Ranking method of additive consistent fuzzy judgment matrix considering scale

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@ 38 G@ 7 ffi u_o$xmd:) Vol.38, No.7 2018 Π 7 J Systems Egieerig Theory & Practice July, 2018 doi: 10.12011/1000-6788(2018)07-1836-06 jas_wξ: O223; C934 m}ffibo: A flψz ~ 6>L8"ffl(ΛρI~&: Π wffl`, T q, zoff, fl! ()0iscffiftfi& tfi&, )0 450015) G = flhξuω μe.mοχ-ρ l~ p(}ff^]ysψ +V'Y fi, _ Ua fi #% d±*.y$ _Q[W-Y,,U xφ"% 0 1 Q[Ymοχ-ρ l~ p(. k, /ν* v 0.1 0.9 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 450015, 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 0.1 0.9, 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$, ffiψibx [2,3]»7Mffl>FRaam* Ψ1:3 1Offi X [3,4], O E@ßM1 : f, Y @V:i^+A D$VNiΨI:,c c [3,4], 6,MHρ $i, ffiψibxc $t:fixtw, +)2hHρUTpΠ9=. :ΞU, tb ffiψibx: ±ffxr:!>.=fi9eam:!>". -Mk7 ffiψibx:,c, VΨ±ffXR2hQ0^h av:,v@#ffi,c ffiψibx:±ffxr, *l [5 18];,V@ffffi,c ffiψibx:±ff XR, *l [18 28]. xw] Xml [4 9] itb ffiψibx:#ffi,c 0fl t±ffxr, 7+ot:!> ). l -ff+*: 2017-03-20 PH ο: hkb (1976 ),, Φ 5ο%, d0d, "?YΞ: ffljχ, IffY. ß5$: ~"q#pχρ6 Πρ6 ~ (11501525); Φ 7i=χ BP" ~ (18A110032) Foudatio item: Natioal Natural Sciece Foudatio of Chia (11501525); 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): 1836 1841. 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): 1836 1841.

@ 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:^i: ffiψibx@tb 0.1 0.9 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( 0.1 0.9 flgx:#ffi,c ffiψibx±ffxrπ t3m. tfi, D4ey4 ffiflg, l (y4 ffiflgx:#ffi,c ffiψibx:±ffxrflffl t3m, <9 0 1 flgπ 0.1 0.9 flg x:±ffxrπ t3m"hffly^,.o<9l [7 9] i±ffxrπ t3m: =1:_,. 2 yω% @ 1 [4] @ 2 [4] @ 3 [4] 4BX R =(r ij ), - 0 r ij 1, O!BX R @ ffibx. 4BX R=(r ij ) h ffibx, - r ij +r ji =1,i,j =1, 2,,, O!BX R @ ffiψibx. 4BX R =(r ij ) h ffiψibx, - r ij = r ik r jk +0.5,i,j,k =1, 2,,, O!BX R @#ffi,c ffiψibx. @ 4 [29] 4 w =(w 1,w 2,,w ) T @ ffiψibx R :±ffλc, * K&3 k =1, 2,,, @ r ik r jk, q w i w j, 5U@<= r ik = r jk,k =1, 2,, "_9, @ w i = w j, O!BX R :±ff XR@,xΠff:. @ 5 [11] 4 ffiψibx R =(r ij ), - r ij 0.5, OKB&3 k =1, 2,,, @ r ik r jk ; - r ij =0.5, OKB&3 k =1, 2,,, @ r ik r jk, οu r ik r jk, O!BX R :ff+a:. 3 1.Ψ2 0.5 0.9 0.7 χ 1 [8] 4 R = 0.1 0.5 0.3 h 3 2#ffi,c ffiψibx. 0.3 0.7 0.5 l [8] DX 4.1 ]zeßi α :fi`h α 1 2 ( 1), Mg^iLzßI α< 1 2 ( 1) 9,»ΛΠ ]flm: e.?bg^i =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 = 4 12. 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 = 1 2..1w]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 1 2. 4@, g^i ffi ΨIBX R =(r ij ) ο=:@ 0.1 0.9 flg, O 0 1 flg. 6,, s FR_Π:Vg9g h 0.1( 1) + 0.5 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).

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=:@ 0.1 0.9 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. 4 0.1 0.9 3»7?M9# )ΞffJ ';Φ± *jvlz, 0x2fl:BXJaflGh 0.1 0.9 : ffi ΨIBX. ffi 1 4 R =(r ij ) h ffiψibx, O 0.1( 1) + 0.5 k=1 r ik 0.9( 1) + 0.5. 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, 2,,, @ 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 5 2 5 ( 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 0.5 0.4, χ r ij 0.5 0.4. ^9Mk7 α 2 5 ( 1), s O@ w i w j = 1 α r ij 0.5 5 2( 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 0.1 0.9 qflg0fl 0.1 0.9 @flgv.k ffiflg8 by, mdvt:y Zl(y4 ffiflg:d4. "fi, l(y4 ffiflgx: ffiψibx:±ffxrπ t3m, <9l [7 9] ΠlißB 0.1 0.9 qflg: t3 "hffly^. @ 6 y4 ffiflgd4*x:

@ 7 ffi hkb, =: NlffiH;$ffl-d» fflωjcy; fiys 1839 4 0.5 ffl>%frd/fr^(m*, 0.5+δ i, i =1, 2,,t ^ ffl>%frψ/frm*:$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 0.1 0.9 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 0.9 @flg. 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) + 0.5 k=1 r ik M 0 ( 1) + 0.5. 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, 2,,, @ 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) + 0.5 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 0 0.5 r ij 0.5 M 0 0.5, Mk7 m 0 + M 0 =1,O@ m 0 0.5 r ij 0.5 0.5 m 0, χ r ij 0.5 0.5 m 0. ^9Mk7 α (0.5 m 0 )( 1), s@ w i w j = 1 α r ij 0.5 1 (0.5 m 0)( 1) (0.5 m 0)= 1 1.

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 0.1 0.9 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, 2,,, @ 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 ) @ff+a:, 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 0.1 0.9 flgx#ffi,c ffiψibx:±ffxrflffl t3m. tfi, r Mk7 0 1 Π 0.1 0.9 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, 1990. Wag L F, Xu S B. Itroductio to aalytic hierarchy process[m]. Beijig: Chia Remi Uiversity Press, 1990. [2] οpk.!enf»iffysffi:> [M]. <: 3χ)Ξ3, 2004. Xu Z S. Ucertai multiple attribute decisio makig: Methods ad applicatios[m]. Beijig: Tsighua Uiversity Press, 2004. [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): 33 48. [4] )y, S/. ffl-dcyffi N,Pχj;:> [J]. v`p%, 1997, 15(2): 54 57. Yao M, Zhag S. Fuzzy cosistet matrix ad its applicatios i soft sciece[j]. Systems Egieerig, 1997, 15(2): 54 57. [5] fkχ, οpk. ffl AHP j-lff;ffihs [J]. K'EvY, 1998, 7(2): 37 40. Li J C, Xu Z S. A ew scale i FAHP[J]. Operatios Research ad Maagemet Sciece, 1998, 7(2): 37 40. [6] οpk. ffl ΩJCY fi;-lts [J]. v`p%χ±, 2001, 16(4): 311 314. Xu Z S. Algorithm for priority of fuzzy complemetary judgemet matrix[j]. Joural of Systems Egieerig, 2001, 16(4): 311 314. [7] SfiL. fflfi-_ss (FAHP)[J]. fflv`ejχ, 2000, 14(2): 80 88. Zhag J J. Fuzzy aalytical hierarchy process[j]. Fuzzy Systems ad Mathematics, 2000, 14(2): 80 88. [8] ji9. ρc ffl-dcy; fflfi-_ss; fi [J]. fflv`ejχ, 2002, 16(2): 79 85. Lü Y J. Weight calculatio method of fuzzy aalytical hierarchy process[j]. Fuzzy Systems ad Mathematics, 2002, 16(2): 79 85. [9] SfiL. ffl ΩJCY fi;-lffys [J]. K'EvY, 2005, 14(2): 59 63. Zhag J J. A ew rakig method of fuzzy complemetary judgemet matrix[j]. Operatios Research ad Maagemet Sciece, 2005, 14(2): 59 63. k=1

@ 7 ffi hkb, =: NlffiH;$ffl-d» fflωjcy; fiys 1841 [10] Thfl, fl~`. fflωjcy-d»φ;ffi fiys [J]. K'EvY, 2000, 9(3): 21 25. Fa Z P, Hu G F. The cosistecy approximatio ad rakig method for fuzzy judgemet matrix[j]. Operatios Research ad Maagemet Sciece, 2000, 9(3): 21 25. [11] οpk. z5 ffl-d»cyffi fiys [J]. 5[LYp3χχ±, 2000, 1(6): 97 99. Xu Z S. Geeralized fuzzy cosistet matrix ad its priority method[j]. Joural of PLA Uiversity of Scieces ad Techology, 2000, 1(6): 97 99. [12] οpk. ffl ΩJCY fi;u±yffis [J]. v`p%ye;*, 2001, 21(10): 93 96. Xu Z S. The least variace priority method (LVM) for fuzzy complemetary judgemet matrix[j]. Systems Egieerig Theory & Practice, 2001, 21(10): 93 96. [13].%fi, Thfl. ρc fflωjcy;-ly fiys [J]. F 3χχ± (q#pχξ), 2000, 21(4): 450 452. Jiag Y P, Fa Z P. A method for rakig alteratives based o fuzzy judgemet matrix[j]. Joural of Northeaster Uiversity (Natural Sciece), 2000, 21(4): 450 452. [14] #, Qm1, ZΩfl. ffl CY fiξd;ρ5ts [J]. <8W3χχ± (q#pχξ), 2007, 43(2): 114 119. Xig Y, Zeg W Y, Li H X. A kid of method to calculate the priority vector of fuzzy reciprocal matrix[j]. Joural of Beijig Normal Uiversity (Natural Sciece), 2007, 43(2): 114 119. [15] Qxfl, '8. fflωjcy fiξd;!eys"? [J]. fflv`ejχ, 2004, 18(2): 73 82. Sog G X, Yag D L. Study o approaches for determiig the priority weight vector of fuzzy judgemet matrix[j]. Fuzzy Systems ad Mathematics, 2004, 18(2): 73 82. [16] Qxfl, '8.!E fflωjcy fiξd;bwys [J]. v`p%yys:>, 2004, 13(2): 161 166. Sog G X, Yag D L. Two kids of approaches for determiig the priority weight vector of fuzzy judgemet matrix[j]. Systems Egieerig Theory Methodology Applicatios, 2004, 13(2): 161 166. [17] Michele F, Matteo B. O the priority vector associated with a reciprocal relatio ad a pairwise compariso matrix[j]. Soft Computig: A Fusio of Foudatios, Methodologies ad Applicatios, 2010(14): 639 645. [18] ji9, %ffw, xa;. ρc Lw; ΩJCY fiys [J]. v`p%ye;*, 2011, 31(7): 1328 1333. Lü Y J, Cheg H T, Ta J Y. Complemetary judgemet matrix rakig method based o relative etropy[j]. Systems Egieerig Theory & Practice, 2011, 31(7): 1328 1333. [19] Thfl,.%fi, ΦO. fflωjcy;-d»ffi»g [J]. SeEIff, 2001, 16(1): 69 71. Fa Z P, Jiag Y P, Xiao S H. Cosistecy of fuzzy judgemet matrix ad its properties[j]. Cotrol ad Decisio, 2001, 16(1): 69 71. [20] Thfl, ZΩ&. ρc Fuzzy ffλuv;-lys fiys [J]. F 3χχ± (q#pχξ), 1999, 20(6): 651 653. Fa Z P, Li H Y. A rakig method for alteratives based o fuzzy preferece relatio[j]. Joural of Northeaster Uiversity (Natural Sciece), 1999, 20(6): 651 653. [21] Thfl, ZΩ&, fl~`. -W Fuzzy ΩJCYffiY fi;~ffi»ys [J]. F 3χχ± (q#pχξ), 2000, 21(1): 60 62. Fa Z P, Li H Y, Hu G F. Fuzzy judgemet matrix ad the goal programmig method for rakig alteratives[j]. Joural of Northeaster Uiversity (Natural Sciece), 2000, 21(1): 60 62. [22] Xu Z S, Da Q L. A least deviatio method to obtai a priority vector of a fuzzy preferece relatio[j]. Europea Joural of Operatioal Research, 2005, 164(1): 206 216. [23] οpk. ΩJCY;bl fiys ffi;u±flysffiz[ξds [J]. v`p%ye;*, 2002, 22(7): 71 75. Xu Z S. Two methods for priorities of complemetary judgemet matrices Weighted least-square method ad eigevector method[j]. Systems Egieerig Theory & Practice, 2002, 22(7): 71 75. [24].%fi, Thfl. -l>c fflωjcy fi; χ 2 YS [J]. F 3χχ± (q#pχξ), 2000, 21(5): 573 575. Jiag Y P, Fa Z P. Chi-square rakig method for the fuzzy judgemet matrix[j]. Joural of Northeaster Uiversity (Natural Sciece), 2000, 21(5): 573 575. [25] RPχ, 1ν], οpk. ΩJCY fi;z5 χ 2 S [J]. F 3χχ± (q#pχξ), 2002, 32(4): 659 662. Kog S Q, Da Q L, Xu Z S. Geeralized chi square method for priorities of complemetary judgemet matrices[j]. Joural of Southeast Uiversity (Natural Sciece Editio), 2002, 32(4): 659 662. [26] hkb, ±w. fflωjcy fiξd;u±ffffis [J]. fflv`ejχ, 2012, 26(6): 132 136. Liu W F, He X. A Least deviatio method of priority vector for derivig priority of fuzzy judgemet matrix[j]. Fuzzy Systems ad Mathematics, 2012, 26(6): 132 136. [27] Masamichi S. Fuzzy sets cocept i rak-orderig objects[j]. Joural of Mathematical Aalysis ad Applicatios, 1973, 43(3): 717 733. [28] Tesuzo T. Fuzzy preferece orderigs i group decisio makig[j]. Fuzzy Sets ad Systems, 1984, 12(2): 117 131. [29] Saaty T L, Luis G V. Icosistecy ad rak preservatio[j]. Joural of Mathematical Psychology, 1984, 28(2): 205 214.

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