Grey GM(1, 1, t α ) model with time power and its application
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1 ¼ u N k í r Ò + ë Vol32, No Systems Engineering Theory & Prctice Oct, 2012 : ) !"#$%,-0/ : N95 : A GM1, 1, t α ) &')*+ 1, , :;=<?>=<@, AB 21122; 2 :CD=EFDG;=<IHJKLM=<@, :C ) OQPSR GM1, 1) TSUSVSWYX[Y[, ^S_SỲ[VSWYXSb[c, ospsq dswsesfsgstyh[isjsgyk[lymsn X[eSf GM1, 1, t α ) TSU RSr, TSUYX[gSTSsSt?uSxSy{z?u msn~}{vy~ { Yk~ Yƒ, S α SŜ SŠS SŒS YX TSUYX[SŽ u SWY S =u msn[}svy, SdSW RS S S Y GM1, 1, t 2 ) X[šS SœS SYŸ[ S Yk[S S S S, S Yk[ X[S S S S S S, SsSỲ[VSWS«S Yk[SjSg X[TSUYX[ S S S±S esfy³[ ; GM1,1) TSU ; GM1, 1, t α ) TSU ; gstsµs Grey GM1, 1, t α ) model ith time poer nd its ppliction QIAN Wu-yong 1, DANG Yo-guo 2, LIU Si-feng 2 1 School of Business, Jingnn Uniersity, Wuxi 21122, Chin; 2 College of Economics nd Mngement, Nnjing Uniersity of Aeronutics nd Astronutics, Nnjing , Chin) Abstrct In ie of limittion of the GM1,1) model, constructed GM1, 1, t α ) model ith time poer using the theory of grey system modeling thought ccording to the prcticl ppliction need Studied this model s modeling process, prmeter estimtion, time response formul, nd use the GM1, 1, t 2 ) model to forecst the embnkment on soft ground of some costl highy, high precision of the forecsting nd fitting obtined in the ppliction hs proen this ne model s effectieness Keyords grey systems; GM1,1) model; GM1, 1, t α ) model; modeling mechnism 1 ¹ ºS»S¼S½S¾S SÀSÁSÂSÃS¼SÄSÅSÆSÇSÈ ÉSÊ ÁSËSÌSÍSÎSÏSÐSÑSÒS¼SÓSÔ S S S YÕ ¼SÓSÔS½{¾ S, {Ø ½{¾ Ö { ÙÕ ¼{Ú{Û{Ü{¼{Ý{Þ{ß{à Ó{Ô{½{¾{á{â{ã{ä{å{æ{ç {è{éùõ~ê{ë{ì{í ¼{Ó{Ô{î{»{ï{½{¾{ð{ñ òsósôsõ ¼SöS Sø, ùsúsûsü ôsõ ¼Yý ísþsÿ ß Sß ß Sß S¼SÓSÔS½S¾SáSâ, u, ¼ S ÓSÔ ásâs SÓSÔS½S¾SáSâ GM1,1) Õ ¼ SáSâ ásâsæ Ó, GM1,1) ûsþsÿ ûsü ôsõ ¼ ½S¾ ¼ ø % & Ø ) * ¼ + S,! u#" $ u S ' ës S,, ¼ ìs / ûsþ{ÿ, 1 2 ásâ ) 3 ¼ Ú 0, GM1,1) 5 6 GM1,1) ásâ87 Ö[í + < 9 u;: 9 u ò & A B C D ) 3 ¼ u;= E F GM1,1) ásâ ¼Sî J K GIH ásâs¼ ÝSÞ LSî J M N O S ÌSÍSÎ ÏSÐSÅ P ásâ [1 2 GM1,1) >?SÁSËSæ, GM1,1) ásâ Q D S N O MSÝSÞ, ãsä R Õ í S T Û U V : ásâs¼ WSÝSÞ 1) GM1,1) ásâ87yx WS¼SÝSÞS [ á U S¼SÇSÈ _ ¼SÝSÞ ` C DSÚ b & [3 c d e ¼SÝSÞ fig, GM1,1) ásâs¼sç h i GIH Ý{Þj{ç á{â ô{õjm Li [ GM1,1) Õ, kjl jn{û ρ o, j {û jp ½S¾ x 0) k) q [5 ê ρ x y z { Æ Ü & [6 y ß }SÝSÞ D ~, ^ jq{îjr W S¼, æ GM1,1), æ GM1,1) ásâ W Sû 2, 2) Ó{Ôjt, s b {ë GM1,1) ásâsæ ƒ ˆ Š : Œ ˆŽ : ˆ Š ); š ˆœ 08AJY02); Ÿ ˆœ CX10B-05R); «± ³ µ BCXJ10-13) ¹ : º» ¼ 1979 ), ½, ¾, Ÿ À Á Â, ±, ÃˆÄ : Å ÆÈÇÊÉ Ë Ã Ì ; Í Î 196 ), ½, ¾, Ï «Ÿ Ð Ñ Ò Â, Ó Ô, ± Õ Ö, ÃˆÄ : Ø Ù Ú Û Ü Ý Þ ß à,
2 õ Ð x ô S i D B C ñ P & K ñ ó á µ ¼ 6 Í B 228 ÙáÚáâäãáÛá áüáåáæ 32 GM1,1) ásâs¼ ç è W, é ¼ ô % WSðSñ, õ ð Ø ö ô % W < øsá{â hjx %SðSñ ù úsê, ÿ Sû [ D{ÝSÞ, ÝSÞ b fig : Sí k l ôsõ ¼ Sø ô % WSðSñ ásâsî ìsí ô %SðSñ GM1,1) $ GM1,1) ásâs¼ ê ësç ì, æ ø ö ôsõ í í î ï & [7 ð Ø ÓSÔSáSâ ñ åsæsósôsásâ [8 í ûsû ü ý ôsõ Sî Ø [ S½S¾ S S * ¼ ìsí ò ó jþ P{ç ô Ø nsû 1 2, GM1,1) ásâ [ 2) GM1,1) 5 6 ásâs¼ GM1,1) 5 6 SÚSÛ z 1) k) {, r ûsü {, ö! ¼ S i S¼ 5 6, ásâ ƒ ôsõ r, s ûsü ", ásâ ƒ # # L%$ ê &, ' GM1,1) ásâs¼ ' ), ¼Sá SÆS½S¾ `S ø ûsû Æ b,, ûsû Æ b ¼ * +, ø k lsûsü ôsõ Æ þ / : 5 6, 1 2, ; < = [9 > z 1) A B C D, E GM1,1) 2 3 F G H I C K L M F G H N O P Q, J R S T [10 U V W X Y F G H [ D ^ _ z 1) k) = αx 1) k 1) + 1 α)x 1) k), α 0, 1) α ` _ b c d e K h i j H k e T [11, f g l m n C c K o I p F G H Mq r E, s t u y 5 z K C c K o F G H ƒ c : { } ~, E, f ˆ, > GM1,1) 2 3%Š p Œ ƒ Ž Q C, K x Q Q { } ~ u, { u Wng T [12 %š c œ x K œ x u { } ~ u, œ x u K œ x c I p F G H { } ~ u, Ÿ 3) GM1,1) 2 3 Q I p GM1,1) 2 3 H «K L, ± e ³š KL 23D^_ ` z ¹_ºM»I¼ i, µ ½, ¾, H «À Á, C  c à o Ä ƒ d e i «À M H [13, Å Æ Ç È «À Ç É Ê Ë% H 2 5 ƒ 6, Ì Ê Ë Í Ç É È 2 5 ƒ 6 { { Ê Ë Í Ç É Î Ï W T ƒ «À H Ê Ë Ç É [1 15 ) GM1,1) 2 3 Ð ^ Ñ Ò T [16 Ó% Ô Õ Ö c Q ¹ Ø Ù GM1,1) 2 3 [17 Ú Û 6 Ü Ý T š [18 > GM1,1) : Þ : C D u, ß C c E Ê GM1,1) 2 3 à œ ¹ 2 3, á 2 3 W â i%š ã ä å b æ ç 5 z T [19 è é ê W â i ã ä ì Ù ë u, Õ c K o M ¹ NGM1,1,k), Ÿ W í Õ 2 9 : b c e 6 î ï ð Á ñ U V Ñ Ò ~ ò óôõ Xö c ¹ GM1,1) 2 3 D ^ ¹ , ø ù tú³û ü º, ý GM1,1) 2 3 à í Ä 2 3 þ þ ÿ î C  M t% ü > GM1,1) 2 3 Õ 2 ± Ž K o ã, { } ~ Ç É Õ 2 ƒ Q º, ý GM1,1) 2 3 W â i u ã â i 5z 7 8 ò, GM1,1) 2 3 Ä Á È p 2 3 â i Lä  î _ º D ^ 6 } % ô p ã ã b à _ º D õ t, ^ ± Ž L!! : " # È%$ & ' p! È & ' p! g Í ) * +, â i ä ì _ º Á ñ D ^ ± Ž, ã b t ñ, - Æ W _ º 0/ µ p ã b ð, Õ _ º 1 2 ½ 3 ä ì M 2 3  i Ñ Ò ½ 5 Á, ê W â i 6 ä ì 8 x 9 ä ì b b Ÿ 7 ÑÒ, ÑÒ ä à _ºäì ¹78 e á á :, Ÿ, Õ C ;z á M 3 ¹ , Ÿ W Õ M 2 3 Õ 2 < È 2 3 ä ì b Ñ Ò, = > ^ ¹ ? _, Ð ô ¹ 7 8 e t ú%û GM1, 1, t α ) LBMBNBO P Q K 1 R X 0) = x 0) 1), x 0) 2),, x 0) n)), S X 1) = x 1) 1), x 1) 2),, x 1) 0) n)) X { } ~ 1-AGO), í x 1) k) = k i=1 x0) k), k = 1, 2,, n S 1) = z 1) 2), z 1) 3),, z 1) n)) X T U Î H 1) } ~, í 1) k) = 1 2 x1) k) + x 1) k 1)), k = 2, 3,, n P Q 2 R X 0), X 1), 1) V 1 : W, n S x 0) k) + z 1) k) = bk α + c GM1, 1, t ) Q ½, í α X Y, S dx 1) + x 1) = bt α + c GM1, 1, t ) 2 3 %Š p ƒ Ž, ô
3 } Á f 5 H p 10 [ ^^_, ` : ^b^c^d^e^f^g^h^i GM1, 1, t α ) j^k^l^m^n^o 229 P p 1 R X 0) X : X 0) = x 0) 1), x 0) 2),, x 0) n)), í x 0) k) 0, k = 1, 2,, n; X 1) X 0) 1-AGO, 1) X T U Î H [ 1) } ~ q ˆr = [, b, c T, r x 0) 2) z 1) 2) 2 α 1 x 0) 3) Y =, B = z 1) 3) 3 α 1, x 0) n) z 1) n) n α 1 n GM1, 1, t α ) 2 3 x 0) k) + z 1) k) = bk α + c N [ Ç É s ˆr = B T B) 1 B T Y t u GM1, 1, t α ) 2 3 x 0) k) + z 1) k) = bk α + c, f s Y = Bâ W, b, c K x s N H x 0) 2) + z 1) 2) = 2 α b + c, x 0) 3) + z 1) 3) = 3 α b + c, x 0) n) + z 1) n) = n α b + c,, z 1) k) + bk α + c y x 0) k), k = 2, 3,, n f z Ë ε = Y Bâ R s = ε ε T = Y Bâ T Y Bâ) = n k=2 x0) k) + z 1) k) bk α c) 2 { s Ç É s = 2 n x 0) k) + z 1) k) bk α c)z 1) k) = 0, k=2 s n b = 2 x 0) k) + z 1) k) bk α c) = 0, Á ñ ƒ Ž x P p 2 R B, Y V x k=2 s n c = 2 x 0) k) + z 1) k) bk α c)k α = 0 k=2 f, b, c % Y = Bâ f B T Bˆr = B T Y, ˆr = B T B) 1 B T Y ˆr = b c = B T B) 1 B T Y, b, c 1 : ñ, ˆr = [, b, c T = B T B) 1 B T Y, n Š p ƒ Ž dx 1) + x 1) = bt α + c x 1) t) = be t GM1, 1, t α ) 2 3 x 0) k) + z 1) k) = bk α + c e t t α + c, Š p ƒ Ž x œ p f g 3 GM1, 1, t α ) LBMBGB~B P p 3 α = 0, GM1, 1, t α ) 23 ' x 0) k) + z 1) k) = bk 0 + c = b 0, s GM1, 1, t α ) GM1, 1) 2 3, q B, Y V 1 : ñ, ˆr = [, b, c T = B T B) 1 B T Y, n 1) Š p ƒ Ž Ï dx x 1) + x 1) = b 0 x 1) t) = x 1) 1) b ) 0 e t 1) + b 0 2) GM1, 1) 2 3 x 0) k) + z 1) k) = b 0 3) ˆx 1) k + 1) = x 1) 1) b 0 ) e k + b 0, k = 0, 1, 2,, n ˆx 0) k + 1) = ˆx 1) k + 1) ˆx 1) k) = 1 e ) x 0) 1) b ) 0 e k, k = 1, 2,, n
4 q q Æ ò H H J Ô C 8 g t µ : ` 7 Æ T «ñ 2250 JƒJ H J J JĴ JŠ Œ 1 > V, α = 0, GM1, 1, t α ) 2 3 ; t â i ã Ù ë u : xt) ce t Õ 2 P p α = 1, GM1, 1, t α ) 2 3 ' x 0) k) + z 1) k) = bk + c, s GM1, 1, t α ) ' GM1, 1, t), B, Y V 1 : ñ ˆr = [, b, c T = B T B) 1 B T Y, n 1) Š p ƒ Ž dx 1) + x 1) = bt + c x x 1) t) = x 1) 1) b c b ) 2 e t 1) + b t + c b 2 2) GM1, 1, t) 2 3 x 0) k) + z 1) k) = bk + c x ˆx 1) k + 1) = x 1) 1) b c b ) 2 e k + b k + c b 2, k = 0, 1, 2,, n 3) ˆx 0) k + 1) = ˆx 1) k + 1) ˆx 1) k) = 1 e ) x 0) 1) b c b ) 2 e k + b, k = 1, 2,, n Œ 2 > V, α = 1, GM1, 1, t) 2 3 ; t â i ã Ù ë u : xt) ce t + bd Õ 2 PŽp 5 α = 2,GM1, 1, t α ) 2 3 ' x 0) k)+z 1) k) = bk 2 +c, s GM1, 1, t α ) ' GM1, 1, t 2 ), B, Y V 1 : ñ ˆr = [, b, c T = B T B) 1 B T Y, n 1) Š p ƒ Ž dx 1) + x 1) = bt 2 + c x x 1) t) = x 1) 1) 2 b + 2 ) c 2b + 2b 3 e t 1) + b t2 2b 2 t + 2b + 2 c 3 2) GM1, 1, t 2 ) 2 3 x 0) k) + z 1) k) = bk 2 + c x ˆx 1) k + 1) = x 1) 1) 2 b + 2 ) c 2b + 2b 3 e k + b k2 2b 2 k + 2b + 2 c 3, k = 0, 1, 2,, n 3) ˆx 0) k+1) = ˆx 1) k+1) ˆx 1) k) = 1 e ) x 1) 1) 2 b + 2 ) c 2b + 2b 3 e k + 2b + 2)b k 2, k = 1, 2,, n Œ 3 > V 5, α = 2, GM1, 1, t 2 ) 2 3 ; t â i Ù ë : xt) ce t + bt + d ã Õ 2 α À í H, Ì α â? À H Ñ Ò GM1, 1, t α ) 2 3 Ï 6 Ï ` å 3,, î 0 «8 9 ¹ ñ t%, Ì GM1, 1, t α ) 2 3, t» 2 3 [ C [,, b, c, š α ` œ Q, > V α À H, Ÿ O t α Ÿ I B B B & Pr r 78K Ž K «& Pr K ± õ p È D ^ ± Ž L ± Ž g Ç V, µ > $ & g & g V ' p ± Ž, > & g $ & g V ' p ± Ž, & P r r ± Ž K L ³ ö å_ º d + µ, & P r r ã â i e Ž jh 0/¹ i º «ˆ, f & P r r î8, µ, { f & P r _ º»  C» õ É ¼ ½, ½ ä ì º 7 8 ƒ þ þ ÿ À A, ý f ¾ 7 8 ¹ I À, í ; _ º t É ¼ ½ ' p Á À 78, Ÿ r & P r õ ± Ž  ã â i 6 Ù ë È 6 å äì Ž Â, r Ì U Ñ Ò, & P r õ ± Ž K L > $ &0à &0à V ö + p ± Ž, á ' p ± Ž GM1, 1, t 2 ) 2 3 _ º : p ± Ž Ï ¹ ¾, Ä Å Æ Ç GM1, 1, t 2 ) W & P r b 78 õ, Ÿ t GM1,1) 2 3 b78, W ð È 23789: ð È 23789: ð È 23ä, Ÿ ÊË É} ¾, ì 6 î t ú%û E Ê Ë Ì Í Î Ï Ð ½ & Í 8 Ñ! Í KL Ñ Ò õ Ò W Ó, K! 0/ 8 r Õ 8! Ö á Ô 1, Ô 32
5 N Í Î Ô ë - ë t ẗ Î z Ô ò Ì Î õ š µ Õ å ï 10 [ ^^_, ` : ^b^c^d^e^f^g^h^i GM1, 1, t α ) j^k^l^m^n^o 2251 b^ì 2 ò^ó GM1, 1) ô^õ^ö^ò^ó GM1, 1, t 2 ) ô^õ^ ^ø^ù^ö^ú^à^û^ü GM1,1) j^k GM1,1) j^k^ý GM1, 1, t 2 ) j GM1, 1, t 2 ) j^k ê^ë^î ë ë ^í ý^þ^ˆ^ÿ þ^ˆ^ÿ k^ý^þ^ˆ^ÿ ý^þ^ˆ^ÿ î ë^î %) Ø^Ù^Ú^Û^Ü^Ý^Þ^Ø^ß^à^á^â^ã^ä^å^æ^ç^ßèà^é ê^ë b^ì ê^ë^î ^í cm) î 8 H ï 6 9, Õ Ö ¹ î 8 È 9 ð, Ÿ ñ 23789: ß t Õ ¹ GM1,1) 236 ¹ GM1, 1, t 2 ) 23, ÈL23 f Q6 Q ß : ˆx 1) k + 1) = 227e 0253k 1917, k = 0, 1, 2,, n 6 ˆx 0) k + 1) = 22e 028k + 12k 08, k = 1, 2,, n È L 2 3 W î 7 8 H 5 z 7 8 k Õ 2 > 2 C ¹ GM1, 1, t 2 ) 23 W!r 5z 78 ¹ á zë GM1,1) 23 É, GM1,1) 2 3 Ï Ç ô W z Ë 1569%, GM1, 1, t 2 ) 2 3 Ï Ç ô W ¹i z Ë 28%, r GM1, 1, t 2 ) 2 3 Ï Ç É W ¹ i z Ë 030%, GM1, 1, t 2 ) Ï W z Ë É GM1,1) 2 3, GM1, 1, t 2 ) 2 3 z Ë 228% GM1,1) % ~ GM1, 1, t 2 ) z : GM1,1) 2 3» Â ¾ : á > $ &0Ã &0Ã V ' p Á À, GM1, 1, t 2 ) 2 3 ä ì, ¾, t GM1, 1, t 2 ) 2 3 W b 5 z 78 á â i 9: ä Ä Å Ï , ¹ GM1, 1, t α ) 2 3 i ù Ÿ F G 6, W 2 3 Ñ Ò á, = f Ÿ t 5 W 8 9 ¹ 7 GM1, 1, t α ) 2 3 b c Ñ Ò, Ÿ W 2 3 á Õ 2 ± Ž, 2 3 [ s, 2 3 Q b c Ñ Ò, ê W α? o ä À Ḧ GM1, 1, t α ) 2 3 ä ì à Q b c d e, d e D Â : 1) GM1,1) 2 3 ; z â i ã Ù ë u Õ 2 ; 2) GM1, 1, t) 2 3 ; z â i ã Ù ë u Õ 2 ; 3) GM1, 1, t 2 ) 2 3 ; z â i 6 Ù ë È 6 ã Õ 2 W α í À Ḧ, 2 3 GM1, 1, t α ) : â i ä ì È 2 3 [, à [ α «I p T! K Ñ Ò " # $ % [1 & ' h^ÿ ˆ^h ) * [M +, : - / , 2002: Deng J L Grey Prediction nd Grey Decision[M Wuhn: Press of Huzhong Uniersity of Science Technology, 2002: [2 & ' h^ ^ 6 7 [M +, : - / , 2002: Deng J L The Bsis of Grey Theory[M Wuhn: Press of Huzhong Uniersity of Science & Technology, 2002: [3 8 9 :, & ' GM1,1) j^k^g ;^o < = [J ^ƒ^ ^ ^ ^ ^ˆ^ ^Š, 2000, 205): Liu S F, Deng J L The rnge suitble for GM1,1)[J Systems Engineering Theory & Prctice, 2000, 205): [ Li X C On prmeters in Grey Model GM1,1)[J The Journl of Grey System, 1998, 102): %) %)
6 2252 JƒJ H J J JĴ JŠ, & h^i^ ^ƒ GM1,1) j^k^g A B C D E F [J G H I J, 1997, 152): Wng W P, Deng J L Study of the chotic chrcteristics of GM1,1) model in grey system theory[j Systems Engineering, 1997, 152): [6 > K L, M N O, 8 9 : P Q GM1,1) R S T A B C D U V [J G H I J W X Y [, 2007, 2711): Wng X, Dng Y G, Liu S F Anlysis of chotic chrcteristics of unbised GM1,1)[J Systems Engineering Theory & Prctice, 2007, 2711): [7 ^, + _ `, b c GM R S T d e D f g [J h O i W j k, 20015): 38 heng N, Wu Y Y, Bo H L Morbidity problem in grey model[j Chinese Journl of Mngement Science, 20015): 38 [8 M N O, > K L, 8 9 : l m R S T d e f g E F [J G H I J W X Y [, 2008, 281): Dng Y G, Wng X, Liu S F Study on morbidity problem in grey model[j Systems Engineering Theory & Prctice, 2008, 281): [9 n o p GM1,1) R S T q r s t u x y z 1)[J G H I J W X Y [, 2000, 20): Tn G J The structure method nd ppliction of bckground lue in grey system GM1,1) modeli)[j Systems Engineering Theory & Prctice, 2000, 20): [10 > {, } ~, } GM1,1) T ƒ ˆ Š R [J G H I J W X Y [, 2000, 209): Wng Y N, Liu G, Liu K D A step by step optimum direct modeling method of GM1,1)[J Systems Engineering Theory & Prctice, 2000, 209): [11 M, } Œ, M N O l m R S GM1,1) [J h O I J j k, 2003, 58): Luo D, Liu S F, Dng Y G The optimiztion of grey model GM1,1)[J Engineering Science, 2003, 58): [12 Wng X, Dng Y G, Liu S F, et l The optimiztion of bckground lue in GM1,1) model[j Journl of Grey System, 2007, 102): 69 7 [13 Dng Y G, Liu S F The GM models tht xn) be tken s initil lue[j The Interntionl Journl of Systems & Cyberntics, 200, 332): [1 } Ž, } Œ,, GM1,1) R S y T [J h O i W j k, 2003, 11): 5 57 Liu B, Liu S F, hi J, et l Optimum time response sequence for GM 1,1)[J Chinese Journl of Mngement Science, 2003, 11): 5 57 [15 š, œ GM1,1) R S T Ÿ [J G H I J W X y z, 2001, 101): 72 7 hng H, Hu S G Accurte solution for GM1,1)model[J Systems Engineering Theory Methodology Applictions, 2001, 101): 72 7 [16 h, p, L h «l m GM1,1) R S [J G H I J W X Y [, 2001, 215): Song M, Tong X J, Xio X P Center pproch grey GM1,1) model[j Systems Engineering Theory & Prctice, 2001, 215): [17 l s T P Q l m GM1,1) R S [J k T [ Y ±, 2003, 333): Mu Y An unbised GM1,1) model ith optimum grey deritie s hitening lues[j Mthemtics in Prctice nd Theory, 2003, 333): [18 ³ µ, } Œ GM1,1) R S Y l m ¹ R S Š R º W [J G H I J W X Y [, 2005, 251): Xie N M, Liu S F Discrete GM1,1) nd mechnism of grey forecsting model[j Systems Engineering Theory & Prctice, 2005, 251): [19», M N O, } Œ ƒ L T l m ¹ R S ¼ ½ Š R º W [J ¾ Y À Á, 2009, 211): Cui J, Dng Y G, Liu S F Noel grey forecsting model nd its modeling mechnism[j Control nd Decision, 2009, 211): [5 >? 32
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