Research ArticleA New Fuzzy TOPSIS-TODIM Hybrid Method forGreen Supplier Selection Using Fuzzy Time Function

Alireza Arshadi Khamseh and Mahdi Mahmoodi

Industrial Engineering Department Kharazmi University Tehran 15719-14911 Iran

Correspondence should be addressed to Alireza Arshadi khamseh alirezaarshadikhamsehgmailcom

Received 4 January 2014 Revised 14 February 2014 Accepted 14 February 2014 Published 17 March 2014

Academic Editor M Onder Efe

Copyright copy 2014 A Arshadi khamseh and M MahmoodiThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Today green supply chain is considered all around the world and supplier selection has been changed regarding these green andcarbon emission criteria so green supplier selection has been a major problem in this area In this study we use fuzzy time functionto assist managers in green supplier selection under uncertainty and ambiguity This function will consider derivation from thegoal during the time and by using it and we will be able to have the best supplier in every period after having some modification inlegal limitations for green supplier selection criteria We use a fuzzy TOPSIS to have better initial weighting in TODIM a discretemulticriteria method based on prospect theory in uncertainty (known as TODIM in Portuguese) decision making method Theresults indicated that our proposed approach can easily and effectively accommodate criteria with gains and loss functions duringtime and also by using this method we will have a more reasonable predict of our suppliers ranking in future and that will help usin future investment in these suppliers Finally it has been shown in car industries in Iran

1 Introduction

In recent years the EuropeanUnion (EU) has established var-ious environmental policies including the RoHS (RestrictedUse of Hazardous Substances in Electronics and ElectricalEquipment) as well as WEEE (Waste Electronics and Elec-trical Equipment) directives So far environmental manage-ment has evolved to include boundary-spanning activitiesin the upstream and the downstream supply chains Sirvas-tava defined green supply chain management (GSCM) asa combination of environmental and supply chain manage-ment activities including product design material selectionmanufacturing process final product delivery and end-of-lifeproduct management With GSCM firms can select from awide variety of suppliers and leverage resources throughoutthe firm to eliminate the environmental impact of supplychain activities Tseng [1]

Firms typically expect their suppliers to surpass environ-mental compliance and to develop efficient and green productdesign In addition suppliers are expected to assess the lifecycle of a product Although the qualitative criteria are lit-teredwith subjective perception because theGSCMevolution

criteria tend to be subjective qualitative or described withlinguistic information Thus it is extremely difficult forthe decision makers to express their preference using exactnumerical values Zhang et al [2] so it is important touse linguistic number to distinguish between these criteriaNevertheless a firmrsquos suppliers must satisfy green criteriaunder the constrains of subjective human preferences whichinclude uncertainty in addition to uncertainties of time andplanning for future in this study we consider time in alldecision making processes and this will be more effectivein combination with fuzzy number for linguistic termsMost of studies assess GSCM based on its alignment withthe firmrsquos identified objectives and fulfillment of a set ofassessment criteria In general the evaluation of criteria ishighly subjective and unstructured as it relies significantlyon managerrsquos experience knowledge and intuition How-ever managers cannot consider all relevant criteria due tobounded rationality and limited capacity for informationprocessing see Tseng [1] The fuzzy TODIM approach caneasily and effectively accommodate criteria with gains andloss functions see Tseng et al [3] but with all advantagesof using fuzzy TODIM two problems have been remained

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2014 Article ID 841405 10 pageshttpdxdoiorg1011552014841405

2 Advances in Fuzzy Systems

0

2

4

6

8

10 20 30 40Time (months)

Customer satisfaction during time

Customer satisfaction method A upper boundMiddleLower bound

Fuzz

y sc

ore (0

ndash10

)

Figure 1 Fuzzy time functions

how can we predict our supplier behavior and how muchtime this supplier can be effective in our system and when wemust change our supplier Previous studies also usedMADMtools for selecting suppliers according to criteria which wasmade it easier with using fuzzy approach Izadikhah [4]used TOPSIS method with interval-valued fuzzy numberin supplier selection and selected best supplier accordingto criteria in our study we used TODIM fuzzy because ofits sensitivity to evaluate data and show how much we canimprove our system with changing supplier also we usedfuzzy time function to show changes in other suppliersand finally we can improve our supply chain with respectto this data and estimation of suppliers action Hence theevolution approach often implemented ineffectively becausemanagement does not effectively use appropriate knowledgeand experience on GSCM Due to the impact of this lackmanagement is not confident that supplier selection hasbeen studied and applied to a set of criteria to maximizefirmsrsquo green supply chain benefits How could managementapply knowledge of previously (successful and unsuccessful)supplier selections to support future decision making Theobjective of this study is to create a mechanism that couldassist managers in analyzing and selecting green suppliersduring time horizons and this will help us to predict a picturefor future of the organization activities for these targetswe use a hybrid fuzzy TOPSIS-TODIM decision makingprocess considering fuzzy time function We will continuethis paper by literature review on fuzzy time function andfuzzy TOPSIS and also TODIM in Section 2 and also havesome necessary explanations about the above techniques inSection 3 we show our proposed method in this paper byusing an illustrative example for green supplier selectionFinally in Section 4 conclusions will be presented

2 Material and Methods

21 Fuzzy Logic A fuzzy set is a class of objects with gradesof membership A membership function is between zero and

one Zadeh [5] Fuzzy logic is derived from fuzzy set theory todeal with reasoning that is approximate rather than preciseIt allows the model to easily incorporate various subjectexpertsrsquo advice in developing critical parameter estimatesZimmermann [6] In other words fuzzy logic enables us tohandle uncertainty

There are some kinds of fuzzy numbers Among the vari-ous shapes of fuzzy number triangular fuzzy number (TFN)is the most popular one It is represented with three pointsas 119860 = (119886

1 1198862 1198863) The membership function is illustrated in

(1) Let119860 and 119861 be defined as119860 = (1198861 1198862 1198863) 119861 = (119887

1 1198872 1198873)

Then119862 = (1198861+1198871 1198862+1198872 1198863+1198873) is a sumof these twonumbers

Besides119863 = (1198861minus1198873 1198862minus1198872 1198863minus1198871) is the subtraction of them

120583119886=

0 119909 lt 1198861

119909 minus 1198861

1198862minus 1198861

1198862ge 119909 ge 119886

1

1198863minus 119909

1198863minus 1198862

1198862ge 119909 ge 119886

1

0 otherwise

(1)

22 Fuzzy Time Function (FTF) Fuzzy time function isan approach to considering time as an important factorin uncertainty In many situations we have to change ourapproach because of the uncertainty and changes in criteriaduring time Sometimes these kinds of changes cause extracost for organization With fuzzy time function we couldpredict when some new factors like technology will be usedin our system and how they could affect our decisions deFigueiredo and Perkusich [7] used fuzzy time function infault and timing analysis in real time where fuzzy timefunction used to verify the possibility of the input tokens inplaces Yoneyama [8] used fuzzy set theory for time-delaysystems that considered the uncertain Takgai-Sugeno fuzzymodel with time delay In this paper we used a new aspect offuzzy time function which represents a combination of timeand fuzzy triangular numbers

A fuzzy time function consists of three sections opti-mistic normal and pessimistic related to time and these linesdo not cross each otherrsquos (Figure 1)

In fact this function in each time represents a fuzzytriangular number (3) which considers all three possibilitiesIn many organizations some methods or some criteria willbe out of date after some years or new equipment or methodwill be better than the older ones and these changes will effectin supplier selection methods and factors By using FTF wecould consider them in our decisions and can also upgradeour systems based on expertrsquos viewpoints Considering theabove descriptions we will have FTF for every criterion as

FTF (1198881) =

1198651

for optimistic1198652

for normal1198653

for pessimistic(2)

Advances in Fuzzy Systems 3

According to (2) FTF represents the triangular fuzzynumber during the time

119860 = (1198861 1198862 1198863) 997888rarr FTF = (119865

1 1198652 1198653) (3)

For computing the FTF in this paper we introduce twoways

First In this case we use average of slopes when the deviationbetween slopes of criteria is low and is not very important andafter this we use (2) for these criteria

To get precise decisions it is recommended to use theseparate functions for each period of time

Second In this method we do not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable

In some cases which fuzzy time function has morecomplicated function we must use the combination of thesetwo methods

23 Fuzzy TOPSIS TOPSIS one of the classical multicriteriadecision making methods was developed by Hwang andYoon [9] It is based on the concept that the selected alterna-tive should have the shortest distance from the positive idealsolution (PIS) and the farthest from the negative ideal solu-tion (NIS) TOPSIS also provides an easily understandableand programmable calculation procedure It has the abilityof taking various criteria with different units into accountsimultaneously see Ekmekcioglu et al [10] A number offuzzy TOPSIS methods have been developed in recent yearsChen and Hwang [11] first applied fuzzy numbers to establishfuzzy TOPSIS method in which relative closeness for eachalternative is evaluated based on fuzzy arithmetic operationsChen [12] extends the TOPSIS method to fuzzy groupdecision making situations by considering triangular fuzzynumbers and defining crisp Euclidean between two fuzzynumbers Chu [13] and Chu and Lin [14] further improvedthe methodology proposed by Chen [12] Jahanshahloo et al[15] and Chu and Lin [16] extended the fuzzy TOPSISmethod based on alpha level sets with interval arithmeticFuzzy TOPSIS has been introduced for variousmultiattributedecision-making problems Yong [17] used fuzzy TOPSIS forplanet location selection Chen et al [18] used fuzzy TOPSISfor supplier selection and Kahraman et al [19] used fuzzyTOPSIS to select municipal solid waste disposal methodand site Kutlu and Ekmekcioglu [20] used a modifiedfuzzy TOPSIS integrated with fuzzy AHP to propose newFMEA failure modes and effects analysis which overcomesthe shortcomings of traditional FMEA Kaya and Kahraman[21] proposed a modified fuzzy TOPSIS for the best energytechnology selection

In the following Chenrsquos fuzzy TOPSIS method isexplained Chen [12] extends the TOPSIS method to fuzzygroup decision making situations by considering triangularfuzzy numbers and defining crisp Euclidean distance betweentwo fuzzy numbers In Chenrsquos fuzzy TOPSIS linguisticpreferences can easily be converted to fuzzy numbers which

Table 1 Linguistic variable representing triangular fuzzy numbers

Fuzzy evaluation scores for alternativesLinguistic terms Fuzzy scoreVery poor (VP) (0 0 1)Poor (P) (0 1 3)Medium poor (MP) (1 3 5)Fair (F) (3 5 7)Medium good (MG) (5 7 9)Good (G) (7 9 10)Very good (VG) (9 10 10)

are allowed to be used in calculations see Ekmekcioglu et al[10] Kutlu and Ekmekcioglu [22] introduced fuzzy FMEAusing TOPSIS and they used linguistic data for FMEA whereeach of the linguistic variables represents triangular fuzzynumbers (Table 1)

In this study the linguistic variables used for each cri-terion and we will make FTF for each criterion duringtimes At the beginning weights of the criteria and fuzzyratings of alternatives with respect to each criterion have beencalculated and the fuzzy multicriteria of decision-makingproblem can be expressed in matrix format as

119863 = (

11990911 119909

1119899

d

1199091198981

sdot sdot sdot 119909119898119899

)

119882 = [1199081 1199082 119908

119899] 119895 = 1 2 119899

(4)

where 119909119894119895is the rating of the alternative 119860

119894according to

criterion 119895 (ie119862119895) and 119908

119895denotes the importance weight of

119862119895 These linguistic variables can be described by triangular

fuzzy numbers 119883119894119895= (119886119894119895 119887119894119895 119888119894119895) To avoid the complicated

normalization formulawe used in classical TOPSIS the linearscale transformation is used here to transform the variouscriteria scales into a comparable scale Therefore we canobtain the normalized fuzzy decision matrix denoted by 119877

= [119903119894119895]119898times119899 (5)

where 119861 and 119862 are the set of benefit criteria and cost criteriarespectively and

119903 = (

119886119894119895

119888lowast

119895

119887119894119895

119888lowast

119895

119888119894119895

119888lowast

119895

) 119895 isin 119861

119903 = (

119886minus

119895

119888119894119895

119887minus

119895

119887119894119895

119888minus

119895

119886119894119895

) 119895 isin 119862

119888lowast

119895= max119894

119888119894119895

if 119895 isin 119861

119886minus

119895= min119894

119886119894119895

if 119894 isin 119862

(6)

The above normalization method preserves the ranges ofnormalized fuzzy decision matrix as

= [119881119894119895]119898times119899 119894 = 1 2 119898 119895 = 1 2 119899 (7)

4 Advances in Fuzzy Systems

where

119881119894119895= 119903119894119895sdot 119889 (119862

119895) (8)

According to the weighted normalized fuzzy decisionmatrix (119881

119894119895) we know that the element V

119894119895is positive

normalized triangular fuzzy numbers and will be in [0 1]Then we can define the fuzzy positive-ideal solution (FPIS119860lowast) and fuzzy negative-ideal solution (FPIS 119860minus) as

119860lowast= (Vlowast1 Vlowast2 Vlowast3)

119860minus= (Vminus1 Vminus2 Vminus3)

(9)

where

Vlowast119895= (1 1 1) Vminus

119895= (0 0 0) 119895 = 1 2 119899 (10)

The distance of each alternative from 119860lowast and 119860minus can be

currently calculated as

119889lowast

119894=

119899

sum

119895=1

120575 (V119894119895 Vlowast119895)

119889minus

119894=

119899

sum

119895=1

120575 (V119894119895 Vminus119895)

(11)

where 119889(sdot) is the distance between two fuzzy numberscalculated as follows

119889 (119901 120591) = radic1

3[(1199011minus 1205911)2

+ (1199012minus 1205912)2

+ (1199013minus 1205913)2

]

(12)

where 119901 = (1199011 1199012 1199013) and 120591 = (120591

1 1205912 1205913) are two triangular

fuzzy numbersThe closeness coefficient of each alternative iscalculated as

119862119862119894=

119889minus

119895

119889lowast

119895+ 119889minus

119895

(13)

Obviously when 119862119862119894is near to 1 alternative 119860

119894is closer

to the (FPIS 119860lowast) and farther from (FPIS 119860minus) Thereforeaccording to the closeness coefficient we can determine theranking order of all alternatives and select the best one amonga set of feasible alternatives

24 TODIM

241 TODIM Method TODIM is a discrete multicriteriamethod founded on prospect theory The TODIM methodhas been successfully used and empirically validated in dif-ferent applications This is an experimental method based onhow people make effective decisions in risky conditions Theshape of the value function of TODIM is identical to prospecttheoryrsquos gain and loss function The global multicriteriavalue function of TODIM aggregates all measures of gainsand losses by considering all criteria Gomes and Rangel[23] apply TODIM to investigate and recommend options

for upstream projects for the natural gas reserves recentlydiscovered in the Mexilho field in the Santos Basin BrazilIn addition Gomes and Rangel presented an evaluation ofresidential properties with real estate agents in Brazil anddefine a reference value for the rents of these propertiesrsquocharacteristics using the TODIM method for multicriteriadecisions This approach can assist professionals in the realestate market to evaluate alternatives clearly using the criteriadefined by specialists In general TODIM can be used forqualitative and quantitative criteria Verbal scales of qualita-tive criteria are converted into cardinal scales and both typesof scales are normalized The relative measure of dominanceof one alternative over another alternative is determined foreach pair of alternatives This measure is computed as thesum of all criteria of relative gain and loss values for thesealternativesThis sumwill be a gain a loss or zero dependingon the performance of each alternative with respect to eachcriterion Tseng et al [3] apply TODIM and TFNs to selectgreen supplier chain They used TODIM to find the bestsupplier but the first weighting for TODIM with using fuzzyset theory does not consider the relationship between thecriteria more obvious than fuzzy TOPSIS with combinationof these two approaches the TODIM input will be moreaccurate and the result will be more correct consequentlyWeuse a new combination of TODIM and fuzzy TOPSIS andFTF in our proposed method

In previous methods used in TODIM or other MCDMmethods data have been collected based on alternative andcriteria comparisons but we use interval-valued triangularfuzzy numbers which consider the criteria with respect toalternatives in a deterministic time and these data collectionwill be continued in other times which are important fordecision makers or when we have some changes in criteriaor alternatives Table 2 shows transformation of linguisticcriteria to fuzzy-interval triangular numbers

Let 119860 and 119861 be defined as TFN 119886 = (1198861 1198862 1198863) 119887 =

(1198871 1198872 1198873) The distance between 119886 and 119887 is

120575 (119886 + ) = radic1

3[(1198861minus 1198871)2

+ (1198862minus 1198872)2

+ (1198863minus 1198873)2

]

(14)

The TFN is based on a three-value judgment the mini-mum possible value 119886

1 the mean value 119886

2 and the maximum

possible value 1198863 But the interval-TFN values that have been

used in this paper have 5 parameters on the other hand theseare two TFN numbers which have the same middle point(Figure 2)119888119894119895is expressed as an interval-value TFN where

119862 = [(1198861 1198861015840

1) 1198862 (1198861015840

3 1198863)] (15)

Given 119862119894119895= [(1198861 1198861015840

1) 1198862 (1198861015840

3 1198863)] the normalized perfor-

mance rating is

Advances in Fuzzy Systems 5

0

02

04

06

08

1

12

a1 a3a998400

1a998400

2a998400

3

Figure 2 An interval-valued TFN

Table 2 Corresponding TFNS for linguistic preferences

Linguistic preferences Interval-valued TFNSVery poor [(0 0) 0 (1 15)]Poor [(0 05) 1 (25 35)]Medium poor [(0 15) 3 (45 55)]Fair [(25 35) 5 (65 75)]Medium good [(45 55) 7 (8 95)]Good [(55 75) 9 (95 10)]Very good [(85 95) 10 (10 10)]

119903119894119895= [(

119886119894119895

119889+

119895

1198861015840

119894119895

119889+

119895

)

119887119894119895

119889+

119895

(

1198861015840

119894119895

119889+

119895

119886119894119895

119889+

119895

)] 119894 = 1 2 119898

119895 = 1 2 119899 for 119895 isin 119868

119903119894119895= [(

119886minus

119895

119889119894119895

119886minus

119895

1198891015840

119894119895

)

119886minus

119895

119887119894119895

(

119886minus

119895

1198861015840

119894119895

119886minus

119895

119886119894119895

)] 119894 = 1 2 119898

119895 = 1 2 119899 for 119895 isin 119869(16)

where 119889+119895= max119888

119894119895 119894 = 1 119898 and 119886minus

119895= min119886

119894119895 119894 =

1 119898 119903119894119895= [(119897

119894119895 1198971015840

119894119895) 119898119894119895 (119906119894119895 1199061015840

119894119895)] 119877 = [119903

119894119895]119898lowast119899

and1198770= (11990301 11990302 119903

0119899) = ([(1 1) 1 (1 1)][(1 1) 1 (1 1)]

[(1 1) 1 (1 1)])The distance between the reference value and each com-

parison value can be calculated by using definition (2) asfollows

120575(1)

119894119895= radic

1

3[(1198971015840

119894119895minus 1)2

+ (119898119894119895minus 1)2

+ (1199061015840

119894119895minus 1)2

]

120575(2)

119894119895= radic

1

3[(119897119894119895minus 1)2

+ (119898119894119895minus 1)2

+ (119906119894119895minus 1)2

]

(17)

These calculations are used to determine the distancebetween the reference value and the comparison value in theinterval after calculation we have a new interval TFN foreveryTFNas 120575

119894119895= [120575(1)

119894119895 120575(2)

119894119895]Theweight vector of the criteria

is calculated according to Zhnag et al [2]

119908119895=

sum119898

119894=1(120575(1)

119894119895+ 120575(2)

119894119895)

sum119898

119894=1sum119899

119894=1(120575(1)

119894119895+ 120575(2)

119894119895)

(18)

0 2 4 6 8 10minus10 minus8 minus6 minus4 minus2

Gains

Value

Losses

minus15

minus1

minus05

0

05

1

15

Figure 3 TODIM value function

242 Prospect Theory The value function used in theprospect theory is described in the form of a power lawexpressed as

V (119909) = 119909120572 if 119909 ge 0

minus120579(minus119909)120573 if 119909 lt 0

(19)

where 120572 and 120573 are parameters related to gains and lossesrespectivelyThe parameter 120579 is a risk factor that is consideredin model and must be greater than one Figure 3 showsprospect value function that must be concave and has an 119878shape form

243 TODIM Formulation The TODIM method uses pair-wise comparisons between the criteria by using technicallysimple resources to eliminate occasional inconsistenciesresulting from these comparisons TODIM allows valuejudgments to be performed on a verbal scale using hier-archy of criteria fuzzy value judgments and interdepen-dence relationships among the alternatives The decisionmatrix consists of alternatives and criteria The alternatives1198601 1198602 119860

119898are viable alternatives 119888

1 1198882 119888

119899are crite-

ria and 119909119894119895indicates the rating of alternative 119860

119894according

to the criteria 119888119895 The weight vector 119908 = (119908

1 1199082 119908

119899)

comprises the individual weights 119908119895(119895 = 1 119899) for each

criterion 119888119895satisfying sum119899

119894=1119908119895= 1 The data of decision

matrix119860 originate fromdifferent sourcesThematrixmust benormalized to be dimensionless and allows various criteria tobe compared with each otherThis study uses the normalizeddecision matrix 119877 = [119903

119894119895]119898times119899

with 119894 = 1 119898 and 119895 =1 119899

119860 = (

11990911 119909

1119899

d

1199091198981

sdot sdot sdot 119909119898119899

) (20)

TODIM then calculates the partial dominance matricesand the final dominance matrix The first calculation that thedecisionmakers must define is a reference criterion (typicallythe criterion with the greatest importance weight)Thereforew119903119888indicates the weight of the criterion 119888 by the reference

criterion 119903 TODIM is expressed by the following equations

6 Advances in Fuzzy Systems

The dominance of an alternative over the other is asfollows

120575 (119860119894 119860119895) =

119898

sum

119888=1

120601119888(119860119894 119860119895)forall(119894119895)

(21)

where

120601 (119860119894 119860119895) =

radic119908119903119888(119909119894119888minus 119909119895119888)

sum119898

119888=1119908119903119888

if (119909119894119888minus 119909119895119888) gt 0

0

if (119909119894119888minus 119909119895119888) = 0

minus1

120579

radic(sum119898

119888=1119908119903119888) (119909119894119888minus 119909119895119888)

119908119903119888

if (119909119894119888minus 119909119895119888) lt 0

(22)

The term 120593119888(119860119894 119860119895) represents the contribution of cri-

terion 119888 (119888 = 1 119898) to the function 120575 (119860119894 119860119895) while

comparing alternative 119894 with alternative 119895 The parameter 120579represents the attention factor of the losses whose mitigationdepends on the specific problem A positive (119909

119894119888minus 119909119895119888) repre-

sents a gain whereas a nil or a negative (119909119894119888minus119909119895119888) represents a

loss The final matrix of dominance is obtained by summingthe partial matrices of dominance for each criterion seeTseng et al [3] The global value of the alternative 119868 isdetermined by normalizing the final matrix of dominanceaccording to the following expression

120585119894=

sum119899

119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

maxsum119899119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

(23)

Ordering the values 120585119894provides the rank of each alterna-

tive and better alternatives have higher values of 120585119894The use of

numerical values in rating alternativesmay be limited in theircapacity to address uncertainties Therefore an extensionof TODIM is proposed to solve problems with uncertaindata resulting in fuzzy TODIM In practical applications thetriangular shape of the membership function is often usedto represent fuzzy numbers Fuzzy models that used TFNsproved highly effective for solving decision making problemswhen information is imprecise Hence this study providessome basic definitions of fuzzy set theory see Tseng [24]

In this study we first use Fuzzy TOPSIS for weighting thecriteria considering time factor and then we use this datato combine with data from TODIM which calculate fromcomparisons between alternatives regarding to criteria andfinally we will able to determine which alternative will bemore reliable and more effective in any duration of time

Table 3 Criteria for green supply chain

Annual growth in green products (1198621)

Cost of revenue extent that it remains flat to decreases each year(1198622)

Industry leadership green market share (1198623)

Customer retentionpercentage of growth with existingcustomers (119862

4)

Customer acquisition the number of new green customerstotalrevenue to new green customers (119862

5)

Life cycle assessment (1198626)

Table 4 Experts data collection

Criteria 10 months 20 months 30 months1198621

P MP VP MP F MP MG MP F1198622

MP VP F F MP MP MP F MG1198623

VP F G MP MP G F MG VG1198624

F G F MP VG G MG G MP1198625

G F P G VG G P F F1198626

F MP G G F F MP MP P

3 Illustrative Method and Example

31 Proposed Approach Weuse the abovementionedmethodto find out the best supplier in green supply chain manage-ment considering time variations as follows (Figure 4)

Step 1 A group of decision makers identified the criteria inGSCM which are important and also will be changed duringtime horizons

Step 2 Collect the opinion of decisionmakers with linguisticvariables (Table 1) and define FTF for each important crite-rion

Step 3 Use fuzzy TOPSIS to evaluate the criteria during timeand initial weight of each criterion

Step 4 Collect the opinion of decision makers on alterna-tives respectively with linguistic variables (Table 2)

Step 5 Use TODIM for evaluating the final weight of eachcriterion against alternatives and the relationship betweenthem

Step 6 Combine the results of TODIM and fuzzy TOPSISto find out which supplier will be more effective from ourcompanyrsquos imagination now and in the future regarding thecondition changes

32 An Illustrative Example

Step 1 In this section we study on green supplier selectionproblem based on time factor in a Tier company in IranIn this company regarding the expert researches we have 6important criteria The data have been collected from three

Advances in Fuzzy Systems 7

Data collection for criteria with

respect to time

Fuzzy TOPSIS

Data collection for evaluating alternative

with respect to criteria

TODIM

ExpertsImportant criteria of GSCM

Combination of TOPSIS and TODIM

Experts (TFNs)

Experts (interval TFNs)

Figure 4 Flowchart of the method

Table 5 Green supply chain criteria considering time

15 months 25 months1198621

(364 558 797) (497 708 9825)1198622

(182 416 632) (5 866 899)1198623

(283 516 699) (95 1082 1166)1198624

(833 982 1049) (033 284 551)1198625

(982 114 116) (0 0 0)1198626

(135 155 1723) (0 0 0)

expert decision makers who have more than 10 years ofexperience in this area and also have the ability to predict themarket and its requirements in future Table 3 represents thecriteria of GSCM which is considered in this study for ourthree suppliers Table 4 represents data and then uses themto produce three functions for each criterion but because oflack of data we consider just three points of time 10 20 and30 months after data collection

Table 6 Criteria weights using fuzzy TOPSIS considering timeperiods

15 months 25 months1198621

05412 053221198622

055 05331198623

05437 0521198624

05256 0551198625

0523 01198626

051 0

Table 7 Criteria ranking for green supply chain considering time

15 months 25 months1198621

3 31198622

1 21198623

2 41198624

4 11198625

5 51198626

6 5

Step 2 The data in Table 4 were prepared based on linguisticfuzzy variable from Table 1 and data are collected from threeexperts to find out their opinion on criteria with respect totime After collecting data with linguistic variable we canchange them to fuzzy triangular number as represented inTable 1 and then FTF must be represented There are twomethods to calculate the FTF

First In this case we use average of slopes when the deviationbetween slopes is low and is not very important in ourexample for criterion 1 we have the following

Criterion 1 (033133) (163656) (357)

pessimistic FTF (1198621) = 119910 = 0127119905 10 lt 119905 lt 20

119910 = 014119905 20 lt 119905 lt 30

then 997904rArr pessimistic FTF (1198621) = 01335119905

(24)

The FTF from (2) for criterion 1 is

FTF (1198621) =

119910 = 0185119905 + 52 optimistic119910 = 015119905 + 333 normal119910 = 01335119905 + 164 pessimistic

(25)

To get precise decisions it is recommended to use theseparate functions for each period of time For example forcriterion 1 using the function between 10 and 20 months ismore exact than considering the FTF for any time Calcula-tions for finding the best green supplier in our study for the15th and 25th months are presented in Table 5 for examplefor Criterion 1 we have

FTF (1198621) =

119910 = 0185 times 15 + 52 = 7975 optimistic119910 = 015 times 15 + 333 = 558 normal119910 = 01335 times 15 + 164 = 364 pessimistic

(26)

8 Advances in Fuzzy Systems

Table 8 Interval-valued TFNs decision matrix

Criteriaalternative 1198601

1198602

1198603

1198621

[(55 75) 9 (95 10)] [(45 55) 7 (8 95)] [(45 55) 7 (8 95)]1198622

[(55 75) 9 (95 10)] [(85 95) 10 (10 10)] [(0 15) 3 (45 55)]1198623

[(45 55) 7 (8 95)] [(25 35) 5 (65 75)] [(45 55) 7 (8 95)]1198624

[(55 75) 9 (95 10)] [(0 15) 3 (45 55)] [(25 35) 5 (65 75)]1198625

[(85 95) 10 (10 10)] [(55 75) 9 (95 10)] [(45 55) 7 (8 95)]1198626

[(45 55) 7 (8 95)] [(0 0) 0 (1 15)] [(85 95) 10 (10 10)]

By using FTF we will have the variance for calculationsthat is useful in some other analysis which could not beachieved by ordinary triangular fuzzy numbers

Second In this method we did not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable For example for criterion 1198624 we have

FTF (1198624) =

optimistic

119910 = 0266119905 + 8

10 lt 119905 lt 20

119910 = minus0266119905 + 966

20 lt 119905 lt 30

normal

119910 = 0233119905 + 633

10 lt 119905 lt 20

119910 = minus0233119905 + 866

20 lt 119905 lt 30

pessimistic

119910 = 0166119905 + 433

10 lt 119905 lt 20

119910 = minus0166119905 + 7

20 lt 119905 lt 30

(27)

We use the first method for C1 and second method forother criteria

Step 3 Table 5 shows that calculated numbers come fromFTF for criteria these data will be the input for fuzzy TOPSISin (12) and (13) The results of calculations are representedin Table 6 This data will be the initial weight of TODIMapproach

By using the fuzzy TOPSIS method we have Table 6As shown in Table 6 we will have different ranking for

criteria with respect to expert opinion changes during thetime that will be measurable by FTFs This evaluation helpsus to know about the importance of each criterion in thesetimes and gives us the perspective for more effective actionsin our company in future and Table 7 presents the ranking ofcriteria with respect to time

Step 4 Table 8 shows the expertsrsquo opinion on alternatives ineach criteria these data come from the linguistic variablesand then change to interval valued triangular fuzzy numbersby Table 2

Table 9 Matrix of alternative scores with respect to criteria

1198601

1198602

1198603

1198621

04232 06933 069331198622

04232 01155 145981198623

06933 10532 011551198624

04232 14598 105321198625

01155 04232 069331198626

06933 19204 01155

Step 5 Use TODIMmethods see Tseng et al [3]

By applying TODIM approach first 119889minus 119889lowast have beencalculated from (17) which is shown in Table 9 after thatthe weight of criteria has been calculated from (18) which isshown in Table 10

After calculating weight of each criterion the data mustbe normalized for calculation dominance weight of criteriaThe normalized data is shown in Table 11

Equation (23) calculates the overall value of alternative bynormalizing the corresponding dominance measurementsThe rank of each alternative derives from ordering the alter-natives values The global measures computed the completerank ordering of all alternatives In addition a sensitivityanalysis should then be applied to verify the stability of theresults based on the decision makersrsquo preferencesThe resultsare presented in Table 12

Considering results in Table 12 we will find out whichcriterion is the most important one we can also find outranking of alternatives regarding each criterion

Step 6 Now the final conclusion will be conducted fromTables 12 and 6 and we can find out in next 15 months and25 months which criteria will be more effective and finallywhich alternative will be the first

By combination of Fuzzy TOPSIS which represents theweight of criteria of green supply chain with respect to timeand TODIM method which shows the weight of alternativesaccording to criteria of supply chain we will have weights ofcriteria and alternatives in Tables 13 and 14

4 Conclusions

This study represents a newway to select supplier during timeperiods by using a hybrid MCDM Using fuzzy TOPSIS tohave more accurate weighting method for TODIM and then

Advances in Fuzzy Systems 9

Table 10 Criteria weights

1198621

1198622

1198623

1198624

1198625

1198626

119908119895

0144 0159 0148 0233 0098 0217

Table 11 Normalized scores

1198601

1198602

1198603

1198621

02338 03831 038311198622

02118 0057 0731198623

03723 05657 00621198624

01441 04972 03581198625

00937 03435 05621198626

0254 07037 00423

Table 12 Final weight of criteria by using TODIM

1198621

1198622

1198623

1198624

1198625

1198626

120585 minus059 minus044 minus099 minus052 minus1083 minus0711Normalization 041 056 001 048 0 03Negative numbers explain losses from goal

Table 13 Criteria weightrsquos changes during time

1198621

1198622

1198623

1198624

1198625

1198626

119882 15 022 038 0005 0252 0 0153119882 25 021 029 0005 0264 0 0

Table 14 Alternative weights during time

1198601

1198602

1198603

15 months 047 086 09825 months 032 056 084

combining it with fuzzy time function (FTF) helped us torank the criteria and alternatives in several time periods Theimportant aim of this study was to use fuzzy time functionwith a new approach to consider time for each triangularfuzzy number that helped us to find out which supplier in thefuture according to criteria is suitable for our green supplychain and when we should change our supplier According tothe results shown in Table 14 supplier A3 has a decreasingrate from month 15 to 25 but its decreasing rate is lessthan supplier 2rsquos Regarding these results we can assign ourfuture improvement programs and orders to these suppliersconsidering criteria changes in green supplier selections andalso have some repairing programs due to our vision andmanagerial decisions in future for these companies Also wecan assign our orders to these companies and we proposed anorder plan for these suppliers regarding their portions

Conflict of Interests

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

References

[1] M L Tseng ldquoAn assessment of cause and effect decision-making model for firm environmental knowledge manage-ment capacities in uncertaintyrdquo Environmental Monitoring andAssessment vol 161 no 1ndash4 pp 549ndash564 2010

[2] S Zhang S Liu and R Zhai ldquoAn extended GRA method forMCDM with interval-valued triangular fuzzy assessments andunknown weightsrdquo Computers and Industrial Engineering vol61 no 4 pp 1336ndash1341 2011

[3] M L Tseng K H Tan R J Lina and Y Gengb ldquoMulticriteriaanalysis of green supply chain management using interval-valued fuzzy TODIMrdquo Knowledge-Based Systems 2012

[4] M Izadikhah ldquoGroup decision making process for supplierselectionwith TOPSISmethod under interval-valued intuition-istic fuzzy numbersrdquo Advances in Fuzzy Systems vol 2012Article ID 407942 14 pages 2012

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[6] H J Zimmermann Fuzzy Set Theory and Its ApplicationInternational Thomson Publishing Norwell Mass USA 2001

[7] J C A de Figueiredo and A Perkusich ldquoFaults and timinganalysis in real-time distributed systems a fuzzy time Petri-net-based approachrdquo Fuzzy Sets and Systems vol 83 no 2 pp 143ndash168 1996

[8] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[9] C L Hwang and K Yoon Multiple Attributes Decision MakingMethods and Applications Springer Berlin 1981

[10] M Ekmekcioglu T Kaya and C Kahraman ldquoFuzzy multicri-teria disposal method and site selection for municipal solidwasterdquoWaste Management vol 30 no 8-9 pp 1729ndash1736 2010

[11] S J Chen and C L Hwang ldquoFuzzy multi attribute decisionmakingrdquo vol 375 of lecture notes in economics andmathematicalsystem Springer New York 1992

[12] C Chen ldquoExtensions of the TOPSIS for group decision-makingunder fuzzy environmentrdquo Fuzzy Sets and Systems vol 114 no1 pp 1ndash9 2000

[13] T Chu ldquoSelecting plant location via a fuzzy TOPSIS approachrdquoInternational Journal of Advanced Manufacturing Technologyvol 20 no 11 pp 859ndash864 2002

[14] T Chu and Y Lin ldquoImproved extensions of the TOPSIS forgroup decision making under fuzzy environmentrdquo Journal ofInformation and Optimization Sciences vol 23273 286 pages2002

[15] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoExtension ofthe TOPSIS method for decision-making problems with fuzzydatardquo Applied Mathematics and Computation vol 181 no 2 pp1544ndash1551 2006

[16] T C Chu and Y C Lin ldquoAn interval arithmetic based fuzzyTOPSIS modelrdquo Expert Systems with Applications vol 36 no8 pp 10870ndash10876 2009

[17] D Yong ldquoPlant location selection based on fuzzy TOPSISrdquoInternational Journal of Advanced Manufacturing Technologyvol 28 no 7-8 pp 839ndash844 2006

[18] C T Chen C T Lin and S F Huang ldquoA fuzzy approach forsupplier evaluation and selection in supply chainmanagementrdquoInternational Journal of Production Economics vol 102 no 2 pp289ndash301 2006

10 Advances in Fuzzy Systems

[19] C Kahraman S Cevik N Y Ates andMGulbay ldquoFuzzymulti-criteria evaluation of industrial robotic systemsrdquoComputers andIndustrial Engineering vol 52 no 4 pp 414ndash433 2007

[20] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS integrated with fuzzyAHPrdquo in Proceedings of the 1st International Symposium onComputing in Science and Engineering pp 706ndash713 2010

[21] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

[22] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[23] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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Advances in

FuzzySystems

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Volume 2014

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ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Advances in Fuzzy Systems

0

2

4

6

8

10 20 30 40Time (months)

Customer satisfaction during time

Customer satisfaction method A upper boundMiddleLower bound

Fuzz

y sc

ore (0

ndash10

)

Figure 1 Fuzzy time functions

how can we predict our supplier behavior and how muchtime this supplier can be effective in our system and when wemust change our supplier Previous studies also usedMADMtools for selecting suppliers according to criteria which wasmade it easier with using fuzzy approach Izadikhah [4]used TOPSIS method with interval-valued fuzzy numberin supplier selection and selected best supplier accordingto criteria in our study we used TODIM fuzzy because ofits sensitivity to evaluate data and show how much we canimprove our system with changing supplier also we usedfuzzy time function to show changes in other suppliersand finally we can improve our supply chain with respectto this data and estimation of suppliers action Hence theevolution approach often implemented ineffectively becausemanagement does not effectively use appropriate knowledgeand experience on GSCM Due to the impact of this lackmanagement is not confident that supplier selection hasbeen studied and applied to a set of criteria to maximizefirmsrsquo green supply chain benefits How could managementapply knowledge of previously (successful and unsuccessful)supplier selections to support future decision making Theobjective of this study is to create a mechanism that couldassist managers in analyzing and selecting green suppliersduring time horizons and this will help us to predict a picturefor future of the organization activities for these targetswe use a hybrid fuzzy TOPSIS-TODIM decision makingprocess considering fuzzy time function We will continuethis paper by literature review on fuzzy time function andfuzzy TOPSIS and also TODIM in Section 2 and also havesome necessary explanations about the above techniques inSection 3 we show our proposed method in this paper byusing an illustrative example for green supplier selectionFinally in Section 4 conclusions will be presented

2 Material and Methods

21 Fuzzy Logic A fuzzy set is a class of objects with gradesof membership A membership function is between zero and

one Zadeh [5] Fuzzy logic is derived from fuzzy set theory todeal with reasoning that is approximate rather than preciseIt allows the model to easily incorporate various subjectexpertsrsquo advice in developing critical parameter estimatesZimmermann [6] In other words fuzzy logic enables us tohandle uncertainty

There are some kinds of fuzzy numbers Among the vari-ous shapes of fuzzy number triangular fuzzy number (TFN)is the most popular one It is represented with three pointsas 119860 = (119886

1 1198862 1198863) The membership function is illustrated in

(1) Let119860 and 119861 be defined as119860 = (1198861 1198862 1198863) 119861 = (119887

1 1198872 1198873)

Then119862 = (1198861+1198871 1198862+1198872 1198863+1198873) is a sumof these twonumbers

Besides119863 = (1198861minus1198873 1198862minus1198872 1198863minus1198871) is the subtraction of them

120583119886=

0 119909 lt 1198861

119909 minus 1198861

1198862minus 1198861

1198862ge 119909 ge 119886

1

1198863minus 119909

1198863minus 1198862

1198862ge 119909 ge 119886

1

0 otherwise

(1)

22 Fuzzy Time Function (FTF) Fuzzy time function isan approach to considering time as an important factorin uncertainty In many situations we have to change ourapproach because of the uncertainty and changes in criteriaduring time Sometimes these kinds of changes cause extracost for organization With fuzzy time function we couldpredict when some new factors like technology will be usedin our system and how they could affect our decisions deFigueiredo and Perkusich [7] used fuzzy time function infault and timing analysis in real time where fuzzy timefunction used to verify the possibility of the input tokens inplaces Yoneyama [8] used fuzzy set theory for time-delaysystems that considered the uncertain Takgai-Sugeno fuzzymodel with time delay In this paper we used a new aspect offuzzy time function which represents a combination of timeand fuzzy triangular numbers

A fuzzy time function consists of three sections opti-mistic normal and pessimistic related to time and these linesdo not cross each otherrsquos (Figure 1)

In fact this function in each time represents a fuzzytriangular number (3) which considers all three possibilitiesIn many organizations some methods or some criteria willbe out of date after some years or new equipment or methodwill be better than the older ones and these changes will effectin supplier selection methods and factors By using FTF wecould consider them in our decisions and can also upgradeour systems based on expertrsquos viewpoints Considering theabove descriptions we will have FTF for every criterion as

FTF (1198881) =

1198651

for optimistic1198652

for normal1198653

for pessimistic(2)

Advances in Fuzzy Systems 3

According to (2) FTF represents the triangular fuzzynumber during the time

119860 = (1198861 1198862 1198863) 997888rarr FTF = (119865

1 1198652 1198653) (3)

For computing the FTF in this paper we introduce twoways

First In this case we use average of slopes when the deviationbetween slopes of criteria is low and is not very important andafter this we use (2) for these criteria

To get precise decisions it is recommended to use theseparate functions for each period of time

Second In this method we do not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable

In some cases which fuzzy time function has morecomplicated function we must use the combination of thesetwo methods

23 Fuzzy TOPSIS TOPSIS one of the classical multicriteriadecision making methods was developed by Hwang andYoon [9] It is based on the concept that the selected alterna-tive should have the shortest distance from the positive idealsolution (PIS) and the farthest from the negative ideal solu-tion (NIS) TOPSIS also provides an easily understandableand programmable calculation procedure It has the abilityof taking various criteria with different units into accountsimultaneously see Ekmekcioglu et al [10] A number offuzzy TOPSIS methods have been developed in recent yearsChen and Hwang [11] first applied fuzzy numbers to establishfuzzy TOPSIS method in which relative closeness for eachalternative is evaluated based on fuzzy arithmetic operationsChen [12] extends the TOPSIS method to fuzzy groupdecision making situations by considering triangular fuzzynumbers and defining crisp Euclidean between two fuzzynumbers Chu [13] and Chu and Lin [14] further improvedthe methodology proposed by Chen [12] Jahanshahloo et al[15] and Chu and Lin [16] extended the fuzzy TOPSISmethod based on alpha level sets with interval arithmeticFuzzy TOPSIS has been introduced for variousmultiattributedecision-making problems Yong [17] used fuzzy TOPSIS forplanet location selection Chen et al [18] used fuzzy TOPSISfor supplier selection and Kahraman et al [19] used fuzzyTOPSIS to select municipal solid waste disposal methodand site Kutlu and Ekmekcioglu [20] used a modifiedfuzzy TOPSIS integrated with fuzzy AHP to propose newFMEA failure modes and effects analysis which overcomesthe shortcomings of traditional FMEA Kaya and Kahraman[21] proposed a modified fuzzy TOPSIS for the best energytechnology selection

In the following Chenrsquos fuzzy TOPSIS method isexplained Chen [12] extends the TOPSIS method to fuzzygroup decision making situations by considering triangularfuzzy numbers and defining crisp Euclidean distance betweentwo fuzzy numbers In Chenrsquos fuzzy TOPSIS linguisticpreferences can easily be converted to fuzzy numbers which

Table 1 Linguistic variable representing triangular fuzzy numbers

Fuzzy evaluation scores for alternativesLinguistic terms Fuzzy scoreVery poor (VP) (0 0 1)Poor (P) (0 1 3)Medium poor (MP) (1 3 5)Fair (F) (3 5 7)Medium good (MG) (5 7 9)Good (G) (7 9 10)Very good (VG) (9 10 10)

are allowed to be used in calculations see Ekmekcioglu et al[10] Kutlu and Ekmekcioglu [22] introduced fuzzy FMEAusing TOPSIS and they used linguistic data for FMEA whereeach of the linguistic variables represents triangular fuzzynumbers (Table 1)

In this study the linguistic variables used for each cri-terion and we will make FTF for each criterion duringtimes At the beginning weights of the criteria and fuzzyratings of alternatives with respect to each criterion have beencalculated and the fuzzy multicriteria of decision-makingproblem can be expressed in matrix format as

119863 = (

11990911 119909

1119899

d

1199091198981

sdot sdot sdot 119909119898119899

)

119882 = [1199081 1199082 119908

119899] 119895 = 1 2 119899

(4)

where 119909119894119895is the rating of the alternative 119860

119894according to

criterion 119895 (ie119862119895) and 119908

119895denotes the importance weight of

119862119895 These linguistic variables can be described by triangular

fuzzy numbers 119883119894119895= (119886119894119895 119887119894119895 119888119894119895) To avoid the complicated

normalization formulawe used in classical TOPSIS the linearscale transformation is used here to transform the variouscriteria scales into a comparable scale Therefore we canobtain the normalized fuzzy decision matrix denoted by 119877

= [119903119894119895]119898times119899 (5)

where 119861 and 119862 are the set of benefit criteria and cost criteriarespectively and

119903 = (

119886119894119895

119888lowast

119895

119887119894119895

119888lowast

119895

119888119894119895

119888lowast

119895

) 119895 isin 119861

119903 = (

119886minus

119895

119888119894119895

119887minus

119895

119887119894119895

119888minus

119895

119886119894119895

) 119895 isin 119862

119888lowast

119895= max119894

119888119894119895

if 119895 isin 119861

119886minus

119895= min119894

119886119894119895

if 119894 isin 119862

(6)

The above normalization method preserves the ranges ofnormalized fuzzy decision matrix as

= [119881119894119895]119898times119899 119894 = 1 2 119898 119895 = 1 2 119899 (7)

4 Advances in Fuzzy Systems

where

119881119894119895= 119903119894119895sdot 119889 (119862

119895) (8)

According to the weighted normalized fuzzy decisionmatrix (119881

119894119895) we know that the element V

119894119895is positive

normalized triangular fuzzy numbers and will be in [0 1]Then we can define the fuzzy positive-ideal solution (FPIS119860lowast) and fuzzy negative-ideal solution (FPIS 119860minus) as

119860lowast= (Vlowast1 Vlowast2 Vlowast3)

119860minus= (Vminus1 Vminus2 Vminus3)

(9)

where

Vlowast119895= (1 1 1) Vminus

119895= (0 0 0) 119895 = 1 2 119899 (10)

The distance of each alternative from 119860lowast and 119860minus can be

currently calculated as

119889lowast

119894=

119899

sum

119895=1

120575 (V119894119895 Vlowast119895)

119889minus

119894=

119899

sum

119895=1

120575 (V119894119895 Vminus119895)

(11)

where 119889(sdot) is the distance between two fuzzy numberscalculated as follows

119889 (119901 120591) = radic1

3[(1199011minus 1205911)2

+ (1199012minus 1205912)2

+ (1199013minus 1205913)2

]

(12)

where 119901 = (1199011 1199012 1199013) and 120591 = (120591

1 1205912 1205913) are two triangular

fuzzy numbersThe closeness coefficient of each alternative iscalculated as

119862119862119894=

119889minus

119895

119889lowast

119895+ 119889minus

119895

(13)

Obviously when 119862119862119894is near to 1 alternative 119860

119894is closer

to the (FPIS 119860lowast) and farther from (FPIS 119860minus) Thereforeaccording to the closeness coefficient we can determine theranking order of all alternatives and select the best one amonga set of feasible alternatives

24 TODIM

241 TODIM Method TODIM is a discrete multicriteriamethod founded on prospect theory The TODIM methodhas been successfully used and empirically validated in dif-ferent applications This is an experimental method based onhow people make effective decisions in risky conditions Theshape of the value function of TODIM is identical to prospecttheoryrsquos gain and loss function The global multicriteriavalue function of TODIM aggregates all measures of gainsand losses by considering all criteria Gomes and Rangel[23] apply TODIM to investigate and recommend options

for upstream projects for the natural gas reserves recentlydiscovered in the Mexilho field in the Santos Basin BrazilIn addition Gomes and Rangel presented an evaluation ofresidential properties with real estate agents in Brazil anddefine a reference value for the rents of these propertiesrsquocharacteristics using the TODIM method for multicriteriadecisions This approach can assist professionals in the realestate market to evaluate alternatives clearly using the criteriadefined by specialists In general TODIM can be used forqualitative and quantitative criteria Verbal scales of qualita-tive criteria are converted into cardinal scales and both typesof scales are normalized The relative measure of dominanceof one alternative over another alternative is determined foreach pair of alternatives This measure is computed as thesum of all criteria of relative gain and loss values for thesealternativesThis sumwill be a gain a loss or zero dependingon the performance of each alternative with respect to eachcriterion Tseng et al [3] apply TODIM and TFNs to selectgreen supplier chain They used TODIM to find the bestsupplier but the first weighting for TODIM with using fuzzyset theory does not consider the relationship between thecriteria more obvious than fuzzy TOPSIS with combinationof these two approaches the TODIM input will be moreaccurate and the result will be more correct consequentlyWeuse a new combination of TODIM and fuzzy TOPSIS andFTF in our proposed method

In previous methods used in TODIM or other MCDMmethods data have been collected based on alternative andcriteria comparisons but we use interval-valued triangularfuzzy numbers which consider the criteria with respect toalternatives in a deterministic time and these data collectionwill be continued in other times which are important fordecision makers or when we have some changes in criteriaor alternatives Table 2 shows transformation of linguisticcriteria to fuzzy-interval triangular numbers

Let 119860 and 119861 be defined as TFN 119886 = (1198861 1198862 1198863) 119887 =

(1198871 1198872 1198873) The distance between 119886 and 119887 is

120575 (119886 + ) = radic1

3[(1198861minus 1198871)2

+ (1198862minus 1198872)2

+ (1198863minus 1198873)2

]

(14)

The TFN is based on a three-value judgment the mini-mum possible value 119886

1 the mean value 119886

2 and the maximum

possible value 1198863 But the interval-TFN values that have been

used in this paper have 5 parameters on the other hand theseare two TFN numbers which have the same middle point(Figure 2)119888119894119895is expressed as an interval-value TFN where

119862 = [(1198861 1198861015840

1) 1198862 (1198861015840

3 1198863)] (15)

Given 119862119894119895= [(1198861 1198861015840

1) 1198862 (1198861015840

3 1198863)] the normalized perfor-

mance rating is

Advances in Fuzzy Systems 5

0

02

04

06

08

1

12

a1 a3a998400

1a998400

2a998400

3

Figure 2 An interval-valued TFN

Table 2 Corresponding TFNS for linguistic preferences

Linguistic preferences Interval-valued TFNSVery poor [(0 0) 0 (1 15)]Poor [(0 05) 1 (25 35)]Medium poor [(0 15) 3 (45 55)]Fair [(25 35) 5 (65 75)]Medium good [(45 55) 7 (8 95)]Good [(55 75) 9 (95 10)]Very good [(85 95) 10 (10 10)]

119903119894119895= [(

119886119894119895

119889+

119895

1198861015840

119894119895

119889+

119895

)

119887119894119895

119889+

119895

(

1198861015840

119894119895

119889+

119895

119886119894119895

119889+

119895

)] 119894 = 1 2 119898

119895 = 1 2 119899 for 119895 isin 119868

119903119894119895= [(

119886minus

119895

119889119894119895

119886minus

119895

1198891015840

119894119895

)

119886minus

119895

119887119894119895

(

119886minus

119895

1198861015840

119894119895

119886minus

119895

119886119894119895

)] 119894 = 1 2 119898

119895 = 1 2 119899 for 119895 isin 119869(16)

where 119889+119895= max119888

119894119895 119894 = 1 119898 and 119886minus

119895= min119886

119894119895 119894 =

1 119898 119903119894119895= [(119897

119894119895 1198971015840

119894119895) 119898119894119895 (119906119894119895 1199061015840

119894119895)] 119877 = [119903

119894119895]119898lowast119899

and1198770= (11990301 11990302 119903

0119899) = ([(1 1) 1 (1 1)][(1 1) 1 (1 1)]

[(1 1) 1 (1 1)])The distance between the reference value and each com-

parison value can be calculated by using definition (2) asfollows

120575(1)

119894119895= radic

1

3[(1198971015840

119894119895minus 1)2

+ (119898119894119895minus 1)2

+ (1199061015840

119894119895minus 1)2

]

120575(2)

119894119895= radic

1

3[(119897119894119895minus 1)2

+ (119898119894119895minus 1)2

+ (119906119894119895minus 1)2

]

(17)

These calculations are used to determine the distancebetween the reference value and the comparison value in theinterval after calculation we have a new interval TFN foreveryTFNas 120575

119894119895= [120575(1)

119894119895 120575(2)

119894119895]Theweight vector of the criteria

is calculated according to Zhnag et al [2]

119908119895=

sum119898

119894=1(120575(1)

119894119895+ 120575(2)

119894119895)

sum119898

119894=1sum119899

119894=1(120575(1)

119894119895+ 120575(2)

119894119895)

(18)

0 2 4 6 8 10minus10 minus8 minus6 minus4 minus2

Gains

Value

Losses

minus15

minus1

minus05

0

05

1

15

Figure 3 TODIM value function

242 Prospect Theory The value function used in theprospect theory is described in the form of a power lawexpressed as

V (119909) = 119909120572 if 119909 ge 0

minus120579(minus119909)120573 if 119909 lt 0

(19)

where 120572 and 120573 are parameters related to gains and lossesrespectivelyThe parameter 120579 is a risk factor that is consideredin model and must be greater than one Figure 3 showsprospect value function that must be concave and has an 119878shape form

243 TODIM Formulation The TODIM method uses pair-wise comparisons between the criteria by using technicallysimple resources to eliminate occasional inconsistenciesresulting from these comparisons TODIM allows valuejudgments to be performed on a verbal scale using hier-archy of criteria fuzzy value judgments and interdepen-dence relationships among the alternatives The decisionmatrix consists of alternatives and criteria The alternatives1198601 1198602 119860

119898are viable alternatives 119888

1 1198882 119888

119899are crite-

ria and 119909119894119895indicates the rating of alternative 119860

119894according

to the criteria 119888119895 The weight vector 119908 = (119908

1 1199082 119908

119899)

comprises the individual weights 119908119895(119895 = 1 119899) for each

criterion 119888119895satisfying sum119899

119894=1119908119895= 1 The data of decision

matrix119860 originate fromdifferent sourcesThematrixmust benormalized to be dimensionless and allows various criteria tobe compared with each otherThis study uses the normalizeddecision matrix 119877 = [119903

119894119895]119898times119899

with 119894 = 1 119898 and 119895 =1 119899

119860 = (

11990911 119909

1119899

d

1199091198981

sdot sdot sdot 119909119898119899

) (20)

TODIM then calculates the partial dominance matricesand the final dominance matrix The first calculation that thedecisionmakers must define is a reference criterion (typicallythe criterion with the greatest importance weight)Thereforew119903119888indicates the weight of the criterion 119888 by the reference

criterion 119903 TODIM is expressed by the following equations

6 Advances in Fuzzy Systems

The dominance of an alternative over the other is asfollows

120575 (119860119894 119860119895) =

119898

sum

119888=1

120601119888(119860119894 119860119895)forall(119894119895)

(21)

where

120601 (119860119894 119860119895) =

radic119908119903119888(119909119894119888minus 119909119895119888)

sum119898

119888=1119908119903119888

if (119909119894119888minus 119909119895119888) gt 0

0

if (119909119894119888minus 119909119895119888) = 0

minus1

120579

radic(sum119898

119888=1119908119903119888) (119909119894119888minus 119909119895119888)

119908119903119888

if (119909119894119888minus 119909119895119888) lt 0

(22)

The term 120593119888(119860119894 119860119895) represents the contribution of cri-

terion 119888 (119888 = 1 119898) to the function 120575 (119860119894 119860119895) while

comparing alternative 119894 with alternative 119895 The parameter 120579represents the attention factor of the losses whose mitigationdepends on the specific problem A positive (119909

119894119888minus 119909119895119888) repre-

sents a gain whereas a nil or a negative (119909119894119888minus119909119895119888) represents a

loss The final matrix of dominance is obtained by summingthe partial matrices of dominance for each criterion seeTseng et al [3] The global value of the alternative 119868 isdetermined by normalizing the final matrix of dominanceaccording to the following expression

120585119894=

sum119899

119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

maxsum119899119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

(23)

Ordering the values 120585119894provides the rank of each alterna-

tive and better alternatives have higher values of 120585119894The use of

numerical values in rating alternativesmay be limited in theircapacity to address uncertainties Therefore an extensionof TODIM is proposed to solve problems with uncertaindata resulting in fuzzy TODIM In practical applications thetriangular shape of the membership function is often usedto represent fuzzy numbers Fuzzy models that used TFNsproved highly effective for solving decision making problemswhen information is imprecise Hence this study providessome basic definitions of fuzzy set theory see Tseng [24]

In this study we first use Fuzzy TOPSIS for weighting thecriteria considering time factor and then we use this datato combine with data from TODIM which calculate fromcomparisons between alternatives regarding to criteria andfinally we will able to determine which alternative will bemore reliable and more effective in any duration of time

Table 3 Criteria for green supply chain

Annual growth in green products (1198621)

Cost of revenue extent that it remains flat to decreases each year(1198622)

Industry leadership green market share (1198623)

Customer retentionpercentage of growth with existingcustomers (119862

4)

Customer acquisition the number of new green customerstotalrevenue to new green customers (119862

5)

Life cycle assessment (1198626)

Table 4 Experts data collection

Criteria 10 months 20 months 30 months1198621

P MP VP MP F MP MG MP F1198622

MP VP F F MP MP MP F MG1198623

VP F G MP MP G F MG VG1198624

F G F MP VG G MG G MP1198625

G F P G VG G P F F1198626

F MP G G F F MP MP P

3 Illustrative Method and Example

31 Proposed Approach Weuse the abovementionedmethodto find out the best supplier in green supply chain manage-ment considering time variations as follows (Figure 4)

Step 1 A group of decision makers identified the criteria inGSCM which are important and also will be changed duringtime horizons

Step 2 Collect the opinion of decisionmakers with linguisticvariables (Table 1) and define FTF for each important crite-rion

Step 3 Use fuzzy TOPSIS to evaluate the criteria during timeand initial weight of each criterion

Step 4 Collect the opinion of decision makers on alterna-tives respectively with linguistic variables (Table 2)

Step 5 Use TODIM for evaluating the final weight of eachcriterion against alternatives and the relationship betweenthem

Step 6 Combine the results of TODIM and fuzzy TOPSISto find out which supplier will be more effective from ourcompanyrsquos imagination now and in the future regarding thecondition changes

32 An Illustrative Example

Step 1 In this section we study on green supplier selectionproblem based on time factor in a Tier company in IranIn this company regarding the expert researches we have 6important criteria The data have been collected from three

Advances in Fuzzy Systems 7

Data collection for criteria with

respect to time

Fuzzy TOPSIS

Data collection for evaluating alternative

with respect to criteria

TODIM

ExpertsImportant criteria of GSCM

Combination of TOPSIS and TODIM

Experts (TFNs)

Experts (interval TFNs)

Figure 4 Flowchart of the method

Table 5 Green supply chain criteria considering time

15 months 25 months1198621

(364 558 797) (497 708 9825)1198622

(182 416 632) (5 866 899)1198623

(283 516 699) (95 1082 1166)1198624

(833 982 1049) (033 284 551)1198625

(982 114 116) (0 0 0)1198626

(135 155 1723) (0 0 0)

expert decision makers who have more than 10 years ofexperience in this area and also have the ability to predict themarket and its requirements in future Table 3 represents thecriteria of GSCM which is considered in this study for ourthree suppliers Table 4 represents data and then uses themto produce three functions for each criterion but because oflack of data we consider just three points of time 10 20 and30 months after data collection

Table 6 Criteria weights using fuzzy TOPSIS considering timeperiods

15 months 25 months1198621

05412 053221198622

055 05331198623

05437 0521198624

05256 0551198625

0523 01198626

051 0

Table 7 Criteria ranking for green supply chain considering time

15 months 25 months1198621

3 31198622

1 21198623

2 41198624

4 11198625

5 51198626

6 5

Step 2 The data in Table 4 were prepared based on linguisticfuzzy variable from Table 1 and data are collected from threeexperts to find out their opinion on criteria with respect totime After collecting data with linguistic variable we canchange them to fuzzy triangular number as represented inTable 1 and then FTF must be represented There are twomethods to calculate the FTF

First In this case we use average of slopes when the deviationbetween slopes is low and is not very important in ourexample for criterion 1 we have the following

Criterion 1 (033133) (163656) (357)

pessimistic FTF (1198621) = 119910 = 0127119905 10 lt 119905 lt 20

119910 = 014119905 20 lt 119905 lt 30

then 997904rArr pessimistic FTF (1198621) = 01335119905

(24)

The FTF from (2) for criterion 1 is

FTF (1198621) =

119910 = 0185119905 + 52 optimistic119910 = 015119905 + 333 normal119910 = 01335119905 + 164 pessimistic

(25)

To get precise decisions it is recommended to use theseparate functions for each period of time For example forcriterion 1 using the function between 10 and 20 months ismore exact than considering the FTF for any time Calcula-tions for finding the best green supplier in our study for the15th and 25th months are presented in Table 5 for examplefor Criterion 1 we have

FTF (1198621) =

119910 = 0185 times 15 + 52 = 7975 optimistic119910 = 015 times 15 + 333 = 558 normal119910 = 01335 times 15 + 164 = 364 pessimistic

(26)

8 Advances in Fuzzy Systems

Table 8 Interval-valued TFNs decision matrix

Criteriaalternative 1198601

1198602

1198603

1198621

[(55 75) 9 (95 10)] [(45 55) 7 (8 95)] [(45 55) 7 (8 95)]1198622

[(55 75) 9 (95 10)] [(85 95) 10 (10 10)] [(0 15) 3 (45 55)]1198623

[(45 55) 7 (8 95)] [(25 35) 5 (65 75)] [(45 55) 7 (8 95)]1198624

[(55 75) 9 (95 10)] [(0 15) 3 (45 55)] [(25 35) 5 (65 75)]1198625

[(85 95) 10 (10 10)] [(55 75) 9 (95 10)] [(45 55) 7 (8 95)]1198626

[(45 55) 7 (8 95)] [(0 0) 0 (1 15)] [(85 95) 10 (10 10)]

By using FTF we will have the variance for calculationsthat is useful in some other analysis which could not beachieved by ordinary triangular fuzzy numbers

Second In this method we did not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable For example for criterion 1198624 we have

FTF (1198624) =

optimistic

119910 = 0266119905 + 8

10 lt 119905 lt 20

119910 = minus0266119905 + 966

20 lt 119905 lt 30

normal

119910 = 0233119905 + 633

10 lt 119905 lt 20

119910 = minus0233119905 + 866

20 lt 119905 lt 30

pessimistic

119910 = 0166119905 + 433

10 lt 119905 lt 20

119910 = minus0166119905 + 7

20 lt 119905 lt 30

(27)

We use the first method for C1 and second method forother criteria

Step 3 Table 5 shows that calculated numbers come fromFTF for criteria these data will be the input for fuzzy TOPSISin (12) and (13) The results of calculations are representedin Table 6 This data will be the initial weight of TODIMapproach

By using the fuzzy TOPSIS method we have Table 6As shown in Table 6 we will have different ranking for

criteria with respect to expert opinion changes during thetime that will be measurable by FTFs This evaluation helpsus to know about the importance of each criterion in thesetimes and gives us the perspective for more effective actionsin our company in future and Table 7 presents the ranking ofcriteria with respect to time

Step 4 Table 8 shows the expertsrsquo opinion on alternatives ineach criteria these data come from the linguistic variablesand then change to interval valued triangular fuzzy numbersby Table 2

Table 9 Matrix of alternative scores with respect to criteria

1198601

1198602

1198603

1198621

04232 06933 069331198622

04232 01155 145981198623

06933 10532 011551198624

04232 14598 105321198625

01155 04232 069331198626

06933 19204 01155

Step 5 Use TODIMmethods see Tseng et al [3]

By applying TODIM approach first 119889minus 119889lowast have beencalculated from (17) which is shown in Table 9 after thatthe weight of criteria has been calculated from (18) which isshown in Table 10

After calculating weight of each criterion the data mustbe normalized for calculation dominance weight of criteriaThe normalized data is shown in Table 11

Equation (23) calculates the overall value of alternative bynormalizing the corresponding dominance measurementsThe rank of each alternative derives from ordering the alter-natives values The global measures computed the completerank ordering of all alternatives In addition a sensitivityanalysis should then be applied to verify the stability of theresults based on the decision makersrsquo preferencesThe resultsare presented in Table 12

Considering results in Table 12 we will find out whichcriterion is the most important one we can also find outranking of alternatives regarding each criterion

Step 6 Now the final conclusion will be conducted fromTables 12 and 6 and we can find out in next 15 months and25 months which criteria will be more effective and finallywhich alternative will be the first

By combination of Fuzzy TOPSIS which represents theweight of criteria of green supply chain with respect to timeand TODIM method which shows the weight of alternativesaccording to criteria of supply chain we will have weights ofcriteria and alternatives in Tables 13 and 14

4 Conclusions

This study represents a newway to select supplier during timeperiods by using a hybrid MCDM Using fuzzy TOPSIS tohave more accurate weighting method for TODIM and then

Advances in Fuzzy Systems 9

Table 10 Criteria weights

1198621

1198622

1198623

1198624

1198625

1198626

119908119895

0144 0159 0148 0233 0098 0217

Table 11 Normalized scores

1198601

1198602

1198603

1198621

02338 03831 038311198622

02118 0057 0731198623

03723 05657 00621198624

01441 04972 03581198625

00937 03435 05621198626

0254 07037 00423

Table 12 Final weight of criteria by using TODIM

1198621

1198622

1198623

1198624

1198625

1198626

120585 minus059 minus044 minus099 minus052 minus1083 minus0711Normalization 041 056 001 048 0 03Negative numbers explain losses from goal

Table 13 Criteria weightrsquos changes during time

1198621

1198622

1198623

1198624

1198625

1198626

119882 15 022 038 0005 0252 0 0153119882 25 021 029 0005 0264 0 0

Table 14 Alternative weights during time

1198601

1198602

1198603

15 months 047 086 09825 months 032 056 084

combining it with fuzzy time function (FTF) helped us torank the criteria and alternatives in several time periods Theimportant aim of this study was to use fuzzy time functionwith a new approach to consider time for each triangularfuzzy number that helped us to find out which supplier in thefuture according to criteria is suitable for our green supplychain and when we should change our supplier According tothe results shown in Table 14 supplier A3 has a decreasingrate from month 15 to 25 but its decreasing rate is lessthan supplier 2rsquos Regarding these results we can assign ourfuture improvement programs and orders to these suppliersconsidering criteria changes in green supplier selections andalso have some repairing programs due to our vision andmanagerial decisions in future for these companies Also wecan assign our orders to these companies and we proposed anorder plan for these suppliers regarding their portions

Conflict of Interests

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

References

[1] M L Tseng ldquoAn assessment of cause and effect decision-making model for firm environmental knowledge manage-ment capacities in uncertaintyrdquo Environmental Monitoring andAssessment vol 161 no 1ndash4 pp 549ndash564 2010

[2] S Zhang S Liu and R Zhai ldquoAn extended GRA method forMCDM with interval-valued triangular fuzzy assessments andunknown weightsrdquo Computers and Industrial Engineering vol61 no 4 pp 1336ndash1341 2011

[3] M L Tseng K H Tan R J Lina and Y Gengb ldquoMulticriteriaanalysis of green supply chain management using interval-valued fuzzy TODIMrdquo Knowledge-Based Systems 2012

[4] M Izadikhah ldquoGroup decision making process for supplierselectionwith TOPSISmethod under interval-valued intuition-istic fuzzy numbersrdquo Advances in Fuzzy Systems vol 2012Article ID 407942 14 pages 2012

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[6] H J Zimmermann Fuzzy Set Theory and Its ApplicationInternational Thomson Publishing Norwell Mass USA 2001

[7] J C A de Figueiredo and A Perkusich ldquoFaults and timinganalysis in real-time distributed systems a fuzzy time Petri-net-based approachrdquo Fuzzy Sets and Systems vol 83 no 2 pp 143ndash168 1996

[8] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[9] C L Hwang and K Yoon Multiple Attributes Decision MakingMethods and Applications Springer Berlin 1981

[10] M Ekmekcioglu T Kaya and C Kahraman ldquoFuzzy multicri-teria disposal method and site selection for municipal solidwasterdquoWaste Management vol 30 no 8-9 pp 1729ndash1736 2010

[11] S J Chen and C L Hwang ldquoFuzzy multi attribute decisionmakingrdquo vol 375 of lecture notes in economics andmathematicalsystem Springer New York 1992

[12] C Chen ldquoExtensions of the TOPSIS for group decision-makingunder fuzzy environmentrdquo Fuzzy Sets and Systems vol 114 no1 pp 1ndash9 2000

[13] T Chu ldquoSelecting plant location via a fuzzy TOPSIS approachrdquoInternational Journal of Advanced Manufacturing Technologyvol 20 no 11 pp 859ndash864 2002

[14] T Chu and Y Lin ldquoImproved extensions of the TOPSIS forgroup decision making under fuzzy environmentrdquo Journal ofInformation and Optimization Sciences vol 23273 286 pages2002

[15] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoExtension ofthe TOPSIS method for decision-making problems with fuzzydatardquo Applied Mathematics and Computation vol 181 no 2 pp1544ndash1551 2006

[16] T C Chu and Y C Lin ldquoAn interval arithmetic based fuzzyTOPSIS modelrdquo Expert Systems with Applications vol 36 no8 pp 10870ndash10876 2009

[17] D Yong ldquoPlant location selection based on fuzzy TOPSISrdquoInternational Journal of Advanced Manufacturing Technologyvol 28 no 7-8 pp 839ndash844 2006

[18] C T Chen C T Lin and S F Huang ldquoA fuzzy approach forsupplier evaluation and selection in supply chainmanagementrdquoInternational Journal of Production Economics vol 102 no 2 pp289ndash301 2006

10 Advances in Fuzzy Systems

[19] C Kahraman S Cevik N Y Ates andMGulbay ldquoFuzzymulti-criteria evaluation of industrial robotic systemsrdquoComputers andIndustrial Engineering vol 52 no 4 pp 414ndash433 2007

[20] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS integrated with fuzzyAHPrdquo in Proceedings of the 1st International Symposium onComputing in Science and Engineering pp 706ndash713 2010

[21] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

[22] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[23] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

Submit your manuscripts athttpwwwhindawicom

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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Advances in Fuzzy Systems 3

According to (2) FTF represents the triangular fuzzynumber during the time

119860 = (1198861 1198862 1198863) 997888rarr FTF = (119865

1 1198652 1198653) (3)

For computing the FTF in this paper we introduce twoways

First In this case we use average of slopes when the deviationbetween slopes of criteria is low and is not very important andafter this we use (2) for these criteria

To get precise decisions it is recommended to use theseparate functions for each period of time

Second In this method we do not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable

In some cases which fuzzy time function has morecomplicated function we must use the combination of thesetwo methods

23 Fuzzy TOPSIS TOPSIS one of the classical multicriteriadecision making methods was developed by Hwang andYoon [9] It is based on the concept that the selected alterna-tive should have the shortest distance from the positive idealsolution (PIS) and the farthest from the negative ideal solu-tion (NIS) TOPSIS also provides an easily understandableand programmable calculation procedure It has the abilityof taking various criteria with different units into accountsimultaneously see Ekmekcioglu et al [10] A number offuzzy TOPSIS methods have been developed in recent yearsChen and Hwang [11] first applied fuzzy numbers to establishfuzzy TOPSIS method in which relative closeness for eachalternative is evaluated based on fuzzy arithmetic operationsChen [12] extends the TOPSIS method to fuzzy groupdecision making situations by considering triangular fuzzynumbers and defining crisp Euclidean between two fuzzynumbers Chu [13] and Chu and Lin [14] further improvedthe methodology proposed by Chen [12] Jahanshahloo et al[15] and Chu and Lin [16] extended the fuzzy TOPSISmethod based on alpha level sets with interval arithmeticFuzzy TOPSIS has been introduced for variousmultiattributedecision-making problems Yong [17] used fuzzy TOPSIS forplanet location selection Chen et al [18] used fuzzy TOPSISfor supplier selection and Kahraman et al [19] used fuzzyTOPSIS to select municipal solid waste disposal methodand site Kutlu and Ekmekcioglu [20] used a modifiedfuzzy TOPSIS integrated with fuzzy AHP to propose newFMEA failure modes and effects analysis which overcomesthe shortcomings of traditional FMEA Kaya and Kahraman[21] proposed a modified fuzzy TOPSIS for the best energytechnology selection

In the following Chenrsquos fuzzy TOPSIS method isexplained Chen [12] extends the TOPSIS method to fuzzygroup decision making situations by considering triangularfuzzy numbers and defining crisp Euclidean distance betweentwo fuzzy numbers In Chenrsquos fuzzy TOPSIS linguisticpreferences can easily be converted to fuzzy numbers which

Table 1 Linguistic variable representing triangular fuzzy numbers

Fuzzy evaluation scores for alternativesLinguistic terms Fuzzy scoreVery poor (VP) (0 0 1)Poor (P) (0 1 3)Medium poor (MP) (1 3 5)Fair (F) (3 5 7)Medium good (MG) (5 7 9)Good (G) (7 9 10)Very good (VG) (9 10 10)

are allowed to be used in calculations see Ekmekcioglu et al[10] Kutlu and Ekmekcioglu [22] introduced fuzzy FMEAusing TOPSIS and they used linguistic data for FMEA whereeach of the linguistic variables represents triangular fuzzynumbers (Table 1)

In this study the linguistic variables used for each cri-terion and we will make FTF for each criterion duringtimes At the beginning weights of the criteria and fuzzyratings of alternatives with respect to each criterion have beencalculated and the fuzzy multicriteria of decision-makingproblem can be expressed in matrix format as

119863 = (

11990911 119909

1119899

d

1199091198981

sdot sdot sdot 119909119898119899

)

119882 = [1199081 1199082 119908

119899] 119895 = 1 2 119899

(4)

where 119909119894119895is the rating of the alternative 119860

119894according to

criterion 119895 (ie119862119895) and 119908

119895denotes the importance weight of

119862119895 These linguistic variables can be described by triangular

fuzzy numbers 119883119894119895= (119886119894119895 119887119894119895 119888119894119895) To avoid the complicated

normalization formulawe used in classical TOPSIS the linearscale transformation is used here to transform the variouscriteria scales into a comparable scale Therefore we canobtain the normalized fuzzy decision matrix denoted by 119877

= [119903119894119895]119898times119899 (5)

where 119861 and 119862 are the set of benefit criteria and cost criteriarespectively and

119903 = (

119886119894119895

119888lowast

119895

119887119894119895

119888lowast

119895

119888119894119895

119888lowast

119895

) 119895 isin 119861

119903 = (

119886minus

119895

119888119894119895

119887minus

119895

119887119894119895

119888minus

119895

119886119894119895

) 119895 isin 119862

119888lowast

119895= max119894

119888119894119895

if 119895 isin 119861

119886minus

119895= min119894

119886119894119895

if 119894 isin 119862

(6)

The above normalization method preserves the ranges ofnormalized fuzzy decision matrix as

= [119881119894119895]119898times119899 119894 = 1 2 119898 119895 = 1 2 119899 (7)

4 Advances in Fuzzy Systems

where

119881119894119895= 119903119894119895sdot 119889 (119862

119895) (8)

According to the weighted normalized fuzzy decisionmatrix (119881

119894119895) we know that the element V

119894119895is positive

normalized triangular fuzzy numbers and will be in [0 1]Then we can define the fuzzy positive-ideal solution (FPIS119860lowast) and fuzzy negative-ideal solution (FPIS 119860minus) as

119860lowast= (Vlowast1 Vlowast2 Vlowast3)

119860minus= (Vminus1 Vminus2 Vminus3)

(9)

where

Vlowast119895= (1 1 1) Vminus

119895= (0 0 0) 119895 = 1 2 119899 (10)

The distance of each alternative from 119860lowast and 119860minus can be

currently calculated as

119889lowast

119894=

119899

sum

119895=1

120575 (V119894119895 Vlowast119895)

119889minus

119894=

119899

sum

119895=1

120575 (V119894119895 Vminus119895)

(11)

where 119889(sdot) is the distance between two fuzzy numberscalculated as follows

119889 (119901 120591) = radic1

3[(1199011minus 1205911)2

+ (1199012minus 1205912)2

+ (1199013minus 1205913)2

]

(12)

where 119901 = (1199011 1199012 1199013) and 120591 = (120591

1 1205912 1205913) are two triangular

fuzzy numbersThe closeness coefficient of each alternative iscalculated as

119862119862119894=

119889minus

119895

119889lowast

119895+ 119889minus

119895

(13)

Obviously when 119862119862119894is near to 1 alternative 119860

119894is closer

to the (FPIS 119860lowast) and farther from (FPIS 119860minus) Thereforeaccording to the closeness coefficient we can determine theranking order of all alternatives and select the best one amonga set of feasible alternatives

24 TODIM

241 TODIM Method TODIM is a discrete multicriteriamethod founded on prospect theory The TODIM methodhas been successfully used and empirically validated in dif-ferent applications This is an experimental method based onhow people make effective decisions in risky conditions Theshape of the value function of TODIM is identical to prospecttheoryrsquos gain and loss function The global multicriteriavalue function of TODIM aggregates all measures of gainsand losses by considering all criteria Gomes and Rangel[23] apply TODIM to investigate and recommend options

for upstream projects for the natural gas reserves recentlydiscovered in the Mexilho field in the Santos Basin BrazilIn addition Gomes and Rangel presented an evaluation ofresidential properties with real estate agents in Brazil anddefine a reference value for the rents of these propertiesrsquocharacteristics using the TODIM method for multicriteriadecisions This approach can assist professionals in the realestate market to evaluate alternatives clearly using the criteriadefined by specialists In general TODIM can be used forqualitative and quantitative criteria Verbal scales of qualita-tive criteria are converted into cardinal scales and both typesof scales are normalized The relative measure of dominanceof one alternative over another alternative is determined foreach pair of alternatives This measure is computed as thesum of all criteria of relative gain and loss values for thesealternativesThis sumwill be a gain a loss or zero dependingon the performance of each alternative with respect to eachcriterion Tseng et al [3] apply TODIM and TFNs to selectgreen supplier chain They used TODIM to find the bestsupplier but the first weighting for TODIM with using fuzzyset theory does not consider the relationship between thecriteria more obvious than fuzzy TOPSIS with combinationof these two approaches the TODIM input will be moreaccurate and the result will be more correct consequentlyWeuse a new combination of TODIM and fuzzy TOPSIS andFTF in our proposed method

In previous methods used in TODIM or other MCDMmethods data have been collected based on alternative andcriteria comparisons but we use interval-valued triangularfuzzy numbers which consider the criteria with respect toalternatives in a deterministic time and these data collectionwill be continued in other times which are important fordecision makers or when we have some changes in criteriaor alternatives Table 2 shows transformation of linguisticcriteria to fuzzy-interval triangular numbers

Let 119860 and 119861 be defined as TFN 119886 = (1198861 1198862 1198863) 119887 =

(1198871 1198872 1198873) The distance between 119886 and 119887 is

120575 (119886 + ) = radic1

3[(1198861minus 1198871)2

+ (1198862minus 1198872)2

+ (1198863minus 1198873)2

]

(14)

The TFN is based on a three-value judgment the mini-mum possible value 119886

1 the mean value 119886

2 and the maximum

possible value 1198863 But the interval-TFN values that have been

used in this paper have 5 parameters on the other hand theseare two TFN numbers which have the same middle point(Figure 2)119888119894119895is expressed as an interval-value TFN where

119862 = [(1198861 1198861015840

1) 1198862 (1198861015840

3 1198863)] (15)

Given 119862119894119895= [(1198861 1198861015840

1) 1198862 (1198861015840

3 1198863)] the normalized perfor-

mance rating is

Advances in Fuzzy Systems 5

0

02

04

06

08

1

12

a1 a3a998400

1a998400

2a998400

3

Figure 2 An interval-valued TFN

Table 2 Corresponding TFNS for linguistic preferences

Linguistic preferences Interval-valued TFNSVery poor [(0 0) 0 (1 15)]Poor [(0 05) 1 (25 35)]Medium poor [(0 15) 3 (45 55)]Fair [(25 35) 5 (65 75)]Medium good [(45 55) 7 (8 95)]Good [(55 75) 9 (95 10)]Very good [(85 95) 10 (10 10)]

119903119894119895= [(

119886119894119895

119889+

119895

1198861015840

119894119895

119889+

119895

)

119887119894119895

119889+

119895

(

1198861015840

119894119895

119889+

119895

119886119894119895

119889+

119895

)] 119894 = 1 2 119898

119895 = 1 2 119899 for 119895 isin 119868

119903119894119895= [(

119886minus

119895

119889119894119895

119886minus

119895

1198891015840

119894119895

)

119886minus

119895

119887119894119895

(

119886minus

119895

1198861015840

119894119895

119886minus

119895

119886119894119895

)] 119894 = 1 2 119898

119895 = 1 2 119899 for 119895 isin 119869(16)

where 119889+119895= max119888

119894119895 119894 = 1 119898 and 119886minus

119895= min119886

119894119895 119894 =

1 119898 119903119894119895= [(119897

119894119895 1198971015840

119894119895) 119898119894119895 (119906119894119895 1199061015840

119894119895)] 119877 = [119903

119894119895]119898lowast119899

and1198770= (11990301 11990302 119903

0119899) = ([(1 1) 1 (1 1)][(1 1) 1 (1 1)]

[(1 1) 1 (1 1)])The distance between the reference value and each com-

parison value can be calculated by using definition (2) asfollows

120575(1)

119894119895= radic

1

3[(1198971015840

119894119895minus 1)2

+ (119898119894119895minus 1)2

+ (1199061015840

119894119895minus 1)2

]

120575(2)

119894119895= radic

1

3[(119897119894119895minus 1)2

+ (119898119894119895minus 1)2

+ (119906119894119895minus 1)2

]

(17)

These calculations are used to determine the distancebetween the reference value and the comparison value in theinterval after calculation we have a new interval TFN foreveryTFNas 120575

119894119895= [120575(1)

119894119895 120575(2)

119894119895]Theweight vector of the criteria

is calculated according to Zhnag et al [2]

119908119895=

sum119898

119894=1(120575(1)

119894119895+ 120575(2)

119894119895)

sum119898

119894=1sum119899

119894=1(120575(1)

119894119895+ 120575(2)

119894119895)

(18)

0 2 4 6 8 10minus10 minus8 minus6 minus4 minus2

Gains

Value

Losses

minus15

minus1

minus05

0

05

1

15

Figure 3 TODIM value function

242 Prospect Theory The value function used in theprospect theory is described in the form of a power lawexpressed as

V (119909) = 119909120572 if 119909 ge 0

minus120579(minus119909)120573 if 119909 lt 0

(19)

where 120572 and 120573 are parameters related to gains and lossesrespectivelyThe parameter 120579 is a risk factor that is consideredin model and must be greater than one Figure 3 showsprospect value function that must be concave and has an 119878shape form

243 TODIM Formulation The TODIM method uses pair-wise comparisons between the criteria by using technicallysimple resources to eliminate occasional inconsistenciesresulting from these comparisons TODIM allows valuejudgments to be performed on a verbal scale using hier-archy of criteria fuzzy value judgments and interdepen-dence relationships among the alternatives The decisionmatrix consists of alternatives and criteria The alternatives1198601 1198602 119860

119898are viable alternatives 119888

1 1198882 119888

119899are crite-

ria and 119909119894119895indicates the rating of alternative 119860

119894according

to the criteria 119888119895 The weight vector 119908 = (119908

1 1199082 119908

119899)

comprises the individual weights 119908119895(119895 = 1 119899) for each

criterion 119888119895satisfying sum119899

119894=1119908119895= 1 The data of decision

matrix119860 originate fromdifferent sourcesThematrixmust benormalized to be dimensionless and allows various criteria tobe compared with each otherThis study uses the normalizeddecision matrix 119877 = [119903

119894119895]119898times119899

with 119894 = 1 119898 and 119895 =1 119899

119860 = (

11990911 119909

1119899

d

1199091198981

sdot sdot sdot 119909119898119899

) (20)

TODIM then calculates the partial dominance matricesand the final dominance matrix The first calculation that thedecisionmakers must define is a reference criterion (typicallythe criterion with the greatest importance weight)Thereforew119903119888indicates the weight of the criterion 119888 by the reference

criterion 119903 TODIM is expressed by the following equations

6 Advances in Fuzzy Systems

The dominance of an alternative over the other is asfollows

120575 (119860119894 119860119895) =

119898

sum

119888=1

120601119888(119860119894 119860119895)forall(119894119895)

(21)

where

120601 (119860119894 119860119895) =

radic119908119903119888(119909119894119888minus 119909119895119888)

sum119898

119888=1119908119903119888

if (119909119894119888minus 119909119895119888) gt 0

0

if (119909119894119888minus 119909119895119888) = 0

minus1

120579

radic(sum119898

119888=1119908119903119888) (119909119894119888minus 119909119895119888)

119908119903119888

if (119909119894119888minus 119909119895119888) lt 0

(22)

The term 120593119888(119860119894 119860119895) represents the contribution of cri-

terion 119888 (119888 = 1 119898) to the function 120575 (119860119894 119860119895) while

comparing alternative 119894 with alternative 119895 The parameter 120579represents the attention factor of the losses whose mitigationdepends on the specific problem A positive (119909

119894119888minus 119909119895119888) repre-

sents a gain whereas a nil or a negative (119909119894119888minus119909119895119888) represents a

loss The final matrix of dominance is obtained by summingthe partial matrices of dominance for each criterion seeTseng et al [3] The global value of the alternative 119868 isdetermined by normalizing the final matrix of dominanceaccording to the following expression

120585119894=

sum119899

119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

maxsum119899119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

(23)

Ordering the values 120585119894provides the rank of each alterna-

tive and better alternatives have higher values of 120585119894The use of

numerical values in rating alternativesmay be limited in theircapacity to address uncertainties Therefore an extensionof TODIM is proposed to solve problems with uncertaindata resulting in fuzzy TODIM In practical applications thetriangular shape of the membership function is often usedto represent fuzzy numbers Fuzzy models that used TFNsproved highly effective for solving decision making problemswhen information is imprecise Hence this study providessome basic definitions of fuzzy set theory see Tseng [24]

In this study we first use Fuzzy TOPSIS for weighting thecriteria considering time factor and then we use this datato combine with data from TODIM which calculate fromcomparisons between alternatives regarding to criteria andfinally we will able to determine which alternative will bemore reliable and more effective in any duration of time

Table 3 Criteria for green supply chain

Annual growth in green products (1198621)

Cost of revenue extent that it remains flat to decreases each year(1198622)

Industry leadership green market share (1198623)

Customer retentionpercentage of growth with existingcustomers (119862

4)

Customer acquisition the number of new green customerstotalrevenue to new green customers (119862

5)

Life cycle assessment (1198626)

Table 4 Experts data collection

Criteria 10 months 20 months 30 months1198621

P MP VP MP F MP MG MP F1198622

MP VP F F MP MP MP F MG1198623

VP F G MP MP G F MG VG1198624

F G F MP VG G MG G MP1198625

G F P G VG G P F F1198626

F MP G G F F MP MP P

3 Illustrative Method and Example

31 Proposed Approach Weuse the abovementionedmethodto find out the best supplier in green supply chain manage-ment considering time variations as follows (Figure 4)

Step 1 A group of decision makers identified the criteria inGSCM which are important and also will be changed duringtime horizons

Step 2 Collect the opinion of decisionmakers with linguisticvariables (Table 1) and define FTF for each important crite-rion

Step 3 Use fuzzy TOPSIS to evaluate the criteria during timeand initial weight of each criterion

Step 4 Collect the opinion of decision makers on alterna-tives respectively with linguistic variables (Table 2)

Step 5 Use TODIM for evaluating the final weight of eachcriterion against alternatives and the relationship betweenthem

Step 6 Combine the results of TODIM and fuzzy TOPSISto find out which supplier will be more effective from ourcompanyrsquos imagination now and in the future regarding thecondition changes

32 An Illustrative Example

Step 1 In this section we study on green supplier selectionproblem based on time factor in a Tier company in IranIn this company regarding the expert researches we have 6important criteria The data have been collected from three

Advances in Fuzzy Systems 7

Data collection for criteria with

respect to time

Fuzzy TOPSIS

Data collection for evaluating alternative

with respect to criteria

TODIM

ExpertsImportant criteria of GSCM

Combination of TOPSIS and TODIM

Experts (TFNs)

Experts (interval TFNs)

Figure 4 Flowchart of the method

Table 5 Green supply chain criteria considering time

15 months 25 months1198621

(364 558 797) (497 708 9825)1198622

(182 416 632) (5 866 899)1198623

(283 516 699) (95 1082 1166)1198624

(833 982 1049) (033 284 551)1198625

(982 114 116) (0 0 0)1198626

(135 155 1723) (0 0 0)

expert decision makers who have more than 10 years ofexperience in this area and also have the ability to predict themarket and its requirements in future Table 3 represents thecriteria of GSCM which is considered in this study for ourthree suppliers Table 4 represents data and then uses themto produce three functions for each criterion but because oflack of data we consider just three points of time 10 20 and30 months after data collection

Table 6 Criteria weights using fuzzy TOPSIS considering timeperiods

15 months 25 months1198621

05412 053221198622

055 05331198623

05437 0521198624

05256 0551198625

0523 01198626

051 0

Table 7 Criteria ranking for green supply chain considering time

15 months 25 months1198621

3 31198622

1 21198623

2 41198624

4 11198625

5 51198626

6 5

Step 2 The data in Table 4 were prepared based on linguisticfuzzy variable from Table 1 and data are collected from threeexperts to find out their opinion on criteria with respect totime After collecting data with linguistic variable we canchange them to fuzzy triangular number as represented inTable 1 and then FTF must be represented There are twomethods to calculate the FTF

First In this case we use average of slopes when the deviationbetween slopes is low and is not very important in ourexample for criterion 1 we have the following

Criterion 1 (033133) (163656) (357)

pessimistic FTF (1198621) = 119910 = 0127119905 10 lt 119905 lt 20

119910 = 014119905 20 lt 119905 lt 30

then 997904rArr pessimistic FTF (1198621) = 01335119905

(24)

The FTF from (2) for criterion 1 is

FTF (1198621) =

119910 = 0185119905 + 52 optimistic119910 = 015119905 + 333 normal119910 = 01335119905 + 164 pessimistic

(25)

To get precise decisions it is recommended to use theseparate functions for each period of time For example forcriterion 1 using the function between 10 and 20 months ismore exact than considering the FTF for any time Calcula-tions for finding the best green supplier in our study for the15th and 25th months are presented in Table 5 for examplefor Criterion 1 we have

FTF (1198621) =

119910 = 0185 times 15 + 52 = 7975 optimistic119910 = 015 times 15 + 333 = 558 normal119910 = 01335 times 15 + 164 = 364 pessimistic

(26)

8 Advances in Fuzzy Systems

Table 8 Interval-valued TFNs decision matrix

Criteriaalternative 1198601

1198602

1198603

1198621

[(55 75) 9 (95 10)] [(45 55) 7 (8 95)] [(45 55) 7 (8 95)]1198622

[(55 75) 9 (95 10)] [(85 95) 10 (10 10)] [(0 15) 3 (45 55)]1198623

[(45 55) 7 (8 95)] [(25 35) 5 (65 75)] [(45 55) 7 (8 95)]1198624

[(55 75) 9 (95 10)] [(0 15) 3 (45 55)] [(25 35) 5 (65 75)]1198625

[(85 95) 10 (10 10)] [(55 75) 9 (95 10)] [(45 55) 7 (8 95)]1198626

[(45 55) 7 (8 95)] [(0 0) 0 (1 15)] [(85 95) 10 (10 10)]

By using FTF we will have the variance for calculationsthat is useful in some other analysis which could not beachieved by ordinary triangular fuzzy numbers

Second In this method we did not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable For example for criterion 1198624 we have

FTF (1198624) =

optimistic

119910 = 0266119905 + 8

10 lt 119905 lt 20

119910 = minus0266119905 + 966

20 lt 119905 lt 30

normal

119910 = 0233119905 + 633

10 lt 119905 lt 20

119910 = minus0233119905 + 866

20 lt 119905 lt 30

pessimistic

119910 = 0166119905 + 433

10 lt 119905 lt 20

119910 = minus0166119905 + 7

20 lt 119905 lt 30

(27)

We use the first method for C1 and second method forother criteria

Step 3 Table 5 shows that calculated numbers come fromFTF for criteria these data will be the input for fuzzy TOPSISin (12) and (13) The results of calculations are representedin Table 6 This data will be the initial weight of TODIMapproach

By using the fuzzy TOPSIS method we have Table 6As shown in Table 6 we will have different ranking for

criteria with respect to expert opinion changes during thetime that will be measurable by FTFs This evaluation helpsus to know about the importance of each criterion in thesetimes and gives us the perspective for more effective actionsin our company in future and Table 7 presents the ranking ofcriteria with respect to time

Step 4 Table 8 shows the expertsrsquo opinion on alternatives ineach criteria these data come from the linguistic variablesand then change to interval valued triangular fuzzy numbersby Table 2

Table 9 Matrix of alternative scores with respect to criteria

1198601

1198602

1198603

1198621

04232 06933 069331198622

04232 01155 145981198623

06933 10532 011551198624

04232 14598 105321198625

01155 04232 069331198626

06933 19204 01155

Step 5 Use TODIMmethods see Tseng et al [3]

By applying TODIM approach first 119889minus 119889lowast have beencalculated from (17) which is shown in Table 9 after thatthe weight of criteria has been calculated from (18) which isshown in Table 10

After calculating weight of each criterion the data mustbe normalized for calculation dominance weight of criteriaThe normalized data is shown in Table 11

Equation (23) calculates the overall value of alternative bynormalizing the corresponding dominance measurementsThe rank of each alternative derives from ordering the alter-natives values The global measures computed the completerank ordering of all alternatives In addition a sensitivityanalysis should then be applied to verify the stability of theresults based on the decision makersrsquo preferencesThe resultsare presented in Table 12

Considering results in Table 12 we will find out whichcriterion is the most important one we can also find outranking of alternatives regarding each criterion

Step 6 Now the final conclusion will be conducted fromTables 12 and 6 and we can find out in next 15 months and25 months which criteria will be more effective and finallywhich alternative will be the first

By combination of Fuzzy TOPSIS which represents theweight of criteria of green supply chain with respect to timeand TODIM method which shows the weight of alternativesaccording to criteria of supply chain we will have weights ofcriteria and alternatives in Tables 13 and 14

4 Conclusions

This study represents a newway to select supplier during timeperiods by using a hybrid MCDM Using fuzzy TOPSIS tohave more accurate weighting method for TODIM and then

Advances in Fuzzy Systems 9

Table 10 Criteria weights

1198621

1198622

1198623

1198624

1198625

1198626

119908119895

0144 0159 0148 0233 0098 0217

Table 11 Normalized scores

1198601

1198602

1198603

1198621

02338 03831 038311198622

02118 0057 0731198623

03723 05657 00621198624

01441 04972 03581198625

00937 03435 05621198626

0254 07037 00423

Table 12 Final weight of criteria by using TODIM

1198621

1198622

1198623

1198624

1198625

1198626

120585 minus059 minus044 minus099 minus052 minus1083 minus0711Normalization 041 056 001 048 0 03Negative numbers explain losses from goal

Table 13 Criteria weightrsquos changes during time

1198621

1198622

1198623

1198624

1198625

1198626

119882 15 022 038 0005 0252 0 0153119882 25 021 029 0005 0264 0 0

Table 14 Alternative weights during time

1198601

1198602

1198603

15 months 047 086 09825 months 032 056 084

combining it with fuzzy time function (FTF) helped us torank the criteria and alternatives in several time periods Theimportant aim of this study was to use fuzzy time functionwith a new approach to consider time for each triangularfuzzy number that helped us to find out which supplier in thefuture according to criteria is suitable for our green supplychain and when we should change our supplier According tothe results shown in Table 14 supplier A3 has a decreasingrate from month 15 to 25 but its decreasing rate is lessthan supplier 2rsquos Regarding these results we can assign ourfuture improvement programs and orders to these suppliersconsidering criteria changes in green supplier selections andalso have some repairing programs due to our vision andmanagerial decisions in future for these companies Also wecan assign our orders to these companies and we proposed anorder plan for these suppliers regarding their portions

Conflict of Interests

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

References

[1] M L Tseng ldquoAn assessment of cause and effect decision-making model for firm environmental knowledge manage-ment capacities in uncertaintyrdquo Environmental Monitoring andAssessment vol 161 no 1ndash4 pp 549ndash564 2010

[2] S Zhang S Liu and R Zhai ldquoAn extended GRA method forMCDM with interval-valued triangular fuzzy assessments andunknown weightsrdquo Computers and Industrial Engineering vol61 no 4 pp 1336ndash1341 2011

[3] M L Tseng K H Tan R J Lina and Y Gengb ldquoMulticriteriaanalysis of green supply chain management using interval-valued fuzzy TODIMrdquo Knowledge-Based Systems 2012

[4] M Izadikhah ldquoGroup decision making process for supplierselectionwith TOPSISmethod under interval-valued intuition-istic fuzzy numbersrdquo Advances in Fuzzy Systems vol 2012Article ID 407942 14 pages 2012

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[6] H J Zimmermann Fuzzy Set Theory and Its ApplicationInternational Thomson Publishing Norwell Mass USA 2001

[7] J C A de Figueiredo and A Perkusich ldquoFaults and timinganalysis in real-time distributed systems a fuzzy time Petri-net-based approachrdquo Fuzzy Sets and Systems vol 83 no 2 pp 143ndash168 1996

[8] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[9] C L Hwang and K Yoon Multiple Attributes Decision MakingMethods and Applications Springer Berlin 1981

[10] M Ekmekcioglu T Kaya and C Kahraman ldquoFuzzy multicri-teria disposal method and site selection for municipal solidwasterdquoWaste Management vol 30 no 8-9 pp 1729ndash1736 2010

[11] S J Chen and C L Hwang ldquoFuzzy multi attribute decisionmakingrdquo vol 375 of lecture notes in economics andmathematicalsystem Springer New York 1992

[12] C Chen ldquoExtensions of the TOPSIS for group decision-makingunder fuzzy environmentrdquo Fuzzy Sets and Systems vol 114 no1 pp 1ndash9 2000

[13] T Chu ldquoSelecting plant location via a fuzzy TOPSIS approachrdquoInternational Journal of Advanced Manufacturing Technologyvol 20 no 11 pp 859ndash864 2002

[14] T Chu and Y Lin ldquoImproved extensions of the TOPSIS forgroup decision making under fuzzy environmentrdquo Journal ofInformation and Optimization Sciences vol 23273 286 pages2002

[15] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoExtension ofthe TOPSIS method for decision-making problems with fuzzydatardquo Applied Mathematics and Computation vol 181 no 2 pp1544ndash1551 2006

[16] T C Chu and Y C Lin ldquoAn interval arithmetic based fuzzyTOPSIS modelrdquo Expert Systems with Applications vol 36 no8 pp 10870ndash10876 2009

[17] D Yong ldquoPlant location selection based on fuzzy TOPSISrdquoInternational Journal of Advanced Manufacturing Technologyvol 28 no 7-8 pp 839ndash844 2006

[18] C T Chen C T Lin and S F Huang ldquoA fuzzy approach forsupplier evaluation and selection in supply chainmanagementrdquoInternational Journal of Production Economics vol 102 no 2 pp289ndash301 2006

10 Advances in Fuzzy Systems

[19] C Kahraman S Cevik N Y Ates andMGulbay ldquoFuzzymulti-criteria evaluation of industrial robotic systemsrdquoComputers andIndustrial Engineering vol 52 no 4 pp 414ndash433 2007

[20] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS integrated with fuzzyAHPrdquo in Proceedings of the 1st International Symposium onComputing in Science and Engineering pp 706ndash713 2010

[21] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

[22] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[23] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Advances in Fuzzy Systems

where

119881119894119895= 119903119894119895sdot 119889 (119862

119895) (8)

According to the weighted normalized fuzzy decisionmatrix (119881

119894119895) we know that the element V

119894119895is positive

normalized triangular fuzzy numbers and will be in [0 1]Then we can define the fuzzy positive-ideal solution (FPIS119860lowast) and fuzzy negative-ideal solution (FPIS 119860minus) as

119860lowast= (Vlowast1 Vlowast2 Vlowast3)

119860minus= (Vminus1 Vminus2 Vminus3)

(9)

where

Vlowast119895= (1 1 1) Vminus

119895= (0 0 0) 119895 = 1 2 119899 (10)

The distance of each alternative from 119860lowast and 119860minus can be

currently calculated as

119889lowast

119894=

119899

sum

119895=1

120575 (V119894119895 Vlowast119895)

119889minus

119894=

119899

sum

119895=1

120575 (V119894119895 Vminus119895)

(11)

where 119889(sdot) is the distance between two fuzzy numberscalculated as follows

119889 (119901 120591) = radic1

3[(1199011minus 1205911)2

+ (1199012minus 1205912)2

+ (1199013minus 1205913)2

]

(12)

where 119901 = (1199011 1199012 1199013) and 120591 = (120591

1 1205912 1205913) are two triangular

fuzzy numbersThe closeness coefficient of each alternative iscalculated as

119862119862119894=

119889minus

119895

119889lowast

119895+ 119889minus

119895

(13)

Obviously when 119862119862119894is near to 1 alternative 119860

119894is closer

to the (FPIS 119860lowast) and farther from (FPIS 119860minus) Thereforeaccording to the closeness coefficient we can determine theranking order of all alternatives and select the best one amonga set of feasible alternatives

24 TODIM

241 TODIM Method TODIM is a discrete multicriteriamethod founded on prospect theory The TODIM methodhas been successfully used and empirically validated in dif-ferent applications This is an experimental method based onhow people make effective decisions in risky conditions Theshape of the value function of TODIM is identical to prospecttheoryrsquos gain and loss function The global multicriteriavalue function of TODIM aggregates all measures of gainsand losses by considering all criteria Gomes and Rangel[23] apply TODIM to investigate and recommend options

for upstream projects for the natural gas reserves recentlydiscovered in the Mexilho field in the Santos Basin BrazilIn addition Gomes and Rangel presented an evaluation ofresidential properties with real estate agents in Brazil anddefine a reference value for the rents of these propertiesrsquocharacteristics using the TODIM method for multicriteriadecisions This approach can assist professionals in the realestate market to evaluate alternatives clearly using the criteriadefined by specialists In general TODIM can be used forqualitative and quantitative criteria Verbal scales of qualita-tive criteria are converted into cardinal scales and both typesof scales are normalized The relative measure of dominanceof one alternative over another alternative is determined foreach pair of alternatives This measure is computed as thesum of all criteria of relative gain and loss values for thesealternativesThis sumwill be a gain a loss or zero dependingon the performance of each alternative with respect to eachcriterion Tseng et al [3] apply TODIM and TFNs to selectgreen supplier chain They used TODIM to find the bestsupplier but the first weighting for TODIM with using fuzzyset theory does not consider the relationship between thecriteria more obvious than fuzzy TOPSIS with combinationof these two approaches the TODIM input will be moreaccurate and the result will be more correct consequentlyWeuse a new combination of TODIM and fuzzy TOPSIS andFTF in our proposed method

In previous methods used in TODIM or other MCDMmethods data have been collected based on alternative andcriteria comparisons but we use interval-valued triangularfuzzy numbers which consider the criteria with respect toalternatives in a deterministic time and these data collectionwill be continued in other times which are important fordecision makers or when we have some changes in criteriaor alternatives Table 2 shows transformation of linguisticcriteria to fuzzy-interval triangular numbers

Let 119860 and 119861 be defined as TFN 119886 = (1198861 1198862 1198863) 119887 =

(1198871 1198872 1198873) The distance between 119886 and 119887 is

120575 (119886 + ) = radic1

3[(1198861minus 1198871)2

+ (1198862minus 1198872)2

+ (1198863minus 1198873)2

]

(14)

The TFN is based on a three-value judgment the mini-mum possible value 119886

1 the mean value 119886

2 and the maximum

possible value 1198863 But the interval-TFN values that have been

used in this paper have 5 parameters on the other hand theseare two TFN numbers which have the same middle point(Figure 2)119888119894119895is expressed as an interval-value TFN where

119862 = [(1198861 1198861015840

1) 1198862 (1198861015840

3 1198863)] (15)

Given 119862119894119895= [(1198861 1198861015840

1) 1198862 (1198861015840

3 1198863)] the normalized perfor-

mance rating is

Advances in Fuzzy Systems 5

0

02

04

06

08

1

12

a1 a3a998400

1a998400

2a998400

3

Figure 2 An interval-valued TFN

Table 2 Corresponding TFNS for linguistic preferences

Linguistic preferences Interval-valued TFNSVery poor [(0 0) 0 (1 15)]Poor [(0 05) 1 (25 35)]Medium poor [(0 15) 3 (45 55)]Fair [(25 35) 5 (65 75)]Medium good [(45 55) 7 (8 95)]Good [(55 75) 9 (95 10)]Very good [(85 95) 10 (10 10)]

119903119894119895= [(

119886119894119895

119889+

119895

1198861015840

119894119895

119889+

119895

)

119887119894119895

119889+

119895

(

1198861015840

119894119895

119889+

119895

119886119894119895

119889+

119895

)] 119894 = 1 2 119898

119895 = 1 2 119899 for 119895 isin 119868

119903119894119895= [(

119886minus

119895

119889119894119895

119886minus

119895

1198891015840

119894119895

)

119886minus

119895

119887119894119895

(

119886minus

119895

1198861015840

119894119895

119886minus

119895

119886119894119895

)] 119894 = 1 2 119898

119895 = 1 2 119899 for 119895 isin 119869(16)

where 119889+119895= max119888

119894119895 119894 = 1 119898 and 119886minus

119895= min119886

119894119895 119894 =

1 119898 119903119894119895= [(119897

119894119895 1198971015840

119894119895) 119898119894119895 (119906119894119895 1199061015840

119894119895)] 119877 = [119903

119894119895]119898lowast119899

and1198770= (11990301 11990302 119903

0119899) = ([(1 1) 1 (1 1)][(1 1) 1 (1 1)]

[(1 1) 1 (1 1)])The distance between the reference value and each com-

parison value can be calculated by using definition (2) asfollows

120575(1)

119894119895= radic

1

3[(1198971015840

119894119895minus 1)2

+ (119898119894119895minus 1)2

+ (1199061015840

119894119895minus 1)2

]

120575(2)

119894119895= radic

1

3[(119897119894119895minus 1)2

+ (119898119894119895minus 1)2

+ (119906119894119895minus 1)2

]

(17)

These calculations are used to determine the distancebetween the reference value and the comparison value in theinterval after calculation we have a new interval TFN foreveryTFNas 120575

119894119895= [120575(1)

119894119895 120575(2)

119894119895]Theweight vector of the criteria

is calculated according to Zhnag et al [2]

119908119895=

sum119898

119894=1(120575(1)

119894119895+ 120575(2)

119894119895)

sum119898

119894=1sum119899

119894=1(120575(1)

119894119895+ 120575(2)

119894119895)

(18)

0 2 4 6 8 10minus10 minus8 minus6 minus4 minus2

Gains

Value

Losses

minus15

minus1

minus05

0

05

1

15

Figure 3 TODIM value function

242 Prospect Theory The value function used in theprospect theory is described in the form of a power lawexpressed as

V (119909) = 119909120572 if 119909 ge 0

minus120579(minus119909)120573 if 119909 lt 0

(19)

where 120572 and 120573 are parameters related to gains and lossesrespectivelyThe parameter 120579 is a risk factor that is consideredin model and must be greater than one Figure 3 showsprospect value function that must be concave and has an 119878shape form

243 TODIM Formulation The TODIM method uses pair-wise comparisons between the criteria by using technicallysimple resources to eliminate occasional inconsistenciesresulting from these comparisons TODIM allows valuejudgments to be performed on a verbal scale using hier-archy of criteria fuzzy value judgments and interdepen-dence relationships among the alternatives The decisionmatrix consists of alternatives and criteria The alternatives1198601 1198602 119860

119898are viable alternatives 119888

1 1198882 119888

119899are crite-

ria and 119909119894119895indicates the rating of alternative 119860

119894according

to the criteria 119888119895 The weight vector 119908 = (119908

1 1199082 119908

119899)

comprises the individual weights 119908119895(119895 = 1 119899) for each

criterion 119888119895satisfying sum119899

119894=1119908119895= 1 The data of decision

matrix119860 originate fromdifferent sourcesThematrixmust benormalized to be dimensionless and allows various criteria tobe compared with each otherThis study uses the normalizeddecision matrix 119877 = [119903

119894119895]119898times119899

with 119894 = 1 119898 and 119895 =1 119899

119860 = (

11990911 119909

1119899

d

1199091198981

sdot sdot sdot 119909119898119899

) (20)

TODIM then calculates the partial dominance matricesand the final dominance matrix The first calculation that thedecisionmakers must define is a reference criterion (typicallythe criterion with the greatest importance weight)Thereforew119903119888indicates the weight of the criterion 119888 by the reference

criterion 119903 TODIM is expressed by the following equations

6 Advances in Fuzzy Systems

The dominance of an alternative over the other is asfollows

120575 (119860119894 119860119895) =

119898

sum

119888=1

120601119888(119860119894 119860119895)forall(119894119895)

(21)

where

120601 (119860119894 119860119895) =

radic119908119903119888(119909119894119888minus 119909119895119888)

sum119898

119888=1119908119903119888

if (119909119894119888minus 119909119895119888) gt 0

0

if (119909119894119888minus 119909119895119888) = 0

minus1

120579

radic(sum119898

119888=1119908119903119888) (119909119894119888minus 119909119895119888)

119908119903119888

if (119909119894119888minus 119909119895119888) lt 0

(22)

The term 120593119888(119860119894 119860119895) represents the contribution of cri-

terion 119888 (119888 = 1 119898) to the function 120575 (119860119894 119860119895) while

comparing alternative 119894 with alternative 119895 The parameter 120579represents the attention factor of the losses whose mitigationdepends on the specific problem A positive (119909

119894119888minus 119909119895119888) repre-

sents a gain whereas a nil or a negative (119909119894119888minus119909119895119888) represents a

loss The final matrix of dominance is obtained by summingthe partial matrices of dominance for each criterion seeTseng et al [3] The global value of the alternative 119868 isdetermined by normalizing the final matrix of dominanceaccording to the following expression

120585119894=

sum119899

119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

maxsum119899119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

(23)

Ordering the values 120585119894provides the rank of each alterna-

tive and better alternatives have higher values of 120585119894The use of

numerical values in rating alternativesmay be limited in theircapacity to address uncertainties Therefore an extensionof TODIM is proposed to solve problems with uncertaindata resulting in fuzzy TODIM In practical applications thetriangular shape of the membership function is often usedto represent fuzzy numbers Fuzzy models that used TFNsproved highly effective for solving decision making problemswhen information is imprecise Hence this study providessome basic definitions of fuzzy set theory see Tseng [24]

In this study we first use Fuzzy TOPSIS for weighting thecriteria considering time factor and then we use this datato combine with data from TODIM which calculate fromcomparisons between alternatives regarding to criteria andfinally we will able to determine which alternative will bemore reliable and more effective in any duration of time

Table 3 Criteria for green supply chain

Annual growth in green products (1198621)

Cost of revenue extent that it remains flat to decreases each year(1198622)

Industry leadership green market share (1198623)

Customer retentionpercentage of growth with existingcustomers (119862

4)

Customer acquisition the number of new green customerstotalrevenue to new green customers (119862

5)

Life cycle assessment (1198626)

Table 4 Experts data collection

Criteria 10 months 20 months 30 months1198621

P MP VP MP F MP MG MP F1198622

MP VP F F MP MP MP F MG1198623

VP F G MP MP G F MG VG1198624

F G F MP VG G MG G MP1198625

G F P G VG G P F F1198626

F MP G G F F MP MP P

3 Illustrative Method and Example

31 Proposed Approach Weuse the abovementionedmethodto find out the best supplier in green supply chain manage-ment considering time variations as follows (Figure 4)

Step 1 A group of decision makers identified the criteria inGSCM which are important and also will be changed duringtime horizons

Step 2 Collect the opinion of decisionmakers with linguisticvariables (Table 1) and define FTF for each important crite-rion

Step 3 Use fuzzy TOPSIS to evaluate the criteria during timeand initial weight of each criterion

Step 4 Collect the opinion of decision makers on alterna-tives respectively with linguistic variables (Table 2)

Step 5 Use TODIM for evaluating the final weight of eachcriterion against alternatives and the relationship betweenthem

Step 6 Combine the results of TODIM and fuzzy TOPSISto find out which supplier will be more effective from ourcompanyrsquos imagination now and in the future regarding thecondition changes

32 An Illustrative Example

Step 1 In this section we study on green supplier selectionproblem based on time factor in a Tier company in IranIn this company regarding the expert researches we have 6important criteria The data have been collected from three

Advances in Fuzzy Systems 7

Data collection for criteria with

respect to time

Fuzzy TOPSIS

Data collection for evaluating alternative

with respect to criteria

TODIM

ExpertsImportant criteria of GSCM

Combination of TOPSIS and TODIM

Experts (TFNs)

Experts (interval TFNs)

Figure 4 Flowchart of the method

Table 5 Green supply chain criteria considering time

15 months 25 months1198621

(364 558 797) (497 708 9825)1198622

(182 416 632) (5 866 899)1198623

(283 516 699) (95 1082 1166)1198624

(833 982 1049) (033 284 551)1198625

(982 114 116) (0 0 0)1198626

(135 155 1723) (0 0 0)

expert decision makers who have more than 10 years ofexperience in this area and also have the ability to predict themarket and its requirements in future Table 3 represents thecriteria of GSCM which is considered in this study for ourthree suppliers Table 4 represents data and then uses themto produce three functions for each criterion but because oflack of data we consider just three points of time 10 20 and30 months after data collection

Table 6 Criteria weights using fuzzy TOPSIS considering timeperiods

15 months 25 months1198621

05412 053221198622

055 05331198623

05437 0521198624

05256 0551198625

0523 01198626

051 0

Table 7 Criteria ranking for green supply chain considering time

15 months 25 months1198621

3 31198622

1 21198623

2 41198624

4 11198625

5 51198626

6 5

Step 2 The data in Table 4 were prepared based on linguisticfuzzy variable from Table 1 and data are collected from threeexperts to find out their opinion on criteria with respect totime After collecting data with linguistic variable we canchange them to fuzzy triangular number as represented inTable 1 and then FTF must be represented There are twomethods to calculate the FTF

First In this case we use average of slopes when the deviationbetween slopes is low and is not very important in ourexample for criterion 1 we have the following

Criterion 1 (033133) (163656) (357)

pessimistic FTF (1198621) = 119910 = 0127119905 10 lt 119905 lt 20

119910 = 014119905 20 lt 119905 lt 30

then 997904rArr pessimistic FTF (1198621) = 01335119905

(24)

The FTF from (2) for criterion 1 is

FTF (1198621) =

119910 = 0185119905 + 52 optimistic119910 = 015119905 + 333 normal119910 = 01335119905 + 164 pessimistic

(25)

To get precise decisions it is recommended to use theseparate functions for each period of time For example forcriterion 1 using the function between 10 and 20 months ismore exact than considering the FTF for any time Calcula-tions for finding the best green supplier in our study for the15th and 25th months are presented in Table 5 for examplefor Criterion 1 we have

FTF (1198621) =

119910 = 0185 times 15 + 52 = 7975 optimistic119910 = 015 times 15 + 333 = 558 normal119910 = 01335 times 15 + 164 = 364 pessimistic

(26)

8 Advances in Fuzzy Systems

Table 8 Interval-valued TFNs decision matrix

Criteriaalternative 1198601

1198602

1198603

1198621

[(55 75) 9 (95 10)] [(45 55) 7 (8 95)] [(45 55) 7 (8 95)]1198622

[(55 75) 9 (95 10)] [(85 95) 10 (10 10)] [(0 15) 3 (45 55)]1198623

[(45 55) 7 (8 95)] [(25 35) 5 (65 75)] [(45 55) 7 (8 95)]1198624

[(55 75) 9 (95 10)] [(0 15) 3 (45 55)] [(25 35) 5 (65 75)]1198625

[(85 95) 10 (10 10)] [(55 75) 9 (95 10)] [(45 55) 7 (8 95)]1198626

[(45 55) 7 (8 95)] [(0 0) 0 (1 15)] [(85 95) 10 (10 10)]

By using FTF we will have the variance for calculationsthat is useful in some other analysis which could not beachieved by ordinary triangular fuzzy numbers

Second In this method we did not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable For example for criterion 1198624 we have

FTF (1198624) =

optimistic

119910 = 0266119905 + 8

10 lt 119905 lt 20

119910 = minus0266119905 + 966

20 lt 119905 lt 30

normal

119910 = 0233119905 + 633

10 lt 119905 lt 20

119910 = minus0233119905 + 866

20 lt 119905 lt 30

pessimistic

119910 = 0166119905 + 433

10 lt 119905 lt 20

119910 = minus0166119905 + 7

20 lt 119905 lt 30

(27)

We use the first method for C1 and second method forother criteria

Step 3 Table 5 shows that calculated numbers come fromFTF for criteria these data will be the input for fuzzy TOPSISin (12) and (13) The results of calculations are representedin Table 6 This data will be the initial weight of TODIMapproach

By using the fuzzy TOPSIS method we have Table 6As shown in Table 6 we will have different ranking for

criteria with respect to expert opinion changes during thetime that will be measurable by FTFs This evaluation helpsus to know about the importance of each criterion in thesetimes and gives us the perspective for more effective actionsin our company in future and Table 7 presents the ranking ofcriteria with respect to time

Step 4 Table 8 shows the expertsrsquo opinion on alternatives ineach criteria these data come from the linguistic variablesand then change to interval valued triangular fuzzy numbersby Table 2

Table 9 Matrix of alternative scores with respect to criteria

1198601

1198602

1198603

1198621

04232 06933 069331198622

04232 01155 145981198623

06933 10532 011551198624

04232 14598 105321198625

01155 04232 069331198626

06933 19204 01155

Step 5 Use TODIMmethods see Tseng et al [3]

By applying TODIM approach first 119889minus 119889lowast have beencalculated from (17) which is shown in Table 9 after thatthe weight of criteria has been calculated from (18) which isshown in Table 10

After calculating weight of each criterion the data mustbe normalized for calculation dominance weight of criteriaThe normalized data is shown in Table 11

Equation (23) calculates the overall value of alternative bynormalizing the corresponding dominance measurementsThe rank of each alternative derives from ordering the alter-natives values The global measures computed the completerank ordering of all alternatives In addition a sensitivityanalysis should then be applied to verify the stability of theresults based on the decision makersrsquo preferencesThe resultsare presented in Table 12

Considering results in Table 12 we will find out whichcriterion is the most important one we can also find outranking of alternatives regarding each criterion

Step 6 Now the final conclusion will be conducted fromTables 12 and 6 and we can find out in next 15 months and25 months which criteria will be more effective and finallywhich alternative will be the first

By combination of Fuzzy TOPSIS which represents theweight of criteria of green supply chain with respect to timeand TODIM method which shows the weight of alternativesaccording to criteria of supply chain we will have weights ofcriteria and alternatives in Tables 13 and 14

4 Conclusions

This study represents a newway to select supplier during timeperiods by using a hybrid MCDM Using fuzzy TOPSIS tohave more accurate weighting method for TODIM and then

Advances in Fuzzy Systems 9

Table 10 Criteria weights

1198621

1198622

1198623

1198624

1198625

1198626

119908119895

0144 0159 0148 0233 0098 0217

Table 11 Normalized scores

1198601

1198602

1198603

1198621

02338 03831 038311198622

02118 0057 0731198623

03723 05657 00621198624

01441 04972 03581198625

00937 03435 05621198626

0254 07037 00423

Table 12 Final weight of criteria by using TODIM

1198621

1198622

1198623

1198624

1198625

1198626

120585 minus059 minus044 minus099 minus052 minus1083 minus0711Normalization 041 056 001 048 0 03Negative numbers explain losses from goal

Table 13 Criteria weightrsquos changes during time

1198621

1198622

1198623

1198624

1198625

1198626

119882 15 022 038 0005 0252 0 0153119882 25 021 029 0005 0264 0 0

Table 14 Alternative weights during time

1198601

1198602

1198603

15 months 047 086 09825 months 032 056 084

combining it with fuzzy time function (FTF) helped us torank the criteria and alternatives in several time periods Theimportant aim of this study was to use fuzzy time functionwith a new approach to consider time for each triangularfuzzy number that helped us to find out which supplier in thefuture according to criteria is suitable for our green supplychain and when we should change our supplier According tothe results shown in Table 14 supplier A3 has a decreasingrate from month 15 to 25 but its decreasing rate is lessthan supplier 2rsquos Regarding these results we can assign ourfuture improvement programs and orders to these suppliersconsidering criteria changes in green supplier selections andalso have some repairing programs due to our vision andmanagerial decisions in future for these companies Also wecan assign our orders to these companies and we proposed anorder plan for these suppliers regarding their portions

Conflict of Interests

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

References

[1] M L Tseng ldquoAn assessment of cause and effect decision-making model for firm environmental knowledge manage-ment capacities in uncertaintyrdquo Environmental Monitoring andAssessment vol 161 no 1ndash4 pp 549ndash564 2010

[2] S Zhang S Liu and R Zhai ldquoAn extended GRA method forMCDM with interval-valued triangular fuzzy assessments andunknown weightsrdquo Computers and Industrial Engineering vol61 no 4 pp 1336ndash1341 2011

[3] M L Tseng K H Tan R J Lina and Y Gengb ldquoMulticriteriaanalysis of green supply chain management using interval-valued fuzzy TODIMrdquo Knowledge-Based Systems 2012

[4] M Izadikhah ldquoGroup decision making process for supplierselectionwith TOPSISmethod under interval-valued intuition-istic fuzzy numbersrdquo Advances in Fuzzy Systems vol 2012Article ID 407942 14 pages 2012

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[6] H J Zimmermann Fuzzy Set Theory and Its ApplicationInternational Thomson Publishing Norwell Mass USA 2001

[7] J C A de Figueiredo and A Perkusich ldquoFaults and timinganalysis in real-time distributed systems a fuzzy time Petri-net-based approachrdquo Fuzzy Sets and Systems vol 83 no 2 pp 143ndash168 1996

[8] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[9] C L Hwang and K Yoon Multiple Attributes Decision MakingMethods and Applications Springer Berlin 1981

[10] M Ekmekcioglu T Kaya and C Kahraman ldquoFuzzy multicri-teria disposal method and site selection for municipal solidwasterdquoWaste Management vol 30 no 8-9 pp 1729ndash1736 2010

[11] S J Chen and C L Hwang ldquoFuzzy multi attribute decisionmakingrdquo vol 375 of lecture notes in economics andmathematicalsystem Springer New York 1992

[12] C Chen ldquoExtensions of the TOPSIS for group decision-makingunder fuzzy environmentrdquo Fuzzy Sets and Systems vol 114 no1 pp 1ndash9 2000

[13] T Chu ldquoSelecting plant location via a fuzzy TOPSIS approachrdquoInternational Journal of Advanced Manufacturing Technologyvol 20 no 11 pp 859ndash864 2002

[14] T Chu and Y Lin ldquoImproved extensions of the TOPSIS forgroup decision making under fuzzy environmentrdquo Journal ofInformation and Optimization Sciences vol 23273 286 pages2002

[15] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoExtension ofthe TOPSIS method for decision-making problems with fuzzydatardquo Applied Mathematics and Computation vol 181 no 2 pp1544ndash1551 2006

[16] T C Chu and Y C Lin ldquoAn interval arithmetic based fuzzyTOPSIS modelrdquo Expert Systems with Applications vol 36 no8 pp 10870ndash10876 2009

[17] D Yong ldquoPlant location selection based on fuzzy TOPSISrdquoInternational Journal of Advanced Manufacturing Technologyvol 28 no 7-8 pp 839ndash844 2006

[18] C T Chen C T Lin and S F Huang ldquoA fuzzy approach forsupplier evaluation and selection in supply chainmanagementrdquoInternational Journal of Production Economics vol 102 no 2 pp289ndash301 2006

10 Advances in Fuzzy Systems

[19] C Kahraman S Cevik N Y Ates andMGulbay ldquoFuzzymulti-criteria evaluation of industrial robotic systemsrdquoComputers andIndustrial Engineering vol 52 no 4 pp 414ndash433 2007

[20] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS integrated with fuzzyAHPrdquo in Proceedings of the 1st International Symposium onComputing in Science and Engineering pp 706ndash713 2010

[21] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

[22] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[23] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

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RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 5

0

02

04

06

08

1

12

a1 a3a998400

1a998400

2a998400

3

Figure 2 An interval-valued TFN

Table 2 Corresponding TFNS for linguistic preferences

Linguistic preferences Interval-valued TFNSVery poor [(0 0) 0 (1 15)]Poor [(0 05) 1 (25 35)]Medium poor [(0 15) 3 (45 55)]Fair [(25 35) 5 (65 75)]Medium good [(45 55) 7 (8 95)]Good [(55 75) 9 (95 10)]Very good [(85 95) 10 (10 10)]

119903119894119895= [(

119886119894119895

119889+

119895

1198861015840

119894119895

119889+

119895

)

119887119894119895

119889+

119895

(

1198861015840

119894119895

119889+

119895

119886119894119895

119889+

119895

)] 119894 = 1 2 119898

119895 = 1 2 119899 for 119895 isin 119868

119903119894119895= [(

119886minus

119895

119889119894119895

119886minus

119895

1198891015840

119894119895

)

119886minus

119895

119887119894119895

(

119886minus

119895

1198861015840

119894119895

119886minus

119895

119886119894119895

)] 119894 = 1 2 119898

119895 = 1 2 119899 for 119895 isin 119869(16)

where 119889+119895= max119888

119894119895 119894 = 1 119898 and 119886minus

119895= min119886

119894119895 119894 =

1 119898 119903119894119895= [(119897

119894119895 1198971015840

119894119895) 119898119894119895 (119906119894119895 1199061015840

119894119895)] 119877 = [119903

119894119895]119898lowast119899

and1198770= (11990301 11990302 119903

0119899) = ([(1 1) 1 (1 1)][(1 1) 1 (1 1)]

[(1 1) 1 (1 1)])The distance between the reference value and each com-

parison value can be calculated by using definition (2) asfollows

120575(1)

119894119895= radic

1

3[(1198971015840

119894119895minus 1)2

+ (119898119894119895minus 1)2

+ (1199061015840

119894119895minus 1)2

]

120575(2)

119894119895= radic

1

3[(119897119894119895minus 1)2

+ (119898119894119895minus 1)2

+ (119906119894119895minus 1)2

]

(17)

These calculations are used to determine the distancebetween the reference value and the comparison value in theinterval after calculation we have a new interval TFN foreveryTFNas 120575

119894119895= [120575(1)

119894119895 120575(2)

119894119895]Theweight vector of the criteria

is calculated according to Zhnag et al [2]

119908119895=

sum119898

119894=1(120575(1)

119894119895+ 120575(2)

119894119895)

sum119898

119894=1sum119899

119894=1(120575(1)

119894119895+ 120575(2)

119894119895)

(18)

0 2 4 6 8 10minus10 minus8 minus6 minus4 minus2

Gains

Value

Losses

minus15

minus1

minus05

0

05

1

15

Figure 3 TODIM value function

242 Prospect Theory The value function used in theprospect theory is described in the form of a power lawexpressed as

V (119909) = 119909120572 if 119909 ge 0

minus120579(minus119909)120573 if 119909 lt 0

(19)

where 120572 and 120573 are parameters related to gains and lossesrespectivelyThe parameter 120579 is a risk factor that is consideredin model and must be greater than one Figure 3 showsprospect value function that must be concave and has an 119878shape form

243 TODIM Formulation The TODIM method uses pair-wise comparisons between the criteria by using technicallysimple resources to eliminate occasional inconsistenciesresulting from these comparisons TODIM allows valuejudgments to be performed on a verbal scale using hier-archy of criteria fuzzy value judgments and interdepen-dence relationships among the alternatives The decisionmatrix consists of alternatives and criteria The alternatives1198601 1198602 119860

119898are viable alternatives 119888

1 1198882 119888

119899are crite-

ria and 119909119894119895indicates the rating of alternative 119860

119894according

to the criteria 119888119895 The weight vector 119908 = (119908

1 1199082 119908

119899)

comprises the individual weights 119908119895(119895 = 1 119899) for each

criterion 119888119895satisfying sum119899

119894=1119908119895= 1 The data of decision

matrix119860 originate fromdifferent sourcesThematrixmust benormalized to be dimensionless and allows various criteria tobe compared with each otherThis study uses the normalizeddecision matrix 119877 = [119903

119894119895]119898times119899

with 119894 = 1 119898 and 119895 =1 119899

119860 = (

11990911 119909

1119899

d

1199091198981

sdot sdot sdot 119909119898119899

) (20)

TODIM then calculates the partial dominance matricesand the final dominance matrix The first calculation that thedecisionmakers must define is a reference criterion (typicallythe criterion with the greatest importance weight)Thereforew119903119888indicates the weight of the criterion 119888 by the reference

criterion 119903 TODIM is expressed by the following equations

6 Advances in Fuzzy Systems

The dominance of an alternative over the other is asfollows

120575 (119860119894 119860119895) =

119898

sum

119888=1

120601119888(119860119894 119860119895)forall(119894119895)

(21)

where

120601 (119860119894 119860119895) =

radic119908119903119888(119909119894119888minus 119909119895119888)

sum119898

119888=1119908119903119888

if (119909119894119888minus 119909119895119888) gt 0

0

if (119909119894119888minus 119909119895119888) = 0

minus1

120579

radic(sum119898

119888=1119908119903119888) (119909119894119888minus 119909119895119888)

119908119903119888

if (119909119894119888minus 119909119895119888) lt 0

(22)

The term 120593119888(119860119894 119860119895) represents the contribution of cri-

terion 119888 (119888 = 1 119898) to the function 120575 (119860119894 119860119895) while

comparing alternative 119894 with alternative 119895 The parameter 120579represents the attention factor of the losses whose mitigationdepends on the specific problem A positive (119909

119894119888minus 119909119895119888) repre-

sents a gain whereas a nil or a negative (119909119894119888minus119909119895119888) represents a

loss The final matrix of dominance is obtained by summingthe partial matrices of dominance for each criterion seeTseng et al [3] The global value of the alternative 119868 isdetermined by normalizing the final matrix of dominanceaccording to the following expression

120585119894=

sum119899

119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

maxsum119899119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

(23)

Ordering the values 120585119894provides the rank of each alterna-

tive and better alternatives have higher values of 120585119894The use of

numerical values in rating alternativesmay be limited in theircapacity to address uncertainties Therefore an extensionof TODIM is proposed to solve problems with uncertaindata resulting in fuzzy TODIM In practical applications thetriangular shape of the membership function is often usedto represent fuzzy numbers Fuzzy models that used TFNsproved highly effective for solving decision making problemswhen information is imprecise Hence this study providessome basic definitions of fuzzy set theory see Tseng [24]

In this study we first use Fuzzy TOPSIS for weighting thecriteria considering time factor and then we use this datato combine with data from TODIM which calculate fromcomparisons between alternatives regarding to criteria andfinally we will able to determine which alternative will bemore reliable and more effective in any duration of time

Table 3 Criteria for green supply chain

Annual growth in green products (1198621)

Cost of revenue extent that it remains flat to decreases each year(1198622)

Industry leadership green market share (1198623)

Customer retentionpercentage of growth with existingcustomers (119862

4)

Customer acquisition the number of new green customerstotalrevenue to new green customers (119862

5)

Life cycle assessment (1198626)

Table 4 Experts data collection

Criteria 10 months 20 months 30 months1198621

P MP VP MP F MP MG MP F1198622

MP VP F F MP MP MP F MG1198623

VP F G MP MP G F MG VG1198624

F G F MP VG G MG G MP1198625

G F P G VG G P F F1198626

F MP G G F F MP MP P

3 Illustrative Method and Example

31 Proposed Approach Weuse the abovementionedmethodto find out the best supplier in green supply chain manage-ment considering time variations as follows (Figure 4)

Step 1 A group of decision makers identified the criteria inGSCM which are important and also will be changed duringtime horizons

Step 2 Collect the opinion of decisionmakers with linguisticvariables (Table 1) and define FTF for each important crite-rion

Step 3 Use fuzzy TOPSIS to evaluate the criteria during timeand initial weight of each criterion

Step 4 Collect the opinion of decision makers on alterna-tives respectively with linguistic variables (Table 2)

Step 5 Use TODIM for evaluating the final weight of eachcriterion against alternatives and the relationship betweenthem

Step 6 Combine the results of TODIM and fuzzy TOPSISto find out which supplier will be more effective from ourcompanyrsquos imagination now and in the future regarding thecondition changes

32 An Illustrative Example

Step 1 In this section we study on green supplier selectionproblem based on time factor in a Tier company in IranIn this company regarding the expert researches we have 6important criteria The data have been collected from three

Advances in Fuzzy Systems 7

Data collection for criteria with

respect to time

Fuzzy TOPSIS

Data collection for evaluating alternative

with respect to criteria

TODIM

ExpertsImportant criteria of GSCM

Combination of TOPSIS and TODIM

Experts (TFNs)

Experts (interval TFNs)

Figure 4 Flowchart of the method

Table 5 Green supply chain criteria considering time

15 months 25 months1198621

(364 558 797) (497 708 9825)1198622

(182 416 632) (5 866 899)1198623

(283 516 699) (95 1082 1166)1198624

(833 982 1049) (033 284 551)1198625

(982 114 116) (0 0 0)1198626

(135 155 1723) (0 0 0)

expert decision makers who have more than 10 years ofexperience in this area and also have the ability to predict themarket and its requirements in future Table 3 represents thecriteria of GSCM which is considered in this study for ourthree suppliers Table 4 represents data and then uses themto produce three functions for each criterion but because oflack of data we consider just three points of time 10 20 and30 months after data collection

Table 6 Criteria weights using fuzzy TOPSIS considering timeperiods

15 months 25 months1198621

05412 053221198622

055 05331198623

05437 0521198624

05256 0551198625

0523 01198626

051 0

Table 7 Criteria ranking for green supply chain considering time

15 months 25 months1198621

3 31198622

1 21198623

2 41198624

4 11198625

5 51198626

6 5

Step 2 The data in Table 4 were prepared based on linguisticfuzzy variable from Table 1 and data are collected from threeexperts to find out their opinion on criteria with respect totime After collecting data with linguistic variable we canchange them to fuzzy triangular number as represented inTable 1 and then FTF must be represented There are twomethods to calculate the FTF

First In this case we use average of slopes when the deviationbetween slopes is low and is not very important in ourexample for criterion 1 we have the following

Criterion 1 (033133) (163656) (357)

pessimistic FTF (1198621) = 119910 = 0127119905 10 lt 119905 lt 20

119910 = 014119905 20 lt 119905 lt 30

then 997904rArr pessimistic FTF (1198621) = 01335119905

(24)

The FTF from (2) for criterion 1 is

FTF (1198621) =

119910 = 0185119905 + 52 optimistic119910 = 015119905 + 333 normal119910 = 01335119905 + 164 pessimistic

(25)

To get precise decisions it is recommended to use theseparate functions for each period of time For example forcriterion 1 using the function between 10 and 20 months ismore exact than considering the FTF for any time Calcula-tions for finding the best green supplier in our study for the15th and 25th months are presented in Table 5 for examplefor Criterion 1 we have

FTF (1198621) =

119910 = 0185 times 15 + 52 = 7975 optimistic119910 = 015 times 15 + 333 = 558 normal119910 = 01335 times 15 + 164 = 364 pessimistic

(26)

8 Advances in Fuzzy Systems

Table 8 Interval-valued TFNs decision matrix

Criteriaalternative 1198601

1198602

1198603

1198621

[(55 75) 9 (95 10)] [(45 55) 7 (8 95)] [(45 55) 7 (8 95)]1198622

[(55 75) 9 (95 10)] [(85 95) 10 (10 10)] [(0 15) 3 (45 55)]1198623

[(45 55) 7 (8 95)] [(25 35) 5 (65 75)] [(45 55) 7 (8 95)]1198624

[(55 75) 9 (95 10)] [(0 15) 3 (45 55)] [(25 35) 5 (65 75)]1198625

[(85 95) 10 (10 10)] [(55 75) 9 (95 10)] [(45 55) 7 (8 95)]1198626

[(45 55) 7 (8 95)] [(0 0) 0 (1 15)] [(85 95) 10 (10 10)]

By using FTF we will have the variance for calculationsthat is useful in some other analysis which could not beachieved by ordinary triangular fuzzy numbers

Second In this method we did not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable For example for criterion 1198624 we have

FTF (1198624) =

optimistic

119910 = 0266119905 + 8

10 lt 119905 lt 20

119910 = minus0266119905 + 966

20 lt 119905 lt 30

normal

119910 = 0233119905 + 633

10 lt 119905 lt 20

119910 = minus0233119905 + 866

20 lt 119905 lt 30

pessimistic

119910 = 0166119905 + 433

10 lt 119905 lt 20

119910 = minus0166119905 + 7

20 lt 119905 lt 30

(27)

We use the first method for C1 and second method forother criteria

Step 3 Table 5 shows that calculated numbers come fromFTF for criteria these data will be the input for fuzzy TOPSISin (12) and (13) The results of calculations are representedin Table 6 This data will be the initial weight of TODIMapproach

By using the fuzzy TOPSIS method we have Table 6As shown in Table 6 we will have different ranking for

criteria with respect to expert opinion changes during thetime that will be measurable by FTFs This evaluation helpsus to know about the importance of each criterion in thesetimes and gives us the perspective for more effective actionsin our company in future and Table 7 presents the ranking ofcriteria with respect to time

Step 4 Table 8 shows the expertsrsquo opinion on alternatives ineach criteria these data come from the linguistic variablesand then change to interval valued triangular fuzzy numbersby Table 2

Table 9 Matrix of alternative scores with respect to criteria

1198601

1198602

1198603

1198621

04232 06933 069331198622

04232 01155 145981198623

06933 10532 011551198624

04232 14598 105321198625

01155 04232 069331198626

06933 19204 01155

Step 5 Use TODIMmethods see Tseng et al [3]

By applying TODIM approach first 119889minus 119889lowast have beencalculated from (17) which is shown in Table 9 after thatthe weight of criteria has been calculated from (18) which isshown in Table 10

After calculating weight of each criterion the data mustbe normalized for calculation dominance weight of criteriaThe normalized data is shown in Table 11

Equation (23) calculates the overall value of alternative bynormalizing the corresponding dominance measurementsThe rank of each alternative derives from ordering the alter-natives values The global measures computed the completerank ordering of all alternatives In addition a sensitivityanalysis should then be applied to verify the stability of theresults based on the decision makersrsquo preferencesThe resultsare presented in Table 12

Considering results in Table 12 we will find out whichcriterion is the most important one we can also find outranking of alternatives regarding each criterion

Step 6 Now the final conclusion will be conducted fromTables 12 and 6 and we can find out in next 15 months and25 months which criteria will be more effective and finallywhich alternative will be the first

By combination of Fuzzy TOPSIS which represents theweight of criteria of green supply chain with respect to timeand TODIM method which shows the weight of alternativesaccording to criteria of supply chain we will have weights ofcriteria and alternatives in Tables 13 and 14

4 Conclusions

This study represents a newway to select supplier during timeperiods by using a hybrid MCDM Using fuzzy TOPSIS tohave more accurate weighting method for TODIM and then

Advances in Fuzzy Systems 9

Table 10 Criteria weights

1198621

1198622

1198623

1198624

1198625

1198626

119908119895

0144 0159 0148 0233 0098 0217

Table 11 Normalized scores

1198601

1198602

1198603

1198621

02338 03831 038311198622

02118 0057 0731198623

03723 05657 00621198624

01441 04972 03581198625

00937 03435 05621198626

0254 07037 00423

Table 12 Final weight of criteria by using TODIM

1198621

1198622

1198623

1198624

1198625

1198626

120585 minus059 minus044 minus099 minus052 minus1083 minus0711Normalization 041 056 001 048 0 03Negative numbers explain losses from goal

Table 13 Criteria weightrsquos changes during time

1198621

1198622

1198623

1198624

1198625

1198626

119882 15 022 038 0005 0252 0 0153119882 25 021 029 0005 0264 0 0

Table 14 Alternative weights during time

1198601

1198602

1198603

15 months 047 086 09825 months 032 056 084

combining it with fuzzy time function (FTF) helped us torank the criteria and alternatives in several time periods Theimportant aim of this study was to use fuzzy time functionwith a new approach to consider time for each triangularfuzzy number that helped us to find out which supplier in thefuture according to criteria is suitable for our green supplychain and when we should change our supplier According tothe results shown in Table 14 supplier A3 has a decreasingrate from month 15 to 25 but its decreasing rate is lessthan supplier 2rsquos Regarding these results we can assign ourfuture improvement programs and orders to these suppliersconsidering criteria changes in green supplier selections andalso have some repairing programs due to our vision andmanagerial decisions in future for these companies Also wecan assign our orders to these companies and we proposed anorder plan for these suppliers regarding their portions

Conflict of Interests

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

References

[1] M L Tseng ldquoAn assessment of cause and effect decision-making model for firm environmental knowledge manage-ment capacities in uncertaintyrdquo Environmental Monitoring andAssessment vol 161 no 1ndash4 pp 549ndash564 2010

[2] S Zhang S Liu and R Zhai ldquoAn extended GRA method forMCDM with interval-valued triangular fuzzy assessments andunknown weightsrdquo Computers and Industrial Engineering vol61 no 4 pp 1336ndash1341 2011

[3] M L Tseng K H Tan R J Lina and Y Gengb ldquoMulticriteriaanalysis of green supply chain management using interval-valued fuzzy TODIMrdquo Knowledge-Based Systems 2012

[4] M Izadikhah ldquoGroup decision making process for supplierselectionwith TOPSISmethod under interval-valued intuition-istic fuzzy numbersrdquo Advances in Fuzzy Systems vol 2012Article ID 407942 14 pages 2012

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[6] H J Zimmermann Fuzzy Set Theory and Its ApplicationInternational Thomson Publishing Norwell Mass USA 2001

[7] J C A de Figueiredo and A Perkusich ldquoFaults and timinganalysis in real-time distributed systems a fuzzy time Petri-net-based approachrdquo Fuzzy Sets and Systems vol 83 no 2 pp 143ndash168 1996

[8] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[9] C L Hwang and K Yoon Multiple Attributes Decision MakingMethods and Applications Springer Berlin 1981

[10] M Ekmekcioglu T Kaya and C Kahraman ldquoFuzzy multicri-teria disposal method and site selection for municipal solidwasterdquoWaste Management vol 30 no 8-9 pp 1729ndash1736 2010

[11] S J Chen and C L Hwang ldquoFuzzy multi attribute decisionmakingrdquo vol 375 of lecture notes in economics andmathematicalsystem Springer New York 1992

[12] C Chen ldquoExtensions of the TOPSIS for group decision-makingunder fuzzy environmentrdquo Fuzzy Sets and Systems vol 114 no1 pp 1ndash9 2000

[13] T Chu ldquoSelecting plant location via a fuzzy TOPSIS approachrdquoInternational Journal of Advanced Manufacturing Technologyvol 20 no 11 pp 859ndash864 2002

[14] T Chu and Y Lin ldquoImproved extensions of the TOPSIS forgroup decision making under fuzzy environmentrdquo Journal ofInformation and Optimization Sciences vol 23273 286 pages2002

[15] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoExtension ofthe TOPSIS method for decision-making problems with fuzzydatardquo Applied Mathematics and Computation vol 181 no 2 pp1544ndash1551 2006

[16] T C Chu and Y C Lin ldquoAn interval arithmetic based fuzzyTOPSIS modelrdquo Expert Systems with Applications vol 36 no8 pp 10870ndash10876 2009

[17] D Yong ldquoPlant location selection based on fuzzy TOPSISrdquoInternational Journal of Advanced Manufacturing Technologyvol 28 no 7-8 pp 839ndash844 2006

[18] C T Chen C T Lin and S F Huang ldquoA fuzzy approach forsupplier evaluation and selection in supply chainmanagementrdquoInternational Journal of Production Economics vol 102 no 2 pp289ndash301 2006

10 Advances in Fuzzy Systems

[19] C Kahraman S Cevik N Y Ates andMGulbay ldquoFuzzymulti-criteria evaluation of industrial robotic systemsrdquoComputers andIndustrial Engineering vol 52 no 4 pp 414ndash433 2007

[20] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS integrated with fuzzyAHPrdquo in Proceedings of the 1st International Symposium onComputing in Science and Engineering pp 706ndash713 2010

[21] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

[22] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[23] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Advances in Fuzzy Systems

The dominance of an alternative over the other is asfollows

120575 (119860119894 119860119895) =

119898

sum

119888=1

120601119888(119860119894 119860119895)forall(119894119895)

(21)

where

120601 (119860119894 119860119895) =

radic119908119903119888(119909119894119888minus 119909119895119888)

sum119898

119888=1119908119903119888

if (119909119894119888minus 119909119895119888) gt 0

0

if (119909119894119888minus 119909119895119888) = 0

minus1

120579

radic(sum119898

119888=1119908119903119888) (119909119894119888minus 119909119895119888)

119908119903119888

if (119909119894119888minus 119909119895119888) lt 0

(22)

The term 120593119888(119860119894 119860119895) represents the contribution of cri-

terion 119888 (119888 = 1 119898) to the function 120575 (119860119894 119860119895) while

comparing alternative 119894 with alternative 119895 The parameter 120579represents the attention factor of the losses whose mitigationdepends on the specific problem A positive (119909

119894119888minus 119909119895119888) repre-

sents a gain whereas a nil or a negative (119909119894119888minus119909119895119888) represents a

loss The final matrix of dominance is obtained by summingthe partial matrices of dominance for each criterion seeTseng et al [3] The global value of the alternative 119868 isdetermined by normalizing the final matrix of dominanceaccording to the following expression

120585119894=

sum119899

119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

maxsum119899119895=1120575 (119894 119895) minusminsum119899

119895=1120575 (119894 119895)

(23)

Ordering the values 120585119894provides the rank of each alterna-

tive and better alternatives have higher values of 120585119894The use of

numerical values in rating alternativesmay be limited in theircapacity to address uncertainties Therefore an extensionof TODIM is proposed to solve problems with uncertaindata resulting in fuzzy TODIM In practical applications thetriangular shape of the membership function is often usedto represent fuzzy numbers Fuzzy models that used TFNsproved highly effective for solving decision making problemswhen information is imprecise Hence this study providessome basic definitions of fuzzy set theory see Tseng [24]

In this study we first use Fuzzy TOPSIS for weighting thecriteria considering time factor and then we use this datato combine with data from TODIM which calculate fromcomparisons between alternatives regarding to criteria andfinally we will able to determine which alternative will bemore reliable and more effective in any duration of time

Table 3 Criteria for green supply chain

Annual growth in green products (1198621)

Cost of revenue extent that it remains flat to decreases each year(1198622)

Industry leadership green market share (1198623)

Customer retentionpercentage of growth with existingcustomers (119862

4)

Customer acquisition the number of new green customerstotalrevenue to new green customers (119862

5)

Life cycle assessment (1198626)

Table 4 Experts data collection

Criteria 10 months 20 months 30 months1198621

P MP VP MP F MP MG MP F1198622

MP VP F F MP MP MP F MG1198623

VP F G MP MP G F MG VG1198624

F G F MP VG G MG G MP1198625

G F P G VG G P F F1198626

F MP G G F F MP MP P

3 Illustrative Method and Example

31 Proposed Approach Weuse the abovementionedmethodto find out the best supplier in green supply chain manage-ment considering time variations as follows (Figure 4)

Step 1 A group of decision makers identified the criteria inGSCM which are important and also will be changed duringtime horizons

Step 2 Collect the opinion of decisionmakers with linguisticvariables (Table 1) and define FTF for each important crite-rion

Step 3 Use fuzzy TOPSIS to evaluate the criteria during timeand initial weight of each criterion

Step 4 Collect the opinion of decision makers on alterna-tives respectively with linguistic variables (Table 2)

Step 5 Use TODIM for evaluating the final weight of eachcriterion against alternatives and the relationship betweenthem

Step 6 Combine the results of TODIM and fuzzy TOPSISto find out which supplier will be more effective from ourcompanyrsquos imagination now and in the future regarding thecondition changes

32 An Illustrative Example

Step 1 In this section we study on green supplier selectionproblem based on time factor in a Tier company in IranIn this company regarding the expert researches we have 6important criteria The data have been collected from three

Advances in Fuzzy Systems 7

Data collection for criteria with

respect to time

Fuzzy TOPSIS

Data collection for evaluating alternative

with respect to criteria

TODIM

ExpertsImportant criteria of GSCM

Combination of TOPSIS and TODIM

Experts (TFNs)

Experts (interval TFNs)

Figure 4 Flowchart of the method

Table 5 Green supply chain criteria considering time

15 months 25 months1198621

(364 558 797) (497 708 9825)1198622

(182 416 632) (5 866 899)1198623

(283 516 699) (95 1082 1166)1198624

(833 982 1049) (033 284 551)1198625

(982 114 116) (0 0 0)1198626

(135 155 1723) (0 0 0)

expert decision makers who have more than 10 years ofexperience in this area and also have the ability to predict themarket and its requirements in future Table 3 represents thecriteria of GSCM which is considered in this study for ourthree suppliers Table 4 represents data and then uses themto produce three functions for each criterion but because oflack of data we consider just three points of time 10 20 and30 months after data collection

Table 6 Criteria weights using fuzzy TOPSIS considering timeperiods

15 months 25 months1198621

05412 053221198622

055 05331198623

05437 0521198624

05256 0551198625

0523 01198626

051 0

Table 7 Criteria ranking for green supply chain considering time

15 months 25 months1198621

3 31198622

1 21198623

2 41198624

4 11198625

5 51198626

6 5

Step 2 The data in Table 4 were prepared based on linguisticfuzzy variable from Table 1 and data are collected from threeexperts to find out their opinion on criteria with respect totime After collecting data with linguistic variable we canchange them to fuzzy triangular number as represented inTable 1 and then FTF must be represented There are twomethods to calculate the FTF

First In this case we use average of slopes when the deviationbetween slopes is low and is not very important in ourexample for criterion 1 we have the following

Criterion 1 (033133) (163656) (357)

pessimistic FTF (1198621) = 119910 = 0127119905 10 lt 119905 lt 20

119910 = 014119905 20 lt 119905 lt 30

then 997904rArr pessimistic FTF (1198621) = 01335119905

(24)

The FTF from (2) for criterion 1 is

FTF (1198621) =

119910 = 0185119905 + 52 optimistic119910 = 015119905 + 333 normal119910 = 01335119905 + 164 pessimistic

(25)

To get precise decisions it is recommended to use theseparate functions for each period of time For example forcriterion 1 using the function between 10 and 20 months ismore exact than considering the FTF for any time Calcula-tions for finding the best green supplier in our study for the15th and 25th months are presented in Table 5 for examplefor Criterion 1 we have

FTF (1198621) =

119910 = 0185 times 15 + 52 = 7975 optimistic119910 = 015 times 15 + 333 = 558 normal119910 = 01335 times 15 + 164 = 364 pessimistic

(26)

8 Advances in Fuzzy Systems

Table 8 Interval-valued TFNs decision matrix

Criteriaalternative 1198601

1198602

1198603

1198621

[(55 75) 9 (95 10)] [(45 55) 7 (8 95)] [(45 55) 7 (8 95)]1198622

[(55 75) 9 (95 10)] [(85 95) 10 (10 10)] [(0 15) 3 (45 55)]1198623

[(45 55) 7 (8 95)] [(25 35) 5 (65 75)] [(45 55) 7 (8 95)]1198624

[(55 75) 9 (95 10)] [(0 15) 3 (45 55)] [(25 35) 5 (65 75)]1198625

[(85 95) 10 (10 10)] [(55 75) 9 (95 10)] [(45 55) 7 (8 95)]1198626

[(45 55) 7 (8 95)] [(0 0) 0 (1 15)] [(85 95) 10 (10 10)]

By using FTF we will have the variance for calculationsthat is useful in some other analysis which could not beachieved by ordinary triangular fuzzy numbers

Second In this method we did not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable For example for criterion 1198624 we have

FTF (1198624) =

optimistic

119910 = 0266119905 + 8

10 lt 119905 lt 20

119910 = minus0266119905 + 966

20 lt 119905 lt 30

normal

119910 = 0233119905 + 633

10 lt 119905 lt 20

119910 = minus0233119905 + 866

20 lt 119905 lt 30

pessimistic

119910 = 0166119905 + 433

10 lt 119905 lt 20

119910 = minus0166119905 + 7

20 lt 119905 lt 30

(27)

We use the first method for C1 and second method forother criteria

Step 3 Table 5 shows that calculated numbers come fromFTF for criteria these data will be the input for fuzzy TOPSISin (12) and (13) The results of calculations are representedin Table 6 This data will be the initial weight of TODIMapproach

By using the fuzzy TOPSIS method we have Table 6As shown in Table 6 we will have different ranking for

criteria with respect to expert opinion changes during thetime that will be measurable by FTFs This evaluation helpsus to know about the importance of each criterion in thesetimes and gives us the perspective for more effective actionsin our company in future and Table 7 presents the ranking ofcriteria with respect to time

Step 4 Table 8 shows the expertsrsquo opinion on alternatives ineach criteria these data come from the linguistic variablesand then change to interval valued triangular fuzzy numbersby Table 2

Table 9 Matrix of alternative scores with respect to criteria

1198601

1198602

1198603

1198621

04232 06933 069331198622

04232 01155 145981198623

06933 10532 011551198624

04232 14598 105321198625

01155 04232 069331198626

06933 19204 01155

Step 5 Use TODIMmethods see Tseng et al [3]

By applying TODIM approach first 119889minus 119889lowast have beencalculated from (17) which is shown in Table 9 after thatthe weight of criteria has been calculated from (18) which isshown in Table 10

After calculating weight of each criterion the data mustbe normalized for calculation dominance weight of criteriaThe normalized data is shown in Table 11

Equation (23) calculates the overall value of alternative bynormalizing the corresponding dominance measurementsThe rank of each alternative derives from ordering the alter-natives values The global measures computed the completerank ordering of all alternatives In addition a sensitivityanalysis should then be applied to verify the stability of theresults based on the decision makersrsquo preferencesThe resultsare presented in Table 12

Considering results in Table 12 we will find out whichcriterion is the most important one we can also find outranking of alternatives regarding each criterion

Step 6 Now the final conclusion will be conducted fromTables 12 and 6 and we can find out in next 15 months and25 months which criteria will be more effective and finallywhich alternative will be the first

By combination of Fuzzy TOPSIS which represents theweight of criteria of green supply chain with respect to timeand TODIM method which shows the weight of alternativesaccording to criteria of supply chain we will have weights ofcriteria and alternatives in Tables 13 and 14

4 Conclusions

This study represents a newway to select supplier during timeperiods by using a hybrid MCDM Using fuzzy TOPSIS tohave more accurate weighting method for TODIM and then

Advances in Fuzzy Systems 9

Table 10 Criteria weights

1198621

1198622

1198623

1198624

1198625

1198626

119908119895

0144 0159 0148 0233 0098 0217

Table 11 Normalized scores

1198601

1198602

1198603

1198621

02338 03831 038311198622

02118 0057 0731198623

03723 05657 00621198624

01441 04972 03581198625

00937 03435 05621198626

0254 07037 00423

Table 12 Final weight of criteria by using TODIM

1198621

1198622

1198623

1198624

1198625

1198626

120585 minus059 minus044 minus099 minus052 minus1083 minus0711Normalization 041 056 001 048 0 03Negative numbers explain losses from goal

Table 13 Criteria weightrsquos changes during time

1198621

1198622

1198623

1198624

1198625

1198626

119882 15 022 038 0005 0252 0 0153119882 25 021 029 0005 0264 0 0

Table 14 Alternative weights during time

1198601

1198602

1198603

15 months 047 086 09825 months 032 056 084

combining it with fuzzy time function (FTF) helped us torank the criteria and alternatives in several time periods Theimportant aim of this study was to use fuzzy time functionwith a new approach to consider time for each triangularfuzzy number that helped us to find out which supplier in thefuture according to criteria is suitable for our green supplychain and when we should change our supplier According tothe results shown in Table 14 supplier A3 has a decreasingrate from month 15 to 25 but its decreasing rate is lessthan supplier 2rsquos Regarding these results we can assign ourfuture improvement programs and orders to these suppliersconsidering criteria changes in green supplier selections andalso have some repairing programs due to our vision andmanagerial decisions in future for these companies Also wecan assign our orders to these companies and we proposed anorder plan for these suppliers regarding their portions

Conflict of Interests

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

References

[1] M L Tseng ldquoAn assessment of cause and effect decision-making model for firm environmental knowledge manage-ment capacities in uncertaintyrdquo Environmental Monitoring andAssessment vol 161 no 1ndash4 pp 549ndash564 2010

[2] S Zhang S Liu and R Zhai ldquoAn extended GRA method forMCDM with interval-valued triangular fuzzy assessments andunknown weightsrdquo Computers and Industrial Engineering vol61 no 4 pp 1336ndash1341 2011

[3] M L Tseng K H Tan R J Lina and Y Gengb ldquoMulticriteriaanalysis of green supply chain management using interval-valued fuzzy TODIMrdquo Knowledge-Based Systems 2012

[4] M Izadikhah ldquoGroup decision making process for supplierselectionwith TOPSISmethod under interval-valued intuition-istic fuzzy numbersrdquo Advances in Fuzzy Systems vol 2012Article ID 407942 14 pages 2012

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[6] H J Zimmermann Fuzzy Set Theory and Its ApplicationInternational Thomson Publishing Norwell Mass USA 2001

[7] J C A de Figueiredo and A Perkusich ldquoFaults and timinganalysis in real-time distributed systems a fuzzy time Petri-net-based approachrdquo Fuzzy Sets and Systems vol 83 no 2 pp 143ndash168 1996

[8] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[9] C L Hwang and K Yoon Multiple Attributes Decision MakingMethods and Applications Springer Berlin 1981

[10] M Ekmekcioglu T Kaya and C Kahraman ldquoFuzzy multicri-teria disposal method and site selection for municipal solidwasterdquoWaste Management vol 30 no 8-9 pp 1729ndash1736 2010

[11] S J Chen and C L Hwang ldquoFuzzy multi attribute decisionmakingrdquo vol 375 of lecture notes in economics andmathematicalsystem Springer New York 1992

[12] C Chen ldquoExtensions of the TOPSIS for group decision-makingunder fuzzy environmentrdquo Fuzzy Sets and Systems vol 114 no1 pp 1ndash9 2000

[13] T Chu ldquoSelecting plant location via a fuzzy TOPSIS approachrdquoInternational Journal of Advanced Manufacturing Technologyvol 20 no 11 pp 859ndash864 2002

[14] T Chu and Y Lin ldquoImproved extensions of the TOPSIS forgroup decision making under fuzzy environmentrdquo Journal ofInformation and Optimization Sciences vol 23273 286 pages2002

[15] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoExtension ofthe TOPSIS method for decision-making problems with fuzzydatardquo Applied Mathematics and Computation vol 181 no 2 pp1544ndash1551 2006

[16] T C Chu and Y C Lin ldquoAn interval arithmetic based fuzzyTOPSIS modelrdquo Expert Systems with Applications vol 36 no8 pp 10870ndash10876 2009

[17] D Yong ldquoPlant location selection based on fuzzy TOPSISrdquoInternational Journal of Advanced Manufacturing Technologyvol 28 no 7-8 pp 839ndash844 2006

[18] C T Chen C T Lin and S F Huang ldquoA fuzzy approach forsupplier evaluation and selection in supply chainmanagementrdquoInternational Journal of Production Economics vol 102 no 2 pp289ndash301 2006

10 Advances in Fuzzy Systems

[19] C Kahraman S Cevik N Y Ates andMGulbay ldquoFuzzymulti-criteria evaluation of industrial robotic systemsrdquoComputers andIndustrial Engineering vol 52 no 4 pp 414ndash433 2007

[20] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS integrated with fuzzyAHPrdquo in Proceedings of the 1st International Symposium onComputing in Science and Engineering pp 706ndash713 2010

[21] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

[22] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[23] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 7

Data collection for criteria with

respect to time

Fuzzy TOPSIS

Data collection for evaluating alternative

with respect to criteria

TODIM

ExpertsImportant criteria of GSCM

Combination of TOPSIS and TODIM

Experts (TFNs)

Experts (interval TFNs)

Figure 4 Flowchart of the method

Table 5 Green supply chain criteria considering time

15 months 25 months1198621

(364 558 797) (497 708 9825)1198622

(182 416 632) (5 866 899)1198623

(283 516 699) (95 1082 1166)1198624

(833 982 1049) (033 284 551)1198625

(982 114 116) (0 0 0)1198626

(135 155 1723) (0 0 0)

expert decision makers who have more than 10 years ofexperience in this area and also have the ability to predict themarket and its requirements in future Table 3 represents thecriteria of GSCM which is considered in this study for ourthree suppliers Table 4 represents data and then uses themto produce three functions for each criterion but because oflack of data we consider just three points of time 10 20 and30 months after data collection

Table 6 Criteria weights using fuzzy TOPSIS considering timeperiods

15 months 25 months1198621

05412 053221198622

055 05331198623

05437 0521198624

05256 0551198625

0523 01198626

051 0

Table 7 Criteria ranking for green supply chain considering time

15 months 25 months1198621

3 31198622

1 21198623

2 41198624

4 11198625

5 51198626

6 5

Step 2 The data in Table 4 were prepared based on linguisticfuzzy variable from Table 1 and data are collected from threeexperts to find out their opinion on criteria with respect totime After collecting data with linguistic variable we canchange them to fuzzy triangular number as represented inTable 1 and then FTF must be represented There are twomethods to calculate the FTF

First In this case we use average of slopes when the deviationbetween slopes is low and is not very important in ourexample for criterion 1 we have the following

Criterion 1 (033133) (163656) (357)

pessimistic FTF (1198621) = 119910 = 0127119905 10 lt 119905 lt 20

119910 = 014119905 20 lt 119905 lt 30

then 997904rArr pessimistic FTF (1198621) = 01335119905

(24)

The FTF from (2) for criterion 1 is

FTF (1198621) =

119910 = 0185119905 + 52 optimistic119910 = 015119905 + 333 normal119910 = 01335119905 + 164 pessimistic

(25)

To get precise decisions it is recommended to use theseparate functions for each period of time For example forcriterion 1 using the function between 10 and 20 months ismore exact than considering the FTF for any time Calcula-tions for finding the best green supplier in our study for the15th and 25th months are presented in Table 5 for examplefor Criterion 1 we have

FTF (1198621) =

119910 = 0185 times 15 + 52 = 7975 optimistic119910 = 015 times 15 + 333 = 558 normal119910 = 01335 times 15 + 164 = 364 pessimistic

(26)

8 Advances in Fuzzy Systems

Table 8 Interval-valued TFNs decision matrix

Criteriaalternative 1198601

1198602

1198603

1198621

[(55 75) 9 (95 10)] [(45 55) 7 (8 95)] [(45 55) 7 (8 95)]1198622

[(55 75) 9 (95 10)] [(85 95) 10 (10 10)] [(0 15) 3 (45 55)]1198623

[(45 55) 7 (8 95)] [(25 35) 5 (65 75)] [(45 55) 7 (8 95)]1198624

[(55 75) 9 (95 10)] [(0 15) 3 (45 55)] [(25 35) 5 (65 75)]1198625

[(85 95) 10 (10 10)] [(55 75) 9 (95 10)] [(45 55) 7 (8 95)]1198626

[(45 55) 7 (8 95)] [(0 0) 0 (1 15)] [(85 95) 10 (10 10)]

By using FTF we will have the variance for calculationsthat is useful in some other analysis which could not beachieved by ordinary triangular fuzzy numbers

Second In this method we did not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable For example for criterion 1198624 we have

FTF (1198624) =

optimistic

119910 = 0266119905 + 8

10 lt 119905 lt 20

119910 = minus0266119905 + 966

20 lt 119905 lt 30

normal

119910 = 0233119905 + 633

10 lt 119905 lt 20

119910 = minus0233119905 + 866

20 lt 119905 lt 30

pessimistic

119910 = 0166119905 + 433

10 lt 119905 lt 20

119910 = minus0166119905 + 7

20 lt 119905 lt 30

(27)

We use the first method for C1 and second method forother criteria

Step 3 Table 5 shows that calculated numbers come fromFTF for criteria these data will be the input for fuzzy TOPSISin (12) and (13) The results of calculations are representedin Table 6 This data will be the initial weight of TODIMapproach

By using the fuzzy TOPSIS method we have Table 6As shown in Table 6 we will have different ranking for

criteria with respect to expert opinion changes during thetime that will be measurable by FTFs This evaluation helpsus to know about the importance of each criterion in thesetimes and gives us the perspective for more effective actionsin our company in future and Table 7 presents the ranking ofcriteria with respect to time

Step 4 Table 8 shows the expertsrsquo opinion on alternatives ineach criteria these data come from the linguistic variablesand then change to interval valued triangular fuzzy numbersby Table 2

Table 9 Matrix of alternative scores with respect to criteria

1198601

1198602

1198603

1198621

04232 06933 069331198622

04232 01155 145981198623

06933 10532 011551198624

04232 14598 105321198625

01155 04232 069331198626

06933 19204 01155

Step 5 Use TODIMmethods see Tseng et al [3]

By applying TODIM approach first 119889minus 119889lowast have beencalculated from (17) which is shown in Table 9 after thatthe weight of criteria has been calculated from (18) which isshown in Table 10

After calculating weight of each criterion the data mustbe normalized for calculation dominance weight of criteriaThe normalized data is shown in Table 11

Equation (23) calculates the overall value of alternative bynormalizing the corresponding dominance measurementsThe rank of each alternative derives from ordering the alter-natives values The global measures computed the completerank ordering of all alternatives In addition a sensitivityanalysis should then be applied to verify the stability of theresults based on the decision makersrsquo preferencesThe resultsare presented in Table 12

Considering results in Table 12 we will find out whichcriterion is the most important one we can also find outranking of alternatives regarding each criterion

Step 6 Now the final conclusion will be conducted fromTables 12 and 6 and we can find out in next 15 months and25 months which criteria will be more effective and finallywhich alternative will be the first

By combination of Fuzzy TOPSIS which represents theweight of criteria of green supply chain with respect to timeand TODIM method which shows the weight of alternativesaccording to criteria of supply chain we will have weights ofcriteria and alternatives in Tables 13 and 14

4 Conclusions

This study represents a newway to select supplier during timeperiods by using a hybrid MCDM Using fuzzy TOPSIS tohave more accurate weighting method for TODIM and then

Advances in Fuzzy Systems 9

Table 10 Criteria weights

1198621

1198622

1198623

1198624

1198625

1198626

119908119895

0144 0159 0148 0233 0098 0217

Table 11 Normalized scores

1198601

1198602

1198603

1198621

02338 03831 038311198622

02118 0057 0731198623

03723 05657 00621198624

01441 04972 03581198625

00937 03435 05621198626

0254 07037 00423

Table 12 Final weight of criteria by using TODIM

1198621

1198622

1198623

1198624

1198625

1198626

120585 minus059 minus044 minus099 minus052 minus1083 minus0711Normalization 041 056 001 048 0 03Negative numbers explain losses from goal

Table 13 Criteria weightrsquos changes during time

1198621

1198622

1198623

1198624

1198625

1198626

119882 15 022 038 0005 0252 0 0153119882 25 021 029 0005 0264 0 0

Table 14 Alternative weights during time

1198601

1198602

1198603

15 months 047 086 09825 months 032 056 084

combining it with fuzzy time function (FTF) helped us torank the criteria and alternatives in several time periods Theimportant aim of this study was to use fuzzy time functionwith a new approach to consider time for each triangularfuzzy number that helped us to find out which supplier in thefuture according to criteria is suitable for our green supplychain and when we should change our supplier According tothe results shown in Table 14 supplier A3 has a decreasingrate from month 15 to 25 but its decreasing rate is lessthan supplier 2rsquos Regarding these results we can assign ourfuture improvement programs and orders to these suppliersconsidering criteria changes in green supplier selections andalso have some repairing programs due to our vision andmanagerial decisions in future for these companies Also wecan assign our orders to these companies and we proposed anorder plan for these suppliers regarding their portions

Conflict of Interests

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

References

[1] M L Tseng ldquoAn assessment of cause and effect decision-making model for firm environmental knowledge manage-ment capacities in uncertaintyrdquo Environmental Monitoring andAssessment vol 161 no 1ndash4 pp 549ndash564 2010

[2] S Zhang S Liu and R Zhai ldquoAn extended GRA method forMCDM with interval-valued triangular fuzzy assessments andunknown weightsrdquo Computers and Industrial Engineering vol61 no 4 pp 1336ndash1341 2011

[3] M L Tseng K H Tan R J Lina and Y Gengb ldquoMulticriteriaanalysis of green supply chain management using interval-valued fuzzy TODIMrdquo Knowledge-Based Systems 2012

[4] M Izadikhah ldquoGroup decision making process for supplierselectionwith TOPSISmethod under interval-valued intuition-istic fuzzy numbersrdquo Advances in Fuzzy Systems vol 2012Article ID 407942 14 pages 2012

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[6] H J Zimmermann Fuzzy Set Theory and Its ApplicationInternational Thomson Publishing Norwell Mass USA 2001

[7] J C A de Figueiredo and A Perkusich ldquoFaults and timinganalysis in real-time distributed systems a fuzzy time Petri-net-based approachrdquo Fuzzy Sets and Systems vol 83 no 2 pp 143ndash168 1996

[8] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[9] C L Hwang and K Yoon Multiple Attributes Decision MakingMethods and Applications Springer Berlin 1981

[10] M Ekmekcioglu T Kaya and C Kahraman ldquoFuzzy multicri-teria disposal method and site selection for municipal solidwasterdquoWaste Management vol 30 no 8-9 pp 1729ndash1736 2010

[11] S J Chen and C L Hwang ldquoFuzzy multi attribute decisionmakingrdquo vol 375 of lecture notes in economics andmathematicalsystem Springer New York 1992

[12] C Chen ldquoExtensions of the TOPSIS for group decision-makingunder fuzzy environmentrdquo Fuzzy Sets and Systems vol 114 no1 pp 1ndash9 2000

[13] T Chu ldquoSelecting plant location via a fuzzy TOPSIS approachrdquoInternational Journal of Advanced Manufacturing Technologyvol 20 no 11 pp 859ndash864 2002

[14] T Chu and Y Lin ldquoImproved extensions of the TOPSIS forgroup decision making under fuzzy environmentrdquo Journal ofInformation and Optimization Sciences vol 23273 286 pages2002

[15] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoExtension ofthe TOPSIS method for decision-making problems with fuzzydatardquo Applied Mathematics and Computation vol 181 no 2 pp1544ndash1551 2006

[16] T C Chu and Y C Lin ldquoAn interval arithmetic based fuzzyTOPSIS modelrdquo Expert Systems with Applications vol 36 no8 pp 10870ndash10876 2009

[17] D Yong ldquoPlant location selection based on fuzzy TOPSISrdquoInternational Journal of Advanced Manufacturing Technologyvol 28 no 7-8 pp 839ndash844 2006

[18] C T Chen C T Lin and S F Huang ldquoA fuzzy approach forsupplier evaluation and selection in supply chainmanagementrdquoInternational Journal of Production Economics vol 102 no 2 pp289ndash301 2006

10 Advances in Fuzzy Systems

[19] C Kahraman S Cevik N Y Ates andMGulbay ldquoFuzzymulti-criteria evaluation of industrial robotic systemsrdquoComputers andIndustrial Engineering vol 52 no 4 pp 414ndash433 2007

[20] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS integrated with fuzzyAHPrdquo in Proceedings of the 1st International Symposium onComputing in Science and Engineering pp 706ndash713 2010

[21] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

[22] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[23] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Advances in Fuzzy Systems

Table 8 Interval-valued TFNs decision matrix

Criteriaalternative 1198601

1198602

1198603

1198621

[(55 75) 9 (95 10)] [(45 55) 7 (8 95)] [(45 55) 7 (8 95)]1198622

[(55 75) 9 (95 10)] [(85 95) 10 (10 10)] [(0 15) 3 (45 55)]1198623

[(45 55) 7 (8 95)] [(25 35) 5 (65 75)] [(45 55) 7 (8 95)]1198624

[(55 75) 9 (95 10)] [(0 15) 3 (45 55)] [(25 35) 5 (65 75)]1198625

[(85 95) 10 (10 10)] [(55 75) 9 (95 10)] [(45 55) 7 (8 95)]1198626

[(45 55) 7 (8 95)] [(0 0) 0 (1 15)] [(85 95) 10 (10 10)]

By using FTF we will have the variance for calculationsthat is useful in some other analysis which could not beachieved by ordinary triangular fuzzy numbers

Second In this method we did not use mean of the slopesbecause of more deviation in numbers and in these caseswe use separate functions in calculations that will be morereasonable For example for criterion 1198624 we have

FTF (1198624) =

optimistic

119910 = 0266119905 + 8

10 lt 119905 lt 20

119910 = minus0266119905 + 966

20 lt 119905 lt 30

normal

119910 = 0233119905 + 633

10 lt 119905 lt 20

119910 = minus0233119905 + 866

20 lt 119905 lt 30

pessimistic

119910 = 0166119905 + 433

10 lt 119905 lt 20

119910 = minus0166119905 + 7

20 lt 119905 lt 30

(27)

We use the first method for C1 and second method forother criteria

Step 3 Table 5 shows that calculated numbers come fromFTF for criteria these data will be the input for fuzzy TOPSISin (12) and (13) The results of calculations are representedin Table 6 This data will be the initial weight of TODIMapproach

By using the fuzzy TOPSIS method we have Table 6As shown in Table 6 we will have different ranking for

criteria with respect to expert opinion changes during thetime that will be measurable by FTFs This evaluation helpsus to know about the importance of each criterion in thesetimes and gives us the perspective for more effective actionsin our company in future and Table 7 presents the ranking ofcriteria with respect to time

Step 4 Table 8 shows the expertsrsquo opinion on alternatives ineach criteria these data come from the linguistic variablesand then change to interval valued triangular fuzzy numbersby Table 2

Table 9 Matrix of alternative scores with respect to criteria

1198601

1198602

1198603

1198621

04232 06933 069331198622

04232 01155 145981198623

06933 10532 011551198624

04232 14598 105321198625

01155 04232 069331198626

06933 19204 01155

Step 5 Use TODIMmethods see Tseng et al [3]

By applying TODIM approach first 119889minus 119889lowast have beencalculated from (17) which is shown in Table 9 after thatthe weight of criteria has been calculated from (18) which isshown in Table 10

After calculating weight of each criterion the data mustbe normalized for calculation dominance weight of criteriaThe normalized data is shown in Table 11

Equation (23) calculates the overall value of alternative bynormalizing the corresponding dominance measurementsThe rank of each alternative derives from ordering the alter-natives values The global measures computed the completerank ordering of all alternatives In addition a sensitivityanalysis should then be applied to verify the stability of theresults based on the decision makersrsquo preferencesThe resultsare presented in Table 12

Considering results in Table 12 we will find out whichcriterion is the most important one we can also find outranking of alternatives regarding each criterion

Step 6 Now the final conclusion will be conducted fromTables 12 and 6 and we can find out in next 15 months and25 months which criteria will be more effective and finallywhich alternative will be the first

By combination of Fuzzy TOPSIS which represents theweight of criteria of green supply chain with respect to timeand TODIM method which shows the weight of alternativesaccording to criteria of supply chain we will have weights ofcriteria and alternatives in Tables 13 and 14

4 Conclusions

This study represents a newway to select supplier during timeperiods by using a hybrid MCDM Using fuzzy TOPSIS tohave more accurate weighting method for TODIM and then

Advances in Fuzzy Systems 9

Table 10 Criteria weights

1198621

1198622

1198623

1198624

1198625

1198626

119908119895

0144 0159 0148 0233 0098 0217

Table 11 Normalized scores

1198601

1198602

1198603

1198621

02338 03831 038311198622

02118 0057 0731198623

03723 05657 00621198624

01441 04972 03581198625

00937 03435 05621198626

0254 07037 00423

Table 12 Final weight of criteria by using TODIM

1198621

1198622

1198623

1198624

1198625

1198626

120585 minus059 minus044 minus099 minus052 minus1083 minus0711Normalization 041 056 001 048 0 03Negative numbers explain losses from goal

Table 13 Criteria weightrsquos changes during time

1198621

1198622

1198623

1198624

1198625

1198626

119882 15 022 038 0005 0252 0 0153119882 25 021 029 0005 0264 0 0

Table 14 Alternative weights during time

1198601

1198602

1198603

15 months 047 086 09825 months 032 056 084

combining it with fuzzy time function (FTF) helped us torank the criteria and alternatives in several time periods Theimportant aim of this study was to use fuzzy time functionwith a new approach to consider time for each triangularfuzzy number that helped us to find out which supplier in thefuture according to criteria is suitable for our green supplychain and when we should change our supplier According tothe results shown in Table 14 supplier A3 has a decreasingrate from month 15 to 25 but its decreasing rate is lessthan supplier 2rsquos Regarding these results we can assign ourfuture improvement programs and orders to these suppliersconsidering criteria changes in green supplier selections andalso have some repairing programs due to our vision andmanagerial decisions in future for these companies Also wecan assign our orders to these companies and we proposed anorder plan for these suppliers regarding their portions

Conflict of Interests

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

References

[1] M L Tseng ldquoAn assessment of cause and effect decision-making model for firm environmental knowledge manage-ment capacities in uncertaintyrdquo Environmental Monitoring andAssessment vol 161 no 1ndash4 pp 549ndash564 2010

[2] S Zhang S Liu and R Zhai ldquoAn extended GRA method forMCDM with interval-valued triangular fuzzy assessments andunknown weightsrdquo Computers and Industrial Engineering vol61 no 4 pp 1336ndash1341 2011

[3] M L Tseng K H Tan R J Lina and Y Gengb ldquoMulticriteriaanalysis of green supply chain management using interval-valued fuzzy TODIMrdquo Knowledge-Based Systems 2012

[4] M Izadikhah ldquoGroup decision making process for supplierselectionwith TOPSISmethod under interval-valued intuition-istic fuzzy numbersrdquo Advances in Fuzzy Systems vol 2012Article ID 407942 14 pages 2012

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[6] H J Zimmermann Fuzzy Set Theory and Its ApplicationInternational Thomson Publishing Norwell Mass USA 2001

[7] J C A de Figueiredo and A Perkusich ldquoFaults and timinganalysis in real-time distributed systems a fuzzy time Petri-net-based approachrdquo Fuzzy Sets and Systems vol 83 no 2 pp 143ndash168 1996

[8] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[9] C L Hwang and K Yoon Multiple Attributes Decision MakingMethods and Applications Springer Berlin 1981

[10] M Ekmekcioglu T Kaya and C Kahraman ldquoFuzzy multicri-teria disposal method and site selection for municipal solidwasterdquoWaste Management vol 30 no 8-9 pp 1729ndash1736 2010

[11] S J Chen and C L Hwang ldquoFuzzy multi attribute decisionmakingrdquo vol 375 of lecture notes in economics andmathematicalsystem Springer New York 1992

[12] C Chen ldquoExtensions of the TOPSIS for group decision-makingunder fuzzy environmentrdquo Fuzzy Sets and Systems vol 114 no1 pp 1ndash9 2000

[13] T Chu ldquoSelecting plant location via a fuzzy TOPSIS approachrdquoInternational Journal of Advanced Manufacturing Technologyvol 20 no 11 pp 859ndash864 2002

[14] T Chu and Y Lin ldquoImproved extensions of the TOPSIS forgroup decision making under fuzzy environmentrdquo Journal ofInformation and Optimization Sciences vol 23273 286 pages2002

[15] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoExtension ofthe TOPSIS method for decision-making problems with fuzzydatardquo Applied Mathematics and Computation vol 181 no 2 pp1544ndash1551 2006

[16] T C Chu and Y C Lin ldquoAn interval arithmetic based fuzzyTOPSIS modelrdquo Expert Systems with Applications vol 36 no8 pp 10870ndash10876 2009

[17] D Yong ldquoPlant location selection based on fuzzy TOPSISrdquoInternational Journal of Advanced Manufacturing Technologyvol 28 no 7-8 pp 839ndash844 2006

[18] C T Chen C T Lin and S F Huang ldquoA fuzzy approach forsupplier evaluation and selection in supply chainmanagementrdquoInternational Journal of Production Economics vol 102 no 2 pp289ndash301 2006

10 Advances in Fuzzy Systems

[19] C Kahraman S Cevik N Y Ates andMGulbay ldquoFuzzymulti-criteria evaluation of industrial robotic systemsrdquoComputers andIndustrial Engineering vol 52 no 4 pp 414ndash433 2007

[20] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS integrated with fuzzyAHPrdquo in Proceedings of the 1st International Symposium onComputing in Science and Engineering pp 706ndash713 2010

[21] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

[22] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[23] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 9

Table 10 Criteria weights

1198621

1198622

1198623

1198624

1198625

1198626

119908119895

0144 0159 0148 0233 0098 0217

Table 11 Normalized scores

1198601

1198602

1198603

1198621

02338 03831 038311198622

02118 0057 0731198623

03723 05657 00621198624

01441 04972 03581198625

00937 03435 05621198626

0254 07037 00423

Table 12 Final weight of criteria by using TODIM

1198621

1198622

1198623

1198624

1198625

1198626

120585 minus059 minus044 minus099 minus052 minus1083 minus0711Normalization 041 056 001 048 0 03Negative numbers explain losses from goal

Table 13 Criteria weightrsquos changes during time

1198621

1198622

1198623

1198624

1198625

1198626

119882 15 022 038 0005 0252 0 0153119882 25 021 029 0005 0264 0 0

Table 14 Alternative weights during time

1198601

1198602

1198603

15 months 047 086 09825 months 032 056 084

combining it with fuzzy time function (FTF) helped us torank the criteria and alternatives in several time periods Theimportant aim of this study was to use fuzzy time functionwith a new approach to consider time for each triangularfuzzy number that helped us to find out which supplier in thefuture according to criteria is suitable for our green supplychain and when we should change our supplier According tothe results shown in Table 14 supplier A3 has a decreasingrate from month 15 to 25 but its decreasing rate is lessthan supplier 2rsquos Regarding these results we can assign ourfuture improvement programs and orders to these suppliersconsidering criteria changes in green supplier selections andalso have some repairing programs due to our vision andmanagerial decisions in future for these companies Also wecan assign our orders to these companies and we proposed anorder plan for these suppliers regarding their portions

Conflict of Interests

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

References

[1] M L Tseng ldquoAn assessment of cause and effect decision-making model for firm environmental knowledge manage-ment capacities in uncertaintyrdquo Environmental Monitoring andAssessment vol 161 no 1ndash4 pp 549ndash564 2010

[2] S Zhang S Liu and R Zhai ldquoAn extended GRA method forMCDM with interval-valued triangular fuzzy assessments andunknown weightsrdquo Computers and Industrial Engineering vol61 no 4 pp 1336ndash1341 2011

[3] M L Tseng K H Tan R J Lina and Y Gengb ldquoMulticriteriaanalysis of green supply chain management using interval-valued fuzzy TODIMrdquo Knowledge-Based Systems 2012

[4] M Izadikhah ldquoGroup decision making process for supplierselectionwith TOPSISmethod under interval-valued intuition-istic fuzzy numbersrdquo Advances in Fuzzy Systems vol 2012Article ID 407942 14 pages 2012

[5] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[6] H J Zimmermann Fuzzy Set Theory and Its ApplicationInternational Thomson Publishing Norwell Mass USA 2001

[7] J C A de Figueiredo and A Perkusich ldquoFaults and timinganalysis in real-time distributed systems a fuzzy time Petri-net-based approachrdquo Fuzzy Sets and Systems vol 83 no 2 pp 143ndash168 1996

[8] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[9] C L Hwang and K Yoon Multiple Attributes Decision MakingMethods and Applications Springer Berlin 1981

[10] M Ekmekcioglu T Kaya and C Kahraman ldquoFuzzy multicri-teria disposal method and site selection for municipal solidwasterdquoWaste Management vol 30 no 8-9 pp 1729ndash1736 2010

[11] S J Chen and C L Hwang ldquoFuzzy multi attribute decisionmakingrdquo vol 375 of lecture notes in economics andmathematicalsystem Springer New York 1992

[12] C Chen ldquoExtensions of the TOPSIS for group decision-makingunder fuzzy environmentrdquo Fuzzy Sets and Systems vol 114 no1 pp 1ndash9 2000

[13] T Chu ldquoSelecting plant location via a fuzzy TOPSIS approachrdquoInternational Journal of Advanced Manufacturing Technologyvol 20 no 11 pp 859ndash864 2002

[14] T Chu and Y Lin ldquoImproved extensions of the TOPSIS forgroup decision making under fuzzy environmentrdquo Journal ofInformation and Optimization Sciences vol 23273 286 pages2002

[15] G R Jahanshahloo F H Lotfi andM Izadikhah ldquoExtension ofthe TOPSIS method for decision-making problems with fuzzydatardquo Applied Mathematics and Computation vol 181 no 2 pp1544ndash1551 2006

[16] T C Chu and Y C Lin ldquoAn interval arithmetic based fuzzyTOPSIS modelrdquo Expert Systems with Applications vol 36 no8 pp 10870ndash10876 2009

[17] D Yong ldquoPlant location selection based on fuzzy TOPSISrdquoInternational Journal of Advanced Manufacturing Technologyvol 28 no 7-8 pp 839ndash844 2006

[18] C T Chen C T Lin and S F Huang ldquoA fuzzy approach forsupplier evaluation and selection in supply chainmanagementrdquoInternational Journal of Production Economics vol 102 no 2 pp289ndash301 2006

10 Advances in Fuzzy Systems

[19] C Kahraman S Cevik N Y Ates andMGulbay ldquoFuzzymulti-criteria evaluation of industrial robotic systemsrdquoComputers andIndustrial Engineering vol 52 no 4 pp 414ndash433 2007

[20] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS integrated with fuzzyAHPrdquo in Proceedings of the 1st International Symposium onComputing in Science and Engineering pp 706ndash713 2010

[21] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

[22] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[23] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Advances in Fuzzy Systems

[19] C Kahraman S Cevik N Y Ates andMGulbay ldquoFuzzymulti-criteria evaluation of industrial robotic systemsrdquoComputers andIndustrial Engineering vol 52 no 4 pp 414ndash433 2007

[20] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS integrated with fuzzyAHPrdquo in Proceedings of the 1st International Symposium onComputing in Science and Engineering pp 706ndash713 2010

[21] T Kaya and C Kahraman ldquoMulticriteria decision making inenergy planning using a modified fuzzy TOPSIS methodologyrdquoExpert Systems with Applications vol 38 no 6 pp 6577ndash65852011

[22] A C Kutlu and M Ekmekcioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash67 2012

[23] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014