Ranking method of additive consistent fuzzy judgment matrix considering scale

Transcript

1 @ 38 7 ffi u_o$xmd:) Vol.38, No Π 7 J Systems Egieerig Theory & Practice July, 2018 doi: / (2018) jas_wξ: O223; C934 m}ffibo: A flψz ~ 6>L8"ffl(ΛρI~&: Π wffl`, T q, zoff, fl! ()0iscffiftfi& tfi&, ) ) G = flhξuω μe.mοχ-ρ l~ p(}ff^]ysψ +V'Y fi, _ Ua fi #% d±*.y$ _Q[W-Y,,U xφ"% 0 1 Q[Ymοχ-ρ l~ p(. k, /ν* v Q[ Ymοχ-ρ l~rp(y}ff^]jμe x. 1k, Zψvgψ lq[, RbUgψ lq[ mοχ-ρ l~ p(y}ff^]j μex, ΠXμe.}ff^]jμexΞ»ß±ΛYffiχ. flμ} l~ p(; }ff^]; Q[; mοχ-ρ Rakig method of additive cosistet fuzzy judgmet matrix cosiderig scale LIU Weifeg, CHANG Jua, MENG Jitao, YANG Yog (College of Sciece, Zhegzhou Uiversity of Aeroautics, Zhegzhou , Chia) Abstract It is foud from the umerical example that the rage of parameter i the rakig method of additive cosistet fuzzy judgmet matrix i the related literatures is ot appropriate, ad by aalyzig the process of proof, the problem existed i the rakig method is due to idiscrimiate scale. Hereby, it is poited out that the series of results i the related literatures are oly suited to additive cosistet fuzzy judgmet matrix i scale 0 1. The, based o scale , the rakig method ad results about additive cosistet fuzzy judgmet matrix are reproved. Fially, geeralized fuzzy scale is defied, ad the rakig method ad results about additive cosistet fuzzy judgmet matrix with geeralized fuzzy scale are discussed, which realized the rakig methods ad results i the related literatures uificatio i forms. 1 C< Keywords fuzzy judgmet matrix; rakig method; scale; additive cosistecy KB UΨIBX [1] O$, ffiψibx [2,3]»7Mffl>FRaam* Ψ1:3 1Offi X [3,4], O E@ßM1 : f, D$VNiΨI:,c c [3,4], 6,MHρ $i, ffiψibxc $t:fixtw, +)2hHρUTpΠ9=. :ΞU, tb ffiψibx: ±ffxr:!>.=fi9eam:!>". -Mk7 ffiψibx:,c, VΨ±ffXR2hQ0^h av:,v@#ffi,c ffiψibx:±ffxr, *l [5 18];,V@ffffi,c ffiψibx:±ff XR, *l [18 28]. xw] Xml [4 9] itb ffiψibx:#ffi,c 0fl t±ffxr, 7+ot:!> ). l -ff+*: PH ο: hkb (1976 ),, Φ 5ο%, d0d, "?YΞ: ffljχ, IffY. ß5$: ~"q#pχρ6 Πρ6 ~ ( ); Φ 7i=χ BP" ~ (18A110032) Foudatio item: Natioal Natural Sciece Foudatio of Chia ( ); Key Scietific Research Projects i Colleges ad Uiversities of Hea (18A110032) N0BEfi,: hkb, ψf, u6w, =. NlffiH;$ffl-d» fflωjcy; fiys [J]. v`p%ye;*, 2018, 38(7): D0BEfi,: Liu W F, Chag J, Meg J T, et al. Rakig method of additive cosistet fuzzy judgmet matrix cosiderig scale[j]. Systems Egieerig Theory & Practice, 2018, 38(7):

2 @ 7 ffi hkb, =: NlffiH;$ffl-d» fflωjcy; fiys 1837 [4] ]ze#ffi,c c $VHρNi:,c, Xme#ffi,c ffiψibx: f, h$th ρi<= ffiψibxcdexmß*; l [5, 6]!>e»^flGx: ffiψibx, l(e,k]= : ffiψibx:±ffxr; l [7] 4Be=#ffi,c ffiψibxffl>6r(aam* Ψ1: X, O pm!>effl>6r(aam* Ψ1:#ffi,c ffiψibxdffl>6rm* $Gflm_( :tu, χ r ij = α(w i w j )+0.5, ffli 0 <α 0.5, K,8 eψj:iffi]z; l [8] ] ß4 ffil(e#ffi,c ffiψibx:,kflm±ffq=, e α :fi`vgh α 0.5( 1); l [9] a(l [8] i±ffxr7a=b#ffi,c ffiψibx, h,3 l [6], l(e,k ffiψibx: ±ffxr. l [4 9] :!>3mKBQRd0 ffiψibx:±ffxre@m*:xm34π9='`. 4@, l [7 9]MXm ffiψibx:±ffxrflffl t3m9, r@ψjff^flg[, O@i_ >/flg}'h 0 1 flg, 4 t:î: ffiψibx@tb flg:.?:1, ffiψib X:±ffXRflffl t3mdflg:fl=v»q^. h,, Cy],kψS:^Lzl [8, 9] ißifi `Vg/M[, ^re[( :G6MBr@ff^flG. "fi, Kl [7 9] i: t3m8 mωe X, l( flgx:#ffi,c ffiψibx±ffxrπ t3m. tfi, D4ey4 ffiflg, l (y4 ffiflgx:#ffi,c ffiψibx:±ffxrflffl t3m, <9 0 1 flgπ flg x:±ffxrπ t3m"hffly^,.o<9l [7 9] i±ffxrπ t3m: =1:_, [4] 4BX R =(r ij ), - 0 r ij 1, O!BX ffibx. 4BX R=(r ij ) h ffibx, - r ij +r ji =1,i,j =1, 2,,, O!BX ffiψibx. 4BX R =(r ij ) h ffiψibx, - r ij = r ik r jk +0.5,i,j,k =1, 2,,, O!BX 4 [29] 4 w =(w 1,w 2,,w ) ffiψibx R :±ffλc, * K&3 k =1, r ik r jk, q w i w j, 5U@<= r ik = r jk,k =1, 2,, w i = w j, O!BX R :±ff 5 [11] 4 ffiψibx R =(r ij ), - r ij 0.5, OKB&3 k =1, r ik r jk ; - r ij =0.5, OKB&3 k =1, r ik r jk, οu r ik r jk, O!BX R :ff+a:. 3 1.Ψ χ 1 [8] 4 R = h 3 2#ffi,c ffiψibx l [8] DX 4.1 ]zeßi α :fi`h α 1 2 ( 1), MgîLzßI α< 1 2 ( 1) 9,»ΛΠ ]flm: e.?bgî =3,Lzα 1, χ α»λπb 1, 4@5 α =0.8 < 1 9, Q0ψS(flm^ h w 1 = 7 12, w 2 = 1 12, w 3 = g^lzl [8, 9] i±ffq=ißi α :fi`vg» ψ. xw^rl [8] DX 4.1 :]zi, 6c α 1 2 ( 1) :G6, l( α : ψfi`vg. l [8] ] z α 1 2 ( 1) : $*x: Mk7 w i : e,.o97 1 α ( 1 X, s 0.5 k=1 r ik 0.5, B@ α max{ 2 k=1 r ik} = 2 k=1 r ik 1 5 ) 1, χ α 2 k=1 r ik.?b R h ffiψib 0.5 = w]zQ0Q, l [8] Mb6 α :fi`vg9, 7=egl :7X 2.1, Og7X&4 ffiψ IBX R =(r ij ) ο= 0 1 flg,.o@ 0.5 k=1 r ik 0.5, 8Ob6( α @, gî ffi ΨIBX R =(r ij ) ο=:@ flg, O 0 1 flg. 6,, s FR_Π:Vg9g h 0.1( 1) k=1 r ik 0.9( 1) + 0.5, so97ßi α :fi`vgh α max{ 2 k=1 r ik} = 2 [0.1( 1) + 0.5] = 2 5 ( 1).

3 1838 v`p%ye ; 38 G?1w:^rQ09^, flg:»^ν< 7flmq=i:ßIfi`. 4l [8, 9] i r@/flgv ψjff^, O@}' ffiψibxh 0 1 flg,.ob6(ßi α 1 2, 6,l [8, 9] i: t3m@ tb 0 1 flg:. 4@^ 1 i: ffiψibxο=:@ flg, so/ßifi`h α 1 2, / ffl=bflmq=» A,.Oνν( 5ßI α =0.8 9, ±ffq=("q0<=:μt. 6,, 9gWK»^:flG^ Xm±ffq=:ψS, ν»ν( ^ 1 i:μt »7?M9# )ΞffJ ';Φ± *jvlz, 0x2fl:BXJaflGh : ffi ΨIBX. ffi 1 4 R =(r ij ) h ffiψibx, O 0.1( 1) k=1 r ik 0.9( 1) ffi 2 4 R =(r ij ) h ffiψibx, O R h#ffi,c ffiψibx5 75, /M 2 e}, Λc w =(w 1,w 2,,w ) T 0fl I α, <9 i, j =1, r ij = α(w i w j )+0.5. K! (&^ ) Dl [8] DX 2.3 ]z ^. (Ω* ) 4 R h#ffi,c ffiψibx, fid α 2 5 ( 1). w i :}, ]zdl [8] DX 2.3 i ^, V[7μ]z w i : e. g w i = 1 1 5α 2( 1) 5 k=1 r ik,? 0.1( 1)+0.5 k=1 r ik Q^, w i 1 1. A^ α 2 5 ( 1), O w i ( 1) 2( 1) 5 =0. [0.1( 1)+0.5] = ffi 3 4 ffiψibx R =(r ij ) ifrdfflflmpstu= r ij = α(w i w j )+0.5, OfflflmΩ "? w i = 1 1 k=1 r ik l(. ffi 4 4 R =(r ij ) h#ffi,c ffiψibx, OfflflmQ? w i = 1 1 k=1 r ik ψs. ffi 5 4 R =(r ij ) h ffiψibx, w =(w 1,w 2,,w ) T?tΠP#RψD, χv@xw ~ [:4 mi z = i=1 j=1 [0.5+α(w i w j ) r ij ] 2, s.t. w i =1,w i 0,i=1, 2,,. q(@ w i = 1 1 k=1 r ik. K! Dl [8] DX 3.3 ^. ffi 6 4 R =(r ij ) h ffiψibx, -ffl e}, flmλcq? w i = 1 1 i=1 k=1 r ik ψ S, OßI α Ωνps α 2 5 ( 1). K! +li? 3 μ^[^r. ffi 7 4 R =(r ij ) h ffiψibx, -fflflmλcpstu= r ij = α(w i w j +0.5), O R i& 3akFR:flm_flDßI α "UΨ, 1 1 w i w j 1 1. K!? r ij = α(w i w j +0.5) Q^, w i w j = 1 α (r ij 0.5), χ w i w j DßI α "UΨ.?B 0.1 r ij 0.9,.O@ 0.4 r ij , χ r ij ^9Mk7 α 2 5 ( 1), s O@ w i w j = 1 α r ij ( 1) 0.4 = 1 1. ffi 8 4 A =(a ij ) h ffiψibx, g r i = j=1 a ij,i=1, 2,,, v*xfiμ r ij = ri rj 2( 1) + 0.5,i,j =1, 2,,, 97#ffi,c ffiψibx R =(r ij ), O? R :FR r ij Dflm w i :tu= r ij = α(w i w j +0.5), ß9:flmΛc w =(w 1,w 2,,w ) T ps w i = 1 1, 2,,, fflißi α 2 5 ( 1). K! Dl [9] idx 1 ]z ^. 5 ffia 3»7?M9# )ΞffJ ';Φ± 4α( 1) + 1 2α( 1) k=1 a ik,i= xw/ 0 1 flg qflg0fl 0.1 ffiflg8 by, mdvt:y Zl(y4 ffiflg:d4. "fi, l(y4 ffiflgx: ffiψibx:±ffxrπ t3m, <9l [7 9] ΠlißB qflg: t3 6 y4 ffiflgd4*x:

4 @ 7 ffi hkb, =: NlffiH;$ffl-d» fflωjcy; fiys ffl>%frd/fr^(m*, 0.5+δ i, i =1, 2,,t ^ ffl>%frψ/frm*:$g, 0.5+δ i H2, fflz%frψ/frm*:$gh2; 0.5 δ i, i =1, 2,,t ^ ffl>%fr»ψ/fr m*:$g, 0.5 δ i HΠ, fflz%frψ/fr»m*:$gh2, ffli δ i [0, 0.5], i =1, 2,,t δ 1 δ 2 δ t. z", y4 ffiflgi@ 2t +1kflG`, 0.5 ± δ i [0, 1], i =1, 2,,t. ffi 9?y4 ffiflgl(ψibx R =(r ij ) E@xw f: 1)r ij =0.5; 2) r ij + r ji =1. z",?y4 ffiflgl(ψibx R =(r ij ) h ffiψibx. O 1 5 t =1,δ 1 =0.5 9, y4 ffiflgχh 0 1.flG; 5 t =4,δ 1 =0.1, δ 2 =0.2, δ 3 =0.3, δ 4 =0.4 9, y4 ffiflgχh qflg; 5 t =4,δ 1 =0.061, δ 2 =0.175, δ 3 =0.362, δ 4 =0.4 9, y4 ffiflgχh 0.1 O 2 /y4 ffiflgi: 2t +1kflG` 0.5, 0.5+δ i, i =1, 2,,t!hGflG, fflv=o*@hρ U-_ ffiψibx9<=. Mk7μ^l imß4flmλc9, ffk ffiψibxviffifi 97, c ffiψibx, Ψ*l [6] iveiffifi r ij = ri rj 2( 1) +0.5,,9 r ij,λ»lfi`h 0.5, 0.5+δ i, i =1, 2,,t, 4 r ij [0.5 δ t, 0.5+δ t ], h,ot»z! r ij hgflg:φ6flg. heffbmg, ot/gflg:t2` 0.5+δ t ΠtΠ` 0.5 δ t ^!h M 0,m 0. "fi, My4 ffi flgx, l( ffiψibx:±ffxrπ t3m. ffi 10 4 R =(r ij ) h ffiψibx, O m 0 ( 1) k=1 r ik M 0 ( 1) ffi 11 4 R =(r ij ) h ffiψibx, O R h#ffi,c ffiψibx5 75, /M 2 e}, Λc w =(w 1,w 2,,w ) T 0fl I α, <9 i, j =1, r ij = α(w i w j )+0.5. K! (&^ ) Dl [8] DX 2.3 ]z ^. (Ω* ) 4 R h#ffi,c ffiψibx, fid α (0.5 m 0 )( 1). w i :}, ]zdl [8] D X 2.3 i ^, V[7μ]z w i : e. g w i = 1 1 2α+(2m 0 1)( 1) 2 k=1 r ik,? m 0 ( 1)+0.5 k=1 r ik Q^, w i 1 1. A^ α (0.5 m 0 )( 1), O w i 2(0.5 m0)( 1)+(2m0 1)( 1) 2 =0. [m 0( 1)+0.5] = ffi 12 4 ffiψibx R =(r ij ) ifrdfflflmpstu= r ij = α(w i w j )+0.5, OfflflmΩ "? w i = 1 1 k=1 r ik l(. ffi 13 4 R =(r ij ) h#ffi,c ffiψibx, OfflflmQ? w i = 1 1 k=1 r ik ψs. ffi 14 4 R =(r ij ) h ffiψibx,w =(w 1,w 2,,w ) T?tΠP#RψD, χv@xw ~ [:4 mi z = i=1 j=1 [0.5+α(w i w j ) r ij ] 2, s.t. w i =1,w i 0,i=1, 2,,. q(@ w i = 1 1 k=1 r ik. K! Dl [8] DX 3.3 ^. ffi 15 4 R =(r ij ) h ffiψibx, -ffl e}, flmλcq? w i = 1 1 k=1 r ik ψs, OßI α Ωνps α (0.5 m 0 )( 1). K!?flm: e Q^, w i = 1 1 i=1 k=1 r ik 0, χ@ 1 α ( 1 k=1 r ik 1 5 ) 1,.O@ α 2 k=1 r ik.?b R =(r ij ) h ffiψibx, s@ m 0 ( 1) k=1 r ik M 0 ( 1) + 0.5, B@@ α max{ 2 k=1 r ik} = 2 [m 0( 1) + 0.5] = (0.5 m 0 ) +(m 0 0.5) = (0.5 m 0 )( 1). ffi 16 4 R =(r ij ) h ffiψibx, -fflflmλcpstu= r ij = α(w i w j +0.5), O R i& 3akFR:flm_flDßI α "UΨ, 1 1 w i w j 1 1. K!? r ij = α(w i w j +0.5) Q^, w i w j = 1 α (r ij 0.5), χ w i w j DßI α "UΨ.?B m 0 r ij M 0,.O@ m r ij 0.5 M 0 0.5, Mk7 m 0 + M 0 =1,O@ m r ij m 0, χ r ij m 0. ^9Mk7 α (0.5 m 0 )( 1), s@ w i w j = 1 α r ij (0.5 m 0)( 1) (0.5 m 0)= 1 1.

5 1840 v`p%ye ; 38 G ffi 17 4 A =(a ij ) h ffiψibx, g r i = j=1 a ij,i=1, 2,,, v*xfiμ r ij = ri rj 2( 1) + 0.5,i,j =1, 2,,, 97#ffi,c ffiψibx R =(r ij ), O? R :FR r ij Dflm w i :tu= 4α( 1) + 1 r ij = α(w i w j +0.5), ß9:flmΛc w =(w 1,w 2,,w ) T ps w i = 1 2α( 1) k=1 a ik,i= 1, 2,,, fflißi α (0.5 m 0 )( 1). K! Dl [9] idx 1 ]z ^. O 3 5 m 0,M 0 ^ 0, 1 Π 0.1, 0.9, χ ffiψibx^ h 0 1 flgπ flg9, DX 10 DX 17 ^ c hl [8, 9] i t3mπlidx 1 8. xwxmy4 ffiflgx±ffλc: f. ffi 18 4 A =(a ij ) h ffiψibx, Offl±ffΛc@,xΠff:, ffli w i = 1 4α( 1) + 1 2α( 1) a ik,i=1, 2,,,α (0.5 m 0 )( 1). K! 4 w =(w 1,w 2,,w ffiψibx A =(a ij ) :±ffλc, O@ w i = 1 4α( 1) + 1 2α( 1) k=1 k=1 a ik,w j = 1 4α( 1) + 1 2α( 1) a jk. -KB k =1, a ik a jk, O@ w i, w j :ffl0=q^ w i w j. A5U@<= a ik = a jk (k =1, 2,,) "_9, z"@ w i = w j. 6,, ±ffλc@,xπff:. ffi 19 ffiψibx A =(a ij χ- a ij 0.5, O@ w i w j ; - a ij =0.5, OοU w i w j, οu w i w j. 6 νf ] ^rψs:^lze, flg:»^k ffiψibx:±ffq=ßifi`vg@pψ2:<. " fi, Xme flgx#ffi,c ffiψibx:±ffxrflffl t3m. tfi, r Mk7 0 1 Π flg:r^yz, D4ey4 ffiflg,!>ey4 ffiflgx: ffiψibx:±ffxrπ t:3m, : el [7 9] i±ffxrπ t3m =1:_,. li!>3 X»e»^flGx:#ffi,c ffiψibx:±ffxr, afπqre ffiψibx±ffxmπxr. fi/4 [1] e`], ρh. fi-_ss8 [M]. <: j~%x3χ)ξ3, Wag L F, Xu S B. Itroductio to aalytic hierarchy process[m]. Beijig: Chia Remi Uiversity Press, [2] οpk.!enf»iffysffi:> [M]. <: 3χ)Ξ3, Xu Z S. Ucertai multiple attribute decisio makig: Methods ad applicatios[m]. Beijig: Tsighua Uiversity Press, [3] Chiclaa F, Herrera F, Herrera-Viedma E. Itegratig three represetatio models i fuzzy multipurpose decisio makig based o fuzzy preferece relatios[j]. Fuzzy Sets ad Systems, 1998, 97(1): [4] )y, S/. ffl-dcyffi N,Pχj;:> [J]. v`p%, 1997, 15(2): Yao M, Zhag S. Fuzzy cosistet matrix ad its applicatios i soft sciece[j]. Systems Egieerig, 1997, 15(2): [5] fkχ, οpk. ffl AHP j-lff;ffihs [J]. K'EvY, 1998, 7(2): Li J C, Xu Z S. A ew scale i FAHP[J]. Operatios Research ad Maagemet Sciece, 1998, 7(2): [6] οpk. ffl ΩJCY fi;-lts [J]. v`p%χ±, 2001, 16(4): Xu Z S. Algorithm for priority of fuzzy complemetary judgemet matrix[j]. Joural of Systems Egieerig, 2001, 16(4): [7] SfiL. fflfi-_ss (FAHP)[J]. fflv`ejχ, 2000, 14(2): Zhag J J. Fuzzy aalytical hierarchy process[j]. Fuzzy Systems ad Mathematics, 2000, 14(2): [8] ji9. ρc ffl-dcy; fflfi-_ss; fi [J]. fflv`ejχ, 2002, 16(2): Lü Y J. Weight calculatio method of fuzzy aalytical hierarchy process[j]. Fuzzy Systems ad Mathematics, 2002, 16(2): [9] SfiL. ffl ΩJCY fi;-lffys [J]. K'EvY, 2005, 14(2): Zhag J J. A ew rakig method of fuzzy complemetary judgemet matrix[j]. Operatios Research ad Maagemet Sciece, 2005, 14(2): k=1

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