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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 920865 9 pageshttpdxdoiorg1011552013920865

Research ArticleMinimum Porosity Formation in Pressure Die Casting byTaguchi Method

Quang-Cherng Hsu and Anh Tuan Do

Department of Mechanical Engineering National Kaohsiung University of Applied Sciences 415 Chien-Kung Road80778 Kaohsiung City Taiwan

Correspondence should be addressed to Anh Tuan Do 1099403120kuasedutw

Received 17 September 2013 Accepted 15 October 2013

Academic Editor Teen-Hang Meen

Copyright copy 2013 Q-C Hsu and A T Do 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

Die casting process is significantly used in the industry for its high productivity and less postmachining requirement Due tolight weight and good formability aluminum die casting plays an important role in the production of transportation and vehiclecomponents In the current study of die casting for Automobile starter motor casing the following issues are focused shot pistonsimulation defect analysis and finally the use of the Taguchi multiquality analytical method to find the optimal parameters andfactors to increase the aluminium ADC10 die casting quality and efficiency Experiments were conducted by varying molten alloytemperature die temperature plunger velocities in the first and second stage and multiplied pressure in the third stage usingL27orthogonal array of Taguchi method After conducting a series of initial experiments in a controlled environment significant

factors for pressure die casting processes are selected to construct an appropriate multivariable linear regression analysis modelfor developing a robust performance for pressure die casting processes The appropriate multivariable linear model is a useful andefficient method to find the optimal process conditions in pressure die casting associated with the minimum shrinkage porositypercent

1 Introduction

High pressure die casting for nonferrous casting applicationsis increasingly used in the foundries today as an economicallyviable casting process High pressure die casting (HPDC)process has beenwidely used tomanufacture a large variety ofproducts with high dimensional accuracy and productivitiesIt has a much faster production rate in comparison toother methods and it is an economical and efficient methodfor producing components with low surface roughness andhigh dimensional accuracy All major aluminium automotivecomponents can be processed with this technology

High Pressure Die Casting process is rapid and dependson many factors So to capture the problem it requires a lotof time and experience including testing and simulationTheconventional trial and error based die design and processdevelopment is expensive and time consuming Such aprocedure also might lead to higher casting rejections TheHPDC castings production process has many defects such

as shrinkage porosity misrun cold-shut blister scab hot-tear Several previous studies of defects in aluminum alloyby the method of HPDC and disability solutions (Shen etal 2007 [1] Dargusch et al 2006 [2] Verran et al 2006[3] Mousavi Anijdan et al 2006 [4] Tsoukalas et al 20042008 [5 6]) However the study to optimize aluminum alloycasting process in the condition of production casting factoryis essential This study focused on analysis of shrinkageporosity defect with mold design and put into productioncasting by foundry factory conditions

Shrinkage porosity is one of the most common defectsleading to rejection of aluminium die casting often onlyshowing up after much value has been added to the castingvia operations such asmachining polishing and coatingTheadded value of the casting at the point of rejection can bevery high If you find out the causes and how to reduce thedefects of castings will be of great significance in reducingthe production cost of die casting However optimizing theconditions to render aluminium die castings of minimum

2 Mathematical Problems in Engineering

(a) (b)

(c) (d)

Figure 1 Casting image

porosity percent is costly and time consuming because manyexperiments are necessary to find the optimal parameters

Taguchi method is one of the efficient problems solvingtools to upgrade the performance of products and processeswith a significant reduction in cost and time involvedTaguchirsquos parameter design offers a systematic approach foroptimization of various parameters with regard to perfor-mance quality and cost (Syrcos 2003 [7] Taguchi 1986 [8])

2 Materials and Methods

The die casting part product of this study is providedthrough aluminium die casting factory so the casting bodyno changes A major factor in the successful development ofcastings is the design of the die and design of gates biscuitand runner system A well-designed gating and runnersystem should avoid turbulence in metal flow and to reduceincidence of inclusions and air entrapment in the castingThedie design is required to avoid solidification related defectslike shrinkage micro-porosities hot-tear and so forth Diedesign process is very much dependent on the experienceand skill of the design engineer The die for this study isthe result of collaboration between the foundry factory andDepartment of Mechanical Engineering-National KaohsiungUniversity of Applied Sciences The casting with full of thegating runner system and biscuit is shown in Figure 1 Thedie casting is designed in CATIA V5R19 software shown inFigure 2 Moreover the die casting material selection is veryimportant The nature of the material will directly affect thequality of the casting and die casting parameters configura-tion this study selects castingmaterial as the aluminium alloy

Table 1 Chemical composition of the alloy ADC10 used in theexperiment

Element Si Fe Cu Mg Mn Ni Zn Smwt 75sim95 13 30sim40 01 05 05 3 035

ADC10 The chemical composition of the aluminum alloyused in the experimental procedure is given in Table 1

Shrinkage porosity formation in pressure die casting isthe result of a so much number of parameters Figure 3shows a cause and effect diagram that was constructed toidentify the casting process parameters that may affect diecasting porosity (Tsoukalas et al 2004 2008 [5 6]) In thiscase holding furnace temperature die temperature plungervelocity in the first stage plunger velocity in the secondstage and multiplied pressure in the third stage were selectedas the most critical in the experimental design The otherparameters were kept constant in the entire experimentationThe range of holding furnace temperature was selected as640sim700∘C the range of die temperature as 180sim260∘C therange of plunger velocity in the first stage as 005sim035msand in the second stage as 15sim35ms and the range ofmultiplied pressure in the third stage was chosen as 200sim280 bars The selected casting process parameters along withits ranges are given in Table 2

Taguchi method based design of experiment has beenused to study the effects of five casting process parameters(holding furnace temperature A die temperature B plungervelocity in the first stage C plunger velocity in the secondstage D multiplied pressure in the third stage E on an

Mathematical Problems in Engineering 3

(a) (b)

(c) (d)

Figure 2 Part product is designed by CATIA software

Molten alloy

Die casting machine

Cavity filling time

Fast shot Plunger

stage

TemperatureLubricant

GateVenting system

Cooling system

Die

Shot sleeve

Filling level

Diameter

Length

Lubricant

Temperature

Composition

Condition

Shrinkage porosity type The smaller the better

Plungervelocity (1st)

Pressure during 3rd

set point velocity (2nd)

Figure 3 Cause and effect diagram

Table 2 The parameter and its value at three levels

Process parameters Parameters range Level 1 Level 2 Level 3Holding furnace temperature (∘C) 640sim700 640 670 700Die temperature (∘C) 180sim260 180 220 260Plunger velocity 1st stage (ms) 005sim035 005 02 035Plunger velocity 2nd stage (ms) 15sim35 15 25 35Multiplied pressure (bars) 200sim280 200 240 280

4 Mathematical Problems in Engineering

Table 3 Experimental layout using an L27orthogonal array

Trials Holding furnacetemperature A

Die temperatureB

Plunger velocity 1ststage C

Plunger velocity 2ndstage D

Multiplied pressureE

1 1 1 1 1 12 1 1 2 2 23 1 1 3 3 34 1 2 1 2 25 1 2 2 3 36 1 2 3 1 17 1 3 1 3 38 1 3 2 1 19 1 3 3 2 210 2 1 1 2 311 2 1 2 3 112 2 1 3 1 213 2 2 1 3 114 2 2 2 1 215 2 2 3 2 316 2 3 1 1 217 2 3 2 2 318 2 3 3 3 119 3 1 1 3 220 3 1 2 1 321 3 1 3 2 122 3 2 1 1 323 3 2 2 2 124 3 2 3 3 225 3 3 1 2 126 3 3 2 3 227 3 3 3 1 3

important output parameter (Shrinkage porosity) For select-ing appropriate orthogonal array degree of freedom (numberof fair and independent comparisons needed for optimizationof process parameters is one less than the number of levels ofparameter) of the array is calculated

In the experimental layout planwith five factors and threelevels using L

27orthogonal array 27 experimentswere carried

out to study the effect of casting input parameters shown inTable 3 The input parameters are installed in the ProCASTsoftware to conduct 27 simulation experiments

Computer simulation procedure-based process develop-ment and die design can be used for rapid process devel-opment and die design in a shorter time Such a computersimulation based procedure often using FINITE ELEMENTANALYSIS based software systems can improve the qualityand enhance productivity of the enterprise by way of fasterdevelopment of new product Analysis software is used as aProCAST commercial with finite element method analysisfor a casting process In this study all parameters can beable to affect the analysis process choice of material isaluminum alloy die casting ADC10 and cold chamber die

casting method with molding material is H13 FEM basedsimulation software systems help the designer to visualize themetal flow in the die cavity the temperature variation thesolidification progress and the evolution of defects such asshrinkage porosity cold-shut hot-tear

ProCAST a FEM simulation-based virtual casting envi-ronment for analysis of the casting process is used as a tool fordie design and process optimization ProCAST with Visual-Viewer module can provide temperature field thermal crack-ing flow field solidification time and shrinkage analysisThis paper focused on the analysis of shrinkage porosity byProCAST software base on parameters input from Table 3

The analysis of defects simulated by ProCAST softwarewith Visual-Viewer module can detect many types of disabil-ities castingThe defective products do not necessarily reflectthe loss of the original function for example the internal poretrims acceptable However with large structural castingsdefect analysis of this study focuses on maximum porosity inthe selection casting and the important parts of the castingshrinkage analysis (an important component) casting defectanalysis are described as follows

Mathematical Problems in Engineering 5

(a) (b)

(c) (d)

Figure 4 Casting measurement area

The Solid Fraction Solid fraction may be available shrinkageprediction casting position the present study is in accordancewith the theory prediction of defect and ProCAST manualreferred to in the final period of solidification Shrinkage solidfraction prone is greater than 07 here as the reference valueof 07 solid fractions When the solid fraction area is belowthis value and the area around the solid phase rate is ratherthan this value we can predict this area shrinkage porosityoccurred

Maximum Porosity The maximum porosity analysis usingthe Shrinkage Porosity function of the Visual-Viewer comesdefined in the manual According to the ProCAST usermanual shrinkage definition andwith the solid fraction it canbe used to analyse the basis of the maximum porosity

Shrinkage Analysis For the amount of inspection shrinkagecasting part used for the Visual-Viewer module functionfor quantitative analysis In each experiment we took fiveelements with the coordinates determined at the importantpositions in the working conditions of automobile startermotor casing Each experiment was repeated five times inorder to reduce experimental errors as shown in Figure 4Data from 27 experiments with five sampling times in eachsimulation are summarized as in Table 4 From this table weconducted quality characteristics analysis

Quality Characteristics The parameter design study involvescontrol and noise factors The measure of interactionsbetween these factors with regard to robustness is signal-to-noise (119878119873) ratio 119878119873 characteristics formulated for three

different categories are as follows the bigger the better andthe smaller the better the nominal the best This paperfocused on studying the effects of five input parameters(119860 119861 119862119863 119864) to defect shrinkage porosity in the process ofcasting so the criteria ldquothe smaller the betterrdquo is selected

The smaller the better (for making the system response assmall as possible) is as follows

119878

119873

119878

= minus10 log(1119899

119899

sum

119894=1

119910

2

119894) (1)

where 119899 is the number of sampling (Each experiment wasrepeated five times sampling so that 119899 = 5) 119910

119894 value of

Shrinkage porosity at each time samplingThe responding graph shown in Figure 5 learned that the

best combination for this studywith shrinkage porosity defectvalue minimum is 119860

3119861

3119862

3119863

1119864

3

Process Parameter Optimization Using MVLR The objectiveof the process optimization is to select the optimal controlvariables in aluminium die casting process in order to obtainthe minimum porosity In this work the fitness functionused in the optimization procedure was based on the MVLRmodel

Multivariable Linear Regression Analysis In most cases theform of the relationship between the response and theindependent variables is usually unknown Multiple linearregression (MLR) is a method used to model the linearrelationship between a dependent variable and one or more

6 Mathematical Problems in Engineering

Table4Sh

rinkage

porosityresults

oftheL27arraydesig

n(Fulltable)

Trials

Holding

Furnace

Temperature

ADie

temperature

B

Plun

ger

velocity

1ststage

C

Plun

ger

Velocity

2ndsta

geD

Multip

lied

Pressure

E

Shrin

kage

porosity(

)Av

erage

MSD

119878119873

Repetition

Repetition

Repetition

Repetition

Repetition

12

34

51

640

180

005

15200

1973

1988

1984

1978

1954

19754

3902346minus591326

264

0180

02

25

240

1912

1893

1948

1908

1912

19146

366

6021minus564

195

364

0180

035

35

280

1846

1853

1814

1813

1828

18308

3352095minus525316

464

0220

005

25

240

1862

1867

1861

1878

1871

18678

3488716minus54266

65

640

220

02

35

280

1793

1804

1761

1764

1775

17794

3166541minus500585

664

0220

035

15200

1916

1908

1912

1887

1903

19052

3629888minus559893

764

0260

005

35

280

1786

1797

1754

1757

1768

17724

3141679minus497162

864

0260

02

15200

1909

1903

1889

1898

1893

18984

3603973minus556782

964

0260

035

25

240

1852

1859

182

1819

1836

18372

3375568minus528347

10670

180

005

25

280

1799

181

1767

177

1781

17854

318793minus503509

11670

180

02

35

200

1957

1968

1971

1951

1965

19624

3851068minus585581

12670

180

035

15240

1857

1864

1825

1824

1841

18422

3393965minus530707

13670

220

005

35

200

1924

1935

1923

1928

1909

19238

3701079minus568328

14670

220

02

15240

1803

1814

1771

1774

1785

17894

3202229minus505452

15670

220

035

25

280

1751

1744

1713

1731

1725

17328

3002778minus477523

16670

260

005

15240

1796

1807

1764

1767

1778

17824

3177227minus50204

817

670

260

02

25

280

1738

1726

1724

1745

1715

17296

2991629minus475908

18670

260

035

35

200

1881

1878

1898

1893

1903

18906

35744

61minus553211

19700

180

005

35

240

1876

1871

1895

1887

1893

18844

3551052minus550357

20700

180

02

15280

1755

1743

1741

1762

1732

1746

63050725minus484403

21700

180

035

25

200

1921

1901

1956

1916

1928

19244

3703644minus568629

22700

220

005

15280

1708

1696

1694

1715

1685

16996

2888753minus46071

23700

220

02

25

200

1864

1898

1881

1876

1885

18808

3537532minus5487

24700

220

035

35

240

1825

1832

1793

1792

1807

18098

3275642minus515296

25700

260

005

25

200

1856

1861

1855

1872

1865

18618

346

6338minus539871

26700

260

02

35

240

1814

1821

1782

1781

1796

17988

3235948minus510001

27700

260

035

15280

1692

1694

1706

1683

1712

16974

2881274minus459585

Mathematical Problems in Engineering 7

Table 5 The results after analysing by Intercooled Stada 82 software

(a)

reg Fx A B C D E

Source SS df MS Number of obs = 27119865(5 21) = 14674

Model 0155044498 5 00310089 Prob gt 119865 = 00000

Residual 0004437799 21 0000211324 119877-squared = 09722Adj 119877-squared = 09655

Total 0159482297 26 0006133935 Root MSE = 001454

(b)

Fx Coef Std Err 119905 119875 gt |119905| [95 Conf Interval]119860 minus00008844 00001142 minus774 0000 minus0001122 minus00006469119861 minus000083 00000857 minus969 0000 minus00010081 minus00006519119862 minus00305925 07284277 minus0042 0000 minus00780965 00169114119863 00175444 00034264 512 0000 00104189 002467119864 minus00020122 00000857 minus2349 0000 minus00021904 minus00018341cons 3054569 00820708 3722 0000 2883893 3225244

SN

(dB)

Plots of factor effectsminus46

minus48

minus50

minus52

minus54

minus56

minus58

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3

Figure 5 119878119873 Response graphs

independent variables MLR is based on least squares themodel is fitted such that the sum-of-squares of differences ofobserved and predicted values is minimized

Let 1199091 119909

2 119909

119903be a set of 119903 predictors believed to be

related to a response variable 119884 The linear regression modelfor the 119895th sample unit has the form

119884

119895= 120573

0+ 120573

1119909

1198951+ 120573

2119909

1198952+ sdot sdot sdot + 120573

119903119909

119895119903+ 120576

119895 (2)

where 120576 is a random error and the 120573119894 119894 = 0 1 119903 are

unknown regression coefficientsIn this paper there are five independent variables and one

dependent variableThe relationships between these variablesare of the following form

119865 (119909) = 120573

0+ 120573

1119860 + 120573

2119861 + 120573

3119862 + 120573

4119863 + 120573

5119864 (3)

In which

119865(119909) Dependence variable119860 (∘C) Holding furnace temperature

119861 (∘C) Die temperature

119862 (ms) Plunger velocity 1st stage

119863 (ms) Plunger velocity 2nd stage

119864 (bars) Multiplied pressure during the third phase

The results after analysing by Intercooled Stada 82 Soft-were as shown in Table 5

The final MVLRmodel equation for porosity after substi-tuting regression coefficients is as follows

119865 (119909) = 3054569 minus 08844 lowast 10

minus3119860 minus 083 lowast 10

minus3119861

minus 003059119862 + 001754119863 minus 000201119864

(4)

3 Results and Discussion

We applied multivariable linear regression analysis (MVLR)to seek the optimal parameter in the casting process ofindependent parameter variables in this study A stationarypoint for the optimal performance was obtained by usingthe multivariable linear regression method in this linearregression equation and the result is presented in Figure 6A very good fit was observed and substantiated by thecoefficient of determination 1198772 = 09722 That is the 1198772value indicates that the polynomial model explains almost9722 of variability in the casting process

Figure 6 shows the efficacy of the optimization scheme bycomparing the MVLR results with the experimental valuesThere is a convincing agreement between experimental valuesand predicted values for shrinkage porosity percent

Matlab code for finding optimization shrinkage porosityvalue

8 Mathematical Problems in Engineering

clc

clear all

close all

f = (x) 3054569-08844e-3lowastx(1)-083e-3lowastx(2)-

003059lowastx(3) + 001754lowastx(4)-000201lowastx(5)

options = optimset(lsquoGradObjrsquo lsquoonrsquo)[xfvalexitflagoutput] =

fmincon(f[6702200225240][ ][ ][ ][ ][60018000515

200][70026003535280][ ]optimset(lsquoDisplayrsquo lsquoiterrsquo))x

fval

Algorithm 1

15155

16165

17175

18185

19195

2

Number of tests

Shrin

kage

por

osity

()

PredictedExperimental

Std dev = 0014537

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

R2 = 9722 adjustedR

2 = 9655

Figure 6 Experimental and predicted values of shrinkage porositypercent

Program in Matlab (see Algorithm 1)Results after running in Matlab are as follows

119909 = 7000000 so that 997888rarr 119860 = 700∘C

2600000 119861 = 260

∘C

03500 119862 = 035ms

15000 119863 = 15ms

2800000 119864 = 280 bar

fval = 16725 Shrinkage porosity 16725

(5)

By the Program in Matlab we are known as the bestcombination in the 27 experimental configurations

This result is similar to quality characteristics and is thebest combination for this study 119860

3119861

3119862

3119863

1119864

3

4 Conclusion

In this paper the optimum process parameters valuespredicted for casting of minimum shrinkage porosity

(16725) and the best combination parameters given asfollows

holding furnace temperature 700∘Cdie temperature 260∘Cplunger velocity 1st stage 035msplunger velocity 2nd stage 15msmultiplied pressure 280 bar

The model proposed in this paper gives satisfactoryresults in the optimization of pressure die casting processThepredicted values of the process parameters and the calculatedare in convincing agreement with the experimental values

The experiments which are conducted to determine thebest levels are based on ldquoOrthogonal Arraysrdquo and are bal-anced with respect to all control factors and yet areminimumin number This in turn implies that the resources (materialssaving time andmoney) required for the experiments are alsominimized

References

[1] C Shen L Wang and Q Li ldquoOptimization of injection mold-ing process parameters using combination of artificial neuralnetwork and genetic algorithm methodrdquo Journal of MaterialsProcessing Technology vol 183 no 2-3 pp 412ndash418 2007

[2] M S Dargusch G Dour N Schauer C M Dinnis and GSavage ldquoThe influence of pressure during solidification of highpressure die cast aluminium telecommunications componentsrdquoJournal of Materials Processing Technology vol 180 no 1ndash3 pp37ndash43 2006

[3] G O Verran R P K Mendes and M A Rossi ldquoInfluenceof injection parameters on defects formation in die castingAl12Si13Cu alloy experimental results and numeric simula-tionrdquo Journal of Materials Processing Technology vol 179 no 1ndash3 pp 190ndash195 2006

[4] S H Mousavi Anijdan A Bahrami H R Madaah Hosseiniand A Shafyei ldquoUsing genetic algorithm and artificial neuralnetwork analyses to design an Al-Si casting alloy of minimumporosityrdquoMaterials and Design vol 27 no 7 pp 605ndash609 2006

[5] V D Tsoukalas S A Mavrommatis N G Orfanoudakis andA K Baldoukas ldquoA study of porosity formation in pressuredie casting using the Taguchi approachrdquo Journal of EngineeringManufacture vol 218 no 1 pp 77ndash86 2004

Mathematical Problems in Engineering 9

[6] V D Tsoukalas ldquoOptimization of porosity formation inAlSi9Cu3pressure die castings using genetic algorithm analy-

sisrdquoMaterials and Design vol 29 no 10 pp 2027ndash2033 2008[7] G P Syrcos ldquoDie casting process optimization using Taguchi

methodsrdquo Journal of Materials Processing Technology vol 135no 1 pp 68ndash74 2003

[8] G Taguchi Introduction to Quality Engineering Asian Produc-tivity Organization UNIPUB 1986

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

(a) (b)

(c) (d)

Figure 1 Casting image

porosity percent is costly and time consuming because manyexperiments are necessary to find the optimal parameters

Taguchi method is one of the efficient problems solvingtools to upgrade the performance of products and processeswith a significant reduction in cost and time involvedTaguchirsquos parameter design offers a systematic approach foroptimization of various parameters with regard to perfor-mance quality and cost (Syrcos 2003 [7] Taguchi 1986 [8])

2 Materials and Methods

The die casting part product of this study is providedthrough aluminium die casting factory so the casting bodyno changes A major factor in the successful development ofcastings is the design of the die and design of gates biscuitand runner system A well-designed gating and runnersystem should avoid turbulence in metal flow and to reduceincidence of inclusions and air entrapment in the castingThedie design is required to avoid solidification related defectslike shrinkage micro-porosities hot-tear and so forth Diedesign process is very much dependent on the experienceand skill of the design engineer The die for this study isthe result of collaboration between the foundry factory andDepartment of Mechanical Engineering-National KaohsiungUniversity of Applied Sciences The casting with full of thegating runner system and biscuit is shown in Figure 1 Thedie casting is designed in CATIA V5R19 software shown inFigure 2 Moreover the die casting material selection is veryimportant The nature of the material will directly affect thequality of the casting and die casting parameters configura-tion this study selects castingmaterial as the aluminium alloy

Table 1 Chemical composition of the alloy ADC10 used in theexperiment

Element Si Fe Cu Mg Mn Ni Zn Smwt 75sim95 13 30sim40 01 05 05 3 035

ADC10 The chemical composition of the aluminum alloyused in the experimental procedure is given in Table 1

Shrinkage porosity formation in pressure die casting isthe result of a so much number of parameters Figure 3shows a cause and effect diagram that was constructed toidentify the casting process parameters that may affect diecasting porosity (Tsoukalas et al 2004 2008 [5 6]) In thiscase holding furnace temperature die temperature plungervelocity in the first stage plunger velocity in the secondstage and multiplied pressure in the third stage were selectedas the most critical in the experimental design The otherparameters were kept constant in the entire experimentationThe range of holding furnace temperature was selected as640sim700∘C the range of die temperature as 180sim260∘C therange of plunger velocity in the first stage as 005sim035msand in the second stage as 15sim35ms and the range ofmultiplied pressure in the third stage was chosen as 200sim280 bars The selected casting process parameters along withits ranges are given in Table 2

Taguchi method based design of experiment has beenused to study the effects of five casting process parameters(holding furnace temperature A die temperature B plungervelocity in the first stage C plunger velocity in the secondstage D multiplied pressure in the third stage E on an

Mathematical Problems in Engineering 3

(a) (b)

(c) (d)

Figure 2 Part product is designed by CATIA software

Molten alloy

Die casting machine

Cavity filling time

Fast shot Plunger

stage

TemperatureLubricant

GateVenting system

Cooling system

Die

Shot sleeve

Filling level

Diameter

Length

Lubricant

Temperature

Composition

Condition

Shrinkage porosity type The smaller the better

Plungervelocity (1st)

Pressure during 3rd

set point velocity (2nd)

Figure 3 Cause and effect diagram

Table 2 The parameter and its value at three levels

Process parameters Parameters range Level 1 Level 2 Level 3Holding furnace temperature (∘C) 640sim700 640 670 700Die temperature (∘C) 180sim260 180 220 260Plunger velocity 1st stage (ms) 005sim035 005 02 035Plunger velocity 2nd stage (ms) 15sim35 15 25 35Multiplied pressure (bars) 200sim280 200 240 280

4 Mathematical Problems in Engineering

Table 3 Experimental layout using an L27orthogonal array

Trials Holding furnacetemperature A

Die temperatureB

Plunger velocity 1ststage C

Plunger velocity 2ndstage D

Multiplied pressureE

1 1 1 1 1 12 1 1 2 2 23 1 1 3 3 34 1 2 1 2 25 1 2 2 3 36 1 2 3 1 17 1 3 1 3 38 1 3 2 1 19 1 3 3 2 210 2 1 1 2 311 2 1 2 3 112 2 1 3 1 213 2 2 1 3 114 2 2 2 1 215 2 2 3 2 316 2 3 1 1 217 2 3 2 2 318 2 3 3 3 119 3 1 1 3 220 3 1 2 1 321 3 1 3 2 122 3 2 1 1 323 3 2 2 2 124 3 2 3 3 225 3 3 1 2 126 3 3 2 3 227 3 3 3 1 3

important output parameter (Shrinkage porosity) For select-ing appropriate orthogonal array degree of freedom (numberof fair and independent comparisons needed for optimizationof process parameters is one less than the number of levels ofparameter) of the array is calculated

In the experimental layout planwith five factors and threelevels using L

27orthogonal array 27 experimentswere carried

out to study the effect of casting input parameters shown inTable 3 The input parameters are installed in the ProCASTsoftware to conduct 27 simulation experiments

Computer simulation procedure-based process develop-ment and die design can be used for rapid process devel-opment and die design in a shorter time Such a computersimulation based procedure often using FINITE ELEMENTANALYSIS based software systems can improve the qualityand enhance productivity of the enterprise by way of fasterdevelopment of new product Analysis software is used as aProCAST commercial with finite element method analysisfor a casting process In this study all parameters can beable to affect the analysis process choice of material isaluminum alloy die casting ADC10 and cold chamber die

casting method with molding material is H13 FEM basedsimulation software systems help the designer to visualize themetal flow in the die cavity the temperature variation thesolidification progress and the evolution of defects such asshrinkage porosity cold-shut hot-tear

ProCAST a FEM simulation-based virtual casting envi-ronment for analysis of the casting process is used as a tool fordie design and process optimization ProCAST with Visual-Viewer module can provide temperature field thermal crack-ing flow field solidification time and shrinkage analysisThis paper focused on the analysis of shrinkage porosity byProCAST software base on parameters input from Table 3

The analysis of defects simulated by ProCAST softwarewith Visual-Viewer module can detect many types of disabil-ities castingThe defective products do not necessarily reflectthe loss of the original function for example the internal poretrims acceptable However with large structural castingsdefect analysis of this study focuses on maximum porosity inthe selection casting and the important parts of the castingshrinkage analysis (an important component) casting defectanalysis are described as follows

Mathematical Problems in Engineering 5

(a) (b)

(c) (d)

Figure 4 Casting measurement area

The Solid Fraction Solid fraction may be available shrinkageprediction casting position the present study is in accordancewith the theory prediction of defect and ProCAST manualreferred to in the final period of solidification Shrinkage solidfraction prone is greater than 07 here as the reference valueof 07 solid fractions When the solid fraction area is belowthis value and the area around the solid phase rate is ratherthan this value we can predict this area shrinkage porosityoccurred

Maximum Porosity The maximum porosity analysis usingthe Shrinkage Porosity function of the Visual-Viewer comesdefined in the manual According to the ProCAST usermanual shrinkage definition andwith the solid fraction it canbe used to analyse the basis of the maximum porosity

Shrinkage Analysis For the amount of inspection shrinkagecasting part used for the Visual-Viewer module functionfor quantitative analysis In each experiment we took fiveelements with the coordinates determined at the importantpositions in the working conditions of automobile startermotor casing Each experiment was repeated five times inorder to reduce experimental errors as shown in Figure 4Data from 27 experiments with five sampling times in eachsimulation are summarized as in Table 4 From this table weconducted quality characteristics analysis

Quality Characteristics The parameter design study involvescontrol and noise factors The measure of interactionsbetween these factors with regard to robustness is signal-to-noise (119878119873) ratio 119878119873 characteristics formulated for three

different categories are as follows the bigger the better andthe smaller the better the nominal the best This paperfocused on studying the effects of five input parameters(119860 119861 119862119863 119864) to defect shrinkage porosity in the process ofcasting so the criteria ldquothe smaller the betterrdquo is selected

The smaller the better (for making the system response assmall as possible) is as follows

119878

119873

119878

= minus10 log(1119899

119899

sum

119894=1

119910

2

119894) (1)

where 119899 is the number of sampling (Each experiment wasrepeated five times sampling so that 119899 = 5) 119910

119894 value of

Shrinkage porosity at each time samplingThe responding graph shown in Figure 5 learned that the

best combination for this studywith shrinkage porosity defectvalue minimum is 119860

3119861

3119862

3119863

1119864

3

Process Parameter Optimization Using MVLR The objectiveof the process optimization is to select the optimal controlvariables in aluminium die casting process in order to obtainthe minimum porosity In this work the fitness functionused in the optimization procedure was based on the MVLRmodel

Multivariable Linear Regression Analysis In most cases theform of the relationship between the response and theindependent variables is usually unknown Multiple linearregression (MLR) is a method used to model the linearrelationship between a dependent variable and one or more

6 Mathematical Problems in Engineering

Table4Sh

rinkage

porosityresults

oftheL27arraydesig

n(Fulltable)

Trials

Holding

Furnace

Temperature

ADie

temperature

B

Plun

ger

velocity

1ststage

C

Plun

ger

Velocity

2ndsta

geD

Multip

lied

Pressure

E

Shrin

kage

porosity(

)Av

erage

MSD

119878119873

Repetition

Repetition

Repetition

Repetition

Repetition

12

34

51

640

180

005

15200

1973

1988

1984

1978

1954

19754

3902346minus591326

264

0180

02

25

240

1912

1893

1948

1908

1912

19146

366

6021minus564

195

364

0180

035

35

280

1846

1853

1814

1813

1828

18308

3352095minus525316

464

0220

005

25

240

1862

1867

1861

1878

1871

18678

3488716minus54266

65

640

220

02

35

280

1793

1804

1761

1764

1775

17794

3166541minus500585

664

0220

035

15200

1916

1908

1912

1887

1903

19052

3629888minus559893

764

0260

005

35

280

1786

1797

1754

1757

1768

17724

3141679minus497162

864

0260

02

15200

1909

1903

1889

1898

1893

18984

3603973minus556782

964

0260

035

25

240

1852

1859

182

1819

1836

18372

3375568minus528347

10670

180

005

25

280

1799

181

1767

177

1781

17854

318793minus503509

11670

180

02

35

200

1957

1968

1971

1951

1965

19624

3851068minus585581

12670

180

035

15240

1857

1864

1825

1824

1841

18422

3393965minus530707

13670

220

005

35

200

1924

1935

1923

1928

1909

19238

3701079minus568328

14670

220

02

15240

1803

1814

1771

1774

1785

17894

3202229minus505452

15670

220

035

25

280

1751

1744

1713

1731

1725

17328

3002778minus477523

16670

260

005

15240

1796

1807

1764

1767

1778

17824

3177227minus50204

817

670

260

02

25

280

1738

1726

1724

1745

1715

17296

2991629minus475908

18670

260

035

35

200

1881

1878

1898

1893

1903

18906

35744

61minus553211

19700

180

005

35

240

1876

1871

1895

1887

1893

18844

3551052minus550357

20700

180

02

15280

1755

1743

1741

1762

1732

1746

63050725minus484403

21700

180

035

25

200

1921

1901

1956

1916

1928

19244

3703644minus568629

22700

220

005

15280

1708

1696

1694

1715

1685

16996

2888753minus46071

23700

220

02

25

200

1864

1898

1881

1876

1885

18808

3537532minus5487

24700

220

035

35

240

1825

1832

1793

1792

1807

18098

3275642minus515296

25700

260

005

25

200

1856

1861

1855

1872

1865

18618

346

6338minus539871

26700

260

02

35

240

1814

1821

1782

1781

1796

17988

3235948minus510001

27700

260

035

15280

1692

1694

1706

1683

1712

16974

2881274minus459585

Mathematical Problems in Engineering 7

Table 5 The results after analysing by Intercooled Stada 82 software

(a)

reg Fx A B C D E

Source SS df MS Number of obs = 27119865(5 21) = 14674

Model 0155044498 5 00310089 Prob gt 119865 = 00000

Residual 0004437799 21 0000211324 119877-squared = 09722Adj 119877-squared = 09655

Total 0159482297 26 0006133935 Root MSE = 001454

(b)

Fx Coef Std Err 119905 119875 gt |119905| [95 Conf Interval]119860 minus00008844 00001142 minus774 0000 minus0001122 minus00006469119861 minus000083 00000857 minus969 0000 minus00010081 minus00006519119862 minus00305925 07284277 minus0042 0000 minus00780965 00169114119863 00175444 00034264 512 0000 00104189 002467119864 minus00020122 00000857 minus2349 0000 minus00021904 minus00018341cons 3054569 00820708 3722 0000 2883893 3225244

SN

(dB)

Plots of factor effectsminus46

minus48

minus50

minus52

minus54

minus56

minus58

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3

Figure 5 119878119873 Response graphs

independent variables MLR is based on least squares themodel is fitted such that the sum-of-squares of differences ofobserved and predicted values is minimized

Let 1199091 119909

2 119909

119903be a set of 119903 predictors believed to be

related to a response variable 119884 The linear regression modelfor the 119895th sample unit has the form

119884

119895= 120573

0+ 120573

1119909

1198951+ 120573

2119909

1198952+ sdot sdot sdot + 120573

119903119909

119895119903+ 120576

119895 (2)

where 120576 is a random error and the 120573119894 119894 = 0 1 119903 are

unknown regression coefficientsIn this paper there are five independent variables and one

dependent variableThe relationships between these variablesare of the following form

119865 (119909) = 120573

0+ 120573

1119860 + 120573

2119861 + 120573

3119862 + 120573

4119863 + 120573

5119864 (3)

In which

119865(119909) Dependence variable119860 (∘C) Holding furnace temperature

119861 (∘C) Die temperature

119862 (ms) Plunger velocity 1st stage

119863 (ms) Plunger velocity 2nd stage

119864 (bars) Multiplied pressure during the third phase

The results after analysing by Intercooled Stada 82 Soft-were as shown in Table 5

The final MVLRmodel equation for porosity after substi-tuting regression coefficients is as follows

119865 (119909) = 3054569 minus 08844 lowast 10

minus3119860 minus 083 lowast 10

minus3119861

minus 003059119862 + 001754119863 minus 000201119864

(4)

3 Results and Discussion

We applied multivariable linear regression analysis (MVLR)to seek the optimal parameter in the casting process ofindependent parameter variables in this study A stationarypoint for the optimal performance was obtained by usingthe multivariable linear regression method in this linearregression equation and the result is presented in Figure 6A very good fit was observed and substantiated by thecoefficient of determination 1198772 = 09722 That is the 1198772value indicates that the polynomial model explains almost9722 of variability in the casting process

Figure 6 shows the efficacy of the optimization scheme bycomparing the MVLR results with the experimental valuesThere is a convincing agreement between experimental valuesand predicted values for shrinkage porosity percent

Matlab code for finding optimization shrinkage porosityvalue

8 Mathematical Problems in Engineering

clc

clear all

close all

f = (x) 3054569-08844e-3lowastx(1)-083e-3lowastx(2)-

003059lowastx(3) + 001754lowastx(4)-000201lowastx(5)

options = optimset(lsquoGradObjrsquo lsquoonrsquo)[xfvalexitflagoutput] =

fmincon(f[6702200225240][ ][ ][ ][ ][60018000515

200][70026003535280][ ]optimset(lsquoDisplayrsquo lsquoiterrsquo))x

fval

Algorithm 1

15155

16165

17175

18185

19195

2

Number of tests

Shrin

kage

por

osity

()

PredictedExperimental

Std dev = 0014537

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

R2 = 9722 adjustedR

2 = 9655

Figure 6 Experimental and predicted values of shrinkage porositypercent

Program in Matlab (see Algorithm 1)Results after running in Matlab are as follows

119909 = 7000000 so that 997888rarr 119860 = 700∘C

2600000 119861 = 260

∘C

03500 119862 = 035ms

15000 119863 = 15ms

2800000 119864 = 280 bar

fval = 16725 Shrinkage porosity 16725

(5)

By the Program in Matlab we are known as the bestcombination in the 27 experimental configurations

This result is similar to quality characteristics and is thebest combination for this study 119860

3119861

3119862

3119863

1119864

3

4 Conclusion

In this paper the optimum process parameters valuespredicted for casting of minimum shrinkage porosity

(16725) and the best combination parameters given asfollows

holding furnace temperature 700∘Cdie temperature 260∘Cplunger velocity 1st stage 035msplunger velocity 2nd stage 15msmultiplied pressure 280 bar

The model proposed in this paper gives satisfactoryresults in the optimization of pressure die casting processThepredicted values of the process parameters and the calculatedare in convincing agreement with the experimental values

The experiments which are conducted to determine thebest levels are based on ldquoOrthogonal Arraysrdquo and are bal-anced with respect to all control factors and yet areminimumin number This in turn implies that the resources (materialssaving time andmoney) required for the experiments are alsominimized

References

[1] C Shen L Wang and Q Li ldquoOptimization of injection mold-ing process parameters using combination of artificial neuralnetwork and genetic algorithm methodrdquo Journal of MaterialsProcessing Technology vol 183 no 2-3 pp 412ndash418 2007

[2] M S Dargusch G Dour N Schauer C M Dinnis and GSavage ldquoThe influence of pressure during solidification of highpressure die cast aluminium telecommunications componentsrdquoJournal of Materials Processing Technology vol 180 no 1ndash3 pp37ndash43 2006

[3] G O Verran R P K Mendes and M A Rossi ldquoInfluenceof injection parameters on defects formation in die castingAl12Si13Cu alloy experimental results and numeric simula-tionrdquo Journal of Materials Processing Technology vol 179 no 1ndash3 pp 190ndash195 2006

[4] S H Mousavi Anijdan A Bahrami H R Madaah Hosseiniand A Shafyei ldquoUsing genetic algorithm and artificial neuralnetwork analyses to design an Al-Si casting alloy of minimumporosityrdquoMaterials and Design vol 27 no 7 pp 605ndash609 2006

[5] V D Tsoukalas S A Mavrommatis N G Orfanoudakis andA K Baldoukas ldquoA study of porosity formation in pressuredie casting using the Taguchi approachrdquo Journal of EngineeringManufacture vol 218 no 1 pp 77ndash86 2004

Mathematical Problems in Engineering 9

[6] V D Tsoukalas ldquoOptimization of porosity formation inAlSi9Cu3pressure die castings using genetic algorithm analy-

sisrdquoMaterials and Design vol 29 no 10 pp 2027ndash2033 2008[7] G P Syrcos ldquoDie casting process optimization using Taguchi

methodsrdquo Journal of Materials Processing Technology vol 135no 1 pp 68ndash74 2003

[8] G Taguchi Introduction to Quality Engineering Asian Produc-tivity Organization UNIPUB 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

(a) (b)

(c) (d)

Figure 2 Part product is designed by CATIA software

Molten alloy

Die casting machine

Cavity filling time

Fast shot Plunger

stage

TemperatureLubricant

GateVenting system

Cooling system

Die

Shot sleeve

Filling level

Diameter

Length

Lubricant

Temperature

Composition

Condition

Shrinkage porosity type The smaller the better

Plungervelocity (1st)

Pressure during 3rd

set point velocity (2nd)

Figure 3 Cause and effect diagram

Table 2 The parameter and its value at three levels

Process parameters Parameters range Level 1 Level 2 Level 3Holding furnace temperature (∘C) 640sim700 640 670 700Die temperature (∘C) 180sim260 180 220 260Plunger velocity 1st stage (ms) 005sim035 005 02 035Plunger velocity 2nd stage (ms) 15sim35 15 25 35Multiplied pressure (bars) 200sim280 200 240 280

4 Mathematical Problems in Engineering

Table 3 Experimental layout using an L27orthogonal array

Trials Holding furnacetemperature A

Die temperatureB

Plunger velocity 1ststage C

Plunger velocity 2ndstage D

Multiplied pressureE

1 1 1 1 1 12 1 1 2 2 23 1 1 3 3 34 1 2 1 2 25 1 2 2 3 36 1 2 3 1 17 1 3 1 3 38 1 3 2 1 19 1 3 3 2 210 2 1 1 2 311 2 1 2 3 112 2 1 3 1 213 2 2 1 3 114 2 2 2 1 215 2 2 3 2 316 2 3 1 1 217 2 3 2 2 318 2 3 3 3 119 3 1 1 3 220 3 1 2 1 321 3 1 3 2 122 3 2 1 1 323 3 2 2 2 124 3 2 3 3 225 3 3 1 2 126 3 3 2 3 227 3 3 3 1 3

important output parameter (Shrinkage porosity) For select-ing appropriate orthogonal array degree of freedom (numberof fair and independent comparisons needed for optimizationof process parameters is one less than the number of levels ofparameter) of the array is calculated

In the experimental layout planwith five factors and threelevels using L

27orthogonal array 27 experimentswere carried

out to study the effect of casting input parameters shown inTable 3 The input parameters are installed in the ProCASTsoftware to conduct 27 simulation experiments

Computer simulation procedure-based process develop-ment and die design can be used for rapid process devel-opment and die design in a shorter time Such a computersimulation based procedure often using FINITE ELEMENTANALYSIS based software systems can improve the qualityand enhance productivity of the enterprise by way of fasterdevelopment of new product Analysis software is used as aProCAST commercial with finite element method analysisfor a casting process In this study all parameters can beable to affect the analysis process choice of material isaluminum alloy die casting ADC10 and cold chamber die

casting method with molding material is H13 FEM basedsimulation software systems help the designer to visualize themetal flow in the die cavity the temperature variation thesolidification progress and the evolution of defects such asshrinkage porosity cold-shut hot-tear

ProCAST a FEM simulation-based virtual casting envi-ronment for analysis of the casting process is used as a tool fordie design and process optimization ProCAST with Visual-Viewer module can provide temperature field thermal crack-ing flow field solidification time and shrinkage analysisThis paper focused on the analysis of shrinkage porosity byProCAST software base on parameters input from Table 3

The analysis of defects simulated by ProCAST softwarewith Visual-Viewer module can detect many types of disabil-ities castingThe defective products do not necessarily reflectthe loss of the original function for example the internal poretrims acceptable However with large structural castingsdefect analysis of this study focuses on maximum porosity inthe selection casting and the important parts of the castingshrinkage analysis (an important component) casting defectanalysis are described as follows

Mathematical Problems in Engineering 5

(a) (b)

(c) (d)

Figure 4 Casting measurement area

The Solid Fraction Solid fraction may be available shrinkageprediction casting position the present study is in accordancewith the theory prediction of defect and ProCAST manualreferred to in the final period of solidification Shrinkage solidfraction prone is greater than 07 here as the reference valueof 07 solid fractions When the solid fraction area is belowthis value and the area around the solid phase rate is ratherthan this value we can predict this area shrinkage porosityoccurred

Maximum Porosity The maximum porosity analysis usingthe Shrinkage Porosity function of the Visual-Viewer comesdefined in the manual According to the ProCAST usermanual shrinkage definition andwith the solid fraction it canbe used to analyse the basis of the maximum porosity

Shrinkage Analysis For the amount of inspection shrinkagecasting part used for the Visual-Viewer module functionfor quantitative analysis In each experiment we took fiveelements with the coordinates determined at the importantpositions in the working conditions of automobile startermotor casing Each experiment was repeated five times inorder to reduce experimental errors as shown in Figure 4Data from 27 experiments with five sampling times in eachsimulation are summarized as in Table 4 From this table weconducted quality characteristics analysis

Quality Characteristics The parameter design study involvescontrol and noise factors The measure of interactionsbetween these factors with regard to robustness is signal-to-noise (119878119873) ratio 119878119873 characteristics formulated for three

different categories are as follows the bigger the better andthe smaller the better the nominal the best This paperfocused on studying the effects of five input parameters(119860 119861 119862119863 119864) to defect shrinkage porosity in the process ofcasting so the criteria ldquothe smaller the betterrdquo is selected

The smaller the better (for making the system response assmall as possible) is as follows

119878

119873

119878

= minus10 log(1119899

119899

sum

119894=1

119910

2

119894) (1)

where 119899 is the number of sampling (Each experiment wasrepeated five times sampling so that 119899 = 5) 119910

119894 value of

Shrinkage porosity at each time samplingThe responding graph shown in Figure 5 learned that the

best combination for this studywith shrinkage porosity defectvalue minimum is 119860

3119861

3119862

3119863

1119864

3

Process Parameter Optimization Using MVLR The objectiveof the process optimization is to select the optimal controlvariables in aluminium die casting process in order to obtainthe minimum porosity In this work the fitness functionused in the optimization procedure was based on the MVLRmodel

Multivariable Linear Regression Analysis In most cases theform of the relationship between the response and theindependent variables is usually unknown Multiple linearregression (MLR) is a method used to model the linearrelationship between a dependent variable and one or more

6 Mathematical Problems in Engineering

Table4Sh

rinkage

porosityresults

oftheL27arraydesig

n(Fulltable)

Trials

Holding

Furnace

Temperature

ADie

temperature

B

Plun

ger

velocity

1ststage

C

Plun

ger

Velocity

2ndsta

geD

Multip

lied

Pressure

E

Shrin

kage

porosity(

)Av

erage

MSD

119878119873

Repetition

Repetition

Repetition

Repetition

Repetition

12

34

51

640

180

005

15200

1973

1988

1984

1978

1954

19754

3902346minus591326

264

0180

02

25

240

1912

1893

1948

1908

1912

19146

366

6021minus564

195

364

0180

035

35

280

1846

1853

1814

1813

1828

18308

3352095minus525316

464

0220

005

25

240

1862

1867

1861

1878

1871

18678

3488716minus54266

65

640

220

02

35

280

1793

1804

1761

1764

1775

17794

3166541minus500585

664

0220

035

15200

1916

1908

1912

1887

1903

19052

3629888minus559893

764

0260

005

35

280

1786

1797

1754

1757

1768

17724

3141679minus497162

864

0260

02

15200

1909

1903

1889

1898

1893

18984

3603973minus556782

964

0260

035

25

240

1852

1859

182

1819

1836

18372

3375568minus528347

10670

180

005

25

280

1799

181

1767

177

1781

17854

318793minus503509

11670

180

02

35

200

1957

1968

1971

1951

1965

19624

3851068minus585581

12670

180

035

15240

1857

1864

1825

1824

1841

18422

3393965minus530707

13670

220

005

35

200

1924

1935

1923

1928

1909

19238

3701079minus568328

14670

220

02

15240

1803

1814

1771

1774

1785

17894

3202229minus505452

15670

220

035

25

280

1751

1744

1713

1731

1725

17328

3002778minus477523

16670

260

005

15240

1796

1807

1764

1767

1778

17824

3177227minus50204

817

670

260

02

25

280

1738

1726

1724

1745

1715

17296

2991629minus475908

18670

260

035

35

200

1881

1878

1898

1893

1903

18906

35744

61minus553211

19700

180

005

35

240

1876

1871

1895

1887

1893

18844

3551052minus550357

20700

180

02

15280

1755

1743

1741

1762

1732

1746

63050725minus484403

21700

180

035

25

200

1921

1901

1956

1916

1928

19244

3703644minus568629

22700

220

005

15280

1708

1696

1694

1715

1685

16996

2888753minus46071

23700

220

02

25

200

1864

1898

1881

1876

1885

18808

3537532minus5487

24700

220

035

35

240

1825

1832

1793

1792

1807

18098

3275642minus515296

25700

260

005

25

200

1856

1861

1855

1872

1865

18618

346

6338minus539871

26700

260

02

35

240

1814

1821

1782

1781

1796

17988

3235948minus510001

27700

260

035

15280

1692

1694

1706

1683

1712

16974

2881274minus459585

Mathematical Problems in Engineering 7

Table 5 The results after analysing by Intercooled Stada 82 software

(a)

reg Fx A B C D E

Source SS df MS Number of obs = 27119865(5 21) = 14674

Model 0155044498 5 00310089 Prob gt 119865 = 00000

Residual 0004437799 21 0000211324 119877-squared = 09722Adj 119877-squared = 09655

Total 0159482297 26 0006133935 Root MSE = 001454

(b)

Fx Coef Std Err 119905 119875 gt |119905| [95 Conf Interval]119860 minus00008844 00001142 minus774 0000 minus0001122 minus00006469119861 minus000083 00000857 minus969 0000 minus00010081 minus00006519119862 minus00305925 07284277 minus0042 0000 minus00780965 00169114119863 00175444 00034264 512 0000 00104189 002467119864 minus00020122 00000857 minus2349 0000 minus00021904 minus00018341cons 3054569 00820708 3722 0000 2883893 3225244

SN

(dB)

Plots of factor effectsminus46

minus48

minus50

minus52

minus54

minus56

minus58

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3

Figure 5 119878119873 Response graphs

independent variables MLR is based on least squares themodel is fitted such that the sum-of-squares of differences ofobserved and predicted values is minimized

Let 1199091 119909

2 119909

119903be a set of 119903 predictors believed to be

related to a response variable 119884 The linear regression modelfor the 119895th sample unit has the form

119884

119895= 120573

0+ 120573

1119909

1198951+ 120573

2119909

1198952+ sdot sdot sdot + 120573

119903119909

119895119903+ 120576

119895 (2)

where 120576 is a random error and the 120573119894 119894 = 0 1 119903 are

unknown regression coefficientsIn this paper there are five independent variables and one

dependent variableThe relationships between these variablesare of the following form

119865 (119909) = 120573

0+ 120573

1119860 + 120573

2119861 + 120573

3119862 + 120573

4119863 + 120573

5119864 (3)

In which

119865(119909) Dependence variable119860 (∘C) Holding furnace temperature

119861 (∘C) Die temperature

119862 (ms) Plunger velocity 1st stage

119863 (ms) Plunger velocity 2nd stage

119864 (bars) Multiplied pressure during the third phase

The results after analysing by Intercooled Stada 82 Soft-were as shown in Table 5

The final MVLRmodel equation for porosity after substi-tuting regression coefficients is as follows

119865 (119909) = 3054569 minus 08844 lowast 10

minus3119860 minus 083 lowast 10

minus3119861

minus 003059119862 + 001754119863 minus 000201119864

(4)

3 Results and Discussion

We applied multivariable linear regression analysis (MVLR)to seek the optimal parameter in the casting process ofindependent parameter variables in this study A stationarypoint for the optimal performance was obtained by usingthe multivariable linear regression method in this linearregression equation and the result is presented in Figure 6A very good fit was observed and substantiated by thecoefficient of determination 1198772 = 09722 That is the 1198772value indicates that the polynomial model explains almost9722 of variability in the casting process

Figure 6 shows the efficacy of the optimization scheme bycomparing the MVLR results with the experimental valuesThere is a convincing agreement between experimental valuesand predicted values for shrinkage porosity percent

Matlab code for finding optimization shrinkage porosityvalue

8 Mathematical Problems in Engineering

clc

clear all

close all

f = (x) 3054569-08844e-3lowastx(1)-083e-3lowastx(2)-

003059lowastx(3) + 001754lowastx(4)-000201lowastx(5)

options = optimset(lsquoGradObjrsquo lsquoonrsquo)[xfvalexitflagoutput] =

fmincon(f[6702200225240][ ][ ][ ][ ][60018000515

200][70026003535280][ ]optimset(lsquoDisplayrsquo lsquoiterrsquo))x

fval

Algorithm 1

15155

16165

17175

18185

19195

2

Number of tests

Shrin

kage

por

osity

()

PredictedExperimental

Std dev = 0014537

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

R2 = 9722 adjustedR

2 = 9655

Figure 6 Experimental and predicted values of shrinkage porositypercent

Program in Matlab (see Algorithm 1)Results after running in Matlab are as follows

119909 = 7000000 so that 997888rarr 119860 = 700∘C

2600000 119861 = 260

∘C

03500 119862 = 035ms

15000 119863 = 15ms

2800000 119864 = 280 bar

fval = 16725 Shrinkage porosity 16725

(5)

By the Program in Matlab we are known as the bestcombination in the 27 experimental configurations

This result is similar to quality characteristics and is thebest combination for this study 119860

3119861

3119862

3119863

1119864

3

4 Conclusion

In this paper the optimum process parameters valuespredicted for casting of minimum shrinkage porosity

(16725) and the best combination parameters given asfollows

holding furnace temperature 700∘Cdie temperature 260∘Cplunger velocity 1st stage 035msplunger velocity 2nd stage 15msmultiplied pressure 280 bar

The model proposed in this paper gives satisfactoryresults in the optimization of pressure die casting processThepredicted values of the process parameters and the calculatedare in convincing agreement with the experimental values

The experiments which are conducted to determine thebest levels are based on ldquoOrthogonal Arraysrdquo and are bal-anced with respect to all control factors and yet areminimumin number This in turn implies that the resources (materialssaving time andmoney) required for the experiments are alsominimized

References

[1] C Shen L Wang and Q Li ldquoOptimization of injection mold-ing process parameters using combination of artificial neuralnetwork and genetic algorithm methodrdquo Journal of MaterialsProcessing Technology vol 183 no 2-3 pp 412ndash418 2007

[2] M S Dargusch G Dour N Schauer C M Dinnis and GSavage ldquoThe influence of pressure during solidification of highpressure die cast aluminium telecommunications componentsrdquoJournal of Materials Processing Technology vol 180 no 1ndash3 pp37ndash43 2006

[3] G O Verran R P K Mendes and M A Rossi ldquoInfluenceof injection parameters on defects formation in die castingAl12Si13Cu alloy experimental results and numeric simula-tionrdquo Journal of Materials Processing Technology vol 179 no 1ndash3 pp 190ndash195 2006

[4] S H Mousavi Anijdan A Bahrami H R Madaah Hosseiniand A Shafyei ldquoUsing genetic algorithm and artificial neuralnetwork analyses to design an Al-Si casting alloy of minimumporosityrdquoMaterials and Design vol 27 no 7 pp 605ndash609 2006

[5] V D Tsoukalas S A Mavrommatis N G Orfanoudakis andA K Baldoukas ldquoA study of porosity formation in pressuredie casting using the Taguchi approachrdquo Journal of EngineeringManufacture vol 218 no 1 pp 77ndash86 2004

Mathematical Problems in Engineering 9

[6] V D Tsoukalas ldquoOptimization of porosity formation inAlSi9Cu3pressure die castings using genetic algorithm analy-

sisrdquoMaterials and Design vol 29 no 10 pp 2027ndash2033 2008[7] G P Syrcos ldquoDie casting process optimization using Taguchi

methodsrdquo Journal of Materials Processing Technology vol 135no 1 pp 68ndash74 2003

[8] G Taguchi Introduction to Quality Engineering Asian Produc-tivity Organization UNIPUB 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Table 3 Experimental layout using an L27orthogonal array

Trials Holding furnacetemperature A

Die temperatureB

Plunger velocity 1ststage C

Plunger velocity 2ndstage D

Multiplied pressureE

In the experimental layout planwith five factors and threelevels using L

27orthogonal array 27 experimentswere carried

Mathematical Problems in Engineering 5

(a) (b)

(c) (d)

Figure 4 Casting measurement area

The smaller the better (for making the system response assmall as possible) is as follows

119878

119873

119878

= minus10 log(1119899

119899

sum

119894=1

119910

2

119894) (1)

119894 value of

Shrinkage porosity at each time samplingThe responding graph shown in Figure 5 learned that the

best combination for this studywith shrinkage porosity defectvalue minimum is 119860

3119861

3119862

3119863

1119864

3

6 Mathematical Problems in Engineering

Table4Sh

rinkage

porosityresults

oftheL27arraydesig

n(Fulltable)

Trials

Holding

Furnace

Temperature

ADie

temperature

B

Plun

ger

velocity

1ststage

C

Plun

ger

Velocity

2ndsta

geD

Multip

lied

Pressure

E

Shrin

kage

porosity(

)Av

erage

MSD

119878119873

Repetition

Repetition

Repetition

Repetition

Repetition

12

34

51

640

180

005

15200

1973

1988

1984

1978

1954

19754

3902346minus591326

264

0180

02

25

240

1912

1893

1948

1908

1912

19146

366

6021minus564

195

364

0180

035

35

280

1846

1853

1814

1813

1828

18308

3352095minus525316

464

0220

005

25

240

1862

1867

1861

1878

1871

18678

3488716minus54266

65

640

220

02

35

280

1793

1804

1761

1764

1775

17794

3166541minus500585

664

0220

035

15200

1916

1908

1912

1887

1903

19052

3629888minus559893

764

0260

005

35

280

1786

1797

1754

1757

1768

17724

3141679minus497162

864

0260

02

15200

1909

1903

1889

1898

1893

18984

3603973minus556782

964

0260

035

25

240

1852

1859

182

1819

1836

18372

3375568minus528347

10670

180

005

25

280

1799

181

1767

177

1781

17854

318793minus503509

11670

180

02

35

200

1957

1968

1971

1951

1965

19624

3851068minus585581

12670

180

035

15240

1857

1864

1825

1824

1841

18422

3393965minus530707

13670

220

005

35

200

1924

1935

1923

1928

1909

19238

3701079minus568328

14670

220

02

15240

1803

1814

1771

1774

1785

17894

3202229minus505452

15670

220

035

25

280

1751

1744

1713

1731

1725

17328

3002778minus477523

16670

260

005

15240

1796

1807

1764

1767

1778

17824

3177227minus50204

817

670

260

02

25

280

1738

1726

1724

1745

1715

17296

2991629minus475908

18670

260

035

35

200

1881

1878

1898

1893

1903

18906

35744

61minus553211

19700

180

005

35

240

1876

1871

1895

1887

1893

18844

3551052minus550357

20700

180

02

15280

1755

1743

1741

1762

1732

1746

63050725minus484403

21700

180

035

25

200

1921

1901

1956

1916

1928

19244

3703644minus568629

22700

220

005

15280

1708

1696

1694

1715

1685

16996

2888753minus46071

23700

220

02

25

200

1864

1898

1881

1876

1885

18808

3537532minus5487

24700

220

035

35

240

1825

1832

1793

1792

1807

18098

3275642minus515296

25700

260

005

25

200

1856

1861

1855

1872

1865

18618

346

6338minus539871

26700

260

02

35

240

1814

1821

1782

1781

1796

17988

3235948minus510001

27700

260

035

15280

1692

1694

1706

1683

1712

16974

2881274minus459585

Mathematical Problems in Engineering 7

Table 5 The results after analysing by Intercooled Stada 82 software

(a)

reg Fx A B C D E

Source SS df MS Number of obs = 27119865(5 21) = 14674

Model 0155044498 5 00310089 Prob gt 119865 = 00000

Residual 0004437799 21 0000211324 119877-squared = 09722Adj 119877-squared = 09655

Total 0159482297 26 0006133935 Root MSE = 001454

(b)

SN

(dB)

Plots of factor effectsminus46

minus48

minus50

minus52

minus54

minus56

minus58

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3

Figure 5 119878119873 Response graphs

Let 1199091 119909

2 119909

119903be a set of 119903 predictors believed to be

119884

119895= 120573

0+ 120573

1119909

1198951+ 120573

2119909

1198952+ sdot sdot sdot + 120573

119903119909

119895119903+ 120576

119895 (2)

where 120576 is a random error and the 120573119894 119894 = 0 1 119903 are

unknown regression coefficientsIn this paper there are five independent variables and one

dependent variableThe relationships between these variablesare of the following form

119865 (119909) = 120573

0+ 120573

1119860 + 120573

2119861 + 120573

3119862 + 120573

4119863 + 120573

5119864 (3)

In which

119865(119909) Dependence variable119860 (∘C) Holding furnace temperature

119861 (∘C) Die temperature

119862 (ms) Plunger velocity 1st stage

119863 (ms) Plunger velocity 2nd stage

119864 (bars) Multiplied pressure during the third phase

The results after analysing by Intercooled Stada 82 Soft-were as shown in Table 5

The final MVLRmodel equation for porosity after substi-tuting regression coefficients is as follows

119865 (119909) = 3054569 minus 08844 lowast 10

minus3119860 minus 083 lowast 10

minus3119861

minus 003059119862 + 001754119863 minus 000201119864

(4)

3 Results and Discussion

Matlab code for finding optimization shrinkage porosityvalue

8 Mathematical Problems in Engineering

clc

clear all

close all

f = (x) 3054569-08844e-3lowastx(1)-083e-3lowastx(2)-

003059lowastx(3) + 001754lowastx(4)-000201lowastx(5)

options = optimset(lsquoGradObjrsquo lsquoonrsquo)[xfvalexitflagoutput] =

fmincon(f[6702200225240][ ][ ][ ][ ][60018000515

200][70026003535280][ ]optimset(lsquoDisplayrsquo lsquoiterrsquo))x

fval

Algorithm 1

15155

16165

17175

18185

19195

2

Number of tests

Shrin

kage

por

osity

()

PredictedExperimental

Std dev = 0014537

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

R2 = 9722 adjustedR

2 = 9655

Figure 6 Experimental and predicted values of shrinkage porositypercent

Program in Matlab (see Algorithm 1)Results after running in Matlab are as follows

119909 = 7000000 so that 997888rarr 119860 = 700∘C

2600000 119861 = 260

∘C

03500 119862 = 035ms

15000 119863 = 15ms

2800000 119864 = 280 bar

fval = 16725 Shrinkage porosity 16725

(5)

By the Program in Matlab we are known as the bestcombination in the 27 experimental configurations

This result is similar to quality characteristics and is thebest combination for this study 119860

3119861

3119862

3119863

1119864

3

4 Conclusion

(16725) and the best combination parameters given asfollows

References

Mathematical Problems in Engineering 9

methodsrdquo Journal of Materials Processing Technology vol 135no 1 pp 68ndash74 2003

[8] G Taguchi Introduction to Quality Engineering Asian Produc-tivity Organization UNIPUB 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

(a) (b)

(c) (d)

Figure 4 Casting measurement area

The smaller the better (for making the system response assmall as possible) is as follows

119878

119873

119878

= minus10 log(1119899

119899

sum

119894=1

119910

2

119894) (1)

119894 value of

Shrinkage porosity at each time samplingThe responding graph shown in Figure 5 learned that the

best combination for this studywith shrinkage porosity defectvalue minimum is 119860

3119861

3119862

3119863

1119864

3

6 Mathematical Problems in Engineering

Table4Sh

rinkage

porosityresults

oftheL27arraydesig

n(Fulltable)

Trials

Holding

Furnace

Temperature

ADie

temperature

B

Plun

ger

velocity

1ststage

C

Plun

ger

Velocity

2ndsta

geD

Multip

lied

Pressure

E

Shrin

kage

porosity(

)Av

erage

MSD

119878119873

Repetition

Repetition

Repetition

Repetition

Repetition

12

34

51

640

180

005

15200

1973

1988

1984

1978

1954

19754

3902346minus591326

264

0180

02

25

240

1912

1893

1948

1908

1912

19146

366

6021minus564

195

364

0180

035

35

280

1846

1853

1814

1813

1828

18308

3352095minus525316

464

0220

005

25

240

1862

1867

1861

1878

1871

18678

3488716minus54266

65

640

220

02

35

280

1793

1804

1761

1764

1775

17794

3166541minus500585

664

0220

035

15200

1916

1908

1912

1887

1903

19052

3629888minus559893

764

0260

005

35

280

1786

1797

1754

1757

1768

17724

3141679minus497162

864

0260

02

15200

1909

1903

1889

1898

1893

18984

3603973minus556782

964

0260

035

25

240

1852

1859

182

1819

1836

18372

3375568minus528347

10670

180

005

25

280

1799

181

1767

177

1781

17854

318793minus503509

11670

180

02

35

200

1957

1968

1971

1951

1965

19624

3851068minus585581

12670

180

035

15240

1857

1864

1825

1824

1841

18422

3393965minus530707

13670

220

005

35

200

1924

1935

1923

1928

1909

19238

3701079minus568328

14670

220

02

15240

1803

1814

1771

1774

1785

17894

3202229minus505452

15670

220

035

25

280

1751

1744

1713

1731

1725

17328

3002778minus477523

16670

260

005

15240

1796

1807

1764

1767

1778

17824

3177227minus50204

817

670

260

02

25

280

1738

1726

1724

1745

1715

17296

2991629minus475908

18670

260

035

35

200

1881

1878

1898

1893

1903

18906

35744

61minus553211

19700

180

005

35

240

1876

1871

1895

1887

1893

18844

3551052minus550357

20700

180

02

15280

1755

1743

1741

1762

1732

1746

63050725minus484403

21700

180

035

25

200

1921

1901

1956

1916

1928

19244

3703644minus568629

22700

220

005

15280

1708

1696

1694

1715

1685

16996

2888753minus46071

23700

220

02

25

200

1864

1898

1881

1876

1885

18808

3537532minus5487

24700

220

035

35

240

1825

1832

1793

1792

1807

18098

3275642minus515296

25700

260

005

25

200

1856

1861

1855

1872

1865

18618

346

6338minus539871

26700

260

02

35

240

1814

1821

1782

1781

1796

17988

3235948minus510001

27700

260

035

15280

1692

1694

1706

1683

1712

16974

2881274minus459585

Mathematical Problems in Engineering 7

Table 5 The results after analysing by Intercooled Stada 82 software

(a)

reg Fx A B C D E

Source SS df MS Number of obs = 27119865(5 21) = 14674

Model 0155044498 5 00310089 Prob gt 119865 = 00000

Residual 0004437799 21 0000211324 119877-squared = 09722Adj 119877-squared = 09655

Total 0159482297 26 0006133935 Root MSE = 001454

(b)

SN

(dB)

Plots of factor effectsminus46

minus48

minus50

minus52

minus54

minus56

minus58

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3

Figure 5 119878119873 Response graphs

Let 1199091 119909

2 119909

119903be a set of 119903 predictors believed to be

119884

119895= 120573

0+ 120573

1119909

1198951+ 120573

2119909

1198952+ sdot sdot sdot + 120573

119903119909

119895119903+ 120576

119895 (2)

where 120576 is a random error and the 120573119894 119894 = 0 1 119903 are

unknown regression coefficientsIn this paper there are five independent variables and one

dependent variableThe relationships between these variablesare of the following form

119865 (119909) = 120573

0+ 120573

1119860 + 120573

2119861 + 120573

3119862 + 120573

4119863 + 120573

5119864 (3)

In which

119865(119909) Dependence variable119860 (∘C) Holding furnace temperature

119861 (∘C) Die temperature

119862 (ms) Plunger velocity 1st stage

119863 (ms) Plunger velocity 2nd stage

119864 (bars) Multiplied pressure during the third phase

The results after analysing by Intercooled Stada 82 Soft-were as shown in Table 5

The final MVLRmodel equation for porosity after substi-tuting regression coefficients is as follows

119865 (119909) = 3054569 minus 08844 lowast 10

minus3119860 minus 083 lowast 10

minus3119861

minus 003059119862 + 001754119863 minus 000201119864

(4)

3 Results and Discussion

Matlab code for finding optimization shrinkage porosityvalue

8 Mathematical Problems in Engineering

clc

clear all

close all

f = (x) 3054569-08844e-3lowastx(1)-083e-3lowastx(2)-

003059lowastx(3) + 001754lowastx(4)-000201lowastx(5)

options = optimset(lsquoGradObjrsquo lsquoonrsquo)[xfvalexitflagoutput] =

fmincon(f[6702200225240][ ][ ][ ][ ][60018000515

200][70026003535280][ ]optimset(lsquoDisplayrsquo lsquoiterrsquo))x

fval

Algorithm 1

15155

16165

17175

18185

19195

2

Number of tests

Shrin

kage

por

osity

()

PredictedExperimental

Std dev = 0014537

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

R2 = 9722 adjustedR

2 = 9655

Figure 6 Experimental and predicted values of shrinkage porositypercent

Program in Matlab (see Algorithm 1)Results after running in Matlab are as follows

119909 = 7000000 so that 997888rarr 119860 = 700∘C

2600000 119861 = 260

∘C

03500 119862 = 035ms

15000 119863 = 15ms

2800000 119864 = 280 bar

fval = 16725 Shrinkage porosity 16725

(5)

By the Program in Matlab we are known as the bestcombination in the 27 experimental configurations

This result is similar to quality characteristics and is thebest combination for this study 119860

3119861

3119862

3119863

1119864

3

4 Conclusion

(16725) and the best combination parameters given asfollows

References

Mathematical Problems in Engineering 9

methodsrdquo Journal of Materials Processing Technology vol 135no 1 pp 68ndash74 2003

[8] G Taguchi Introduction to Quality Engineering Asian Produc-tivity Organization UNIPUB 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table4Sh

rinkage

porosityresults

oftheL27arraydesig

n(Fulltable)

Trials

Holding

Furnace

Temperature

ADie

temperature

B

Plun

ger

velocity

1ststage

C

Plun

ger

Velocity

2ndsta

geD

Multip

lied

Pressure

E

Shrin

kage

porosity(

)Av

erage

MSD

119878119873

Repetition

Repetition

Repetition

Repetition

Repetition

12

34

51

640

180

005

15200

1973

1988

1984

1978

1954

19754

3902346minus591326

264

0180

02

25

240

1912

1893

1948

1908

1912

19146

366

6021minus564

195

364

0180

035

35

280

1846

1853

1814

1813

1828

18308

3352095minus525316

464

0220

005

25

240

1862

1867

1861

1878

1871

18678

3488716minus54266

65

640

220

02

35

280

1793

1804

1761

1764

1775

17794

3166541minus500585

664

0220

035

15200

1916

1908

1912

1887

1903

19052

3629888minus559893

764

0260

005

35

280

1786

1797

1754

1757

1768

17724

3141679minus497162

864

0260

02

15200

1909

1903

1889

1898

1893

18984

3603973minus556782

964

0260

035

25

240

1852

1859

182

1819

1836

18372

3375568minus528347

10670

180

005

25

280

1799

181

1767

177

1781

17854

318793minus503509

11670

180

02

35

200

1957

1968

1971

1951

1965

19624

3851068minus585581

12670

180

035

15240

1857

1864

1825

1824

1841

18422

3393965minus530707

13670

220

005

35

200

1924

1935

1923

1928

1909

19238

3701079minus568328

14670

220

02

15240

1803

1814

1771

1774

1785

17894

3202229minus505452

15670

220

035

25

280

1751

1744

1713

1731

1725

17328

3002778minus477523

16670

260

005

15240

1796

1807

1764

1767

1778

17824

3177227minus50204

817

670

260

02

25

280

1738

1726

1724

1745

1715

17296

2991629minus475908

18670

260

035

35

200

1881

1878

1898

1893

1903

18906

35744

61minus553211

19700

180

005

35

240

1876

1871

1895

1887

1893

18844

3551052minus550357

20700

180

02

15280

1755

1743

1741

1762

1732

1746

63050725minus484403

21700

180

035

25

200

1921

1901

1956

1916

1928

19244

3703644minus568629

22700

220

005

15280

1708

1696

1694

1715

1685

16996

2888753minus46071

23700

220

02

25

200

1864

1898

1881

1876

1885

18808

3537532minus5487

24700

220

035

35

240

1825

1832

1793

1792

1807

18098

3275642minus515296

25700

260

005

25

200

1856

1861

1855

1872

1865

18618

346

6338minus539871

26700

260

02

35

240

1814

1821

1782

1781

1796

17988

3235948minus510001

27700

260

035

15280

1692

1694

1706

1683

1712

16974

2881274minus459585

Mathematical Problems in Engineering 7

Table 5 The results after analysing by Intercooled Stada 82 software

(a)

reg Fx A B C D E

Source SS df MS Number of obs = 27119865(5 21) = 14674

Model 0155044498 5 00310089 Prob gt 119865 = 00000

Residual 0004437799 21 0000211324 119877-squared = 09722Adj 119877-squared = 09655

Total 0159482297 26 0006133935 Root MSE = 001454

(b)

SN

(dB)

Plots of factor effectsminus46

minus48

minus50

minus52

minus54

minus56

minus58

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3

Figure 5 119878119873 Response graphs

Let 1199091 119909

2 119909

119903be a set of 119903 predictors believed to be

119884

119895= 120573

0+ 120573

1119909

1198951+ 120573

2119909

1198952+ sdot sdot sdot + 120573

119903119909

119895119903+ 120576

119895 (2)

where 120576 is a random error and the 120573119894 119894 = 0 1 119903 are

unknown regression coefficientsIn this paper there are five independent variables and one

dependent variableThe relationships between these variablesare of the following form

119865 (119909) = 120573

0+ 120573

1119860 + 120573

2119861 + 120573

3119862 + 120573

4119863 + 120573

5119864 (3)

In which

119865(119909) Dependence variable119860 (∘C) Holding furnace temperature

119861 (∘C) Die temperature

119862 (ms) Plunger velocity 1st stage

119863 (ms) Plunger velocity 2nd stage

119864 (bars) Multiplied pressure during the third phase

The results after analysing by Intercooled Stada 82 Soft-were as shown in Table 5

The final MVLRmodel equation for porosity after substi-tuting regression coefficients is as follows

119865 (119909) = 3054569 minus 08844 lowast 10

minus3119860 minus 083 lowast 10

minus3119861

minus 003059119862 + 001754119863 minus 000201119864

(4)

3 Results and Discussion

Matlab code for finding optimization shrinkage porosityvalue

8 Mathematical Problems in Engineering

clc

clear all

close all

f = (x) 3054569-08844e-3lowastx(1)-083e-3lowastx(2)-

003059lowastx(3) + 001754lowastx(4)-000201lowastx(5)

options = optimset(lsquoGradObjrsquo lsquoonrsquo)[xfvalexitflagoutput] =

fmincon(f[6702200225240][ ][ ][ ][ ][60018000515

200][70026003535280][ ]optimset(lsquoDisplayrsquo lsquoiterrsquo))x

fval

Algorithm 1

15155

16165

17175

18185

19195

2

Number of tests

Shrin

kage

por

osity

()

PredictedExperimental

Std dev = 0014537

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

R2 = 9722 adjustedR

2 = 9655

Figure 6 Experimental and predicted values of shrinkage porositypercent

Program in Matlab (see Algorithm 1)Results after running in Matlab are as follows

119909 = 7000000 so that 997888rarr 119860 = 700∘C

2600000 119861 = 260

∘C

03500 119862 = 035ms

15000 119863 = 15ms

2800000 119864 = 280 bar

fval = 16725 Shrinkage porosity 16725

(5)

By the Program in Matlab we are known as the bestcombination in the 27 experimental configurations

This result is similar to quality characteristics and is thebest combination for this study 119860

3119861

3119862

3119863

1119864

3

4 Conclusion

(16725) and the best combination parameters given asfollows

References

Mathematical Problems in Engineering 9

methodsrdquo Journal of Materials Processing Technology vol 135no 1 pp 68ndash74 2003

[8] G Taguchi Introduction to Quality Engineering Asian Produc-tivity Organization UNIPUB 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 5 The results after analysing by Intercooled Stada 82 software

(a)

reg Fx A B C D E

Source SS df MS Number of obs = 27119865(5 21) = 14674

Model 0155044498 5 00310089 Prob gt 119865 = 00000

Residual 0004437799 21 0000211324 119877-squared = 09722Adj 119877-squared = 09655

Total 0159482297 26 0006133935 Root MSE = 001454

(b)

SN

(dB)

Plots of factor effectsminus46

minus48

minus50

minus52

minus54

minus56

minus58

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3

Figure 5 119878119873 Response graphs

Let 1199091 119909

2 119909

119903be a set of 119903 predictors believed to be

119884

119895= 120573

0+ 120573

1119909

1198951+ 120573

2119909

1198952+ sdot sdot sdot + 120573

119903119909

119895119903+ 120576

119895 (2)

where 120576 is a random error and the 120573119894 119894 = 0 1 119903 are

unknown regression coefficientsIn this paper there are five independent variables and one

dependent variableThe relationships between these variablesare of the following form

119865 (119909) = 120573

0+ 120573

1119860 + 120573

2119861 + 120573

3119862 + 120573

4119863 + 120573

5119864 (3)

In which

119865(119909) Dependence variable119860 (∘C) Holding furnace temperature

119861 (∘C) Die temperature

119862 (ms) Plunger velocity 1st stage

119863 (ms) Plunger velocity 2nd stage

119864 (bars) Multiplied pressure during the third phase

The results after analysing by Intercooled Stada 82 Soft-were as shown in Table 5

The final MVLRmodel equation for porosity after substi-tuting regression coefficients is as follows

119865 (119909) = 3054569 minus 08844 lowast 10

minus3119860 minus 083 lowast 10

minus3119861

minus 003059119862 + 001754119863 minus 000201119864

(4)

3 Results and Discussion

Matlab code for finding optimization shrinkage porosityvalue

8 Mathematical Problems in Engineering

clc

clear all

close all

f = (x) 3054569-08844e-3lowastx(1)-083e-3lowastx(2)-

003059lowastx(3) + 001754lowastx(4)-000201lowastx(5)

options = optimset(lsquoGradObjrsquo lsquoonrsquo)[xfvalexitflagoutput] =

fmincon(f[6702200225240][ ][ ][ ][ ][60018000515

200][70026003535280][ ]optimset(lsquoDisplayrsquo lsquoiterrsquo))x

fval

Algorithm 1

15155

16165

17175

18185

19195

2

Number of tests

Shrin

kage

por

osity

()

PredictedExperimental

Std dev = 0014537

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

R2 = 9722 adjustedR

2 = 9655

Figure 6 Experimental and predicted values of shrinkage porositypercent

Program in Matlab (see Algorithm 1)Results after running in Matlab are as follows

119909 = 7000000 so that 997888rarr 119860 = 700∘C

2600000 119861 = 260

∘C

03500 119862 = 035ms

15000 119863 = 15ms

2800000 119864 = 280 bar

fval = 16725 Shrinkage porosity 16725

(5)

By the Program in Matlab we are known as the bestcombination in the 27 experimental configurations

This result is similar to quality characteristics and is thebest combination for this study 119860

3119861

3119862

3119863

1119864

3

4 Conclusion

(16725) and the best combination parameters given asfollows

References

Mathematical Problems in Engineering 9

methodsrdquo Journal of Materials Processing Technology vol 135no 1 pp 68ndash74 2003

[8] G Taguchi Introduction to Quality Engineering Asian Produc-tivity Organization UNIPUB 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

clc

clear all

close all

f = (x) 3054569-08844e-3lowastx(1)-083e-3lowastx(2)-

003059lowastx(3) + 001754lowastx(4)-000201lowastx(5)

options = optimset(lsquoGradObjrsquo lsquoonrsquo)[xfvalexitflagoutput] =

fmincon(f[6702200225240][ ][ ][ ][ ][60018000515

200][70026003535280][ ]optimset(lsquoDisplayrsquo lsquoiterrsquo))x

fval

Algorithm 1

15155

16165

17175

18185

19195

2

Number of tests

Shrin

kage

por

osity

()

PredictedExperimental

Std dev = 0014537

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

R2 = 9722 adjustedR

2 = 9655

Figure 6 Experimental and predicted values of shrinkage porositypercent

Program in Matlab (see Algorithm 1)Results after running in Matlab are as follows

119909 = 7000000 so that 997888rarr 119860 = 700∘C

2600000 119861 = 260

∘C

03500 119862 = 035ms

15000 119863 = 15ms

2800000 119864 = 280 bar

fval = 16725 Shrinkage porosity 16725

(5)

By the Program in Matlab we are known as the bestcombination in the 27 experimental configurations

This result is similar to quality characteristics and is thebest combination for this study 119860

3119861

3119862

3119863

1119864

3

4 Conclusion

(16725) and the best combination parameters given asfollows

References

Mathematical Problems in Engineering 9

methodsrdquo Journal of Materials Processing Technology vol 135no 1 pp 68ndash74 2003

[8] G Taguchi Introduction to Quality Engineering Asian Produc-tivity Organization UNIPUB 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

methodsrdquo Journal of Materials Processing Technology vol 135no 1 pp 68ndash74 2003

[8] G Taguchi Introduction to Quality Engineering Asian Produc-tivity Organization UNIPUB 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of