Expert Systems with Applications 38 (2011) 9557–9563

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Expert Systems with Applications

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A goal programming-TOPSIS approach to multiple response optimizationusing the concepts of non-dominated solutions and prediction intervals

Majid Ramezani a, Mahdi Bashiri a,⇑, Anthony C. Atkinson b

a Department of Industrial Engineering, Shahed University, P.O. Box 18155/159, Tehran, Iranb Department of Statistics, London School of Economics and Political Sciences, UK

a r t i c l e i n f o

Keywords:Multi-response optimizationConfidence and prediction intervalsNon-dominated solutionGPTOPSIS

0957-4174/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.eswa.2011.01.139

⇑ Corresponding author. Tel.: +98 9123150355; faxE-mail addresses: majid.ramezani@ymail.com (M.

ac.ir (M. Bashiri), A.C.Atkinson@lse.ac.uk (A.C. Atkinso

a b s t r a c t

Multiple response problems include three stages: data gathering, modeling and optimization. Mostapproaches to multiple response optimization ignore the effects of the modeling stage; the model istaken as given and subjected to multi-objective optimization. Moreover, these approaches use subjectivemethods for the trade off between responses to obtain one or more solutions. In contradistinction, in thispaper we use the Prediction Intervals (PIs) from the model building stage to trade off between responsesin an objective manner. Our new method combines concepts from the goal programming approach withnormalization based on negative and positive ideal solutions as well as the use of prediction intervals forobtaining a set of non-dominated and efficient solutions. Then, the non-dominated solutions (alterna-tives) are ranked by the TOPSIS (Technique for Order Preference by Similarity to the Ideal Solution)approach. Since some suggested settings of the input variables may not be possible in practice or maylead to unstable operating conditions, this ranking can be extremely helpful to Decision Makers (DMs).The consideration of statistical results together with the selection of the preferred solution among theefficient solutions by Multiple Attribute Decision Making (MADM) distinguishes our approach from oth-ers in the literature. We also show, through a numerical example, how the solutions of other methods canbe obtained by modifying the relevant approach according to the DM’s requirements.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The control of production processes in an industrial environ-ment requires the selection and correct setting of the input vari-ables, so that on-specification product is produced at minimumcost. But first, the relationship between input and output variablesmust be determined. The series of techniques used in the empiricalstudy of the association between response variables and several in-put variables is called Response Surface Methodology (RSM). See,for example, (Box & Draper, 1987 or Myers & Montgomery, 2002).

Much of the emphasis in RSM has been on building models forone response, whereas industrial processes often have many re-sponses, the values of which ideally require simultaneous optimi-zation. Usually optimizing all responses simultaneously wouldcause conflicts of interest. Our method of resolution of this conflictis based on realizing that the problem has three stages: datagathering, model building and optimization. At first the dataare collected, using an experimental design. Then the techniquesof RSM are applied for estimating the relation between response

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: +98 2151212021.Ramezani), Bashiri@shahed.

n).

(output) and explanatory (input) variables and the model is con-structed. Finally, the model is optimized. At this stage the purposeis to obtain optimum conditions on the input variables so that allresponses concurrently will be as near as possible to their optima.Khuri (1996) discusses such multiple response problems.

Most research in multi-response optimization ignores the sta-tistical uncertainty in the results of the modeling stage; the modelis taken as given and subjected to multi-objective optimization.When conflicts exist between optimization of the various re-sponses, subjective methods are typically used for trade off be-tween responses to obtain a variety of alternative solutions. Onthe contrary, in our paper we use the Prediction Intervals (PIs) fromthe model building stage to trade off between responses in anobjective manner. We then find the non-dominated solutions ofthe problem by modified goal programming. An advantage of ourmethod is that the continuous region of solutions is transformedinto a discrete region. Using the Technique for Order Preferenceby Similarity to the Ideal Solution (TOPSIS) we rank the non-dom-inated solutions from this discrete region. This automatic proce-dure sometimes suggests impractical settings of the explanatoryvariables. But the ranking is a powerful tool for Decision Makers(DMs) who may need to modify the solutions for practical pur-poses. Our approach is distinguished from others in this field by

9558 M. Ramezani et al. / Expert Systems with Applications 38 (2011) 9557–9563

the incorporation of statistical properties and by the selection ofthe preferred solution among the efficient solutions by MultipleAttribute Decision Making (MADM).

The structure of the paper is as follows. Section 2 reviews theexisting work on multiple response problems and provides defini-tions relevant to the proposed method. The details of the methodare expounded in Section 3. Section 4 clarifies the method by anumerical example. Finally, implementation issues are discussedin Section 5 with some conclusions in Section 6.

2. Literature review

This section first reviews some work on multi-response optimi-zation (MRO). Then we present definitions necessary for thedescription of our proposed method.

2.1. A summary of existing work on MRO

The majority of approaches to the multi-response optimizationproblem in the literature fall in one of the following categories:

The first approach, loss function, considers three properties:bias, robustness and the quality of prediction. Bias measures sys-tematic deviation, robustness refers to the low sensitivity of theresponse variable to nuisance factors and quality of predictionmeasures the variance of predictions. Originally Taguchi, Elsayed,and Hsiang (1989) presented a univariate loss function whichensured that the response converged to the target value withsmall variance. Pignatiello (1993) expanded Taguchi’s method

Fig. 1. The procedure of

to a loss function for multi-response problems that consideredboth bias and robustness. Vining (1998) introduced a loss func-tion that considered bias and quality. Finally, Ko, Kim, and Jun(2005) proposed a new loss function embracing all threeproperties.

The second approach, Multiple Objective Optimization (MOO),can itself be divided into three classes depending on the form ofthe preference information from the decision maker: prior, pro-gressive, or posterior articulation methods (Hwang, Masud, Paidy,& Yoon, 1979). The prior method takes all of the preference infor-mation from the DM before solving the problem (e.g. Kazemzadeh,Bashiri, Atkinson, & Noorossana, 2008 who use goal programmingto find the optimal setting of the controllable variables). In the pro-gressive method, the solver and DM are in contact so that the solu-tion is obtained interactively. As an example, Jeong and Kim (2009)proposed an interactive desirability function approach (IDFA) inwhich the shape, bound and target of the desirability function issubjectively changed. Further papers using interactive methods in-clude (Köksalan & Plante, 2003; Mollaghasemi & Evans, 1994; Park& Kim, 2005). The posterior method finds all (or most) efficientsolutions and then allows the DM to select the best one from theefficient solutions. We have failed to find any references to workemploying the posterior method in MRO. Also in this categoryare the methods of Köksoy (2006) and of Köksoya and Yalcinozb(2006) which incorporate statistics such as Mean Squared Error(MSE) for robust design. For more details about MOO (see Collette& Siarry, 2003; Figueira, Greco, & Ehrgott, 2005). A summary re-view of MRO methods is in Table 1.

proposed method.

Table 1review of existing methods.

Reference First category Second category

Bias Robustness Quality Method Prior Interactive a

Taguchi et al. (1989) �Pignatiello (1993) � �Vining (1998) � �Ko et al. (2005) � � �Derringer and Suich (1980) DF �Mollaghasemi and Evans (1994) STEM �Kim and Lin (2000) DF �Köksalan and Plante (2003) PSAP �Park and Kim (2005) GDF �Köksoya and Yalcinozb (2006) GA � �Köksoy (2006) NIMBUS � �Kim and Lin (2006) DF � �Kazemzadeh et al. (2008) GP � �Jeong and Kim (2009) DF �This paper GP & TOPSIS � � �

a: Incorporating the statistical measures such as MSE, variance, prediction interval of responses into model building, DF: Desirability Function, STEM: Step method, PSAP:Parametric Achievement-Scalarizing Program, GDF: Geoffrion–Dyer–Feinberg method, GA: Genetic Algorithm, NIMBUS: Non-differentiable Interactive Multi-objectiveBundle-based Optimization system, GP: Goal Programming.

M. Ramezani et al. / Expert Systems with Applications 38 (2011) 9557–9563 9559

2.2. The multiple response optimization problem

Generally, a multiple response optimization problem is formu-lated as follows:

Optimize ff1ðxÞ; f2ðxÞ; . . . ; frðxÞgSubject to x 2 X

ð1Þ

Here ‘‘optimize’’ denotes minimization or maximization, fu(x),u = 1,2, . . . ,r are responses to be optimized and defines x 2X thefeasible region for solution of the problem.

Usually, procedures for solution of this problem are classifiedinto two categories. Methods in the first category simplify theproblem, choosing the most important response and disregardingthe other responses or treating them as model constraints. For in-stance, the method proposed by Coello Coello (2000), the e-con-straint method, belongs to this class. In the second category, theattempt is made to aggregate the responses into a single objectivefunction. For example, use of the differentially weighted sum of allobjective functions falls in this category as well as the desirabilityfunction approach, in which the desirability functions for each re-sponse are combined in one desirability function using variousoperators. Most methods in this category consider the importanceof responses and hence assigning a weight to each response is nec-essary. All of the proposed methods present solutions that dependon the preferences of the decision maker. An important property ofthe solutions is Pareto optimality of which the generalizeddefinitions (Yu, 1985) are given as follows (for the minimizationproblem):

Definition 1. A point x⁄ 2X is a Pareto optimal solution if and onlyif there does not exist another x 2X such that fu(x) 6 fu(x⁄) "u withstrict inequality holding for at least one u.

Definition 2. A point x⁄ 2X is a weak Pareto optimal solution ifand only if there does not exist another x 2X such that fu(x) < fu(x⁄)"u.

2.3. Confidence intervals and prediction intervals

The (1 � a)% confidence interval (CI) is the range in which theprocess average is expected to fall (1 � a)% of the time. However,the prediction interval (PI) is the range in which any individualvalue should fall (1 � a)% of the time. The prediction interval is

wider than the confidence interval because of the added uncer-tainty involved in predicting a single response versus the mean re-sponse. The formulas are as follows:

ð1� aÞ%CI ¼ y0 � ta=2;n�p SE mean; SE mean ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixT

0ðXT XÞ�1x0

qð2Þ

ð1� aÞ%PI ¼ y0 � ta=2;n�p SEpred; SE pred ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ xT

0ðXT XÞ�1x0

qð3Þ

Here a = chosen alpha value, n = number of observations,p = number of predictors, r2 ¼ mean squared error, (XT X)�1 isthe variance–covariance matrix of the coefficients and x0 is the va-lue of input variables corresponding to cy0 as well as a vector. Weconventionally denote (ta/2,n�p SEpred) as TSEP.

Since we are concerned with possible values of the response, weuse the prediction interval to provide intervals for the change ineach response during the solution procedure. Using TSEP values,the responses are altered to obtain a new non-dominated solution.We use TSEP as a base for changing responses instead of the hap-hazard changes of some other methods.

3. The proposed method

The proposed method has three phases as shown in Fig. 1. Inphase 0, the data are collected. Then, relations between input vari-ables and output variable are estimated by one of the RSM tech-niques and the values of TSEP are calculated for each of theresponses using the mean square error (MSE) from the ANOVA ta-ble. In phase 1, we combine the goal programming approach (Char-nes, Cooper, & Ferguson, 1955), concepts of ‘‘negative idealsolution’’ and ‘‘positive ideal solution’’, which is used in interactivesequence goal programming-II (ISGP-II) approach (Masud &Hwang, 1980), as well as prediction intervals for obtaining a setof non-dominated solutions. In phase 2, the alternative non-domi-nated solutions obtained from the previous phase are ranked bythe TOPSIS approach and the preferred solution is determined.

3.1. Detailed procedure of proposed method

Phase 0Step 1: Select a proper experimental design considering the

views of experts and then collect the data according tothis design.

9560 M. Ramezani et al. / Expert Systems with Applications 38 (2011) 9557–9563

Step 2: Obtain the ANOVA table for the collected data. After-wards, following deletion of the non-significant effectsof input variables, calculate the modified ANOVA table.

Step 3: Find the relations between input variables and outputvariable for every response in accordance with the modi-fied ANOVA table.

Phase 1Step 4: Obtain the positive ideal solutions (PIS), fþu , and negative

ideal solutions (NIS), f�u ; u ¼ 1; . . . ; r and construct thePIS and NIS payoff Table. In Tables 2 and 3, fij is the valueof response j when response i is equal respectively to thePIS and NIS. These solutions show the lower and upperbounds of the response values and provide suitable infor-mation for the decision maker (s) (DM) in selecting thepreferred solution.

Step 5: Determine the weight, wu "u, which are the relativeimportance of every response which is assigned by theDM.

Step 6: Obtain the first solution solving the problem:

Table 2PIS pay

f1(x)f2(x)

..

.

fr(x)

Table 3NIS pay

f1(x)f2(x)

..

.

fr(x)

Min Z1 ¼Xr

k¼1

wkd�k ð4Þ

s:t:fþj � fjðxÞfþj � f�j

� d�j � dþj� �

¼fþj � bj

fþj � f�j8j 2 J ð5Þ

fiðxÞ � fþif�i � fþi

þ d�i � dþi� �

¼ bi � fþif�i � fþi

8i 2 I ð6Þ

x 2 X ð7Þ

Here I and J denote the responses to be minimized and maximized,respectively, bi and bj denote the goals of the ith and jth response,respectively and di, dj are auxiliary variables which show deviationfrom the target. We show the values of responses corresponding tothe resulted solution as (p11,p12, . . . ,p1r).Step 7: Check to see if the resulting solution is non-dominated. A

value of the objective function equal to zero (Z = 0), indi-cates that the solution is dominated. Then increase bj anddecrease bi subjectively (or in agreement with DM) and goback to step6. If the objective function is not equal to zero(Z – 0), then take the obtained solution as a non-domi-nated solution and go to step8.

Step 8: Compute the values of TSEP for each response by Eq. (3) atpoints resulting from the previous step which are shownas TSEPu u = 1, . . . ,r.

Step 9: Solve the following model (Eqs. (8)–(13)) r times andobtain the r non-dominated solutions considering eachresponse in one of Eq. (11) or Eq. (12). The problems are:

off table.

f1(x) f2(x) � � � fr(x)

fþ1 f12 � � � f1r

p21 fþ2 � � � f2r

..

. ... � � � ..

.

fr1 fr2 � � � fþr

off table.

f1(x) f2(x) � � � fr(x)

f�1 f12 � � � f1r

p21 f�2 � � � f2r

..

. ... � � � ..

.

fr1 fr2 � � � f�r

Table 4The dec

A1

A2

..

.

Ar+1

Min Zu ¼Xr

k¼1;k–u

wkd�k ð8Þ

S:t:fþj � fjðxÞfþj � f�j

� d�j � dþj� �

¼fþj � bj

fþj � f�j8j 2 J � fug ð9Þ

fiðxÞ � fþif�i � fþi

þ d�i � dþi� �

¼ bi � fþif�i � fþi

8i 2 I � fug ð10Þ

Or f uðxÞP f �u � TSEPu; u 2 J ð11Þf uðxÞ 6 f �u þ TSEPu; u 2 I ð12Þx 2 X ð13Þ

Here j 2 J � {u}, i 2 I � {u} and f �u is the optimum value of the uth re-sponse from the previous step. The values of response correspond-ing to resulting solutions are denoted as (pu1,pu2, . . . ,pur),u = 2,3, . . . , r + 1.Step 10: Check to see if the resulting solutions are non-dominated.

If Z = 0, which would imply domination, increase bj anddecrease bi subjectively and go to step 9. If Z – 0, takethe obtained solution as a non-dominated solution andgo to step 11.

Phase 2Step 11: Construct the decision matrix using the resulted non-

dominated solutions as Table 4. The non-dominatedsolutions from previous steps and the objective func-tions are treated as alternatives and criteria, respec-tively. Here we turn the continuous region of solutionsinto a discrete region of non-dominated solutions.

Step 12: Normalize the decision matrix by

nij ¼ pij

� ffiffiffiffiffiffiffiffiffiffiffiffiffiXrþ1

i¼1

p2ij

vuut ð14Þ

Here index i and index j denote alternatives and criteria,respectively.Step 13: Construct the weighted matrix by multiplying the nor-

malized decision matrix by the weight of each response:

W ¼ w1;w2; . . . ;wrf g ð15Þ

V ¼ ND:Wn�n ¼

v11 � � � v1j � � � v1r

..

. ... ..

.

v ðrþ1Þ1 � � � v ðrþ1Þj � � � v ðrþ1Þr

��������

��������ð16Þ

ND is the normalized matrix and Wn�n is a diagonal matrix.Step 14: Determine the positive ideal solution (PIS) and negative

ideal solution (NIS) from the definitions

Sþ ¼ fðmaxi

V ijjj 2 JÞ; ðmini

V ijjj 2 J0Þji¼ 1; . . . ;rþ1g ¼ Vþ1 ;Vþ2 ; . . . ;V

þr

ð17Þ

S� ¼ fðmini

V ijjj 2 JÞ; ðmaxi

V ijjj 2 J0Þji¼ 1; . . . ;rþ1g ¼ V�1 ;V�2 ; . . . ;V

�r

ð18Þ

Here J and J0 denote the responses which should be maximizedand minimized, respectively. Also the number of non-dominated

ision matrix.

f1(x) f2(x) � � � fr(x)

p11 p12 � � � p1r

p21 p22 � � � p2r

..

. ... � � � ..

.

p(r+1)1 p(r+1)2 � � � p(r+1)r

Table 5NIS payoff table.

f1(x) f2(x) f3(x) f4(x)

Minf1 50.38 279.65 565.49 66.25

M. Ramezani et al. / Expert Systems with Applications 38 (2011) 9557–9563 9561

solutions is specified by r + 1 (A solution of step 6 plus the r solu-tions from step 9).Step 15: Calculate the distance of every alternative, i.e. every

non-dominated solution, from PIS and NIS as follows:

Minf2 70.13 7.3 665.19 64.9Maxf3 94.8 882.84 846.91 60.29Minf3 204.37 2693.27 152.58 73.47Maxf4 123.05 1227.04 427.43 87.72Min f4 100.46 1097.84 665.25 57.66

dþi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXr

j¼1

Vij � Vþj� �2

vuut ; i ¼ 1;2; . . . ; r þ 1; d�i

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXr

j¼1

Vij � V�j� �2

vuut ; i ¼ 1;2; . . . ; r þ 1 ð19Þ

Step 16: compute the relative closeness of every alternative withrespect to the ideal solution defined as

Cþi ¼d�i

d�i þ dþi; 0 6 Cþi 6 1; i ¼ 1; . . . ; r þ 1 ð20Þ

Step 17: Finally, rank the alternatives in descending order of Cþiand select the alternative with the highest value as thesolution of the proposed method.

4. Numerical example

We use the ‘‘tire tread compound’’ problem (Derringer & Suich,1980) to illustrate the proposed method. The purpose of the exper-iment is to improve the tire tread performance as measured by fourresponses via controlling three chemical ingredients (input vari-ables). The responses are: PICO abrasion index (f1), 200% modulus(f2), elongation at break (f3) and hardness (f4). The input variablesinclude: silica (x1), silane (x2) and sulfur (x4). Step by Step solutionof the problem is given below.

Phase 0Step 1: The data comes from the experiment of Derringer and

Suich (1980).Step 2: The data are analyzed and the final ANOVA table is

obtained using Design Expert software.Step 3: The estimated relation between input variables and

responses is given below.

1:57x23 þ 5:13x1x2 þ 7:13x1x3 þ 7:88x2x3;

21 � 124:79x2

2 þ 199:17x23 þ 69:38x1x2 þ 94:13x1x3 þ 104:38x2x3;

1x22 þ 0:43x2

3 þ 8:75x1x2 þ 6:25x1x3 þ 1:25x2x3;

0:32x23 � 1:63x1x2 þ 0:13x1x3 � 0:25x2x3;

f1ðxÞ ¼ 139:12þ 16:49x1 þ 17:88x2 þ 10:91x3 � x21 � 3:45x2

2 �f2ðxÞ ¼ 1261:11þ 268:15x1 þ 246:5x2 þ 1391:48x3 � 83:55x

f3ðxÞ ¼ 400:38� 99:67x1 � 31:4x2 � 73:92x3 þ 7:93x21 þ 17:3

f4ðxÞ ¼ 68:91� 1:41x1 þ 4:32x2 þ 1:63x3 þ 1:56x21 þ 0:06x2

2 �s:t: � 1:633 6 xi 6 1:633 i ¼ 1;2;3

R21 ¼ 0:97; R2

2 ¼ 0:74; R23 ¼ 0:98; R2

4 ¼ 0:95

The responses and should be maximized. The targets for and arerespectively 500 and 67.5.

Phase 1Step 4: The NISs and PISs are computed are shown in Tables 5 and

6. Since two negative values exist for responses 3 and 4,the NIS values are given in the manner shown below.

Step 5: We assume that the responses have the same importancefor the decision. Therefore we take the weights as equal.

Step 6: From Tables 5 and 6 and according to Eqs. (4)–(7), wesolve the following problem:

min Z ¼ d�1 þ d�2 þ d�3 þ d�4 þ d�5 þ d�6s:t: f 1 þ ðd

�1 � dþ1 Þð242:68� 50:38Þ ¼ 242:68

f 2 þ d�2 � dþ2� �

ð3019:23� 7:3Þ ¼ 3019:23f 3 þ d�3 � dþ3

� �ð500� 152:58Þ ¼ 500

f 3 � d�4 � dþ4� �

ð846:91� 500Þ ¼ 500f 4 þ d�5 � dþ5

� �ð67:5� 57:66Þ ¼ 67:5

f 4 � d�6 � dþ6� �

ð87:7� 67:5Þ ¼ 67:5d�i � dþi ¼ 0 and d�i ;d

þi 2 ½0;1� 8ix 2 X

Step 7: The solution of the above equation isx⁄ = (0.95,0.48,�1.633), f⁄ = (119.4,1585.28,500,67.5)and Z = 0.64. Therefore the solution is non-dominated.We go to the next step.

Step 8: The values of TSEP for each response with a = 0.05 atpoint (0.95,0.48,�1.633) are respectively 12.68, 742.65,46.42 and 2.86.

Step 9: The other non-dominated solutions are calculated fromthe same equations. For instance, the second alternativeis obtained by solving the problem:

min Z2 ¼ d�1 þ d�3 þ d�4 þ d�5 þ d�6s:t: f 1 þ d�1 � dþ1

� �ð242:68� 50:3Þ ¼ 242:68

f 3 þ d�3 � dþ3� �

ð500� 152:58Þ ¼ 500

f 3 � d�4 � dþ4� �

ð846:91� 500Þ ¼ 500

f 4 þ d�5 � dþ5� �

ð67:5� 57:66Þ ¼ 67:5

f 4 � d�6 � dþ6� �

ð87:7� 67:5Þ ¼ 67:5f 2 P 1585:28� 742:65d�i � dþi ¼ 0 and d�i ; d

þi 2 ½0;1� 8ix 2 X

Step 10: Since we find that the objectives of all equation are great-er than zero, the solutions are non-dominated. It is notedtwo solutions are the same and thus we attain four dis-tinct solutions as follows

A1 ¼ x11; x

12; x

13

� �¼ ð0:957;0:488;�1:633Þ

A2 ¼ x21; x

22; x

23

� �¼ ð�0:267;�0:021;�0:942Þ

A3 ¼ x31; x

32; x

33

� �¼ ð0:54;0:65;�1:633Þ

A4 ¼ x41; x

42; x

43

� �¼ ð0:115;0:369;�1:633Þ

Phase 2Step 11: We now construct the decision matrix as shown in Table

7.Step 12: The decision matrix is normalized by the formulation of

Eq. (14).Step 13: Assuming identical weights for the responses, the

weighted matrix is obtained as in Table 8.Step 14: The PIS and NIS are computed as:

Sþ ¼ f0:128;0:132;0:13;0:125gS� ¼ f0:121;0:103;0:107;0:124g

Step 15: The distance of all alternatives from PIS and NIS iscalculated.

Table 6PIS payoff table.

f1(x) f2(x) f3(x) f4(x)

Maxf1 242.68 3019.23 177.41 75.12Maxf2 242.68 3019.23 177.41 75.12f3 = 500 119.4 1585.28 500 67.5f4 = 67.5 119.4 1585.28 500 67.5

Table 7Decision matrix.

f1(x) f2(x) f3(x) f4(x)

A1 119.4 1585.28 500 67.5A2 124.34 1249.912 500 67.5A3 122.17 1623.12 453.58 67.5A4 119.22 1590.62 500 66.91

Table 8The weighted decision matrix.

f1(x) f2(x) f3(x) f4(x)

A1 0.1230 0.1303 0.1278 0.1252A2 0.1281 0.1027 0.1278 0.1252A3 0.1258 0.1334 0.1159 0.1252A4 0.1228 0.1308 0.1278 0.1241

Table 10Optimization results of the CGA example.

DS KL IDFA PM

f1 129.5 157.79 139.82 119.22f2 1300 1689.5 1239.1 1590.62f3 465.7 346.62 446.51 500f4 68 76.43 73.93 66.91

9562 M. Ramezani et al. / Expert Systems with Applications 38 (2011) 9557–9563

Step 16: The relative closeness of each alternative is measured.Step 17: The rank of alternatives (solutions) is A4, A1, A3, A2 so the

preferred solution is A4. The results of last three steps areillustrated in Table 9.

5. Issues for implementation

The resulting numerical solution may not satisfy the decisionmaker and we may require further solutions for discussion withthe DM. Therefore, as future research, we intend to investigatethe interactive usage of our method. In this case, we could changebi or bj, j 2 J � {u} i 2 I � {u}, in Step 6 of each iteration and repeatStep 6 to Step 17 or we could use different confidence levels(a = 0.1,0.05,0.01) to obtain a preferred solution according to thejudgment of the DM.

To be sure of the obtained solution, we numerically comparedour Proposed Method (PM) with three others: DS – the conven-tional desirability function method (Derringer & Suich, 1980); KL– the method of maximizing a desirability function (Kim & Lin,2000) and IDFA – the interactive desirability function approach(Jeong & Kim, 2009). Briefly, the DS method aims to find x⁄ to max-imize,[d1(f1),d2(f2), . . . ,dr(fr)]1/r, the KL method seeks to find x⁄ tomaximize the minimum of {d1(f1),d2(f2), . . . ,dr(fr)} and the IDFAmethod seeks to maximize the minimum of the individual desir-ability functions via changing the shape, bound, and target of themin an interactive manner, where du(fu) is the desirability function ofthe uth response.

As Table 10 shows, the PM solution is not dominated by theother solutions – our proposed method is not invalid. Moreover,

Table 9The final results of problem.

d+ d� C+

A1 0.0059 0.0300 0.8343A2 0.0306 0.0130 0.2981A3 0.0120 0.0308 0.7187A4 0.0060 0.0304 0.8349

we can show that the other solutions in Table 10 can be obtainedfrom our proposed approach. For instance, as suggested before,we can change the goals of the responses in step 6 according tothe DM’s point of view. Specifically, assume the DM sets her goalsas b1 = 140, b2 = 1300, b3 = 465, b4 = 68. The related model is thenconstructed as follows

min Z ¼ d�1 þ d�2 þ d�3 þ d�4 þ d�5 þ d�6s:t: f 1 þ d�1 � dþ1

� �ð242:68� 50:38Þ ¼ 140

f 2 þ d�2 � dþ2� �

ð3019:23� 7:3Þ ¼ 1300f 3 þ d�3 � dþ3

� �ð500� 152:58Þ ¼ 465

f 3 � d�4 � dþ4� �

ð846:91� 500Þ ¼ 465f 4 þ d�5 � dþ5

� �ð67:5� 57:66Þ ¼ 68

f 4 � d�6 � dþ6� �

ð87:7� 67:5Þ ¼ 68d�i � dþi ¼ 0 and d�i ;d

þi 2 ½0;1� 8ix 2 X

After using the proposed method, the resulting solution is(�0.456,0.143,�0.863) and the corresponding response is(129.53,1300,465,68), equal to that for the DS method.

6. Conclusion

Multiple response problems include three stages: data gather-ing, modeling and optimization. Multiple response optimizationoften entails convicting and incommensurate responses whichhas led to a variety of approaches to the solution of these problems.Most approaches to MRO ignore the effects of the modeling stage,treating the problem as one with multiple objectives. In addition,these approaches use subjective methods for trade off between re-sponses to obtain one or more alternative. In contradistinction, inthis paper we have used the PI results from the model buildingstage to provide an objective trade off between responses in thestatistically indicated model. We have then found a non-domi-nated solution of the problem.

Our new method combines concepts from the goal program-ming approach with normalization based on negative and positiveideal solutions as well as the use of prediction intervals for obtain-ing a set of non-dominated and efficient solutions. Specifically, weused this hybrid method for changing the continuous feasible re-gion to a discrete efficient region. Then, the non-dominated solu-tions (alternatives) were ranked by the TOPSIS approach. Sincesome suggested settings of the input variables may not be possiblein practice or may lead to unstable operating conditions, this rank-ing can be extremely helpful to DMs. The consideration of statisti-cal results together with selection of the preferred solution amongthe efficient solutions by Multiple Attribute Decision Making(MADM) distinguishes our approach from others in the literature.We have also shown, through a numerical example, how the solu-tions of other methods may be obtained by modifying the relevantmodel according to the DM’s requirements.

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