Some q-rung orthopair linguistic Heronian mean operators with their application to multi-attribute group decision making

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1 10.445/acs Archves of Cotrol Sceces Volume 8LXIV) 018 No. 4 pages Some q-rug orthopar lgustc Heroa mea operators wth ther applcato to mult-attrbute group decso makg LI LI RUNTONG ZHANG JUN WANG ad XIAOPU SHANG The recetly proposed q-rug orthopar fuzzy set q-rofs) characterzed by a membershp degree ad a o-membershp degree s powerful tool for hadlg ucertaty ad vagueess. Ths paper proposes the cocept of q-rug orthopar lgustc set q-rols) by combg the lgustc term sets wth q-rofss. Thereafter we vestgate mult-attrbute group decso makg MAGDM) wth q-rug orthopar lgustc formato. To aggregate q-rug orthopar lgustc umbers q-rolns) we exted the Heroa mea HM) to q-rolss ad propose a famly of q-rug orthopar lgustc Heroa mea operators such as the q-rug orthopar lgustc Heroa mea q-rolhm) operator the q-rug orthopar lgustc weghted Heroa mea q-rolwhm) operator the q-rug orthopar lgustc geometrc Heroa mea q-rolghm) operator ad the q-rug orthopar lgustc weghted geometrc Heroa mea q-rolwghm) operator. Some desrable propertes ad specal cases of the proposed operators are dscussed. Further we develop a ovel approach to MAGDM wth q-rug orthopar lgustc cotext based o the proposed operators. A umercal stace s provded to demostrate the effectveess ad superortes of the proposed method. Key words: q-rug orthopar fuzzy set q-rug orthopar lgustc set Heroa mea q-rug orthopar lgustc Heroa mea mult-attrbute group decso makg 1. Itroducto Mult-attrbute decso makg s a actvty that ams to select the best alteratve from a set of caddates wth respect to a set of attrbutes. Due to the crease of complexty decso makg we have to face the dffcultes of represetg the attrbute values dfferet complcated ad fuzzy evromets. Zadeh s fuzzy set FS) theory [1] s a powerful tool to descrbe ad depct fuzzess ad ucertaty. Thereafter Ataassov [] pro- The authors are wth School of Ecoomcs ad Maagemet Beg Jaotog Uversty Beg Cha. The correspodg author s Rutog Zhag e-mal: rtzhag@btu.edu.c. Ths work was partally supported by Natoal Natural Scece Foudato of Cha Grat umber ) ad a key proect of Beg Socal Scece Foudato Research Base Grat umber 18JDGLA017). Rceved

2 55 L. LI R. ZHANG J. WANG X. SHANG posed the cocept of tutostc fuzzy set IFS) whch has a membershp degree ad a o-membershp degree. IFS ca be vewed as a exteso of the classc FS theory ad t ca cope wth ucertaty more comprehesvely. Sce the troducto of IFS t has draw much scholars atteto ad qute a few works have bee reported. For stace Xu [3] proposed a famly of tutostc fuzzy ordered weghted average operators by extedg the ordered weghted average operator to IFSs. Jag et al. [4] developed the etropy-based tutostc fuzzy power operator ad appled t to MAGDM. Re et al. [5] proposed a thermodyamc method for multple crtera decso makg wth tutostc fuzzy umbers IFNs). Zhag [6] proposed a famly of tutostc fuzzy Este hybrd weghted operators based o Este operatos for IFNs. I addto IFSs have bee wdely appled to medcal decso makg [7 8] cluster aalyss [9 10] ad patter recogto [11 1]. From above aalyss we ca fd that IFSs are a effectve tool decso makg. However there are qute a few stuatos that IFSs caot effectvely deal wth. For stace f the membershp degree ad the o-membershp degree provded by a decso maker are 0.6 ad 0.7 respectvely the t s ot vald for IFNs. I other words the ordered par ) caot be deoted by a IFN. I order to address these kds of crcumstaces Yager [13] put forward the cocept of Pythagorea fuzzy set PFS) whch also has a membershp degree ad a o-membershp degree. The promet feature of PFS s that the sum of ts membershp ad o-membershp degrees s allowed to be greater tha oe wth ther square sum s less tha or equal to oe. Therefore PFS s more geeral tha IFS ad all tutostc membershp degrees are part of Pythagorea fuzzy membershp degrees. Sce ts appearace t has receved more ad more atteto. Garg [14 15] ad Rahma et al. [16] proposed a famly of Pythagorea fuzzy Este operators based o Este t-orm ad t-coorm respectvely. To cosder the relatoshp betwee Pythagorea fuzzy umbers PFNs) We ad Lu [17] proposed some Pythagorea fuzzy power aggregato operators by extedg Yager s power average operator [18] to PFSs. To process the teractos betwee membershp ad o-membershp degrees of PFSs We [19] developed a seres of Pythagorea fuzzy teracto operators. To capture the terrelatoshp betwee aggregated PFNs Lag et al. [0] ad Zhag et al. [1] proposed some Pythagorea fuzzy Boferro mea operators. We ad Lu [] developed a seres of Pythagorea fuzzy Maclaur symmetrc mea operators. Cosderg there are stuatos whch decso makers may be hestat whe determg the membershp degrees betwee a set of possble values Lag et al. [3] Kha et al. [4] ad Lu et al. [5] proposed the cocept of hestat Pythagorea fuzzy set HPFS) respectvely. Cocretely Lag et al. [3] employed the techque for order preferece by smlarty to deal soluto TOPSIS) method to MAGDM based o HPFSs. Kha

3 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 553 et al. [4] proposed a famly of hestat Pythagorea fuzzy aggregato operators based o algebrac t-orm ad t-coorm. Lu et al. [5] proposed some ovel hestat Pythagorea fuzzy aggregato operators based o Hamacher t-orm ad t-coorm. Recetly We ad Lu [6] proposed the cocept of dual hestat Pythagorea fuzzy sets as well as ther aggregato operators based o Hamacher t-orm ad t-coorm. More recetly Yager [7] troduced a ew exteso of FS called q-rofs. The promet feature of the q-rofs s that the sum ad square sum of membershp ad o-membershp degrees are allowed to be greater tha oe wth ther sum of qth power of the membershp degree ad the qth power of the degree of o-membershp beg equal to or less tha 1. Thus q-rofs s more geeral ad powerful tha IFS ad PFS. To effectvely aggregate q-rug orthopar fuzzy formato Lu ad Wag [8] proposed a famly of q-rug orthopar fuzzy weghted aggregato operators. To capture the terrelatoshp betwee q-rug orthopar fuzzy umbers q-rofns) Lu PD ad Lu JL [9] put forward a famly of q-rug orthopar fuzzy Boferro mea operators. There are some stuatos whch decso makers prefer to make qualtatve decsos stead of quattatve decsos due to a lack of tme ad expertse. Zadeh s [30] lgustc varables are effectve tools to make qualtatve decsos. However the tradtoal lgustc varables ca oly reflect decso makers qualtatve prefereces ad the membershp ad o-membershp degrees of a elemet to a partcular cocept are gored. Therefore motvated by the cocept of IFS Wag ad L [31] proposed the cocept of tutostc lgustc set ILS) by combg lgustc term set wth IFS. Du et al. [3] proposed the terval-valued Pythagorea fuzzy lgustc set as well as some aggregato operators. I ths paper we frst gve the defto of q-rols as well as ther operatos. To effectvely aggregate q-rug orthopar lgustc formato we vestgate HM uder q-rolss ad propose some q-rug orthopar lgustc Heroa mea operators. Moreover we apply the proposed operators to solve MAGDM. I order to do ths the remader of the paper s orgazed as follows. Sectos brefly recalls some otos. Secto 3 develops some q-rug orthopar lgustc Heroa meas. I addto we also vestgate some propertes of the proposed operators. Secto 4 presets a ovel approach to q-rug orthopar lgustc MAGDM ad a umercal expermet s coducted Secto 5. Coclusos are gve Secto 6.. Prelmares I ths secto we brefly revew cocepts about q-rofs lgustc term set ad HM.

4 554 L. LI R. ZHANG J. WANG X. SHANG.1. The q-rug orthopar fuzzy set Defto 1 [7] Let X be a ordary fxed set a q-rofs A defed o X s gve by A={xu A x)v A x) x X} 1) where u A x) ad v A x) represet the membershp degree ad o-membershp degree respectvely satsfyg u A x) [01] v A x) [01] ad 0 u A x) q + v A x) q 1 q 1). The determacy degree of A s defed as π A x) = ua x) q + v A x) q u A x) q v A x) q) 1/q. For coveece ua x)v A x)) s called a q-rofn by Lu ad Wag [8] whch ca be deoted as A=u A v A ). Lu ad Wag [8] also proposed some operatos for q-rofns. Defto [8] Let ã 1 = u 1 v 1 ) ad ã = u v ) be two q-rofns λ be a postve real umber the u q 1) ã 1 ã = 1 + uq ) ) 1/qv1 uq 1 uq v ; ) ã 1 ã = u 1 u v q 1 + vq ) ) 1/q vq 1 vq ; 3) λã 1 = u q ) ) ) λ 1/qv λ 1 1 ; 4) ã1 λ = u1 λ v q ) ) ) λ 1/q 1. To compare two q-rofns Lu ad Wag [8] proposed a comparso method for q-rofns. Defto 3 [8] Let ã = u a v a ) be a q-rofn the the score fucto of ã s defed as Sã) = µ q a ν q a the accuracy fucto of ã s defed as Hã) = µ q a + ν q a. For ay two q-rofns ã 1 =u 1 v 1 ) ad ã =u v ). The 1) If Sã 1 )>Sã ) the ã 1 > ã ; ) If Sã 1 )>Sã ) the If Hã 1 )>Hã ) the ã 1 > ã ; If Hã 1 )=Hã ) the ã 1 = ã... Lgustc term set ad q-rug orthopar lgustc set Let S = {s...t} be a lgustc term set wth odd cardalty ad t s the cardalty of S. The label s represets a possble value for a lgustc varable. For stace a possble lgustc term set ca be defed as follows: S=s 1 s s 3 s 4 s 5 s 6 s 7 ) ={very poor poor slghtly poor far slghtly good good very good}.

5 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 555 Motvated by the cocept of tutostc lgustc set ILS) [31] we propose the cocept of q-rofls by combg the lgustc term set wth q-rofs. Defto 4 Let X be a ordary fxed set ad S be a cotuous lgustc term set of S ={s...t} the a q-rug orthopar lgustc set q-rols) A o X ca be gve as follows A= { xs θx) u A x)v A x)) x X } ) where s θx) S u A x): X [01] ad v A x): X [01] satsfyg 0 u A x)) q +v A x)) q 1 q 1) the s θx) u A x)v A x)) s called a q- rug orthopar lgustc umber q-roln) whch ca be smply deoted by α =s θ uv). Based o the operatos for q-rofns we provde some operatos for q- ROLNs. Defto 5 Let α 1 = s θ1 u 1 v 1 ) ad α = s θ u v ) be ay two q-rolns ad λ be a postve real umber the u q 1) α 1 α = s θ1 +θ 1 + uq ) ) 1/qv1 uq 1 uq v ; ) α 1 α = s θ1 θ u 1 u v q 1 + vq ) ) 1/q vq 1 vq ; 3) λα 1 = s λ θ1 u q ) ) ) λ 1/qv λ 1 1 ; 4) α1 s λ = θ λ u λ1 v q ) ) ) λ 1/q 1 1. To compare two q-rolns we frstly propose the cocepts of score fucto ad accuracy fucto of a q-roln. The based o the two cocepts we propose a comparso rule for q-rolns. Defto 6 Let α =s θ uv) be a q-roln the score fucto of α s gve by Sα)=u q + v q ) θ. 3) Defto 7 Let α =s θ uv) be a q-roln the the accuracy fucto of α s defed as Hα)=u q + v q ) θ. 4) The we provde a comparso law for q-rolns. Defto 8 Let α 1 = s θ1 u 1 v 1 ) ad α = s θ u v ) be ay two q- ROLNs Sα 1 ) ad Sα ) be the score fuctos of α 1 ad α respectvely Hα 1 ) ad Hα ) be the accuracy fuctos of α 1 ad α respectvely the

6 556 L. LI R. ZHANG J. WANG X. SHANG 1) If Sα 1 )>Sα ) the α 1 > α ; ) If Sα 1 )=Sα ) the If Hα 1 )>Hα ) the α 1 > α ; If Hα 1 )=Hα ) the α 1 = α..3. Heroa mea Defto 9 [33 34] Let a...) be a collecto of crsp umbers ad s t > 0 the the Heroa mea HM) s defed as follows: HM st a 1 a...a )= +1) a s at ) 1/). 5) Defto 10 [35] Let a...) be a collecto of crsp umbers ad s t > 0 the the geometrc Heroa mea GHM) s defed as follows: GHM st a 1 a...a )= 1 sa +ta ) 1 +). 6) 3. The q-rug orthopar lgustc Heroa mea operators I ths secto we exted HM to q-rolss ad proposed a seres of q-rug orthopar lgustc Heroa mea operators The q-rug orthopar lgustc Heroa mea q-rolhm) operator Defto 11 Let α...) be a collecto of q-rolns ad s t > 0. If q-rolhm st α 1 α...α )= +1) α s α t ) 1/) 7) the q-rolhm st s called the q-rug orthopar lgustc Heroa mea q- ROLHM) operator. Accordg to the operatos for q-rolns the followg theorem ca be obtaed. Theorem 1 Let α...) be a collecto of q-rolns the the aggregated value by usg q-rolhm s also a q-roln ad

7 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 557 q-rolhm st α 1 α...α )= =1 =1 v q s +1) ) u sq u tq +1) ) s q) ) q t +1) v ) 1/) θ sθ t ) 1/q) ) 1/) 1/q. 8) Proof. Accordg to the operatos for q-rolns we ca obta the followgs α s = s θ s u s v q ) ) ) s 1/q α t s = θ t u t v q ) ) ) t 1/q. Therefore Further α s α t = α s α t = s θ s θ t u s ut v q I addto s θ sθ t α s α t = ) ) 1/q u sq u tq s θ sθ t ) )) 1/q u sq u tq ) s v q) )) t. v q v q ) s q) ) t v ) s q) ) t v..

8 558 L. LI R. ZHANG J. WANG X. SHANG Thus +1) So +1) α s α t = s +1) θ sθ t ) ) u sq u tq +1) v q q-rolhm st α 1 α...α )= +1) = s ) 1/) θ sθ t v q 1/q ) s q) ) ) +1) t v α s α t ) 1/) ) ) u sq u tq +1) ) s q) ) ) q 1/) 1/q +1) t v. 1/q). I addto the q-rolhm operator has the followg propertes. Theorem Mootocty) Let α ad β...) be two collectos of q-rolns f α β for all... the q-rolhm st α 1 α...α ) q-rolhm st β 1 β...β ). 9) Proof. As α = α for all we ca obta Sce α β ad α β for = 1... ad = we have α sαt β sβ t. The +1) α s α t +1) β s β t.

9 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 559.e. So +1) 1/) α s α ) t +1) β s β t ) 1/) q-rolhm st α 1 α...α ) q-rolhm st β 1 β...β ). Theorem 3 Idempotecy) Let α = 1...) be a collecto of q-rolns f α = α for all... the q-rolhm st α 1 α...α )=α. 10) Proof. Sce α = α for all we have q-rolhm st α 1 α...α )= +1) = α ) 1/) = α. α s α t ) 1/) Theorem 4 Boudedess) The q-rolhm operator les betwee the max ad m operators mα 1 α...α ) q-rolhm st α 1 α...α ) maxα 1 α...α ). 11) Proof. Let a=mα 1 α...α ) b=maxα 1 α...α ) accordg to Theorem we have.e. Further So q-rolhm st aa...a) q-rolhm st α 1 α...α ) q-rolhm st bb...b). q-rolhm st aa...a)=a ad q-rolhm st bb...b)=b. a q-rolhm st α 1 α...α ) b mα 1 α...α ) q-rolhm st α 1 α...α ) maxα 1 α...α ). The parameters s ad t play a very mportat role the aggregated results. I the followgs we dscuss some specal cases of the q-rolhm operator wth respect to the parameters s ad t.

10 560 L. LI R. ZHANG J. WANG X. SHANG Case 1 Whe t 0 the the q-rolhm operator reduces to the followgs q-rolhm s0 α 1 α...α )=lm t 0 = v q s ) 1/s +1) +)θ s s +1) ) ) u sq u tq +1) ) 1/) θ sθ t 1/q) ) s q) ) ) q 1/) 1/q +1) t v ) +1) sq) + u ) q 1/s 1/q +1) q) s ) + v 1/qs 1) whch s a q-rug orthopar lgustc geeralzed lear descedg weghted mea operator. Evdetly t s equvalet to weght the formato α s 1 αs...αs ) wth 1...1). Case Whe s 0 the the q-rolhm operator reduces to the followgs q-rolhm 0t α 1 α...α )= lm s 0 v q s +1) ) ) u sq u tq +1) ) 1/) θ sθ t 1/q) ) s q) ) ) q 1/) 1/q +1) t v

11 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 561 = s ) 1/t +1) θ t ) +1) tq) u 1/s ) q 1/t 1/q v q ) ) t +1) 13) whch s a q-rug orthopar lgustc geeralzed lear ascedg weghted mea operator. Obvously t s equvalet to weght the formato α t 1 αt...αt ) wth 1...).e. whe t = 0 or s=0 the q-rolhm operator has the lear weghted fucto for put data. Case 3 Whe s=t = 1 the the q-rolhm operator reduces to the followgs q-rolhm 11 α 1 α...α )= s v q +1) θ θ ) 1) u u ) q)) +1) 1/q ) v q) )) q 1/ 1/q +1) 14) whch s a q-rug orthopar lgustc le Heroa mea operator. Case 4 Whe s=t= 1/ the the q-rolhm operator reduces to the followgs q-rolhm 1 1 α1 α...α )= s +1) θ θ ) ) u q +1) uq v q 1/q ) v q) )) +1) whch s a q-rug orthopar lgustc basc Heroa mea operator. 15)

12 56 L. LI R. ZHANG J. WANG X. SHANG Case 5 Whe q= the the q-rolhm operator reduces to the followgs q-rolhm st α 1 α...α )= s u s +1) =1 u t ) 1 θ sθ t ) ) +1) 1/) v ) s ) v t ) ) 1/) 1/ 4 +1) 16) whch s the Pythagorea lgustc Heroa mea operator. Case 6 Whe q=1 the the q-rolhm operator reduces to the followgs q-rolhm st α 1 α...α )= =1 s =1 +1) =1 ) 1 θ sθ t u s u t ) ) +1) 1 ) s ) ) ) +1) t v v 1 17) whch s the tutostc lgustc Heroa mea operator. 3.. The q-rug orthopar lgustc weghted Heroa mea q-rolwhm) operator It s oted that the proposed q-rolhm operator does ot cosder the weghts of the aggregated argumets. Therefore we put forward the weghted Heroa mea for q-rolns.

13 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 563 Defto 1 Let α = 1...)be a collecto of q-rolns ad s t > 0 w=w 1 w...w ) T be the weght vector satsfyg w [01] ad w = 1. If q-rolwhm st α 1 α...α )= +1) the q-rolwhm st α 1 α...α ) s called the q-rolwhm. w α ) s w α ) t ) 1/) 18) Accordg to the operatos for q-rolns the followg theorem ca be obtaed. Theorem 5 Let α = 1...) be a collecto of q-rolns w = w 1 w...w ) T be the weght vector satsfyg w [01] ad the aggregated value by usg q-rolwhm s also a q-roln ad q-rolwhm st α 1 α...α )= s u q ) w ) s +1) v w q +1) ) s v w q w θ ) s w θ ) t ) 1/) u q ) ) t w +1) ) ) ) 1/) t +1) w = 1 the ) ) 1/)q 1/q. 19) The proof of Theorem 5 s smlar to that of Theorem 1 whch s omtted here. To llustrate the performace of q-rolwhm operator we provde a example the followgs. Example 1 Let α 1 = s ) α = s ) α 3 = s ) ad α 3 = s ) be four q-rolns wth the weght vector s w = ) T. Let α = s θ uv) be the comprehesve value ad f we utlze q-rolwhm to aggregate the four q-rolns we ca obta suppose q=3 s= t = 3): θ = ) ) ) ) ) ) ) ) ) ) ) ) ) 3 = /+3)

14 564 L. LI R. ZHANG J. WANG X. SHANG Smlarly we ca obta u= ad v= It s oted that the parameters s ad t play a sgfcat role the score of the comprehesve value. Detals ca be foud Fg. 1. Fgure 1: Score values of the alteratve whe s t [16] ad q=3 usg q-rolwhm operator Smlarly q-rolwhm has the followg propertes. Theorem 6 Mootocty) Let α ad β...) be two collectos of q-rolns f α β for all the q-rolwhm st α 1 α...α ) q-rolwhm st β 1 β...β ). 0) Theorem 7 Boudedess) The q-rolwhm operator les betwee the max ad m operators mα 1 α...α ) q-rolwhm st α 1 α...α ) maxα 1 α...α ). 1) Evdetly the q-rolwhm operator does ot has the property of dempotecy The q-rug orthopar lgustc geometrc Heroa mea q-rolghm) operator Defto 13 Let α...) be a collecto of q-rolns ad s t > 0. If q-rolghm st α 1 α...α )= 1 sα +tα ) +1) ) the q-rolghm st s called the q-rug orthopar lgustc geometrc Heroa mea q-rolghm) operator.

15 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 565 Smlarly the followg theorem ca be obtaed accordg to Defto 5. Theorem 8 Let α...) be a collecto of q-rolns the the aggregated value by usg q-rolghm s also a q-roln ad q-rolghm st α 1 α...α )= s 1 u q sθ +tθ ) +1) ) s q) ) t +1) u ) 1 ) 1 ) v sq v tq )q +1) 1/q. 3) The proof of Theorem 8 s smlar to that of Theorem 1. I the followg we preset some desrable propertes of the q-rolghm operator. Theorem 9 Idempotecy) Let α =s θ u v )...) be a collecto of q-rolns f all the q-rolns are equal.e. α = α for all the q-rolghm st α 1 α...α )=α. 4) The proof of Theorem 9 s smlar to that of Theorem. Theorem 10 Mootocty) Let α ad β...) be two collectos of q-rolns f α β for all the q-rolghm st α 1 α...α ) q-rolghm st β 1 β...β ). 5) The proof of Theorem 10 s smlar to that of Theorem 3. Theorem 11 Boudedess) Let α =s θ u v )...) be a collecto of q-rolns the mα 1 α...α ) q-rolghm st α 1 α...α ) maxα 1 α...α ). 6) The proof of Theorem 11 s smlar to that of Theorem 4. I the followg we dscuss some specal cases of the q-rolghm operator.

16 566 L. LI R. ZHANG J. WANG X. SHANG Case 1 Whe t 0 the the q-rolghm operator reduces to the followgs q-rolghm s0 α 1 α...α )=lm t 0 = s ) 1 s sθ ) + +1) u q s 1 ) +1) sθ +tθ ) 1 ) s q) ) t +1) u ) 1 ) v sq v tq )q +1) v sq 1/q ) +1) u q) s ) + ) +1) ) + 1 sq 11 q s 7) whch s a q-rug orthopar lgustc geeralzed geometrc lear descedg weghted mea operator. Case Whe s 0 the the q-rolghm operator reduces to the followgs q-rolghm 0t α 1 α...α )= lm s 0 u q s 1 sθ +tθ ) +1) ) 1 ) s q) ) t +1) u ) 1 ) v sq v tq )q +1) 1/q

17 = s SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 567 ) 1 t tθ ) +1) ) u q ) ) t +1) v tq ) +1) ) 1 tq 11/q t 8) whch s a q-rug orthopar lgustc geeralzed geometrc lear ascedg weghted mea operator. Case 3 Whe s= t= 1 the the q-rolghm operator reduces to the followgs q-rolghm 11 α 1 α...α )= s 1 u q ) +1) θ +θ ) q) ) +1) u ) ) v q +1) vq 1 )1 q 1 q 9) whch s a q-rug orthopar lgustc geometrc le Heroa mea operator. Case 4 Whe s=t = 1/ the the q-rolghm operator reduces to the followgs q-rolghm 1 1 α1 α...α )= s u q 1 θ + 1 θ ) ) u q ) ) ) +1) )1 q +1) ) v q vq +1) )1 q 30) whch s a q-rug orthopar lgustc basc geometrc Heroa mea operator.

18 568 L. LI R. ZHANG J. WANG X. SHANG Case 5 Whe q= the the q-rolghm operator reduces to the followgs q-rolghm st α 1 α...α )= s 1 sθ +tθ ) =1 ) +1) ) 1 u ) s ) u t ) +1) v s v t ) 1 ) ) +1) whch s the Pythagorea lgustc geometrc Heroa mea operator. 1/ 31) Case 6 Whe q=1 the the q-rolghm operator reduces to the followgs q-rolghm st α 1 α...α )= =1 s 1 =1 ) +1) sθ +tθ ) u ) s u ) t)) +1) =1 v s v t ) ) +1) whch s the tutostc lgustc geometrc Heroa mea operator The q-rug orthopar lgustc weghted geometrc Heroa mea q-rolwghm) operator 1 1 3) Defto 14 Let α = 1...) be a collecto of q-rolns ad s t > 0 w=w 1 w...w ) T be the weght vector satsfyg w [01] ad q-rolwghm st α 1 α...α )= 1 the q-rolwghm st s called the q-rolwghm. sa w w = 1. If ) +ta w +1) 33)

19 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 569 The followg theorem ca be easly obtaed. Theorem 1 Let α...) be a collecto of q-rolns w=w 1 w...w ) T be the weght vector satsfyg w [01] ad the the aggregated value by usg q-rolwghm s also a q-roln ad q-rolwghm st α 1 α...α )= = s 1 u w q θ w ) s u w q +tθ w ) +1) ) 1 ) ) t +1) ) v q s ) ) ) 1 )q )w v q t +1) )w 1/q w = 1. 34) The proof of Theorem 1 s smlar to that of Theorem 1 whch s omtted here. Example We utlze the values provded Example 1 to demostrate the performace of q-rolwghm operator. Let α =s θ uv) be the comprehesve value ad suppose q = 3 s = t = 3. If we utlze q-rolwghm operator to aggregate the four q-rolns we ca obta α =s ) Sα)= The calculato process s smlar to that of Example 1. Subsequetly we vestgate the fluece of the parameters s ad t o the score fucto of the comprehesve value. Detals ca be foud Fg. I addto the q-rolwghm operator has the followg propertes. Theorem 13 Mootocty) Let α ad β...) be two collectos of q-rolns f α β for all the q-rolwghm st α 1 α...α ) q-rolwghm st β 1 β...β ). 35) Theorem 14 Boudedess) The q-rolwghm operator les betwee the max ad m operators mα 1 α...α ) q-rolwghm st α 1 α...α ) maxα 1 α...α ). 36) Evdetly the q-rolwghm operator does ot has the property of dempotecy.

20 570 L. LI R. ZHANG J. WANG X. SHANG Fgure : Score values of the alteratve whe s t [16] ad q=3 usg q-rolwghm operator 4. A ovel approach to MAGDM based o the proposed operators I ths secto we shall propose a ovel decso makg method wth q-rug orthopar lgustc formato. Cosderg a MAGDM process uder q-rug orthopar lgustc evromet: let X ={x 1 x...x m } be a set of all alteratves ad Y ={y 1 y...y } be a set of attrbutes wth the weght vector beg w=w 1 w...w ) T satsfyg w = 1. Several experts D k are orgazed to make the assessmet for every attrbute y =1...) of all alteratves by q-rolns α k = s k θ u k vk ) ad λ = λ 1 λ...λ k ) s the weght vector of decso makers {D 1 D...D p }. Therefore the q-rug orthopar lgustc decso matrces ca be obtaed by A k =α k ) m. The ma steps to solve the MAGDM problems based o the proposed operators are gve as follows. Step 1 Stadardze the orgal decso matrces. There are two types of attrbutes beeft ad cost attrbutes. Therefore the orgal decso matrx should be ormalzed by α k = ) s k θ u k vk ) s k θ v k uk y I 1 y I 37) where I 1 ad I represet the beeft attrbutes ad cost attrbutes respectvely.

21 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 571 Step Utlze the q-rolwhm operator α = q-rolwhm st ) α 1 α...α p 38) or the q-rolwghm operator α = q-rolwghm st ) α 1 α...α p 39) to aggregate all the decso matrces A k k = 1... p) to a collectve decso matrx A=α ) m. Step 3 Utlze the q-rolwhm operator or the q-rolwghm operator α = q-rolwhm st α 1 α...α ) 40) α = q-rolwghm st α 1 α...α ) 41) to aggregate the assessmets α = 1...) for each A so that the overall preferece values α...m) of alteratves ca be obtaed. Step 4 Calculate the score fuctos of the overall values α...m). Step 5 Rak all alteratves accordg to the score fuctos of the correspodg overall values ad select the best oes). Step 6 Ed. 5. Numercal example I ths secto to verfy the proposed method we provde a umercal stace adopted from [36]. A vestmet compay wats vest ts moey to a compay. After prmary evaluato there are four possble compaes remaed o the caddates lst ad they are: 1) A 1 s a car compay; ) A s a computer compay; 3) A 3 s a TV compay; 4) A 4 s a food compay. Three experts are vted to evaluate the four caddates uder four attrbutes they are 1) C 1 s the rsk aalyss; ) C s the growth aalyss; 3) C 3 s the socal-poltcal mpact aalyss; 4) C 4 s the evrometal mpact aalyss. Weght vector of the four attrbute s w= ) T. The decso makers are requred to use the lgustc term set S={s 0 = extremely poor s 1 = very poor s = poor s 3 = far s 4 = good s 5 = very good s 6 = extremely good}to express ther preferece formato. Decso makers weght vector s λ = ) T.) After evaluato the dvdual tutostc lgustc decso matrx A k = α k 4 4 k=13) ca be obtaed whch are show Tables 1 ad 3.

22 57 L. LI R. ZHANG J. WANG X. SHANG Table 1: Itutostc lgustc decso matrx R 1 C 1 C C 3 C 4 A 1 s ) s ) s ) s ) A s ) s ) s ) s ) A 3 s ) s ) s ) s ) A 4 s ) s 0.0.8) s ) s ) Table : Itutostc lgustc decso matrx R C 1 C C 3 C 4 A 1 s ) s ) s ) s ) A s ) s ) s ) s ) A 3 s ) s ) s ) s ) A 4 s ) s ) s ) s ) Table 3: Itutostc lgustc decso matrx R 3 C 1 C C 3 C 4 A 1 s ) s ) s ) s ) A s ) s ) s ) s ) A 3 s ) s ) s ) s ) A 4 s ) s ) s ) s ) 5.1. The decso makg process Step 1 As the fve attrbutes are beeft types the orgal decso matrces do ot eed ormalzato. Step Utlze Eq. 38) to calculate the comprehesve value α of each attrbute for every alteratve. The collectve decso matrx A=α ) 4 4 s show Table 4 suppose s=t = 1 q=3). Table 4: Collectve tutostc lgustc decso matrx by q-rolwhm operator) C 1 C C 3 C 4 A 1 s ) s ) s ) s ) A s ) s ) s ) s ) A 3 s ) s ) s ) s ) A 4 s ) s ) s ) s )

23 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 573 Step 3 Utlze Eq. 40) to obta the overall values of each alteratve we ca get α 1 =s ) α =s ) α 3 =s ) α 4 =s ). Step 4 Compute the score fuctos of the overall values whch are show as follows: Sα 1 )=3.33 Sα )= Sα 3 )=.616 Sα 4 )= Step 5 The the rak of the four alteratves s obtaed A 1 A A 4 A 3. Therefore the optmal alteratve s A 1. I step f we utlze Eq. 39) to aggregate the assessmets the we ca derve the followg collectve decso matrx Table 5 suppose s = t = 1 q=3). Table 5: Collectve tutostc lgustc decso matrx by q-rolwghm operator) C 1 C C 3 C 4 A 1 s ) s ) s ) s ) A s ) s ) s ) s ) A 3 s ) s ) s ) s ) A 4 s ) s ) s ) s ) The we utlze Eq. 41) to obta the followg overall values of alteratves: α 1 =s ) α =s ) α 3 =s ) α 4 =s ). I addto we calculate the score fuctos of the overall assessmets ad we ca get Sα 1 )= Sα )= Sα 3 )=.5551 Sα 4 )= Therefore the rak of the four alteratves s A 1 A A 4 A 3 ad the best alteratve s A 1.

24 574 L. LI R. ZHANG J. WANG X. SHANG 5.. The fluece of the parameters o the rakg results The parameters q s ad t pay a sgfcat role the fal rakg results. I the followg we shall vestgate the fluece of the parameters o the overall assessmets of alteratves ad the fal rakg results. Frst we dscuss the effects of the parameters q o the rakg results suppose s=t = 1). Detals ca be foud Fgs 3 ad 4. Fgure 3: Score values of the alteratves whe q [110] s= t = 1 usg q-rolwhm operator Fgure 4: Score values of the alteratves whe q [110] s= t = 1 usg q-rolwghm operator

25 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 575 The promet feature of the q-rofs s that the sum of membershp ad o-membershp degrees s allowed to be greater tha oe wth ther q-th power of the membershp degree ad the q-th power of the degree of o-membershp beg equal to or less tha 1. Ths feature makes q-rofs more geeralzed ad powerful tha IFS ad PFS. I addto q-rofs ca descrbe ad depct wder formato rage ad cota more formato tha IFS ad PFS. The proposed q-rols herts the advatages of q-rofs. I other word q-rols ca obta more formato tha ILS ad Pythagorea lgustc set PLS). For stace the argumet s ) s ot vald for tutostc lgustc umbers or Pythagorea lgustc umbers whereas s t vald for q-rolns. Moreover as see Fgs 3 ad 4 the score values of the overall assessmets crease wth the crease of the values of q ad subsequetly result dfferet rakg results. I addto the value of q ca be vewed as decso makers atttude to optmsm ad pessmsm. The more optmstc decso makers are the greater value should be assged to q whereas the more pessmstc the less value should be assged to q. I the followgs we vestgate fluece of the parameters s ad t o the score fuctos ad rakg orders respectvely suppose q = 3). Detals ca be foud Fgs 5 8. Fgure 5: Score values of the alteratves whe s [19] t = 1 q=3 usg q-rolwhm operator As see Fgs 5 8 dfferet values are assged to the parameters s ad t resultg dfferet score values ad varyg the rakg results. Especally q-rolwhm operator the crease of the parameters s ad t leads to crease of

26 576 L. LI R. ZHANG J. WANG X. SHANG Fgure 6: Score values of the alteratves whe s [19] t = 1 q=3 usg q-rolwghm operator Fgure 7: Score values of the alteratves whe t [1 9] s = 1 q = 3 usg q-rolwhm operator the score fuctos whereas decrease of the score fuctos usg q-rolwghm operator. Therefore the parameters s ad t ca be also vewed a decso makers optmstc or pessmstc atttude to ther assessmets. Ths demostrates the flexblty the aggregato processes usg the proposed operators.

27 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 577 Fgure 8: Score values of the alteratves whe t =[1 9] s = 1 q = 3 usg q-rolwghm operator 5.3. Comparatve aalyss I ths secto we coduct some comparsos from a quattatve perspectve. We utlze some extg methods to solve the same example ad compare ther fal rakg results. We compare our method wth the method proposed by Ju et al. [37] based o weghted tutostc lgustc Maclaur symmetrc mea WILMSM) operator the method proposed by Wag et al. [38] based o tutostc lgustc hybrd ILH) operator the method proposed by Lu et al. [39] based o the tutostc fuzzy lgustc umbers hybrd geometrc IFLNHG) operator the method proposed by Lu et al. [40] based o the tutostc lgustc weghted Boferro mea ILWBM) operator ad the method proposed by Zhag et al. [41] based o the tutostc lgustc geeralzed weghted Heroa mea ILGWHM) operator. The score fuctos ad rakg results are show Table 6. Frst of all all the methods except our method are based o ILSs. As we metoed before ILS s oly a specal case of q-rols whe q=1). Recetly Yager [7] proposed the cocept of PFS ad f we combe PFS wth lgustc varables we ca obta PLS whch s also a specal case of q-rols whe q=). Therefore our method s more geeralzed tha the other methods. Ju et al. s [37] method s based o WILMSM operator ad whe k= the terrelatoshp betwee ay two argumets ca be cosdered whch s the same as our proposed method. However our method s based o the q-rolwhm or

28 578 L. LI R. ZHANG J. WANG X. SHANG Table 6: Score fuctos ad rakg results usg dfferet methods Method Score fuctos Rakg result Ju et al. s [37] method based o WILMSM operator k=) Wag et al. s [38] method based o ILH operator Lu et al. s [39] method based o IFLNHG operator Lu et al. s [40] method based o the ILWBM operator Zhag et al. s [41] method based o the ILGWHM operator The proposed method based o q- ROLWHM operator ths paper The proposed method based o q- ROLWGHM operator ths paper Sα 1 )= Sα )= Sα 3 )= Sα 4 )=0.189 Sα 1 )= Sα )=0.186 Sα 3 )=0.149 Sα 4 )= Sα 1 )= Sα )= Sα 3 )= Sα 4 )= Sα 1 )=.4355 Sα )=.5011 Sα 3 )= Sα 4 )=.3713 Sα 1 )=.5760 Sα )=.6877 Sα 3 )=.0378 Sα 4 )=.5539 Sα 1 )=3.33 Sα )= Sα 3 )=.616 Sα 4 )= Sα 1 )= Sα )= Sα 3 )=.5551 Sα 4 )=.9538 A A 4 A 1 A 3 A A 4 A 1 A 3 A A 4 A 1 A 3 A A 1 A 4 A 3 A A 1 A 4 A 3 A 1 A A 4 A 3 A 1 A A 4 A 3 q-rolwghm) operator whch has two parameters s ad t). The promet of the method s that we ca cotrol the degree of the teractos of attrbute values that are emphaszed. The crease of values of the parameters s ad t) meas the teractos of attrbute values are more emphaszed. Therefore the decso makg commttee ca properly select the desrable alteratve accordg to ther terests ad the actual eeds by determg the values of parameters. Moreover the WILMSM operator proposed by Ju et al. [37] the balacg coeffcet s ot cosdered leadg to some ureasoable results. I our proposed operator the coeffcet s cosdered so that our method s more relable ad reasoable. Wag et al. s [38] ad Lu et al. s [39] methods are based o hybrd averagg operator whch caot cosder the terrelatoshp amog attrbute values. I most real decso makg problems attrbutes are correlated so that the terrelatoshp amog attrbutes should be take to cosderato. Therefore our proposed method s more reasoable tha Wag et al. s [38] ad Lu et al. s [39] methods. Lu et al. s [40] ad Zhag et al. s methods [41] are based o BM ad HM respectvely whch ca cope wth the terrelatoshp betwee augmets. However as Ya ad Wu [4] poted out that HM has some advatages over BM our method s better tha Lu et al. s [40] method. As Zhag et al. s method [41] s based o ILS whch s a specal case of q-rolsq=1) our proposed method s also better tha Zhag et al. s [41] method.

29 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING Coclusos I ths paper we propose the q-rols whch s a powerful ad effectve tool for copg wth ucertaty ad vagueess. Subsequetly we vestgate MAGDM problems q-rug orthopar lgustc evromet. To aggregate q-rolns we exted the HM to q-rolss ad propose a famly of q-rug orthopar lgustc Heroa mea operators such as the q-rolhm operator the q-rolwhm operator the q-rolghm operator ad the q-rolwghm operator. The promet characterstc of these proposed operators s that they ca capture the terrelatoshp betwee q-rolns. Moreover we have studed some desrable propertes ad specal cases of the proposed operators. Thereafter we utlze the proposed operators to establsh a ovel method to MAGDM problems. To llustrate the valdty of the proposed method we utlze the method to solve a vestmet proect selecto problem. I addto we coduct some comparatve aalyss to demostrate the effectveess ad superortes of the proposed method. I the future we wll utlze the proposed method to solve some other practcal decso-makg problems. Lst of symbols A a q-rug orthopar fuzzy set q-rofs) ã...) a collecto of q-rug orthopar fuzzy umber q-roln) a...) a set of crsp umbers Hã) the accuracy fucto of ã dex of a cotrol volume q power of q-rug orthopar fuzzy set s postve umber S a lgustc term set S a cotuous lgustc term set of S={s...t} s a lgustc varable the lgustc term set S s θx) lgustc varable of A= { xs θx) u A x)v A x)) x X } s θ lgustc varable of the q-roln α =s θ uv) s θ1 lgustc varable of the q-roln α 1 = s θ1 u 1 v 1 ) s θ lgustc varable of the q-roln α = s θ u v ) Sã) the score fucto of ã t postve umber u membershp degree of q-roln α =s θ uv) u A x) membershp degree of q-rofs A={xu A x)v A x) x X} u 1 membershp degree of q-rofn ã 1 =u 1 v 1 ) v o-membershp degree of q-roln α =s θ uv)

30 580 L. LI R. ZHANG J. WANG X. SHANG v A x) o-membershp degree of q-rofs A={xu A x)v A x) x X} v 1 o-membershp degree of q-rofn ã 1 =u 1 v 1 ) w weght vector x varable fxed set X X a ordary fxed set λ a postve real umber α β a q-roln Refereces [1] L.A. ZADEH: Fuzzy sets Iformato Cotrol ) [] K.T. ATANASSOV: Itutostc fuzzy sets Fuzzy sets ad Systems 01) 1986) [3] Z.S. XU: Itutostc fuzzy aggregato operators IEEE Trasactos o fuzzy systems 156) 007) [4] W. JIANG B. WEI X. LIU X.Y. LI ad H.Q. ZHENG: Itutostc fuzzy power aggregato operator based o etropy ad ts applcato decso makg Iteratoal Joural of Itellget Systems 331) 018) [5] P.J. REN Z.S. XU H.C. LIAO ad X.J. ZENG: A thermodyamc method of tutostc fuzzy MCDM to assst the herarchcal medcal system Cha Iformato Sceces ) [6] Z.M. ZHANG: Mult-crtera group decso-makg methods based o ew tutostc fuzzy Este hybrd weghted aggregato operators Neural Computg ad Applcatos 81) 017) [7] S. MAHESHWARI ad A. SRIVASTAVA: Study o dvergece measures for tutostc fuzzy sets ad ts applcato medcal dagoss Joural of Appled Aalyss ad Computato 63) 016) [8] C.P. WEI P. WANG ad Y.Z. ZHANG: Etropy smlarty measure of terval-valued tutostc fuzzy sets ad ther applcatos Iformato Sceces 18119) 011) [9] Z. WANG Z.S. XU S.S. LIU ad Z.Q. YAO: Drect clusterg aalyss based o tutostc fuzzy mplcato Appled Soft Computg 3 014) 1 8. [10] Z. WANG Z.S. XU S.S. LIU ad J. TANG: A ettg clusterg aalyss method uder tutostc fuzzy evromet Appled Soft Computg 118) 011)

31 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 581 [11] S.M. CHEN S.H. CHENG ad T.C. LAN: A ovel smlarty measure betwee tutostc fuzzy sets based o the cetrod pots of trasformed fuzzy umbers wth applcatos to patter recogto Iformato Sceces ) [1] C.P. WEI P. WANG ad Y.Z. ZHANG: Etropy smlarty measure of terval-valued tutostc fuzzy sets ad ther applcatos Iformato Sceces 18119) 011) [13] R.R. YAGER: Pythagorea membershp grades mult-crtera decso makg IEEE Trasactos o Fuzzy Systems 014) [14] H. GARG: A ew geeralzed Pythagorea fuzzy formato aggregato usg Este operatos ad ts applcato to decso makg Iteratoal Joural of Itellget Systems 319) 016) [15] H. GARG: Geeralzed Pythagorea fuzzy geometrc aggregato operators usg Este t-orm ad t-coorm for multcrtera decso-makg process Iteratoal Joural of Itellget Systems 36) 017) [16] K. RAHMAN S. ABDULLAH R. AHMED ad U. MURAD: Pythagorea fuzzy Este weghted geometrc aggregato operator ad ther applcato to multple attrbute group decso makg Joural of Itellget & Fuzzy Systems 331) 017) [17] G.W. WEI ad M. LU: Pythagorea fuzzy power aggregato operators multple attrbute decso makg Iteratoal Joural of Itellget Systems 331) 018) [18] R.R. YAGER: The power average operator IEEE Trasactos o Systems Ma ad Cyberetcs-Part A: Systems ad Humas 316) 001) [19] G.W. WEI: Pythagorea fuzzy teracto aggregato operators ad ther applcato to multple attrbute decso makg Joural of Itellget & Fuzzy Systems 334) 017) [0] D.C. LIANG Z.S. XU ad A.P. DARKO: Proecto model for fusg the formato of Pythagorea fuzzy multcrtera group decso makg based o geometrc Boferro mea Iteratoal Joural of Itellget Systems 39) 017) [1] R.T. ZHANG J. WANG X.M. ZHU M.M. XIA ad M. YU: Some geeralzed Pythagorea fuzzy Boferro mea aggregato operators wth ther applcato to multattrbute group decso-makg Complexty ) Artcle ID

32 58 L. LI R. ZHANG J. WANG X. SHANG [] G.W. WEI ad M. LU: Pythagorea fuzzy Maclaur symmetrc mea operators multple attrbute decso makg Iteratoal Joural of Itellget Systems 017) do: /t [3] D.C. LIANG ad Z.S. XU: The ew exteso of TOPSIS method for multple crtera decso makg wth hestat Pythagorea fuzzy sets Appled Soft Computg ) [4] M.S.A. KHAN S. ABDULLAH A. ALI N. SIDDIQUI ad F. AMIN: Pythagorea hestat fuzzy sets ad ther applcato to group decso makg wth complete weght formato Joural of Itellget & Fuzzy Systems 336) 017) [5] M. LU G.W. WEI F.E. ALSAADI T. HAYAT ad A. ALSAEDI: Hestat Pythagorea fuzzy Hamacher aggregato operators ad ther applcato to multple attrbute decso makg Joural of Itellget & Fuzzy Systems 33) 017) [6] G.W. WEI ad M. LU: Dual hestat Pythagorea fuzzy Hamacher aggregato operators multple attrbute decso makg Archves of Cotrol Sceces 73) 017) [7] R.R. YAGER: Geeralzed orthopar fuzzy sets IEEE Trasactos o Fuzzy Systems 55) 017) [8] P.D. LIU ad P. WANG: Some q-rug orthopar fuzzy aggregato operators ad ther applcatos to multple-attrbute decso makg Iteratoal Joural of Itellget Systems 33) 018) [9] P.D. LIU ad J.L. LIU: Some q-rug orthopar fuzzy Boferro mea operators ad ther applcato to mult-attrbute group decso makg Iteratoal Joural of Itellget Systems 33) 018) [30] L.A. ZADEH: The cocept of a lgustc varable ad ts applcato to approxmate reasog Part II Iformato Sceces ) [31] J.Q. WANG ad J.J. LI: The mult-crtera group decso makg method based o mult-graularty tutostc two sematcs Scece Techology ad Iformato ) 8 9. [3] Y.Q. DU F.J. HOU W. ZAFAR Q. YU ad Y.B. ZHAI: A ovel method for mult-attrbute decso makg wth terval-valued Pythagorea fuzzy lgustc formato Iteratoal Joural of Itellget Systems 310) 017)

33 SOME q-rung ORTHOPAIR LINGUISTIC HERONIAN MEAN OPERATORS WITH THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING 583 [33] S. SYKORA: Mathematcal meas ad averages: geeralzed Heroa meas 009) do: /SL3Math [34] P.D. LIU ad L.L. SHI: Some eutrosophc ucerta lgustc umber Heroa mea operators ad ther applcato to mult-attrbute group decso makg Neural Computg ad Applcatos 85) 017) [35] D.J. YU: Itutostc fuzzy geometrc Heroa mea aggregato operators Appled Soft Computg 13) 013) [36] P.D. LIU ad Y.M. WANG: Multple attrbute group decso makg methods based o tutostc lgustc power geeralzed aggregato operators Appled Soft Computg ) [37] Y.B. JU X.Y. LIU ad D.W. JU: Some ew tutostc lgustc aggregato operators based o Maclaur symmetrc mea ad ther applcatos to multple attrbute group decso makg Soft Computg 011) 016) [38] X.F. WANG J.Q. WANG ad W.E. YANG: Mult-crtera group decso makg method based o tutostc lgustc aggregato operators Joural of Itellget & Fuzzy Systems 61) 014) [39] P.D. LIU C. LIU ad L.L. RONG: Itutostc fuzzy lgustc umber geometrc aggregato operators ad ther applcato to group decso makg Ecoomc Computato & Ecoomc Cyberetcs Studes & Research 481) 014) [40] P.D. LIU L.L. RONG Y.C. CHU ad Y.W. LI: Itutostc lgustc weghted Boferro mea operator ad ts applcato to multple attrbute decso makg The Scetfc World Joural 014) Artcle ID [41] C.H. ZHANG W.H. SU ad S.Z. ZENG: Itutostc lgustc multple attrbute decso-makg based o Heroa mea method ad ts applcato to evaluato of scetfc research capacty Eurasa Joural of Mathematcs Scece ad Techology Educato 131) 017) [4] D.J. YU ad Y.Y. WU: Iterval-valued tutostc fuzzy Heroa mea operators ad ther applcato mult-crtera decso makg Afrca Joural of Busess Maagemet 611) 01) 4158.