min. z = 602.5x x 2 + 2

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1 ? 1 LP (intger programming problem) x 1 = / /

2 min. z = 602.5x x x x x x x 7 s.t. 24x x x x x x x 7 = x 1 + 5x x x x x x x x x x x x x x j 0, j = 1,..., 7 (4.1) 1 x 1 x 3 x 5 x 7 x 2 x 4 x 6 x 1 = x 2 = 0, x 3 = 19.4, x 4 = 14.3, x 5 = 13.1, x 6 = x 7 = 0, z = ( ) x j 250, j = 1,..., 7 x 1 = 0.1, x 2 = 0, x 3 = 17.6, x 4 = 13.9, x 5 = 6.9, x 6 = 6.0, x 7 = 0, z =

3 ( 250 ) x j /? LP 128 LP / x j / z j (j = 1,..., 7) 0 1 z j = 0 j z j = 1 j / (z j = 1 or 0) 24x 1 1 z 1 = 0 24x 1 = 0 z 1 = x z 1 24x 1 250z 1 / min. z = 602.5x x x x x x x 7 s.t. 24x x x x x x x 7 = x 1 + 5x x x x x x x x x x x x x z 1 24x 1 250z 1, 100z 2 12x 2 250z 2 100z 3 14x 3 250z 3, 100z 4 18x 4 250z 4 100z 5 36x 5 250z 5, 100z 6 42x 6 250z 6 100z 7 24x 7 250z 7 x j 0, j = 1,..., 7 z j {0, 1}, j = 1,..., 7 (4.2)

4 ( ) z 1 = 1 z 2 = 0 z 3 = 1 z 4 = 1 z 5 = 1 z 6 = 1 z 7 = 0 x 1 = 4.2 x 2 = 0 x 3 = 17.9 x 4 = 13.9 x 5 = 6.9 x 6 = 3.6 x 7 = maximize subject to n j=1 c jx j n j=1 a ijx j = b i, x j 0, i = 1,..., m (1.5) maximize subject to n j=1 c jx j n j=1 a ijx j = b i, x j 0, x j Z, i = 1,..., m (4.3) ( ) ((mixed) integer programming problem) 2 Z (1.5) (4.3) (1.5) x (4.3) x c x c x (1.5) (4.3) x x maximize x 1 + x 2 + x 3 subject to 0.27x x 2 + 3x x x x 1 + x 2 x 3 1 x j 0, j = 1,..., 3 (4.4) x 1 = 2.80, x 2 = 1.51 x 3 = ( )

5 ( 3 ) (4.4) maximize x 1 + x 2 + x 3 subject to 0.27x x 2 + 3x x x x 1 + x 2 x 3 1 x j 0, j = 1,..., 3 x j Z, j = 1,..., 3 (4.5) x 1 = 3, x 2 = 0 x 3 = 0 x x = x LP IP [0, 1] [1, 2] 10 4, OS: Linux CPU: Intel Pentium GHz lp solve (Version 4.0) ( ) IP LP (branch and bounding method) 3 x x IP 1

6 LP IP ( )

7 LP IP ( ) 1400 LP IP time(sec) n max c x s.t. Ax = b P 0 l 0 j x j u 0 j, (4.6) x Z n P 0 Z n n l u P 0 P 0 P 0 P 0 (4.7) P 0 s.t. Ax = b max c x l 0 j x j u 0 j, P 0 P 0 P 0 P 0 P 0 P 0 (x 0 1,..., x0 n) x 0 s ξ ξ ( )

8 112 4 P 0 P 1 P 1 x 0 s 2 P 1 max s.t. c x Ax = b l 0 j x j u 0 j, l s x s x 0 s x Z n ; j s P 2 max s.t. c x Ax = b l 0 j x j u 0 j, ; j s x 0 s + 1 x s u 0 s x Z n P 1 P 2 P 0 P 1 P 2 P 0 P 1 P 2 ( ) P j P j 3 case 1) P j case 2) P j case 3) P j x j ^x c x j c ^x P j c x j > c ^x ( )

9 P1 P11: P0 P12 P121: P2:»» P122:»ˆ Œ ^x x c ^x c x ɛ ɛ 2 max 5x 1 + 2x 2 s. t. 6x 1 + 2x x 1 + 4x 2 15 x 1 + 2x 2 5 x 1, x 2 0 x 1, x 2 Z (4.8) (4.8) P 0 max 5x 1 + 2x 2 s. t. 6x 1 + 2x x 1 + 4x 2 15 x 1 + 2x 2 5 x 1, x 2 0 (4.9) P 0 (x 0 1, x0 2 ) ( x 0 1 x 0 2) = ( ) z 0 = (4.8)

10 (4.8) 3 2 x x 1 P 0 P 1 P 2 x 0 1 = x 2 x 1 1 max 5x 1 + 2x 2 max 5x 1 + 2x 2 s. t. 6x 1 + 2x 2 15 s. t. 6x 1 + 2x x 1 + 4x x 1 + 4x 2 15 P 1 P 2 (4.10) x 1 + 2x 2 5 x 1 + 2x 2 5 x 1 2 x 1 1 x 1, x 2 0 x 1, x 2 0 P 1 P 1 : ( ) ( ) x 1 1 x 1 2 = z 1 = 13 P 2 P 1 P 11 P 12 x 1 2 = 1.5 P 1

11 P 1 P x 2 1 P 2 P x 1 x 2 2 x 2 1 max 5x 1 + 2x 2 s. t. 6x 1 + 2x x 1 + 4x 2 15 P 11 x 1 + 2x 2 5 x 1 2 x 2 2 x 1, x 2 0 P 12 max 5x 1 + 2x 2 s. t. 6x 1 + 2x x 1 + 4x 2 15 x 1 + 2x 2 5 x 1 2 x 2 1 x 1, x 2 0 (4.11) P 11 P 12 P 12 : ( ) ( ) x 12 1 x 12 2 = z 12 = P 11 P 12 P 121 P 122

12 116 4 P 121 max 5x 1 + 2x 2 s. t. 6x 1 + 2x x 1 + 4x 2 15 x 1 + 2x 2 5 x 1 3 x 2 1 x 1, x 2 0 P 122 max 5x 1 + 2x 2 s. t. 6x 1 + 2x x 1 + 4x 2 15 x 1 + 2x x 1 2 x 2 1 x 1, x 2 0 (4.12) P 121 P 122 P 122 : ( ) ( ) x x = 2 1 z 122 = 12 ( 2 1 ) P 2 P 2 : ( x 2 1 x 2 2) = ( ) z 2 = 10.5 z 2 = P 2 ( 2 1 ) (4.8) 4.2 P 0 x 0 1 x ( ) ( )

13 j (j = 1,..., 4) 0-1 x j x j = 1 j x j = 0 max 100x x x x 4 s.t. 70x x x x x j {0, 1}, j = 1,..., 4 (4.13) (4.13) (napsack problem) (4.13) ( 4.9) max 100x x x x x x 6 s.t. 70x x x x x x x x x 1 + x 5 1, x 4 + x 6 1 x 1 x 7 0 x 2 x 7 0 x 5 x 7 0 x 3 x 8 0 x 4 x 8 0 x 6 x 8 0 x j {0, 1}, j = 1,..., 8 (4.14)

14 118 4 (4.13) (4.14) (x 1, x 2, x 3, x 4 ) = (1, 0, 0, 1) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8 ) = (1, 0, 1, 0, 0, 1, 1, 1) ( ) ( ) A 20 B 18 C 32 D 9 E 22 5 (greedy method)

15 {1 x j (j A, B, C, D, E) min 20x A + 18x B + 32x C + 9x D + 22x E s.t. x A + x C + x D + x E 1 x A + x B 1 x C + x E 1 x A + x C + x E 1 x C + x D + x E 1 x A + x C 1 x A + x D + x E 1 x B + x C + x D 1 x j {0, 1}, j {A, B, C, D, E} x A = x D = x E = 1 x B = x C = (69 ) LP LP (set covering problem) A , B C D E F

16 kg 1kg 500g 200g 6 (4.13) x 1 2 x 2 U(x 1, x 2 ) max U(x 1, x 2 ) s.t. 20x x x 1 = 100z 1 x 2 = 80z 2 x j 0 j = 1, 2 z j Z j = 1, max U(x 1, x 2 ) s.t. 20x x f f x 1 = 100z 1 x 2 = 80z 2 z j Mf j, j = 1, 2 x j 0, j = 1, 2 z j Z, j = 1, 2 f j {0, 1}, j = 1, 2 M ( M 50 ) z 1 = 0 f 1 =

17 d 3 d 2 transacton cost d 1 q q q quantity: x x j v j L ( )c j p j min s.t. n j=1 p jx j n j=1 c jx j + L x j 0, q 3 j y j x j 0-1 z j0 z j1 z j2 λ j0 λ j1 λ j2 λ j3 6 7

18 122 4 min s.t. n j=1 p jx j + n j=1 y j n j=1 c jx j + L x j = q 1 λ j1 + q 2 λ j2 + q 3 λ j3, y j = d 1 λ j1 + d 2 λ j2 + d 3 λ j3, λ j0 + λ j1 + λ j2 + λ j3 = 1, λ 0j z 1j, λ 1j z 1j + z 2j, λ 2j z 2j + z 3j, λ 3j z 3j, z 1j + z 2j + z 3j = 1, x j 0, y j 0, λ 0j, λ 1j, λ 2j, λ 3j 0, z 1j, z 2j, z 3j {0, 1}, (2 ) 8

19 (1 0 ) 7/5 Sun. 7/6 Mon. 7/18 Sat A B C D E W X A { 1 B { 1 C { { D { { E { {.. W { X { i j

20 124 4 x ij (i = A, B,, X; j = 1, 2,..., 70) x i,j 0-1 x ij = { 1, i j 0, i j (4.15) x A1 = A A x A4 = 0, x A5 = A x A1 + 4x A2 + 5x A3 + 7x A4 + 3x A5 + 5x A6 + 4x A x A69 + 3x A x A1 + 4x A2 + 5x A3 + 7x A4 + 3x A5 + 5x A6 + 4x A x A69 + 3x A70 70 (4.16) A x A1 + 4x A2 + 5x A3 12, 4x A2 + 5x A3 + 7x A4 12, 5x A3 + 7x A4 + 3x A5 12, 7x A4 + 3x A5 + 5x A6 12, 3x A5 + 5x A6 + 4x A7 + 5x A8 12 5x A6 + 4x A7 + 5x A8 12 (4.17). 4x A67 + 5x A68 + 7x A69 12, 5x A68 + 7x A69 + 3x A70 12 A 12 A 7 5 (1 x A2 ) + (1 x A3 ) + (1 x A4 ) 3(x A1 x A2 ) (x A1 = 1) (x A2 = 0) 3 (x A2 = x A3 = x A4 = 0) (x A1 = 1) (x A2 = 1) 0 8 x A4 x A5

21 (1 x A2 ) + (1 x A3 ) + (1 x A4 ) 3(x A1 x A2 ) (1 x A3 ) + (1 x A4 ) 2(x A2 x A3 ) (1 x A4 ) + (1 x A5 ) + (1 x A6 ) 3(x A3 x A4 ). (1 x A68 ) + (1 x A69 ) 2(x A67 x A68 ) (4.18) 3 A X (4.16) (4.17) (4.18) W W A B G H x W1 x A1 + x B1 + x G1 + x H1,. (4.19) x W70 x A70 + x B70 + x G70 + x H70 X x A1 + x B1 + + x X1 2 x A2 + x B2 + + x X2 3 x A70 + x B x X70 2 (4.20) M 50(5x A1 + 4x A2 + 5x A3 + 7x A4 + 3x A5 + 5x A6 + 4x A x A69 + 3x A70 ) + 80(5x B1 + 4x B2 + 5x B3 + 7x B4 + 3x B5 + 5x B6 + 4x B x B69 + 3x B70 ) 50(5x X1 + 4x X2 + 5x X3 + 7x X4 + 3x X5 + 5x X6 + 4x X x X69 + 3x X70 ) M (4.21) = ( ) = 3499

22 A 4.6 A 12

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