Optimization of probabilistic multiple response surfaces

Optimization of probabilistic multiple response surfaces

Taha Hossein Hejazi

^a

, Mahdi Bashiri

^a,⇑

, José A. Dı´az-Garcı´a

^b

, Kazem Noghondarian

^c

aShahed University, Department of Industrial Engineering, P.O. Box 18155/159, Tehran, Iran

bUniversidad Autónoma Agraria Antonio Narro, Department of Statistics and Computation, 25315 Buenavista, Saltillo, Coahuila, Mexico

cIran University of Science and Technology, Faculty of Industrial Engineering, Tehran, Iran

a r t i c l e i n f o

Article history:

Received 9 April 2010

Received in revised form 16 July 2011 Accepted 25 July 2011

Available online xxxx Keywords:

Multiple response surfaces (MRS) Robust design

Probabilistic optimization Group Decision Making Goal Programming (GP)

a b s t r a c t

Response surface methodology (RSM) is a statistical–mathematical method used for ana- lyzing and optimizing the experiments. In analysis process, experts usually face several input variables having effect on several outputs called response variables. Simultaneous optimization of the correlated response variables has become more important in complex systems. In this paper multi-response surfaces and their related stochastic nature have been modeled and optimized by Goal Programming (GP) in which the weights of response variables have been obtained through a Group Decision Making (GDM) process. Because of existing uncertainty in the stochastic model, some stochastic optimization methods have been applied to find robust optimum results. At the end, the proposed method is described numerically and analytically.

Ó2011 Published by Elsevier Inc.

1. Introduction

Response surface methodology (RSM) is useful in improving and optimizing process by finding the analytical relationship between input and output variables considered in experiments. RSM also has the ability to produce an approximate function using a smaller amount of data and fewer numbers of experiments runs Yeh[1]. However, most previous applications based on RSM have only dealt with a single-response problem and multi-response problems have received only limited attention.

Studies have shown that the optimal factor settings for one performance characteristic are not necessarily compatible with those of other performance characteristics. In more general situations, finding compromising conditions on the input vari- ables might be considered that are somewhat favorable to all responses Koksoy[2]. More details on RSM, related designs and optimization of response surfaces are given in Kleijnen[3], Myers and Montgomery[4]and Kleijnen[5]. This paper is orga- nized as following sections. In Sections1.1 and 1.2some major works on Multiple Response Optimization (MRO) and uncer- tainty in multi-response problems are reviewed respectively. The proposed methods and its related definitions are represented in Section2. In order to show the efficiencies and computational steps of the proposed methods, a real case from the literature is analyzed in Section3and the results are compared with some previous works.

1.1. Multi-response optimization

MRO problems have been studied in several areas from different aspects and could be categorized into three general categories:

0307-904X/$ - see front matterÓ2011 Published by Elsevier Inc.

doi:10.1016/j.apm.2011.07.067

⇑Corresponding author.

E-mail addresses:t.h.hejazi@aut.ac.ir,hejazi@shahed.ac.ir(T.H. Hejazi),bashiri@shahed.ac.ir(M. Bashiri),jadiaz@uaaan.mx,jadiazg@ugr.es(J.A. Dı´az- Garcı´a),noghondarian@iust.ac.ir(K. Noghondarian).

Contents lists available atSciVerse ScienceDirect

Applied Mathematical Modelling

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a p m

(a) Desirability viewpoints: in this category, researchers try to aggregate information of all responses into one response and then an optimization method is performed on a single objective called total desirability function.

(b) Priority based methods: some cases have responses with different importance degrees, in such problems, we must consider most important response for optimization and if solutions were not unique, then find the best solution by comparing the status of other responses for alternative solutions and the aforesaid steps are repeated till all the responses are considered or a unique optimal solution is found.

(c) Loss function: in this category, based on Taguchi loss function, all responses values are aggregated and converted to a single one. Wide range of researches, have been studied to develop and generalize Taguchi loss function with respect to special trait of its cases.

Some earlier works in multiresponse optimization are: Derringer and Suich[6]transformed each response function of a desirability function and then maximize the geometric mean of individual desirability functions for a compromise solution.

Layne[7]presented a procedure that simultaneously considers three functions – the weighted loss function, the desirability function, and a distance function – to determine the optimum parameter combination. Pignatiello[8]utilized a variance com- ponent and a squared deviation-from-target to form an expected loss function to optimize a multiple response problem. This method is difficult to implement. The first reason is that a cost matrix must be initially obtained, and the second reason is that it needs more experimental data. He et al.[9]proposed desirability based approach considering robustness which made balance between robustness and optimization for multiple response problems. Simplex search method is used to search for the most robust optimal point in the feasible operating region. Saurav et al.[10]applied a combination of Taguchi method and Principal Component Analysis (PCA) to optimize a real case in the field of submerged arc welding. They also compared the proposed ap- proach with Gray–Taguchi method. Chang et al.[11]generalized Taguchi function to more general situations. The generalized model was represented by using a weighted convex loss function. Mak and Nebebe[12]considered parameter design in off-line quality control and the proposed approach tried to find production factors that minimized expected loss function. The expected loss function was minimized based on a distributional free procedure using the empirical distribution of the standardized resid- uals. Bashiri and Hejazi[13]used Multiple Attribute Decision Making (MADM) methods such as VIKOR, PROMETHEE II, ELECTRE III and TOPSIS in converting multi-response to single response to analyze the robust experimental design. The main advantage of their method was the consideration of standard deviation that contributed to robust experimental design and also because of fitting only one response regression function; the proposed method decreased statistical error. Alvarez et al.[14]applied RSM to design and optimization of a capacitive accelerometer. A faced-centered cube design was used in the experimentation and the data were obtained conducting computer simulations using finite element design techniques. Hsieh[15]used neural networks to estimate relation between control variables and responses. Tong et al.[16]used VIKOR methods in converting Taguchi cri- teria to single response and then found regression model and related optimal setting. Tong et al.[17]also considered correlation of responses and used PCA and TOPSIS methods to find the best variable setting. Kazemzadeh et al.[18]proposed a general framework for multiresponse optimization problems based on Goal Programming and studied some existing works and at- tempted to aggregate all characteristics into one approach. Shah et al.[19]Illustrated and Seemingly unrelated regressions (SUR) method for estimating the regression parameters that it can be useful when response variables in MRS problems are cor- related and can lead to a more precise estimate of the optimum variable setting.

1.2. Uncertainty in MRO problems

There are two main approaches for characterizing uncertainty in design of experiments including fuzzy theory, and prob- abilistic analysis. Fuzzy theory facilitates uncertainty analysis of systems where uncertainty is due to vagueness or fuzziness rather than randomness alone and can be applied to uncertainty analysis with imprecise observation or with verbal expres- sions. Probabilistic analysis is the most widely used method for characterizing uncertainty in real systems and can describe uncertainty arising from stochastic disturbances, variability and has capability for risk considerations, especially when esti- mates of input variables probability distributions are available.

Nan[20]combined some of the well-established methods in multiresponse optimization such as genetic algorithm, Artificial Neural Network (ANN), Taguchi loss function and desirability function to optimize stencil printing operations. In the proposed approach, fuzzy quality loss function was applied for analyzing qualitative measures. Chiao and Hamada[21]considered exper- iments with correlated multiple responses whose means, variances, and correlations depend on experimental factors. Analysis of these experiments consists of modeling distributional parameters in terms of the experimental factors and finding factor set- tings which maximize the probability of being in a specification region, i.e., all responses are simultaneously meeting their respective specifications. Amiri et al.[22]used genetic algorithm to find best solution of multiresponse problem in fuzzy envi- ronment. Their proposed method was a combination of simulation approach, fuzzy Goal Programming and genetic algorithm.

Khuri and Conlon[23]and Khuri and Cornell[24]considered multivariate RSM and the stochastic character when optimizing several response surfaces simultaneously, in which case they applied the minimax technique to analyze the stochastic aspect and the distance based method was applied to solve the multiple objective problem. Dı´az-Garcı´a et al.[25]studied application of some stochastic programming methods in response surface optimization. They assumed normal distribution for coefficients of response surface and compared the results of some methods such as E-Model, V-Model, and Lexicography. Hejazi et al.[26]

represented a novel method based on goal programming to find the best combination of factors and stochastic covariates to optimize multiresponse-multicovariate surfaces with consideration of location and dispersion effects. The method also

considered covariate probable values as an objective function which should be maximized.Table 1shows a summary of MRO studies that considered uncertain and probabilistic aspects.

2. Proposed approach

In this paper, multi-response surfaces with probabilistic coefficients and responses weights have been modeled and opti- mized so as to ensure robustness in results. For this purpose, several steps are considered through the integrated approach.

The proposed approach has the following steps:

(a) Identifying the experimental variables such as responses and factors.

(b) Applying a proper design, running experiments and fitting the response surfaces.

(c) Getting information about the importance weights of response variables from an expert team through a GDM process.

(d) Forming a multiresponse Goal Programming model subjected to the technical limitations and robustness consideration.

(e) Solving the model using some stochastic programming methods.

Comparisons of the proposed model with other techniques and related studies are summarized inTable 2.

Table 2shows that the proposed model has some capabilities to analyze multi-response model in which most of the ma- jor work’s characteristics are considered simultaneously. For instance, multiple response surfaces according to the proposed model considered the assumptions mentioned in Dı´az-Garcı´a et al.[25]in addition to some modifications listed below:

Multiple response variables are analyzed and optimized simultaneously.

Correlations between response variables are considered.

Correlations between response regression coefficientsbjare considered.

Group Decision Making approach is applied in identifying the importance of response variables.

2.1. Model description

In this section the proposed approach will be explained by using mathematical formulations and related assumptions will be described. The aim of this section is to develop stochastic GP model in terms of controllable variables.

Table 1

Summary of some major works on MRO with uncertainty.

Method Type of

uncertainty

Approach/method Non-deterministic

parameters

Layne and Chang[27] Fuzzy Fuzzy regression possibility distribution Response variables

Ip et al.[28] Fuzzy Fuzzy regression Response variables

Lin and Lin[29] Fuzzy Fuzzy logic analysis of variance Response variables

Dı´az-Garcı´a et al.

[25]

Probabilistic Stochastic programming deterministic equivalent model Response variables Chiao and Hamada

[21]

Probabilistic Maximization of probability with consideration of correlation Response variables Kazemzadeh et al.

[18]

Probabilistic Maximization of probability with consideration of correlation Goal Programming

Response variables Hejazi et al.[26] Probabilistic Achieving most probable value of covariates in Goal Programming Covariates

Table 2

Comparisons of the proposed approach with other existing methods.

Method Type model Robustness Correlation Stochastic approach GDM

Dı´az-Garcı´a et al.[25] Single response ⁄ ⁄

Chiao and Hamada[21] Multiple responses ⁄ ⁄

Kazemzadeh et al.[18] Multiple responses ⁄ ⁄

Shah et al.[19] Multiple responses ⁄

Ko et al.[30] Multiple responses ⁄

Fung and Kang[31] Multiple responses ⁄ ⁄

Kovach and Cho[32] Multiple responses ⁄

Sadjadi et al.[33] Multiple responses ⁄ ⁄

Hejazi et al.[26] Multiple responses ⁄ ⁄

Proposed method Multiple responses ⁄ ⁄ ⁄ ⁄

2.1.1. Notation

LetNbe the number of experimental runs andpbe the number of response variables which can be measured for each setting of a group ofncoded variablesx₁,x₂,. . .,x_n. We assume that the response variables can be modeled by second order polynomials regression model in terms ofx^s_j. Hence, thekth response model can be written as

Rk¼Xbkþ

e

k; ð1Þ

whereRkis anN1 vector of observations on thekth response,Xis anNqmatrix of rankqtermed design or regression matrix,q= 1 +n+n(n+ 1)/2,bkis aq1 vector of unknown constant parameters, and

e

kis a random error vector associated with thekth response. Note that(1)can be represented as

R¼XBþE; ð2Þ

whereR¼ ½R1... R2...

...

RpŠ; B¼ ½b1... b2...

...

bpŠandE¼ ½

e

1...

e

2... ...

e

pŠ, such thatE NNpð0;INRÞi.e.,Ehas anNpmatrix variate normal distribution with EðEÞ ¼0 and CovðvecE⁰Þ ¼INR, where R ^{is pp} positive definite matrix, vecT¼T⁰₁;T⁰₂;. . .;T⁰_p0

withT= [T₁... T2...

...

Tp] andis a symbol for the direct (or Kronecker) product of matrices, see Muir- head[34, Theorem 3.2.2, p. 79]. Also

x= (x1,x2,. . .,xn)⁰: Vector of controllable variables.

Bb¼ ½^b1...^b2...

...^bpŠ: The least squares estimator ofBgiven byBb¼ ðX⁰XÞ ¹X⁰R, form whereb^k¼ ðX⁰XÞ ¹X⁰Rk; k¼1;2;. . .;p.

Moreover, under the assumption thatE NNpð0;INRÞ, thenBb NqpðB;ðX⁰XÞ ¹RÞ, with CovðvecbB⁰Þ ¼ ðX⁰XÞ ¹R. zðxÞ ¼ 1;x1;x2;. . .;xn;x²₁;x²₂;. . .;x²_n;x1x2;x1x3;. . .;xn 1xn

0

.

bR_kðxÞ ¼z⁰ðxÞb^k: Response surface or predictor equation at the pointxforkth response variable.

RðxÞ ¼ ðbb R1ðxÞ;bR2ðxÞ;. . .;bRpðxÞÞ⁰¼Bb⁰zðxÞ: Response surface or predicted response vector at the pointx.

w= (w1,w2,. . .,wp)⁰: Vector of response weights.

Rw: Variance–covariance matrix of weights.

bR¼^{R0ðIN XðX}_{N q}^0XÞ1X0ÞR: Estimator of variance–covariance matrixR^{such thatðN qÞb}Rhas a Wishart distribution with (N q) freedom degrees and parameterR; denoted this fact asðN qÞRb WpðN q;RÞ. Here,Imdenotes a identity matrix of orderm.

l= (l1,l2,. . .,ln)⁰: Vector of lower bounds of factors.

u= (u1,u2,. . .,un)⁰: Vector of upper bounds of factors.

t= (t1,t₂,. . .,t_p)⁰: Vector of target values for response vector.

The MRO problem in general is proposed as

minx RbðxÞ ¼min

x

bR1ðxÞ bR2ðxÞ

... bRpðxÞ 0 BB BB B@

1 CC CC CA

subject to l<x<u;

ð3Þ

which is a nonlinear problem of the multi-objective optimization, see Steuer[35], Rı´os et al.[36]and Miettinen[37]; and wherel<x<udenotesli<xi<ui,i= 1, 2,. . .,n.

In the RMS context, observe that in multi-objective optimization problems, there rarely exists a pointx^⁄which is consid- ered as an optimum, i.e., few cases satisfy the requirement thatbRkðxÞis minimum for allk= 1, 2,. . .,p. From a point view of multi-objective optimization, this justifies the following notion of the Pareto point, which is more weakly defined than an optimum point:

We say that bRðxÞ is a Pareto point of RðxÞ, if there is not other pointb Rb¹ðxÞ such that bR¹ðxÞ6RbðxÞ, i.e., for all k; bR¹_kðxÞ6Rb_kðxÞandRb¹ðxÞ–RbðxÞ.

Existence criteria for Pareto points of a multi-objective optimization problem are established in Ríos et al.[36]and Miet- tinen[37]. In particular we have:

GivenRðxÞb :Rⁿ!R^pand let us consider a nonempty compactN Nⁿsuch thatN is the set of all possible values ofxdeter- mined by the restrictions in(3).IfbRkðxÞis an upper semi-continuous for each k= 1,. . .,p,then the problem(3)has a Pareto optimal solution.

On the other hand, Steuer[35], Rı´os et al.[36]and Miettinen[37]studied the extension of scalar optimization (Kuhn–

Tucker’s conditions) to the vectorial case. In particular, they proposed necessary conditions for Pareto solutions, which be- come sufficient conditions if:N is convex; the functionsRbkðxÞ; k¼1;. . .;pare convex; and the Lagrange generalized mul- tipliersdk, associated with each functionbRkðxÞ, are positive,dk> 0 for allk.

Methods for solving a multi-objective optimization problem are based on the information possessed about a particular problem. There are three possible scenarios; when the investigator possesses: complete, partial or null information, see Rı´os et al.[36], Miettinen[37]and Steuer[35]. In a RMS context, complete information means that, the investigator knows the pop- ulation in such a way that it is possible to propose avalue functionreflecting the importance of each response variables, where

Value function is a function/:Rⁿ!Rsuch thatminRðxb Þ<minbRðx1Þ () /ðbRðxÞÞ</ðbRðx1ÞÞ; x–x1.

In partial information, the investigator knows the main response variable of the study very well and this is sufficient sup- port for the research. Finally, under null information, the researcher only possesses information about the estimators of the parameter of response surfaces, and with this material an appropriate solution can be found.

Then, observe that the three general categories described in Section1.1are particular cases of the method of value func- tion. In particular we have the following alternative approach in terms of GP model of the multi-objective problem(3)

minxFðxÞ ¼minx

X^p

k¼1

wkdkþd⁰_k subject to

bRkðxÞ þdk d⁰_k¼tk; k¼1;2;. . .;p;

l<x<u;

ð4Þ

where dk¼1

2 bRkðxÞ tk

þ bRkðxÞ tk

; d⁰_k¼1

2 bRkðxÞ tk

bRkðxÞ tk

;

Consider multi-response model in form of second order polynomial function bRkðxÞ ¼bb0kþXⁿ

i¼1

bbikxiþXⁿ

i¼1

^biikx²_i þXⁿ

i¼1

Xⁿ

j>i

b^ijkxixj¼z⁰ðxÞ^bk 8k¼1;2;. . .;p: ð5Þ

Hence, see Khuri and Cornell[24, p. 289]

EðRbðxÞÞ ¼EðBb⁰zðxÞÞ ¼B⁰zðxÞ ð6Þ

and

CovðRðxÞÞ ¼b z⁰ðxÞðX⁰XÞ ¹zðxÞR: ð7Þ

An unbiased estimator of VarðRðxÞÞb is given by d

CovðbRðxÞÞ ¼z⁰ðxÞðX⁰XÞ ¹zðxÞbR: ð8Þ

Finally, observing that VarðRb_kðxÞ þd_k d⁰_kÞ ¼0 and taking into account thatBandRbis a random matrix and assuming thatw is an independent random vector, we have the following stochastic optimization problem version of(4)

minx FðxÞ ¼min

x

P^p

k¼1

wkdkþd⁰_k subject to

bRkðxÞ þdk d⁰_k¼tk; k¼1;2;. . .;p;

l<x<u

w2Xw; EðwÞand CovðwÞ ¼Rwand their estimators are known;

ðB;RÞ 2X_B;R ^{Bb NqpðB;ðX}

0XÞ ¹RÞ; and;

ðN qÞbR WpðN q;RÞ;

(

ð9Þ

whereXwandXB;Rare definition regions of response variables.

In addition, for incorporating weights in the model and developing the GP form (Eq.(4)), some other notations about mul- tiplication of probabilistic variables have to be considered. Following relations show expected value and variance of goal function.

Lemma 1. Assume that Y and X are random variables(not necessarily independent). If Z₁and Z₂are defined as Z₁=X+Y and Z2=XY.Then

EðZ1Þ ¼EðXÞ þEðYÞ; ð10Þ

VarðZ1Þ ¼VarðXÞ þVarðYÞ þ2CovðX;YÞ; ð11Þ

EðZ2Þ ¼EðXÞEðYÞ þCovðX;YÞ; ð12Þ

VarðZ2Þ ¼E²ðYÞVarðXÞ þE²ðXÞVarðYÞ þ2EðXÞEðYÞCovðX;YÞ Cov²ðX;YÞ þE½ðX EðXÞÞ²ðY EðYÞÞ²Š

þ2EðYÞE½ðX EðXÞÞ²ðY EðYÞފ þ2EðXÞE½ðX EðXÞÞðY EðYÞÞ²Š: ð13Þ According to Eqs.(10)–(13)written based on Taylor series in several variables[38], the general GP model(9)could be con- verted into deterministic model by use of some stochastic programming techniques termed deterministic equivalent model, Dı´az-Garcı´a et al.[25]. Furthermore, it is reasonable to assume that the weights and responses regression coefficients are independent so equivalent problem for GP model could be developed in the next step.

2.2. Deterministic equivalent problem

In context of stochastic programming there are some deterministic equivalents for stochastic optimization models such as E-model, modified V-model and Lexicographic method, see Charnes and Cooper[39]. A brief description of these methods is represented below (for more details see Dı´az-Garcı´a et al.[25]).

Also it is reasonable to assume that the weights and responses regression coefficients are independent then equivalent problem for GP model could be developed as follow.

2.2.1. Modified E-model

In this method a linear combination of expected value and standard deviation of objective function is used to model and analyze the stochastic nature of(9), see Prekopa[40].

minxr1bEðFðxÞÞ þr2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VarðFðxÞÞd q

subject to

RbkðxÞ þdk d⁰_k¼tk; k¼1;2;. . .;p;

l<x<u;

ð14Þ

wherer1andr2are non-negative constants representing importance of the expectation and variance, which in general are such thatr1+r2= 1; we are denotedbEðFðxÞÞ EðFðxÞÞd andVarðFðxÞÞ d VarðFðxÞÞd and

bEðFðxÞÞ ¼X^p

k¼1

bEðwkÞbE d kþd⁰_k

þCovwk;dkþd⁰_k

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼0

; ð15Þ

VarðFðxÞÞ ¼d X^p

k¼1

Vardwkdkþd⁰_k þ2X^p

i¼1

X^p

j¼iþ1

Covd widiþd⁰_i

;wjdjþd⁰_j

h i

: ð16Þ

Note that the type ofF(x) in this E-model is Smaller-The-Better (STB).

2.2.2. Modified V-model

Under this technique the deterministic equivalent model is minx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VarðFðxÞÞd q

subject to

RbkðxÞ þdk d⁰_k¼tk; k¼1;2;. . .;p;

bEðFðxÞÞ 2 fdesired regiong l<x<u:

ð17Þ

By Dı´az-Garcı´a et al.[25]we have that the solutions given by the modified E-model and modified V-model techniques are also solution of the following deterministic multi-objective problem

minx

bEðFðxÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VarðFðxÞÞd q

2 4

3 5

subject to

RbkðxÞ þdk d⁰_k¼tk; k¼1;2;. . .;p;

l<x<u;

ð18Þ

These equivalences allow us the use of other multi-objective optimization techniques for solved(18)and propose these as stochastic solutions of(9).

In the next section, a numerical example is analyzed by some aforesaid methods and results are compared with other previous works.

3. Numerical example

In this section a real case from the literature is analyzed by the proposed procedure steeply. The case is analyzed step by step in order to illustrate the proposed method more intelligible.

Step a: In this example there are two response variables R= (R₁,R₂)⁰ with 4 replicates and three setting variables x= (x1,x2,x3)⁰. The experimental data is shown inTable 3. In this example it is assumed that the targets of the responses are 103 and 73 and that the specification regions are (97, 109), (70, 76) forR1,R2, respectively, see Pignati- ello[8].

Step b: Next, Eqs.(19) and (20)are response surfaces forR1andR2.

bR1ðxÞ ¼104:86 3:147x1 0:142x2 0:199x3þ2:379x1x2 0:35x1x3 0:106x2x3; ð19Þ

bR2ðxÞ ¼70:45 0:348x1þ3:59x2þ0:28x3þ0:323x1x2 0:45x1x3þ0:614x2x3: ð20Þ From where the multi-response optimization problem is given as

minx RðxÞ ¼b min

x

bR1ðxÞ bR2ðxÞ

!

subject to

1<x1;x2;x3<1;

ð21Þ

Step c: In this section, the importance of each response variable must be assessed from the decision makers’ viewpoint.

Table 4shows the hypothetical importance of two response variables from the viewpoint of 20 decision makers.

Step d: Now the optimization model can be developed in form of Goal Programming as in(4).

minx FðxÞ ¼min

x w1d1þd⁰₁

þw2d2þd⁰₂

subject to

bR1ðxÞ þd1 d⁰₁¼t1; bR2ðxÞ þd2 d⁰₂¼t2; 1<x1;x2;x3<1:

Now fromTable 4we have bEðwÞ

l

^w¼ 0:285 0:715

and CovðwÞ d Rbw¼ 0:0073 0:0073 0:0073 0:0073

and from(19) and (20)

Bb¼ b^⁰1

b^⁰₂

" #0

¼

b^0 ^b1 b^2 ^b3 ^b12 ^b13 ^b23

104:86 3:147 0:142 0:199 2:379 0:35 0:106 70:45 0:348 3:59 0:28 0:323 0:45 0:614 2

64

3 75

0

:

Also,

Table 3

Experimental data for the numerical example.

Replicate 1 2 3 4 1 2 3 4

ID x1 x2 x3 R1 R2

8 1 1 1 104.45 105.03 99.79 104.92 76.90 77.03 67.99 75.77

4 1 1 1 104.12 104.80 104.20 104.34 72.99 74.25 73.94 73.28

6 1 1 1 98.73 99.36 102.84 94.24 67.10 63.61 68.65 62.42

2 1 1 1 100.19 99.63 100.27 100.60 67.03 66.18 66.58 67.94

7 1 1 1 103.15 106.96 107.62 103.44 71.68 76.27 77.50 76.37

3 1 1 1 106.08 105.64 105.67 105.39 72.94 72.85 72.58 72.38

5 1 1 1 113.52 111.12 112.85 106.67 68.29 68.47 68.96 64.71

1 1 1 1 109.90 109.76 110.70 109.77 67.70 67.24 67.96 66.93

Rb¼ 4:190 3:546 3:546 4:666

:

From where

Covðvecd BÞ ¼b bR ðX⁰XÞ ¹¼ 4:190 3:546 3:546 4:666

0:03125I7: In particular,Covð^d b1Þ ¼0:131I7andCovð^d b2Þ ¼0:145I7.

Before analyzing the example in next step, let us compute the estimator of covariance matrix of response surfaces according to Eq.(8)is

CovðRðxÞÞ ¼d 1þx²₁þx²₂þx²₃þx²₁x²₂þx²₁x²₃þx²₂x²₃ 0:131 0:111 0:111 0:145

:

Step e: In this step by using of some stochastic programming techniques, the stochastic model could be converted into a deterministic one. From(18)and according to Eqs.(19) and (20)the deterministic multiobjective model for this problem can be written as follows:

minx

bEðFðxÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VarðFðxÞÞd q

2 4

3 5

subject to

Rb1ðxÞ þd1 d⁰₁¼103;

Rb2ðxÞ þd1 d⁰₁¼73;

1<x1;x2;x3<1 with

bEðFðxÞÞ ¼0:285jbR1ðxÞ 103j þ0:715jbR2ðxÞ 73j and

VarðFðxÞÞ ¼ ½ð0:285Þd ²Varðd bR1ðxÞÞ þ0:0073ðbR1ðxÞ 103Þ²þ0:0073Varðd bR1ðxÞÞ þ ð0:715Þ²Varðd bR2ðxÞÞ þ0:0073ðbR2ðxÞ 73Þ²þ0:0073Varðd bR2ðxÞފ 0:0073jbR1ðxÞ 103jjRb2ðxÞ 73j þ ð0:285Þ

ð0:715Þb

q

_b

R1ðxÞ;bR2ðxÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðd bR1ðxÞÞVarðd bR2ðxÞÞ

q

;

where b

q

_b

R1ðxÞ;b^R2ðxÞ¼ Covðd bR1ðxÞ;bR1ðxÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðd bR1ðxÞÞVarðd bR1ðxÞÞ

q :

As mentioned in Section2.2there are several methods in stochastic programming that can solve such problems. In this paper another method is also represented according to the context of robust optimization. Minimax approach is one of the well- established methods in robust optimization (Matos[41]) so the multiobjective model in the case study is solved by combi- nation of modified E-model and Minimax approach. The robust E-model could represent aforesaid multiobjective problem in following objective and constraints:

minx bEðFðxÞÞ þza

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VarðFðxÞÞd q

subject to

RbkðxÞ þdk d⁰_k¼tk; k¼1;2;. . .;p;

l<x<u;

ð22Þ

Table 4

Result of GDM for assessment of response importance.

DM 1 2 3 4 5 6 7 8 9 10 11 12 13

w1 0.2 0.3 0.3 0.2 0.3 0.4 0.3 0.3 0.3 0.2 0.4 0.2 0.4

w₂ 0.8 0.7 0.7 0.8 0.7 0.6 0.7 0.7 0.7 0.8 0.6 0.8 0.6

DM 14 15 16 17 18 19 20 Mean Variance

w1 0.3 0.3 0.2 0.4 0.1 0.4 0.2 0.285 0.0073

w2 0.7 0.7 0.8 0.6 0.9 0.6 0.8 0.715 0.0073

As represented above, the model(22)tries to minimize upper bound of (1

a

)% confidence interval for expected value of the goal function. Table 5 shows optimal results for proposed approach in comparison with some other methods. It is considerable that because of some existing different between the parameters of those methods, the model has been analyzed without consideration of the GDM phase and the weights of the response variables assumed to be equal and the results have been compared with some other methods.

As shown inTable 5, the proposed method has considered both location and dispersion effects in presence of correlated responses. One other benefit of the proposed method is to analyze the response surfaces with stochastic terms.

For incorporating the GDM phase into the analysis process, by using the model represented in step e, the case is analyzed and optimized too.Table 6shows the results of proposed approach with consideration of probabilistic weights (seeTable 4).

Table 5

Comparison between the results of the proposed model and other methods.

Stochastic programming method x1 x2 x3 bR1ðxÞ bR2ðxÞ VarðdbR1ðxÞÞ dVarðbR2ðxÞÞ

Kazemzadeh et al.[18] – 1.000 0.265 1.000 108.504 69.367 0.185 0.130

Chiao and Hamada[21] – 1.000 1.000 1.000 104.612 73.574 0.012 0.448

Vining[42] – 0.848 0.659 0.333 103.247 72.806 0.355 0.394

Proposed approach Distance based^a 0.953 0.709 0.407 103.247 73.000 0.428 0.476

Robust E-model 1.000 0.707 0.483 103.332 73.000 0.470 0.523

Lexicofraphic (FirstbEðFðxÞÞ) 1.000 0.707 0.483 103.000 73.000 0.470 0.523 Lexicofraphic (FirstVarðFðxÞÞ)d 0.000 0.000 0.000 104.865 70.453 0.131 0.146

Modified V-model 0.000 0.000 0.000 104.865 70.453 0.131 0.146

aSquared euclidean distance.

Table 6

Optimum results of the proposed model and other models.

Stochastic programming method x1 x2 x3 bR1ðxÞ bR2ðxÞ VarðbdR1ðxÞÞ dVarðbR2ðxÞÞ

Proposed approach Distance based 0.836 0.703 0.369 103.325^a 73.000 0.372 0.414

Robust E-model 0.832 0.703 0.368 103.332 73.000 0.370 0.412

Lexicofraphic (FirstbEðFðxÞÞ) 1.000 0.707 0.483 103.000 73.000 0.470 0.523 Lexicofraphic (FirstVarðFðxÞÞ)d 0.113 0.250 0.039 104.533 71.333 0.141 0.156

Modified V-model 0.418 0.693 0.213 104.053 73.000 0.238 0.265

aBold values represent the better results.

Fig. 1.Sensitivity analysis on location and dispersion effects forRb1ðxÞ:x⁰¼ ðx1;x2¼0:693;x3Þ.

Fig. 1provides a graph in order to represent the confliction in considering the location and dispersion effect (the dimen- sions are homogenized).

4. Conclusions

RSM is an important method in process improvements. It is based on statistical and mathematical concepts so its stochas- tic aspects have to be considered in analysis process. In this paper multiple correlated response surfaces have been studied with the stochastic nature. In addition, in multiple response problems, the analyst must consider the relative preference of the responses. In this study importance degree of each response variable has been obtained by Group Decision Making pro- cess. The weights and regression coefficients in the proposed model have been considered with probabilistic view so fuzzy analysis of such problem in modeling and optimization phase could be included in future research. At the end, through a numerical example, it was shown that the proposed method would produce more reliable results that those of other major methods and also it has a capability to consider different output weights by sampling from expert population.

Acknowledgments

This research work was supported by University of Medellı´n (Medellı´n, Colombia) and Universidad Autónoma Agraria Antonio Narro (México), joint Grant No. 469, SUMMA group. Also, the third author was partially supported IDI-Spain, Grants No. FQM2006-2271.

References

[1] W.C. Yeh, A MCS-RSM approach for network reliability to minimize the total cost, Int. J. Adv. Manuf. Technol. 22 (2003) 618–688, doi:10.1007/s00170- 003-1560-y.

[2] O. Koksoy, A nonlinear programming solution to robust multi-response quality problem, Appl. Math. Comput. 196 (2) (2008) 603–612, doi:10.1016/

j.amc.2007.06.023.

[3] J.P.C. Kleijnen, Response surface methodology for constrained simulation optimization: an overview, Simul. Model. Practices Theory 16 (1) (2008) 50–

64, doi:10.1016/j.simpat.2007.10.001.

[4] R.H. Myers, D.C. Montgomery, Response Surface Methodology: Process and Product Optimization Using Designed Experiments, second ed., Wiley, New York, 2002.

[5] J.P.C. Kleijnen, DASE: Design and Analysis of Simulation Experiments, Springer, New York, 2008.

[6] G. Derringer, R. Suich, Simultaneous optimisation of several response variables, J. Qual. Technol. 12 (1980) 214–219.

[7] K.L. Layne, Methods to determine optimum factor levels for multiple response in designed experimentation, Qual. Eng. 7 (4) (1995) 649–656, doi:10.1080/08982119508918813.

[8] J. Pignatiello, Strategies for robust multiresponse quality engineering, IIE Trans. 25 (1993) 5–15, doi:10.1080/07408179308964286.

[9] Z. He, J. Wang, O. Jinho, S.H. Park, Robust optimization for multiple responses using response surface methodology, Appl. Stoch. Models Bus. Ind.

(2009), doi:10.1002/asmb.788.

[10] D. Saurav, N. Goutam, A. vBandyopadhyay, K.P. Pradip, Application of PCA-based hybrid Taguchi method for correlated multicriteria optimization of submerged arc weld: a case study, Int. J. Adv. Manuf. Technol. 45 (3–4) (2009) 276–286:, doi:10.1007/s00170-009-1976-0.

[11] Ch.Y. Chang, L.Ch. Ti, H.W. Liang, Optimization of process parameters using weighted convex loss functions, Eur. J. Oper. Res. 196 (2009) 752–763, doi:10.1016/j.ejor.2008.04.013.

[12] T.K. Mak, F. Nebebe, Minimizing a general loss function in off-line quality control, Appl. Stoch Models Bus. Ind. 18 (1) (2002) 75–85, doi:10.1002/

asmb.455.

[13] M. Bashiri, T.H. Hejazi, An extension of multiresponse optimization in MADM view, J. Appl. Sci. 9 (9) (2009) 1695–1702, doi:10.3923/

jas.2009.1695.1702.

[14] M.J. Alvarez, N. Gil-Negrete, L. Ilzarbe, M. Tanco, E. Viles, A. Asensio, A computer experiment application to the design and optimization of a capacitive accelerometer, Appl. Stoch Models Bus. Ind. 25 (2) (2009) 151–162, doi:10.1002/asmb.737.

[15] K.L. Hsieh, Parameter optimization of a multi-response process for lead frame manufacturing by employing artificial neural networks, Int. J. Adv.

Manuf. Technol. 28 (2006) 584–591, doi:10.1007/s00170-004-2383-1.

[16] L.I. Tong, H.C. Chen, C.H. Wang, Optimization of multi-response processes using the VIKOR method, Int. J. Adv. Manuf. Technol. 31 (11–12) (2007) 1049–1057, doi:10.1007/s00170-005-0284-6 (9).

[17] L.I. Tong, C.H. Wang, H.C. Chen, Optimization of multiple responses using principal component analysis and technique for order preference by similarity to ideal solution, Int. J. Adv. Manuf. Technol. 27 (2005) 407–414.

[18] R.B. Kazemzadeh, M. Bashiri, A.C. Atkinson, R. Noorossana, A general framework for multiresponse optimization problems based on goal programming, Eur. J. Oper. Res. 189 (2008) 421–429.

[19] H.K. Shah, D.C. Montgomery, W.M. Carlyle, Response surface modeling and optimization in multiresponse experiments using seemingly unrelated regressions, Qual. Eng. 16 (2004) 387–397.

[20] T.T. Nan, Modeling and optimization of stencil printing operations: a comparison study, Comput. Ind. Eng. 54 (2008) 374–389, doi:10.1016/

j.cie.2007.08.001.

[21] C. Chiao, M. Hamada, Analyzing experiments with correlated multiple responses, J. Qual. Technol. 33 (2001) 451–465.

[22] M. Amiri, N. Karimi, S.F. Jamshidi, A methodology for optimizing statistical multi-response problems using genetic local search algorithm through fuzzy goal programming, J. Appl. Sci. 8 (18) (2008) 3199–3206.

[23] A.I. Khuri, J.A. Conlon, Simultaneous optimization of multiple responses represented by polynomial regression functions, Technometrics 23 (1981) 363–375.

[24] A.I. Khuri, J.A. Cornell, Response Surfaces: Designs and Analyses, Marcel Dekker Inc., NewYork, 1987.

[25] J.A. Dı´az-Garcı´a, R. Ramos-Quiroga, E. Cabrera-Vicencio, Stochastic programming methods in the response surface methodology, Comput. Statist. Data Anal. 49 (2005) 837–848, doi:10.1016/j.csda.2004.06.007.

[26] T.H. Hejazi, M. Bashiri, K. Noghondarian, and A.C. Atkinson, Multiresponse optimization with consideration of probabilistic covariates, Qual. Reli. Eng.

Int. 27 (2011) 437–449, doi:10.1002/qre.1133.

[27] Y. Lai, S. Chang, A fuzzy approach for multiresponse optimization: an off-line quality engineering problem, Fuzzy Set System 63 (2) (1994) 117–129.

[28] K.W. Ip, C.K. Kwong, Y.W. Wong, Fuzzy regression approach to modeling transfer moulding for microchip encapsulation, J. Mater. Process. Technol. 140 (1–3) (2003) 147–151. 22.

[29] J.L. Lin, C.L. Lin, The use of grey-fuzzy logic for the optimization of the manufacturing process, J. Materials Process. Technol. 160 (2005) 9–14.

[30] Y.H. Ko, K.J. Kim, C.H. Jun, A new loss function-based method for multiresponse optimization, J. Qual. Technol. 37 (2005) 50–59.

[31] Ch.P. Fung, P.Ch. Kang, Multi-response optimization in friction properties of PBT composites using Taguchi method and principle component analysis, J. Mater. Process. Technol. 170 (2005) 602–610, doi:10.1016/j.jmatprotec.2005.06.040.

[32] J. Kovach, B.R. Cho, Development of a multidisciplinary–multiresponse robust design optimization model, Eng. Optim. 40 (2008) 805–819.

[33] S.J. Sadjadi, M. Habibian, V. Khaledi, A multi-objective decision making approach for solving quadratic multiple response surface problems, Int. J.

Contemp. Math. Sci. 3 (32) (2008) 1595–1606.

[34] R.J. Muirhead, Aspects of multivariate statistical theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc, 1982.

[35] R.E. Steuer, Multiple Criteria Optimization: Theory, Computation and Applications, John Wiley, New York, 1986.

[36] S. Rı´os, S. Rı´os Insua, y M.J. Rı´os Insua, Procesos de decisión Multicriterio, EUDEMA, Madrid, 1989 (in Spanish).

[37] K.M. Miettinen, Non Linear Multiobjective Optimization, Kluwer Academic Publishers, Boston, 1999.

[38] A.M. Mood, F.A. Graybill, D.C. Boes, Introduction to Theory of Statistics, third ed., McGraw-Hill Publishing Co., 1974.

[39] A. Charnes, W.W. Cooper, Deterministic equivalents for optimizing and satisficing under chance constraints, Oper. Res. 11 (1963) 18–39.

[40] A. Prekopa, Stochastic Programming, Serie Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1995.

[41] M.A. Matos, Decision under risk as a multicriteria problem, Eur. J. Oper. Res. 181 (2007) 1516–1529.

[42] G.G. Vining, A compromise approach to multiresponse optimization, J. Qual. Technol. 30 (1998) 309–313.