MATHEMATICAL MODELING SPONSORED BY: SHUMO.COM COMPILED BY: Mathematical Modeling Editors Group HOMEPAGE:

MATHEMATICAL MODELING S H U M O SHUMO.COM 2004. 1 Vol. 1 No. 1

MATHEMATICAL MODELING SPONSORED BY: SHUMO.COM COMPILED BY: Mathematical Modeling Editors Group www.shumo.com HOMEPAGE: www.shumo.com mmjournal@yeah.net EMAIL: mmjournal@yeah.net ADDRESS: Science College, NUDT, Changsha Hunan 410073 POSTALCODE: 410073

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1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 410073 2000 Floyd Internet 2001 5 9 24 SHUMO.COM 2001

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1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 430072 CUMCM GRE 2004 MCM 2004

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1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 410073 1 9 80 100 OK SHUMO.COM 2001

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10 2004 http://csiam.edu.cn/mcm/mcm01/ab01.htm BMP BMP C BMP MATLAB imread 100 BMP BMP 512 512 0 1 save ASCII txt C BMP 0 1 2 99 01,02,03,04,...99 CT ACDSEE 100 BMP Windows console DOS Turbo C 512 512 DOS 32 Windows 4 2

1 11 Word 1. 2. 3. 5 [2] MATLAB C++ [1]. [J]. 2004 1:1-2. [2]. [J]. 2004 1:12-14

1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 100876 1 1. 2. MATLAB 3. Lindo Lingo 4. 5. 6. 7. 8. 9. 10. MATLAB 2 2.1 97 A 108 SHUMO.COM 2002

1 13 2.2 98 A 94 A MATLAB MATLAB 2.3 98 B Lindo Lingo 2.4 98 B 00 B 95 Dijkstra Floyd Prim Bellman-Ford 2.5 92 B 97 B 98 B ACM 2.6 97 A 00 B 01 B 89 A BP 86 BP 89 03 B 2.7 N [a, b] M + 1 a, a + (b a)/m, a + 2 (b a)/m,, b (M + 1) N 97 A 99 B MATLAB 2002

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1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 100876 1, 2 3 10 4 SHUMO.COM 2002

16 2004 5 6 B 01 02 1 6 7 99 A MATLAB Mathematica C/C++ Floyd 8 2002 12

1 17 9 MATLAB MATLAB MATLAB 10 B

1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 430072 Huffman Huffman MATLAB 1. R 2 1 2. 982.4MW 303 495 3. 1052.8MW 356 39.2% 510 Huffman 2004

1 19 1 2 1. 2. AGC AGC 3. 4. 5. 6. 7. 8. 3 g i i MW i = 1, 2,..., 8 g i0 i MW l i j MW j = 1, 2,..., 6 a ij j i b j j m j j MW r j j η j j v i i MW / g mi i MW p i (g i ) i g i /MWh p M : /MWh P L MW E t 1/4 4 4.1

20 2004 4.2 [1] 32 15 4.3 1. 2. 3. 4.4 1. 2. 0 3. 4. 5. 0 0 1

1 21 4.5 1. 8 g i = P L i=1 2. g i0 v i t g i g i0 + v i t, i = 1, 2,..., 8 3. g mi g i g mi, i = 1, 2,..., 8 4. l j m j < 0, j = 1, 2,..., 6; 5. 0 l j (1 + r j )m j 0, j = 1, 2,..., 6 5 5.1 33 1 2 5.1.1 8 l j = a ij g i + ε j (1) i=1 8 l j = a ij g i + b j + ε j (2) i=1 ε j ε j N ( 0, σ 2 j ) j = 1, 2,..., 6 a ij b j 0 32

22 2004 (1) L j = ( ) T l (0) j, l (1) j,, l (32) j, G = A j = (a 1j, a 2j,, a 8j ) T, E j = g (0) 1 g (0) 2 g (0) 8 g (32) 1 g (32) 2 g (32) 8 ( ) T ε (0) j, ε (1) j,, ε (32) j G T GA j = G T L j (3) (3) A j A j (2) G = [N.G] G N = (1, 1,..., 1) T 1 33 1 A j = [b j, A j ] A j A j (1) 5.1.2 l j g 1, g 2,..., g 8 α j (j = 1, 2,..., 6) [2] 1. F Q Q = 32 H0 : a 1j = a 2j = a 3j = a 4j = a 5j = a 6j = a 7j = a 8j = 0 H1 : i = 1, 2,..., 8 a ij 0 k=0 (l (k) j (k) ˆl j ) 2 (k) ˆl j = 8 a ij g (k) i i=1 U U = 32 (ˆl (k) j l j ) 2 l j = (1/33) l (k) j k=0 k F = n = 8 m = 33 F F (n, m n 1) U/n Q/(m n 1) α H0 W = {F > F 1 α (n, m n 1)} F > F 1 α (n, m n 1) H0 α 2. R 2 R = U/(U + Q), R 2 R 2 1 l j g 1, g 2,..., g 8 α H0 5.1.3 W = {R 2 > R 2 0}, R 2 0 = nf 1 α (n, m n 1) m n 1 + nf 1 α (n, m n 1) MATLAB α = 5% 1 2 (2)

1 23 5.1.4 1 2 (2) (1) (2) 0 (2) 1. 2. (2) (1) 5.2 5.2.1 1. i k g (0) i,k p(0) p (0) M (p(0) i,k < p(0) M i,k ) g(1) i,k g(0) i,k > g(1) i,k E i,k = (p (0) M p(0) i,k )(g(0) i,k g(1) i,k ) t (4) 2. i k g (0) i,k = 0 p(0) i,k > p(0) M g (1) i,k > 0 E i,k = (p (0) i,k p(0) M )(g(1) i,k g(0) i,k ) t = (p(0) M E = i E = γ E p(0) i,k )(g(0) i,k g(1) i,k k E i,k ) t (5) γ 0 1 γ = 1 (4) (5) g (1) E i,k = g(0) i,k 5.2.2

24 2004 g t g t = P L p (0) M p(1) M p (2) M E = t ( ) P (2) M P (0) M P L p (2) M = (1 θ)p(0) M + θp(1) M θ θ = 1 5.2.3 1. 2. 3. p i,k = p M E i,k = 0 p i,k 4. 5.3 Step1 P L Step2 g i g i g i0 g i g i0 v i t g i = g i0 + v i t flag i = 1 g i g i0 v i t g i = g i0 v i t flag i = 1 g i flag i = 0 Step3 flag i 0 g i i Step2 g = g i i Step4 g = P L g i i g > P L flag i 1 g < P L flag i -1 Step5 Step5 g < P L flag i 1 Step2 g > P L flag i 1 Step2

1 25 5.4 5.4.1 1. l j = 8 a ij g i + b j 2. i=1 l j m j < 0, j = 1, 2,..., 6 (6) min E 8 g i = P L, i=1 s.t. l j m j < 0, g i0 v i t g i g i0 + v i t, g i g mi i = 1, 2,..., 8, j = 1, 2,..., 6 (7) 7 7 3. 7 j η j η j = l j /m j 1 l j = (1 + r j η j )m j r j η j (a) { } min max η j j g i = P L i l j (1 + η j r j )m j < 0 s.t. g i0 v i t g i g i0 + v i t i = 1, 2,..., 8, j = 1, 2,..., 6 (8) g i g mi η j 1

26 2004 (8) {ηj }6 j=1 (8) (b) (8) (8) s.t. min E g i = P L i l j (1 + η j r j )m j < 0 g i0 v i t g i g i0 + v i t (9) g i g mi η j ηj (9) (8) (9) (8) (9) 4. (7) max i g i0 v i t g i g i0 + v i t s.t. g i g mi l j (1 + r j )m j 0 g i i = 1, 2,..., 8, j = 1, 2,..., 6 (10) 5.4.2 1. (7) E = max{(p i (g i ) p (0) i M ) P L}, G max p i (g i ) i j a ij a ij i = 1, 2,..., 8 a ij { min max i 8 g i = P L, i=1 s.t. l j m j < 0, g i0 v i t g i g i0 + v i t g i g mi } p i (g i ) i = 1, 2,..., 8, j = 1, 2,..., 6

1 27 MATLAB [3] 0 (a) (b) [6] (c) i. ii. a ij j l j > m j {a ij } G 0 p (0) M Step1 G 0 0 G 0 i g i Step2 g i g i max i g i l j m j s.t. g i0 v i t g i g i0 + v i t g i g i Step3 g i P L Step4 Step2 i Step4 g i = P L i MATLAB 4 5 2. (a) 8

28 2004 8 a ij s.t. g i = P L i min η l j (1 + η j r j )m j < 0 g i0 v i t g i g i0 + v i t g i g mi η j η 0 η 1 i = 1, 2,..., 8; j = 1, 2,..., 6 15 (g 1, g 2,..., g 8, η 1, η 2,..., η 6, η) (b) 9 7 g i = P L i { min max i l j (1 + η j r j )m j < 0 s.t. g i0 v i t g i g i0 + v i t } p i (g i ) i = 1, 2,..., 8, j = 1, 2,..., 6 g i g mi η j η j 3. 10 max i g i g i0 v i t g i0 g i0 + v i t s.t. g i g mi l j (1 + r j )m j 0 i = 1, 2,..., 8, j = 1, 2,..., 6 8 (g 1, g 2,..., g 8 ) 5.5 4 5.5.1 Huffman

1 29 1. (6) 2. (7) 1 2 3 4 3. (8) 1 2 3 4 4. (10) 4 1 1: 1 1 2 (6) 3 (7) 4 (8) 5 (10) 6 7 8 9 10 n (n = 6, 7,..., 10) P n P n n Huffman [4] Huffman Huffman Step0 {p n } 10 n=6 5 F Step1 F F F Step2 Step1 F 24 60/15 = 96 P n P n Huffman 5.5.2 Huffman Huffman (6) (7) (8) (10) Huffman

30 2004 2: 3: A 1 B 2 C 3 D 4 L0 Y N L1 g i = P L i L2 g i0 v i t g i g i0 + v i t L3 8 a ij g i + b j m j i=1 L4 8 a ij g i + b j (1 + r j )m j 1. L 1 &L 2 &L 4 = N N 2. L 1 &L 2 &L 4 = Y L 1 &L 2 &L 3 = N 3. L 1 &L 2 &L 3 = Y 2 L1, L2 L1&L2 3 L0, L1, L2, L3, L4 Haffman A,B,C,D Huffman 2 3 D D D i=1

1 31 6 6.1 6.1.1 (1) 1: a ij (i = 1, 2, 8)(10 2 ) R 2 F α : 10 10 1 [19.382,32.389,14.084,24.792,10.393,32.064,6.7494,13.063] 0.7307 9.6882 85053 2 [7.7276,46.115,10.453,18.548,24.079,12.39,-8.3441,25.564] 0.6213 5.8596 4196100. 3 [-17.89-21.49,-24.34,-13.62,-0.322,-19.396,5.098,-33.156] 0.8008 14.357 240.85 4 [4.3359,9.4364,26.689,6.9081,7.905,14.548,10.693,16.917] 0.8750 24.996 8.7098 5 [13.411,58.101,4.1388,11.312,9.0764,30.982,-6.9572,15.007] 0.6036 5.4375 6979500 6 [35.901,24.627,1.823,23.298,18.844,21.738,10.674,14.494] 0.7797 12.638 7992.1 1 j j 1 α 6.1.2 (2) 2: a ij (i = 1, 2, 8)(10 2 ) b j R 2 F α 1 [8.2607,4.7764,5.2794,11.986,-2.5705,12.165,12.199,-0.15179] 110.48 0.99944 5376.8 0 2 [-5.4717,12.75,0.014644,3.3244,8.6667,-11.269,-1.8644, 9.8528] 131.35 0.99957 6970.2 0 3 [-6.9387,6.1985,-15.65,-0.9871,12.467,0.23561,-0.2787, -20.119] -109 0.99986 21788 0 4 [-3.4632,-10.278,20.504,-2.0882,-1.2018,0.56932,14.522, 7.6336] 77.612 0.99988 24424 0 5 [0.03271,24.283,-6.471,-4.1202,-6.5452,7.0026,-0.38961, -0.917] 133.13 0.99953 6433.9 0 6 [23.757,-6.0693,-7.8055,9.2897,4.6634,-0.029128,16.64, 0.0388] 120.85 0.99981 16029 0 6.2 6.3 982.4MW 3 3: 1 2 3 4 5 6 7 8 120 73 180 80 125 125 81.1 90 150 79 180 99.5 125 140 95 113.9 252 300 233 302 215 252 260 303 33 15 48 19.5 27 30 21 27 30 6 0 19.5 0 15 13.9 23.9 303 6.4 982.4MW 3 4 5

32 2004 4: 1 2 3 4 5 6 165 150 160 155 132 162 173.3047 141.0049-150.9235 120.9114 136.8265 168.519 8.3047-8.9951-9.0765-34.0886 4.8265 6.519 5: [153,86.87, 228, 90.1124, 152, 95.3222, 60.1,117] [33,15,48,19.5,27,30,21,27] [33,15,48,10.11,27,-29.69,-21,27] [165,150,160,155,132,162] [165,150,155.26,124.51,131.51,159.53] 495 1 3183.1 2 47155 1 2 6.5 1052.8MW 6 7 8 6: 1 2 3 4 5 6 7 8 120 73 180 80 125 125 81.1 90 150 81 218.2 99.5 135 150 102.1 117 252 320 356 302 310 305 306 303 356 7: 1 2 3 4 5 6 165 150 160 155 132 162 177.24 141.17 156.15 129.74 134.83 167.06 12.24-8.83-3.85-25.26 2.83 5.06 7 1. 2. 3.

1 33 8: ( [153,88,228,99.5,152,155,60.3,117] ( ) [489,495,356,302,510,380,120,303] ( ) [33,15,48,19.5,27,30,21,27] ( ) [33,15,48,19.5,27,30,20.8,27] ( ) [165,150,160,155,132,162] ( ) [173.4093,143.5833,155.2113,124.6828,135.2969,160.4221] [39.2%,12.3%,4.25%,-85.16%,30.23%,19.29%] ( ) 510 1( ) 1962.3 2( ) 40533 4. 5. 6. 7. Huffman 8. [7] Pool 8 [1]. [M]. 2003. [2]. [M]. 2002. [3]. [M]. 2000. [4]. [M]. 1997. [5]. [M]. 2001. [6]. - [M]. 2000. [7]. [J]. 2002(5) 10 12.

1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 410073 982.4MW 303 /MWh 1052.8MW 356 /MWh 983.511MW 1094.6MW. 982.4MW 982.4MW < 983.511MW 1473.5 /(15 ) 1094.6MW>1052.8MW>983.511MW 682.52 /(15 ), 1 2 1. 2. 2004

1 35 3. 1 1: 1., 2. 3 3.1 1. 2. 3. 4. 3.2 Q q ij (i = 1,..., 8 j = 1,..., 10) i j Q d ij (i = 1,..., 8 j = 1,..., 10) i j d ij,1 (i = 1,..., 8 j = 1,..., 10) 982.4 W d ij,2 (i = 1,..., 8 j = 1,..., 10) 1052.8 W M m ij (i = 1,..., 8 j = 1,..., 10) i j M w ij (i = 1,..., 8 j = 1,..., 10) i j

36 2004 V i (i = 1,..., 8) i P i (i = 1,..., 6) i P i max (i = 1,..., 6) i η i (i = 1,..., 6) i C c ij (i = 1,..., 8 j = 1,..., 10) i j S w 0 T ij (i = 0,..., 32 j = 1,..., 8) i j 4 4.1 4.1.1 T k (k = 1,..., 32) T 0 T 1 T 4 1 T 5 T 8 2 T 29 T 32 8 T 1 T 4 1 1 2 2: 1 1 T 5 T 8 2 1 3 1 1 1 8 1

1 37 3: 2 1 x i (i = 1,..., 8) 1 a 1i (i = 1,..., 8) 1 8 P 1 = a 10 + a 11 x 1 + a 12 x 2 + + a 18 x 8 = a 10 + a 1i x i i=1 P 1 = a 10 + a 1 X, X = (x 1, x 2,..., x 8 ) T a 1 = (a 11, a 12,..., a 18 ) 95% 0.082607 0.047764 0.052794 a T 1 = 0.11986 0.025705, a 10 = 110.48 0.12165 0.12199 0.0015179 4.1.2 1. R 2 0.99944 2. F 5376.8!F 0.05 (8, 33 8 1) 3. p 0.0 0.5 4. r r

38 2004 2 6 P = (P 1, P 2,..., P 6 ) T P = b + Ax A = 1 10 2 b = (a 10, a 20,..., a 80 ) T A = (a 1, a 2,..., a 6 ) T 8.2607 4.7764 5.2794 11.986 2.5705 12.165 12.199 0.1518 5.4717 12.75 0.0146 3.3224 8.6667 11.269 1.8644 9.8528 6.9387 6.1985 15.65 0.9871 12.467 0.2356 0.2787 20.119 3.4632 10.278 20.504 2.0882 1.2018 0.5693 14.522 7.6336 0.0327 24.283 6.471 4.1202 6.5452 7.0026 0.3896 0.9170 23.757 6.0693 7.8055 9.2897 4.6634 0.0291 16.636 0.0388 110.48 131.35 108.99 b = 77.612 133.13 120.85 4.2 AGC w 0 s 1 λ w ij w 0 s 2 (1) (S)= λ s 1 + s 2 λ < 1 λ = 0.1 λ 8 10 s 1 = (d ij q ij ) w 0, d ij q ij i=1 j=1 8 10 s 2 = (w ij w 0 ) (q ij d ij ), w ij w 0, q ij d ij i=1 j=1

1 39 (2) w 0 : λ s 2 = (S ) = λ (s 1)+ (s 2) s 1 = 8 ( i=1 10 j=1 d ij 10 j=1 q ij ) w 0, 8 (max(w i1, w i2,..., w i10 ) w 0 ) i=1 10 j=1 q ij 10 j=1 d ij 10 d ij j=1, 10 j=1 q ij 0 10 j=1 q ij 10 j=1 d ij 0 λ [1] φ (q, t) = ( a + bq + cq 2) t q 10 q ij a, b, c 0. j=1 ϕ (q) = ( a + bq + cq 2) /q a c. [2] (1) 80% -90% (2) (1) a (2) c S 4.3 982.4 MW 10 q ij T 0,i 15 V i (i = 1, 2,..., 8) j=1

40 2004 0 q ij m ij (i = 1, 2,..., 8; j = 1, 2,..., 10) 982.4 MW D 1 1 1: 982.4 MW / 1 2 3 4 5 6 7 8 9 10 1 70 0 50 0 0 30 0 0 0 0 150 2 30 0 20 8 15 6 0 0 0 0 79 3 110 0 40 0 30 0 0 0 0 0 180 4 55 5 10 10 10 9.5 0 0 0 0 99.5 5 75 5 15 0 15 15 0 0 0 0 125 6 95 0 10 20 0 15 0 0 0 0 140 7 50 15 5 15 10 0 0 0 0 0 95 8 70 0 20 0 20 0 3.9 0 0 0 113.9 982.4 303 / MWh, C 1 = 982.4 303 15/60 = 74416.8( /15 ) 1052.8 MW D 2 2 2: 1052.8MW / 1 2 3 4 5 6 7 8 9 10 1 70 0 50 0 0 30 0 0 0 0 150 2 30 0 20 8 15 6 2 0 0 0 81 3 110 0 40 0 30 0 20 18.2 0 0 2182 4 55 5 10 10 10 9.5 0 0 0 0 99.5 5 75 5 15 0 15 15 0 10 0 0 135 6 95 0 10 20 0 15 10 0 0 0 150 7 50 15 5 15 10 7.1 0 0 0 0 102.1 8 70 0 20 0 20 0 7 0 0 0 117 1052.8 356 /MWh C 2 = 1052.8 356 15/60 = 93699.2( /15 ) 4.4 : 982.4MW 982.4MW P = A 10 d 1j,1, 10 d 2j,1,..., j=1 j=1 j=1 10 d 8j,1 + b

1 41 3: 982.4MW 1 2 3 4 5 6 (MW) 165 150 160 155 132 162 (MW) 173.3074 141.0023-151.5798 120.9124 136.8228 168.5210 0.0503 0 0 0 0.0365 0.0403 P P 1, P 2,..., P 6. 3 s.t. 8 max 10 q ij i=1 j=1 0 q ij m ij (i = 1, 2,..., 8; j = 1, 2,..., 10) 10 q ij T 0,i 15V i (i = 1, 2,..., 8) j=1 P i P i,max (i = 1, 2,..., 6) 983.511MW. 982.4MW s.t. min S 0 q ij m ij (i = 1, 2,..., 8; j = 1, 2,..., 10) 10 q ij T 0,i 15V i (i = 1, 2,..., 8) j=1 P i P i,max (i = 1, 2,..., 6) 8 10 q ij = 982.4 i=1 j=1 w 0 = 303 4 1473.5 /(15 ) 5 4.5 1052.8MW 4.5.1 1052.8MW 983.511MW max 8 10 q ij i=1 j=1

42 2004 4: 982.4MW / 1 2 3 4 5 6 7 8 9 10 1 70 0 50 0 0 30 0 0 0 0 ()150 2 30 0 20 8 15 6 2 0 0 6.7365 87.7365 3 110 0 40 0 30 0 20 28 0 0 228 4 49.887 4.75 9.7499 9.7499 9.7499 6.9918 0 0 0 0 90.8782 5 75 5 15 0 15 15 0 10 10 7 152 6 77.871 0 0 18.814 0 0 0 0 0 0 96.6853 7 15.241 15 4.8592 15 10 0 0 0 0 0 60.1000 8 70 0 20 0 20 0 7 0 0 0 117 982.4 5: 982.4MW 1 2 3 4 5 6 (MW) 165 150 160 155 132 162 (MW) 164.97 149.996-155.52 124.63 131.5 158.91 0 0 0 0 0 0 s.t. 0 q ij m ij (i = 1, 2,..., 8; j = 1, 2,..., 10) 10 q ij T 0,i 15V i (i = 1, 2,..., 8) j=1 P i P i,max (1 + η i ) (i = 1, 2,..., 6) 1094.6MW 1052.8MW, : 6 I 1 = F (P i P i,max )/P i,max i=1 I 2 = max(f (P i P i,max )/P i,max ) min(f (P i P i,max )/P i,max ) { x x > 0 F (x) = 0 x 0 I 1 0 I 1 0 I 1 I 2 I 2 L L = S/C 2

1 43 4.5.2 min µ 1 L + µ 2 I 1 + µ 3 I 2 0 q ij m ij (i = 1, 2,..., 8; j = 1, 2,..., 10) 10 q ij T 0,i 15V i (i = 1, 2,..., 8) j=1 P s.t. i P i,max (1 + η i ) (i = 1, 2,..., 6) 8 10 q ij = 1052.8 i=1 j=1 w 0 = 356 λ = 0.1 µ 1, µ 2, µ 3 I 1 I 2 L S 6 6: λ = 0.1 µ 1 : µ 2 : µ 3 S /(15 ) I 1 (%) I 2 (%) 2 5 : 2 5 : 1 5 682.5200 6.15% 0 2 5 : 1 5 : 2 5 608.0199 6.78% 5.1% 1 3 : 1 3 : 1 3 664.6200 6.60% 4.38% λ λ = 0.2 7 7: λ = 0.2 µ 1 : µ 2 : µ 3 S /(15 ) I 1 I 2 2 5 : 2 5 : 1 5 853.03 6.23% 0 2 5 : 1 5 : 2 5 846.54 6.78% 5.1% 1 3 : 1 3 : 1 3 936.96 6.60% 4.38% λ λ I 1 I 2 S 5% S I 1 I 2 λ µ 1 : µ 2 : µ 3 = 1 : 1 : 0 1052.8MW 682.52 /(15 ) 8 9 5 S S 10 11

44 2004 4: 8: 1052.8MW / 1 2 3 4 5 6 7 8 9 10 1 70 0 33.63 0 0 30 0 0 0 0 133.63 2 30 0 13.984 8 15 6 2 0 0 0 74.9840 3 110 0 40 0 30 0 20 28 0 0 228 4 55 5 10 10 10 9.5 1e-007 0 0 0 99.5 5 75 5 15 0 15 15 0 10 10 7 152 6 95 0 10 20 0 15 10 5 0 0 155 7 50 15 2.6861 15 10 0 0 0 0 0 92.6861 8 70 0 20 0 20 0 7 1e-007 0 0 117 1052.8 9: 1052.8MW 1 2 3 4 5 6 (MW) 165 150 160 155 132 162 (MW) 175 1412 142 3772-155.4914 131 3955 131.9999 161 98 0 0615 0 0 0 0 0 10: λ = 0.1 µ 1 : µ 2 : µ 3 S /(15 ) S /(15 ) 2 5 : 2 5 : 1 5 853.03 5182.4 2 5 : 1 5 : 2 5 846.54 6090.5 1 3 : 1 3 : 1 3 936.96 7041.7

1 45 11: λ = 0.2 µ 1 : µ 2 : µ 3 S /(15 ) S /(15 ) 2 5 : 2 5 : 1 5 853.03 1249.5 2 5 : 1 5 : 2 5 846.54 6329.0 1 3 : 1 3 : 1 3 936.96 1230.7 S 6 - - 96 5: 5 5 a 24 p 0 ρ 0 b h 1 p 1 ρ 1 h i > h j ρ i < ρ j. ρ 0, ρ 1,...,

46 2004 1. 2. 3. 7 1. 2. 3. 4. 5. 8 : [1]. [J]. 2002 26(9). [2]. [J]. 2001 21(12). [3]. [J]. 2002 26(11). [4]. MATLAB 6.5 [M]. 2003.3

1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 410073 1 2 3 4 5 6 7 8 982.4MW (MW) 150 79 180 99.5 125 140 95 113.9 ( /MWh) 303 1052.8MW (MW) 150 81 218.2 99.5 135 150 102.1 117 ( /MWh) 356 1 2 3 4 5 6 982.4MW 1052.8MW (MW) 165.0 149.9-155.1 124.6 131.5 158.9 (MW) 165 150 160 155 132 162 15430 89846.8 (MW) 175.1 142.4-154.8 131.2 132.0 162.0 (MW) 165 150 160 155 132 162 6923.5 100622.7 982.4MW 1052.8MW 1 5 1 5 41.31% 14.88%. 1 2004

48 2004 ATC ATC 0 2 1. 2. 3. 4. 5. 6. 3 Γ(P ) P 8 1 PL 1 PL 2 P r (t) t C(t) t δ(x) µ M Q r Q s t 0 F i i F = (F 1, F 2,..., F 6 ) T 6 ω i i Ω = (ω 1, ω 2,..., ω 6 ) T 6

1 49 P k (t) t k P (t) = (P 1 (t), P 2 (t),..., P 8 (t)) T t 8 v k k v = (v 1, v 2,..., v 8 ) T 8 F i i F = ( F 1, F 2,..., F 6 ) T 6 Fi 0 i F 0 = ( ) F1 0, F2 0,..., F6 0 T 6 η i i η = (η 1, η 2,..., η 6 ) T 6 ηi 0 i η 0 = ( η1, 0 η2, 0..., η6) 0 T 6 C k k P k (t) a ij i j a 11 a 12 a 18 a 21 a 22 a 28 A = (a ij ) 6 8 =...... a 61 a 62 a 68 4 4.1 F j = 8 a ij P i +ω j P F F = A P + Ω A A ij i j (i = 1, 2,..., 8), (j = 1, 2,..., 6) F j (j = 1, 2,..., 6) A Ω 4.2 4.2.1 i=1 Γ = Γ 1 + Γ 2 + Γ 3 8 Γ 1 = P k (t) (C k (P k (t) + P k (t)) C(t)) δ (C k (P k (t) + P k (t)) C(t)) k=1, P k (t) 0

50 2004 Γ 2 = 8 k=1, P k (t) 0 P k (t) (C k (P k (t) + P k (t)) C(t)) δ (C k (P k (t) + P k (t)) C(t)) Γ 3 = µ C(t) 8 k=1, P k 0 ( P k (t)) { 1 (x 0) δ(x) = 0 (x < 0) Γ 1 Γ 2 Γ 3 Γ 1 = 8 k=1, P k (t) 0 P k (t) (C k (P k (t) + P k (t)) C(t)) δ (C k (P k (t) + P k (t)) C(t)) C k (P k (t) + P k (t)) C ( t) P k (t) P k (t) 0 δ(x) Γ 2 = 8 k=1, P k (t) 0 P k (t) (C k (P k (t) + P k (t)) C(t)) δ (C k (P k (t) + P k (t)) C(t)) C k (P k (t) + P k (t)) C k (P k (t) + P k (t)) C ( t) P k (t) 0 δ(x) µ µ Γ 3 = µ C(t) 8 k=1, P k (t) 0 8 k=1, P k 0 ( P k (t)) ( P k (t)) µ µ Γ = Γ 1 + Γ 2 + Γ 3

1 51 4.2.2 + * Γ 2 L M Γ = Γ 2 L + M (Q r Q s ) 4.3 P k (t 1) P k (t) t 0 v k Step 1 Step 2 n 1 n n 1 n Step 3 Step 4 Step 5 Step 1 4.4 F = (F 1, F 2,..., F 6 ) T F 0 = ( ) F1 0, F2 0,..., F6 0 T F k Fk 0, k

52 2004 F > F 0, k 8 C(t) (P k + P k ) + Γ k=1 C(t) 8 (P k + P k ) C(t) 8 (P k + P k ) = 8 P k C(t) 8 (P k + P k ) k=1 [1] k=1 1 k=1 k=1 s.t. min Γ A (P (t) + P (t)) + Ω F 0 P (t 1) P (t) t 0 v P (t) P (t 1) t 0 v 8 P k + P k P r (t) k=1 P + P 0 C(t) t C k (P k (t) + P k (t)) C k (k = 1, 2,..., 8) 505 (0 P 1 (t) 70) 124 (70 < P 1 (t) 120) C 1 (P 1 (t)) = 252 (120 < P 1 (t) 150) 489 (150 < P 1 (t) 190) P + P 1 P L 1 = 983.4829MW P L 1 2 2 s.t. min Γ A (P (t) + P (t)) + Ω = F 0 + F P (t 1) P (t) t 0 v P (t) P (t 1) t 0 v 8 P k + P k P r (t) k=1 P + P 0 F 6 η i η 0 i=1 i σ = 1 i<j 6 ( η i η 0 i η j η 0 j ) 2 1 2

1 53 1: 2 2 P L 2 1094.6MW P L 2 3 3 2 PL 2 2 5 5.1 j 8 F j = a ij P i + ω j i=1 F = A P + Ω F j, ω j A = 0.0826 0.0478 0.0528 0.1199 0.0257 0.1216 0.1220 0.0015 0.0547 0.1275 0.0001 0.0332 0.0867 0.1127 0.0186 0.0985 0.0694 0.0620 0.1565 0.0099 0.1247 0.0024 0.0028 0.2012 0.0346 0.1028 0.2050 0.0209 0.0120 0.0057 0.1452 0.0763 0.0003 0.2428 0.0647 0.0412 0.0655 0.0700 0.0039 0.0092 0.2376 0.0607 0.0781 0.0929 0.0466 0.0003 0.1664 0.0004 Ω = (110.4775, 131.3521, 108.9928, 77.6116, 133.1334, 120.8481) T [2] 1 2 3 4 5 6 0.0340 0.0252 0.0266 0.0251 0.0279 0.0363 5.2 Γ 1 Γ 2

54 2004 Γ 3 µ µ µ = 5% 8 Γ 3 = 5% C(t) ( P k (t)) k=1, P k 0 Γ = Γ 1 + Γ 2 + Γ 3 5.3 0 1 2 3 4 5 6 7 8 (MW) 153 88 228 99.5 152 155 102.1 117 (MW) 87 58 132 60.5 98 95 60.1 63 982.4MW 1 2 3 4 5 6 7 8 (MW) 150 79 180 99.5 125 140 95 113.9 ( /MWh) 303 5.4 1: 1 2 3 4 5 6 (MW) 173.31 141.02-150.92 120.90 136.81 168.51 (MW) 165 150 160 155 132 162 1 5 6 982.4 < PL 1 = 983.4829 1 1 2 3 4 5 6 7 8 (MW) 150 79 180 99.5 125 140 95 113.9 (MW) 150.000 87.7856 228 90.6504 152 96.8641 60.100 117 0 (MW) 120 73 180 80 125 125 81.1 90 (MW/m) 2 0.98 3.2 0.71 1.8 1.88 1.4 1.8 (MW/m) 2.2 1 3.2 1.3 1.8 2 1.4 1.8 i = i 0 i

1 55 : 1 2 3 4 5 6 165.000 149.995-155.0613 124.6244 131.5200 158.8757 165 150 160 155 132 162 15430 982.4*303*1/4+15430 89846.8 5.5 1052.8MW : 1 2 3 4 5 6 7 8 (MW) 150 81 218.2 99.5 135 150 102.1 117 ( /MWh) 356 1 2 3 4 5 6 (MW) 177.2415 141.1811-156.1459 129.7333 134.8112 167.0558 (MW) 165 150 160 155 132 162 PL 1 < 1052.8MW < P L 2 6 η i η 0 i=1 i 1 i<j 6 ( η i η 0 i η j η 0 j ) 2 1 2 2

56 2004 1 2 3 4 5 6 7 8 (MW) 150 81 218.2 99.5 135 150 102.1 117 (MW) 134.2667 75.1407 228 99.5 152 155 91.8926 117 0 (MW) 120 73 180 80 125 125 81.1 90 (MW/m) 0.95 0.14 3.2 1.3 1.8 2 0.72 1.8 (MW/m) 2.2 1 3.2 1.3 1.8 2 1.4 1.8 1 2 3 4 5 6 (MW) 175.1051 142.3939-154.7905 131.2314 132.0261 162.0000 (MW) 165 150 160 155 132 162 13% 18% 9% 11% 15% 14% 41.31% 0 0 0 14.88% 0 2: 1 5 1 41.31% 5 14.88%. 6923.5 1052.8 356 1/4 + 6923.5 = 100622.7 MATLAB C MATLAB C C MATLAB C X...... 6 1. F [3] H 0 H 1

1 57 3: U/k F F = Q F (k, n k 1) U Q e/(n k 1) e n k α α = P {F > F 1 α (k, n k 1) H 0 } F > F 1 α (k, n k 1) H 0 MATLAB regress(y X α) α 0.05 stats r 2 r 2 1 F F F p p α 1 1 2 3 4 5 6 r 2 0.9994 0.9996 0.9999 0.9999 0.9996 0.9998 F 5377 6970 26788 24424 6434 16029 P 0 0 0 0 0 0 F 0.95 (8, 24) = (F 0.05 (24, 8)) 1 = 1/3.12 α = 0.05 F F 0.95 (8, 24) 2. 0.5 /KWh 982.4MW 0.5 /KW h 982400KW 0.25h = 122800 89846.8 122800 > 89846.8 7 µ 0.05 982.4MW µ 0.01 0.03 0.05 0.07 0.09 15166 15298 15430 15561 15693

58 2004 µ 0.01 0.03 0.05 0.07 0.09 6810.3 6866.9 6923.5 6980.1 7036.7 1052.8WM µ 8 1. Γ = Γ 2 + Γ 3 982.4MW 1052.8MW 1 2 3 4 5 6 7 8 (MW) 150.0 87.8 228.0 90.7 152.0 96.8 60.1 117.0 2784 982.4 303 1/4 + 2784 = 77200.8 1 2 3 4 5 6 7 8 (MW) 134.4 75.1 228.0 99.5 152.0 155.0 91.9 117.0 939.2 1052.8 356 1/4 + 939.2 = 94638.4 2. 1 8 Γ 3 = (C(t) C k (P k (t) + P k (t))) ( P k (t)) k=1, P k 0 Γ = Γ 2 + Γ 3 982.4MW 3. (a) Price Elasticity of Demand

1 59 1 2 3 4 5 6 (MW) 165.00 149.33-155.01 126.29 132.00 159.62 (MW) 165 150 160 155 132 162 4851 79275.375 (b) FACTS [4] (c) 4. (a) (b) [5] 280 340 340 300 /kwh. 9 1. 2. 3. 1. 2. [1]. [M]. 1990. [2]. [M]. 2002. [3]. [M]. 2000. [4]. FACTS [J]. 4 5 16 18 2002. [5]. [J]. 28 8 32 33 2004.

1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 410073 1. 2. 1 MS Huff MS σ 2 MS MS MS MS LMS 2 MS 1: A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 5.87% 4.05% 4.93% 5.81% 6.70% 13.8% 6.70% 5.81% 4.93% 4.05% B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 3.71% 3.44% 5.58% 3.44% 3.71% 6.94% 0.78% 3.35% 0.78% 5.60% 2: LMS MS A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 LMS 3 1 2 2 3 6 3 2 2 1 MS 2 2 1 1 0 0 0 1 1 2 B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 LMS 2 2 4 2 2 3 1 3 1 4 MS 0 0 2 1 1 1 1 2 1 2 2004

1 61 1 2 2008 1 4 14 15 [1] 33 137 2006 2007 3 10600 MS MS MS MS 4 1. 2008 2. 3. MS 62%

62 2004 75% 4. MS 50 100 MS 50 MS [2] 5. MS MS 20 6. MS 20 7. MS 8. 16.5 7.26 7.17 70 110 80 80 5 A1-A10 B1-B6 C1-C4 A1-A10 A B1-B6 B C1-C4 C i 20 i = 1 20 30 i = 2 30 50 i = 3 50 i = 4 j j = 1 j = 2 k k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 s s = 1 s = 2 s = 3 t 0 100 t = 1 100 200 t = 2 200 300 t = 3 300 400 t = 4 400 500 t = 5 500 t = 6 P (k, s) k s LMS MS N 20 u i i i = 1, 2, 3,..., 20 A MS B MS ω i α 6 6.1 10600

1 63 1. 20 30 58.02% 20 50 11.08% 10.72% 2. 52 17% 47 83% 3. 3. 3: k 1 2 3 4 5 6 P (k) 17 49% 17 09% 19 43% 8 80% 18 43% 18 77% 33.98% 17 49%+17 09% 38.00% 18 43%+18 77% 19.43% 9.4% 50 35.82% 1 1: 4. 4. 4: s 1 2 3 P (s) 22 37% 52 49% 25 14% 52.52% 30 30 2 5. 5. 200 300 45.43% 300 500 0.97%

64 2004 2: 5: t 1 2 3 4 5 6 P (t) 19 51% 23 80% 45 43% 8 94% 3 14% 0 97% 50 300 400 20 30 3 3: 6. (a) 200 300 200-300 4 4 5 6 (b) 200 300 200 300

1 65 4: 6.2 6.2.1 1. 2. 5 5 [4] 5: 3. 2000

66 2004 6:4 10 6 4 4. 5 5. 1 2 7 10 7 4 4 7 7 10 6.2.2 A8 6 7 A8 6 A8 A 6: A8 A

1 67 6 A8 u 1 8 u 1 8 = P (1, 1) 1 2 + P (5, 1) 1 2 + [P (3, 2) + P (3, 3) + P (4, 2) + P (4, 3)] 10 10 + 6 + 4 1 2 7 A8 A 7: A8 A A6 A A1 A10 A9 A8 A8 4/10 A1 A A8 A7 A6 A8 A6 1/2 A8 2/10+1/10 1/2=5/20 A8 u 2 8 u 2 8 = [P (5, 2) + P (5, 3)] 4 5 + [P (3, 1) + P (4, 1)] 10 20 A8 u 8 = 5.81% u 1 = 5.87% u 2 = 4.05% u 3 = 4.93% u 4 = 5.81% u 5 = 6.70% u 6 = 13.8% u 7 = 6.70% u 8 = 5.81% u 9 = 4.93% u 10 = 4.05% u 11 = 3.71% u 12 = 3.44% u 13 = 5.58% u 14 = 3.44% u 15 = 3.71% u 16 = 6.94% u 17 = 0.78% u 18 = 3.35% u 19 = 0.78% u 20 = 5.60% 8 8 1. A1 A6 B3 B6 C2 C4

68 2004 8: 2. A6>A1 B6>B3 C4>C2 3. 2 A6 A5 A7 2 A6 A6 A5 A7. 6.3 20 MS A B C A MS MS MS MS MS LMS 6.3.1 1. Huff [4] Huff Huff D Huff

1 69 P ij = S j F (d ij )/ m S j F (d ij ) j=1 i = 1,..., n; j = 1,..., m F (d ij ) = d λ ij P ij i j S j j d ij i j λ (a) A6 A8 Huff Huff 9 9: (b) A8 10 10: 10

70 2004 11 11: 11 ω i 2. 6 C = 209.785 q i = ω i u i NαC (1) i ω i u i N α 11 ω i 6 6: i 1 2 3 4 5 ω i 0.1089 0 099 0 099 0 099 0 099 i 6 7 8 9 10 ω i 0.1089 0 099 0 099 0 099 0 099 i 11 12 13 14 15 ω i 0 1639 0 1639 0.1803 0 1639 0 1639 i 16 17 18 19 20 ω i 0.1803 0.2439 0.2683 0.2439 0.2683 1 12 6.3.2 MS MS MS MS LMS MS MS [ LMS MS qi ] MS n i = + 1 MS x i = q i A n i

1 71 12: MS 7 7: MS A i 1 2 3 4 5 6 7 8 9 10 n i 9 6 7 7 9 18 9 7 7 6 6.3.3 MS MS [3] MS IRS = C RE RF IRS RE C RF MS MS IRS = x i. σ 2 MS σ 2 = 1 n (xi ˆx) 2. σ 2 MS MS MS 9.1 MS MS x i x j, i, j = 1,..., n

72 2004 {x 1, x 2,..., x n } x 1 < x 2 n = 2 y 1 = x 1 + x y 2 = x 2 x x x 1, x 2 y 1 + y 2 = x 1 + x 2 = M σ 2 1/8(x 1 x 2 + 2 x) 2 1/8(x 1 x 2 + 2 x) 2 1/8(x 1 x 2 ) 2. 1/2 x(x 1 x 2 + x) 0 1/2 x(x 1 x 2 ) 0. x x x 1 x 2 = 0 x 1 > x 2 x i x j n MS 1. 2. 3. max : qi n i B (2) min : σ 2 = 1 n (xi ˆx) 2 ˆx = xi n n = n i (3) s.t. ni A q i 2 MS q i MS 3 σ 2 U(n) U(n) = λ i f(n i ) min {U(n) = λ 1 ( n i B q i ) + λ 2 σ 2 } λ 1 + λ 2 = 1 λ 1 MS λ 2 MS MS λ 1 = 0.8 λ 2 = 0.2

1 73 MS MS MS MS S 0 MSNo(S) S MS C(S ) S MSNo A MS Step 1 S 0 S 0 S 0 S0 Step 2 S0 MS S i n 1 i n MSNo(S i ) MSNo (1 i n) S i S i (1 i n) Step 3 C(Sj ) (0 j n) C(S j ) S k j = k C(S j ) Sk C(S k ) MS 8 8: MS A i 1 2 3 4 5 6 7 8 9 10 ni σ 2 n i 9 6 7 7 9 18 9 7 7 6 85 1.41593 10 8 8 5 7 7 9 19 9 7 7 6 85 1.09264 10 8 8 6 7 7 9 18 9 7 7 6 84 1.53641 10 8 MS 9 9: MS A i 1 2 3 4 5 6 7 8 9 10 ni σ 2 n i 8 5 7 7 9 18 9 7 7 5 82 1.40176 10 8 1. MS 2. 8 MS MS σ 2 3. 8 MS σ 2 4. 9 MS 6.3.4 LMS MS LMS MS

74 2004 G LMS : G MS = p : q. S LMS : S MS = m : n. F LMS : F MS = c : d. MS ω x LMS y MS min (x c + y d) s.t. { x m + y n ω n x p + y q ω q MS LMS MS 1. LMS MS LMS MS 2. LMS MS p : q = c : d LMS p q MS LMS MS 3. LMS MS p : q > c : d LMS MS LMS LMS y = [ ] ni, x = n i y p/q 4. LMS MS p : q < c : d LMS MS LMS MS G LMS : G MS = 3 : 1 F LMS : F MS = 2 : 1 LMS MS 10 10: A LMS MS A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 LMS 3 1 2 2 3 6 3 2 2 1 MS 2 2 1 1 0 0 0 1 1 2 B C LMS MS 11: B C LMS MS B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 LMS 2 2 4 2 2 3 1 3 1 4 MS 0 0 2 1 1 1 1 2 1 2 6.4 1. (a)

1 75 10600 (b) (c) MS 2. 20 20 7 Huff MS MS 20 MS MS MS

76 2004 LMS MS 8 9 10 MS [5] MS MS 13: MS 14:

1 77 9 1. 2. 3. MS LMS MS 10 [1]. 2008 [J]. 2001(2) 50 53. [2]. [J]. 2002(4) 46 50. [3] [C]. 2002 [4]. GIS [J]. 2003(18) 144 148. [5] Olympic Stadium, www.athens2004.com/en, 2004.09.19. [6]. [M]. 2002.6 [7]. Matlab [M]. 2003.9

1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 410073 MS MS MS A6 15, 000 m 2 5 MS 450 m 2 17 MS 150 m 2 4, 800 m 2 1 2 1. 10 6 4 1 2. 3. 4. 5. 2008 2004

1 79 6. 3 P k R 0 P (C j ) MS k C j 4 20 MS 5 5.1

80 2004 5.1.1 1 2 1: C1 C2 C3 C4 C5 C6 E1 E2 E3 0.1749 0.1709 0.1943 0.0880 0.1843 0.1877 0.2237 0.5249 0.2514 0.1681 0.1744 0.1859 0.0919 0.1891 0.1906 0.2263 0.5225 0.2513 0.1600 0.1723 0.1885 0.0913 0.1938 0.1941 0.2244 0.5277 0.2479 0.1677 0.1725 0.1896 0.0909 0.1891 0.1908 0.2248 0.5250 0.2502 2: M1 M2 M3 M4 M5 M6 0.1966 0.2550 0.4350 0.0841 0.0188 0.0106 0.1951 0.2380 0.4549 0.0894 0.0134 0.0097 0.1918 0.2513 0.4323 0.1028 0.0128 0.0090 0.1945 0.2481 0.4407 0.0921 0.0150 0.0098 C1 C6 E1 E3 : M1 M6 0 100 100 200 200 300 300 400 400 500 500 1 3 1. 2. 1:2:1 3. 300 300 5.1.2 1. (a) 2:1 1:2 (b) 2. (a) 20 30

1 81 3: 20 20 30 30 50 50 0.6275 0.3725 0.1552 0.5866 0.1503 0.1078 0.6572 0.3428 0.0953 0.4850 0.2341 0.1856 0.3397 0.6603 0.0882 0.5882 0.2250 0.0985 0.3312 0.6688 0.1331 0.5974 0.2078 0.0617 0.5256 0.4744 0.1039 0.6016 0.2078 0.0868 0.5738 0.4262 0.0944 0.6210 0.1948 0.0898 0.5262 0.4738 0.0498 0.4125 0.3436 0.1941 0.5346 0.4654 0.0985 0.6848 0.1600 0.0566 0.4909 0.5091 0.1841 0.5091 0.1682 0.1386 (b) P (C j ) = 6 n i M i i=1 N(C j ) P (C j ) C j M i i 500 550 n i C j i N(C j ) C j 4: 20 20 30 30 50 50 1.8215 2.2306 1.4948 2.2940 1.9993 1.0926 4 5.1.3 1. 2. 3.

82 2004 4. 5. 5.2 5.2.1 1 2 1: 2: 1. 2. 1 2008

1 83 2008 5.2.2 1. 2. 3. 4. 1. A1 A4 A9 2. 5 5: A6 B6 A1 A5 A7 A4 A8 B3 A3 A9 11.98% 8.51% 7.28% 5.99% 5.99% 5.48% 5.48% 5.28% 4.95% 4.95% A2 A10 B1 B5 C4 B2 B4 C3 C2 C1 4.43% 4.43% 3.97% 3.97% 3.78% 3.45% 3.25% 2.42% 2.38% 2.00% 250.936 / 5.3 5.3.1, 6 40.3429 / 5 6 B3 A4 A8

84 2004 6: A6 B6 A1 A5 A7 B3 A4 A8 A3 A9 12.14% 8.65% 6.72% 6.02% 6.02% 5.47% 5.41% 5.41% 4.81% 4.81% A2 A10 B1 B5 C4 B2 B4 C3 C2 C1 4.21% 4.21% 4.14% 4.14% 3.93% 3.54% 3.40% 2.48% 2.30% 1.99% C B3 C B3 B3 A4 A8 7 7: 1.8474 2.0847 2.0272 5.48% 5.28% 52.50% B3 A4 A8 5.3.2 1. 33 [1] 30 1.5 15, 000 m 2 2. 18% 25% [2] MS 18% MS 25%. 3. 120 400 m 2[3] MS 500 m 2 MS 50 m 2 MS MS 5.3.3 MS MS MS MS MS MS MS MS MS MS MS MS MS MS

1 85 1. MS S 1 MS S 2 1 = k s S 2 k s k s N x MS y MS f(x) = max(xr 0 S 1 α + yr 0 S 2 β) (1) P k R 0 = k P k k xs 1 + ys 2 R 0 α, β MS k f = α β < 1 k f = 0.95 2. ( ) xs1 C 1 k p1 + ys 2 C 2 k p2 g(x) = max S 1 + S 2 C 1 MS C 2 MS k p1 MS k p2 MS C 1 k p1 > C 2 k p2 C 1 > k p2 = 25% C 2 k p1 18% 1.3890 (2) (1) 3. [2] = : α s = 1 α s 30% 20%-30% 10%-20% 10% [2] α s 0.3 α s = 1 γ = 1 = 1 R 0(0.18xS 1 + 0.25yS 2 )γ P k 18% γ α s 1 R 0(0.18xS 1 + 0.25yS 2 )γ 0.18P k 0.3 (3) (2)

86 2004 4. MS MS 1 : c = = C 1R 0 xs 1 + C 2 R 0 ys 2 k c (4) P k k c C 1, C 2 (1) 1 C 1 C 1 = k o C 2, k o > 1.3890 5. χ = 1 1 n C 1 R 0 x i S 1 + C 2 R 0 y i S 2 2 j=1 n Q i Q k x (5) C 1 R 0 x i S 1 + C 2 R 0 y i S 2 j=1 C 1 R 0 x i S 1 +C 2 R 0 y i S 2 n C 1 R 0 x i S 1 + C 2 R 0 y i S 2 Q i i Q k x χ [5] 6. xs 1 + ys 2 k s S 0 (6) S 0 k s k s = 1 f(x) = max(xr 0 S 1 α + yr 0 S 2 β) (1) j=1 g(x) = max( xs 1 C 1 k p1 + ys 2 C 2 k p2 S 1 + S 2 ) (2) α s = 1 R 0(0.18xS 1 + 0.25yS 2 )γ 0.18P k 0.3 (3) c = C 1R 0 xs 1 + C 2 R 0 ys 2 k c (4) P k χ = 1 1 n C 1 R 0 x i S 1 + C 2 R 0 y i S 2 2 j=1 n Q i Q k x (5) C 1 R 0 x i S 1 + C 2 R 0 y i S 2 j=1 xs 1 + ys 2 k s S 0 (6) 5.3.4 S 1 = 450, S 2 = 150, α = 0.95, β = 1, γ = 0.5, k c = 0.9, k s = 1, k x = 0.9

1 87 8: A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 MS 3 2 2 3 3 5 3 3 2 2 MS 9 5 7 6 7 17 7 6 7 5 R 0 1033.66 1028.90 995.23 970.53 1011.13 1020.19 1011.13 970.53 995.226 1028.90 B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 MS 2 2 3 2 2 4 1 1 1 2 MS 5 3 6 3 5 12 2 3 3 4 R 0 1013.19 1058.35 982.36 1017.24 1013.19 991.68 1069.94 1029.4 1109.47 1056.80 6 6.1 20 48 MS 122 MS 3.99 13.3% A6 5 MS 17 MS 4800m 2 32% 6.2 1. (4) 2. (5) A A A 0.9803 B 0.9720 C 0.9695. MS 3. R 0 20 R 0 3

88 2004 3: R 0 R 0 4. 7 7.1 1. 7% 4 7% 2008 (1 + 7%) 4 1.3108 2. 10% 100 1:8 (100 8 201.7185 1.3108) 0.1 = 53.5587 2008 1.5763 9 20 72 MS 209 MS 6.375 21.25% A6 8 MS 28 MS 7800m 2 52%

1 89 9: A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 MS 5 3 3 4 4 8 4 4 3 3 MS 14 9 10 11 14 28 14 11 10 9 B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 MS 3 3 4 3 3 6 2 2 2 3 MS 9 6 12 5 9 20 2 4 4 8 7.2 d D 1 = 1 + D 1 1 d P 1, P 2 70%-95% [4] MS MS P 1 P 2 MS MS MS 8 2008 1.

90 2004 2. 3. 2008 [1].. http://house.sohu.com 2004-9-17 [2].. http://www.lzrs.net 2004-9-17 [3].. http://bfsp.cn 2004-9-17 [4]. [J] 2004(7) [5]. [M]. 2000 [6] Berry B.J.L. Geography of Market Centers and Retail Distribution[M]. N.J. Prentice Hall, 1967

1 1 Vol.1 No.1 2004 11 Mathematical Modeling 2004. 11 2004 2004 2004 11 19 5304 172 404 1577 56 137 http://mcm.edu.cn 2004 11 27 30