2007 3 3 100026788 (2007) 0320082209 1,2 2, (11, 330047 ;21, 200433),, ogt, Nah,, ;Nah ; ; F22411 ;C93111 A A Mut2commodty Fow uppy Chan Network Equbrum Mode wth tochatc Choce XU Bng 1,2, ZHU Dao2 2 (11Management cence and Engneerng Department, Nanchang Unverty, Nanchang 330047, Chna ; 21Management choo, Fudan Unverty, hangha 200433, Chna) Abtract A three2eve compettve uppy chan network equbrum mode wth mut2commodty tuded n conderaton of the product dfferentaton of regon and brand tochatc utty theory and mutnoma ogt mode are ued to anayze tochatc choce n demand market, whe the nfuence of regon and brand treated a tochatc varabe Compettve behavor of manufacture and retaer anayzed by ung Nah equbrum theory equbrum mode of each eve and whoe network are deveoped by varatona nequaty method, aong wth ther equbrum condton and economc nterpretaton Fnay, an agorthm put forward and apped to ovng a numerca exampe Key word uppy chan network ; Nah equbrum ; tochatc choce ; varatona nequaty The 1, [1 ] ( ),, ( ),, [2,3, ] Nagurney, [46 ], [7 ] [8 ogt, 2005212227 (70432001) ; (20060400584) (1972 - ),,,, ; (1945 - ),,,
3 83, [9 ], [8,9 ],, 1) [47 ], Nah, ;2) [47 ],,,,,,,, [8 ], ogt, Nah, ;, ; 2 211,,, j,, j, ( ), Nah,, Nah Nah [10 ] N Nah X R n, u X R, X = X 1 X 2 X N = {1,2,, N}, x - = ( x 1,, x - 1, x + 1,, x N ), x = ( x 1,, x N ) = ( x, x - ), X - X N, X = X X - 1 x 3 = ( x 3 1,, x 3 N ) Nah, = X 1 X - 1 X + 1 u ( x 3, x 3 - ) u ( x, x 3 - ), Π, Π x X (1) 1, u X x 3 = ( x 3 1,, x 3 N ) Nah,x 3 F( x) = - 5 u 1 ( x),, 5 u N ( x) 5 x 1 5 x N x - ) x,(2) F( x 3 ) T ( x - x 3 ) 0, Π x X, (2) T, x - X -, u ( x,,,,,,,,
84 2007 3 212 q, q = ( q 1,, q,, q ), q = ( q 1,, q,, q ),, f ( q) q j, p j, c j ( q j ) j ( ) f ( q), c j ( q j) Q = ( q 1 1,, q j,, q ), Q = ( Q 1,, Q,, Q ) R ( q, Q) = p j q j - f ( q) - c j ( q j), (3) Nah, (4) (5) = ( q, Q ) q 0, Q 0, q = q = q j, Π,, (4) q j 0, Π, j,, (5) q j, Π,= 1 2 ( q 3 (Nah ),, Q 3 ) R ( q 3, Q 3, q 3 -, Q 3 - ) R ( q, Q, q 3 -, Q 3 - ), Π( q, Q ),,2,,, (6) q - = ( q 1,, q - 1, q + 1,, q ), Q - = ( Q 1,, Q - 1, Q + 1,, Q ) (6),, R ( q, Q), 1, (6) 5f ( q 3 ), 5 q q - q 3 + 5 c j ( q 3 j ) - p j, q j - q 3 j 0, Π( q, Q), (7), (7) ( q 3, Q 3 [11 ) ater Π, j,, ] q 3 5f ( q 3 ) 5 q q 3 q 3 j 5f ( q 3 ) 5 q 0 5 c j ( q 3 j ) q 3 j - u 0 5 c j ( q 3 j ) 0 - u = 0 + u - p j = 0 + u - p j 0, (8), (9) u (4), (8), ;, (9), ;, (8) (9), 213 q, p j j
3 85 gq j = ( q 1 1 j,, q j,, q j, q 11 j1,, q,, q j ), gq = ( gq 1,, gq j,, gq ), c j ( q j ), q j = q j, c j ( q j ) j R j ( gq) = p q - p jq j - c j ( q j ), (10) Nah, (11) (12) g j = gq j gq j 0, q j = q q j = q q j, Π, j,, (11) 0, q 0, Π, j,, (12), Π,, g= g 1 g 2 g gq 3 g (Nah ), R j ( gq 3 j, gq 3 - j ) R j ( gq j, gq 3 - j), Π gq j g j,,2,,, (13) gq - j = ( gq 1,, gq j - 1, gq j + 1,, gq ) (13),, R j ( gq) g, 1, (13) p j + 5 c j ( q 3 j ), q j - q 3 j - p, q - q 3 0, Π gq g, (14), (14) gq 3 [11 ater Π, j,,, ] q 3 p j q 3 j j p j + 5 c j ( q 3 + 5 c j ( q 3 j ) 0 j ) - u j 0 q 3 ( - p + u j) = 0 - p + u j 0 q 3 0 - u j = 0, (15), (16) u j (11), j (15), ;, (16), (15), (16),, 214 q j, q = q j, c = c ( q ) j,,, U U,
86 2007 3 U = - ( p + c ) +, Π, j,,, (17) > 0, E[ ] = 0, E[ U ] = - ( p + c ), j w, j, w = Prob{ U U j, j= 1,2,, ; = 1,2,, }, (18) 0 w 1, Π, j,,, (19) w, Gumbe = 1, Π, (20) F( Y) = Prob{ Y} = exp ( - e - Y+ E ), Π, j,,, (21) Y, E = 015708, ogt gp = p w = exp[ - ( p + c ) ], Π, j,, (22) exp[ - ( p j + c j ) ] = 1 j = 1 + c j ( ), ( ) p = wgp = w ( p + c ), Π, (23),, d ( p ), j d gp,, d d = w d ( p ), Π, j,,, (24) = q q,gp 0, Π, j,,, (25),gp = 0, [12 ] (24), (25) ogt,+,, (24), (25) p p = mn { Π, j p + c ( q ) }, Π,, (26) + c d ( p ) = p, q p 0, Π, j,,, (27), q = 0 = q, p 0 q, p = 0, Π,, (28) [7 ], ( ) = - 1 n exp [ - ( p + c ) ], p = ( gp 11 11,, gp,, gp ), d = ( d 1 1,, d,, d ), gq = ( q 11 11,, q,, q ) [13 ],,
3 87 1 ogt, ( gq 3, d 3, p 3 ) g= { ( gq, d, p) 0 d = q, Π, j,, }, Π( gq, d, p) g, 3 p ( + c ( q 3 ) ) + 1 3 (n d - n d 3 ) -, q - q 3 3 +q - d 3,gp - gp 3 0 ogt, ( gq 3, d 3, p 3 ) g (24) (25), (30) (31) d 3 = d 3 q 3 - d 3 = 1 j = 1 (30),, (29) exp[ - ( p 3 + c ( q 3 ) ) ], Π, j,,, (30) exp[ - ( p j + c j ) ] ( p 3 + c ( q 3 ) ) + 1 3 (n d - n d 3 (31) (32), ( gq 3, d 3, p 3 ) (29), (29),gp - gp 3 0, Π, j,,, (31) q 3 ( p 3 + c ( q 3 ) ) + 1 3 (n d - n d 3 ) - = 0, Π, j,,, (32) ) - = 0 ( p 3 + c ( q 3 ) ) + 1 3 (n d - n d 3 ) - 0, (33) q 3 0 gp 3 ( q 3 - d 3 ) = 0 q 3 - d 3 gp 3 0 0, (34) (34) (25) (33), ( p 3 + c ( q 3 ) ) + 1 3 (n d - n d 3 d 3 = d 3 exp [ - ( p 3 + c ( q 3 ) + ) ] = d 3 ) - = 0, exp [ - ( p 3 + c ( q 3 ) ) ] exp [ - ( p j + c j ) ] = 1 j = 1 215,,, (7), (14), (29), ogt, 2 ogt, (7), (14), (29) X = { q, q j, q,gp } ^= q, q j, q,gp 0 q = q j, q j = q, d = q, Π, j,,,,
2 ogt, X 3 ^, 5f 1 ( q 3 ) 5 q 3 (n d - n d 3, q - q 3 + 5 c j ( q 3 j ) ) -, q - q 3 +q 3 - d 3 + 5 c j ( q 3 j ), q j - q 3 j + = 1 {c ( q 3 ) +,gp - gp 3 } 0, Π X ^ (35) (7), (14), (29),, (35), (35) q 3 88 2007 3 {p j, q j - q 3 j -p j, q j - q 3 j } {p 3 }, (7), (14), (29), q - q 3 -p 3 +, p (26), (27), (28), (35) [7 ] 5f ( q 3 ), 5 q q - q 3 c ( q 3 ) - p 3 + 5 c, q - q 3 j ( q 3 + j ) t Π q, q j, q, p = q, q j, q, p 0 q = + 5 c j ( q 3 j ) q 3, q j - q 3 j + - d 3 ( p ), p - p 3 q j, q j = 0, q j, d = q, q - (36) 3 (35) (8), (9), (15), (16) (24), (25), (35),q > 0, q j > 0, q > 0, p j = 5f ( q) 5 q + 5 c j ( q j), p = p j + 5 c j ( q j ) gp = 5f ( q) 5 q, + 5 c j ( q j) + 5 c j ( q j ) + c, (37),, q,gp, [12 ] (MA), 1 n = 1, q n > 0, ( Π, j,, ) ; 2 q n j, q n ( Π, j, ),(37) gp n 3 (22) w n,(23) p n, (24) d n d n ; 4 q n + 1 1 = q n + ( n + ) ( d 10,1 > 015 ( [8 ]) ; 2 n - q n ),, 5 d n - q n + 1,, q n + 1 ; n = n + 1,, j,,, d n < 0, d n = 0 ;, (22) w n
3 89 w n = = 1 j = 1 1 exp[ - ( gp n j - gp n ) ] 1 [7 ] 2,2 2 2, 2, 1) q, q j, q, gp 2) f ( q) f 1 1 ( q) = 015 ( q 1 1) 2 + q 1 1 q 1 2 + 2 q 1 1, f 1 2 ( q) = 215 ( q 1 2) 2 + q 1 1 q 1 2 + 2 q 1 2, f 2 1 ( q) = 3 ( q 2 1) 2 + 2 q 2 1 q 2 2 + q 2 1, f 2 2 ( q) = 3 ( q 2 2) 2 + 2 q 2 1 q 2 2 + q 2 1 j ( ) c j ( q j) c 1 11 ( q 1 11) = 012 ( q 1 11) 2 + q 1 11, c 1 12 ( q 1 12) = 015 ( q 1 12) 2 + 315 q 1 12, c 1 21 ( q 1 21) = 015 ( q 1 21) 2 + 315 q 1 21, c 1 22 ( q 1 22) = 015 ( q 1 22) 2 + 315 q 1 22, c 2 11 ( q 2 11) = ( q 2 11) 2 + 3 q 2 11, c 2 12 ( q 2 12) = ( q 2 12) 2 + 3 q 2 12, c 2 21 ( q 2 21) = ( q 2 21) 2 + 3 q 2 21, c 2 22 ( q 2 22) = ( q 2 22) 2 + 3 q 2 22 j c j = c j ( q j ) = c j q j c 1 1 = 011 ( q 1 11 + q 1 21) 2, c 1 2 = 015 ( q 1 12 + q 1 22) 2, c 2 1 = ( q 2 11 + q 2 21) 2, c 2 2 = ( q 2 12 + q 2 22) 2 d 2 d 1 1 ( p 1 ) = - 2 p 1 1-115 p 1 2 + 1000, d 1 2 ( p 1 ) = - 2 p 1 2-115 p 1 1 + 1000, d 2 1 ( p 2 ) = - 115 p 2 1-2 p 2 2 + 1000, d 2 2 ( p 2 ) = - 115 p 2 2-2 p 2 1 + 1000 j gc = c ( q ) = c q gc 1 11 = 015 ( q 11 11 + q 21 11) + 1, gc 1 12 = ( q 11 12 + q 21 12) + 5, gc 1 21 = ( q 11 21 + q 21 21) + 5, gc 1 22 = ( q 11 22 + q 21 22) + 5, gc 2 11 = 2 ( q 12 11 + q 22 11) + 3, c 2 12 = 2 ( q 12 12 + q 22 12) + 3, gc 2 21 = 2 ( q 12 21 + q 22 21) + 3, c 2 22 = 2 ( q 12 22 + q 22 22) + 3 MA, 1 4 2, 1, 2 q 1 2 2, 1 144 17 22 68 2, 2 9 26 22 68 2, 2 1, 1 2, ; 1 1 2, 1 ; 1 1 2, 1 ; 1 1 2 1, 1 ; 1 1, 1 1 2,, ( ),, ( 1), 2
90 2007 3 q j 2 1 2 1 2 1 2 1 106 04 11 34 38 13 11 34 2 5 78 11 34 3 48 11 34 3 q 1 2 1 2 1 2 1 63 46 5 67 42 58 5 67 1 2 3 46 5 67 2 32 5 76 1 12 50 5 67 25 63 5 67 2 2 1 14 5 67 2 34 5 67 4 gp 1 2 1 2 1 2 1 255 67 279 23 271 12 279 23 1 2 258 58 279 23 274 03 279 23 1 257 30 279 23 271 63 279 23 2 2 259 69 279 23 274 02 279 23 256 11 279 23 271 48 279 23 4, ogt,,, 2, [ 1 ] Baou R H [M],, 2002 1-19 Baou R H, Bune ogtc Management Pannng, Organzaton and Controng the uppy Chan [ M ] Bejng Chna Machne Pre, 2003 1-19 [ 2 ] Nagurney A Network Economc A Varatona nequate Approach,econd and Reved ed [M] Kuwer Academc Pubher, Dordrecht, 1999 [ 3 ], [M],2003 194-203 Gao Z Y, un H Modern ogtc and Tranportaton ytem [M] Bejng Chna Communcaton Pre,2003 194-203 [ 4 ] Nagurney A, Zhao Varatona nequate and network n the formaton and computaton of market equbra and dequbra The cae of drect demand functon [ ] Tranportaton cence, 1993, 27(1) 4-15 ( 104 )
104 2007 3 Engneerng, 2003,14(13) 1130-1132 [ 4 ],, [ ],2005, (8) 13-15 ang ang, u an, Pan huangxa upport vector machne2baed method for machnng error predcton modeng[ ] Moduar Machne Too & Automatc Manufacturng Technque, 2005,(8) 13-15 [ 5 ] Metaxot K, Kaganna A, Akoun A, et a Artfca ntegence n hort term eectrc oad forecatng A tate2of2the2art urvey for the reearcher[ ] Energy Converon and Management, 2003,44 1525-1534 [ 6 ] Vapnk V N The Nature of tattca earnng Theory[M] New York prng2verag, 1999 [ 7 ] Vapnk V N An overvew of tattca earnng theory[ ] EEE Tranacton Neura Network, 1999,10(5) 988-999 [ 8 ] uyken A K, Vandewae eat quare upport vector machne cafer[ ] Neura Proceng etter, 1999,9(3) 293-300 [ 9 ] uyken A K, Vandewae pare eat quare upport vector machne cafer [ C ]ΠΠEuropean ympoum on Artfca Neura Network, Bruge Begum,2000,37-42 [10 ] n C F, Wang D Fuzzy upport vector machne[ ] EEE Tranacton Neura Network, 2002,13(3) 466-471 ( 90 ) [ 5 ] Nagurney A, Dong, Zhang D A uppy chan network equbrum mode [ ] Tranportaton Reearch Part E, 2002, 38(5) 281-303 [ 6 ] Nagurney A, Toyaak F uppy chan upernetwork and emvronmenta crtera [ ] Tranportaton Reearch Part D, 2003, 8 (3) 185-213 [ 7 ],,, [ ], 2005,25(7) 61-66 Zhang T Z, u Z Y, Teng C X, et a A mut2commodty fow uppy chan networkequbrum mode [ ] ytem Engneerng - Theory & Practce,2005,25(7) 61-66 [ 8 ] [ ], 1997,17(12) 74-78 Zhang N tochatc pata prce equbrum wth tranportaton competton and congeton [ ] ytem Engneerng - Practce, 1997,17(12) 74-78 [ 9 ], [ ], 1997,17(10) 14-17,90 Theory & Zhang N, u X P A market equbrum mode for dfferentated mut2productwth tochatc choce [ ] ytem Engneerng - Theory & Practce, 1997,17(10) 14-17,90 [10 ],, Nah [ ],2005, 8(3) 1-7 Xu Q, Zhu D, U Q H Reaton among Nah equbrum, varatona nequate and generazed equbrum probem [ ] ourna of Management cence n Chna, 2005, 8(3) 1-7 [11 ] Harker P T, Pang Fnte2dmenona varatona nequaty and nonnear compementarty probem A urvey of theory, agorthm and appcaton [ ] Mathematca Programmng, 1990,(48) 161-220 [12 ] heff Y Urban Tranportaton Network Equbrum Anay wth Mathematca Programmng Method [M] Prentce2Ha, NC, Engewood Cff, New erey, 1985 324-331 [13 ] [ ],2003, 23(1) 120-127 Zhou tochatc uer equbrum and t varatona nequaty probem [ ] ourna of ytem cence &Mathematca cence, 2003, 23(1) 120-127