Fuzzy -discountingmethod for multi-criteriadecision-makingAtilla Karamana* and MetinDagdevirenbaDepartmentof Operations Research, Defense Sciences Institute,Turkish Military Academy, Ankara, Turkey;bFaculty of Engineering,Department of IndustrialEngineering, Gazi University, Ankara, Turkey(Received 25 February 2014; accepted27 July 2014)The-DiscountingMethodwas developedtobeanalternativetoandextensionof theAnalytical HierarchyProcess(AHP) tosolvemulti-criteriadecision-making(MCDM) problemswithnon-commensurableandconictingcriteria. Incontrast totheAHP, thismethodworksnot onlyfor pairwisecomparisonsbut alsofor n-wisecomparisonsif relativeimportance of criteria can be expressed in a system of linear homogenous equations. This method also has a comparativeadvantage asitcantransformthoseMCDM problems,classiedasinconsistentby theAHP,into aconsistentform.ThisstudybrieycomparesthetwomethodsandthendevelopstheFuzzy-DiscountingMethodforMulti-CriteriaDecisionMaking(F-DMMCDM). TwoillustrativefuzzyMCDMproblemsfromtheliteraturehavebeensolvedtoshowhowthe F-DMMCDM works.Keywords: MCDM; -Discounting Method; consistency in fuzzydecision problems; fuzzyset theory1. IntroductionIt is not an easy task for a decision maker to choosefromamongthe alternatives s/hehas becausethedeci-sion-making process mostly includes multiple criteriawithdifferent directionsinsteadofamono-criterionwithone direction. The conict situationwhichthe decisionmaker faces emerges while reaching a comprehensivejudgment as a result of balancing the criteria that arecompeting with each other and which are orientedtowardmultiplegoals(BeltonandStewart2003).Thisisafundamental probleminmulti-criteriadecision-making(MCDM) known as the aggregation problem (Roy2005).Asdecision-makingproblemsinvolvecriteriathatare non-commensurable (different units) andconicting(competingwith eachother), the MCDMgains impor-tancefromtheviewpoint of decisionmakersasit pro-vides the tools and techniques that balance thosemultiple criteria(Beltonand Stewart2003).TheMCDMmethodsproposedintheliteraturehavebothadvantagesanddisadvantages. Analytical HierarchyProcess (AHP), as oneof themost usedMCDMmeth-ods, facilitates the structuring of the decision-makingproblems through a hierarchical approach; yet, it hasreceived criticisms including that by Dodd, Donegan,andMcMaster (1995) for thearbitrarymeasureof con-sistency proposed in the method. In the scope of thispaper, thedevelopment of Fuzzy-DiscountingMethodforMulti-CriteriaDecisionMaking(F-DMMCDM)forpairwisecomparisonsusinglinguisticvariables(variableswhoseunderlyingvaluesarederivedfromverbal prefer-ence statements) has been motivated by this criticismfromDodd, Donegan, andMcMaster(1995). Inthecaseof exceedingtheinconsistencythresholdof 10%intheAHP, the analysts need to return back to the decisionmaker(s) andrepeat all the procedures toget correctedpreferencestatementsofthedecisionmaker(s)topresentan acceptablenal output.Besides theburdenof overcomingtheinconsistencyissue, the analysts may sometimes need to work withdecision makers who prefer to express relationshipsamongcriteriathroughverbal (fuzzy)statementsinsteadof crisp numerical values. In this respect, the main objec-tive of this paper is to develop a novel method thatensuresconsistencyfor thepreferencestatementsof thedecisionmakersthatarereectedindecisionmatricesofthefuzzyMCDMproblems withpairwisecomparisons.To reach this objective, we have chosen to widen thescope of the -Discounting Method (-D) for MCDMdevelopedbySmarandache(2010), analternativetoandextensionoftheAHP(Saaty1972)tosolvelineardeter-ministic MCDMproblems by assuring consistency forthe decision matrices of concern, as a basis for thedevelopment of the aforementioned method for fuzzydecisionmatrices constructedbothinsinglepersonandgroupdecision-makingsettings. Anotherobjectiveofthispaper is toalsointegrate a revisedversion of a noveldefuzzication method (Converting Fuzzy Data intoCrispScoresMethodCFCSdevelopedbyOpricovicand Tzeng (2003)) in the aforementioned method toobtainthecrispweightsthat areassignedtothecriteriaintheend.Applicabletoboththemixedsetofcrispandfuzzy numbers, the defuzzication method CFCS has*Corresponding author. Email: tllkrmn@gmail.com 2015The ChineseInstituteof EngineersJournal of the ChineseInstitute of Engineers, 2015http://dx.doi.org/10.1080/02533839.2015.1037352Downloaded by [85.98.133.61] at 11:17 17 May 2015 advantages in treating the triangular fuzzy numbers(TFNs) with the same mean.The F-DM MCDM proposed in this papercontributes to the MCDM literature by expandingSmarandaches (2010) methodtoincludefuzzyMCDMproblems andtoensure consistencyfor the constructedfuzzy decision matrices which was only possible forweakly inconsistent linear deterministic MCDMprob-lems with pairwise comparisons. The F-DMMCDMdealswiththefuzzyMCDMproblems, constructedwithTFNswithinthescopeofthispaper, byrst decompos-ingthe fuzzydecisionmatrices intothree separate (l),(m), and(u)matricesthat constitutetherespectivefuzzynumbers from the lower boundary, most likely, andupper boundaryfuzzyvalues, respectively. These fuzzymatrices are thencheckedfor consistencybytestingifthedeterminant (det) of therespectivematrixequals 0.Inthecaseofdet =0, thematrixissaidtobeaconsis-tentmatrix.Inthecontrarysituation(wheredet0),thematrixisfoundtobe aninconsistentmatrixandneedstobeeither discountedor increasedwiththecoefcient.During the formation of the fuzzy decision matrixthrough the linguistic variables assigned to the criteriabythedecisionmaker, it is not expectedthat thedeci-sion maker assigns linguistic variables fully consistentwithall pairwisecomparisons. This is either duetothesize of the MCDMproblem, environmental conditionssurroundingthedecisionmaker that mayinuenceher/hisrationality,orher/hispartialknowledgeofthefactorsconcerningtheMCDMproblemthatmayhinderher/himinassigninglinguistic variables tocriteria pairs consis-tent with others in the fuzzy decision matrix. In thisrespect, consistency is needed to ensure valid resultsfromthesolutionoftheMCDMproblemofconcern.Byapplying F-DMMCDMin inconsistent linear fuzzydecisionmatrices, thereis ahighlikelihoodof assuringconsistencyforthedecomposedfuzzymatricesatanini-tialstagewhilesolvingthefuzzyMCDMproblem.Afterthe multiplication of the matrix elements above thediagonalwithtestingthecoefcient, thenewmatrixisrecheckedtosee if it gives det =0. Providedthat allmatricesgivedet =0, defuzzicationisappliedtohavethecrispcriteriaweights. Inthenext section, weintro-duce, in detail, the consistency/inconsistency of anMCDMproblemwithin the context of this paper. Webrieycomparetheoriginal -DMCDMandAHP, andthenintroducetheF-DMMCDMforpairwisecompar-isonsinafuzzysetting. Then, wesolvetwoillustrativefuzzyMCDMproblemsfromtheliteraturetoshowhowthe F-DMMCDMworks.2. -Discounting Method for MCDMDeveloped by Smarandache (2010), the -DiscountingMethod for Multi-Criteria Decision Making (-DMCDM) is analternativetoandextensionof theAHPwhose foundations were laid by Miller (1966) whoformulatedcriteria under a hierarchyfor the rst time.ThiswasthenadvancedbySaaty(1972)underthenameAHP. Incontrast totheAHPwhichcanonlydeal withpairwise comparisons, -D MCDM can handle anyn-wise(withn2) comparisonsof criteriaintheformof linear homogeneousequations.2.1. Classication of linear decision-makingproblemsIn-DMCDM, decision-makingproblemswhichcanbeexpressedas linear homogeneous equations are dividedinto three groups based on their consistency levels(Smarandache 2010):(i) Consistent lineardecision-makingproblems: Anysubstitutionofavariablexifromanequationintoanother equation returns a result in agreementwith all equations.(ii) Weaklyinconsistent lineardecision-makingprob-lems(WD): Inthistypeofproblems, at least onesubstitutionofavariablexifromanequationintoanother equationreturnsaresult indisagreementwith at least another equation in the followingways:WD(1)xi = k1 xj; k [1;xi = k2 xj; k2[1; k2 ,= k1_ _(1)orWD(2)xi = k1 xj; 0\k1\1;xi = k2 xj; 0\k2\1; k2 ,= k1_ _(2)orWD(3) xi = kxi; k ,= 1 : (3)(iii) Strongly inconsistent linear decision-makingproblems(SD):Inthistypeofproblems, atleastonesubstitutionof avariable xifromanequa-tion into another equation results in disagree-ment with at least another equation in thefollowing way:SD(4)xi = k1 xj;xi = k2 xj;_ _ while 0\k1\1\k2or 0\k2\1\k1:(4)2.2. Procedural steps for -D MCDM-DMCDMtransformsaninconsistent decision-makingproblemintoaconsistent statebydiscounting/increasingthelinear judgment scalecoefcientsat/tosomevalues.Theprocedural steps for -DMCDMcanbedescribedas follows (Smarandache2010):2 A. Karaman and M.DagdevirenDownloaded by [85.98.133.61] at 11:17 17 May 2015 (1) Let C={C1, C2, , Cn}, n2, beaset forthehierarchical components(criteriaandalterna-tives). Get the relationships fromthe decisionmaker(s)rstforthecriteriaand thenthealterna-tivesandformthepreferencestatementsset, P={P1, P2, , Pm}, m 1. Each preferencePi isalinear homogeneous equationof the criteria C1,C2, , Cn: Pi=f(C1, C2, , Cn). For theweightsofthehierarchycomponents, let uscon-struct abasicbelief assignment (bba): m: C[0, 1] such that m(Ci) = xi, 0 < xi< 1 and

ni=1 mCi( ) =

ni=1 xi = 1. Consequently, thetotalweight ofthehierarchycomponentsintheirown groupequals1.(2) Findallvariablesxiinaccordancewiththesetofpreferences Pto get an mn linear homoge-neous system of equations whose associatedmatrixisA=(aij), 1i mand1j n. Inorderforthissystemtohavenontrivialsolutions,therankof thematrixAshouldbestrictlylessthann.(3) Calculate thedeterminant ofmatrix A, det(A).(4) Ifdet(A)=0,thenthedecisionproblemisfoundto be consistent as the systemof equations isdependent. Thereisnoneedtoparameterizethesystemwith. Inthe case of parameterization,use the Fairness Principle to set all parametersequal a1 = a2 == ap = a [0. Solve thesystemto nd a general solution. Replace parameters andsecondaryvariables,1toreachaparticularsolution. Obtainthepriorityvector;i.e.corresponding eigenvector for the maximumeigenvalue, by normalizing the particular solu-tion.(5) Ifdet(A)0,thenthedecisionproblemisfoundtobeinconsistentasthehomogeneouslinearsys-temhasonlyatrivialsolution.Basedonthetypeofinconsistency, go to either step 6 or step7.(6) If thelinear decision-makingproblemis weaklyinconsistent, then parameterize the right-hand sidecoefcients, anddenotethesystemmatrixA().Compute the values which makes det(A())equal 0 in order to get the parametric equation (IftheFairnessPrincipleisused, set all parametersequaltoeachotherandsolvefor>0).Replace in A(), and solve the resulting dependenthomogeneouslinear system. Similarlyasinstep4,replaceeachsecondaryvariableby1,andnor-malize theparticular solutiontoget thepriorityvector.(7) If the linear decision-making problem isstronglyinconsistent, get moreinformationfromthe decision-maker(s) to remove inconsistenciesfound in the decision-making problem. Instrong inconsistency situation, the FairnessPrinciple may not work properly. Anotherprinciple shouldbe consideredtodeal withtheinconsistencies.In weak inconsistent decision-making problems,where-DMCDM/FairnessPrincipleisused, thedegreeof consistencyandinconsistencyof thedecision-makingproblemat theinitial stageiscalculatedbasedonthevalue obtained asfollows (Smarandache2010): If 0 0 Multiplication (): 1 2 = (l1l2, m1m2, u1u2) Multiplication of a real number r with a TFN:r 1 = (rl1, rm1, ru1) r, li, mi, ui > 0 Division (): 1 2 = (l1/u2, m1/m2, u1/l2) Inversion: 11= (1/u1, 1/m1, 1/l1) li, mi, ui > 03.2. Single persondecision-making withF-DMMCDMInthis section, weintroducethe F-DMMCDMalgo-rithm for decision-making problems with one singledecision maker. Within the concept of single persondecision-making with F-DMMCDM, the preferencestatements of a single decision makeris obtained throughaquestionnairethatincludesquestionsforpairwisecom-parisons amongthe criteria of concern. As the numberof criteria/alternatives increases, there is an inclinationforaninconsistentoutcome(Olsonetal.1995)basedonthe increasingcomplexityof the decision-makingprob-lem. In this respect, the algorithms proposed in thispaperarebothforthesinglepersonandgroupdecision-makingsettings.Theprobabilityofreturningtothedeci-sionmaker(s)togivepreferencestatementsasaresultofaninconsistent outcomeishighlyreducedthankstothe coefcient(s) calculated within the procedure of thealgorithms. Inthecaseofendingupwithmorethanonecoefcient, theproposedalgorithmsalsodonot differ-entiatebetweenthemandtakeanaverageoftheweightsthat are obtained afterthemultiplicationof each coef-cient with the respective fuzzy decision matrix. Aftermultiplication with coefcients, the elements of thefuzzydecisionmatricesshouldhavevaluesbetween1/9and 9 to be consistentwith the intensityscale.Theprocedural steps for theF-DMMCDMareasfollows:(1) Structure the criteria of the decision-makingproblem undera hierarchy.(2) Given the linguistic variables with determinedscales, andtheir correspondingfuzzyimportancegrades; collect the subjective preference state-ments of the decisionmaker amongthe criteriaand construct the pairwise fuzzy decisionmatrix.(3) Assuming that the fuzzy importance grades areTFN, group the pairwise fuzzy decision matrixinthreematrices; i.e. (l) lower bound, (m) mostlikely, and (u) upper bound value matrix,respectively.(4) Reorder the elements of these (l), (m), and(u)matrices suchthat theabovecells of thediago-nal contain the lower bound, most likely, andupper bound values in lij, mij, and uij, respec-tivelyoftherespectiveTFN.(5) Calculate determinants for (l), (m), and (u)matrices. If obtaineddeterminant valuesequal 0,thentherespectivematricesarefoundtobecon-sistent. Calculate the weights of the criteria byrst normalizing the column values and thentakingaveragesoftherowcolumnvaluesoftherespectivenormalizedmatrices.(6) If det 0inanyof thematricescreatedinstep5, then the respective matrix/matrices is/areinconsistent. Findthe >0values resultingindet =0ininconsistent matrix/matricesbasedontheFairnessPrinciple.(7) If thesolutionset for thecoefcient obtainedinstep6has:(a) Onlyoneelement, multiplytheupper partof the matrixdiagonal withandbottomwith1/. Test if the newvalues result indet =0. If the matrix is consistent, thencalculatethe weightsofthe criteria.(b) Morethanoneelement,choosethevaluebased on the following: For 0 import matrixlibrary>>> equalize x as an unknown variable>>>construct the matrix M with elements from(m)thataremultipliedbyxforthosefoundintheupperdiagonal and by 1/xfor the lower diagonal>>> solve forthe coefcients making det(M) = 0Thesolutionset for is foundas [14/9, 35/9, 7/6,and7/15]. Amongthem7/6is thevalueafter whosemultiplication with matrix elements (matrix elementsoverthediagonal aremultipliedwith7/6andbelowthediagonal with 6/7) yields only one matrix element toexceedthe19scaleandthecriteriaweights=[0.2558,0.0898, 0.6119, and0.0425]. Aftertheapplicationofthesameprocedurefor(u),thesolutionsetforisfoundas[2, 3/8, 32/9, and2/3]amongwhich2/3ischosenasitsmultiplicationwithmatrixelementsdoesnot causethemtoexceedthe19scale. Thecriteriaweights=[0.2056,0.0855,0.6470, and 0.0620].Table4. Matrix(m)constructedfromthefuzzypairwisecom-parison.(m) D A R1 R2D 1 3 1/5 7A 1/3 1 1/7 2R1 5 7 1 9R2 1/7 1/9 16 A. Karaman and M.DagdevirenDownloaded by [85.98.133.61] at 11:17 17 May 2015 Defuzzify the obtained weights for each criterionwiththe revisedCFCS(RCFCS)method:(1) After sorting in an ascending order, normalizethefuzzy criteriaweightsgiven inTable 6:imax=0.66380imin=0.0425Dmaxmin=0.6213(Table7)(2) Compute leftand rightscores(ls, rs):Table 8(3) Compute total normalizedcrisp value:xcrispj= [0.2767,0.0687,0.9697,and 0.0306](4) Compute crispweight values:fij= [0.2144, 0.0851, 0.6450, and 0.0615]and wij = [0.2131,0.0846, 0.6412, and 0.0611]Intheoriginal problem, TzengandHuangusedthegeometricmeanmethodtondthefuzzyweight vectorandthe center of area methodtodefuzzifythe weightvector into crisp values. They computed the weightsassigned to each criterion as [0.2278, 0.0898, 0.6572,and0.0498] amounting toa total of 1.0246. Althoughthe weights computed with F-DMMCDM, [0.2131,0.0846, 0.6412, and0.0611], result inthesameranking(R1>D>A>R2)asFAHPdid, theweightsarelowerfor D, A, andR1, andonlyhigher for R2whichis anindicationofdifferent methodsyieldingdifferent results.However, it shouldbe noted that due tothe relativelysmallsizeoftheillustrativeproblemcoveredinthissec-tion, the criteria rankings happened to be in the sameorder inbothmethods. Also, theweights obtainedas aresult of F-DMMCDMdonot exceedthe maximumvalue of1.3.3. Group decision-making with F-DM MCDMF-DMMCDMforagroupdecision-makingsituationisapplied in the same way as single person decision-making with F-DM MCDM, with an additionalaggregationprocedureforpairwisecomparisonsbetweencriteria froma groupof decisionmakers. It shouldbenotedthat theillustrativeexamplecoveredinthispapercontainsfullinformationfromthedecisionmakers;how-ever, theF-DMMCDMforgroupdecision-makinghasthe potential to deal with missing information.Theprocedural stepsfor F-DMMCDMinagroupdecisionsettingare identical tothe procedural steps insingleperson decision setting withthe followingadditionin Step2: Given the linguistic variables with determinedscales, and their corresponding fuzzy importancegrades; collect thesubjectivepreferencestatementsfroma groupof decision makers among thecriteriato construct the pairwise fuzzy decision matrix.Then, aggregate these matrices into a fuzzy syn-thetic pairwise comparison matrix using the geo-metricmeanmethodsuggested by Buckley(1985):aij~=_a1ij~ anij~ _1=n: (7)3.3.1. Illustrative exampleTo illustrate the application of F-DM MCDM in a groupdecision-making situation, we took another examplefoundintheworkofTzengandHuang(2011)that wasTable5. Matrix(u) constructedfromthefuzzypairwisecom-parison.(u) D A R1 R2D 1 4 1/4 8A 1/4 1 1/6 3R1 4 6 1 9R2 1/8 1/3 1/9 1Table 6. Fuzzy criteriaweights.l m u = rD 0.1974 0.2056 0.2558A 0.0748 0.0855 0.0898R1 0.6119 0.6470 0.6638R2 0.0425 0.0620 0.0640Table 7. Normalizedfuzzy criteriaweights.Normalized matrix l m rD 0.2494 0.2625 0.3433A 0.0521 0.0693 0.0762R1 0.9166 0.9729 1.0000R2 0.0000 0.0314 0.0346Table 8. Leftand right scores.ls rsD 0.2591 0.3177A 0.0681 0.0757R1 0.9210 0.9736R2 0.0304 0.0345Table 9. Linguistic variables and corresponding fuzzy numbers.Linguistic variables LAb LVs LEs LWk LEq Eq Wk Es Vs AbFuzzy numbers91~71~51~31~11~1~3~5~7~9~TFN values19 ; 19 ; 17_ _19 ; 17 ; 15_ _17 ; 15 ; 13_ _15 ; 13 ; 1_ _13, 1, 1_ _(1, 1, 3) (1, 3, 5) (3, 5, 7) (5, 7, 9) (7, 9, 9)Journalofthe ChineseInstituteof Engineers 7Downloaded by [85.98.133.61] at 11:17 17 May 2015 solvedusingfuzzyAHP. Tocomputethevalues, weusedMATLAB(2012), anumerical computingenviron-ment and fourth-generation programming language. Inbrief,theproblemisabouttheselectionofarm,afteratender process, for theprovisionof engineeringservicesfor the construction ofa branch station of the TaipeiCityPolice Bureau. Five rms submitted proposals which willbeevaluatedagainst sixdifferent criteriabythreedeci-sion-makinggroupscomposedofvepersonseach. Thedecisionmakerswant tondtheweightstobeassignedto these criteria. The linguistic variables used by the deci-sionmakersaregiveninTable9andthefuzzypairwisecomparisonmatrices andcorrespondingtriangular fuzzynumbers for the ve rms against six criteria are given inTables 10 and 11. Toconstruct thefuzzysyntheticpairwisecompar-ison matrix fromthe matrices given above, thegeometric mean method is used:aij~ = a1ij~a2ij~a3ij~a4ij~a5ij~_ _1=5. Asanexample,a~12in the fuzzy synthetic pairwise comparisonmatrixiscalculatedasfollows:a12~ = 1;3;5 ( )15;13;1_ _17;15;13_ _15;13;1_ _1;3;5 ( )_ _1=5= 11517151_ _1=5; 31315133_ _1=5; 511315_ _1=5_ _=0:356;0:725;1:528 ( )(8)The fuzzy synthetic pairwise comparison matrix isshownin Table 12.Table 10. Linguistic variable assignment to criteria for eachrm.C1 C2 C3 C4 C5 C6Firm1C1 1 Wk Es Wk Eq EqC2 1 Wk Eq Eq WkC3 1 Eq LEs LEsC4 1 LWk LWkC5 1 EqC6 1Firm2C1 1 LWk Wk Eq Eq LWkC2 1 Wk Wk Wk LWkC3 1 LVs LVs LVsC4 1 LWk LWkC5 1 LWkC6 1Firm3C1 1 LEs Eq Wk Eq LVsC2 1 Wk Ab Vs LEsC3 1 Es Wk LVsC4 1 Eq LAbC5 1 LVsC6 1Firm4C1 1 LWk LEs Wk LVs EsC2 1 Es Vs LAb VsC3 1 Wk LAb WkC4 1 LAb WkC5 1 AbC6 1Firm5C1 1 Wk Es LWk LVs LEqC2 1 Wk LEs LVs LWkC3 1 LVs LAb LEsC4 1 LWk WkC5 1 EsC6 1Table 11. Corresponding fuzzy number assignment to eachrm.C1 C2 C3 C4 C5 C6Firm1C1 1 3~5~3~1~1~C2 1 3~1~1~3~C3 1 1~51~51~C4 1 31~31~C5 1 1~C6 1Firm2C1 1 31~3~1~1~31~C2 1 3~3~3~31~C3 1 71~71~71~C4 1 31~31~C5 1 31~C6 1Firm3C1 1 51~1~3~1~71C2 1 3~9~7~51~C3 1 5~3~71~C4 1 1~91~C5 1 71~C6 1Firm4C1 1 31~51~3~71~5~C2 1 5~7~91~7~C3 1 3~91~3~C4 1 91~3~C5 1 9~C6 1Firm5C1 1 3~5~31~71~11~C2 1 3~31~71~31~C3 1 71~91~51~C4 1 31~3~C5 1 5~C6 18 A. Karaman and M.DagdevirenDownloaded by [85.98.133.61] at 11:17 17 May 2015 Applysteps from5to7as describedabove anddefuzzify the computed fuzzy criteria weights inTable13 by applying the RCFCS.Acomparisonforthecrispcriteriaweightscomputedthrough F-DMMCDMand FAHPis givenin Table14.Asintherankingresultinthesinglepersondecision-makingcase, therankingresult ingroupdecision-makingfor F-DM MCDM and FAHP has been observed to be thesame. Both the FAHP and F-DM MCDM rank the criteriaas D5 > D2 > D6 > D1 > D4 > D3. As to the weightsassigned to criteria by FAHP, it has been observed that theyarehigherthantheweightsassignedbyF-DMMCDM.Thereasonforhavingsuchadifferenceinweightassign-ment is based on the different approaches in processing offuzzynumbersandapplicationofdifferentdefuzzicationmethods. It is a fact that Tzeng and Huang used best non-fuzzy performance as a defuzzication method as a result ofwhichtheassigned weightsamounted to1.289, exceedingthe total maximum value of 1.4. Conclusionand further researchBeyond the perimeters of academic circles, the AHPhas been widely used by non-academicians due to itsstructuring the decision-making problems in simple,hierarchical application, usage of verbal preferencestatements to denote the numerical scale of 19, andsetting a consistency level in MCDMproblems. How-ever, there are the criticisms of Barzilai (2010), Beltonand Gear (1983), and Dodd, Donegan, and McMaster(1995) against AHP which we previously mentioned.Furthermore, AHP can only handle pairwise compar-isons and the number of these pairwise comparisonsincreases enormously as the number of criteria/alterna-tives increases. -DMCDMnot only responds to thisshortcoming by getting n-wise comparisons fromthedecision-maker but alsohandlesweakinconsistent deci-sion-making problems (even above the AHP consis-tency ratio (CR) of 10%). This study developed theF-DMMCDMbothfor singlepersonandgroupdeci-sion-making situations, which groups the TFNs of thecorresponding linguistic variables in (l), (m), and (u)matrices inanascendingorder byconsideringthe val-ues abovethematrixdiagonal. Althoughit might seemunlikelyinthe rst place, this emerges as a necessarycondition to ensure consistent allocation of the TFNsfrom corresponding linguistic variables in relevantmatrices. F-DMMCDMthen ensures the consistencyof the relevant matrix by applying the aforementionedsteps to reach a matrix determinant value of 0. As alast step, RCFCS defuzzication method is applied toobtaincrispvalues fromthe(l), (m), and(u) matrices.Additionally, for thegroupdecision-makingsituationinF-DMMCDM, the geometric meanmethodhas beenused to produce an aggregate synthetic fuzzy decisionmatrix from the decision matrices obtained throughdecision makers. Two MCDM problems previouslysolvedwithFAHPtakenfromthe literature have beenresolved with F-DMMCDM. Considering the solu-tions, F-DMMCDMandFAHPhave giventhe samerankingbut withdifferent criteriaweightsinbothsingleandgroupdecision-makingsettings. Thereasonfor dif-ferent criteria weights is due to the different handlingof fuzzy numbers and defuzzication methods used inbothmethods.As afurther researchvenue, theF-DMMCDMiscapable of dealing with fuzzy MCDMproblems withmissing information, and even in a group decision-makingsetting, this novel method can include the preferenceTable 12. Fuzzy synthetic pairwise comparison matrix.SM C1 C2 C3 C4 C5 C6C1 1 1 1 0.356 0.725 1.528 1.052 1.719 3.005 0.725 1.552 3.272 0.415 0.459 1.016 0.467 0.750 1.332C2 1 1 1 1.246 3.323 5.348 1.380 2.068 3.323 0.573 0.803 1.310 0.491 0.859 1.719C3 1 1 1 0.517 0.789 1.332 0.181 0.254 0.369 0.191 0.300 0.467C4 1 1 1 0.245 0.333 0.844 0.339 0.644 1.290C5 1 1 1 0.859 1.165 2.068C6 1 1 1Table 13. Fuzzy criteriaweights computed with F-D MCDM.l m u = rC1 0.1407 0.1608 0.1991C2 0.2052 0.2256 0.2267C3 0.0705 0.0726 0.0845C4 0.1004 0.1031 0.1076C5 0.2299 0.2351 0.2745C6 0.1641 0.1908 0.2086Table 14. Crisp criteria weights according to F-DMCDMand FAHP.F-D MCDM FAHPC1 0.1664 0.2000C2 0.2237 0.2720C3 0.0739 0.0940C4 0.1043 0.1380C5 0.2415 0.3220C6 0.1902 0.2630Journalofthe ChineseInstituteof Engineers 9Downloaded by [85.98.133.61] at 11:17 17 May 2015 statementsinamixedstatebothwithperfectinformation(i.e. all pairwisecomparisonshavebeenprovidedbythedecision makers), and with missing information beforeaggregatingthemintoafuzzysyntheticdecisionmatrix.We also consider the MCDM problems with interval num-bers to be included within the scope of further research.NomenclatureAHP AnalyticalHierarchy ProcessCFCS Converting Fuzzy Data into CrispScores Methoddet determinantFAHP FuzzyAnalyticalHierarchy ProcessF-DMMCDM Fuzzy -Discounting Method forMulti-CriteriaDecision Making(l) matrix for lower boundary fuzzyvaluesls leftnormalized value/score(m) matrixformost likely fuzzyvaluesMCDM Multi-CriteriaDecision MakingRCFCS RevisedConvertingFuzzyDataintoCrisp ScoresMethodrs rightnormalized value/scoreSD stronglyinconsistentlineardecision-makingproblemsTFN triangular fuzzynumber(u) matrix for upper boundary fuzzyvaluesWD weaklyinconsistent linear decision-makingproblems-D MCDM -Discounting Method for Multi-CriteriaDecision MakingDisclosure statementNo potentialconict of interestwas reported by the authors.Note1. Secondaryvariables areusedtoindicatethevalues of themain variables in a decision-making problemwith linearpreference statements. Assume that in a linear decision-making problemwith three criteria x, y, and z thedegreesofimportanceassignedtocriteriaxandy, asmainvariables, canbeindicatedthroughz, asecondaryvariable,based on the transitivity rule. An example fromSmaran-dache(2010)showsthisasfollows: Supposethat C1(x)isfourtimesasimportant asC2(y); C2(y)isthreetimesasimportant asC3(z); andC3(z)isone-twelfthasimportantas C1 (x). The linear homogeneous system associated to thisdecision-makingproblemis:x=4y;y=3z;z=x/12.Usingzassecondaryvariable, theweightstobeassignedtomainvariables xandycanbeobtainedeasily. Solvingthishomogeneouslinearsystem, wegetitsgeneralsolutionthatwesetasavector[12z3zz],wherezcanbeanyrealnum-ber(zis consideredasecondaryvariable,whilex=12zandy = 3z are main variables).ORCIDAtilla Karaman http://orcid.org/0000-0002-8328-4103Metin Dagdeviren http://orcid.org/0000-0003-2121-5978ReferencesBarzilai, J. 2010. PreferenceFunctionModeling: TheMathe-matical Foundations of Decision Theory. In Trends inMultipleCriteriaDecisionAnalysis, editedbyM. Ehrgott,J. R. Figueira, and S. Greco, 5786. Halifax: Springer.Belton, V., andT. Gear. 1983. OnaShort-comingof SaatysMethodofAnalyticHierarchies.Omega11(3): 228230.doi:10.1016/0305-0483(83)90047-6.Belton, V., andT. J. Stewart. 2003. MultipleCriteriaDecisionAnalysis: An Integrated Approach. 2nd ed. Boston, MA:Kluwer Academic.Buckley, J. J. 1985. FuzzyHierarchical Analysis.FuzzySetsandSystems 17(3): 233247. doi:10.1016/1065-0114(85)90090-9.Chen, S.-J., andC.-L. Hwang. 1992. FuzzyMultipleAttributeDecision Making, Methods &Applications. New York:Springer Verlag.Demirel, T., N. C. Demirel, andC. Kahraman. 2008. FuzzyAnalyticHierarchyProcessandItsApplication.InFuzzyMulti-criteria Decision Making: Theory and ApplicationswithRecent Developments, editedbyC. Kahraman, 5383.New York: Springer Science+BusinessMedia.Dodd, F. J., H. A. Donegan, and T. B. M.McMaster. 1995.Reassessment of ConsistencyCriteriainJudgment Matri-ces.JournaloftheRoyalStatisticalSociety,SeriesD(TheStatistician)44 (1): 3141.Dubois, D., and H. Prade. 1978. Operations on FuzzyNumbers.International Journal of SystemsScience9(6):613626. doi:10.1080/00207727808941724.The MathWorks. 2012. MATLAB 7.14. Natick, MA: TheMathWorks.Miller, J. 1966. The Assessment of Worth: A SystematicProcedure and Its Experimental Validation. PhD diss.,Massachusetts Institute of Technology Sloan School ofManagement.Olson, D. L., H. M. Moshkovich, R. Schellenberger, andA.Mechitov. 1995. Consistency and Accuracy in DecisionAids: Experiments with Four Multiattribute Systems.Decision Sciences 26 (6): 723747. doi:10.1111/j.1540-5915.1995.tb01573.x.Opricovic, S., andG.-H. Tzeng. 2003. Defuzzicationwithina Multicriteria DecisionModel. International Journal ofUncertainty, Fuzziness and Knowledge-based Systems 11(5): 635652. doi:10.1142/S0218488503002387.Roy, B. 2005. ParadigmsandChallenges.InMultipleCrite-ria DecisionAnalysis:StateoftheArtSurveys, editedbyJ.Figueira, S. Greco, and M. Ehrgott, 324. New York:Springer Science+BusinessMedia.Saaty, T. L. 1972. AnEigenvalueAllocationModel forPrior-itization and Planning. PA: University of PennsylvaniaEnergy Management and Policy Center.Smarandache, F. 2010. -DiscountingMethodfor Multi-crite-riaDecisionMaking(-DMCDM).InProceedingsofthe10 A. Karaman and M.DagdevirenDownloaded by [85.98.133.61] at 11:17 17 May 2015 Fusion 2010 International Conference, Edinburgh, 2629July 2010: 17. Washington, DC: IEEE. doi:10.1109/ICIF.2010.5712044.SymPy Development Team. 2012. SymPy 0.7.1. http://www.sympy.org.Tzeng, G. H., and J. J. Huang. 2011. Multiple AttributeDecision Making: Methods &Applications. Boca Raton,FL: CRC Press.Zadeh, L. A. 1965. FuzzySetsandSystems.InProceedingsof theSymposiumonSystemTheory, editedbyJ.Fox, NewYork, 20April 1965: 2937. NewYork: PolytechnicPressof the PolytechnicInstitute of Brooklyn.Zadeh, L. A. 1975. Theconcept of alinguisticvariableandits applicationtoapproximatereasoningI.InformationSciences 8 (3): 199249. doi:10.1016/0020-0255(75)90036-5.Journalofthe ChineseInstituteof Engineers 11Downloaded by [85.98.133.61] at 11:17 17 May 2015