Some Scoring Functions of Intuitionistic Fuzzy Sets with Parameters and Their Application to Multiple Attribute Decision Making

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1 JOURNL OF COMPUTER VOL 8 NO JNURY 3 55 ome corg Fuctos of Itutostc Fuzzy ets wth Parameters ad Ther pplcato to Multple ttrbute Decso Makg Zhehua Zhag Csco chool of Iformatcs Guagdog Uversty of Foreg tudes Guagzhou Cha Emal: statstcs_zhag@yahoocomc zhagzhehua@malgdufseduc Jyu Yag ad Youpe Ye chool of Computer cece ad Techology Najg Uversty of cece ad Techology Najg Cha Emal: yagjy@maljusteduc youpeyezyl@yahoocomc Yog Hu ad Qasheg Zhag chool of maagemet Csco chool of Iformatcs Guagdog Uversty of Foreg tudes Guagzhou Cha Emal: heryhu@gmalcom zhqash@6com bstract ovel tutostc fuzzy sets wth parameters (IFP) s troduced ths paper Compared wth covetoal tutostc fuzzy sets (IF) IFP ca provde more choces whe t s appled to multple attrbute decso makg By aalyzg the degree of hestacy a tutostc fuzzy sets wth double parameters (IFDP) model s preseted d the ths paper proposes some weghted scorg fuctos based o IF ad IFP Fally a multple attrbute decso makg eample appled to cty plag s gve to demostrate the applcato of IFDP ad ts weghted scorg fuctos The smulato results show that the weghted scorg fucto method based o IFDP s more comprehesve ad fleble tha the tradtoal tutostc fuzzy sets method Ide Terms tutostc fuzzy sets tutostc fuzzy sets wth parameters tutostc fuzzy sets wth double parameters scorg fucto multple attrbute decso makg I INTRODUCTION I 986 taassov troduced membershp fucto o-membershp fucto ad hestacy fucto ad preseted IF ([]) Hece may scholars appled IF to decso-makg aalyss ad patter recogto wdely I the research feld of IF Yager dscussed the cut set characterstcs of IF [3] zmdt ad Kacprzyk appled t to medcal dagoss [4 5 6] some scholars (Z Xu H Zhao G W We Lu ad X Y Yue et al) Mauscrpt receved Jauary ; revsed Jue ; accepted eptember The Natoal Natural cece Foudato of Cha (No 776 No676) Zhehua Zhag Csco chool of Iformatcs Guagdog Uversty of Foreg tudes Guagzhou Hgher Educato Mega Ceter Guagzhou Cha 56 Tel: appled t to decso-makg aalyss [ ] some researchers ( P Xu D F L Y H L ad W L Hug et al) appled t to patter recogto [7 8 9 ] Le studed tutostc fuzzy reasog [7] ad Y Q Zhag ad X B Yag appled t to attrbute reductos based o rough sets [9] By troducg membershp fucto μ () omembershp fucto ν () ad hestacy fucto π () the IF theory s establshed whch geeralzes Zadeh s fuzzy sets (F) ([ 3]) ccordg to the IF defto μ () deotes the proporto of the support party ν () deotes the proporto of the opposto party ad π () deotes the proporto of the abset party Though may scholars studed IF ad appled t to decso makg ([ ]) ther methods are sutable for statc model ad usutable for dyamc model The Xu ad Yager ([]) preseted a dyamc decso makg model whch was also studed by We u et al ([4 6]) However tradtoal decso makg researches based o IF do ot volve the detachmet of the abset party whch meas that the abset party s ot specfcally aalyzed covetoal decso makg model of IF Moreover the abset party may chage over tme practce whle covetoal IF method caot deal wth ths kd of dyamc decso makg model Thus the research o the varato of the abset party wll play a mportat role dyamc decso makg Takg to accout ths we preset a ovel IFP model by aalyzg the hestacy fucto We frst assume that μ () s the frm support party of evet ν () s the frm opposto party of evet π () s the mamum abset party of evet π ()(- λ ()) π () s the frm abset party of evet ad π ()-π ()λ ()π () deotes the covertble abset part where λ () s the proporto of the covertble abset dvduals all the abset dvduals Obvously do:434/jcp855-6

2 56 JOURNL OF COMPUTER VOL 8 NO JNURY 3 we have μ () ν () π () ad λ () We dvde the covertble abset part to two parts: λ ()λ ()π () beg the abset party whch ca be coverted to the support party ad λ () (-λ ())π () beg the abset party whch ca be coverted to the opposto party where λ () s the proporto of the covertble abset dvduals beg coverted to the support party ad -λ () s the proporto of the covertble abset dvduals beg coverted to the opposto party d we also have λ () ccordg to the detachmet of the abset party we propose a seres of deftos ad costructo methods of IFP ad cocetrate o the model of IFDP The we troduce a type of scorg fucto o the bass of IF ad geeralze t to IFP d the takg advatage of the scorg fucto IFDP s appled to multple attrbute decso makg We ca adjust the parameters to approprate values to obta all the feasble results Therefore the IFP method ca be appled to the dyamc decso makg feld ad we ca predct all the possble decso makg results the future accordg to the varato of membershp fucto o-membershp fucto ad hestacy fucto I summary the ew method rased ths paper ca epad the scope of IF appled to decso makg The smulato results show that the method troduced ths paper s more comprehesve ad fleble tha the covetoal IF method Hece ths paper ca provde valuable cocluso for the feld of applcato research of IF ad the model of IFP s also useful to tutostc fuzzy reasog Furthermore ths method ca be geeralzed to terval-valued tutostc fuzzy sets as [8 3] II CONTRUCTION OF IFP Defto IF uverse X s gve as follows (taassov [ ]): {< μ () ν () > X} () Where μ : X [ ]ν : X [ ] wth the codto μ () ν () for each X The umbers μ () ν () [ ] deote membershp fucto ad omembershp fucto of to respectvely For each IF X we call π () μ () ν () hestacy fucto of to π () for each X Defto Let X be a uverse of dscourse Beg the epaso of IF a IFDP X s a object havg the form: { < μ( ) ν ( ) > X} Let μ( ) μ( ) α ( ) ν ( ) ν ( ) β( ) where μ () ν () ad π () are the same as defto d we have: α( ) β( ) π ( ) π ( ) where α( ) β( ) π ( ) π ( ) ad the μ() ν() π() μ() ν() α() β() π() μ() ν() π() Thus IF s a specal case of IFDP whe α( ) β( ) Theorem Let be a IFDP as defto the μ ( ) μ ( ) ν ( ) ν ( ) π ( ) π ( ) () From defto we have formula () ccordg to defto let all sample data be dvded to three parts μ () beg the frm support party of evet ν () represetg the frm opposto party of evet ad π () showg all the abset party I the abset party π ( ) s the frm abset party ad π( ) π ( ) s the covertble abset party whch each sample may become oe of the support party ad the opposto party If there s α ( ) sample supportg evet ad β ( ) sample opposg evet we have IFDP as defto If the proporto of the abset party coverted to the support party s λ ( ) ad that coverted to the opposto party s λ ( ) the model wll become tutostc fuzzy sets wth sgle parameter where α( ) λ ( )( π( ) π( )) β( ) ( λ ( ))( π( ) π( )) If the frm abset party s π ( ) ( λ( )) π ( ) the π ( ) π ( ) λ( ) π ( ) ad the we wll obta the other IFDP defto as follows: Defto 3 IFDP whch s the epaso of IF uverse X s gve as follows: { < μ( ) ν ( ) > X} Where μ( ) ν ( ) π ( ) represet membershp fucto o-membershp fucto ad hestacy fucto of to respectvely d we have: μ () μ () λ () λ () π () ν () ν () λ ()( λ ()) π () π λ π () ( ()) () Where λ ( ) ad μ () ν () ad π () are the same as defto Whe λ ( ) ad λ ( ) are radom varables the IFDP model becomes a radom IFDP model ad they possess the correspodg probablty dstrbutos For eample f P s a probablty fucto ad we have: If P( λ ( ) ) the μ () μ () ν () ν () π () π () d f P( λ( ) λ λ ( ) λ) the μ () μ () λλπ () ν () ν () λ ( λ) π () π () ( λ ) π() It s clear that we wll get the followg coclusos: whe λ ( ) IFDP s IF as defto ; whe

3 JOURNL OF COMPUTER VOL 8 NO JNURY 3 57 λ ( ) IFDP s fuzzy sets; whe < λ ( ) < ad λ ( ) or λ ( ) t s a severe skewess IF; whe < λ ( ) < ad λ ( ) 5 IFDP s a compromsg oe If λ ( ) λ ad λ s costat the IFDP s a tutostc fuzzy sets wth fed double parameters otherwse t s a varable model III WEIGHTED CORING FUNCTION uppose T ad F are two types of etreme IF X where T{<> X} meas μ T () ad ν T () ad F{<> X} meas μ F () ad ν F () We ote mf ( ) to be a weghted scorg mf ( ) mt ( ) fucto where m ( T) ad m ( F) deote dstace measures I ths paper the followg weghted scorg fucto based o IF wll be used the smulato w[ μ( ) ( ν ( ) ) π ( ) ] w μ ν π w μ ν π (3) [ ( ) ( ( ) ) ( ) ] [( ( ) ) ( ) ( ) ] The we have the followg formula (4): w[ μ ( ) ν ( )] 5 (4) 4 w[( μ( )) ( ν ( )) μ( ) ν ( ) 5 μ( ) 5 ν ( ) ] Obvously we have If F the F ad f T the T ce F dcates that all the eample data s the frm opposto party of evet we defe F whch meas that the score of F decso s zero ad the result of F caot be selected mlarly we have T whch meas that the result of T s perfect ad should be selected uppose that s a IFDP beg the epaso of IF ad the we ca defe ts weghted scorg fucto as follows: m ( F) m ( F) m ( T) ( ) [ μ( ) ( ν ( ) ) π ( ) ] m F w ( ) [( μ( ) ) ν ( ) π ( ) ] m T w The we have: (5) U 5 (6) L U w[( μ ( ) ν ( )) π ( ) λ ( ) λ ( ) π ( ) λ ( )] [4( μ( ) ν ( ) μ( ) ν ( ) 5 μ( ) 5 ν ( ) ) L w 4 π ( ) λ ( )( λ ( ) λ ( ) ) 4 π ( )( μ( ) ν ( )) λ( ) λ ( ) (4 μ ( ) 8 ν ( ) 6) π ( ) λ ( )] ccordg to formula (5 6) we ca draw a cocluso that s a specal case of whe λ ( ) ad that they both have some smlar propertes as follows Defto4 Let ad B be two IFs uverse X B f ad oly f μ ( ) μb( ) ν ( ) ν B( ) for each X mlarly suppose that ad B are two IFDPs as defto (or defto3) we also have B f ad oly f μ( ) μb( ) ν ( ) ν B( ) for each X Theorem Let ad B be two IFs ad the we have: If B the core( ) core( B) mlarly let ad B be two IFDPs we also have: If B the core( ) core( B ) Proof: ccordg to formula (4) B thus we have μ ( ) μb( ) ν ( ) νb( ) μ( ) ν ( ) μb( ) ν B( ) ad the we have core( ) core( B) We ca also obta the same cocluso for IFDPs Defto5 Let be a IF uverse X {< μ () ν () > X} the complemet of s defed by: {< ν () μ ()> X} for each X mlarly suppose that {< μ ( t) ν ( t) > X} s a IFDP whch s the epaso of IF ad the the complemet of s defed by: ( ) {< ν ( t) μ ( t)> X} Theorem3 Let be a IF as metoed above the core( ) core( ) mlarly we also have core( ) core(( )) for each IFDP whch s the epaso of IF Proof: w[ μ( ) ν ( )] 5 4 [( ( )) ( ( )) ( ) ( ) 5 ( ) 5 ( ) ] w μ ν μ ν μ ν w[ ν( ) μ( )] w ν μ μ ν μ ν 5 4 [( ( )) ( ( )) ( ) ( ) 5 ( ) 5 ( ) ] We ca also obta the same cocluso for IFDP ccordg to theorem 3 ad formula (4) t s easy to get theorem 4 Theorem4 Let be a IF uverse X the core( ) 5 f ad oly f μ ( ) ν ( ) core( ) < 5 f ad oly f μ ( ) < ν ( ) ad core( ) > 5 f ad oly f μ ( ) > ν ( ) mlarly we have the same property for each IFDP uverse X too From the defto of the radom IFDP model we ca costruct the radom scorg fucto model For eample: If P( λ ( ) ) the ssume that P s a probablty fucto λ ( ) ad λ ( ) are both radom varables Let

4 58 JOURNL OF COMPUTER VOL 8 NO JNURY 3 λ ( ) λ λ ( ) λ If the jot probablty dstrbuto P( λ λ ) s a cotuous dstrbuto for each λ [] ad for each λ [] the we obta the epectato of all the radom decso-makgs as follows: E P( λ λ ) dλ dλ (7) If we kow the margal probablty dstrbuto of ( λ λ ) the we also have the margal epectato of all the radom decso-makgs as follows: Eλ P( λ λ ) dλ (8) μ ( ) ( ν ( ) ) π ( ) μ ν π μ ν π ( ) ( ( ) ) ( ) ( ( ) ) ( ) ( ) IV PPLICTION TO INGLE TTRIBUTE DECIION MKING I the followg we wll apply the scorg fucto of IF ad IFDP above to decso makg uppose that s feasble optos set { } ccordg to the followg data from we wll make a choce amog ssume that ( ) are represeted by IF show as follows: {< 7 3>} {< 6 35>} 3 {< 6 4>} 4 {< 5 4>} 5 {< 4 4>} 6 {< 4 5>} 7 {< 4 6>} 8 {< 35 6>} 9 {< 3 7>} where {< μ () ν ()> X} For eample meas that the degree of membershp s μ () 7 ad that the degree of o-membershp s ν () 3 thus the degree of hestacy s π () for X ccordg to ther practcal sgfcace we have: I the followg we wll compare the results calculated by covetoal rakg fuctos of IF wth the results calculated by the scorg fucto of IF ad IFDP ccordg to the practcal sgfcace of IF we ca make decso applyg the degree of membershp Thus we have the followg membershp fucto ([]): For each X RM( ) w ( ) μ ( ) (9) X Che ad Ta proposed the followg rakg fucto to make decso ([]): For each X RCT ( ) w ( )( μ ( ) ( )) ν () X Hog ad Cho troduced the followg rakg fucto to make decso ([3]): For each X RHC ( ) w ( )( μ ( ) ( )) ν () X From formulas (4 9 ) we ca get the results as Fgure For eample R ( ) R ( ) ( ) CT R HC ad are calculated by formulas (4 9 ) ad the we have: R μ ( ) 7 R μ ( ) ν ( ) 4 R μ ( ) ν ( ) M CT HC Fgure Result comparsos betwee covetoal rakg fuctos ad scorg fucto Let {< μ () ν ()> X} { I } { < ma( μ ( )) m( ν ( )) > X} I I { < m( μ ( ))ma( ν ( )) > X} I I μ ( ) ma( μ ( )) ν ( ) m( ν ( )) I I μ ( ) m( μ ( )) ν ( ) ma( ν ( )) I π ( ) μ ( ) ν ( ) π ( ) μ ( ) ν ( ) d the Xu preseted the followg rakg fucto ([]): m ( ) RXu ( ) () m ( ) m ( ) Xu made use of four dstace measures ad appled them to formula () to make decso ccordg to Xu [] let p calculated by Xu s formulas ad the we have: p p m ( ) [5 w ( )( μ ( ) μ ( ) ( ) ( ) ν ν I X p / p π ( ) π ( ) )] p p ( ) [5 ( )( μ ( ) μ ( ) ( ) ( ) ν ν X p / p π ( ) π ( ) )] p p p / p [ w( )( μ ( ) μ ( ) ( ) ( ) ( ) ( ) )] ν ν π π X p p p / p [ w( )( μ ( ) μ ( ) ( ) ( ) ( ) ( ) )] ν ν π π X p p p / p [ w( )( μ ( ) μ ( ) ( ) ( ) ( ) ( ) )] ν ν π π X p p p / p [ w ( )( μ ( ) μ ( ) ( ) ( ) ( ) ( ) )] ν ν π π X m w m ( ) m ( ) X X w ( )[m( ( ) μ ( )) m( ( ) ν ( )) m( ( ) π ( ))] μ ν π X m3 ( ) w ( )[ma( μ ( ) μ ( )) ma( ν ( ) ν ( )) ma( π ( ) π ( ))] w ( )[m( μ ( ) μ( )) m( ν ( ) ν ( )) m( π ( ) π ( ))] m3 ( ) w ( )[ma( μ ( ) μ ( )) ma( ν ( ) ν ( )) ma( π ( ) π ( ))] X w ( )[ μ ( ) μ ( ) ν ( ) ν ( ) π ( ) π ( )] m ( ) ma[ w ( )(( ( ) ( ) ( )) w( )( ( ) ( ) ( ))] w ( )[ μ ( ) μ ( ) ν ( ) ν ( ) π ( ) π ( )] m ( ) ma[ w ( )(( ( ) ( ) ( )) w ( )( ( ) ( ) ( ))] We defe:

5 JOURNL OF COMPUTER VOL 8 NO JNURY 3 59 k mk( ) RXu ( ) k 34; 345 (3) mk( ) mk( ) From formula (3) we have Fgure For eample { < 73 > X} { < 37 > X} Let p the we have: m ( ) [5( μ ( ) μ ( ) ν ( ) ν ( ) / π ( ) π ( ) )] m ( ) [5( μ ( ) μ ( ) ν ( ) ν ( ) π ( ) π ( ) )] / / [5( )] 6 4 λ ( ) λ ( ) λ ( ) 5 6 ( )( ( ) ( ) ) 4 ( ) 5 λ λ λ λ λ ( ) λ ( ) λ ( ) 5 4 ( )( ( ) ( ) ) 4 ( ) ( ) 4 ( ) 4 6 λ λ λ λ λ λ 35 λ ( ) λ ( ) 5 λ ( ) 5 ( )( ( ) ( ) ) 7 ( ) ( ) ( ) 3 8 λ λ λ λ λ λ d the we have the followg Fgure 3 From Fgure 3 accordg to IFDP we have the same cocluso: 9 p 8 p 7 p 6 p 5 p 4 p 3 p p R R Xu Xu m ( ) ( ) 65 m ( ) m ( ) 6 m ( ) 95 ( ) 64 m ( ) m ( ) R R m ( ) 9 ( ) 565 ( ) ( ) Xu 3 m 3 m 3 m ( ) 868 ( ) 59 ( ) ( ) Xu 4 m 4 m 4 mlarly we ca obta other results as Fgure Fgure3 Three-dmesoal fgure ( ( k )λ () λ ()) of IFDP From Fgure 3 f λ () the we have Fgure 4 ad f λ () ad λ () the we have From ther practcal sgfcace we also have {< 4 6>} f λ () ad λ () From Fgure 4 we also have: p p p p p p p p Fgure Result comparsos betwee Xu s rakg fuctos ad scorg fucto ccordg to the rakg of Xu s methods Che s method ad scorg fucto method are more effectve tha Hog s method ad membershp method ad they are appromate o the fal decso results whch s: 9 p 8 p 7 p 6 p 5 p 4 p 3 p p However there are some dffereces amog Xu s methods Che s method ad scorg fucto method ad the dfferece amog the slops of them are the most sgfcat dfferece Obvously the slops of them are raked as follows: Che s method > scorg fucto method> Xu3> Xu Xu > X4 From formula (6) we get the followg formulas λ( ) λ ( ) 5 λ( ) 5 λ ( )( λ ( ) λ ( ) ) 7 λ ( ) λ ( ) 5 λ ( ) 3 λ ( ) λ ( ) λ ( ) 5 4 ( )( ( ) ( ) ) 4 ( ) ( ) 8 ( ) 4 4 λ λ λ λ λ λ Fgure4 ecto fgure f λ () based o IFDP VII PPLICTION TO MULTIPLE TTRIBUTE DECIION MKING I the followg we wll apply the scorg fucto above to multple attrbute decso makg based o IF ad IFP We llustrate the advatage of IFP by the followg eample from [] cty s plag to buld a mucpal lbrary Oe of the problems facg the cty developmet commssoer s to determe what kd of ar-codtog system should be stalled the lbrary ([]) The cotractor

6 6 JOURNL OF COMPUTER VOL 8 NO JNURY 3 offers fve feasble optos ( 3 4 5) whch mght be adapted to the physcal structure of the lbrary uppose that three attrbutes C (ecoomc) C (fuctoal) ad C 3 (operatoal) are take to cosderato the stallato problem the weght vector of the attrbutes C j (j3) s w(35) T ssume that the characterstcs of the optos ( 345) are represeted by IF show as follows: {<C 4> <C 7 > <C 3 6 3>} {<C 4 > <C 5 > <C 3 8 >} 3 {<C 5 4> <C 6 > <C 3 9 >} 4 {<C 3 5> <C 8 > <C 3 7 >} 5 {<C 8 > <C 7 > <C 3 6>} We wll compare the results calculated by covetoal dstace measures of IF ([]) wth the results calculated by the score fucto of IF ad IFDP Calculated by formulas (4 6) above we ca obta the results as follows We oly calculate the results usg score fucto ad the results of Xu s methods are from [] For eample: By formula (4) we calculate the followg scores: mlarly we have the followg results Table From formula (4) we have 3 f 4 f 5 f f we have the followg formulas: 4 λ ( )( λ ( ) λ ( ) ) 564 λ ( ) λ ( ) 49 λ ( ) 8 ( )( ( ) ( ) ) 68 ( ) ( ) 54 ( ) 4 λ λ λ λ λ λ 9 λ ( )( λ ( ) λ ( ) ) 746 λ ( ) λ ( ) 658 λ ( ) 7 38 ( )( ( ) ( ) ) 33 ( ) ( ) 736 ( ) 44 λ λ λ λ λ λ 5 λ ( )( λ ( ) λ ( ) ) 4 λ ( ) λ ( ) 86 λ ( ) 7 ( )( ( ) ( ) ) 44 ( ) ( ) 7 ( ) 3 3 λ λ λ λ λ λ 38 λ ( )( λ ( ) λ ( ) ) 36 λ ( ) λ ( ) λ ( ) 7 76 ( )( ( ) ( ) ) 3 ( ) ( ) 8 ( ) λ λ λ λ λ λ 6 λ ( )( λ ( ) λ ( ) ) 57 λ ( ) λ ( ) 474 λ ( ) 6 5 ( )( ( ) ( ) ) 3 ( ) ( ) 58 ( ) 59 5 λ λ λ λ λ λ ccordg to the formulas ad 3 we 4 5 descrbe the three-dmesoal space fgure (Fgure 5) ad the secto fgure (Fgure 6) whe λ () 5 Where we ote λ λ () λ λ () From Fgure5 3 s the optmal decso most cases oly ecept the rght upper corer where λ () ad λ () ccordg to defto 3 λ () dcates that the proporto of the abset party beg TBLE I MULTIPLE TTRIBUTE DECIION MKING BED ON RNKING FUNCTION OF IF Rakg fucto Xu Xu Xu3 Xu4 corg fucto ad ths result s the same as that [] From Table we have the optmal decsos Table TBLE II THE OPTIML DECIION ON RNKING FUNCTION OF IF Rakg fucto Optmal decso Xu Xu Xu3 Xu4 corg fucto From defto 3 we wll apply the scorg fucto of IFDP to multple attrbuto decso makg ccordg to defto 3 we have formula (6) ad the Fgure5 Three-dmesoal fgure (( k )λ () λ ()) coverted to aother party s hgh ad λ () dcates that the proporto of the covertble abset party beg coverted to the support party s hgh Thus we kow that whe the majorty of the abset party ca be coverted ad most of them are coverted to the support party s the optmal decso Based o the deftos of ad 3 {<C 4 > <C 5 > <C 3 8 >} ad 3 {<C 5 4> <C 6 > <C 3 9 >} There are oly slght dffereces betwee ad 3 We have 3 > for attrbute C ad attrbute C 3 whle t s dffcult to judge whch oe s the better betwee ad 3 for attrbute C The hestacy degree of s bgger tha that of 3 thus s mpacted more tha 3 by the etreme varato of the abset party decso-makg ad s less stable If most of the abset party s coverted to the support party > 3 ; otherwse 3 > Cosderg that 3 s more stable tha 3 should be the optmal decso the actual decso-makg From Fgure 6 the kow codto s λ () 5 whch meas that half of the covertble abset party s coverted to the support party ad the other s

7 JOURNL OF COMPUTER VOL 8 NO JNURY 3 6 coverted to the opposto party The we have the followg results: Fgure6 ecto fgure f λ () 5 If λ () <45 the 3 f 4 f 5 f f ad ths result s the same as IF If 45<λ () <73 the f f f f ; If λ () >73 the f f f f Compared wth these results 3 s the optmal decso for λ () [ ] mlarly we ca also study the varato of k wth λ () whe λ () s kow If λ () the IFP s equvalet to IF ad the we have μ( ) μ( ) ν ( ) ν ( ) π ( ) π ( ) whch meas that the result of the left plae Fgure 5 s the result calculated by IF mlarly the result of the left coordate as Fgure 6 s also the result calculated by IF I [] professor Xu apples four kds of dstace measures to make decsos I ths paper we use oly oe scorg fucto to make decsos ad the results are the same as Xu s Furthermore by aalyzg the varato of the determacy degree we reveal a potetal decso makg result ad the reaso for selectg Because t s the frst tme for us to eplore the applcato of IFP to multple attrbute decso makg we assume that λ () s the same value for all the attrbutes ad λ () s smlar to λ () I practcal decso-makg sce the proporto of the covertble abset party may be dfferet for all the attrbutes researcher ca set more parameters to meet the eeds of actual problem realty The epermet results above show that there s much dfferece betwee multple attrbute decso makg results of IFP ad that of IF Covetoal IF method s smple but ts decso makg results are fed whe t s calculated by covetoal rakg fuctos Thus t s dffcult to reveal the potetal law from all the avalable formato whe usg the IF method d the results of the IFP method are varable whch ca be adjusted to possble results wth the varato of the parameters Furthermore for supervsed models f the fed decso makg results of IF are dfferet from the results of the practcal data ad the actual decsos the IF method wll fall to fal However whe usg the IFP method we ca meet the eeds of the practcal data ad the actual decso by adjustg the parameters to approprate values ll the results above show that the IFP method s more comprehesve ad fleble tha the IF method VIII EXPECTTION CORING FUNCTION PPLIED TO MULTIPLE TTRIBUTE DECIION MKING From formula (8) we obta the margal epectato rakg fuctos as follows: Eλ P( λ λ ) dλ (4) ( ) Eλ P λ λ dλ (5) Where P( λ λ ) represets the jot probablty dstrbuto of the radom vector ( λ λ ) If the jot probablty dstrbuto of ( λ λ ) s a cotuous uform dstrbuto for each λ [] ad for each λ [] the we obta the formulas as follows: λ λ λ λ E d (6) E d (7) Cosderg formula (6) we have: aλ a 5 a3λ a4λ a5 (8) bλ b 5 bλ b λ b (9) Where a w[ π ( ) λ π ( )] a w( μ ( ) ν ( )) 3 π λ λ a w 4 ( )( ) a w[4 π ( )( μ ( ) ν ( )) λ (4 μ ( ) 8 ν ( ) 6) π ( )] 4 5 μ ν μ ν μ ν a w4( ( ) ( ) ( ) ( ) 5 ( ) 5 ( ) ) b w π ( ) λ b w( μ ( ) ν ( ) π ( ) λ ) 3 4 π ( ) λ 4 4 π ( )[( μ( ) ν ( )) λ π ( ) λ] 5 μ ν μ ν μ ν 4 π( ) λ (4 μ( ) 8 ν ( ) 6) π( ) λ] b w b w b w[4( ( ) ( ) ( ) ( ) 5 ( ) 5 ( ) ) Formula (8) s substtuted to formula (6) ad we get: aλ a E 5 dλ λ a3λ a4λ a5 () mlarty we have formula () from formula (7) ad formula (9):

8 6 JOURNL OF COMPUTER VOL 8 NO JNURY 3 bλ b E 5 dλ λ b3λ b4λ b5 () Obvously formula () ad formula () ca be calculated from the tegral method of ratoal fucto IX CONCLUION We propose a ovel IFP model accordg to IF ad apply t to multple attrbute decso makg The IFP method ot oly volves membershp fucto ad omembershp fucto but also volves the detachmet of hestacy fucto Therefore t s more effectve tha the IF method CKNOWLEDGMENT Ths work was supported part by a grat from the Natoal Natural cece Foudato of Cha (No776 No676) the Foudato of Guagdog Uversty of Foreg tudes (NoGWJYYB) Project of phlosophy ad socal sceces research Mstry of Educato (NoYJC7446 NoYJCZH8) cece ad Techology Plag Project of Guagdog Provce (No B634) REFERENCE [] k taassov Itutostc fuzzy sets Fuzzyets ad ystems vol pp [] K taassov Itutostc Fuzzy ets Theory ad pplcatos New York: Hedelberg Physca-verl 999 [3] Roald R Yager ome aspects of tutostc fuzzy sets Fuzzy Optm Decs Makg vol 8 pp [4] E zmdt ad J Kacprzyk smlarty measure for tutostc fuzzy sets ad ts applcato supportg medcal dagostc reasog ICIC pp [5] E zmdt ad J Kacprzyk Dstaces betwee tutostc fuzzy sets ad ther applcatos reasog tudes Computatoal Itellgece vol pp -6 5 [6] E zmdt ad J Kacprzyk Dlemmas wth dstaces betwee tutostc fuzzy sets: straghtforward approaches may ot work tudes Computatoal Itellgece vol 9 pp [7] P Xu C Q L L Jag ad X P Lu mlarty measures for cotet-based mage retreval based o tutostc fuzzy set theory Joural of Computers vol 7(7) pp [8] D F L ad C T Cheg New smlarty measures of tutostc fuzzy sets ad applcatos to patter recogtos Patter Recogto Letters vol 3 pp -5 [9] Y H L Davd L Olso ad Z Q mlarty measures betwee tutostc fuzzy (vague) sets: comparatve aalyss Patter Recogto Letters vol 8 pp [] W L Hug M Yag mlarty measure of tutostc fuzzy sets based o L p metrc Iteratoal Joural of ppromate Reasog vol 46 pp [] Z Xu ome smlarty measures of tutostc fuzzy sets ad ther applcatos to multple attrbute decso makg Fuzzy Optm Decs Makg vol 6 pp 9-7 [] Z Xu Roald R Yager Dyamc tutostc fuzzy mult-attrbute decso makg Iteratoal Joural of ppromate Reasog vol 48 pp [3] H Zhao Z Xu M F N ad Lu Geeralzed aggregato operators for tutostc fuzzy sets Iteratoal Joural of Itellget ystems vol 5 pp - 3 [4] G W We ome geometrc aggregato fuctos ad ther applcato to dyamc attrbute decso makg tutostc fuzzy settg Iteratoal Joural of Ucertaty Fuzzess ad Kowledge-Based ystems vol 7 pp [5] G W We ome duced geometrc aggregato operators wth tutostc fuzzy formato ad ther applcato to group decso makg ppled oft Computg vol pp [6] Z X u M Y Che G P Xa ad L Wag teractve method for dyamc tutostc fuzzy multattrbute group decso makg Epert ystems wth pplcatos vol 38() pp [7] Y Le Y J Le J X Hua W W Kog ad R Ca Techques for target recogto based o adaptve tutostc fuzzy ferece ystems Egeerg ad Electrocs vol 3 pp [8] Z Xu method based o dstace measure for terval-valued tutostc fuzzy group decso makg Iformato ceces vol 8 pp 8-9 [9] Y Q Zhag ad X B Yag Itutostc fuzzy domace based rough set approach: model ad attrbute reductos Joural of oftware vol 7(3) pp [] Lu Research o selecto of cooperato parters supply cha of agrcultural products based o IL WG Joural of oftware vol 6() pp 67-7 [] G W We ad R L Models for electg a ERP ystem wth Itutostc Trapezodal Fuzzy Iformato Joural of oftware vol 5(3) pp [] X Y Yue G K Xa ad Y P L Mult-attrbute group decso-makg method based o tragular tutostc fuzzy umber ad -tuple lgustc formato Joural of oftware vol 7(7) pp [3] Q Zhag Y Jag B G Ja ad H Luo ome formato measures for terval-valued tutostc fuzzy sets Iformato ceces vol 8 pp Zhehua Zhag bor 97 Ph D caddate lecturer He works at Csco school of formatcs Guagdog Uversty of foreg studes Hs research terests clude fuzzy reasog fuzzy decso makg ad tellget computg Jgyu Yag bor 94 professor Ph D supervsor Hs research terests clude patter recogto computer vso tellget robots ad data fuso Youpe Ye bor 944 professor Ph D supervsor Hs research terests clude fuzzy reasog ad cryptography Yog Hu Ph D assocate professor Hs research terests clude data mg ad tellget computg Qasheg Zhag bor 975 Ph D assocate professor Hs research terests clude fuzzy reasog fuzzy automata ad quatum logc