Multi-attribute group decision making based on Choquet integral under interval-valued intuitionistic fuzzy environment

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1 Iteratioal Joural of Computatioal Itelligece Systems, Vol. 9, No ) Multi-attribute group decisio makig based o Choquet itegral uder iterval-valued ituitioistic fuzzy eviromet Jidog Qi 1, Xiwag Liu 1, Witold Pedrycz 2 1 School of Ecoomics ad Maagemet, Southeast Uiversity, Najig, , Chia qijidogseu@126.com, xwliu@seu.edu.c 2 Departmet of Electrical ad Computer Egieerig, Uiversity of Alberta, Edmoto, T6G 2G7, Caada wpedrycz@ualberta.ca Received 14 November 2014 Accepted 5 October 2015 Abstract I this paper, we propose ew methods to represet iterdepedece amog alterative attributes ad experts opiios by costructig Choquet itegral usig iterval-valued ituitioistic fuzzy umbers. I the sequel, we apply these methods to solve the multiple attribute group decisio-makig MAGDM) problems uder iterval-valued ituitioistic fuzzy eviromet. First, the cocept of iterval-valued ituitioistic fuzzy Choquet itegral is defied, ad some elemetary properties are studied i detail. Next, a axiomatic system of iterval-valued ituitioistic fuzzy measure is established by deliverig a series of mathematical proofs. The, with fuzzy etropy ad Shapely-values i game theory, we propose the iterval-valued ituitioistic fuzzy measure developmet methods i order to form the importace measure of attributes ad correlatio measure of the experts, respectively. Based o the results of theoretical aalysis, a ew method is proposed to hadle the iterval-valued ituitioistic fuzzy group decisio makig problems. A umerical example illustrates the procedure of the proposed methods ad verifies the validity ad effectiveess of our ew proposed methods. Keywords: Iterval-valued ituitioistic fuzzy Choquet itegral, iterval-valued ituitioistic fuzzy sets, iterval-valued ituitioistic fuzzy measure, multiple attribute group decisio makig. 1. Itroductio With the icreasig complexity ad ucertaity of the social-ecoomic eviromet ad various limitatios, such as time pressure, lack of kowledge of problem domai, difficulties i iformatio extractio etc., there are umerous ucertai pheomea existig i our daily life. Therefore, i order to better uderstad the vagueess ad ucertaity of the real world ad beig able to explai it, Zadeh 1 proposed fuzzy set theory i 1960s. Sice its iceptio, the fuzzy set theory has show coveiece as a powerful tool for modelig vagueess ad ucertaity i a variety domais, such as ecoomics 2,3,4, maagemet 5,6,7,8, artificial itelligece 16, processig cotrol 10,11, patter recogitio 12, decisio makig 13,14,15,16 etc. I 1980s, Ataassov 17,18 geeralized fuzzy set theory by brigig a idea of ituitioistic fuzzy set IFS). IFS is characterized by three importat otatios: membership de- Correspodig author, xwliu@seu.edu.c X.Liu). Co-published by Atlatis Press ad Taylor & Fracis 133

2 J. Qi et al. / Multi-attribute group decisio makig gree, o-membership degree ad hesitacy degree. O the basis of these three idexes, IFS ca cope with the vagueess ad ucertaity characteristics of complex systems. Now, IFS has become a importat area of research, ad attracted more attetio i various fields. For example, Xu 19,20 has completed some research i the field of IFS, especially i aggregatio operators, developed a series of ituitioistic fuzzy iformatio aggregatio operators for aggregatig ituitioistic fuzzy iformatio ad applied these operators to multiple attribute decisio makig MADM) problems. Wag ad Liu 22 proposed the ituitioistic fuzzy aggregatio operators realized through Eistei operatios ad aalyzed the relatios betwee these operators ad some existig ituitioistic fuzzy aggregatio operators. Li 23 developed the cocept of IFS, ad further exteded mathematical programmig methodology of matrix games with payoffs represeted by IFS. Furthermore, some desirable properties were discussed i detail. Farhadiia ad Ba 24 developed some ew similarity measures to deal with geeralized ituitioistic fuzzy umbers. However, i may practical situatios, there are limitatios give various factors, such as a lack of iformatio, ucertaity of the decisio makig eviromet, difficulties i iformatio extractio etc. Therefore, it is ot easy for decisio maker to determie exact values of membership degree, omembership degree ad hesitacy degree. To overcome these limitatios, Ataassov ad Gargov 18 proposed the cocept of the iterval-valued ituitioistic fuzzy set IVIFS), i which the attributes are take the form with iterval-valued ituitioistic fuzzy umber IVIFN). May studies were completed i recet years ad various approaches to solve MADM problems uder iterval-valued ituitioistic fuzzy eviromet have bee developed 24,25,26,27,28,29,30,31,32,33,34. For example, Farhadiia 31 geeralized recet results for the etropy of iterval-valued fuzzy set IVFS) based o the ituitioistic distace measure ad its relatioship with the similarity measure, ad the applied them to iterval-valued ituitioistic fuzzy decisio makig problem. Liu 32 developed some geometric aggregatio operators based o iterval-valued ituitioistic ucertai liguistic variables ad some desirable properties were discussed as well. Wag ad Li 34 proposed a framework to hadle multi-attribute group decisio makig MAGDM) problems with icomplete pair-wise compariso preferece over decisio alteratives. Wu ad Chiclaa 33 defied iterval-valued ituitioistic fuzzy COWA operator ad a ew score fuctio, the proposed some o-domiace ad attitudial prioritizatio decisio makig methods for ituitioistic fuzzy preferece relatios. As the most importat brach of fuzzy mathematics, fuzzy itegrals were origially ivestigated by Choquet 36 i 1950s, as a powerful tool for modelig iteractio pheomea amog various factors. The mai differece betwee the fuzzy itegral ad the classic itegral is that fuzzy itegral focuses o o-additive cases while the classical itegral oly cosider additive cases. O the basis of classic sets theory ad measure theory, it ca be easily proved the fact that the Choquet itegral is a special case of classical itegral whe Choquet itegral satisfies the coditio of additivity, I this case Choquet itegral reduces to the Lebesgue itegral. Therefore, the Choquet itegral is more geeral. Compared with the Sugeo itegral 37, which is also kow as a fuzzy itegral for aggregatig discrete sets, Choquet itegral exhibits iterestig mathematical properties, such as boudary coditios, mootoicity, cotiuity from below, cotiuity from above. Therefore, Choquet itegral is more suitable to cope with fuzzy ucertaity problems i practical applicatios. Recetly, Choquet itegral has bee widely used i decisio makig 35,38,45,46,48, especially i ituitioistic fuzzy MADM problems. Xu 40 ivestigated Choquet itegral to propose some ituitioistic fuzzy aggregatio operators, ad the applied them to solve ituitioistic fuzzy multiple attribute decisio makig problems. Ta ad Che 42,43 developed a weight solutio method based o Choquet itegral ad further proposed a ew Choquet itegral-topsis approach to hadle MADM problems. Meg ad Tag 44 developed a itervalvalued ituitioistic fuzzy multi-attribute group decisio makig approach based o cross etropy measure ad Choquet itegral. Compared with other Co-published by Atlatis Press ad Taylor & Fracis 134

3 J. Qi et al. / Multi-attribute group decisio makig existig methods, this algorithm ca ot oly deal with the relevace amog the idexes, but also ca cope with dyamic fuzzy systems. Meg et al. 45 defied some ew hybrid Choquet itegral aggregatio operators, ad proposed a method to solve multiple criteria group decisio makig MCGDM) based o ituitioistic fuzzy Choquet itegral with respect to the geeralized λ-shapely idex. Wu et al. 47 preseted a detailed discussio o the itegratio properties of the iterval ituitioistic fuzzy cojugate Choquet itegral, ad applied these theoretical aalysis results to software developmet risks decisio makig. I additio, some authors have also applied Choquet itegral to other fields, such as hesitat fuzzy set 49, game theory 50, eural etworks 51,52, statistical learig 53, ad combiatorial optimizatio 54,55,56. Accordig to the previous overview, ituitioistic fuzzy Choquet itegral ad its property has bee widely used i decisio makig problems. However, there has bee a limited research o Choquet itegral uder iterval-valued ituitioistic fuzzy decisio makig eviromet i the literatures. Whe Choquet itegral is used to hadle MADM problems uder iterval-valued ituitioistic fuzzy eviromet, the most challegig difficulty is how to determie fuzzy measure with a effective ad accurate method. For a MADM problem which cotais attributes, we should determie 2 2 elemets of fuzzy measures i decisio makig process. Whe the problem dimesioality icreases, the computatioal complexity icreases rapidly. So far, i the curret literature 42,43,44,45,57,58,59, the fuzzy measures are usually provided by decisio makers DMs) i advace, which maybe difficult to the DMs ad is closely related the subjective prefereces of the DMs, or ca eve lead to icosistet results. If the fuzzy measures ca be determied from the problem formulatio or the decisio maker s koformatio directly, the decisio makig model solutio ca be more reliable ad objective tha the curret oes. Therefore, how to determie the fuzzy measure based o the expert decisio makig iformatio becomes a very importat issue both i theory ad practice. The motivatio of this study is to develop a ew method of costructig fuzzy measures based o iterval-valued ituitioistic iformatio ad apply them to the group decisio makig problems. The remaider of this paper is orgaized as follows. I Sectio 2, we briefly itroduce some basic cocepts ad related operatioal laws of itervalvalued ituitioistic fuzzy set IVIFS) ad the Choquet itegral. I Sectio 3, we give the defiitio of iterval-valued ituitioistic fuzzy measure with Choquet itegral ad propose some useful theoretical backgroud. I Sectio 4, we develop ew methods to determie fuzzy measure i real-world decisio markig situatios, by establishig itervalvalued ituitioistic fuzzy measure to determie the measure of expert weights ad attribute weights, ad strict mathematical proof process is give. I Sectio 5, a ew approach based o iterval-valued ituitioistic fuzzy Choquet itegral is proposed to solve MAGDM problems. I Sectio 6, a illustrative example is provided to illustrate the proposed method. Fially, we come up with some coclusios ad poit out the future research i Sectio Prelimiaries I this sectio, we briefly review some basic cocepts of iterval-valued ituitioistic fuzzy set IV- IFS) ad fuzzy measure, which are exteded to iterval-valued ituitioistic fuzzy measure i ext Sectio Iterval-valued ituitioistic fuzzy set Defiitio 1 18 Let X be a uiverse of discourse. The A { x, [μax), L μa R x)],[νax),ν L A R x)] ) x X } 1) is a IVIFS, where μ A x)[μa Lx), μr A x)],ν Ax) [νa Lx),νR A x)] is called as iterval membership degree ad o-membership degree of x. The followig two coditios are satisfied: 1) For all x X,[μA Lx), μr A x)] [0,1], ad [νa Lx),νR A x)] [0,1]; 2) x X,0 μa Rx)+νR A x) 1. Co-published by Atlatis Press ad Taylor & Fracis 135

4 J. Qi et al. / Multi-attribute group decisio makig I particular, if μa Lx) μr A x),νl A x) νr A x), for each x X, the the IVIFS is reduced to IFS. For coveiece, Xu 20 called α [a L,a R ],[b L,b R ]) a iterval-valued ituitioistic fuzzy umber IVIFN), where [a L,a R ] [0,1],[b L,b R ] [0,1],a R + b b 1, ad let Θ be the set of all IVIFNs. Obviously, α + [1,1],[0,0]) is the largest IVIFN, ad α [0,0],[1,1]) is the smallest IVIFN. Defiitio 2 20 Let α [a L,a R ],[b L,b R ]),β [c L,c R ],[d L,d R ]) be two IVIFNs. The operatioal laws are show as follows: 1) α β [a L + c L a L c L,a R + c R a R c R ],[b L d L,b R d R ]); 2) α β [a L c L,a R c R ],[b L + d L b L d L,b R + d R b R d R ]); 3) α C [b L,b R ],[a L,a R ]); 4) a λ [a L ) λ,a R ) λ ],[1 1 b L ) λ,1 1 b R ) λ ]), λ > 0. Defiitio 3 19 Let α [a L,a R ],[b L,b R ]) be a IV- IFN. The the score fuctio of α is defied as follows: sα) al + a R b L b R 2) 2 where sα) [ 1, 1]. Obviously, the greater sα) is, the larger IVIFN α becomes. Defiitio 4 21 Let A { x i, [a L i,ar i ],[bl i,br i ]) x X } i 1,2,,) be a discrete IVIFS. The etropy of A is defied i the followig form: EA) 1 b L cos i a L i + b R i a R i 2 πi L + πi R + 2 ) π 3) The iterval-valued ituitioistic fuzzy etropy satisfies the followig properties: 1) EA)0 if ad oly if a L i a R i 0,b L i b R i 1 or a L i a R i 1,b L i b R i 0; 2) EA) 1 if ad oly if [a L i,ar i ][bl i,br i ], for ay x i X; 3) EA)EA c ); 4) EA) EB), x i X, if a L B bl B,aR B br B, the a L A al B,aR A ar B,bL A bl B,bR A br B or a L B bl B,aR B br B, the al A al B,aR A ar B,bL A b L B,bR A br B Fuzzy measure ad Choquet itegral Defiitio 5 60 Let X be a uiverse of discourse, PX) be the power set of X. A fuzzy measure o X is a mappig μ: PX) [0,1] which satisfies the followig properties: 1) μφ)0, μx)1 boudary coditios); 2) if A B implies μa) μb), for all A,B X,mootoicity); 3) μa + B) μa)+ μb)+ λμa)μb), where A,B PX) ad A B φ,λ 1; 4) μ A ) lim μa ), wheever A 1 A +1,A X, Ncotiuity from below); 5) μ A ) lim μa ), wheever A 1 A +1,A X, Ncotiuity from above). Defiitio 6 37 Let X {x 1,x 2,,x } be a uiverse of discourse, ad X x i, the λ f uzzy measure μ satisfies followig coditio: 1 λ [1 + λμx i )] 1) ifλ 0 μx) μx i ) ifλ 0 4) where λ > 1. From Eq.4), the value of λ ca be uiquely determied by settig μx) 1, which is equivalet by solvig the followig equatio: λ λμx i )) 5) Defiitio 7 35 Let μ be a fuzzy measure o X, f is a oegative, real-valued, ad measurable fuctio. The the Choquet itegral is defied i the form: c) fdμ μ{f α }dl 6) 0 where F α {x f x) α} deotes the α cut of the fuctio. Whe X {x 1,x 2,,x } be a discrete Co-published by Atlatis Press ad Taylor & Fracis 136

5 J. Qi et al. / Multi-attribute group decisio makig set, the discrete form of Choquet itegral with respect to a fuzzy measure μ is defied as: c) f dμ f xi ) f xi 1))μ { xi,xi+1,,x }) 7) Where A {x 1,x 2,,x } is a mootoic odecrease permutatio of X {x 1,x 2,,x } such that f x 1 ) f x 2 ) f x ) ad A +1 /0. 3. Iterval-valued ituitioistic fuzzy Choquet itegral I this sectio, we ivestigate the defiitio of the iterval-valued ituitioistic fuzzy measure ad the iterval-valued ituitioistic fuzzy Choquet itegral with such fuzzy measure. Some properties ad characteristics of them are proposed ad discussed i detail. Defiitio 8 Let X be a uiverse of discourse, ad X, Σ, μ) be a iterval ituitioistic fuzzy measure space, Σ is a σ algebra ad μ [μ 1, μ 2 ],[μ 3, μ 4 ]), where fuzzy measure μ satisfies the followig properties: 1) μφ) [μ 1 φ), μ 2 φ)],[μ 3 φ), μ 4 φ)]) [0,0],[1,1]); 2) μx) [μ 1 X), μ 2 X)],[μ 3 X), μ 4 X)]) [1,1],[0,0]); 3) If E,F Σ, ad E F, the μ 1 E) μ 1 F), μ 2 E) μ 2 F) ad μ 3 E) μ 3 F), μ 4 E) μ 4 F) ; 4) For ay E Σ, μ 2 E)+μ 4 E) 1. Especially, whe μ 1 E) μ 2 E), μ 3 E) μ 4 E), for ay E Σ, the iterval-valued ituitioistic fuzzy measure reduces to the commoly ecoutered ituitioistic fuzzy measure. Based o the decompositio theorem for the origial Choquet itegral ad the exteded ituitioistic fuzzy Choquet itegral, the iterval-valued ituitioistic fuzzy Choquet itegral of f with respect to μ is defied as: c) f dμ [ a L i dμ 1, a R i dμ 2 ], [ b L i dμ 3, b R i dμ 4 ]) 8) where f [ a L ] [ i,ar i, b L i,b R ]) i, μ [μ1, μ 2 ],[μ 3, μ 4 ]). For ay set A Σ, which satisfies μ 1 A) μ 2 A), μ 3 A) μ 4 A), μ 2 A)+μ 4 A) 1, ad - deotes the dual operatio, i.e., f x)1 f x) μa)1 μa) 9) Theorem 1 Let μ be a iterval-valued ituitioistic fuzzy measure defied o X, ad A is a itervalvalued ituitioistic fuzzy set o PX). The the iterval-valued ituitioistic fuzzy Choquet itegral with respect to μ is also a IVIFN. Proof. Based o the defiitio of IVIFS, we have a L a R,b L b R. Accordig to the mootoicity of fuzzy measure, we ca prove that a L i dμ 1 a R i dμ 1 a R i dμ 2 ad b L i dμ 3 1 b L i dμ b L i )dμ b R i )dμ b R i )dμ 4 1 b R i dμ 4 b R i dμ 4. Thus, Eq. 8) satisfies the partial order of the IVIFS. For coveiece, we deote f 1 a L i, f 2 a R i, f 3 b L i, f 4 b R i i the followig proof process. Based o the defiitio of discrete Choquet itegral described i Sectio 2, we have: c) c) f 1 μ 1 f 2 μ 2 μ 1 f 1 x) α 1 )dα 1 10) μ 2 f 2 x) α 2 )dα 2 So we oly eed to prove the followig relatioships: c) c) f 3 μ 3 f 4 μ 4 μ 3 f 3 x) α 3 )dα 3 11) μ 4 f 4 x) α 4 )dα 4 Co-published by Atlatis Press ad Taylor & Fracis 137

6 J. Qi et al. / Multi-attribute group decisio makig Based o Eq. 10), we have 1 f 3 μ 3 1 μ 3 f 3 x) α 3 )dα [1 μ 3 f 3 x) α 3 )]dα μ 3 f 3 x) α 3 )dα μ 3 ) f 3 x) 1 α 3 )dα μ 3 f 3 x) 1 α 3 )dα μ 3 f 3 x) α 3 )dα ) c) a L f i x i)μ 1 A i ) μ 1 A i+1 )), dμ a R i x, i)μ 2 A i ) μ 2 A i+1 )) b L i x i)μ 1 B i+1 ) μ 1 B i )), b R i x i)μ 4 B i+1 ) μ 4 B i )) 16) where A i { x i,,x a L + a R) x i+1 ) a L + a R) x i ) } ad B i { x i,,x b L + b R) x i+1 ) b L + b R) x i ) } i 1,2,,) I a similar way, it is easy to prove that c) f 4 μ 4 μ 4 f 4 x) α 4 )dα 4 holds. Next, Based o the defiitio of iterval ituitioistic fuzzy sets described i Sectio 2, we oly eed to prove that the followig iequality holds: c) f 2 μ 2 +c) f 4 μ ) Sice f 2 + f 4 1, μ 2 + μ 4 1, based o the mootoicity of Choquet itegral, the followig iequality esues: c) The c) c) f 4 μ 4 c) f 2 μ 2 +c) f 4 μ 4 c) f 2 μ 4 c) f 4 μ 4 c) f 2 μ c) The proof has bee completed. f 2 μ 2 14) f 2 μ f 2 μ ) Theorem 2 Let X {x 1,x 2,,x } be the uiverse of discourse, the the iterval-valued ituitioistic fuzzy Choquet itegral of f with respect to μ ca be expressed as the followig form: Proof. Based o the defiitio of Choquet itegral described i Sectio 2, it ca be easily show that the followig two relatioships are satisfied: c) c) f 1 dμ 1 f 2 dμ 2 f 1 x i )μ 1 A i ) μ 1 A i+1 )) a L i x i )μ 1 A i ) μ 1 A i+1 )) f 2 x i )μ 2 A i ) μ 2 A i+1 )) a R i x i )μ 2 A i ) μ 2 A i+1 )) Thus, we should oly prove the followig formulas: c) c) f 3 dμ 3 f 4 dμ 4 f 3 x i )μ 3 B i+1 ) μ 3 B i )) b L i x i )μ 3 B i+1 ) μ 3 B i )) f 4 x i )μ 4 B i+1 ) μ 4 B i )) b R i x i )μ 4 B i+1 ) μ 4 B i )) Co-published by Atlatis Press ad Taylor & Fracis 138

7 J. Qi et al. / Multi-attribute group decisio makig Based o Eqs.9,10,12), we obtai c) b L dμ 3 f 3 dμ 3 1 f 3 dμ f 3 x i )[μ 3 B i ) μ 3 B i+1 )] f 3 x i )[1 μ 3 B i )) 1 μ 3 B i+1 ))] μ 3 B i+1 ) μ 3 B i )) μ 3 B i+1 ) μ 3 B i ))1 f 3 x i )) f 3 x i )μ 3 B i+1 ) μ 3 B i )) f 3 x i )μ 3 B i+1 ) μ 3 B i )) Similarly, it is easy to verify that the followig formula holds: c) b R dμ 4 b R i x i )μ 4 B i+1 ) μ 4 B i )) Thus, the proof has bee completed. I what follows, we ivestigate two useful theorems of the iterval-valued ituitioistic fuzzy Choquet itegral proposed above. Theorem 3 Let X be a uiverse of discourse, f [ f 1, f 2 ],[ f 3. f 4 ]) be a iterval ituitioistic fuzzyvalued fuctio o X. 0 [0,0],[1,1]) ad 1 [1,1],[0,0]) are the smallest ad largest iterval ituitioistic fuzzy-valued fuctio, respectively. The 1. Idempotecy). c0,0,,0)0,c1,1,,1) 1,c f, f,, f ) f; 2. Boudary). mi f 1), f 2),, f ) ) c f 1), f 2), f ) ) max f 1), f 2),, f ) ); 3. Mootoicity). If f k) f k ), the c f 1),, f k),, f ) ) c f 1),, f k ),, f ) ). Proof. 1) Based o the defiitio of Choquet itegral, we have c)0,0,,0)0 c)1,1,,1)1 c) f, f,, f ) f μx i ) μx i+1 )) μx i ) μx i+1 )) μx i ) μx i+1 )) f 1 f 2) Based o Theorems 1 ad 2, we obtai c) mi f i),, mi f i) ) 1 i 1 i c) f 1),, f ) ) c) max f i),, max f i) ). 1 i 1 i The accordig to the property of idempotecy, we have c) mi f i),, mi f i) ) 1 i 1 i mi f i) ad c) max f i),, max f i) ) 1 i 1 i 1 i max f i). Therefore, mi f 1), f 2),, f ) ) 1 i c) f 1), f 2),, f ) ) max f 1), f 2),, f ) ). 3) Based o the mootoicity of fuzzy itegral ad Theorem 2, the coclusio directly esues. which completes the proof of Theorem 3. Theorem 4 Let X be a uiverse of discourse, PX) is a power set of X, set fuctio μ is a itervalvalued ituitioistic fuzzy measure o X,let f [ f 1, f 2 ],[ f 3, f 4 ]) be a iterval-valued ituitioistic fuzzy-valued fuctio X. 0 [0,0],[1,1]) ad 1 [1, 1],[0, 0]) are the smallest ad largest itervalvalued ituitioistic fuzzy measure, respectively. The 1. c) f 1 max x X f; 2. c) f 0 mi x X f; 3. For ay iterval ituitioistic fuzzy measure μ o X, mi f c) f μ max f. x X x X Proof. 1) For ay α [α 1,α 2 ],[α 3,α 4 ]) X, based o the defiitio of iterval ituitioistic fuzzy Co-published by Atlatis Press ad Taylor & Fracis 139

8 J. Qi et al. / Multi-attribute group decisio makig Choquet itegral described i Sectio 2, we have c) f μ [ 1 0 μ 1 f 1 α 1 )dα 1, 1 0 μ 2 f 2 α 2 )dα 2 ], [ 1 0 μ 3 f 3 α 3 )dα 3, ] 1 0 μ 4 f 4 α 4 )dα 4 where { f α} {x X f x) α} {x X f 1 x) α 1 f 2 x) α 2 f 3 x) α 3 f 4 x) α 4 }. Whe α β, the {x f α} {x f β}, which implies μ{x f α} μ{x f β}. If μ{x f α} 1 μ{x f β} 1. The, c) f μ [ 1 0 μ 1 f 1 α 1 )dα 1, 1 0 μ 2 f 2 α 2 )dα 2 ], [ 1 0 μ 3 f 3 α 3 )dα 3, ] 1 0 μ 4 f 4 α 4 )dα 4 c) f 1 sup μ f α)1 [ 1 0 μ 1 f 1 β 1 )dβ 1, 1 0 μ 2 f 2 β 2 )dβ 2 ], [ 1 0 μ 3 f 3 β 3 )dβ 3, 1 0 μ 4 f 4 β 4 )dβ 4 ] sup μ f α)1 [α 1,α 2 ],[α 3,α 4 ]) sup μ f α)1 α sup μ f α)1 if sup sup f max f μ f α)1 μ f α)1 μ f α)1 f 2) The proof of property 2 is similar to property 1, omitted. 3) Based o the mootoicity of fuzzy measure, we have 0 μ 1, so we ca easily prove the followig iequality c) f 0 c) f μ c) f 1 Accordig to Theorem 3, we have mi x X f c) f μ max x X f. which completes the proof of Theorem A ew method to determie fuzzy measure uder iterval-valued ituitioistic fuzzy eviromet I most real-world decisio makig situatios, where the measure of attributes usually does ot satisfy the coditio of o-additive property. However, there exist some cases where the attributes may iteract. Therefore, how to derive the fuzzy measures to reflect the relatioship of attributes becomes a vital problem i such practical problems. From the curret literature 57,58,59, we ca oly fid research o the determiatio of the fuzzy measures with the decisio makers assigmet directly, which ot oly raises difficulties for large dimesioal problems ad may also lead to iformatio losses, distortio ad icosistecies. Here, we propose a ew method to determie the fuzzy measures from the koformatio, which ca avoid the shortcomigs of the curret direct assigmet methods. For coveiece, we first itroduce some ew otatios. Let D {D 1,D 2,,D p } be the set of experts, where D i idicates the i-th expert, ad E i idicates the average iterval-valued ituitioistic fuzzy etropy provided by D i, which is calculated by Eq. 3) Experts importace measure based o iterval-valued ituitioistic fuzzy etropy Defiitio 9 Let μd i ) deote the importace measure of expert D i, ad μd i1,d i2,,d i ) deotes the joit absolute importace measure of experts D i1,d i2,,d i, the experts absolute importace measure ca be defied as: μd i )1 E i 17) μd i,d j )1 EE i E j ) 18) μd i1,d i2,,d i )1 EE i1 E i2 E i ) 19) where {i 1,i 2,,i } is a subset of {1,2,, p}, E ki j is the iterval-valued ituitioistic fuzzy etropy provided by expert D k for the alterative A i with respect to the attribute C j. It should be oted that EE i1 E i2 E i ) idicate the average itervalvalued ituitioistic fuzzy etropy provided by ex- Co-published by Atlatis Press ad Taylor & Fracis 140

9 J. Qi et al. / Multi-attribute group decisio makig perts D i1,d i2,,d i, where E i1 E i2 E i is a real-valued etropy matrix. Remark 1 This defiitio is established based o iterval-valued ituitioistic fuzzy etropy. As we kow, the etropy ca be regarded as a importat measure that reflects the decisio makig iformatio to some extet. The smaller the etropy is, the greater the iformatio will be. Therefore, cosider the characteristics of the group decisio makig problems uder iterval-valued ituitioistic fuzzy eviromet, we utilize the miimize etropy priciple to fuse the decisio makig iformatio amog the experts ad establish the model above to obtai the experts absolute importace measure. Due to the fact that importace measure should satisfy ormalizatio coditio, we should ormalize the absolute measure before calculatio. Hece, we defie a ew cocept as follows: Defiitio 10 Let μd i ) deote the relative importace measure of D i, ad μd i1,d i2,,d i ) deotes the joit relative importace measure of D i1,d i2,,d i, the experts relative importace measure is defied as follows: μd i1,d i2,,d i ) μd i 1,D i2,,d i ) μd 1,D 2,,D ) 20) Theorem 5 μd i1,d i2,,d i ) satisfies the mootoicity of the fuzzy measure. Proof. Sice {D i1,d i2,,d i } {D i1,d i2,,d i+1 }, the μd i1,d i2,,d i,d i+1 ) μd i1,d i2,,d i ) μd i 1,D i2,,d i,d i+1 ) μd i1,d i2,,d i ) μd 1,D 2,,D ) 1 EE i 1 E i2 E i E i+1 ) 1 EE i1 E i2 E i )) μd 1,D 2,,D ) EE i 1 E i2 E i ) EE i1 E i2 E i E i+1 ) 0 μd 1,D 2,,D ) 21) So μd i1,d i2,,d i ) satisfies the mootoicity of the fuzzy measure. The proof has bee completed. Remark 2 Theorem 5 idicates the fact that with the icreasig umber of experts or decisio makers), the iformatio proposed by experts or decisio makers) is also icreasig. It also states that the ucertaity of iformatio reduces. Therefore, the experts relative importace measure proposed here is reasoable i real life decisio makig situatios Attributes importace measure based o weight iformatio matrix Attribute importace measure is a key issue of the decisio makig problems i practice. As a useful tool to solve the attribute importace measure, Shao etropy was widely used to trade off the importace amog the attributes i real life decisio makig. Let C i deote the ith attribute, H i deote the Shao etropy with respect to C i. We usually measure the importace of C i based o the value of H i. The smaller value of H i, the great importace of C i is. I what follows, we defie some ew joit attributes iformatio etropy based o origial Shao etropy to measure the importace of the attributes. Defiitio 11 Let H i idicates the Shao etropy with respect to the attribute C i. Three types of joit attributes iformatio etropy are defied as follows: 1) EΔ 1 i ) i 1 H i ) π 4 ; 2) EΔ 2 i )si i 1 H i ) π 4 3) EΔ 3 i )ta i 1 H i ) π 4 ) ; ). where I {i 1,i 2,,i } is a idex sequece such that i 1 i,1 i p. It is clear that 0 EΔ 1 i ),EΔ 2 i ),EΔ 3 i ) < 1, ad EΔ 1 i ),EΔ 2 i ),EΔ 3 i ) are all cotiuous, mootoic ad decreasig fuctios of I. That is to say, if I < I +1 where I is the umber of I ), the EΔ j i +1 ) < EΔ j i ), j 1,2,3, which meas, with a icreasig of the umber of attributes, the ucertaity of the iformatio is reduced. Therefore, we ca use these three formulas to measure the importace of the attributes. Let C I {{i 1,i 2,,i } i i 1,1 i p} be a set of idex sequece, the we derive the followig Theorem 6. Co-published by Atlatis Press ad Taylor & Fracis 141

10 J. Qi et al. / Multi-attribute group decisio makig Theorem 6 For ay I C I,EΔ 2 I ) < EΔ 1 I ) < EΔ 3 I ). Proof. Cosiderig the sie fuctio y six, we ca clearly see that whe x 0, π 2 ),the sie fuctio is a cotiuous, mootoic icreasig with respect to x. Based o the theory of trigoometric fuctio, we ca easily ifer the fact that if x 0, π 2 ), the six < x < tax. Let x i 1 H i ) π 4, it is clear to see that 0 < x i 1 H i ) π 4 < π 4 < π 2. Thus, we obtai the followig iequalities: ) ) i si H i ) 1 π i < 4 1 H i ) π i 4 < ta H i ) 1 π 4 Therefore, we have EΔ 2 i ) < EΔ 1 i ) < EΔ 3 i ). The proof has bee completed. To determie the importace of attributes, Xu 41 proposed a useful weight solvig method with icomplete iformatio. A weight iformatio matrix W j ) m is established based o a multiobjective optimizatio model, whose elemets j is the optimal weight solutio correspodig to the alteratives. I what follows, we cosider a multiattribute group decisio-makig problem with attributes iteractio iformatio, ad give a formulatio based o weight iformatio matrix ad the three types of joit attributes iformatio etropy we proposed to determie the attributes importat measure uder iterval-valued ituitioistic fuzzy eviromet. Defiitio 12 Let W w ji ) m be a weight iformatio matrix, w ji idicates the optimal weight for the alterative A j with respect to the attribute C i, μc i ) idicates the attribute importace measure with respect to the attribute C i. μc i1,c i2,,c i ) idicates the attribute importace measure with respect to the attributes C i1,c i2,,c i. The the attributes importace measures are defied as: μc i ) [ ] w 1 λ i,w 1 λ)eδ 3 i 1 ) i, [ 1 wi 1 λ)eδ 2 i 1 ) ) 2,1 wi 1 λ)eδ 1 i 1 )] 22) μc i1,c [ i2,,c i ) ) i 1 λ, i 1 i 1 ) ] 1 λ)eδ3 i i ), ) 1 λ)eδ2 i ) ) 2 ) 1 λ)eδ1 i ), 23) where λ idicates a iteractio coefficiet. Theorem 7 μc i1,c i2,,c i ) satisfies the mootoicity of the iterval-valued ituitioistic fuzzy measure. Proof. Based o the theory of iequality, sice 1 λ 1,0 H 1, ad 0 < i 1, so we have 0 < 1 H i ) < 1,0 < 1 H i ) π 4 < π 4, ad ) 0 < ta 1 H i ) π 4 < ta π 4 1. Sice i ii 1 ) ii 1 ad 1 λ) > 1 λ)ta 1 H i ) π 4, ) ) i 1 λ ) i 1 λ)ta 1 H i ) π 4 the <, ii 1 ) 1 i i i i 1 ) 1 λ)si 1 H i ) π 4 ) ) 1 λ)si 1 H i ) π 4 ) ) 1 λ)ta 1 H i ) π 4 ) 1 λ) 1 H i ) π 4 i +1 2 < 1 i +1 i < 1 ) ) 1 λ) 1 H i ) π 4 ) 1 λ) 1 H i ) π 4 < ) 1 λ) 1 H i ) π 4 The we prove the mootoicity of the itervalvalued ituitioistic fuzzy measure as follows: ) i 1 λ 1) Let μ 1 i), the we have μ 1 i + Co-published by Atlatis Press ad Taylor & Fracis 142

11 J. Qi et al. / Multi-attribute group decisio makig 1) μ 1 i) l e i+1 ) i+1 1 λ l e i ) 1 λ ) 1 λ ) i 1 λ > 0. Thus, μ 1 i) is a icreasig fuctio with respect to i. 2) Let μ 2 i) The i ) 1 λ)ta 1 H i ) π 4 i l μ 2 i + 1) l μ 2 i) i+1 ) i+1 ) 1 λ)ta 1 H i ) π 4 l ) ) i 1 λ)ta 1 H i ) π i 4 l ii 1 i+1 ) i+1 ) 1 λ) ta 1 H i ) π 4 l ) )) i ta 1 H i ) π i 4 l ii 1 ) i+1 i+1 ) ta 1 H i ) π 4 ii 1 λ) l w 1 i l i 1 λ)l i ) ta 1 H i ) π 4 ) ) i+1 ta i +1 1 H i ) π 4 ii w 1 i ) ta i 1 H i ) π i 4 ii1 > 0 From l μ 2 i + 1) l μ 2 i) > 0 μ 2 i + 1) > μ 2 i). So μ 2 i) is a icreasig fuctio with respect to i. ). i 3) Let μ 3 i) 1 μ 3 i + 1) μ 3 i) i+1 1 i 1 2 i ) 1 λ)si 1 H i ) π 4 ) 1 λ)si i+1 1 H i ) π 4 i ) 1 λ)si 1 H i ) π 4 ii w 1 i i+1 i i+1 ) 2 ) 2 ) i+1 ) 1 λ)si 1 H i ) π 4 ) 1 λ)si i+1 1 H i ) π 4 i ) 1 λ)si ) 1 H i ) π 4 ) i+1 ) ) 1 λ)si 1 H i ) π 4 < 0 ) From μ 3 i+1) μ 3 i) < 0, we ca obtai μ 3 i+ 1) < μ 3 i). Hece, μ 3 i) is a decreasig fuctio with respect to i. 4) The proof of 4) is similar to the proofs of 1)- 3), omitted. Hece, we verify the proposed fuzzy measure satisfies the properties of the iterval-valued ituitioistic fuzzy sets ad whe i 1 i 2 i, Eq. 23) is reduced to Eq. 22). Thus, the proof is complete. 2 I this sectio, we give a ew method to solve experts importace measure ad attributes importace measure uder iterval-valued ituitioistic fuzzy eviromet. Sice the give approach based o iterval-valued ituitioistic fuzzy Choquet itegral, which greatly reduces the complexity of determi- Co-published by Atlatis Press ad Taylor & Fracis 143

12 J. Qi et al. / Multi-attribute group decisio makig ig the fuzzy measures, ad improves the practicality of the method. 5. A approach to multi-attribute group decisio-makig based o iterval-valued ituitioistic fuzzy Choquet itegral Iterval-valued ituitioistic fuzzy multiple attribute group decisio makig task is a special case of fuzzy multiple attribute group decisio makig problems. I the sequel, based o the iterval-valued ituitioistic fuzzy etropy ad the iterval-valued ituitioistic fuzzy measure we derived above, a ew approach is proposed to solve multiple attribute group decisio makig problem uder iterval-valued ituitioistic fuzzy eviromet. We propose two methods to determie the fuzzy measure which are used to calculate the iteractive importace measure of attributes ad its correlatio measure of experts e.g. absolute measure, relative measure). Firstly, we describe the iterval-valued ituitioistic fuzzy multiple attribute group decisio makig MAGDM) problems i this paper The descriptio of MAGDM problems For a iterval-valued ituitioistic fuzzy multiple attribute group decisio makig problem. Let A {A 1,A 2,,A m } be a discreet set of alteratives, C {C 1,C 2,,C } be a set of attributes, ad D {D 1,D 2,,D p } be a set of experts. The performace of the alterative A i with respect to the attribute C j which is provided by expert D k is expressed by a IVIFN A k) ij [ ] [ ]) [ ] a L ijk,ar ijk, b L ijk,br ijk, where a L ijk,ar ijk idicates the iterval-valued degrees that [ the alterative A i satisfies the attribute C j, ad b L ijk,br ijk ] idicates the iterval-valued degrees that the alterative [ A i does ] ot satisfies [ the ] attribute C j, such that a L ijk,ar ijk [0,1], b L ijk,br ijk [0,1] ad 0 a R ijk +br ijk 1. For all elemets Ak) ij are cotaied i the decisio matrix DM k) which is provided by expert D k. Based o the aalysis preseted before, we develop a ew method for MAGDM problems uder iterval-valued ituitioistic fuzzy eviromet. Based o these ecessary coditios, the rakig of alteratives is required Decisio makig steps The iterval-valued ituitioistic fuzzy Choquet itegral method is illustrated i what follows. The algorithm ivolves the followig steps: Step 1 Costruct the idividual decisio matrix DM k) k 1,2,, p). Step 2 Calculate the etropy of each decisio matrix E k k 1,2,, p). Step 3 Calculate the importace measure based o the etropy of experts i accordace with decisio matrix. 1) Calculate the absolute measure With the use of Defiitio 9 ad Eqs ), we obtai the experts absolute importace measure. 2) Calculate the relative measure Usig Defiitio 10 ad Eq. 20), we determie the experts relative importace measure. Step 4 Calculate the weights of experts. Based o game theory, we get the expert weights with Shapely value 60. p k )! k 1)! ) σ k v) μd S ) μd p! S {i} ) 24) where σ k v) idicates the expert weight of D k. Step 5 Aggregate the decisio iformatio of each expert to collect ito group decisio matrix G by usig the IVIFWA operator. Utilize the IVIFWA operator to arrage the idividual decisio matrices ito a collective aggregate) decisio matrix as follows: ] ]) G [g L μij,g L μij, [g L νij,g L νij m 25) Step 6 Calculate the group decisio score matrix ad the etropy matrix, respectively. With Eqs. 2) ad 3), we calculate the score matrix S G ad etropy matrix E G respectively, which Co-published by Atlatis Press ad Taylor & Fracis 144

13 J. Qi et al. / Multi-attribute group decisio makig are defied as follows: S G E G s 11 s 12 s 1 s 21 s 22 s 2... s m1 s m2 s m E 11 E 12 E 1 E 21 E 22 E 2... E m1 E m2 E m 26) Step 7 Costruct the programmig model of attribute weights based o the relative etropy measure. Based o the defiitio of relative etropy, ad with the purpose to determie the optimal weight vector, we have the basic idea to maximize the score fuctio of each alterative, ad miimize the etropy fuctio of each alterative. O the basis of the above aalysis, we establish the followig liear programmig model correspodig to the i-th alterative: miw i j l E ij,i 1,2, m j1 s ij 27) j 1 s.t. j1 j 0 By solvig the above sated liear programmig model, we produce the solutio of the attribute weight vectors W i) ) 1,wi) 2, wi) ), where i 1,2,,m. Step 8 Solve the optimal attribute weights vector. Based o the defiitio of Shao etropy, we ca determie the attribute weight vector as follows: H j 1 ml2 m ) ij l) ij j 1,2,,) 28) w j 1 H j j 1,2,,) 29) H k k1 Step 9 Calculate the iteractio coefficiet λ. I virtue of Eq. 5), calculate the value of parameter λ by solvig the followig o-liear multiobjective programmig problem: Accordig to the priciple of optimizatio, we ca costruct a sigle-objective programmig model to solve this multiple-objective programmig: 2 mi λh i ) λ + 1)) H i + ) ))] 1 1 λ [1 + λh i ] 1 l λ [1 + λh i ] 1 { 1 λ 1 s.t. λ 0 30) Step 10 Calculate the importace measure of attribute weights. Based o Eqs ), form the fuzzy measure of attribute weights. Step 11 Utilize the Choquet itegral to aggregate the iterval-valued ituitioistic fuzzy iformatio of decisio-makig matrix. Based o Defiitio 8 ad Eq. 16), the aggregate value of the alteratives is formed. Step 12 Use the score fuctio of IVIFS to obtai the score of overall alteratives. Based o Eq. 2), we form the score value of the alteratives. Step 13 Utilize the score value to rak the alteratives i the descedig order. Rak all the alteratives by Eq. 2), the larger the score value is, the better the alterative is. Step 14 Ed. 6. Numerical example I this sectio, we provide a example to show the applicatio of the proposed method for supplier selectio problem The supplier selectio problem descriptio With the cotiuous developmet of ecoomic globalizatio, the supply chai maagemet has played a importat role i marketig ecoomic ad become the most hot research topic i moder maagemet sciece, which directly impact o the mau- Co-published by Atlatis Press ad Taylor & Fracis 145

14 J. Qi et al. / Multi-attribute group decisio makig factures performace. Cosider a problem i a shipbuildig compay, which aims to search for the best supplier for purchasig ship equipmets. Three potetial ship equipmets suppliers have bee idetified. The attributes to be cosidered i the evaluatio process are: C 1 : busiess maagemet capacity; C 2 : quality; C 3 : trasportatio ad delivery ability; C 4 : service ability. The three shipbuildig compaies are to be evaluated usig the itervalvalued ituitioistic fuzzy iformatio by three experts D k k 1,2,3) Illustratio of the proposed method Step 1 Costruct the idividual decisio matrix DM k) k 1,2,3), respectively. [0.3,0.4],[0.4,0.6]) [0.5,0.6],[0.1,0.2]) [0.6,0.8],[0.1,0.2]) [0.6,0.7],[0.2,0.3]) DM 1) [0.5,0.8],[0.1,0.2]) [0.7,0.8],[0.0,0.2]) [0.6,0.7],[0.2,0.3]) [0.7,0.8],[0.0,0.1]) [0.2,0.3],[0.4,0.6]) [0.5,0.6],[0.1,0.3]) [0.5,0.5],[0.4,0.5]) [0.2,0.3],[0.2,0.4]) [0.4,0.5],[0.3,0.4]) [0.5,0.6],[0.1,0.2]) [0.6,0.8],[0.1,0.2]) [0.5,0.6],[0.3,0.4]) DM 2) [0.5,0.6],[0.3,0.4]) [0.5,0.7],[0.1,0.2]) [0.6,0.7],[0.2,0.3]) [0.7,0.8],[0.1,0.2]) [0.4,0.5],[0.3,0.4]) [0.4,0.6],[0.3,0.4]) [0.5,0.6],[0.3,0.4]) [0.3,0.4],[0.2,0.5]) [0.4,0.6],[0.3,0.4]) [0.5,0.7],[0.0,0.2]) [0.5,0.8],[0.1,0.2]) [0.3,0.5],[0.2,0.3]) DM 3) [0.5,0.6],[0.0,0.1]) [0.5,0.8],[0.1,0.2]) [0.5,0.6],[0.2,0.4]) [0.6,0.8],[0.1,0.2]) [0.3,0.6],[0.2,0.4]) [0.4,0.5],[0.2,0.4]) [0.4,0.7],[0.2,0.3]) [0.2,0.4],[0.2,0.3]) Step 2 Calculate the etropy of each idividual decisio matrix E k k 1,2,3), respectively E E E Step 3 Calculate the importace measure based o the etropy of expert i accordace with the obtaied decisio matrix. Based o Defitio 10 ad Eqs.11-13), we calculate the experts importace measure. The obtaied results are show as follows: μd 1 )0.1551, μd 2 )0.1009, μd 3 ) μd 1,D 2 ) 1 E Similarly, we form other fuzzy measures based o the etropy of iterval-valued ituitioistic fuzzy iformatio. The results are calculated as follows: μd 1,D 3 )0.1664, μd 2,D 3 )0.1245, μd 1,D 2,D 3 ) The we use Eq. 20) to ormalize the absolute measure, ad obtai the experts importace measures: μd 1 )0.9292, μd 2 )0.6045, μd 3 )0.6093, μd 1,D 2 )0.9508, μd 1,D 3 )0.9970, μd 2,D 3 )0.7459, μd 1,D 2,D 3 )1 Step 4 Calculate the weight of experts. Accordig to the results of game theory, we get the expert weight based o Shapely-value. The re- Co-published by Atlatis Press ad Taylor & Fracis 146

15 J. Qi et al. / Multi-attribute group decisio makig sults read show as follows: 3 1)!1 1)! σ 1 v) μd 1 ) 3! 3 2)!2 1)! + μd 1,D 2 ) μd 2 )) 3! 3 2)!2 1)! + μd 1,D 3 ) μd 3 )) 3! 3 3)!3 1)! + μd 1,D 2,D 3 ) μd 2,D 3 )) 3! ) ) ) σ 2 v) ) ) ) σ 3 v) ) ) ) Normalize the results of σ i v)i 1,2,3), which leads to the weight iformatio correspodig to the three experts: σ 1 v) w D1 σ 1 v)+σ 2 v)+σ 3 v) σ 2 v) w D2 σ 1 v)+σ 2 v)+σ 3 v) σ 3 v) w D3 σ 1 v)+σ 2 v)+σ 3 v) Step 5 Aggregate the decisio iformatio of each expert to collect a group decisio matrix G by usig the IVIFWA operator. Utilize the IVIFWA operator, ad the structure the decisio iformatio i the form of the group decisio matrix. G [0.35,0.48],[0.34,0.48]) [0.50,0.63],[0.00,0.20]) [0.57,0.80],[0.10,0.20]) [0.51,0.63],[0.22,0.32]) [0.50,0.71],[0.00,0.19]) [0.61,0.77],[0.00,0.20]) [0.57,0.68],[0.20,0.32]) [0.67,0.80],[0.00,0.14]) [0.36,0.44],[0.31,0.48]) [0.45,0.57],[0.16,0.35]) [0.47,0.59],[0.30,0.41]) [0.25,0.35],[0.20,0.39]) Step 6 Calculate the group decisio score matrix ad the etropy matrix, respectively. Based o the Defiitios 3 ad 4, the score matrix S G ad etropy matrix E G come i the form: S G E G Step 7 Costruct the programmig model of attribute weights based o relative etropy. The iformatio about attribute weights give by the decisio makers ca be described as follows: Ω D1 :0.1 w 1 0.3,w 2 w Ω D2 :0.1 w 2 0.4,0.2 w Ω D3 : w 3 w 4,w 4 w 2 w 3 w 1 The we establish a liear programmig problem: miw i 5.29w w w w 4 { w1 + w s.t. 2 + w 3 + w 4 1 0,Ω Dk,k 1,2,3 By solvig this problem, we ca get the optimal weight vector is W 1) 0.10,0.25,0.25,0.40). Similarly, we get W 2) 0.23,0.37,0.20,0.20),W 3) 0.20,0.40,0.20,0.20). Hece, the optimal weight iformatio matrix becomes: W Step 8 Solve the optimal attribute weights vector. Based o Eq. 28), we derive the value of H i i 1,2,3,4) as follows: H ,H ,H 3 Co-published by Atlatis Press ad Taylor & Fracis 147

16 J. Qi et al. / Multi-attribute group decisio makig 0.399,H Utilize Eq. 29) to calculate the weights of attributes: w 1 1 H k1 w 2 1 H k1 w 3 1 H k1 w H k1 H k H k H k H k , , , Step 9 Calculate the iteractio coefficiet λ. Based o Eq. 5), we calculate the iteractio coefficiet to be λ Step 10 Establish the importace measure of attribute weights. Based o Eqs ), we obtai the fuzzy measures directly. The results are show as follows: μc 1 )[0.0806,0.2800],[0.4318,0.6773]), μc 2 )[0.1232,0.2261],[0.4463,0.7014]), μc 3 )[0.0876,0.2696],[0.4367,0.6831]), μc 4 )[0.0935,0.2619],[0.4394,0.6869]), μc 1,C 2 )[0.3264,0.6813],[0.0927,0.3090]), μc 1,C 3 )[0.2718,0.6973],[0.0861,0.2965]), μc 1,C 4 )[0.2811,0.6940],[0.0877,0.2993]), μc 2,C 3 )[0.3387,0.6761],[0.0948,0.3131]), μc 2,C 4 )[0.3489,0.6726],[0.0961,0.3157]), μc 3,C 4 )[0.2927,0.6889],[0.0900,0.3036]), μc 1,C 2,C 3 )[0.6192,0.9081],[0.0127,0.0907]), μc 1,C 2,C 4 )[0.6322,0.9086],[0.0126,0.0901]), μc 1,C 3,C 4 )[0.7098,0.8143],[0.0714,0.1855]), μc 2,C 3,C 4 )[0.6483,0.9091],[0.0124,0.0896]), μc 1,C 2,C 3,C 4 )[1,1],[0,0]). Step 11 Utilize Choquet itegral to aggregate the iterval-valued ituitioistic fuzzy iformatio of decisio matrix. Take A 1 as a example. From Defiitio 8 ad Eq. 16), we have: μa L μc) μc 2,C 3,C 4 )) μc 2,C 3,C 4 ) μc 3,C 4 )) μc 3,C 4 ) μc 4 )) μc 4 ) μφ)) ) ) ) ) μa R μc) μc 2,C 3,C 4 )) μc 2,C 3,C 4 ) μc 3,C 4 )) μc 3,C 4 ) μc 4 )) μc 4 ) μφ)) ) ) ) ) νa L μc 1,C 2,C 4 ) μc)) μc 2,C 4 ) μc 1,C 2,C 4 )) + 0 μc 4 ) μc 2,C 4 )) + 0 μφ) μc 4 )) ) ) ) ) νa R μc 2,C 3,C 4 ) μc)) μc 2,C 4 ) μc 2,C 3,C 4 )) μc 4 ) μc 2,C 4 )) μφ) μc 4 )) ) ) ) ) So the aggregatio value of A 1 based o itervalvalued ituitioistic fuzzy Choquet itegral is: c)a 1 [0.4771,0.6843],[0.0210,0.2334]) Co-published by Atlatis Press ad Taylor & Fracis 148

17 J. Qi et al. / Multi-attribute group decisio makig Similarly, we ca calculate the aggregatio values of A 2 ad A 3, respectively: c)a 2 [0.4738,0.6607],[0.2652,0.3055]) c)a 3 [0.4631,0.6524],[0.2266,0.3287]) Step 12 Utilize the score fuctio to get the score value of the overall alteratives. Based o Eq. 2), we ca calculate the score value of overall alteratives, respectively: SA 1 )0.4518,SA 2 )0.2819,SA 3 ) Step 13 Utilize the score value to rak the alteratives. Sice SA 1 ) > SA 2 ) > SA 3 ), we have A 1 A 2 A 3. Thus, we choose A 1 as the best alterative Comparisos ad further discussio I what follows, we compare our method with other existig methods icludig iterval-valued ituitioistic fuzzy group decisio makig with Choquet itegral-based TOPSIS 42, ad the itervalvalued ituitioistic fuzzy Choquet itegral with respect to the geeralized lambda-shapley idex method 50. The results are sho Table 1. Table 1. Comparasios with Choquet itegral- TOPSIS ad geeralized lambda-shapley idex methods. Methods Order of alteratives The proposed method A 1 A 2 A 3 Choquet itegral-based TOPSIS A 1 A 2 A 3 lambda-shapley idex method A 1 A 2 A 3 From Table 1, it is clear to see that three methods have the same rakig results. This verifies the method we proposed is reasoable ad validity i this study. 1) Compared with the Choquet itegral-based TOPSIS which was proposed by Ta 42. The mai advatage of our method is that we develop a ew method to determie the fuzzy measures directly based o decisio iformatio, while i Ta s method, the fuzzy measures eed provided by the decisio maker which maybe difficult to the DMs ad is closely related the subjective prefereces of the DMs, or eve lead to icosistet results. I additio, our method makes full use of weights iformatio to characterize the iterrelatioships amog the attributes, which esures the rakig results are more objective ad accurate. 2) Compared with the geeralized lambda-shapley idex method which was proposed by Meg et al. 50. The computatioal complexity of our method is much lower tha Megs method. The reaso is that our method is a costructive method, which oly eeds to determie a sigle parameter by solvig a simple liear programmig model, ad the the fuzzy measures ca be easily obtaied, whereas Meg s method eeds to solve complex programmig model, which leads to high computatioal complexity ad iformatio loss. Moreover, our method ca solve the MAGDM with icomplete koformatio, while geeralized lambda-shapley idex method does ot deal with this problem. Therefore, our method is more geeral. Accordig to the comparisos ad aalysis above, the method we proposed is better tha the other two methods. Therefore, our method is more suitable to solve iterval-valued ituitioistic fuzzy multiple attribute group decisio makig. 7. Coclusios I this study, we have preseted a ew approach to determie the fuzzy measure of Choquet itegral o the basis of iterval-valued ituitioistic fuzzy decisio matrix. Cosiderig the iterval-valued ituitioistic fuzzy etropy theory, we give a simple solutio method to determie the iterval-valued ituitioistic fuzzy measures ad preset a method to obtai the experts weights ad attributes weights with fuzzy measures directly. The strict mathematical reasoig shows that our ew proposed method satisfies the axioms of iterval-valued ituitioistic fuzzy measures. The, the experts weights ad attributes weights are aggregated by usig the Shapely values. The promiet advatage of such fuzzy Co-published by Atlatis Press ad Taylor & Fracis 149

18 J. Qi et al. / Multi-attribute group decisio makig measure method is that it ca ease the burde of the decisio maker, avoid the iformatio losig ad distortio ad improve the accuracy of the decisio makig results. Based o our theoretical aalysis results, we develop a ew Choquet itegral-based approach to solve the iterval-valued ituitioistic fuzzy multiple attribute decisio makig problems. The fuzzy measures to determie the attribute weights ad experts weights are geerated by relative etropy models. A solutio process is proposed to solve the itervalcvalued ituitioistic fuzzy multiple attribute group decisio makig problems. Our ew method is simple prerequisite coditios ad ca be easily itegrated ito other multi-attribute group decisio-makig techiques, which are effective expasios o the curret iterval-valued ituitioistic fuzzy decisio makig models ad solutio methods. Furthermore, our ew methods ca also be applied to the similar problems i the forms of liguistic variables hesitate fuzzy sets ad type-2 fuzzy sets. These will be cosidered i the future research. Ackowledgmets The work is supported by the Natioal Natural Sciece Foudatio of Chia NSFC) uder Projects ad , Ph.D. Program Foudatio of Chiese Miistry of Educatio , the Scietific Research ad Iovatio Project for College Graduates of Jiagsu Provice CXZZ , ad the Scietific Research Foudatio of Graduate School of Southeast Uiversity YBJJ1454. Refereces 1. L.A. Zadeh, Fuzzy sets, Iformatio ad cotrol, ) D.-C. Li, J.-S. Yao, Fuzzy ecoomic productio for productio ivetory, Fuzzy sets ad systems, ) Z. Hu, C. Rao, Y. Zheg, D. Huag, Optimizatio Decisio of Supplier Selectio i Gree Procuremet uder the Mode of Low Carbo Ecoomy, Iteratioal Joural of Computatioal Itelligece Systems, ) X. Huag, Mea-etropy models for fuzzy portfolio selectio, IEEE Trasactios o Fuzzy Systems, ) A. Boetti, S. Bortot, M. Fedrizzi, R.M. Pereira, A. Moliari, Modellig group processes ad effort estimatio i project maagemet usig the Choquet itegral: A MCDM approach, Expert Systems with Applicatios, ) I. Saad, S. Hammadi, M. Berejeb, P. Bore, Choquet itegral for criteria aggregatio i the flexible jobshop schedulig problems, Mathematics ad Computers i Simulatio, ) D. Wu, G. Zhag, J. Lu, A fuzzy preferece treebased recommeder system for persoalized busiessto-busiess e-services, IEEE Trasactios o Fuzzy Systems, ) S. Ha, J.M. Medel, A ew method for maagig the ucertaities i evaluatig multi-perso multi-criteria locatio choices, usig a perceptual computer, Aals of Operatios Research, ) W. Pedrycz, From fuzzy data aalysis ad fuzzy regressio to graular fuzzy data aalysis, Fuzzy Sets ad Systems, ) S. Srivastava, M. Sigh, V.K. Madasu, M. Hamadlu, Choquet fuzzy itegral based modelig of oliear system, Applied Soft Computig, ) S. Dhompogsa, A. Kaewkhao, S. Saejug, O topological properties of the Choquet weak covergece of capacity fuctioals of radom sets, Iformatio Scieces, ) S. Wag, W. Pedrycz, Q. Zhu, W. Zhu, Subspace learig for usupervised feature selectio via matrix factorizatio, Patter Recogitio, ) R.E. Bellma, L.A. Zadeh, Decisio-makig i a fuzzy eviromet, Maagemet sciece, ) J. Qi, X. Liu, Multi-attribute group decisio makig usig combied rakig value uder iterval type-2 fuzzy eviromet, Iformatio Scieces, ) W. Pedrycz, P. Ekel, R. Parreiras, Fuzzy multicriteria decisio-makig: models, methods ad applicatios, Joh Wiley & Sos, W. Pedrycz, Allocatio of iformatio graularity i optimizatio ad decisio-makig models: towards buildig the foudatios of graular computig, Europea Joural of Operatioal Research, ) K.T. Ataassov, Ituitioistic fuzzy sets, Fuzzy sets ad Systems, ) K. Ataassov, G. Gargov, Iterval valued ituitioistic fuzzy sets, Fuzzy sets ad systems, ) Z. Xu, R.R. Yager, Some geometric aggregatio operators based o ituitioistic fuzzy sets, Iteratioal joural of geeral systems, ) Z. Xu, Ituitioistic fuzzy aggregatio operators, IEEE Trasactios o Fuzzy Systems, ) Co-published by Atlatis Press ad Taylor & Fracis 150