Hesitant Probabilistic Fuzzy Linguistic Sets with Applications in Multi-Criteria Group Decision Making Problems

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1 Artcle estat Probablstc Fuzzy Lgustc Sets wth Alcatos Mult-Crtera Grou Decso Mag Problems Dheeraj Kumar Josh Ismat Beg * ad Sajay Kumar Deartmet of Mathematcs Statstcs ad Comuter Scece G. B. Pat Uversty of Agrculture ad Techology Patagar Uttarahad 6345 Ida; maths.dj4400@gmal.com D.K.J.; sruhela@hotmal.com S.K. Cetre for Mathematcs ad Statstcal Sceces Lahore School of Ecoomcs Lahore 5300 Pasta * Corresodece: beg@lahoreschool.edu. Receved: 3 February 08; Acceted: 3 March 08; Publshed: 6 March 08 Abstract: Ucertates due to radomess ad fuzzess comrehesvely est cotrol ad decso suort systems. I the reset study we troduce oto of occurrg robablty of ossble values to hestat fuzzy lgustc elemet FLE ad defe hestat robablstc fuzzy lgustc set PFLS for ll structured ad comle decso mag roblem. PFLS rovdes a sgle framewor where both stochastc ad o-stochastc ucertates ca be effcetly hadled alog wth hestato. We have also roosed eected mea varace score ad accuracy fucto ad basc oeratos for PFLS. Weghted ad ordered weghted aggregato oerators for PFLS are also defed the reset study for ts alcatos mult-crtera grou decso mag MCGDM roblems. We roose a MCGDM method wth PFL formato whch s llustrated by a eamle. A real case study s also tae the reset study to ra State Ba of Ida IfoTech Eterrses I.T.C..D.F.C. Ba Tata Steel Tata Motors ad Bajaj Face usg real data. Proosed PFLS-based MCGDM method s also comared wth two FL-based decso mag methods. Keywords: hestat fuzzy set; hestat robablstc fuzzy lgustc set; score ad accuracy fucto; mult-crtera grou decso mag; aggregato oerator. Itroducto Ucertates decso mag roblems are due to ether radomess or fuzzess or by both ad ca be classfed to stochastc ad o-stochastc ucertaty []. Stochastc ucertates every system may be well catured by the robablstc modelg [3]. Although several theores have bee roosed the lterature to deal wth o-stochastc ucertates but amog them fuzzy set theory [45] s etesvely researched ad successfully aled decso mag [6 0]. A etesve lterature s due to Marda et al. [] o the varous fuzzy aggregato oerators roosed last thrty years. Tye- fuzzy sets [5] terval-valued fuzzy set IVFS [4] tutostc fuzzy sets IFS [] ad terval-valued tutostc fuzzy sets IVIFS [3] Pythagorea fuzzy set [4] ad eutrosohc sets [5] are few other etesos of fuzzy sets ractced MCGDM roblems to clude o-stochastc ucertaty ad hestato. Ofte decso maers DMs mult-crtera grou decso mag MCGDM roblems are ot favor of the same assessmet o decso crtera ad rovde dfferet assessmet formato o each crtero. Dffculty of agreeg o a commo assessmet s ot because of marg of error or some ossble dstrbuto as case of IFS ad tye- fuzzy sets. To address ths ssue MCGDM roblems Torra ad Naruawa [6] ad Torra [7] troduced hestat fuzzy set Mathematcs ; do:0.3390/math

2 Mathematcs of 0 FS ad aled MCGDM roblems [89]. Varous etesos of FS e.g. tragular hestat fuzzy set TFS geeralzed hestat fuzzy set GFS terval valued hestat fuzzy set IVFS dual hestat fuzzy set DFS terval valued tutostc hestat fuzzy set IVIFS ad hestat ythagorea fuzzy set were used decso mag roblems [0 8] cosderg decso hestacy ad rortzato amog decso crtera ad develoed a fuzzy grou decso mag method to evaluate comle emergecy resose sustaable develomet. Recetly Garg ad Arora [9] roosed dstace ad smlarty measures-based MCDM method usg dual hestat fuzzy soft set. Qualtatve ad quattatve aalyss of decso crtera wth hestat ad ucerta formato has always bee a mortat ssue for researchers MCGDM roblems. Lmted owledge of decso maers DMs ature of cosdered alteratves ad uredctablty of evets are ma costrats gettg suffcet ad accurate formato about the decso refereces ad decso crtera. May crtera whch are dffcult to be aalyzed quattatvely ca be aalyzed usg lgustc varables [5]. Lgustc varables mrove cosstecy ad fleblty of tradtoal decso mag methods [30] ad hece may researchers [3 45] have roosed use of lgustc varable decso mag roblems. Koba et al. [46] roosed few robablstc lgustc aggregato oerators for decso mag roblem. Garg ad Kumar [47] Lu et al. [48] ad Garg [49] roosed varous aggregato oerators rortzed aggregato oerators for lgustc IFS ad lgustc eutrosohc set ad aled them to MCGDM roblems. L et al. [50] tegrated lgustc sets wth FS to defe hestat fuzzy lgustc set FLS to clude hestacy ad cossteces amog DMs assessmet of a alteratve wth resect to a certa crtero. Re et al. [5] ad Josh & Kumar [5] roosed TOPSIS method MCGDM usg hestat fuzzy lgustc ad IVIFL formato. Recetly few researchers [53 55] have roosed geeralzed sgle-valued eutrosohc hestat fuzzy rortzed aggregato oerators ad lgustc dstrbuto-based decso mag methods usg hestat fuzzy lgustc assessmet for decso mag methods. Probablstc ad fuzzy aroach-based MCGDM method rocess oly ether stochastc or o stochastc ucertaty. Oe of ther major lmtatos s ot to hadle both tyes of ucertates smultaeously. Comrehesve cocurrece of stochastc ad o stochastc ucertaty real lfe roblems attracted researchers to cororate robablty theory wth fuzzy logc. Idea of tegratg fuzzy set theory wth robablstc theory was tated by Lag ad Sog [56] ad Meghdad ad Abarzadeh []. I 005 Lu ad L [57] defed robablstc fuzzy set PFS to hadle both stochastc ad o stochastc ucertates a sgle framewor. To hadle smultaeous occurrece of both stochastc ad o stochastc ucertates wth hestato Xu ad Zhou [58] troduced robablstc hestat fuzzy set PFS. PFS ermts more tha oe membersh degree of a elemet wth dfferet robabltes. Recetly may alcatos of PFS are foud MCGDM roblems [58 65]. Earler all FL-based decso mag methods robabltes of occurrece of elemets are assumed to be equal. Assumto of equal robabltes FL s too hard to be followed by DMs real lfe roblems of decso mag due to ther hestato. For eamle a decso maer rovdes hestat fuzzy lgustc elemet FLE { s } to evaluate the safety level of a vehcle. e or she ths that the safety level assocated wth 0.6 ad 0.4 are the most ad least sutable. owever he or she cotradcts wth ow decso by assocatg equal robablty to each ece FLE { s } wth equal robabltes caot rereset DM s accurate assessmet of decso crtera. Wth ths lmtato reset form of FLS we troduce oto of hestat robablstc fuzzy lgustc set PFLS. Ths ew class of set udertaes both ucertates caused by radomess ad fuzzess the evromet of hestato a sgle framewor. I the reset study we have roosed PFLS wth eected mea varace score ad accuracy fucto ad a few oeratos o ts elemets. We also develo ovel hestat robablstc fuzzy lgustc weghted averagg PFLWA hestat robablstc fuzzy lgustc weghted geometrc PFLWG hestat robablstc fuzzy lgustc ordered weghted averagg

3 Mathematcs of 0 PFLOWA ad hestat robablstc fuzzy lgustc ordered weghted geometrc PFLOWG aggregato oerators to aggregate the PFL formato. A MCGDM method wth PFL formato s roosed. Methodology of roosed MCGDM method s llustrated by a umercal eamle ad also aled o a real case study to ra the orgazatos.. Prelmares I ths secto we brefly revew fudametal cocets ad deftos of hestat fuzzy set lgustc varables hestat fuzzy lgustc set ad hestat robablstc fuzzy set. Defto. [67] Let X be a referece set. A FS A o X s defed usg a fucto returs a subset of [0 ]. Mathematcally t s symbolzed usg followg eresso: where I A X A { I A X} I A X s hestat fuzzy elemet FE havg a set of dfferet values les betwee [0 ]. Defto. [3] Let S { s... t} be a fte dscrete LTS. ere for a lgustc varable ad satsfes the followg characterstcs:. The set s ordered:. Ma{ s s j} s f s s j j j f j s that reresets a ossble value 3. M{ s s j} s f j Xu [66] eteded fte dscrete LTS S { s... t} to cotuous LTS S { s s0 s st [0 t]} to coserve all the rovded formato. A LTS s orgal f otherwse t s called vrtual. Defto 3. [50] Let X be the referece set ad mathematcal object of followg form: ere membersh degrees that belogs to s. s S s S.A hestat fuzzy lgustc set A X s a A { s ha X} h A s a set of ossble fte umber of values belogg to [0] ad deotes the ossble Defto 4. [58] Let X be the referece set. A PFS F o X s a mathematcal object of followg form: ere h s set of elemets eressg the hestat fuzzy formato wth robabltes to the set 0...# h umber of ossble elemets h [0] are P corresodg robabltes wth codto P # h s. } { h X 3. estat Probablstc Fuzzy Lgustc Set PFLS ad estat Probablstc Fuzzy Lgustc Elemet PFLE Qualtatve ad quattatve aalyss of decso crtera wth hestat s always bee a mortat ssue for researchers MCGDM roblems. Earler classfcato of fuzzy sets hestat fuzzy set [67] hestat fuzzy lgustc set [50] ad robablstc hestat fuzzy [58] are ot caable to deal wth fuzzess hestacy ad ucertaty both qualtatvely ad quattatvely. Keeg md the lmtatos of FLEs ad to fully descrbe recous formato rovded by DMs; our am s to roose a ew class of set called PFLS. Ths set ca easly descrbe stochastc 3

4 Mathematcs of 0 ad o-stochastc ucertates wth hestat formato usg both qualtatve ad quattatve terms. I ths secto we also develo eected mea varace score ad accuracy fucto of PFLEs alog wth a comarso method. Some basc oeratos of PFLEs are also defed ths secto. Defto 5. Let X ad S be the referece set ad lgustc term set. A PFLS o X s a mathematcal object of followg form: h s h ere { h X} 4 s S deotg the hestat fuzzy lgustc formato wth robabltes to the set. ere # # h h. s the umber of ossble elemets h ad [0] We call h PFLE ad s set of all FEs. h s set of some elemets 0 = # s the hestat robablty of As a llustrato of Defto 5 we assume two PFLEs h [ s { }] h [ s { }] o referece set X { y}. y 3 A object [ s { } y s3 { }] reresets a PFLS. It s mortat to ote that f the robabltes of the ossble values PFLEs are equal.e. = the PFLE reduced to FLE. = #h 3.. Some Basc Oeratos o estat Probablstc Fuzzy Lgustc Elemet PFLEs Based o oeratoal rules of hestat fuzzy lgustc set [50] ad hestat robablstc set [6] we roose followg oeratoal laws for j j j j y h s y h the h s h for some 0 j j j j h s h for some 0 j j j j y 3 h h s h 4 h h s h y 5 h h s y h j j Ma j 6 h h s y h j j M j j j j j j h ad h s h ad Usg defto of ad t ca be easly roved that h h ad h h are commutatve. I order to show that h h h h h h h h ad h h are aga PFLE we assume that h [ s { }] ad h y [ s3 { }] are two PFLEs o referece set X { y} ad erform the oerato laws as follows: [ s h h [ s 5 h 4 h [ s4 { }] { }] { }]

5 Mathematcs of 0 h [ s 5 h h [ s 5 h h [ s 5 h { }] { }] { }] 3.. Score ad Accuracy Fucto for estat Probablstc Fuzzy Lgustc Elemet PFLE Comarso s a dsesable ad s requred f we ted to aly PFLE decso mag ad otmzato roblems. ece we defe eected mea varace score ad accuracy fucto of PFLE ths sub secto as follows: Defto 6. Eected mea E h ad varace h s h are defed as follows: E h # h # h V h for a PFLE 5 V # h h E h 6 Defto 7. Score fucto S h ad accuracy fucto A h fora PFLE h s h are defed as follows: S h E h s 7 A h V h s 8 Usg score ad accuracy fuctos two PFLEs h h ca be comared as follows: S h S h y the h h y S h S h the h h If y y If 3 If S h S h A y h A h y the h h y h A h y the h h y h A h y the h h y a If A b If A c If y As a llustrato of Deftos 6 ad 7 we comare two PFLEs h [ s { }] ad h y [ s3 { }] usg score ad accuracy fuctos as follows: E h 0.* *0. 0.5*0.5 / 3 = 0.3 E h 0.4*0. 0.5*0.4 0.*0.4 / 3 = 0. y * V h * *0.5 / V h y * * *0.4 / S h s *0.* *0. 0.5*0.5/ 3 s0. 6

6 Mathematcs of 0 S s 3 / s S h y * 0. 4* * * A s * * * *0.5 / 3 s A y s 3* * * *0.4 / 3 s h y S h h y h Sce therefore. Dfferet PFLEs may have dfferet umber of PFNs. To mae them equal umbers we eted PFLEs utl they have same umber of PFNs. It ca be eteded accordg to DMs rs behavor. 4. Aggregato Oerators for estat Probablstc Fuzzy Lgustc Set PFLS I grou decso mag roblems a meratve tas s to aggregate the assessmet formato obtaed from DMs about alteratves agast each crtero. Varous aggregato oerators for FLS [850] ad PFS [ ] have bee develoed the ast few decades. As we roose PFLS for MCGDM roblems we also develo few aggregatos oerators to aggregate formato the form of PFLEs. I ths secto we defe PFLW ad PFLOW oerators. 4.. estat Probablstc Lgustc Fuzzy Weghted Aggregato Oerators h s h.. be collecto of PFLEs. estat Let robablstc fuzzy lgustc weghted averagg PFLWA oerator ad hestat robablstc fuzzy lgustc weghted geometrc PFLWG oerator are defed as follows: Defto 8. PFLWA s a mag such that s PFLWA Defto 9. PFLWG oerator s a mag such that s PFLWG where... s weght vector of... wth [0 ] ad the robablty of s the PFLEs.... I artcular f... the PFLWA ad PFLWG oerator are reduced to followg hestat robablstc fuzzy lgustc averagg PFLA oerator ad hestat robablstc fuzzy lgustc geometrc PFLG oerator resectvely: T PFLA...

7 Mathematcs of s PFLG s Lemma. [7] Let ad the ad equalty holds f ad oly f.... Theorem. Let.. } { h s h be collecto of PFLEs. Let... be weght vector of... wth [0] ad the PFLWG... PFLWA... PFLG... PFLA... Proof. Usg Lemma l we have followg equalty for ay... Thus we ca obta the followg equalty: s s Usg defto of score fucto # # h s h h S we have PFLWG.... PFLWA... Smlarly t ca be roved that PFLG.... PFLA estat Probablstc Fuzzy Lgustc Ordered Weghted Aggregato Oerators Xu ad Zhou [58] defed ordered weghted averagg ad geometrc aggregato oerators to aggregate hestat robablstc fuzzy formato for MCGDM roblems. I ths sub secto we roose hestat robablstc fuzzy lgustc ordered weghted averagg PFLOWA oerator ad hestat robablstc fuzzy lgustc ordered weghted geometrc PFLOWG oerators.

8 Mathematcs of 0 Let h { s h }..... the PFLEs... be collecto of PFLEs s weght vector of wth [0] ad th be the largest of. Let s the robablty of s the robablty of be the largest of. We develo the followg two ordered weghted aggregato oerators: ad Defto 0. PFLOWA oerator s a mag such that s PFLOWA Defto. PFLOWG oerator s a mag such that PFLOWG..... s... 4 Smlar to Theorem the above ordered weghted oerators have the relatosh below: PFLOWG... PFLOWA Proertes of Proosed Weghted ad Ordered Weghted Aggregato Oerators Followg are few roertes of roosed weghted ad ordered weghted aggregato oerators that mmedately follow from ther deftos. Proerty. Mootocty. Let... ad... be two collectos of PFLNs f for all I = the PFLWA... PFLWA... PFLWG... PFLWG... PFLOWA... PFLOWA... PFLOWG... PFLOWG... Proerty. Idemotecy. Let.. the PFLWA... PFLWG... PFLOWA... PFLOWG... Proerty 3. Boudedess. All aggregato oerators le betwee the ma ad m oerators: m... PFLWA... ma... m... PFLWG... ma... m... PFLOWA... ma... m... PFLOWG... ma...

9 Mathematcs of 0 5. Alcato of estat Probablstc Fuzzy Lgustc Set to Mult-Crtera Grou Decso Mag MCGDM I ths secto we roose a MCGDM method wth hestat robablstc fuzzy lgustc formato. Let } be set of alteratves to be raed by a grou of DMs { D D... D } T agast crtera { C C... C}. w w w... s the weght vector of crtera wth the codto 0 w j j ad { A A j w j A m w ~. j m } t.. s PFL decso matr where { s h T deotes PFLE whe alteratve thdm uder the crtera t t C j t t A s evaluated by. If two or more decso maers rovde the same value the the value comes oly oce decso matr. Algorthm of roosed PFLS-based MCGDM method cludes followg stes: Ste : Costruct PFL decso matrces =... m; j =... ~ j m accordg to the refereces formato rovded by the DMs about the alteratve A uder the crtera Cj deoted by PFLE h { s h } t.. T t t t t. Ste : Use the roosed aggregato oerators PFLWA ad PFLWG gve Secto 3 to aggregate dvdual hestat robablstc fuzzy lgustc decso matr formato rovded by each decso maer to a sgle PFL decso matr ~ j m =... m; j =.... Ste 3: Calculate the overall crtera value for each alteratve A.. m by alyg the PFLWA ad PFLWG aggregato oerator as follows:... m... m PFLWA C C PFLWA C PFLWG C PFLWG C C C...C C...C...C...C C C C C...C...C...C...C Ste 4: Use score or accuracy fuctos to calculate the score values S h ad accuracy values A h of the aggregated hestat robablstc fuzzy lgustc referece values.. m. Ste 5: Ra all the alteratves A.. m accordace wth S h or A h.. m. 6. Illustratve Eamle m m C C m m...c...c m m A eamle s udertae ths secto to uderstad the mlemetato methodology of roosed MCGDM method wth PFL formato. Further a real case study s doe to ra orgazatos usg roosed MCGDM method. We also comare roosed method wth estg PFL-based MCGDM methods roosed by L et al. [50] ad Zhou et al. [68]. Eamle. Suose that a grou of three decso maers D D D3ted to ra four alteratves A A A3 A4o the bass of three crtera C C C3. All DMs are cosdered equally mortat ad equal weghts are assged to them. Each DM rovdes evaluato formato of each alteratve uder each crtero form of PFLEs wth followg LTS: S{s0 = etremely oor s = very oor s = oor s 5 = very good = etremly good}. s 6 s 3 = far s 4 = good

10 Mathematcs of 0 Ste : PFL decso matrces are costructed accordg to refereces formato rovded by DMsD D ad D3 about the alteratve A = 3 4 uder the crtera Cj = 3. Tables 3 rereset PFL evaluato matrces rovded by DMsD D ad D3. Table. estat robablstc fuzzy lgustc PFL decso matr ~ rovded by D. C C C3 A {s } {s } {s 0.4.0} A {s } {s } {s } A3 {s } {s } {s } A4 {s } {s 0..0} {s } Table. PFL decso matr ~ rovded by D. C C C3 A {s 0.4.0} {s } {s 0.8.0} A {s } {s } {s } A3 {s } {s } {s 0..0} A4 {s } {s } {s } Table 3. PFL decso matr ~ 3 rovded by D3. ~ Ste : Aggregate C C C3 A {s } {s4 0..0} {s } A {s } {s } {s } A3 {s } {s } {s } A4 {s } {s } {s } ~ ~ ad ~ 3 to a sgle PFL decso matr =...4; j=...3 usg PFLWA ad PFLWG oerators. j 43 Followg s the samle comutato rocess of aggregato of PFLEs usg roosed PFLWA ad PFLWG oerators. h 3 h h to a sgle 3 PFLWA h h h [{s }{s 0.4.0}{s {s /3 {[ ^ /3 * - 0.4^ /3 * - 0.^ /3] {[ ^ /3 * - 0.4^ /3 * - 0.^ /3] {[ ^ /3 * - 0.4^ /3 * - 0.4^ /3] {[ ^ /3 * - 0.4^ /3 * - 0.4^ /3] {[ ^ /3 * - 0.4^ /3 * - 0.^ /3] {[ ^ /3 * - 0.4^ /3 * - 0.4^ /3] {[ ^ /3 * - 0.4^ /3 * - 0.^ /3] {[ ^ /3 * - 0.4^ /3 * - 0.4^ /3] }] 0.3 ** 0.4} 0.3 ** 0.4} 0.3 ** 0.4} 0.3 ** 0.6} 0.7 ** 0.4} 0.7 ** 0.6} 0.7 ** 0.4} 0.7 ** 0.6} [{s }] [{s }] 3 PFLWG h h h [{s }{s 0.4.0}{s }]

11 Mathematcs of 0 {s ^/3 ^/3 ^/3 [{s 3.3 {[0.4^ /3 * 0.4^ /3 * 0.^ {[0.4^ /3 * 0.4^ /3 * 0.^ {[0.4^ /3 * 0.4^ /3 * 0.4^ {[0.4^ /3 * 0.4^ /3 * 0.4^ {[0.5^ /3 * 0.4^ /3 * 0.^ {[0.5^ /3 * 0.4^ /3 * 0.4^ {[0.5^ /3 * 0.4^ /3 * 0.^ {[0.5^ /3 * 0.4^ /3 * 0.4^ /3] /3] /3] /3] /3] /3] /3] /3] 0.3 ** 0.4} 0.3 ** 0.4} 0.3 ** 0.6} 0.7 ** 0.4} 0.7 ** 0.6} 0.7 ** 0.4} 0.7 ** 0.6} 0.3 ** 0.4} }] [{s }] Smlarly other PFLEs of PFL decso matrces Tables 3 are aggregated to the sgle PFL decso matr usg PFLWA ad PFLWG oerators ad show Tables 4 ad 5. Table 4. Aggregated hestat robablstc fuzzy lgustc elemet PFLE grou decso matr usg hestat robablstc fuzzy lgustc weghted averagg PFLWA oerator. A A A3 A4 C C C3 {s } {s } {s } {s } {s } {s } {s } {s } {s 0.6.0} {s } {s } {s } Table 5. Aggregated PFLE grou decso matr usg hestat robablstc fuzzy lgustc weghted geometrc PFLWG oerator. A A A3 A4 C C C3 {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } Ste 3: Aggregate assessmet of each alteratve A 34 agast each crtera s calculated usg the PFLWA ad PFLWG aggregato oerators wth crtera weghts w 0.4 w 0.3 w3 0.4 as follows: PFLWA CCC3 s s [.3.7 s s 0.6.0] PFLWG C CC3

12 Mathematcs of 0 [s s s3.6 s ] Smlarly other elemets of PFL decso matrces Tables 4 ad 5 are aggregated to the overall PFL decso matr usg PFLWA ad PFLWG oerators ad show Tables 6 ad 7. Table 6. Collectve PFLE grou decso matr usg PFLWA oerator..a A A3 A4 {s {s {s } {s } Table 7. Collectve PFLE grou decso matr usg PFLWG oerator. A A A3 A4 {s {s {s } {s

13 Mathematcs of } Ste 4: The score values S h 34 ad show as follows Table 8: of the alteratves A 34 are calculated Table 8. Score values for the alteratves usg PFLWA ad PFLWG oerators. Score PFLWA PFLWG S S S h S S S h S S S h h 3 4 S S0.090 S Ste 5. Fally alteratves A = 3 4 are raed accordace wth score values S h ad show Table 9. Table 9. Rag of alteratves usg roosed PFLWA ad PFLWG oerators. Method Rag Best/Worst Usg PFLWA oerator A > A4 > A3 > A A/A Usg PFLWG oerator A > A4 > A3 > A A/A Table 9 cofrms that usg both roosed PFLWA ad PFLWG oerators best ad worst alteratves are A ad A resectvely. 6.. A Real Case Study A real case study s udertae to ra seve orgazatos; State Ba of Ida A IfoTech Eterrses A ITC A3.D.F.C. Ba A4 Tata Steel A5 Tata Motors A6 ad Bajaj Face A7 o the bass of ther erformace agast followg four crtera.. Eargs er share EPS of comay C. Face value C 3. Boo value C3 4. P/C rato Put-Call Rato of comay C4 I ths real case study C C ad C3 are beeft crtera whle C4 s cost crtero. Real data for each alteratve agast each crtero are retreved from htt:// from date to Table 0 shows ther average values. Table 0. Average of actual umercal value of crtera. C C C3 C4 A A A A A A A

14 Mathematcs of 0 To costruct hestat fuzzy decso matr Table we use the method roosed by Bsht ad Kumar [69] ad fuzzfy Table 0 usg tragular ad Gaussa membersh fuctos. Table. estat fuzzy decso matr C C C3 C4 A A A A A A A Probabltes are assocated wth elemets of hestat fuzzy decso matr Table to covert t to robablstc hestat fuzzy decso matr I [ I P ]. Probabltes whch are assocated wth frst row of hestat fuzzy decso matr Table are as follows: Smlarly all elemets of hestat fuzzy decso matr are assocated wth robabltes ad robablstc hestat fuzzy decso matr Table s obtaed. Table. Probablstc estat fuzzy decso matr. j j j m A A A3 A4 A5 A6 A7 C C C3 C4 { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } Followg table Table 3 shows hestat robablstc fuzzy lgustc decso matr. Table 3. estat robablstc fuzzy lgustc decso matr. A A C C C3 C4 {s } {s } {s } {s } {s } {s } {s } {s }

15 Mathematcs of 0 A3 A4 A5 A6 A7 {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } {s } Ste : Assessmet of each alteratve A = agast each crtera Cj = 3 4 s aggregated usg PFLWA aggregato oerator Equato 9 as follows: PFLWA CC C3C4 {s }{s {s } {s {s } } I the aggregato of assessmet of the alteratves all crtera are cosdered of equal weght of 0.5. Smlarly other elemets of PFL decso matr Table 3 are aggregated ad followg collectve PFL decso matr Table 4 s obtaed. Table 4. Collectve hestat robablstc fuzzy lgustc decso matr A A A3 A4 A5 A6 A7 {s } {s } {s } {s } {s } {s } {s } Ste 3: The score values S h of the alteratves A are calculated usg Equato 7 ad are show as follows: S h S0. 09 S h S 3 S h S0. 09

16 Mathematcs of S h S S h S Ste 4: Fally alteratves A are raed as S h S S h S A 4 A6 A A A5 A7 A 3 accordace wth score values S h. 6.. Comaratve Aalyss I ths secto we comare roosed PFL-based MCGDM methods wth estg FL-based methods. We aly the roosed method o two dfferet roblems whch are adated from Zhou et al. 06 ad L et al. 04 ad comare the rag results. I order to aly the roosed PFL-based MCGDM o the eamles tae by both L et al. 04 ad Zhou et al. 06 we have cosdered robablty of each elemet of FL decso matrces as uty Comarso I comarso methodology of roosed PFL-based MCGDM method s aled o the followg FL decso matr Table 5 of the roblem tae by L et al. [50]. Table 5. estat fuzzy lgustc decso matr [50]. G G G3 G4 A <s > <s > <s > <s > A <s > <s > <s > <s > A3 <s > <s > <s > <s > A4 <s > <s > <s > <s > A5 <s > <s > <s > <s > Followg table Table 6 shows the rag results of the alteratves whch are obtaed usg roosed PFL ad estg FL-based MCDM method of L et al. [50]. Table 6. Comarso of rag of alteratves. Method Rag Best Alteratve/Worst Alteratve Proosed A4 > A > A3 > A > A5 A4/A5 L et al. [50] A4 > A3 > A > A > A5 A4/A5 O alyg the roosed MCGDM method o rag roblem whch s adated from L et al. 04 A4 ad A5 are raed aga as the best ad the worst alteratves resectvely Comarso I comarso the methodology of roosed PFL-based MCGDM method s aled o the followg FL decso matr Table 7 of the roblem tae by Zhou et al. [67]. Table 7. The secal lgustc hestat fuzzy decso matr [67]. C C C3 C4 X <s > <s > <s > <s4 0.4> X <s > <s > <s4 0.7> <s > X3 <s > <s5 0.4> <s > <s3 0.6> X4 <s > <s4 0.6> <s > <s3 0.8> X5 <s > <s > <s > <s5 0.7> Followg table Table 8 shows the rag results of the alteratves whch are obtaed usg roosed PFL ad estg FL-based MCDM method of Zhou et al. [67].

17 Mathematcs of 0 Table 8. Comarso of rag of alteratves. Method Rag Best Alteratve/Worst Alteratve Proosed X5 > X3 > X > X > X4 X5/X4 Zhou et al. [67] X5 > X3 > X > X > X4 X5/X4 O alyg the roosed MCGDM method o rag roblem adated from Zhou et al. [67] X5 ad X4 are raed aga as the best ad the worst alteratves resectvely. As there s o chage foud the rag results of the alteratves both the comarsos t cofrms that the roosed PFL-based MCGDM method s also sutable wth FL formato. 7. Coclusos Ucertates due to radomess ad fuzzess both occur the system smultaeously. I certa decso mag roblem DMs refer to aalyze the alteratves agast decso crtera qualtatvely usg lgustc terms. I ths aer we have roosed hestat robablstc fuzzy lgustc set PFLS to tegrate hestat fuzzy lgustc formato wth robablty theory. Promet characterstc of PFLS s to assocate occurrg robabltes to FLEs whch maes t more effectve tha FLS. We have vestgated the eected mea varace score ad accuracy fucto ad basc oeratos for PFLEs. We have also defed PFLWA PFLWG PFLOWA ad PFLOWG aggregato oerators to aggregate hestat robablstc fuzzy lgustc formato. A ovel MCGDM method usg PFLWA PFLWG PFLOWA ad PFLOWG s also roosed the reset study. Advatage of roosed PFLS-based MCGDM method s that t assocates robabltes to FLE whch maes t cometet eough to hadle both stochastc ad o-stochastc ucertates wth hestat formato usg both qualtatve ad quattatve terms. Aother advatage of roosed MCGDM method s that t allows DMs to use ther tutve ablty to judge alteratves agast crtera usg robabltes. Ths s also mortat to ote that the roosed method ca also be used wth FL formato f DMs assocate equal robabltes to FLE. Methodology of roosed PFL-based MCGDM method s llustrated by a eamle. A real case study to ra the orgazatos s also udertae the reset wor. Eve though roosed PFL-based MCGDM method cludes both stochastc ad o-stochastc ucertates alog wth hestato but to determe robabltes of membersh grades lgustc fuzzy set s very dffcult real lfe roblem of decso mag. Proosed PFL-based MCGDM method wll be effectve whe ether DMs are eert of ther feld or they have re-defed robablty dstrbuto fucto so that the arorate robabltes could be assged. Alcatos of roosed PFLS wth Pythagorea membersh grades ca also be see as the scoe of future research decso mag roblems as a ehacemet of the methods roosed by Garg [49]. Author Cotrbutos: Dheeraj Kumar Josh ad Sajay Kumar defed PFLS ad studed ts roertes. They together develoed MCGDM method usg PFL formato. Ismat Beg cotrbuted verfyg the roof of Theorem ad the roertes of aggregato oerators. All authors equally cotrbuted the research aer. Coflcts of Iterest: Authors declare o coflcts of terest. Refereces. Meghdad A..; Abarzadeh-T M.R. Probablstc fuzzy logc ad robablstc fuzzy systems. I Proceedgs of the 0th IEEE Iteratoal Coferece o Fuzzy Systems Melboure Australa 5 December 00; Volume Valavas K.P.; Sards G.N. Probablstc modelg of tellget robotc systems. IEEE Tras. Robot. Autom Pdre J.C.; Carrllo C.J.; Lorezo A.E.F. Probablstc model for mechacal ower fluctuatos asychroous wd ars. IEEE Tras. Power Syst Zadeh L.A. Fuzzy sets. If. Cotrol

18 Mathematcs of 0 5. Zadeh L.A. Fuzzy logc ad aromate reasog. Sythese Lee L.W.; Che S.M. Fuzzy decso mag ad fuzzy grou decso mag based o lelhood-based comarso relatos of hestat fuzzy lgustc term sets. J. Itell. Fuzzy Syst Wag.; Xu Z. Admssble orders of tycal hestat fuzzy elemets ad ther alcato ordered formato fuso mult-crtera decso mag. If. Fuso Lu J.; Che.; Zhou L.; Tao Z. Geeralzed lgustc ordered weghted hybrd logarthm averagg oerators ad alcatos to grou decso mag. It. J. Ucerta. Fuzz. Kowl.-Based Syst Lu J.; Che.; Xu Q.; Zhou L.; Tao Z. Geeralzed ordered modular averagg oerator ad ts alcato to grou decso mag. Fuzzy Sets Syst Yoo K.P.; wag C.L. Multle Attrbute Decso Mag: A Itroducto; Sage Publcatos: New Yor NY USA 995; Volume 04.. Marda A.; Nlach M.; Zavadsas E.K.; Awag S.R.; Zare.; Jamal N.M. Decso mag methods based o fuzzy aggregato oerators: Three decades of revew from 986 to 08. It. J. If. Techol. Decs. Ma. 07 do:0.4/s x.. Ataassov K.T. Itutostc fuzzy sets. Fuzzy Sets Syst Ataassov K.T.; Gargov G. Iterval-valued tutostc fuzzy sets. Fuzzy Sets Syst Yager R.R. Pythagorea membersh grades multcrtera decso mag. IEEE Tras. Fuzzy Syst Majumdar P. Neutrosohc Sets ad Its Alcatos to Decso Mag. I Comutatoal Itellgece for Bg Data Aalyss. Adatato Learg ad Otmzato; Acharjya D. Dehur S. Sayal S. Eds.; Srger: Cham Swtzerlad 05; Volume Torra V.; Naruawa Y. O hestat fuzzy sets ad decso. I Proceedgs of the 8th IEEE Iteratoal Coferece o Fuzzy Systems Jeju Islad Korea 0 4 August 009; Torra V. estat fuzzy sets. It. J. Itell. Syst Xa M.; Xu Z. estat fuzzy formato aggregato decso mag. It. J. Aro. Reaso Farhada B.; Xu Z. Dstace ad aggregato-based methodologes for hestat fuzzy decso mag. Cog. Comut Qa G.; Wag.; Feg X. Geeralzed hestat fuzzy sets ad ther alcato decso suort system. Kowl.-Based Syst Peg J.J.; Wag J.Q.; Wag J.; Yag L.J.; Che X.. A eteso of ELECTRE to mult-crtera decso-mag roblems wth mult-hestat fuzzy sets. If. Sc Che S.W.; Ca L.N. Iterval-valued hestat fuzzy sets. Fuzzy Syst. Math Yu D. Tragular hestat fuzzy set ad ts alcato to teachg qualty evaluato. J. If. Comut. Sc Zhu B.; Xu Z.; Xa M. Dual hestat fuzzy sets. J. Al. Math Zhag Z. Iterval-valued tutostc hestat fuzzy aggregato oerators ad ther alcato grou decso-mag. J. Al. Math Josh D.; Kumar S. Iterval-valued tutostc hestat fuzzy Choquet tegral based TOPSIS method for mult-crtera grou decso mag. Eur. J. Oer. Res Garg. estat Pythagorea fuzzy sets ad ther aggregato oerators multle attrbute decso mag. It. J. Ucerta. Quatf. 08 do:0.65/it.j.ucertatyquatfcato Q X.-W.; Zhag J.-L.; Zhao S.-P.; Lag C.-Y. Taclg comle emergecy resose solutos evaluato roblems sustaable develomet by fuzzy grou decso mag aroaches wth cosderg decso hestacy ad rortzato amog assessg crtera. It. J. Evro. Res. Publc ealth do:0.3390/jerh4065.

19 Mathematcs of 0 9. Garg.; Arora R. Dstace ad smlarty measures for dual hestat fuzzy soft sets ad ther alcatos multcrtera decso mag roblem. It. J. Ucerta. Quatf Martíez L.; Rua D.; errera F.; Wag P.P. Lgustc decso mag: Tools ad alcatos. If. Sc Xu Z. A teractve rocedure for lgustc multle attrbute decso mag wth comlete weght formato. Fuzzy Otm. Decs. Ma errera F.; Martíez L. A -tule fuzzy lgustc reresetato model for comutg wth words. IEEE Tras. Fuzzy Syst errera F.; errera-vedma E.; Aloso S.; Chclaa F. Comutg wth words ad decso mag. Fuzzy Otm. Decs. Ma La J.; Su Q.; Che Q.; Wag Q. Grou decso mag based o duced ucerta lgustc OWA oerators. Decs. Suort Syst Beg I.; Rashd T. TOPSIS for hestat fuzzy lgustc term sets. It. J. Itell. Syst Rodríguez R.M; Martıez L.; errera F. A grou decso mag model dealg wth comaratve lgustc eressos based o hestat fuzzy lgustc term sets. If. Sc Yue K.K.F. Combg comoud lgustc ordal scale ad cogtve arwse comarso the rectfed fuzzy TOPSIS method for grou decso mag. Fuzzy Otm. Decs. Ma Zhag Z.; Wu C. estat fuzzy lgustc aggregato oerators ad ther alcatos to multle attrbute grou decso mag. J. Itell. Fuzzy Syst Beg I.; Rashd T. Grou decso mag usg comaratve lgustc eresso based o hestat tutostc fuzzy sets. Al. Al. Math. It. J Wag J.Q.; Wag D.D.; Zhag.Y.; Che X.. Mult-crtera grou decso mag method based o terval -tule lgustc formato ad Choquet tegral aggregato oerators. Soft Comut Mergó J.M.; Palacos-Marqués D.; Zeg S. Subjectve ad objectve formato lgustc mult-crtera grou decso mag. Eur. J. Oer. Res Beg I.; Rashd T. estat -tule lgustc formato multle attrbutes grou decso mag. J. Itell. Fuzzy Syst Zhou W.; Xu Z. Geeralzed asymmetrc lgustc term set ad ts alcato to qualtatve decso mag volvg rs aettes. Eur. J. Oer. Res De Mao C.; Feza G.; Loa V.; Orcuol F. Lgustc fuzzy cosesus model for collaboratve develomet of fuzzy cogtve mas: A case study software develomet rss. Fuzzy Otm. Decs. Ma. 07 ress do:0.007/s Gao J.; Xu Z.; Lao. A dyamc referece ot method for emergecy resose uder hestat robablstc fuzzy evromet. It. J. Fuzzy Syst Koba A.; Lag D.; e X. Probablstc lgustc ower aggregato oerators for mult-crtera grou decso mag. Symmetry do:0.3390/sym Garg.; Kumar K. Some aggregato oerators for lgustc tutostc fuzzy set ad ts alcato to grou decso-mag rocess usg the set ar aalyss. Arba J. Sc. Eg. 07 do:0.007/s Lu P.; Mahmood T.; Kha Q. Mult-attrbute decso-mag based o rortzed aggregato oerator uder hestat tutostc fuzzy lgustc evromet. Symmetry do:0.3390/sym Garg. Lgustc Pythagorea fuzzy sets ad ts alcatos mult attrbute decso mag rocess. It. J. Itell. Syst. 08 do:0.00/t L R.; Zhao X.; We G. Models for selectg a ERP system wth hestat fuzzy lgustc formato. J. Itell. Fuzzy Syst Re F.; Kog M.; Pe Z. A ew hestat fuzzy lgustc toss method for grou mult-crtera lgustc decso mag. Symmetry do:0.3390/sym9089.

20 Mathematcs of 0 5. Josh D.; Kumar S. Traezum cloud TOPSIS method wth terval-valued tutostc hestat fuzzy lgustc formato. Graul. Comut. 07 do:0.007/s Wu Y.; L C.-C.; Che X.; Dog Y. Grou decso mag based o lgustc dstrbuto ad hestat assessmet: Mamzg the suort degree wth a accuracy costrat. If. Fuso Wag R.; L Y. Geeralzed sgle-valued eutrosohc hestat fuzzy rortzed aggregato oerators ad ther alcatos to multle crtera decso-mag. Iformato do:0.3390/fo Garg.; Nacy. Lgustc sgle-valued eutrosohc rortzed aggregato oerators ad ther alcatos to multle-attrbute grou decso-mag. J. Ambet Itell. umaz. Comut. 08 do:0.007/s Lag P.; Sog F. What does a robablstc terretato of fuzzy sets mea? IEEE Tras. Fuzzy Syst Lu Z.; L.X. A robablstc fuzzy logc system for modelg ad cotrol. IEEE Tras. Fuzzy Syst Xu Z.; Zhou W. Cosesus buldg wth a grou of decso maers uder the hestat robablstc fuzzy evromet. Fuzzy Otm. Decs. Ma ao Z.; Xu Z.; Zhao.; Su Z. Probablstc dual hestat fuzzy set ad ts alcato rs evaluato. Kowl.-Based Syst Zhou W.; Xu Z. Grou cosstecy ad grou decso mag uder ucerta robablstc hestat fuzzy referece evromet. If. Sc. 07 do:0.06/j.s ress. 6. Zhou W.; Xu Z. Eected hestat VaR for tal decso mag uder robablstc hestat fuzzy evromet. Al. Soft Comut Dg J.; Xu Z.; Zhao N. A teractve aroach to robablstc hestat fuzzy mult-attrbute grou decso mag wth comlete weght formato. J. Itell. Fuzzy Syst L J.; Wag J.Q. Mult-crtera outrag methods wth hestat robablstc fuzzy sets. Cog. Comut Zhag S.; Xu Z.; e Y. Oeratos ad tegratos of robablstc hestat fuzzy formato decso mag. If. Fuso Wag Z.-X.; L J. Correlato coeffcets of robablstc hestat fuzzy elemets ad ther alcatos to evaluato of the alteratves. Symmetry Xu Z. A method based o lgustc aggregato oerators for grou decso mag wth gustc referece relatos. If. Sc Gou X.; Lao.; Xu Z.; errera F. Double herarchy hestat fuzzy lgustc term set ad MULTIMOORA method: A case of study to evaluate the mlemetato status of haze cotrollg measures. If. Fuso Zhou.; Wag J.Q.; Zhag.Y.; Che X.. Lgustc hestat fuzzy mult-crtera decso-mag method based o evdetal reasog. It. J. Syst. Sc Bsht K.; Kumar S. Fuzzy tme seres forecastg method based o hestat fuzzy sets. Eert Syst. Al by the authors. Submtted for ossble oe access ublcato uder the terms ad codtos of the Creatve Commos Attrbuto CC BY lcese htt://creatvecommos.org/lceses/by/4.0/.