2 1/2 1/4 x 1 x 2 x 1, x 2 9 3x 1 + 2x 2 9 (1.1) 1/3 RDA 1 15 x /4 RDA 1 6 x /6 1 x 1 3 x 2 15 x (1.2) (1.3) (1.4) 1 2 (1.5) x 1
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1 1 1 [1] TS 9 1/3 RDA 1/4 RDA 1 1/2 1/ /15 RDA 2/15 RDA 1/6 RDA 1 1/
2 2 1/2 1/4 x 1 x 2 x 1, x 2 9 3x 1 + 2x 2 9 (1.1) 1/3 RDA 1 15 x /4 RDA 1 6 x /6 1 x 1 3 x 2 15 x (1.2) (1.3) (1.4) 1 2 (1.5) x 1 x 2 x 1 0 (1.6) x 2 0 (1.7) (1.1) (1.7) (x 1, x 2 ) x 1 x 2 z = 50x x 2 (1.1) (1.7) z = 50x x 2 (x 1, x 2 )
3 3 z = 50x x 2 3x 1 + 2x x x x x 1 3x 2 0 2x 1 x 2 0 x 1 0 x 2 0 (1.8) z (x 1, x 2 ) (1.8) (1.1) (1.7) x 1 x x 1 +65x 2 = z z (2.0, 1.5) / ,000 1, [2]
4 4 x 1 (2.0, 1.5) The cost increases. O x 2 1.1: 1,000 [1] C. Rorres and H. Anton ( : ), (1980) [2],, (1995).
5 5 1.2 c 1, c 2,..., c n x 1, x 2,..., x n f(x 1, x 2,..., x n ) = c 1 x 1 + c 2 x c n x n (linear function) f b f(x 1, x 2,..., x n ) = b (linear equality) f(x 1, x 2,..., x n ) b, f(x 1, x 2,..., x n ) b (linear inequality) (linear programming problem; abbr, LP) (x 1, x 2,..., x n ) c 1 x 1 + c 2 x c n x n a (P) i1 x 1 + a i2 x a in x n = b i, i = 1,..., k a i1 x 1 + a i2 x a in x n b i, i = k + 1,..., l a i1 x 1 + a i2 x a in x n b i, i = l + 1,...,m c 1 x 1 + c 2 x c n x n (P) (objective function) (x 1, x 2,..., x n ) m (linear constraint) (feasible solution) a i1 x 1 + a i2 x a in x n = b i, i = 1,..., k D = x R n a i1 x 1 + a i2 x a in x n b i, i = k + 1,..., l a i1 x 1 + a i2 x a in x n b i, i = l + 1,...,m
6 6 (feasible region) x 1 65x 2 3x 1 + 2x x x x (1.9) x 1 3x 2 0 2x 1 x 2 0 x 1 0 x (x 1, x 2 ) = (1.5, 3.0), (1.5, 2.25), (2.0, 1.5), (5.0, 5.0) x = (x 1, x 2..., x n ) D (1.8) (optimal solution) 1.1 (1.9) (x 1, x 2 ) = (2.0, 1.5) (1.9) x 1 + x 2 + x 3 2x 1 + 2x 2 x 3 6 2x 1 + 4x 3 4 4x 1 + 3x 2 x 3 1 x 1 0, x 2 0, x 3 0 (1.10)
7 7 3x 1 + x 2 + 5x 3 + 2x 4 4x 1 + 2x 2 4x 3 + x 4 = 11 2x 1 + 3x 2 2x 3 4x x 1 2x 2 + 3x 4 1 x 3 0, x 4 0 (1.11) x 1 + 5x 2 x 1 + x 2 6 (1.12) x 1 x 2 4 D = {(x 1, x 2 ) x 1 + x 2 6, x 1 x 2 4} (infeasible) (feasible) 2x 1 x 2 x 1 + x 2 6 x 1 3x 2 4 (1.13) x 2 x 1 x 1 + M > M < M (unbounded) (bounded)
8 [ ] x 1 x 1 < 5 (1.14) x 1 = 5 exp(x 1 ) x 1 0 (1.15) 1.3 [3] Markowitz Nash (polynomial-time algorithm)
9 9 Gordan (1873) Fourier (1923) Farkas (1923) 20 Leontief (1936) Motzkin (1923) Von Neuman & (1947) Dantzig (1947) Morgenstern (1944) Koopmans (1948) Von Neuman (1947) (1975) Koopmans Kantrovich ( ) Khachian (1979) (1990) Markowitz Sharpe Miller Karmarkar (1984) ( ) (1994) Nash ( ) [3] V. Chvátal ( : ), ( ), (1986).
10 (a) (b) (c) s, t x 1 + x 2 sx 1 + tx 2 1 x 1, x c 1 x 1 + c 2 x c n x n (P) a i1 x 1 + c i2 + + a in x n b i, i = 1,..., m k x k (P k ) a i1 x 1 + c i2 + + a in x n b i, i = 1,..., m 1.3 x 1 = (x 1 1, x1 2,..., x1 n ), x2 = (x 2 1, x2 2,..., x2 n ) x 3 = (1 λ)x 1 + λx 2, λ 0 λ (transportation problem) 40, 20, 40 25, 10, 20, 30, 15 /
11 (product mix problem) M1, M2, M3 R1, R2, R3, R4 1 kg R1 R2 R3 R4 ( ) M1 20 % M1 20 % M M2 M3 35 % 45 % M ( /kg)
12 12 2 LP ( ) (simplex method) 2.1 LP (canonical form) LP : c 1 x 1 + c 2 x c n x n a i1 x 1 + a i2 x a in x n b i, i = 1,..., m x 1 0, x 2 0,... x n 0 LP x 1, x 2,..., x n 1.2 (1.10) 2x 1 + x 2 + x 3 2x 1 + 2x 2 x 3 6 2x 1 + 4x 3 4 4x 1 + 3x 2 x 3 1 x 1 0, x 2 0, x 3 0 (2.1)
13 13 (1.9) (1.11) : a i1 + + a in x n = b i = a i1 x a in x n b i a i1 x 1 a in x n b i x j = x j : x j = x j x j 0, x j 0 : a i1 x a in x n b i = a i1 x 1 a in x n b i c 1 x c n x n = c 1 x 1 c n x n 2.2 LP (2.1) 2x 1 + x 2 + x x 1 2x 2 + x x 1 4x x 1 3x 2 + x 3 x 1, x 2, x 3 0 z = 2x 1 + x 2 + x 3
14 14 x 4, x 5, x 6 LP z x 4 = 6 2x 1 2x 2 + x 3 x 5 = 4 2x 1 4x 3 x 6 = 1 + 4x 1 3x 2 + x 3 z = 0 + 2x 1 + x 2 + x 3 x 1, x 2, x 3, x 4, x 5, x 6 0 (2.2) (2.1) (2.2) (2.1) (x 1, x 2, x 3 ) (2.2) x 4, x 5, x 6, z x 4, x 5, x 6, z (x 1, x 2, x 3, x 4, x 5, x 6, z ) (2.2) (2.2) (x 1, x 2, x 3, x 4, x 5, x 6, z ) x 4, x 5, x 6 x 5, x 6 (2.1) (x 1, x 2, x 3 ) x 4, (2.1) (2.2) (2.2) x 1, x 2, x 3 x 1, x 2, x 3 x 4, x 5, x 6, z (x 1, x 2, x 3, x 4, x 5, x 6, z) = (0, 0, 0, 6, 4, 1, 0) (2.3) x 1, x 2, x 3, x 4, x 5, x 6 (2.3) z
15 15 (2.3) x 1, x 2, x 3 (2.3) x 1 t x 4 = 6 2t x 5 = 4 2t x 6 = 1 + 4t z = 0 + 2t t = 1 : x 4 = 4, x 5 = 2, x 6 = 5, z = 2 t = 2 : x 4 = 2, x 5 = 0, x 6 = 9, z = 4 t = 3 : x 4 = 0, x 5 = 2, x 6 = 13, z = 6 x 1 t z z x 1 (= 2) x 1 t 2 (x 1, x 2, x 3, x 4, x 5, x 6, z) = (2, 0, 0, 2, 0, 9, 4) (2.4) x 1 (2.4) x 1 > 0, x 5 = 0 (2.3) x 1 x 5 (2.2) x 5 = 4 2x 1 4x 3 x 1 x 4 = 2 + x 5 2x 2 + 5x 3 x 1 = 2 1/2x 5 2x 3 (2.5) x 6 = 9 2x 5 3x 2 7x 3 z = 4 x 5 + x 2 3x 3
16 16 (2.5) (2.2) LP (2.4) (2.5) z x 5, x 2, x 3 z x 2 x 2 2/2 = 1 x 4 (x 1, x 2, x 3, x 4, x 5, x 6, z) = (2, 1, 0, 0, 0, 6, 5) (2.6) x 2 x 4 x 2 = 1 + 1/2x 5 1/2x 4 + 5/2x 3 x 1 = 2 1/2x 5 2x 3 x 6 = 6 7/2x 5 3/2x 4 29/2x 3 z = 5 1/2x 5 1/2x 4 1/2x 3 (2.7) ( 3.2) b i
17 17 x 3 1 X (0, 0, 0) 1/3 x 2 x 1 2 (2, 0, 0) (2, 1, 0) 2.1: (2.1) X (2.1) x 1 x 2 x 3 LP 2.1 m + n n (n = 2 n = 3 ) (polyhedron) n (vertex) LP (1.11) LP
18 LP c 1 x 1 + c 2 x c n x n a 11 x 1 + c 12 x a 1n x n b 1... a m1 x 1 + c m2 x a mn x n b m x 1, x 2,,..., x n 0 (2.8) (2.1) b i 0, i = 1,..., m (2.9) (2.1) (2.2) (2.8) z x n+1 = b 1 a 11 x 1 a 12 x 2 a 1n x n.. x n+m = b m a m1 x 1 a m2 x 2 a mn x n z = c 0 + c 1 x 1 + c 2 x c m x n x 1, x 2,..., x n 0.. (2.10) c 0 = 0 x n+i (i = 1,..., m) (slack variable) (2.8) (2.10) (2.10) 1,..., n + m N(1) = 1, N(2) = 2,..., N(n) = n B(1) = n + 1, B(2) = n + 2,..., B(m) = n + m
19 19 (2.10) (E 1 ) : x B(1) = b 1 a 11 x N(1) a 12 x N(2) a 1n x N(n).... (E m ) : x B(m) = b m a m1 x N(1) a m2 x N(2) a mn x N(n) (2.11) (E m+1 ) : z = c 0 + c 1 x N(1) + c 2 x N(2) + + c m x N(n) (2.8) (dictionary) m + 1 x B(i), z (basic variable) n x N(j) (nonbasic variable) (2.11) x N(1) = x N(2) =... = x N(n) = 0 x B(1) = b 1, x B(2) = b 2,..., x B(m) = b m, z = c 0 (basic solution) (2.11) (feasible dictionary) (2.9) (2.8) b i (2.8) (2.11) ( ) a rs (1 r m, 1 s n) ( (r, s), (pivoting operation) procedure (r, s) begin (E r ) x N(s) (E r ) ; for i r do (E r ) (E i ) x N(s) (E i ) ;
20 20 B(r) N(s) tmp := B(r); B(r) := N(s); N(s) := tmp end; procedure (2.8) b i LP algorithm : (2.11) N(1) = 1,..., N(n) = n, B(1) = n + 1,..., B(m) = n + m. begin stop := false; while stop = false do begin { 1: } if j c j 0 then stop := true { } else begin { 2: } c s > 0 s (1 s n) ; if i a is 0 then stop := true { LP } else
21 21 begin { 3: } end end; end end x N(s) { b r bi = min a rs a is r ; } a is > 0, i = 1, 2,..., m procedure (r, s) (r, s) (2.11) algorithm 2.1. LP algorithm LP (2.9) (r, s) LP LP algorithm LP
22 LP (a), (b), (c) 3x 1 + 2x 2 + 4x 3 x 1 + x 2 + 2x 3 4 (a) 2x 1 + 3x 3 5 2x 1 + x 2 + 3x 3 7 x 1, x 2, x 3 0. (b) (c) 5x 1 + 6x 2 + 9x x 4 x 1 + 2x x x 4 5 x 1 + x 2 + 2x 3 + 3x 4 3 2x 1 + x 2 2x 1 + 3x 2 3 x 1 + 5x 2 1 2x 1 + x 2 4 4x 1 + x 2 5 x 1, x 2 0. x 1, x 2, x 3, x LP 2.4 (a) (b) 2.5
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