Probabilistic Linguistic Power Aggregation Operators for Multi-Criteria Group Decision Making

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1 Artcle Probablstc Lgustc Power Aggregato Operators for Mult-Crtera Group Decso Makg Agbodah Koba 1,2 ID, Decu Lag 1,2, * ad X He 1 1 School of Maagemet ad Ecoomcs, Uversty of Electroc Scece ad Techology of Cha, Chegdu , Cha; kobaagbodah@yahoo.com (A.K.); amhex@163.com (X.H.) 2 Ceter for West Afrca Studes, Uversty of Electroc Scece ad Techology of Cha, Chegdu , Cha * Correspodece: deculag@126.com Receved: 4 November 2017; Accepted: 13 December 2017; Publshed: 19 December 2017 Abstract: As a effectve aggregato tool, power average (PA) allows the put argumets beg aggregated to support ad reforce each other, whch provdes more versatlty the formato aggregato process. Uder the probablstc lgustc term evromet, we deeply vestgate the ew power aggregato (PA) operators for fusg the probablstc lgustc term sets (PLTSs). I ths paper, we frstly develop the probablstc lgustc power average (PLPA), the weghted probablstc lgustc power average (WPLPA) operators, the probablstc lgustc power geometrc (PLPG) ad the weghted probablstc lgustc power geometrc (WPLPG) operators. At the same tme, we carefully aalyze the propertes of these ew aggregato operators. Wth the ad of the WPLPA ad WPLPG operators, we further desg the approaches for the applcato of mult-crtera group decso-makg (MCGDM) wth PLTSs. Fally, we use a llustrated example to expoud our proposed methods ad verfy ther performaces. Keywords: power average operator; probablstc lgustc term sets; mult-crtera decso makg; group decso makg 1. Itroducto Yager [1] troduced a operator of power average (PA) to provde more versatlty the formato aggregato process. PA s a olear weghted average aggregato tool for whch the weght vector depeds o the put argumets ad that allows the values beg aggregated to support ad reforce each other [2]. It has receved a large amout of atteto the lterature. For stace, Xu ad Yager [2] developed power geometrc operator o the bass of a geometrc mea (GM) ad power average (PA). Uder the lgustc evromet, Xu et al. [3] developed ew lgustc aggregato operators based o the power average (PA) to address the relatoshp of put argumets. Zhou ad Che [4] dscussed a geeralzato of the power aggregato operators for lgustc evromet ad ts applcato group decso makg (GDM). By extedg the PA to the lgustc hestat fuzzy evromet, Zhu et al. [5] establshed a seres of lgustc hestat fuzzy power aggregato operators. Wth the above-metoed lterature, PA has successfully bee exteded to may complex ad real stuatos. Oe of the useful theores dealg wth the mult-crtera decso makg (MCDM) problems s the theory of probablstc lgustc term sets (PLTSs). Ths theory proposed by Pag et al. [6] plays a key role the decso process where experts express ther prefereces [7 9]. Nowadays, PLTSs have become a hot topc the area of hestat fuzzy lgustc term sets (HFLTSs) [10 12] ad hestat fuzzy sets (HFSs) [13,14]. For example, Pag et al. [6] establshed a framework for rakg PLTSs ad they coducted a comparso method va the score or devato degree of each PLTS. Symmetry 2017, 9, 320; do: /sym

2 Symmetry 2017, 9, of 21 Ba et al. [7] stated that the exstg approaches assocated wth PLTSs are lmted or hghly complex real applcatos. Thus they establshed more approprate comparso method ad developed a more effcet way to hadle PLTSs. Gou ad Xu [15] defed ovel operatoal laws for the probablty formato. He et al. [16] proposed a algorthm for mult-crtera group decso makg (MCGDM) wth probablstc terval preferece ordergs. Wu ad Xu [17] defed the cocept of possblty dstrbuto ad preseted a ew framework model to address MCDM. Zhag et al. [18] troduced the cocept of probablstc lgustc preferece relatos to preset the DMs prefereces. Uder the hestat probablstc fuzzy evromet, Zhou ad Xu [19] studed the cosesus buldg wth a group of decso makers. PLTSs geeralze the exstg models of HFLTSs ad HFSs so as to cota hestatos ad probabltes. Compared wth HFLTSs, the PLTSs have strog ablty to express the formato vagueess ad ucertaty the hestat stuatos uder qualtatve settg. Wth respect to the PLTSs, the decso makers (DMs) ca ot oly provde several possble lgustc values over a object (alteratve or attrbute), but also reflect the probablstc formato of the set of values [6]. I the exstg lterature, most aggregato operators developed for PLTSs are based o the depedece assumpto ad do ot take to accout formato about the terrelatoshp betwee PLTSs beg aggregated. For the PLTSs, t also ca ecouter the relatoshp pheomeo betwee the put argumets. Meawhle, PA provdes a versatlty the aggregato process ad has the ablty to depct the terrelatoshp of put argumets,.e., t allows the put argumet beg aggregated to support ad reforce each other. However, t rarely dscusses the research works of PLTSs. Hece, we troduce PA to PLTSs ad come out wth ew operators that wll mproved upo the exstg aggregato operators of PLTSs. I ths paper, we frstly develop four ew aggregato operators based o the Power Average (PA) ad the Power Geometrc (PG),.e., probablstc lgustc power average (PLPA), weghted probablstc lgustc power average (WPLPA), probablstc lgustc power geometrc (PLPG) ad weghted probablstc lgustc power geometrc (WPLPG). These operators take to accout all the decso argumets ad ther relatoshps. O the bass of probablstc lgustc GDM, we utlze the WPLPA or WPLPG operator to aggregate the formato ad desg the correspodg approach. I a word, the desrable advatages of our research work are summarzed as follows: (1) We volve the probablstc formato. Our proposed methods ca allow the collecto of a few dfferet lgustc terms evaluated by the DMs ad the opos of the DMs wll stll rema the same. (2) Our proposed methods also cosder the terrelatoshp of the dvdual evaluato. The rest of the paper s structured as follows: Some basc cocepts ad operatos relato to PLTSs ad PA are troduced Secto 2. I Secto 3, we develop the PLPA operator, PLPG operator ad ther ow correspodg weghted forms. Meawhle, we also study several desred propertes of these operators. I Secto 4, we desg the approaches for the applcato of MCGDM utlzg the WPLPG ad WPLPA operators. I Secto 5, we gve a llustratve example to elaborate ad verfy our proposed methods. Secto 6 cocludes the paper ad elaborates o future studes. 2. Prelmares I ths secto, we maly revew some basc cocepts ad operatos relato to PLTSs ad PA Probablstc Lgustc Term Sets (PLTSs) The cocept of PLTSs [6] s a exteso of the cocepts of HFLTSs. I the followg, we revew some basc cocepts of PLTSs ad the correspodg operatos. Defto 1. [6] Let S s t t 0, 1,, τ be a lgustc term set. The a probablstc lgustc term set (PLTS) s defed as: L(p) (p (k) ) S, r (k) t, p (k) #L(p) 0, k 1, 2,, #L(p), p (k) 1, (1)

3 Symmetry 2017, 9, of 21 where (p (k) ) s the lgustc term assocated wth the probablty p (k), r (k) s the subscrpt of ad #L(p) s the umber of all lgustc terms L(p). Sce the postos of elemets a set ca be swapped arbtrarly, Pag et al. [6] proposed the ordered PLTSs to esure that the operatoal results amog PLTSs ca be straghtforwardly determed. It s descrbed as: Defto 2. Gve a PLTS L(p) (p (k) ) k 1, 2,, #L(p), ad r (k) s the subscrpt of lgustc term. L(p) s called a ordered PLTS, f the lgustc terms (p (k) ) are arraged accordg to the values of r (k) p (k) descedg order. Defto 3. Let S s t t 0, 1,, τ be a lgustc term set. Gve three PLTSs L(p), L 1 (p) ad L 2 (p), ther basc operatos are summarzed as follows [6]: (1) L 1 (p) L 2 (p) 1 L 1(p), p (k) 2 L 2(p) 1 L(k) 1 p (k) 2 L(k) 2 ; (2) L 1 (p) L 2 (p) 1 L 1(p), 2 L 2(p) (3) λ(l(p)) L(p) λp (k) ad λ 0; (4) (L(p)) λ L(p) ( ) λp(k) ad λ 0. ( 1 )p(k) 1 ( 2 )p(k) 2 ; To compare the PLTSs, Pag et al. [6] defed the score ad the devato degree of a PLTS: Defto 4. Let L(p) (p (k) ) k 1, 2,, #L(p) be a PLTS, ad r (k) s the subscrpt of lgustc term. The, the score of L(p) s defed as follows: where ᾱ #L(p) r (k) p (k) / #L(p) p (k). The devato degree of L(p) s: E(L(p)) sᾱ, (2) σ(l(p)) ( #L(p) (p(k) (r (k) ᾱ)) 2 ) 0.5 #L(p) p (k). (3) Based o the score ad the devato degree of a PLTS, Pag et al. [6] further proposed the followg laws to compare them. Defto 5. Gve two PLTSs L 1 (p) ad L 2 (p). E(L 1 (p)) ad E(L 2 (p)) are the scores of L 1 (p) ad L 2 (p), respectvely. σ(l 1 (p)) ad σ(l 2 (p)) deote the devato degrees of L 1 (p) ad L 2 (p). The, we have: (1) If E(L 1 (p)) > E(L 2 (p)), the L 1 (p) s bgger tha L 2 (p), deoted by L 1 (p) > L 2 (p); (2) If E(L 1 (p)) < E(L 2 (p)), the L 1 (p) s smaller tha L 2 (p), deoted by L 1 (p) < L 2 (p); (3) If E(L 1 (p)) E(L 2 (p)), the we eed to compare ther devato degrees: (a) (b) (c) If σ(l 1 (p)) σ(l 2 (p)), the L 1 (p) s equal to L 2 (p), deoted by L 1 (p) L 2 (p); If σ(l 1 (p)) > σ(l 2 (p)), the L 1 (p) s smaller tha L 2 (p), deoted by L 1 (p) < L 2 (p); If σ(l 1 (p)) < σ(l 2 (p)), the L 1 (p) s bgger tha L 2 (p), deoted by L 1 (p) > L 2 (p). Whe we aalyze ad dscuss the comparso of PLTSs, we may realse that the umber of ther correspodg umber of the lgustc terms may ot be equal. To solve ths problem, Pag et al. [6] ormalzed the PLTSs by creasg the umbers of lgustc terms for PLTSs. The ormalzed Defto of PLTSs s the followg.

4 Symmetry 2017, 9, of 21 Defto 6. Let L 1 (p) 1 (p(k) 1 ) k 1, 2,, #L 1(p) ad L 2 (p) 2 (p(k) 2 ) k 1, 2,, #L 2 (p) be ay two PLTSs. #L 1 (p) ad #L 2 (p) are the umbers of the lgustc terms L 1 (p) ad L 2 (p). If #L 1 (p) > #L 2 (p), the we wll add #L 1 (p) #L 2 (p) lgustc terms to L 2 (p) so that the umbers of lgustc terms L 1 (p) ad L 2 (p) are detcal. The added lgustc terms are the smallest oes L 2 (p) ad the probabltes of all the lgustc terms are zero. Aalogously, f #L 1 (p) < #L 2 (p), we ca use the smlar method. Based o the ormalzed PLTSs, Pag et al. [6] further defed the devato degree betwee PLTSs. The result s show as follows. Defto 7. [6] Let L 1 (p) 1 (p(k) 1 ) k 1, 2,, #L 1(p) ad L 2 (p) 2 (p(k) 2 ) k 1, 2,, #L 2 (p) be ay two PLTSs, f #L 1 (p) #L 2 (p), the the devato degree betwee PLTSs s defed as: 2.2. Power Average (PA) d(l 1 (p), L 2 (p)) #L 1(p) (r (k) 1 p(k) 1 r (k) 2 p(k) 2 )2 /#L 1 (p). (4) Iformato fuso s a process of aggregatg data operators from dfferet resources by proper aggregatg operators. Power average (PA) operator, as a techque of fusg formato, was troduced by Yager [1], whch allows the argumets to support each other the aggregato process. Defto 8. [1] Let A a 1, a 2,, a be a collecto of o-egatve umbers. The power aggregato s defed as follows: where PA(a 1, a 2,, a ) (1 + T(a ))a (1 + T(a )), (5) T(a ) sup(a, a j ). (6) j1,j I ths case, sup(a, a j ) s deoted as the support for a from a j, whch satsfes the followg three propertes: (1) sup(a, a j ) [0, 1]; (2) sup(a, a j ) sup(a j, a ); (3) sup(a, a j ) sup(a, a k ), f a a j < a a k. From the result of Defto 7, the supports amog the put argumets are volved the PA. I geeral, sup(a, a j ) ca be measured by the dstace betwee the argumets, e.g., d(a, a j ). By troducg geometrc mea (GM), Xu ad Yager [2] defed a power geometrc (PG) operator as follows: PG(a 1, a 2,, a ) a (1+T(a )) (1+T(a )), (7) where a ( 1, 2,, ) are a collecto of argumets, ad T(a ) satsfes the codto above.

5 Symmetry 2017, 9, of Probablstc Lgustc Power Aggregato Operators Uder the probablstc lgustc evromet, we assume that the put argumets are PLTSs ad we maly study the exteso of power average (PA) ad power geometrc (PG) aggregato operators Probablstc Lgustc Power Average (PLPA) Aggregato Operators I ths secto, we dscuss the exteso of power average (PA) aggregato operators to accommodate the probablstc lgustc evromet. I the followg, some probablstc lgustc power average aggregato operators should be developed, whch allow the put data to support each other the aggregato process,.e., Probablstc Lgustc Power Average (PLPA) ad Weghted Probablstc Lgustc Power Average (WPLPA) PLPA Based o the results of Deftos 1 ad 7, we preset the Defto of the PLPA aggregato operator as follows: Defto 9. Let L(p) (p (k) ) k 1, 2,, #L (p) ( 1, 2,..., ) be a collecto of PLTSs. A probablstc lgustc power average (PLPA) s a mappg L (p) L(p) such that: where: PLPA(L 1 (p), L 2 (p),, L (p)) T(L (p)) (1 + T(L (p)))l (p) (1 + T(L (p))), (8) sup(l (p), L j (p)). (9) j1,j ad sup(l (p), L j (p)) s cosdered to be the support for L (p) from L j (p) whch satsfes the followg propertes: (1) sup(l (p), L j (p)) [0, 1]; (2) sup(l (p), L j (p)) sup(l j (p), L (p)); (3) sup(l (p), L j (p)) sup(l (p), L k (p)) f d(l (p), L j (p)) < d(l (p), L k (p)). I lght of the operatos law (1) of Defto 3, Defto 8 ca be trasformed to the followg form: PLPA(L 1 (p), L 2 (p),, L (p)) (1 + T(L 1 (p))) (1 + T(L (p))) L 1(p) (1 + T(L 2(p))) (1 + T(L (p))) L 2(p) (1 + T(L (p))) (1 + T(L (p))) L (p). Hece, we ca deduce the followg result from Defto 8. Proposto 1. Let L(p) A probablstc lgustc power average (PLPA) s calculated as: PLPA(L 1 (p), L 2 (p),, L (p)) 1 L 1(p) v 1 p (k) 1 L(k) 1 (p (k) ) k 1, 2,, #L (p) ( 1, 2,..., ) be a collecto of PLTSs. v L (p) 2 L 2(p) v 2 p (k) 2 L(k) 2 L (p) v p (k), (10)

6 Symmetry 2017, 9, of 21 where v (1+T(L (p))) j1 (1+T(L ( 1, 2,..., ). j(p))) O the bass of Defto 8 ad Proposto 1, t ca easly be prove that the PLPA aggregato operator has the followg desrable propertes. Theorem 1. (Commutatvty) Let (L 1 (p), L 2 (p),, L (p) ) be ay permutato of (L 1 (p), L 2 (p),, L (p)), the PLPA(L 1 (p), L 2 (p),, L (p)) PLPA(L 1 (p), L 2 (p),, L (p) ). Proof. Accordg to the result of Defto 2, L (p) s called a ordered PLTS ( 1, 2,..., ). By Proposto 1 ad the operatos law (1) of Defto 3, we ca coclude that: PLPA(L 1 (p), L 2 (p),, L (p)) PLPA(L 1 (p), L 2 (p),, L (p) ). Therefore, we complete the proof of Theorem 1. Theorem 2. (Idempotecy) Let L (p) ( 1, 2,, ) be a collecto of PLTSs. If all L (p) ( 1, 2,, ) are equal,.e., L (p) L(p), the PLPA(L 1 (p), L 2 (p),, L (p)) L(p). Proof. If L (p) L(p) for all, the PLPA(L 1 (p), L 2 (p),, L (p)) s computed as: PLPA(L 1 (p), L 2 (p),, L (p)) (1 + T(L (p))) (1 + T(L (p))) L (p) 1 L (p) L (p). Hece, the statemet of Theorem 2 holds. Theorem 3. (Boudedess) Let L (p) ( 1, 2,, ) be a collecto of PLTSs, the we have: #L (p) m m p(k) where L PLPA(L 1 (p), L 2 (p),, L (p)). L max #L (p) max p(k) Proof. Accordg to the result of Proposto 1, PLPA(L 1 (p), L 2 (p),, L (p)) s computed as: PLPA(L 1 (p), L 2 (p),, L (p)) 1 L 1(p) v 1 p (k) 1 L(k) 1 v L (p) 2 L 2(p) The, we ca deduce the followg relatoshp: #L (p) m m p(k) p (k), v 2 p (k) 2 L(k) 2 max #L (p) max p(k) L (p). By utlzg the result of Theorem 2, we ca easly fsh the proof of Theorem 1. v p (k). Theorem 4. (Mootocty) Let L (p) ad L (p) be two sets of PLTSs ad the umbers of lgustc terms L (p) ad L (p) are detcal ( 1, 2,, ). If (p (k) ) (p (k) ) for all,.e., L (p) L (p), the PLPA(L 1 (p), L 2 (p),, L (p)) PLPA(L 1 (p), L 2 (p),, L (p) ).

7 Symmetry 2017, 9, of 21 Theorem 5. Let sup(l (p), L j (p)) k for all j, the PLPA(L 1 (p), L 2 (p),, L (p)) 1 L (p). Proof. If sup(l (p), L j (p)) k for all j, t dcates that all the supports are the same. I ths stuato, the PLPA operator s computed as follows: PLPA(L 1 (p), L 2 (p),, L (p)) (1 + T(L (p))) (1 + T(L (p))) L (p) (1 + ( 1)k) (1 + ( 1)k) L (p) 1 L (p). It s a smple probablstc lgustc averagg operator. Hece, the statemet of Theorem 5 holds WPLPA Wth respect to the PLPA operator, the weghts of the argumets should be cosdered, because each argumet that s beg aggregated has a weght dcatg ts mportace [3]. Based o ths dea, we exted the PLPA ad gve the Defto of the weghted probablstc lgustc power average (WPLPA) operator as follows: Defto 10. Let L (p) be a collecto of PLTSs. w (w 1, w 2,..., w ) T deotes the weghtg vector of L (p) ad w [0, 1], w 1. Gve the value of the weght vector w (w 1, w 2,..., w ) T, we defe weghted probablstc lgustc power average (WPLA) operator as follows: WPLPA(L 1 (p), L 2 (p),, L (p)) I ths case, T (L (p)) j1,j w jsup(l (p), L j (p)). w (1 + T (L (p)))l (p) w (1 + T (L (p))). (11) Based o the operatos of the PLTSs descrbed Defto 3, we ca derve the followg Proposto 2. Proposto 2. Let L (p) ( 1, 2,, ) be a collecto of PLTSs, the ther aggregated values by usg the WPLPA operator s also a PLTS, ad: where v WPLPA(L 1 (p), L 2 (p),, L (p)) w (1+T (L (p))) j1 w j(1+t ( 1, 2,, ). (L j (p))) 1 L 1(p) v 1 p(k) 1 L(k) 1 L (p) v p (k) 2 L 2(p) v 2 p(k) 2 L(k) 2. (12) Especally, f sup(l (p), L j (p)) 0 for all j, the T(L (p) 0. Thus, WPLPA(L 1 (p), L 2 (p),, L (p)) w L (p). Uder ths stuato, the WPLPA operator reduces to PLWA proposed by Ref. [6]. If the weght vector w (w 1, w 2,..., w ) T ( 1, 1,, 1 )T, of Proposto 2 s computed as: v v w (1 + T (L (p))) j1 w j(1 + T (L j (p))) (1 + T (L (p))) (1 + T (L (p))) v.

8 Symmetry 2017, 9, of 21 Thus, the WPLPA operator s computed as: WPLPA(L 1 (p), L 2 (p),, L (p)) v 1 p (k) 1 L(k) 1 1 L 1(p) 2 L 2(p) PLPA(L 1 (p), L 2 (p),, L (p)). v 2 p (k) 2 L(k) 2 L (p) v p (k) It dcates that the WPLPA reduces to the PLPA operator. Accordg to the results of Deftos 3 ad 10, t ca easly prove that the WPLPA operator has the followg propertes. Theorem 6. (Idempotecy) Let L (p) ( 1, 2,, ) be a collecto of PLTSs, f all L (p) ( 1, 2,, ) are equal,.e., L (p) L(p), the WPLPA(L 1 (p), L 2 (p),, L (p)) L(p). Proof. If L (p) L(p) for all, the WPLPA(L 1 (p), L 2 (p),, L (p)) s computed as: WPLPA(L 1 (p), L 2 (p),, L (p)) w (1 + T (L (p)))l (p) w (1 + T (L (p))) 1 L (p) L (p). Thus, the statemet of Theorem 6 holds. Theorem 7. (Boudedess) Let L (p) ( 1, 2,, ) be a collecto of PLTSs, the we have: #L (p) m m p(k) where L WPLPA(L 1 (p), L 2 (p),, L (p)). L max #L (p) max p(k) If we let sup(l (p), L j (p)) k for all j, we have: T (L (p)) j1,j w jsup(l (p), L j (p)) k j1,j w j ( 1, 2,, ). Based o the result of Defto 10, we have: WPLPA(L 1 (p), L 2 (p),, L (p)). w (1 + k j1,j w j) w (1 + k j1,j w j) L (p). I ths case, WPLPA(L 1 (p), L 2 (p),, L (p)) s ot equvalet to PLPA(L 1 (p), L 2 (p),, L (p)) 1 L (p). Theorem 8. Let (L 1 (p), L 2 (p),, L (p) ) be ay permutato of (L 1 (p), L 2 (p),, L (p)), the we ca deduce the followg relatoshp: WPLPA(L 1 (p), L 2 (p),, L (p) ) WPLPA(L 1 (p), L 2 (p),, L (p)). Proof. Accordg to the result of Defto 10, we ca obta: T (L p) w j sup(l (p), L j (p) ). j1,j

9 Symmetry 2017, 9, of 21 The, we ca deduce: WPLPA(L 1 (p), L 2 (p),, L (p)) w (1 + T (L (p) )) w (1 + T (L (p) )) L (p). Sce (T (L 1 (p) ), T (L 2 (p) ),, T (L 2 (p) )) may ot be the permutato of (T (L 1 (p)), T (L 2 (p)),, T (L (p))), we ca judge that the WPLPA operator s ot commutatve. Therefore, we complete the proof of Theorem Probablstc Lgustc Power Geometrc (PLPG) Aggregato Operators I ths secto, we vestgate the exteso of power geometrc (PG) aggregato operators uder the probablstc lgustc evromet,.e., the probablstc lgustc power geometrc (PLPG) ad weghted probablstc lgustc power geometrc (WPLPG) PLPG By utlzg the results of Defto 1 ad Equato (7), we preset the Defto of the PLPG operator as follows. Defto 11. Let L(p) (p k) k 1, 2,, #L (p) ( 1, 2,, ) be a collecto of PLTSs. A probablstc lgustc power geometrc (PLPG) operator s a mappg L (p) L(p) such that: PLPG(L 1 (p), L 2 (p),, L (p)) (1+T(L (p))) (L (p)) (1+T(L (p))), (13) where T(L (p)) j1,j sup(l (p), L j (p)). sup(l (p), L j (p)) s cosdered to be the support of L (p) from L j (p) whch also satsfes the followg propertes: (1) sup(l (p), L j (p)) [0, 1]; (2) sup(l (p), L j (p)) sup(l j (p), L (p)); (3) sup(l (p), L j (p)) sup(l j (p), L (p)) f d(l (p), L j (p)) < d(l (p), L k (p)). By the operatos law (2) of Defto 3, Defto 11 ca be trasformed to the followg form: PLPG(L 1 (p), L 2 (p),, L (p)) (1+T(L 1 (p))) (L 1 (p)) (1+T(L (p))) (L 2 (p)) (1+T(L (p))) (L (p)) (1+T(L 2 (p))) Therefore, we ca deduce the followg based o the results of Defto 9: (1+T(L(p))) (1+T(L (p))). Proposto 3. Let L(p) (p k) k 1, 2,, #L (p) ( 1, 2,, ) be a collecto of PLTSs. A probablstc lgustc power geometrc (PLPG) operator s calculated as: where v PLPG(L 1 (p), L 2 (p),, L (p)) 1 L 1(p) ( 1 )v 1 p (k) 1 (1+T(L (p))) j1 (1+T(L ( 1, 2,..., ). j(p))) (L (p)) v 2 L 2(p) ( 2 )v 2 p (k) 2 L (p) ( ) v p (k), (14) O the bass of Defto 11 ad Proposto 3, t ca be proved that the PLPG operator has the followg desrable propertes:

10 Symmetry 2017, 9, of 21 Theorem 9. (Commutatvty) Let (L 1 (p), L 2 (p),, L (p) ) be ay permutato of (L 1 (p), L 2 (p),, L (p)) the PLPG(L 1 (p), L 2 (p),, L (p))plpg(l 1 (p), L 2 (p),, L (p) ). Proof. Accordg to the result of Defto 2, L (p) s called a ordered PLTS ( 1, 2,, ). By the results of Proposto 3 ad the operatos laws (2) of Defto 3, we ca coclude that: PLPG(L 1 (p), L 2 (p),, L (p)) PLPG(L 1 (p), L 2 (p),, L (p) ). Therefore, we complete the proof of Theorem 9. Theorem 10. (Idempotecy) Let L (p) ( 1, 2,, ) be a collecto of PLTSs. If all L (p) ( 1, 2,, ) are equal,.e., L (p) L(p), the PLPG(L 1 (p), L 2 (p),, L (p)) L(p). Proof. If L (p) L(p) for all, the PLPG(L 1 (p), L 2 (p),, L (p)) s computed as: PLPG(L 1 (p), L 2 (p),, L (p)) (L (p)) (1+T(L (p))) (1+T(L (p))) (L (p)) 1 L(p). Hece, the statemet of Theorem 10 holds. Theorem 11. (boudedess) Let L (p) ( 1, 2,, ) be a collecto of PLTSs, the we have: #L (p) m m (L(k) ) p(k) where L PLPG(L 1 (p), L 2 (p),, L (p)). L max #L (p) max (L(k) ) p(k), Proof. Accordg to the result of Proposto 3, PLPG(L 1 (p), L 2 (p),, L (p)) s computed as: PLPG(L 1 (p), L 2 (p),, L (p)) 1 L 1(p) ( 1 )v 1 p (k) 1 (L (p)) v 2 L 2(p) The, we ca deduce the followg relatoshp: #L (p) m m (L(k) ) p(k) ( ) p(k) ( 2 )v 2 p (k) 2 max #L (p) max (L(k) L (p) ) p(k). ( I lght of the results of Theorem 10, we ca easly fsh the proof of Theorem WPLPG ) v p (k). Cosderg the mportace of the aggregated argumets, we exted the PLPG ad gve the Defto of the weghted probablstc lgustc power geometrc (WPLPG) operator as followg.

11 Symmetry 2017, 9, of 21 Defto 12. Let L (p) be a collecto of PLTSs. w (w 1, w 2,, w ) T deotes the weghtg vector of L (p), w [0, 1] ad w 1. Gve the value of the weght vector w (w 1, w 2,, w ) T, we defe weghted probablstc lgustc power geometrc (WPLPG) operator as follows: WPLPG(L 1 (p), L 2 (p),, L (p)) w (1+T (L (p))) (L (p)) w (1+T (L (p))). (15) I ths case, T (L (p)) j1,j w jsup(l (p), L j (p)). Based o the operatos of the PLPTs descrbed Defto 3, we ca derve the followg Proposto: Proposto 4. Let L (p) ( 1, 2,, ) be a collecto of PLTSs, the ther aggregated values by usg the WPLPG operator s also a PLTS, ad: where v WPLPG(L 1 (p), L 2 (p),, L (p)) w (1+T (L (p))) j1 w j(1+t ( 1, 2,, ). (L j (p))) 1 L 1(p) ( 1 )v 1 p(k) 1 L (p) ( 2 L 2(p) ) v p (k) ( 2 )v 2 p(k) 2. (16) For the result of Proposto 4, f sup(l (p), L j (p)) 0 for all j, the T(L (p)) 0. Thus, we have: WPLPG(L 1 (p), L 2 (p),, L (p)) (L (p)) w. Uder ths stuato, the WPLPG operator reduces to PLWG proposed by Ref. [6]. If the weght vector w (w 1, w 2,..., w ) T ( 1, 1,, 1 )T, v of Proposto 4 s computed as: v w (1 + T (L (p))) j1 w j(1 + T (L j (p))) (1 + T (L (p))) (1 + T (L (p))) v. Hece, the WPLPG operator s computed as: WPLPG(L 1 (p), L 2 (p),, L (p)) 1 L 1(p) ( 1 )v 1 p(k) 1 L (p) ( 2 L 2(p) ) v p (k) PLPG(L 1 (p), L 2 (p),, L (p)). ( 2 )v 2 p(k) 2 Thus, t dcates that the WPLPG ca be reduced to the PLPG operator. Accordg to the results of Deftos 3 ad 12, t ca easly prove that the WPLPG operator has the followg propertes. Theorem 12. (Idempotecy) Let L (p) ( 1, 2,, ) be a collecto of PLTSs, f all L (p) ( 1, 2,, ) are equal,.e., L (p) L(p), the WPLPG(L 1 (p), L 2 (p),, L (p)) L(p).

12 Symmetry 2017, 9, of 21 Proof. If L (p) L(p) for all, the WPLPG(L 1 (p), L 2 (p),, L (p)) s computed as: WPLPG(L 1 (p), L 2 (p),, L (p)) (L (p)) w (1+T (L (p))) w (1+T (L (p))) (L (p)) 1 L(p). Thus, the statemet of Theorem 12 holds. Theorem 13. (Boudedess) Let L (p) ( 1, 2,, ) be a collecto of PLTSs, the we have: #L (p) m m p(k) where L WPLPG(L 1 (p), L 2 (p),, L (p)). L max #L (p) max p(k) Theorem 14. Let (L 1 (p), L 2 (p),, L (p) ) s ay permutato of (L 1 (p), L 2 (p),, L (p)), the we ca deduce the followg relatoshp:, WPLPG(L 1 (p), L 2 (p),, L (p) ) WPLPG(L 1 (p), L 2 (p),, L (p)). Proof. Accordg to the result of Defto 12, we ca obta: The, we ca deduce: T (L p) WPLPG(L 1 (p), L 2 (p),, L (p)) w j sup(l (p), L j (p) ). j1,j (L (p) ) w (1+T (L (p) ) w (1+T (L (p) ). Sce (T (L 1 (p) ), T (L 2 (p) ),, T (L 2 (p) )) may ot be the permutato of (T (L 1 (p)), T (L 2 (p)),, T (L (p))), we ca judge that the WPLPG operator s ot commutatve. Hece, we complete the proof of Theorem Approaches to Mult-Crtera Group Decso Makg wth Probablstc Lgustc Power Aggregato Operators I ths secto, we frstly preset a MCGDM problem whch the evaluato formato may be expressed by PLTSs. The, we utlze the WPLPA or WPLPG operator to support our decso. Let X x 1, x 2,, x m be a fte set of m alteratves ad C c 1, c 2,, c be a set of attrbutes. Suppose that D d 1, d 2,, d e deotes the set of DMs. By usg the lgustc scale S s α α 0, 1,, τ, each DM d q provdes hs or her lgustc evaluatos over the alteratve x wth respect to the attrbute a j,.e., A q (L q j ) m ( 1, 2,, m; j 1, 2,, ; q 1, 2,, e). The, we determe the collectve evaluatos of DMs for each alteratve terms of PLTSs. I the cotext of GDM, the lgustc evaluato values j (k 1, 2,, #L j (p)) wth the correspodg probablty p (k) j are descrbed as the PLTS L j (p) j (p (k) j ) k 1, 2,, #L j (p) ad #L j (p) s the umber of lgustc terms L j (p). The PLTS L j (p) deotes the evaluato values over the alteratve x ( 1, 2,, m) wth respect to the attrbutes c j (j 1, 2,, ), where j ad p (k) j s the probablty of j (k 1, 2,, #L j (p)). I the case, p (k) j s the k th value of L j (p), 0 ad #L j(p) p (k) j 1.

13 Symmetry 2017, 9, of 21 All the PLTSs are cotaed the probablstc lgustc decso matrx R. Hece, the result s show as follows: L 11 (p) L 12 (p) L 1 (p) L 21 (p) L 22 (p) L 2 (p) R (L j (p)) m..... (17) L m1 (p) L m2 (p) L m (p) Wthout loss of geeralty, we assume that each PLTS L j (p) s a ordered PLTS. w (w 1, w 2,, w ) T deotes the weghtg vector of the attrbutes C ad w j [0, 1], j1 w j 1. Based o the above results, we wll use the WPLPA or WPLPG aggregato operator to develop the correspodg approach for MCGDM wth probablstc lgustc formato. Ths approach s desged as follows: Step 1: Accordg to the practcal decso-makg problem, we determe the alteratves X x 1, x 2,, x m ad a set of the attrbutes C c 1, c 2,, c. The, we ca obta the decso matrx A q (L q j ) m provded by the DM d q. By usg the PLTSs, we costruct the collectve matrx R (L j (p)) m. Step 2: Wth respect to the collectve matrx R (L j (p)) m, we ca ormalze the etres of R as stated Defto 6. Step 3: Based o the matrx R ad the result of Defto 7, the devato degree betwee PLTSs L j (p) ad L t (p) s calculated below ( 1, 2,, m; j, t 1, 2,, ): #L j(p) d(l j (p), L t (p)) (p (k) j r (k) j #L j (p) p (k) t r (k) t ) 2. Step 4: By usg the results of Deftos 7 ad 8, we calculate the support of the alteratve x as follows: sup(l j (p), L t (p)) 1 d(l j (p), L t (p)) g1,g j d(l j(p), L g (p)), (18) whch satsfes the support codtos (1) (3) of Defto 9. Step 5: Accordg to the result of Defto 10, we ca calculate the support T (L j (p)) of L j (p) by all of other L t (p) (j, t 1, 2,, ; t j): T (L j (p)) w t sup(l j (p), L t (p)). t1,t j Step 6: Wth the ad of Proposto 2, we further compute the weght v j assocated wth the PLTS L j (p): v j w j(1 + T (L j (p))) j1 w j(1 + T (L j (p))).

14 Symmetry 2017, 9, of 21 Step 7: If the DM prefers the WPLPA operator, the the aggregated value of the alteratve x s determed based o Equato (12). The result s: WPLPA(L 1 (p), L 2 (p),, L (p)) 1 L 1(p) v 1 p(k) 1 L(k) 1 L (p) v p(k) L(k) 2 L 2(p). v 2 p(k) 2 L(k) 2 If the DM uses the WPLPG operator, the the aggregated value of the alteratve x s determed based o Equato (16). The result s: WPLPG(L 1 (p), L 2 (p),, L (p)) 1 L 1(p) ( 1 )v 1 p(k) 1 L (p) ( )v p(k) 2 L 2(p). ( 2 )v 2 p(k) 2 I ths case, we deote the aggregated value of the alteratve x as Z. Step 8: Based o the results of Defto 4, the score ad the devato degree of Z of the alteratve x are computed,.e., E(Z ) ad σ(z ) ( 1, 2,, m). Step 9: Rak all of the alteratves accordace wth the rakg results of Defto A Illustratve Example I recet years, there has bee cosderable cocer regardg problems assocated wth udergraduate school rakgs, graduate school rakgs, evaluatg ad rewardg uversty professors Cha ad other coutres of the world. Katz et al. [20] metoed that these problems always exsted ad poltcal actvsm together wth varous ecoomc recesso have worse them. Katz ad hs parters were cocered wth the crtera for evaluatg them. They came out wth multple regresso aalyss to determe the factors mportat salary ad promoto decso-makg at the uversty level ad developed a more ratoal meas of evaluatg ad rewardg uversty professors. They were motvated by the fact that there s a dscrmatory polcy rak ad reward the uverstes whch s ot ecessarly justfable. They wet further to state that rewardg professors goes through a arbtrary ad chaotc process ad a more equtable system could be sttuted to ehace decso-makg process. Aother cocer rased was that decsos o salares ad promotos were made a tutve maer such a way that the weghts attached to the varous crtera for classfcato lack clear uderstadg. I ths secto, we llustrate our proposed approach by evaluatg some uversty faculty for teure ad promoto Cha adapted from Bryso et al. [21]. Hece, we frstly preset a MCGDM problem whch the evaluato formato may be expressed by PLTSs. The, we utlze the WPLPA ad WPLPG operator to support our decso. I lght of the results of Ref. [21], the crtera cosdered for the assessmet of the decso problem are summarzed as follows: (1) teachg (c 1 ); (2) research (c 2 ); (3) servce (c 3 ). Let X x 1, x 2, x 3, x 4, x 5 be the set of fve alteratves ad C c 1, c 2, c 3 be the set of three attrbutes. The lgustc scale s S s α α 0, 1,, 8. Suppose that D d 1, d 2, d 3, d 4 deotes the set of DMs. Based o the results of Ref. [22], ther evaluatos are show Tables 1 4.

15 Symmetry 2017, 9, of 21 Table 1. Decso matrx A 1 provded by d 1. c 1 c 2 c 3 x 1 s 8 s 6 s 6 x 2 s 6 s 7 s 7 x 3 s 5 s 8 s 7 x 4 s 7 s 4 s 6 x 5 s 8 s 6 s 7 Table 2. Decso matrx A 2 provded by d 2. c 1 c 2 c 3 x 1 s 6 s 8 s 5 x 2 s 5 s 6 s 7 x 3 s 7 s 6 s 7 x 4 s 8 s 6 s 7 x 5 s 8 s 7 s 6 Table 3. Decso matrx A 3 provded by d 3. c 1 c 2 c 3 x 1 s 7 s 8 s 6 x 2 s 4 s 5 s 6 x 3 s 8 s 7 s 6 x 4 s 7 s 5 s 8 x 5 s 6 s 7 s 6 Table 4. Decso matrx A 4 provded by d 4. c 1 c 2 c 3 x 1 s 6 s 7 s 6 x 2 s 8 s 7 s 7 x 3 s 7 s 6 s 8 x 4 s 5 s 7 s 6 x 5 s 5 s 6 s Decso Aalyss wth Our Proposed Approaches Based o the proposed approaches of Secto 4, we eed to fuse the formato preseted the decso matrces A 1 A 4 by (17). I the cotext of GDM, all the PLTSs are cotaed the probablstc lgustc decso matrx R. Hece, the result s show Table 5. Table 5. The probablstc lgustc decso matrx R. c 1 c 2 c 3 x 1 s 8 (0.25), s 6 (0.5), s 7 (0.25) s 6 (0.25), s 8 (0.5), s 7 (0.25) s 6 (0.75), s 5 (0.25) x 2 s 6 (0.25), s 5 (0.25), s 4 (0.25), s 8 (0.25) s 7 (0.5), s 6 (0.25), s 5 (0.25) s 7 (0.75), s 6 (0.25) x 3 s 5 (0.25), s 7 (0.5), s 8 (0.25) s 8 (0.25), s 6 (0.5), s 7 (0.25) s 7 (0.5), s 6 (0.25), s 8 (0.25) x 4 s 7 (0.5), s 8 (0.25), s 5 (0.25) s 4 (0.25), s 6 (0.25), s 5 (0.25), s 7 (0.25) s 6 (0.5), s 7 (0.25), s 8 (0.25) x 5 s 8 (0.5), s 6 (0.25), s 5 (0.25) s 6 (0.5), s 7 (0.5) s 7 (0.25), s 6 (0.5), s 5 (0.25)

16 Symmetry 2017, 9, of 21 For Table 5, each PLTS L j (p) s assumed to be a ordered PLTS ( 1, 2, 3, 4, 5; j 1, 2, 3). I ths case, the weghtg vector of the attrbutes C s w (w 1, w 2, w 3 ) T (0.3, 0.4, 0.3) T. We use the WPLPA or WPLPG aggregato operator to aalyze the results of Table 5. Based o the above results ad the proposed methods of Secto 4, the detaled steps are show as follows: Step 2: Wth respect to the collectve matrx R (L j (p)) 5 3, we ca fd that the umber of ther correspodg umber of the lgustc terms s ot equal. Thus, we ormalze the etres of R as stated Defto 6. The ormalzed probablstc lgustc decso matrx s show Table 6. Table 6. The ormalzed probablstc lgustc decso matrx. c 1 c 2 c 3 x 1 s 6 (0.5), s 8 (0.25), s 7 (0.25), s 6 (0) s 8 (0.5), s 7 (0.25), s 6 (0.25), s 6 (0) s 6 (0.75), s 5 (0.25), s 5 (0), s 5 (0) x 2 s 8 (0.25), s 6 (0.25), s 5 (0.25), s 4 (0.25) s 7 (0.5), s 6 (0.25), s 5 (0.25), s 5 (0) s 7 (0.75), s 6 (0.25), s 6 (0), s 6 (0) x 3 s 7 (0.5), s 8 (0.25), s 5 (0.25), s 5 (0) s 6 (0.5), s 8 (0.25), s 7 (0.25), s 6 (0) s 7 (0.5), s 8 (0.25), s 6 (0.25), s 6 (0) x 4 s 7 (0.5), s 8 (0.25), s 5 (0.25), s 5 (0) s 7 (0.25), s 6 (0.25), s 5 (0.25), s 4 (0.25) s 6 (0.5), s 8 (0.25), s 7 (0.25), s 6 (0) x 5 s 8 (0.5), s 6 (0.25), s 5 (0.25), s 5 (0) s 7 (0.5), s 6 (0.5), s 6 (0), s 6 (0) s 6 (0.5), s 7 (0.25), s 5 (0.25), s 5 (0) Step 3: Accordg to the results of Defto 7 ad Table 6, the devato degree betwee PLTSs L j (p) ad L t (p) ( 1, 2, 3, 4, 5; j, t 1, 2, 3) ca be calculated by the followg equato: #L j(p) d(l j (p), L t (p)) (p (k) j r (k) j #L j (p) p (k) t r (k) The, we ca calculate the devato degree of ay two L j (p), respectvely. For alteratve x 1, the devato degrees are show as follows: t ) 2 d(l 11, L 12 ) ; d(l 12, L 13 ) ; d(l 11, L 13 ) For alteratve x 2, the devato degrees are show as follows: d(l 21, L 22 ) ; d(l 22, L 23 ) ; d(l 21, L 23 ) For alteratve x 3, the devato degrees are show as follows: d(l 31, L 32 ) ; d(l 32, L 33 ) ; d(l 31, L 33 ) For alteratve x 4, the devato degrees are show as follows: d(l 41, L 42 ) ; d(l 42, L 43 ) 0.875; d(l 41, L 43 ) For alteratve x 5, the devato degrees are show as follows: d(l 51, L 52 ) ; d(l 52, L 53 ) ; d(l 51, L 53 ) Step 4: Based o the results of Deftos 7 ad 8, we ca calculate the support of the alteratve x by usg (18) ( 1, 2, 3, 4, 5). The results are summarzed as follows: sup(l 11, L 12 ) ; sup(l 12, L 13 ) ; sup(l 11, L 13 )

17 Symmetry 2017, 9, of 21 sup(l 21, L 22 ) ; sup(l 22, L 23 ) ; sup(l 21, L 23 ) sup(l 31, L 32 ) ; sup(l 32, L 33 ) ; sup(l 31, L 33 ) sup(l 41, L 42 ) ; sup(l 42, L 43 ) ; sup(l 41, L 43 ) sup(l 51, L 52 ) ; sup(l 52, L 53 ) ; sup(l 51, L 53 ) Step 5: I lght of the result of Defto 10, we ca calculate the support T (L j (p)) of L j (p) by all of other L t (p) (j, t 1, 2, 3; t j) by the followg equato: T (L j (p)) These results are show as the followg matrx: T (L j (p)) 3 w t sup(l j (p), L t (p)). t1,t j Step 6: Wth the ad of Proposto 2, we further compute the weght v j assocated wth the PLTS L j (p) by the followg equato ( 1, 2, 3, 4, 5; j 1, 2, 3): v j w j(1 + T (L j (p))) 3 j1 w j(1 + T (L j (p))). These results are show as the followg matrx: v j Step 7: If the DM prefers the WPLPA operator, the the aggregated value of the alteratve x s determed based o Equato (12) ( 1, 2, 3, 4, 5). We deote the aggregated value of the alteratve x as Z. The results are: Z 1 WPLPA(L 11 (p), L 12 (p), L 13 (p)). ((0.9171, , , 0); (1.5924, , , 0); (1.3325, , 0, 0)) (3.8420, , , 0), Z 2 WPLPA(L 21 (p), L 22 (p), L 23 (p)) ((0.6052, , , ); (1.3951, , , 0); (1.5687, , 0, 0)) (3.569, 1.5, , ),

18 Symmetry 2017, 9, of 21 Z 3 WPLPA(L 31 (p), L 32 (p), L 33 (p)) ((0.6052, , , ); (1.3951, , , 0); (1.5687, , 0, 0)) (3.3113, 2, , 0), Z 4 WPLPA(L 41 (p), L 42 (p), L 43 (p)) ((0.6052, , , ); (1.3951, , , 0); (1.5687, , 0, 0)) (2.6824, , , ), Z 5 WPLPA(L 51 (p), L 52 (p), L 53 (p)) ((1.2326, , , 0); (1.3322, , 0, 0); (0.9336, , , 0)) (3.4985, , , 0). If the DM uses the WPLPG operator, the the aggregated value of the alteratve x s determed based o Equato (16) ( 1, 2, 3, 4, 5). I the same way, we deote the aggregated value of the alteratve x as Z. The results are: Z 1 WPLPG(L 11 (p), L 12 (p), L 13 (p)) ((1.3150, , , 1); (1.5127, , , 1); (1.4887, , 1, 1)) (2.9613, , , 1), Z 2 WPLPG(L 21 (p), L 22 (p), L 23 (p)) ((1.1703, , , ); (1.4738, , , 1); (1.5466, 1.132, 1, 1)) (2.6676, , , ), Z 3 WPLPG(L 31 (p), L 32 (p), L 33 (p)) ((1.3482, , , 1); (1.4024, , , 1); (1.3592, , , 1)) (2.5698, , , 1), Z 4 WPLPG(L 41 (p), L 42 (p), L 43 (p)) ((1.3500, , , 1); (1.2012, , , ); (1.3256, , , 1)) (2.1496, , , ), Z 5 WPLPG(L 51 (p), L 52 (p), L 53 (p)) ((1.3777, , , 1); (1.4482, , 1, 1); (1.3216, , , 1)) (2.6367, , , 1). Step 8: Based o the results of Defto 4, the scores of the alteratve x ca be computed,.e., E(Z ). If the DM uses WPLPA operator to aggregate the decso formato, the scores are determed as follows: E(Z 1 ) ; E(Z 2 ) ; E(Z 3 ) ; E(Z 4 ) ; E(Z 5 )

19 Symmetry 2017, 9, of 21 If the the DM uses WPLPG operator to aggregate the decso formato, the scores are determed as follows: E(Z 1 ) ; E(Z 2 ) ; E(Z 3 ) ; E(Z 4 ) ; E(Z 5 ) Step 9: If the DM uses WPLPA operator, we ca determe the rakg of the scores of the alteratves based o the results of the Step 8. It s show as follows: E(Z 3 ) > E(Z 1 ) > E(Z 5 ) > E(Z 4 ) > E(Z 2 ). That s to say, the orderg of the alteratves s: x 3 > x 1 > x 5 > x 4 > x 2. If the DM uses WPLPG operator, we ca obta the rakg of the scores of the alteratves as follows: E(Z 1 ) > E(Z 3 ) > E(Z 5 ) > E(Z 2 ) > E(Z 4 ). I ths stuato, the orderg of the alteratves s: 5.2. Comparso Aalyss x 1 > x 3 > x 5 > x 2 > x 4. Uder the probablstc lgustc formato, Pag et al. [6] have developed a aggregato-based method for MAGDM. I order to verfy the performace of our proposed methods, we compare our decso results wth Pag et al. [6] based o our llustratve example. Torra [13], Mergó et al. [22] ad Zhag et al. [23] also developed some methods for the lgustc formato ad GDM. Thus, we also compare our results wth the methods of Refs. [12,22,23]. The decso results are show Table 7. Table 7. The decso results of dfferet methods. Method Rak Aggregato-based method of Ref. [6] x 3 > x 1 > x 5 > x 2 > x 4 The method wth HFLWA of Ref. [23] x 3 > x 1 > x 5 > x 2 x 4 The method wth HFLWG of Ref. [23] x 1 > x 2 > x 5 > x 3 > x 4 Max lower operator of Ref. [12] x 3 > x 2 x 5 x 4 > x 1 ILGCIA wth group decso makg of Ref. [22] x 3 > x 2 > x 1 > x 4 > x 5 Our proposed method wth WPLPA x 3 > x 1 > x 5 > x 4 > x 2 Our proposed method wth WPLPG x 1 > x 3 > x 5 > x 2 > x 4 I Table 7, we ca fd the rak result of the method proposed Ref. [6] s: x 3 > x 1 > x 5 > x 2 > x 4. Compared wth the decso results of our proposed method wth WPLPA, the aggregato-based method wth PLTSs ca select the same best caddate,.e., x 3. Meawhle, for the WPLPG, the best caddate s x 1. Uder the result of Ref. [23], HFLWA has the rak: x 3 > x 1 > x 5 > x 2 x 4. Meawhle, the rakg of HFLWG s x 1 > x 2 > x 5 > x 3 > x 4. By usg the max lower operator of Ref. [12], we ca fd the rak s: x 3 > x 2 x 5 x 4 > x 1. For ILGCIA wth group decso makg of Ref. [22], the result s x 3 > x 2 > x 1 > x 4 > x 5. O the MCGDM problems uder lgustc evromet, we troduced our model to acheve the same acceptable performace wth the exstg techques or to mprove upo them. Ulke the exstg models cosdered ths paper, our model cotas probabltes whch ormally help gettg a comprehesve ad accurate preferece formato of the DMs [6]. I Ref. [3], for stace, the developed approaches take all the decsos ad ther relatoshps to accout, ad the decso argumets reforce ad support each

20 Symmetry 2017, 9, of 21 other, but sce probabltes were ot cosdered, the accuracy of preferece formato of the DMs mght be questoable. I addto, wthout the PLTS, t mght ot be easy for the DMs to provde several possble lgustc values over a alteratve or a attrbute. Ths stuato traslates to some kd of lmtato of the model proposed Ref. [3] spte of the power average (PA) volvemet the aggregato process.the PLTSs tself as a theory has some lmtatos. I geeral, WPLPG apples to the average of the rato data ad s maly used to calculate the average growth (or chage) rate of the data. From the trat of Table 6, the WPLPA s much better tha WPLPG. 6. Coclusos Wth respect to the support ad reforcemet amog put argumets wth PLTSs, we troduce PA to the probablstc lgustc evromet. Meawhle, we develop the correspodg ew operators,.e., the PLPA, PLPG, WPLPA ad WPLPG operators. I lght of the PLMCGDM, we descrbe the decso-makg problem ad desg correspodg approaches by employg the WPLPA ad WPLPG. I ths paper, we expaded the appled feld of the orgal PA ad erch the research work of PLTSs. Future research work may focus o explorg the decso-makg mechasms whe the weght formato s ukow or complete ad developg some ew geeralzed aggregato operators of PLTSs. I addto, we also deeply vestgate a more complex case study wth more alteratves ad crtera. Ackowledgmets: Ths work s partally supported by the Natoal Scece Foudato of Cha (Nos , , ), the Fudametal Research Fuds for the Cetral Uverstes of Cha (No. ZYGX2014J100), the Socal Scece Plag Project of the Schua Provce (No. SC15C009) ad the Schua Youth Scece ad Techology Iovato Team (2016TD0013). Author Cotrbutos: Decu Lag desged the reaserach work ad the basc dea. Agbodah Koba aalyzed the data ad fshed the deducto procedure. X He also aalyzed the data ad modfed the expresso. Coflcts of Iterest: The authors declare o coflct of terest. Refereces 1. Yager, R.R. The power average operator. IEEE Tras. Syst. Ma Cyber. Part A Syst. Hum. 2001, 31, Xu, Z.S.; Yager, R.R. Power-Geometrc operators ad ther use group decso makg. IEEE Tras. Fuzzy Syst. 2010, 18, Xu, Y.J.; Mergó, J.M.; Wag, H.M. Lgustc power aggregato operators ad ther applcato to multple attrbute group decso makg. Appl. Math. Model. 2012, 36, Zhou, L.G.; Che, H.Y. A geeralzato of the power aggregato operators for lgustc evromet ad ts applcato group decso makg. Kowl. Based Syst. 2012, 26, Zhu, C.; Zhu, L.; Zhag, X. Lgustc hestat fuzzy power aggregato operators ad ther applcatos multple attrbute decso-makg. If. Sc. 2016, , Pag, Q.; Wag, H.; Xu, Z.S. Probablstc lgustc term sets mult-attrbute group decso makg. If. Sc. 2016, 369, Ba, C.Z.; Zhag, R.; Qa, L.X.; Wu, Y.N. Comparsos of probablstc lgustc term sets for mult-crtera decso makg. Kowl. Based Syst. 2017, 119, Mergó, J.M.; Casaovas, M.; Martíez, L. Lgustc aggregato operators for lgustc decso makg based o the Dempster-Shafer theory of evdece. It. J. Ucerta. Fuzzess Kowl. Based Syst. 2010, 18, Zha, Y.L.; Xu, Z.S.; Lao, H.C. Probablstc lgustc vector-term set ad ts applcato group decso makg wth mult-graular lgustc formato. Appl. Soft Comput. 2016, 49, Lao, H.C.; Xu, Z.S.; Zeg, X.J.; Mergó, J.M. Qualtatve decso makg wth correlato coeffcets of hestat fuzzy lgustc term sets. Kowl. Based Syst. 2015, 76, Lao, H.C.; Xu, Z.S.; Zeg, X.J. Hestat fuzzy lgustc vkor method ad ts applcato qualtatve multple crtera decso makg. IEEE Tras. Fuzzy Syst. 2015, 23, Rodrguez, R.M.; Martez, L.; Herrera, F. Hestat fuzzy lgustc term sets for decso makg. IEEE Tras. Fuzzy Syst. 2012, 20,

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