.
.
- , ( -301 1- , -301 ). , , , , . , , .
. ( ) : , , - . , a - .
- . .
m (): 1, 2, , m, - () 1, 2,,m. , n (): B1, B2, ,Bn, b1, b2, , bn .
cij Ai Bj. () cij :
c11 12 13... 1n
c21 c22 c23... c2n
........
ci1 ci2 ci3... cin
........
cm1 cm2 cm3... cmn
, , .
. xij - , i- ( i- ) i j- (j- ) Bj (i=1,m; j=1,n). xi1, xi2, , xmn ( mxn) :
1. , , ( ). m -:
x11 + x12 +... + x1j +... + x1n = a1,
x21 + x22 +... + x2j +... + x2n = a2,
.....
xi1 + xi2 +... + xij +... + xin = ai,
.....
xm1 + xm2 +... + xmj +... + xmn = am,
, , xij = ai (i = 1, m), (2.1)
2. , , , . n -:
x11 + x21 +... + xi1 +... + xm1 = b1,
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x12 + x22 +... + xi2 +... + xm2 = b2,
.....
x1j + x2j +... + xij +... + xmj = bj,
.....
x1n + x2n +... + xin +... + xmn = bn,
, , xij = bj (j = 1, n), (2.2)
(2.1), (2.2) .
3. , xij, cij, :
F = c11x11 + c12x12 +... +... +c1jx1j +... + c1nx1n +
+ c21x21 + c22x22 +... +... +c2jx2j +... + c2nx2n +... +
+ cm1xm1 + cm2xm2 +... +... +cmjxmj +... + cmnxmn → min,
, , F = (ij xij) → min (2.3)
, (i=1,m; j=1,n), .
(2.3) , - (2.1), (2.2) . - (), .
xij, (i = 1, m; j = 1, n), (2.1), (2.2) F (2.3).
, (2.1)-(2.3), :
) ( , );
) ;
) : i j;
,
ai = bj, (2.4)
, , ( ). ( ).
.
.
, , , -. , , .
( - , , ), , .
, (2.1), (2.2) . , . , + . , (2.1) (2.2),
xij = ai (2.5)
xij = bj, (2.6)
(2.4) (2.5) (2.6) . , xij , . , m n . , (2.1), (2.2) , +- 1, + .
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:
2.1. R (2.1), (2.2) (2.4) +n-1.
, (2.1), (2.2) +n-1 , , , : r= mn - (+-1) = - ( -1) = m(-1)- (-1) = (m-1)(n-1)
xij , i j,- (i,j).
(ij) (i=1,..., ; j = 1,..., ) , .
(xij) , (2.1), (2.2) ( ), , , .
( ), R=+-1 xij , ( r=(n-1)(m-1)) .
(xij) , , , .
, , , , , r . , , , (-1)(-1) xij .
() -, , . . :
- () (), ;
- , ,( );
- , , ( );
- ( ).
( ) , xij (i,j).
. 2.0.
2.0
1 | 2 | ... | j | ... | n | aj | |
A1 | 11 | 12 | ... | 1j | ... | 1n | a1 |
A2 | 21 | 22 | ... | 2j | ... | 2n | a2 |
... | ... | ... | ... | ... | ... | ... | ... |
i | i1 | i2 | ... | ij | ... | in | ai |
... | ... | ... | ... | ... | ... | ... | ... |
Am | m1 | m2 | ... | mj | ... | mn | am |
bj | b1 | b2 | ... | bj | ... | bn |
, .
2.1. - .
. , - ( 2.1). (i,j)- 2.1 ( i- , j- ) , , i- j‑ . , (1,4) 3, , 1- 4- 3 .
2.1.
1 | 2 | 3 | 4 | () - i | |||||||||||||
1 |
| 1 | 1 | ||||||||||||||
2 | 21 | | | ||||||||||||||
3 | 31 | | |||||||||||||||
() -bj |
.
. i- j- - (i, j) ij. ij (i, j) . ij. , , , 1- , 60 , , 11+ 12 + 13 + 14 = 60 ( ).
, ,
11 + 12 + 13 + 14 = 60,
21 + 22 + 23 + 24 = 120, (2.1.1)
31 + 32 + 33 + 34 = 100,
, , :
11 + 21 + 31 = 20,
12 + 22 + 32 = 110, (2.1.2)
13 + 23 + 33 = 40,
14+ 24 + 34 = 110,
, , , (2.1.3)
F :
F = 1∙ 11 +2 ∙ 12 +5 ∙ 13 +3 ∙ 14 +1 ∙ 21 +6 ∙ 22 +5 ∙ 23 +2 ∙ 24 +
+ 6∙ 31 +3 ∙ 32 +7 ∙ 33 +4 ∙ 34 (2.1.4)
( ):
(2.1.1) (2.1.2) *=(*11, *12, *13,... *33, *34), (2.1.4) .
* * * * * * * * * * * * * * * * * * * * * * * * *
, - R= + -1, - , n - . , , , , + -1 .
, , , ()- .
, : , , , :
1) ( ) , , ();
2) () , , ();
3) - .
2.1.
.
, , , , . , .
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. - .
.
2.2. ( ) 2.1.
. (1,1) - , , 11 : 11 = min(60,20) = 20. 11 = 20 (1,1) . 1- , . - (1,2) , 1- 20 60-20 = 40 , , 12 = min(60-20,110) = 40. 1- 1- . - .. ( 2.2).
2.2.
1 | 2 | 3 | 4 | i | |||||||||||||||
1 | |||||||||||||||||||
2 | 70 | 40 | |||||||||||||||||
3 | |||||||||||||||||||
bj |
: , . , , .
, , . , .
m+n-1= =3+4-1=6, , () . , , . , , , , , , . ( ) , , m+n-1. 2.1 - , .
- , . , , .