research article a new method based on topsis and response...

Research ArticleA New Method Based on TOPSIS and Response SurfaceMethod for MCDM Problems with Interval Numbers

Peng Wang1 Yang Li1 Yong-Hu Wang2 and Zhou-Quan Zhu1

1School of Marine Science and Technology Northwestern Polytechnical University Xirsquoan 710072 China2Flight Technology College Civil Aviation Flight University of China Guanghan Sichuan 618307 China

Correspondence should be addressed to Peng Wang wangpeng305nwpueducn

Received 8 December 2014 Revised 30 March 2015 Accepted 9 April 2015

Academic Editor Marek Lefik

Copyright copy 2015 Peng Wang 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

As the preference of design maker (DM) is always ambiguous we have to face many multiple criteria decision-making (MCDM)problems with interval numbers in our daily life Though there have been some methods applied to solve this sort of problem itis always complex to comprehend and sometimes difficult to implement The calculation processes are always ineffective when anew alternative is added or removed In view of the weakness like this this paper presents a new method based on TOPSIS andresponse surface method (RSM) for MCDM problems with interval numbers RSM-TOPSIS-IN for short The key point of thisapproach is the application of deviation degree matrix which ensures that the DM can get a simple response surface (RS) modelto rank the alternatives In order to demonstrate the feasibility and effectiveness of the proposed method three illustrative MCMDproblems with interval numbers are analysed including (a) selection of investment program (b) selection of a right partner and(c) assessment of road transport technologies The contrast of ranking results shows that the RSM-TOPSIS-IN method is in goodagreement with those derived by earlier researchers indicating it is suitable to solve MCDM problems with interval numbers

1 Introduction

MCDM which is short for multiple criteria decision makingis a problem about how to find the best option from all of thefeasible alternatives on the basis of two or more attributesIn order to deal with this sort of problem many methodshave been developed for instance simple additive weighting(SAW) [1] order preference by similarity to ideal solution(TOPSIS) method [2] analytic hierarchy process (AHP)[3] grey relational analysis (GRA) [4] and multiobjectiveoptimization on the basis of ratio analysis (MOORA)method[5] All these approaches have been applied in solving avariety of MCDM problems including plant layout designproblem [6] optimization [7] material selection [8] andforestry [9] Among all the related methods TOPSIS hasgained popularity in the field of MCDM because of itssimplicity and practicality [10ndash13]

However since human judgments including preferencesare often vague and cannot be estimated with an exactnumerical value therefor these data may be expressed by a

number of ways such as bounded data ordinal data intervaldata and fuzzy data [14] Sincemost criteria of these problemshave interdependent and interactive features they cannotbe evaluated by conventional measures method Bellman ampZadeh [15] are the first to study on the decision-making prob-lemunder a fuzzy environment-watershedThey heralded theinitiation of fuzzy MCDM and this analysis method is thenwidely used to deal with decision-making problems involvingmultiple criteria evaluationselection of alternatives In manypractical MCDM fuzzy MCDM approaches using intervalnumbers instead of exact values now have been introducedand illustrated in many literatures For example a multicri-teria group decision making for evaluation of supplier usingintuitionistic fuzzy TOPSIS is presented by Boran et al [16]Lee et al [17] constructed an approach based on the fuzzyanalytic hierarchy process (FAHP) and balanced scorecard(BSC) to evaluate an IT department An example whoseattributes weights and ratings are expressed by interval num-bers is successfully solved through the GRA-based TOPSISdecision-making approach proposed by Peng andWang [18]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 938535 11 pageshttpdxdoiorg1011552015938535

2 Mathematical Problems in Engineering

and also there is a good survey done by Chen et al [19] whomake distinctions between fuzzy ranking methods and fuzzymultiple attribute decision-making methods

Compared with the traditional MCDM models thefuzzy versions of MCDM applications are very complex tocomprehend and sometimes difficult to implement [20] Onone hand while the MCDM model consisted of intervalnumbers too much mathematical calculation leads to morecomputation time making it impracticable and ineffectiveto the decision makers (DMs) without a strong capability inmathematics At the same time the range of intervals maybecome more and more wide along with the calculationwhich will be more difficult for the DMs to make the finalchoice On the other hand if the decision alternatives arechanged or an alternative is added or removed the DMshave to analyse the problem from the beginning again Itis really a waste of time So how to find some simple waysto deal with the MCDM problems with interval numbersbecomes an important study topic for the researchers Theonewhich involves a unique expression to fulfill differing taskrequirements may be an ideal choice

In this paper we present a newmethod based on TOPSISand response surface method (RSM) for MCDM problemswith interval numbers (RSM-TOPSIS-IN) In experimentaldesign the best and worst intervals of each attribute arecombined to represent all the possibilities of alternativesapproximately With the application of deviation degreematrix the experimental data can be used to get a responsesurface (RS) model helping the DMs to acquire the rankingresults through the established equation simply What ismore if some new alternatives are considered later we canonly use the established RS model to get the ranking resultseasily reducing both time and effort a lot Three practicalMCDM problems that consisted of interval numbers will besolved by the RSM-TOPSIS-IN method we proposed Thecompared results will show that it can help the DMs to get themost appropriate alternative in complex MCDM problemsusing both less time and less calculation

The reminder of this paper is organized as followsSection 2 focuses onpresenting a brief reviewof relatedmeth-ods involving interval numbers TOPSIS method exper-imental design and response surface method Then thedetailed processes of the RSM-TOPSIS-IN method are statedin Section 3 In Section 4 two more illustrative examples aredescribed to demonstrate the capabilities of the proposedmethod Finally conclusions and future research areas arediscussed in Section 5

2 Backgrounds

21 Interval Numbers In the real world the decision-makingproblems are always very vague and uncertain which maybe expressed by a variety of ways such as bounded num-ber ordinal number interval number and fuzzy numberConsidering these problems are not like the conventionalissueswith exact values they cannot be evaluated by commonmethods BellmanampZadeh [15] are the first ones who studiedthe decision-making problem under a fuzzy environment-watershed and they heralded the initiation of fuzzy MCDM

Whereafter this analysis method is widely used to dealwith decision-making problems involving multiple criteriaevaluation of alternatives

Among all the fuzzy MCDM problems interval numbersand fuzzy numbers are used the most to construct thedecision matrix Interval numbers are a set of real numberswith the property that any number that lies between twonumbers is also included in the set [21] Fuzzy numbers areexpressed by a fuzzy subset of real numbers representingthe expansion of the idea of the confidence interval [22]for the fuzzy numbers are developed based on the intervalnumbers In this paper we only talk about the special partof fuzzy MCDM problems which are the problems withinterval numbers actually Though there have been somearticles aiming at the study on interval numbers in recentyears finding a more complete and robust methodology isstill needed It is worth mentioning that transforming theuncertain problem to a deterministic problem seems a goodway for this sort of problems [23]

In this section some basic definitions [24] used to solvethe interval numbers will be introduced briefly helping usto know more about the interval numbers Suppose 119886 =

[119886minus

119886+

] = 119909 | 0 le 119886minus

le 119909 le 119886+

119887 = [119887minus 119887+] = 119909 |0 le 119887minus

le 119909 le 119887+

are two positive interval numbers and thenone can find the following

Definition 1 Consider

119886 + 119887 = [119886minus

+ 119887minus

119886+

+ 119887+

] (1)


119886 minus 119887 = [119886minus

minus 119887+

119886+

minus 119887minus

] (2)


119886 times 119887 = [min (119886minus119887minus 119886minus119887+ 119886+119887minus 119886+119887+)

max (119886minus119887minus 119886minus119887+ 119886+119887minus 119886+119887+)] (3)


119886

119887

= [119886minus

119886+

] sdot [

1

119887+

1

119887minus

] (4)


119863 (119886 119887) =

radic2

2

radic(119886minus

minus 119887minus

)2

+ (119886+

minus 119887+

)2

(5)

It is worth mentioning that the application of distancedefinition is the key point of the proposed method in thispaper Actually the distance definition is the fifth definitiongiven above In different articles the main idea of distancedefinition is the same but the expressions are always not likeeach other Particularly separations measured by Hammingdistance and Euclidean distance are discussed in paper [25]

In order to get the deviation degree matrix we mustmake some change to the decision plan Suppose 119883 is thedecision matrix of a MCDM problem with interval numbers1198601

1198602

119860119899

included representing 119899 alternatives that

Mathematical Problems in Engineering 3

need to be evaluated or 119899 decision plans to be studied119860119894

= ([1199091198941

minus

1199091198941

+

] [1199091198942

minus

1199091198942

+

] [119909119894119898

minus

119909119894119898

+

]) is the 119894thalternative and 119909

119894119895

= [1199091198941

minus

1199091198941

+

] is the 119895th attribute ofalternative 119860

119894

According to all alternatives in 119883 an idealalternative 119868 = ([119861

1

1198611

] [1198612

1198612

] [119861119898

119861119898

]) will bedecided [119861

119895

119861119895

] has the same lower and upper values whichis the best value that the 119895th attribute can reach Thenby calculating the distance between each attribute and itsideal interval that can be reached the decision matrix thatconsisted of interval attributes will be transformed to anormal matrix with true values just as shown below where119889119894119895

= (radic22)radic(119909119894119895

minus

minus 119861minus

119895

)2

+ (119909119894119895

+

minus 119861+

119895

)2 based on (5)

119883 =

[

[

[

[

[

[

[

[

[119909minus

11

119909+

11

] [119909minus

12

119909+

12

] sdot sdot sdot [119909+

1119898

119909minus

1119898

]

[119909minus

21

119909+

21

] [119909minus

22

119909+

22

] sdot sdot sdot [119909minus

2119898

119909+

2119898

]

[119909minus

1198991

119909+

1198991

] [119909minus

1198992

119909+

1198992


119899119898

119909+

119899119898

]

]

]

]

]

]

]

]

]

997888rarr 119863

=

[

[

[

[

[

[

[

11988911

11988912

sdot sdot sdot 1198891119898

11988921

11988922


1198891198991

1198891198992

sdot sdot sdot 119889119899119898

]

]

]

]

]

]

]

(6)

22 TOPSIS Method TOPSIS method a widely usedapproach for MCDM problems is developed by Hwangand Yoon in 1981 [26] and further developed by Yoon [27]and Hwang et al [28] The basic principle of this methodis that the best decision should be the closest to the idealsolution and farthest from the nonideal solution [29] Itassumes that each attribute is monotonically increasingor decreasing making it easy to locate the best and worstselection visually Normalization is usually required as theparameters or criteria are often of incongruous dimensionsin multicriteria problems [30] Euclidean distance which isgiven by the Pythagorean formula [31] is applied to measurethe alternatives Considering the Euclidian Distance Metricmay produce many symmetric solutions the weights whichrepresent the importance of each criterion are added toits criterion after the process of normalization The finalrank is reached by comparing the Euclidean distances of allalternatives

There exist some disadvantages in the traditional TOPSISmethod such that (1) the Euclidean distance algorithm it usesin principle does not consider the correlation of attributesand (2) generally the weight coefficients acquired by expertinvestigation method or AHPmethod both have subjectivityIt is also considered as a good choice for MCDM problemsbecause of the following reasons (1) It is relatively easyand fast (2) It is useful for qualitative and quantitativedata (3) The output can be a preferential ranking usingboth negative and positive criteria [32] Because of theseadvantages TOPSIS has been widely used and developed todeal with MCDM problems in fuzzy environment [14 33]In view of its simpleness and wide application we choose

TOPSIS to solve theMCDMproblems with interval numbersalso

Suppose that MCDM problem is composed of119898 alterna-tives (119860

1

1198602

119860119898

) and 119899 criteria (1198621

1198622

119862119899

) Matrix119883 = [119909

119894119895

]119898times119899

shows all values assigned to the alternatives con-cerning each criterion The related weight of each criterionhas been denoted by119882 = [119908

1

1199082

119908119899

]with the conditionsum119899

119895=1

119908119895

= 1The detailed steps of TOPSIS [34] are carried out as

follows

Step 1 Normalize the decision matrix

119903119894119895

=

119909119894119895

radicsum119898

119896=1

1199092

119896119895

119894 = 1 119898 119895 = 1 119899 (7)

where 119903119894119895

denotes the normalized value of 119895th criteria for the119894th alternative 119860

119894

Step 2 Calculate the weighted normalized decision matrix

V119894119895

= 119908119895

119903119894119895

119894 = 1 119898 119895 = 1 119899 (8)

where 119908119895

is the weight of the 119895th criteria or attribute

Step 3 Determine the positive ideal and negative idealsolutions

119860+

= V+1

V+119899

119860minus

= Vminus1

Vminus119899

(9)

where 119860+ denotes the positive ideal solution and 119860minus denotesthe negative ideal solution If the 119895th criterion is a beneficialcriterion V

119895

+

= maxV119894119895

119894 = 1 119898 and V119895

minus

= minV119894119895

119894 =

1 119898 On the contrary if the 119895th criterion is cost criterionV119895

+

= minV119894119895

119894 = 1 119898 and V119895

minus

= maxV119894119895

119894 = 1 119898

Step 4 Calculate the distances of each alternative from thepositive ideal solution and the negative ideal solution

119878+

119894

= radic

119899

sum

119895=1

(V119894119895

minus V+119895

)

2

119894 = 1 119898

119878minus

119894

= radic

119899

sum

119895=1

(V119894119895

minus Vminus119895

)

2

119894 = 1 119898

(10)

where 119878+119894

denotes the distance between the 119894th alternativeand the positive ideal solution and 119878minus

119894

denotes the distancebetween the 119894th alternative and the negative ideal solution

Step 5 Calculate the relative closeness to the ideal solution

119877119894

=

119878minus

119894

119878+

119894

+ 119878minus

119894

(11)

23 Experimental Design and Response Surface MethodExperimental design (also called design of experiment DoE


for short) is a statistical method used to determine the indi-vidual and interactive effects of many factors simultaneouslyon the system response [35] It not only provides a full insightof the interaction between design elements but also helpsturning any standard design into a robust oneThere are somekey terms in DoE Experimental domain is the experimentalfield defined by the minimum and maximum limits of theexperimental variables Factors are experimental variablesthat need to be studied and changed independently with eachother Levels are different values of each factor at which theexperiments must be carried out Response is the value ofresults measured from experiments [36]

It is important to choose an appropriate DoE methodin the experimental design application Though full factorialdesign orthogonal array technique Latin hypercube designand optimal Latin hypercube design are the four active onesthe simplest and most common one is the factorial designthat uses two levels 119899 factors that is 2119899 factorial designIt can reduce the number of experimental conditions Butdue to its inability to distinguish between linear and higherorder effects the disadvantages exist at the same time So it isjudicious to decide the number of levels in a factorial designexperiment

After the indispensable procedure mentioned above theRSM can be applied to solve the problem further In additionto analyzing the effects of each factor RSM can also generatea mathematical model The model collecting mathematicaland statistical techniques is based on the fit of a polynomialequation to the experimental data The concept of a responsesurface involves a dependent variable 119891 called the responsevariable and several independent variables 119909

1

1199092

119909119899

[37]The relationship between the response surface and the

experimental variables is given in the following equation

119891 = 1205730

+

119899

sum

119894=1

120573119894

119909119894

+

119899

sum

119894=1

119899

sum

119895=1

120573119894119895

119909119894

119909119895

+ sdot sdot sdot + 120576 119894 lt 119895 (12)

where 119899 is the number of experimental variables 120573 are theregression coefficients and 120576 is the statistical error whichrepresents other sources of variability [38]

In this paper we pick out the best and worst interval ofeach attribute to do the 2119899 factorial design After transform-ing all interval numbers to real values the data of DoE will becombinedwith TOPSISmethod Later we can use the data setthey created to generate a RS model It will help the DMs toevaluate all the alternatives simply and quickly

3 Illustration of the Proposed Method

31 Framework of the RSM-TOPSIS-IN Method There arefour basic stages in the RSM-TOPSIS-IN method In the firststage the DMs get the decision matrix of MCDM problemIn stage 2 the DMs have to acquire the deviation degreematrix of experimental design First input attributes andtheir levels are determined according to the decision matrixThen the experimental design of intervals is carried outWith the aforementioned concept presented in Section 21the deviation degreematrix of experimental design is realizedfinally The third stage of the proposed methodology is

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Getting the decision matrix

Experimental design

Transforming the interval matrix intoa deviation degree matrix

Calculating TOPSIS scores

Constructing the RS model

Ranking the alternatives

Stage 1

Stage 2

Stage 3

Stage 4

Figure 1 Steps of the RSM-TOPSIS-IN method

constructing the RS model based on the TOPSIS scoreswhich are calculated by the TOPSIS method introduced inSection 22 In the last stage the DMs are able to evaluatealternatives and determine the final ranking results using theestablished RS model simply Flow diagram of the proposedapproach is illustrated in Figure 1

In order to introduce the detailed processes of the RSM-TOPSIS-IN method a simple but persuasive problem aboutwhat investment to choose is demonstrated in the followingsection Design Expert and MATLAB two essential softwareprograms we use make the proposed method feasible Thedesign of experiment and the establishment of RS modelare realized under the help of Design Expert MATLABundertakes all the calculations during the application

32 Detailed Steps of the RSM-TOPSIS-IN Method In thissection we present a practical example to illustrate thedetailed processes of the RSM-TOPSIS-INmethod It is abouthow to select a proper investment program [39] A companyis going to develop a new product and it has formulated fiveinvestment alternatives to choose Each alternative consists ofinvestment (IN) profit of expectation (EP) profit of risk (RP)and the cost of risk (RC) whose attributes are all expressedby interval numbers instead of real values Among the fourattributes EP and RP are two beneficial attributes while INand RC are nonbeneficial ones Beneficial attribute meanshigher value is desirable whereas nonbeneficial attributeprefers lower value Next various steps will be carried out toshow the details of the RSM-TOPSIS-IN application

Step 1 (get the decision matrix) According to the specificdecision matrix shown in Table 1 IN with minimum interval[5 6] and maximum interval [10 11] EP with minimuminterval [3 5] and maximum interval [6 7] RP with min-imum interval [3 4] and maximum interval [5 7] and RC


Table 1 Decision matrix for investment program

Alternatives IN EP RP RC1198601

[5 7] [4 5] [4 6] [04 06]1198602

[10 11] [6 7] [5 6] [15 20]1198603

[5 6] [4 5] [3 4] [04 07]1198604

[9 11] [5 6] [5 7] [13 15]1198605

[6 8] [3 5] [3 4] [08 10]Minmax Min Max Max MinBest [5 6] [6 7] [5 7] [04 06]Worst [10 11] [3 5] [3 4] [15 20]

with minimum interval [04 06] and maximum interval[15 20] are determined as factor levels affecting the invest-ment program selection ldquoMaxrdquo means the attribute is bene-ficial and ldquoMinrdquo represents that the attribute is nonbeneficialFor five different alternatives 119860

1

has the lowest cost of riskthe advantage of 119860

2

is the high profit of expectation theinvestment of 119860

3

is in a relatively higher interval profit ofrisk provided by 119860

4

is the most attractive one and though1198605

does not have a best attribution some of them are betterthan others So it is really a hard work to make a decision

Step 2 (experimental design) As shown in Table 1 it is easyto pick out the best interval and the worst interval of eachattribute Because the impacts of each factor are different onranking results we utilize the experimental design to assesstheir influence In this example a two-level factorial designwas used to create a data set with all combination of the bestinterval and the worst interval of each attribute Generallythe orthogonal arrays with two-level factors are expressed by119871119860

(2119861

) where 119860 = 2119873 is the number of experiments 119873 is apositive integer which is more than 1 119861 denotes the numberof dependent factors and 2 means the experiment is taken asa two-level factorial design The letter ldquo119871rdquo comes from Latinwhich has been associated with Latin square designs from theoutset using orthogonal arrays for experimental design In theorthogonal table based on two levels four factors are given inTable 2

Step 3 (transform the orthogonal table to a deviation degreematrix) For it is not easy to handle a fuzzy MCDM problemwith interval numbers a creative method which turns theinterval number to a real value is developed It calculates thedistance between two intervals of the same attribute so theorthogonal table can be transformed to a deviation degreematrix

First determine an ideal alternative that consists of theintervals with the same upper and lower bounds based on thedecisionmatrix Because 5 7 7 04 are the best limits that canbe reached to five alternatives ([5 5] [7 7] [7 7] [04 04])is chosen as the ideal alternative in this issue Then calculatethe distance between each attribute shown in Table 2 and theideal alternative according to (5)The deviation degreematrixof experimental design is shown in Table 3

Step 4 (calculate the TOPSIS scores) In this procedurethe TOPSIS scores of all the alternatives in DoE should be

Table 2 Experimental design results of attribute factors

Exp number Factor levelsIN EP RP RC

1 [5 6] [6 7] [5 7] [04 06]2 [5 6] [6 7] [5 7] [15 20]3 [5 6] [6 7] [3 4] [04 06]4 [5 6] [6 7] [3 4] [15 20]5 [5 6] [3 5] [5 7] [04 06]6 [5 6] [3 5] [5 7] [15 20]7 [5 6] [3 5] [3 4] [04 06]8 [5 6] [3 5] [3 4] [15 20]9 [10 11] [6 7] [5 7] [04 06]10 [10 11] [6 7] [5 7] [15 20]11 [10 11] [6 7] [3 4] [04 06]12 [10 11] [6 7] [3 4] [15 20]13 [10 11] [3 5] [5 7] [04 06]14 [10 11] [3 5] [5 7] [15 20]15 [10 11] [3 5] [3 4] [04 06]16 [10 11] [3 5] [3 4] [15 20]

calculated As introduced in Section 22 the TOPSIS methodis applied to get the experimental results with twoweight sets1205961

= (028 017 023 032) is defined by article [39] and 1205962

=

(015 020 020 045) comes from article [40] The TOPSISscores of experimental design are presented in Table 3 also

Step 5 (construct the RS model) In addition to the maineffects of four factors interactions between two factors arealso included in the RS model as shown in the followingequation

119877 = 1205730

+

4

sum

119894=1

120573119894

119909119894

+

4

sum

119894=1

4

sum

119895=1

120573119894119895

119909119894

119909119895

+ 120576 (13)

where 119877 is the relevant 119877119894

of sixteen alternatives 1205730

is theintercept coefficient 120573

119894

means the first-order effect of factor119894 and 120573

119894119895

shows the two-factor interaction between factors 119894and 119895 when 119894 = 119895 just as introduced before

In order to check out whether the established RS modelcan describe the real relationship or not it must be analyzedby the analysis of variance (ANOVA) procedureTheANOVAresults can be achieved with the help of the software calledDesign Expert It gives a summary of the main effects andinteraction effects of factors which helps us to get the final RSmodel fitting to the experimental data In the ANOVA tablevalues of ldquo119901 valuerdquo less than 00500 indicate model terms aresignificant If the value of ldquoPred 119877-Squaredrdquo is in reasonableagreement with the value of ldquoAdj119877-Squaredrdquo it will prove thatthis model can be used to navigate the design space

As shown inTable 4 for the RSmodel with1205961

IN EP andRC are three significant model terms The value of ldquoPred 119877-Squaredrdquo is 07988 and the value of ldquoAdj119877-Squaredrdquo is 08825indicating an adequate signal that this model can be used toget the final ranking results The same analysis is also donefor the RS model with 120596

2

In this model four factors areall available and the value of ldquoPred 119877-Squaredrdquo is in good


Table 3 The deviation degree matrix and TOPSIS scores of experimental design

Exp number Factor levels TOPSIS scoresIN EP RP RC 120596

1

1205962

1 07071 07071 14142 01414 0 02 07071 07071 14142 13730 04912 064463 07071 07071 35355 01414 02445 020034 07071 07071 35355 13730 05401 068545 07071 31623 14142 01414 02166 022876 07071 31623 14142 13730 05277 072047 07071 31623 35355 01414 03083 028748 07071 31623 35355 13730 05781 075729 07071 07071 14142 01414 04170 0229110 55227 07071 14142 13730 06881 0710711 55227 07071 35355 01414 04665 0288312 55227 07071 35355 13730 07808 0769513 55227 31623 14142 01414 04543 0306214 55227 31623 14142 13730 07525 0797915 55227 31623 35355 01414 05021 0348716 55227 31623 35355 13730 09818 09781

Table 4 The ANOVA for two RS models

RS model with 1205961


Source Coefficient 119901 value Source Coefficient 119901 valueIntercept minus010875 00000 Intercept minus012387 00000IN +322986 00013 IN +262822 00175EP +245232 00886 EP +266299 00209RP +254492 00482 RP +250448 00466RC +345764 00004 RC +374815 lt00001IN lowast EP minus122932 09619 IN lowast EP minus142959 09653IN lowast RC minus013025 09896 IN lowast RC +041041 09693EP lowast RP minus229479 09624 EP lowast RP minus315194 09344EP lowast RC minus084065 09693 EP lowast RC minus106550 09201RP lowast RC minus067275 09716 RP lowast RC minus116626 09252Adj 119877-Squared 08825 Adj 119877-Squared 09823Pred 119877-Squared 07988 Pred 119877-Squared 09469

agreement with the value of ldquoAdj 119877-Squaredrdquo Based on theANOVA results 119877

1

and 1198772

are established according to 1205961

and 1205962

respectively

1198771

= minus010875 + 322986 lowast IN + 254492 lowast RP

+ 345764 lowast RC(14)

1198772

= minus012387 + 262822 lowast IN + 266299 lowast EP

+ 250448 lowast RP + 374815 lowast RC(15)

Step 6 (rank the alternatives) After two RSmodels have beenestablished we can simply use (14) and (15) to get the rankingscores of each alternative The deviation degree matrix of

decision matrix is realized by repeating the steps of 1 3 4and 5 introduced above

The ranking results of the RSM-TOPSIS-IN method arethen compared with the ones obtained from two methodsbrought up by the same author but in different articles [3940] respectively The results shown in Table 5 provide theinformation that the ranking results are exactly the same asthe two other existing methods which lead to the conclusionthat the RSM-TOPSIS-INmethod could be used to determinethe final choice of investment program

When applying the proposed method the DMs make theproblem a common case and easy to solve by turning thedecision matrix with interval numbers to a deviation degreematrix with exact values For the calculation between twointervals costs is twice the time of the calculation between


Table 5 The ranking results with two weight sets

Alternatives Ranking results with 1205961

Ranking results with 1205962

Scores RSM-TOPSIS-IN Zhu and Chen [39] Scores RSM-TOPSIS-IN Zhu [40]1198601

01262 1 1 02059 1 11198602

07055 5 5 07618 5 51198603

01728 2 2 02772 2 21198604

05598 4 4 06170 4 41198605

03510 3 3 04868 3 3

Table 6 The performance characteristics of suppliers

Suppliers Delivery lead time (A) Product price (B) Product quality (C) Assortment flexibility (D)1198781

[089 093] [233 239] [16 20] [094 098]1198782

[096 100] [240 246] [23 27] [089 093]1198783

[090 094] [197 203] [19 23] [090 094]1198784

[095 099] [225 231] [21 25] [092 096]1198785

[093 097] [212 218] [18 22] [095 099]Minmax Max Min Min MaxBest [096 100] [197 203] [16 20] [095 099]Worst [089 093] [240 246] [23 27] [089 093]

Table 7 The ANOVA for RS model

Source Coefficient 119901 value Source Coefficient 119901 valueIntercept minus016128 00000 AC 836119864 minus 16 10000A 337115 00011 AD 172119864 minus 14 10000B 352842 00005 BC 882119864 minus 15 10000C 271392 00259 BD 506119864 minus 15 10000D 218969 02018 CD 789119864 minus 14 10000AB 000000 10000Adj 119877-Squared 08795 Pred 119877-Squared 08723

two numbers the translation will save half of the time Whena new alternative is added or removed the DMs have no needto deal with it step by step again but they only need to get theranking scores by the RSmodel established before saving halfan hour that the whole procedures usually costs As a resultthe more alternatives are added the more time will be saved

4 Application and Discussion

It is not easy to make sure that a new proposed methodis more reasonable and reliable than the other existingmethods in solving different MCDM problems In order todemonstrate the applicability and potentiality of the methodexplained in our paper it is appropriate to make the finaldecision by applying several MCDM approaches to comparetheir ranking results for the same problemThe following twoillustrative examples are taken to fulfill the task

41 Selection of a Right Partner As a manufacturer selectinga right partner is as important as making a high qualityproduct In this case study a practical problem with four

Table 8 The ranking results

Alternatives Ranking resultsScores RSM-TOPSIS-IN Zhou [41]

1198781

05826 5 51198782

04932 4 41198783

03310 2 31198784

03745 3 11198785

02962 1 2

performance attributes and five alternative partners is pre-sented [41]Themanufacturer has to choose themost suitablesupplier to assure the performance of product Deliverylead time product price product quality and assortmentflexibility are the four factors concerned We would like tominimize both the delivery lead time and product pricewhile maximizing product quality and assortment flexibilityThe author obtained the normalized weight of these criteriaas 120596 = (036 030 021 013) [41] and the performancecharacteristics of suppliers are stated in Table 6


Table 9 Decision matrix for road transport technologies

Technologies GHG (A) PM10 (B) NO119909

(C) CO (D) HCs (E) Cost (F)HEV [100 400] [0001 0035] [010 030] [010 05] [030 110] [1755 1875]Av BEV [125 300] [0035 0135] [018 070] [010 04] [018 060] [180 1905]Re BEV [25 80] [0005 0018] [005 015] [005 01] [005 015] [188 194]LPG [125 450] [0007 0035] [017 037] [050 12] [018 065] [192 222]CNG [120 420] [0010 0025] [005 020] [049 10] [050 150] [148 166]Petrol [125 500] [0010 0035] [010 040] [049 10] [040 142] [168 192]Diesel [115 430] [0010 0060] [025 062] [020 05] [015 062] [125 139]Bioethanol [80 350] [0023 0120] [038 120] [100 35] [019 048] [143 170]Biodiesel [70 300] [0005 0070] [044 110] [02 075] [038 120] [125 138]Minmax Min Min Min Min Min MinBest [25 80] [0001 0035] [005 015] [005 01] [005 015] [125 138]Worst [125 500] [0035 0135] [038 120] [100 35] [050 150] [192 222]

Table 10 Twoweight sets for assessment of road transport technolo-gies

Weight Emission criteriaGHG PM10 NO

119909

CO HCs Cost1205961

016 016 016 016 016 021205962

004 004 004 004 004 08

In order to deal with this case ([100 100] [197 197][16 16] [099 099]) is considered as the ideal alternativeWith the proposed method we are able to generate a RSmodel expressed in the following equation to evaluate thealternatives

119877 = minus016128 + 337115 lowast 119860 + 352842 lowast 119861

+ 271392 lowast 119862 + 218969 lowast 119863

(16)

The ANOVA results are shown in Table 7 specificallyassuring the RS model is available We obtain the ranking ofthe alternatives as 5-4-2-3-1 which are almost the same as theresults obtained from [41] The detailed results are tabulatedin Table 8

42 Assessment of Road Transport Technologies It is men-tioned that around a quarter of the European Union green-house gas emissions are caused by transport so the paperaims at assessing energy technologies in road transport sectorin terms of atmospheric emissions and cost [42] A TOPSISmethod for interval data is proposed and applied to indicatewhat the most competitive and environmentally friendlytransport technology is

There are nine alternative transports and six performanceattributes including five regulated pollutants and the costFor the problem we would like to minimize both regulatedpollutants and the cost The parameters of different modes ofcars are given in Table 9

Using the proposed methodology in our paper a six-factor mathematical model is developed The detailed cal-culation processes are omitted here As shown in Table 10

the road transport technologies were assessed in terms oftwo weight sets 120596

1

is the preference under environmentalconcern and 120596

2

represents the opinions of consumers whichare all chosen from paper [42] According to the ANOVAresults shown in Table 11 we can get the following two RSmodels respectively

1198771

= minus0060525 + 186822119860 + 185112119861 + 191909119862

+ 203517119863 + 194725119864 + 221114119865

(17)

1198772

= minus0016954 + 146548119860 + 144383119861 + 150011119862

+ 159839119863 + 152191119864 + 348365119865

(18)

The comparative results presented in Table 12 show thatthe ranks are almost the same which prove the acceptabilityof the RSM-TOPSIS-IN method in getting the rank of roadtransport technologies

5 Conclusion

In this paper a hybrid method called RSM-TOPSIS-IN isproposed to solve MCDM problems with interval data BothTOPSIS and RSM are applied within the framework Itsmain thought distinguished from other methods is that ittransforms the interval matrix into a deviation degreematrixmaking thematter easy to solve almost like a normal problemwith the exact values There are two main advantages of theRSM-TOPSIS-IN compared with the other existing methods

(1) For RSM greatly reduces the cost time and amountof calculation steps when applying the proposedmethod DMs can use the obtained RS model tochoose and analyse factors and attributes easily nomatter whether they have good numerical capabilitiesor not So simplicity is treated as its first superiority

(2) If a new alternative is added to or removed from theMCDM problem the DMs have no need to take theprocedures step by step from the beginning but theycan only use the RS model established to get the final


Table 11 The ANOVA results for two RS models



Source Coefficient 119901 value Source Coefficient 119901 valueIntercept minus0060525 00000 Intercept minus0016954 00000A +186822 lt00001 A +146548 lt00001B +185112 lt00001 B +144383 lt00001C +191909 lt00001 C +150011 lt00001D +203517 lt00001 D +159839 lt00001E +194725 lt00001 E +152191 lt00001F +221114 lt00001 F +348365 lt00001AB +125119864 minus 14 10000 AB +167119864 minus 13 10000AC minus107119864 minus 13 10000 AC +290119864 minus 13 10000AD minus165119864 minus 14 10000 AD minus121119864 minus 13 10000AE minus876119864 minus 14 10000 AE minus122119864 minus 13 10000BC +219119864 minus 14 10000 BC +141119864 minus 13 10000BD minus885119864 minus 15 10000 BD +599119864 minus 14 10000BE +697119864 minus 14 10000 BE +131119864 minus 13 10000BF +644119864 minus 14 10000 BF +529119864 minus 14 10000CD +482119864 minus 15 10000 CD +234119864 minus 13 10000CE +307119864 minus 14 10000 CE +526119864 minus 14 10000CF minus603119864 minus 14 10000 CF minus551119864 minus 15 10000DE minus260119864 minus 14 10000 DE +604119864 minus 15 10000DF minus120119864 minus 14 10000 DF minus507119864 minus 14 10000EF +127119864 minus 14 10000 EF minus260119864 minus 14 10000Adj 119877-Squared 08965 Adj 119877-Squared 09909Pred 119877-Squared 08398 Pred 119877-Squared 09860


Technologies Ranking results with 1205961


Scores RSM-TOPSIS-IN Streimikiene et al [42] Scores RSM-TOPSIS-IN Streimikiene et al [42]HEV 05678 3 2 06270 6 5Av BEV 06970 8 8 06352 7 8Re BEV 03193 1 1 06473 8 7LPG 06830 6 4 06862 9 9CNG 05730 4 5 05745 3 3Petrol 06945 7 6 06259 5 6Diesel 04793 2 3 05210 1 1Bioethanol 09303 9 9 05772 4 4Biodiesel 06202 5 7 05225 2 2

results reducing both time and effort This makesefficiency the other advantage

Three examples illustrated later verify the capability ofthis method indicating it is simple to comprehend andeasy to implement Due to its convenient and practicabilityit is easy to draw a conclusion that the RSM-TOPSIS-INmethod is suitable for solvingMCDMproblems with intervalnumbers The same hybrid idea can also be applied to othermethodologies which may become a new direction of ourfuture research

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and the anonymousreferees for their insightful and constructive comments andsuggestions which have been very helpful in improving thispaper This research was supported by the National Natural


Science Foundation of China under Grant no 51375389Fundamental Research Funds for the Central Universitiesunder Grant no 3102014JCQ01007 and Civil Aviation JointFund under Grant no U1333133

References

[1] A Afshari M Mojahed and R M Yusuff ldquoSimple additiveweighting approach to personnel selection problemrdquo Interna-tional Journal of Innovation Management and Technology vol1 pp 511ndash515 2010

[2] M F Chen and G H Tzeng ldquoCombining grey relation andTOPSIS concepts for selecting an expatriate host countryrdquoMathematical and ComputerModelling vol 40 no 13 pp 1473ndash1490 2004

[3] T L Saaty ldquoDecision making with the analytic hierarchyprocessrdquo International Journal of Services Sciences vol 1 pp 83ndash98 2008

[4] P Wang P Meng J-Y Zhai and Z-Q Zhu ldquoA hybridmethodusing experiment design and grey relational analysis formultiple criteria decision making problemsrdquo Knowledge-BasedSystems vol 53 pp 100ndash107 2013

[5] W K M Brauers and E K Zavadskas ldquoThe MOORA methodand its application to privatization in a transition economyrdquoControl and Cybernetics vol 35 no 2 pp 445ndash469 2006

[6] T Yang and C-C Hung ldquoMultiple-attribute decision mak-ing methods for plant layout design problemrdquo Robotics andComputer-Integrated Manufacturing vol 23 no 1 pp 126ndash1372007

[7] M Ehrgott K Klamroth and C Schwehm ldquoAn MCDMapproach to portfolio optimizationrdquo European Journal of Oper-ational Research vol 155 no 3 pp 752ndash770 2004

[8] L Anojkumar M Ilangkumaran and V Sasirekha ldquoCompara-tive analysis of MCDM methods for pipe material selection insugar industryrdquo Expert Systems with Applications vol 41 no 6pp 2964ndash2980 2014

[9] J Kangas A Kangas P Leskinen and J Pykalainen ldquoMCDMmethods in strategic planning of forestry on state-owned landsin finland applications and experiencesrdquo Journal of Multi-Criteria Decision Analysis vol 10 pp 257ndash271 2001

[10] T-C Wang and T-H Chang ldquoApplication of TOPSIS inevaluating initial training aircraft under a fuzzy environmentrdquoExpert Systems with Applications vol 33 no 4 pp 870ndash8802007

[11] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

[12] M A Abo-Sinna and A H Amer ldquoExtensions of TOPSIS formulti-objective large-scale nonlinear programming problemsrdquoApplied Mathematics and Computation vol 162 no 1 pp 243ndash256 2005

[13] Y T Ic ldquoAn experimental design approach using TOPSISmethod for the selection of computer-integratedmanufacturingtechnologiesrdquo Robotics and Computer-Integrated Manufactur-ing vol 28 no 2 pp 245ndash256 2012

[14] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoAn algorith-mic method to extend TOPSIS for decision-making problemswith interval datardquo Applied Mathematics and Computation vol175 no 2 pp 1375ndash1384 2006

[15] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[16] F E Boran S Genc M Kurt and D Akay ldquoA multi-criteriaintuitionistic fuzzy groupdecisionmaking for supplier selectionwith TOPSISmethodrdquoExpert Systemswith Applications vol 36no 8 pp 11363ndash11368 2009

[17] A H I Lee W-C Chen and C-J Chang ldquoA fuzzy AHP andBSC approach for evaluating performance of IT departmentin the manufacturing industry in Taiwanrdquo Expert Systems withApplications vol 34 no 1 pp 96ndash107 2008

[18] A Peng and Z Wang ldquoGRA-based TOPSIS decision-makingapproach to supplier selection with interval numberrdquo in Pro-ceedings of the Chinese Control and Decision Conference (CCDCrsquo11) pp 1742ndash1747 May 2011

[19] S Chen C Hwang and F Hwang Fuzzy Multiple AttributeDecision Making vol 375 of Lecture Notes in Economics andMathematical Systems Springer Berlin Germany 1992

[20] C Carlsson and R Fuller ldquoFuzzy multiple criteria decisionmaking recent developmentsrdquo Fuzzy Sets and Systems vol 78no 2 pp 139ndash153 1996

[21] E Kaucher ldquoInterval analysis in the extended interval spaceIRrdquo in Fundamentals of Numerical Computation ComputerOriented Numerical Analysis pp 33ndash49 Springer BerlinGermany 1980

[22] V Y C Chen H-P Lien C-H Liu J J H Liou G-H Tzengand L-S Yang ldquoFuzzy MCDM approach for selecting the bestenvironment-watershed planrdquo Applied Soft Computing vol 11no 1 pp 265ndash275 2011

[23] C Jiang X Han F J Guan and Y H Li ldquoAn uncertainstructural optimization method based on nonlinear intervalnumber programming and interval analysis methodrdquo Engineer-ing Structures vol 29 no 11 pp 3168ndash3177 2007

[24] X Zhang and P Liu ldquoMethod for multiple attribute decision-making under riskwith interval numbersrdquo International Journalof Fuzzy Systems vol 12 no 3 pp 237ndash242 2010

[25] G-H Tzeng and J-J Huang Multiple Attribute Decision Mak-ing Methods and Applications CRC Press New York NY USA2011

[26] C Hwang and K Yoon Multiple Attribute Decision MakingMethods and Application Springer New York NY USA 1981

[27] K Yoon ldquoA reconciliation among discrete compromise solu-tionsrdquo Journal of the Operational Research Society vol 38 no3 pp 277ndash286 1987

[28] C-L Hwang Y-J Lai and T-Y Liu ldquoA new approach formultiple objective decision makingrdquo Computers amp OperationsResearch vol 20 no 8 pp 889ndash899 1993

[29] P Karande and S Chakraborty ldquoApplication of multi-objectiveoptimization on the basis of ratio analysis (MOORA) methodfor materials selectionrdquoMaterials amp Design vol 37 pp 317ndash3242012

[30] K P Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction Sage Thousand Oaks Calif USA1995

[31] M M Deza and E Deza Encyclopedia of Distances SpringerNew York NY USA 2009

[32] M F Aly H A Attia and A M Mohammed ldquoIntegrated fuzzy(GMM)-TOPSIS model for best design concept and materialselection processrdquo International Journal of Innovative Researchin Science Engineering and Technology vol 2 pp 6464ndash64862013

[33] X Zhang and Z Xu ldquoExtension of TOPSIS to multiple criteriadecision making with pythagorean fuzzy setsrdquo InternationalJournal of Intelligent Systems vol 29 pp 1061ndash1078 2014


[34] J R San Cristobal Multi Criteria Analysis in the RenewableEnergy Industry Springer Berlin Germany 2012

[35] R E Kirk Experimental Design JohnWiley amp Sons New YorkNY USA 1982

[36] M A Bezerra R E Santelli E P Oliveira L S Villar and L AEscaleira ldquoResponse surface methodology (RSM) as a tool foroptimization in analytical chemistryrdquo Talanta vol 76 no 5 pp965ndash977 2008

[37] A Kannan K Esakkiraja and M Nataraj ldquoModeling andanalysis for cutting temperature in turning of aluminium6063 using response surface methodologyrdquo IOSR Journal ofMechanical and Civil Engineering vol 9 pp 59ndash64 2013

[38] Y Lian and M-S Liou ldquoMultiobjective optimization usingcoupled response surface model and evolutionary algorithmrdquoAIAA Journal vol 43 no 6 pp 1316ndash1325 2005

[39] F-X Zhu and H-Y Chen ldquoA method of entropy for obtainingthe attribute weights of interval numbers decision-makingmatrixrdquo Journal of Anhui University (Natural Science Edition)vol 5 pp 4ndash6 2006

[40] F-X Zhu ldquoAnewmethod based on deviation degree for priorityof decision-making matrixrdquo Journal of Hefei University vol 15pp 1ndash4 2005

[41] C Zhou ldquoApproach to incidence decision making modelof multi-attribute interval number on supplier selectionrdquo inProceedings of the International Conference on Intelligent Controland Information Processing (ICICIP rsquo10) pp 78ndash82 IEEEAugust 2010

[42] D Streimikiene T Balezentis and L Balezentiene ldquoCompara-tive assessment of road transport technologiesrdquo Renewable andSustainable Energy Reviews vol 20 pp 611ndash618 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and also there is a good survey done by Chen et al [19] whomake distinctions between fuzzy ranking methods and fuzzymultiple attribute decision-making methods

Compared with the traditional MCDM models thefuzzy versions of MCDM applications are very complex tocomprehend and sometimes difficult to implement [20] Onone hand while the MCDM model consisted of intervalnumbers too much mathematical calculation leads to morecomputation time making it impracticable and ineffectiveto the decision makers (DMs) without a strong capability inmathematics At the same time the range of intervals maybecome more and more wide along with the calculationwhich will be more difficult for the DMs to make the finalchoice On the other hand if the decision alternatives arechanged or an alternative is added or removed the DMshave to analyse the problem from the beginning again Itis really a waste of time So how to find some simple waysto deal with the MCDM problems with interval numbersbecomes an important study topic for the researchers Theonewhich involves a unique expression to fulfill differing taskrequirements may be an ideal choice

In this paper we present a newmethod based on TOPSISand response surface method (RSM) for MCDM problemswith interval numbers (RSM-TOPSIS-IN) In experimentaldesign the best and worst intervals of each attribute arecombined to represent all the possibilities of alternativesapproximately With the application of deviation degreematrix the experimental data can be used to get a responsesurface (RS) model helping the DMs to acquire the rankingresults through the established equation simply What ismore if some new alternatives are considered later we canonly use the established RS model to get the ranking resultseasily reducing both time and effort a lot Three practicalMCDM problems that consisted of interval numbers will besolved by the RSM-TOPSIS-IN method we proposed Thecompared results will show that it can help the DMs to get themost appropriate alternative in complex MCDM problemsusing both less time and less calculation

The reminder of this paper is organized as followsSection 2 focuses onpresenting a brief reviewof relatedmeth-ods involving interval numbers TOPSIS method exper-imental design and response surface method Then thedetailed processes of the RSM-TOPSIS-IN method are statedin Section 3 In Section 4 two more illustrative examples aredescribed to demonstrate the capabilities of the proposedmethod Finally conclusions and future research areas arediscussed in Section 5

2 Backgrounds

21 Interval Numbers In the real world the decision-makingproblems are always very vague and uncertain which maybe expressed by a variety of ways such as bounded num-ber ordinal number interval number and fuzzy numberConsidering these problems are not like the conventionalissueswith exact values they cannot be evaluated by commonmethods BellmanampZadeh [15] are the first ones who studiedthe decision-making problem under a fuzzy environment-watershed and they heralded the initiation of fuzzy MCDM

Whereafter this analysis method is widely used to dealwith decision-making problems involving multiple criteriaevaluation of alternatives

Among all the fuzzy MCDM problems interval numbersand fuzzy numbers are used the most to construct thedecision matrix Interval numbers are a set of real numberswith the property that any number that lies between twonumbers is also included in the set [21] Fuzzy numbers areexpressed by a fuzzy subset of real numbers representingthe expansion of the idea of the confidence interval [22]for the fuzzy numbers are developed based on the intervalnumbers In this paper we only talk about the special partof fuzzy MCDM problems which are the problems withinterval numbers actually Though there have been somearticles aiming at the study on interval numbers in recentyears finding a more complete and robust methodology isstill needed It is worth mentioning that transforming theuncertain problem to a deterministic problem seems a goodway for this sort of problems [23]

In this section some basic definitions [24] used to solvethe interval numbers will be introduced briefly helping usto know more about the interval numbers Suppose 119886 =

[119886minus

119886+

] = 119909 | 0 le 119886minus

le 119909 le 119886+

119887 = [119887minus 119887+] = 119909 |0 le 119887minus

le 119909 le 119887+

are two positive interval numbers and thenone can find the following


119886 + 119887 = [119886minus

+ 119887minus

119886+

+ 119887+

] (1)


119886 minus 119887 = [119886minus

minus 119887+

119886+

minus 119887minus

] (2)


119886 times 119887 = [min (119886minus119887minus 119886minus119887+ 119886+119887minus 119886+119887+)

max (119886minus119887minus 119886minus119887+ 119886+119887minus 119886+119887+)] (3)


119886

119887

= [119886minus

119886+

] sdot [

1

119887+

1

119887minus

] (4)


119863 (119886 119887) =

radic2

2

radic(119886minus

minus 119887minus

)2

+ (119886+

minus 119887+

)2

(5)

It is worth mentioning that the application of distancedefinition is the key point of the proposed method in thispaper Actually the distance definition is the fifth definitiongiven above In different articles the main idea of distancedefinition is the same but the expressions are always not likeeach other Particularly separations measured by Hammingdistance and Euclidean distance are discussed in paper [25]

In order to get the deviation degree matrix we mustmake some change to the decision plan Suppose 119883 is thedecision matrix of a MCDM problem with interval numbers1198601

1198602

119860119899

included representing 119899 alternatives that



= ([1199091198941

minus

1199091198941

+

] [1199091198942

minus

1199091198942

+

] [119909119894119898

minus

119909119894119898

+


119894119895

= [1199091198941

minus

1199091198941

+


119894


1

1198611

] [1198612

1198612

] [119861119898

119861119898


119895

119861119895


= (radic22)radic(119909119894119895

minus

minus 119861minus

119895

)2

+ (119909119894119895

+

minus 119861+

119895

)2 based on (5)

119883 =

[

[

[

[

[

[

[

[

[119909minus

11

119909+

11

] [119909minus

12

119909+

12


1119898

119909minus

1119898

]

[119909minus

21

119909+

21

] [119909minus

22

119909+

22


2119898

119909+

2119898

]

[119909minus

1198991

119909+

1198991

] [119909minus

1198992

119909+

1198992


119899119898

119909+

119899119898

]

]

]

]

]

]

]

]

]

997888rarr 119863

=

[

[

[

[

[

[

[

11988911

11988912


11988921

11988922


1198891198991

1198891198992

sdot sdot sdot 119889119899119898

]

]

]

]

]

]

]

(6)





1

1198602

119860119898

) and 119899 criteria (1198621

1198622

119862119899

) Matrix119883 = [119909

119894119895

]119898times119899


1

1199082

119908119899


119895=1

119908119895


follows


119903119894119895

=

119909119894119895

radicsum119898

119896=1

1199092

119896119895

119894 = 1 119898 119895 = 1 119899 (7)

where 119903119894119895


119894


V119894119895

= 119908119895

119903119894119895

119894 = 1 119898 119895 = 1 119899 (8)

where 119908119895



119860+

= V+1

V+119899

119860minus

= Vminus1

Vminus119899

(9)


119895

+

= maxV119894119895

119894 = 1 119898 and V119895

minus

= minV119894119895

119894 =


+

= minV119894119895

119894 = 1 119898 and V119895

minus

= maxV119894119895

119894 = 1 119898


119878+

119894

= radic

119899

sum

119895=1

(V119894119895

minus V+119895

)

2

119894 = 1 119898

119878minus

119894

= radic

119899

sum

119895=1

(V119894119895

minus Vminus119895

)

2

119894 = 1 119898

(10)

where 119878+119894


119894



119877119894

=

119878minus

119894

119878+

119894

+ 119878minus

119894

(11)






1

1199092

119909119899



119891 = 1205730

+

119899

sum

119894=1

120573119894

119909119894

+

119899

sum

119894=1

119899

sum

119895=1

120573119894119895

119909119894

119909119895

+ sdot sdot sdot + 120576 119894 lt 119895 (12)





Step 1

Step 2

Step 3

Step 4

Step 5

Step 6


Experimental design





Stage 1

Stage 2

Stage 3

Stage 4









[5 7] [4 5] [4 6] [04 06]1198602

[10 11] [6 7] [5 6] [15 20]1198603

[5 6] [4 5] [3 4] [04 07]1198604

[9 11] [5 6] [5 7] [13 15]1198605



1


2


3


4




(2119861







1 [5 6] [6 7] [5 7] [04 06]2 [5 6] [6 7] [5 7] [15 20]3 [5 6] [6 7] [3 4] [04 06]4 [5 6] [6 7] [3 4] [15 20]5 [5 6] [3 5] [5 7] [04 06]6 [5 6] [3 5] [5 7] [15 20]7 [5 6] [3 5] [3 4] [04 06]8 [5 6] [3 5] [3 4] [15 20]9 [10 11] [6 7] [5 7] [04 06]10 [10 11] [6 7] [5 7] [15 20]11 [10 11] [6 7] [3 4] [04 06]12 [10 11] [6 7] [3 4] [15 20]13 [10 11] [3 5] [5 7] [04 06]14 [10 11] [3 5] [5 7] [15 20]15 [10 11] [3 5] [3 4] [04 06]16 [10 11] [3 5] [3 4] [15 20]



=



119877 = 1205730

+

4

sum

119894=1

120573119894

119909119894

+

4

sum

119894=1

4

sum

119895=1

120573119894119895

119909119894

119909119895

+ 120576 (13)




119894


119894119895





2





1

1205962

1 07071 07071 14142 01414 0 02 07071 07071 14142 13730 04912 064463 07071 07071 35355 01414 02445 020034 07071 07071 35355 13730 05401 068545 07071 31623 14142 01414 02166 022876 07071 31623 14142 13730 05277 072047 07071 31623 35355 01414 03083 028748 07071 31623 35355 13730 05781 075729 07071 07071 14142 01414 04170 0229110 55227 07071 14142 13730 06881 0710711 55227 07071 35355 01414 04665 0288312 55227 07071 35355 13730 07808 0769513 55227 31623 14142 01414 04543 0306214 55227 31623 14142 13730 07525 0797915 55227 31623 35355 01414 05021 0348716 55227 31623 35355 13730 09818 09781






1

and 1198772


and 1205962

respectively

1198771


+ 345764 lowast RC(14)

1198772












01262 1 1 02059 1 11198602

07055 5 5 07618 5 51198603

01728 2 2 02772 2 21198604

05598 4 4 06170 4 41198605

03510 3 3 04868 3 3



[089 093] [233 239] [16 20] [094 098]1198782

[096 100] [240 246] [23 27] [089 093]1198783

[090 094] [197 203] [19 23] [090 094]1198784

[095 099] [225 231] [21 25] [092 096]1198785










1198781

05826 5 51198782

04932 4 41198783

03310 2 31198784

03745 3 11198785

02962 1 2








119909

CO HCs Cost1205961

016 016 016 016 016 021205962

004 004 004 004 004 08



+ 271392 lowast 119862 + 218969 lowast 119863

(16)






1


2


1198771

= minus0060525 + 186822119860 + 185112119861 + 191909119862

+ 203517119863 + 194725119864 + 221114119865

(17)

1198772

= minus0016954 + 146548119860 + 144383119861 + 150011119862

+ 159839119863 + 152191119864 + 348365119865

(18)


5 Conclusion

















Acknowledgments




References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= ([1199091198941

minus

1199091198941

+

] [1199091198942

minus

1199091198942

+

] [119909119894119898

minus

119909119894119898

+


119894119895

= [1199091198941

minus

1199091198941

+


119894


1

1198611

] [1198612

1198612

] [119861119898

119861119898


119895

119861119895


= (radic22)radic(119909119894119895

minus

minus 119861minus

119895

)2

+ (119909119894119895

+

minus 119861+

119895

)2 based on (5)

119883 =

[

[

[

[

[

[

[

[

[119909minus

11

119909+

11

] [119909minus

12

119909+

12


1119898

119909minus

1119898

]

[119909minus

21

119909+

21

] [119909minus

22

119909+

22


2119898

119909+

2119898

]

[119909minus

1198991

119909+

1198991

] [119909minus

1198992

119909+

1198992


119899119898

119909+

119899119898

]

]

]

]

]

]

]

]

]

997888rarr 119863

=

[

[

[

[

[

[

[

11988911

11988912


11988921

11988922


1198891198991

1198891198992

sdot sdot sdot 119889119899119898

]

]

]

]

]

]

]

(6)





1

1198602

119860119898

) and 119899 criteria (1198621

1198622

119862119899

) Matrix119883 = [119909

119894119895

]119898times119899


1

1199082

119908119899


119895=1

119908119895


follows


119903119894119895

=

119909119894119895

radicsum119898

119896=1

1199092

119896119895

119894 = 1 119898 119895 = 1 119899 (7)

where 119903119894119895


119894


V119894119895

= 119908119895

119903119894119895

119894 = 1 119898 119895 = 1 119899 (8)

where 119908119895



119860+

= V+1

V+119899

119860minus

= Vminus1

Vminus119899

(9)


119895

+

= maxV119894119895

119894 = 1 119898 and V119895

minus

= minV119894119895

119894 =


+

= minV119894119895

119894 = 1 119898 and V119895

minus

= maxV119894119895

119894 = 1 119898


119878+

119894

= radic

119899

sum

119895=1

(V119894119895

minus V+119895

)

2

119894 = 1 119898

119878minus

119894

= radic

119899

sum

119895=1

(V119894119895

minus Vminus119895

)

2

119894 = 1 119898

(10)

where 119878+119894


119894



119877119894

=

119878minus

119894

119878+

119894

+ 119878minus

119894

(11)






1

1199092

119909119899



119891 = 1205730

+

119899

sum

119894=1

120573119894

119909119894

+

119899

sum

119894=1

119899

sum

119895=1

120573119894119895

119909119894

119909119895

+ sdot sdot sdot + 120576 119894 lt 119895 (12)





Step 1

Step 2

Step 3

Step 4

Step 5

Step 6


Experimental design





Stage 1

Stage 2

Stage 3

Stage 4









[5 7] [4 5] [4 6] [04 06]1198602

[10 11] [6 7] [5 6] [15 20]1198603

[5 6] [4 5] [3 4] [04 07]1198604

[9 11] [5 6] [5 7] [13 15]1198605



1


2


3


4




(2119861







1 [5 6] [6 7] [5 7] [04 06]2 [5 6] [6 7] [5 7] [15 20]3 [5 6] [6 7] [3 4] [04 06]4 [5 6] [6 7] [3 4] [15 20]5 [5 6] [3 5] [5 7] [04 06]6 [5 6] [3 5] [5 7] [15 20]7 [5 6] [3 5] [3 4] [04 06]8 [5 6] [3 5] [3 4] [15 20]9 [10 11] [6 7] [5 7] [04 06]10 [10 11] [6 7] [5 7] [15 20]11 [10 11] [6 7] [3 4] [04 06]12 [10 11] [6 7] [3 4] [15 20]13 [10 11] [3 5] [5 7] [04 06]14 [10 11] [3 5] [5 7] [15 20]15 [10 11] [3 5] [3 4] [04 06]16 [10 11] [3 5] [3 4] [15 20]



=



119877 = 1205730

+

4

sum

119894=1

120573119894

119909119894

+

4

sum

119894=1

4

sum

119895=1

120573119894119895

119909119894

119909119895

+ 120576 (13)




119894


119894119895





2





1

1205962

1 07071 07071 14142 01414 0 02 07071 07071 14142 13730 04912 064463 07071 07071 35355 01414 02445 020034 07071 07071 35355 13730 05401 068545 07071 31623 14142 01414 02166 022876 07071 31623 14142 13730 05277 072047 07071 31623 35355 01414 03083 028748 07071 31623 35355 13730 05781 075729 07071 07071 14142 01414 04170 0229110 55227 07071 14142 13730 06881 0710711 55227 07071 35355 01414 04665 0288312 55227 07071 35355 13730 07808 0769513 55227 31623 14142 01414 04543 0306214 55227 31623 14142 13730 07525 0797915 55227 31623 35355 01414 05021 0348716 55227 31623 35355 13730 09818 09781






1

and 1198772


and 1205962

respectively

1198771


+ 345764 lowast RC(14)

1198772












01262 1 1 02059 1 11198602

07055 5 5 07618 5 51198603

01728 2 2 02772 2 21198604

05598 4 4 06170 4 41198605

03510 3 3 04868 3 3



[089 093] [233 239] [16 20] [094 098]1198782

[096 100] [240 246] [23 27] [089 093]1198783

[090 094] [197 203] [19 23] [090 094]1198784

[095 099] [225 231] [21 25] [092 096]1198785










1198781

05826 5 51198782

04932 4 41198783

03310 2 31198784

03745 3 11198785

02962 1 2








119909

CO HCs Cost1205961

016 016 016 016 016 021205962

004 004 004 004 004 08



+ 271392 lowast 119862 + 218969 lowast 119863

(16)






1


2


1198771

= minus0060525 + 186822119860 + 185112119861 + 191909119862

+ 203517119863 + 194725119864 + 221114119865

(17)

1198772

= minus0016954 + 146548119860 + 144383119861 + 150011119862

+ 159839119863 + 152191119864 + 348365119865

(18)


5 Conclusion

















Acknowledgments




References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1

1199092

119909119899



119891 = 1205730

+

119899

sum

119894=1

120573119894

119909119894

+

119899

sum

119894=1

119899

sum

119895=1

120573119894119895

119909119894

119909119895

+ sdot sdot sdot + 120576 119894 lt 119895 (12)





Step 1

Step 2

Step 3

Step 4

Step 5

Step 6


Experimental design





Stage 1

Stage 2

Stage 3

Stage 4









[5 7] [4 5] [4 6] [04 06]1198602

[10 11] [6 7] [5 6] [15 20]1198603

[5 6] [4 5] [3 4] [04 07]1198604

[9 11] [5 6] [5 7] [13 15]1198605



1


2


3


4




(2119861







1 [5 6] [6 7] [5 7] [04 06]2 [5 6] [6 7] [5 7] [15 20]3 [5 6] [6 7] [3 4] [04 06]4 [5 6] [6 7] [3 4] [15 20]5 [5 6] [3 5] [5 7] [04 06]6 [5 6] [3 5] [5 7] [15 20]7 [5 6] [3 5] [3 4] [04 06]8 [5 6] [3 5] [3 4] [15 20]9 [10 11] [6 7] [5 7] [04 06]10 [10 11] [6 7] [5 7] [15 20]11 [10 11] [6 7] [3 4] [04 06]12 [10 11] [6 7] [3 4] [15 20]13 [10 11] [3 5] [5 7] [04 06]14 [10 11] [3 5] [5 7] [15 20]15 [10 11] [3 5] [3 4] [04 06]16 [10 11] [3 5] [3 4] [15 20]



=



119877 = 1205730

+

4

sum

119894=1

120573119894

119909119894

+

4

sum

119894=1

4

sum

119895=1

120573119894119895

119909119894

119909119895

+ 120576 (13)




119894


119894119895





2





1

1205962

1 07071 07071 14142 01414 0 02 07071 07071 14142 13730 04912 064463 07071 07071 35355 01414 02445 020034 07071 07071 35355 13730 05401 068545 07071 31623 14142 01414 02166 022876 07071 31623 14142 13730 05277 072047 07071 31623 35355 01414 03083 028748 07071 31623 35355 13730 05781 075729 07071 07071 14142 01414 04170 0229110 55227 07071 14142 13730 06881 0710711 55227 07071 35355 01414 04665 0288312 55227 07071 35355 13730 07808 0769513 55227 31623 14142 01414 04543 0306214 55227 31623 14142 13730 07525 0797915 55227 31623 35355 01414 05021 0348716 55227 31623 35355 13730 09818 09781






1

and 1198772


and 1205962

respectively

1198771


+ 345764 lowast RC(14)

1198772












01262 1 1 02059 1 11198602

07055 5 5 07618 5 51198603

01728 2 2 02772 2 21198604

05598 4 4 06170 4 41198605

03510 3 3 04868 3 3



[089 093] [233 239] [16 20] [094 098]1198782

[096 100] [240 246] [23 27] [089 093]1198783

[090 094] [197 203] [19 23] [090 094]1198784

[095 099] [225 231] [21 25] [092 096]1198785










1198781

05826 5 51198782

04932 4 41198783

03310 2 31198784

03745 3 11198785

02962 1 2








119909

CO HCs Cost1205961

016 016 016 016 016 021205962

004 004 004 004 004 08



+ 271392 lowast 119862 + 218969 lowast 119863

(16)






1


2


1198771

= minus0060525 + 186822119860 + 185112119861 + 191909119862

+ 203517119863 + 194725119864 + 221114119865

(17)

1198772

= minus0016954 + 146548119860 + 144383119861 + 150011119862

+ 159839119863 + 152191119864 + 348365119865

(18)


5 Conclusion

















Acknowledgments




References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












[5 7] [4 5] [4 6] [04 06]1198602

[10 11] [6 7] [5 6] [15 20]1198603

[5 6] [4 5] [3 4] [04 07]1198604

[9 11] [5 6] [5 7] [13 15]1198605



1


2


3


4




(2119861







1 [5 6] [6 7] [5 7] [04 06]2 [5 6] [6 7] [5 7] [15 20]3 [5 6] [6 7] [3 4] [04 06]4 [5 6] [6 7] [3 4] [15 20]5 [5 6] [3 5] [5 7] [04 06]6 [5 6] [3 5] [5 7] [15 20]7 [5 6] [3 5] [3 4] [04 06]8 [5 6] [3 5] [3 4] [15 20]9 [10 11] [6 7] [5 7] [04 06]10 [10 11] [6 7] [5 7] [15 20]11 [10 11] [6 7] [3 4] [04 06]12 [10 11] [6 7] [3 4] [15 20]13 [10 11] [3 5] [5 7] [04 06]14 [10 11] [3 5] [5 7] [15 20]15 [10 11] [3 5] [3 4] [04 06]16 [10 11] [3 5] [3 4] [15 20]



=



119877 = 1205730

+

4

sum

119894=1

120573119894

119909119894

+

4

sum

119894=1

4

sum

119895=1

120573119894119895

119909119894

119909119895

+ 120576 (13)




119894


119894119895





2





1

1205962

1 07071 07071 14142 01414 0 02 07071 07071 14142 13730 04912 064463 07071 07071 35355 01414 02445 020034 07071 07071 35355 13730 05401 068545 07071 31623 14142 01414 02166 022876 07071 31623 14142 13730 05277 072047 07071 31623 35355 01414 03083 028748 07071 31623 35355 13730 05781 075729 07071 07071 14142 01414 04170 0229110 55227 07071 14142 13730 06881 0710711 55227 07071 35355 01414 04665 0288312 55227 07071 35355 13730 07808 0769513 55227 31623 14142 01414 04543 0306214 55227 31623 14142 13730 07525 0797915 55227 31623 35355 01414 05021 0348716 55227 31623 35355 13730 09818 09781






1

and 1198772


and 1205962

respectively

1198771


+ 345764 lowast RC(14)

1198772












01262 1 1 02059 1 11198602

07055 5 5 07618 5 51198603

01728 2 2 02772 2 21198604

05598 4 4 06170 4 41198605

03510 3 3 04868 3 3



[089 093] [233 239] [16 20] [094 098]1198782

[096 100] [240 246] [23 27] [089 093]1198783

[090 094] [197 203] [19 23] [090 094]1198784

[095 099] [225 231] [21 25] [092 096]1198785










1198781

05826 5 51198782

04932 4 41198783

03310 2 31198784

03745 3 11198785

02962 1 2








119909

CO HCs Cost1205961

016 016 016 016 016 021205962

004 004 004 004 004 08



+ 271392 lowast 119862 + 218969 lowast 119863

(16)






1


2


1198771

= minus0060525 + 186822119860 + 185112119861 + 191909119862

+ 203517119863 + 194725119864 + 221114119865

(17)

1198772

= minus0016954 + 146548119860 + 144383119861 + 150011119862

+ 159839119863 + 152191119864 + 348365119865

(18)


5 Conclusion

















Acknowledgments




References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1

1205962

1 07071 07071 14142 01414 0 02 07071 07071 14142 13730 04912 064463 07071 07071 35355 01414 02445 020034 07071 07071 35355 13730 05401 068545 07071 31623 14142 01414 02166 022876 07071 31623 14142 13730 05277 072047 07071 31623 35355 01414 03083 028748 07071 31623 35355 13730 05781 075729 07071 07071 14142 01414 04170 0229110 55227 07071 14142 13730 06881 0710711 55227 07071 35355 01414 04665 0288312 55227 07071 35355 13730 07808 0769513 55227 31623 14142 01414 04543 0306214 55227 31623 14142 13730 07525 0797915 55227 31623 35355 01414 05021 0348716 55227 31623 35355 13730 09818 09781






1

and 1198772


and 1205962

respectively

1198771


+ 345764 lowast RC(14)

1198772












01262 1 1 02059 1 11198602

07055 5 5 07618 5 51198603

01728 2 2 02772 2 21198604

05598 4 4 06170 4 41198605

03510 3 3 04868 3 3



[089 093] [233 239] [16 20] [094 098]1198782

[096 100] [240 246] [23 27] [089 093]1198783

[090 094] [197 203] [19 23] [090 094]1198784

[095 099] [225 231] [21 25] [092 096]1198785










1198781

05826 5 51198782

04932 4 41198783

03310 2 31198784

03745 3 11198785

02962 1 2








119909

CO HCs Cost1205961

016 016 016 016 016 021205962

004 004 004 004 004 08



+ 271392 lowast 119862 + 218969 lowast 119863

(16)






1


2


1198771

= minus0060525 + 186822119860 + 185112119861 + 191909119862

+ 203517119863 + 194725119864 + 221114119865

(17)

1198772

= minus0016954 + 146548119860 + 144383119861 + 150011119862

+ 159839119863 + 152191119864 + 348365119865

(18)


5 Conclusion

















Acknowledgments




References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














01262 1 1 02059 1 11198602

07055 5 5 07618 5 51198603

01728 2 2 02772 2 21198604

05598 4 4 06170 4 41198605

03510 3 3 04868 3 3



[089 093] [233 239] [16 20] [094 098]1198782

[096 100] [240 246] [23 27] [089 093]1198783

[090 094] [197 203] [19 23] [090 094]1198784

[095 099] [225 231] [21 25] [092 096]1198785










1198781

05826 5 51198782

04932 4 41198783

03310 2 31198784

03745 3 11198785

02962 1 2








119909

CO HCs Cost1205961

016 016 016 016 016 021205962

004 004 004 004 004 08



+ 271392 lowast 119862 + 218969 lowast 119863

(16)






1


2


1198771

= minus0060525 + 186822119860 + 185112119861 + 191909119862

+ 203517119863 + 194725119864 + 221114119865

(17)

1198772

= minus0016954 + 146548119860 + 144383119861 + 150011119862

+ 159839119863 + 152191119864 + 348365119865

(18)


5 Conclusion

















Acknowledgments




References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119909

CO HCs Cost1205961

016 016 016 016 016 021205962

004 004 004 004 004 08



+ 271392 lowast 119862 + 218969 lowast 119863

(16)






1


2


1198771

= minus0060525 + 186822119860 + 185112119861 + 191909119862

+ 203517119863 + 194725119864 + 221114119865

(17)

1198772

= minus0016954 + 146548119860 + 144383119861 + 150011119862

+ 159839119863 + 152191119864 + 348365119865

(18)


5 Conclusion

















Acknowledgments




References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Acknowledgments




References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a new method based on topsis and response...

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