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1. .. . ˳ , . .: . , 1998. . 214-234.
2. .. : .: , 2003 . . 436-458.
3. .., .. . .:, 2003 . . 184-252.
4. .. ˳ : . . .: , 2001. . 202-227.
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2. ϳ (m + n 1) , m ; n , . . (n + m 1), . , . , , .
(m + n 1), . , , , .
3. ui vj .
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( ). * = (xij *) ui vj,
ui + vj = cij, xij > 0,
ui + vj £ cij, xij = 0
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ui + vj = cij, .
4. ui + vj £ cij . , ui + vj > cij, .
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5. . 3 .
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1. 1, 2, 3, 150, 60 80 . . . 1, 2, 3, 4, 110, 40, 60 80 . . . 1000 . .
1000 . | ||||
1 | 2 | 3 | 4 | |
1 | ||||
2 | ||||
3 |
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. xij , - j - . ,
: .
: , , . :
, , .
Z = 4 × x 11 + 4 × x 12 + 2 × x 13 + 5 × x 14 + 5 × x 21 + 3 × x 22 + x 23 +
+ 2 × x 24 + 2 × x 31 + x 32 +4 × x 33 +2 × x 34 min.
:
Z = 4 x 11 + 4 x 12 + 2 x 13 + 5 x 14 + 5 x 21 + 3 x 22 + x 23 + 2 x 24 + 2 x 31 + x 32 +4 x 33 +2 x 34 min
. , . .
Aj | Bj | ui | |||
B 1= 110 | B 2= 40 | B 3= 60 | B 4= 80 | ||
A 1= 150 | 2 + | 40 | u 1= 5 | ||
A 2= 60 | 60 | 0+ | u 2= 2 | ||
A 3= 80 | u 3= 2 | ||||
vj | v 1= 1 | v 2= 1 | v 3= 1 | v 4= 0 |
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Z = 4 × 110 + 5 × 40 + 1 × 60 + 1 × 40 + 2 × 40 = 820 . .
, , (m + n 1) = 3 + 4 1 = 6. , , - , . , 2 4. , .
ui + vj = cij :
, , , , v 4 = 0. : u 1 = 5, u 2 = 2, u 3 = 2, v 1 = 1, v 2 = 1, v 3 = 1.
ui + vj ≤ cij ( ):
1 B 2: u 1 + v 2 = 5 + (1) = 4 = 4;
1 B 3: u 1 + v 3 = 5 + (1) = 4 > 2;
2 B 1: u 2 + v 1 = 2 + (1) = 1 < 5;
2 B 2: u 2 + v 2 = 2 + (1) = 1 < 3;
3 B 1: u 3 + v 1 = 2 + (1) = 1 < 2;
3 B 3: u 3 + v 3 = 2 + (1) = 1 < 4.
1 B 3. D13 = (u 1 + v 3) c 13 = 4 2 = 2 .
. , .
1 B 3, . +. , , , 1 B 3, +. . 1 B 3 ij, . ij , +, , , .
, . , . , , . ʳ , (n + m 1).
, :
Aj | Bj | ui | |||
B 1= 110 | B 2= 40 | B 3= 60 | B 4= 80 | ||
A 1= 150 | 110 | 40+ | u 1= 0 | ||
A 2= 60 | 20 | 40+ | u 2= 1 | ||
A 3= 80 | 1 + | 40 | u 3= 1 | ||
vj | v 1= 4 | v 2= 2 | v 3= 2 | v 4= 3 |
Z 2 = 4 × 110 + 2 × 40 + 1 × 20 + 2 × 40 + 1 × 40 + 2 × 40 =
740 . .
, 䳿. ( 3 B 1). :
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Aj | Bj | ui | |||
B 1= 110 | B 2= 40 | B 3= 60 | B 4= 80 | ||
A 1= 150 | u 1= 2 | ||||
A 2= 60 | u 2= 0 | ||||
A 3= 80 | u 3= 0 | ||||
vj | v 1= 2 | v 2= 1 | v 3= 0 | v 4= 2 |
Z 3 = 4 × 90 + 2 × 60 + 2 × 60 + 2 × 20 + 1 × 40 + 2 × 20 = 720 . .
, .
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90 . . 20 . . . 40 . . . . 720 . .
2. ᒺ 1, 2, 3, . 50, 30 20 , B 1, B 2, B 3, B 4. 30, 30, 10 20 . 1 .
1 | ||||
1 | 2 | 3 | 4 | |
1 | ||||
2 | ||||
3 |
, ᒺ .
. ij , - j - (; ). - :
Z = 2 x 11 + 3 x 12 + 4 x 13 + 2 x 14 + 5 x 21 + 7 x 22 + x 23 +
+ 4 x 24 + 9 x 31 + 4 x 32 +3 x 33 +2 x 34 min,
≤ , :
, , , .
. , , . , B 5 B 5 = 100 90 = 10 . .
.
Aj | Bj | ui | ||||
B 1 = 30 | B 2= 30 | B 3= 10 | B 4= 20 | B 5= 10 | ||
A 1= 50 | u 1= 4 | |||||
A 2= 30 | 10 | 0+ | u 2= 0 | |||
A 3= 20 | 1 + | 20 | u 3= 2 | |||
vj | v 1= 6 | v 2= 7 | v 3= 1 | v 4= 0 | v 5= 0 |
, , A 2 B 4, .
, . A 3 B 2 . , (D21 = D32 = 1) , ( 32 < c 21).
:
Aj | Bj | ui | ||||
B 1= 30 | B 2= 30 | B 3= 10 | B 4= 20 | B 5= 10 | ||
A 1= 50 | u 1= 3 | |||||
A 2= 30 | u 2= 0 | |||||
A 3= 20 | u 3= 2 | |||||
vj | v 1= 5 | v 2= 6 | v 3= 1 | v 4= 4 | v 5= 0 |
, :
;
Z min = 2 × 30 + 3 × 20 + 1 × 10 + 4 × 10 + 4 × 10 + 2 × 10 = 320 . .
( 10 ). 230 . .
, . : ui + vj = cij. A 2 B 1: u 1 + v 1 = 0 + 5 = 21 = 5.
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, , :
, .
Aj | Bj | ui | ||||
B 1 = 30 | B 2 = 30 | B 3 = 10 | B 4 = 20 | B 5 = 10 | ||
1 = 50 | u 1 = 3 | |||||
A 2 = 30 | u 2 = 0 | |||||
A 3 = 20 | u 3 = 2 | |||||
vj | v 1 = 5 | v 2 = 6 | v 3 = 1 | v 4 = 4 | v 5 = 0 |
;
Z min = 2 × 20 + 3 × 30 + 5 × 10 + 1 × 10 + 2 × 20 = 230 . .
. 20 30 ; 10 10 , 10 , 20 . 230 . . .
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