Research ArticleMultiple Attribute Decision Making Based on Cross-Evaluationwith Uncertain Decision Parameters

Tao Ding1 Liang Liang2 Min Yang1 and Huaqing Wu3

1School of Management Hefei University of Technology 193 Tunxi Road Hefei Anhui 230009 China2Hefei University of Technology 193 Tunxi Road Hefei Anhui 230009 China3School of Economics Hefei University of Technology 193 Tunxi Road Hefei Anhui 230009 China

Correspondence should be addressed to Min Yang yangminhfuteducn

Received 27 December 2015 Accepted 31 March 2016

Academic Editor Victor Shi

Copyright copy 2016 Tao Ding et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Multiple attribute decision making (MADM) problem is one of the most common and popular research fields in the theory ofdecision science A variety of methods have been proposed to deal with such problems Nevertheless many of them assumed thatattribute weights are determined by different types of additional preference information which will result in subjective decisionmaking In order to solve such problems in this paper we propose a novel MADM approach based on cross-evaluation withuncertain parameters Specifically the proposed approach assumes that all attribute weights are uncertain It can overcome thedrawback in prior research that the alternativesrsquo ranking may be determined by a single attribute with an overestimated weight Inaddition the proposedmethod can also balance themean and deviation of each alternativersquos cross-evaluation score to guarantee thestability of evaluation Then this method is extended to a more generalized situation where the attribute values are also uncertainFinally we illustrate the applicability of the proposedmethod by revisiting two reported studies and by a case study on the selectionof community service companies in the city of Hefei in China

1 Introduction

Multiple attribute decision making (MADM) is an importantpart of modern decision science [1ndash4] It assumes that thereexists a set of alternatives with multiple attributes which adecision maker (DM) should evaluate and analyze The aimofMADM is to find themost desirable alternative or rank thefeasible alternatives for supporting decision makings As anactive research area MADM problems have been tried to besolved by some classical methods such as the simple additiveweighting method (SAW) the analytic hierarchy process(AHP) and the technique for order preference by similarityto ideal solution (TOPSIS) [5ndash7] It also has been extensivelyapplied to many aspects such as economics managementengineering and technology [8ndash15]

In MADM problems the needed decision making infor-mation includes attribute values and attribute weights Theattribute values describe characteristics qualities and per-formances of alternatives The attribute weights are used tomeasure the importance of attributes Due to the complexity

of the real world and limited knowledge and perception capa-bility of human beings both attribute values and attributeweights are uncertain [16 17] The decision makers are notable to express their preferences or assessments explicitly [18]In order to deal with such problems many researchers try todescribe attribute values and attribute weights accurately inuncertain environments Generally the researches in this areacan be classified into three categories as follows

Firstly fuzzy set theory that appeared in 1965 has gradu-ally become the mainstream in the field of representing andhandling uncertain attribute values [19] Since the inceptionof the fuzzy set theory various extensions have been pro-posed such as interval-valued fuzzy set [20] fuzzy multiset[21] intuitionistic fuzzy set (IFS) [22] linguistic fuzzy set[23] and hesitant fuzzy set (HFS) [24]

Secondly several approaches have been proposed todetermine attribute weights Generally they fall into threecategories subjective objective and hybrid methods [25]The subjective methods utilize the preferences of a decision

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4313247 10 pageshttpdxdoiorg10115520164313247

2 Mathematical Problems in Engineering

maker to select attribute weights [26ndash28] The objectivemethods determine attribute weights based on the objectiveinformation [29 30] Hybrid methods [31ndash34] combine boththe preferences of a decisionmaker and objective informationto determine attribute weights

Thirdly as a nonparametric programming efficiency-rating technique for evaluating a set of decision making units(DMUs) with multiple inputs and outputs date envelopmentanalysis (DEA) requires relatively less additional informa-tion [35] DEA has been considered as an effective tool tosolve MADM problems without information on weights Forinstance Doyle [36] presents an approach based on DEAmethod which derives a set of attribute weights from thedata itself Bernroider and Stix [37] introduce an approachbased onDEAwith restrictedmultipliers for accountable andunderstandable MADM Wu and Liang [38] develop a newmultiple attribute rankingmethod based onDEAgame cross-efficiency model in which each alternative is viewed as aplayer that seeks to maximize its own performance

Although existing methods handle uncertainty to someextent some drawbacks are also exposed which are listed asfollows First both different defuzzy approaches and weightdetermination methods may generate different attributeweights and produce different optimal solutions [39 40]Therefore it is difficult to identify which optimal solutionis the best one Second behavior factors are not consideredin existing MADM methods which may lead to deviationresults in decision making Specially they do not avoid thebad situation that the ranking results may be determined bya single attribute with an exorbitant weight In other wordsthe alternative rank order is consistent with the rank order ofoverestimated attribute which means that other attributes aretotally useless in evaluation Obviously such result is unfairand unreasonable because the optimal alternative should bedetermined by multiple attributes instead of an individualattribute Third although DEA models can evaluate MADMproblems without weight information it also suffers from thefollowing three drawbacks One is that the optimal weightsderived by DEA may not be consistent with the opinionsof the decision maker [41] Another shortage is that DEAidentifies too many efficient alternatives and cannot welldistinguish them [42 43] Furthermore the nonuniquenesssolution phenomenon is also common inDEA basedMADMmethods [44]

Considering the above analysis we attempt to develop across-evaluation based approach to solve MADM problemswith uncertain decision information To be specific we firstlydefine and analyze the attribute weight space which containsall feasible weights Specially to avoid the single dominatingattribute situation we calculate the upper bound weightof each attribute Then considering a condition where theattribute values are certain and the attribute weights areuncertain the return and risk of closeness indexes are cal-culated based on cross-evaluation in which each alternativeseeks to maximize its closeness index by selecting a set ofoptimal weights And the coefficient of variation is utilizedto integrate the results and produce an alternative rankorder In a more generalized condition where the attributevalues are denoted by interval values or ordinal values

the cross-evaluationmodels consideringDMrsquos risk preferenceare developed

The main contributions of the current paper can besummarized as follows (1) we firstly defined the attributeweight space and determined the upper bound weight ofeach attribute to conduct the uncertain information (2) bothmean and deviationmeasures of closeness index under cross-evaluation are considered to integrate a unique solution tothe MADM problem (3) the proposed method is extendedto more generalized situations where the attribute values aredescribed by interval values and ordinal values respectively

The remainder of this paper is organized as follows InSection 2 we briefly review the relative works In Section 3we propose a cross-evaluation basedmethod to solveMADMproblems with uncertain weight information In Section 4the proposed method is extended to more generalized formswhere the attribute information is expressed by intervalvalues and ordinal values Section 5 illustrates the methodswith reportedMADMstudies and a case study of the selectionof community service companies Conclusions and futureresearch will be displayed in Section 6

2 Preliminaries

21 Formulation of MADM Problems Suppose 119883 = 1198831

1198832 119883

119898 is a discrete set of alternatives and 119884 = 119884

1

1198842 119884

119899 is a set of 119899 attributes with the weight vector 119908 =

(1199081 1199082 119908

119899)119879 where 119908

119895gt 0 sum119899

119895=1119908119895

= 1 Let 119860 =

(119886119894119895)119898times119899

be a decision matrix where 119886119894119895denotes the attribute

value that the 119894th alternative119883119894isin 119883 takes with respect to the

119895th attribute 119884119895isin 119884

In general there are both benefit type attributes (ie thelarger the better) and cost type attributes (ie the smallerthe better) in a MADA problem We transform the originaldecision matrix 119860 = (119886

119894119895)119898times119899

into a normalized matrix 119877 =

(119903119894119895)119898times119899

by using the method from Hwang and Yoon [7]

119903119894119895=

119886119894119895

max119894119886119894119895

for benefit attribute 119884119895

min119894119886119894119895

119886119894119895

for cost attribute 119884119895

119894 = 1 2 119898 119895 = 1 2 119899

(1)

22 Characteristics of the TOPSIS Method The basic prin-ciple of TOPSIS method is that the optimal alternativeshould have both the shortest distance from the positive idealsolution and the furthest distance from the negative idealsolution [14]

The TOPSIS procedure is performed in six stages asfollows

(a) Normalize the decision matrix

119903119894119895=

119886119894119895

radicsum119899

119895=11198862

119894119895

119894 = 1 2 119898 119895 = 1 2 119899 (2)

where 119886119894119895is the 119895th attribute value of alternative 119894

Mathematical Problems in Engineering 3

(b) Calculate the weighted normalized decision matrix

V119894119895= 119908119895119903119894119895 119894 = 1 2 119898 119895 = 1 2 119899 (3)

where 119908119895is the weight of the 119895th attribute and

sum119899

119895=1119908119895= 1

(c) Determine the positive ideal and negative ideal solu-tionPositive ideal solution 119860

+= V+1 V+2 V+

119899

= (max119894

V119894119895| 119895 isin 119868

1015840) (min

119894

V119894119895| 119895 isin 119868

10158401015840)

Negative ideal solution 119860minus= Vminus1 Vminus2 Vminus

119899

= (min119894

V119894119895| 119895 isin 119868

1015840) (max

119894

V119894119895| 119895 isin 119868

10158401015840)

(4)

where 1198681015840 indicate the index set of the benefit type

attributes and 11986810158401015840 the index set of the cost type

attributes(d) Measure the distances from the positive ideal solution

and negative ideal solution respectively

119863+

119894= radic

119899

sum

119895=1

(V119894119895minus V+119895)2

119894 = 1 2 119898

119863minus

119894= radic

119899

sum

119895=1

(V119894119895minus Vminus119895)2

119894 = 1 2 119898

(5)

(e) Calculate the relative closeness to the ideal solutions

119862lowast

119894=

119863minus

119894

(119863+

119894+ 119863minus

119894) 119894 = 1 2 119898 (6)

(f) Rank the alternatives

The larger the closeness index of an alternative the higherthe rank of the alternative

23 DEA Cross-Efficiency Evaluation We assume that thereare 119899 DMUs to be evaluated in terms of 119898 inputs and 119899

outputs The 119894th input and 119903th output of DMU119895are denoted

by 119909119894119895(119894 = 1 119898) and 119910

119903119895 respectively The efficiency of

a DMU119889can be calculated by the following CCR model in

linear programming formulation [35]

max119904

sum

119903=1

119906119903119910119903119889

= 120579119889

st119898

sum

119894=1

119908119894119909119894119895minus

119904

sum

119903=1

119906119903119910119903119895ge 0 119895 = 1 2 119899

119898

sum

119894=1

119908119894119909119894119889

= 1

119908119894ge 0 119894 = 1 2 119898

119906119903ge 0 119903 = 1 2 119904

(7)

When the above model is solved a set of optimal weights119908lowast

1119889 119908

lowast

119898119889 119906lowast

1119889 119906

lowast

119904119889is obtained Then the 119889 cross-

efficiency of any DMU119895is computed as [38]

119864119889119895

=

sum119904

119903=1119906lowast

119903119889119910119903119895

sum119898

119894=1119908lowast

119894119889119909119894119895

119889 119895 = 1 119899 (8)

For DMU119895(119895 = 1 119899) the average of all 119864

119889119895(119889 =

1 119899) denoted by 119864119895

= (1119899)sum119899

119889=1119864119889119895 is considered as

the cross-efficiency score of DMU119895

3 Proposed Method

In this section we will introduce the so-called single dom-inating attribute issue and propose a model to supportdecisions by incorporating considerations on this issue inSection 31 furthermore in Section 32 we will introducethe cross-evaluation framework and the accompanying riskconsideration finally in Section 33 we propose a proce-dure summarizing all the aforementioned considerations forMADM

31 Determination of the Weight Space We observe that inMADM problems an individual attribute can determine thefinal ranks of all alternatives when the corresponding weightis greater than some threshold value In other words therank order of the alternatives is completely in accordancewith the rank order of the dominating attribute and therest of the attributes are totally ignored in the evaluationIt should be noted that the ignored attributes are takeninto the evaluation only because the decision maker regardsthem as import factors or they could have excluded themfrom the decision process in the first place Therefore weargue this is unacceptable In order to address this issueit is necessary to define and compute the threshold upperbound of each attribute weight The upper bound can aid thedecision process by suggesting meaningful weights

For the simplicity of exposition if the attribute values119903119894119896

(119894 = 1 119898) in the 119896th column of the decision matrix119877 have the relation 119903

1198941119896

ge 1199031198942119896

ge sdot sdot sdot ge 119903119894119898119896(119894119904

isin

1119898 119904 = 1 119898) then we denote the ranks of thealternatives according to the values of the 119896th attribute 119884

119896

as 1198831198941

≻1198961198831198942

≻119896sdot sdot sdot ≻119896119883119894119898

With this notation in position thesingle dominating attribute happens if the final rank orderof all alternates is the same as some 119883

1198941

≻1198961198831198942

≻119896sdot sdot sdot ≻119896119883119894119898

no matter how the decision maker distributes the relativeweights of the other attributes We formulize this notion asDefinition 1 as below

Definition 1 Let 119896 be the index of some attribute 119908lowast119896is the

threshold upper bound of 119908119896if and only if the following

conditions are satisfied

(a) If 119908119896ge 119908lowast

119896 then forall1 le 119894 119905 le 119898 119903

119894119896ge 119903119905119896if and only

if sum119899119895=1

119908119895119903119894119895ge sum119899

119895=1119908119895119903119905119895

(b) If 119908119896lt 119908lowast

119896 then exist1 le 119894 119905 le 119898 such that 119903

119894119896ge 119903119905119896

andsum119899

119895=1119908119895119903119894119895le sum119899

119895=1119908119895119903119905119895hold true simultaneously

4 Mathematical Problems in Engineering

To implement Definition 1 we provide Theorem 2 belowto determine the threshold upper bound

Theorem 2 The threshold upper bound of the 119896th attribute(119908lowast119896) is given by the minimization program

min 119908119896

st 11990811989611990311989411+

119899

sum

119895=1119895 =119896

119908(119891)

1198951199031198941119895ge 11990811989611990311989421+

119899

sum

119895=1119895 =119896

119908(119891)

1198951199031198942119895

ge sdot sdot sdot

ge 1199081198961199031198941198981+

119899

sum

119895=1119895 =119896

119908(119891)

119895119903119894119898119895

(9)

119899

sum

119895=1

119908119895= 1

119908119895ge 0

119895 = 1 2 119899

119891 = 1 2 119899 minus 1

(10)

where 119908(119891)

119895= 1 minus 119908

119896if 119895 is the 119891th element of the set

1 119873 119896 in ascending order otherwise 119908(119891)119895

= 0 Letus take 119896 = 1 as an example We have (119908(1)

2 119908(1)

3 119908

(1)

119899) =

(1 minus 1199081 0 0) (119908(2)

2 119908(2)

3 119908

(2)

119899) = (0 1 minus 119908

1 0 0)

(119908(119899minus1)

2 119908(119899minus1)

3 119908

(119899minus1)

119899) = (0 0 1 minus 119908

1) The sub-

script 119894ℎ(ℎ isin 1 119898) of 119903 in the above model indicates the

119894ℎpreferred alternative determined by the preference relation

1198831198941

≻1198961198831198942

≻119896sdot sdot sdot ≻119896119883119894119898

Proof Clearly there are (119899 minus 1) constraints in (9) When119896 = 1 the weight combinations (119908

(1)

2 119908(1)

3 119908

(1)

119899) =

(1 minus 1199081 0 0) (119908(2)

2 119908(2)

3 119908

(2)

119899) = (0 1 minus 119908

1 0 0)

(119908(119899minus1)

2 119908(119899minus1)

3 119908

(119899minus1)

119899) = (0 0 1 minus 119908

1) are applied

to the 119899minus1 constraints in orderThefirst constraint guaranteesthat the alternatives rank result has nothing to do withattribute 119884

2 Similarly the remaining 119899 minus 2 constraints ensure

that the alternatives rank result has nothing to do with theother attributes 119884

3 119884

119899 So the optimal minimization 119908

1

satisfies (a) in Definition 1Conversely we suggest that a constraint belonging to the

119897th (119897 = 1 119899 minus 1) group is removed from the model Thenattribute 119884

119897may play a part in the ranking because it is not

completely suppressed In other words (b) in Definition 1holds

Similarly the upper bound of each attribute weight(119908lowast

119896 119896 = 2 119899) can be calculated by the model in

Theorem 2 with 119896 = 2 119899 This proof is completed

Example 3 Ma et al [31] consider a case of a robot userexpecting to select a robot and there are four alternatives forhim to choose There are four attributes to evaluate them

that is cost velocity repeatability and load capacityThe nor-malized decision matrix is as follows

119860 =

(((

(

0 15

6

1

35

12

3

51 0

1 0 0 1

2

3

2

5

2

3

2

3

)))

)

(11)

Suppose that the information on attribute weights iscompletely unknown The upper bound weight of the firstattribute cost can be calculated by the above model

By calculating the above model provided in Theorem 2we obtain the optimal value119908lowast

1= 23 denoting the threshold

upper bound of the first attribute1199081 Analogously each upper

boundof attributeweight can be calculated by similarmodelsand the values are 119908lowast

2= 1013 119908

lowast

3= 45 119908

lowast

4= 914

32 Mean and Deviation of Closeness Index Based Cross-Evaluation Models In this subsection in the environmentof uncertain attribute weights we develop cross-evaluationmodels to calculate the mean and deviation of closenessindex for each assessed alternative respectively Then thecomputed indexes are integrated to obtain the evaluationresults

Cross-efficiency evaluation method since being pro-posed by Sexton et al [45] has been used to evaluate andrankDMUs Itsmain idea is to use cross-evaluation instead ofa self-evaluation There are at least two advantages for cross-evaluation method One is that it obtains a unique rankingamong the DMUs The other is that the cross-efficiencymeans can act effectively to differentiate between good andbad alternatives [46] Hence the cross-evaluation methodis widely used for ranking performance of alternatives forexample equipment design [47] supplier selection [48] andstock market [49]

To calculate the self-evaluation closeness index for eachalternative we consider the following mathematical pro-gramming problem for a given alternative119883

119894

max119863minus

119894

119863minus

119894+ 119863+

119894

= 119862max119894

119908 isin Ω (119908)

(12)

Here 119863minus119894and 119863

+

119894represent the distance from negative

ideal point and ideal point (defined in (5)) respectively Forease of calculation the ideal point and negative point aredefined as 119860+ = 119908

1 1199082 119908

119899 and 119860

minus= 0 0 0 Then

119863+

119894and 119863

minus

119894are denoted by 119863

+

119894= radicsum

119899

119895=1(119908119895119903119894119895minus 119908119895)2 and

119863minus

119894= radicsum

119899

119895=1(119908119895119903119894119895)2 respectively The constraint 119908 isin Ω(119908)

depicts the possible set of attribute weights Obviously whenthe weights information of attributes is completely unknownΩ(119908) = 119908

119895gt 0 sum

119899

119895=1119908119895= 1 Introducing the upper bound

weight information calculated in Section 31 Ω(119908) = 119908119895lt

119908lowast

119895 sum119899

119895=1119908119895= 1

Mathematical Problems in Engineering 5

In many practical situations although the weight valuesare difficult to obtain the weight information is partiallyknown and predefined by the weight space As followswe introduce two types of partial weight information asreferences to decision maker

Type 1 (pairwise comparisonweights information) InMADMattribute weights 119908

119895are mainly introduced to reflect the

importance to attributes 119884119895(119895 = 1 2 119899) As stated in

Section 1 pairwise comparison is a widely accepted subjectivemethod to determine attribute weight

Attribute 1198842is at least two times and not more than three

times as important as attribute 1198841

In notational jargons this is denoted by

21199081le 1199082le 31199081 (13)

In general terms we compare 1198841with 119884

119895(119895 = 2 3 119899)

respectively The bound linear constraints are denoted by

1198971198951199081le 119908119895le 1199061198951199081 (14)

in which 119897119895and 119906

119895represent the lower and upper bounds of

times119908119895compared with119908

1 respectively Hence the attribute

weights space is obtained by the above pairwise comparisoninformation which is expressed as Ω(119908) = 119908

119895gt 0 119897

1198951199081le

119908119895le 1199061198951199081 119895 = 1 2 119899 Clearly 119897

1= 1199061= 1 holds

Compared with conventional subjective weight determi-nation approaches such as the AHP weights space is morefeasible and applicative because the decision makers neednot express their preferences explicitly In many situationsexperts are pleased to give such interval information ratherthan exact value due to the complexity of actual problems

Type 2 (ordinal weights information) Outranking relation[50] usually denoted by 119878 was proposed by Roy whoseaim was to establish a realistic representation of the threebasic preference situations strict preference indifferenceand incomparability Motivated by the widely accepted out-ranking relation we introduce the concept of rank-orderweights information In many situations although the weightvalues are difficult to give by decision makers the rankingof each attribute is easy to obtain according to importanceTake the Olympic Games as an example the achievementsof athletes and nations are judged by gold silver and bronzemedals Apparently a gold medal is more important than asilver medal and a silver medal is more important than abronze medal Then the weights space can be denoted byΩ(119908) = 119908

1gt 1199082

gt 1199083 1199081 1199082 1199083

gt 0 in which1199081 1199082 1199083represent the weight of gold silver and bronze

medal respectivelyThe above self-evaluation model (see (12)) shows that all

alternatives choose their own attribute weights in order tomaximize their own self-evaluation closeness index Similarto the formofDEA cross-efficiency we obtain a set of optimalattribute weights 119882lowast

119894= (119908lowast

1198941 119908

lowast

119894119899) according to the above

model When these optimal weights are applied to evaluatealternative119883

119896 a cross score denoted by119862

119894119896= 119863minus

119894119896(119863minus

119894119896+119863+

119894119896)

is obtained

For alternative 119896 (119896 = 1 119898) we use the average ofall 119862119894119896to measure the expected return of cross-evaluation

closeness index which is written as

119903 (119883119896) =

1

119898

119898

sum

119894=1

119862119894119896 (15)

Furthermore we use the standard variance

120575 (119883119896) =

radicsum119898

119894=1(119862119894119896minus 119903 (119883

119896))2

119899

(16)

tomeasure the risk caused by different sets of optimal weightsproduced by each alternative

In general there are both return of closeness index(defined by the mean of cross-evaluation score) and risk ofcloseness index (defined by the deviation between return andeach cross-evaluation closeness index) in cross-evaluationframework It is not hard to see that the larger return scorethe better and the smaller risk score the better Specificallygiven two alternatives 119883

119904and 119883

119905 if 119877(119883

119904) gt 119877(119883

119905) and

120575(119883119904) lt 120575(119883

119905) then 119883

119904≻ 119883119905(119883119904is better than 119883

119905) clearly

holds However when 119877(119883119904) gt 119877(119883

119905) and 120575(119883

119904) gt 120575(119883

119905) it

is not easy to compare the two alternatives directlyThen we integrate the return and risk measures by the

well-known coefficient of variation [51]

CV (119883119896) =

120575 (119883119896)

119903 (119883119896)times 100 (17)

For two given alternatives one having a smaller coeffi-cient of variation is considered to be better than the otherone Thus all the alternatives are ranked by the comparisonof their corresponding coefficient of variation

33 Procedure of the Proposed Approach Based on the aboveanalysis the procedure for generating a solution to theMADM problem is elaborated in the following

Step 1 (formulation of a MADM problem) The decisionmaker identifies 119899 attributes to evaluate 119898 alternatives Theoriginal attribute values are standardized by (1)

Step 2 (obtainment of the attribute weight space) First theupper bounds of each attribute are calculated byTheorem 2 toguarantee the effectiveness of all the attributesThen differenttypes of constraint of attribute weights are determined by thedecision maker to reflect the importance of each attributeLast the weight space including all the feasible weightcombinations is obtained

Step 3 (calculation of the mean and deviation measures)Firstly the maximized closeness index of each alternativeis calculated by (12) Secondly the mean measure of eachalternativersquos cross closeness index is computed by (15) Thirdthe deviation measure of each alternativersquos cross closenessindex is obtained by (16)

6 Mathematical Problems in Engineering

Step 4 (evaluation of the alternatives via coefficient ofvariation) Since the mean and deviation measures of eachalternativersquos cross closeness index are obtained in Step 3 thecoefficient of variation defined in (17) is utilized to integratethe results

Step 5 (generation of rank order of the alternatives) Throughcomparing the coefficients of variation among the alterna-tives the rank order can be directly obtained

4 Extension with Uncertain Attribute Values

41 Model with Interval Attribute Value It is widely acknowl-edged that uncertainty is associated with attribute values dueto complexity of real world and limited knowledge of humanbeings In many situations the attribute value 119903

119894119895is uncertain

but we know that the feasible attribute value 119903119894119895

isin [119903minus

119894119895 119903+

119894119895]

Here we present the following self-evaluation model withinterval attribute values [52] and uncertain attribute weights

max 120572

radicsum119899

119895=1(119908119895119903+

119894119895)2

radicsum119899

119895=1(119908119895119903+

119894119895)2

+ radicsum119899

119895=1(119908119895119903+

119894119895minus 119908119895)2

+ 120573

radicsum119899

119895=1(119908119895119903minus

119894119895)2

radicsum119899

119895=1(119908119895119903minus

119894119895)2

+ radicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2

= 119862max119894

119908 isin Ω (119908)

120572 + 120573 = 1 120572 ge 0 120573 ge 0

(18)

in which radicsum119899

119895=1(119908119895119903+

119894119895)2 radicsum

119899

119895=1(119908119895119903+

119894119895minus 119908119895)2 radicsum

119899

119895=1(119908119895119903minus

119894119895)2

andradicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2 represent distance of upper bounds

from ideal point distance of upper bounds from negativeideal point distance of lower bounds from ideal pointand distance of lower bounds from negative ideal pointrespectively 120572 and 120573 reflect the risk coefficients that combineupper closeness index and lower closeness respectively Inpractice120572 and120573 are decided by the decisionmaker accordingto his or her risk preference Specifically the decision makeris neutral if 120572 = 120573 = 12 120572 gt 120573 and 120572 lt 120573 indicate that he isoptimistic and pessimistic respectively

42 Model with Ordinal Attribute Value Another form ofuncertain preferential information is expressed by ordinalattribute value (eg ldquomore preferablerdquo and ldquoless preferablerdquo)It is argued to be more stable and more reliable than cardinalvalue in some situations [53] In this paper an approach ofthreshold value is introduced to transform ordinal value intolinear formulation For instance given four attributes 119886 119887 119888and 119889 the ordinal values are 1 2 3 and 4 respectively Byspecifying a threshold value 01 the attribute values can beconverted to a cardinal attribute space Ω(119886 119887 119888 119889) = 119886 = 1

119886 minus 119887 ge 01 119887 minus 119888 ge 01 119888 minus 119889 ge 01 119889 = 0 In generallet the ordinal attribute values 119886

119894119895(119894 = 1 119898) have relation

1198861198941119895gt 1198861198942119895gt sdot sdot sdot gt 119886

119894119898119895(119894119896isin [1119898] 119896 = 1 119898) and then

the cardinal attribute spaceΩ(119903119894119895) = 119886

119894119898119895= 1 119886119894119898119895minus119886119894119898minus1119895ge 120576

119886119894119898minus1119895minus119886119894119898minus2119895ge 120576 119886

1198942119895minus1198861198941119895ge 120576 1198861198941119895= 0 120576 gt 0 is obtained

(120576 is the threshold value given by the decision maker) Herewe present the following self-evaluation model with ordinalattribute values and uncertain attribute weights

maxradicsum119899

119895=1(119908119895119903119894119895)2

radicsum119899

119895=1(119908119895119903119894119895)2

+ radicsum119899

119895=1(119908119895119903119894119895minus 119908119895)2

= 119862max119894

st 119903 isin Ω (119903)

119908 isin Ω (119908)

(19)

119894 = 1 119898 (20)

119895 = 1 119899 (21)

For each alternative 119883119896(119896 = 1 119898) under evaluation

we obtain a group of optimal attribute values and a set ofattribute weights They are denoted by 119860

lowast

119896= (119903119896

119894119895)119898times119899

and119882lowast

119896= (119908

119896

1 119908119896

2 119908

119896

119899) respectively When the optimal

decision information is applied to evaluate alternative119883119897(119897 =

1 119898) a cross score denoted by 119862119896119897

= radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2

(radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2+ radicsum

119899

119895=1(119908119896

119895119903119896

119897119895minus 119908119896

119895)2) is obtained

Similar to that defined by (14) and (15) the mean anddeviation measures of alternative 119883

119897are denoted by 119903(119883

119897) =

sum119898

119896=1119862119896119897and 120575(119883

119897) = radicsum

119898

119896=1(119862119896119897minus 119903(119883

119897))2119899 respectively

Note that (12) (18) and (19) are nonlinear models theycan be transformed into linear forms according to practicalcases

5 Illustrative Examples

In order to illustrate the applications of our proposed meth-ods three practical MADM problems are presented

51 Application to House Selection with Cardinal AttributeValues Xu [54] stated a purchase house problem there arefour possible houses to be selected and the attributes are asfollows (1)119884

1is the house price (ten thousand yuan) (2)119884

2is

the usable area (square meter) (3) 1198843is the distance between

house and the center of city (kilometers) (4) 1198844is the nature

environment (scores) where 1198842and 119884

4are benefit attributes

and 1198841and 119884

3are cost attributes The attribute weights are

completely unknown The decision making matrix 119860 andnormalized decision making matrix 119877 are listed in Tables 1and 2 respectively Step 1 is completed

By utilizing (9) and (10) the upper boundof each attributeweight is obtained And as a result the attribute weight spaceis Ω(119908) = 0 lt 119908

1lt 077 0 lt 119908

2lt 078 0 lt 119908

3lt 062

0 lt 1199084lt 062 119908

1+ 1199082+ 1199083+ 1199084= 1 Step 2 is completed

Mathematical Problems in Engineering 7

Table 1 Decision making matrix 119860 of house evaluation

1198841

1198842

1198843

1198844

1198831

30 100 10 71198832

25 80 8 51198833

18 50 20 111198834

22 70 12 9

Table 2 Normalized decisionmakingmatrix119877 of house evaluation

1198841

1198842

1198843

1198844

1198831

06 10 08 06361198832

072 08 10 04551198833

10 05 04 101198834

0818 07 0667 0818

Table 3 The cross-evaluation measurements

Alternatives 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

119903 (119883119894) 120575 (119883

119894)

1198831

096 096 082 061 061 075 0151198832

092 080 092 066 066 076 0111198833

1 052 051 1 1 076 0241198834

0818 070 070 0818 0818 076 006

Table 4 Alternative efficiency matrix

Alternatives 1198831

1198832

1198833

1198834

CV (119883119894) 020 014 032 008

In terms of Step 3 in Section 33 we obtain the cross-evaluation measurements of each alternative which arepresented in Table 3 Step 3 is completed

Based on Table 3 and the calculation of coefficient ofvariation the results are shown in Table 4

Therefore we can rank the houses in accordance with theCV values presented in Table 4 to get a complete rank order1198834

≻ 1198832

≻ 1198831

≻ 1198833 And 119883

4is the best house under

evaluation Step 4 is completedIn order to understand our approach more clearly we

compare our approachwith other three similarmethods [54]namely average attribute value method maximizing devia-tion method and information entropy method respectivelywith the same example The corresponding results of suchthree methods are listed in Table 5

According to Tables 4 and 5 the result derived from ourcross-evaluation approach is quite different from that of theexisting three weight determination methods In fact Table 5shows that the three weight determination methods generatethree attribute weight vectors and produce three differentoptimal solutions However there is no single method thatcan guarantee more convincing weight vector than othersHence we treat the attribute weight as a weight space rather

Table 5 The results of three methods

Method Weight vector Rank order

Average attributevalue

(025 025 025 025) 1198831≻ 1198834≻ 1198832≻ 1198833

Maximizingdeviation

(020 024 029 027) 1198831≻ 1198832≻ 1198834≻ 1198833

Informationentropy

(013 022 036 029) 1198831≻ 1198832≻ 1198834≻ 1198833

Table 6 Attribute weights

1198841

1198842

1198843

[119908119871

1 119908119880

1] [119908

119871

2 119908119880

2] [119908

119871

3 119908119880

3]

[035 050] [015 030] [030 035]

Table 7 Attribute values of alternatives

1198841

1198842

1198843

119860 [065 085] [068 086] [045 078]119861 [075 1] [055 078] [064 080]119862 [054 068] [046 055] [078 1]119863 [077 090] [067 078] [056 077]119864 [025 045] [089 1] [075 094]

than a certain value in this paperThe comparison result vali-dates that our proposed approach can overcome the shortagethat we cannot obtain themost dominant result caused by thediversity of the way of weight vector determined

52 Application to University Faculty with Interval AttributeValue Consider the example in Bryson and Mobolurinrsquos[55] study that is five faculty candidates (119860 119861 119862 119863119864) are evaluated for tenure and promotion through threeattributes teaching research and service According to theDMrsquos preference the information for both attribute andalternatives is given by interval numbers as shown in Tables6 and 7 respectively (for the purpose of illustration our datais different from the literature)

Table 8 shows the computed results of the proposedapproach with interval values The self-evaluation score ofeach alternative is listed in the second column and the cross-evaluation scores are listed in columns 3ndash7 Columns 8 and9 show the mean and deviation measures respectively Thenthe results in column 10 are applied to generate a rank orderof the five candidates119860 ≻ 119862 ≻ 119864 ≻ 119861 ≻ 119863 which is a solutionto the problem

53 Application to the Selection of Community Service Com-panies with Ordinal Attribute Values In this subsection weapply our approach to the evaluation and selection of fivecommunity service companies that are located in Hefei acity in Anhui province of China These companies providevarious kinds of service for the public such as property

8 Mathematical Problems in Engineering

Table 8 Evaluation results for the comparison of faculty candidates

Candidates 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

1198625119894

119903119894

120575119894

CV(119883119894)

119860 0717 0717 0712 0706 0717 0703 0711 0006 0008119861 0792 0792 0792 0788 0792 0746 0782 0018 0023119862 0672 0651 0661 0672 0651 0660 0659 0008 0012119863 0774 0774 0771 0764 0774 0737 0764 0014 0461119864 0645 0525 0526 0540 0525 0645 0552 0047 0018

Table 9 Ordinal values of alternatives

1198841

1198842

1198843

1198844

1198831

4 2 4 21198832

1 4 3 31198833

2 5 1 11198834

5 3 2 51198835

3 1 5 4

management catering health care and entertainment Inthe case study the decision maker is an official from thedevelopment and reform commission of Hefei with thehelp of a group of experts from the governments relevantenterprises and community organizations Four attributesare used to evaluate the six companies They are quality ofservice (119884

1) diversity of service (119884

2) profitability (119884

3) and

development potential (1198844) The original attribute values are

given by ordinal values as shown in Table 9 The decisionprocedures are as follows

After considering the four attributes the decision makerviews that quality of service (119884

1) is the most important

attribute in evaluation The development potential (1198844) is the

second most important one profitability (1198843) is the third

most important one and diversity of service (1198842) is the

least important one respectively And the difference betweentwo adjacent attribute weights should be more than 01 Inaddition each weight value should be nomore than 05Thenthe weight space is denoted by Ω(119908) = 119908

1minus 01 gt 119908

4

1199084minus 01 gt 119908

3 1199083minus 01 gt 119908

2 1199081le 05 119908

2le 05 119908

3le 05

1199084le 05 119908

1+ 1199082+ 1199083+ 1199084= 1

By specifying a threshold value of 01 ordinal attributevalues are transformed into four sets of attribute value spacesThey are denoted by

Ω(1198841) = 119903

21= 1 11990321

minus 11990331

ge 01 11990331

minus 11990351

ge 01 11990351

minus 11990311

ge 01 11990311

minus 11990341

ge 01 11990341

= 0

Ω (1198842) = 119903

52= 1 11990352

minus 11990312

ge 01 11990312

minus 11990342

ge 01 11990342

minus 11990322

ge 01 11990322

minus 11990332

ge 01 11990332

= 0

Ω (1198843) = 119903

33= 1 11990333

minus 11990343

ge 01 11990343

minus 11990323

ge 01 11990323

minus 11990313

ge 01 11990313

minus 11990353

ge 01 11990353

= 0

Ω (1198844) = 119903

34= 1 11990334

minus 11990314

ge 01 11990314

minus 11990324

ge 01 11990324

minus 11990354

ge 01 11990354

minus 11990344

ge 01 11990344

= 0

(22)

Table 10 shows the computed results of the proposedapproachThe second column shows the self-evaluation scoreof each alternative computed by (19) And columns 3ndash7 listthe cross-evaluation scores Then the mean and deviationmeasures of each alternative are presented in columns 8 and9 respectively Consequently the results in column 10 areapplied to generate a rank order of the five community servicecompanies 119883

5≻ 1198831≻ 1198832≻ 1198834≻ 1198833 The decision maker

is pleased to select company 1198835with the best performance

among the five alternatives

6 Conclusions and Future Research

Due to the complexity and uncertainty of MADM problemsin real world the decision maker is not able to provide theexact attribute weights And different weight determinationmethods are difficult to generate a most satisfaction solutionThe current paper presents a new MADM method basedon cross-evaluation in which the weight information isexpressed by the so-called weight space It attempts to rectifysome shortcomings of existingMADMmethods by providingthe following important features (1) it is more feasibleto provide a weight space rather than specifying a set ofweight values (2) the single dominating attribute drawbackcan be avoided by the restriction of the upper bound ofeach attribute (3) compared with DEA based methods thevariable attribute weights are consistent with the opinionof decision maker (4) both mean score and the deviationscore are considered in our cross-evaluation models and (5)the proposed method is also successfully extended to moregeneralized situations in which attribute values are expressedby interval values and ordinal values respectively

We have illustrated the applicability of the proposedmethods by revisiting two reported MADM studies and bydepicting a case study on the selection of community servicecompanies Although our proposed method makes someimprovement in solving MADM problems it only can dealwith the individual decision making case Future research

Mathematical Problems in Engineering 9

Table 10 Evaluation results for the selection of community service companies

Candidates 119862max119897

1198621119897

1198622119897

1198623119897

1198624119897

1198625119897

119903119894

120575119894

CV(119883119894)

1198831

0489 0489 0317 0295 0317 0317 0347 0072 02071198832

0532 0433 0532 0508 0458 0504 0487 0036 00741198833

0535 0526 0526 0535 0526 0526 0528 0004 00081198834

0223 0202 0223 0211 0223 0223 0216 0009 00421198835

0422 0349 0349 0103 0349 0422 0314 0109 0347

may attempt to employ our cross-evaluation methods tohandle group decision making problem

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research is supported by National Key Technology Sup-port Program (no 2014BAH27F00) National Natural Sci-ence Funds of China (no 71471053) and the FundamentalResearch Fund for the Central Universities (no JZ2014-HGBZ0339)

References

[1] R H Green and J R Doyle ldquoOn maximizing discriminationin multiple criteria decision makingrdquo Journal of the OperationalResearch Society vol 46 no 2 pp 192ndash204 1995

[2] R E Steuer and P Na ldquoMultiple criteria decision makingcombined with finance a categorized bibliographic studyrdquoEuropean Journal of Operational Research vol 150 no 3 pp496ndash515 2003

[3] J Wallenius J S Dyer P C Fishburn R E Steuer S Ziontsand K Deb ldquoMultiple criteria decision making multi-attributeutility theory recent accomplishments and what lies aheadrdquoManagement Science vol 54 no 7 pp 1336ndash1349 2008

[4] W Ho X Xu and P K Dey ldquoMulti-criteria decision makingapproaches for supplier evaluation and selection a literaturereviewrdquo European Journal of Operational Research vol 202 no1 pp 16ndash24 2010

[5] C W Churchman R L Ackoff and N M Smith ldquoAn approx-imate measure of valuerdquo Journal of the Operational ResearchSociety of America vol 2 no 2 pp 172ndash187 1954

[6] T L Saaty ldquoThe analytic hierarchy processrdquo in Proceedings ofthe Second International Seminar on Operational Research in theBasque Provinces vol 4 pp 189ndash234 1980

[7] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[8] H Deng C-H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[9] P K Dey ldquoIntegrated project evaluation and selection usingmultiple-attribute decision-making techniquerdquo InternationalJournal of Production Economics vol 103 no 1 pp 90ndash1032006

[10] G-D Li D Yamaguchi and M Nagai ldquoA grey-based decision-making approach to the supplier selection problemrdquoMathemat-ical amp Computer Modelling vol 46 no 3-4 pp 573ndash581 2007

[11] A Shanian and O Savadogo ldquoTOPSIS multiple-criteria deci-sion support analysis for material selection of metallic bipolarplates for polymer electrolyte fuel cellrdquo Journal of Power Sourcesvol 159 no 2 pp 1095ndash1104 2006

[12] G-H Tzeng Y-P O Yang C-T Lin and C-B Chen ldquoHier-archical MADM with fuzzy integral for evaluating enterpriseintranet web sitesrdquo Information Sciences vol 169 no 3-4 pp409ndash426 2005

[13] 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

[14] A Kelemenis and D Askounis ldquoA new TOPSIS-based multi-criteria approach to personnel selectionrdquo Expert Systems withApplications vol 37 no 7 pp 4999ndash5008 2010

[15] B Bairagi B Dey B Sarkar and S K Sanyal ldquoADeNovomulti-approaches multi-criteria decision making technique with anapplication in performance evaluation of material handlingdevicerdquo Computers amp Industrial Engineering vol 87 pp 267ndash282 2015

[16] J S H Kornbluth ldquoDynamic multi-criteria decision makingrdquoJournal of Multi-Criteria Decision Analysis vol 1 no 2 pp 81ndash92 1992

[17] Y-H Lin P-C Lee T-P Chang and H-I Ting ldquoMulti-attribute group decision making model under the condition ofuncertain informationrdquoAutomation in Construction vol 17 no6 pp 792ndash797 2008

[18] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multicrite-ria acceptability analysis for groupdecisionmakingrdquoOperationsResearch vol 49 no 3 pp 444ndash454 2001

[19] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[20] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[21] S Miyamoto ldquoMultisets and fuzzymultisetsrdquo in Soft Computingand Human-Centered Machines Z Q Liu and S MiyamotoEds pp 9ndash33 Springer Berlin Germany 2000

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[23] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[25] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling Simulation and Computation for Engineering andEnvironmental Systems vol 34 no 7 pp 1779ndash1787 2010

10 Mathematical Problems in Engineering

[26] D Horsky andM R Rao ldquoEstimation of attribute weights frompreference comparisonsrdquoManagement Science vol 30 no 7 pp801ndash822 1984

[27] P A Bottomley and J R Doyle ldquoA comparison of three weightelicitation methods good better and bestrdquo Omega vol 29 no6 pp 553ndash560 2001

[28] J Figueira and B Roy ldquoDetermining the weights of criteria inthe ELECTRE type methods with a revised Simosrsquo procedurerdquoEuropean Journal ofOperational Research vol 139 no 2 pp 317ndash326 2002

[29] L E Shirland R R Jesse R L Thompson and C L IacovouldquoDetermining attribute weights using mathematical program-mingrdquo Omega vol 31 no 6 pp 423ndash437 2003

[30] Y-M Wang and Y Luo ldquoIntegration of correlations withstandard deviations for determining attribute weights in mul-tiple attribute decision makingrdquo Mathematical amp ComputerModelling vol 51 no 1-2 pp 1ndash12 2010

[31] J Ma Z-P Fan and L-H Huang ldquoA subjective and objectiveintegrated approach to determine attribute weightsrdquo EuropeanJournal of Operational Research vol 112 no 2 pp 397ndash4041999

[32] X Xu ldquoA note on the subjective and objective integratedapproach to determine attribute weightsrdquo European Journal ofOperational Research vol 156 no 2 pp 530ndash532 2004

[33] P J H Schoemaker and C C Waid ldquoAn experimental compar-ison of different approaches to determining weights in additiveutility modelsrdquoManagement Science vol 28 no 2 pp 182ndash1961982

[34] Y-M Wang and C Parkan ldquoA general multiple attributedecision-making approach for integrating subjective prefer-ences and objective informationrdquo Fuzzy Sets and Systems vol157 no 10 pp 1333ndash1345 2006

[35] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[36] J R Doyle ldquoMultiattribute choice for the lazy decision makerlet the alternatives deciderdquoOrganizational Behavior andHumanDecision Processes vol 62 no 1 pp 87ndash100 1995

[37] E Bernroider and V Stix ldquoA method using weight restrictionsin data envelopment analysis for ranking and validity issues indecisionmakingrdquoComputersampOperations Research vol 34 no9 pp 2637ndash2647 2007

[38] J Wu and L Liang ldquoA multiple criteria ranking method basedon game cross-evaluation approachrdquo Annals of OperationsResearch vol 197 no 1 pp 191ndash200 2012

[39] C Fu and K-S Chin ldquoRobust evidential reasoning approachwith unknown attribute weightsrdquo Knowledge-Based Systemsvol 59 no 2 pp 9ndash20 2014

[40] B S Ahn ldquoExtreme point-based multi-attribute decision anal-ysis with incomplete informationrdquo European Journal of Opera-tional Research vol 240 no 3 pp 748ndash755 2015

[41] WW Cooper J L Ruiz and I Sirvent ldquoChoosing weights fromalternative optimal solutions of dualmultipliermodels inDEArdquoEuropean Journal of Operational Research vol 180 no 1 pp443ndash458 2007

[42] G R Jahanshahloo and P F Shahmirzadi ldquoNew methods forranking decision making units based on the dispersion ofweights and Norm 1 in Data Envelopment Analysisrdquo Computersamp Industrial Engineering vol 65 no 2 pp 187ndash193 2013

[43] V E Krivonozhko O B Utkin M M Safin and A V LychevldquoOn some generalization of the DEA modelsrdquo Journal of theOperational Research Society vol 60 no 11 pp 1518ndash1527 2009

[44] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[45] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo New Directions for ProgramEvaluation vol 1986 no 32 pp 73ndash105 1986

[46] A Boussofiane R G Dyson and E Thanassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[47] K Wang and F Wei ldquoGame cross efficiency DEA based projectreview of equipment designrdquo Journal of Beijing University ofAeronautics and Astronautics vol 35 no 10 pp 1278ndash12822009

[48] T Wu and J Blackhurst ldquoSupplier evaluation and selection anaugmented DEA approachrdquo International Journal of ProductionResearch vol 47 no 16 pp 4593ndash4608 2009

[49] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Korean stockmarketrdquo European Journal of Operational Research vol 236 no1 pp 361ndash368 2014

[50] B Roy ldquoTheoutranking approach and the foundations of electremethodsrdquoTheory and Decision vol 31 no 1 pp 49ndash73 1991

[51] G F Reed F Lynn and B D Meade ldquoUse of coefficient ofvariation in assessing variability of quantitative assaysrdquo Clinicalamp Diagnostic Laboratory Immunology vol 9 no 6 pp 1235ndash1239 2002

[52] 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

[53] H M Moshkovich A I Mechitov and D L Olson ldquoOrdinaljudgments in multiattribute decision analysisrdquo European Jour-nal of Operational Research vol 137 no 3 pp 625ndash641 2002

[54] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Tsinghua Press Beijing China 2005

[55] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

maker to select attribute weights [26ndash28] The objectivemethods determine attribute weights based on the objectiveinformation [29 30] Hybrid methods [31ndash34] combine boththe preferences of a decisionmaker and objective informationto determine attribute weights

Thirdly as a nonparametric programming efficiency-rating technique for evaluating a set of decision making units(DMUs) with multiple inputs and outputs date envelopmentanalysis (DEA) requires relatively less additional informa-tion [35] DEA has been considered as an effective tool tosolve MADM problems without information on weights Forinstance Doyle [36] presents an approach based on DEAmethod which derives a set of attribute weights from thedata itself Bernroider and Stix [37] introduce an approachbased onDEAwith restrictedmultipliers for accountable andunderstandable MADM Wu and Liang [38] develop a newmultiple attribute rankingmethod based onDEAgame cross-efficiency model in which each alternative is viewed as aplayer that seeks to maximize its own performance

Although existing methods handle uncertainty to someextent some drawbacks are also exposed which are listed asfollows First both different defuzzy approaches and weightdetermination methods may generate different attributeweights and produce different optimal solutions [39 40]Therefore it is difficult to identify which optimal solutionis the best one Second behavior factors are not consideredin existing MADM methods which may lead to deviationresults in decision making Specially they do not avoid thebad situation that the ranking results may be determined bya single attribute with an exorbitant weight In other wordsthe alternative rank order is consistent with the rank order ofoverestimated attribute which means that other attributes aretotally useless in evaluation Obviously such result is unfairand unreasonable because the optimal alternative should bedetermined by multiple attributes instead of an individualattribute Third although DEA models can evaluate MADMproblems without weight information it also suffers from thefollowing three drawbacks One is that the optimal weightsderived by DEA may not be consistent with the opinionsof the decision maker [41] Another shortage is that DEAidentifies too many efficient alternatives and cannot welldistinguish them [42 43] Furthermore the nonuniquenesssolution phenomenon is also common inDEA basedMADMmethods [44]

Considering the above analysis we attempt to develop across-evaluation based approach to solve MADM problemswith uncertain decision information To be specific we firstlydefine and analyze the attribute weight space which containsall feasible weights Specially to avoid the single dominatingattribute situation we calculate the upper bound weightof each attribute Then considering a condition where theattribute values are certain and the attribute weights areuncertain the return and risk of closeness indexes are cal-culated based on cross-evaluation in which each alternativeseeks to maximize its closeness index by selecting a set ofoptimal weights And the coefficient of variation is utilizedto integrate the results and produce an alternative rankorder In a more generalized condition where the attributevalues are denoted by interval values or ordinal values

the cross-evaluationmodels consideringDMrsquos risk preferenceare developed

The main contributions of the current paper can besummarized as follows (1) we firstly defined the attributeweight space and determined the upper bound weight ofeach attribute to conduct the uncertain information (2) bothmean and deviationmeasures of closeness index under cross-evaluation are considered to integrate a unique solution tothe MADM problem (3) the proposed method is extendedto more generalized situations where the attribute values aredescribed by interval values and ordinal values respectively

The remainder of this paper is organized as follows InSection 2 we briefly review the relative works In Section 3we propose a cross-evaluation basedmethod to solveMADMproblems with uncertain weight information In Section 4the proposed method is extended to more generalized formswhere the attribute information is expressed by intervalvalues and ordinal values Section 5 illustrates the methodswith reportedMADMstudies and a case study of the selectionof community service companies Conclusions and futureresearch will be displayed in Section 6

2 Preliminaries

21 Formulation of MADM Problems Suppose 119883 = 1198831

1198832 119883

119898 is a discrete set of alternatives and 119884 = 119884

1

1198842 119884

119899 is a set of 119899 attributes with the weight vector 119908 =

(1199081 1199082 119908

119899)119879 where 119908

119895gt 0 sum119899

119895=1119908119895

= 1 Let 119860 =

(119886119894119895)119898times119899

be a decision matrix where 119886119894119895denotes the attribute

value that the 119894th alternative119883119894isin 119883 takes with respect to the

119895th attribute 119884119895isin 119884

In general there are both benefit type attributes (ie thelarger the better) and cost type attributes (ie the smallerthe better) in a MADA problem We transform the originaldecision matrix 119860 = (119886

119894119895)119898times119899

into a normalized matrix 119877 =

(119903119894119895)119898times119899

by using the method from Hwang and Yoon [7]

119903119894119895=

119886119894119895

max119894119886119894119895

for benefit attribute 119884119895

min119894119886119894119895

119886119894119895

for cost attribute 119884119895

119894 = 1 2 119898 119895 = 1 2 119899

(1)

22 Characteristics of the TOPSIS Method The basic prin-ciple of TOPSIS method is that the optimal alternativeshould have both the shortest distance from the positive idealsolution and the furthest distance from the negative idealsolution [14]

The TOPSIS procedure is performed in six stages asfollows

(a) Normalize the decision matrix

119903119894119895=

119886119894119895

radicsum119899

119895=11198862

119894119895

119894 = 1 2 119898 119895 = 1 2 119899 (2)

where 119886119894119895is the 119895th attribute value of alternative 119894

Mathematical Problems in Engineering 3

(b) Calculate the weighted normalized decision matrix

V119894119895= 119908119895119903119894119895 119894 = 1 2 119898 119895 = 1 2 119899 (3)

where 119908119895is the weight of the 119895th attribute and

sum119899

119895=1119908119895= 1

(c) Determine the positive ideal and negative ideal solu-tionPositive ideal solution 119860

+= V+1 V+2 V+

119899

= (max119894

V119894119895| 119895 isin 119868

1015840) (min

119894

V119894119895| 119895 isin 119868

10158401015840)

Negative ideal solution 119860minus= Vminus1 Vminus2 Vminus

119899

= (min119894

V119894119895| 119895 isin 119868

1015840) (max

119894

V119894119895| 119895 isin 119868

10158401015840)

(4)

where 1198681015840 indicate the index set of the benefit type

attributes and 11986810158401015840 the index set of the cost type

attributes(d) Measure the distances from the positive ideal solution

and negative ideal solution respectively

119863+

119894= radic

119899

sum

119895=1

(V119894119895minus V+119895)2

119894 = 1 2 119898

119863minus

119894= radic

119899

sum

119895=1

(V119894119895minus Vminus119895)2

119894 = 1 2 119898

(5)

(e) Calculate the relative closeness to the ideal solutions

119862lowast

119894=

119863minus

119894

(119863+

119894+ 119863minus

119894) 119894 = 1 2 119898 (6)

(f) Rank the alternatives

The larger the closeness index of an alternative the higherthe rank of the alternative

23 DEA Cross-Efficiency Evaluation We assume that thereare 119899 DMUs to be evaluated in terms of 119898 inputs and 119899

outputs The 119894th input and 119903th output of DMU119895are denoted

by 119909119894119895(119894 = 1 119898) and 119910

119903119895 respectively The efficiency of

a DMU119889can be calculated by the following CCR model in

linear programming formulation [35]

max119904

sum

119903=1

119906119903119910119903119889

= 120579119889

st119898

sum

119894=1

119908119894119909119894119895minus

119904

sum

119903=1

119906119903119910119903119895ge 0 119895 = 1 2 119899

119898

sum

119894=1

119908119894119909119894119889

= 1

119908119894ge 0 119894 = 1 2 119898

119906119903ge 0 119903 = 1 2 119904

(7)

When the above model is solved a set of optimal weights119908lowast

1119889 119908

lowast

119898119889 119906lowast

1119889 119906

lowast

119904119889is obtained Then the 119889 cross-

efficiency of any DMU119895is computed as [38]

119864119889119895

=

sum119904

119903=1119906lowast

119903119889119910119903119895

sum119898

119894=1119908lowast

119894119889119909119894119895

119889 119895 = 1 119899 (8)

For DMU119895(119895 = 1 119899) the average of all 119864

119889119895(119889 =

1 119899) denoted by 119864119895

= (1119899)sum119899

119889=1119864119889119895 is considered as

the cross-efficiency score of DMU119895

3 Proposed Method

In this section we will introduce the so-called single dom-inating attribute issue and propose a model to supportdecisions by incorporating considerations on this issue inSection 31 furthermore in Section 32 we will introducethe cross-evaluation framework and the accompanying riskconsideration finally in Section 33 we propose a proce-dure summarizing all the aforementioned considerations forMADM

31 Determination of the Weight Space We observe that inMADM problems an individual attribute can determine thefinal ranks of all alternatives when the corresponding weightis greater than some threshold value In other words therank order of the alternatives is completely in accordancewith the rank order of the dominating attribute and therest of the attributes are totally ignored in the evaluationIt should be noted that the ignored attributes are takeninto the evaluation only because the decision maker regardsthem as import factors or they could have excluded themfrom the decision process in the first place Therefore weargue this is unacceptable In order to address this issueit is necessary to define and compute the threshold upperbound of each attribute weight The upper bound can aid thedecision process by suggesting meaningful weights

For the simplicity of exposition if the attribute values119903119894119896

(119894 = 1 119898) in the 119896th column of the decision matrix119877 have the relation 119903

1198941119896

ge 1199031198942119896

ge sdot sdot sdot ge 119903119894119898119896(119894119904

isin

1119898 119904 = 1 119898) then we denote the ranks of thealternatives according to the values of the 119896th attribute 119884

119896

as 1198831198941

≻1198961198831198942

≻119896sdot sdot sdot ≻119896119883119894119898

With this notation in position thesingle dominating attribute happens if the final rank orderof all alternates is the same as some 119883

1198941

≻1198961198831198942

≻119896sdot sdot sdot ≻119896119883119894119898

no matter how the decision maker distributes the relativeweights of the other attributes We formulize this notion asDefinition 1 as below

Definition 1 Let 119896 be the index of some attribute 119908lowast119896is the

threshold upper bound of 119908119896if and only if the following

conditions are satisfied

(a) If 119908119896ge 119908lowast

119896 then forall1 le 119894 119905 le 119898 119903

119894119896ge 119903119905119896if and only

if sum119899119895=1

119908119895119903119894119895ge sum119899

119895=1119908119895119903119905119895

(b) If 119908119896lt 119908lowast

119896 then exist1 le 119894 119905 le 119898 such that 119903

119894119896ge 119903119905119896

andsum119899

119895=1119908119895119903119894119895le sum119899

119895=1119908119895119903119905119895hold true simultaneously

4 Mathematical Problems in Engineering

To implement Definition 1 we provide Theorem 2 belowto determine the threshold upper bound

Theorem 2 The threshold upper bound of the 119896th attribute(119908lowast119896) is given by the minimization program

min 119908119896

st 11990811989611990311989411+

119899

sum

119895=1119895 =119896

119908(119891)

1198951199031198941119895ge 11990811989611990311989421+

119899

sum

119895=1119895 =119896

119908(119891)

1198951199031198942119895

ge sdot sdot sdot

ge 1199081198961199031198941198981+

119899

sum

119895=1119895 =119896

119908(119891)

119895119903119894119898119895

(9)

119899

sum

119895=1

119908119895= 1

119908119895ge 0

119895 = 1 2 119899

119891 = 1 2 119899 minus 1

(10)

where 119908(119891)

119895= 1 minus 119908

119896if 119895 is the 119891th element of the set

1 119873 119896 in ascending order otherwise 119908(119891)119895

= 0 Letus take 119896 = 1 as an example We have (119908(1)

2 119908(1)

3 119908

(1)

119899) =

(1 minus 1199081 0 0) (119908(2)

2 119908(2)

3 119908

(2)

119899) = (0 1 minus 119908

1 0 0)

(119908(119899minus1)

2 119908(119899minus1)

3 119908

(119899minus1)

119899) = (0 0 1 minus 119908

1) The sub-

script 119894ℎ(ℎ isin 1 119898) of 119903 in the above model indicates the

119894ℎpreferred alternative determined by the preference relation

1198831198941

≻1198961198831198942

≻119896sdot sdot sdot ≻119896119883119894119898

Proof Clearly there are (119899 minus 1) constraints in (9) When119896 = 1 the weight combinations (119908

(1)

2 119908(1)

3 119908

(1)

119899) =

(1 minus 1199081 0 0) (119908(2)

2 119908(2)

3 119908

(2)

119899) = (0 1 minus 119908

1 0 0)

(119908(119899minus1)

2 119908(119899minus1)

3 119908

(119899minus1)

119899) = (0 0 1 minus 119908

1) are applied

to the 119899minus1 constraints in orderThefirst constraint guaranteesthat the alternatives rank result has nothing to do withattribute 119884

2 Similarly the remaining 119899 minus 2 constraints ensure

that the alternatives rank result has nothing to do with theother attributes 119884

3 119884

119899 So the optimal minimization 119908

1

satisfies (a) in Definition 1Conversely we suggest that a constraint belonging to the

119897th (119897 = 1 119899 minus 1) group is removed from the model Thenattribute 119884

119897may play a part in the ranking because it is not

completely suppressed In other words (b) in Definition 1holds

Similarly the upper bound of each attribute weight(119908lowast

119896 119896 = 2 119899) can be calculated by the model in

Theorem 2 with 119896 = 2 119899 This proof is completed

Example 3 Ma et al [31] consider a case of a robot userexpecting to select a robot and there are four alternatives forhim to choose There are four attributes to evaluate them

that is cost velocity repeatability and load capacityThe nor-malized decision matrix is as follows

119860 =

(((

(

0 15

6

1

35

12

3

51 0

1 0 0 1

2

3

2

5

2

3

2

3

)))

)

(11)

Suppose that the information on attribute weights iscompletely unknown The upper bound weight of the firstattribute cost can be calculated by the above model

By calculating the above model provided in Theorem 2we obtain the optimal value119908lowast

1= 23 denoting the threshold

upper bound of the first attribute1199081 Analogously each upper

boundof attributeweight can be calculated by similarmodelsand the values are 119908lowast

2= 1013 119908

lowast

3= 45 119908

lowast

4= 914

32 Mean and Deviation of Closeness Index Based Cross-Evaluation Models In this subsection in the environmentof uncertain attribute weights we develop cross-evaluationmodels to calculate the mean and deviation of closenessindex for each assessed alternative respectively Then thecomputed indexes are integrated to obtain the evaluationresults

Cross-efficiency evaluation method since being pro-posed by Sexton et al [45] has been used to evaluate andrankDMUs Itsmain idea is to use cross-evaluation instead ofa self-evaluation There are at least two advantages for cross-evaluation method One is that it obtains a unique rankingamong the DMUs The other is that the cross-efficiencymeans can act effectively to differentiate between good andbad alternatives [46] Hence the cross-evaluation methodis widely used for ranking performance of alternatives forexample equipment design [47] supplier selection [48] andstock market [49]

To calculate the self-evaluation closeness index for eachalternative we consider the following mathematical pro-gramming problem for a given alternative119883

119894

max119863minus

119894

119863minus

119894+ 119863+

119894

= 119862max119894

119908 isin Ω (119908)

(12)

Here 119863minus119894and 119863

+

119894represent the distance from negative

ideal point and ideal point (defined in (5)) respectively Forease of calculation the ideal point and negative point aredefined as 119860+ = 119908

1 1199082 119908

119899 and 119860

minus= 0 0 0 Then

119863+

119894and 119863

minus

119894are denoted by 119863

+

119894= radicsum

119899

119895=1(119908119895119903119894119895minus 119908119895)2 and

119863minus

119894= radicsum

119899

119895=1(119908119895119903119894119895)2 respectively The constraint 119908 isin Ω(119908)

depicts the possible set of attribute weights Obviously whenthe weights information of attributes is completely unknownΩ(119908) = 119908

119895gt 0 sum

119899

119895=1119908119895= 1 Introducing the upper bound

weight information calculated in Section 31 Ω(119908) = 119908119895lt

119908lowast

119895 sum119899

119895=1119908119895= 1

Mathematical Problems in Engineering 5

In many practical situations although the weight valuesare difficult to obtain the weight information is partiallyknown and predefined by the weight space As followswe introduce two types of partial weight information asreferences to decision maker

Type 1 (pairwise comparisonweights information) InMADMattribute weights 119908

119895are mainly introduced to reflect the

importance to attributes 119884119895(119895 = 1 2 119899) As stated in

Section 1 pairwise comparison is a widely accepted subjectivemethod to determine attribute weight

Attribute 1198842is at least two times and not more than three

times as important as attribute 1198841

In notational jargons this is denoted by

21199081le 1199082le 31199081 (13)

In general terms we compare 1198841with 119884

119895(119895 = 2 3 119899)

respectively The bound linear constraints are denoted by

1198971198951199081le 119908119895le 1199061198951199081 (14)

in which 119897119895and 119906

119895represent the lower and upper bounds of

times119908119895compared with119908

1 respectively Hence the attribute

weights space is obtained by the above pairwise comparisoninformation which is expressed as Ω(119908) = 119908

119895gt 0 119897

1198951199081le

119908119895le 1199061198951199081 119895 = 1 2 119899 Clearly 119897

1= 1199061= 1 holds

Compared with conventional subjective weight determi-nation approaches such as the AHP weights space is morefeasible and applicative because the decision makers neednot express their preferences explicitly In many situationsexperts are pleased to give such interval information ratherthan exact value due to the complexity of actual problems

Type 2 (ordinal weights information) Outranking relation[50] usually denoted by 119878 was proposed by Roy whoseaim was to establish a realistic representation of the threebasic preference situations strict preference indifferenceand incomparability Motivated by the widely accepted out-ranking relation we introduce the concept of rank-orderweights information In many situations although the weightvalues are difficult to give by decision makers the rankingof each attribute is easy to obtain according to importanceTake the Olympic Games as an example the achievementsof athletes and nations are judged by gold silver and bronzemedals Apparently a gold medal is more important than asilver medal and a silver medal is more important than abronze medal Then the weights space can be denoted byΩ(119908) = 119908

1gt 1199082

gt 1199083 1199081 1199082 1199083

gt 0 in which1199081 1199082 1199083represent the weight of gold silver and bronze

medal respectivelyThe above self-evaluation model (see (12)) shows that all

alternatives choose their own attribute weights in order tomaximize their own self-evaluation closeness index Similarto the formofDEA cross-efficiency we obtain a set of optimalattribute weights 119882lowast

119894= (119908lowast

1198941 119908

lowast

119894119899) according to the above

model When these optimal weights are applied to evaluatealternative119883

119896 a cross score denoted by119862

119894119896= 119863minus

119894119896(119863minus

119894119896+119863+

119894119896)

is obtained

For alternative 119896 (119896 = 1 119898) we use the average ofall 119862119894119896to measure the expected return of cross-evaluation

closeness index which is written as

119903 (119883119896) =

1

119898

119898

sum

119894=1

119862119894119896 (15)

Furthermore we use the standard variance

120575 (119883119896) =

radicsum119898

119894=1(119862119894119896minus 119903 (119883

119896))2

119899

(16)

tomeasure the risk caused by different sets of optimal weightsproduced by each alternative

In general there are both return of closeness index(defined by the mean of cross-evaluation score) and risk ofcloseness index (defined by the deviation between return andeach cross-evaluation closeness index) in cross-evaluationframework It is not hard to see that the larger return scorethe better and the smaller risk score the better Specificallygiven two alternatives 119883

119904and 119883

119905 if 119877(119883

119904) gt 119877(119883

119905) and

120575(119883119904) lt 120575(119883

119905) then 119883

119904≻ 119883119905(119883119904is better than 119883

119905) clearly

holds However when 119877(119883119904) gt 119877(119883

119905) and 120575(119883

119904) gt 120575(119883

119905) it

is not easy to compare the two alternatives directlyThen we integrate the return and risk measures by the

well-known coefficient of variation [51]

CV (119883119896) =

120575 (119883119896)

119903 (119883119896)times 100 (17)

For two given alternatives one having a smaller coeffi-cient of variation is considered to be better than the otherone Thus all the alternatives are ranked by the comparisonof their corresponding coefficient of variation

33 Procedure of the Proposed Approach Based on the aboveanalysis the procedure for generating a solution to theMADM problem is elaborated in the following

Step 1 (formulation of a MADM problem) The decisionmaker identifies 119899 attributes to evaluate 119898 alternatives Theoriginal attribute values are standardized by (1)

Step 2 (obtainment of the attribute weight space) First theupper bounds of each attribute are calculated byTheorem 2 toguarantee the effectiveness of all the attributesThen differenttypes of constraint of attribute weights are determined by thedecision maker to reflect the importance of each attributeLast the weight space including all the feasible weightcombinations is obtained

Step 3 (calculation of the mean and deviation measures)Firstly the maximized closeness index of each alternativeis calculated by (12) Secondly the mean measure of eachalternativersquos cross closeness index is computed by (15) Thirdthe deviation measure of each alternativersquos cross closenessindex is obtained by (16)

6 Mathematical Problems in Engineering

Step 4 (evaluation of the alternatives via coefficient ofvariation) Since the mean and deviation measures of eachalternativersquos cross closeness index are obtained in Step 3 thecoefficient of variation defined in (17) is utilized to integratethe results

Step 5 (generation of rank order of the alternatives) Throughcomparing the coefficients of variation among the alterna-tives the rank order can be directly obtained

4 Extension with Uncertain Attribute Values

41 Model with Interval Attribute Value It is widely acknowl-edged that uncertainty is associated with attribute values dueto complexity of real world and limited knowledge of humanbeings In many situations the attribute value 119903

119894119895is uncertain

but we know that the feasible attribute value 119903119894119895

isin [119903minus

119894119895 119903+

119894119895]

Here we present the following self-evaluation model withinterval attribute values [52] and uncertain attribute weights

max 120572

radicsum119899

119895=1(119908119895119903+

119894119895)2

radicsum119899

119895=1(119908119895119903+

119894119895)2

+ radicsum119899

119895=1(119908119895119903+

119894119895minus 119908119895)2

+ 120573

radicsum119899

119895=1(119908119895119903minus

119894119895)2

radicsum119899

119895=1(119908119895119903minus

119894119895)2

+ radicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2

= 119862max119894

119908 isin Ω (119908)

120572 + 120573 = 1 120572 ge 0 120573 ge 0

(18)

in which radicsum119899

119895=1(119908119895119903+

119894119895)2 radicsum

119899

119895=1(119908119895119903+

119894119895minus 119908119895)2 radicsum

119899

119895=1(119908119895119903minus

119894119895)2

andradicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2 represent distance of upper bounds

from ideal point distance of upper bounds from negativeideal point distance of lower bounds from ideal pointand distance of lower bounds from negative ideal pointrespectively 120572 and 120573 reflect the risk coefficients that combineupper closeness index and lower closeness respectively Inpractice120572 and120573 are decided by the decisionmaker accordingto his or her risk preference Specifically the decision makeris neutral if 120572 = 120573 = 12 120572 gt 120573 and 120572 lt 120573 indicate that he isoptimistic and pessimistic respectively

42 Model with Ordinal Attribute Value Another form ofuncertain preferential information is expressed by ordinalattribute value (eg ldquomore preferablerdquo and ldquoless preferablerdquo)It is argued to be more stable and more reliable than cardinalvalue in some situations [53] In this paper an approach ofthreshold value is introduced to transform ordinal value intolinear formulation For instance given four attributes 119886 119887 119888and 119889 the ordinal values are 1 2 3 and 4 respectively Byspecifying a threshold value 01 the attribute values can beconverted to a cardinal attribute space Ω(119886 119887 119888 119889) = 119886 = 1

119886 minus 119887 ge 01 119887 minus 119888 ge 01 119888 minus 119889 ge 01 119889 = 0 In generallet the ordinal attribute values 119886

119894119895(119894 = 1 119898) have relation

1198861198941119895gt 1198861198942119895gt sdot sdot sdot gt 119886

119894119898119895(119894119896isin [1119898] 119896 = 1 119898) and then

the cardinal attribute spaceΩ(119903119894119895) = 119886

119894119898119895= 1 119886119894119898119895minus119886119894119898minus1119895ge 120576

119886119894119898minus1119895minus119886119894119898minus2119895ge 120576 119886

1198942119895minus1198861198941119895ge 120576 1198861198941119895= 0 120576 gt 0 is obtained

(120576 is the threshold value given by the decision maker) Herewe present the following self-evaluation model with ordinalattribute values and uncertain attribute weights

maxradicsum119899

119895=1(119908119895119903119894119895)2

radicsum119899

119895=1(119908119895119903119894119895)2

+ radicsum119899

119895=1(119908119895119903119894119895minus 119908119895)2

= 119862max119894

st 119903 isin Ω (119903)

119908 isin Ω (119908)

(19)

119894 = 1 119898 (20)

119895 = 1 119899 (21)

For each alternative 119883119896(119896 = 1 119898) under evaluation

we obtain a group of optimal attribute values and a set ofattribute weights They are denoted by 119860

lowast

119896= (119903119896

119894119895)119898times119899

and119882lowast

119896= (119908

119896

1 119908119896

2 119908

119896

119899) respectively When the optimal

decision information is applied to evaluate alternative119883119897(119897 =

1 119898) a cross score denoted by 119862119896119897

= radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2

(radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2+ radicsum

119899

119895=1(119908119896

119895119903119896

119897119895minus 119908119896

119895)2) is obtained

Similar to that defined by (14) and (15) the mean anddeviation measures of alternative 119883

119897are denoted by 119903(119883

119897) =

sum119898

119896=1119862119896119897and 120575(119883

119897) = radicsum

119898

119896=1(119862119896119897minus 119903(119883

119897))2119899 respectively

Note that (12) (18) and (19) are nonlinear models theycan be transformed into linear forms according to practicalcases

5 Illustrative Examples

In order to illustrate the applications of our proposed meth-ods three practical MADM problems are presented

51 Application to House Selection with Cardinal AttributeValues Xu [54] stated a purchase house problem there arefour possible houses to be selected and the attributes are asfollows (1)119884

1is the house price (ten thousand yuan) (2)119884

2is

the usable area (square meter) (3) 1198843is the distance between

house and the center of city (kilometers) (4) 1198844is the nature

environment (scores) where 1198842and 119884

4are benefit attributes

and 1198841and 119884

3are cost attributes The attribute weights are

completely unknown The decision making matrix 119860 andnormalized decision making matrix 119877 are listed in Tables 1and 2 respectively Step 1 is completed

By utilizing (9) and (10) the upper boundof each attributeweight is obtained And as a result the attribute weight spaceis Ω(119908) = 0 lt 119908

1lt 077 0 lt 119908

2lt 078 0 lt 119908

3lt 062

0 lt 1199084lt 062 119908

1+ 1199082+ 1199083+ 1199084= 1 Step 2 is completed

Mathematical Problems in Engineering 7

Table 1 Decision making matrix 119860 of house evaluation

1198841

1198842

1198843

1198844

1198831

30 100 10 71198832

25 80 8 51198833

18 50 20 111198834

22 70 12 9

Table 2 Normalized decisionmakingmatrix119877 of house evaluation

1198841

1198842

1198843

1198844

1198831

06 10 08 06361198832

072 08 10 04551198833

10 05 04 101198834

0818 07 0667 0818

Table 3 The cross-evaluation measurements

Alternatives 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

119903 (119883119894) 120575 (119883

119894)

1198831

096 096 082 061 061 075 0151198832

092 080 092 066 066 076 0111198833

1 052 051 1 1 076 0241198834

0818 070 070 0818 0818 076 006

Table 4 Alternative efficiency matrix

Alternatives 1198831

1198832

1198833

1198834

CV (119883119894) 020 014 032 008

In terms of Step 3 in Section 33 we obtain the cross-evaluation measurements of each alternative which arepresented in Table 3 Step 3 is completed

Based on Table 3 and the calculation of coefficient ofvariation the results are shown in Table 4

Therefore we can rank the houses in accordance with theCV values presented in Table 4 to get a complete rank order1198834

≻ 1198832

≻ 1198831

≻ 1198833 And 119883

4is the best house under

evaluation Step 4 is completedIn order to understand our approach more clearly we

compare our approachwith other three similarmethods [54]namely average attribute value method maximizing devia-tion method and information entropy method respectivelywith the same example The corresponding results of suchthree methods are listed in Table 5

According to Tables 4 and 5 the result derived from ourcross-evaluation approach is quite different from that of theexisting three weight determination methods In fact Table 5shows that the three weight determination methods generatethree attribute weight vectors and produce three differentoptimal solutions However there is no single method thatcan guarantee more convincing weight vector than othersHence we treat the attribute weight as a weight space rather

Table 5 The results of three methods

Method Weight vector Rank order

Average attributevalue

(025 025 025 025) 1198831≻ 1198834≻ 1198832≻ 1198833

Maximizingdeviation

(020 024 029 027) 1198831≻ 1198832≻ 1198834≻ 1198833

Informationentropy

(013 022 036 029) 1198831≻ 1198832≻ 1198834≻ 1198833

Table 6 Attribute weights

1198841

1198842

1198843

[119908119871

1 119908119880

1] [119908

119871

2 119908119880

2] [119908

119871

3 119908119880

3]

[035 050] [015 030] [030 035]

Table 7 Attribute values of alternatives

1198841

1198842

1198843

119860 [065 085] [068 086] [045 078]119861 [075 1] [055 078] [064 080]119862 [054 068] [046 055] [078 1]119863 [077 090] [067 078] [056 077]119864 [025 045] [089 1] [075 094]

than a certain value in this paperThe comparison result vali-dates that our proposed approach can overcome the shortagethat we cannot obtain themost dominant result caused by thediversity of the way of weight vector determined

52 Application to University Faculty with Interval AttributeValue Consider the example in Bryson and Mobolurinrsquos[55] study that is five faculty candidates (119860 119861 119862 119863119864) are evaluated for tenure and promotion through threeattributes teaching research and service According to theDMrsquos preference the information for both attribute andalternatives is given by interval numbers as shown in Tables6 and 7 respectively (for the purpose of illustration our datais different from the literature)

Table 8 shows the computed results of the proposedapproach with interval values The self-evaluation score ofeach alternative is listed in the second column and the cross-evaluation scores are listed in columns 3ndash7 Columns 8 and9 show the mean and deviation measures respectively Thenthe results in column 10 are applied to generate a rank orderof the five candidates119860 ≻ 119862 ≻ 119864 ≻ 119861 ≻ 119863 which is a solutionto the problem

53 Application to the Selection of Community Service Com-panies with Ordinal Attribute Values In this subsection weapply our approach to the evaluation and selection of fivecommunity service companies that are located in Hefei acity in Anhui province of China These companies providevarious kinds of service for the public such as property

8 Mathematical Problems in Engineering

Table 8 Evaluation results for the comparison of faculty candidates

Candidates 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

1198625119894

119903119894

120575119894

CV(119883119894)

119860 0717 0717 0712 0706 0717 0703 0711 0006 0008119861 0792 0792 0792 0788 0792 0746 0782 0018 0023119862 0672 0651 0661 0672 0651 0660 0659 0008 0012119863 0774 0774 0771 0764 0774 0737 0764 0014 0461119864 0645 0525 0526 0540 0525 0645 0552 0047 0018

Table 9 Ordinal values of alternatives

1198841

1198842

1198843

1198844

1198831

4 2 4 21198832

1 4 3 31198833

2 5 1 11198834

5 3 2 51198835

3 1 5 4

management catering health care and entertainment Inthe case study the decision maker is an official from thedevelopment and reform commission of Hefei with thehelp of a group of experts from the governments relevantenterprises and community organizations Four attributesare used to evaluate the six companies They are quality ofservice (119884

1) diversity of service (119884

2) profitability (119884

3) and

development potential (1198844) The original attribute values are

given by ordinal values as shown in Table 9 The decisionprocedures are as follows

After considering the four attributes the decision makerviews that quality of service (119884

1) is the most important

attribute in evaluation The development potential (1198844) is the

second most important one profitability (1198843) is the third

most important one and diversity of service (1198842) is the

least important one respectively And the difference betweentwo adjacent attribute weights should be more than 01 Inaddition each weight value should be nomore than 05Thenthe weight space is denoted by Ω(119908) = 119908

1minus 01 gt 119908

4

1199084minus 01 gt 119908

3 1199083minus 01 gt 119908

2 1199081le 05 119908

2le 05 119908

3le 05

1199084le 05 119908

1+ 1199082+ 1199083+ 1199084= 1

By specifying a threshold value of 01 ordinal attributevalues are transformed into four sets of attribute value spacesThey are denoted by

Ω(1198841) = 119903

21= 1 11990321

minus 11990331

ge 01 11990331

minus 11990351

ge 01 11990351

minus 11990311

ge 01 11990311

minus 11990341

ge 01 11990341

= 0

Ω (1198842) = 119903

52= 1 11990352

minus 11990312

ge 01 11990312

minus 11990342

ge 01 11990342

minus 11990322

ge 01 11990322

minus 11990332

ge 01 11990332

= 0

Ω (1198843) = 119903

33= 1 11990333

minus 11990343

ge 01 11990343

minus 11990323

ge 01 11990323

minus 11990313

ge 01 11990313

minus 11990353

ge 01 11990353

= 0

Ω (1198844) = 119903

34= 1 11990334

minus 11990314

ge 01 11990314

minus 11990324

ge 01 11990324

minus 11990354

ge 01 11990354

minus 11990344

ge 01 11990344

= 0

(22)

Table 10 shows the computed results of the proposedapproachThe second column shows the self-evaluation scoreof each alternative computed by (19) And columns 3ndash7 listthe cross-evaluation scores Then the mean and deviationmeasures of each alternative are presented in columns 8 and9 respectively Consequently the results in column 10 areapplied to generate a rank order of the five community servicecompanies 119883

5≻ 1198831≻ 1198832≻ 1198834≻ 1198833 The decision maker

is pleased to select company 1198835with the best performance

among the five alternatives

6 Conclusions and Future Research

Due to the complexity and uncertainty of MADM problemsin real world the decision maker is not able to provide theexact attribute weights And different weight determinationmethods are difficult to generate a most satisfaction solutionThe current paper presents a new MADM method basedon cross-evaluation in which the weight information isexpressed by the so-called weight space It attempts to rectifysome shortcomings of existingMADMmethods by providingthe following important features (1) it is more feasibleto provide a weight space rather than specifying a set ofweight values (2) the single dominating attribute drawbackcan be avoided by the restriction of the upper bound ofeach attribute (3) compared with DEA based methods thevariable attribute weights are consistent with the opinionof decision maker (4) both mean score and the deviationscore are considered in our cross-evaluation models and (5)the proposed method is also successfully extended to moregeneralized situations in which attribute values are expressedby interval values and ordinal values respectively

We have illustrated the applicability of the proposedmethods by revisiting two reported MADM studies and bydepicting a case study on the selection of community servicecompanies Although our proposed method makes someimprovement in solving MADM problems it only can dealwith the individual decision making case Future research

Mathematical Problems in Engineering 9

Table 10 Evaluation results for the selection of community service companies

Candidates 119862max119897

1198621119897

1198622119897

1198623119897

1198624119897

1198625119897

119903119894

120575119894

CV(119883119894)

1198831

0489 0489 0317 0295 0317 0317 0347 0072 02071198832

0532 0433 0532 0508 0458 0504 0487 0036 00741198833

0535 0526 0526 0535 0526 0526 0528 0004 00081198834

0223 0202 0223 0211 0223 0223 0216 0009 00421198835

0422 0349 0349 0103 0349 0422 0314 0109 0347

may attempt to employ our cross-evaluation methods tohandle group decision making problem

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research is supported by National Key Technology Sup-port Program (no 2014BAH27F00) National Natural Sci-ence Funds of China (no 71471053) and the FundamentalResearch Fund for the Central Universities (no JZ2014-HGBZ0339)

References

[1] R H Green and J R Doyle ldquoOn maximizing discriminationin multiple criteria decision makingrdquo Journal of the OperationalResearch Society vol 46 no 2 pp 192ndash204 1995

[2] R E Steuer and P Na ldquoMultiple criteria decision makingcombined with finance a categorized bibliographic studyrdquoEuropean Journal of Operational Research vol 150 no 3 pp496ndash515 2003

[3] J Wallenius J S Dyer P C Fishburn R E Steuer S Ziontsand K Deb ldquoMultiple criteria decision making multi-attributeutility theory recent accomplishments and what lies aheadrdquoManagement Science vol 54 no 7 pp 1336ndash1349 2008

[4] W Ho X Xu and P K Dey ldquoMulti-criteria decision makingapproaches for supplier evaluation and selection a literaturereviewrdquo European Journal of Operational Research vol 202 no1 pp 16ndash24 2010

[5] C W Churchman R L Ackoff and N M Smith ldquoAn approx-imate measure of valuerdquo Journal of the Operational ResearchSociety of America vol 2 no 2 pp 172ndash187 1954

[6] T L Saaty ldquoThe analytic hierarchy processrdquo in Proceedings ofthe Second International Seminar on Operational Research in theBasque Provinces vol 4 pp 189ndash234 1980

[7] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[8] H Deng C-H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[9] P K Dey ldquoIntegrated project evaluation and selection usingmultiple-attribute decision-making techniquerdquo InternationalJournal of Production Economics vol 103 no 1 pp 90ndash1032006

[10] G-D Li D Yamaguchi and M Nagai ldquoA grey-based decision-making approach to the supplier selection problemrdquoMathemat-ical amp Computer Modelling vol 46 no 3-4 pp 573ndash581 2007

[11] A Shanian and O Savadogo ldquoTOPSIS multiple-criteria deci-sion support analysis for material selection of metallic bipolarplates for polymer electrolyte fuel cellrdquo Journal of Power Sourcesvol 159 no 2 pp 1095ndash1104 2006

[12] G-H Tzeng Y-P O Yang C-T Lin and C-B Chen ldquoHier-archical MADM with fuzzy integral for evaluating enterpriseintranet web sitesrdquo Information Sciences vol 169 no 3-4 pp409ndash426 2005

[13] 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

[14] A Kelemenis and D Askounis ldquoA new TOPSIS-based multi-criteria approach to personnel selectionrdquo Expert Systems withApplications vol 37 no 7 pp 4999ndash5008 2010

[15] B Bairagi B Dey B Sarkar and S K Sanyal ldquoADeNovomulti-approaches multi-criteria decision making technique with anapplication in performance evaluation of material handlingdevicerdquo Computers amp Industrial Engineering vol 87 pp 267ndash282 2015

[16] J S H Kornbluth ldquoDynamic multi-criteria decision makingrdquoJournal of Multi-Criteria Decision Analysis vol 1 no 2 pp 81ndash92 1992

[17] Y-H Lin P-C Lee T-P Chang and H-I Ting ldquoMulti-attribute group decision making model under the condition ofuncertain informationrdquoAutomation in Construction vol 17 no6 pp 792ndash797 2008

[18] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multicrite-ria acceptability analysis for groupdecisionmakingrdquoOperationsResearch vol 49 no 3 pp 444ndash454 2001

[19] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[20] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[21] S Miyamoto ldquoMultisets and fuzzymultisetsrdquo in Soft Computingand Human-Centered Machines Z Q Liu and S MiyamotoEds pp 9ndash33 Springer Berlin Germany 2000

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[23] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[25] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling Simulation and Computation for Engineering andEnvironmental Systems vol 34 no 7 pp 1779ndash1787 2010

10 Mathematical Problems in Engineering

[26] D Horsky andM R Rao ldquoEstimation of attribute weights frompreference comparisonsrdquoManagement Science vol 30 no 7 pp801ndash822 1984

[27] P A Bottomley and J R Doyle ldquoA comparison of three weightelicitation methods good better and bestrdquo Omega vol 29 no6 pp 553ndash560 2001

[28] J Figueira and B Roy ldquoDetermining the weights of criteria inthe ELECTRE type methods with a revised Simosrsquo procedurerdquoEuropean Journal ofOperational Research vol 139 no 2 pp 317ndash326 2002

[29] L E Shirland R R Jesse R L Thompson and C L IacovouldquoDetermining attribute weights using mathematical program-mingrdquo Omega vol 31 no 6 pp 423ndash437 2003

[30] Y-M Wang and Y Luo ldquoIntegration of correlations withstandard deviations for determining attribute weights in mul-tiple attribute decision makingrdquo Mathematical amp ComputerModelling vol 51 no 1-2 pp 1ndash12 2010

[31] J Ma Z-P Fan and L-H Huang ldquoA subjective and objectiveintegrated approach to determine attribute weightsrdquo EuropeanJournal of Operational Research vol 112 no 2 pp 397ndash4041999

[32] X Xu ldquoA note on the subjective and objective integratedapproach to determine attribute weightsrdquo European Journal ofOperational Research vol 156 no 2 pp 530ndash532 2004

[33] P J H Schoemaker and C C Waid ldquoAn experimental compar-ison of different approaches to determining weights in additiveutility modelsrdquoManagement Science vol 28 no 2 pp 182ndash1961982

[34] Y-M Wang and C Parkan ldquoA general multiple attributedecision-making approach for integrating subjective prefer-ences and objective informationrdquo Fuzzy Sets and Systems vol157 no 10 pp 1333ndash1345 2006

[35] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[36] J R Doyle ldquoMultiattribute choice for the lazy decision makerlet the alternatives deciderdquoOrganizational Behavior andHumanDecision Processes vol 62 no 1 pp 87ndash100 1995

[37] E Bernroider and V Stix ldquoA method using weight restrictionsin data envelopment analysis for ranking and validity issues indecisionmakingrdquoComputersampOperations Research vol 34 no9 pp 2637ndash2647 2007

[38] J Wu and L Liang ldquoA multiple criteria ranking method basedon game cross-evaluation approachrdquo Annals of OperationsResearch vol 197 no 1 pp 191ndash200 2012

[39] C Fu and K-S Chin ldquoRobust evidential reasoning approachwith unknown attribute weightsrdquo Knowledge-Based Systemsvol 59 no 2 pp 9ndash20 2014

[40] B S Ahn ldquoExtreme point-based multi-attribute decision anal-ysis with incomplete informationrdquo European Journal of Opera-tional Research vol 240 no 3 pp 748ndash755 2015

[41] WW Cooper J L Ruiz and I Sirvent ldquoChoosing weights fromalternative optimal solutions of dualmultipliermodels inDEArdquoEuropean Journal of Operational Research vol 180 no 1 pp443ndash458 2007

[42] G R Jahanshahloo and P F Shahmirzadi ldquoNew methods forranking decision making units based on the dispersion ofweights and Norm 1 in Data Envelopment Analysisrdquo Computersamp Industrial Engineering vol 65 no 2 pp 187ndash193 2013

[43] V E Krivonozhko O B Utkin M M Safin and A V LychevldquoOn some generalization of the DEA modelsrdquo Journal of theOperational Research Society vol 60 no 11 pp 1518ndash1527 2009

[44] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[45] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo New Directions for ProgramEvaluation vol 1986 no 32 pp 73ndash105 1986

[46] A Boussofiane R G Dyson and E Thanassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[47] K Wang and F Wei ldquoGame cross efficiency DEA based projectreview of equipment designrdquo Journal of Beijing University ofAeronautics and Astronautics vol 35 no 10 pp 1278ndash12822009

[48] T Wu and J Blackhurst ldquoSupplier evaluation and selection anaugmented DEA approachrdquo International Journal of ProductionResearch vol 47 no 16 pp 4593ndash4608 2009

[49] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Korean stockmarketrdquo European Journal of Operational Research vol 236 no1 pp 361ndash368 2014

[50] B Roy ldquoTheoutranking approach and the foundations of electremethodsrdquoTheory and Decision vol 31 no 1 pp 49ndash73 1991

[51] G F Reed F Lynn and B D Meade ldquoUse of coefficient ofvariation in assessing variability of quantitative assaysrdquo Clinicalamp Diagnostic Laboratory Immunology vol 9 no 6 pp 1235ndash1239 2002

[52] 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

[53] H M Moshkovich A I Mechitov and D L Olson ldquoOrdinaljudgments in multiattribute decision analysisrdquo European Jour-nal of Operational Research vol 137 no 3 pp 625ndash641 2002

[54] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Tsinghua Press Beijing China 2005

[55] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

(b) Calculate the weighted normalized decision matrix

V119894119895= 119908119895119903119894119895 119894 = 1 2 119898 119895 = 1 2 119899 (3)

where 119908119895is the weight of the 119895th attribute and

sum119899

119895=1119908119895= 1

(c) Determine the positive ideal and negative ideal solu-tionPositive ideal solution 119860

+= V+1 V+2 V+

119899

= (max119894

V119894119895| 119895 isin 119868

1015840) (min

119894

V119894119895| 119895 isin 119868

10158401015840)

Negative ideal solution 119860minus= Vminus1 Vminus2 Vminus

119899

= (min119894

V119894119895| 119895 isin 119868

1015840) (max

119894

V119894119895| 119895 isin 119868

10158401015840)

(4)

where 1198681015840 indicate the index set of the benefit type

attributes and 11986810158401015840 the index set of the cost type

attributes(d) Measure the distances from the positive ideal solution

and negative ideal solution respectively

119863+

119894= radic

119899

sum

119895=1

(V119894119895minus V+119895)2

119894 = 1 2 119898

119863minus

119894= radic

119899

sum

119895=1

(V119894119895minus Vminus119895)2

119894 = 1 2 119898

(5)

(e) Calculate the relative closeness to the ideal solutions

119862lowast

119894=

119863minus

119894

(119863+

119894+ 119863minus

119894) 119894 = 1 2 119898 (6)

(f) Rank the alternatives

The larger the closeness index of an alternative the higherthe rank of the alternative

23 DEA Cross-Efficiency Evaluation We assume that thereare 119899 DMUs to be evaluated in terms of 119898 inputs and 119899

outputs The 119894th input and 119903th output of DMU119895are denoted

by 119909119894119895(119894 = 1 119898) and 119910

119903119895 respectively The efficiency of

a DMU119889can be calculated by the following CCR model in

linear programming formulation [35]

max119904

sum

119903=1

119906119903119910119903119889

= 120579119889

st119898

sum

119894=1

119908119894119909119894119895minus

119904

sum

119903=1

119906119903119910119903119895ge 0 119895 = 1 2 119899

119898

sum

119894=1

119908119894119909119894119889

= 1

119908119894ge 0 119894 = 1 2 119898

119906119903ge 0 119903 = 1 2 119904

(7)

When the above model is solved a set of optimal weights119908lowast

1119889 119908

lowast

119898119889 119906lowast

1119889 119906

lowast

119904119889is obtained Then the 119889 cross-

efficiency of any DMU119895is computed as [38]

119864119889119895

=

sum119904

119903=1119906lowast

119903119889119910119903119895

sum119898

119894=1119908lowast

119894119889119909119894119895

119889 119895 = 1 119899 (8)

For DMU119895(119895 = 1 119899) the average of all 119864

119889119895(119889 =

1 119899) denoted by 119864119895

= (1119899)sum119899

119889=1119864119889119895 is considered as

the cross-efficiency score of DMU119895

3 Proposed Method

In this section we will introduce the so-called single dom-inating attribute issue and propose a model to supportdecisions by incorporating considerations on this issue inSection 31 furthermore in Section 32 we will introducethe cross-evaluation framework and the accompanying riskconsideration finally in Section 33 we propose a proce-dure summarizing all the aforementioned considerations forMADM

31 Determination of the Weight Space We observe that inMADM problems an individual attribute can determine thefinal ranks of all alternatives when the corresponding weightis greater than some threshold value In other words therank order of the alternatives is completely in accordancewith the rank order of the dominating attribute and therest of the attributes are totally ignored in the evaluationIt should be noted that the ignored attributes are takeninto the evaluation only because the decision maker regardsthem as import factors or they could have excluded themfrom the decision process in the first place Therefore weargue this is unacceptable In order to address this issueit is necessary to define and compute the threshold upperbound of each attribute weight The upper bound can aid thedecision process by suggesting meaningful weights

For the simplicity of exposition if the attribute values119903119894119896

(119894 = 1 119898) in the 119896th column of the decision matrix119877 have the relation 119903

1198941119896

ge 1199031198942119896

ge sdot sdot sdot ge 119903119894119898119896(119894119904

isin

1119898 119904 = 1 119898) then we denote the ranks of thealternatives according to the values of the 119896th attribute 119884

119896

as 1198831198941

≻1198961198831198942

≻119896sdot sdot sdot ≻119896119883119894119898

With this notation in position thesingle dominating attribute happens if the final rank orderof all alternates is the same as some 119883

1198941

≻1198961198831198942

≻119896sdot sdot sdot ≻119896119883119894119898

no matter how the decision maker distributes the relativeweights of the other attributes We formulize this notion asDefinition 1 as below

Definition 1 Let 119896 be the index of some attribute 119908lowast119896is the

threshold upper bound of 119908119896if and only if the following

conditions are satisfied

(a) If 119908119896ge 119908lowast

119896 then forall1 le 119894 119905 le 119898 119903

119894119896ge 119903119905119896if and only

if sum119899119895=1

119908119895119903119894119895ge sum119899

119895=1119908119895119903119905119895

(b) If 119908119896lt 119908lowast

119896 then exist1 le 119894 119905 le 119898 such that 119903

119894119896ge 119903119905119896

andsum119899

119895=1119908119895119903119894119895le sum119899

119895=1119908119895119903119905119895hold true simultaneously

4 Mathematical Problems in Engineering

To implement Definition 1 we provide Theorem 2 belowto determine the threshold upper bound

Theorem 2 The threshold upper bound of the 119896th attribute(119908lowast119896) is given by the minimization program

min 119908119896

st 11990811989611990311989411+

119899

sum

119895=1119895 =119896

119908(119891)

1198951199031198941119895ge 11990811989611990311989421+

119899

sum

119895=1119895 =119896

119908(119891)

1198951199031198942119895

ge sdot sdot sdot

ge 1199081198961199031198941198981+

119899

sum

119895=1119895 =119896

119908(119891)

119895119903119894119898119895

(9)

119899

sum

119895=1

119908119895= 1

119908119895ge 0

119895 = 1 2 119899

119891 = 1 2 119899 minus 1

(10)

where 119908(119891)

119895= 1 minus 119908

119896if 119895 is the 119891th element of the set

1 119873 119896 in ascending order otherwise 119908(119891)119895

= 0 Letus take 119896 = 1 as an example We have (119908(1)

2 119908(1)

3 119908

(1)

119899) =

(1 minus 1199081 0 0) (119908(2)

2 119908(2)

3 119908

(2)

119899) = (0 1 minus 119908

1 0 0)

(119908(119899minus1)

2 119908(119899minus1)

3 119908

(119899minus1)

119899) = (0 0 1 minus 119908

1) The sub-

script 119894ℎ(ℎ isin 1 119898) of 119903 in the above model indicates the

119894ℎpreferred alternative determined by the preference relation

1198831198941

≻1198961198831198942

≻119896sdot sdot sdot ≻119896119883119894119898

Proof Clearly there are (119899 minus 1) constraints in (9) When119896 = 1 the weight combinations (119908

(1)

2 119908(1)

3 119908

(1)

119899) =

(1 minus 1199081 0 0) (119908(2)

2 119908(2)

3 119908

(2)

119899) = (0 1 minus 119908

1 0 0)

(119908(119899minus1)

2 119908(119899minus1)

3 119908

(119899minus1)

119899) = (0 0 1 minus 119908

1) are applied

to the 119899minus1 constraints in orderThefirst constraint guaranteesthat the alternatives rank result has nothing to do withattribute 119884

2 Similarly the remaining 119899 minus 2 constraints ensure

that the alternatives rank result has nothing to do with theother attributes 119884

3 119884

119899 So the optimal minimization 119908

1

satisfies (a) in Definition 1Conversely we suggest that a constraint belonging to the

119897th (119897 = 1 119899 minus 1) group is removed from the model Thenattribute 119884

119897may play a part in the ranking because it is not

completely suppressed In other words (b) in Definition 1holds

Similarly the upper bound of each attribute weight(119908lowast

119896 119896 = 2 119899) can be calculated by the model in

Theorem 2 with 119896 = 2 119899 This proof is completed

Example 3 Ma et al [31] consider a case of a robot userexpecting to select a robot and there are four alternatives forhim to choose There are four attributes to evaluate them

that is cost velocity repeatability and load capacityThe nor-malized decision matrix is as follows

119860 =

(((

(

0 15

6

1

35

12

3

51 0

1 0 0 1

2

3

2

5

2

3

2

3

)))

)

(11)

Suppose that the information on attribute weights iscompletely unknown The upper bound weight of the firstattribute cost can be calculated by the above model

By calculating the above model provided in Theorem 2we obtain the optimal value119908lowast

1= 23 denoting the threshold

upper bound of the first attribute1199081 Analogously each upper

boundof attributeweight can be calculated by similarmodelsand the values are 119908lowast

2= 1013 119908

lowast

3= 45 119908

lowast

4= 914

32 Mean and Deviation of Closeness Index Based Cross-Evaluation Models In this subsection in the environmentof uncertain attribute weights we develop cross-evaluationmodels to calculate the mean and deviation of closenessindex for each assessed alternative respectively Then thecomputed indexes are integrated to obtain the evaluationresults

Cross-efficiency evaluation method since being pro-posed by Sexton et al [45] has been used to evaluate andrankDMUs Itsmain idea is to use cross-evaluation instead ofa self-evaluation There are at least two advantages for cross-evaluation method One is that it obtains a unique rankingamong the DMUs The other is that the cross-efficiencymeans can act effectively to differentiate between good andbad alternatives [46] Hence the cross-evaluation methodis widely used for ranking performance of alternatives forexample equipment design [47] supplier selection [48] andstock market [49]

To calculate the self-evaluation closeness index for eachalternative we consider the following mathematical pro-gramming problem for a given alternative119883

119894

max119863minus

119894

119863minus

119894+ 119863+

119894

= 119862max119894

119908 isin Ω (119908)

(12)

Here 119863minus119894and 119863

+

119894represent the distance from negative

ideal point and ideal point (defined in (5)) respectively Forease of calculation the ideal point and negative point aredefined as 119860+ = 119908

1 1199082 119908

119899 and 119860

minus= 0 0 0 Then

119863+

119894and 119863

minus

119894are denoted by 119863

+

119894= radicsum

119899

119895=1(119908119895119903119894119895minus 119908119895)2 and

119863minus

119894= radicsum

119899

119895=1(119908119895119903119894119895)2 respectively The constraint 119908 isin Ω(119908)

depicts the possible set of attribute weights Obviously whenthe weights information of attributes is completely unknownΩ(119908) = 119908

119895gt 0 sum

119899

119895=1119908119895= 1 Introducing the upper bound

weight information calculated in Section 31 Ω(119908) = 119908119895lt

119908lowast

119895 sum119899

119895=1119908119895= 1

Mathematical Problems in Engineering 5

In many practical situations although the weight valuesare difficult to obtain the weight information is partiallyknown and predefined by the weight space As followswe introduce two types of partial weight information asreferences to decision maker

Type 1 (pairwise comparisonweights information) InMADMattribute weights 119908

119895are mainly introduced to reflect the

importance to attributes 119884119895(119895 = 1 2 119899) As stated in

Section 1 pairwise comparison is a widely accepted subjectivemethod to determine attribute weight

Attribute 1198842is at least two times and not more than three

times as important as attribute 1198841

In notational jargons this is denoted by

21199081le 1199082le 31199081 (13)

In general terms we compare 1198841with 119884

119895(119895 = 2 3 119899)

respectively The bound linear constraints are denoted by

1198971198951199081le 119908119895le 1199061198951199081 (14)

in which 119897119895and 119906

119895represent the lower and upper bounds of

times119908119895compared with119908

1 respectively Hence the attribute

weights space is obtained by the above pairwise comparisoninformation which is expressed as Ω(119908) = 119908

119895gt 0 119897

1198951199081le

119908119895le 1199061198951199081 119895 = 1 2 119899 Clearly 119897

1= 1199061= 1 holds

Compared with conventional subjective weight determi-nation approaches such as the AHP weights space is morefeasible and applicative because the decision makers neednot express their preferences explicitly In many situationsexperts are pleased to give such interval information ratherthan exact value due to the complexity of actual problems

Type 2 (ordinal weights information) Outranking relation[50] usually denoted by 119878 was proposed by Roy whoseaim was to establish a realistic representation of the threebasic preference situations strict preference indifferenceand incomparability Motivated by the widely accepted out-ranking relation we introduce the concept of rank-orderweights information In many situations although the weightvalues are difficult to give by decision makers the rankingof each attribute is easy to obtain according to importanceTake the Olympic Games as an example the achievementsof athletes and nations are judged by gold silver and bronzemedals Apparently a gold medal is more important than asilver medal and a silver medal is more important than abronze medal Then the weights space can be denoted byΩ(119908) = 119908

1gt 1199082

gt 1199083 1199081 1199082 1199083

gt 0 in which1199081 1199082 1199083represent the weight of gold silver and bronze

medal respectivelyThe above self-evaluation model (see (12)) shows that all

alternatives choose their own attribute weights in order tomaximize their own self-evaluation closeness index Similarto the formofDEA cross-efficiency we obtain a set of optimalattribute weights 119882lowast

119894= (119908lowast

1198941 119908

lowast

119894119899) according to the above

model When these optimal weights are applied to evaluatealternative119883

119896 a cross score denoted by119862

119894119896= 119863minus

119894119896(119863minus

119894119896+119863+

119894119896)

is obtained

For alternative 119896 (119896 = 1 119898) we use the average ofall 119862119894119896to measure the expected return of cross-evaluation

closeness index which is written as

119903 (119883119896) =

1

119898

119898

sum

119894=1

119862119894119896 (15)

Furthermore we use the standard variance

120575 (119883119896) =

radicsum119898

119894=1(119862119894119896minus 119903 (119883

119896))2

119899

(16)

tomeasure the risk caused by different sets of optimal weightsproduced by each alternative

In general there are both return of closeness index(defined by the mean of cross-evaluation score) and risk ofcloseness index (defined by the deviation between return andeach cross-evaluation closeness index) in cross-evaluationframework It is not hard to see that the larger return scorethe better and the smaller risk score the better Specificallygiven two alternatives 119883

119904and 119883

119905 if 119877(119883

119904) gt 119877(119883

119905) and

120575(119883119904) lt 120575(119883

119905) then 119883

119904≻ 119883119905(119883119904is better than 119883

119905) clearly

holds However when 119877(119883119904) gt 119877(119883

119905) and 120575(119883

119904) gt 120575(119883

119905) it

is not easy to compare the two alternatives directlyThen we integrate the return and risk measures by the

well-known coefficient of variation [51]

CV (119883119896) =

120575 (119883119896)

119903 (119883119896)times 100 (17)

For two given alternatives one having a smaller coeffi-cient of variation is considered to be better than the otherone Thus all the alternatives are ranked by the comparisonof their corresponding coefficient of variation

33 Procedure of the Proposed Approach Based on the aboveanalysis the procedure for generating a solution to theMADM problem is elaborated in the following

Step 1 (formulation of a MADM problem) The decisionmaker identifies 119899 attributes to evaluate 119898 alternatives Theoriginal attribute values are standardized by (1)

Step 2 (obtainment of the attribute weight space) First theupper bounds of each attribute are calculated byTheorem 2 toguarantee the effectiveness of all the attributesThen differenttypes of constraint of attribute weights are determined by thedecision maker to reflect the importance of each attributeLast the weight space including all the feasible weightcombinations is obtained

Step 3 (calculation of the mean and deviation measures)Firstly the maximized closeness index of each alternativeis calculated by (12) Secondly the mean measure of eachalternativersquos cross closeness index is computed by (15) Thirdthe deviation measure of each alternativersquos cross closenessindex is obtained by (16)

6 Mathematical Problems in Engineering

Step 4 (evaluation of the alternatives via coefficient ofvariation) Since the mean and deviation measures of eachalternativersquos cross closeness index are obtained in Step 3 thecoefficient of variation defined in (17) is utilized to integratethe results

Step 5 (generation of rank order of the alternatives) Throughcomparing the coefficients of variation among the alterna-tives the rank order can be directly obtained

4 Extension with Uncertain Attribute Values

41 Model with Interval Attribute Value It is widely acknowl-edged that uncertainty is associated with attribute values dueto complexity of real world and limited knowledge of humanbeings In many situations the attribute value 119903

119894119895is uncertain

but we know that the feasible attribute value 119903119894119895

isin [119903minus

119894119895 119903+

119894119895]

Here we present the following self-evaluation model withinterval attribute values [52] and uncertain attribute weights

max 120572

radicsum119899

119895=1(119908119895119903+

119894119895)2

radicsum119899

119895=1(119908119895119903+

119894119895)2

+ radicsum119899

119895=1(119908119895119903+

119894119895minus 119908119895)2

+ 120573

radicsum119899

119895=1(119908119895119903minus

119894119895)2

radicsum119899

119895=1(119908119895119903minus

119894119895)2

+ radicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2

= 119862max119894

119908 isin Ω (119908)

120572 + 120573 = 1 120572 ge 0 120573 ge 0

(18)

in which radicsum119899

119895=1(119908119895119903+

119894119895)2 radicsum

119899

119895=1(119908119895119903+

119894119895minus 119908119895)2 radicsum

119899

119895=1(119908119895119903minus

119894119895)2

andradicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2 represent distance of upper bounds

from ideal point distance of upper bounds from negativeideal point distance of lower bounds from ideal pointand distance of lower bounds from negative ideal pointrespectively 120572 and 120573 reflect the risk coefficients that combineupper closeness index and lower closeness respectively Inpractice120572 and120573 are decided by the decisionmaker accordingto his or her risk preference Specifically the decision makeris neutral if 120572 = 120573 = 12 120572 gt 120573 and 120572 lt 120573 indicate that he isoptimistic and pessimistic respectively

42 Model with Ordinal Attribute Value Another form ofuncertain preferential information is expressed by ordinalattribute value (eg ldquomore preferablerdquo and ldquoless preferablerdquo)It is argued to be more stable and more reliable than cardinalvalue in some situations [53] In this paper an approach ofthreshold value is introduced to transform ordinal value intolinear formulation For instance given four attributes 119886 119887 119888and 119889 the ordinal values are 1 2 3 and 4 respectively Byspecifying a threshold value 01 the attribute values can beconverted to a cardinal attribute space Ω(119886 119887 119888 119889) = 119886 = 1

119886 minus 119887 ge 01 119887 minus 119888 ge 01 119888 minus 119889 ge 01 119889 = 0 In generallet the ordinal attribute values 119886

119894119895(119894 = 1 119898) have relation

1198861198941119895gt 1198861198942119895gt sdot sdot sdot gt 119886

119894119898119895(119894119896isin [1119898] 119896 = 1 119898) and then

the cardinal attribute spaceΩ(119903119894119895) = 119886

119894119898119895= 1 119886119894119898119895minus119886119894119898minus1119895ge 120576

119886119894119898minus1119895minus119886119894119898minus2119895ge 120576 119886

1198942119895minus1198861198941119895ge 120576 1198861198941119895= 0 120576 gt 0 is obtained

(120576 is the threshold value given by the decision maker) Herewe present the following self-evaluation model with ordinalattribute values and uncertain attribute weights

maxradicsum119899

119895=1(119908119895119903119894119895)2

radicsum119899

119895=1(119908119895119903119894119895)2

+ radicsum119899

119895=1(119908119895119903119894119895minus 119908119895)2

= 119862max119894

st 119903 isin Ω (119903)

119908 isin Ω (119908)

(19)

119894 = 1 119898 (20)

119895 = 1 119899 (21)

For each alternative 119883119896(119896 = 1 119898) under evaluation

we obtain a group of optimal attribute values and a set ofattribute weights They are denoted by 119860

lowast

119896= (119903119896

119894119895)119898times119899

and119882lowast

119896= (119908

119896

1 119908119896

2 119908

119896

119899) respectively When the optimal

decision information is applied to evaluate alternative119883119897(119897 =

1 119898) a cross score denoted by 119862119896119897

= radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2

(radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2+ radicsum

119899

119895=1(119908119896

119895119903119896

119897119895minus 119908119896

119895)2) is obtained

Similar to that defined by (14) and (15) the mean anddeviation measures of alternative 119883

119897are denoted by 119903(119883

119897) =

sum119898

119896=1119862119896119897and 120575(119883

119897) = radicsum

119898

119896=1(119862119896119897minus 119903(119883

119897))2119899 respectively

Note that (12) (18) and (19) are nonlinear models theycan be transformed into linear forms according to practicalcases

5 Illustrative Examples

In order to illustrate the applications of our proposed meth-ods three practical MADM problems are presented

51 Application to House Selection with Cardinal AttributeValues Xu [54] stated a purchase house problem there arefour possible houses to be selected and the attributes are asfollows (1)119884

1is the house price (ten thousand yuan) (2)119884

2is

the usable area (square meter) (3) 1198843is the distance between

house and the center of city (kilometers) (4) 1198844is the nature

environment (scores) where 1198842and 119884

4are benefit attributes

and 1198841and 119884

3are cost attributes The attribute weights are

completely unknown The decision making matrix 119860 andnormalized decision making matrix 119877 are listed in Tables 1and 2 respectively Step 1 is completed

By utilizing (9) and (10) the upper boundof each attributeweight is obtained And as a result the attribute weight spaceis Ω(119908) = 0 lt 119908

1lt 077 0 lt 119908

2lt 078 0 lt 119908

3lt 062

0 lt 1199084lt 062 119908

1+ 1199082+ 1199083+ 1199084= 1 Step 2 is completed

Mathematical Problems in Engineering 7

Table 1 Decision making matrix 119860 of house evaluation

1198841

1198842

1198843

1198844

1198831

30 100 10 71198832

25 80 8 51198833

18 50 20 111198834

22 70 12 9

Table 2 Normalized decisionmakingmatrix119877 of house evaluation

1198841

1198842

1198843

1198844

1198831

06 10 08 06361198832

072 08 10 04551198833

10 05 04 101198834

0818 07 0667 0818

Table 3 The cross-evaluation measurements

Alternatives 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

119903 (119883119894) 120575 (119883

119894)

1198831

096 096 082 061 061 075 0151198832

092 080 092 066 066 076 0111198833

1 052 051 1 1 076 0241198834

0818 070 070 0818 0818 076 006

Table 4 Alternative efficiency matrix

Alternatives 1198831

1198832

1198833

1198834

CV (119883119894) 020 014 032 008

In terms of Step 3 in Section 33 we obtain the cross-evaluation measurements of each alternative which arepresented in Table 3 Step 3 is completed

Based on Table 3 and the calculation of coefficient ofvariation the results are shown in Table 4

Therefore we can rank the houses in accordance with theCV values presented in Table 4 to get a complete rank order1198834

≻ 1198832

≻ 1198831

≻ 1198833 And 119883

4is the best house under

evaluation Step 4 is completedIn order to understand our approach more clearly we

compare our approachwith other three similarmethods [54]namely average attribute value method maximizing devia-tion method and information entropy method respectivelywith the same example The corresponding results of suchthree methods are listed in Table 5

According to Tables 4 and 5 the result derived from ourcross-evaluation approach is quite different from that of theexisting three weight determination methods In fact Table 5shows that the three weight determination methods generatethree attribute weight vectors and produce three differentoptimal solutions However there is no single method thatcan guarantee more convincing weight vector than othersHence we treat the attribute weight as a weight space rather

Table 5 The results of three methods

Method Weight vector Rank order

Average attributevalue

(025 025 025 025) 1198831≻ 1198834≻ 1198832≻ 1198833

Maximizingdeviation

(020 024 029 027) 1198831≻ 1198832≻ 1198834≻ 1198833

Informationentropy

(013 022 036 029) 1198831≻ 1198832≻ 1198834≻ 1198833

Table 6 Attribute weights

1198841

1198842

1198843

[119908119871

1 119908119880

1] [119908

119871

2 119908119880

2] [119908

119871

3 119908119880

3]

[035 050] [015 030] [030 035]

Table 7 Attribute values of alternatives

1198841

1198842

1198843

119860 [065 085] [068 086] [045 078]119861 [075 1] [055 078] [064 080]119862 [054 068] [046 055] [078 1]119863 [077 090] [067 078] [056 077]119864 [025 045] [089 1] [075 094]

than a certain value in this paperThe comparison result vali-dates that our proposed approach can overcome the shortagethat we cannot obtain themost dominant result caused by thediversity of the way of weight vector determined

52 Application to University Faculty with Interval AttributeValue Consider the example in Bryson and Mobolurinrsquos[55] study that is five faculty candidates (119860 119861 119862 119863119864) are evaluated for tenure and promotion through threeattributes teaching research and service According to theDMrsquos preference the information for both attribute andalternatives is given by interval numbers as shown in Tables6 and 7 respectively (for the purpose of illustration our datais different from the literature)

Table 8 shows the computed results of the proposedapproach with interval values The self-evaluation score ofeach alternative is listed in the second column and the cross-evaluation scores are listed in columns 3ndash7 Columns 8 and9 show the mean and deviation measures respectively Thenthe results in column 10 are applied to generate a rank orderof the five candidates119860 ≻ 119862 ≻ 119864 ≻ 119861 ≻ 119863 which is a solutionto the problem

53 Application to the Selection of Community Service Com-panies with Ordinal Attribute Values In this subsection weapply our approach to the evaluation and selection of fivecommunity service companies that are located in Hefei acity in Anhui province of China These companies providevarious kinds of service for the public such as property

8 Mathematical Problems in Engineering

Table 8 Evaluation results for the comparison of faculty candidates

Candidates 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

1198625119894

119903119894

120575119894

CV(119883119894)

119860 0717 0717 0712 0706 0717 0703 0711 0006 0008119861 0792 0792 0792 0788 0792 0746 0782 0018 0023119862 0672 0651 0661 0672 0651 0660 0659 0008 0012119863 0774 0774 0771 0764 0774 0737 0764 0014 0461119864 0645 0525 0526 0540 0525 0645 0552 0047 0018

Table 9 Ordinal values of alternatives

1198841

1198842

1198843

1198844

1198831

4 2 4 21198832

1 4 3 31198833

2 5 1 11198834

5 3 2 51198835

3 1 5 4

management catering health care and entertainment Inthe case study the decision maker is an official from thedevelopment and reform commission of Hefei with thehelp of a group of experts from the governments relevantenterprises and community organizations Four attributesare used to evaluate the six companies They are quality ofservice (119884

1) diversity of service (119884

2) profitability (119884

3) and

development potential (1198844) The original attribute values are

given by ordinal values as shown in Table 9 The decisionprocedures are as follows

After considering the four attributes the decision makerviews that quality of service (119884

1) is the most important

attribute in evaluation The development potential (1198844) is the

second most important one profitability (1198843) is the third

most important one and diversity of service (1198842) is the

least important one respectively And the difference betweentwo adjacent attribute weights should be more than 01 Inaddition each weight value should be nomore than 05Thenthe weight space is denoted by Ω(119908) = 119908

1minus 01 gt 119908

4

1199084minus 01 gt 119908

3 1199083minus 01 gt 119908

2 1199081le 05 119908

2le 05 119908

3le 05

1199084le 05 119908

1+ 1199082+ 1199083+ 1199084= 1

By specifying a threshold value of 01 ordinal attributevalues are transformed into four sets of attribute value spacesThey are denoted by

Ω(1198841) = 119903

21= 1 11990321

minus 11990331

ge 01 11990331

minus 11990351

ge 01 11990351

minus 11990311

ge 01 11990311

minus 11990341

ge 01 11990341

= 0

Ω (1198842) = 119903

52= 1 11990352

minus 11990312

ge 01 11990312

minus 11990342

ge 01 11990342

minus 11990322

ge 01 11990322

minus 11990332

ge 01 11990332

= 0

Ω (1198843) = 119903

33= 1 11990333

minus 11990343

ge 01 11990343

minus 11990323

ge 01 11990323

minus 11990313

ge 01 11990313

minus 11990353

ge 01 11990353

= 0

Ω (1198844) = 119903

34= 1 11990334

minus 11990314

ge 01 11990314

minus 11990324

ge 01 11990324

minus 11990354

ge 01 11990354

minus 11990344

ge 01 11990344

= 0

(22)

Table 10 shows the computed results of the proposedapproachThe second column shows the self-evaluation scoreof each alternative computed by (19) And columns 3ndash7 listthe cross-evaluation scores Then the mean and deviationmeasures of each alternative are presented in columns 8 and9 respectively Consequently the results in column 10 areapplied to generate a rank order of the five community servicecompanies 119883

5≻ 1198831≻ 1198832≻ 1198834≻ 1198833 The decision maker

is pleased to select company 1198835with the best performance

among the five alternatives

6 Conclusions and Future Research

Due to the complexity and uncertainty of MADM problemsin real world the decision maker is not able to provide theexact attribute weights And different weight determinationmethods are difficult to generate a most satisfaction solutionThe current paper presents a new MADM method basedon cross-evaluation in which the weight information isexpressed by the so-called weight space It attempts to rectifysome shortcomings of existingMADMmethods by providingthe following important features (1) it is more feasibleto provide a weight space rather than specifying a set ofweight values (2) the single dominating attribute drawbackcan be avoided by the restriction of the upper bound ofeach attribute (3) compared with DEA based methods thevariable attribute weights are consistent with the opinionof decision maker (4) both mean score and the deviationscore are considered in our cross-evaluation models and (5)the proposed method is also successfully extended to moregeneralized situations in which attribute values are expressedby interval values and ordinal values respectively

We have illustrated the applicability of the proposedmethods by revisiting two reported MADM studies and bydepicting a case study on the selection of community servicecompanies Although our proposed method makes someimprovement in solving MADM problems it only can dealwith the individual decision making case Future research

Mathematical Problems in Engineering 9

Table 10 Evaluation results for the selection of community service companies

Candidates 119862max119897

1198621119897

1198622119897

1198623119897

1198624119897

1198625119897

119903119894

120575119894

CV(119883119894)

1198831

0489 0489 0317 0295 0317 0317 0347 0072 02071198832

0532 0433 0532 0508 0458 0504 0487 0036 00741198833

0535 0526 0526 0535 0526 0526 0528 0004 00081198834

0223 0202 0223 0211 0223 0223 0216 0009 00421198835

0422 0349 0349 0103 0349 0422 0314 0109 0347

may attempt to employ our cross-evaluation methods tohandle group decision making problem

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research is supported by National Key Technology Sup-port Program (no 2014BAH27F00) National Natural Sci-ence Funds of China (no 71471053) and the FundamentalResearch Fund for the Central Universities (no JZ2014-HGBZ0339)

References

[1] R H Green and J R Doyle ldquoOn maximizing discriminationin multiple criteria decision makingrdquo Journal of the OperationalResearch Society vol 46 no 2 pp 192ndash204 1995

[2] R E Steuer and P Na ldquoMultiple criteria decision makingcombined with finance a categorized bibliographic studyrdquoEuropean Journal of Operational Research vol 150 no 3 pp496ndash515 2003

[3] J Wallenius J S Dyer P C Fishburn R E Steuer S Ziontsand K Deb ldquoMultiple criteria decision making multi-attributeutility theory recent accomplishments and what lies aheadrdquoManagement Science vol 54 no 7 pp 1336ndash1349 2008

[4] W Ho X Xu and P K Dey ldquoMulti-criteria decision makingapproaches for supplier evaluation and selection a literaturereviewrdquo European Journal of Operational Research vol 202 no1 pp 16ndash24 2010

[5] C W Churchman R L Ackoff and N M Smith ldquoAn approx-imate measure of valuerdquo Journal of the Operational ResearchSociety of America vol 2 no 2 pp 172ndash187 1954

[6] T L Saaty ldquoThe analytic hierarchy processrdquo in Proceedings ofthe Second International Seminar on Operational Research in theBasque Provinces vol 4 pp 189ndash234 1980

[7] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[8] H Deng C-H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[9] P K Dey ldquoIntegrated project evaluation and selection usingmultiple-attribute decision-making techniquerdquo InternationalJournal of Production Economics vol 103 no 1 pp 90ndash1032006

[10] G-D Li D Yamaguchi and M Nagai ldquoA grey-based decision-making approach to the supplier selection problemrdquoMathemat-ical amp Computer Modelling vol 46 no 3-4 pp 573ndash581 2007

[11] A Shanian and O Savadogo ldquoTOPSIS multiple-criteria deci-sion support analysis for material selection of metallic bipolarplates for polymer electrolyte fuel cellrdquo Journal of Power Sourcesvol 159 no 2 pp 1095ndash1104 2006

[12] G-H Tzeng Y-P O Yang C-T Lin and C-B Chen ldquoHier-archical MADM with fuzzy integral for evaluating enterpriseintranet web sitesrdquo Information Sciences vol 169 no 3-4 pp409ndash426 2005

[13] 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

[14] A Kelemenis and D Askounis ldquoA new TOPSIS-based multi-criteria approach to personnel selectionrdquo Expert Systems withApplications vol 37 no 7 pp 4999ndash5008 2010

[15] B Bairagi B Dey B Sarkar and S K Sanyal ldquoADeNovomulti-approaches multi-criteria decision making technique with anapplication in performance evaluation of material handlingdevicerdquo Computers amp Industrial Engineering vol 87 pp 267ndash282 2015

[16] J S H Kornbluth ldquoDynamic multi-criteria decision makingrdquoJournal of Multi-Criteria Decision Analysis vol 1 no 2 pp 81ndash92 1992

[17] Y-H Lin P-C Lee T-P Chang and H-I Ting ldquoMulti-attribute group decision making model under the condition ofuncertain informationrdquoAutomation in Construction vol 17 no6 pp 792ndash797 2008

[18] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multicrite-ria acceptability analysis for groupdecisionmakingrdquoOperationsResearch vol 49 no 3 pp 444ndash454 2001

[19] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[20] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[21] S Miyamoto ldquoMultisets and fuzzymultisetsrdquo in Soft Computingand Human-Centered Machines Z Q Liu and S MiyamotoEds pp 9ndash33 Springer Berlin Germany 2000

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[23] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[25] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling Simulation and Computation for Engineering andEnvironmental Systems vol 34 no 7 pp 1779ndash1787 2010

10 Mathematical Problems in Engineering

[26] D Horsky andM R Rao ldquoEstimation of attribute weights frompreference comparisonsrdquoManagement Science vol 30 no 7 pp801ndash822 1984

[27] P A Bottomley and J R Doyle ldquoA comparison of three weightelicitation methods good better and bestrdquo Omega vol 29 no6 pp 553ndash560 2001

[28] J Figueira and B Roy ldquoDetermining the weights of criteria inthe ELECTRE type methods with a revised Simosrsquo procedurerdquoEuropean Journal ofOperational Research vol 139 no 2 pp 317ndash326 2002

[29] L E Shirland R R Jesse R L Thompson and C L IacovouldquoDetermining attribute weights using mathematical program-mingrdquo Omega vol 31 no 6 pp 423ndash437 2003

[30] Y-M Wang and Y Luo ldquoIntegration of correlations withstandard deviations for determining attribute weights in mul-tiple attribute decision makingrdquo Mathematical amp ComputerModelling vol 51 no 1-2 pp 1ndash12 2010

[31] J Ma Z-P Fan and L-H Huang ldquoA subjective and objectiveintegrated approach to determine attribute weightsrdquo EuropeanJournal of Operational Research vol 112 no 2 pp 397ndash4041999

[32] X Xu ldquoA note on the subjective and objective integratedapproach to determine attribute weightsrdquo European Journal ofOperational Research vol 156 no 2 pp 530ndash532 2004

[33] P J H Schoemaker and C C Waid ldquoAn experimental compar-ison of different approaches to determining weights in additiveutility modelsrdquoManagement Science vol 28 no 2 pp 182ndash1961982

[34] Y-M Wang and C Parkan ldquoA general multiple attributedecision-making approach for integrating subjective prefer-ences and objective informationrdquo Fuzzy Sets and Systems vol157 no 10 pp 1333ndash1345 2006

[35] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[36] J R Doyle ldquoMultiattribute choice for the lazy decision makerlet the alternatives deciderdquoOrganizational Behavior andHumanDecision Processes vol 62 no 1 pp 87ndash100 1995

[37] E Bernroider and V Stix ldquoA method using weight restrictionsin data envelopment analysis for ranking and validity issues indecisionmakingrdquoComputersampOperations Research vol 34 no9 pp 2637ndash2647 2007

[38] J Wu and L Liang ldquoA multiple criteria ranking method basedon game cross-evaluation approachrdquo Annals of OperationsResearch vol 197 no 1 pp 191ndash200 2012

[39] C Fu and K-S Chin ldquoRobust evidential reasoning approachwith unknown attribute weightsrdquo Knowledge-Based Systemsvol 59 no 2 pp 9ndash20 2014

[40] B S Ahn ldquoExtreme point-based multi-attribute decision anal-ysis with incomplete informationrdquo European Journal of Opera-tional Research vol 240 no 3 pp 748ndash755 2015

[41] WW Cooper J L Ruiz and I Sirvent ldquoChoosing weights fromalternative optimal solutions of dualmultipliermodels inDEArdquoEuropean Journal of Operational Research vol 180 no 1 pp443ndash458 2007

[42] G R Jahanshahloo and P F Shahmirzadi ldquoNew methods forranking decision making units based on the dispersion ofweights and Norm 1 in Data Envelopment Analysisrdquo Computersamp Industrial Engineering vol 65 no 2 pp 187ndash193 2013

[43] V E Krivonozhko O B Utkin M M Safin and A V LychevldquoOn some generalization of the DEA modelsrdquo Journal of theOperational Research Society vol 60 no 11 pp 1518ndash1527 2009

[44] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[45] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo New Directions for ProgramEvaluation vol 1986 no 32 pp 73ndash105 1986

[46] A Boussofiane R G Dyson and E Thanassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[47] K Wang and F Wei ldquoGame cross efficiency DEA based projectreview of equipment designrdquo Journal of Beijing University ofAeronautics and Astronautics vol 35 no 10 pp 1278ndash12822009

[48] T Wu and J Blackhurst ldquoSupplier evaluation and selection anaugmented DEA approachrdquo International Journal of ProductionResearch vol 47 no 16 pp 4593ndash4608 2009

[49] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Korean stockmarketrdquo European Journal of Operational Research vol 236 no1 pp 361ndash368 2014

[50] B Roy ldquoTheoutranking approach and the foundations of electremethodsrdquoTheory and Decision vol 31 no 1 pp 49ndash73 1991

[51] G F Reed F Lynn and B D Meade ldquoUse of coefficient ofvariation in assessing variability of quantitative assaysrdquo Clinicalamp Diagnostic Laboratory Immunology vol 9 no 6 pp 1235ndash1239 2002

[52] 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

[53] H M Moshkovich A I Mechitov and D L Olson ldquoOrdinaljudgments in multiattribute decision analysisrdquo European Jour-nal of Operational Research vol 137 no 3 pp 625ndash641 2002

[54] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Tsinghua Press Beijing China 2005

[55] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

To implement Definition 1 we provide Theorem 2 belowto determine the threshold upper bound

Theorem 2 The threshold upper bound of the 119896th attribute(119908lowast119896) is given by the minimization program

min 119908119896

st 11990811989611990311989411+

119899

sum

119895=1119895 =119896

119908(119891)

1198951199031198941119895ge 11990811989611990311989421+

119899

sum

119895=1119895 =119896

119908(119891)

1198951199031198942119895

ge sdot sdot sdot

ge 1199081198961199031198941198981+

119899

sum

119895=1119895 =119896

119908(119891)

119895119903119894119898119895

(9)

119899

sum

119895=1

119908119895= 1

119908119895ge 0

119895 = 1 2 119899

119891 = 1 2 119899 minus 1

(10)

where 119908(119891)

119895= 1 minus 119908

119896if 119895 is the 119891th element of the set

1 119873 119896 in ascending order otherwise 119908(119891)119895

= 0 Letus take 119896 = 1 as an example We have (119908(1)

2 119908(1)

3 119908

(1)

119899) =

(1 minus 1199081 0 0) (119908(2)

2 119908(2)

3 119908

(2)

119899) = (0 1 minus 119908

1 0 0)

(119908(119899minus1)

2 119908(119899minus1)

3 119908

(119899minus1)

119899) = (0 0 1 minus 119908

1) The sub-

script 119894ℎ(ℎ isin 1 119898) of 119903 in the above model indicates the

119894ℎpreferred alternative determined by the preference relation

1198831198941

≻1198961198831198942

≻119896sdot sdot sdot ≻119896119883119894119898

Proof Clearly there are (119899 minus 1) constraints in (9) When119896 = 1 the weight combinations (119908

(1)

2 119908(1)

3 119908

(1)

119899) =

(1 minus 1199081 0 0) (119908(2)

2 119908(2)

3 119908

(2)

119899) = (0 1 minus 119908

1 0 0)

(119908(119899minus1)

2 119908(119899minus1)

3 119908

(119899minus1)

119899) = (0 0 1 minus 119908

1) are applied

to the 119899minus1 constraints in orderThefirst constraint guaranteesthat the alternatives rank result has nothing to do withattribute 119884

2 Similarly the remaining 119899 minus 2 constraints ensure

that the alternatives rank result has nothing to do with theother attributes 119884

3 119884

119899 So the optimal minimization 119908

1

satisfies (a) in Definition 1Conversely we suggest that a constraint belonging to the

119897th (119897 = 1 119899 minus 1) group is removed from the model Thenattribute 119884

119897may play a part in the ranking because it is not

completely suppressed In other words (b) in Definition 1holds

Similarly the upper bound of each attribute weight(119908lowast

119896 119896 = 2 119899) can be calculated by the model in

Theorem 2 with 119896 = 2 119899 This proof is completed

Example 3 Ma et al [31] consider a case of a robot userexpecting to select a robot and there are four alternatives forhim to choose There are four attributes to evaluate them

that is cost velocity repeatability and load capacityThe nor-malized decision matrix is as follows

119860 =

(((

(

0 15

6

1

35

12

3

51 0

1 0 0 1

2

3

2

5

2

3

2

3

)))

)

(11)

Suppose that the information on attribute weights iscompletely unknown The upper bound weight of the firstattribute cost can be calculated by the above model

By calculating the above model provided in Theorem 2we obtain the optimal value119908lowast

1= 23 denoting the threshold

upper bound of the first attribute1199081 Analogously each upper

boundof attributeweight can be calculated by similarmodelsand the values are 119908lowast

2= 1013 119908

lowast

3= 45 119908

lowast

4= 914

32 Mean and Deviation of Closeness Index Based Cross-Evaluation Models In this subsection in the environmentof uncertain attribute weights we develop cross-evaluationmodels to calculate the mean and deviation of closenessindex for each assessed alternative respectively Then thecomputed indexes are integrated to obtain the evaluationresults

Cross-efficiency evaluation method since being pro-posed by Sexton et al [45] has been used to evaluate andrankDMUs Itsmain idea is to use cross-evaluation instead ofa self-evaluation There are at least two advantages for cross-evaluation method One is that it obtains a unique rankingamong the DMUs The other is that the cross-efficiencymeans can act effectively to differentiate between good andbad alternatives [46] Hence the cross-evaluation methodis widely used for ranking performance of alternatives forexample equipment design [47] supplier selection [48] andstock market [49]

To calculate the self-evaluation closeness index for eachalternative we consider the following mathematical pro-gramming problem for a given alternative119883

119894

max119863minus

119894

119863minus

119894+ 119863+

119894

= 119862max119894

119908 isin Ω (119908)

(12)

Here 119863minus119894and 119863

+

119894represent the distance from negative

ideal point and ideal point (defined in (5)) respectively Forease of calculation the ideal point and negative point aredefined as 119860+ = 119908

1 1199082 119908

119899 and 119860

minus= 0 0 0 Then

119863+

119894and 119863

minus

119894are denoted by 119863

+

119894= radicsum

119899

119895=1(119908119895119903119894119895minus 119908119895)2 and

119863minus

119894= radicsum

119899

119895=1(119908119895119903119894119895)2 respectively The constraint 119908 isin Ω(119908)

depicts the possible set of attribute weights Obviously whenthe weights information of attributes is completely unknownΩ(119908) = 119908

119895gt 0 sum

119899

119895=1119908119895= 1 Introducing the upper bound

weight information calculated in Section 31 Ω(119908) = 119908119895lt

119908lowast

119895 sum119899

119895=1119908119895= 1

Mathematical Problems in Engineering 5

In many practical situations although the weight valuesare difficult to obtain the weight information is partiallyknown and predefined by the weight space As followswe introduce two types of partial weight information asreferences to decision maker

Type 1 (pairwise comparisonweights information) InMADMattribute weights 119908

119895are mainly introduced to reflect the

importance to attributes 119884119895(119895 = 1 2 119899) As stated in

Section 1 pairwise comparison is a widely accepted subjectivemethod to determine attribute weight

Attribute 1198842is at least two times and not more than three

times as important as attribute 1198841

In notational jargons this is denoted by

21199081le 1199082le 31199081 (13)

In general terms we compare 1198841with 119884

119895(119895 = 2 3 119899)

respectively The bound linear constraints are denoted by

1198971198951199081le 119908119895le 1199061198951199081 (14)

in which 119897119895and 119906

119895represent the lower and upper bounds of

times119908119895compared with119908

1 respectively Hence the attribute

weights space is obtained by the above pairwise comparisoninformation which is expressed as Ω(119908) = 119908

119895gt 0 119897

1198951199081le

119908119895le 1199061198951199081 119895 = 1 2 119899 Clearly 119897

1= 1199061= 1 holds

Compared with conventional subjective weight determi-nation approaches such as the AHP weights space is morefeasible and applicative because the decision makers neednot express their preferences explicitly In many situationsexperts are pleased to give such interval information ratherthan exact value due to the complexity of actual problems

Type 2 (ordinal weights information) Outranking relation[50] usually denoted by 119878 was proposed by Roy whoseaim was to establish a realistic representation of the threebasic preference situations strict preference indifferenceand incomparability Motivated by the widely accepted out-ranking relation we introduce the concept of rank-orderweights information In many situations although the weightvalues are difficult to give by decision makers the rankingof each attribute is easy to obtain according to importanceTake the Olympic Games as an example the achievementsof athletes and nations are judged by gold silver and bronzemedals Apparently a gold medal is more important than asilver medal and a silver medal is more important than abronze medal Then the weights space can be denoted byΩ(119908) = 119908

1gt 1199082

gt 1199083 1199081 1199082 1199083

gt 0 in which1199081 1199082 1199083represent the weight of gold silver and bronze

medal respectivelyThe above self-evaluation model (see (12)) shows that all

alternatives choose their own attribute weights in order tomaximize their own self-evaluation closeness index Similarto the formofDEA cross-efficiency we obtain a set of optimalattribute weights 119882lowast

119894= (119908lowast

1198941 119908

lowast

119894119899) according to the above

model When these optimal weights are applied to evaluatealternative119883

119896 a cross score denoted by119862

119894119896= 119863minus

119894119896(119863minus

119894119896+119863+

119894119896)

is obtained

For alternative 119896 (119896 = 1 119898) we use the average ofall 119862119894119896to measure the expected return of cross-evaluation

closeness index which is written as

119903 (119883119896) =

1

119898

119898

sum

119894=1

119862119894119896 (15)

Furthermore we use the standard variance

120575 (119883119896) =

radicsum119898

119894=1(119862119894119896minus 119903 (119883

119896))2

119899

(16)

tomeasure the risk caused by different sets of optimal weightsproduced by each alternative

In general there are both return of closeness index(defined by the mean of cross-evaluation score) and risk ofcloseness index (defined by the deviation between return andeach cross-evaluation closeness index) in cross-evaluationframework It is not hard to see that the larger return scorethe better and the smaller risk score the better Specificallygiven two alternatives 119883

119904and 119883

119905 if 119877(119883

119904) gt 119877(119883

119905) and

120575(119883119904) lt 120575(119883

119905) then 119883

119904≻ 119883119905(119883119904is better than 119883

119905) clearly

holds However when 119877(119883119904) gt 119877(119883

119905) and 120575(119883

119904) gt 120575(119883

119905) it

is not easy to compare the two alternatives directlyThen we integrate the return and risk measures by the

well-known coefficient of variation [51]

CV (119883119896) =

120575 (119883119896)

119903 (119883119896)times 100 (17)

For two given alternatives one having a smaller coeffi-cient of variation is considered to be better than the otherone Thus all the alternatives are ranked by the comparisonof their corresponding coefficient of variation

33 Procedure of the Proposed Approach Based on the aboveanalysis the procedure for generating a solution to theMADM problem is elaborated in the following

Step 1 (formulation of a MADM problem) The decisionmaker identifies 119899 attributes to evaluate 119898 alternatives Theoriginal attribute values are standardized by (1)

Step 2 (obtainment of the attribute weight space) First theupper bounds of each attribute are calculated byTheorem 2 toguarantee the effectiveness of all the attributesThen differenttypes of constraint of attribute weights are determined by thedecision maker to reflect the importance of each attributeLast the weight space including all the feasible weightcombinations is obtained

Step 3 (calculation of the mean and deviation measures)Firstly the maximized closeness index of each alternativeis calculated by (12) Secondly the mean measure of eachalternativersquos cross closeness index is computed by (15) Thirdthe deviation measure of each alternativersquos cross closenessindex is obtained by (16)

6 Mathematical Problems in Engineering

Step 4 (evaluation of the alternatives via coefficient ofvariation) Since the mean and deviation measures of eachalternativersquos cross closeness index are obtained in Step 3 thecoefficient of variation defined in (17) is utilized to integratethe results

Step 5 (generation of rank order of the alternatives) Throughcomparing the coefficients of variation among the alterna-tives the rank order can be directly obtained

4 Extension with Uncertain Attribute Values

41 Model with Interval Attribute Value It is widely acknowl-edged that uncertainty is associated with attribute values dueto complexity of real world and limited knowledge of humanbeings In many situations the attribute value 119903

119894119895is uncertain

but we know that the feasible attribute value 119903119894119895

isin [119903minus

119894119895 119903+

119894119895]

Here we present the following self-evaluation model withinterval attribute values [52] and uncertain attribute weights

max 120572

radicsum119899

119895=1(119908119895119903+

119894119895)2

radicsum119899

119895=1(119908119895119903+

119894119895)2

+ radicsum119899

119895=1(119908119895119903+

119894119895minus 119908119895)2

+ 120573

radicsum119899

119895=1(119908119895119903minus

119894119895)2

radicsum119899

119895=1(119908119895119903minus

119894119895)2

+ radicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2

= 119862max119894

119908 isin Ω (119908)

120572 + 120573 = 1 120572 ge 0 120573 ge 0

(18)

in which radicsum119899

119895=1(119908119895119903+

119894119895)2 radicsum

119899

119895=1(119908119895119903+

119894119895minus 119908119895)2 radicsum

119899

119895=1(119908119895119903minus

119894119895)2

andradicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2 represent distance of upper bounds

from ideal point distance of upper bounds from negativeideal point distance of lower bounds from ideal pointand distance of lower bounds from negative ideal pointrespectively 120572 and 120573 reflect the risk coefficients that combineupper closeness index and lower closeness respectively Inpractice120572 and120573 are decided by the decisionmaker accordingto his or her risk preference Specifically the decision makeris neutral if 120572 = 120573 = 12 120572 gt 120573 and 120572 lt 120573 indicate that he isoptimistic and pessimistic respectively

42 Model with Ordinal Attribute Value Another form ofuncertain preferential information is expressed by ordinalattribute value (eg ldquomore preferablerdquo and ldquoless preferablerdquo)It is argued to be more stable and more reliable than cardinalvalue in some situations [53] In this paper an approach ofthreshold value is introduced to transform ordinal value intolinear formulation For instance given four attributes 119886 119887 119888and 119889 the ordinal values are 1 2 3 and 4 respectively Byspecifying a threshold value 01 the attribute values can beconverted to a cardinal attribute space Ω(119886 119887 119888 119889) = 119886 = 1

119886 minus 119887 ge 01 119887 minus 119888 ge 01 119888 minus 119889 ge 01 119889 = 0 In generallet the ordinal attribute values 119886

119894119895(119894 = 1 119898) have relation

1198861198941119895gt 1198861198942119895gt sdot sdot sdot gt 119886

119894119898119895(119894119896isin [1119898] 119896 = 1 119898) and then

the cardinal attribute spaceΩ(119903119894119895) = 119886

119894119898119895= 1 119886119894119898119895minus119886119894119898minus1119895ge 120576

119886119894119898minus1119895minus119886119894119898minus2119895ge 120576 119886

1198942119895minus1198861198941119895ge 120576 1198861198941119895= 0 120576 gt 0 is obtained

(120576 is the threshold value given by the decision maker) Herewe present the following self-evaluation model with ordinalattribute values and uncertain attribute weights

maxradicsum119899

119895=1(119908119895119903119894119895)2

radicsum119899

119895=1(119908119895119903119894119895)2

+ radicsum119899

119895=1(119908119895119903119894119895minus 119908119895)2

= 119862max119894

st 119903 isin Ω (119903)

119908 isin Ω (119908)

(19)

119894 = 1 119898 (20)

119895 = 1 119899 (21)

For each alternative 119883119896(119896 = 1 119898) under evaluation

we obtain a group of optimal attribute values and a set ofattribute weights They are denoted by 119860

lowast

119896= (119903119896

119894119895)119898times119899

and119882lowast

119896= (119908

119896

1 119908119896

2 119908

119896

119899) respectively When the optimal

decision information is applied to evaluate alternative119883119897(119897 =

1 119898) a cross score denoted by 119862119896119897

= radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2

(radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2+ radicsum

119899

119895=1(119908119896

119895119903119896

119897119895minus 119908119896

119895)2) is obtained

Similar to that defined by (14) and (15) the mean anddeviation measures of alternative 119883

119897are denoted by 119903(119883

119897) =

sum119898

119896=1119862119896119897and 120575(119883

119897) = radicsum

119898

119896=1(119862119896119897minus 119903(119883

119897))2119899 respectively

Note that (12) (18) and (19) are nonlinear models theycan be transformed into linear forms according to practicalcases

5 Illustrative Examples

In order to illustrate the applications of our proposed meth-ods three practical MADM problems are presented

51 Application to House Selection with Cardinal AttributeValues Xu [54] stated a purchase house problem there arefour possible houses to be selected and the attributes are asfollows (1)119884

1is the house price (ten thousand yuan) (2)119884

2is

the usable area (square meter) (3) 1198843is the distance between

house and the center of city (kilometers) (4) 1198844is the nature

environment (scores) where 1198842and 119884

4are benefit attributes

and 1198841and 119884

3are cost attributes The attribute weights are

completely unknown The decision making matrix 119860 andnormalized decision making matrix 119877 are listed in Tables 1and 2 respectively Step 1 is completed

By utilizing (9) and (10) the upper boundof each attributeweight is obtained And as a result the attribute weight spaceis Ω(119908) = 0 lt 119908

1lt 077 0 lt 119908

2lt 078 0 lt 119908

3lt 062

0 lt 1199084lt 062 119908

1+ 1199082+ 1199083+ 1199084= 1 Step 2 is completed

Mathematical Problems in Engineering 7

Table 1 Decision making matrix 119860 of house evaluation

1198841

1198842

1198843

1198844

1198831

30 100 10 71198832

25 80 8 51198833

18 50 20 111198834

22 70 12 9

Table 2 Normalized decisionmakingmatrix119877 of house evaluation

1198841

1198842

1198843

1198844

1198831

06 10 08 06361198832

072 08 10 04551198833

10 05 04 101198834

0818 07 0667 0818

Table 3 The cross-evaluation measurements

Alternatives 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

119903 (119883119894) 120575 (119883

119894)

1198831

096 096 082 061 061 075 0151198832

092 080 092 066 066 076 0111198833

1 052 051 1 1 076 0241198834

0818 070 070 0818 0818 076 006

Table 4 Alternative efficiency matrix

Alternatives 1198831

1198832

1198833

1198834

CV (119883119894) 020 014 032 008

In terms of Step 3 in Section 33 we obtain the cross-evaluation measurements of each alternative which arepresented in Table 3 Step 3 is completed

Based on Table 3 and the calculation of coefficient ofvariation the results are shown in Table 4

Therefore we can rank the houses in accordance with theCV values presented in Table 4 to get a complete rank order1198834

≻ 1198832

≻ 1198831

≻ 1198833 And 119883

4is the best house under

evaluation Step 4 is completedIn order to understand our approach more clearly we

compare our approachwith other three similarmethods [54]namely average attribute value method maximizing devia-tion method and information entropy method respectivelywith the same example The corresponding results of suchthree methods are listed in Table 5

According to Tables 4 and 5 the result derived from ourcross-evaluation approach is quite different from that of theexisting three weight determination methods In fact Table 5shows that the three weight determination methods generatethree attribute weight vectors and produce three differentoptimal solutions However there is no single method thatcan guarantee more convincing weight vector than othersHence we treat the attribute weight as a weight space rather

Table 5 The results of three methods

Method Weight vector Rank order

Average attributevalue

(025 025 025 025) 1198831≻ 1198834≻ 1198832≻ 1198833

Maximizingdeviation

(020 024 029 027) 1198831≻ 1198832≻ 1198834≻ 1198833

Informationentropy

(013 022 036 029) 1198831≻ 1198832≻ 1198834≻ 1198833

Table 6 Attribute weights

1198841

1198842

1198843

[119908119871

1 119908119880

1] [119908

119871

2 119908119880

2] [119908

119871

3 119908119880

3]

[035 050] [015 030] [030 035]

Table 7 Attribute values of alternatives

1198841

1198842

1198843

119860 [065 085] [068 086] [045 078]119861 [075 1] [055 078] [064 080]119862 [054 068] [046 055] [078 1]119863 [077 090] [067 078] [056 077]119864 [025 045] [089 1] [075 094]

than a certain value in this paperThe comparison result vali-dates that our proposed approach can overcome the shortagethat we cannot obtain themost dominant result caused by thediversity of the way of weight vector determined

52 Application to University Faculty with Interval AttributeValue Consider the example in Bryson and Mobolurinrsquos[55] study that is five faculty candidates (119860 119861 119862 119863119864) are evaluated for tenure and promotion through threeattributes teaching research and service According to theDMrsquos preference the information for both attribute andalternatives is given by interval numbers as shown in Tables6 and 7 respectively (for the purpose of illustration our datais different from the literature)

Table 8 shows the computed results of the proposedapproach with interval values The self-evaluation score ofeach alternative is listed in the second column and the cross-evaluation scores are listed in columns 3ndash7 Columns 8 and9 show the mean and deviation measures respectively Thenthe results in column 10 are applied to generate a rank orderof the five candidates119860 ≻ 119862 ≻ 119864 ≻ 119861 ≻ 119863 which is a solutionto the problem

53 Application to the Selection of Community Service Com-panies with Ordinal Attribute Values In this subsection weapply our approach to the evaluation and selection of fivecommunity service companies that are located in Hefei acity in Anhui province of China These companies providevarious kinds of service for the public such as property

8 Mathematical Problems in Engineering

Table 8 Evaluation results for the comparison of faculty candidates

Candidates 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

1198625119894

119903119894

120575119894

CV(119883119894)

119860 0717 0717 0712 0706 0717 0703 0711 0006 0008119861 0792 0792 0792 0788 0792 0746 0782 0018 0023119862 0672 0651 0661 0672 0651 0660 0659 0008 0012119863 0774 0774 0771 0764 0774 0737 0764 0014 0461119864 0645 0525 0526 0540 0525 0645 0552 0047 0018

Table 9 Ordinal values of alternatives

1198841

1198842

1198843

1198844

1198831

4 2 4 21198832

1 4 3 31198833

2 5 1 11198834

5 3 2 51198835

3 1 5 4

management catering health care and entertainment Inthe case study the decision maker is an official from thedevelopment and reform commission of Hefei with thehelp of a group of experts from the governments relevantenterprises and community organizations Four attributesare used to evaluate the six companies They are quality ofservice (119884

1) diversity of service (119884

2) profitability (119884

3) and

development potential (1198844) The original attribute values are

given by ordinal values as shown in Table 9 The decisionprocedures are as follows

After considering the four attributes the decision makerviews that quality of service (119884

1) is the most important

attribute in evaluation The development potential (1198844) is the

second most important one profitability (1198843) is the third

most important one and diversity of service (1198842) is the

least important one respectively And the difference betweentwo adjacent attribute weights should be more than 01 Inaddition each weight value should be nomore than 05Thenthe weight space is denoted by Ω(119908) = 119908

1minus 01 gt 119908

4

1199084minus 01 gt 119908

3 1199083minus 01 gt 119908

2 1199081le 05 119908

2le 05 119908

3le 05

1199084le 05 119908

1+ 1199082+ 1199083+ 1199084= 1

By specifying a threshold value of 01 ordinal attributevalues are transformed into four sets of attribute value spacesThey are denoted by

Ω(1198841) = 119903

21= 1 11990321

minus 11990331

ge 01 11990331

minus 11990351

ge 01 11990351

minus 11990311

ge 01 11990311

minus 11990341

ge 01 11990341

= 0

Ω (1198842) = 119903

52= 1 11990352

minus 11990312

ge 01 11990312

minus 11990342

ge 01 11990342

minus 11990322

ge 01 11990322

minus 11990332

ge 01 11990332

= 0

Ω (1198843) = 119903

33= 1 11990333

minus 11990343

ge 01 11990343

minus 11990323

ge 01 11990323

minus 11990313

ge 01 11990313

minus 11990353

ge 01 11990353

= 0

Ω (1198844) = 119903

34= 1 11990334

minus 11990314

ge 01 11990314

minus 11990324

ge 01 11990324

minus 11990354

ge 01 11990354

minus 11990344

ge 01 11990344

= 0

(22)

Table 10 shows the computed results of the proposedapproachThe second column shows the self-evaluation scoreof each alternative computed by (19) And columns 3ndash7 listthe cross-evaluation scores Then the mean and deviationmeasures of each alternative are presented in columns 8 and9 respectively Consequently the results in column 10 areapplied to generate a rank order of the five community servicecompanies 119883

5≻ 1198831≻ 1198832≻ 1198834≻ 1198833 The decision maker

is pleased to select company 1198835with the best performance

among the five alternatives

6 Conclusions and Future Research

Due to the complexity and uncertainty of MADM problemsin real world the decision maker is not able to provide theexact attribute weights And different weight determinationmethods are difficult to generate a most satisfaction solutionThe current paper presents a new MADM method basedon cross-evaluation in which the weight information isexpressed by the so-called weight space It attempts to rectifysome shortcomings of existingMADMmethods by providingthe following important features (1) it is more feasibleto provide a weight space rather than specifying a set ofweight values (2) the single dominating attribute drawbackcan be avoided by the restriction of the upper bound ofeach attribute (3) compared with DEA based methods thevariable attribute weights are consistent with the opinionof decision maker (4) both mean score and the deviationscore are considered in our cross-evaluation models and (5)the proposed method is also successfully extended to moregeneralized situations in which attribute values are expressedby interval values and ordinal values respectively

We have illustrated the applicability of the proposedmethods by revisiting two reported MADM studies and bydepicting a case study on the selection of community servicecompanies Although our proposed method makes someimprovement in solving MADM problems it only can dealwith the individual decision making case Future research

Mathematical Problems in Engineering 9

Table 10 Evaluation results for the selection of community service companies

Candidates 119862max119897

1198621119897

1198622119897

1198623119897

1198624119897

1198625119897

119903119894

120575119894

CV(119883119894)

1198831

0489 0489 0317 0295 0317 0317 0347 0072 02071198832

0532 0433 0532 0508 0458 0504 0487 0036 00741198833

0535 0526 0526 0535 0526 0526 0528 0004 00081198834

0223 0202 0223 0211 0223 0223 0216 0009 00421198835

0422 0349 0349 0103 0349 0422 0314 0109 0347

may attempt to employ our cross-evaluation methods tohandle group decision making problem

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research is supported by National Key Technology Sup-port Program (no 2014BAH27F00) National Natural Sci-ence Funds of China (no 71471053) and the FundamentalResearch Fund for the Central Universities (no JZ2014-HGBZ0339)

References

[1] R H Green and J R Doyle ldquoOn maximizing discriminationin multiple criteria decision makingrdquo Journal of the OperationalResearch Society vol 46 no 2 pp 192ndash204 1995

[2] R E Steuer and P Na ldquoMultiple criteria decision makingcombined with finance a categorized bibliographic studyrdquoEuropean Journal of Operational Research vol 150 no 3 pp496ndash515 2003

[3] J Wallenius J S Dyer P C Fishburn R E Steuer S Ziontsand K Deb ldquoMultiple criteria decision making multi-attributeutility theory recent accomplishments and what lies aheadrdquoManagement Science vol 54 no 7 pp 1336ndash1349 2008

[4] W Ho X Xu and P K Dey ldquoMulti-criteria decision makingapproaches for supplier evaluation and selection a literaturereviewrdquo European Journal of Operational Research vol 202 no1 pp 16ndash24 2010

[5] C W Churchman R L Ackoff and N M Smith ldquoAn approx-imate measure of valuerdquo Journal of the Operational ResearchSociety of America vol 2 no 2 pp 172ndash187 1954

[6] T L Saaty ldquoThe analytic hierarchy processrdquo in Proceedings ofthe Second International Seminar on Operational Research in theBasque Provinces vol 4 pp 189ndash234 1980

[7] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[8] H Deng C-H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[9] P K Dey ldquoIntegrated project evaluation and selection usingmultiple-attribute decision-making techniquerdquo InternationalJournal of Production Economics vol 103 no 1 pp 90ndash1032006

[10] G-D Li D Yamaguchi and M Nagai ldquoA grey-based decision-making approach to the supplier selection problemrdquoMathemat-ical amp Computer Modelling vol 46 no 3-4 pp 573ndash581 2007

[11] A Shanian and O Savadogo ldquoTOPSIS multiple-criteria deci-sion support analysis for material selection of metallic bipolarplates for polymer electrolyte fuel cellrdquo Journal of Power Sourcesvol 159 no 2 pp 1095ndash1104 2006

[12] G-H Tzeng Y-P O Yang C-T Lin and C-B Chen ldquoHier-archical MADM with fuzzy integral for evaluating enterpriseintranet web sitesrdquo Information Sciences vol 169 no 3-4 pp409ndash426 2005

[13] 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

[14] A Kelemenis and D Askounis ldquoA new TOPSIS-based multi-criteria approach to personnel selectionrdquo Expert Systems withApplications vol 37 no 7 pp 4999ndash5008 2010

[15] B Bairagi B Dey B Sarkar and S K Sanyal ldquoADeNovomulti-approaches multi-criteria decision making technique with anapplication in performance evaluation of material handlingdevicerdquo Computers amp Industrial Engineering vol 87 pp 267ndash282 2015

[16] J S H Kornbluth ldquoDynamic multi-criteria decision makingrdquoJournal of Multi-Criteria Decision Analysis vol 1 no 2 pp 81ndash92 1992

[17] Y-H Lin P-C Lee T-P Chang and H-I Ting ldquoMulti-attribute group decision making model under the condition ofuncertain informationrdquoAutomation in Construction vol 17 no6 pp 792ndash797 2008

[18] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multicrite-ria acceptability analysis for groupdecisionmakingrdquoOperationsResearch vol 49 no 3 pp 444ndash454 2001

[19] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[20] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[21] S Miyamoto ldquoMultisets and fuzzymultisetsrdquo in Soft Computingand Human-Centered Machines Z Q Liu and S MiyamotoEds pp 9ndash33 Springer Berlin Germany 2000

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[23] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[25] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling Simulation and Computation for Engineering andEnvironmental Systems vol 34 no 7 pp 1779ndash1787 2010

10 Mathematical Problems in Engineering

[26] D Horsky andM R Rao ldquoEstimation of attribute weights frompreference comparisonsrdquoManagement Science vol 30 no 7 pp801ndash822 1984

[27] P A Bottomley and J R Doyle ldquoA comparison of three weightelicitation methods good better and bestrdquo Omega vol 29 no6 pp 553ndash560 2001

[28] J Figueira and B Roy ldquoDetermining the weights of criteria inthe ELECTRE type methods with a revised Simosrsquo procedurerdquoEuropean Journal ofOperational Research vol 139 no 2 pp 317ndash326 2002

[29] L E Shirland R R Jesse R L Thompson and C L IacovouldquoDetermining attribute weights using mathematical program-mingrdquo Omega vol 31 no 6 pp 423ndash437 2003

[30] Y-M Wang and Y Luo ldquoIntegration of correlations withstandard deviations for determining attribute weights in mul-tiple attribute decision makingrdquo Mathematical amp ComputerModelling vol 51 no 1-2 pp 1ndash12 2010

[31] J Ma Z-P Fan and L-H Huang ldquoA subjective and objectiveintegrated approach to determine attribute weightsrdquo EuropeanJournal of Operational Research vol 112 no 2 pp 397ndash4041999

[32] X Xu ldquoA note on the subjective and objective integratedapproach to determine attribute weightsrdquo European Journal ofOperational Research vol 156 no 2 pp 530ndash532 2004

[33] P J H Schoemaker and C C Waid ldquoAn experimental compar-ison of different approaches to determining weights in additiveutility modelsrdquoManagement Science vol 28 no 2 pp 182ndash1961982

[34] Y-M Wang and C Parkan ldquoA general multiple attributedecision-making approach for integrating subjective prefer-ences and objective informationrdquo Fuzzy Sets and Systems vol157 no 10 pp 1333ndash1345 2006

[35] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[36] J R Doyle ldquoMultiattribute choice for the lazy decision makerlet the alternatives deciderdquoOrganizational Behavior andHumanDecision Processes vol 62 no 1 pp 87ndash100 1995

[37] E Bernroider and V Stix ldquoA method using weight restrictionsin data envelopment analysis for ranking and validity issues indecisionmakingrdquoComputersampOperations Research vol 34 no9 pp 2637ndash2647 2007

[38] J Wu and L Liang ldquoA multiple criteria ranking method basedon game cross-evaluation approachrdquo Annals of OperationsResearch vol 197 no 1 pp 191ndash200 2012

[39] C Fu and K-S Chin ldquoRobust evidential reasoning approachwith unknown attribute weightsrdquo Knowledge-Based Systemsvol 59 no 2 pp 9ndash20 2014

[40] B S Ahn ldquoExtreme point-based multi-attribute decision anal-ysis with incomplete informationrdquo European Journal of Opera-tional Research vol 240 no 3 pp 748ndash755 2015

[41] WW Cooper J L Ruiz and I Sirvent ldquoChoosing weights fromalternative optimal solutions of dualmultipliermodels inDEArdquoEuropean Journal of Operational Research vol 180 no 1 pp443ndash458 2007

[42] G R Jahanshahloo and P F Shahmirzadi ldquoNew methods forranking decision making units based on the dispersion ofweights and Norm 1 in Data Envelopment Analysisrdquo Computersamp Industrial Engineering vol 65 no 2 pp 187ndash193 2013

[43] V E Krivonozhko O B Utkin M M Safin and A V LychevldquoOn some generalization of the DEA modelsrdquo Journal of theOperational Research Society vol 60 no 11 pp 1518ndash1527 2009

[44] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[45] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo New Directions for ProgramEvaluation vol 1986 no 32 pp 73ndash105 1986

[46] A Boussofiane R G Dyson and E Thanassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[47] K Wang and F Wei ldquoGame cross efficiency DEA based projectreview of equipment designrdquo Journal of Beijing University ofAeronautics and Astronautics vol 35 no 10 pp 1278ndash12822009

[48] T Wu and J Blackhurst ldquoSupplier evaluation and selection anaugmented DEA approachrdquo International Journal of ProductionResearch vol 47 no 16 pp 4593ndash4608 2009

[49] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Korean stockmarketrdquo European Journal of Operational Research vol 236 no1 pp 361ndash368 2014

[50] B Roy ldquoTheoutranking approach and the foundations of electremethodsrdquoTheory and Decision vol 31 no 1 pp 49ndash73 1991

[51] G F Reed F Lynn and B D Meade ldquoUse of coefficient ofvariation in assessing variability of quantitative assaysrdquo Clinicalamp Diagnostic Laboratory Immunology vol 9 no 6 pp 1235ndash1239 2002

[52] 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

[53] H M Moshkovich A I Mechitov and D L Olson ldquoOrdinaljudgments in multiattribute decision analysisrdquo European Jour-nal of Operational Research vol 137 no 3 pp 625ndash641 2002

[54] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Tsinghua Press Beijing China 2005

[55] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

In many practical situations although the weight valuesare difficult to obtain the weight information is partiallyknown and predefined by the weight space As followswe introduce two types of partial weight information asreferences to decision maker

Type 1 (pairwise comparisonweights information) InMADMattribute weights 119908

119895are mainly introduced to reflect the

importance to attributes 119884119895(119895 = 1 2 119899) As stated in

Section 1 pairwise comparison is a widely accepted subjectivemethod to determine attribute weight

Attribute 1198842is at least two times and not more than three

times as important as attribute 1198841

In notational jargons this is denoted by

21199081le 1199082le 31199081 (13)

In general terms we compare 1198841with 119884

119895(119895 = 2 3 119899)

respectively The bound linear constraints are denoted by

1198971198951199081le 119908119895le 1199061198951199081 (14)

in which 119897119895and 119906

119895represent the lower and upper bounds of

times119908119895compared with119908

1 respectively Hence the attribute

weights space is obtained by the above pairwise comparisoninformation which is expressed as Ω(119908) = 119908

119895gt 0 119897

1198951199081le

119908119895le 1199061198951199081 119895 = 1 2 119899 Clearly 119897

1= 1199061= 1 holds

Compared with conventional subjective weight determi-nation approaches such as the AHP weights space is morefeasible and applicative because the decision makers neednot express their preferences explicitly In many situationsexperts are pleased to give such interval information ratherthan exact value due to the complexity of actual problems

Type 2 (ordinal weights information) Outranking relation[50] usually denoted by 119878 was proposed by Roy whoseaim was to establish a realistic representation of the threebasic preference situations strict preference indifferenceand incomparability Motivated by the widely accepted out-ranking relation we introduce the concept of rank-orderweights information In many situations although the weightvalues are difficult to give by decision makers the rankingof each attribute is easy to obtain according to importanceTake the Olympic Games as an example the achievementsof athletes and nations are judged by gold silver and bronzemedals Apparently a gold medal is more important than asilver medal and a silver medal is more important than abronze medal Then the weights space can be denoted byΩ(119908) = 119908

1gt 1199082

gt 1199083 1199081 1199082 1199083

gt 0 in which1199081 1199082 1199083represent the weight of gold silver and bronze

medal respectivelyThe above self-evaluation model (see (12)) shows that all

alternatives choose their own attribute weights in order tomaximize their own self-evaluation closeness index Similarto the formofDEA cross-efficiency we obtain a set of optimalattribute weights 119882lowast

119894= (119908lowast

1198941 119908

lowast

119894119899) according to the above

model When these optimal weights are applied to evaluatealternative119883

119896 a cross score denoted by119862

119894119896= 119863minus

119894119896(119863minus

119894119896+119863+

119894119896)

is obtained

For alternative 119896 (119896 = 1 119898) we use the average ofall 119862119894119896to measure the expected return of cross-evaluation

closeness index which is written as

119903 (119883119896) =

1

119898

119898

sum

119894=1

119862119894119896 (15)

Furthermore we use the standard variance

120575 (119883119896) =

radicsum119898

119894=1(119862119894119896minus 119903 (119883

119896))2

119899

(16)

tomeasure the risk caused by different sets of optimal weightsproduced by each alternative

In general there are both return of closeness index(defined by the mean of cross-evaluation score) and risk ofcloseness index (defined by the deviation between return andeach cross-evaluation closeness index) in cross-evaluationframework It is not hard to see that the larger return scorethe better and the smaller risk score the better Specificallygiven two alternatives 119883

119904and 119883

119905 if 119877(119883

119904) gt 119877(119883

119905) and

120575(119883119904) lt 120575(119883

119905) then 119883

119904≻ 119883119905(119883119904is better than 119883

119905) clearly

holds However when 119877(119883119904) gt 119877(119883

119905) and 120575(119883

119904) gt 120575(119883

119905) it

is not easy to compare the two alternatives directlyThen we integrate the return and risk measures by the

well-known coefficient of variation [51]

CV (119883119896) =

120575 (119883119896)

119903 (119883119896)times 100 (17)

For two given alternatives one having a smaller coeffi-cient of variation is considered to be better than the otherone Thus all the alternatives are ranked by the comparisonof their corresponding coefficient of variation

33 Procedure of the Proposed Approach Based on the aboveanalysis the procedure for generating a solution to theMADM problem is elaborated in the following

Step 1 (formulation of a MADM problem) The decisionmaker identifies 119899 attributes to evaluate 119898 alternatives Theoriginal attribute values are standardized by (1)

Step 2 (obtainment of the attribute weight space) First theupper bounds of each attribute are calculated byTheorem 2 toguarantee the effectiveness of all the attributesThen differenttypes of constraint of attribute weights are determined by thedecision maker to reflect the importance of each attributeLast the weight space including all the feasible weightcombinations is obtained

Step 3 (calculation of the mean and deviation measures)Firstly the maximized closeness index of each alternativeis calculated by (12) Secondly the mean measure of eachalternativersquos cross closeness index is computed by (15) Thirdthe deviation measure of each alternativersquos cross closenessindex is obtained by (16)

6 Mathematical Problems in Engineering

Step 4 (evaluation of the alternatives via coefficient ofvariation) Since the mean and deviation measures of eachalternativersquos cross closeness index are obtained in Step 3 thecoefficient of variation defined in (17) is utilized to integratethe results

Step 5 (generation of rank order of the alternatives) Throughcomparing the coefficients of variation among the alterna-tives the rank order can be directly obtained

4 Extension with Uncertain Attribute Values

41 Model with Interval Attribute Value It is widely acknowl-edged that uncertainty is associated with attribute values dueto complexity of real world and limited knowledge of humanbeings In many situations the attribute value 119903

119894119895is uncertain

but we know that the feasible attribute value 119903119894119895

isin [119903minus

119894119895 119903+

119894119895]

Here we present the following self-evaluation model withinterval attribute values [52] and uncertain attribute weights

max 120572

radicsum119899

119895=1(119908119895119903+

119894119895)2

radicsum119899

119895=1(119908119895119903+

119894119895)2

+ radicsum119899

119895=1(119908119895119903+

119894119895minus 119908119895)2

+ 120573

radicsum119899

119895=1(119908119895119903minus

119894119895)2

radicsum119899

119895=1(119908119895119903minus

119894119895)2

+ radicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2

= 119862max119894

119908 isin Ω (119908)

120572 + 120573 = 1 120572 ge 0 120573 ge 0

(18)

in which radicsum119899

119895=1(119908119895119903+

119894119895)2 radicsum

119899

119895=1(119908119895119903+

119894119895minus 119908119895)2 radicsum

119899

119895=1(119908119895119903minus

119894119895)2

andradicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2 represent distance of upper bounds

from ideal point distance of upper bounds from negativeideal point distance of lower bounds from ideal pointand distance of lower bounds from negative ideal pointrespectively 120572 and 120573 reflect the risk coefficients that combineupper closeness index and lower closeness respectively Inpractice120572 and120573 are decided by the decisionmaker accordingto his or her risk preference Specifically the decision makeris neutral if 120572 = 120573 = 12 120572 gt 120573 and 120572 lt 120573 indicate that he isoptimistic and pessimistic respectively

42 Model with Ordinal Attribute Value Another form ofuncertain preferential information is expressed by ordinalattribute value (eg ldquomore preferablerdquo and ldquoless preferablerdquo)It is argued to be more stable and more reliable than cardinalvalue in some situations [53] In this paper an approach ofthreshold value is introduced to transform ordinal value intolinear formulation For instance given four attributes 119886 119887 119888and 119889 the ordinal values are 1 2 3 and 4 respectively Byspecifying a threshold value 01 the attribute values can beconverted to a cardinal attribute space Ω(119886 119887 119888 119889) = 119886 = 1

119886 minus 119887 ge 01 119887 minus 119888 ge 01 119888 minus 119889 ge 01 119889 = 0 In generallet the ordinal attribute values 119886

119894119895(119894 = 1 119898) have relation

1198861198941119895gt 1198861198942119895gt sdot sdot sdot gt 119886

119894119898119895(119894119896isin [1119898] 119896 = 1 119898) and then

the cardinal attribute spaceΩ(119903119894119895) = 119886

119894119898119895= 1 119886119894119898119895minus119886119894119898minus1119895ge 120576

119886119894119898minus1119895minus119886119894119898minus2119895ge 120576 119886

1198942119895minus1198861198941119895ge 120576 1198861198941119895= 0 120576 gt 0 is obtained

(120576 is the threshold value given by the decision maker) Herewe present the following self-evaluation model with ordinalattribute values and uncertain attribute weights

maxradicsum119899

119895=1(119908119895119903119894119895)2

radicsum119899

119895=1(119908119895119903119894119895)2

+ radicsum119899

119895=1(119908119895119903119894119895minus 119908119895)2

= 119862max119894

st 119903 isin Ω (119903)

119908 isin Ω (119908)

(19)

119894 = 1 119898 (20)

119895 = 1 119899 (21)

For each alternative 119883119896(119896 = 1 119898) under evaluation

we obtain a group of optimal attribute values and a set ofattribute weights They are denoted by 119860

lowast

119896= (119903119896

119894119895)119898times119899

and119882lowast

119896= (119908

119896

1 119908119896

2 119908

119896

119899) respectively When the optimal

decision information is applied to evaluate alternative119883119897(119897 =

1 119898) a cross score denoted by 119862119896119897

= radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2

(radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2+ radicsum

119899

119895=1(119908119896

119895119903119896

119897119895minus 119908119896

119895)2) is obtained

Similar to that defined by (14) and (15) the mean anddeviation measures of alternative 119883

119897are denoted by 119903(119883

119897) =

sum119898

119896=1119862119896119897and 120575(119883

119897) = radicsum

119898

119896=1(119862119896119897minus 119903(119883

119897))2119899 respectively

Note that (12) (18) and (19) are nonlinear models theycan be transformed into linear forms according to practicalcases

5 Illustrative Examples

In order to illustrate the applications of our proposed meth-ods three practical MADM problems are presented

51 Application to House Selection with Cardinal AttributeValues Xu [54] stated a purchase house problem there arefour possible houses to be selected and the attributes are asfollows (1)119884

1is the house price (ten thousand yuan) (2)119884

2is

the usable area (square meter) (3) 1198843is the distance between

house and the center of city (kilometers) (4) 1198844is the nature

environment (scores) where 1198842and 119884

4are benefit attributes

and 1198841and 119884

3are cost attributes The attribute weights are

completely unknown The decision making matrix 119860 andnormalized decision making matrix 119877 are listed in Tables 1and 2 respectively Step 1 is completed

By utilizing (9) and (10) the upper boundof each attributeweight is obtained And as a result the attribute weight spaceis Ω(119908) = 0 lt 119908

1lt 077 0 lt 119908

2lt 078 0 lt 119908

3lt 062

0 lt 1199084lt 062 119908

1+ 1199082+ 1199083+ 1199084= 1 Step 2 is completed

Mathematical Problems in Engineering 7

Table 1 Decision making matrix 119860 of house evaluation

1198841

1198842

1198843

1198844

1198831

30 100 10 71198832

25 80 8 51198833

18 50 20 111198834

22 70 12 9

Table 2 Normalized decisionmakingmatrix119877 of house evaluation

1198841

1198842

1198843

1198844

1198831

06 10 08 06361198832

072 08 10 04551198833

10 05 04 101198834

0818 07 0667 0818

Table 3 The cross-evaluation measurements

Alternatives 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

119903 (119883119894) 120575 (119883

119894)

1198831

096 096 082 061 061 075 0151198832

092 080 092 066 066 076 0111198833

1 052 051 1 1 076 0241198834

0818 070 070 0818 0818 076 006

Table 4 Alternative efficiency matrix

Alternatives 1198831

1198832

1198833

1198834

CV (119883119894) 020 014 032 008

In terms of Step 3 in Section 33 we obtain the cross-evaluation measurements of each alternative which arepresented in Table 3 Step 3 is completed

Based on Table 3 and the calculation of coefficient ofvariation the results are shown in Table 4

Therefore we can rank the houses in accordance with theCV values presented in Table 4 to get a complete rank order1198834

≻ 1198832

≻ 1198831

≻ 1198833 And 119883

4is the best house under

evaluation Step 4 is completedIn order to understand our approach more clearly we

compare our approachwith other three similarmethods [54]namely average attribute value method maximizing devia-tion method and information entropy method respectivelywith the same example The corresponding results of suchthree methods are listed in Table 5

According to Tables 4 and 5 the result derived from ourcross-evaluation approach is quite different from that of theexisting three weight determination methods In fact Table 5shows that the three weight determination methods generatethree attribute weight vectors and produce three differentoptimal solutions However there is no single method thatcan guarantee more convincing weight vector than othersHence we treat the attribute weight as a weight space rather

Table 5 The results of three methods

Method Weight vector Rank order

Average attributevalue

(025 025 025 025) 1198831≻ 1198834≻ 1198832≻ 1198833

Maximizingdeviation

(020 024 029 027) 1198831≻ 1198832≻ 1198834≻ 1198833

Informationentropy

(013 022 036 029) 1198831≻ 1198832≻ 1198834≻ 1198833

Table 6 Attribute weights

1198841

1198842

1198843

[119908119871

1 119908119880

1] [119908

119871

2 119908119880

2] [119908

119871

3 119908119880

3]

[035 050] [015 030] [030 035]

Table 7 Attribute values of alternatives

1198841

1198842

1198843

119860 [065 085] [068 086] [045 078]119861 [075 1] [055 078] [064 080]119862 [054 068] [046 055] [078 1]119863 [077 090] [067 078] [056 077]119864 [025 045] [089 1] [075 094]

than a certain value in this paperThe comparison result vali-dates that our proposed approach can overcome the shortagethat we cannot obtain themost dominant result caused by thediversity of the way of weight vector determined

52 Application to University Faculty with Interval AttributeValue Consider the example in Bryson and Mobolurinrsquos[55] study that is five faculty candidates (119860 119861 119862 119863119864) are evaluated for tenure and promotion through threeattributes teaching research and service According to theDMrsquos preference the information for both attribute andalternatives is given by interval numbers as shown in Tables6 and 7 respectively (for the purpose of illustration our datais different from the literature)

Table 8 shows the computed results of the proposedapproach with interval values The self-evaluation score ofeach alternative is listed in the second column and the cross-evaluation scores are listed in columns 3ndash7 Columns 8 and9 show the mean and deviation measures respectively Thenthe results in column 10 are applied to generate a rank orderof the five candidates119860 ≻ 119862 ≻ 119864 ≻ 119861 ≻ 119863 which is a solutionto the problem

53 Application to the Selection of Community Service Com-panies with Ordinal Attribute Values In this subsection weapply our approach to the evaluation and selection of fivecommunity service companies that are located in Hefei acity in Anhui province of China These companies providevarious kinds of service for the public such as property

8 Mathematical Problems in Engineering

Table 8 Evaluation results for the comparison of faculty candidates

Candidates 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

1198625119894

119903119894

120575119894

CV(119883119894)

119860 0717 0717 0712 0706 0717 0703 0711 0006 0008119861 0792 0792 0792 0788 0792 0746 0782 0018 0023119862 0672 0651 0661 0672 0651 0660 0659 0008 0012119863 0774 0774 0771 0764 0774 0737 0764 0014 0461119864 0645 0525 0526 0540 0525 0645 0552 0047 0018

Table 9 Ordinal values of alternatives

1198841

1198842

1198843

1198844

1198831

4 2 4 21198832

1 4 3 31198833

2 5 1 11198834

5 3 2 51198835

3 1 5 4

management catering health care and entertainment Inthe case study the decision maker is an official from thedevelopment and reform commission of Hefei with thehelp of a group of experts from the governments relevantenterprises and community organizations Four attributesare used to evaluate the six companies They are quality ofservice (119884

1) diversity of service (119884

2) profitability (119884

3) and

development potential (1198844) The original attribute values are

given by ordinal values as shown in Table 9 The decisionprocedures are as follows

After considering the four attributes the decision makerviews that quality of service (119884

1) is the most important

attribute in evaluation The development potential (1198844) is the

second most important one profitability (1198843) is the third

most important one and diversity of service (1198842) is the

least important one respectively And the difference betweentwo adjacent attribute weights should be more than 01 Inaddition each weight value should be nomore than 05Thenthe weight space is denoted by Ω(119908) = 119908

1minus 01 gt 119908

4

1199084minus 01 gt 119908

3 1199083minus 01 gt 119908

2 1199081le 05 119908

2le 05 119908

3le 05

1199084le 05 119908

1+ 1199082+ 1199083+ 1199084= 1

By specifying a threshold value of 01 ordinal attributevalues are transformed into four sets of attribute value spacesThey are denoted by

Ω(1198841) = 119903

21= 1 11990321

minus 11990331

ge 01 11990331

minus 11990351

ge 01 11990351

minus 11990311

ge 01 11990311

minus 11990341

ge 01 11990341

= 0

Ω (1198842) = 119903

52= 1 11990352

minus 11990312

ge 01 11990312

minus 11990342

ge 01 11990342

minus 11990322

ge 01 11990322

minus 11990332

ge 01 11990332

= 0

Ω (1198843) = 119903

33= 1 11990333

minus 11990343

ge 01 11990343

minus 11990323

ge 01 11990323

minus 11990313

ge 01 11990313

minus 11990353

ge 01 11990353

= 0

Ω (1198844) = 119903

34= 1 11990334

minus 11990314

ge 01 11990314

minus 11990324

ge 01 11990324

minus 11990354

ge 01 11990354

minus 11990344

ge 01 11990344

= 0

(22)

Table 10 shows the computed results of the proposedapproachThe second column shows the self-evaluation scoreof each alternative computed by (19) And columns 3ndash7 listthe cross-evaluation scores Then the mean and deviationmeasures of each alternative are presented in columns 8 and9 respectively Consequently the results in column 10 areapplied to generate a rank order of the five community servicecompanies 119883

5≻ 1198831≻ 1198832≻ 1198834≻ 1198833 The decision maker

is pleased to select company 1198835with the best performance

among the five alternatives

6 Conclusions and Future Research

Due to the complexity and uncertainty of MADM problemsin real world the decision maker is not able to provide theexact attribute weights And different weight determinationmethods are difficult to generate a most satisfaction solutionThe current paper presents a new MADM method basedon cross-evaluation in which the weight information isexpressed by the so-called weight space It attempts to rectifysome shortcomings of existingMADMmethods by providingthe following important features (1) it is more feasibleto provide a weight space rather than specifying a set ofweight values (2) the single dominating attribute drawbackcan be avoided by the restriction of the upper bound ofeach attribute (3) compared with DEA based methods thevariable attribute weights are consistent with the opinionof decision maker (4) both mean score and the deviationscore are considered in our cross-evaluation models and (5)the proposed method is also successfully extended to moregeneralized situations in which attribute values are expressedby interval values and ordinal values respectively

We have illustrated the applicability of the proposedmethods by revisiting two reported MADM studies and bydepicting a case study on the selection of community servicecompanies Although our proposed method makes someimprovement in solving MADM problems it only can dealwith the individual decision making case Future research

Mathematical Problems in Engineering 9

Table 10 Evaluation results for the selection of community service companies

Candidates 119862max119897

1198621119897

1198622119897

1198623119897

1198624119897

1198625119897

119903119894

120575119894

CV(119883119894)

1198831

0489 0489 0317 0295 0317 0317 0347 0072 02071198832

0532 0433 0532 0508 0458 0504 0487 0036 00741198833

0535 0526 0526 0535 0526 0526 0528 0004 00081198834

0223 0202 0223 0211 0223 0223 0216 0009 00421198835

0422 0349 0349 0103 0349 0422 0314 0109 0347

may attempt to employ our cross-evaluation methods tohandle group decision making problem

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research is supported by National Key Technology Sup-port Program (no 2014BAH27F00) National Natural Sci-ence Funds of China (no 71471053) and the FundamentalResearch Fund for the Central Universities (no JZ2014-HGBZ0339)

References

[1] R H Green and J R Doyle ldquoOn maximizing discriminationin multiple criteria decision makingrdquo Journal of the OperationalResearch Society vol 46 no 2 pp 192ndash204 1995

[2] R E Steuer and P Na ldquoMultiple criteria decision makingcombined with finance a categorized bibliographic studyrdquoEuropean Journal of Operational Research vol 150 no 3 pp496ndash515 2003

[3] J Wallenius J S Dyer P C Fishburn R E Steuer S Ziontsand K Deb ldquoMultiple criteria decision making multi-attributeutility theory recent accomplishments and what lies aheadrdquoManagement Science vol 54 no 7 pp 1336ndash1349 2008

[4] W Ho X Xu and P K Dey ldquoMulti-criteria decision makingapproaches for supplier evaluation and selection a literaturereviewrdquo European Journal of Operational Research vol 202 no1 pp 16ndash24 2010

[5] C W Churchman R L Ackoff and N M Smith ldquoAn approx-imate measure of valuerdquo Journal of the Operational ResearchSociety of America vol 2 no 2 pp 172ndash187 1954

[6] T L Saaty ldquoThe analytic hierarchy processrdquo in Proceedings ofthe Second International Seminar on Operational Research in theBasque Provinces vol 4 pp 189ndash234 1980

[7] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[8] H Deng C-H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[9] P K Dey ldquoIntegrated project evaluation and selection usingmultiple-attribute decision-making techniquerdquo InternationalJournal of Production Economics vol 103 no 1 pp 90ndash1032006

[10] G-D Li D Yamaguchi and M Nagai ldquoA grey-based decision-making approach to the supplier selection problemrdquoMathemat-ical amp Computer Modelling vol 46 no 3-4 pp 573ndash581 2007

[11] A Shanian and O Savadogo ldquoTOPSIS multiple-criteria deci-sion support analysis for material selection of metallic bipolarplates for polymer electrolyte fuel cellrdquo Journal of Power Sourcesvol 159 no 2 pp 1095ndash1104 2006

[12] G-H Tzeng Y-P O Yang C-T Lin and C-B Chen ldquoHier-archical MADM with fuzzy integral for evaluating enterpriseintranet web sitesrdquo Information Sciences vol 169 no 3-4 pp409ndash426 2005

[13] 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

[14] A Kelemenis and D Askounis ldquoA new TOPSIS-based multi-criteria approach to personnel selectionrdquo Expert Systems withApplications vol 37 no 7 pp 4999ndash5008 2010

[15] B Bairagi B Dey B Sarkar and S K Sanyal ldquoADeNovomulti-approaches multi-criteria decision making technique with anapplication in performance evaluation of material handlingdevicerdquo Computers amp Industrial Engineering vol 87 pp 267ndash282 2015

[16] J S H Kornbluth ldquoDynamic multi-criteria decision makingrdquoJournal of Multi-Criteria Decision Analysis vol 1 no 2 pp 81ndash92 1992

[17] Y-H Lin P-C Lee T-P Chang and H-I Ting ldquoMulti-attribute group decision making model under the condition ofuncertain informationrdquoAutomation in Construction vol 17 no6 pp 792ndash797 2008

[18] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multicrite-ria acceptability analysis for groupdecisionmakingrdquoOperationsResearch vol 49 no 3 pp 444ndash454 2001

[19] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[20] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[21] S Miyamoto ldquoMultisets and fuzzymultisetsrdquo in Soft Computingand Human-Centered Machines Z Q Liu and S MiyamotoEds pp 9ndash33 Springer Berlin Germany 2000

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[23] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[25] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling Simulation and Computation for Engineering andEnvironmental Systems vol 34 no 7 pp 1779ndash1787 2010

10 Mathematical Problems in Engineering

[26] D Horsky andM R Rao ldquoEstimation of attribute weights frompreference comparisonsrdquoManagement Science vol 30 no 7 pp801ndash822 1984

[27] P A Bottomley and J R Doyle ldquoA comparison of three weightelicitation methods good better and bestrdquo Omega vol 29 no6 pp 553ndash560 2001

[28] J Figueira and B Roy ldquoDetermining the weights of criteria inthe ELECTRE type methods with a revised Simosrsquo procedurerdquoEuropean Journal ofOperational Research vol 139 no 2 pp 317ndash326 2002

[29] L E Shirland R R Jesse R L Thompson and C L IacovouldquoDetermining attribute weights using mathematical program-mingrdquo Omega vol 31 no 6 pp 423ndash437 2003

[30] Y-M Wang and Y Luo ldquoIntegration of correlations withstandard deviations for determining attribute weights in mul-tiple attribute decision makingrdquo Mathematical amp ComputerModelling vol 51 no 1-2 pp 1ndash12 2010

[31] J Ma Z-P Fan and L-H Huang ldquoA subjective and objectiveintegrated approach to determine attribute weightsrdquo EuropeanJournal of Operational Research vol 112 no 2 pp 397ndash4041999

[32] X Xu ldquoA note on the subjective and objective integratedapproach to determine attribute weightsrdquo European Journal ofOperational Research vol 156 no 2 pp 530ndash532 2004

[33] P J H Schoemaker and C C Waid ldquoAn experimental compar-ison of different approaches to determining weights in additiveutility modelsrdquoManagement Science vol 28 no 2 pp 182ndash1961982

[34] Y-M Wang and C Parkan ldquoA general multiple attributedecision-making approach for integrating subjective prefer-ences and objective informationrdquo Fuzzy Sets and Systems vol157 no 10 pp 1333ndash1345 2006

[35] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[36] J R Doyle ldquoMultiattribute choice for the lazy decision makerlet the alternatives deciderdquoOrganizational Behavior andHumanDecision Processes vol 62 no 1 pp 87ndash100 1995

[37] E Bernroider and V Stix ldquoA method using weight restrictionsin data envelopment analysis for ranking and validity issues indecisionmakingrdquoComputersampOperations Research vol 34 no9 pp 2637ndash2647 2007

[38] J Wu and L Liang ldquoA multiple criteria ranking method basedon game cross-evaluation approachrdquo Annals of OperationsResearch vol 197 no 1 pp 191ndash200 2012

[39] C Fu and K-S Chin ldquoRobust evidential reasoning approachwith unknown attribute weightsrdquo Knowledge-Based Systemsvol 59 no 2 pp 9ndash20 2014

[40] B S Ahn ldquoExtreme point-based multi-attribute decision anal-ysis with incomplete informationrdquo European Journal of Opera-tional Research vol 240 no 3 pp 748ndash755 2015

[41] WW Cooper J L Ruiz and I Sirvent ldquoChoosing weights fromalternative optimal solutions of dualmultipliermodels inDEArdquoEuropean Journal of Operational Research vol 180 no 1 pp443ndash458 2007

[42] G R Jahanshahloo and P F Shahmirzadi ldquoNew methods forranking decision making units based on the dispersion ofweights and Norm 1 in Data Envelopment Analysisrdquo Computersamp Industrial Engineering vol 65 no 2 pp 187ndash193 2013

[43] V E Krivonozhko O B Utkin M M Safin and A V LychevldquoOn some generalization of the DEA modelsrdquo Journal of theOperational Research Society vol 60 no 11 pp 1518ndash1527 2009

[44] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[45] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo New Directions for ProgramEvaluation vol 1986 no 32 pp 73ndash105 1986

[46] A Boussofiane R G Dyson and E Thanassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[47] K Wang and F Wei ldquoGame cross efficiency DEA based projectreview of equipment designrdquo Journal of Beijing University ofAeronautics and Astronautics vol 35 no 10 pp 1278ndash12822009

[48] T Wu and J Blackhurst ldquoSupplier evaluation and selection anaugmented DEA approachrdquo International Journal of ProductionResearch vol 47 no 16 pp 4593ndash4608 2009

[49] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Korean stockmarketrdquo European Journal of Operational Research vol 236 no1 pp 361ndash368 2014

[50] B Roy ldquoTheoutranking approach and the foundations of electremethodsrdquoTheory and Decision vol 31 no 1 pp 49ndash73 1991

[51] G F Reed F Lynn and B D Meade ldquoUse of coefficient ofvariation in assessing variability of quantitative assaysrdquo Clinicalamp Diagnostic Laboratory Immunology vol 9 no 6 pp 1235ndash1239 2002

[52] 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

[53] H M Moshkovich A I Mechitov and D L Olson ldquoOrdinaljudgments in multiattribute decision analysisrdquo European Jour-nal of Operational Research vol 137 no 3 pp 625ndash641 2002

[54] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Tsinghua Press Beijing China 2005

[55] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Step 4 (evaluation of the alternatives via coefficient ofvariation) Since the mean and deviation measures of eachalternativersquos cross closeness index are obtained in Step 3 thecoefficient of variation defined in (17) is utilized to integratethe results

Step 5 (generation of rank order of the alternatives) Throughcomparing the coefficients of variation among the alterna-tives the rank order can be directly obtained

4 Extension with Uncertain Attribute Values

41 Model with Interval Attribute Value It is widely acknowl-edged that uncertainty is associated with attribute values dueto complexity of real world and limited knowledge of humanbeings In many situations the attribute value 119903

119894119895is uncertain

but we know that the feasible attribute value 119903119894119895

isin [119903minus

119894119895 119903+

119894119895]

Here we present the following self-evaluation model withinterval attribute values [52] and uncertain attribute weights

max 120572

radicsum119899

119895=1(119908119895119903+

119894119895)2

radicsum119899

119895=1(119908119895119903+

119894119895)2

+ radicsum119899

119895=1(119908119895119903+

119894119895minus 119908119895)2

+ 120573

radicsum119899

119895=1(119908119895119903minus

119894119895)2

radicsum119899

119895=1(119908119895119903minus

119894119895)2

+ radicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2

= 119862max119894

119908 isin Ω (119908)

120572 + 120573 = 1 120572 ge 0 120573 ge 0

(18)

in which radicsum119899

119895=1(119908119895119903+

119894119895)2 radicsum

119899

119895=1(119908119895119903+

119894119895minus 119908119895)2 radicsum

119899

119895=1(119908119895119903minus

119894119895)2

andradicsum119899

119895=1(119908119895119903minus

119894119895minus 119908119895)2 represent distance of upper bounds

from ideal point distance of upper bounds from negativeideal point distance of lower bounds from ideal pointand distance of lower bounds from negative ideal pointrespectively 120572 and 120573 reflect the risk coefficients that combineupper closeness index and lower closeness respectively Inpractice120572 and120573 are decided by the decisionmaker accordingto his or her risk preference Specifically the decision makeris neutral if 120572 = 120573 = 12 120572 gt 120573 and 120572 lt 120573 indicate that he isoptimistic and pessimistic respectively

42 Model with Ordinal Attribute Value Another form ofuncertain preferential information is expressed by ordinalattribute value (eg ldquomore preferablerdquo and ldquoless preferablerdquo)It is argued to be more stable and more reliable than cardinalvalue in some situations [53] In this paper an approach ofthreshold value is introduced to transform ordinal value intolinear formulation For instance given four attributes 119886 119887 119888and 119889 the ordinal values are 1 2 3 and 4 respectively Byspecifying a threshold value 01 the attribute values can beconverted to a cardinal attribute space Ω(119886 119887 119888 119889) = 119886 = 1

119886 minus 119887 ge 01 119887 minus 119888 ge 01 119888 minus 119889 ge 01 119889 = 0 In generallet the ordinal attribute values 119886

119894119895(119894 = 1 119898) have relation

1198861198941119895gt 1198861198942119895gt sdot sdot sdot gt 119886

119894119898119895(119894119896isin [1119898] 119896 = 1 119898) and then

the cardinal attribute spaceΩ(119903119894119895) = 119886

119894119898119895= 1 119886119894119898119895minus119886119894119898minus1119895ge 120576

119886119894119898minus1119895minus119886119894119898minus2119895ge 120576 119886

1198942119895minus1198861198941119895ge 120576 1198861198941119895= 0 120576 gt 0 is obtained

(120576 is the threshold value given by the decision maker) Herewe present the following self-evaluation model with ordinalattribute values and uncertain attribute weights

maxradicsum119899

119895=1(119908119895119903119894119895)2

radicsum119899

119895=1(119908119895119903119894119895)2

+ radicsum119899

119895=1(119908119895119903119894119895minus 119908119895)2

= 119862max119894

st 119903 isin Ω (119903)

119908 isin Ω (119908)

(19)

119894 = 1 119898 (20)

119895 = 1 119899 (21)

For each alternative 119883119896(119896 = 1 119898) under evaluation

we obtain a group of optimal attribute values and a set ofattribute weights They are denoted by 119860

lowast

119896= (119903119896

119894119895)119898times119899

and119882lowast

119896= (119908

119896

1 119908119896

2 119908

119896

119899) respectively When the optimal

decision information is applied to evaluate alternative119883119897(119897 =

1 119898) a cross score denoted by 119862119896119897

= radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2

(radicsum119899

119895=1(119908119896

119895119903119896

119897119895)2+ radicsum

119899

119895=1(119908119896

119895119903119896

119897119895minus 119908119896

119895)2) is obtained

Similar to that defined by (14) and (15) the mean anddeviation measures of alternative 119883

119897are denoted by 119903(119883

119897) =

sum119898

119896=1119862119896119897and 120575(119883

119897) = radicsum

119898

119896=1(119862119896119897minus 119903(119883

119897))2119899 respectively

Note that (12) (18) and (19) are nonlinear models theycan be transformed into linear forms according to practicalcases

5 Illustrative Examples

In order to illustrate the applications of our proposed meth-ods three practical MADM problems are presented

51 Application to House Selection with Cardinal AttributeValues Xu [54] stated a purchase house problem there arefour possible houses to be selected and the attributes are asfollows (1)119884

1is the house price (ten thousand yuan) (2)119884

2is

the usable area (square meter) (3) 1198843is the distance between

house and the center of city (kilometers) (4) 1198844is the nature

environment (scores) where 1198842and 119884

4are benefit attributes

and 1198841and 119884

3are cost attributes The attribute weights are

completely unknown The decision making matrix 119860 andnormalized decision making matrix 119877 are listed in Tables 1and 2 respectively Step 1 is completed

By utilizing (9) and (10) the upper boundof each attributeweight is obtained And as a result the attribute weight spaceis Ω(119908) = 0 lt 119908

1lt 077 0 lt 119908

2lt 078 0 lt 119908

3lt 062

0 lt 1199084lt 062 119908

1+ 1199082+ 1199083+ 1199084= 1 Step 2 is completed

Mathematical Problems in Engineering 7

Table 1 Decision making matrix 119860 of house evaluation

1198841

1198842

1198843

1198844

1198831

30 100 10 71198832

25 80 8 51198833

18 50 20 111198834

22 70 12 9

Table 2 Normalized decisionmakingmatrix119877 of house evaluation

1198841

1198842

1198843

1198844

1198831

06 10 08 06361198832

072 08 10 04551198833

10 05 04 101198834

0818 07 0667 0818

Table 3 The cross-evaluation measurements

Alternatives 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

119903 (119883119894) 120575 (119883

119894)

1198831

096 096 082 061 061 075 0151198832

092 080 092 066 066 076 0111198833

1 052 051 1 1 076 0241198834

0818 070 070 0818 0818 076 006

Table 4 Alternative efficiency matrix

Alternatives 1198831

1198832

1198833

1198834

CV (119883119894) 020 014 032 008

In terms of Step 3 in Section 33 we obtain the cross-evaluation measurements of each alternative which arepresented in Table 3 Step 3 is completed

Based on Table 3 and the calculation of coefficient ofvariation the results are shown in Table 4

Therefore we can rank the houses in accordance with theCV values presented in Table 4 to get a complete rank order1198834

≻ 1198832

≻ 1198831

≻ 1198833 And 119883

4is the best house under

evaluation Step 4 is completedIn order to understand our approach more clearly we

compare our approachwith other three similarmethods [54]namely average attribute value method maximizing devia-tion method and information entropy method respectivelywith the same example The corresponding results of suchthree methods are listed in Table 5

According to Tables 4 and 5 the result derived from ourcross-evaluation approach is quite different from that of theexisting three weight determination methods In fact Table 5shows that the three weight determination methods generatethree attribute weight vectors and produce three differentoptimal solutions However there is no single method thatcan guarantee more convincing weight vector than othersHence we treat the attribute weight as a weight space rather

Table 5 The results of three methods

Method Weight vector Rank order

Average attributevalue

(025 025 025 025) 1198831≻ 1198834≻ 1198832≻ 1198833

Maximizingdeviation

(020 024 029 027) 1198831≻ 1198832≻ 1198834≻ 1198833

Informationentropy

(013 022 036 029) 1198831≻ 1198832≻ 1198834≻ 1198833

Table 6 Attribute weights

1198841

1198842

1198843

[119908119871

1 119908119880

1] [119908

119871

2 119908119880

2] [119908

119871

3 119908119880

3]

[035 050] [015 030] [030 035]

Table 7 Attribute values of alternatives

1198841

1198842

1198843

119860 [065 085] [068 086] [045 078]119861 [075 1] [055 078] [064 080]119862 [054 068] [046 055] [078 1]119863 [077 090] [067 078] [056 077]119864 [025 045] [089 1] [075 094]

than a certain value in this paperThe comparison result vali-dates that our proposed approach can overcome the shortagethat we cannot obtain themost dominant result caused by thediversity of the way of weight vector determined

52 Application to University Faculty with Interval AttributeValue Consider the example in Bryson and Mobolurinrsquos[55] study that is five faculty candidates (119860 119861 119862 119863119864) are evaluated for tenure and promotion through threeattributes teaching research and service According to theDMrsquos preference the information for both attribute andalternatives is given by interval numbers as shown in Tables6 and 7 respectively (for the purpose of illustration our datais different from the literature)

Table 8 shows the computed results of the proposedapproach with interval values The self-evaluation score ofeach alternative is listed in the second column and the cross-evaluation scores are listed in columns 3ndash7 Columns 8 and9 show the mean and deviation measures respectively Thenthe results in column 10 are applied to generate a rank orderof the five candidates119860 ≻ 119862 ≻ 119864 ≻ 119861 ≻ 119863 which is a solutionto the problem

53 Application to the Selection of Community Service Com-panies with Ordinal Attribute Values In this subsection weapply our approach to the evaluation and selection of fivecommunity service companies that are located in Hefei acity in Anhui province of China These companies providevarious kinds of service for the public such as property

8 Mathematical Problems in Engineering

Table 8 Evaluation results for the comparison of faculty candidates

Candidates 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

1198625119894

119903119894

120575119894

CV(119883119894)

119860 0717 0717 0712 0706 0717 0703 0711 0006 0008119861 0792 0792 0792 0788 0792 0746 0782 0018 0023119862 0672 0651 0661 0672 0651 0660 0659 0008 0012119863 0774 0774 0771 0764 0774 0737 0764 0014 0461119864 0645 0525 0526 0540 0525 0645 0552 0047 0018

Table 9 Ordinal values of alternatives

1198841

1198842

1198843

1198844

1198831

4 2 4 21198832

1 4 3 31198833

2 5 1 11198834

5 3 2 51198835

3 1 5 4

management catering health care and entertainment Inthe case study the decision maker is an official from thedevelopment and reform commission of Hefei with thehelp of a group of experts from the governments relevantenterprises and community organizations Four attributesare used to evaluate the six companies They are quality ofservice (119884

1) diversity of service (119884

2) profitability (119884

3) and

development potential (1198844) The original attribute values are

given by ordinal values as shown in Table 9 The decisionprocedures are as follows

After considering the four attributes the decision makerviews that quality of service (119884

1) is the most important

attribute in evaluation The development potential (1198844) is the

second most important one profitability (1198843) is the third

most important one and diversity of service (1198842) is the

least important one respectively And the difference betweentwo adjacent attribute weights should be more than 01 Inaddition each weight value should be nomore than 05Thenthe weight space is denoted by Ω(119908) = 119908

1minus 01 gt 119908

4

1199084minus 01 gt 119908

3 1199083minus 01 gt 119908

2 1199081le 05 119908

2le 05 119908

3le 05

1199084le 05 119908

1+ 1199082+ 1199083+ 1199084= 1

By specifying a threshold value of 01 ordinal attributevalues are transformed into four sets of attribute value spacesThey are denoted by

Ω(1198841) = 119903

21= 1 11990321

minus 11990331

ge 01 11990331

minus 11990351

ge 01 11990351

minus 11990311

ge 01 11990311

minus 11990341

ge 01 11990341

= 0

Ω (1198842) = 119903

52= 1 11990352

minus 11990312

ge 01 11990312

minus 11990342

ge 01 11990342

minus 11990322

ge 01 11990322

minus 11990332

ge 01 11990332

= 0

Ω (1198843) = 119903

33= 1 11990333

minus 11990343

ge 01 11990343

minus 11990323

ge 01 11990323

minus 11990313

ge 01 11990313

minus 11990353

ge 01 11990353

= 0

Ω (1198844) = 119903

34= 1 11990334

minus 11990314

ge 01 11990314

minus 11990324

ge 01 11990324

minus 11990354

ge 01 11990354

minus 11990344

ge 01 11990344

= 0

(22)

Table 10 shows the computed results of the proposedapproachThe second column shows the self-evaluation scoreof each alternative computed by (19) And columns 3ndash7 listthe cross-evaluation scores Then the mean and deviationmeasures of each alternative are presented in columns 8 and9 respectively Consequently the results in column 10 areapplied to generate a rank order of the five community servicecompanies 119883

5≻ 1198831≻ 1198832≻ 1198834≻ 1198833 The decision maker

is pleased to select company 1198835with the best performance

among the five alternatives

6 Conclusions and Future Research

Due to the complexity and uncertainty of MADM problemsin real world the decision maker is not able to provide theexact attribute weights And different weight determinationmethods are difficult to generate a most satisfaction solutionThe current paper presents a new MADM method basedon cross-evaluation in which the weight information isexpressed by the so-called weight space It attempts to rectifysome shortcomings of existingMADMmethods by providingthe following important features (1) it is more feasibleto provide a weight space rather than specifying a set ofweight values (2) the single dominating attribute drawbackcan be avoided by the restriction of the upper bound ofeach attribute (3) compared with DEA based methods thevariable attribute weights are consistent with the opinionof decision maker (4) both mean score and the deviationscore are considered in our cross-evaluation models and (5)the proposed method is also successfully extended to moregeneralized situations in which attribute values are expressedby interval values and ordinal values respectively

We have illustrated the applicability of the proposedmethods by revisiting two reported MADM studies and bydepicting a case study on the selection of community servicecompanies Although our proposed method makes someimprovement in solving MADM problems it only can dealwith the individual decision making case Future research

Mathematical Problems in Engineering 9

Table 10 Evaluation results for the selection of community service companies

Candidates 119862max119897

1198621119897

1198622119897

1198623119897

1198624119897

1198625119897

119903119894

120575119894

CV(119883119894)

1198831

0489 0489 0317 0295 0317 0317 0347 0072 02071198832

0532 0433 0532 0508 0458 0504 0487 0036 00741198833

0535 0526 0526 0535 0526 0526 0528 0004 00081198834

0223 0202 0223 0211 0223 0223 0216 0009 00421198835

0422 0349 0349 0103 0349 0422 0314 0109 0347

may attempt to employ our cross-evaluation methods tohandle group decision making problem

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research is supported by National Key Technology Sup-port Program (no 2014BAH27F00) National Natural Sci-ence Funds of China (no 71471053) and the FundamentalResearch Fund for the Central Universities (no JZ2014-HGBZ0339)

References

[1] R H Green and J R Doyle ldquoOn maximizing discriminationin multiple criteria decision makingrdquo Journal of the OperationalResearch Society vol 46 no 2 pp 192ndash204 1995

[2] R E Steuer and P Na ldquoMultiple criteria decision makingcombined with finance a categorized bibliographic studyrdquoEuropean Journal of Operational Research vol 150 no 3 pp496ndash515 2003

[3] J Wallenius J S Dyer P C Fishburn R E Steuer S Ziontsand K Deb ldquoMultiple criteria decision making multi-attributeutility theory recent accomplishments and what lies aheadrdquoManagement Science vol 54 no 7 pp 1336ndash1349 2008

[4] W Ho X Xu and P K Dey ldquoMulti-criteria decision makingapproaches for supplier evaluation and selection a literaturereviewrdquo European Journal of Operational Research vol 202 no1 pp 16ndash24 2010

[5] C W Churchman R L Ackoff and N M Smith ldquoAn approx-imate measure of valuerdquo Journal of the Operational ResearchSociety of America vol 2 no 2 pp 172ndash187 1954

[6] T L Saaty ldquoThe analytic hierarchy processrdquo in Proceedings ofthe Second International Seminar on Operational Research in theBasque Provinces vol 4 pp 189ndash234 1980

[7] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[8] H Deng C-H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[9] P K Dey ldquoIntegrated project evaluation and selection usingmultiple-attribute decision-making techniquerdquo InternationalJournal of Production Economics vol 103 no 1 pp 90ndash1032006

[10] G-D Li D Yamaguchi and M Nagai ldquoA grey-based decision-making approach to the supplier selection problemrdquoMathemat-ical amp Computer Modelling vol 46 no 3-4 pp 573ndash581 2007

[11] A Shanian and O Savadogo ldquoTOPSIS multiple-criteria deci-sion support analysis for material selection of metallic bipolarplates for polymer electrolyte fuel cellrdquo Journal of Power Sourcesvol 159 no 2 pp 1095ndash1104 2006

[12] G-H Tzeng Y-P O Yang C-T Lin and C-B Chen ldquoHier-archical MADM with fuzzy integral for evaluating enterpriseintranet web sitesrdquo Information Sciences vol 169 no 3-4 pp409ndash426 2005

[13] 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

[14] A Kelemenis and D Askounis ldquoA new TOPSIS-based multi-criteria approach to personnel selectionrdquo Expert Systems withApplications vol 37 no 7 pp 4999ndash5008 2010

[15] B Bairagi B Dey B Sarkar and S K Sanyal ldquoADeNovomulti-approaches multi-criteria decision making technique with anapplication in performance evaluation of material handlingdevicerdquo Computers amp Industrial Engineering vol 87 pp 267ndash282 2015

[16] J S H Kornbluth ldquoDynamic multi-criteria decision makingrdquoJournal of Multi-Criteria Decision Analysis vol 1 no 2 pp 81ndash92 1992

[17] Y-H Lin P-C Lee T-P Chang and H-I Ting ldquoMulti-attribute group decision making model under the condition ofuncertain informationrdquoAutomation in Construction vol 17 no6 pp 792ndash797 2008

[18] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multicrite-ria acceptability analysis for groupdecisionmakingrdquoOperationsResearch vol 49 no 3 pp 444ndash454 2001

[19] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[20] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[21] S Miyamoto ldquoMultisets and fuzzymultisetsrdquo in Soft Computingand Human-Centered Machines Z Q Liu and S MiyamotoEds pp 9ndash33 Springer Berlin Germany 2000

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[23] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[25] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling Simulation and Computation for Engineering andEnvironmental Systems vol 34 no 7 pp 1779ndash1787 2010

10 Mathematical Problems in Engineering

[26] D Horsky andM R Rao ldquoEstimation of attribute weights frompreference comparisonsrdquoManagement Science vol 30 no 7 pp801ndash822 1984

[27] P A Bottomley and J R Doyle ldquoA comparison of three weightelicitation methods good better and bestrdquo Omega vol 29 no6 pp 553ndash560 2001

[28] J Figueira and B Roy ldquoDetermining the weights of criteria inthe ELECTRE type methods with a revised Simosrsquo procedurerdquoEuropean Journal ofOperational Research vol 139 no 2 pp 317ndash326 2002

[29] L E Shirland R R Jesse R L Thompson and C L IacovouldquoDetermining attribute weights using mathematical program-mingrdquo Omega vol 31 no 6 pp 423ndash437 2003

[30] Y-M Wang and Y Luo ldquoIntegration of correlations withstandard deviations for determining attribute weights in mul-tiple attribute decision makingrdquo Mathematical amp ComputerModelling vol 51 no 1-2 pp 1ndash12 2010

[31] J Ma Z-P Fan and L-H Huang ldquoA subjective and objectiveintegrated approach to determine attribute weightsrdquo EuropeanJournal of Operational Research vol 112 no 2 pp 397ndash4041999

[32] X Xu ldquoA note on the subjective and objective integratedapproach to determine attribute weightsrdquo European Journal ofOperational Research vol 156 no 2 pp 530ndash532 2004

[33] P J H Schoemaker and C C Waid ldquoAn experimental compar-ison of different approaches to determining weights in additiveutility modelsrdquoManagement Science vol 28 no 2 pp 182ndash1961982

[34] Y-M Wang and C Parkan ldquoA general multiple attributedecision-making approach for integrating subjective prefer-ences and objective informationrdquo Fuzzy Sets and Systems vol157 no 10 pp 1333ndash1345 2006

[35] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[36] J R Doyle ldquoMultiattribute choice for the lazy decision makerlet the alternatives deciderdquoOrganizational Behavior andHumanDecision Processes vol 62 no 1 pp 87ndash100 1995

[37] E Bernroider and V Stix ldquoA method using weight restrictionsin data envelopment analysis for ranking and validity issues indecisionmakingrdquoComputersampOperations Research vol 34 no9 pp 2637ndash2647 2007

[38] J Wu and L Liang ldquoA multiple criteria ranking method basedon game cross-evaluation approachrdquo Annals of OperationsResearch vol 197 no 1 pp 191ndash200 2012

[39] C Fu and K-S Chin ldquoRobust evidential reasoning approachwith unknown attribute weightsrdquo Knowledge-Based Systemsvol 59 no 2 pp 9ndash20 2014

[40] B S Ahn ldquoExtreme point-based multi-attribute decision anal-ysis with incomplete informationrdquo European Journal of Opera-tional Research vol 240 no 3 pp 748ndash755 2015

[41] WW Cooper J L Ruiz and I Sirvent ldquoChoosing weights fromalternative optimal solutions of dualmultipliermodels inDEArdquoEuropean Journal of Operational Research vol 180 no 1 pp443ndash458 2007

[42] G R Jahanshahloo and P F Shahmirzadi ldquoNew methods forranking decision making units based on the dispersion ofweights and Norm 1 in Data Envelopment Analysisrdquo Computersamp Industrial Engineering vol 65 no 2 pp 187ndash193 2013

[43] V E Krivonozhko O B Utkin M M Safin and A V LychevldquoOn some generalization of the DEA modelsrdquo Journal of theOperational Research Society vol 60 no 11 pp 1518ndash1527 2009

[44] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[45] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo New Directions for ProgramEvaluation vol 1986 no 32 pp 73ndash105 1986

[46] A Boussofiane R G Dyson and E Thanassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[47] K Wang and F Wei ldquoGame cross efficiency DEA based projectreview of equipment designrdquo Journal of Beijing University ofAeronautics and Astronautics vol 35 no 10 pp 1278ndash12822009

[48] T Wu and J Blackhurst ldquoSupplier evaluation and selection anaugmented DEA approachrdquo International Journal of ProductionResearch vol 47 no 16 pp 4593ndash4608 2009

[49] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Korean stockmarketrdquo European Journal of Operational Research vol 236 no1 pp 361ndash368 2014

[50] B Roy ldquoTheoutranking approach and the foundations of electremethodsrdquoTheory and Decision vol 31 no 1 pp 49ndash73 1991

[51] G F Reed F Lynn and B D Meade ldquoUse of coefficient ofvariation in assessing variability of quantitative assaysrdquo Clinicalamp Diagnostic Laboratory Immunology vol 9 no 6 pp 1235ndash1239 2002

[52] 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

[53] H M Moshkovich A I Mechitov and D L Olson ldquoOrdinaljudgments in multiattribute decision analysisrdquo European Jour-nal of Operational Research vol 137 no 3 pp 625ndash641 2002

[54] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Tsinghua Press Beijing China 2005

[55] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 Decision making matrix 119860 of house evaluation

1198841

1198842

1198843

1198844

1198831

30 100 10 71198832

25 80 8 51198833

18 50 20 111198834

22 70 12 9

Table 2 Normalized decisionmakingmatrix119877 of house evaluation

1198841

1198842

1198843

1198844

1198831

06 10 08 06361198832

072 08 10 04551198833

10 05 04 101198834

0818 07 0667 0818

Table 3 The cross-evaluation measurements

Alternatives 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

119903 (119883119894) 120575 (119883

119894)

1198831

096 096 082 061 061 075 0151198832

092 080 092 066 066 076 0111198833

1 052 051 1 1 076 0241198834

0818 070 070 0818 0818 076 006

Table 4 Alternative efficiency matrix

Alternatives 1198831

1198832

1198833

1198834

CV (119883119894) 020 014 032 008

In terms of Step 3 in Section 33 we obtain the cross-evaluation measurements of each alternative which arepresented in Table 3 Step 3 is completed

Based on Table 3 and the calculation of coefficient ofvariation the results are shown in Table 4

Therefore we can rank the houses in accordance with theCV values presented in Table 4 to get a complete rank order1198834

≻ 1198832

≻ 1198831

≻ 1198833 And 119883

4is the best house under

evaluation Step 4 is completedIn order to understand our approach more clearly we

compare our approachwith other three similarmethods [54]namely average attribute value method maximizing devia-tion method and information entropy method respectivelywith the same example The corresponding results of suchthree methods are listed in Table 5

According to Tables 4 and 5 the result derived from ourcross-evaluation approach is quite different from that of theexisting three weight determination methods In fact Table 5shows that the three weight determination methods generatethree attribute weight vectors and produce three differentoptimal solutions However there is no single method thatcan guarantee more convincing weight vector than othersHence we treat the attribute weight as a weight space rather

Table 5 The results of three methods

Method Weight vector Rank order

Average attributevalue

(025 025 025 025) 1198831≻ 1198834≻ 1198832≻ 1198833

Maximizingdeviation

(020 024 029 027) 1198831≻ 1198832≻ 1198834≻ 1198833

Informationentropy

(013 022 036 029) 1198831≻ 1198832≻ 1198834≻ 1198833

Table 6 Attribute weights

1198841

1198842

1198843

[119908119871

1 119908119880

1] [119908

119871

2 119908119880

2] [119908

119871

3 119908119880

3]

[035 050] [015 030] [030 035]

Table 7 Attribute values of alternatives

1198841

1198842

1198843

119860 [065 085] [068 086] [045 078]119861 [075 1] [055 078] [064 080]119862 [054 068] [046 055] [078 1]119863 [077 090] [067 078] [056 077]119864 [025 045] [089 1] [075 094]

than a certain value in this paperThe comparison result vali-dates that our proposed approach can overcome the shortagethat we cannot obtain themost dominant result caused by thediversity of the way of weight vector determined

52 Application to University Faculty with Interval AttributeValue Consider the example in Bryson and Mobolurinrsquos[55] study that is five faculty candidates (119860 119861 119862 119863119864) are evaluated for tenure and promotion through threeattributes teaching research and service According to theDMrsquos preference the information for both attribute andalternatives is given by interval numbers as shown in Tables6 and 7 respectively (for the purpose of illustration our datais different from the literature)

Table 8 shows the computed results of the proposedapproach with interval values The self-evaluation score ofeach alternative is listed in the second column and the cross-evaluation scores are listed in columns 3ndash7 Columns 8 and9 show the mean and deviation measures respectively Thenthe results in column 10 are applied to generate a rank orderof the five candidates119860 ≻ 119862 ≻ 119864 ≻ 119861 ≻ 119863 which is a solutionto the problem

53 Application to the Selection of Community Service Com-panies with Ordinal Attribute Values In this subsection weapply our approach to the evaluation and selection of fivecommunity service companies that are located in Hefei acity in Anhui province of China These companies providevarious kinds of service for the public such as property

8 Mathematical Problems in Engineering

Table 8 Evaluation results for the comparison of faculty candidates

Candidates 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

1198625119894

119903119894

120575119894

CV(119883119894)

119860 0717 0717 0712 0706 0717 0703 0711 0006 0008119861 0792 0792 0792 0788 0792 0746 0782 0018 0023119862 0672 0651 0661 0672 0651 0660 0659 0008 0012119863 0774 0774 0771 0764 0774 0737 0764 0014 0461119864 0645 0525 0526 0540 0525 0645 0552 0047 0018

Table 9 Ordinal values of alternatives

1198841

1198842

1198843

1198844

1198831

4 2 4 21198832

1 4 3 31198833

2 5 1 11198834

5 3 2 51198835

3 1 5 4

management catering health care and entertainment Inthe case study the decision maker is an official from thedevelopment and reform commission of Hefei with thehelp of a group of experts from the governments relevantenterprises and community organizations Four attributesare used to evaluate the six companies They are quality ofservice (119884

1) diversity of service (119884

2) profitability (119884

3) and

development potential (1198844) The original attribute values are

given by ordinal values as shown in Table 9 The decisionprocedures are as follows

After considering the four attributes the decision makerviews that quality of service (119884

1) is the most important

attribute in evaluation The development potential (1198844) is the

second most important one profitability (1198843) is the third

most important one and diversity of service (1198842) is the

least important one respectively And the difference betweentwo adjacent attribute weights should be more than 01 Inaddition each weight value should be nomore than 05Thenthe weight space is denoted by Ω(119908) = 119908

1minus 01 gt 119908

4

1199084minus 01 gt 119908

3 1199083minus 01 gt 119908

2 1199081le 05 119908

2le 05 119908

3le 05

1199084le 05 119908

1+ 1199082+ 1199083+ 1199084= 1

By specifying a threshold value of 01 ordinal attributevalues are transformed into four sets of attribute value spacesThey are denoted by

Ω(1198841) = 119903

21= 1 11990321

minus 11990331

ge 01 11990331

minus 11990351

ge 01 11990351

minus 11990311

ge 01 11990311

minus 11990341

ge 01 11990341

= 0

Ω (1198842) = 119903

52= 1 11990352

minus 11990312

ge 01 11990312

minus 11990342

ge 01 11990342

minus 11990322

ge 01 11990322

minus 11990332

ge 01 11990332

= 0

Ω (1198843) = 119903

33= 1 11990333

minus 11990343

ge 01 11990343

minus 11990323

ge 01 11990323

minus 11990313

ge 01 11990313

minus 11990353

ge 01 11990353

= 0

Ω (1198844) = 119903

34= 1 11990334

minus 11990314

ge 01 11990314

minus 11990324

ge 01 11990324

minus 11990354

ge 01 11990354

minus 11990344

ge 01 11990344

= 0

(22)

Table 10 shows the computed results of the proposedapproachThe second column shows the self-evaluation scoreof each alternative computed by (19) And columns 3ndash7 listthe cross-evaluation scores Then the mean and deviationmeasures of each alternative are presented in columns 8 and9 respectively Consequently the results in column 10 areapplied to generate a rank order of the five community servicecompanies 119883

5≻ 1198831≻ 1198832≻ 1198834≻ 1198833 The decision maker

is pleased to select company 1198835with the best performance

among the five alternatives

6 Conclusions and Future Research

Due to the complexity and uncertainty of MADM problemsin real world the decision maker is not able to provide theexact attribute weights And different weight determinationmethods are difficult to generate a most satisfaction solutionThe current paper presents a new MADM method basedon cross-evaluation in which the weight information isexpressed by the so-called weight space It attempts to rectifysome shortcomings of existingMADMmethods by providingthe following important features (1) it is more feasibleto provide a weight space rather than specifying a set ofweight values (2) the single dominating attribute drawbackcan be avoided by the restriction of the upper bound ofeach attribute (3) compared with DEA based methods thevariable attribute weights are consistent with the opinionof decision maker (4) both mean score and the deviationscore are considered in our cross-evaluation models and (5)the proposed method is also successfully extended to moregeneralized situations in which attribute values are expressedby interval values and ordinal values respectively

We have illustrated the applicability of the proposedmethods by revisiting two reported MADM studies and bydepicting a case study on the selection of community servicecompanies Although our proposed method makes someimprovement in solving MADM problems it only can dealwith the individual decision making case Future research

Mathematical Problems in Engineering 9

Table 10 Evaluation results for the selection of community service companies

Candidates 119862max119897

1198621119897

1198622119897

1198623119897

1198624119897

1198625119897

119903119894

120575119894

CV(119883119894)

1198831

0489 0489 0317 0295 0317 0317 0347 0072 02071198832

0532 0433 0532 0508 0458 0504 0487 0036 00741198833

0535 0526 0526 0535 0526 0526 0528 0004 00081198834

0223 0202 0223 0211 0223 0223 0216 0009 00421198835

0422 0349 0349 0103 0349 0422 0314 0109 0347

may attempt to employ our cross-evaluation methods tohandle group decision making problem

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research is supported by National Key Technology Sup-port Program (no 2014BAH27F00) National Natural Sci-ence Funds of China (no 71471053) and the FundamentalResearch Fund for the Central Universities (no JZ2014-HGBZ0339)

References

[1] R H Green and J R Doyle ldquoOn maximizing discriminationin multiple criteria decision makingrdquo Journal of the OperationalResearch Society vol 46 no 2 pp 192ndash204 1995

[2] R E Steuer and P Na ldquoMultiple criteria decision makingcombined with finance a categorized bibliographic studyrdquoEuropean Journal of Operational Research vol 150 no 3 pp496ndash515 2003

[3] J Wallenius J S Dyer P C Fishburn R E Steuer S Ziontsand K Deb ldquoMultiple criteria decision making multi-attributeutility theory recent accomplishments and what lies aheadrdquoManagement Science vol 54 no 7 pp 1336ndash1349 2008

[4] W Ho X Xu and P K Dey ldquoMulti-criteria decision makingapproaches for supplier evaluation and selection a literaturereviewrdquo European Journal of Operational Research vol 202 no1 pp 16ndash24 2010

[5] C W Churchman R L Ackoff and N M Smith ldquoAn approx-imate measure of valuerdquo Journal of the Operational ResearchSociety of America vol 2 no 2 pp 172ndash187 1954

[6] T L Saaty ldquoThe analytic hierarchy processrdquo in Proceedings ofthe Second International Seminar on Operational Research in theBasque Provinces vol 4 pp 189ndash234 1980

[7] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[8] H Deng C-H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[9] P K Dey ldquoIntegrated project evaluation and selection usingmultiple-attribute decision-making techniquerdquo InternationalJournal of Production Economics vol 103 no 1 pp 90ndash1032006

[10] G-D Li D Yamaguchi and M Nagai ldquoA grey-based decision-making approach to the supplier selection problemrdquoMathemat-ical amp Computer Modelling vol 46 no 3-4 pp 573ndash581 2007

[11] A Shanian and O Savadogo ldquoTOPSIS multiple-criteria deci-sion support analysis for material selection of metallic bipolarplates for polymer electrolyte fuel cellrdquo Journal of Power Sourcesvol 159 no 2 pp 1095ndash1104 2006

[12] G-H Tzeng Y-P O Yang C-T Lin and C-B Chen ldquoHier-archical MADM with fuzzy integral for evaluating enterpriseintranet web sitesrdquo Information Sciences vol 169 no 3-4 pp409ndash426 2005

[13] 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

[14] A Kelemenis and D Askounis ldquoA new TOPSIS-based multi-criteria approach to personnel selectionrdquo Expert Systems withApplications vol 37 no 7 pp 4999ndash5008 2010

[15] B Bairagi B Dey B Sarkar and S K Sanyal ldquoADeNovomulti-approaches multi-criteria decision making technique with anapplication in performance evaluation of material handlingdevicerdquo Computers amp Industrial Engineering vol 87 pp 267ndash282 2015

[16] J S H Kornbluth ldquoDynamic multi-criteria decision makingrdquoJournal of Multi-Criteria Decision Analysis vol 1 no 2 pp 81ndash92 1992

[17] Y-H Lin P-C Lee T-P Chang and H-I Ting ldquoMulti-attribute group decision making model under the condition ofuncertain informationrdquoAutomation in Construction vol 17 no6 pp 792ndash797 2008

[18] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multicrite-ria acceptability analysis for groupdecisionmakingrdquoOperationsResearch vol 49 no 3 pp 444ndash454 2001

[19] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[20] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[21] S Miyamoto ldquoMultisets and fuzzymultisetsrdquo in Soft Computingand Human-Centered Machines Z Q Liu and S MiyamotoEds pp 9ndash33 Springer Berlin Germany 2000

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[23] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[25] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling Simulation and Computation for Engineering andEnvironmental Systems vol 34 no 7 pp 1779ndash1787 2010

10 Mathematical Problems in Engineering

[26] D Horsky andM R Rao ldquoEstimation of attribute weights frompreference comparisonsrdquoManagement Science vol 30 no 7 pp801ndash822 1984

[27] P A Bottomley and J R Doyle ldquoA comparison of three weightelicitation methods good better and bestrdquo Omega vol 29 no6 pp 553ndash560 2001

[28] J Figueira and B Roy ldquoDetermining the weights of criteria inthe ELECTRE type methods with a revised Simosrsquo procedurerdquoEuropean Journal ofOperational Research vol 139 no 2 pp 317ndash326 2002

[29] L E Shirland R R Jesse R L Thompson and C L IacovouldquoDetermining attribute weights using mathematical program-mingrdquo Omega vol 31 no 6 pp 423ndash437 2003

[30] Y-M Wang and Y Luo ldquoIntegration of correlations withstandard deviations for determining attribute weights in mul-tiple attribute decision makingrdquo Mathematical amp ComputerModelling vol 51 no 1-2 pp 1ndash12 2010

[31] J Ma Z-P Fan and L-H Huang ldquoA subjective and objectiveintegrated approach to determine attribute weightsrdquo EuropeanJournal of Operational Research vol 112 no 2 pp 397ndash4041999

[32] X Xu ldquoA note on the subjective and objective integratedapproach to determine attribute weightsrdquo European Journal ofOperational Research vol 156 no 2 pp 530ndash532 2004

[33] P J H Schoemaker and C C Waid ldquoAn experimental compar-ison of different approaches to determining weights in additiveutility modelsrdquoManagement Science vol 28 no 2 pp 182ndash1961982

[34] Y-M Wang and C Parkan ldquoA general multiple attributedecision-making approach for integrating subjective prefer-ences and objective informationrdquo Fuzzy Sets and Systems vol157 no 10 pp 1333ndash1345 2006

[35] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[36] J R Doyle ldquoMultiattribute choice for the lazy decision makerlet the alternatives deciderdquoOrganizational Behavior andHumanDecision Processes vol 62 no 1 pp 87ndash100 1995

[37] E Bernroider and V Stix ldquoA method using weight restrictionsin data envelopment analysis for ranking and validity issues indecisionmakingrdquoComputersampOperations Research vol 34 no9 pp 2637ndash2647 2007

[38] J Wu and L Liang ldquoA multiple criteria ranking method basedon game cross-evaluation approachrdquo Annals of OperationsResearch vol 197 no 1 pp 191ndash200 2012

[39] C Fu and K-S Chin ldquoRobust evidential reasoning approachwith unknown attribute weightsrdquo Knowledge-Based Systemsvol 59 no 2 pp 9ndash20 2014

[40] B S Ahn ldquoExtreme point-based multi-attribute decision anal-ysis with incomplete informationrdquo European Journal of Opera-tional Research vol 240 no 3 pp 748ndash755 2015

[41] WW Cooper J L Ruiz and I Sirvent ldquoChoosing weights fromalternative optimal solutions of dualmultipliermodels inDEArdquoEuropean Journal of Operational Research vol 180 no 1 pp443ndash458 2007

[42] G R Jahanshahloo and P F Shahmirzadi ldquoNew methods forranking decision making units based on the dispersion ofweights and Norm 1 in Data Envelopment Analysisrdquo Computersamp Industrial Engineering vol 65 no 2 pp 187ndash193 2013

[43] V E Krivonozhko O B Utkin M M Safin and A V LychevldquoOn some generalization of the DEA modelsrdquo Journal of theOperational Research Society vol 60 no 11 pp 1518ndash1527 2009

[44] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[45] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo New Directions for ProgramEvaluation vol 1986 no 32 pp 73ndash105 1986

[46] A Boussofiane R G Dyson and E Thanassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[47] K Wang and F Wei ldquoGame cross efficiency DEA based projectreview of equipment designrdquo Journal of Beijing University ofAeronautics and Astronautics vol 35 no 10 pp 1278ndash12822009

[48] T Wu and J Blackhurst ldquoSupplier evaluation and selection anaugmented DEA approachrdquo International Journal of ProductionResearch vol 47 no 16 pp 4593ndash4608 2009

[49] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Korean stockmarketrdquo European Journal of Operational Research vol 236 no1 pp 361ndash368 2014

[50] B Roy ldquoTheoutranking approach and the foundations of electremethodsrdquoTheory and Decision vol 31 no 1 pp 49ndash73 1991

[51] G F Reed F Lynn and B D Meade ldquoUse of coefficient ofvariation in assessing variability of quantitative assaysrdquo Clinicalamp Diagnostic Laboratory Immunology vol 9 no 6 pp 1235ndash1239 2002

[52] 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

[53] H M Moshkovich A I Mechitov and D L Olson ldquoOrdinaljudgments in multiattribute decision analysisrdquo European Jour-nal of Operational Research vol 137 no 3 pp 625ndash641 2002

[54] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Tsinghua Press Beijing China 2005

[55] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 8 Evaluation results for the comparison of faculty candidates

Candidates 119862max119894

1198621119894

1198622119894

1198623119894

1198624119894

1198625119894

119903119894

120575119894

CV(119883119894)

119860 0717 0717 0712 0706 0717 0703 0711 0006 0008119861 0792 0792 0792 0788 0792 0746 0782 0018 0023119862 0672 0651 0661 0672 0651 0660 0659 0008 0012119863 0774 0774 0771 0764 0774 0737 0764 0014 0461119864 0645 0525 0526 0540 0525 0645 0552 0047 0018

Table 9 Ordinal values of alternatives

1198841

1198842

1198843

1198844

1198831

4 2 4 21198832

1 4 3 31198833

2 5 1 11198834

5 3 2 51198835

3 1 5 4

management catering health care and entertainment Inthe case study the decision maker is an official from thedevelopment and reform commission of Hefei with thehelp of a group of experts from the governments relevantenterprises and community organizations Four attributesare used to evaluate the six companies They are quality ofservice (119884

1) diversity of service (119884

2) profitability (119884

3) and

development potential (1198844) The original attribute values are

given by ordinal values as shown in Table 9 The decisionprocedures are as follows

After considering the four attributes the decision makerviews that quality of service (119884

1) is the most important

attribute in evaluation The development potential (1198844) is the

second most important one profitability (1198843) is the third

most important one and diversity of service (1198842) is the

least important one respectively And the difference betweentwo adjacent attribute weights should be more than 01 Inaddition each weight value should be nomore than 05Thenthe weight space is denoted by Ω(119908) = 119908

1minus 01 gt 119908

4

1199084minus 01 gt 119908

3 1199083minus 01 gt 119908

2 1199081le 05 119908

2le 05 119908

3le 05

1199084le 05 119908

1+ 1199082+ 1199083+ 1199084= 1

By specifying a threshold value of 01 ordinal attributevalues are transformed into four sets of attribute value spacesThey are denoted by

Ω(1198841) = 119903

21= 1 11990321

minus 11990331

ge 01 11990331

minus 11990351

ge 01 11990351

minus 11990311

ge 01 11990311

minus 11990341

ge 01 11990341

= 0

Ω (1198842) = 119903

52= 1 11990352

minus 11990312

ge 01 11990312

minus 11990342

ge 01 11990342

minus 11990322

ge 01 11990322

minus 11990332

ge 01 11990332

= 0

Ω (1198843) = 119903

33= 1 11990333

minus 11990343

ge 01 11990343

minus 11990323

ge 01 11990323

minus 11990313

ge 01 11990313

minus 11990353

ge 01 11990353

= 0

Ω (1198844) = 119903

34= 1 11990334

minus 11990314

ge 01 11990314

minus 11990324

ge 01 11990324

minus 11990354

ge 01 11990354

minus 11990344

ge 01 11990344

= 0

(22)

Table 10 shows the computed results of the proposedapproachThe second column shows the self-evaluation scoreof each alternative computed by (19) And columns 3ndash7 listthe cross-evaluation scores Then the mean and deviationmeasures of each alternative are presented in columns 8 and9 respectively Consequently the results in column 10 areapplied to generate a rank order of the five community servicecompanies 119883

5≻ 1198831≻ 1198832≻ 1198834≻ 1198833 The decision maker

is pleased to select company 1198835with the best performance

among the five alternatives

6 Conclusions and Future Research

Due to the complexity and uncertainty of MADM problemsin real world the decision maker is not able to provide theexact attribute weights And different weight determinationmethods are difficult to generate a most satisfaction solutionThe current paper presents a new MADM method basedon cross-evaluation in which the weight information isexpressed by the so-called weight space It attempts to rectifysome shortcomings of existingMADMmethods by providingthe following important features (1) it is more feasibleto provide a weight space rather than specifying a set ofweight values (2) the single dominating attribute drawbackcan be avoided by the restriction of the upper bound ofeach attribute (3) compared with DEA based methods thevariable attribute weights are consistent with the opinionof decision maker (4) both mean score and the deviationscore are considered in our cross-evaluation models and (5)the proposed method is also successfully extended to moregeneralized situations in which attribute values are expressedby interval values and ordinal values respectively

We have illustrated the applicability of the proposedmethods by revisiting two reported MADM studies and bydepicting a case study on the selection of community servicecompanies Although our proposed method makes someimprovement in solving MADM problems it only can dealwith the individual decision making case Future research

Mathematical Problems in Engineering 9

Table 10 Evaluation results for the selection of community service companies

Candidates 119862max119897

1198621119897

1198622119897

1198623119897

1198624119897

1198625119897

119903119894

120575119894

CV(119883119894)

1198831

0489 0489 0317 0295 0317 0317 0347 0072 02071198832

0532 0433 0532 0508 0458 0504 0487 0036 00741198833

0535 0526 0526 0535 0526 0526 0528 0004 00081198834

0223 0202 0223 0211 0223 0223 0216 0009 00421198835

0422 0349 0349 0103 0349 0422 0314 0109 0347

may attempt to employ our cross-evaluation methods tohandle group decision making problem

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research is supported by National Key Technology Sup-port Program (no 2014BAH27F00) National Natural Sci-ence Funds of China (no 71471053) and the FundamentalResearch Fund for the Central Universities (no JZ2014-HGBZ0339)

References

[1] R H Green and J R Doyle ldquoOn maximizing discriminationin multiple criteria decision makingrdquo Journal of the OperationalResearch Society vol 46 no 2 pp 192ndash204 1995

[2] R E Steuer and P Na ldquoMultiple criteria decision makingcombined with finance a categorized bibliographic studyrdquoEuropean Journal of Operational Research vol 150 no 3 pp496ndash515 2003

[3] J Wallenius J S Dyer P C Fishburn R E Steuer S Ziontsand K Deb ldquoMultiple criteria decision making multi-attributeutility theory recent accomplishments and what lies aheadrdquoManagement Science vol 54 no 7 pp 1336ndash1349 2008

[4] W Ho X Xu and P K Dey ldquoMulti-criteria decision makingapproaches for supplier evaluation and selection a literaturereviewrdquo European Journal of Operational Research vol 202 no1 pp 16ndash24 2010

[5] C W Churchman R L Ackoff and N M Smith ldquoAn approx-imate measure of valuerdquo Journal of the Operational ResearchSociety of America vol 2 no 2 pp 172ndash187 1954

[6] T L Saaty ldquoThe analytic hierarchy processrdquo in Proceedings ofthe Second International Seminar on Operational Research in theBasque Provinces vol 4 pp 189ndash234 1980

[7] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[8] H Deng C-H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[9] P K Dey ldquoIntegrated project evaluation and selection usingmultiple-attribute decision-making techniquerdquo InternationalJournal of Production Economics vol 103 no 1 pp 90ndash1032006

[10] G-D Li D Yamaguchi and M Nagai ldquoA grey-based decision-making approach to the supplier selection problemrdquoMathemat-ical amp Computer Modelling vol 46 no 3-4 pp 573ndash581 2007

[11] A Shanian and O Savadogo ldquoTOPSIS multiple-criteria deci-sion support analysis for material selection of metallic bipolarplates for polymer electrolyte fuel cellrdquo Journal of Power Sourcesvol 159 no 2 pp 1095ndash1104 2006

[12] G-H Tzeng Y-P O Yang C-T Lin and C-B Chen ldquoHier-archical MADM with fuzzy integral for evaluating enterpriseintranet web sitesrdquo Information Sciences vol 169 no 3-4 pp409ndash426 2005

[13] 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

[14] A Kelemenis and D Askounis ldquoA new TOPSIS-based multi-criteria approach to personnel selectionrdquo Expert Systems withApplications vol 37 no 7 pp 4999ndash5008 2010

[15] B Bairagi B Dey B Sarkar and S K Sanyal ldquoADeNovomulti-approaches multi-criteria decision making technique with anapplication in performance evaluation of material handlingdevicerdquo Computers amp Industrial Engineering vol 87 pp 267ndash282 2015

[16] J S H Kornbluth ldquoDynamic multi-criteria decision makingrdquoJournal of Multi-Criteria Decision Analysis vol 1 no 2 pp 81ndash92 1992

[17] Y-H Lin P-C Lee T-P Chang and H-I Ting ldquoMulti-attribute group decision making model under the condition ofuncertain informationrdquoAutomation in Construction vol 17 no6 pp 792ndash797 2008

[18] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multicrite-ria acceptability analysis for groupdecisionmakingrdquoOperationsResearch vol 49 no 3 pp 444ndash454 2001

[19] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[20] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[21] S Miyamoto ldquoMultisets and fuzzymultisetsrdquo in Soft Computingand Human-Centered Machines Z Q Liu and S MiyamotoEds pp 9ndash33 Springer Berlin Germany 2000

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[23] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[25] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling Simulation and Computation for Engineering andEnvironmental Systems vol 34 no 7 pp 1779ndash1787 2010

10 Mathematical Problems in Engineering

[26] D Horsky andM R Rao ldquoEstimation of attribute weights frompreference comparisonsrdquoManagement Science vol 30 no 7 pp801ndash822 1984

[27] P A Bottomley and J R Doyle ldquoA comparison of three weightelicitation methods good better and bestrdquo Omega vol 29 no6 pp 553ndash560 2001

[28] J Figueira and B Roy ldquoDetermining the weights of criteria inthe ELECTRE type methods with a revised Simosrsquo procedurerdquoEuropean Journal ofOperational Research vol 139 no 2 pp 317ndash326 2002

[29] L E Shirland R R Jesse R L Thompson and C L IacovouldquoDetermining attribute weights using mathematical program-mingrdquo Omega vol 31 no 6 pp 423ndash437 2003

[30] Y-M Wang and Y Luo ldquoIntegration of correlations withstandard deviations for determining attribute weights in mul-tiple attribute decision makingrdquo Mathematical amp ComputerModelling vol 51 no 1-2 pp 1ndash12 2010

[31] J Ma Z-P Fan and L-H Huang ldquoA subjective and objectiveintegrated approach to determine attribute weightsrdquo EuropeanJournal of Operational Research vol 112 no 2 pp 397ndash4041999

[32] X Xu ldquoA note on the subjective and objective integratedapproach to determine attribute weightsrdquo European Journal ofOperational Research vol 156 no 2 pp 530ndash532 2004

[33] P J H Schoemaker and C C Waid ldquoAn experimental compar-ison of different approaches to determining weights in additiveutility modelsrdquoManagement Science vol 28 no 2 pp 182ndash1961982

[34] Y-M Wang and C Parkan ldquoA general multiple attributedecision-making approach for integrating subjective prefer-ences and objective informationrdquo Fuzzy Sets and Systems vol157 no 10 pp 1333ndash1345 2006

[35] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[36] J R Doyle ldquoMultiattribute choice for the lazy decision makerlet the alternatives deciderdquoOrganizational Behavior andHumanDecision Processes vol 62 no 1 pp 87ndash100 1995

[37] E Bernroider and V Stix ldquoA method using weight restrictionsin data envelopment analysis for ranking and validity issues indecisionmakingrdquoComputersampOperations Research vol 34 no9 pp 2637ndash2647 2007

[38] J Wu and L Liang ldquoA multiple criteria ranking method basedon game cross-evaluation approachrdquo Annals of OperationsResearch vol 197 no 1 pp 191ndash200 2012

[39] C Fu and K-S Chin ldquoRobust evidential reasoning approachwith unknown attribute weightsrdquo Knowledge-Based Systemsvol 59 no 2 pp 9ndash20 2014

[40] B S Ahn ldquoExtreme point-based multi-attribute decision anal-ysis with incomplete informationrdquo European Journal of Opera-tional Research vol 240 no 3 pp 748ndash755 2015

[41] WW Cooper J L Ruiz and I Sirvent ldquoChoosing weights fromalternative optimal solutions of dualmultipliermodels inDEArdquoEuropean Journal of Operational Research vol 180 no 1 pp443ndash458 2007

[42] G R Jahanshahloo and P F Shahmirzadi ldquoNew methods forranking decision making units based on the dispersion ofweights and Norm 1 in Data Envelopment Analysisrdquo Computersamp Industrial Engineering vol 65 no 2 pp 187ndash193 2013

[43] V E Krivonozhko O B Utkin M M Safin and A V LychevldquoOn some generalization of the DEA modelsrdquo Journal of theOperational Research Society vol 60 no 11 pp 1518ndash1527 2009

[44] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[45] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo New Directions for ProgramEvaluation vol 1986 no 32 pp 73ndash105 1986

[46] A Boussofiane R G Dyson and E Thanassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[47] K Wang and F Wei ldquoGame cross efficiency DEA based projectreview of equipment designrdquo Journal of Beijing University ofAeronautics and Astronautics vol 35 no 10 pp 1278ndash12822009

[48] T Wu and J Blackhurst ldquoSupplier evaluation and selection anaugmented DEA approachrdquo International Journal of ProductionResearch vol 47 no 16 pp 4593ndash4608 2009

[49] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Korean stockmarketrdquo European Journal of Operational Research vol 236 no1 pp 361ndash368 2014

[50] B Roy ldquoTheoutranking approach and the foundations of electremethodsrdquoTheory and Decision vol 31 no 1 pp 49ndash73 1991

[51] G F Reed F Lynn and B D Meade ldquoUse of coefficient ofvariation in assessing variability of quantitative assaysrdquo Clinicalamp Diagnostic Laboratory Immunology vol 9 no 6 pp 1235ndash1239 2002

[52] 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

[53] H M Moshkovich A I Mechitov and D L Olson ldquoOrdinaljudgments in multiattribute decision analysisrdquo European Jour-nal of Operational Research vol 137 no 3 pp 625ndash641 2002

[54] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Tsinghua Press Beijing China 2005

[55] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 10 Evaluation results for the selection of community service companies

Candidates 119862max119897

1198621119897

1198622119897

1198623119897

1198624119897

1198625119897

119903119894

120575119894

CV(119883119894)

1198831

0489 0489 0317 0295 0317 0317 0347 0072 02071198832

0532 0433 0532 0508 0458 0504 0487 0036 00741198833

0535 0526 0526 0535 0526 0526 0528 0004 00081198834

0223 0202 0223 0211 0223 0223 0216 0009 00421198835

0422 0349 0349 0103 0349 0422 0314 0109 0347

may attempt to employ our cross-evaluation methods tohandle group decision making problem

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research is supported by National Key Technology Sup-port Program (no 2014BAH27F00) National Natural Sci-ence Funds of China (no 71471053) and the FundamentalResearch Fund for the Central Universities (no JZ2014-HGBZ0339)

References

[1] R H Green and J R Doyle ldquoOn maximizing discriminationin multiple criteria decision makingrdquo Journal of the OperationalResearch Society vol 46 no 2 pp 192ndash204 1995

[2] R E Steuer and P Na ldquoMultiple criteria decision makingcombined with finance a categorized bibliographic studyrdquoEuropean Journal of Operational Research vol 150 no 3 pp496ndash515 2003

[3] J Wallenius J S Dyer P C Fishburn R E Steuer S Ziontsand K Deb ldquoMultiple criteria decision making multi-attributeutility theory recent accomplishments and what lies aheadrdquoManagement Science vol 54 no 7 pp 1336ndash1349 2008

[4] W Ho X Xu and P K Dey ldquoMulti-criteria decision makingapproaches for supplier evaluation and selection a literaturereviewrdquo European Journal of Operational Research vol 202 no1 pp 16ndash24 2010

[5] C W Churchman R L Ackoff and N M Smith ldquoAn approx-imate measure of valuerdquo Journal of the Operational ResearchSociety of America vol 2 no 2 pp 172ndash187 1954

[6] T L Saaty ldquoThe analytic hierarchy processrdquo in Proceedings ofthe Second International Seminar on Operational Research in theBasque Provinces vol 4 pp 189ndash234 1980

[7] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications Springer Berlin Germany 1981

[8] H Deng C-H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[9] P K Dey ldquoIntegrated project evaluation and selection usingmultiple-attribute decision-making techniquerdquo InternationalJournal of Production Economics vol 103 no 1 pp 90ndash1032006

[10] G-D Li D Yamaguchi and M Nagai ldquoA grey-based decision-making approach to the supplier selection problemrdquoMathemat-ical amp Computer Modelling vol 46 no 3-4 pp 573ndash581 2007

[11] A Shanian and O Savadogo ldquoTOPSIS multiple-criteria deci-sion support analysis for material selection of metallic bipolarplates for polymer electrolyte fuel cellrdquo Journal of Power Sourcesvol 159 no 2 pp 1095ndash1104 2006

[12] G-H Tzeng Y-P O Yang C-T Lin and C-B Chen ldquoHier-archical MADM with fuzzy integral for evaluating enterpriseintranet web sitesrdquo Information Sciences vol 169 no 3-4 pp409ndash426 2005

[13] 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

[14] A Kelemenis and D Askounis ldquoA new TOPSIS-based multi-criteria approach to personnel selectionrdquo Expert Systems withApplications vol 37 no 7 pp 4999ndash5008 2010

[15] B Bairagi B Dey B Sarkar and S K Sanyal ldquoADeNovomulti-approaches multi-criteria decision making technique with anapplication in performance evaluation of material handlingdevicerdquo Computers amp Industrial Engineering vol 87 pp 267ndash282 2015

[16] J S H Kornbluth ldquoDynamic multi-criteria decision makingrdquoJournal of Multi-Criteria Decision Analysis vol 1 no 2 pp 81ndash92 1992

[17] Y-H Lin P-C Lee T-P Chang and H-I Ting ldquoMulti-attribute group decision making model under the condition ofuncertain informationrdquoAutomation in Construction vol 17 no6 pp 792ndash797 2008

[18] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multicrite-ria acceptability analysis for groupdecisionmakingrdquoOperationsResearch vol 49 no 3 pp 444ndash454 2001

[19] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[20] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[21] S Miyamoto ldquoMultisets and fuzzymultisetsrdquo in Soft Computingand Human-Centered Machines Z Q Liu and S MiyamotoEds pp 9ndash33 Springer Berlin Germany 2000

[22] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[23] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[25] C Kao ldquoWeight determination for consistently ranking alterna-tives in multiple criteria decision analysisrdquo Applied Mathemati-cal Modelling Simulation and Computation for Engineering andEnvironmental Systems vol 34 no 7 pp 1779ndash1787 2010

10 Mathematical Problems in Engineering

[26] D Horsky andM R Rao ldquoEstimation of attribute weights frompreference comparisonsrdquoManagement Science vol 30 no 7 pp801ndash822 1984

[27] P A Bottomley and J R Doyle ldquoA comparison of three weightelicitation methods good better and bestrdquo Omega vol 29 no6 pp 553ndash560 2001

[28] J Figueira and B Roy ldquoDetermining the weights of criteria inthe ELECTRE type methods with a revised Simosrsquo procedurerdquoEuropean Journal ofOperational Research vol 139 no 2 pp 317ndash326 2002

[29] L E Shirland R R Jesse R L Thompson and C L IacovouldquoDetermining attribute weights using mathematical program-mingrdquo Omega vol 31 no 6 pp 423ndash437 2003

[30] Y-M Wang and Y Luo ldquoIntegration of correlations withstandard deviations for determining attribute weights in mul-tiple attribute decision makingrdquo Mathematical amp ComputerModelling vol 51 no 1-2 pp 1ndash12 2010

[31] J Ma Z-P Fan and L-H Huang ldquoA subjective and objectiveintegrated approach to determine attribute weightsrdquo EuropeanJournal of Operational Research vol 112 no 2 pp 397ndash4041999

[32] X Xu ldquoA note on the subjective and objective integratedapproach to determine attribute weightsrdquo European Journal ofOperational Research vol 156 no 2 pp 530ndash532 2004

[33] P J H Schoemaker and C C Waid ldquoAn experimental compar-ison of different approaches to determining weights in additiveutility modelsrdquoManagement Science vol 28 no 2 pp 182ndash1961982

[34] Y-M Wang and C Parkan ldquoA general multiple attributedecision-making approach for integrating subjective prefer-ences and objective informationrdquo Fuzzy Sets and Systems vol157 no 10 pp 1333ndash1345 2006

[35] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[36] J R Doyle ldquoMultiattribute choice for the lazy decision makerlet the alternatives deciderdquoOrganizational Behavior andHumanDecision Processes vol 62 no 1 pp 87ndash100 1995

[37] E Bernroider and V Stix ldquoA method using weight restrictionsin data envelopment analysis for ranking and validity issues indecisionmakingrdquoComputersampOperations Research vol 34 no9 pp 2637ndash2647 2007

[38] J Wu and L Liang ldquoA multiple criteria ranking method basedon game cross-evaluation approachrdquo Annals of OperationsResearch vol 197 no 1 pp 191ndash200 2012

[39] C Fu and K-S Chin ldquoRobust evidential reasoning approachwith unknown attribute weightsrdquo Knowledge-Based Systemsvol 59 no 2 pp 9ndash20 2014

[40] B S Ahn ldquoExtreme point-based multi-attribute decision anal-ysis with incomplete informationrdquo European Journal of Opera-tional Research vol 240 no 3 pp 748ndash755 2015

[41] WW Cooper J L Ruiz and I Sirvent ldquoChoosing weights fromalternative optimal solutions of dualmultipliermodels inDEArdquoEuropean Journal of Operational Research vol 180 no 1 pp443ndash458 2007

[42] G R Jahanshahloo and P F Shahmirzadi ldquoNew methods forranking decision making units based on the dispersion ofweights and Norm 1 in Data Envelopment Analysisrdquo Computersamp Industrial Engineering vol 65 no 2 pp 187ndash193 2013

[43] V E Krivonozhko O B Utkin M M Safin and A V LychevldquoOn some generalization of the DEA modelsrdquo Journal of theOperational Research Society vol 60 no 11 pp 1518ndash1527 2009

[44] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[45] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo New Directions for ProgramEvaluation vol 1986 no 32 pp 73ndash105 1986

[46] A Boussofiane R G Dyson and E Thanassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[47] K Wang and F Wei ldquoGame cross efficiency DEA based projectreview of equipment designrdquo Journal of Beijing University ofAeronautics and Astronautics vol 35 no 10 pp 1278ndash12822009

[48] T Wu and J Blackhurst ldquoSupplier evaluation and selection anaugmented DEA approachrdquo International Journal of ProductionResearch vol 47 no 16 pp 4593ndash4608 2009

[49] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Korean stockmarketrdquo European Journal of Operational Research vol 236 no1 pp 361ndash368 2014

[50] B Roy ldquoTheoutranking approach and the foundations of electremethodsrdquoTheory and Decision vol 31 no 1 pp 49ndash73 1991

[51] G F Reed F Lynn and B D Meade ldquoUse of coefficient ofvariation in assessing variability of quantitative assaysrdquo Clinicalamp Diagnostic Laboratory Immunology vol 9 no 6 pp 1235ndash1239 2002

[52] 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

[53] H M Moshkovich A I Mechitov and D L Olson ldquoOrdinaljudgments in multiattribute decision analysisrdquo European Jour-nal of Operational Research vol 137 no 3 pp 625ndash641 2002

[54] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Tsinghua Press Beijing China 2005

[55] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[26] D Horsky andM R Rao ldquoEstimation of attribute weights frompreference comparisonsrdquoManagement Science vol 30 no 7 pp801ndash822 1984

[27] P A Bottomley and J R Doyle ldquoA comparison of three weightelicitation methods good better and bestrdquo Omega vol 29 no6 pp 553ndash560 2001

[28] J Figueira and B Roy ldquoDetermining the weights of criteria inthe ELECTRE type methods with a revised Simosrsquo procedurerdquoEuropean Journal ofOperational Research vol 139 no 2 pp 317ndash326 2002

[29] L E Shirland R R Jesse R L Thompson and C L IacovouldquoDetermining attribute weights using mathematical program-mingrdquo Omega vol 31 no 6 pp 423ndash437 2003

[30] Y-M Wang and Y Luo ldquoIntegration of correlations withstandard deviations for determining attribute weights in mul-tiple attribute decision makingrdquo Mathematical amp ComputerModelling vol 51 no 1-2 pp 1ndash12 2010

[31] J Ma Z-P Fan and L-H Huang ldquoA subjective and objectiveintegrated approach to determine attribute weightsrdquo EuropeanJournal of Operational Research vol 112 no 2 pp 397ndash4041999

[32] X Xu ldquoA note on the subjective and objective integratedapproach to determine attribute weightsrdquo European Journal ofOperational Research vol 156 no 2 pp 530ndash532 2004

[33] P J H Schoemaker and C C Waid ldquoAn experimental compar-ison of different approaches to determining weights in additiveutility modelsrdquoManagement Science vol 28 no 2 pp 182ndash1961982

[34] Y-M Wang and C Parkan ldquoA general multiple attributedecision-making approach for integrating subjective prefer-ences and objective informationrdquo Fuzzy Sets and Systems vol157 no 10 pp 1333ndash1345 2006

[35] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[36] J R Doyle ldquoMultiattribute choice for the lazy decision makerlet the alternatives deciderdquoOrganizational Behavior andHumanDecision Processes vol 62 no 1 pp 87ndash100 1995

[37] E Bernroider and V Stix ldquoA method using weight restrictionsin data envelopment analysis for ranking and validity issues indecisionmakingrdquoComputersampOperations Research vol 34 no9 pp 2637ndash2647 2007

[38] J Wu and L Liang ldquoA multiple criteria ranking method basedon game cross-evaluation approachrdquo Annals of OperationsResearch vol 197 no 1 pp 191ndash200 2012

[39] C Fu and K-S Chin ldquoRobust evidential reasoning approachwith unknown attribute weightsrdquo Knowledge-Based Systemsvol 59 no 2 pp 9ndash20 2014

[40] B S Ahn ldquoExtreme point-based multi-attribute decision anal-ysis with incomplete informationrdquo European Journal of Opera-tional Research vol 240 no 3 pp 748ndash755 2015

[41] WW Cooper J L Ruiz and I Sirvent ldquoChoosing weights fromalternative optimal solutions of dualmultipliermodels inDEArdquoEuropean Journal of Operational Research vol 180 no 1 pp443ndash458 2007

[42] G R Jahanshahloo and P F Shahmirzadi ldquoNew methods forranking decision making units based on the dispersion ofweights and Norm 1 in Data Envelopment Analysisrdquo Computersamp Industrial Engineering vol 65 no 2 pp 187ndash193 2013

[43] V E Krivonozhko O B Utkin M M Safin and A V LychevldquoOn some generalization of the DEA modelsrdquo Journal of theOperational Research Society vol 60 no 11 pp 1518ndash1527 2009

[44] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[45] T R Sexton RH Silkman andA JHogan ldquoData envelopmentanalysis critique and extensionsrdquo New Directions for ProgramEvaluation vol 1986 no 32 pp 73ndash105 1986

[46] A Boussofiane R G Dyson and E Thanassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[47] K Wang and F Wei ldquoGame cross efficiency DEA based projectreview of equipment designrdquo Journal of Beijing University ofAeronautics and Astronautics vol 35 no 10 pp 1278ndash12822009

[48] T Wu and J Blackhurst ldquoSupplier evaluation and selection anaugmented DEA approachrdquo International Journal of ProductionResearch vol 47 no 16 pp 4593ndash4608 2009

[49] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Korean stockmarketrdquo European Journal of Operational Research vol 236 no1 pp 361ndash368 2014

[50] B Roy ldquoTheoutranking approach and the foundations of electremethodsrdquoTheory and Decision vol 31 no 1 pp 49ndash73 1991

[51] G F Reed F Lynn and B D Meade ldquoUse of coefficient ofvariation in assessing variability of quantitative assaysrdquo Clinicalamp Diagnostic Laboratory Immunology vol 9 no 6 pp 1235ndash1239 2002

[52] 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

[53] H M Moshkovich A I Mechitov and D L Olson ldquoOrdinaljudgments in multiattribute decision analysisrdquo European Jour-nal of Operational Research vol 137 no 3 pp 625ndash641 2002

[54] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Tsinghua Press Beijing China 2005

[55] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of