Research Article Hybrid Integration of Taguchi Parametric

Research ArticleHybrid Integration of Taguchi Parametric Design GreyRelational Analysis and Principal Component AnalysisOptimization for Plastic Gear Production

Nik Mizamzul Mehat12 Shahrul Kamaruddin2 and Abdul Rahim Othman2

1 Department of Mould Technology Kolej Kemahiran Tinggi MARA Balik Pulau Genting 11000 Balik Pulau Penang Malaysia2 School of Mechanical Engineering Universiti Sains Malaysia Engineering Campus 14300 Nibong Tebal Penang Malaysia

Correspondence should be addressed to Nik Mizamzul Mehat nik miza78yahoocom

Received 27 September 2013 Accepted 17 November 2013 Published 14 January 2014

Academic Editors M Chen C Del Gaudio H Lu S Simani and B Sun

Copyright copy 2014 Nik Mizamzul Mehat et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The identification of optimal processing parameters is an important practice in the plastic injection moulding industry because ofthe significant effect of such parameters on plastic part quality and cost However the optimization design of injection mouldingprocess parameters can be difficult because more than one quality characteristic is used in the evaluationThis study systematicallydevelops a hybrid optimization method for multiple quality characteristics by integrating the Taguchi parameter design greyrelational analysis and principal component analysis A plastic gear is used to demonstrate the efficiency and validity of theproposed hybrid optimization method in controlling all influential injection moulding processing parameters during plastic gearmanufacturing To minimize the shrinkage behaviour in tooth thickness addendum circle and dedendum circle of moulded gearthe optimal combination of different process parameters is determinedThe case study demonstrates that the proposed optimizationmethod can produce plastic-moulded gear with minimum shrinkage behaviour of 18 153 and 242 in tooth thicknessaddendum circle and dedendum circle respectively these values are less than the values in the main experiment Thereforeshrinkage-related defects that lead to severe failure in plastic gears can be effectively minimized while satisfying the demand ofthe global plastic gear industry

1 Introduction

Injection moulding is a complex process because of itsrequirements for numerous delicate adjustmentsThe qualityof an injection-moulded part significantly depends on theselection of appropriate materials parts mould designsand processing parameters A review of previous worksshows that the setting of processing parameters significantlyaffects the quality of plastic parts [1 2] The selection ofinjection moulding process parameters previously involves atrial-and-error method [3] However obtaining an optimalparameter setting for complex-manufacturing processes isdifficult because the trial-and-error method is a ldquoone changeat a timerdquo test [4] This tuning exercise is repeated untilthe quality of the moulded part is found satisfactory thusincurring high production costs and long setup times [5]Moreover the adjustments and modifications of processing

parameters rely heavily on the experience and intuition of themoulding personnel [6] Nevertheless the growing demandin the industry for expert moulding personnel exceeds thesupply moreover amateurmoulding personnel requiremorethan 10 years of experience to be considered experts [7]

In a fiercely competitive market the trial-and-errorapproach cannot meet the challenges of globalization par-ticularly at the point where the disadvantages outweigh theadvantages Thus the interest in substituting the trial-and-error method with a fast and reliable optimization approachsuch as the Taguchi method continues to increase TheTaguchi method considerably reduces the time and effortneeded to obtain optimal process conditions and to deter-mine the important factors that affect plastic part qualityTheapplication of the Taguchi method has attracted considerableattention in the literature over the past 20 years and hasbeen widely applied to various fields such as manufacturing

Hindawi Publishing CorporationChinese Journal of EngineeringVolume 2014 Article ID 351206 11 pageshttpdxdoiorg1011552014351206

2 Chinese Journal of Engineering

systems [8] mechanical component design [9] and parame-ter optimization [10 11] Li et al [12] exploited the Taguchimethod to eliminate the weld line of the right door of acopy machine by optimizing the melt temperature injectionspeed and injection pressure Wu and Liang [13] furtherimproved the weld line strength of an injection-mouldedpart by optimizing the mould temperature packing pressureinjection acceleration and packing time By focusing onwarpage in the thin-walled plastic part many researchersused the experimental design of the Taguchi method to opti-mize the injection moulding parameters packing pressurewas found to be the most influential factor on the warpageproblem [14ndash16] Oktem et al [17] improved the shrinkageand warpage in a plastic part by approximately 217 and07 respectively by determining the optimal packing timepacking pressure injection time and cooling time by usingthe Taguchi method

A number of studies have shown that the Taguchimethod is a robust experimental design for obtaining thebest combination of factors or levels with the lowest costsolution to achieve the quality requirements of plastic partsNevertheless the Taguchi method exhibits a number oflimitations The optimization design of injection mouldingprocess parameters can be difficult because more than onequality characteristic is used in the evaluation Problemsarise when the optimal process parameters contradict oneanother because of different mechanisms [18] To addressthis problem many studies devised new experiment designmethodologies that optimize multiple quality characteristicssimultaneously while providing accurate results by integrat-ing the Taguchi method with other techniques One suchtechnique is grey relational analysis (GRA) This techniquewas first proposed byDeng [19] to optimize themultiresponseproblem by using the grey relational coefficient and greyrelational grade The grey relational coefficient can expressthe relationship between the desired and actual experimentresults whereas the grey relational grade is simultaneouslycomputed to correspond to each quality characteristic

Few works have studied the optimization of injectionmoulding process parameters by using the integration ofTaguchimethod andGRA Fung [20] adopted the integrationmethod to improve the wear volume losses in both theparallel and perpendicular directions and inspected theextent in which process parameters influence each qualitycharacteristic by using the comparability sequence with alarge value of grey relational grade Another two studiesconducted by Fung et al [21] and Yang [22] used the sameprocedure to optimize the processing parameters formultiplequality characteristicsThe former focused on the concurrentimprovement of yield stress and elongation of the PCABSblend whereas the latter examined the effects of processingparameters on the mechanical and tribological properties ofPC composites including ultimate stress surface roughnessand friction coefficient

Another technique that can be applied with the Taguchimethod in addressing multiple quality characteristic prob-lems is principal component analysis (PCA) By using PCAa set of original responses is transformed into a set ofuncorrelated components to find the optimal factor or level

combination PCA does nothing when responses are uncor-related thus the best results are obtained when responses orquality characteristics are highly correlated (either positivelyor negatively) [23] The application of PCA involves a seriesof steps that are capable of solving the weakness of thestandalone Taguchi method which requires engineeringjudgment to handle multiple quality characteristics becausethe judgment of an engineer increases uncertainty during thedecision-making process [24] Fung and Kang [25] appliedPCA to the Taguchi method to improve friction propertiesincluding friction coefficients and surface roughness indifferent sliding directions (P-type andAP-type)The authorsextracted the principal components and the coefficient ofdetermination to establish a comprehensive index and obtainthe final optimal parameter setting

Although both techniques are widely integrated with theTaguchi method to overcome multiple quality characteristicproblems both integration techniques have several limita-tions In combining the Taguchimethod andGRA a problemarises when calculating the value of the grey relational gradein GRA An engineering judgment or subjective estimation isrequired to determine the weighting values for each qualitycharacteristic thus resulting in an increase in uncertaintyduring the decision-making process This approach cannotobjectively reveal the relative importance of multiple qualitycharacteristic performance For the integration of the Taguchimethod and PCA the primary problem is the difficulty ofinterpretation because the original variables are substitutedand the principal components are heavily affected by the scal-ing of the variables In both methods no scientific analysishas been performed in terms of performance under optimalprocess conditions to verify the accuracy of the optimizationresults Therefore the effectiveness and reliability of theintegration of the Taguchi method with GRA or PCA aloneremain uncertain Further validation experiments on theoptimization results are needed to prove the effectiveness ofthe integrated approach

This study develops a hybrid optimization method byintegrating the robust parameter design of Taguchi GRA andPCA to solve problems with multiple quality characteristicsAn injection-moulded plastic gear is used to demonstrate theefficiency and validity of the proposed hybrid optimizationmethod in controlling all influential injection mouldingprocessing parameters during plastic gear manufacturing Byfocusing onminimizing the shrinkage behaviour in the tooththickness addendumcircle and dedendumcircle ofmouldedgear the optimal combination of four process parametersnamely melt temperature packing pressure packing timeand cooling time at three different levels is determined byusing the proposed hybrid optimization method

2 TaguchiGRAPCA Hybrid Method forthe Optimization of Injection MouldingProcess Parameters

21 Taguchi Method The Taguchi method is used to designexperiments based on the orthogonal arrays (OAs) TaguchiOAs are highly fractional orthogonal designs that can be used

Chinese Journal of Engineering 3

to study the whole parameter space with a small number ofexperiments In analyzing the results the Taguchi methoduses a statistical measure of performance known as signal-to-noise (SN) ratio The SN ratio is a measure of performanceto develop products or processes that are insensitive tonoise factors in a controlled manner [26] Noise factorsare uncontrollable factors that influence product or processuncertainty These factors include humidity and weatherDepending on the objective three different methods can beused to calculate the SN ratio in the Taguchi method

(1) smaller-the-better quality characteristic

120578119894119895= minus10 log(1

119899

119899

sum

119895=1

1199102

119894119895) (1)

(2) bigger-the-better quality characteristic

120578119894119895= minus10 log(1

119899

119899

sum

119895=1

1

1199102119894119895

) (2)

(3) nominal-the-better quality characteristic

120578119894119895= minus10 log( 1

119899119904

119899

sum

119895=1

1199102

119894119895) (3)

where 119910119894119895is the 119894th test at the 119895th trial 119899 is the total

number of tests and 119904 is the standard deviation

22 GRA In the analysis of the processing parameters ofthe plastic gear in the case study an appropriate mathematicmodel is established to study the relationship between targetvalues and the quality characteristics of the plastic gearobtained from the experiment The primary concern is toanalyze the differences among the quality characteristics ofplastic gear because of various processing parameters and tounderstand the relationship between quality characteristicsand target values GRA is a method that measures thecorrelation degree among factors based on the similarity ordifference among factors GRA is characterized by small datarequirements and multifactor analysis The procedure of theGRA is presented below

Grey relational generation involves data preprocessingand calculation according to the quality characteristics Thecomputing method of the grey relational generation is asfollows

(1) The-larger-the-better (the higher the target value thebetter)

119909lowast

119894(119896) =

119909(119874)

119894(119896) minusmin119909(119874)

119894(119896)

max 119909(119874)119894(119896) minusmin119909(119874)

119894(119896)

(4)

(2) The-smaller-the-better (the smaller the target valuethe better)

119909lowast

119894(119896) =

max 119909(119874)119894(119896) minus 119909

(119874)

119894(119896)

max 119909(119874)119894(119896) minusmin119909(119874)

119894(119896)

(5)

(3) The-nominal-the-better characteristic (if the target isa specific value set the target value as OB)

119909lowast

119894(119896) = 1 minus

10038161003816100381610038161003816119909(119874)

119894(119896) minusOB10038161003816100381610038161003816

max max 119909(119874)119894(119896) minus OBOB minusmin119909(119874)

119894(119896)

(6)

119909(119874)

119894(119896) is the measurement of the quality characteristic

max 119909(119874)119894(119896) is the largest value of 119909(119874)

119894(119896) and min119909(119874)

119894(119896) is

the smallest value of 119909(119874)119894(119896)

We determine the difference sequence Δ0119894(119896) and the

minimum value Δmin and maximum value Δmax in thedifference sequence including the parameter values requiredto set the reference sequence

Δ0119894 (119896) =

1003816100381610038161003816119909lowast

0(119896) minus 119909

lowast

119894(119896)1003816100381610038161003816

Δmax = maxforall119895isin119894

maxforall119896

10038161003816100381610038161003816119909lowast

0(119896) minus 119909

lowast

119895(119896)10038161003816100381610038161003816

Δmin = minforall119895isin119894

minforall119896

10038161003816100381610038161003816119909lowast

0(119896) minus 119909

lowast

119895(119896)10038161003816100381610038161003816

(7)

We set the identification coefficient 119885 which generally has avalue of 05

After the data preprocessing a grey relational coefficientis calculated to express the relationship between the ideal andactual normalized experimental results The grey relationalcoefficient can be expressed as follows

120576119894(119896) (119896) =

Δmin + 120595ΔmaxΔ0119894+ 120595Δmax

0 lt 120576 [119909lowast

0(119896) 119909

lowast

119894(119896)] le 1

(8)

The average of the grey relational coefficient is then calculatedto obtain the grey relational grade The grey relational gradeis defined as follows

120574119894=1

119899

119899

sum

119896=1

120576119894 (119896) (9)

However the effect of each factor on the system is not exactlythe same in real applications Thus (9) can be modified asfollows

120574119894=

119899

sum

119896=1

119908119896sdot 120576119894 (119896) (10)

where119908119896represents the normalized weighting value of factor

119896 Given the same weights (9) and (10) are equal In GRA thegrey relational grade is used to show the relationship amongsequences If two sequences are identical the value of thegrey relational grade is equal to oneThe grey relational gradealso indicates the degree of influence that the comparabilitysequence can exert over the reference sequence Therefore ifa particular comparability sequence is more important thanthe other comparability sequences to the reference sequencethe grey relational grade for that comparability sequence andreference sequence will be higher than other grey relationalgrades [27] In this study the corresponding weighting values119908119896are obtained from PCA


23 PCA PCA was developed by Pearson [28] and Hotelling[29] This approach explains the structure of variance-covariance by the linear combinations of each quality char-acteristic The procedures are described as follows

(1) Original Multiple Quality Characteristics Array Consider

119909119894(119895) 119894 = 1 2 119898 119895 = 1 2 119899

119909119894=

[[[[

[

1199091 (1) 119909

1 (2) 1199091 (3) sdot sdot sdot sdot sdot sdot 119909

1 (119899)

1199092 (1) 119909


2 (119899)

sdot sdot sdot sdot sdot sdot

119909119898 (1) 119909119898 (2) 119909119898 (3) sdot sdot sdot sdot sdot sdot 119909119898 (119899)

]]]]

]

(11)

where 119898 is the number of experiments and 119899 is the numberof quality characteristics In this paper 119909 is the grey relationalcoefficient of each quality characteristic and119898 = 9 and 119899 = 3

(2) Correlation Coefficient Array The correlation coefficientarray is evaluated as follows

119877119895119897= (

Cov (119909119894(119895) 119909

119894 (119897))

120590119909119894(119895)119883120590

119909119895 (119897)) 119895 = 1 2 3 119899

119897 = 1 2 3 119899

(12)

where Cov(119909119894(119895) 119909119894(119897)) are the covariance of sequences 119909

119894(119895)

and 119909119894(119897) respectively 120590

119909119894(119895) is the standard deviation of

sequence 119909119894(119895) 120590119909119895(119897) is the standard deviation of sequence

119909119894(119897)

(3) Determining the Eigenvalues and Eigenvectors The eigen-values and eigenvectors are determined from the correlationcoefficient array

(119877 minus 120582119896119868119898) 119881119894119896= 0 (13)

where 120582119896is an eigenvalue sum119899

119896=1120582119896= 119899 and 119896 = 1 2 119899

119881119894119896= [1198861198961 1198861198962 119886

119896119899]119879 correspond to eigenvalue 120582

119896

(4) Principal Components The uncorrelated principal com-ponent is formulated as follows

119884119898119896=

119899

sum

119894=1

119883119898 (119894) sdot 119881119894119896 (14)

where 1198841198981

is the first principal component 1198841198982

is the secondprincipal component and so on The entire technical line ofthe hybrid TaguchiGRAPCA process optimization methodfor plastic injection moulding is summarized and illustratedin Figure 1

3 Implementation of the Proposed HybridTaguchiGRAPCA Optimization Proceduresfor Plastic Gear

31 Determination of Quality Characteristics In this papera plastic gear is used to demonstrate the efficiency and

Table 1 General properties of PP

PropertiesDensity (gcm3) 090 to 091Raw density (gcm3) 0805Melt flow index (g10min) 1078Modulus of elasticity (MPa) 4100Charpy impact toughness (kJm2) 14 to 18

validity of the proposed hybrid optimization method Withthe capability to run without lubrication and corrosion plas-tic gears have been widely used in the automotive industryofficemachines household utensils food and textile machin-ery and a host of other applications [30] Plastic-mouldedgears offer some material properties that are not achievablewith metal-based gears including unique advantages inweight noise modulus self-lubrication chemical resistanceand low cost However the practicability of injection mould-ing in producing low-cost plastic-moulded gears remainsrestricted by the occurrence of shrinkage in the final partsAs one of themost common and prominent defects of plasticsevere shrinkage leads to the deflection or warpage of themoulded part and negatively affects the dimensional stabilityand accuracy of the involute profile concentricity roundnesstooth spacing uniformity and gear size Sever shrinkagedefect is often forgotten and regarded lightly in many worksrelated to plastic-moulded gears However this defect criti-cally affects the quality of the final moulded gear noise andvibration and shortens the service life of the gear because ofdifferent damage mechanisms such as tooth fatigue creepexcessive wear and plastic deformation Hence shrinkageminimization is the optimization objective of this paperThispaper focuses on the optimization of processing parametersvia the integration of the TaguchiGRAPCA tominimize theshrinkage behaviour of plastic injection-moulded gears Toinspect shrinkage behaviour on the dimensional stability ofthe moulded gear three quality characteristics are selectedtooth thickness addendum circle and dedendum circle

32 Selection of Taguchi OA In this study the crystallinethermoplastic polypropylene (PP) is specified for the gearThe PP used in this study is obtained from PropyleneMalaysia Sdn Bhd Malaysia The general properties of PPare shown in Table 1 The spur gear design used is compliantwith the American Gears Manufacturers Association stan-dards The detailed geometry and specifications of the gearare shown in Figure 2

ThePPgears are injected by using aBattenfeldTM750210injection moulding machine The experiment is conductedwith four controllable three-level processing parametersmelttemperature packing pressure packing time and coolingtime Other processing parameters such as mould tempera-ture (25∘C) injection pressure (80 bar) and stroke distance(60mm) are kept constant during the experiment Theselected processing parameters and their levels are shownin Table 2 The L

9(34) OA (Table 3) is conducted to study


Start

End

Determination of quality characteristics

Selection of Taguchirsquos orthogonal array

Conduct designed experiment

Quality testing

Computation of SN ratio

Nominal-the-betterBigger-the-betterSmaller-the-better

Grey generation of raw data

Computation of grey relational coefficient of response variables

Computation of the contribution of the respective quality characteristics using PCA

Determination of covariance of normalised grey relational coefficient and correlation coefficient array

Determination of the eigenvalues and eigenvectors

Computation of grey relational grades

Identification of optimal processing parameters and levels using the main effect graph

Identification of most influential processing parameters using the ANOVA

Verification test

Normalization of grey relational coefficient between 0 and 1

Figure 1 Technical line of hybrid TaguchiGRAPCA process optimization method

the four processing parameters Nine trials of PP gears withfive repetitions are produced by using the OA

33 Shrinkage Measurement Rax Vision DC 3000 Mitutoyoprofile projector is used to inspect the accuracy of the

specific profile of the involute gear teeth after mouldingIn this study the profile projector is used to measure 2Dtooth thickness as well as addendum and dedendum circlesby using the coordinates of selected points along the gearprofile With large magnifications and micrometer readouts


R1650

R1350

empty10

10

empty12

Figure 2 Geometry and specification of the spur gear module =15 pressure angle = 20∘ number of teeth = 20 face width = 10mm

Table 2 Processing parameters and levels studied

Column Factors Level 1 Level 2 Level 3A Melt temperature (∘C) 200 220 240B Packing pressure () 60 80 100C Packing time (s) 5 10 15D Cooling time (s) 30 40 50

Table 3 OA L9 (34) of the experimental runs

Factorstrialno

A B C DMelt

temperature(∘C)

Packingpressure()

Packingtime (s)

Coolingtime (s)

1 200 60 5 302 200 80 10 403 200 100 15 504 220 60 10 505 220 80 15 306 220 100 5 407 240 60 15 408 240 80 5 509 240 100 10 30

the profile projector can ensure fairly accurate measurementscompared with a Vernier caliper and micrometer In theprocess of addendum and dedendum circle measurement 20edge points of each tooth gear are set in the profile projectorTo ensure the integrity of the comparison procedure 5injection-moulded gears from the same batch are measuredto determine the repeatability of the part geometry Therelative shrinkage of the selected quality characteristics iscalculated by using the following equation

119878 =(119863 minus 119863

119898)

119863119898

(15)

where 119878 is shrinkage 119863 is the reading of the diametermeasurement by using the profile projector and 119863

119898is the

mould dimension

34 SN Analysis For constant processing parameters theaverage value of five repeated results for shrinkage behaviourin tooth thickness addendum circle and dedendum circleis calculated and is considered the final result In processingoptimization all the results of shrinkage behaviour in tooththickness addendum circle and dedendum circle are trans-formed into SN ratio The conversion of the results into SNratio involves a series of calculations of the mean squareddeviation For this case the smaller-the-better category isused to characterize the shrinkage behaviour of tooth thick-ness addendum circle and dedendum circle (see (1)) Thefinal measured results and SN ratios for the three qualitycharacteristics are shown in Table 4

35 Grey Generation of Raw Data In the GRA the experi-mental results for the SN ratios of shrinkage behaviour intooth thickness addendum circle and dedendum circle inTable 4 are first normalized according to the-smaller-the-better characteristic of the sequence by using (5) The valuesof tooth thickness addendum circle and dedendum circleare set as the reference sequence 119909(119874)

0(119896) 119896 = 1 2 3 and

the comparability sequences 119909(119874)119894(119896) 119894 = 1 2 3 9 119896 =

1 2 3 Table 5 lists all the sequences after data preprocessingAccording to Deng (1989) [19] a larger value of the nor-malized results corresponds to better performance and themaximum normalized results that are equal to one indicatethe best performance

According to Table 5 the deviation sequences Δ01(119896) can

be calculated as followsΔ01 (1) =

1003816100381610038161003816119909lowast

0(1) minus 119909

lowast

1(1)1003816100381610038161003816 = |10000 minus 00000| = 10000

Δ01 (2) =

1003816100381610038161003816119909lowast

0(2) minus 119909

lowast

1(2)1003816100381610038161003816 = |10000 minus 08743| = 01257

Δ01 (3) =

1003816100381610038161003816119909lowast

0(3) minus 119909

lowast

1(3)1003816100381610038161003816 = |10000 minus 09757| = 00243

(16)

Therefore Δ01= (10000 01257 00243)

The same calculating method is performed for 119894 =

1 9 and the results of all Δ0119894for 119894 = 1 9 are listed

in Table 6 By investigating the data presented in Table 6Δmax(119896) and Δmin(119896) can be expressed as follows

Δmax = Δ 08 (1) = Δ 05 (2) = Δ02 (3) = 10000

Δmin = Δ 01 (1) = Δ 08 (2) = Δ09 (3) = 00000(17)

36 Computation of theGrey Relational Coefficient of ResponseVariables The grey relational coefficients for each qualitycharacteristic have been calculated by substituting the distin-guishing coefficient 120595 = 05 by using (8) Examples on greyrelational coefficient 120576

1(119896)are provided as follows

1205761(1)

=00000 + (05) (10000)

10000 + (05) (10000)= 03333

1205761(2)

=00000 + (05) (10000)

01257 + (05) (10000)= 07991

1205761(3)

=00000 + (05) (10000)

00243 + (05) (10000)= 09536

(18)


Table 4 Average shrinkage and SN ratios for each trial

TrialsAverage shrinkage SN ratio

Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

Tooth thickness(dB)

Addendum circle(dB)

Dedendum circle(dB)

1 minus00285 minus00211 minus00197 291568 335159 3408012 minus00542 minus00213 minus00198 250132 334284 3404643 minus00718 minus00202 minus00188 222234 338712 3448914 minus01067 minus00200 minus00196 192536 339651 3415435 minus01045 minus00215 minus00205 191704 333515 3375066 minus01174 minus00211 minus00192 184870 335269 3430387 minus00942 minus00202 minus00186 204134 338720 3460278 minus01298 minus00185 minus00174 176876 346592 3519619 minus00998 minus00197 minus00169 199179 340946 354325

Table 5 Sequences after data preprocessing (grey generation)

Trials Tooththickness

Addendumcircle

Dedendumcircle

Reference sequence 10000 10000 10000Comparability sequence

1 00000 08743 097572 03613 09412 100003 06045 06026 068064 08635 05308 092215 08707 10000 121336 09303 08659 081437 07623 06020 059878 10000 00000 017069 08055 04317 00000

Table 6 Deviation sequences

Deviation sequences Δ01

Δ02

Δ03

Trial 1 10000 01257 00243Trial 2 06387 00588 00000Trial 3 03955 03974 03194Trial 4 01365 04692 00779Trial 5 01293 00000 02133Trial 6 00697 01341 01857Trial 7 02377 03980 04013Trial 8 00000 10000 08294Trial 9 01945 05683 10000

Thus 1205761(119896)

= (03333 07991 09536) 119896 = 1 2 3 A similarprocedure is applied for 119894 = 1 9 Table 7 lists the greyrelational coefficient for each trial of the L

9OA

37 Computation of the Contribution of Respective QualityCharacteristics by Using PCA In optimizing a problem con-cerning multiple quality characteristics or performances anengineering judgment or subjective estimation is needed to

Table 7 Calculated grey relational coefficient for nine comparabil-ity sequences

TrialsGrey relational coefficient

Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

1 03333 07991 095362 04391 08948 100003 05584 05572 061024 07855 05159 086525 07946 10000 070096 08777 07885 072917 06778 05568 055478 10000 03333 037619 07200 04680 03333

Table 8 Eigenvalues and explained variation for principal compo-nents

Principal component Eigenvalue Explained variation ()First 21402 7134Second 05926 1975Third 02672 8908

determine the weighting values for each quality character-istic The use of the conventional method to determine thevalues for each quality characteristics is heavily reliant onexperience and trial-and-error thus resulting in an increasein uncertainty during the decision-making process To revealthe relative importance for each quality characteristic inGRA objectively the PCA is introducedThe PCA is adoptedto determine the corresponding weighting values for eachquality characteristic

The elements of the array for the multiple quality char-acteristics listed in Table 7 represent the grey relationalcoefficient of each quality characteristic These data areused to evaluate the correlation coefficient matrix and todetermine the corresponding eigenvalues from (13) (Table 8)The eigenvector corresponding to each eigenvalue is listed


Table 9 Eigenvectors for principal components

Qualitycharacteristic

EigenvectorFirst principalcomponent

Second principalcomponent

Third principalcomponent

Tooththickness minus05365 07716 03419

Addendumcircle 05678 06297 minus05302

Dedendumcircle 06244 00904 07759

Table 10 Contribution of each individual quality characteristic forthe principal component

Quality characteristic ContributionTooth thickness 02878Addendum circle 03224Dedendum circle 03898

Table 11 Grey relational grade and its order

Trial Grey relational grade Order1 07253 52 08046 23 05782 74 07297 45 08243 16 07910 37 05908 68 05419 89 04880 9

in Table 9 and the square of the eigenvector can representthe contribution of the corresponding quality characteristicto the principal component The contribution of shrinkagebehaviour in the tooth thickness addendum circle anddedendumcircle of the PP-moulded gear is shown inTable 10these contributions are listed as 02878 03224 and 03898respectively Moreover the variance contribution for thefirst principal component characterizing the three qualitycharacteristics is as high as 7134 Therefore for this studythe squares of the respective eigenvectors are selected asthe weighting values of the related quality characteristicCoefficients 119908

1 1199082 and 119908

3in (10) are set as 02878 03224

and 03898 respectively

38 ComputationGrey Relational Grades On the basis of (10)and the data listed in Table 7 the grey relational grades arecalculated as follows

1205741= (02878 times 03333) + (03224 times 07991)

+ (03898 times 09536) = 07232

(19)

By using the same procedure the grey relational grade ofthe comparability sequence for 119894 = 1ndash9 can also be obtainedand is presented in Table 11 The processing parameters were

05

055

06

065

07

075

08

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

Gre

y re

latin

al g

rade

Injection moulding process parameters

Packing time Cooling timeMelt

temperaturePacking pressure

Figure 3 Effect of injection moulding parameter levels on multiplequality characteristics

optimized with respect to a single grey relational grade ratherthan complicated multiple quality characteristics

4 Results and Discussion

41 Optimal Combination of Injection Moulding ProcessingParameters andTheir Levels To determine the optimal com-bination of injection moulding processing parameters forshrinkage behaviour in the tooth thickness addendum circleand dedendumcircle of the studiedmoulded gear the averagegrey relational grade for each injection moulding parameterlevel is calculated by employing the main effect analysis ofthe Taguchi method This process is performed by sortingthe grey relational grades corresponding to the levels of theinjectionmoulding parameters in each column of theOA andthen taking the average of parameters with the same levelsFor instance for factor A (Table 3) experiments 1 2 and 3are set to level 1Therefore by using the data listed in Table 11the average grey relational grade for 119860

1can be calculated as

follows

1198601=(07253 + 08046 + 05782)

3= 07027 (20)

The average grey relational grade for 1198602and 119860

3is

calculated as follows

1198602=(07297 + 08243 + 07910)

3= 07817

1198603=(05908 + 05419 + 04880)

3= 05402

(21)

By using a similarmethod calculations are performed foreach injection moulding parameter level and the main effectanalysis is constructed (Table 12 and Figure 3)

Considering that the grey relational grade represents thelevel of correlation between the reference and comparabilitysequences a larger grey relational grade indicates that thecomparability sequence exhibits a stronger correlation withthe reference sequence A larger grey relational grade resultsin better multiple quality characteristics Figure 3 clearlyshows that the multiple quality characteristics of the PP-moulded gear are significantly affected by the adjustments


Table 12 Main effects table for grey relational grades

Symbol Process parameter Level 1 Level 2 Level 3A Melt temperature (∘C) 07027 07817 05403B Packing pressure () 06819 07236 06191C Packing time (s) 06861 06741 06644D Cooling time (s) 06792 07288 06166

Table 13 ANOVA table for the grey relational grade of ninecomparability sequences

Column Parameters DOF S V PA Melt temperature (∘C) 2 00909 00454 7147B Packing pressure () 2 00166 00083 1306C Packing time (s) 2 00007 00004 055D Cooling time (s) 2 00190 00095 1492Allotherserror 0 00000 00000 000

Total 8 01272 10000

of the processing parameters The increment of melt temper-ature packing pressure and cooling time initially increasesthe grey relational grades thus resulting in the reduction ofshrinkage behaviour in the tooth thickness addendum circleand dedendum circle of the moulded gear However therelational grades for the three quality characteristics are lowerwhen the melt temperature packing pressure and coolingtime are 240∘C 100 and 50 s respectively By contrastthe increment of packing time reduces the value of the greyrelational grade thus increasing the shrinkage behaviour intooth thickness addendum circle and dedendum circle

In this case the best combination of processing param-eters and levels can easily be obtained from the main effectanalysis by selecting the level of each parameter with thehighest grey relational grade A

2 B2 C1 and D

2show the

largest value of grey relational grade for factors A B C andDrespectively (Figure 3) Thus the optimal parameter settingthat statistically results in the minimum shrinkage behaviourin the tooth thickness addendumcircle and dedendumcircleof the PP-moulded gear is predicted to be A

2 B2 C1 and

D2 The melt temperature is 220∘C packing pressure is 80

packing time is 5 s and cooling time is 40 s

42 Effects of Processing Parameters onQuality CharacteristicsTo examine the extent in which injection moulding parame-ters significantly influence the performance of moulded gearANOVA is performed on the Taguchi method for the greyrelational grade of nine comparability sequences (Table 11)The computed quantity of degrees of freedom (DOF) sum ofsquare (119878) variance (119881) and percentage contribution (119875) arepresented in Table 13

In this case the percentage contribution of each pro-cessing parameter is directly calculated from S because theDOF for the error term is equal to zero [31] The significanceof each processing parameter in the shrinkage behaviour inthe tooth thickness addendum circle and dedendum circleof the moulded gear can be determined by the percentage

Table 14 Results of the verification test

Verificationtest

Tooth thickness(mm)

Addendumcircle (mm)

Dedendumcircle (mm)

1 minus00187 minus00194 minus001602 minus00391 minus00186 minus001483 minus00245 minus00165 minus001594 minus00164 minus00194 minus001505 minus00221 minus00162 minus00148Average minus00180 minus00153 minus00242

contribution Roy [32] suggested an alternative by using the10 rule that is a parameter is considered insignificantwhen its influence is less than 10 of the highest parameterinfluence From the results of ANOVA in Table 13 melttemperature appears to be the most decisive processingparameter in reducing the shrinkage behaviour in the tooththickness addendum circle and dedendum circle of themoulded gear with the highest percentage contribution of7147 thus outweighing the other process variables Theanalysis also reveals that packing pressure and cooling timeare significant because their percentages are more than 10of the highest parameter influence (715) Cooling time andpacking pressure achieve 1492 and 1306 respectivelyPacking time results in only 055 which is significantlyless than 10 of the highest parameter influence In thiscase the packing time is considered insignificant in theshrinkage behaviour of the tooth thickness addendum circleand dedendum circle of the moulded gear

43 Verification Test Once the optimal levels of the injectionmoulding process parameters are identified the subsequentstep is to verify the improvements in the quality characteris-tics by using this optimal combination The verification testcan be used to assess the accuracy of the proposed hybridTaguchiGRAPCA optimization method An experimentalverification test is conducted by using the same procedures asprevious runs under the optimal process conditions namelyA2 B2 C1 and D

2 to produce the plastic-moulded gear

Table 14 lists the results of five repetitions of the verificationtests by using the optimal process conditions obtained by theproposed hybrid optimization method

After optimization the minimum shrinkage behaviourin tooth thickness addendum circle and dedendum circleis averaged as 18 153 and 242 respectively Theshrinkage behaviour for the three quality characteristicsdecreases with respect to those attained in the main exper-iment presented in Table 4

5 Conclusion

In this study a hybrid TaguchiGRAPCA optimizationmethod for injection moulding process parameters has beendeveloped systematically to overcome the shortcomings ofindividual methods in multiple quality characteristics prob-lems The problem that arises when calculating the weight-ing value for each quality characteristic in GRA has been


addressed by integrating the PCA to determine the greyrelational grades Thus the conventional approach of usingengineering judgment or subjective estimation to determinethe weighting values for each quality characteristics canbe avoided A plastic gear is employed in this study asa case study to demonstrate the efficiency and validity ofthe proposed hybrid optimization method in controlling allinfluential injection moulding processing parameters duringplastic gearmanufacturingThe shrinkage behaviour in tooththickness addendumcircle and dedendumcircle is chosen asthe optimization objectives Through a series of analyses andoptimizations of selected multiple quality characteristics forthe case of a plastic gear the minimum shrinkage behaviourin tooth thickness addendum circle and dedendum circleafter optimization by the proposed hybrid optimizationmethod is averaged as 18 153 and 242 respectivelywhich are less than the values in the main experiment

Conflict of Interests

The authors would like to declare that all financial andmaterial support for the conduct of this research and thepreparation of this paper is clearly termed in the Acknowl-edgment and there will be no conflict of interests in relationto the financial gain subject to the publication of this paper

Acknowledgment

The authors acknowledge the Research Grant provided byUniversiti Sains Malaysia Pulau Pinang for funding thestudy that resulted in this paper

References

[1] H Kurtaran and T Erzurumlu ldquoEfficient warpage optimizationof thin shell plastic parts using response surface methodologyand genetic algorithmrdquo International Journal of AdvancedMan-ufacturing Technology vol 27 no 5-6 pp 468ndash472 2006

[2] K-T Chiang and F-P Chang ldquoAnalysis of shrinkage andwarpage in an injection-molded part with a thin shell featureusing the response surface methodologyrdquo International Journalof AdvancedManufacturingTechnology vol 35 no 5-6 pp 468ndash479 2007

[3] F Shi Z L Lou J G Lu and Y Q Zhang ldquoOptimisationof plastic injection moulding process with soft computingrdquoInternational Journal of Advanced Manufacturing Technologyvol 21 no 9 pp 656ndash661 2003

[4] J-R Shie ldquoOptimization of injection-molding process formechanical properties of polypropylene components via ageneralized regression neural networkrdquo Polymers for AdvancedTechnologies vol 19 no 1 pp 73ndash83 2008

[5] Y C Lam L Y Zhai K Tai and S C Fok ldquoAn evolutionaryapproach for cooling system optimization in plastic injectionmouldingrdquo International Journal of Production Research vol 42no 10 pp 2047ndash2061 2004

[6] W-C Chen G-L Fu P-H Tai and W-J Deng ldquoProcessparameter optimization forMIMOplastic injectionmolding viasoft computingrdquo Expert Systems with Applications vol 36 no 2pp 1114ndash1122 2009

[7] S L Mok C K Kwong and W S Lau ldquoReview of research inthe determination of process parameters for plastic injectionmoldingrdquo Advances in Polymer Technology vol 18 no 3 pp225ndash236 1999

[8] A Mahfouz S A Hassan and A Arisha ldquoPractical simula-tion application evaluation of process control parameters inTwisted-Pair Cables manufacturing systemrdquo Simulation Mod-elling Practice and Theory vol 18 no 5 pp 471ndash482 2010

[9] H-J Shim and J-K Kim ldquoCause of failure and optimization of aV-belt pulley considering fatigue life uncertainty in automotiveapplicationsrdquo Engineering Failure Analysis vol 16 no 6 pp1955ndash1963 2009

[10] N M Mehat and S Kamaruddin ldquoInvestigating the effects ofinjection molding parameters on the mechanical properties ofrecycled plastic parts using the Taguchi methodrdquoMaterials andManufacturing Processes vol 26 no 2 pp 202ndash209 2011

[11] A Akbarzadeh S Kouravand and B M Imani ldquoRobust designof a bimettallic micro thermal sensor using Taguchi methodrdquoJournal of Optimization Theory and Applications vol 157 no 1pp 188ndash198 2013

[12] H Li Z Guo and D Li ldquoReducing the effects of weldlineson appearance of plastic products by Taguchi experimentalmethodrdquo International Journal of Advanced ManufacturingTechnology vol 32 no 9-10 pp 927ndash931 2007

[13] C-H Wu and W-J Liang ldquoEffects of geometry and injection-molding parameters on weld-line strengthrdquo Polymer Engineer-ing and Science vol 45 no 7 pp 1021ndash1030 2005

[14] M-C Huang and C-C Tai ldquoEffective factors in the warpageproblem of an injection-molded part with a thin shell featurerdquoJournal ofMaterials Processing Technology vol 110 no 1 pp 1ndash92001

[15] B Ozcelik and T Erzurumlu ldquoComparison of the warpage opti-mization in the plastic injectionmolding using ANOVA neuralnetwork model and genetic algorithmrdquo Journal of MaterialsProcessing Technology vol 171 no 3 pp 437ndash445 2006

[16] B Ozcelik and I Sonat ldquoWarpage and structural analysis ofthin shell plastic in the plastic injectionmoldingrdquoMaterials andDesign vol 30 no 2 pp 367ndash375 2009

[17] H Oktem T Erzurumlu and I Uzman ldquoApplication of Taguchioptimization technique in determining plastic injection mold-ing process parameters for a thin-shell partrdquo Materials andDesign vol 28 no 4 pp 1271ndash1278 2007

[18] S J Liao D Y Chang H J Chen et al ldquoOptimal process con-ditions of shrinkage and warpage of thin-wall partsrdquo PolymerEngineering and Science vol 44 no 5 pp 917ndash928 2004

[19] J Deng ldquoIntroduction to grey system theoryrdquo Journal of GreySystem vol 1 pp 1ndash24 1989

[20] C-P Fung ldquoManufacturing process optimization for wearproperty of fiber-reinforced polybutylene terephthalate com-posites with grey relational analysisrdquoWear vol 254 no 3-4 pp298ndash306 2003

[21] C-P Fung C-H Huang and J-L Doong ldquoThe study onthe optimization of injection molding process parameters withgray relational analysisrdquo Journal of Reinforced Plastics andComposites vol 22 no 1 pp 51ndash66 2003

[22] Y-K Yang ldquoOptimization of injection-molding process formechanical and tribological properties of short glass fiber andpolytetrafluoroethylene reinforced polycarbonate compositeswith grey relational analysis a case studyrdquo Polymer vol 45 no7 pp 769ndash777 2006


[23] B F J ManlyMultivariate Statistical Methods A Primer Chap-man and HallCRC press Boca Raton Fla USA 3rd edition2005

[24] H C Lio and Y K Chen ldquoOptimizing multi-response problemin the Taguchi method by DEA based ranking methodrdquo Inter-national Journal of Quality and Reliability Management vol 19no 7 pp 825ndash837 2002

[25] C-P Fung and P-C Kang ldquoMulti-response optimization infriction properties of PBT composites using Taguchi methodand principle component analysisrdquo Journal of Materials Process-ing Technology vol 170 no 3 pp 602ndash610 2005

[26] YHChen S C TamW L Chen andHY Zheng ldquoApplicationof the Taguchi method in the optimization of laser micro-engraving of photomasksrdquo International Journal of Materialsand Product Technology vol 11 no 3-4 pp 333ndash344 1996

[27] Y-K Yang J-R Shie and C-H Huang ldquoOptimization of drymachining parameters for high-purity graphite in end-millingprocessrdquo Materials and Manufacturing Processes vol 21 no 8pp 832ndash837 2006

[28] K Pearson ldquoOn lines and planes of closest fit to systems ofpoints in spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash572 1901

[29] H Hotelling ldquoAnalysis of a complex of statistical variables intoprincipal componentsrdquo Journal of Educational Psychology vol24 no 6 pp 417ndash441 1933

[30] K Mao W Li C J Hooke and D Walton ldquoFriction and wearbehaviour of acetal and nylon gearsrdquoWear vol 267 no 1ndash4 pp639ndash645 2009

[31] S M Mousavi S A Shojaosadati J Golestani and F YazdianldquoCFD simulation and optimization of effective parameters forbiomass production in a horizontal tubular loop bioreactorrdquoChemical Engineering and Processing vol 49 no 12 pp 1249ndash1258 2010

[32] R K Roy Design of Experiment Using the Taguchi Approach 16Steps to Product and Process Improvement JohnWiley and SonsNew York NY USA 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



systems [8] mechanical component design [9] and parame-ter optimization [10 11] Li et al [12] exploited the Taguchimethod to eliminate the weld line of the right door of acopy machine by optimizing the melt temperature injectionspeed and injection pressure Wu and Liang [13] furtherimproved the weld line strength of an injection-mouldedpart by optimizing the mould temperature packing pressureinjection acceleration and packing time By focusing onwarpage in the thin-walled plastic part many researchersused the experimental design of the Taguchi method to opti-mize the injection moulding parameters packing pressurewas found to be the most influential factor on the warpageproblem [14ndash16] Oktem et al [17] improved the shrinkageand warpage in a plastic part by approximately 217 and07 respectively by determining the optimal packing timepacking pressure injection time and cooling time by usingthe Taguchi method

A number of studies have shown that the Taguchimethod is a robust experimental design for obtaining thebest combination of factors or levels with the lowest costsolution to achieve the quality requirements of plastic partsNevertheless the Taguchi method exhibits a number oflimitations The optimization design of injection mouldingprocess parameters can be difficult because more than onequality characteristic is used in the evaluation Problemsarise when the optimal process parameters contradict oneanother because of different mechanisms [18] To addressthis problem many studies devised new experiment designmethodologies that optimize multiple quality characteristicssimultaneously while providing accurate results by integrat-ing the Taguchi method with other techniques One suchtechnique is grey relational analysis (GRA) This techniquewas first proposed byDeng [19] to optimize themultiresponseproblem by using the grey relational coefficient and greyrelational grade The grey relational coefficient can expressthe relationship between the desired and actual experimentresults whereas the grey relational grade is simultaneouslycomputed to correspond to each quality characteristic

Few works have studied the optimization of injectionmoulding process parameters by using the integration ofTaguchimethod andGRA Fung [20] adopted the integrationmethod to improve the wear volume losses in both theparallel and perpendicular directions and inspected theextent in which process parameters influence each qualitycharacteristic by using the comparability sequence with alarge value of grey relational grade Another two studiesconducted by Fung et al [21] and Yang [22] used the sameprocedure to optimize the processing parameters formultiplequality characteristicsThe former focused on the concurrentimprovement of yield stress and elongation of the PCABSblend whereas the latter examined the effects of processingparameters on the mechanical and tribological properties ofPC composites including ultimate stress surface roughnessand friction coefficient

Another technique that can be applied with the Taguchimethod in addressing multiple quality characteristic prob-lems is principal component analysis (PCA) By using PCAa set of original responses is transformed into a set ofuncorrelated components to find the optimal factor or level

combination PCA does nothing when responses are uncor-related thus the best results are obtained when responses orquality characteristics are highly correlated (either positivelyor negatively) [23] The application of PCA involves a seriesof steps that are capable of solving the weakness of thestandalone Taguchi method which requires engineeringjudgment to handle multiple quality characteristics becausethe judgment of an engineer increases uncertainty during thedecision-making process [24] Fung and Kang [25] appliedPCA to the Taguchi method to improve friction propertiesincluding friction coefficients and surface roughness indifferent sliding directions (P-type andAP-type)The authorsextracted the principal components and the coefficient ofdetermination to establish a comprehensive index and obtainthe final optimal parameter setting

Although both techniques are widely integrated with theTaguchi method to overcome multiple quality characteristicproblems both integration techniques have several limita-tions In combining the Taguchimethod andGRA a problemarises when calculating the value of the grey relational gradein GRA An engineering judgment or subjective estimation isrequired to determine the weighting values for each qualitycharacteristic thus resulting in an increase in uncertaintyduring the decision-making process This approach cannotobjectively reveal the relative importance of multiple qualitycharacteristic performance For the integration of the Taguchimethod and PCA the primary problem is the difficulty ofinterpretation because the original variables are substitutedand the principal components are heavily affected by the scal-ing of the variables In both methods no scientific analysishas been performed in terms of performance under optimalprocess conditions to verify the accuracy of the optimizationresults Therefore the effectiveness and reliability of theintegration of the Taguchi method with GRA or PCA aloneremain uncertain Further validation experiments on theoptimization results are needed to prove the effectiveness ofthe integrated approach

This study develops a hybrid optimization method byintegrating the robust parameter design of Taguchi GRA andPCA to solve problems with multiple quality characteristicsAn injection-moulded plastic gear is used to demonstrate theefficiency and validity of the proposed hybrid optimizationmethod in controlling all influential injection mouldingprocessing parameters during plastic gear manufacturing Byfocusing onminimizing the shrinkage behaviour in the tooththickness addendumcircle and dedendumcircle ofmouldedgear the optimal combination of four process parametersnamely melt temperature packing pressure packing timeand cooling time at three different levels is determined byusing the proposed hybrid optimization method

2 TaguchiGRAPCA Hybrid Method forthe Optimization of Injection MouldingProcess Parameters

21 Taguchi Method The Taguchi method is used to designexperiments based on the orthogonal arrays (OAs) TaguchiOAs are highly fractional orthogonal designs that can be used




120578119894119895= minus10 log(1

119899

119899

sum

119895=1

1199102

119894119895) (1)


120578119894119895= minus10 log(1

119899

119899

sum

119895=1

1

1199102119894119895

) (2)


120578119894119895= minus10 log( 1

119899119904

119899

sum

119895=1

1199102

119894119895) (3)






119909lowast

119894(119896) =

119909(119874)

119894(119896) minusmin119909(119874)

119894(119896)

max 119909(119874)119894(119896) minusmin119909(119874)

119894(119896)

(4)


119909lowast

119894(119896) =

max 119909(119874)119894(119896) minus 119909

(119874)

119894(119896)

max 119909(119874)119894(119896) minusmin119909(119874)

119894(119896)

(5)


119909lowast

119894(119896) = 1 minus

10038161003816100381610038161003816119909(119874)

119894(119896) minusOB10038161003816100381610038161003816


119894(119896)

(6)

119909(119874)



119894(119896) and min119909(119874)

119894(119896) is




Δ0119894 (119896) =

1003816100381610038161003816119909lowast

0(119896) minus 119909

lowast

119894(119896)1003816100381610038161003816


maxforall119896

10038161003816100381610038161003816119909lowast

0(119896) minus 119909

lowast

119895(119896)10038161003816100381610038161003816


minforall119896

10038161003816100381610038161003816119909lowast

0(119896) minus 119909

lowast

119895(119896)10038161003816100381610038161003816

(7)



120576119894(119896) (119896) =

Δmin + 120595ΔmaxΔ0119894+ 120595Δmax

0 lt 120576 [119909lowast

0(119896) 119909

lowast

119894(119896)] le 1

(8)


120574119894=1

119899

119899

sum

119896=1

120576119894 (119896) (9)


120574119894=

119899

sum

119896=1

119908119896sdot 120576119894 (119896) (10)






119909119894(119895) 119894 = 1 2 119898 119895 = 1 2 119899

119909119894=

[[[[

[

1199091 (1) 119909


1 (119899)

1199092 (1) 119909


2 (119899)



]]]]

]

(11)



119877119895119897= (

Cov (119909119894(119895) 119909

119894 (119897))

120590119909119894(119895)119883120590

119909119895 (119897)) 119895 = 1 2 3 119899

119897 = 1 2 3 119899

(12)


119894(119895)




119909119894(119897)


(119877 minus 120582119896119868119898) 119881119894119896= 0 (13)


119896=1120582119896= 119899 and 119896 = 1 2 119899

119881119894119896= [1198861198961 1198861198962 119886


119896


119884119898119896=

119899

sum

119894=1

119883119898 (119894) sdot 119881119894119896 (14)

where 1198841198981












Start

End




Quality testing











Verification test







R1650

R1350

empty10

10

empty12





Factorstrialno

A B C DMelt

temperature(∘C)

Packingpressure()

Packingtime (s)

Coolingtime (s)

1 200 60 5 302 200 80 10 403 200 100 15 504 220 60 10 505 220 80 15 306 220 100 5 407 240 60 15 408 240 80 5 509 240 100 10 30


119878 =(119863 minus 119863

119898)

119863119898

(15)


119898is the

mould dimension



0(119896) 119896 = 1 2 3 and





1003816100381610038161003816119909lowast

0(1) minus 119909

lowast

1(1)1003816100381610038161003816 = |10000 minus 00000| = 10000

Δ01 (2) =

1003816100381610038161003816119909lowast

0(2) minus 119909

lowast

1(2)1003816100381610038161003816 = |10000 minus 08743| = 01257

Δ01 (3) =

1003816100381610038161003816119909lowast

0(3) minus 119909

lowast

1(3)1003816100381610038161003816 = |10000 minus 09757| = 00243

(16)

Therefore Δ01= (10000 01257 00243)




Δmax = Δ 08 (1) = Δ 05 (2) = Δ02 (3) = 10000

Δmin = Δ 01 (1) = Δ 08 (2) = Δ09 (3) = 00000(17)



1205761(1)

=00000 + (05) (10000)

10000 + (05) (10000)= 03333

1205761(2)

=00000 + (05) (10000)

01257 + (05) (10000)= 07991

1205761(3)

=00000 + (05) (10000)

00243 + (05) (10000)= 09536

(18)




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

Tooth thickness(dB)

Addendum circle(dB)

Dedendum circle(dB)




Addendumcircle

Dedendumcircle


1 00000 08743 097572 03613 09412 100003 06045 06026 068064 08635 05308 092215 08707 10000 121336 09303 08659 081437 07623 06020 059878 10000 00000 017069 08055 04317 00000



Δ02

Δ03


Thus 1205761(119896)


9OA




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

1 03333 07991 095362 04391 08948 100003 05584 05572 061024 07855 05159 086525 07946 10000 070096 08777 07885 072917 06778 05568 055478 10000 03333 037619 07200 04680 03333



















1 1199082 and 119908

3in (10) are set as 02878 03224



1205741= (02878 times 03333) + (03224 times 07991)

+ (03898 times 09536) = 07232

(19)


05

055

06

065

07

075

08

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

Gre

y re

latin

al g

rade









follows

1198601=(07253 + 08046 + 05782)

3= 07027 (20)


3is


1198602=(07297 + 08243 + 07910)

3= 07817

1198603=(05908 + 05419 + 04880)

3= 05402

(21)








Total 8 01272 10000



2 B2 C1 and D

2show the


2 B2 C1 and






Verificationtest

Tooth thickness(mm)

Addendumcircle (mm)

Dedendumcircle (mm)







5 Conclusion






Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120578119894119895= minus10 log(1

119899

119899

sum

119895=1

1199102

119894119895) (1)


120578119894119895= minus10 log(1

119899

119899

sum

119895=1

1

1199102119894119895

) (2)


120578119894119895= minus10 log( 1

119899119904

119899

sum

119895=1

1199102

119894119895) (3)






119909lowast

119894(119896) =

119909(119874)

119894(119896) minusmin119909(119874)

119894(119896)

max 119909(119874)119894(119896) minusmin119909(119874)

119894(119896)

(4)


119909lowast

119894(119896) =

max 119909(119874)119894(119896) minus 119909

(119874)

119894(119896)

max 119909(119874)119894(119896) minusmin119909(119874)

119894(119896)

(5)


119909lowast

119894(119896) = 1 minus

10038161003816100381610038161003816119909(119874)

119894(119896) minusOB10038161003816100381610038161003816


119894(119896)

(6)

119909(119874)



119894(119896) and min119909(119874)

119894(119896) is




Δ0119894 (119896) =

1003816100381610038161003816119909lowast

0(119896) minus 119909

lowast

119894(119896)1003816100381610038161003816


maxforall119896

10038161003816100381610038161003816119909lowast

0(119896) minus 119909

lowast

119895(119896)10038161003816100381610038161003816


minforall119896

10038161003816100381610038161003816119909lowast

0(119896) minus 119909

lowast

119895(119896)10038161003816100381610038161003816

(7)



120576119894(119896) (119896) =

Δmin + 120595ΔmaxΔ0119894+ 120595Δmax

0 lt 120576 [119909lowast

0(119896) 119909

lowast

119894(119896)] le 1

(8)


120574119894=1

119899

119899

sum

119896=1

120576119894 (119896) (9)


120574119894=

119899

sum

119896=1

119908119896sdot 120576119894 (119896) (10)






119909119894(119895) 119894 = 1 2 119898 119895 = 1 2 119899

119909119894=

[[[[

[

1199091 (1) 119909


1 (119899)

1199092 (1) 119909


2 (119899)



]]]]

]

(11)



119877119895119897= (

Cov (119909119894(119895) 119909

119894 (119897))

120590119909119894(119895)119883120590

119909119895 (119897)) 119895 = 1 2 3 119899

119897 = 1 2 3 119899

(12)


119894(119895)




119909119894(119897)


(119877 minus 120582119896119868119898) 119881119894119896= 0 (13)


119896=1120582119896= 119899 and 119896 = 1 2 119899

119881119894119896= [1198861198961 1198861198962 119886


119896


119884119898119896=

119899

sum

119894=1

119883119898 (119894) sdot 119881119894119896 (14)

where 1198841198981












Start

End




Quality testing











Verification test







R1650

R1350

empty10

10

empty12





Factorstrialno

A B C DMelt

temperature(∘C)

Packingpressure()

Packingtime (s)

Coolingtime (s)

1 200 60 5 302 200 80 10 403 200 100 15 504 220 60 10 505 220 80 15 306 220 100 5 407 240 60 15 408 240 80 5 509 240 100 10 30


119878 =(119863 minus 119863

119898)

119863119898

(15)


119898is the

mould dimension



0(119896) 119896 = 1 2 3 and





1003816100381610038161003816119909lowast

0(1) minus 119909

lowast

1(1)1003816100381610038161003816 = |10000 minus 00000| = 10000

Δ01 (2) =

1003816100381610038161003816119909lowast

0(2) minus 119909

lowast

1(2)1003816100381610038161003816 = |10000 minus 08743| = 01257

Δ01 (3) =

1003816100381610038161003816119909lowast

0(3) minus 119909

lowast

1(3)1003816100381610038161003816 = |10000 minus 09757| = 00243

(16)

Therefore Δ01= (10000 01257 00243)




Δmax = Δ 08 (1) = Δ 05 (2) = Δ02 (3) = 10000

Δmin = Δ 01 (1) = Δ 08 (2) = Δ09 (3) = 00000(17)



1205761(1)

=00000 + (05) (10000)

10000 + (05) (10000)= 03333

1205761(2)

=00000 + (05) (10000)

01257 + (05) (10000)= 07991

1205761(3)

=00000 + (05) (10000)

00243 + (05) (10000)= 09536

(18)




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

Tooth thickness(dB)

Addendum circle(dB)

Dedendum circle(dB)




Addendumcircle

Dedendumcircle


1 00000 08743 097572 03613 09412 100003 06045 06026 068064 08635 05308 092215 08707 10000 121336 09303 08659 081437 07623 06020 059878 10000 00000 017069 08055 04317 00000



Δ02

Δ03


Thus 1205761(119896)


9OA




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

1 03333 07991 095362 04391 08948 100003 05584 05572 061024 07855 05159 086525 07946 10000 070096 08777 07885 072917 06778 05568 055478 10000 03333 037619 07200 04680 03333



















1 1199082 and 119908

3in (10) are set as 02878 03224



1205741= (02878 times 03333) + (03224 times 07991)

+ (03898 times 09536) = 07232

(19)


05

055

06

065

07

075

08

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

Gre

y re

latin

al g

rade









follows

1198601=(07253 + 08046 + 05782)

3= 07027 (20)


3is


1198602=(07297 + 08243 + 07910)

3= 07817

1198603=(05908 + 05419 + 04880)

3= 05402

(21)








Total 8 01272 10000



2 B2 C1 and D

2show the


2 B2 C1 and






Verificationtest

Tooth thickness(mm)

Addendumcircle (mm)

Dedendumcircle (mm)







5 Conclusion






Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119909119894(119895) 119894 = 1 2 119898 119895 = 1 2 119899

119909119894=

[[[[

[

1199091 (1) 119909


1 (119899)

1199092 (1) 119909


2 (119899)



]]]]

]

(11)



119877119895119897= (

Cov (119909119894(119895) 119909

119894 (119897))

120590119909119894(119895)119883120590

119909119895 (119897)) 119895 = 1 2 3 119899

119897 = 1 2 3 119899

(12)


119894(119895)




119909119894(119897)


(119877 minus 120582119896119868119898) 119881119894119896= 0 (13)


119896=1120582119896= 119899 and 119896 = 1 2 119899

119881119894119896= [1198861198961 1198861198962 119886


119896


119884119898119896=

119899

sum

119894=1

119883119898 (119894) sdot 119881119894119896 (14)

where 1198841198981












Start

End




Quality testing











Verification test







R1650

R1350

empty10

10

empty12





Factorstrialno

A B C DMelt

temperature(∘C)

Packingpressure()

Packingtime (s)

Coolingtime (s)

1 200 60 5 302 200 80 10 403 200 100 15 504 220 60 10 505 220 80 15 306 220 100 5 407 240 60 15 408 240 80 5 509 240 100 10 30


119878 =(119863 minus 119863

119898)

119863119898

(15)


119898is the

mould dimension



0(119896) 119896 = 1 2 3 and





1003816100381610038161003816119909lowast

0(1) minus 119909

lowast

1(1)1003816100381610038161003816 = |10000 minus 00000| = 10000

Δ01 (2) =

1003816100381610038161003816119909lowast

0(2) minus 119909

lowast

1(2)1003816100381610038161003816 = |10000 minus 08743| = 01257

Δ01 (3) =

1003816100381610038161003816119909lowast

0(3) minus 119909

lowast

1(3)1003816100381610038161003816 = |10000 minus 09757| = 00243

(16)

Therefore Δ01= (10000 01257 00243)




Δmax = Δ 08 (1) = Δ 05 (2) = Δ02 (3) = 10000

Δmin = Δ 01 (1) = Δ 08 (2) = Δ09 (3) = 00000(17)



1205761(1)

=00000 + (05) (10000)

10000 + (05) (10000)= 03333

1205761(2)

=00000 + (05) (10000)

01257 + (05) (10000)= 07991

1205761(3)

=00000 + (05) (10000)

00243 + (05) (10000)= 09536

(18)




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

Tooth thickness(dB)

Addendum circle(dB)

Dedendum circle(dB)




Addendumcircle

Dedendumcircle


1 00000 08743 097572 03613 09412 100003 06045 06026 068064 08635 05308 092215 08707 10000 121336 09303 08659 081437 07623 06020 059878 10000 00000 017069 08055 04317 00000



Δ02

Δ03


Thus 1205761(119896)


9OA




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

1 03333 07991 095362 04391 08948 100003 05584 05572 061024 07855 05159 086525 07946 10000 070096 08777 07885 072917 06778 05568 055478 10000 03333 037619 07200 04680 03333



















1 1199082 and 119908

3in (10) are set as 02878 03224



1205741= (02878 times 03333) + (03224 times 07991)

+ (03898 times 09536) = 07232

(19)


05

055

06

065

07

075

08

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

Gre

y re

latin

al g

rade









follows

1198601=(07253 + 08046 + 05782)

3= 07027 (20)


3is


1198602=(07297 + 08243 + 07910)

3= 07817

1198603=(05908 + 05419 + 04880)

3= 05402

(21)








Total 8 01272 10000



2 B2 C1 and D

2show the


2 B2 C1 and






Verificationtest

Tooth thickness(mm)

Addendumcircle (mm)

Dedendumcircle (mm)







5 Conclusion






Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Start

End




Quality testing











Verification test







R1650

R1350

empty10

10

empty12





Factorstrialno

A B C DMelt

temperature(∘C)

Packingpressure()

Packingtime (s)

Coolingtime (s)

1 200 60 5 302 200 80 10 403 200 100 15 504 220 60 10 505 220 80 15 306 220 100 5 407 240 60 15 408 240 80 5 509 240 100 10 30


119878 =(119863 minus 119863

119898)

119863119898

(15)


119898is the

mould dimension



0(119896) 119896 = 1 2 3 and





1003816100381610038161003816119909lowast

0(1) minus 119909

lowast

1(1)1003816100381610038161003816 = |10000 minus 00000| = 10000

Δ01 (2) =

1003816100381610038161003816119909lowast

0(2) minus 119909

lowast

1(2)1003816100381610038161003816 = |10000 minus 08743| = 01257

Δ01 (3) =

1003816100381610038161003816119909lowast

0(3) minus 119909

lowast

1(3)1003816100381610038161003816 = |10000 minus 09757| = 00243

(16)

Therefore Δ01= (10000 01257 00243)




Δmax = Δ 08 (1) = Δ 05 (2) = Δ02 (3) = 10000

Δmin = Δ 01 (1) = Δ 08 (2) = Δ09 (3) = 00000(17)



1205761(1)

=00000 + (05) (10000)

10000 + (05) (10000)= 03333

1205761(2)

=00000 + (05) (10000)

01257 + (05) (10000)= 07991

1205761(3)

=00000 + (05) (10000)

00243 + (05) (10000)= 09536

(18)




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

Tooth thickness(dB)

Addendum circle(dB)

Dedendum circle(dB)




Addendumcircle

Dedendumcircle


1 00000 08743 097572 03613 09412 100003 06045 06026 068064 08635 05308 092215 08707 10000 121336 09303 08659 081437 07623 06020 059878 10000 00000 017069 08055 04317 00000



Δ02

Δ03


Thus 1205761(119896)


9OA




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

1 03333 07991 095362 04391 08948 100003 05584 05572 061024 07855 05159 086525 07946 10000 070096 08777 07885 072917 06778 05568 055478 10000 03333 037619 07200 04680 03333



















1 1199082 and 119908

3in (10) are set as 02878 03224



1205741= (02878 times 03333) + (03224 times 07991)

+ (03898 times 09536) = 07232

(19)


05

055

06

065

07

075

08

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

Gre

y re

latin

al g

rade









follows

1198601=(07253 + 08046 + 05782)

3= 07027 (20)


3is


1198602=(07297 + 08243 + 07910)

3= 07817

1198603=(05908 + 05419 + 04880)

3= 05402

(21)








Total 8 01272 10000



2 B2 C1 and D

2show the


2 B2 C1 and






Verificationtest

Tooth thickness(mm)

Addendumcircle (mm)

Dedendumcircle (mm)







5 Conclusion






Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










R1650

R1350

empty10

10

empty12





Factorstrialno

A B C DMelt

temperature(∘C)

Packingpressure()

Packingtime (s)

Coolingtime (s)

1 200 60 5 302 200 80 10 403 200 100 15 504 220 60 10 505 220 80 15 306 220 100 5 407 240 60 15 408 240 80 5 509 240 100 10 30


119878 =(119863 minus 119863

119898)

119863119898

(15)


119898is the

mould dimension



0(119896) 119896 = 1 2 3 and





1003816100381610038161003816119909lowast

0(1) minus 119909

lowast

1(1)1003816100381610038161003816 = |10000 minus 00000| = 10000

Δ01 (2) =

1003816100381610038161003816119909lowast

0(2) minus 119909

lowast

1(2)1003816100381610038161003816 = |10000 minus 08743| = 01257

Δ01 (3) =

1003816100381610038161003816119909lowast

0(3) minus 119909

lowast

1(3)1003816100381610038161003816 = |10000 minus 09757| = 00243

(16)

Therefore Δ01= (10000 01257 00243)




Δmax = Δ 08 (1) = Δ 05 (2) = Δ02 (3) = 10000

Δmin = Δ 01 (1) = Δ 08 (2) = Δ09 (3) = 00000(17)



1205761(1)

=00000 + (05) (10000)

10000 + (05) (10000)= 03333

1205761(2)

=00000 + (05) (10000)

01257 + (05) (10000)= 07991

1205761(3)

=00000 + (05) (10000)

00243 + (05) (10000)= 09536

(18)




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

Tooth thickness(dB)

Addendum circle(dB)

Dedendum circle(dB)




Addendumcircle

Dedendumcircle


1 00000 08743 097572 03613 09412 100003 06045 06026 068064 08635 05308 092215 08707 10000 121336 09303 08659 081437 07623 06020 059878 10000 00000 017069 08055 04317 00000



Δ02

Δ03


Thus 1205761(119896)


9OA




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

1 03333 07991 095362 04391 08948 100003 05584 05572 061024 07855 05159 086525 07946 10000 070096 08777 07885 072917 06778 05568 055478 10000 03333 037619 07200 04680 03333



















1 1199082 and 119908

3in (10) are set as 02878 03224



1205741= (02878 times 03333) + (03224 times 07991)

+ (03898 times 09536) = 07232

(19)


05

055

06

065

07

075

08

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

Gre

y re

latin

al g

rade









follows

1198601=(07253 + 08046 + 05782)

3= 07027 (20)


3is


1198602=(07297 + 08243 + 07910)

3= 07817

1198603=(05908 + 05419 + 04880)

3= 05402

(21)








Total 8 01272 10000



2 B2 C1 and D

2show the


2 B2 C1 and






Verificationtest

Tooth thickness(mm)

Addendumcircle (mm)

Dedendumcircle (mm)







5 Conclusion






Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

Tooth thickness(dB)

Addendum circle(dB)

Dedendum circle(dB)




Addendumcircle

Dedendumcircle


1 00000 08743 097572 03613 09412 100003 06045 06026 068064 08635 05308 092215 08707 10000 121336 09303 08659 081437 07623 06020 059878 10000 00000 017069 08055 04317 00000



Δ02

Δ03


Thus 1205761(119896)


9OA




Tooth thickness(mm)

Addendum circle(mm)

Dedendum circle(mm)

1 03333 07991 095362 04391 08948 100003 05584 05572 061024 07855 05159 086525 07946 10000 070096 08777 07885 072917 06778 05568 055478 10000 03333 037619 07200 04680 03333



















1 1199082 and 119908

3in (10) are set as 02878 03224



1205741= (02878 times 03333) + (03224 times 07991)

+ (03898 times 09536) = 07232

(19)


05

055

06

065

07

075

08

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

Gre

y re

latin

al g

rade









follows

1198601=(07253 + 08046 + 05782)

3= 07027 (20)


3is


1198602=(07297 + 08243 + 07910)

3= 07817

1198603=(05908 + 05419 + 04880)

3= 05402

(21)








Total 8 01272 10000



2 B2 C1 and D

2show the


2 B2 C1 and






Verificationtest

Tooth thickness(mm)

Addendumcircle (mm)

Dedendumcircle (mm)







5 Conclusion






Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation























1 1199082 and 119908

3in (10) are set as 02878 03224



1205741= (02878 times 03333) + (03224 times 07991)

+ (03898 times 09536) = 07232

(19)


05

055

06

065

07

075

08

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

Gre

y re

latin

al g

rade









follows

1198601=(07253 + 08046 + 05782)

3= 07027 (20)


3is


1198602=(07297 + 08243 + 07910)

3= 07817

1198603=(05908 + 05419 + 04880)

3= 05402

(21)








Total 8 01272 10000



2 B2 C1 and D

2show the


2 B2 C1 and






Verificationtest

Tooth thickness(mm)

Addendumcircle (mm)

Dedendumcircle (mm)







5 Conclusion






Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Total 8 01272 10000



2 B2 C1 and D

2show the


2 B2 C1 and






Verificationtest

Tooth thickness(mm)

Addendumcircle (mm)

Dedendumcircle (mm)







5 Conclusion






Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Hybrid Integration of Taguchi Parametric