.38 0.38 0.06
Error 17 108.40 6.38
Total 23 281.96
7.2 The various sEepsfor the selection of 3“orthogonalarray design are given below:
a)
b)
c)
Under the given situation, estimate the totaldegrees offieedom required. The total degreesof freedom is equal to the sum of degrees offreedom for main effects and interactionswhich are required to Le estimated;
Decide, depending upon the number ofdegrees of freedom, as to which of theorthogonal tables will be used, namely,L9(34)or Lz7(313);
Depending upon the situation, that is, which
d)
e)
o
g)
IS 10427( Part 2 ): 2006
of the main effects and interactions arerequired, draw the required linear graph;
Select a standard linear graph from Annex Dwhich is closest to the required linear graph;
Make the required changes, if any, in thestandard linear graph so that the requiredlinear graph is obtained. Write down columnnumbers to various main effects andinteractions and obtain the theoreticaldesign;
Translate the theoretical design into physicallayout; and
Construct a random sequence of experimentsto be used while carrying out the totalexperiment.
7.3 Example
In an investigation for obtaining required colour in awatch dial at plating stage, following four factors werestudied, each at three levels. Design an experimentfor studying the main effect and interactions AB, ACand BC.
S1 No. Factor
~ A = Temperature of bath
ii) B = Voltage
iii) C= Time of immersion
iv) D = Concentration ofbath
7.3.1 Design
Levels
400c,47.50c,550c
3.5V,4.5V,5.5V
30s,40s,50s
30%,35%,40%
The various steps in giving the layout of the designare as follows:
a)
b)
c)
d)
Total degrees of freedom = A(2) + B(2) +C(2) + D(2)+ AB(4) + AC(4) + BC(4) =20
Since there are 20 degrees of freedom, thisexperiment can be tried in L27 ( 3‘3).Theremaining 6 degrees of freedom will be usedfor error.
The required linear graph is as follows:
An
AAB Ac
oB BC c D
The standard linear graph (1) for L27(313 ),given in Annex D, matches the aboverequirement completely.
7
——— —,. .- —-— -—.—-
IS 10427( Part 2 ): 2006
e) The allocation of main effects and interactionsto various columns is as follows:
A:l
AAB:3,4 AC:6,7
o;;;
B:2 BC:8,11 C:5 D:g — Error—
o The theoretical design is as follows
Experiment A B C D AB BC AC ErrorNo. 1 2 5 9 3,4 8,11 6,7 10,12,13
12345
678910
1112131415
1617181920
21z232425
2627
11111122113312121223
12311313132213312112
21232131221322212232
23112322233331133121
31323211322232333312
33233331
g) The actual experiment maybe translated insimilar way from the above theoretical design,as has been done in 6.4 for 2“ series.
h) The degrees of freedom for error= 26- 20=6. So it is not necessary to go for secondreplication.
J> Select a random sequence from experiment1 to 27. For this purpose reference maybemade to IS 4905. The sequence of theexperiments to be conducted shall be as perthe random sequence obtained and not fromexperiment 1to 27.
7.3.2 Analysis
The various steps in the analysis of above designedexperiment are as follows:
a)
b)
c)
d)
e)
o
g)
h)
J]
k)
Letyl, ye, . .. .. . . . . . . . Y27,be the test responsefor experiments 1,2 , . ....27 respectively
Grand totai(G)=yl +yz+ .. . .. . . . . . . . +Y27,
Correction factor (Cm = $
Total sum of squares (TS$ = Z# – CF,
For obtaining the response totals due to themain effects, reference may be made to thetable of theoretical design given in 7.3.l(fi.This tabb gives the information as to in whichof the experiments, a particular factor is atwhich level.
The response total for the factor A is obtainedas follows:
‘*l=(y] +y~+ . .. . . . . . . . . . . . . . . . ..+Y9)
‘*2=( y]~+y], + . . . . . . . . . . . . . . . . . . ..+Y1*)
TA3=(y,9+y20 + . .. . .. .. . . . . . . . . . . ..+y27)
where A,, Az and As represent the three levelsof factor A.
Sum of squares due to the Factor A, ( SSA)T2A, + ~A2 + ~A3
=9
- CF
Sum of squares due to interaction AB, SSA~
‘{[T*A1 ~,+ ~2A, B2 + ~Ar B3 + ‘A2 B1 +
T2A2 ~2 + T2A2 ~3+ T2A3 ~1 + ~ A3 B2+
‘A3B3 1/3} - CF - SSA- SSB
Similarly, the sum of squares due to otherfactors and other interactions may beobtained.
Sum of squared due to error ( SSF ) =TSS – SSA- – SSB – SSC - SSD – Sfm –SSBC - SSAC
7.4 Example
For determining the surface finish in a reamingoperation ( .Cr – Mo alloy steel ), the following fourfactors were studied, each at three levels:
A: core drill size
B: speed
C : feed
D : coolant
For this purpose, an Lg [ 34) experiment wasconducted. Since the 8 degrees of freedom availablefor this design are consumed by the 4 factors ( eachhaving 2 degrees of freedom ), WOreplications ofthkexperiment were conducted. The test responsesobtained are given in Table 10.
8
—
IS 10427( Part 2 ): 2006
Table 10 Test Responses on Surface Finish
( cl~~~~ 7.4 )
Experiment Column NumberNo.
Surface Finish
1 2 3 4 Replicate Replicate Totalc .
Factor 1 2rA
\B c D
(1) (2) (3) (4) (5) (6) (7) (8)
I .1 1 1 1 0.8 0.7 1.5
2
3
4
5
6
7
8
9
1
I
2
2
2
3
3
3
2
3
1
2
3
1
2
3
2
3
2
3
I
3
1
2
2
3
3
1
2
2
3
1
1.8
1.0
0.7
0.9
1.1
2.2
1.5
1.7
1.9
0.9
0.7
0.9
I .4
1.8
1.6
1.3
3.7
1.9
1.4
1.8
2.5
4.0
3.1
3.0
Set-up an ANOVA table and examine the significanceof these four factors:
a)
b)
c)
d)
e)
9
g)
Grandtotal(@= l.5+3.7+ . .. . ....+9.0 =22.9
Correction Factor (C~ = ( 22.9)2‘29.13
18
Total sum of squares ( TSS ) = Z# - CF =3.90
Response totals for various main effects aregiven below:
TA1= 7.1 T~l = 6.9
TA2=5.7 T~2= 8.6
T~3=10.1 T~3= 7.4
Tc, = 7.1 T~l = 6.3
TC2=8.1 T~2= 10.2
TC3= 7.7 T~3= 6.4
Sum of squares due to main effect A, ( SSA),
= (7.1)2+(5.7)2+(10.1)2
6-CF=l .68
Similarly the sums of squares due to othermain effects may be calculated. The valuesof sum of squares of the other three maineffects are as follows:
SS~-=0.25
SSC= 0.08
SS~= 1.65
The sum of squares due to error ( SS~ ) =TSS– SSA - SS~ - SSC – SS~ = 0.24
7.4.1 The above sum of squares may be enteredin the following analysis of variance table( see Table 11).
9
7.4.2 The tabulated values of F for ( 2,9 ) degrees offreedom at 5 and 1 percent level of significance are4.26 and 8.02 respectively [ see IS 6200 ( Part 1 )].Since the calculated value of F for the factors Aand D is more than 8.02, they are highlysignificant. Factor -B is significant ( at 5 percentlevel of significance ).
8 FACTORS WITH UNEQUAL NUMBER OFLEVEE
8.1 In most of the practical situations, all the-factors to be studied in an experiment do not havesame number of levels. Usually an experimentconsists of the combination of two-level factors,three-level factors or four-level factors. In suchsituations, the three-level or four-level factors maybe incorporated in 2“ orthogonal array designsby suitably modifying the 2“ orthogonal tables.This arrangement generally gives orthogonalmain effect plans and as such some precautions areto be taken when estimation of interaction effectsare necessary, though in few cases it is possibleto estimate the same.
8.2 Experiments with Two-Level and Three-LevelFactors Only
The experiments wherein some of the factors areat two-levels and the others are at three-levels canbe incorporated in 2“ orthogonal array designsby-modifying 2“ orthogonal tables or 3northogonalarray designs by modifying 3n orthogonal tables.The procedure of suitably modifying theorthogonal tables is illustrated with the help ofexample given in 8.3.
IS 10427( Part 2 ) :2006 .
Table 11 Analysis of Variance Table
(Clause 7.4. 1)
Source of Degree of Sum of Mean FVariation Freedom Squares Square
.(1) (2) (3) (4) (5)
Core drill 2 1.68 0.840 31.11size (A)
Speed (B) 2 0.25 0.125 4.63
Feed (C) 2 0.08 0.040 1.48
Coolant (D) 2 1.65 0.825 30.56
Error 9 0.24 0.027
Total 17 3.90
8.3 Example
In an investigation on the establishment of thermalcharacteristics of certain type of heat sink, it wasdecided to examine the following four factors:
Factors Levels
Thyristor type (A) A,, Az, AjCooling device (B). B], B2, B3Tier arrangement (~ c,, c1Heat sink type (D) D,, Dz
It is required to examinethe main effects only.
8.3.1 This example is of 22x 32 type. The totalnumber of degrees of freedom required are= A ( 2 ) +B ( 2 ) + C ( 1) + D ( 1) =6. Therefore, this experimentmay either be tried in L8( 27) or in Lg( 34).
8.3.2 Idle Column Method
In this method, thr~e-level factors are introducedin two-level series. Further the number ofexperiments for each level of these three-level mainfactors will not be same in this design and theacceptable optimum level will be repeated morenumber of times. Any one column in two-levelorthogonal table is earmarked as idle column andno factor is allotted to it. If it is acceptable thatfactor A at level 2 is optimum and factor B at level 1 isoptimum, then the following procedure is adopted.
8.3.2.1 Idle CGlumn Level Compare
1 A, with Az, BI with Bz
2 A2with As, B1 with B3
This means that no change is required in the columnsrepresenting the three-level factors when idle columnis at level 1. When the idle column is at level 2, replacethe level 1 in column representing factor A by level 3and replace the level 2 in column representing factorB by level 3.
8.3.2.2 The required linear graph and the allocation
are as follows:
IDLE
A1
3 5
A B 002 4 C6 D:?
8.3.2.3 In the above required linear graph, column 1is allotted to idle column, column 2 to factor A andcolumn 4 to factor B. Since factor A has two degreesof freedom, column 3 ( which represents theinteraction between columns 1 and 2 ) is omittedcompletely from the design. Similarly colum 5whichrepresents the interaction between columns 1 and 4is omitted, so as to create additional degree offreedom of factor B, The node representing theinteraction is therefore encircled in the above lineargraph.
8.3.2.4 The theoretical design as obtained is givenbelow:
Experiment IDLE A A B B C DNo, (1) (2) (3) (4) (~ (q (~
1
DD
11111112 1 1.122223 12211224 12222115 24311126 2 4“3 2 3 2 17 222112.18 2222312
As this design becomes saturated, two replicationsof the experiment will be necessary to generate 8 degreesof freedom for error.
8.3.2.5 The various steps in the analysis of abovedesigned experiment are as follows:
a)
b)
c)
10
Let y,, yz, .. . .. yg, be the teSt responsesin the first replication andy ’l,y’2, . .. . ...y ’8.in the second replication for experiments 1,2,...... ,8 respectively. Denote Ti=yi+Y’i =total response from ith experiment for boththe replications.
Sum of squares due to idle column
=~ [( T~+TG+TT+ Tg)–(T1+T2+T3+16 T4) ]2
Sum of squares due to factor A is obtainedas follows:
Sum of squares due to ( A, - A2 )
=: [T3+T4-TI-T212
d)
e)
9
g)
Sum of squares due to (A2 – A3 )
=~[T7+Tg–T~–Tc]2
Sum of squares due to factor A, ( SSA)= Sumof squares due to ( A, – A2 ) + Sum of squaresdueto(A2– A3)
— [T,+ T~-= ~ [TJ+TQ-T1-TZ]2+ ~
T~– TG]2
Similarly the sum of squares due to factor B
( ss~ )
= Sum of squares due to (B, – B2) + Sum ofsquares due to ( B] – B3 )
L [Tb+ T8-= ~ [T2+Td-Tl-T3]2+ ~
T~– T7]2
Sum of squares due to factor C ( SSC)
‘~ [ T2+,T3+ T6+ T7– TI– T4– T~– T8J2
Sum of squares due to factor D ( SS~)
1‘~ [T2+T3+Tj+Ts–T~-T4–Tb–TT]2
Sum of squares due to error ( SS~)
=+ [(Y,–y ’1)2+(y2–y ’2)2+........+(y~-y’g)1
8.3.3 Dummy Level Method
In this method, two-level factors are introduced in three-Ievel series by suitably modify the three-levelorthogonal tables. The level ( first or second ) of atwo-level factors, which is expected to fare better, isrepeated in more number of experiments as compared-to the other level. This is done by changing the levelnumber 3 in a column, assigned to a two-level factor,by a level, which is expected to fare better, in a three-Ievel orthogonal tables. Suppose in example in 8.3,C2 and D, are expected to fare better, then these arerepeated in more number of experiments than C, andD2 respectively.
The three-level standard orthogonal table L9( 34)willbe modified as follows:
Experiment A B CDNo. (1) (2) (3) (4)
1 1 1 1 12 1 2 2 23, 1 3 r ,,
4 2 1 2 1’5 2 2 2’ 16 2 3 1 27 3 1 ~ 28 3 2 1 1’9 3 3 2 1
IS 19427( Part 2 ): 2006
Where 2’ denotes the level 2 of factor C repeated inplace of level 3 in column 3 and is the dummy level.Similarly 1‘ denotes the first level of factor D repeatedin place of level 3 in column 4.
8.3.3.-1 The various steps in the analysis of the abovedesigned experiment are as follows:
a)
b)
c)
d)
e)
9
g)
h)
J)
Letyl, y2, .. . .. yg be the test respons~ forexperiments 1,2 .. . . .. ...9 respectively.
Grand total-(G) ‘y, +y2 + . . ..,Yg
Correction factor ( CF ) = G2/9
Total sum of squares ( TSS ) = Z# - CF
The response totals are as follows:
TA1‘YI + Y2+ Y3; TB1‘Yl + Y4+ Y7
TA2“y4 + Y5+ yb; TB2‘Y2+ Y5+ Y8
TA3‘y7 + yg+ y9; ‘B3=y3+ yb+ y9
Tcl ‘yl + yb+ yg; TD,=Y1+Y5+Y9
TC2‘Y2 + Y4+ Y9; ‘D,.=Y3 + y4+ y8
TC2,‘y3+ y~+ Y7; TD2‘y2 ‘Y6+Y7
Where A,, A2 and A3 represents the threelevels of factor A. Similarly other levels offactors may be defined.
Sum of-squares due to factor A ( 55A),
=*(rA, + T*A2+ ~A3 ) – CF
Similarly, the sum of squares due to factor Bmay be obtained.
Sum of squares due to factor C ( SSC)
(2 TC, -TC2-TC2’)2=—
18
Sum of squares due to factor D ( SS~ )
(2 T~2-T~, -TD,’)2=18
Sum of squares to error (SS,)
A (TC2-TC2’)2= ~ (TD, -T~l’ )2+ 6
8.4 Experiments with Two-Level and Four-LevelFactors
The experiments wherein some factors are at twolevels and others at four levels can be incorporatedin 2“orthogonal array designs by suitably modi&ing2“ orthogonal tables. Since the degrees of freedomfor any four-level factor is three and each columnin 2“ orthogonal tables has one degree of freedom,three columns will be used for each four-levelfactor. In the linear graph, two nodes and the edgejoining them make the representation of a four-level factor. The procedure is to choose any twocolumns in 2“ orthogonal tables and for the pairs
11
IS 10427( Part 2 ) :2006
(1, 1), (1, 2), (2, 1) and (2, 2) make the followingtransformation for four level factor:
(1,1)+ 1,(1,2)+ 2,(2,1 )+3, and(2,2)+4
The corresponding interaction column is also deletedso as to generate three degrees of freedom.
8.4.1 Example
The following experiment consisting of 8 factorsrelates to the hydraulic design of impeller of a pump.Give the layout of the design to estimate all the maineffects and interactions A x B, A x C and A x G.
S1 No. Factors Levels
~ Outlet angle of impeller (A) 2
ii) Inlet angle of impeller (B) 4
iii) Outlet width (C) 2
iv) Inlet width(D) 2
v) Impeller eye dia (E) 2
vi) Vane thickness (~ 2
vii) Shape of vanes (G) 2
viii) Outlet tip length (~ 2
8.4.1.1 The degrees of freedom for main effects is 10and for itieractions is 5, making a total of 15. Hence,
the smallest array in which this design can beaccommodated isL16( 45). The required linear graphis shown below:
0000DEFH
8.4.1.2 The design matrix is obtahed from the standardlinear graph 3 of the L,b( 4* ) array. This is shown asbelow:
2 6
K3 7
110
9 13
8 12
G6
K3 7
11 &l—s 10
9 13
C12
o n
15:14? 00004 04 E5 F14 M15
8.4.1.3 The resulting theoretical design is given asfollows:
1
Experiment .A B AxB c AxC D E F G AxG HNo, 1 2,8,10 3,9,11 12 13 4 5 14 6 7 15
1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 2 1 2 2 2 2 1 1 2 1 1 2
3 1 1 1 1 1 2 2 2 2 2 2 2 2
4 1 2 1 2 2 1 1 2. 2 1 2 2 1
5 1 3 2 1 2 1 1 1 I 2 2 2 2
6 1 4 2 2 1 2 2 1 1 1 2 2 1
7 1 3 2 1 2 2 2 2 2 1 1 1 1
8 1 4 2 2 1 1 1 2 2 2 1 1 2
9 2 1 2 2 2 1 2 1 2 1 1 2 “2
10 2 2 2 2 1 2 1 1 2 2 1 -2 1
11 2 1 2 2 2 2 1 2 1 2 2 1 1
12 2 2 2 1 1 1 2 2 1 1 2 1 2
13 2 3 1 2 1 1 2 1 2 2 2 1 1
14 2 4 1 1 2 2 1 1 2 1 2 1 2
15 2 3 1 2 1 2 1 2 1 1 1 2 2
16 2 4 1 1 2 1 2 2 1 2 1 2 1
9 ADDITIONAL OA TABLES
For level 2 experiments, one may refer to L ,2(2] I )
(see Annex E). For asymmetrical factorial with factorsat 2 and 3 levels, one may refer to L,g( 21 x 37 )( see Annex F).
12
IS -10427( Part 2 ) :2006
ANNEX A
[ Clauses 4.3,6.2 and 6.4.l(g) ]
ORTHOGONAL ARRAY TABLES FOR 2“ SERIES
O.A. ( 4,3,2,2 ) LA( 23 )
No. 1 2 3
1 1 1 12 1 2 23 2 1 24 2 2 1
Group 1 2
O.A. ( 8,7,2,2 ) L8( 27)
No. 1 2 3 4 5 6 7
1 1 1 1 1 1 1 12 1 1 1 2 2 2 23 1 2 2 1 1 2 24 1 2 2 2 2 1 -15 .2 1 2 1 1 26 2 1 2 2 .; .2 17 2 2 1 1 2 2 18 2 2 1 2 -1 1 2
IGroup 1 2 3
Interaction Between ~o Columns in L8( 27 )
CO1 1 2 3 4 5 6 7
(1) 2 5 4 7 6(:) 6 7 4 5
(:) 6 5 4(:) 2 3
(:) 2(;)
()
13
IS 10427( Part 2 ): 2006
0.A. ( 16,15, 2,2) L,6(21S)
SINO.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1234
5678
9101112
13141516
I111
1111
2222
2222
1111
2222
1111
2222
1=111
2222
2222
1111
1122
1122
1122
1122
Interaction Between Columns
1122
1122
2211
2211
1122
2211
1122
2211
1122
2211
2211
1122
1212
1212
1212
1212
1212
.1212
2121
2121
1212
2121
I212
2121
1212
2121
2121
1212
1221
1221
1221
1221
I221
1221
2112
2112
1221
2112
1221
2112
1221
2112
2112
1221
Col 1 2 3 4 5 6 7 -8 9 10 11 12 13 14 15,
(1) 3 2 5 4 7(2) 1 6 7 4
(3) 7 6 5(4) 1 2
(5) 3(6)
65432
(7;
91011121314
(;
8 11 1011 8 910 9 813 14 1512 15 1415 12 1314 13 12
23(9; 3 2
(lo)(11;
13141589
1011456
(12;
12 1515 1214 139 108 11
11 810 956477465
2(13; 3
(14)
111312111098765432
(15;
14
0. A.(32,31, 2,2) L3Z(23’)
SINO. 12345 6 7 89 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
12
34
5678
91011D
131415
u 16
17181920
21222324
29303132
1111
1111
1111
1111
2222
2222
2222
2222
1111
1111
2222
2222
1111
1111
2222
2222
1111
1111
2222
2222
2222
2222
11
11
11
11
1111
2222
1111
2222
1111
2222
1
111
2222
1
1
11
2
2~
2
1111
2222
2222
1111
222
2
111
1
1111
2
222
2222
1111
1111
2222
?.2
22
1
111
1
1I
1
2222
2222
1111
2222
1111
1
111
2222
1112
112
2
1122
1122
1122
1122
112
2
112
2
11
12
1
122
1122
1122
2211
2211
2
2i1
2211
1
112
1
122
2211
2211
1122
1122
2
211
2211
11
12
11
22
2211
2211
2211
2211
112
2
1122
11
12
2
211
1122
2211
1122
2211
11
22
2211
11
12
221
1
1122
2211
2211
1122
22
11
1122
1112
2211
2211
1122
11
22
2211
2
211
11
22
1
112
221
1
2211
11
22
2211
1122
1122
2211
11
12
1212
1212
1212
1212
1212
121
2
121
2
121
2
1212
1212
1212
2121
2121
21
21
2
121
1
2
12
1212
21
1
2121
1212
1212
2121
21
21
1
2
1
2
1212
21
1
21
21
2121
2121
1
21
2
1
212
12
1
2
21
21
12
2
2121
1212
2121
12
12
212
1
1
21
2
21
21
12
2
2121
2121
1212
2
12
1
121
2
1
2
1
2
2
1
21
21
1
1212
1212
2121
2121
121
2
12
1
2
2
12
1
2i
1
1212
2121
1212
1212
2121
1
22
1
1
221
1222
1221
1221
1221
12
21
1
221
1
22
1
12
21
1
222
1221
2112
2112
21
12
2
112
1
2
21
12
21
2111
2112
1221
1221
21
12
21
12
1
2
21
12
21
21
11
2112
2112
2112
122
1
1
22
1
12
2
1
21
1
2
1222
2112
1221
211
2
12
21
21
12
12
2
1
2
112
12
22
2112
2112
1221
2
11
2
1
221
12
2
1
2
112
2111
1221
1221
2112
211
2
1
221
1
2
11
21
12
2111
1221
2112
1221
1 E~-
02e
12~
2q
1s1=2Z
0a
Interaction Between No Columns O. A. ( 32,31,2,2 ) EH
Col 1 2 3 4 5 6 7 89 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 =
3254 7 6 98 11 101312151417161918 212023222524272629 28313:167 4 5 1011 8 9 141512131819171722 232021262724253031 2829P
76 5 4 1110 9 8 151413121918171623 2221 ~ 27X25243130 3283
1 231213 14 15 8 9 101120212223 16171819%29303124 252627:3 2 13 12 15 14 8 11 1021 ~23D 17161918 18 B283130U 2427X :
1 14 15 u 13 10 11 892223 ti21 181916173031282926 272425:15 14 13 12 11 10 9823222120191817 16313029282726Z24
12345 6724 Z 2627282930 31 16171819 ~21D23
3254 7625242726 W2831 301716191821 ~2322
1 67 45262724X303128 W 1819161722232021
765 42726 Z243130W28 191817 i62322212012 328 W 30312425X27 2021222316171819
32 W283130Z 2427 2621 ~23D 171619181303128 W26272425 Z 23202118191617
3130 W 282726252423 Z 21201918171612345678 9 10 11 12 13 14 15
3254769 8 11 10 13 12 15 1416745 10 11 8 91415U13
765411 10 9 8 15 14 13 12123 U1314158 9 10 11
32 13 12 15 14 9 8 11 101 14151213101189
15 14 13 12 11 10981234 567
3 25 47616 745
7 654123
321
IS 10427( Pati 2 ): 2006
ANNEX B
( Clauses 4.4 and 7.1)
ORTHOGONAL ARRAY TABLES FOR 3“SERIES L9( 34)
Experiment No. Column
12
3
456
78
9
Experiment No.
123
4
56
789
101112
13141516
1718
192021
222324
252627
1111
222
333
212
3
123
123
3123
231
312
L27(3’3)Column
41
23
312
231
‘1111
-1
.11
111
222
2222
2z
333
333
333
2
111
2
22
33
3
1
11
2223
31
11
1
22
2
33
3
3111
2
22
333
222
3331
12
333
111
222
4
111
2
2
2
33
3
3
3
3
1
112
23
222
333
1
11
512
3
1
23
12
3
1
2
3
1231
21
12
3
22
3
12
3
6123
1
23
123
231
2312
32
312
312
312
7123
1
2
3
1
2
3
3
12
312
3
12
221
23
“1
2
31
8123
2
31
31
2
1
23
2313
13
13
3
2
31
312
9123
2
31
312
231
3121
21
312
123
231
10123
2
31
312
31
2
12
32
31
221
312
1
2
3
11123
3
12
231
123
3122
32
133
312
231
12123
3
12
231
231
1233
12
312
231
123
13 ‘123
3
12
2
i
312
2311
23
231’
123
312
17
IS 10427 (Pti2 ) :2006
~W~ON TABLE ~R3” DESIGN
Columnr A 3
2 3456789(;) 32265598
4 3 7 7 6 10 lo;) 1 1 8 9]0 5 6
3 11 12 13 11 12(:) 1 9 10 8 7 5
13 11 12 n 13(~) 10 8 9 6 7
13 11 13 11;; 1123
11 13(~!42
13 12
;34
(; ?
:;
10 11 u 138 12 11 119 13 13 n756713 8 9 10667511 10 3 9575612 9 10 84243U 8 10 9332411 10 9 8243213 9 8 10123495761423
65(:0) : 4 2
5 “7(:1) 1 1
12(R) 1
(13)
18
IS 10427( Pati 2):2006
ANNEX C
( Clauses 5 and 6.3.4 j
STANDARD LINEAR GRAPHS FOR 2“ SERIES
L4 (23) 1 2
(1) o3
L8 (27)1
A(1)
3 5
60
2 4 7
L15(2’5) 1
‘(1)
414
8
(3)
2 6
10
8 12
(2)2 6
Y317
5
4
4 10 6
(2)
2 12 15
(4)
10
6
Q
12 14
4
19
IS 10427( Pati 2 ): 2006
(5)
10 ~23
40 0812
50 01015
6~ 0914
70 01113
L32(23’)
(1)
(2)
18
16
(6)
4 8
10
6 9
2( )3
)1
13
? 15
12
5 14
19 21
26
22 25
22 24
41-
8
20
IS 10427( Pati 2 ): 2006
(3)
18 20
(4)
18 20
3
2
(5)
12 26
3
2
28
023
11
8I
15
25 ~ 16
B
22’
17 3124
7
8’ -15
20 23 19
31
( [
28
0010 18 11 21 14
21
IS 10427( Part 2 ): 2006
2
(6)
259
16
H
-22
17 31
24
78
I 15014
023
(7)
2 2018
12
14
004 10 13 28 15
01 16
(8)
1
22
(9)
IS 10427(Pati 2) :-2006
5 8
28
30
(lo)24
26
480 16
iv::
12 2917
9026
19 13 3118
10030 3
205
f4 2721
11028
23 70 1525
22
1
(11)
26
24
16
17
22
20
07
21 11
23
IS 10427( Part 2 ): 2006
(12)6 9
10026
20
4
1222
8 5
018
(13)6 17
,
...
24
IS 10427(Pati 2): 2W
ANNEX D
[ Clauses 5, 7.2(d) and7.3.l ]
STANDARD LINEAR GRAPHS FOR 3“ SERIES
LO (34)1
1
L27(3’3)
(1)
(2)
009 10
012
013
2
+
3,4
512,13 ,1
,:,10
25
IS 10427( Part 2 ) :2006
.
ANNEX E
( Clause 9 )
ORTHOGONAL TABLE FOR LIZ ( 211)
0. A.(-12,11,2,2)
No. 3 4 5 6 7 8 9 10 11
1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 2 2 2 2 2 23 1 1 2 2 2 1 1 1 2 2 24 1 2 1 2 2 1 2 2 1 1 2
5 1 2 2 1 2 2 1 2 1 2 16 1 2 2 2 1 2 2 1 2 1 .17 2 1 2 2 1 1 2 2 1 2 18 2 1 2 1 2 2 2 1 1 1 2
9 2 1 1 2 2 2 1 2 2 1 110 2 2 2 1 1 1 1 2 2 1 111 2 2 1 2 1 2 1 1 1 2 212 2 2 1 1 2 1 2 1 2 2 1
Group 1 2
NOTE— Interaction between two columnsare to some extent mixed up with parts of other columns. To determinetheir interactions, it is necessary to analyse them one by one. Therefore should not be used for experimentswherethere is interaction.
ANNEX F
( Clause 9 )
ORTHOGONAL TABLE FOR L1~( 21 X 37)
123
456
789
10
11
12
131415
16H18
1
111
111
1I1
222
222
222
0. A.( 18,’
2
111
222
333
111
222
333
-
3
123
123
123
123
123
123
3,2)
14123
123
231
312
231
312
5
123
23I
123
3I3
3I2
231
6
123
231
312
231
123
312
7
1
23
312
231
233
312
123
8
123
312
312
122
231
231
NOTE— The Interactionbetweentwo columnsof level3 is partly mingledwith there columns of level 1-3. The sameremark as in case of note for L12 can be given here also.
26
IS 10427( Pati 2 ): 2006
ANNEX G
(Forewora
CO~~ECO~S~ION
Statistical Methods for Qual@ and Reliabili~ Sectional Committee, MSD 3
Organization
Indian Statistical Institute, New Delhi
Bharat Heavy Electrical Ltd, New Delhi
Birla Cellulosic, Bharuch
Continental Devices India Ltd, New Delhi
Defence Research & Development Organization, LaserScience and Technology Centre, Delhi
Directorate General Quality Assurance, Kanpur
Electronics Regional Test Laboratory (North), New Delhi
h personal upacity (B -109 Malviya Naga~ New Delhi 11001 ~
In personal capacity (20/1 Krishna Naga~ Safdaq”ungEnclave, New Delhi 110 029)
Indian Agricultural Statistics Research Institute, New Delhi
Irrdian Association for Productivity Quality and Reliability,Kolkata
‘Indian Institute of Management, Lucknow
Indian Institute of Management, Kozhikode
Maruti Udyog Limited, Gurgaon
Newage Electrical India Ltd, “Pune
National Institution for Quality and Reliability (NIQR),New Delhi
Polyutrusions Private Limited, Kilpauk
POWERGRID Corporation of India Ltd, New Delhi
Reliance Industries Limited, Surat
Samtel Color Ltd, New Delhi
Sons Koyo Steering Systems Ltd, Gurgaon
SRF Limited, Manali
Tata Motors Ltd, Jamsbedpur
BIS Directorate General
Representative (s)
DR AJWIND SETH( Chairman)PROF S. R. MOHAN ( Alternate )
SHRI S. N. JHA
SHRI A. V. KRISHNAN ( Alternate )
SHRI VAIDYANATHAN
SHRI SANJEEV SAOAVARTI( Alternate )
SHRI NAVIN KAPUR
SHRI VIPUL GUPTA ( Alternate )
DR ASHOK KUMAR
SHRI S. K. SRIVASTAVA
LT-COL C. P. VIIAYAN( Alternate )
SHRI S. K. KIMOTHI
SHRI R. P. SONDHI ( Alternate )
PROF A. N. NANKANA
SHRI D. R. SEN
DR V. K. GUPTASHRI V. K. BHATIA ( Alfernate )
DR BISWANATHDAS
DR DEBABRATARAY ( Alternate )
PROF S. CHAKRABORTY
DR R. P. SURESH
SHRI R. B. MADHEKAR
SHRI NITIN GHAMAND1
SHRI G. W. DATEY
SHRI Y. K. BHAT ( Alternate )
SHRI R. PATTABI
SHRI SAI VENKAT PRASAD ( Afternate )
SHRI K. K. AGARWAL
SHRI DHANANJAYCHAKRABoRTy ( Alternate )
DR S. ARVINDANATH
SHRI A. K. BHATNAGAR( Alternate )
SHRI S. R. PRASAD
SHRI KIRAN DESHMUKH
SHRI DINESH K. SHARMA( Altirnate )
SHRI C. DESIGAN
SHRI SHANTI SARUP
SHRI A. KUMAR ( Alternate )
SHRI P. K. GAMBHIR. Scientist ‘F’ & Head (MSD)[ Representing Director General (Ex-oJjcio) ] ‘
Member Secretary
SHRI LALJT MEHTA
Scientist ‘D’ (MSD), BIS
27
.—
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This Indian Standard has been developed from DOC : No. MSD3 ( 168 ).
Amendments Issued Since Publication
Amend No. Date of Issue Text Affected
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