Research Article Extended VIKOR Method for Intuitionistic Fuzzy Multiattribute Decision-Making Based on a New Distance Measure

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1 Hidawi Mathematical Problems i Egieerig Volume 017, Article ID , 16 pages Research Article Exteded VIKOR Method for Ituitioistic Fuzzy Multiattribute Decisio-Makig Based o a New Distace Measure Xiao Luo 1 ad Xuazi Wag 1 School of Air ad Missile Defese, Air Force Egieerig Uiversity, Xi a , Chia School of Maagemet, Xi a Jiaotog Uiversity, Xi a , Chia Correspodece should be addressed to Xiao Luo; xiao hgz@163.com Received 4 Jue 017; Revised 18 September 017; Accepted 7 September 017; Published 9 October 017 Academic Editor: Aa M. Gil-Lafuete Copyright 017 Xiao Luo ad Xuazi Wag. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. A ituitioistic fuzzy VIKOR (IF-VIKOR) method is proposed based o a ew distace measure cosiderig the waver of ituitioistic fuzzy iformatio. The method aggregates all idividual decisio-makers assessmet iformatio based o ituitioistic fuzzy weighted averagig operator (IFWA), determies the weights of decisio-makers ad attributes objectively usig ituitioistic fuzzy etropy, calculates the group utility ad idividual regret by the ew distace measure, ad the reaches a compromise solutio. It ca be effectively applied to multiattribute decisio-makig (MADM) problems where the weights of decisio-makers ad attributes are completely ukow ad the attribute values are ituitioistic fuzzy umbers (IFNs). The validity ad stability of this method are verified by example aalysis ad sesitivity aalysis, ad its superiority is illustrated by the compariso with the existig method. 1. Itroductio Multiattribute decisio-makig (MADM), which has bee icreasigly studied ad is of cocer to researchers ad admiistrators, is oe of the most importat parts of decisio theory. It aims to provide a comprehesive solutio by evaluatig ad rakig alteratives based o coflictig attributes with respect to decisio-makers (DMs ) prefereces ad has widely bee used i egieerig, ecoomics, ad maagemet. Several classical MADM methods have bee proposed by researchers i literature, such as the TOPSIS (Techique for Order Preferece by Similarity to Ideal Solutio) method [1], the VIKOR (VIseKriterijumska Optimizacija I Kompromiso Reseje i Serbia, meaig multiattribute optimizatio ad compromise solutio) method [], the PROMETHEE (Preferece Rakig Orgaizatio Method for Erichmet Evaluatios) method [3], ad theelectre(elimiatioadchoicetraslatigreality) method [4]. Amog these methods, VIKOR is show to have some advatages over others by several researchers. I [5], Opricovic ad Tzeg compared VIKOR method ad TOPSIS method from the perspective of aggregatio fuctio ad foud that although both methods calculate the distace of a alterative to the ideal solutio, the former method provides a compromise solutio by mutual cocessios, while the latter method obtais a solutio with the farthest distace from the egative-ideal solutio ad the shortest distace from the positive ideal solutio without cosiderig the relative importace degrees of these distaces. Furthermore, Opricovic ad Tzeg compared a extesio of the VIKOR method comprehesively with the TOPSIS, ELECTRE, ad PROMETHEE method ad revealed that the VIKOR method is superior i meetig coflictig ad ocommesurable attributes [6]. At preset, the VIKOR method has wide applicatio i may areas, such as desig, mechaical egieerig, ad maufacturig [7 9], logistics ad supply chai maagemet [10, 11], structural, costructio, ad trasportatio egieerig [1, 13], ad busiess maagemet [14]. Sice Zadah put forward the cocept of fuzzy sets i 1965, fuzzy sets, especially fuzzy umbers (e.g., triagular fuzzy umbers ad trapezoidal fuzzy umbers), have bee widely researched ad applied i MADM problems to deal

2 Mathematical Problems i Egieerig with ucertaity i actual decisio-makig process. Sayadi et al. exteded the cocept of VIKOR method to solve MADM problems with iterval fuzzy umbers [15]. Kaya ad Kahrara developed a itegrated VIKOR-AHP method based o triagular fuzzy umbers i forestatio district selectio sceario [16]. Ashtiai ad Abdollahi Azgomi costructed a trust modelig based o a combiatio of fuzzy aalytic hierarchy process ad VIKOR method with triagular fuzzy umbers [17]. Liu et al. applied the VIKOR method to fid a compromise priority rakig of failure modes accordig to the risk factors, i which the weights of attributes ad ratigs of alteratives are liguistic variables represeted by trapezoidal or triagular fuzzy umbers [18]. Ju ad Wag proposed a ew method to solve multiattribute group decisio-makig (MAGDM) problems based o the traditioal idea of VIKOR method with trapezoidal fuzzy umbers [19]. However, oe of the above-metioed studies cosidered hesitacy. Due to the icreasig complexity ad ucertaity of ecoomic eviromet, decisio-makers perceptio of thigs ivolves ot oly affirmatio ad egatio, butalsohesitatio.therefore,hesitatio,asoeofthemost importat pieces of ucertai iformatio, may have great impact o the fial decisio ad should be cosidered i decisio-makig process. To deal with this, Ataassov exteded traditioal fuzzy sets to ituitioistic fuzzy sets (IFSs) which cosider membership, omembership, ad degree of hesitacy at the same time. I practical applicatio, the use of IFSs ca depict the fuzziess ad ospecificity ofproblemsbyemployigbothmembershipfuctioad omembership fuctio. Therefore, it is cosidered to be more effective tha classical FSs with merely a membership fuctio. Recetly, ituitioistic fuzzy VIKOR (IF- VIKOR) method becomes a popular research topic because it aims to deal with the widespread ucertaity i decisiomakig process, ad some researchers focus o the IF- VIKOR method to solve MADM problems. For istace, Devi [0] first exteded VIKOR method i ituitioistic fuzzy eviromet to deal with robot selectio problem, i which the weights of attributes ad ratigs of alteratives are represeted by triagular ituitioistic fuzzy sets (TIFSs). Lu ad Tag [1] employed the VIKOR method to evaluate auto parts supplier based o IFSs. Roostaee et al. exteded VIKOR method for group decisio-makig to solve the supplier selectio problem uder ituitioistic fuzzy eviromet [].Waetal.putforwardaewVIKORmethodfor MADM problem with TIFSs [3], i which the weights of attributes ad DMs are completely ukow. Park et al. exteded VIKOR method for dyamic ituitioistic fuzzy MADM cocerig uiversity faculty evaluatio problem [4]. Based o similarity measure betwee IFSs, Peg et al. proposed a ovel IF-VIKOR method to deal with multirespose optimizatio problem, i which the importace of eachresposeisgivebyaegieerasifss[5].hashemi et al. proposed a ew compromise method based o classical VIKOR model with iterval-valued ituitioistic fuzzy sets (IVIFSs), illustrated by a applicatio i reservoir flood cotrol operatio [6]. Mousavi et al. preseted a ew group decisio-makig method for MADM problems based o IF- VIKOR, i which the ratigs of alteratives cocerig each attribute ad the weights of criterios provided by DMs are represeted as liguistic variables, characterized by IFSs with multijudge [7]. Distace is a importat fudametal cocept of IFSs adplaysasigificatroleivikormethod.theclassical compromise programig is based o the distace measure which is determied by the closeess of a specific solutio to the ideal/ifeasible solutio, maximizig the group utility ad miimizig the idividual regret simultaeously [8, 9]. Although some existig methods employ ituitioistic fuzzy subtractio ad divisio operator to calculate group utility ad idividual regret, the result is i IFSs form ad eeds to be further sorted accordig to IFSs rakig rule. To some extet, the amout of computatio is icreased, ad the results are ot cocise as the crisp value obtaied by distace measure. Also, some researchers [30 3] reviewed existig distace ad similarity measures betwee IFSs ad tested them with a artificial bechmark. The result showed that may existig distace measures geerate couterituitive cases ad fail to capture waver, the lack of kowledge, whichshouldhavebeeaadvatageofifss.otheoe had, existig distace measures are show to have some limitatios with hidered effectiveess, while, o the other had, the measure choice is crucial to IF-VIKOR method due to its sigificat ifluece o the result. Therefore, this paper s objective is to costruct a ew distace measure, which ca capture IFS iformatio effectively. Based o this, a IF-VIKOR method is further proposed to solve MADM problems with completely ukow weights of DMs ad attributes. The mai features ad ovelties of this paper are as follows: (1) Differet from the VIKOR method based o classical fuzzy sets, such as iterval fuzzy umbers [15], triagular fuzzy umbers [16 18], ad trapezoidal fuzzy umbers [18, 19],themethodproposedithispaperisbasedoIFSs, which adds a omembership o the basis of the traditioal membership ad is able to describe support, oppositio, ad eutrality i huma cogitio. Such a exteded defiitio helps more adequately to represet situatios whe decisio maker abstais or hesitates from expressig their assessmets. () Compared with the method [0, 7] employig ituitioistic fuzzy operator to calculate group utility ad idividual regret, our method geerates crisp value result i a more ituitive way with less computatio. Although Roostaee et al. [] exteded the VIKOR method to ituitioistic fuzzy eviromet based o hammig distace, the method sometimes geerates couterituitive cases ad takes o cosideratio of waver i IFSs. Istead, our method basedoaewdistacemeasurecouldreflectituitioistic fuzzy iformatio effectively. Also, the ew distace does ot geerate couterituitive cases i the artificial bechmark test proposed by [30]. (3) Compared with the VIKOR method [0, 3, 6] based o those extesios of IFSs, IFSs possess soud theoretical foudatio such as basic defiitio ad operatios, compariso rules, iformatio fusio method, ad measures of IFS.

3 Mathematical Problems i Egieerig 3 The proposed IF-VIKOR method ca take advatage of these theories i costructio ad verificatio, which is favorable to practical applicatio. (4) Although IFSs have the advatage of beig able to cosider waver, most of existig IF-VIKOR methods are applied to eviromet with less ospecificity iformatio [7, 33]. Through a example of ERP system selectio, our paper illustrates the effectiveess of proposed method i such situatio. Moreover, a example of material hadlig selectio idicates that our method ca obtai desired rakig result, superior i eviromet with low specificity. (5) I the MADM problem with ituitioistic fuzzy umbers (IFNs), subjective radomess exists because the weights of DMs or attributes are usually give artificially [, 7]. To avoid this issue, these weights are determied objectively usig ituitioistic fuzzy etropy i the proposed method. (6) Compared with similar methods used for MADM [1, 3, 4], the proposed method based o the decisio priciple of the classical VIKOR method ca maximize the group utility ad miimize the idividual regret simultaeously, makig the decisio result more reasoable. Also, the coefficiet of decisio mechaism ca be chaged accordig to actual requiremets to balace group utility ad idividual regret, which ca icrease the flexibility of decisio-makig. The remaider of this paper is orgaized as follows. Sectio reviews some basic cocepts of IFSs. I Sectio 3, a ewdistacemeasureforifssisproposedadacomparative aalysis is coducted. I Sectio 4, a ew distace based IF- VIKOR method is developed to deal with MADM problem. I Sectio 5, two applicatio examples are demostrated to highlight the applicability ad superiority of proposed IF- VIKOR method. The fial sectio summarizes the mai work of this paper with a discussio of implicatios for future research.. Prelimiaries I this sectio, we briefly review some basic otios ad theories related to IFSs..1. Ituitioistic Fuzzy Set Defiitio 1. A ituitioistic fuzzy set A itheuiverseof discourse X is defied as follows [34]: A={ x, u A (x), V A (x) x X}, (1) where u A : X [0, 1] ad V A : X [0,1] are called membership fuctio ad omembership fuctio of x to A, respectively.u A (x) represets the lowest boud of degree of membership derived from proofs of supportig x while V A (x) represets the lowest boud of degree of omembership derived from proofs of opposig x.foray x X, 0 u A +V A 1.Thefuctioπ A (x) = 1 u A (x) V A (x) is called hesitace idex, ad π A (x) [0, 1] represets the degree of hesitacy of x to A. Specially,ifπ A (x) = 0, where x X iskowabsolutely,theituitioisticfuzzyseta degeerates ito a fuzzy set. Defiitio. For coveiece of computatio, let a =(u a, V a ) be a ituitioistic fuzzy umber (IFN). The the score fuctio of a is defied as follows [35]: s ( a) = (u a V a ). () Ad the accuracy fuctio of a is defied as follows [36]: h ( a) =(u a + V a ). (3) Let a 1 ad a be two IFNs; the Xu ad Yager [37] proposed the followig rules for rakig of IFNs: (1) s( a 1 )<s( a ),the a 1 is smaller tha a, deoted by a 1 < a ; () s( a 1 )>s( a ),the a is smaller tha a 1, deoted by a 1 > a ; (3) s( a 1 )=s( a ),the (i) h( a 1 )=h( a ),the a 1 is equal to a, deoted by a 1 = a ; (ii) h( a 1 ) < h( a ),the a 1 is smaller tha a, deoted by a 1 < a ; (iii) h( a 1 ) > h( a ),the a is smaller tha a 1, deoted by a 1 > a. Defiitio 3. Let a i (i = 1,,...,)be the set of IFNs, where a i = (u ai, V ai ). The ituitioistic fuzzy weighted averagig (IFWA) operator ca be defied as follows [33]: IFWA w ( a 1, a,..., a )=w 1 a 1 w a w a =(1 (1 u ai ) w i, V ai w i ), where w i is the weight of a i (i = 1,,...,), w i [0, 1], ad w i =1. Defiitio 4. Let A be a ituitioistic fuzzy set i the uiverse of discourse X. Ituitioistic fuzzy etropy is give as follows [38]: (4) E (A) = 1 mi (u A (x i ),V A (x))+π A (x) max (u A (x i ),V A (x))+π A (x). (5).. Distace Measure betwee IFSs Defiitio 5. For ay A, B, C IFSs(X),letd be a mappig d: IFSs(X) IFSs(X) [0, 1]. If d(a, B) satisfies the followig properties [39]: (DP1) 0 d(a, B) 1; (DP) d(a, B) = 0 if ad oly if A=B; (DP3) d(a, B) = d(b, A);

4 4 Mathematical Problems i Egieerig (DP4) if A B C,thed(B, C) d(a, C) ad d(a, B) d(a, C), the d(a, B) is a distace measure betwee IFSs A ad B. Let X be the uiverse of discourse, where X={x 1,x,...,x },adletaad B be two IFSs i the uiverse of discourse X, wherea = { x i,u A (x i ), V A (x i ) x i X} ad B={ x i,u B (x i ), V B (x i ) x i X}. Several widely used distace measures are reviewed as follows. I [40], Szmidt ad Kacprzyk proposed four distaces betwee IFSs usig the well-kow hammig distace, Euclidea distace, ad their ormalized couterparts as follows: d H (A, B) = 1 [ u A (x i ) u B (x i ) + V A (x i ) V B (x i ) + π A (x i ) π B (x i ) ], d H (A, B) = 1 [ u A (x i ) u B (x i ) + V A (x i ) V B (x i ) + π A (x i ) π B (x i ) ], d E (A, B) = 1 [(u A (x i ) u B (x i )) +(V A (x i ) V B (x i )) +(π A (x i ) π B (x i )) ], (6) d E (A, B) = 1 [(u A (x i ) u B (x i )) +(V A (x i ) V B (x i )) +(π A (x i ) π B (x i )) ]. Wag ad Xi [39] poited out that Szmidt ad Kacprzyk s distace measures [40] have some good geometric properties, but there are some limitatios i the applicatio. Therefore, they proposed several ew distaces as follows. betwee IFSs, ad these suggested distaces are also geeralizatios of the well-kow hammig distace, Euclidea distace, ad their ormalized couterparts. d h (A, B) d 1 (A, B) = 1 = max { u A (x i ) u B (x i ), V A (x i ) V B (x i ) }, d P [ u A (x i ) u B (x i ) + V A (x i ) V B (x i ) 4 + max ( u A (x i ) u B (x i ), V A (x i ) V B (x i ) ) ], (A, B) = 1 p (7) l h (A, B) = 1 e h (A, B) = max { u A (x i ) u B (x i ), V A (x i ) V B (x i ) }, max {(u A (x i ) u B (x i )),(V A (x i ) V B (x i )) }, (8) q h (A, B) p ( u A (x i ) u B (x i ) + V p A (x i ) V B (x i ) ). = 1 max {(u A (x i ) u B (x i )),(V A (x i ) V B (x i )) }. O the basis of the Hausdorff metric, Grzegorzewski [41] put forward some approaches for computig distaces Yag ad Chiclaa [4] suggested that the 3D iterpretatio of IFSs could provide differet cotradistictio results to the oes obtaied with their D couterparts [41] ad itroduced several exteded 3D Hausdorff based distaces. d eh (A, B) = l eh (A, B) = 1 max { u A (x i ) u B (x i ), V A (x i ) V B (x i ), π A (x i ) π B (x i ) }, max { u A (x i ) u B (x i ), V A (x i ) V B (x i ), π A (x i ) π B (x i ) },

5 Mathematical Problems i Egieerig 5 e eh (A, B) = max {(u A (x i ) u B (x i )),(V A (x i ) V B (x i )),(π A (x i ) π B (x i )) }, q eh (A, B) = 1 max {(u A (x i ) u B (x i )),(V A (x i ) V B (x i )),(π A (x i ) π B (x i )) }. (9) 3. New Distace Measure betwee IFSs 3.1. Aalysis o Existig Distace Measures betwee IFSs. Although various types of distace measures betwee IFSs were proposed over the past several years ad most of them satisfies the distace axioms (Defiitio 5), some ureasoable cases ca still be foud [31, 3]. For example, cosider three sigle-elemet IFSs A = (x, 0.4, 0.), B = (x, 0.5, 0.3), ad C = (x, 0.5, 0.).Ithiscase,thedistacebetweeAad B calculated by some existig measures (e.g., d H, d E, l h, l eh, d 1, d P ) is equal to or greater tha that betwee A ad C,which does ot seem to be reasoable sice IFSs A, B, adc are ordered as C>B>Aaccordig to the score fuctio ad accuracy fuctio give i Defiitio, idicatig that the distace betwee A ad B is smaller tha that betwee A ad C. For oe reaso, differet distace measures have differet focus, which may be suitable for differet applicatios. For aother, this could mea the defiitio of distace measures is too weak. Specifically, the defiitio may be more complete adreasoableifithascosideredthefollowigtwoaspects. First, Defiitio 5 just provides the value costraits whe d(a, B) = 0, ad the other edpoit d(a, B) = 1 has ot bee discussed. Secod, the distace measure betwee IFSs could be better coviced if it satisfies the requiremet of the triagle iequality. Aother poit worth otig is that there are two facets of ucertaity of ituitioistic fuzzy iformatio, oe of which isrelatedtofuzziesswhiletheotherisrelatedtolackof kowledge or ospecificity [43]. First, IFSs are a extesio of FSs ad hece are associated with fuzziess. This fuzziess is the first kid of ucertaity that maily reflects two types of distict ad specific iformatio: degree of membership ad degree of omembership. I IFSs there exists aother kid of ucertaity that ca be called lack of kowledge or ospecificity ad might also be called waver [30]. This kid of ucertaity reflects a type of ospecific iformatio: degree of hesitacy, idicatig how much we do ot kow about membership ad omembership. These two kids of ucertaity are obviously differet, the degree of hesitacy with its ow ucertaity because it represets a state of both this ad that. However, most of the existig distace measures oly cosider the differeces betwee umerical values of the IFS parameters ad igore the waver of ituitioistic fuzzy iformatio. As a illustratio, cosider two patters represeted by IFSs A = (x, 0.1, 0.) ad B = (x, 0.1, 0.), i which both hesitace idexes of A ad B are equal to 0.7. I this case, the distace betwee IFSs A ad B calculated by all aforemetioed measures is equal to zero, deotig that the twopattersarethesame.thisistruefromtheperspective of umerical value because the three parameters of IFSs A ad B have the same value. However, from the perspective of ituitioistic fuzzy iformatio, the coclusio draw is ureasoable. The hesitace idex π(x) = 1 u(x) V(x) i both IFSs A ad B is equal to 0.7, but it just meas that i IFSs A ad B theproportiooflackofkowledgeis0.7,ad it caot represet that the patters A ad B are the same. Iotherwords,sicewehavealackofkowledge,weshould ot draw the coclusio that the two patters are the same. As a powerful tool i modelig ucertai iformatio, IFSs have the advatage of beig able to cosider waver mailybecausethehesitatidexcabeusedtodescribea eutrality state of both this ad that. Moreover, it should be oted that the ultimate goal of distace measure is to measure the differece of iformatio carried by IFSs, rather tha the umerical value of IFSs themselves. Therefore, i ourviewthedistacemeasureshouldtakeitoaccout the characteristics of ituitioistic fuzzy iformatio, both fuzziess ad ospecificity, ad have its ow target. 3.. Ituitive Distace for IFSs. Ispired by the characteristics of ituitioistic fuzzy iformatio, we itroduce a otio called ituitive distace to compare the iformatio betwee two IFSs ad compute the degree of differece, ad the axiomatic defiitio of ituitive distace is as follows. Defiitio 6. For ay A, B, C, D IFSs(X),letd be a mappig d : IFSs(X) IFSs(X) [0, 1]. d(a, B) is said to be a ituitive distace betwee A ad B if d(a, B) satisfies the followig properties: (P1) 0 d(a, B) 1; (P) d(a, B) = 0 if ad oly if A=Bad π A (x) = π B (x) = 0; (P3) d(a, B) = 1 if ad oly if both A ad B are crisp sets ad A=B C ; (P4) d(a, B) = d(b, A); (P5) d(a, C) d(a, B)+d(B, C) for ay A, B, C IFSs(X); (P6) if A B C,thed(B, C) d(a, C) ad d(a, B) d(a, C); (P7) if A=Bad C=D,the (i) d(a, B) > d(c, D) whe π A (x) + π B (x) > π C (x) + π D (x); (ii) d(a, B) < d(c, D) whe π A (x) + π B (x) < π C (x) + π D (x);

6 6 Mathematical Problems i Egieerig (iii) d(a, B) = d(c, D) whe π A (x) + π B (x) = π C (x) + π D (x). The above defiitio is a developmet for Defiitio 5. O the oe had, some properties are itroduced to make the defiitio stroger: (P3) is a more precise coditio for the edpoit of distace d(a, B) = 1; (P5)isaewstrog property coditio, which requires distace to meet triagle iequality. O the other had, the defiitio highlights the characteristics of ituitioistic fuzzy iformatio by reiforcig the existig property coditio (P), as well as addig a ew property coditio (P7). The ituitive distace ot oly meets the most geeral properties of traditioal distace, but also satisfies a special property: as log as the hesitat idex is ot zero, eve if two IFSs are equal i value, the distace betwee them is ot zero. Ad the higher the hesitat idex, the greater the distace. The, we propose a ew distace measure. Let A = x, u A (x), V A (x), B = x,u B (x), V B (x) betwoifssithe uiverse of discourse X, ad deote X={x 1,x,...,x }. d L (A, B) = 1 [θ 3 1 (x i )+θ (x i )+θ 3 (x i )], θ 1 (x i )= u A (x i ) u B (x i ) + V A (x i ) V B (x i ) + (u A (x i )+1 V A (x i )) (u B (x i )+1 V B (x i )), θ (x i )= π A (x i )+π B (x i ), (10) θ 3 (x i )=max ( u A (x i ) u B (x i ), V A (x i ) V B (x i ), π A (x i ) π B (x i ) ). The structure of d L (A, B) is maily related to two aspects. Oe is to take the differeces of IFS parameters ito accout, which is similar to the traditioal distace. It icludes differeces betwee memberships u A (x i ) ad u B (x i ), omemberships V A (x i ) ad V B (x i ), hesitace idexes π A (x i ) ad π B (x i ), ad the differeces betwee media values of itervals (u A (x i )+1 V A (x i ))/ ad (u B (x i )+1 V B (x i ))/.The secod is to satisfy the ew requiremets of ituitive distace, takig the degree of hesitacy ito accout. θ (x i ) aims to reflect the waver of ucertai iformatio. The higher the degree of ospecificity of ituitioistic fuzzy iformatio, the greater the possibility of existig differeces betwee them. The, we have the followig theorem. Theorem 7. d L (A, B) is a ituitive distace betwee IFSs A ad B itheuiverseofdiscoursex={x 1,x,...,x }. Proof. It is easy to see that d L (A, B) satisfies (P4) ad (P7) of Defiitio 6. Therefore, we shall prove that d L (A, B) satisfies (P1), (P), (P3), (P5), ad (P6). (P1) Let A ad B be two IFSs; we ca write the relatioal expressio as follows: 0 θ 1 (x i )+θ (x i )=( u A (x i ) u B (x i ) + V A (x i ) V B (x i ) + (u A (x i )+1 V A (x i )) (u B (x i )+1 V B (x i )) +π A (x i )+π B (x i )) () 1 =( u A (x i ) u B (x i ) + V A (x i ) V B (x i ) + (u A (x i ) u B (x i )) (V A (x i ) V B (x i )) + u A (x i ) u B (x i ) V A (x i ) V B (x i )) () 1 ( u A (x i ) u B (x i ) + V A (x i ) V B (x i ) + u A (x i ) u B (x i ) + V A (x i ) V B (x i ) + u A (x i ) u B (x i ) V A (x i ) V B (x i )) () 1 =( u A (x i ) u B (x i ) (u A (x i )+u B (x i )) + V A (x i ) V B (x i ) (V A (x i )+V B (x i )) + ) () 1 ((u A (x i )+u B (x i )) (u A (x i )+u B (x i )) + (V A (x i )+V B (x i )) (V A (x i )+V B (x i )) + ) () 1 = u A (x i )+u B (x i )+V A (x i )+V B (x i )+ = (u A (x i )+V A (x i )) + (u B (x i )+V B (x i )) It is kow that =. 0 θ 3 (x i )=max ( u A (x i ) u B (x i ), V A (x i ) V B (x i ), π A (x i ) π B (x i ) ) 1. (11) (1)

7 Mathematical Problems i Egieerig 7 Takig (11) ad (1) ito accout, it is ot difficult to fid that Ad therefore we have 0 θ 1 (x i ) +θ (x i ) +θ 3 (x i ) 3. (13) 0 d L (A, B) = 1 [θ 3 1 (x i )+θ (x i )+θ 3 (x i )] (14) 3 3 =1. (P) Let A ad B be two IFSs; the followig relatioal expressio ca be writte: d L (A, B) =0 u A (x i )=u B (x i ), V A (x i )=V B (x i ), π A (x i )+π B (x i )=0 A (x) =B(x), π A (x) =π B (x) =0. Thus, d L (A, B) satisfies (P) of Defiitio 6. (15) (P3) Let A ad B be two IFSs. Takig (11) ad (1) ito accout, the followig relatioal expressios ca be writte: d L (A, B) =1 θ 1 (x i )+θ (x i )=, θ 3 (x i )=1 u A (x i )+u B (x i )+V A (x i )+V B (x i )+ =, max ( u A (x i ) u B (x i ), V A (x i ) V B (x i ), π A (x i ) π B (x i ) )=1 A (x) = (1, 0, 0), B (x) = (0, 1, 0) or A (x) = (0, 1, 0), B (x) = (1, 0, 0). Thus, d L (A, B) satisfies (P3) of Defiitio 6. (16) (P5) Let A, B, adc be three IFSs; the distaces betwee A ad B, B ad C,adA ad C are the followig: d L (A, B) = 1 [ u A (x i ) u B (x i ) + V A (x i ) V B (x i ) + (u A (x i )+1 V A (x i )) (u B (x i )+1 V B (x i )) 3 + π A (x i )+π B (x i ) + max ( u A (x i ) u B (x i ), V A (x i ) V B (x i ), π A (x i ) π B (x i ) )], d L (B, C) = 1 [ u B (x i ) u C (x i ) + V B (x i ) V C (x i ) + (u B (x i )+1 V B (x i )) (u C (x i )+1 V C (x i )) 3 + π B (x i )+π C (x i ) d L (A, C) = max ( u B (x i ) u C (x i ), V B (x i ) V C (x i ), π B (x i ) π C (x i ) )], (17) [ u A (x i ) u C (x i ) + V A (x i ) V C (x i ) + (u A (x i )+1 V A (x i )) (u C (x i )+1 V C (x i )) +π A (x i )+π C (x i ) + max ( u A (x i ) u C (x i ), V A (x i ) V C (x i ), π A (x i ) π C (x i ) )].

8 8 Mathematical Problems i Egieerig It is obvious that u A (x i ) u B (x i ) + u B (x i ) u C (x i ) u A (x i ) u B (x i )+u B (x i ) u C (x i ) = u A (x i ) u C (x i ), V A (x i ) V B (x i ) + V B (x i ) V C (x i ) V A (x i ) V B (x i )+V B (x i ) V C (x i ) = V A (x i ) V C (x i ), (u A (x i )+1 V A (x i )) (u B (x i )+1 V B (x i )) + (u B (x i )+1 V B (x i )) (u C (x i )+1 V C (x i )) = u A (x i )+V B (x i ) u B (x i ) V A (x i ) + u B (x i )+V C (x i ) u C (x i ) V A (x i )+V C (x i ) u C (x i ) = (u A (x i )+1 V A (x i )) (u C (x i )+1 V C (x i )), π A (x i ) π B (x i ) + π B (x i ) π C (x i ) π A (x i ) π B (x i )+π B (x i ) π C (x i ) = π A (x i ) π C (x i ), π A (x i )+π B (x i )+π B (x i )+π C (x i ) π A +π C. Therefore, we have d L (A, B) + d L (B, C) d L (A, C). (18) V B (x i ) u A (x i )+V B (x i ) u B (x i ) V A (x i ) +u B (x i )+V C (x i ) u C (x i ) V B (x i ) = u A (x i ) (P6) Let A, B, adc be three IFSs; if A B C,the we have u A (x i ) u B (x i ) u C (x i ), V A (x i ) V B (x i ) V C (x i ). The followig equatios are give out: d L (A, B) = 1 [ u B (x i ) u A (x i )+V A (x i ) V B (x i )+u B (x i ) u A (x i )+V A (x i ) V B (x i )+π A (x i )+π B (x i ) 3 + max (u B (x i ) u A (x i ),V A (x i ) V B (x i ), π A (x i ) π B (x i ) )] = 1 3 [ u B (x i ) u A (x i )+V A (x i ) V B (x i )+u B (x i ) u A (x i )+V A (x i ) V B (x i )+ u A (x i ) u B (x i ) V A (x i ) V B (x i ) + max (u B (x i ) u A (x i ),V A (x i ) V B (x i ), u B (x i ) u A (x i )+V B (x i ) V A (x i ) )] = 1 3 [ u B (x i ) 3u A (x i )+V A (x i ) 3V B (x i ) + max (u B (x i ) u A (x i ),V A (x i ) V B (x i ))], d L (A, C) = 1 [ u C (x i ) u A (x i )+V A (x i ) V C (x i )+u C (x i ) u A (x i )+V A (x i ) V C (x i )+π A (x i )+π C (x i ) 3 + max (u C (x i ) u A (x i ),V A (x i ) V C (x i ), π A (x i ) π C (x i ) )] = 1 3 (19) [ u C (x i ) u A (x i )+V A (x i ) V C (x i )+u C (x i ) u A (x i )+V A (x i ) V C (x i )+ u A (x i ) u C (x i ) V A (x i ) V C (x i ) + max (u C (x i ) u A (x i ),V A (x i ) V C (x i ), [ u C (x i ) 3u A (x i )+V A (x i ) 3V C (x i ) u C (x i ) u A (x i )+V C (x i ) V A (x i ) )] = max (u C (x i ) u A (x i ),V A (x i ) V C (x i ))]. We kow that u C (x i ) 3V C (x i ) u B (x i ) 3V B (x i ), u C (x i ) u A (x i ) u B (x i ) u A (x i ), V A (x i ) V C (x i ) V A (x i ) V B (x i ). It meas that (0) d L (A, B) d L (A, C). (1) Similarly, it is easy to prove that d L (B, C) d L (A, C). () Thus, d L (A, B) satisfies (P6) of Defiitio 6.

9 Mathematical Problems i Egieerig 9 Table 1: Test ituitioistic fuzzy sets (IFSs). Test IFSs A=(u A, V A ) (1, 0) (0, 0) (0.4, 0.) (0.3, 0.3) (0.3, 0.4) (0.1, 0.) (0.4, 0.) B=(u B, V B ) (0, 0) (0.5, 0.5) (0.5, 0.3) (0.4, 0.4) (0.4, 0.3) (0.1, 0.) (0.5, 0.) Table : The compariso of distace measures (couterituitive cases are i bold italic type). Test IFSs d L (A, B) d H (A, B) d E (A, B) l h (A, B) l eh (A, B) d 1 (A, B) (A, B), p = d P That is to say, d L (A, B) is a ituitive distace betwee IFSs A ad B sice it satisfies (P1) (P7) A Compariso of Distace Measures for IFSs Based o a Artificial Bechmark. Whe a ew distace measure is proposed, it is always accompaied with explaatios of overcomig couterituitive cases of other methods ad these cases are usually illustrated by sigle-elemet IFSs. I [30], Li et al. summarized couterituitive cases proposed by previous literature ad costituted a artificial bechmark with six differet pairs of sigle-elemet IFSs, which has bee widely used i the test of distace ad similarity measures [31, 44 49]. Although these cases caot represet all couterituitive situatios, they are typical ad represetative. I [31], Papakostas et al. suggested that ay proposed measures should be tested by the artificial bechmark to avoid couterituitive cases. To illustrate the effectiveess of theproposeddistacemeasure,allthetestifssoftheartificial bechmark are applied to compare the proposed distace measure to the widely used distace measures. I additio, i order to reflect the characteristics of ituitioistic fuzzy iformatio, a ew pair of sigle-elemet IFSs as illustrated i Sectio 3.1 is added to the artificial bechmark test set. Table 1 shows the test IFSs of exteded artificial bechmark. Table provides a comprehesive compariso of the distace measures for IFSs with couterituitive cases. It is apparet that the property coditio (P3) is ot met by d H, d E, l h, l eh, because the distaces calculated by these measures are equal to 1 whe {A = (x, 1, 0), B = (x, 0, 0)}. Similarly, the property coditio (P3) is also ot satisfied by d H, l eh whe {A = (x, 0, 0), B = (x, 0.5, 0.5)}.Thedistacemeasures d H, l eh,add P, p = 1, idicate that the distaces of the 1st test IFSs ad the d test IFSs are idetical, which does ot seem to be reasoable. Distace measures l h, d 1,ad d P, p = 1, claim that the distaces of the 4th test IFSs ad the 5th test IFSs show the same value of 0.1, which idicates that there are ot sufficiet abilities to distiguish positive differece from egative differece. The distace of 3rd test IFSs is equal to or greater tha the distace of the 7th test IFSs whe d H, d E, l h, l eh, d 1,add P, p = 1,areused, which does ot seem to be reasoable sice IFSs A, B, ad C are ordered as C>B>Aaccordig to the score fuctio ad accuracy fuctio give i Defiitio, idicatig that the distace betwee A ad B is smaller tha that betwee A ad C. Furthermore, all of these existig distaces claim that the distace of the 6th test IFSs is equal to zero, which does otseemtobereasoable.asamathematicaltool,ifssca describe the ucertai iformatio greatly, because it adds a hesitat idex to describe the state of both this ad that. I this case, we ideed caot cofirm that there is o differece betwee the iformatio carried by IFSs A ad B, because the hesitace idex icludes some ospecificity iformatio ad the proportio of support ad oppositio is ot sure. BasedoaalysisiTable,itisdeducedthattheexistig distace measures with their ow measurig focus ca meet all or most of properties coditio of distace measure betwee IFSs; however, most distace measures show couterituitive cases ad may fail to distiguish IFSs accurately i some practical applicatios. Besides, the proposed distace measure is the oly oe that has o aforemetioed couterituitivecasesasillustrateditable.furthermore,the proposed distace coforms to all the property requiremets of the ituitive distace ad the potetial differece brought by hesitace idex is cosidered. 4. IF-VIKOR Method for MADM O the basis of ew distace measure, this sectio preset stepwise algorithm for proposed IF-VIKOR method. For a MADM problem with alteratives A i (i = 1,,...,m), the performace of the alterative A i cocerig the attribute C j (j=1,,...,)is assessed by a decisio orgaizatio with several decisio-makers D q (q = 1,,...,l). The correspodig weights of attributes are deoted by w j (j=1,,...,), 0 w j 1, j=1 w j =1,adtheweights of DMs are deoted by λ q (q = 1,,...,l), 0 λ q 1,

10 10 Mathematical Problems i Egieerig l q=1 λ q = 1.IspiredbytheclassicalVIKORmethodad its extesios, the ituitive distace based VIKOR method ca be give for MADM problem with ituitioistic fuzzy iformatio; it icludes seve steps. Step 1. Geerate assessmet iformatio. Assume that DMs D q (q = 1,,...,l)provide their opiio of the alteratives A i cocerig each attribute C j by usig IFNs x q ij = (u q ij, Vq ij,πq ij ) or liguistic values represeted by IFNs. The, the assessmets give by D q ca be expressed as x (q) 11 x (q) 1 x (q) 1 x (q) D (q) 1 x (q) x (q) =.. d.. (3). [ ] [ x (q) m1 x(q) m x(q) m] Step. Acquire the weights of DMs. Accordig to the degree of fuzziess ad ospecificity of assessmets provided by DMs, i this step, DM weight λ q (q = 1,,...,l) ca be acquired by ituitioistic fuzzy etropy measure objectively. The lower the degree of fuzziess ad ospecificity is, the smallertheetropyisadthebiggertheweightofdmis, ad vice versa. By usig (5), the ituitioistic fuzzy etropy of assessmets provided by D q cabeobtaiedasfollows: m E q = j=1 mi (u (q) ij, V(q) ij )+π(q) ij max (u (q) ij, V(q) ij )+π(q) ij. (4) The, the weight of DM D q cabedefiedasfollows: where l is the umber of DMs. 1 E q λ q = l l q=1 E, (5) q Step 3. Establish the aggregated ituitioistic fuzzy decisio matrix. By usig (4), all idividual decisio matrixes D (q) ca becoverteditoaaggregateddecisiomatrixasfollows: x 11 x 1 x 1 x 1 x x D=.. [.. d., (6). ] [ x m1 x m x m ] where x ij = (u ij, V ij,π ij ), u ij = 1 l q=1 (1 uq ij )λ q, V ij = l q=1 (Vq ij )λ q, π ij =1 u ij V ij. Step 4. Acquire the weights of attributes. Similar to Step, the ukow weight of attribute w j (j = 1,,...,) ca be determied by etropy measure to effectively reduce the subjective radomess. By usig (5), we ca obtai the etropy with respect to C j : E j = 1 m mi (u ij, V ij )+π ij. (7) m max (u ij, V ij )+π ij The, the weight of attribute C j cabedefiedasfollows: where is the umber of attributes. 1 E j w j = j=1 E, (8) j Step 5. Fid the best ad worst value. The best value x + j ad the worst value x j for each attribute C j cabedefiedas follows: x + j = { max x ij, for beefit attribute C j,,...,m { mi { x ij, for cos t attribute C j,,,...,m x j = { mi x ij, for beefit attribute C j,,...,m { max { x ij, for cos t attribute C j,,,...,m (j=1,,...,). (j=1,,...,) (9) Step 6. Compute the values S i, R i,adq i.threekeyvaluesof IF-VIKOR method, the group utility value S i,theidividual regret value R i, ad the compromise value Q i,arecomputed i light of the ituitive distace measure for each alterative: S i = j=1 w j ( d(x+ j,x ij) d(x j +,x j ) ), R i = max w j ( d(x+ j,x ij) j d(x j +,x j ) ), Q i =γ( S i S S S )+(1 γ)(r i R R R ), (30) where S = max i S i,s = mi i S i,r = max i R i,r = mi i R i. γ is the coefficiet of decisio mechaism. The compromise solutio ca be elected by majority (γ > 0.5), cosesus (γ = 0.5), or veto (γ < 0.5). Step 7. Rak the alteratives ad derive the compromise solutio. Sort S i, R i,adq i i ascedig order ad geerate three rakig lists S [ ], R [ ],adq [ ].The,thealterativeA (1) that raks the best i Q [ ] (miimum value) ad fulfills followig two coditios simultaeously would be the compromise solutio. Coditio 1 (acceptable advatage). Oe has Q(A () ) Q(A (1) ) 1 m 1, (31) where A (1) ad A () are the top two alteratives i Q i.

11 Mathematical Problems i Egieerig 11 Coditio (acceptable stability). The alterative A (1) should alsobethebestrakedbys i ad R i. If the above coditios caot be satisfied simultaeously, there exist multiple compromise solutios: (1) alteratives A (1) ad A () if oly Coditio is ot satisfied; () alteratives A (1),A (),...,A (u) if Coditio 1 is ot satisfied, where A (u) is established by the relatio Q(A (u) ) Q(A (1) ) < 1/(m 1) for the maximum. Table 3: Liguistic terms for ratig the alteratives with IFNs. Liguistic variables IFNs Extremely poor (EP) (0.05, 0.95, 0.00) Poor (P) (0.0, 0.70, 0.10) Medium poor (MP) (0.35, 0.55, 0.10) Medium (M) (0.50, 0.40, 0.10) Medium good (MG) (0.65, 0.5, 0.10) Good (G) (0.80, 0.10, 0.10) Extremely good (EG) (0.95, 0.05, 0.00) 5. Applicatio Examples 5.1. ERP Selectio Problem. I recet years, eterprise resource plaig (ERP) system has become a powerful tool for eterprises to improve their operatig performace ad competitiveess. However, ERP projects report a uusually high failure rate ad sometimes imperil implemeters core operatio due to their high costs ad wide rage of cofiguratio. Cosequetly, selectig the ERP system which fit the eterprise would be a critical step to success. Cosiderig a situatio that oe high-tech maufacturig eterprise is tryig to implemet ERP system with four alteratives A 1, A, A 3,adA 4,threeDMsofD 1, D, D 3 are employed to evaluate these alteratives from five mai aspects as follows: C 1 : Fuctioality ad reliability, which ivolve suitability, accuracy, security, fuctioality compliace, maturity, recoverability, fault tolerace, ad reliability compliace. C : Usability ad efficiecy, which ivolve uderstadability, learability, operability, attractiveess, usability compliace, time behavior, resource behavior, ad efficiecy compliace. C 3 : Maitaiability ad portability, which ivolve aalyzability, chageability, testability, coexistece, iteroperatio, maitaiability, ad portability compliace. C 4 : Supplier services, which ivolve the quality of traiig staff, techical support ad follow-up services, the level of implemetatio ad stadardizatio, customer satisfactio, supplier credibility, ad stregth. C 5 : The eterprise characteristics, which ivolve eterprise maagemet, employee support, comprehesive ivestmet cost, iteral rate of retur, beefit cost ratio, ad dyamic payback period. Sice the weights of attributes ad DMs are completely ukow, the best alterative would be selected with the iformatio give above. I the followig, the proposed IF-VIKOR method is applied to solve this problem. The operatio process accordig to the algorithm developed i Sectio 4 is give below. Step 1. Each DM assesses alterative A i cocerig attribute C j with liguistic ratig variables i Table 3. Table 4 shows the assessmets by three decisio-makers. Step. By usig (4) ad (5), the weights of DMs ca be obtaied as λ 1 = 0.30, λ = 0.344, λ 3 = Step 3. By usig (6), we ca establish the aggregated decisio matrix as follows. (0.6689, 0.193) (0.8000, ) (0.3500, ) (0.7081, ) (0.559, ) (0.606, 0.717) (0.6495, 0.331) (0.6071, 0.91) (0.5000, ) (0.756, ) D=. (3) (0.554, ) (0.7308, ) (0.8355, 0.13) (0.8465, ) (0.718, ) [ ] [(0.6495, 0.331) (0.6719, 0.168) (0.6074, 0.908) (0.5546, ) (0.543, ) ] Step 4. By usig (7) ad (8), the weights of attributes are obtaied as w 1 = , w = 0.19, w 3 = , w 4 = , w 5 = Step 5. By usig (9), the best ad the worst values of all attribute ratigs ca be calculated ad we have x + 1 =(0.6689, 0.193), x + = (0.8000, ), x+ 3 = (0.8355, 0.13), x+ 4 = (0.8465, ), x + 5 = (0.756, ), x 1 =(0.554,0.3438), x =(0.6495,0.331),x 3 = (0.3500, ), x 4 = (0.5000, ), x 5 =(0.543,0.3491). Step 6. Without loss of geerality, let γ = 0.5. By usig (30), the values of S i, R i,adq i foreachalterativecabeobtaied as listed i Table 5. Step 7. From Table 5, we have Q 3 < Q 1 < Q 4 < Q, which meas A 3 (miimum value) raks best i terms of Q. I additio, Q 1 Q 3 = ad A 3 is also the best raked by S i ad R i, which shows that A 3 is the uique compromise solutio for this problem.

12 1 Mathematical Problems i Egieerig Table4:RatigofthealterativesfromDMs. Attributes DM1 DM3 DM4 A 1 A A 3 A 4 A 1 A A 3 A 4 A 1 A A 3 A 4 C 1 MG M MG MP G MP M MG M G M G C G MP G G G MG M M G G G MG C 3 MP MG M M MP M G MG MP MG EG MG C 4 MG M EG M G M MG MG MG M G M C 5 M G MG MG M G MG P MG MG G MG Table 5: The values ofs, R,ad Q for all alteratives by the proposed IF-VIKOR method. Value A 1 A A 3 A 4 S R Q Table 6: Liguistic terms for ratig the alteratives with trapezoidal fuzzy umbers. Liguistic variables Trapezoidal fuzzy umbers Extremelypoor(EP) (0,0,1,) Poor (P) (1,,, 3) Medium poor (MP) (, 3, 4, 5) Medium (M) (4, 5, 5, 6) Medium good (MG) (5, 6, 7, 8) Good (G) (7, 8, 8, 9) Extremely good (EG) (8, 9, 10, 10) Liu et al. [18] developed the VIKOR method for MADM problem based o trapezoidal fuzzy umbers, which is oe of the most commoly used fuzzy umbers. I this method, the weights of DMs or attributes are give artificially ad the liguistic variables are represeted by trapezoidal fuzzy umbers show i Table 6. A trapezoidal fuzzy umber A ca be deoted as (a 1, a, a 3, a 4 ), where a 1 ad a 4 are called lower ad upper limits of A ad a ad a 3 are two most promisig values. Whe a ad a 3 are the same value, the trapezoidal fuzzy umber degeerated a triagular fuzzy umber. We applythemethod[18]itheerpselectioproblemtoexplai the differece betwee the proposed method ad traditioal VIKOR method with trapezoidal fuzzy umbers. Assume that the weights of DMs ad attributes are λ 1 = 0.30, λ = 0.344, λ 3 = ad w 1 = , w = 0.19, w 3 = , w 4 = , w 5 = , respectively. Through calculatio of liguistic assessmet iformatio ad trapezoidal fuzzy umbers show i Tables 4 ad 6, the values of S, R,adQ for all alteratives are obtaied as i Table 7. The result shows that the rakig order of alteratives obtaied by the method [18] is A 3 A 1 A 4 A,which is i harmoy with our proposed method. Although the results obtaied from the two VIKOR methods are cosistet, the trapezoidal fuzzy umbers are oly characterized by a membership fuctio, while IFNs are characterized by a membership fuctio ad a omembership fuctio, which closely resembles the thikig habit of huma beigs uder the situatio of beig ucertai ad hesitat. Furthermore, the trapezoidal fuzzy umbers eed four values to determie the membership distributio, while the IFNs oly eed two values: membership ad omembership, ad the degree of hesitacy ca be automatically geerated by 1 mius membership ad omembership. I geeral, the computatio of trapezoidal fuzzy umbers is large, ad the applicatio of IFNs is relatively simple. I additio, the method preseted by [18] eeds the weight iformatio of DMs ad attributes predetermied, whereas these weights are determied objectively usig ituitioistic fuzzy etropy i the proposed method, which avoids subjective radomess to some extet. I [], Roostaee et al. put forward a hammig distace based IF-VIKOR method. To further illustrate its effectiveess, the proposed method is compared to the IF-VIKOR method preseted by []. Computatioal results for the hammig distace based IF-VIKOR method are show i Table 8. The result idicates that these two methods reach a cosesus that the third ERP system should be implemeted by the high-tech maufacturig eterprise. I this example, it should be oted that the assessmets provided by DMs are i low degree of hesitacy. As show i Tables 3 ad 4, the maximum hesitace idex with respect to liguistic values is 0.1, deotig the low degree of ospecificity (lack of kowledge) for assessmets. I this case, both methods ca effectively evaluate ad sort the alteratives. However, the hammig distace based IF- VIKOR method is ot always capable of obtaiig valid results especially i MADM problem with high degree of ospecificity. It might geerate some couterituitive cases so that ureasoable results might be obtaied. The followig example further illustrates such cases. 5.. Material Hadlig Selectio Problem. To illustrate the superiority of the proposed methods, a compariso betwee the proposed IF-VIKOR method ad the hammig distace basedif-vikormethodiamaterialhadligselectio problem is made.

13 Mathematical Problems i Egieerig 13 Table 7: The values ofs, R, ad Q for all alteratives by the trapezoidal fuzzy umbers based VIKOR method. Value A 1 A A 3 A 4 S R Q Table 8: The values of S, R,adQ for all alteratives by the hammig distace based IF-VIKOR method. Value A 1 A A 3 A 4 S R Q Table 9: Ratig of the alteratives from decisio orgaizatio. C 1 C C 3 C 4 A 1 (0.1, 0.) (0.4, 0.) (0.5, 0.) (0., 0.4) A (0., 0.1) (0.3, 0.3) (0.3, 0.5) (0.5, 0.5) A 3 (0.0, 0.3) (0.3, 0.4) (0., 0.3) (0.5, 0.) A 4 (0.4, 0.4) (0.5, 0.) (0.5, 0.) (0., 0.4) Table 10: The compromise values ad rakig results obtaied by the hammig distace based IF-VIKOR method ad the proposed method i this paper. Alterative Hammig distace based IF-VIKOR method The method proposed i this paper The values of Q i Rakig orders The values of Q i Rakig orders A A A A Suppose that a maufacturig compay is cosiderig implemetig a material hadlig system. After prelimiary screeig, four alteratives of A 1, A, A 3,adA 4 remai to be further evaluated. Several DMs from the compay s techical committee are arraged to evaluate ad select the appropriate alterative. They assess the four alteratives accordig to four coflictig attributes, icludig ivestmet cost C 1, operatio time C, expasio possibility C 3,ad closeess to market demad C 4. Due to the lack of experiece, time costraits, ad other factors, the ratigs of alteratives cocerig each attribute provided by DMs are represeted as IFNs with high degree of hesitacy, listed i Table 9. By usig (7) ad (8), the weights of attributes are obtaied as w 1 = 0.46, w = 0.490, w 3 = 0.593, w 4 = Without losig geerality, let γ = 0.5. The, the values of S i, R i,adq i for each alterative ca be calculated. Computatioal results obtaied by the hammig distace basedif-vikormethodadtheproposedmethodithis paper are listed i Table 10. The rakig of alteratives calculatedbythehammigdistacebasedif-vikormethod is A 1 A A 3 A 4, reachig a coclusio that A 1 (miimum value) is the best choice ad A 4 is the worst choice for the material hadlig selectio problem. However, accordig to the score fuctio ad accuracy fuctio, each attribute ratig of A 1 is lower tha or equal to that of A 4, idicatig that A 4 is superior to A 1,whichiscotradictedto the rakig results obtaied by hammig distace based IF- VIKORmethod.Istead,ourproposedmethodcaovercome the drawback of traditioal method ad obtai a valid rakig result as A 4 A 1 A A 3,whichmeas that material hadlig system A 4 is the best choice for the maufacturig compay Sesitivity Aalysis. I ituitioistic fuzzy VIKOR method, γ, the coefficiet of decisio mechaism is critical to the rakig results. Hece, a sesitivity aalysis is coducted i order to assess the stability of our method i these examples. For each γ from 0 to 1 at 0.1 itervals, we calculate the correspodig compromise solutio to ivestigate the ifluece of differet γ o the rakig result. Table 11 shows the sesitivity aalysis of ERP selectio example. For all the tested values of γ, three differet rakig results are geerated icludig A 3 A 4 A 1 A, A 3 A 1 A 4 A,adA 3 A 1 A A 4. While the rakig result is ideed affected by γ, A 3 is always the optimal solutio. Table 1 shows the sesitivity aalysis of material hadlig system selectio example. For all the tested values of γ, the rakig result remais A 4 A 1 A A 3 ad

14 14 Mathematical Problems i Egieerig Table 11: Ratig of the alteratives for differet γ values (ERP selectio example). γ Q 1 Q Q 3 Q 4 Rakig Optimal solutio A 3 A 4 A 1 A A A 3 A 1 A 4 A A A 3 A 1 A 4 A A A 3 A 1 A 4 A A A 3 A 1 A 4 A A A 3 A 1 A 4 A A A 3 A 1 A 4 A A A 3 A 1 A 4 A A A 3 A 1 A 4 A A A 3 A 1 A A 4 A A 3 A 1 A A 4 A 3 Table 1: Ratig of the alteratives for differet γ values (material hadlig selectio example). γ Q 1 Q Q 3 Q 4 Rakig Optimal solutio A 4 A 1 A A 3 A A 4 A 1 A A 3 A A 4 A 1 A A 3 A A 4 A 1 A A 3 A A 4 A 1 A A 3 A A 4 A 1 A A 3 A A 4 A 1 A A 3 A A 4 A 1 A A 3 A A 4 A 1 A A 3 A A 4 A 1 A A 3 A A 4 A 1 A A 3 A 4 the optimal solutio is A 4. The sesitivity aalysis illustrates how the decisio-makig strategy would affect the result, ad also it idicates that the coclusio arrived from our method is stable ad effective. 6. Coclusio Sice the VIKOR method is a effective MADM method to reach a compromise solutio, ad IFSs are a effective tool to depict fuzziess ad ospecificity i assessmet iformatio, we combie them ad develop a ew ituitive distace based IF-VIKOR method. This ew method aims at MADM problems with ukow weights of both the DMs ad attributes i ituitioistic fuzzy eviromet. It aggregates assessmet iformatio by ituitioistic fuzzy weighted averagig operator, geerates weights of DMs ad attributes by ituitioistic fuzzy etropy objectively, calculates the group utility ad idividual regret based o ituitive distace measure,adfiallyreachesthecompromisesolutio.two applicatio examples of ERP ad material hadig selectio problem further illustrate each step of this method. Comparedwiththehammigdistacemeasureuseditraditioal IF-VIKOR method, the ew ituitive distace measure i this method focuses o the fuzziess ad ospecificity of ituitioistic fuzzy iformatio, reflectig ot oly the differece amog the values of ituitioistic fuzzy sets, but also the waver of ituitioistic fuzzy sets. Both the artificial bechmark test ad applicatio examples demostrate its effectiveess ad superiority to traditioal method. Also, the determiatio of weights of DMs ad attributes usig ituitioistic fuzzy etropy ca avoid subjective radomess, ad sesitivity aalysis shows the stability of the proposed method. For future work, the compariso of the proposed VIKOR method with other MADM methods uder ituitioistic fuzzy eviromet, such as the TOPSIS method, the PROMETHEE method, ad the ELECTRE method, is worthy of further study ad exploratio. It would also be iterestig to apply the proposed VIKOR method to other MADM problems, such as ivestmet decisio ad supplier selectio. Coflicts of Iterest The authors declare that they have o coflicts of iterest. Ackowledgmets This work was supported by the Natioal Natural Sciece Foudatio of Chia uder Grat

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