Файл: Ху, Т. Целочисленное программирование и потоки в сетях.pdf
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ПРИЛОЖЕНИЕ D |
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Р (Gil. (Ю)) |
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Грани |
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Строка^^\^ |
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7з |
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7б |
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7*о |
7о |
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16 |
30 |
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Вершины Р (Gji, (10)) |
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1. |
(h) |
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= |
(10) |
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18. |
(hi |
^2» h) = |
(h |
1, |
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2. |
(h) |
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= |
(5) |
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19. |
(hi |
h) |
= |
(1. |
1) |
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3. |
(hi |
h) |
= |
(2) |
2) |
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20. |
(h) |
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= |
( 3 ) |
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4. |
(hi |
*з ) |
= |
(1, |
3 ) |
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21. |
(tii |
ta) |
= |
(2, |
1) |
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5. |
(h) |
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= |
(7) |
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22. |
(hi |
ta) |
= |
(1. |
1) |
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6. |
(hi |
h) |
= |
(2, |
1) |
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23. |
(tei |
te) |
= |
( 4 , |
1) |
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7. |
(tu |
h) |
= |
(2, |
2) |
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24. |
(te< |
hi |
tg) = |
(1, |
1, |
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8. |
(tii |
G) |
= |
(1. |
2) |
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25. |
(G) |
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= |
(4) |
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9. |
(h) |
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= |
(8) |
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26. |
(hi |
h) |
= |
(1. |
1) |
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10. |
(h) |
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= |
(2) |
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27. |
(hi |
h) |
= |
(3, |
1) |
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И. |
(G> |
te) |
= |
( 4 , |
1) |
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28. |
(tei |
h) |
= |
(2, |
1) |
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12. |
(hi |
te) |
= |
(2, |
1) |
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29. |
(ti, |
t8i |
h) = |
(1. |
1, |
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13. |
(h, |
tei |
t^) = |
(1, |
1, 1) |
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30. |
(hi |
te) |
= |
(1. |
2) |
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14. |
(tii |
te) |
= |
(1. |
1) |
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31. |
(te i |
h) |
= |
(1. |
3 ) |
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15. |
(t3i |
{e) |
= |
(1. |
3 ) |
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32. |
(te) |
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= |
(6) |
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16. |
(te) |
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= |
(9) |
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33. |
(Go) |
= |
( 1 ) |
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17. |
(tii |
h) |
= |
(3, |
1) |
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Матрица инциденций Р (G1(, (10))
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Г р а н ь |
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В ер ш и н а |
1 |
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0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
2 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
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1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
3 |
1 |
0 1 |
0 |
0 |
0 |
0 0 |
1 0 |
0 |
0 |
0 0 |
0 0 |
0 0 |
1 |
О 0 1 1 1 1 1 1 1 |
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1 |
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1 |
0 |
0 |
0 |
0 |
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0 |
0 |
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1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
5 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
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0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
6 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
7 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
8 |
1 0 |
0 |
0 |
1 1 0 |
1 1 0 |
0 |
0 |
1 0 |
1 0 |
1 |
1 1 |
О 1 0 1 1 1 1 1 1 |
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9 |
0 |
0 |
0 |
0 |
1 |
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0 |
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1 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
10 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
11 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
12 |
1 0 |
1 0 |
0 |
0 1 0 |
1 |
0 |
1 0 |
0 0 0 |
0 1 |
1 1 |
о 1 |
1 1 0 1 1 |
1 |
1 |
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13 |
1 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
14 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
0 |
1 |
1 |
1 |
1 |
15 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
16 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
17 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
18 |
1 0 |
0 |
0 |
0 |
0 |
1 1 0 |
0 |
0 |
0 |
0 0 0 0 |
0 |
1 0 |
о 1 1 1 1 0 1 1 1 |
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19 |
1 |
1 1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
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20 |
0 |
1 0 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
1 1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
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21 |
1 |
0 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
22 |
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
1 о 1 1 1 1 1 0 1 1 |
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23 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
0 |
1 |
1 |
24 |
0 |
1 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
1 |
1 |
25 |
0 |
1 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
26 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
27 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
0 |
1 |
28 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
29 |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
30 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
31 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
0 |
1 |
32 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
33 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
СПИСОК ЛИТЕРАТУРЫ
ACM: Association for Computing Machinery
NRLQ: Naval Research Logistics Quarterly
ORSA: Operations Research Society of America
SIAM: Society of Industrial and Applied Mathematics
1. |
Abadie J. M. (ed.) |
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Non-Linear Programming, North-Holland Publishing Co., Amsterdam, |
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1967. |
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2. |
Ackers J. R. |
Transformation |
in Network Simplification, |
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The Use of Wye-Delta |
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J. ORSA, 8 (3), 311—323 (1960). |
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3. |
Arrow K. J., Hurwicz L. and Uzawa H. |
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Studies in Linear and Nonlinear Programming, Stanford University |
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Press, Stanford, California, 1958. (Русский перевод: Эрроу К. Дж., |
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Гурвиц Л., Удзава X ., Исследования по линейному и нелинейному |
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программированию, ИЛ, М., 1962.) |
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4. |
Ralas Е. |
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An Additive Algorithm for Solving Linear Programs with Zero — One |
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Variables, J. ORSA, 13 (4), 517—546 (1965). (Русский перевод: |
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Балаш |
Э., Аддитивный |
алгоритм |
для |
решения |
задач линейного |
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программирования с переменными, принимающими |
значение 0 |
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|
или 1. Б «Кибернетическом сборнике», |
новая серия, вып. 6, |
изд-во |
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|
«Мир», М., 1969.) |
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5. |
Balas Е. |
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Duality in Discrete Programming, Graduate School of Industrial |
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Administration, Carnegie-Mellon University, Dec. |
1967. |
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6. |
Balinski M. L. |
Methods, |
Uses, |
Computation, |
Man. |
Sci., |
||
|
Integer |
Programming: |
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12(3), 253-313 (Nov. 1965).
7.Balinski M. L. and Comory R. E.
A Mutual Primal-Dual Simplex Method, in R.L. Graves and P. Wolfe (eds.), Recent Advances in Mathematical Programming, McGrawHill, New York, 1963, 17—26.
8. Beale E. M. L.
An Alternate Method of Linear Programming, Proc. Cambridge Phil. Soc., 50 (4), 513—523 (1954). (Русский перевод: Бил E., Альтер нативный метод линейного программирования. В сборнике «Методы решения общей задачи линейного программирования», М., Госстат-
издат, 1963.)
9. Beale Е. М. L.
Cycling in the Dual Simplex Algorithm, NRLQ, 2 (4), 269—276 (Dec. 1955).
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СПИСОК ЛИТЕРАТУРЫ |
497 |
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10. |
Beale Е. М. L. |
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A Method of Solving Linear Programming Problems When Some but |
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Not All of the Variables Must Take Integral Values, Statistical Tech. |
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Research Group, Princeton University, March 1958. |
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11. |
Beale E. |
M. L. |
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|
An Algorithm for Solving the Transportation Problem When the |
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Shipping Cost Over Each Route is Convex, NRLQ, 6 (1), 43—56 (March |
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1959). |
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12. |
Beale E. |
M. L. |
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On Quadratic Programming, NRLQ, 6 (3), 227—243 (sept. 1959). |
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13. |
Beale E. |
M. L. |
Pitman & Son, London, |
1968. |
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Mathematical Programming, Isaac |
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14. |
Beale E. M. L. and Small R. E. |
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Mixed Integer Programming by a Branch and Bound Technique Proc. |
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IFIP Congress, New York, 2 (May 1965).. |
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15. |
Bellman |
R. |
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N.J., |
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Dynamic Programming, Princeton University Press, Princeton, |
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1957. (Русский перевод: Веллман P., Динамическое программиро |
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вание, ИЛ, М., 1959.) |
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Bellman R. Е. and Dreyfus S. Е. |
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Applied Dynamic Programming, Princeton University Press, Prince |
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N .J., 1962. (Русский перевод: |
Веллман P., Дрейфус С., |
При |
кладные задачи динамического программирования, изд-во «Наука»,
М., 1965.)
17.Ben-Israel A. and Charnes А.
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18. |
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Benders |
J. F. |
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Partitioning Procedures for Solving Mixed. Variables Programming |
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Problems, |
Numerische |
Mathematik, |
4, |
238—252 |
(1962). |
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19. |
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Berge |
C. |
and Ghouila-LIouri A. |
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Networks (Translated by |
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Programming, |
Games |
and |
Transportation |
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M. Merrington |
and C. Ramanujacharyula), |
John W iley & Sons, New |
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York, |
1965. |
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20. |
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Blankinship |
W. A. |
the Euclidean Algorithm, Am. Math. Monthly, |
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A |
New |
Version of |
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70 |
(7), |
742—745 |
(1963). |
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21. |
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Boldyreff |
A. W. |
of the Maximal Steady State Flow of Traffic |
Through |
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Determination |
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a Railroad Network, J. ORSA, 3 (4), 443—465 (Nov. 1955). |
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22. |
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Busacker |
R. G. and Gowen P. J. |
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of Minimal-Cost Network |
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A Procedure for Determining a Family |
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Flow Patterns, ORO Technical Report 15, Operations Research Office, |
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Johns |
Hopkins |
University, |
1961. |
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23. |
Busacker R. G. and Saaty T. L. |
McGraw-Hill, New |
York, |
1964. |
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Finite |
Graphs |
and |
Networks, |
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24. |
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Charnes A. |
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and Degeneracy |
in |
Linear |
Programming, Econometrica, |
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Optimality |
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20 |
(2), |
160—170 |
(April |
1952). |
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25. |
Charnes A. and Cooper W. W. |
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Management Models and Industrial Applications of Linear Program |
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ming, |
John W iley |
& |
Sons, |
New |
York, |
1961. |
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V2 32 |
t . xy |
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498 |
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СПИСОК ЛИТЕРАТУРЫ |
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26. Charnes |
A., Cooper W. W. and Henderson A. |
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Wiley |
& Sons, New |
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An |
Introduction |
to Linear Programming, John |
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York, 1953. |
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27. |
Dakin R. J. |
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A Tree-Search Algorithm lor Mixed-Integer Programming Problems, |
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The Computer Journal, 8 (3), 250—255 (1965). |
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28. |
Dantzig |
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G. B. |
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Programming in a Linear Structure, Comptroller, USAF, Washington, |
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D.C., |
Feb. 1948. |
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29. |
Dantzig |
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G. B. |
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Maximization of a Linear Function of Variables Subject to Linear |
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Inequalities, in T. C. Koopmans (ed.), Activity Analysis of Production |
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and |
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Allocation, |
John W iley & |
Sons, |
New |
York, 1951, 339—347. |
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(Русский перевод: Данциг Дж. Б., Максимизация линейной функ |
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ции переменных, подчиненных линейным неравенствам. В сборнике |
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«Методы решения общей задачи линейного программирования» М., |
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Госстатиздат, |
1963.) |
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30. |
Dantzig |
G. В. |
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A Proof of the Equivalence of the Programming Problem and the |
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Game Problem, in T. C. Koopmans (ed.), Activity Analysis of Pro |
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duction |
and Allocation, |
John Wiley & Sons, |
New York, |
1951, 359— |
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373. |
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31. |
Dantzig |
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G. |
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B. |
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Notes on Linear Programming: Part VII. The Dual Simplex Algo |
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rithm, |
RAND Report RM-1270, July |
1954. |
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32. |
Dantzig |
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G. |
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B. |
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Upper Bounds, Secondary Constraints, and Block Triangularity in |
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Linear |
Programming, Econometrica, 23 |
(2), |
174—183 |
(April 1955). |
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33. |
Dantzig G. B. |
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Discrete Variable Extremum Problems, J. ORSA, 5 (2), 266—277 |
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(April 1957). |
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34. |
Dantzig |
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G. |
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B. |
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Solving |
Linear |
Programming Problems with |
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On |
the |
Significance of |
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Some |
Integer |
Variables, |
Econometrica, |
28, (1) |
30—44 |
(Jan. 1960). |
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35. |
Dantzig |
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G. |
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B. |
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On the Shortest Route Through a Network, Man. Sci., 6 (2), 187—190 |
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(Jan. |
I960). |
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36. |
Dantzig |
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G. |
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B. |
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J. Res. Develop., 4 (5), |
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Inductive Proof of the Simplex Method, IBM, |
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505—506 |
(Nov. |
1960). |
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37. |
Dantzig |
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G. |
B. |
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Linear Programming and Extensions, Princeton University Press, Princeton, N .J., 1962. (Русский перевод: Данциг Дж., Линейное программирование, его обобщения и применения, изд-во «Прогресс»,
М., 1966.)
38.Dantzig G. В.
All Shortest Routes in a Graph, Operations Research House, Stanford University Technical Report 66-3, Nov. 1966.
39. Dantzig G. B., Blanttner W. D. and Rao |
M. R. |
|
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All Shortest Routes from a Fixed Origin in a Graph, Operations Research |
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House, Stanford |
University |
Technical |
Report |
66-2, Nov. 1966. |
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40. Dantzig G. B., Ford |
L. R ., |
Jr., |
and Fulkerson D. |
R. |
|
A Primal-Dual Algorithm |
for |
Linear Programs |
in H. W. Kuhn and |
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