htt://www.ewtheory.org ISSN: 249-402 Receve: 0.2.207 Publshe: 4.0.208 Year: 208, Number: 20, Pages: 27-47 Orgal rtcle O Dstaces a Smlarty Measures betwee Two Iterval Neutrosohc Sets Su hattacharyya kas Kol Roy Pak Majumar,* <su.mathematca@gmal.com> <bkashroy235@gmal.com> <majumar2@reffmal.com> M.U.C Wome s College, Mathematcs Deartmet, 7304, urwa, Ia. bstract Iterval Neutrosohc set INS) s a stace of a Neutrosohc set a also a emergg tool for ucerta ata rocessg real scetfc a egeerg alcatos. I ths aer, several stace a smlarty measures betwee two Iterval Neutrosohc sets have bee scusse. Dstaces a smlartes are very useful techques to eterme teractg segmets a ata set. Here we have also show a alcato of our smlarty measures solvg a multcrtera ecso makg metho base o INS s. Fally, we take a llustratve eamle from [4] to aly the roose ecso makg metho. We use the stace as well as the smlarty measures betwee each alteratve a eal alteratve to form a rakg orer a also to f the best alteratve. We comare the obtae results wth the estg result [4] a also reveal the best stace a smlarty measure to f the best alteratve a also ot out the best alteratve. Keywors Iterval Neutrosohc Set, Dstace, Smlarty Measure, Multcrtera Decso Makg.. Itroucto s far as the laws of Mathematcs refer to realty, they are ot certa; a as far they are certa, they o ot referr to realty. lbert Este. Ucertaty s a commo heomeo our aly lfe; because our real or aly lfe we have to take accout a lot of ucertates. From cetures, umerous theores have bee eveloe both Scece a Phlosohy to uersta a rereset the features of ucertaty. Probablty theory a stochastc techques are such theores, whch were eveloe early eghteeth cetury a robablty was the sole techque to hale a certa tye of ucertaty calle Raomess. ut there are several other ks of ucertates, such as vagueess, mrecso, clouess, hazess, ambguty, varety etc. It s geerally agree that the most mortat veto the evoluto of the cocet of ucertaty was mae by Zaeh 965, whe he coe the theory of Fuzzy sets [7], whch was a remarkable ste to eal wth such tyes of ucertates, though some eas resete by hm, were borrowe from the evsos of merca hlosoher Ma lack 937). I hs theory, Zaeh trouce
Joural of New Theory 20 208) 27-47 28 the fuzzy sets, whch have mrecse bouares. Whe s a fuzzy set a s a object of, the the statemet s a member of s ot oly ether true or false as crs sets, but also t s true oly to some egree to whch s actually a member of. The membersh egrees are wth the close terval [0,]. Later, ths theory leas to a hghly commeable theory of Fuzzy logc, whch was ale to egeerg such as washg mache or shftg gears of cars wth great effcecy. fter Zaeh s veto of Fuzzy sets, may other cocets bega to evelo. I 986, K. taassov [], trouce the ea of Itutostc fuzzy sets IFS), whch s a geeralzato of Fuzzy sets. The IFS s a set wth each member havg a egree of beloggess a a egree of o-beloggess as well. There s a restrcto that sum of the membersh grae a o-membersh grae of a elemet s less or equal to. IFS s qute useful to eal wth alcatos lke eert systems, formato fuso etc., where egree of o beloggess of a object s equally mortat as the egree of beloggess. eses IFS, there are other geeralzatos of Fuzzy sets a tutostc fuzzy sets lke L-Fuzzy sets, terval value fuzzy sets, tutostc L-Fuzzy sets, terval value tutostc fuzzy sets [,2] etc. I 995, Smaraache [9, 0], trouce a more geeralze tool to hale Ucertaty, calle as Neutrosohc logc a sets. It s a logc, whch each roosto has a egree of truth T), a egree of etermacy I), a a egree of falsty F). lso a elemet a Neutrosohc set NS) has a truth membersh, a etermacy membersh a a falsty membersh, whch are eeet a whch les betwee [0, ], a sum of them s less or equal to 3. Thus Neutrosohc set s a geeralzato of fuzzy set [7], terval value fuzzy set [], tutostc fuzzy set [], terval value tutostc fuzzy set [2], aracosstet set [9], alethest set [9], araost set [9] a tautologcal set [9]. Though the NS geeralze the above metoe sets, but the geeralzato was oly from hlosohcal ot of vew. For alcato egeerg a other areas of scece, NS eee to be more secfc. Further Wag et. al., 2005, eveloe a stace of NS, calle as sgle value Neutrosohc sets SVNS) [3]. Later they have also trouce the oto of Iterval value eutrosohc sets INS) [2]. The INS s more caable to hale the ucerta, mrecse, comlete a cosstet formato that est real worl. I INS, the egree of truth, etermacy a falsty membersh of a object are eresse close subtervals of [0, ]. I may roblems, t s ofte eee to comare two sets, whch may be fuzzy, tutostc fuzzy, vague etc. We are ofte tereste to reveal the smlarty or the least egree of smlarty of two mages or atters. Dstace a smlarty measures are the effcet tools to o ths. May authors have oe etesve research regarg stace a smlarty of fuzzy a tutostc fuzzy sets a ther terval value versos [7, 8, 5, 6]. Smlarty measures are also a very goo tool for solvg may ecso makg roblems. The oto of stace a smlarty was frst trouce [5,6]. Later roum et. al. [3] has efe several other smlarty measures o Sgle value eutrosohc sets. The oto of smlarty of INS s trouce [4, 4].Ths aer also eals wth stace a smlarty of Iterval eutrosohc sets. However, ths artcle, our motve s to establsh the best sutable stace a smlarty measures by comarg the umercal value of varous staces a smlartes betwee two INSs. We are to also ot out the best alteratve, smlar to the eal alteratve the ecso makg roblem state a solve by Ju Ye [4], by comarg umercal values of staces a smlartes of each alteratve wth the eal alteratve a also comarg wth the estg results [4].
Joural of New Theory 20 208) 27-47 29 The orgazato of the rest of ths aer s as follows: I secto 2, eftos of Fuzzy set, Itutostc Fuzzy set, Neutrosohc Set NS) a Iterval value Neutrosohc set INS) are gve a some oeratos o NS a INS have bee efe a also Set theoretc roertes o INS are also gve. Several staces a Smlartes o INSs are efe secto 3 a 4. ecso makg metho s establshe Iterval Neutrosohc settg by meas of stace a smlarty measures betwee each alteratve a eal alteratve secto 5. I secto 6, a llustratve eamle s aate from [4], to llustrate the roose metho. Fally a comaratve stuy has bee mae wth the estg results secto 7 a at last secto 8 coclues the artcle. 2. Prelmares I ths secto, we gve some useful eftos, eamles a results whch wll be use the rest of ths aer. Defto 2. Tye I Fuzzy set) If s a collecto of objects eote by, the a fuzzy set or tye I fuzzy set) s a set of orere ars: = { µ )) } where µ ) s calle the membersh fucto or grae of membersh also egree of comatblty or egree of truth) of that mas to the membersh sace,.e. µ : M = [0,]. becomes a crs set whe M cotas oly two ots 0 a a µ s the characterstc fucto χ of. Eamle 2.2 s a llustrato, coser the followg eamle. Let, the set P s the set of eole. To each erso P we have to assg a egree of membersh the fuzzy subset YOUTH, whch s efe as follows: ) = { f age ) ) f age ) Youth, 20,, age ) f < age ) 40 / 20, 20 40, 0, > 40 } y.5 0.5 0 0 0 20 30 40 50 The the set YOUTH s a fuzzy set of tye I or a orary fuzzy set.
Joural of New Theory 20 208) 27-47 30 Defto 2.3 Itutostc fuzzy set) Itutostc fuzzy sets geeralze fuzzy sets, sce wth membersh fucto μ, a o-membersh fucto ν s also trouce for each object t. Let us have a fe uverse. Let. Let us costruct the set: * = {, µ ), ν )) & 0 µ ) + ν ) } where µ : [0,], ν : [0,] a set IFS)..We call the set * tutostc fuzzy Eamle 2.4 Let us llustrate the cocet of IFS by a eamle as follows: Let be the set of all Secoary schools a strct. We assume that, for every school, the umber of stuets qualfe the fal eam s kow a say t s P). Let, µ ) ) = P total umber of stuets) Take ν ) = µ ), whch cates the art of stuets coul t qualfy the eam. y Fuzzy set theory, we caot obta that how may stuets have ot gve the eam. ut, f we take ν ) as the umber of stuets fale to qualfy the eam, the we ca easly obta the art of the stuets, have ot gve the eam at all a the value wll be µ ) ν ). Thus we costruct the IFS, {, µ ), ν )) : } a obvously 0 µ ) + ν ) Defto 2.5 Neutrosohc set) Neutrosohc sets NS) further geeralzes the IFS. s NS, the etermacy s elctly efe a also the truth membersh, falsty membersh a etermacy membersh are beyo ay restrcto. Let be a collecto of objects eote by. Neutrosohc set s characterze by a truth membersh fucto T, a Ietermacy membersh fucto I a a falsty membersh fucto F, where, T ), I ) a F ) : [0,] a 0 su T ) + su I ) + su F ) 3. The NS ca be eote as = {, T ), I ), F ) : } Eamle 2.6 If be a elemet of a set a f we take the robablty of s 60%, robablty of ot s 20% a robablty of s ueterme s 0%, the the NS ca be eote as 0.6,0.,0.2). lso to geeralze the eamle, Take be the set of ray ays. Coser be the set toay t wll ra heavly. Let accorg to a observer, robablty of heavy rag s 80%, that of ot rag s 0%, a also the etermacy s 0%. ccorg to aother observer 2, those robabltes are 40%, 50% a 0% resectvely. The NS ca be eote as follows: = 0.8,0.,0. / + 0.4,0.,0.5 / 2
Joural of New Theory 20 208) 27-47 3 Defto 2.7Iterval Neutrosohc set) Let be a sace of objects, whose elemets are eote by. INS s characterze by a truth-membersh fucto. T ), a etermacy-membersh fucto I ) a a falsty-membersh fucto F ). For each ot, we have: a T ) = [f T ),su T )] [0,], I ) = [f I ),su I )] [0,], F ) = [f F ),su F )] [0,] 0 su T ) + su I ) + su F ) 3,. Whe s cotuous, a INS ca be wrtte as : = T ), I ), F ), Whe s screte, a INS ca be wrtte as : = T ), I ), F ), = Eamle 2.8 For eamle, ssume that s qualty, 2 s trustworthess a 3 s rce of a book. The values of, 2 a 3 are [0, ]. They are obtae from some questoares, havg otos as egree of goo, egree of etermacy a egree of ba. Take a are terval eutrosohc sets of efe as: = [0.,0.3],[0,0.2],[0.5,0.7] / + [0.4,0.5],[0.,0.2],[0.6,0.7] / + [0.7,0.8],[0,0.3],[0.,0.2] / 3 2 = [0.2, 0.4],[0., 0.3],[0.6, 0.8] + [0.7, 0.9],[0.4, 0.6],[0.2, 0.4] + [0.3, 0.5],[0.2, 0.4],[0., 0.3] 2 3 Some oeratos o Neutrosohc sets Defto 2.9 ) Comlemet: Let be a Neutrosohc set. The comlemet of s eote by or a s efe by c T ) = F ), I ) = I ), F ) = T ),
Joural of New Theory 20 208) 27-47 32 ) Cotamet: NS s cotae the other NS, eote as oly f: T ) T ) ; I ) I ) ; F ) F ) ;, f a ) Uo: The uo of two NS a s a NS C, wrtte as C =, whose truthmembersh, etermacy-membersh a falsty membersh fuctos are relate to those of a by: T ) = T ) T ), C I ) = I ) I ), C F ) = F ) F ), C v) Itersecto: The tersecto of two NS a s a NS C, eote as C=, whose truth-membersh, etermacy-membersh a falsty membersh fuctos are relate to those of a by: T ) = T ) T ), C I ) = I ) I ), C F ) = F ) F ), C Some oeratos o Iterval Neutrosohc set The oto of IVNS was efe by Wag et. al. [3]. Here we gve some eftos a eamles of IVNS Defto 2.0 Comlemet): Let be a Iterval Neutrosohc set. The comlemet c of s eote by or a s efe by: T ) = F ), f I ) = su I ), su I ) = f I ), F ) = T ) Eamle 2. Let be the terval value Neutrosohc set efe eamle 2.8. The = [0.5, 0.7],[0.8,.0],[0.,0.3] + [0.6, 0.7],[0.8, 0.9],[0.4, 0.5 [0., 0.2],[0.7,.0],[0.7,0.8] 2 3 + Defto 2.2 Cotamet) INS s cotae the other INS, eote as, f a oly f:
Joural of New Theory 20 208) 27-47 33 f T ) f T ), su T ) su T ); f I ) f I ), su I ) su I ); f F ) f F ), suf ) suf ); Two terval eutrosohc sets a are equal, wrtte as =, f a oly f a Eamle 2.3 Let a be two INS efe eamle 3..4, the t ca be easly observe that those INSs o ot satsfy all the requre roertes for cotamet of. So here. Defto 2.4 Uo): The uo of two INS a s a INS C, wrtte as C =, whose truth-membersh, etermacy-membersh a falsty membersh fuctos are relate to those of a by: Eamle 2.5: Coser two INS a efe eamle 2.8. The ther uo C = s C = [0.2, 0.4],[0, 0.2],[0.5, 0.7] + [0.7, 0.9],[0., 0.2],[0.2, 0.4] + [0.7, 0.8],[0, 0.3],[0., 0.2] f T ) = maf T ), f T )), C su T ) = masu T ), su T )), C f I ) = mf I ), f I )), C su I ) = msu I ), su I )), C f F ) = mf F ), f F )), C su F C ) = msuf ), su F )), 3 2 Defto 2.6 Itersecto) The tersecto of two INS a s a INS C, eote as C=, whose truth-membersh, etermacy-membersh a falsty membersh fuctos are relate to those of a by: f T ) = mf T ), f T )), C su T ) = msu T ), su T )), C f I ) = maf I ), f I )), C su I ) = masu I ), su I )), C f F ) = maf F ), f F )), C su F C ) = masuf ), su F )), Eamle 2.7 Take a be two INS efe eamle 2.8. The ther tersecto C= s as follows:
Joural of New Theory 20 208) 27-47 34 C = [0., 0.3],[0., 0.3],[0.6, 0.8] + [0.4, 0.5],[0.4, 0.6],[0.6, 0.7] [0.3, 0.5],[0.2, 0.4],[0., 0.3] 3 2 + Set theoretcal roertes Here we wll gve some roertes of set-theoretc oerators efe o terval eutrosohc sets. Let,, a C be three INSs. The the roertes satsfe by, a C are as follows: Proerty Commutatvty) = = Proerty 2 ssocatvty) C) = ) C C) = ) C Proerty 3 Dstrbutvty) C) = ) C) C) = ) C) Proerty 4 Iemotecy) =, =. Proerty 5 Φ = Φ, =, Where Φ a are resectvely Null set a absolute INS efe below: Proerty 6 f T = sut = 0, Φ f I = su I = f F = su F =, f T = sut =, Φ Φ Φ Φ Φ f I = su I = f F = su F = 0 Φ =, =, Where Φ a are efe above.
Joural of New Theory 20 208) 27-47 35 Proerty 7 bsorto) ) =, ) = Proerty8 Ivoluto) = Here, we otce that by the eftos of comlemet, uo a tersecto of terval eutrosohc set as efe revously, INS satsfes the most roertes of crs set, fuzzy set a tutostc fuzzy set. lso, t oes ot satsfy the rcle of eclue mle, same as fuzzy set a tutostc fuzzy set. 3. Dstace Measure I ths secto, we vestgate several stace measures for two INS s a. lso, we take the weghts of the elemet =, 2,.,) to accout. I the followg, we coser some weghte stace measures betwee INSs. For ths we take w={w,w 2,.,w } as the weght vector of the elemet =,2,.,) a also w [0,], =, 2,...,.We aot some stace a smlarty measures from [5] a ete those INS settg as follows: a. Hammg Dstace :, ) = [ f T ) f T su T ) su T 6 = f I ) f I su I ) su I f F ) f F su F ) su F ) ] b. Normalze Hammg Dstace : 2, ) = [ f T ) f T su T ) su T 6 = f I ) f I su I ) su I f F ) f F su F ) su F ) ] c. Euclea stace : 3, ) = { [ f T ) f T su T ) su T 6 = f I ) f I su I ) su I F F F F f ) f su ) su ) ]} 2
Joural of New Theory 20 208) 27-47 36. Normalze Euclea stace : 4, ) = { [ f T ) f T su T ) su T 6 = f I ) f I su I ) su I F F F F f ) f ) su ) su ) ]} 2 + e. Hausroff stace : 5, ) = = m a [ f T ) f T ), su T ) su T ), f I ) f I ), su I ) su I ), f F ) f F ), su F ) su F ) ] f. Normalze Hausroff stace : 6, ) = m a [ f T ) f T ), s u T ) s u T ), = f I ) f I ), s u I ) s u I ), g. Weghte Hammg Dstace : f F ) f F ), s u F ) s u F ) ] 7, ) = w [ f T ) f T su T ) su T 6 = f I ) f I su I ) su I f F ) f F su F ) su F ) ] h. Weghte ormalze Hammg stace : 8, ) = w[ f T ) f T su T ) su T 6 = f I ) f I su I ) su I f F ) f F su F ) su F ) ]. Weghte Euclea stace : 9, ) = { w[ f T ) f T su T ) su T 6 = f I ) f I su I ) su I f F ) f ) su ) su ) ]} 2 F + F F
Joural of New Theory 20 208) 27-47 37 j. Weghte ormalze Euclea stace 0, ) = { w [ f T ) f T su T ) su T 6 = f I ) f I su I ) su I f F ) f ) su ) su ) ]} 2 F + F F k. Weghte Hausroff stace :, ) = w m a[ f T ) f T ), su T ) su T ), = f I ) f I ), su I ) su I ), f F ) f F ), su F ) su F ) ] l. Weghte ormalze Hausroff stace: 2, ) = w m a[ f T ) f T ), su T ) su T ), = f I ) f I ), su I ) su I ), f F ) f F ), su F ) su F ) ] m. Euclea Hausroff stace : 3, ) = { m a [ f ) f ), su ) su ), = T T T T f I ) f I ), su I ) su I ), f F ) f ), su ) su ) ]} 2 F F F. Weghte Euclea Hausroff stace : 4, ) = { m a[ f ) f ), su ) su ), = w T T T T f I ) f I ), su I ) su I ), f F ) f ), su ) su ) ]} 2 F F F o. Normalze Euclea Hausroff Dstace : T T T T 5, ) = { m a[ f ) f ), su ) su ), = f I ) f I ), su I ) su I ), F F F F f ) f ), su ) su ) ]} 2
Joural of New Theory 20 208) 27-47 38. Normalze Weghte Euclea Hausroff Dstace : w T T T T 6, ) = { m a[ f ) f ), su ) su ), = I I I I f ) f ), su ) su ), f F ) f ), su ) su ) ]} 2 F F F Some other staces betwee two INS s are gve as follows We coser as a ostve teger the followg. q. 7, ) = { [ f T ) f T su T ) su T 6 = f I ) f I su I ) su I f F ) f F su F ) su F ) ]}, > 0 r. 8, ) = { w [ f T ) f T su T ) su T 6 = f I ) f I su I ) su I f F ) f F su F ) su F ) ]}, > 0 s. 9, ) = { [ f T ) f T su T ) su T 6 = f I ) f I su I ) su I f F ) f F su F ) su F ) ]} > 0, t. 20, ) = { w [ f T ) f T ) + su T ) su T ) + 6 = f I ) f I ) + su I ) su I ) + f F ) f F ) + su F ) su F ) ]}, > 0 u., ) = { ma[ f T ) f T ), su T ) su T ), 2 = f I ) f I ), su I ) su I ), f F ) f F ), su F ) su F ) ]}, > 0
Joural of New Theory 20 208) 27-47 39 v., ) = { w ma[ f T ) f T ), su T ) su T ), 22 = f I ) f I ), su I ) su I ), f F ) f F ), su F ) su F ) ]}, > 0 w. 23, ) = { ma[ f T ) f T ), su T ) su T ), = f I ) f I ), su I ) su I ), f F ) f F ), su F ) su F ) ]}, > 0. 24, ) = { w ma[ f T ) f T ), su T ) su T ), = f I ) f I ), su I ) su I ), f F ) f F ), su F ) su F ) ]}, > 0 Proertes of Dstace Measure The above efe stace k, ) k=, 2, 3, )betwee INSs a satsfes the followg roertes D D3) : D: k, ) 0 ; D2: k, ) = 0 f a oly f = D3:, ) =, ) ; k k It ca be easly show that the staces as efe above satsfy the sa roertes. 4. lgorthm Now we reset a algorthm to solve a ecso makg roblem Iterval Neutrosohc Sets by meas of stace a smlarty measures INSs. Let { : =,2,..., m} be a set of alteratves a { C : j =,2,..., } be a set of crtera. ssume that the weght of the crtero C j s w j [0,] a ca be eote as follows: j wj =. I ths case the INS j=
Joural of New Theory 20 208) 27-47 40 where = { C, T C ), I C ), F C )) : C C}, j j j j j T C ) = [f T C ),su T C )] [0,], j j j I C ) = [f I C ), su I C )] [0,], j j j F C ) = [f F C ),su F C )] [0,], j j j a 0 su T C ) + su I C ) + su F C ) 3, =,2,,m a j=,2,,. j j j Now let us coser a INS eote as: α = [ a, b ],[ c, ],[ e, f ]) j j j j j j j where [ a, b ] = [f T C ),su T C )], j j j j [ c, ] = [f I C ),su I C )], j j j j [ e, f ] = [f F C ),su F C )] j j j j Now, a INS s erve from the evaluato of a alteratve wth resect to a crtero C j, by meas of score law a ata rocessg. Therefore, we ca trouce a terval eutrosohc ecso matr D = α ). j m The evaluato crtera are geerally take of two ks, beeft crtera a cost crtera. Let be a collecto of beeft crtera a P be a collecto of cost crtera. The we efe a eal INS for a beeft crtero the eal alteratve * as: α = [ a, b ],[ c, ],[ e, f ]) = [,],[0,0],[0,0]) for j * * * * * * * j j j j j j j a for a cost crtero, we efe the eal alteratve ** as: α = [ a, b ],[ c, ],[ e, f ]) = [0,0],[,],[,]) for j P. ** ** ** ** ** ** ** j j j j j j j lthough, the eal alteratve oes t est real worl, t s oly use to etfy the best alteratve ecso set. Now f we eote the eal alteratve as the INS E, the by the stace measures E, ), =,2,.,m), k=,2,,24) a the smlarty measures s E, ), k =,2,.,m), k=,2,,2) as efe revous secto), betwee each alteratve a the eal alteratve E For beeft crtera E = * a for cost crtera E = ** ), the rakg orer of all alteratves ca be eterme a the best oe ca be easly etfe as well. k
Joural of New Theory 20 208) 27-47 4 5. Problem To llustrate the above algorthm we take a mult-crtera ecso makg roblem of alteratves to aly the roose ecso makg metho. We aat the requre roblem from the artcle by Ju Ye [4], state as follows: There s a vestmet comay, whch wats to vest a sum of moey the best oto. There s a ael wth four ossble alteratves to vest the moey: ) s a car comay; 2) 2 s a foo comay; 3) 3 s a comuter comay; 4) 4 s a arms comay. The vestmet comay must take a ecso accorg to the followg three crtera: ) C s the rsk aalyss; 2) C 2 s the growth aalyss; 3) C 3 s the evrometal mact aalyss, where C a C 2 are beeft crtera a C 3 s a cost crtero. The weght vector of the crtera s gve by :w = 0.35, 0.25, 0.40). The four ossble alteratves are to be evaluate uer the above three crtera by corresog to the INSs, as show the followg terval eutrosohc ecso matr D: D = [0.4, 0.5],[0.2, 0.3],[0.3, 0.4] [0.4, 0.6],[0., 0.3],[0.2, 0.4] [0.7, 0.9],[0.2, 0.3],[0.4, 0.5] [0.6, 0.7],[0., 0.2],[0.2, 0.3] [0.6, 0.7],[0., 0.2],[0.2, 0.3] [0.3, 0.6],[0.3, 0.5],[0.8, 0.9] [0.3,0.6],[0.2,0.3],[0.3, 0.4] [0.5, 0.6],[0.2, 0.3],[0.3, 0.4] [0.4, 0.5],[0.2, 0.4],[0.7, 0.9] [0.7, 0.8],[0.0, 0.],[0., 0.2] [0.6, 0.7],[0., 0.2],[0., 0.3] [0.6, 0.7],[0.3, 0.4],[0.8, 0.9] Now we measure the staces a also the smlartes betwee each alteratve a the eal alteratves E, as efe earler. To calculate the Hammg stace betwee E a we take : E, ) = 6 = [ f T ) f T su T ) su T E E f I ) f I su I ) su I E E f F ) f F su F ) su F ) ] E E =/6[ -0.4 + -0.5 + 0-0.2 + 0-0.3 + 0-0.3 + 0-0.4 + -0.4 + -0.6 + 0-0. + 0-0.3 + 0-0.2 + 0-0.4 + 0-0.7 + 0-0.9 + -0.2 + -0.3 + -0.4 + -0.5 ] =.467 Smlarly,, 2 ) 0.9 E = a E, 4) = 0.86. E =,, 3).25 I ths way, the obtae results are resete tabular form as follows:
Joural of New Theory 20 208) 27-47 42 For Dstace measuremet Dstace E, ) 2 E, ) 3 E, ) 4 E, ) 5 E, ) 6 E, ) 7 E, ) 8 E, ) 9 E, ) 0 E, ) E, ) 2 E, ) 3 E, ) 4 E, ) 5 E, ) 6 E, ) 7 E, ) Obtae Results =.467 2 = 0.9 3 =.25 4 = 0.86 = 0.4722 2 = 0.3 3 = 0.467 4 = 0.2867 = 0.8990 2 = 0.596 3 = 0.7450 4 = 0.6245 = 0.590 2 = 0.346 3 = 0.430 4 = 0.3606 = 2. 2 =.5 3 = 2.0 4 =.4 = 0.7000 2 = 0.5000 3 = 0.6667 4 = 0.4667 = 0.4975 2 = 0.300 3 = 0.4233 4 = 0.3042 = 0.658 2 = 0.033 3 = 0.4 4 = 0.04 = 0.5428 2 = 0.3545 3 = 0.440 4 = 0.3800 = 0.334 2 = 0.2047 3 = 0.254 4 = 0.294 = 0.7200 2 = 0.5200 3 = 0.6900 4 = 0.4850 = 0.2400 2 = 0.733 3 = 0.2300 4 = 0.67 =.2369 2 = 0.9000 3 =.747 4 = 0.8602 = 0.7348 2 = 0.5404 3 = 0.7000 4 = 0.572 = 0.74 2 = 0.596 3 = 0.6782 4 = 0.4966 = 0.4242 2 = 0.320 3 = 0.404 4 = 0.2986 For = 6 = 0.033700 2 = 0.005336 3 = 0.03387 4 = 0.009309 For = 0 = 0.00888288 2 = 0.0005984 3 = 0.00240292 4 = 0.00458 Rak of lteratves esceg orer) > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 4 > 2 2 > 3 > 4 > 2 2 > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 4 > 2 2 > 3 > 4 > 2 2 > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 4 > 2 2 > 3 > 4 > 2 2 est alteratve obtae
Joural of New Theory 20 208) 27-47 43 Dstace 8 E, ) 9 E, ) 20 E, ) 2 E, ) 22 E, ) 23 E, ) 24 E, ) Obtae Results For = 6 = 0.037057 2 = 0.0020276 3 = 0.00524945 4 = 0.00624732 For = 0 = 0.0035354 2 = 0.00023634 3 = 0.00093445 4 = 0.00045777 For = 6 = 0.06740000 2 = 0.006760 3 = 0.0267756 4 = 0.086966 For = 0 = 0.0296096 2 = 0.0097280 3 = 0.00800974 4 = 0.0038729 For =6 = 0.026345 2 = 0.00420552 3 = 0.0049890 4 = 0.0249464 For = 0 = 0.07782 2 = 0.00078782 3 = 0.003483 4 = 0.0052592 For = 6 = 0.04255 2 = 0.0209735 3 = 0.0659030 4 = 0.020423 For = 0 =0.0360776 2 = 0.00284572 3 =0.0365982 4 = 0.00283582 For = 6 = 0.0400950 2 = 0.0082528 3 = 0.024990 4 = 0.0080200 For =0 = 0.4975 2 = 0.300 3 = 0.4233 4 = 0.3042 For = 6 = 0.20825 2 = 0.04947 3 = 0.3806 4 = 0.040824 For = 0 = 0.20825 2 = 0.04947 3 = 0.3806 4 = 0.040824 For = 6 = 0.080900 2 = 0.065057 3 = 0.0499803 4 = 0.06040 For = 0 = 0.0475990 2 = 0.0037873 3 = 0.02630 4 = 0.0037757 Rak of lteratves esceg orer) > 4 > 3 > 2 2 > 3 > 4 > 2 2 > 3 > 4 > 2 2 > 3 > 4 > 2 2 > 4 > 3 > 2 2 > 3 > 4 > 2 2 > 3 > 2 > 4 4 > 3 > > 4 est alteratve obtae 4 > 3 > 2 > 4 4 > 2 > 3 > 4 4 > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 2 > 4 4 > 3 > 2 > 4 4
Joural of New Theory 20 208) 27-47 44 For smlarty measuremet Smlarty s E, ) s 2 E, ) s 3 E, ) s 4 E, ) s 5 E, ) s 6 E, ) s 7 E, ) s 8 E, ) s 9 E, ) s 0 E, ) s E, ) s 2 E, ) s 3 E, ) Obtae Results = 0.6678 2 = 0.7634 3 = 0.7026 4 = 0.7668 = 0.8342 2 = 0.8967 3 = 0.8589 4 = 0.8986 = 0.648 2 = 0.7383 3 = 0.6944 4 = 0.7246 = 0.6866 2 = 0.7953 3 = 0.7459 4 = 0.7806 = 0.584 2 = 0.6579 3 = 0.597 4 = 0.6734 = 0.7600 2 = 0.8267 3 = 0.7700 4 = 0.8383 = 0.52778 2 = 0.70000 3 = 0.60555 4 = 0.7 For = 6 = 0.993246 2 = 0.9997453 3 = 0.99926028 4 = 0.999282 For =0 = 0.99905556 2 = 0.99999644 3 = 0.99998544 4 = 0.99997679 For = 6 = 0.999449 2 = 0.9997025 3 = 0.9998473 4 = 0.9994266 For = 0 = 0.99886727 2 = 0.99999574 3 = 0.99998329 4 = 0.9999725 For = 6 = 0.92097654 2 = 0.98738343 3 = 0.96956463 4 = 0.97780405 For = 0 = 0.9788070 2 = 0.998589 3 = 0.99439330 4 = 0.99725333 = 0.66245024 2 = 0.74828426 3 = 0.67495694 4 = 0.7638054 = 0.6290322 2 = 0.70459388 3 = 0.6260626 4 = 0.9838033 For = 6 = 0.9326000 2 = 0.9893284 3 = 0.9732248 4 = 0.983803 For = 0 = 0.97039038 2 = 0.9980279 3 = 0.9999025 4 = 0.9968270 Rak of lteratves esceg orer) 4 > 2 > 3 > 4 4 > 2 > 3 > 4 2 > 4 > 3 > 2 2> 4 > 3 > 2 4> 2 > 3 > 4 4> 2 > 3 > 4 4> 2 > 3 > 4 2 > 4 > 3 > 2 2 > 3 > 4 > 2 2 > 3 > 4 > 2 2 > 3 > 4 > 2 2 > 4 > 3 > 2 2 > 4 > 3 > 2 4 > 2 > 3 > 4 4 > 2 > 3 > 4 2 > 4 > 3 > 2 2 > 4 > 3 > 2 est alteratve obtae
Joural of New Theory 20 208) 27-47 45 Smlarty s 4 E, ) s 5 E, ) s 6 E, ) s 7 E, ) s 8 E, ) s 9 E, ) s 20 E, ) s 2 E, ) Obtae Results For = 6 = 0.97365884 2 = 0.99579447 3 = 0.9895009 4 = 0.98750536 For = 0 = 0.9882287 2 = 0.999227 3 = 0.9968856 4 = 0.99847407 = 0.296606 2 = 0.35238095 3 = 0.37684 4 = 0.50000000 = 0.300 2 = 0.550 3 = 0.450 4 = 0.483 = 0.43708609 2 = 0.6538465 3 = 0.5493548 4 = 0.66666666 = 0.20283243 2 = 0.37547646 3 = 0.26990699 4 = 0.38997923 = 0.8945738 2 = 0.3027000 3 = 0.22405482 4 = 0.3378245 = 0.425 2 = 0.6375 3 = 0.5250 4 = 0.6500 = 0.40625 2 = 0.222500 3 = 0.83750 4 = 0.226250 Rak of lteratves esceg orer) 2 > 3 > 4 > 2 2 > 4 > 3 > 2 4 > 3 > 2 > 4 2> 4 > 3 > 2 4 > 2 > 3 > 4 4> 2 > 3 > 4 4 > 2 > 3 > 4 4 > 2 > 3 > 4 4 > 2 > 3 > 4 est alteratve obtae 7. Comaratve stuy wth estg work Hece we comare the results gve [4] a the results obtae revous secto secto 6). I the artcle [4], the authors have use the smlarty measures s,)a s 3,) as state the secto 4, where s the eal alteratve E a s the alteratve to be measure), to obta the best alteratves. Usg s,) the best alteratve obtae s 4 a usg s 3,) the best alteratve s 2. lso the smlarty measure of 4 wth eal alteratve s 0.9600 a the same of 2 s 0.9323. However, we have measure usg varous umbers of smlartes a staces as well, betwee the alteratves a the eal alteratve, to obta the best alteratve. ccorg to the results, 4 s the best alteratve both staces a smlarty measures) whe the stace or the smlarty s lear form.e. Hammg stace, Hausroff stace a ther relate stace a smlarty measures, etc. ecet 2,) a ts relate stace measures, where though they are ot lear, the best alteratve obtae s 4 ). Otherwse the best alteratve s 2 ecet s 6,), where beg lear smlarty measure, the best alteratve gve s 2 ). Now, oe ca ece the best alteratve coserg the alteratve obtae as best alteratve accorg to umercal value most umber of cases both stace a smlarty measures a also ths ecso ca be mae coserg more stace a smlartes beses those efe ths aer. So, we suggest that, accorg to the umber of cases, 4 ca be take as the best alteratve.
Joural of New Theory 20 208) 27-47 46 8. Cocluso I ths artcle, at frst we have efe varous staces k, ), k =,2,,24) a smlarty measures sk, ), k =,2,,2), betwee two Iterval Neutrosohc sets. The we have show a alcato of these staces a smlartes solvg a multcrtera ecso-makg roblem. metho, for the soluto of ths tye of roblems, has bee establshe by meas of stace a smlarty measures betwee each alteratve a the resectve eal alteratve. The, as a llustratve eamle, a roblem from [4] has bee recosere a alyg our stace a smlarty measures, the rakg orer of all alteratves has bee calculate a state tabular form a the best alteratve has also bee etfe as well. Fally we have mae a comarso betwee the estg result [4] a the results obtae ths artcle a fally coclue that the result obtae ths aer s more recse a more secfc. The roose smlarty measures are also useful real lfe alcatos of scece a egeerg such as mecal agoss, atter recogtos etc. Furthermore, the roose techques, base o stace a smlarty measures, ca be more useful for ecso makers as t ete the estg ecso makg methos. ckowlegemets The authors are hghly ebte to the revewers a etor--chef for ther valuable commets whch have hele to rewrte the aer ts reset form. Refereces [] K. taassov, Itutostc Fuzzy Sets, Fuzzy Sets a Systems, 20986) 87-96. [2] K. taassov a G.Gargov, Iterval value tutostc fuzzy sets, Fuzzy Sets a Systems 3 989), 343 349. [3] S. roum a F. Smaraache, Several Smlarty Measures of Neutrosohc Sets, Neutrosohc Sets a Systems, Vol., 203, 54-6. [4] S. roum, F. Smaraache, New stace a smlarty measures of Iterval Neutrosohc Sets, IEEE Coferece ublcato 204). [5] P. Majumar, S.K. Samata, O Smlarty a Etroy of Neutrosohc Sets, Joural of Itellget a Fuzzy systems, 204, vol 26, o. 3,204, 245-252 [6] P. Majumar, Neutrosohc Sets a ts alcatos to ecso makg, Comutatoal Itellgece for g Data alyss: Froter vaces & lcatos, Srger- Verlag/Heelberg, D.P. charjee at. al. etors, 205, 97-5 [7] C. P. Pas, N. I. Karacals, comaratve assessmet of measures of smlarty of fuzzy values. Fuzzy Sets a Systems, 56:774, 993. [8] G.. Paakostas,. G. Hatzmchals, V.G. Kaburlasos, Dstace a Smlarty measures betwee Itutostc Fuzzy sets : comaratve aalyss from a atter recogto ot of vew, Patter Recogto Letters, vol. 34, ssue 4,203,609-622. [9] F. Smaraache, Ufyg Fel Logcs. Neutrosohy: Neutrosohc Probablty, Set a Logc, merca Research Press, Rehoboth 999). [0] F. Smaraache,, Neutrosohc set: geeralzato of the tutostc fuzzy set, Iteratoal Joural of Pure a le Mathematcs 24 2005), 287 297.
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