Journal of Manufacturing Systems 28 (2009) 55–63

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Journal of Manufacturing Systems

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Technical paper

Application of entropy measurement technique in grey based Taguchi methodfor solution of correlated multiple response optimization problems: A case studyin weldingSaurav Datta a,∗, Goutam Nandi b, Asish Bandyopadhyay ba Department of Mechanical Engineering, National Institute of Technology (NIT), Rourkela, Orissa-769008, Indiab Department of Mechanical Engineering, Jadavpur University, Raja S. C. Mallik Road, Kolkata- 700032, West Bengal, India

a r t i c l e i n f o

Article history:Received 23 November 2008Received in revised form1 June 2009Accepted 18 August 2009Available online 14 October 2009

a b s t r a c t

In the presentwork, an attempt has beenmade to apply an efficient technique, in order to solve correlatedmultiple response optimization problems, in the field of submerged arc welding. The traditional greybased Taguchi approach has been extended to tackle correlated multi-objective optimization problems.The Taguchi optimization technique is based on the assumption that the quality indices (i.e. responses)are independent or uncorrelated. But, in practical cases, the assumption may not be valid always.However, the common trend in the solution of multi-objective optimization problems is to initiallyconvert thesemulti-objectives into an equivalent single objective function.While deriving this equivalentobjective function, different priority weights are assigned to different responses, according to theirrelative importance. But, there is no specific guideline for assigning these response weights. In thiscontext, the present study aims to apply the entropy measurement technique in order to calculate therelative responseweights from the analysis of entropy of the entire process. Principal Component Analysis(PCA) has been adopted to eliminate correlation that exists among the responses and to convert correlatedresponses into uncorrelated and independent quality indices, called principal components. These havebeen accumulated to calculate the overall grey relational grade, using the theory of grey relationalanalysis. Finally, the grey based Taguchi method has been used to derive an optimal process environmentcapable of producing the desired weld quality. The previously mentioned method has been applied tooptimize bead geometry parameters of submerged arc bead-on-plate weldment. The paper highlights adetailed methodology of the proposed technique and its effectiveness.

© 2009 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Literature shows that there are different approaches to tacklingsimulation modeling, prediction and multi-objective optimizationproblems in various fields. The common approaches include Re-sponse Surface Methodology (RSM), [1,2], Artificial Neural Net-work (ANN), Genetic Algorithm (GA), [3], Fuzzy regression, [4] andDesirability Function (DF) approach, [5,6].Gunaraj andMurugan [1] developed amodel using the five level

factorial technique to relate the important process control vari-ables–welding voltage, wire feed rate, welding speed and nozzleto plate distance – to a few important bead quality parameters –penetration, reinforcement, bead width, total volume of the weldbead and dilution. The model thus developed was checked for itsadequacy with the F-Test. They highlighted quantitatively, as well

∗ Corresponding author.E-mail addresses: s_bppimt@yahoo.com, sdatta@nitrkl.ac.in (S. Datta).

0278-6125/$ – see front matter© 2009 The Society of Manufacturing Engineers. Publidoi:10.1016/j.jmsy.2009.08.001

as graphically, the main effect and interactive effect of the processcontrol variables on important bead geometry parameters.Kim et al. [7] developed an intelligent system of artificial neural

network in GMA welding process using MATLAB/SIMULINK soft-ware. Based onmultiple regressions andneural network, themath-ematical models were derived from extensive experiments withdifferent welding parameters and complex geometrical features.Their developed system was capable of receiving the desired welddimensions as input, and selecting the optimal welding param-eters as outputs. Kim et al. [8] applied an intelligent system forthe determination of welding parameters for each pass and weld-ing position, for pipeline welding, based on one database and a fi-nite element method (FEM) model, and on two back-propagation(BP) neural networkmodels and a corrective neural networkmodel(CNN). Experiments using the predicted welding parameters fromthe developed system proved the feasibility of interface standardsand intelligent control technology to increase productivity, im-prove quality and reduce the cost of system integration.Xue et al. [9] reported the possibilities of the fuzzy regression

method in modeling the bead width in the robotic arc-welding

shed by Elsevier Ltd. All rights reserved.

56 S. Datta et al. / Journal of Manufacturing Systems 28 (2009) 55–63

process. In their paper, they developed a model for proper predic-tion of the process variables for obtaining the optimal bead width.Sathiya et al. [10] proposed a method to decide near optimal set-tings of the process parameters using Genetic Algorithm to opti-mize weld quality in friction welding. Correia et al. [11] stated thepossibility of using Genetic Algorithm as a method to decide near-optimal parameter settings of a GMAW process.In order to solve a multi-objective optimization problem, it is a

common trend to convert multiple objectives into an equivalentsingle objective function. This can be assumed as the overallrepresentative function which needs to be optimized finally. Inmost of the cases, a mathematical model is to be developed torepresent the relationship among the overall quality index and theprocess variables, in which the overall quality index is representedas a function of process control parameters. This mathematicalmodel is then optimized, within an experimental domain. Themethod has an inherent disadvantage. To develop an adequatemodel of statistical importance, an enormous dataset is required,which results in an increase in experimentation cost and loss ofconsiderable time. Moreover, in this case, the search domain isassumed as continuous. Therefore, it may so happen that, afteroptimization, the predicted optimal setting may not be accuratelyadjusted in the machine or setup. As a result, a compromise has tobe made to select a parameter setting which is very close to theoptimal setting. This drawback can be eliminated by the Taguchimethod, [12–15]. This method is very efficient in searching for theoptimal setting within some discrete points in the experimentaldomain. These discrete points are nothing but the different settingsin the provision for adjusting factor values in the machine/setup.Another advantage of the Taguchi method is that it requires alimited number of experiments, as dictated by Orthogonal Array(OA) design. However, the traditional Taguchimethod cannot solvemulti-objective optimization problem. To avoid this shortcoming,it is necessary to convert multiple objectives into a single overallobjective function, called overall quality index. In this way, oncea multi-objective optimization problem can be converted intoa single objective optimization problem, the traditional Taguchimethod can easily be used for evaluating the optimal result. Thereare two popular approaches to calculating the overall qualityindex: (a) grey relational analysis and (b) Desirability Functionapproach. Both have the same purpose, but the way of calculatingthe overall quality index differs. Grey relational analysis is based onthe concept of quality loss. On the contrary, Desirability Functionapproach takes into account the desirability of the quality features.In grey relational analysis, grey relational coefficients of individualresponses have been accumulated to calculate the overall greyrelational grade. In the Desirability Function approach, individualresponse desirability values are combined to calculate the overalldesirability index. The optimal parameter setting is then evaluatedby maximizing (a) overall grey relational grade or (b) overalldesirability. This can be performed by the Taguchi method. Tarnget al. [16] applied grey-based Taguchi methods for optimization ofsubmerged arc welding (SAW) process parameters in hardfacing.They considered multiple weld qualities and determined optimalprocess parameters based on grey relational grade from greyrelational analysis proposed by the Taguchi method.Apart from the Taguchi method coupled with grey analysis and

desirability function, several hybrid Taguchi methods deserve amention. Thesemethods are: Taguchi based fuzzy logy, GA coupledwith Taguchi, DF and RSM in combination with Taguchi etc.The study conducted by Kim and Rhee [17,18] focused on the

definition and optimization of the objective function in the dualresponse approach applied to a Gas Metal Arc Welding (GMAW)process. The objective function was defined using the desirabilityfunction. In their work, first, the regression models of the meanvalue and standard deviation of the depth of penetration were in-duced through regression analysis. Subsequently, an optimization

algorithm (Genetic Algorithm) based on the regressionmodels andconstraints was applied to evaluate the welding process parame-ters, which could generate the desired penetrationwithminimizedvariance. Tarng et al. [19] applied fuzzy logic in the Taguchimethodto optimize the submerged arc welding process with multiple per-formance characteristics. An Orthogonal Array, the signal-to-noiseratio, multi-response performance index and Analysis of Variance(ANOVA) were employed to study the performance characteristicsin the submerged arc welding process. Kim [20] suggested a Ge-netic Algorithm and Response Surface Methodology for determin-ing optimal welding conditions of a GMA welding process. First,in a relatively broad region, near-optimal conditions were deter-mined through a Genetic Algorithm. Then, the optimal conditionsfor welding were evaluated by the investigators over a relativelysmall region around these near-optimal conditions by using Re-sponse Surface Methodology.In the Taguchi approach, it is assumed that all quality features

are independent (i.e. they are not correlated). But in the actualcase, the assumption may deviate. To overcome this, instead ofgrey-Taguchi, some researchers have applied Principal ComponentAnalysis. PCA is a multivariate statistical technique, used forreduction of correlated data dimension. The correlated responsesare transformed into independent components, called principalcomponents. Optimal parametric combination is then determinedby maximizing the principal component, which has highestcontribution to the response variation. But, when more than oneprincipal component contribute significantly to the variation, theproblem of evaluating the composite principal component arises.Another problematic situation in multi-objective optimization

is that, all the response variables may not be equally important.For example, in case of butt welding, the depth of penetration ismore important compared to reinforcement, whereas, in case ofcladding, bead width is a more important feature compared topenetration. So, depending on the relative importance of differentresponse features, different weight values are to be assigned. Butthere is no specific rule for assigning these response weights. Itentirely depends on the decision maker. This can be overcome byentropy measurement technique.Entropy measurement technique was proposed and defined by

Wen et al. [21]. It is a method to evaluate individual responseweights on the basis of entropy of the entire process.In the present report, an attempt has been made to develop a

methodology for solving a correlated multi-response optimizationproblem in submerged arc welding. The correlation among theresponse data – penetration depth, reinforcement, bead width anddilution (i.e. four selected features of bead geometry) – has beeneliminated by using Principal Component Analysis [22]. Correlateddata has been transformed into four independent principalcomponents. These have been accumulated to calculate the overallgrey relational grade, which has been finally optimized. Entropymeasurement technique has been used to estimate individualresponse weights [23]. Finally, the grey based Taguchi method hasbeen used to determine the optimal process environment. Resultsof optimization have been verified through the confirmatory test.

2. Methodology for optimization

Assuming the number of experimental runs in Taguchi’s OAdesign is m, and the number of quality characteristics is n, theexperimental results can be expressed by the following series:X1, X2, X3, . . . , Xi, . . . , Xm.Here,

X1 = {X1(1), X1(2), . . . , X1(k), . . . , X1(n)}.Xi = {Xi(1), Xi(2), . . . , Xi(k), . . . , Xi(n)}.Xm = {Xm(1), Xm(2), . . . , Xm(k), . . . , Xm(n)}.

S. Datta et al. / Journal of Manufacturing Systems 28 (2009) 55–63 57

Here, Xi represents the ith experimental results and is called thecomparative sequence in grey relational analysis.Let, X0 be the reference sequence: X0 = {X0(1), X0(2), . . . ,

X0(k), . . . , X0(n)}.The value of the elements in the reference sequence means the

optimal value of the corresponding quality characteristic. X0 andXi both includes n elements, and X0(k) and Xi(k) represent thenumeric value of kth element in the reference sequence and thecomparative sequence, respectively, k = 1, 2, . . . , n. The followingillustrates the proposed parameter optimization procedures indetail.Step 1: Normalization of the responses (quality characteristics)When the range of the series is too large or the optimal value of

a quality characteristic is too enormous, it will cause the influenceof some factors to be ignored. The original experimental data mustbe normalized to eliminate such effect. There are three differenttypes of data normalization according towhetherwe require the LB(lower-the-better), the HB (higher-the-better) and NB (nominal-the-best). The normalization is taken by the following equations.

(a) LB (lower-the-better)

X∗i (k) =min Xi(k)Xi(k)

. (1)

(b) HB (higher-the-better)

X∗i (k) =Xi(k)

max Xi(k). (2)

(c) NB (nominal-the-best)

X∗i (k) =min{Xi(k), X0b(k)}max{Xi(k), X0b(k)}

. (3)

Here, i = 1, 2, . . . ,m; k = 1, 2, . . . , n.X∗i (k) is the normalized data of the kth element in the ith

sequence.X0b(k) is the desired value of the k th quality characteristic. After

data normalization, the value of X∗i (k)will be between 0 and 1. Theseries X∗i , i = 1, 2, 3, . . . ,m. can be viewed as the comparativesequence used in the grey relational analysis.Step 2: Checking for correlation between two quality characteristics

Qk = {x∗0(k), x∗

1(k), x∗

2(k), . . . , X∗

m(k)}k = 1, 2, . . . , n.

It is the normalized series of the ith quality characteristic.The correlation coefficient between two quality characteristics iscalculated by the following equation:

ρjk =Cov(Qj,Qk)σQj × σQk

, (4)

here, j = 1, 2, 3, n, k = 1, 2, 3, . . . , n, j 6= k.Here, ρjk is the correlation coefficient between quality charac-

teristic j and quality characteristic k; Cov(Qj,Qk) is the covarianceof quality characteristic j and quality characteristic k; σQjand σQkare the standard deviation of quality characteristic j and qualitycharacteristic k, respectively.The correlation is checked by testing the following hypothesis:{H0 : ρjk = 0 (There is no correlation)H1 : ρjk 6= 0 (There is correlation).

Step 3: Calculation of the principal component score

(a) Calculate the eigenvalue λk and the corresponding eigenvectorβk(k = 1, 2, . . . , n) from the correlation matrix formed by allquality characteristics.

(b) Calculate the principal component scores of the normalizedreference sequence and comparative sequences using theequation shown below:

Yi(k) =n∑j=1

X∗i (j)βkj, i = 0, 1, 2, . . . ,m; k = 1, 2, . . . , n (5)

where, Yi(k) is the principal component score of the kth elementin the ith series. X∗i (j) is the normalized value of the jth element inthe ith sequence, and βkj is the jth element of eigenvector βk.Step 4: Calculation of the individual grey relational grades(1) Calculation of the individual grey relational coefficientsUse the following equation to calculate the grey relational

coefficient between X0(k) and Xi(k).

r0,i(k) =∆min + ξ∆max

∆0,i(k)+ ξ∆max, i = 1, 2, . . . ,m; k = 1, 2, . . . , n.(6)

Here, r0,i(k) is the relative difference of kth element betweensequence Xi and the comparative sequence X0 (also called greyrelational grade), and ∆0,i(k) is the absolute value of differencebetween X0(k) and Xi(k).

∆0,i(k) =

∣∣X∗0 (k)− X∗i (k)∣∣ , no significant correlationbetween quality characteristics|Y0(k)− Yi(k)| , Significant correlationbetween quality characteristics.

(7)

∆max =

maximaxk

∣∣X∗0 (k)− X∗i (k)∣∣ , no significant

correlation between quality characteristicsmaximaxk|Y0(k)− Yi(k)| , Significant correlation

between quality characteristics.

(8)

∆min =

minimink

∣∣X∗0 (k)− X∗i (k)∣∣ , no significant

correlation between quality characteristicsminimink|Y0(k)− Yi(k)| , Significant correlation

between quality characteristics.

(9)

Note that ξ is called the distinguishing coefficient, and its valueis in between 0 to 1. In general it is set to 0.5, [24].(2) Calculation of the weight of each quality characteristics by entropymethodIn information theory, entropy is a measure of how disordered

a system is. When applying the concept of entropy to weight mea-surement, an attribute with a large entropy means it has a greatdiversity to responses so the attribute has more significant influ-ence on the response. Recently, entropy measurement method isused to decide the weights in grey relational analysis. Accordingto the definition proposed by Wen, Chang, and You [21], the map-ping function fi : [0, 1] → [0, 1] used in entropy should satisfythree conditions: (1) fi(0) = 0 (2) fi(x) = fi(1 − x) and (3) fi(x) ismonotonic increasing in the range x ∈ (0, 0.5). Thus, the follow-ing functionwe(x) can be used as the mapping function in entropymeasurement.

we(x) = x.e(1−x) + (1− x)ex − 1. (10)

The maximum value of this function occurs at x = 0.5, and thevalue is e0.5 − 1 = 0.6487. In order to get the mapping result inthe range [0, 1], Wen et al. [21] defined new entropy:

W ≡1

(e0.5 − 1)

m∑i=1

we(xi). (11)

Assume there is a sequence ∈i = {ri(1), ri(2), . . . , ri(n)}. Notethat i = 1, 2, . . . ,m; j = 1, 2, . . . , n.The steps for weight calculation are as follows [25].

58 S. Datta et al. / Journal of Manufacturing Systems 28 (2009) 55–63

Table 1Process control parameters and their limits.

Parameters Units Notation Level 1 Level 2 Level 3 Level 4 Level 5

Voltage (OCV) Volts V 25 27 28 29 31Wire feed rate cm/min Wf 340 655 970 1285 1600Traverse speed cm/min Tr 46 72 98 124 150Stick-out mm N 25 27 29 31 33

(a) Calculation of the sum of the grey relational coefficient in allsequences for each quality characteristic

Dj =m∑i=1

ri(j), j = 1, 2, . . . , n. (12)

(b) Evaluation of the normalized coefficient

k =1

(e0.5 − 1)×m=

10.6487×m

. (13)

(c) Calculation of the entropy of each quality characteristics

ej = km∑i=1

we

(ri(j)Dj

), j = 1, 2, . . . , n. (14)

Here,we(x) = x.e(1−x) + (1− x)ex − 1.(d) Calculation of the sum of entropy

E =n∑j=1

ej. (15)

(e) Calculation of the weight of each quality characteristic:

wj =1

n− E.[1− ej]

n∑j=1

1n−E .[1− ej]

, here, j = 1, 2, . . . , n. (16)

(3) Calculation of the overall grey relational gradeAfter the calculation of the grey relational coefficient and the

weight of each quality characteristic, the grey relational grade isdetermined by:

Γ0,i =

n∑k=1

wkr0,i(k), i = 1, 2, . . . ,m. (17)

In this paper, the multiple quality characteristics are combinedto one grey relational grade, thus the traditional Taguchi methodcanbeused to evaluate the optimal parameter combination. Finallythe anticipated optimal process parameters are verified by carryingout the confirmatory experiments.

3. Grey based Taguchi method

Taguchi’s philosophy, developed by Dr. Genichi Taguchi, is anefficient tool for the design of high quality manufacturing systems.It is a method, based on Orthogonal Array experiments, whichprovides much reduced variance for the experiment, resulting inoptimal setting of process control parameters. Orthogonal Arrayprovides a set of well-balanced experiments (with fewer experi-mental runs), and Taguchi’s signal-to-noise ratios (S/N), which arelogarithmic functions of desired output, serve as objective func-tions, in the optimization process. In order to evaluate the optimalparameter setting, Taguchi method uses a statistical measure ofperformance, called signal-to-noise ratio. The S/N ratio takes boththe mean and the variability into account. The S/N ratio is the ra-tio of themean (Signal) to the standard deviation (Noise). The ratiodepends on the quality characteristics of the product/process to be

optimized. The standard S/N ratios generally used are as follows:—Nominal-is-Best (NB), lower-the-better (LB) andHigher-the-Better(HB). The optimal setting is the parametric combination which hasthe highest S/N ratio. However, Traditional Taguchi method can-not solve the multi-objective optimization problem. This can beachieved by the grey based Taguchimethod. In grey relational anal-ysis, experimental data (i.e. measured features of quality charac-teristics of the product) are first normalized, ranging from zeroto one. This process is known as grey relational generation. Next,based on normalized experimental data, the grey relational coeffi-cient is calculated to represent the correlation between the desiredand actual experimental data. Then, overall grey relational gradeis determined by averaging the grey relational coefficient corre-sponding to selected responses. The overall performance charac-teristic of themultiple response process depends on the calculatedoverall grey relational grade. This approach converts a multiple-response process optimization problem into a single response op-timization situation, where the objective function is overall greyrelational grade. Using the grey-Taguchimethod, the optimal para-metric combination is then evaluated by maximizing the S/N ratioof the overall grey relational grade.

4. Experimentation and data collection

Bead-on-plate submerged arc welding on mild steel plates(thickness 10 mm) has been carried out as per Taguchi’s L25 OAdesign, with 25 combinations of voltage (OCV), wire feed rate,traverse speed (also called welding speed) and electrode stick-out (length of electrode extension from the nozzle of SAW setup).Copper coated electrode wire of diameter 3.16 mm (AWS A/S5.17:EH14) has been used during the experiments. Welding hasbeen performedwith flux (AWSA5.17/SFA 5.17)with grain size 0.2to 1.6 mm with basicity index 1.6 (Al2O3 +MnO2 35%, CaO+MgO25% and SiO2 + TiO2 20% and CaF2 15%). The experiments havebeen performed on Submerged Arc Welding Machine- INDARCAUTOWELD MAJOR (Maker: IOL Ltd., India). A weld havingbeen made, the specimens are been prepared for metallographic(macrostructure) test. Features of bead geometry (Fig. 1) havebeen observed in Optical Trinocular Metallurgical Microscope(Make: Leica, GERMANY, Model No. DMLM, S6D & DFC320 andQ win Software). Process parameters and selected domain ofexperimentation is shown in Table 1. From the knowledge acquiredfrom literature [26], it has been found that voltage, wire feed rate,traverse speed and stick-out happen to be the important factorsin influencing weld quality (bead geometry in the present case).Moreover, these factors have been found possible to be controlledmanually in the experimental setup used. Therefore, these factorshave been selected as process variables.The levels of the factors (voltage, wire feed rate, traverse speed)

have been selected based on the availability of factor settingprovision in the setupused. The setuphas theprovision of changingeach of the factors at seven discrete levels, out of which five levelshave been selected for this investigation. Different values of stick-out have been selected based on the knowledge from literature [1].Welding current, as well as voltage, are directly related to theamount of heat input during the welding process, which obviouslyinfluence various bead geometry parameters. The SAW setup thathas been used during experimentation is of a constant voltage type.

S. Datta et al. / Journal of Manufacturing Systems 28 (2009) 55–63 59

Table 2Taguchi’s L25 Orthogonal Array (OA) design.

Sl. no. Levels of factorsV Wf Tr N

1 1 1 1 12 1 2 2 23 1 3 3 34 1 4 4 45 1 5 5 56 2 1 2 37 2 2 3 48 2 3 4 59 2 4 5 110 2 5 1 211 3 1 3 512 3 2 4 113 3 3 5 214 3 4 1 315 3 5 2 416 4 1 4 217 4 2 5 318 4 3 1 419 4 4 2 520 4 5 3 121 5 1 5 422 5 2 1 523 5 3 2 124 5 4 3 225 5 5 4 3

Therefore, it is not possible to vary current whilst keeping voltageconstant, or vice versa. Moreover, the SAW setup does not havethe provision of controlling (to set fixed) welding current, weldingvoltagemanually. Therefore, voltage (OCV) alone has been selectedas the representative of heat input.The design of the experiment, based on Taguchi’s L25 Orthogo-

nal Array [26], and the collected experimental data, related to in-dividual quality indicators of bead geometry, have been listed inTables 2 and 3 respectively.

5. Data analysis and optimization

Experimental data have been normalized using Eqs. (1) and (2).For reinforcement and beadwidth LB; and for depth of penetration,

Table 4Data preprocessing of each performance characteristics (Normalization of experi-mental data).

Sl. no. Normalized dataPenetration Reinforcement Bead width Dilution

Ideal condition 1.0000 1.0000 1.0000 1.00001 0.8510 0.4902 0.9487 0.65492 0.7855 0.6691 0.8865 0.77713 0.7801 0.7214 0.8785 0.79994 0.7930 0.7205 0.8843 0.79715 0.8735 0.5813 0.9410 0.71496 0.7004 0.7871 0.8891 0.76247 0.7047 0.8786 0.8567 0.81808 0.7154 0.8735 0.8643 0.78159 0.6827 0.7611 0.9795 0.875810 1.0000 0.4634 0.6214 0.865811 0.6688 0.9090 0.8598 0.732912 0.6575 0.9398 0.8771 0.866813 0.6590 0.8377 1.0000 0.829714 0.8436 0.5640 0.5912 0.841315 0.8887 0.6270 0.6888 0.869416 0.6438 1.0000 0.8920 0.768817 0.6473 0.8778 0.9898 0.778018 0.7746 0.6222 0.5515 0.805819 0.7593 0.7946 0.6352 0.791520 0.8474 0.6173 0.7526 1.000021 0.6782 0.8383 0.9253 0.641422 0.7198 0.6742 0.4632 0.698823 0.7154 0.7277 0.5791 0.900224 0.7363 0.7633 0.6382 0.886725 0.8217 0.6349 0.7216 0.8708

dilution HB criteria have been selected. The normalized data areshown in Table 4.After normalization, a check has been made to verify whether

the responses (i.e. quality indexes of bead geometry) are correlatedor not. Table 5 indicates the correlation coefficient among theresponses. The coefficient of correlation between two responses,has been calculated using Eq. (4). It has been observed that allresponses are correlated to each other. In order to eliminateresponse correlations, Principal Component Analysis (PCA) hasbeen applied to derive four independent quality indexes (calledprincipal components), using Eq. (5). The analysis of the correlationmatrix is shown in Table 6. The independent quality indexes are

Table 3Experimental data related to features of bead geometry.

Sl. no. Experimental data related to quality features of bead geometryPenetration (mm) Reinforcement (mm) Bead width (mm) Dilution (%)

1 4.1970 2.2773 9.1647 35.38252 3.8740 1.6683 9.8077 41.98253 3.8475 1.5475 9.8975 43.21254 3.9110 1.5493 9.8327 43.06255 4.3080 1.9203 9.2398 38.62256 3.4545 1.4183 9.7800 41.18757 3.4755 1.2705 10.1497 44.19508 3.5285 1.2780 10.0600 42.22009 3.3670 1.4667 8.8767 47.312510 4.9320 2.4087 13.9917 46.772511 3.2985 1.2280 10.1127 39.597512 3.2430 1.1878 9.9132 46.830013 3.2500 1.3325 8.6950 44.825014 4.1605 1.9793 14.7075 45.452515 4.3830 1.7803 12.6232 46.970016 3.1750 1.1163 9.7482 41.532517 3.1925 1.2717 8.7850 42.032518 3.8205 1.7940 15.7650 43.535019 3.7450 1.4048 13.6882 42.762520 4.1795 1.8085 11.5528 54.025021 3.3450 1.3317 9.3967 34.652522 3.5500 1.6557 18.7718 37.752523 3.5285 1.5340 15.0150 48.635024 3.6315 1.4625 13.6247 47.905025 4.0525 1.7582 12.0500 47.0425

60 S. Datta et al. / Journal of Manufacturing Systems 28 (2009) 55–63

Table 5Check for correlation between the responses.

Sl. no. Correlation between responses Coefficient of correlation Comment

1 Penetration and reinforcement −0.571 Both are correlated2 Penetration and bead width −0.194 Both are correlated3 Penetration and dilution +0.368 Both are correlated4 Reinforcement and bead width +0.458 Both are correlated5 Reinforcement and dilution +0.064 Both are correlated6 Bead width and dilution −0.150 Both are correlated

Table 6(Analysis of correlation matrix) Eigenvalues, eigenvectors, accountability proportion (AP) and cumulative accountability proportion (CAP) computed for the four majorquality indicators.

ψ1 ψ2 ψ3 ψ4

Eigenvalue 1.9037 1.0885 0.7834 0.2243

Eigenvector

∣∣∣∣∣∣∣0.589−0.593−0.4730.276

∣∣∣∣∣∣∣∣∣∣∣∣∣∣0.2310.4210.2510.841

∣∣∣∣∣∣∣∣∣∣∣∣∣∣0.511−0.2380.785−0.256

∣∣∣∣∣∣∣∣∣∣∣∣∣∣−0.581−0.6440.3100.389

∣∣∣∣∣∣∣AP (Accountability proportion) 0.476 0.272 0.196 0.056CAP(Cumulative accountability proportion) 0.476 0.748 0.944 1.000

Table 7Principal components in all L25 OA experimental observations.

Sl. no. Principal components ψ1 (1st PC) to ψ4(4th PC)(1st PC) ψ1 (2nd PC) ψ2 (3rd PC) ψ3 (4th PC) ψ4

Ideal condition −0.2010 1.7440 0.8020 −0.52601 −0.0574 1.1918 0.8953 −0.26132 −0.1390 1.3392 0.7391 −0.31023 −0.1631 1.3771 0.7118 −0.33434 −0.1585 1.3788 0.7239 −0.34055 −0.0780 1.2839 0.8637 −0.31216 −0.2643 1.3575 0.6733 −0.34167 −0.2854 1.4356 0.6141 −0.39158 −0.2897 1.4072 0.6361 −0.40629 −0.2708 1.4605 0.7124 −0.242510 0.2592 1.3102 0.6669 −0.350011 −0.3495 1.3694 0.6127 −0.422312 −0.3457 1.4967 0.5789 −0.378213 −0.3526 1.4537 0.7100 −0.289614 0.1150 1.2882 0.5456 −0.342815 0.0658 1.3733 0.6230 −0.368416 −0.4235 1.4402 0.5944 −0.442517 −0.3927 1.4218 0.6997 −0.331918 0.0488 1.2570 0.4744 −0.366319 −0.1060 1.3350 0.4949 −0.448120 0.0531 1.4855 0.6209 −0.267621 −0.3583 1.2813 0.7092 −0.397622 −0.0021 1.1541 0.3921 −0.437023 −0.0356 1.3740 0.4165 −0.354624 −0.0761 1.3973 0.4686 −0.376625 0.0065 1.3706 0.6123 −0.3238

denoted as principal componentsψ1 (1st PC) toψ4 (4th PC). Table 7represents the values of four independent principal components inall experimental runs.∆0i(k) (Table 8) for all principal components have been

evaluated using Eqs. (7)–(9). Grey relational coefficients of all fourprincipal components have been calculated using Eq. (6). Thesehave been furnished in Table 9.The sum of grey relational coefficients Dj, j = 1, 2, 3, 4 for all

principal components, have been calculated using Eq. (12). Theseare shown in Table 10. The value of the normalized coefficienthas been calculated using Eq. (13). In the present case, m =25. Calculated value of the normalized coefficient becomes k =0.061661785.The values of ( ri(j)Dj ) and k×we(

ri(j)Dj) for the four principal com-

ponents have been furnished in Tables 11 and 12, respectively.Entropy of four independent quality indexes (PC), have been calcu-lated using Eq. (14); the values have been furnished in Table 13. The

Table 8Calculation of∆0i(k) for all principal components.

Sl. no. ∆0i (1st PC) ∆0i (2nd PC) ∆0i (3rd PC) ∆0i(4th PC)

1 0.1436 0.5522 0.0933 0.26472 0.0620 0.4048 0.0629 0.21583 0.0379 0.3669 0.0902 0.19174 0.0425 0.3652 0.0781 0.18555 0.1230 0.4601 0.0617 0.21396 0.0633 0.3865 0.1287 0.18447 0.0844 0.3084 0.1879 0.13458 0.0887 0.3368 0.1659 0.11989 0.0698 0.2835 0.0896 0.283510 0.4602 0.4338 0.1351 0.176011 0.1485 0.3746 0.1893 0.103712 0.1447 0.2473 0.2231 0.147813 0.1516 0.2903 0.0920 0.236414 0.3160 0.4558 0.2564 0.183215 0.2668 0.3707 0.1790 0.157616 0.2225 0.3038 0.2076 0.083517 0.1917 0.3222 0.1023 0.194118 0.2498 0.4870 0.3276 0.159719 0.0950 0.4090 0.3071 0.077920 0.2541 0.2585 0.1811 0.258421 0.1573 0.4627 0.0928 0.128422 0.1989 0.5899 0.4099 0.089023 0.1654 0.3700 0.3855 0.171424 0.1249 0.3467 0.3334 0.149425 0.2075 0.3734 0.1897 0.2022

sum of entropy E = 0.657 has been calculated using Eq. (15). Theweights of four principal components (Table 14) have been calcu-lated using Eq. (16). The overall grey relational grade has been cal-culated using Eq. (17), shown in Table 15. Thus, the multi-criteriaoptimization problem has been transformed into a single objectiveoptimization problem using the combination of Taguchi approachand grey relational analyses. The higher the value of grey relationalgrade, the closer the corresponding factor combination is said to beto the optimal.The S/N ratio plot for the overall grey relational grade is

represented graphically in Fig. 2. The S/N ratio for overall greyrelational grade has been calculated using HB (higher-the-better)criterion (Eq. (18)).

SN(Higher − the− better) = −10 log

[1t

t∑i=1

1y2i

]. (18)

Here t is the number of measurements, and yi the measured ithcharacteristic value (i.e. ith quality indicator). With the help of the

S. Datta et al. / Journal of Manufacturing Systems 28 (2009) 55–63 61

Table 9Calculation of individual grey relational coefficients.

Sl. no. Individual grey relational coefficients(1st PC) (2nd PC) (3rd PC) (4th PC)

1 0.7172 0.6401 0.8940 0.54042 0.9175 0.7749 0.9955 0.61433 1.0000 0.8193 0.9034 0.65874 0.9831 0.8214 0.9421 0.67125 0.7590 0.7182 1.0000 0.61766 0.9134 0.7957 0.7992 0.67357 0.8521 0.8987 0.6788 0.79518 0.8407 0.8583 0.7190 0.83989 0.8936 0.9374 0.9053 0.516510 0.3882 0.7441 0.7841 0.691311 0.7079 0.8099 0.6763 0.894912 0.7150 1.0000 0.6229 0.758613 0.7021 0.9265 0.8980 0.580914 0.4908 0.7223 0.5780 0.676015 0.5393 0.8146 0.6945 0.733816 0.5921 0.9056 0.6463 0.975117 0.6354 0.8786 0.8679 0.654018 0.5584 0.6935 0.5007 0.728619 0.8244 0.7703 0.5207 1.000020 0.5535 0.9798 0.6907 0.548921 0.6918 0.7157 0.8955 0.813122 0.6247 0.6128 0.4337 0.951923 0.6776 0.8155 0.4516 0.701424 0.7549 0.8451 0.4953 0.754425 0.6124 0.8113 0.6757 0.6386

Table 10Calculation of Dj (Sum of grey relational coefficients).

Sum of grey relational coefficients of each principal components(1st PC) (2nd PC) (3rd PC) (4th PC)

17.9451 17.8377 16.6466 15.9342

Table 11

Calculation of(ri(j)Dj

).

Sl. no.(ri(j)Dj

)(1st PC) (2nd PC) (3rd PC) (4th PC)

1 0.0400 0.0359 0.0537 0.03392 0.0511 0.0434 0.0598 0.03863 0.0557 0.0459 0.0543 0.04134 0.0548 0.0460 0.0566 0.04215 0.0423 0.0403 0.0601 0.03886 0.0509 0.0446 0.0480 0.04237 0.0475 0.0504 0.0408 0.04998 0.0468 0.0481 0.0432 0.05279 0.0498 0.0526 0.0544 0.032410 0.0216 0.0417 0.0471 0.043411 0.0394 0.0454 0.0406 0.056212 0.0398 0.0561 0.0374 0.047613 0.0391 0.0519 0.0539 0.036514 0.0274 0.0405 0.0347 0.042415 0.0301 0.0457 0.0417 0.046116 0.0330 0.0508 0.0388 0.061217 0.0354 0.0493 0.0521 0.041018 0.0311 0.0389 0.0301 0.045719 0.0459 0.0432 0.0313 0.062820 0.0308 0.0549 0.0415 0.034421 0.0386 0.0401 0.0538 0.051022 0.0348 0.0344 0.0261 0.059723 0.0378 0.0457 0.0271 0.044024 0.0421 0.0474 0.0298 0.047325 0.0341 0.0455 0.0406 0.0401

Fig. 2, optimal parametric combination has been determined. Theoptimal factor setting becomes V1Wf1 Tr5 N5. [Number indicateslevel of factors (Table 1)].

Table 12

Calculation of k× we(ri(j)Dj

).

Sl. no. k× we(ri(j)Dj

)(1st PC) (2nd PC) (3rd PC) (4th PC)

1 0.0064 0.0058 0.0084 0.00552 0.0081 0.0069 0.0093 0.00623 0.0087 0.0073 0.0085 0.00664 0.0086 0.0073 0.0089 0.00675 0.0067 0.0064 0.0094 0.00626 0.0080 0.0071 0.0076 0.00677 0.0075 0.0080 0.0065 0.00798 0.0074 0.0076 0.0069 0.00839 0.0079 0.0083 0.0085 0.005210 0.0035 0.0066 0.0075 0.006911 0.0063 0.0072 0.0065 0.008812 0.0064 0.0088 0.0060 0.007513 0.0063 0.0082 0.0085 0.005914 0.0044 0.0065 0.0056 0.006815 0.0049 0.0073 0.0066 0.007316 0.0053 0.0080 0.0062 0.009517 0.0057 0.0078 0.0082 0.006518 0.0050 0.0062 0.0049 0.007319 0.0073 0.0069 0.0051 0.009820 0.0050 0.0086 0.0066 0.005521 0.0062 0.0064 0.0085 0.008022 0.0056 0.0055 0.0042 0.009323 0.0061 0.0073 0.0044 0.007024 0.0067 0.0075 0.0048 0.007525 0.0055 0.0072 0.0065 0.0064

Table 13Calculation of ej (entropy of each quality indexes).

Entropy of each quality indexes(1st PC) (2nd PC) (3rd PC) (4th PC)

0.1595 0.1807 0.1584 0.1584

Table 14Calculation ofwj (weight value of each quality characteristics).

Weight value of each quality indexes(1st PC) (2nd PC) (3rd PC) (4th PC)

0.2514 0.2451 0.2517 0.2517

Table 15Calculation of overall grey relational grade.

Sl. no. Γ0,i

1 0.69822 0.82583 0.84544 0.85455 0.77406 0.79537 0.80558 0.81419 0.812310 0.651311 0.771912 0.772613 0.775814 0.616115 0.694716 0.778917 0.758118 0.619819 0.778820 0.691321 0.779422 0.656023 0.660424 0.711525 0.6836

62 S. Datta et al. / Journal of Manufacturing Systems 28 (2009) 55–63

Table 16Analysis of variance (ANOVA) of overall grey relational grade.

Source DF Seq SS Adj SS Adj MS F P Rank

V 4 0.0342561 0.0342561 0.0085640 11.02 0.002 2Wf 4 0.0147500 0.0147500 0.0036875 4.75 0.029 3Tr 4 0.0614616 0.0614616 0.0153654 19.78 0.000 1N 4 0.0029761 0.0029761 0.0007440 0.96 0.480 4Error 8 0.0062156 0.0062156 0.0007770Total 24 0.1196594

DF= Degree of freedom, SS= Sum of squared deviation; MS=Mean squared deviation, F= Fisher’s F ratio; P= Probability of significance.

Area of reinforcement

Area of penetration

Penetration

HAZBead width

Weld bead

Reinforcement

Base metal

Fig. 1. Features of bead geometry.

ANOVA of overall grey relational grade (Table 16) is importantto estimate the level of significance and order of significance(rank) of the controllable factors influencing overall quality index(grey relational grade in the present case), representative of thefeatures of bead geometry. In ANOVA, p-value is determined,whichis termed as probability of significance. If p-value for a factorbecomes less than 0.05; then it can be concluded that the factorialinfluence is significant on the response parameter (95% confidencelevel). ANOVA of overall grey relational grade has revealed thatthe important factors influencing overall grey relation grade aretraverse speed, voltage and wire feed rate, in their order ofsignificance. Stick-out has been found to be an insignificant factorin influencing the overall grey relational grade.After evaluating the optimal parameter settings, the next step

is to predict and verify the enhancement of quality characteristicsusing the optimal parametric combination. Table 17 reflects thesatisfactory result of the confirmatory experiment.

Table 17Results of confirmatory experiment.

Optimal settingPrediction Experiment

Level of factors V1Wf1 Tr5 N5 V1 Wf1 Tr5 N5S/N ratio of overall grey relational grade −1.09930 −1.07890Overall grey relational grade 0.8811 0.8832

6. Conclusions

In the present work, the multi-response optimization problemhas been solved by searching an optimal parametric combination,capable of producing desired qualityweld. Four bead geometry fea-tures – depth of penetration, reinforcement, bead width and dilu-tion – have been optimized using grey-based Taguchi method. PCAhas been used to eliminate correlation among the responses and toconvert the correlated responses into independent quality indexes,so as to meet the basic requirement of Taguchi method. Entropymeasurement technique has been found to be efficient in calculat-ing individual response weights, prior to determining the equiva-lent objective function for optimization. The previouslymentionedmethod can be applied for continuous quality improvement of theproduct and off-line quality control.

Acknowledgments

They authors would like to express sincere thanks to the re-viewers for suggesting necessary corrections as well as modifica-tions to make the paper a good contribution.

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-3.2

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1

Levels of factors

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