....... LP...........7....8....8.....7....8....9...8....8....8....9...............................................7....9
......... (ASSIGNMENT PROLEM)....... (INTEGER PROGRAMMING PROLEM).... (KNAPSACK PROLEM)...............8...................7...8...9......
Operao Reearch, OR 9-99 Operaoal Reearch A.P. 9 98 OR OR 9 9 NDRC (Naoal Defee Reearch Commee) 9 Operao Reearch 9 98 99 9 9 OR 97 p
Lear Programmg, LP. LP kg (obecve fuco) z (cora),
, z z z/. z. (mple mehod) z, (lack varable),, Sep z z [ ],,,, z,,,,,,, (bac varable) (o-bac varable)
, z / / < Sep z (pvo operao) (cycle) z z [ ],,,,,, z,, Sep z 9 z 9
,,,,,, z,, Sep z z 7 7,,,,,,,, z, z,, z,,. z - - () z - / / (/) -/ / -/ z -/ 9/ / -/ -/ / () - z 7/ / / -/ -/ / - 7
. A (,) A (,) (,) O C (,), Sep Sep (Cycle ),, O A Sep Sep (Cycle ),, A Sep Sep (Cycle ),, C. z c c L c 8
a a L a a a a L a L a L a m m m,, L, b, b, L, bm b b b m z c a L a L a c L c L c a a r a r L a L a r L a m m m m m,, L, m c c L c m L a L a r L a r c > b a ( a > ) r r r a r r b b a b r r r b bra ar r b b r b a > b a ( b a b a ) a b b r r a < b b z c c L L c c a c b r r c b a r r pvo m r a r 9
c z c > a z, 8 Sep z - - [] - - 8 z -/ / -/ / -/ / z,,, z z. z
,, z, w,,,,, ) w w - w
w z,,,, Sep -w - - - - z - - - [] -w -/ -/ / - z -/ -/ / / [/] -/ / / / -w z -7/ / / / / / -/ 8/ / -/ / / Sep z -7/ / / 8/ / / z 8 -,, z 8 z
,, Sep -w - - - - z - - - [] -w z - -* - -w z - - - Sep z z Sep a a -w - - - z - - - a a - -w -w
.7 z c c L c z c c L c a a L a b < a a b > a L,, a a L a b a a L a b z,, Sep a a7 -w - - -7 z - - - a - a7 -
z Sep - a -w - - - z - - a -.8 z, 8 Sep z - - [] 8 z -/ / / / [/] -/ z / / / / -/ / -/ / / ) z z ( ) (
8 8 c A b 8 c z A b, / / / / A / / 8 b b A c z ( ) z
/ / / A b / / / c c c A c b z c c ( / / ) 8 ( / / ) / A z ( c c z ( c c A) c b A) ( / /) Sep, c z ( c c A b A) c b c c N cn 7
b, N z c b.9. z, z y y y y y y y y, y, y y y z c z by A b,. z Ay c, y z y y y y y y, y z z y y y y y y, y y 8
z z y y y y y y, y, y z, z y y y y y y y y y z c c z by by A A A A b b A A y A y A y y c c y. ) z c z by A b, Ay c, y 9
z c ya yb by z z z y ) z c z by A b, Ay c z c ya yb by z z z y ) z c c ( y A y ( A y b y b y A A ) ( r y A ) y ( A b y b y y A A z ) ). z c z by A b, Ay c z ( c c A) c b A b () * () * A b ()
z* c b () c c A A c c y c () Ay A c c y z* c b yb by z z * z y z z z z z z y c ()... z c A b z by Ay c y A, y, Ay y y b c () ()
,, y, y y y () ya y yb z () Ay y c z z z y y z z y y () () z z ( Ay y z y ( z z) ( y c) Ay y y z y y ) c y z > z. z c A b z by Ay c ya yb z Ay y c z Ay y c () () () y ()
y. reduced co c* c c A c* c A dual prce y* c z * y *b c
.. (7) () () () 8 (9) () () (8) (8) (7) (8) () 8 ( ) c a b z c a b. (orh-we corer mehod) (7) () () () 8 (9) () () (8) 8 (8) (7) (8) ()
8 8 8 oal A A oal oal A A A oal (Houhakker mehod) (u pealy mehod). (7) () () () 8 (9) () () (8) 8 (8) (7) (8) () 8 8 8 888 (,)(,)(,)(,)(,)(,)(,) - - 8 -
Sep (7) () () () [-] 8 (9) () () (8) 8 (8) (7) (8) () - 8 8 8 (,) --[-] (,) --8- (,) 9--7 (,) 8-8- (,) 8-8--7- (,) 7-8- 8888 - (,) (,) (,) (,) Sep (7) () () () 8 (9) () () (8) - (8) (7) (8) () [-] 8 8 8 88 (-) (,) -- (,) -8- (,) 9--7- (,) 8-8- (,) 8-8-7[-] (,) 7-8- (,) 8 Sep (7) () () () 8 (9) () () (8)
[-] - (8) (7) (8) () 8 7 8 8 888 (-) (,) -- (,) -78- (,) 9--7[-] (,) 8--78-- (,) 7--7-87 (,) 8-7-8 (,) Sep (7) () () () 8 (9) () () (8) (8) (7) (8) () 8 8 8 88 (-) 9 (,) -79- (,) -78- (,) -7-9 (,) 8-98- (,) 7-9-8 (,) 8-7-8. u, v 7
ep u (7) () () () -- - -- 8 (9) () () (8) 9--7 8 8-- (8) (7) (8) () 8--7-7-- 8 8 8 v 7 8888 u, c c u v u, c c c c u v c ( u u v c c v ) ( u c c v ) ( u v v ) ( u v v ) ( u v ) Sep u (7) () () () --(-) -- 8 (9) () () (8) 9--7-8-- (8) (7) (8) () 8--7 [-] 7--(-) 8 8 8 v 7-88 (-) 8
Sep u (7) () () () --(-) -- 8 (9) () () (8) 9--7 [-] 8-- - (8) (7) (8) () 8 7--(-) 7 8-- 8 8 v 7-888 (-) Sep u (7) () () () -- -- 8 (9) () () (8) -- (8) (7) (8) () 8 7-- 8-- 8 8 v 7 88 (-) 9. c a b z m m c a b 9
z u m v a u c b v z z z z m m m a u ( c ( u v ) u b v v ) > c u v u, v z z u, v c u v < u, v, c u v oal poeal A c c a u A c c a u oal b b poeal v v z c a b c c c a z a u au bv bv b,,,
u v c u v c u u v c v c z z z ( a u a u b v b v ) z ( ( ( c ( c ) v u ) u ( ( v ) u v ) ( c ) v ( c ) u u v ) u v ). e e e e e e e (Vere) V V {,, L, } (edge)e E { e, e, e, e, e, e e } G G ( V, E), (dreced graph) (udreced graph) N N ( V, E, d, w) d : V R w: E R (, ),(, ),, (, ) k k k
.. (Agme Problem) c c c c c c z ) ( ) ( L L L ),,,, ( } {, ),,, ( ),,, ( L L L L L ) ( c C 8 C!. C
7 () C C () C,},, ma{ L c p },,, ma{ L c q coa ) ( z q p c q p c z z z YES NO () C () C,, -, -
[] [] [] [] [] C ) ( 8 z ) ( c C 7 7 9 8 7 9 C 7 9 7 8 7 8 () C 9 8 8 8 () C 8 9 7 7 9 () C,, -, -
[] [] 8 9 [] 7 [] 7 [] 9 C ) ( 9 7 7 9 z
. (eger programmg problem) (combaoral opmzao problem) z c A b A A b b. (kapack problem). z c c Lc a a L a b,,, L {, }. z 7
,, {, }, 8 c a c z 7 8,,, 8,, z 7 7 z 7 ma zma zm, z 7 8 8 z m z 8 z > zm z m z,, z a, z > zm z m 7
z z < zm z 7 8,,, z 7 < zm, z ma.. 7 8 7 8 7 8 8
z c (,,,, ) (,,,, ) {, } ( ) 7 C 8 7 8 C Ye X 9
8 7 8 7 C C X 7 8 z z m 8 7 8 7 C 9 C 8 C X 8 7 z m z 7 C
& & & & & z 7 C z 7 & C & & & & & & z z
D.P.: Dyamc Programmg. ( ) 7 8 8 z r ) r ( ) r ( ) r ( ) ( f (d) d f () r() () r() () r() f f 7 f ( ) f() r () f ( ) ma{ f() r (), f() r ()} ma{, } f ( ) ma{ f() r (), f() r (), f() r ()} ma{7,, 8} 9 f ( ) ma{ f() r (), f() r (), f() r ()} ma{9,, 8} f (), f ( ) f ( ) f (), f (),
d (, d) (d) (, d) f ( d) r ( ) r ( d) (, d) d f ( d) ma L d ma[ f d [ r ( ) L r ( d ) r ( )] ( ) r ( )] f 7,,, z 7 f (d) L d L d f () () () f f f () f ) f () (
f f f ( ) ma{ f (), f ()} ma{, } ( ) ma{ f (), f (), f ()} ma{,, } ( ) ma{ f (), f (), f (), f ()} ma{,,, } f ) f () f ( ( ) ma{ f(), f ()} ma{, } f ( ) ma{ f(), f(), f ()} ma{,, } f ( ) ma{ f (), f (), f(), f ()} ma{, 8,, } 8 f ( 7) ma{ f (), f(), f(), f()} ma{8, 8,, } 8, z f (7) 8 f f f f m{ f, f } m{, }
f m{ f, f } m{, 8} f m{ f, f, f } m{,, } f m{ f, f } m{, 8} 8 8 f f m { f r }
z, 8 Sep Sep z z [ ] 8 [ ] Sep z, z Sep z - - [] 8 z -/ / / / [/] -/ z / / / / -/ / -/ / /
z z,,,, Sep z - - [] z -/ / [/] -/ / / z / 7/ 8 / -/ -/ / 7
z z,,,,, Sep z - - [] z -/ / 8 / / [/] -/ / -/ z / / 8 / -/ -/ / 7/ -/ 8
M, M P P P kgkg 8 P P P M (kg) 7 M (kg) z, 7, 8 z 7 8,,, z - -8 - [7] z /7 -/7 8/7 /7 /7 /7 /7 /7 /7 [/7] -/7 /7 z / / / 8 / / -/ / -/ 7/ 9
z, Sep Sep -w - - - z - - [] -w z - z - [] z, z
z,, Sep -w - -7 - - z - - - a [] a -w -/ -/ 7/ - z - / / / a [/] / -/ -w z 8/ -/ 9/ / / -/ / -/ / Sep z 8/ / /,, z
oal A 8 A oal Sep oal A 8 - [-] A oal co (,) -8-- (,) -8-[-] (,) Sep oal A 8 - A oal co 8 (-) 9 (,) -8-- (,) -8- (,) Sep oal A 8 A 9 oal co (9-) 9 (,) 8-- (,) --