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

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

2 MATHEMATICAL MODELING SPONSORED BY: SHUMO.COM COMPILED BY: Mathematical Modeling Editors Group HOMEPAGE: ADDRESS: Science College, NUDT, Changsha Hunan POSTALCODE:

3 I

4 II 2004

5 1 1 Vol.1 No Mathematical Modeling Floyd Internet SHUMO.COM 2001

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7 1 1 Vol.1 No Mathematical Modeling CUMCM GRE 2004 MCM 2004

8 GRE CUMCM RA

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10 1 1 Vol.1 No Mathematical Modeling OK SHUMO.COM 2001

11 1 7 Word Microsoft WPS Word Visio SmartDraw db CPU UPS bird 10 2

12 Word Visio Word MM 3 [1]

13 :) 5 MATLAB C UPS Modem MATLAB Visual C++ Microsoft Word Windows 2000 Visio HP Laser Jet 6.0 8:00 Down 8:00 /:) SHUMO.COM

14 BMP BMP C BMP MATLAB imread 100 BMP BMP save ASCII txt C BMP ,02,03,04,...99 CT ACDSEE 100 BMP Windows console DOS Turbo C DOS 32 Windows 4 2

15 1 11 Word [2] MATLAB C++ [1]. [J] :1-2. [2]. [J] :12-14

16 1 1 Vol.1 No Mathematical Modeling MATLAB 3. Lindo Lingo MATLAB A 108 SHUMO.COM 2002

17 A 94 A MATLAB MATLAB B Lindo Lingo B 00 B 95 Dijkstra Floyd Prim Bellman-Ford B 97 B 98 B ACM A 00 B 01 B 89 A BP 86 BP 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

18 MATLAB Mathematica A BMP 98 A 03 B MATLAB

19 1 1 Vol.1 No Mathematical Modeling , SHUMO.COM 2002

20 B A MATLAB Mathematica C/C++ Floyd

21 MATLAB MATLAB MATLAB 10 B

22 1 1 Vol.1 No Mathematical Modeling Huffman Huffman MATLAB 1. R MW MW % 510 Huffman 2004

23 AGC AGC 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/

24 [1]

25 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,..., 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

26 (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 A j = [b j, A j ] A j A j (1) 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 H 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)

27 (2) (1) (2) 0 (2) (2) (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

28 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 = p i,k = p M E i,k = 0 p i,k 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

29 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) 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

30 (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) (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

31 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 (a) 8

32 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 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 ) Huffman

33 (6) 2. (7) (8) (10) 4 1 1: (6) 3 (7) 4 (8) 5 (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 Huffman Huffman (6) (7) (8) (10) Huffman

34 : 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

35 (1) 1: a ij (i = 1, 2, 8)(10 2 ) R 2 F α : [19.382,32.389,14.084,24.792,10.393,32.064,6.7494,13.063] [7.7276,46.115,10.453,18.548,24.079,12.39, ,25.564] [ ,-24.34,-13.62,-0.322, ,5.098, ] [4.3359,9.4364,26.689,6.9081,7.905,14.548,10.693,16.917] [13.411,58.101,4.1388,11.312,9.0764,30.982, ,15.007] [35.901,24.627,1.823,23.298,18.844,21.738,10.674,14.494] j j 1 α (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, ,12.165,12.199, ] [ ,12.75, ,3.3244,8.6667, , , ] [ ,6.1985,-15.65, ,12.467, , , ] [ , ,20.504, , , ,14.522, ] [ ,24.283,-6.471, , ,7.0026, , ] [23.757, , ,9.2897,4.6634, ,16.64, ] MW 3 3: MW 3 4 5

36 : : [153,86.87, 228, , 152, , 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] MW : :

37 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] ( ) [ , , , , , ] [39.2%,12.3%,4.25%,-85.16%,30.23%,19.29%] ( ) 510 1( ) ( ) Huffman 8. [7] Pool 8 [1]. [M] [2]. [M] [3]. [M] [4]. [M] [5]. [M] [6]. - [M] [7]. [J]. 2002(5)

38 1 1 Vol.1 No Mathematical Modeling MW 303 /MWh MW 356 /MWh MW MW MW 982.4MW < MW /(15 ) MW>1052.8MW> MW /(15 ),

39 : 1., 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) W d ij,2 (i = 1,..., 8 j = 1,..., 10) W M m ij (i = 1,..., 8 j = 1,..., 10) i j M w ij (i = 1,..., 8 j = 1,..., 10) i j

40 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 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 : 1 1 T 5 T

41 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 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% a T 1 = , a 10 = R F !F 0.05 (8, ) 3. p r r

42 P = (P 1, P 2,..., P 6 ) T P = b + Ax A = b = (a 10, a 20,..., a 80 ) T A = (a 1, a 2,..., a 6 ) T b = 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= s 2 = (w ij w 0 ) (q ij d ij ), w ij w 0, q ij d ij i=1 j=1

43 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 MW 10 q ij T 0,i 15 V i (i = 1, 2,..., 8) j=1

44 q ij m ij (i = 1, 2,..., 8; j = 1, 2,..., 10) MW D 1 1 1: MW / / MWh, C 1 = /60 = ( /15 ) MW D 2 2 2: MW / /MWh C 2 = /60 = ( /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

45 1 41 3: 982.4MW (MW) (MW) 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) MW MW 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 = i=1 j=1 w 0 = /(15 ) MW MW MW max 8 10 q ij i=1 j=1

46 : 982.4MW / () : 982.4MW (MW) (MW) 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) MW MW, : 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

47 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 = 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 : % 5.1% 1 3 : 1 3 : % 4.38% λ λ = : λ = 0.2 µ 1 : µ 2 : µ 3 S /(15 ) I 1 I : 2 5 : % : 1 5 : % 5.1% 1 3 : 1 3 : % 4.38% λ λ I 1 I 2 S 5% S I 1 I 2 λ µ 1 : µ 2 : µ 3 = 1 : 1 : MW /(15 ) S S 10 11

48 : 8: MW / e e : MW (MW) (MW) : λ = 0.1 µ 1 : µ 2 : µ 3 S /(15 ) S /(15 ) 2 5 : 2 5 : : 1 5 : : 1 3 :

49 : λ = 0.2 µ 1 : µ 2 : µ 3 S /(15 ) S /(15 ) 2 5 : 2 5 : : 1 5 : : 1 3 : S : 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,...,

50 : [1]. [J] (9). [2]. [J] (12). [3]. [J] (11). [4]. MATLAB 6.5 [M]

51 1 1 Vol.1 No Mathematical Modeling MW (MW) ( /MWh) MW (MW) ( /MWh) MW MW (MW) (MW) (MW) (MW) MW MW % 14.88%

52 ATC ATC 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

53 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 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 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

54 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

55 * 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 F = (F 1, F 2,..., F 6 ) T F 0 = ( ) F1 0, F2 0,..., F6 0 T F k Fk 0, k

56 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 = MW P L 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

57 1 53 1: 2 2 P L MW P L PL j 8 F j = a ij P i + ω j i=1 F = A P + Ω F j, ω j A = Ω = ( , , , , , ) T [2] Γ 1 Γ 2

58 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 (MW) (MW) MW (MW) ( /MWh) : (MW) (MW) < PL 1 = (MW) (MW) (MW) (MW/m) (MW/m) i = i 0 i

59 1 55 : *303*1/ MW : (MW) ( /MWh) (MW) (MW) PL 1 < MW < 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 )

60 (MW) (MW) (MW) (MW/m) (MW/m) (MW) (MW) % 18% 9% 11% 15% 14% 41.31% % 0 2: % % / = MATLAB C MATLAB C C MATLAB C X F [3] H 0 H 1

61 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 r F P F 0.95 (8, 24) = (F 0.05 (24, 8)) 1 = 1/3.12 α = 0.05 F F 0.95 (8, 24) /KWh 982.4MW 0.5 /KW h KW 0.25h = > µ MW µ

62 µ WM µ 8 1. Γ = Γ 2 + Γ MW MW (MW) / = (MW) / = Γ 3 = (C(t) C k (P k (t) + P k (t))) ( P k (t)) k=1, P k 0 Γ = Γ 2 + Γ MW 3. (a) Price Elasticity of Demand

63 (MW) (MW) (b) FACTS [4] (c) 4. (a) (b) [5] /kwh [1]. [M] [2]. [M] [3]. [M] [4]. FACTS [J] [5]. [J]

64 1 1 Vol.1 No Mathematical Modeling MS Huff MS σ 2 MS MS MS MS LMS 2 MS 1: A1 A2 A3 A4 A5 A6 A7 A8 A9 A % 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 MS B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 LMS MS

65 [1] MS MS MS MS MS 62%

66 % 4. MS MS 50 MS [2] 5. MS MS MS MS A1-A10 B1-B6 C1-C4 A1-A10 A B1-B6 B C1-C4 C i 20 i = i = 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 t = t = t = t = t = 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

67 % % 10.72% % 47 83% : k 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% % 1 1: : s P (s) 22 37% 52 49% 25 14% 52.52% % %

68 : 5: t P (t) 19 51% 23 80% 45 43% 8 94% 3 14% 0 97% : 6. (a) (b)

69 1 65 4: [4] 5:

70 : A8 6 7 A8 6 A8 A 6: A8 A

71 A8 u 1 8 u 1 8 = P (1, 1) P (5, 1) [P (3, 2) + P (3, 3) + P (4, 2) + P (4, 3)] 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)] [P (3, 1) + P (4, 1)] 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% A1 A6 B3 B6 C2 C4

72 : 2. A6>A1 B6>B3 C4>C A6 A5 A7 2 A6 A6 A5 A MS A B C A MS MS MS MS MS LMS Huff [4] Huff Huff D Huff

73 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) A : 10

74 : 11 ω i 2. 6 C = q i = ω i u i NαC (1) i ω i u i N α 11 ω i 6 6: i ω i i ω i i ω i i ω i MS MS MS MS LMS MS MS [ LMS MS qi ] MS n i = + 1 MS x i = q i A n i

75 : MS 7 7: MS A i n i 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

76 {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 x) 2 1/8(x 1 x 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 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

77 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 ni σ 2 n i MS 9 9: MS A i ni σ 2 n i MS 2. 8 MS MS σ MS σ MS LMS MS LMS MS

78 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 MS B C LMS MS 11: B C LMS MS B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 LMS MS (a)

79 (b) (c) MS Huff MS MS 20 MS MS MS

80 LMS MS MS [5] MS MS 13: MS 14:

81 MS LMS MS 10 [1] [J]. 2001(2) [2]. [J]. 2002(4) [3] [C] [4]. GIS [J]. 2003(18) [5] Olympic Stadium, [6]. [M] [7]. Matlab [M]

82 1 1 Vol.1 No Mathematical Modeling MS MS MS A6 15, 000 m 2 5 MS 450 m 2 17 MS 150 m 2 4, 800 m

83 P k R 0 P (C j ) MS k C j 4 20 MS 5 5.1

84 : C1 C2 C3 C4 C5 C6 E1 E2 E : M1 M2 M3 M4 M5 M C1 C6 E1 E3 : M1 M :2: (a) 2:1 1:2 (b) 2. (a) 20 30

85 1 81 3: (b) P (C j ) = 6 n i M i i=1 N(C j ) P (C j ) C j M i i n i C j i N(C j ) C j 4:

86 : 2:

87 A1 A4 A : A6 B6 A1 A5 A7 A4 A8 B3 A3 A % 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% / , / 5 6 B3 A4 A8

88 : A6 B6 A1 A5 A7 B3 A4 A8 A3 A % 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: % 5.28% 52.50% B3 A4 A [1] , 000 m % 25% [2] MS 18% MS 25% m 2[3] MS 500 m 2 MS 50 m 2 MS MS MS MS MS MS MS MS MS MS MS MS MS MS MS MS

89 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 = ( ) 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% (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 yS 2 )γ P k 18% γ α s 1 R 0(0.18xS yS 2 )γ 0.18P k 0.3 (3) (2)

90 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 > 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 yS 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) 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

91 1 87 8: A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 MS MS R B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 MS MS R MS 122 MS % A6 5 MS 17 MS 4800m 2 32% (4) 2. (5) A A A B C MS 3. R 0 20 R 0 3

92 : R 0 R % 4 7% 2008 (1 + 7%) % 100 1:8 ( ) 0.1 = MS 209 MS % A6 8 MS 28 MS 7800m 2 52%

93 1 89 9: A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 MS MS B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 MS MS d D 1 = 1 + D 1 1 d P 1, P 2 70%-95% [4] MS MS P 1 P 2 MS MS MS

94 [1] [2] [3] [4]. [J] 2004(7) [5]. [M] [6] Berry B.J.L. Geography of Market Centers and Retail Distribution[M]. N.J. Prentice Hall, 1967

95 1 1 Vol.1 No Mathematical Modeling