A Multi2commodity Flow Supply Chain Network Equilibrium Model with Stochastic Choice

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1 (2007) ,2 2, (11, ;21, ),, 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 , Chna ; 21Management choo, Fudan Unverty, hangha , 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, ( ) ; ( ) ( ),,,, ; ( ),,,

2 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,,,,,,,,

3 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 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

4 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 c j ( q 3 j ) 0 j ) - u j 0 q 3 ( - p + u j) = 0 - p + u j 0 q 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,

5 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 = , 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 ],,

6 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 ) ) (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 ) ) (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 ) ) (n d - n d 3 ) - = 0, Π, j,,, (32) ) - = 0 ( p 3 + c ( q 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 ) ) (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,,,,

7 2 ogt, X 3 ^, 5f 1 ( q 3 ) 5 q 3 (n d - n d 3, q - q c j ( q 3 j ) ) -, q - q 3 +q 3 - d 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 {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 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 = 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

8 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 q 1 1, f 1 2 ( q) = 215 ( q 1 2) 2 + q 1 1 q q 1 2, f 2 1 ( q) = 3 ( q 2 1) q 2 1 q q 2 1, f 2 2 ( q) = 3 ( q 2 2) q 2 1 q 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) q 1 12, c 1 21 ( q 1 21) = 015 ( q 1 21) q 1 21, c 1 22 ( q 1 22) = 015 ( q 1 22) q 1 22, c 2 11 ( q 2 11) = ( q 2 11) q 2 11, c 2 12 ( q 2 12) = ( q 2 12) q 2 12, c 2 21 ( q 2 21) = ( q 2 21) q 2 21, c 2 22 ( q 2 22) = ( q 2 22) q 2 22 j c j = c j ( q j ) = c j q j c 1 1 = 011 ( q q 1 21) 2, c 1 2 = 015 ( q q 1 22) 2, c 2 1 = ( q q 2 21) 2, c 2 2 = ( q q 2 22) 2 d 2 d 1 1 ( p 1 ) = - 2 p p , d 1 2 ( p 1 ) = - 2 p p , d 2 1 ( p 2 ) = p p , d 2 2 ( p 2 ) = p p j gc = c ( q ) = c q gc 1 11 = 015 ( q q 21 11) + 1, gc 1 12 = ( q q 21 12) + 5, gc 1 21 = ( q q 21 21) + 5, gc 1 22 = ( q q 21 22) + 5, gc 2 11 = 2 ( q q 22 11) + 3, c 2 12 = 2 ( q q 22 12) + 3, gc 2 21 = 2 ( q q 22 21) + 3, c 2 22 = 2 ( q q 22 22) + 3 MA, 1 4 2, 1, 2 q 1 2 2, , , 2 1, 1 2, ; 1 1 2, 1 ; 1 1 2, 1 ; , 1 ; 1 1, 1 1 2,, ( ),, ( 1), 2

9 q j q gp , ogt,,, 2, [ 1 ] Baou R H [M],, Baou R H, Bune ogtc Management Pannng, Organzaton and Controng the uppy Chan [ M ] Bejng Chna Machne Pre, [ 2 ] Nagurney A Network Economc A Varatona nequate Approach,econd and Reved ed [M] Kuwer Academc Pubher, Dordrecht, 1999 [ 3 ], [M], Gao Z Y, un H Modern ogtc and Tranportaton ytem [M] Bejng Chna Communcaton Pre, [ 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 )

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