Research Article An Alternative Approach of Dual Response...

Research ArticleAn Alternative Approach of Dual Response SurfaceOptimization Based on Penalty Function Method

Ishaq Baba1 Habshah Midi12 Sohel Rana12 and Gafurjan Ibragimov12

1Department of Mathematics Faculty of Science Universiti Putra Malaysia 43400 Serdang Selangor Malaysia2Laboratory of Computational Statistics and Operations Research Institute for Mathematical ResearchUniversiti Putra Malaysia 43400 Serdang Selangor Malaysia

Correspondence should be addressed to Habshah Midi habshahupmedumy

Received 10 January 2015 Accepted 27 April 2015

Academic Editor Mohammed Nouari

Copyright copy 2015 Ishaq Baba et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The dual response surface for simultaneously optimizing the mean and variance models as separate functions suffers somedeficiencies in handling the tradeoffs between bias and variance components of mean squared error (MSE) In this paper theaccuracy of the predicted response is given a serious attention in the determination of the optimum setting conditionsWe considerfour different objective functions for the dual response surface optimization approach The essence of the proposed method is toreduce the influence of variance of the predicted response by minimizing the variability relative to the quality characteristics ofinterest and at the same time achieving the specific target output The basic idea is to convert the constraint optimization functioninto an unconstraint problem by adding the constraint to the original objective function Numerical examples and simulationsstudy are carried out to compare performance of the proposed method with some existing procedures Numerical results show thatthe performance of the proposed method is encouraging and has exhibited clear improvement over the existing approaches

1 Introduction

Response surface methodology (RSM) is a design of experi-mental technique which shows relationship between severaldesigns and response variables The goal of the experimenteris to determine the optimal settings for the design variablesthat minimize or maximize the fitted response For moreexplanation on response surface techniques see [1ndash3] Mostof the early work in RSM is centered on a single responseproblem This methodology works effectively under theassumption of the homogeneous variance of the responseHowever such an assumption may not hold in solving real-life applications Myers and Carter [4] suggested the need fordeveloping statistical methodology known as dual responsesurfacemethodology which can simultaneously optimize themean and the variance function as to achieve the desiredtarget while keeping the variance small Generally theydefined the two responses as primary and secondary Theobjective is to find the condition 119909 on the design factors that

minimize or maximize the primary response function y119901(119909)subject to the secondary response y119904(119909) In order to achievethis three basic strategies are involved experimental designregression fitting and optimization aspect For the regressionfitting the method of the least squares is usually used toobtain the adequate response functions for the processesmean and variance by assuming that the collected data comesfrom a normal distribution While in the optimization stagethe interest is on what to optimize (ie determination of theobjective function) and how to optimize (the optimizationalgorithm) in this paper we propose a new optimizationtechnique in dual response surfacemethodology based on thepenalty functionmethod for simultaneously optimizing boththe location and scale functions The usefulness of our newlyproposed method for estimating the mean and variance ofthe optimal mean response is studied by some well knowndata sets and a simulation study The outline of this paper isorganized as follows In the next section we discussed somebasic concept of the dual response surface followed by the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 450131 6 pageshttpdxdoiorg1011552015450131

2 Mathematical Problems in Engineering

description of the proposed method in Section 3 Numericalexamples and simulation study are given in Sections 4 and 5respectively Finally the conclusion is given in Section 6

2 Dual Response Surface Review

Dual response surface technique consists of finding theoptimum setting condition of the controllable factors in orderto diminish the performance variability and deviation fromthe desired target of the decision maker This method is anextension of the standard ridge analysis procedure whichwas introduced by Myers and Carter [4] Ridge analysis hasbeen used by researchers in searching the optimum settingcondition for a single response problem [1 2 5] The dualresponse used mean and variance as separate functions forthe system under examination Then these functions areoptimized based on the chosen optimization technique todetermine the optimum operating conditions of the systemFollowing the strategy of Vining andMyers [6] themean andthe standard deviation fitted response surfaces can be writtenas

120583 = 1198870 + 1199091015840119887 + 1199091015840119861119909 (1)

where

119909 =

[

[

[

[

[

[

[

1199091

1199092

119909119896

]

]

]

]

]

]

]

119887 =

[

[

[

[

[

[

[

1205731

1205732

120573119896

]

]

]

]

]

]

]

119861 =

1

2

[

[

[

[

[

[

[

[

212057311

12057312 sdot sdot sdot

1205731119896

12057312 2


1205732119896

d

1205731119896

1205732119896 sdot sdot sdot 2

120573119896119896

]

]

]

]

]

]

]

]

(2)

120590 = 1198880 + 1199091015840119888 + 1199091015840119862119909 (3)

where

119909 =

[

[

[

[

[

[

[

1199091

1199092

119909119896

]

]

]

]

]

]

]

119888 =

[

[

[

[

[

[

[

1

2

119896

]

]

]

]

]

]

]

119862 =

1

2

[

[

[

[

[

[

[

211 12 sdot sdot sdot 1119896


d

1119896 2119896 sdot sdot sdot 2119896119896

]

]

]

]

]

]

]

(4)

where 1198870 1198880 119887 119888 119861 and 119862 are estimated vectors and matricesof coefficients obtained from the least squares method Thesimultaneous optimization of (1) and (3) using the Lagrangianmultipliers method was proposed by Vining and Myers [6]Lin and Tu [7] noted that the Vining and Myers approachdoes not always guarantee global optimum solutions due tothe restriction of the optimization to equality constraintsBased on this they proposed the minimization of meansquared error model (MSE) by introducing a slight biasin order to minimize the variability in the responses Thismethod includes twomajor parameters (the bias and the vari-ance) Copeland and Nelson [8] observed that minimizingMSE function does not specify how large the estimated mean120583 might be from the specified target value 119879 Instead theymodified VM model by placing some restriction on 120583 thatminimizes 120590 subject to (120583 minus 119879)

2le Δ2 Kim and Lin [9]

introduced a fuzzy modeling methodology using the idea ofdesirability function method Several other techniques forsolving the dual response surface problem have been pre-sented for example [10ndash12] proposed modification of meansquared errormodel Furthermore [13ndash16] presented a robustdesign for contaminated and nonnormal data using squaredloss optimization scheme a highly efficient and outlierresistant robust design estimator a dual response approachto multiple response robust design problem a multivariaterobust design using MSE and dual response modeling androbust parameter design respectively Recently a biobjectiverobust design model has been developed in [17] and a robustcutting parameter design using computer simulation hasbeen studied in [18] In many real world situations theexperimenter or the decision maker often needs to keep abalance between the process mean and the process varianceto achieve the desired target It is known that getting allefficient solutions with the class of LTmethods is challengingdue to the large resulting process variance However most ofthe preceding optimization schemes are derived from the LTmodel except the Vining andMyers model which is basicallybased on the Lagrangian multipliers approach In this paperan alternative objective function is considered based on thepenalty function method

3 Proposed Optimization Scheme forDual Response Surface

In the present study we present a new optimization techniquefor dual response surface methodology based on the penaltyfunction method The penalty function approach swaps aconstrained optimization problem by a sequence of uncon-strained optimization problems whose approximate solutionideally converges to a true solution of the original constrained

Mathematical Problems in Engineering 3

problemTheunconstrained problem is formulated by addinga penalty term to the original objective function whichconsists of the penalty parameter multiplied by a measureof violation of the constraints [19 20] Consider the generalformulation of the constrained optimization problem givenbelow

Minimize 119891 (119909)

Subject to 119892119895 (119909) le 0 119895 = 1 2 119898

ℎ119894 (119909) = 0 119894 = 1 2 119899

(5)

By applying the penalty function method we can obtain thesolution of (5) using the modified objective function

119865 (119909) = 119891 (119909) +

119898

sum

119895=1

120583 (119892119895 (119909)) +

119899

sum

119894=1

120583 (ℎ119894 (119909)) (6)

where119891(119909) is the original objective function to beminimizedand 119892119895(119909) and ℎ119894(119909) are set of inequality and equalityconstraints respectively This paper specifically considers aquadratic penalty function of the form

119865 (119909 120583) = 119891 (119909) + (

120583

2

)

119899

sum

119894=1

(ℎ119894 (119909))2 (7)

where 120583 is called the penalty constant which penalizes theequality constraints when the constraints relations are notsatisfied For the purpose of clarity we replaced 119891(119909) andℎ119894(119909) in (7) with 120590 and (120583 minus119879) and then write the followingquadratic unconstrained minimization problem as

min119865 (119909 120583) = 120590 + (

120583

2

) [120583 minus 119879]

2

(8)

where 120583 is the fitted response surface for mean 120590 is thefitted response surface for the standard deviation functionand 119879 is the target value (usually specified by experimenter)If 120583 = infin the method gives exact solution Since in this caseit is necessary that 120583 = 119879 which implies the bigger penaltyparameter 120583 thus the more exact solution is achieved For(8) we can apply any unconstrained optimization methodsuch as Newtonrsquos method BFGSmethod Conjugate gradientmethod and Steepest ascent (descent) method Moreoverany nonlinear optimization software may be used to findthe optimal design settings for the dual response surfaceproblem We used the package Rsolnp introduced in [2122] in R language which is open source statistical softwareto perform the numerical computations and analysis Ouraim is to find an optimum solution such that the estimatedmean value will be very close or equal to the target value119879 while the variance is kept small The proposed approachhas some advantages over some existing methods Firstly theproposed method takes into consideration the measure ofviolation of the constraint whereas the VM [6] and the classof LT methods are minimized without regard to the relativemagnitude of violation of constraint Secondly the penaltyparameter in (8) forced the (120583 minus 119879)

2 to be close to zero orequal to zero so as to achieve the target of the experimenter

(decisionmaker)Therefore we anticipate the optimal settingcondition obtained by the proposed method would be moreefficient in terms of the contribution of both bias and variancecomponents of of the estimated mean response comparedwith other existing methods

4 Simulation Study and Results

In this section a simulation study is conducted to assess theperformance of the newly proposed method and compareit with the commonly used methods such as VM LT andWMSE Following [13 23] the five responses (1199101198941 1199101198945) arerandomly generated from a normal distribution with mean120596120583 and standard deviation120596120590 at each control factor of settings119909119894 = (1199091198941 1199091198942 1199091198943) 119894 = 1 27 The mean 120596120583 and thestandard deviation 120596120590 are given as

120596120583 = 500 + (1199091 + 1199092 + 1199093)2+ 1199091 + 1199092 + 1199093

120596120590 = 100 + (1199091 + 1199092 + 1199093)2+ 1199091 + 1199092 + 1199093

(9)

where119879 = 500 All the fourmethods were then applied to thedataThe total of 500 1000 and 2000 iterations is consideredSome summary values such as the estimated mean of theoptimal mean response computed over 119898 iterations aredefined by 120583 = sum

119898

119894=1(120583119898) bias = 120583 minus 500 and var(120583) =

sum119898

119894=1(120583 minus 120583)

2119898 The mean squared error denoted by (MSE)

is written as MSE(120583) = (Bias)2 + var(120583) Hence the rootmean squared error (RMSE) is given by [MSE(120583)]12 Thebias standard error and root mean squared error (RMSE) ofthe estimates of the optimal mean response are exhibited inTable 1

Figures 1 and 2 show the estimated bias andmean squarederror based on the total number iteration for the variousmethods It can be observed that the bias of the VM estimateis smaller than the LT and WMSE estimates However itsRMSE is the largest among the three estimates since thevariance of the VM estimate makes up most of the MSEIt is interesting to see that our proposed method is thebest in terms of the smallest bias and RMSE values Due tospace constraint Figure 3 presents kernel density estimatesof VM LT WMSE and PM for 1000 iterations only Theplotted results indicate that the proposed approach has agood behavior in which it is very close to the desired targetTherefore one can say that the behavior of the constructedobjective function based on the penalty function techniqueis more efficient and robust than other existing methods forsolving dual response surface optimization problem

5 Numerical Examples

51 Printing Process Study Data To show a clear comparisonwe consider the data set used by Vining and Myers [6] andLin and Tu [7] which is given in Table 2 The experimentwas conducted to determine the effect of the three variables1199091 (speed) 1199092 (pressure) and 1199093 (distance) on the qualityof the printing process that is on the machines ability toapply colored inks to package labels The experiment is a 3

3


Table 1 Estimated bias standard error (SE) and RMSE of the optimal mean response

Methods 500 iterations 1000 iterations 2000 iterationsBias SE RMSE Bias SE RMSE Bias SE RMSE

VM 084 438 446 130 441 452 096 453 464LT 103 375 397 146 373 401 148 382 410WMSE 132 382 404 147 378 406 150 387 414PM 007 022 022 007 023 024 007 022 024

500 1000 1500 2000Number of iterations

Bia

s

PM

VM

LT

00

05

10

15 WMSE

Figure 1 Estimated bias for the various methods


MSE

VM

LT

PM0

5

10

15

20

25

WMSE

Figure 2 Estimated MSE for the various methods

factorial design with 3 replicates at each point Firstly theaverage and variance of the 3 responses at each design pointare computed respectively Vining and Myers [6] used the

490 495 500 505 510 515 520x

Den

sity

VMLT PM

00

05

10

15

20

25

WMSE

Figure 3 Kernel density estimates for various methods

least squaresmethod to fit a quadratic response surfacemodelfor mean and standard deviation as follows

120583 = 3276 + 17701199091 + 10941199092 + 13151199093 + 3201199092

1

minus 2241199092

2minus 291119909

2

3+ 66011990911199092 + 75511990911199093

+ 43611990921199093

120590 = 349 + 1151199091 + 1531199092 + 2921199093 + 421199092

1

minus 131199092

2minus 168119909

2

3+ 7711990911199092 + 5111990911199093

+ 14111990921199093

(10)

Based on the models in (10) Table 3 gives the summaryof the results for the four different approaches obtainedusing the cuboidal region minus1 lt 119909119894 lt 1 119894 = 1 2 3The optimal setting estimated mean response estimatedstandard deviation and RMSE are presented in Table 3 TheRMSE is calculated using the formula RMSE = [(120583 minus 119879)

2+

2

120590]12 where 119879 = 500 It can be seen that the VM approach

leads to the optimum setting (1199091 1199092 1199093) = (062 023 01)

which resulted in an estimated mean response of (50157)and root mean squared error of (5194) This optimal setting


Table 2 The printing process study data

Index 1199091 1199092 1199093 1199101199061 1199101199062 1199101199063 119910119906

119904119906

1 minus1 minus1 minus1 34 10 28 240 12492 0 minus1 minus1 115 116 130 1204 8393 1 minus1 minus1 192 186 263 2137 42804 minus1 0 minus1 82 88 88 86 3465 0 0 minus1 44 178 188 1367 80416 1 0 minus1 322 350 350 3407 16177 minus1 1 minus1 141 110 86 1123 27578 0 1 minus1 259 251 259 2563 4629 1 1 minus1 290 280 245 2717 236310 minus1 minus1 0 81 81 81 810 00011 0 minus1 0 90 122 93 1017 176712 1 minus1 0 319 376 376 3570 329113 minus1 0 0 180 180 154 1713 150114 0 0 0 372 372 372 3720 00015 1 0 0 541 568 396 5017 92516 minus1 1 0 288 192 312 2640 635017 0 1 0 432 336 513 4270 886118 1 1 0 713 725 754 7307 210819 minus1 minus1 1 364 99 199 2207 1338020 0 minus1 1 232 221 266 2397 234621 1 minus1 1 408 415 443 4220 185222 minus1 0 1 182 233 182 1990 294523 0 0 1 507 515 434 4853 446424 1 0 1 846 535 640 6737 1582025 minus1 1 1 236 126 168 1767 555126 0 1 1 660 440 403 5010 1389027 1 1 1 878 991 1161 10100 14250

Table 3 Comparison with other methods using printing processstudy data

Method Optimal settings 120583 120590 RMSEVM (062 02300 010) 50157 5192 5194LT (100 007 minus025) 49469 4446 4478WMSE (100 008 minus025) 49644 4467 4481PM (104 002 minus023) 50000 4475 4475

is very close to the target mean response but with largerRMSE The second approach LT produces a target value of(49469) and RMSE of (4478) which is smaller than theRMSE of the VM The third approach is the WMSE whichgives the estimated mean response of (49644) and RMSE of(4481) which indicate a slight increase in RMSE and littleimprovement in mean response The overall performance ofthe proposed approach is better than those three approachesmentioned with approximate mean value of (50000) andRMSE of (4475) Similar procedure is repeated in the nextexample in order to demonstrate a clear advantage of usingthe proposed method in terms of closeness to target meanresponse and the smallest RMSE

Table 4 Comparison with other methods using catapult study data

Method Optimal settings 120583 120590 RMSEVM (009 065 000) 8113 279 300LT (minus012 100 039) 7990 259 259WMSE (minus014 minus100 035) 7868 254 287PM (minus012 minus100 039) 8000 259 259

Table 5 The catapult study data

Index 1199091 1199092 1199093 1199101199061 1199101199062 1199101199063 119910119906

119904119906

1 minus1 minus1 minus1 39 42 42 383 402 minus1 minus1 1 80 91 71 807 103 minus1 1 minus1 52 45 44 47 44 minus1 minus1 1 97 60 68 75 1955 1 minus1 minus1 60 68 53 603 756 1 1 1 113 127 104 1147 1167 1 1 minus1 78 65 64 69 788 minus1682 0 1 130 75 79 94 3079 1682 0 0 59 60 51 567 4910 0 minus1682 0 115 117 102 1113 8111 0 1682 0 50 57 43 500 7012 0 0 minus1682 88 43 49 600 24413 0 0 1682 54 60 50 547 5014 0 0 0 122 119 109 1167 6815 0 0 0 87 89 78 847 5916 0 0 0 86 85 79 833 3817 0 0 0 88 87 81 853 3818 0 0 0 89 87 82 860 3619 0 0 0 86 88 79 843 4720 0 0 0 88 90 79 857 59

52 The Catapult Study Data This example will consider thedata used by Luner [24] and Kim and Lin [9]Three variables1199091 (arm length) 1199092 (stop angle) and 1199093 (pivot height) areunder consideration to predict the distance to the pointwherea projectile landed from the base of the roman style catapultThe experiment is a central composite design with threereplicates as given in Table 5 In [9] the fitted second orderpolynomial regression models for the mean and standarddeviation functions are given by

120583 = 8488 + 15291199091 + 0241199092 + 18801199093 minus 0521199092

1

minus 11801199092

2+ 039119909

2

3+ 02211990911199092 + 36011990911199093

minus 44211990921199093

120590 = 453 + 1841199091 + 421199092 + 3731199093 + 1161199092

1

+ 4401199092

2+ 094119909

2

3+ 12011990911199092 + 07311990911199093

+ 34911990921199093

(11)

Here the assumed target mean value is119879 = 80 Table 4 showsthe estimatedmean andRMSEof the estimated optimalmeanresponse It is evident fromTable 4 that the proposedmethodoutperformed the other existing procedures


6 Conclusion

Numerous procedures have been developed in the literatureto obtain an optimal setting condition for the dual responsemethodology This paper discusses four different objectivefunctions for dual response optimization approach based onthemean and variancemodels as separate response functionsThe proposed objective function is based on the penaltyfunctionmethodWe have proposed a new objective functionwhich is more efficient compared with the other existingmethods Numerical examples and simulations study are car-ried out to compare the performance of the newly proposedmethod with the frequently used methods The numericalresults clearly show an improvement of the proposedmethodover the existing methods in terms of having the smallestbias and RMSE Moreover the proposed approach can beapplied to ridge analysismethod and robust parameter designoptimization

Abbreviations

119879 Desired target for the mean response119909 Vector of control variables119910 Vector of the observed responses119904119906 Standard deviation of the observed responses120583 Penalty constantΔ Desired upper bound for the bias120583 Estimated mean of the optimal response119910119906 Average of the observed responses

120583 Fitted response surface for the mean function120590 Fitted response surface for the standard

deviation functionVM Vining and Myers [6]PM Proposed method in this paperLT Lin and Tu or can be referred to as MSE [7]WMSE Ding et al [10]BFGS Broyden-Fletcher-Goldfarb-ShannoMSE(120583) Mean squared error of the estimated mean

optimal response[MSE(120583)]12 Root Mean squared error of the estimated

mean optimal response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N R Draper ldquolsquoRidge analysisrsquo of response surfacesrdquoTechnomet-rics vol 5 no 4 pp 469ndash479 1963

[2] A I Khuri and R H Myers ldquoModified ridge analysisrdquo Techno-metrics vol 21 no 4 pp 467ndash473 1979

[3] A I Khuri and J A Cornell Response Surfaces Designs andAnalyses vol 152 CRC Press 1996

[4] R HMyers andW H Carter ldquoResponse surface techniques fordual response systemsrdquoTechnometrics vol 15 no 2 pp 301ndash3171973

[5] A E Hoerl ldquoRidge analysisrdquo in Chemical Engineering ProgressSymposium Series vol 60 pp 67ndash77 1964

[6] G G Vining and R H Myers ldquoCombining taguchi andresponse surface philosophies a dual response approachrdquoJournal of Quality Technology vol 22 no 1 1990

[7] D K J Lin and W Tu ldquoDual response surface optimizationrdquoJournal of Quality Technology vol 27 no 1 pp 34ndash39 1995

[8] KA F Copeland andP RNelson ldquoDual response optimizationvia direct function minimizationrdquo Journal of Quality Technol-ogy vol 28 no 3 pp 331ndash336 1996

[9] K-J Kim and D K J Lin ldquoDual response surface optimizationa fuzzy modeling approachrdquo Journal of Quality Technology vol30 no 1 pp 1ndash10 1998

[10] R Ding D K J Lin and D Wei ldquoDual-response surfaceoptimization a weighted MSE approachrdquo Quality Engineeringvol 16 no 3 pp 377ndash385 2004

[11] G Steenackers and P Guillaume ldquoBias-specified robust designoptimization a generalized mean squared error approachrdquoComputers amp Industrial Engineering vol 54 no 2 pp 259ndash2682008

[12] Y Ma and T Tian ldquoOptimal weighted approach in dual-response surface optimizationrdquo in Proceedings of the Interna-tional Conference on Management and Service Science (MASSrsquo09) pp 1ndash4 Wuhan China September 2009

[13] C Park and B R Cho ldquoDevelopment of robust design undercontaminated and non-normal datardquo Quality Engineering vol15 no 3 pp 463ndash469 2003

[14] G M Quesada and E Del Castillo ldquoA dual-response approachto the multivariate robust parameter design problemrdquo Techno-metrics vol 46 no 2 pp 176ndash187 2004

[15] O Koksoy ldquoMultiresponse robust design mean square error(MSE) criterionrdquo Applied Mathematics and Computation vol175 no 2 pp 1716ndash1729 2006

[16] A B Shaibu and B R Cho ldquoAnother view of dual responsesurfacemodeling and optimization in robust parameter designrdquoThe International Journal of Advanced Manufacturing Technol-ogy vol 41 no 7-8 pp 631ndash641 2009

[17] S Shin F Samanlioglu B R Cho and M M Wiecek ldquoCom-puting trade-offs in robust design perspectives of the meansquared errorrdquo Computers amp Industrial Engineering vol 60 no2 pp 248ndash255 2011

[18] A Jeang ldquoRobust cutting parameters optimization for pro-duction time via computer experimentrdquo Applied MathematicalModelling vol 35 no 3 pp 1354ndash1362 2011

[19] S DongMethods for Constrained Optimization MassachusettsInstitute of Technology Cambridge Mass USA 2006

[20] D K Shin Z Gurdal and O H Griffin Jr ldquoA penalty approachfor nonlinear optimization with discrete design variablesrdquoEngineering Optimization vol 16 no 1 pp 29ndash42 1990

[21] Y Ye Solnp Usersrsquo Guide University of Iowa 1989[22] A Ghalanos S Theussl and M A Ghalanos General non-

linear optimization (package rsolnp) pp 1ndash15 2012[23] S B Lee C Park and B-R Cho ldquoDevelopment of a highly

efficient and resistant robust designrdquo International Journal ofProduction Research vol 45 no 1 pp 157ndash167 2007

[24] J J Luner ldquoAchieving continuous improvement with the dualapproach a demonstration of the roman catapultrdquo QualityEngineering vol 6 no 4 pp 691ndash705 1994

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description of the proposed method in Section 3 Numericalexamples and simulation study are given in Sections 4 and 5respectively Finally the conclusion is given in Section 6

2 Dual Response Surface Review

Dual response surface technique consists of finding theoptimum setting condition of the controllable factors in orderto diminish the performance variability and deviation fromthe desired target of the decision maker This method is anextension of the standard ridge analysis procedure whichwas introduced by Myers and Carter [4] Ridge analysis hasbeen used by researchers in searching the optimum settingcondition for a single response problem [1 2 5] The dualresponse used mean and variance as separate functions forthe system under examination Then these functions areoptimized based on the chosen optimization technique todetermine the optimum operating conditions of the systemFollowing the strategy of Vining andMyers [6] themean andthe standard deviation fitted response surfaces can be writtenas

120583 = 1198870 + 1199091015840119887 + 1199091015840119861119909 (1)

where

119909 =

[

[

[

[

[

[

[

1199091

1199092

119909119896

]

]

]

]

]

]

]

119887 =

[

[

[

[

[

[

[

1205731

1205732

120573119896

]

]

]

]

]

]

]

119861 =

1

2

[

[

[

[

[

[

[

[

212057311


1205731119896

12057312 2


1205732119896

d

1205731119896

1205732119896 sdot sdot sdot 2

120573119896119896

]

]

]

]

]

]

]

]

(2)

120590 = 1198880 + 1199091015840119888 + 1199091015840119862119909 (3)

where

119909 =

[

[

[

[

[

[

[

1199091

1199092

119909119896

]

]

]

]

]

]

]

119888 =

[

[

[

[

[

[

[

1

2

119896

]

]

]

]

]

]

]

119862 =

1

2

[

[

[

[

[

[

[



d

1119896 2119896 sdot sdot sdot 2119896119896

]

]

]

]

]

]

]

(4)

where 1198870 1198880 119887 119888 119861 and 119862 are estimated vectors and matricesof coefficients obtained from the least squares method Thesimultaneous optimization of (1) and (3) using the Lagrangianmultipliers method was proposed by Vining and Myers [6]Lin and Tu [7] noted that the Vining and Myers approachdoes not always guarantee global optimum solutions due tothe restriction of the optimization to equality constraintsBased on this they proposed the minimization of meansquared error model (MSE) by introducing a slight biasin order to minimize the variability in the responses Thismethod includes twomajor parameters (the bias and the vari-ance) Copeland and Nelson [8] observed that minimizingMSE function does not specify how large the estimated mean120583 might be from the specified target value 119879 Instead theymodified VM model by placing some restriction on 120583 thatminimizes 120590 subject to (120583 minus 119879)

2le Δ2 Kim and Lin [9]

introduced a fuzzy modeling methodology using the idea ofdesirability function method Several other techniques forsolving the dual response surface problem have been pre-sented for example [10ndash12] proposed modification of meansquared errormodel Furthermore [13ndash16] presented a robustdesign for contaminated and nonnormal data using squaredloss optimization scheme a highly efficient and outlierresistant robust design estimator a dual response approachto multiple response robust design problem a multivariaterobust design using MSE and dual response modeling androbust parameter design respectively Recently a biobjectiverobust design model has been developed in [17] and a robustcutting parameter design using computer simulation hasbeen studied in [18] In many real world situations theexperimenter or the decision maker often needs to keep abalance between the process mean and the process varianceto achieve the desired target It is known that getting allefficient solutions with the class of LTmethods is challengingdue to the large resulting process variance However most ofthe preceding optimization schemes are derived from the LTmodel except the Vining andMyers model which is basicallybased on the Lagrangian multipliers approach In this paperan alternative objective function is considered based on thepenalty function method

3 Proposed Optimization Scheme forDual Response Surface

In the present study we present a new optimization techniquefor dual response surface methodology based on the penaltyfunction method The penalty function approach swaps aconstrained optimization problem by a sequence of uncon-strained optimization problems whose approximate solutionideally converges to a true solution of the original constrained



Minimize 119891 (119909)

Subject to 119892119895 (119909) le 0 119895 = 1 2 119898

ℎ119894 (119909) = 0 119894 = 1 2 119899

(5)


119865 (119909) = 119891 (119909) +

119898

sum

119895=1

120583 (119892119895 (119909)) +

119899

sum

119894=1

120583 (ℎ119894 (119909)) (6)


119865 (119909 120583) = 119891 (119909) + (

120583

2

)

119899

sum

119894=1

(ℎ119894 (119909))2 (7)


min119865 (119909 120583) = 120590 + (

120583

2

) [120583 minus 119879]

2

(8)






120596120583 = 500 + (1199091 + 1199092 + 1199093)2+ 1199091 + 1199092 + 1199093

120596120590 = 100 + (1199091 + 1199092 + 1199093)2+ 1199091 + 1199092 + 1199093

(9)


119898


sum119898

119894=1(120583 minus 120583)






3




VM 084 438 446 130 441 452 096 453 464LT 103 375 397 146 373 401 148 382 410WMSE 132 382 404 147 378 406 150 387 414PM 007 022 022 007 023 024 007 022 024


Bia

s

PM

VM

LT

00

05

10

15 WMSE



MSE

VM

LT

PM0

5

10

15

20

25

WMSE



490 495 500 505 510 515 520x

Den

sity

VMLT PM

00

05

10

15

20

25

WMSE



120583 = 3276 + 17701199091 + 10941199092 + 13151199093 + 3201199092

1

minus 2241199092

2minus 291119909

2

3+ 66011990911199092 + 75511990911199093

+ 43611990921199093

120590 = 349 + 1151199091 + 1531199092 + 2921199093 + 421199092

1

minus 131199092

2minus 168119909

2

3+ 7711990911199092 + 5111990911199093

+ 14111990921199093

(10)


2+

2






Index 1199091 1199092 1199093 1199101199061 1199101199062 1199101199063 119910119906

119904119906








Index 1199091 1199092 1199093 1199101199061 1199101199062 1199101199063 119910119906

119904119906



120583 = 8488 + 15291199091 + 0241199092 + 18801199093 minus 0521199092

1

minus 11801199092

2+ 039119909

2

3+ 02211990911199092 + 36011990911199093

minus 44211990921199093

120590 = 453 + 1841199091 + 421199092 + 3731199093 + 1161199092

1

+ 4401199092

2+ 094119909

2

3+ 12011990911199092 + 07311990911199093

+ 34911990921199093

(11)



6 Conclusion


Abbreviations








References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Minimize 119891 (119909)

Subject to 119892119895 (119909) le 0 119895 = 1 2 119898

ℎ119894 (119909) = 0 119894 = 1 2 119899

(5)


119865 (119909) = 119891 (119909) +

119898

sum

119895=1

120583 (119892119895 (119909)) +

119899

sum

119894=1

120583 (ℎ119894 (119909)) (6)


119865 (119909 120583) = 119891 (119909) + (

120583

2

)

119899

sum

119894=1

(ℎ119894 (119909))2 (7)


min119865 (119909 120583) = 120590 + (

120583

2

) [120583 minus 119879]

2

(8)






120596120583 = 500 + (1199091 + 1199092 + 1199093)2+ 1199091 + 1199092 + 1199093

120596120590 = 100 + (1199091 + 1199092 + 1199093)2+ 1199091 + 1199092 + 1199093

(9)


119898


sum119898

119894=1(120583 minus 120583)






3




VM 084 438 446 130 441 452 096 453 464LT 103 375 397 146 373 401 148 382 410WMSE 132 382 404 147 378 406 150 387 414PM 007 022 022 007 023 024 007 022 024


Bia

s

PM

VM

LT

00

05

10

15 WMSE



MSE

VM

LT

PM0

5

10

15

20

25

WMSE



490 495 500 505 510 515 520x

Den

sity

VMLT PM

00

05

10

15

20

25

WMSE



120583 = 3276 + 17701199091 + 10941199092 + 13151199093 + 3201199092

1

minus 2241199092

2minus 291119909

2

3+ 66011990911199092 + 75511990911199093

+ 43611990921199093

120590 = 349 + 1151199091 + 1531199092 + 2921199093 + 421199092

1

minus 131199092

2minus 168119909

2

3+ 7711990911199092 + 5111990911199093

+ 14111990921199093

(10)


2+

2






Index 1199091 1199092 1199093 1199101199061 1199101199062 1199101199063 119910119906

119904119906








Index 1199091 1199092 1199093 1199101199061 1199101199062 1199101199063 119910119906

119904119906



120583 = 8488 + 15291199091 + 0241199092 + 18801199093 minus 0521199092

1

minus 11801199092

2+ 039119909

2

3+ 02211990911199092 + 36011990911199093

minus 44211990921199093

120590 = 453 + 1841199091 + 421199092 + 3731199093 + 1161199092

1

+ 4401199092

2+ 094119909

2

3+ 12011990911199092 + 07311990911199093

+ 34911990921199093

(11)



6 Conclusion


Abbreviations








References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












VM 084 438 446 130 441 452 096 453 464LT 103 375 397 146 373 401 148 382 410WMSE 132 382 404 147 378 406 150 387 414PM 007 022 022 007 023 024 007 022 024


Bia

s

PM

VM

LT

00

05

10

15 WMSE



MSE

VM

LT

PM0

5

10

15

20

25

WMSE



490 495 500 505 510 515 520x

Den

sity

VMLT PM

00

05

10

15

20

25

WMSE



120583 = 3276 + 17701199091 + 10941199092 + 13151199093 + 3201199092

1

minus 2241199092

2minus 291119909

2

3+ 66011990911199092 + 75511990911199093

+ 43611990921199093

120590 = 349 + 1151199091 + 1531199092 + 2921199093 + 421199092

1

minus 131199092

2minus 168119909

2

3+ 7711990911199092 + 5111990911199093

+ 14111990921199093

(10)


2+

2






Index 1199091 1199092 1199093 1199101199061 1199101199062 1199101199063 119910119906

119904119906








Index 1199091 1199092 1199093 1199101199061 1199101199062 1199101199063 119910119906

119904119906



120583 = 8488 + 15291199091 + 0241199092 + 18801199093 minus 0521199092

1

minus 11801199092

2+ 039119909

2

3+ 02211990911199092 + 36011990911199093

minus 44211990921199093

120590 = 453 + 1841199091 + 421199092 + 3731199093 + 1161199092

1

+ 4401199092

2+ 094119909

2

3+ 12011990911199092 + 07311990911199093

+ 34911990921199093

(11)



6 Conclusion


Abbreviations








References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Index 1199091 1199092 1199093 1199101199061 1199101199062 1199101199063 119910119906

119904119906








Index 1199091 1199092 1199093 1199101199061 1199101199062 1199101199063 119910119906

119904119906



120583 = 8488 + 15291199091 + 0241199092 + 18801199093 minus 0521199092

1

minus 11801199092

2+ 039119909

2

3+ 02211990911199092 + 36011990911199093

minus 44211990921199093

120590 = 453 + 1841199091 + 421199092 + 3731199093 + 1161199092

1

+ 4401199092

2+ 094119909

2

3+ 12011990911199092 + 07311990911199093

+ 34911990921199093

(11)



6 Conclusion


Abbreviations








References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6 Conclusion


Abbreviations








References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article An Alternative Approach of Dual Response...

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