最適化手法 第5回 [3mm] 整数計画法 (5) [3mm]
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- はな わくや
- 5 years ago
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Transcription
1 5 (5) ( ) (5) / 60
2 Gomory Chvátal 2013 ( ) (5) / 60
3 3x 1 + 4x 2 x 1,x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z x 2 3x 1 +11x 2 = 66 ( ) 3 4 3x 1 x 2 = 12 ( ) (5) / 60 x 1
4 x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 R x 2 ( ) 3 4 ( ) (5) / 60 x 1
5 1 2 3 Gomory Chvátal 4 ( ) (5) / 60
6 () 3x 1 + 4x 2 x 1,x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z x 2 3x 1 +11x 2 = 66 ( ) 3 4 3x 1 x 2 = 12 ( ) (5) / 60 x 1
7 x 2 x 1 ( ) (5) / 60
8 x 2 x 1 () ( ) (5) / 60
9 x 2 ( ) 3 4 x 1 ( ) (5) / 60
10 x 2 ( ) 3 4 x 1 ( ) (5) / 60
11 x 2 ( ) 3 4 x 1 3x + 4x x 1,x x 1 x 2 12, 2x 1 + 5x 2 30, x 1 5, x 1 0, x 2 0, x 1, x 2 R ( ) (5) / 60
12 x 2 ( ) 3 4 x 1 3x + 4x x 1,x x 1 x 2 12, 2x 1 + 5x 2 30, x 1 5, x 1 0, x 2 0, x 1, x 2 R ( ) (5) / 60
13 x 2 ( ) 3 4 x 1 3x + 4x x 1,x x 1 x 2 12, 2x 1 + 5x 2 30, x 1 5, x 1 0, x 2 0, x 1, x 2 R ( ) (5) / 60
14 x 2 ( ) 3 4 ( ) (5) / 60 x 1
15 (cutting plane) ( ) (5) / 60
16 (cutting plane) ( ) (5) / 60
17 1 2 3 Gomory Chvátal 4 ( ) (5) / 60
18 1 2 (fractional cut) ( ) (5) / 60
19 () x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z x 1 0, x 2 0 x 1 Z, x 2 Z () () ( ) (5) / 60
20 (1) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z 1 2 3x 1 x x x 2 66 ( ) (5) / 60
21 (2) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z x 1 10x x x 2 66 ( ) (5) / 60
22 (3) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z 2 30x 1 10x x x x 1 + x ( ) (5) / 60
23 (4) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z 33 (x 1 1 ) 33x 1 + x x x ( ) (5) / 60
24 (5) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z x x x 1 + x ( ) (5) / 60
25 (6) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z (186/33 = 563) x x 1 33 ( ) (5) / 60
26 (7) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z x 1 5 ( ) (5) / 60
27 (1) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 R x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1 5, x 1, x 2 R ( ) (5) / 60
28 (2) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 R x 2 ( ) 3 4 x ( ) (5) / 60
29 (3) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1 5, x 1, x 2 R ( ) x 2 ( ) 3 4 x ( ) (5) / 60
30 1 2 3 ( ) 4 ( ) Gomory Chvátal ( ) (5) / 60
31 Gomory Chvátal Gomory Chvátal 4 ( ) (5) / 60
32 Gomory Chvátal 1 2 Gomory Chvátal ( ) ( ) (5) / 60
33 Gomory Chvátal Ralph Gomory Vašek Chvátal Ralph Gomory (1929 ) Vašek Chvátal (1946 ) ( ) (5) / 60
34 Gomory Chvátal Gomory Chvátal x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 Z x 2 3x 1 +11x 2 = 66 ( ) 3 4 3x 1 x 2 = 12 x 1 ( ) (5) / 60
35 Gomory Chvátal Gomory Chvátal 1 (1) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 0, x 2 0, x 1, x 2 R x 2 ( ) 3 4 ( ) (5) / 60 x 1
36 Gomory Chvátal Gomory Chvátal 1 (2) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1, x 2, s 1, s 2 0 s 2 = 0 x 2 ( ) 3 4 x 1 = 0 s 1 = 0 x 2 = 0 x 1, x 2 Z s 1, s 2 Z x 1 ( ) (5) / 60
37 Gomory Chvátal Gomory Chvátal 1 (3) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1, x 2, s 1, s 2 0 s 2 = 0 x 2 ( ) 3 4 x 1 = 0 s 1 = 0 x 2 = 0 x 1 = 11 2, x 2 = 9 2, s 1 = 0, s 2 = 0 2 () x 1 ( ) (5) / 60
38 Gomory Chvátal Gomory Chvátal 1 (4) Gomory Chvátal 0 ( ) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1, x 2, s 1, s 2 0 x 1 s 1, s x 1 11x s 1 = x x 2 + s 2 = 66 36x s 1 + s 2 = 198 x 1 = s s 2 ( ) (5) / 60
39 Gomory Chvátal ( ) (5) / 60 Gomory Chvátal 1 (5) x 1 + x s s x 1 = s s 2 36 s 2 = 11 2 s x x 1 x
40 Gomory Chvátal Gomory Chvátal 2 (1) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 5, x 1 0, x 2 0, x 1, x 2 R s 2 = 0 s x 3 = 0 2 ( ) 3 4 x 1 = 0 s 1 = 0 x 2 = 0 ( ) (5) / 60 x 1
41 Gomory Chvátal Gomory Chvátal 2 (2) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1 + s 3 = 5, x 1, x 2, s 1, s 2, s 3 0 s 2 = 0 s x 3 = 0 2 ( ) 3 4 x 1 = 0 s 1 = 0 x 2 = 0 x 1 x 1, x 2 Z s 1, s 2, s 3 Z ( ) (5) / 60
42 Gomory Chvátal Gomory Chvátal 2 (3) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1 + s 3 = 5, x 1, x 2, s 1, s 2, s 3 0 s 2 = 0 s x 3 = 0 2 ( ) 3 4 x 1 = 0 s 1 = 0 x 2 = 0 x 1 x 1 = 5, x 2 = 51 11, s 1 = 18 11, s 2 = 0, s 3 = 0 x 2 s 2, s 3 ( ) (5) / 60
43 Gomory Chvátal Gomory Chvátal 2 (4) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1 + s 3 = 5, x 1, x 2, s 1, s 2, s x 2 = 66 3x 1 s x 2 = 66 3(5 s 3 ) s 2 = 51 s 2 + 3s 3 x 2 = s s 3 x s s 3 = ( ) (5) / 60
44 Gomory Chvátal Gomory Chvátal 2 (5) x s s 3 = x 2 + s s x 2 s x 2 s 3 11 x 2 s 3 4 ( ) (5) / 60
45 Gomory Chvátal Gomory Chvátal 2 (6) x 2 s 3 4 x 2 (5 x 1 ) 4 x 1 + x 2 9 x 1 + s 3 = 5 ( ) (5) / 60
46 Gomory Chvátal Gomory Chvátal 3 (1) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 5, x 1 + x 2 9, x 1 0, x 2 0, x 1, x 2 R x 2 s 2 = 0 s 4 = 0 s 3 = 0 ( ) 3 4 x 1 = 0 s 1 = 0 x 2 = 0 ( ) (5) / 60 x 1
47 Gomory Chvátal Gomory Chvátal 3 (2) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1 + s 3 = 5, x 1 + x 2 + s 4 = 9, x 2 s 2 = 0 s 4 = 0 s 3 = 0 ( ) 3 4 x 1 = 0 s 1 = 0 x 2 = 0 x 1, x 2, s 1, s 2, s 3, s 4 0 x 1 x 1, x 2 Z s 1, s 2, s 3, s 4 Z ( ) (5) / 60
48 Gomory Chvátal Gomory Chvátal 3 (3) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1 + s 3 = 5, x 1 + x 2 + s 4 = 9, x 2 s 2 = 0 s 4 = 0 s 3 = 0 ( ) 3 4 x 1 = 0 s 1 = 0 x 2 = 0 x 1, x 2, s 1, s 2, s 3, s 4 0 x 1 x 1 = 33 8, x 2 = 39 8, s 1 = 33 8, s 2 = 0, s 3 = 7 8, s 4 = 0 x 2 s 2, s 4 ( ) (5) / 60
49 Gomory Chvátal Gomory Chvátal 3 (4) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1 + s 3 = 5, x 1 + x 2 + s 4 = 9, x 1, x 2, s 1, s 2, s 3, s x x 2 = 66 s x 1 + 3x 2 = 27 3s 4 8x 2 = 39 s 2 + 3s 4 8 x s s 4 = 39 8 ( ) (5) / 60
50 Gomory Chvátal Gomory Chvátal 3 (5) x 2 + x s s 4 = s s x 2 s x 2 s 4 8 x 2 s 4 4 ( ) (5) / 60
51 Gomory Chvátal Gomory Chvátal 3 (6) x 2 s 4 4 x 2 (9 x 1 x 2 ) 4 x 1 + 2x 2 13 x 1 + x 2 + s 4 = 9 ( ) (5) / 60
52 Gomory Chvátal Gomory Chvátal 4 (1) x 1,x 2 3x 1 + 4x 2 3x 1 x 2 12, 3x x 2 66, x 1 5, x 1 + x 2 9, x 1 + 2x 2 13, x 1 0, x 2 0, x 1, x 2 R x 2 s 2 = 0 s 4 = 0 s 3 = 0 ( ) 3 4 x 1 = 0 s 1 = 0 s 5 = 0 x 2 = 0 ( ) (5) / 60 x 1
53 Gomory Chvátal Gomory Chvátal 4 (2) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1 + s 3 = 5, x 1 + x 2 + s 4 = 9, x 1 + 2x 2 + s 5 = 13, x 2 s 2 = 0 s 4 = 0 s 3 = 0 ( ) 3 4 x 1 = 0 s 1 = 0 s 5 = 0 x 2 = 0 x 1, x 2, s 1, s 2, s 3, s 4, s 5 0 x 1 x 1, x 2 Z s 1, s 2, s 3, s 4, s 5 Z ( ) (5) / 60
54 Gomory Chvátal Gomory Chvátal 4 (3) x,s 3x 1 + 4x 2 3x 1 x 2 + s 1 = 12, 3x x 2 + s 2 = 66, x 1 + s 3 = 5, x 1 + x 2 + s 4 = 9, x 1 + 2x 2 + s 5 = 13, x 2 s 2 = 0 s 4 = 0 s 3 = 0 ( ) 3 4 x 1 = 0 s 1 = 0 s 5 = 0 x 2 = 0 x 1, x 2, s 1, s 2, s 3, s 4, s 5 0 x 1 x 1 = 5, x 2 = 4, s 1 = 1, s 2 = 7, s 3 = 0, s 4 = 0, s 5 = 0 ( ) (5) / 60
55 Gomory Chvátal Gomory Chvátal (Gomory 1958, 1963) Gomory Chvátal ( ) Gomory Chvátal ( ) (5) / 60
56 Gomory Chvátal 2 ( ) (5) / 60
57 1 2 3 Gomory Chvátal 4 ( ) (5) / 60
58 Gomory Chvátal ( ) (5) / 60
59 ( ) (5) / 60
60 1 2 3 Gomory Chvátal 4 ( ) (5) / 60
No. 1261 2003. 4. 9 14 14 14 14 15 30 21 19 150 35 464 37 38 40 20 970 90 80 90 181130 a 151731 48 11 151731 42 44 47 63 12 a 151731 47 10 11 16 2001 11000 11 2002 10 151731 46 5810 2795195261998 151731
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š ( š ) 812,488 8,633,171 390,374,410 324,279,452 9,953,269 17,329,976 2,944,796 2,944,796 6,866,917 341,279,452 12,000,000 12,000,000 2,000,000 2,000,000 1,000,000 1,000,000 500,000 600,000 I 1,000,000
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š š ( œ ) 170,000,000 23. 2.14 9,925,843 525,843 9,400,000 9,896,851 9,000,000 9,000,000 400,000 400,000 7,809,518 2,850,232 2,285,616 2,673,670 2,087,333 36,000 1,273,004 373,800 211,050 162,750 40,529
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šœ Ÿ ( Ÿ ) Ÿ 3,658,819,708 612,940,933 1,441,054,976 1,536,693,282 369,033,491 1,167,659,791 68,105,057 25,460 7,803,540,263 1,713,934,550 541,531,413 702,848,302 11,827 1,552,629,488 23,421,737,374 2,572,144,704
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š š ( œ ) 80,000 120,000 100,000 120,000 120,000 80,000 100,000 180,000 60,000 100,000 60,000 120,000 100,000 240,000 120,000 290,000 240,000 100,000 120,000 72,000 300,000 72,000 100,000 100,000 60,000
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( ) 500,000 500,000 320,000 300,000 1,000,000 1,140,000 1,500,000 560,000 640,000 400,000 240,000 600,000 400,000 780,000 300,000 300,000 1,500,000 260,000 420,000 400,000 400,000 300,000 840,000 1,500,000
( ) 3,000,000 1,437, ,003,853 9,003,853 9,003,853 9,000,000 9,000,000 3,853 3,853 8,021,583 4,000, ,302 3,670, ,270
( ) 800,000 300,000 460,000 1,440,000 500,000 1,500,000 1,200,000 400,000 21 900,000 21 1,500,000 820,000 720,000 760,000 400,000 300,000 600,000 1,000,000 440,000 600,000 1,500,000 1,000,000 400,000 980,000
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