4.1.
m A 1, A 2, , Am, , , a 1, a 2,, am. n B 1, B 2,, Bn, ; b 1, b 2, , bn. ( ) Ai (i = 1, 2, , m) Bj (j = 1, 2, , n), cij () C = (cij).
, Ai Bj () .
, Ai Bj, xij. , xij ³ 0. xij X.
:
(15) | |
(16) | |
(17) |
(15) , (16) , (17) .
(15). (16) .
() , . ( ).
X = (xij) m ´ n , C = (cij) m´n , cij (i; j).
X = (xij) m ´ n , (16).
, (15), .
13
, j | 1 | 2 | , i | ||
, i | |||||
c 11 | c 12 | c 1 n | |||
A 1 | x 11 | x 12 | x 1 n | a 1 | |
c 21 | c 22 | c 2 n | |||
A 2 | x 21 | x 22 | x 2 n | a 2 | |
cm 1 | cm 2 | cmn | |||
Am | xm 1 | xm 2 | xmn | am | |
, bj | b 1 | b 2 | bn |
8 ( ). , ,
(18) |
( ).
(18) .
, (18), .
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|
:
(19) |
, | (20) |
.
.
, . . (19), (n + 1)- Bn+ 1 ( ) , , . . , , . , , . , Bn+ 1, .
, . . (20), (m + 1)- Am+ 1( ) .
, , , .
9 ( ). ( (16)) r (A) = m + n 1.
. , . ( , ), ( ). m + n 1 .
, xij, , .
, m + n 1 , -. , m + n 1, .
- . (1; 1), -. , (2; 1); , (1; 2). , , , ( ), , . (m; n). (1; 1)(m; n), - .
- cij, .
( ). . . , , , , , . . , , . , m + n 1 . . , . 0 - ( ), ( , ).
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0 , .
, , ( ), . ( .)
, , , , - .
( , ) , , . .
10 ( ). , m + n , , .
, i - j - .
10 , .
. , , Ai ( ) , Bj ( ) . ui, vj Ai Bj (ui, vj ).
, ui vj , , . .
. | (21) |
m + n 1, ui vj m + n, ui, vj m + n 1 (21) m + n . , , , . .
:
. | (22) |
Sij .
Sij ³ 0, .
Sij < 0, , .
:
1. , Sij , . , .
2. , . . , . . , . , ( ), , ( ) , . . . , ( ), . .
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.
. 7.
. 7
3. + . +; , , +, .
4. , , ( q), , . . xij + ( ) .
, .
, . (z min).
. Sij > 0, . , Sij = 0, .
4.6.
:
1. .
2. (18) ().
3. .
4. ui vj , (21).
5. Sij (22). , , . , .
6. ( 4.5), 4 .
8. , , ai i - , bj j - , C = (cij) :
a 1 = 12; a 2 = 3; a 3 = 10; | b 1 = 14; b 2 = 10; b 3 = 6; | . |
. 14.
14
bj | |||
ai | |||
, (12 + 3 + 10 = 25) (14 + 10 + 6 = 30).
25 < 30, . . , .
.
, . , , . .
.
: .
|
|
, (. 15).
() (. 16).
15
bj | ||||||||
ai | ||||||||
16
bj | ||||||||
ai | ||||||||
X 1 (m + n 1 = 4 + 3 1 = 6).
z 1 = 1 × 7 + 0 × 5 + 1 × 3 + 4 × 7 + 5 × 3 + 0 × 5 = 53.
. 8.
7 | 5 + | |||
7 + | 3 | |||
5 | 5 | + | ||
. 8
:
S 13 = 2 (3 + 5) = 0; | S 32 = 6 (0 + 3) = 3 > 0; |
S 21 = 3 ( 4 + 4) = 3 > 0; | S 41 = 0 (3+ 4) = 1 < 0; |
S 22 = 4 ( 4 + 3) = 5 > 0; | S 43 = 0 (3 + 5) = 2 < 0. |
S 41 < 0 S 43 < 0, . (4; 3). (4; 3) . 8. (4; 3), (3; 3), (3; 1), (1; 1), (1; 2), (4; 2). + , . , , . (. 9).
:
S 13 = 2 (0 + 0) = 2 > 0; | S 32 = 6 (3 + 0) = 3 > 0; |
S 21 = 3 (1 + 1) = 1 > 0; | S 33 = 5 (3 + 0) = 2 > 0; |
S 22 = 4 (1 + 0) = 3 > 0; | S 41 = 0 (0 + 1) = 1 < 0. |
4 | 8 + | u 1 = 0 | ||
u 2 = 1 | ||||
u 3 = 3 | ||||
5 | + | u 4 = 0 | ||
v 1 = 1 | v 2 = 0 | v 3 = 0 |
. 9
S 41 < 0, . (4; 1) . 9. (4; 1), (1; 1), (1; 2), (4; 2). . (4; 1) .
2 (. 10).
u 1 = 0 | ||||
u 2 = 0 | ||||
u 3 = 3 | ||||
u 4 = 1 | ||||
v 1 = 1 | v 2 = 0 | v 3 = 1 |
. 10
, . 10. :
S 13 = 2 (0 + 1) = 1 > 0, | S 32 = 6 (3 + 0) = 3 > 0, |
S 21 = 3 (0 + 1) = 2 > 0, | S 33 = 5 (3 + 1) = 1 > 0, |
S 22 = 4 (0 + 0) = 4 > 0, | S 42 = 0 (1 + 0) = 1 > 0. |
, :
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.
(. .).
x 41 = 2 x 43 = 3 5 . B 1 B 3.
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