04 Iteratoal Joural of Fuzzy Systems Vol. 0 No. Jue 008 Fuzzy Number Itutostc Fuzzy rthmetc ggregato Operators Xfa Wag bstract fuzzy umber tutostc fuzzy set (FNIFS s a geeralzato of tutostc fuzzy set. The fudametal characterstc of FNIFS s that the values of ts membershp fucto ad o-membershp fucto are trgoometrc fuzzy umbers rather tha exact umbers. I ths paper we defe some operatoal laws of fuzzy umber tutostc fuzzy umbers ad based o these operatoal laws develop some ew arthmetc aggregato operators such as the fuzzy umber tutostc fuzzy weghted averagg (FIFW operator the fuzzy umber tutostc fuzzy ordered weghted averagg (FIFOW operator ad the fuzzy umber tutostc fuzzy hybrd aggregato (FIFH operator for aggregatg fuzzy umber tutostc fuzzy formato. Furthermore we gve a applcato of the FIFH operator to multple attrbute decso mag based o fuzzy umber tutostc fuzzy formato. Fally a llustratve example s gve to verfy the developed approach. Keywords: Itutostc fuzzy set fuzzy umber tutostc fuzzy set operatoal laws FIFW operator FIFOW operator FIFH operator.. Itroducto taassov [ ] troduced the cocept of tutostc fuzzy set (IFS whch s a geeralzato of the cocept of fuzzy set [3]. Gau ad Buehrer [4] troduced the cocept of vague set but Buste ad Burllo [5] showed that vague sets are tutostc fuzzy sets. I [6] Xu ad Yager defed some multplcatve operatoal laws of tutostc fuzzy values ad based o these operatoal laws proposed some tutostc fuzzy geometrc aggregato operators such as the tutostc fuzzy weghted Correspodg uthor: Xfa Wag. uthor s wth the School of Scece Hua Uversty of Techology Zhuzhou Hua Cha 4008. Ths wor was supported by Scetfc Research Fud of Hua Provcal Educato Departmet uder grat 07C3. E-mal: zzwxfydm@6.com Mauscrpt receved 7 Ja. 008; revsed 7 May. 008; accepted 7 May. 008. geometrc (IFWG operator the tutotc fuzzy ordered weghted geometrc (IFOWG operator ad the tutotc fuzzy hybrd geometrc (IFHG operator. I [7] Xu defed some addtve operatoal laws of tutostc fuzzy values ad developed some tutostc fuzzy arthmetc aggregato operators such as the tutostc fuzzy weghted averagg (IFW operator the tutostc fuzzy ordered weghted averagg (IFOW operator ad the tutostc fuzzy hybrd aggregato (IFH operator. Later taassov ad Gargov [8-9] further troduced the terval-valued tutostc fuzzy set (IVIFS whch s a geeralzato of the IFS. The fudametal characterstc of the IVIFS s that the values of ts membershp fucto ad o-membershp fucto are tervals rather tha exact umbers. I [0] based o some operatoal laws of terval-valued tutostc fuzzy umbers [] Xu ad Che proposed some terval-valued tutostc fuzzy arthmetc aggregato operators such as the terval-valued tutostc fuzzy weghted averagg (IIFW operator the terval-valued tutostc fuzzy ordered weghted averagg (IIFOW operator ad the terval-valued tutostc fuzzy hybrd aggregato (IIFH operator. O the bass of the terval-valued tutostc fuzzy weghted geometrc (IIFWG operator [] Xu ad Che [] developed some terval-valued tutostc fuzzy geometrc aggregato operators such as the terval-valued tutostc fuzzy ordered weghted geometrc (IIFOWG operator ad the terval-valued tutostc fuzzy hybrd geometrc (IIFHG operator. Recetly Lu [3] also exteded the IFS troduced the fuzzy umber tutostc fuzzy set (FNIFS whose fudametal characterstc s that the values of ts membershp fucto ad o-membershp fucto are trgoometrc fuzzy umbers rather tha exact umbers. However t seems that the lterature there s lttle vestgato o aggregato operators for aggregatg a collecto of FNIFSs [4]. I ths paper we try to defe some operatoal laws of fuzzy umber tutostc fuzzy umbers ad based o these operatoal laws develop some ew arthmetc aggregato operators such as the fuzzy umber tutostc fuzzy weghted averagg (FIFW operator the fuzzy umber tutostc fuzzy ordered weghted averagg (FIFOW operator ad the fuzzy umber tutostc fuzzy hybrd aggregato 008
Xfa Wag: Fuzzy Number Itutostc Fuzzy rthmetc ggregato Operators 05 (FIFH operator.. Basc Cocepts ad Relatos The cocept of tutostc fuzzy set was troduced by taassov [] to deal wth vagueess whch ca be defed as follows. Defto. []. Let X be a uverse of dscourse the a IFS X s gve by: = {( x t( x f( x x X} ( where the fuctos t : X [0 ] ad f : X [0] determe the degree of membershp ad the degree of o-membershp of the elemet x X respectvely ad for every x X : 0 t( x + f( x ( Sometme t s ot approxmate to assume that the membershp degrees for certa elemets of are exactly defed but a value rage ca be gve. I such cases taassov ad Gargov [8] defed the oto of IVIFS as below. Defto. [8]. IVIFS % X s a obect havg the followg form: % = {( x t % ( x f % ( x x X } % % (3 Where t% ( x = ( t% ( x t% ( x [0] ad % % % f% ( x = % ( f % ( x f % ( x [0 ] are tervals ad for every % % x X : 0 t% ( x + f% ( x % % (4 Especally f each of the tervals t% ( x ad % f% ( x % cotas exactly oe elemet.e. f for every x X : t% ( x = t% ( x = t% ( x % % % f% ( x = f% ( x = % % f % ( x % the the gve IVIFS % s trasformed to a ordary tutostc fuzzy set. However terval lacs gravty ceter caot emphass ceter pot whch s most mpossble to be gve value so t s more sutable that the degree of membershp ad the degree of o-membershp are expressed by trgoometrc fuzzy umber [5]. I such cases Lu [3] defed the oto of FNIFS as follow. Defto.3 [3 5]. If = ( l p q F( I I = [0] we call a trgoometrc fuzzy umber I ts membershp fucto μ ( x: I [0] s defed by: μ ( x l ( pl l x p ( x = ( xq ( pq p x q 0 other where x I 0 l p q l s called lower lmt of q s called upper lmt of ad p s called gravty ceter of. O the bass of documet [5] the mea of s defed as below: ( θ ( θ l+ p+ θ q E( = θ [0] (5 where θ s a dex that reflects the decso maer's rs-bearg atttude. If θ > 0.5 the the decso maer s a rs lover. If θ < 0.5 the the decso maer s a rs averter. I geeral letθ = 0.5 the the atttude of the decso maer s eutral to the rs ad ( θ l+ p+ q E( = (6 4 Defto.4 [3]. FNIFS X s a obect havg the followg form: = {( x t ( x f ( x x X} (7 3 t ( x ( t ( x t ( x t ( x F( I 3 ( ( ( ( ( ( where = ad f x = f x f x f x F I are two trgoometrc fuzzy umbers ad for every x X : 3 3 0 t ( x + f ( x (8 Especally f each of the trgoometrc fuzzy umbers t ( x ad f ( x cotas exactly oe elemet.e. f 3 for every x X : t ( x = t ( x = t ( x = t ( x 3 f ( x = f ( x = f ( x = f ( x the the gve FNIFS s trasformed to a ordary tutostc fuzzy set. I [] Xu defed the oto of terval-valued tutostc fuzzy umber (IVIFN ad troduced some operatoal laws of IVIFNs. Based o these we defe the cocept of fuzzy umber tutostc fuzzy umber (FNIFN ad troduce some operatoal laws of FNIFNs. Defto.5. Let = {( x t ( x f ( x x X} be a FNIFS the we call the par t ( x f ( x a fuzzy umber tutostc fuzzy umber (FNIFN. For coveece we deote a FNIFN by ( abc ( l pq where ( abc F( I ( l p q F( I 0 c+ q (9 ad let Ω be the set of all FNIFNs. Defto.6. Let = ( a b c( l p q ad = ( a b c( l p q be ay two FNIFNs the some operatoal laws of ad ca be defed as: ( = ( a+ a aa b+ b bb c+ c cc ( ll pp qq ;
06 Iteratoal Joural of Fuzzy Systems Vol. 0 No. Jue 008 λ λ λ = ( ( a ( b ( c ( λ λ λ ( l p q λ λ > 0. It ca be easly prove that all the above results are also FNIFNs. Based o Defto.6 we ca further obta the followg relato: ( = ; ( λ( = λ λ λ > 0 ; (3 ( λ+ λ = λ λ λ λ > 0. Che ad Ta [6] troduced a score fucto to measure a tutostc fuzzy value. Later Hog ad Cho [7] defed a accuracy fucto to evaluate the accuracy degree of a tutostc fuzzy value. Furthermore Xu [] developed a score fucto ad a accuracy fucto to measure a IVIFN respectvely. I the followg we develop two score fucto to measure a FNIFN. Defto 3.. Let = ( abc ( l pq be a FNIFN the we call a+ b+ c l+ p+ q S ( = (0 4 4 a score fucto of where S ( [ ]. The greater the value of S ( the greater the FNIFN. Especally f S ( = the = ( (000 whch s the largest FNIFN; f S ( = the = (000( whch s the smallest FNIFN. Based o the score fucto L( E( = t ( t f of tutostc fuzzy value developed [8] we defe a ew score fucto. Defto 3.. Let = ( abc ( l pq be a FNIFN the we call a+ b+ c a+ b+ c l+ p+ q L( = ( ( 4 4 4 a exteded score fucto of where L( [0]. The greater the value of L( the better the FNIFN. ccordg to Defto 3. ad Defto 3. we develop a method for comparso betwee two FNIFNs. Defto 3.3. Let ad be ay two FNIFNs the ( If S( < S( the s smaller tha deoted by < ; ( If S( = S( the If L( = L( the ad represet the same formato deoted by = ; If L( < L( the s smaller tha deoted by <. 3. Fuzzy Number Itutostc Fuzzy ggregato Operators Up to ow may operators have bee proposed for aggregatg formato [9]-[6]. The weghted averagg (W operator [9] ad the ordered weghted averagg (OW operator [0] are the most commo operators for aggregatg argumets. Let a ( = L be a collecto of real umbers w= ( w w L w T be the weght vector of a ( = L wth w [0] ad the a W operator s defed as [9]: W w( a a a = w a L = w = = OW operator [0] of dmeso s a mappg OW: R R that has a assocated vector = ( L T such that [0] ad Furthermore OW ( a a L a = b = =. = where b s the th largest of a ( = L. From above we ow that the W operator frst weghts all the gve argumets ad the aggregates all these weghted argumets to a collectve oe. The fudametal aspect of the OW operator s the reorderg step t frst reorders all the gve argumets descedg order ad the weghts these ordered argumets ad fally aggregates all these ordered weghted argumets to a collectve oe. I [7] Xu exteded the W operator ad the OW operator to accommodate the stuatos where the put argumets are tutostc fuzzy values. I [0] Xu ad Che also exteded the W operator ad the OW operator to accommodate the stuatos where the put argumets are terval-valued tutostc fuzzy umbers. I the followg we shall exted the W operator ad the OW operator to accommodate the stuatos where the put argumets are FNIFNs ad develop some ew arthmetc aggregato operators for aggregatg fuzzy umber tutostc fuzzy formato.. Fuzzy Number Itutostc Fuzzy Weghted veragg (FIFW Operator Defto 4.. Let = ( a b c ( l p q
Xfa Wag: Fuzzy Number Itutostc Fuzzy rthmetc ggregato Operators 07 ( = L be a collecto of FNIFNs ad let FIFW: Ω Ω f FIFW w( L = w w L w ( the FIFW s called a fuzzy umber tutostc fuzzy weghted averagg (FIFW operator of dmeso where w= ( w w L w T be the weght vector of ( = L wth w [0] ad w =. = Especally f w= ( L T the the FIFW operator s reduced to a fuzzy umber tutostc fuzzy averagg (FIF operator of dmeso whch s defed as follows: FIF w( L = ( L (3 By Defto.6 ad Defto 4. we ca get the followg result by usg mathematcal ducto o. Theorem 4.. Let = ( a b c( l p q ( = L be a collecto of FNIFNs the ther aggregated value by usg the FIFW operator s also a FNIFN ad FIFW w( L = where w w a b = = w w w = = = = ( ( ( w ( c ( l p q w= ( w w L w T (4 be the weght vector of ( = L wth w [0] ad w =. = Proof: The frst result follows qucly from Defto.6. Below we prove (4 by usg mathematcal ducto o. We frst prove that (4 holds for =. Sce w w w = ( ( a ( b w w w w ( c ( l p q w w w = ( ( a ( b w w w w ( c ( l p q the FIFW w( = w w w w w = ( ( a + ( a ( ( a w w w ( ( a ( ( b + ( b w w w ( ( b ( ( b ( ( c + w w w ( c ( ( c ( ( c w w w w w w ( l l p p q q w w w w = ( ( a ( a ( b ( b w w w w w w w w ( c ( c ( l l p p q q If (4 holds for = that s FIFW w( L = w w ( ( a ( b = = w w w w c l p q = = = = ( ( the whe = + by the operatoal laws Defto.6 we have FIFW w( L + w w + = ( ( a + ( a+ = w w + ( ( a ( ( a+ = w w + ( b + ( b+ = w w + ( ( b ( ( b+ = w w + ( c + ( c+ = w w + ( ( c ( ( c+ = w w w + ( w w + w+ l l+ p p+ q q+ = = = + + w w ( ( a ( b = = = + + + + w w w w c l p q = = = = ( (..e. (4 holds for = +. Therefore (4 holds for all whch completes the proof of Theorem 4.. B. Fuzzy Number Itutostc Fuzzy Ordered Weghted veragg (FIFOW Operator Defto 4.. Let = ( a b c ( l p q ( = L be a collecto of FNIFNs. fuzzy umber tutostc fuzzy ordered weghted averagg (FIFOW operator of dmeso s a mappg FIFOW: Ω Ω that has a assocated vector = ( T L such that [0] ad =. Furthermore FIFOW ( L =
08 Iteratoal Joural of Fuzzy Systems Vol. 0 No. Jue 008 = σ( σ( L (5 σ ( where ( σ ( σ( L σ( s a permutato of ( L such that σ( σ( for all. Especally f = ( L T the the FIFOW operator s reduced to a FIF operator of dmeso. Smlar Theorem 4. we have the followg. Theorem 4.. Let = ( a b c( l p q ( = L be a collecto of FNIFNs the ther aggregated value by usg the FIFOW operator s also a FNIFN ad FIFOW ( = ( ( a L σ ( = bσ ( ( c ( σ = = ( l ( ( p q ( σ σ σ = = = ( ( (6 where = ( T L s the weghtg vector of the FIFOW operator wth [0] ad = whch ca be determed by usg the ormal dstrbuto based method [4]. The FIFOW weghts ca be determed smlar to the OW weghts (for example we ca use the ormal dstrbuto based method to determe the FIFOW weghts [4]. C. Fuzzy Number Itutostc Fuzzy Hybrd ggregato (FIFH Operator Cosder that the FIFW operator weghts oly the FNIFNs whle the FIFOW operator weghts oly the ordered postos of the FNIFNs stead of weghtg the FNIFNs themselves. To overcome ths lmtato motvated by the dea of combg the W ad the OW operator [5 6] what follows we develop a fuzzy umber tutostc fuzzy hybrd aggregato (FIFH operator. Defto 4.3. Let = ( a b c( l p q ( = L be a collecto of FNIFNs. fuzzy umber tutostc fuzzy hybrd aggregato (FIFH Operator of dmeso s a mappg FIFH: Ω Ω whch has a assocated vector = ( T L wth [0] ad = such that FIFH w ( L = σ σ L σ (7 = ( ( ( = where σ ( s the th largest of the weghted FNIFNs ( = w = L w= ( w w L w T s the weght vector of ( = L wth w [0] ad w = ad s the balacg = coeffcet whch plays a role of balace ( ths case f the vector w= ( w w L w T approaches ( L T the the vector ( w w L w approaches ( L. Theorem 4.3. Let = ( a b c( l p q ( = L be a collecto of FNIFNs the ther aggregated value by usg the FIFH operator s also a FNIFN ad satsfes: FIFH w ( = ( ( a L σ ( = b σ ( ( c ( σ = = ( l ( p ( q σ σ σ( = = = ( ( (8 where = ( T L s the weghtg vector of the FIFH operator wth [0] ad σ ( = a ( b ( c ( l ( p σ σ σ σ σ( q σ( = ( ( ( = L s the th largest of the weghted FNIFNs ( = w = L w= ( w s w L w T the weght vector of ( = L wth w = [0] ad w = ad s the balacg coeffcet. = Theorem 4.4. The FIFW operator s a specal case of the FIFH operator. Proof: Let = ( L T the FIFH w ( L = σ( σ( L σ ( = ( σ( σ( L σ( = w w L w = FIFW w( L Ths completes the proof of Theorem 4.4. Theorem 4.5. The FIFOW operator s a specal case of the FIFH operator. Proof: Let w= ( L T the = w
Xfa Wag: Fuzzy Number Itutostc Fuzzy rthmetc ggregato Operators 09 = ( = L thus FIFH w ( L = L σ σ( σ( ( = σ( σ( L σ ( = FIFOW ( L whch completes the proof of Theorem 4.5. Clearly from Theorem 4.4 ad Theorem 4.5 we ow that the FIFH operator geeralzes both the FIFOW ad FIFW operators ad reflects the mportace degrees of both the gve fuzzy umber tutostc fuzzy argumet ad the ordered posto of the argumet. 4. applcato of the FIFH operator to multple attrbute decso mag I the followg we apply the FIFH operator to multple attrbute decso mag based o fuzzy umber tutostc fuzzy formato. Let = { L m } be a set of alteratves ad let B = { B B L B } be a set of attrbutes. w = ( w w L w T s the weght vector of B ( = L wth w [0] ad w =. ssume that the characterstcs of the alteratves ( = L m s represeted by the fuzzy umber tutostc fuzzy set: = {( B ( a b c( l p q ( B ( a b c( l p q L ( B ( a b c( l p q } (9 where trgoometrc fuzzy umber ( a b c dcates the degree that the alteratve satsfes the attrbute B ( l p q dcates the degree that the alteratve does ot satsfy the attrbute B ( a b c F( I ( l p q F( I 0 c + q m. Let = ( a b c ( l p q for all the equato (9 ca wrtte as = {( B B B} (0 To get the best alteratve(s we ca utlze the FIFH operator: = FIFH w ( L = L m ( to derve the overall values = ( a b c ( l p q = ( = L m of the alteratves ( = L m where = ( L T s the weghtg vector of the FIFH operator wth [0] ad = whch ca be determed by the ormal dstrbuto based method (Xu 005 [4]. The by equato (0 we calculate the scores S ( ( = L m of the overall values ( = L m ad utlze the scores S ( ( = L m to ra the alteratves ( = L m ad the select the best oe(s (f there s o dfferece betwee two scores S ( ad S ( the we eed to calculate the exteded scores L( = ad L( of the overall values ad respectvely ad the ra the alteratves ad accordace wth the exteded scores L( ad L(. 5. Illustratve example I ths secto a multple attrbute decso-mag problem volves the prortzato of a set of formato techology mprovemet proects (adapted from [ 7] s used to llustrate the developed procedures. The formato maagemet steerg commttee of Mdwest merca Maufacturg Corp. must prortze for developmet ad mplemetato a set of four formato techology mprovemet proects ( = 3 4 whch have bee proposed by area maagers. The commttee s cocered that the proects are prortzed from hghest to lowest potetal cotrbuto to the frm's strategc goal of gag compettve advatage the dustry. I assessg the potetal of each proect three factors are cosdered B --productvty B -- dfferetato ad B3 --maagemet. The productvty factor assesses the potetal of a proposed proect to crease effectveess ad effcecy of the frm's maufacturg ad servce operatos. The dfferetato factor assesses the potetal of a proposed proect to fudametally dfferetate the frm s products ad servces from ts compettors ad to mae them more desrable to ts customers. The maagemet factor assesses the potetal of a proposed proect to assst maagemet mprovg ther plag cotrollg ad decso-mag actvtes. The followg s the lst of proposed
0 Iteratoal Joural of Fuzzy Systems Vol. 0 No. Jue 008 formato systems proects: --Qualty Maagemet Iformato --Ivetory Cotrol 3 --Customer Order Tracg 4 --Materals 3 4 Purchasg Maagemet. Suppose that the weght vector of B B ad B 3 s w = (0.3440.5000.4058 T ad the commttee represets the characterstcs of the proects ( = 3 4 by the FNIFNs ( = 3 4 = 3 wth respect to the factors B ( = 3 lsted the followg. = {( B (0.30.40.5(0.30.30.4 ( B (0.40.50.6(0.0.30.4 ( B3 (0.0.0.3(0.50.60.7 } = {( B (0.50.60.7(0.0.0.3 ( B (0.60.60.6(0.0.0.3 ( B3 (0.40.60.7(0.0.0. } 3 = {( B (0.30.40.6(0.0.30.4 ( B (0.40.50.6(0.30.30.3 ( B3 (0.70.70.7(0.0.0.3 } 4 = {( B (0.70.70.8(0.0.0. ( B (0.60.60.7(0.0.0.3 ( B3 (0.0.30.4(0.0.0.3 } To ra the gve four proects we frst weght all the FNIFVs ( = 3 4 = 3 by the vector w = (0.3440.5000.4058 T of attrbutes B ( = 3 ad multples these values by the balacg coeffcet = 3 ad gets the weghted FNIFVs where = w ( = 3 4 = 3 : = (0.3080.40990.5(0.8850.8850.388 = (0.3830.40540.4970(0.990.40540.5030 3 = (0.040.3790.35(0.4300.53690.6478 = (0.50.680.75(0.8980.8980.885 = (0.49700.49700.4970(0.7780.990.4054 3 = (0.4630.670.769(0.06060.400.40 3 = (0.3080.40990.68(0.8980.8850.388 3 = (0.3830.40540.4970(0.40540.40540.4054 33 = (0.7690.7690.769(0.06060.400.309 4 = (0.750.750.80(0.0980.8980.898 4 = (0.49700.49700.5946(0.7780.990.4054 43 = (0.3790.350.463(0.06060.400.309 By equato (0 we calculate the scores of ( = 3 4 = 3 : S( = 0.0964 S( = 0.0033 S( 3 = -0.3008 S( = 0.397 S( = 0.07 S( 3 = 0.53 S( 3 = 0.46 S( 3 = 0.00 S( 33 = 0.657 S( 4 = 0.5706 S( 4 = 0.6 S( 43 = 0.080 Thus σ( = σ( = σ(3 = 3 σ( = 3 σ( = σ(3 = 3 σ( = 33 3 σ( = 3 3 σ(3 = 3 = = = 4 σ( 4 4 σ( 4 4 σ(3 43 Suppose that = (0.490.540.49 T (derved by the ormal dstrbuto based method (Xu 005 [4] s the weghtg vector of the FIFH operator the by equato (8 we get the overall values ( = 3 4 of the alteratves ( = 3 4 : = (0.70.36960.4688(0.3380.39960.503 = (0.49640.6030.687(0.460.970.633 3 = (0.4790.5930.6356(0.790.6330.3458 4 = (0.5390.5370.6390(0.690.30.94 By equato (0 we calculate the scores of the overall values ( = 3 4 : S( = -0.0363 S( = 0.3977 S( 3 = 0.80 S( 4 = 0.3403 Sce S( > S( 4 > S( 3 > S( Thus f 4 f 3 f the the most desreble alteratve s. 6. Cocluso I ths paper we have troduced some operatoal laws of FNIFNs ad developed some ew arthmetc aggregato operators cludg the FIFW operator the FIFOW operator ad the FIFH operator whch exted the W operator ad the OW operator to accommodate the stuatos where the gve argumets are fuzzy umber tutostc fuzzy umbers. The FIFH operator geeralzes both the FIFOW ad FIFW operators ad reflects the mportace degrees of both the gve fuzzy umber tutostc fuzzy umber ad the ordered posto of the argumet. Ths paper develops the theores both the aggregato operators ad
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