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8/10/2019 Assignment Problem.pdf
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1
06IP/ IM74 OPERATIONS RESEARCH
, 4:
(By Dr.G.Rajendra, Prof & HOD (IEM), Dr. AIT, Bangalore 560 056)
The assignment problem is a special case of transportation problem in which the objective is to
assign m jobs or workers to n machines such that the cost incurred is minimized.
1 2
1
2
T C I (I,= 1,2,). T
( ) .
T
.
T 1.
11 12 1
21 22 2
1 2
8/10/2019 Assignment Problem.pdf
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2
T I C.
T .
T 1
H .
D A
S. N. T A1 T
T
2 T
I . T
.
3 T
M+N1
M
N
T
4 T
T
5 I
I
8/10/2019 Assignment Problem.pdf
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3
( )
S I : C
.
S II: D
N
) I N = (= )
) I N
8/10/2019 Assignment Problem.pdf
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4
S VII)
. T ()
.
N: H
1.
T A B C . T
. F .
M
J
A 19 28 31
B 11 17 16
C 12 15 13
:
S 1:
R M
0 9 12
0 6 5
0 3 1
C M
0 6 11
0 3 4
0 0 0
8/10/2019 Assignment Problem.pdf
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5
N H
0 6 11
0 3 4
0 0 0
H,
T 3 3, , N
N,
.
0 3 8
0 0 1
3 0 0
T N=
0 3 8
0 0 1
3 0 0
T A = 19
B = 17
C = 13
49
8/10/2019 Assignment Problem.pdf
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6
2
S
I II III IV V
1 11 17 8 16 20
2 9 7 12 6 15
3 13 16 15 12 16
4 21 24 17 28 26
5 14 10 12 11 15
S:
R M
3 9 0 8 12
3 1 6 0 9
1 4 3 0 4
4 7 0 11 9
4 0 2 1 5
C M
2 9 0 8 8
2 1 6 0 1
0 4 3 0 0
3 7 0 11 5
3 0 2 1 1
N
54
2 9 0 8 8
2 1 6 0 1
0 4 3 0 0
3 7 0 11 5
3 0 2 1 1
8/10/2019 Assignment Problem.pdf
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7
T 1,
.
1 9 0 8 7
1 1 6 0 0
0 4 3 0 0
2 7 0 11 4
2 0 2 1 0
A N
0 8 0 7 6
1 1 6 0 0
0 4 3 0 0
1 6 0 10 3
2 0 2 1 0
H, N=
N,
0 8 0 7 6
1 1 6 0 0
0 4 3 0 0
1 6 0 10 3
2 0 2 1 0
A
1 I = 11
2 IV = 6
3 V = 16
4 III = 17
5 II = 10
40
8/10/2019 Assignment Problem.pdf
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8
3
U , ) )
R M
C
JOB
MECHANIC
1 2 3 4 5
A 10 3 3 2 8
B 9 7 8 2 7
C 7 5 6 2 4
D 3 5 8 2 4
E 9 10 9 6 10
8 1 1 0 6
7 5 6 0 5
5 3 4 0 2
1 3 6 0 2
3 4 3 0 4
7 0 0 0 4
6 4 5 0 3
4 2 3 0 0
0 2 5 0 02 3 2 0 2
8/10/2019 Assignment Problem.pdf
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9
D
H, N, 4 5
T, . 2 .
N= ,
A 2 = 3
B 4 = 2C 5 = 4
D 1 = 3
E 3 = 9
21 R. 21
9 0 0 2 6
6 2 2 0 3
4 0 1 0 0
0 0 3 0 0
2 1 0 0 2
7 0 0 0 4
6 4 5 0 3
4 2 3 0 0
0 2 5 0 0
2 3 2 0 2
9 0 0 2 6
6 2 2 0 3
4 0 1 0 0
0 0 3 0 0
2 1 0 0 2
8/10/2019 Assignment Problem.pdf
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10
4
J 4 4 . T
. T . 1 3,
3 4. D H .
J
1 2 3 4
1 R. 50 R.50 R.20
2 R.70 R.40 R.20 R.30
3 R.90 R.30 R.50
4 R.70 R.20 R60 R70
R M
30 30 0
50 20 0 10
60 0 20
50 0 40 50
C
0 30 0
20 20 0 10
30 0 20
20 0 40 50
N
34
T 10
8/10/2019 Assignment Problem.pdf
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11
0 40 0
10 20 0 0
20 0 20
10 0 40 40N
3=4
T 10
0 50 0
10 20 0 0
10 0 10
0 0 30 30
N=
T , .
0 50 0
10 20 0 0
10 0 10
0 0 30 30
1 4 = 20
2 3 = 20
3 2 = 30
4 1 = 70
140
414, 232
T
8/10/2019 Assignment Problem.pdf
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5
A ,
. J
F 1 M
T
+1 1. T
(
12
, F.
. T
M .
.
.
F.2. I
) ;
.
H
. T
8/10/2019 Assignment Problem.pdf
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F 2. N
M 8
15 12
10 M
12 M
18 17
T ,
R
M 7 5 11
8 5 0 M
5 M 0 9
0 M 0 4
5 4 1 M
13
.
6 12 1
7 M 10
5 14 M
12 16 15
14 M 13
H
0
3
M
3
0
8/10/2019 Assignment Problem.pdf
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14
C
N, 45
T
N, 45 T
N=
M 3 5 7 0
8 1 0 M 3
5 M 0 5 M
0 M 0 0 3
5 0 1 M 0
M 3 5 7 0
8 1 0 M 3
5 M 0 5 M
0 M 0 0 3
5 0 1 M 0
M 2 5 6 0
7 0 0 M 3
4 M 0 4 M
0 M 1 0 3
2 0 1 M 0
M 2 5 2 0
3 0 0 M 3
0 M 0 0 M
0 M 1 0 3
2 0 1 M 0
M 2 5 2 0
3 0 0 M 3
0 M 0 0 M0 M 1 0 3
2 0 1 M 0
8/10/2019 Assignment Problem.pdf
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T
,
F 3. S
6.
F
;
J M
M1
J1
J2
J3
J4
5
8
4
10
H
15
F. 3 1
.
P
. T
M2 M3
7 11
5 9
7 10
4 8
4
.
8/10/2019 Assignment Problem.pdf
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16
R
0 2 6 1
3 0 4 1
0 3 6 37 1 5 0
C
0 2 2 1
3 0 0 1
0 3 2 3
7 1 1 0
D
0 2 2 1
3 0 0 1
0 3 2 3
7 1 1 0
N 34
T 1,
0 1 1 1
4 0 0 2
0 2 1 3
7 0 0 0
N
34
T 1,
0 0 0 0
5 0 0 2
0 1 0 2
8 0 0 0
N=
8/10/2019 Assignment Problem.pdf
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17
4=4
0 0 0 0
5 0 0 2
0 1 0 2
8 0 0 0
J1 M4 = 6
J2M2 = 5
J3M1 = 4
J4M3 = 823
A
0 0 0 0
5 0 0 2
0 1 0 2
8 0 0 0
J1 M1 = 5
J2M2 = 5
J3M3 = 10
J4M4 = 3
23
8/10/2019 Assignment Problem.pdf
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18
7
A 5 R.
(=1,2,3,5) (=1,2,3,). A 5 5
.
JOB
M
1 2 3 4 5
1 5 11 10 12 4
2 2 4 6 3 5
3 3 12 5 14 6
4 6 14 4 11 75 7 9 8 12 8
S,
S 1: C O
1
T
JOB
M
1 2 3 4 5
1 5 11 10 12 4
2 2 4 6 3 5
3 3 12 5 14 6
4 6 14 4 11 7
5 7 9 8 12 8
8/10/2019 Assignment Problem.pdf
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19
S2: = ()
I 14. U
. T
.
JOB
M
1 2 3 4 5
1 9 3 4 2 10
2 12 10 8 11 9
3 11 2 9 0 8
4 8 0 10 3 7
5 7 5 6 2 6
E
C11= 5 (14) = 9
S 3: U .. . T
H .
7 1 2 0 8
4 2 0 3 1
11 2 9 0 8
8 0 10 3 7
5 3 4 0 4
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20
3 1 2 0 7
0 2 0 3 0
7 2 9 0 74 0 10 3 6
1 3 4 0 3
N=3, =55
N,
.
2 1 1 0 6
0 3 0 4 0
6 2 8 0 6
3 0 9 3 5
0 3 3 0 2
N=4 =55
N
T 1
2 1 0 0 5
1 4 0 5 0
6 2 7 0 5
3 0 8 3 4
0 3 2 0 1
8/10/2019 Assignment Problem.pdf
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N=,
2 1 0 0 5
1 4 0 5 0
6 2 7 0 5
3 0 8 3 4
0 3 2 0 1
N
13 = 10
25 = 5
34 = 14
42 = 14
51 = 7
50
8/10/2019 Assignment Problem.pdf
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22
7
A 5 5 . C
,
( 1000 ) .
A B C D E
1 32 38 40 28 40
2 40 24 28 21 36
3 41 27 33 30 37
4 22 38 41 36 36
5 29 33 40 35 39
F .
:
T . T ,
. T ..,
41. T
.
A B C D E
1 9 3 1 13 1
2 1 17 13 20 5
3 0 14 8 11 4
4 19 3 0 5 5
5 12 8 1 6 2
U H .
8/10/2019 Assignment Problem.pdf
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23
R
8 2 0 12 0
0 16 12 19 4
0 14 8 11 419 3 0 5 5
11 7 0 5 1
C
8 0 0 7 0
0 14 12 14 4
0 12 8 6 4
19 1 0 0 5
11 5 0 0 1
D
N=4 55
N,
. T 4
8 0 0 7 0
0 10 8 10 0
0 8 4 2 0
19 1 0 0 5
11 5 0 0 1
N=,
8/10/2019 Assignment Problem.pdf
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24
8 0 0 7 0
0 10 8 10 0
0 8 4 2 0
19 1 0 0 511 5 0 0 1
1B = 38
2A = 40
3E = 37
4C = 41
5D = 35
191
8
A . T
;
:
T : I II III IV
A S (R) 60,000 50,000 40,000 30,000
F : ,
, :
S: A B C D
P: 7 5 5 4
I ,
, , ,
.
8/10/2019 Assignment Problem.pdf
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25
:
S 1
B R. 10000/
:
S 10
S P 6 5 4 3
I II III IV
7 A 42 35 28 21
5 B 30 25 20 15
5 C 30 25 20 15
4 D 24 20 16 12
T 11=
= 76=42
I
S 2:
B 1. T :
I II III IV
A 42 35 28 21
B 30 25 20 15
C 30 25 20 15
D 24 20 16 12
S 3: .. 42.
. T
8/10/2019 Assignment Problem.pdf
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I II III IV
A 0 7 14 21
B 12 17 22 27
C 12 17 22 27D 18 22 26 30
N, H
,
R
0 7 14 21
0 5 10 15
0 5 10 15
0 4 8 12
C
0 3 6 9
0 1 2 3
0 1 2 3
0 0 0 0
D
0 3 6 9
0 1 2 3
0 1 2 3
0 0 0 0
N, 24
T 1
8/10/2019 Assignment Problem.pdf
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0 2 5 8
0 0 1 2
0 0 1 2
0 0 0 0
N, 34
T 1
0 2 4 7
0 0 0 1
0 0 0 1
0 0 0 0
N=, 4=4,
0 2 4 7
0 0 0 1
0 0 0 1
0 0 0 0
AI
BII
CIII
DIV
8/10/2019 Assignment Problem.pdf
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28
9:
A C
. G
, .
S (R. 000 P)
P 1 2 3 4
A 50 68 49 62
B 60 70 51 74
C 55 67 53 70
D 58 65 54 69
P (R. 000 P)
P 1 2 3 4
A 49 60 45 61
B 55 63 45 69
C 52 62 49 68
D 55 64 48 66
:
N, .
P =
P
1 2 3 4
A 1 8 4 1
B 5 7 6 5
C 3 5 4 2
D 3 1 6 3
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N , .. 8
.
7 0 4 7
3 1 2 3
5 3 4 6
5 7 2 5
U H
R
7 0 4 7
2 0 1 2
2 0 1 3
3 5 0 3
C
5 0 4 5
0 0 1 0
0 0 1 1
1 5 0 1
D
5 0 4 5
0 0 1 0
0 0 1 1
1 5 0 1
N=,
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5 0 4 5
0 0 1 0
0 0 1 1
1 5 0 1
A2
B4
C1
D3
11
A . C
() 5 , . O
. F
.
D J JD
F N. D A F N. D A
1 7.00 AM 8.00 AM 101 8.00 AM 9.15 AM
2 8.00 AM 9.00 AM 102 8.30 AM 9.45 AM
3 1.30 PM 2.30 P.M 103 12.00 NOON 1.15 PM
4 6.30AM 7.30 PM 104 5.30 PM 6.45 PM
, .
:
S1 D,
15 = 1.
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31
T 1: D
F 101 102 103 104
1 96 98 112 38
2 92 94 108 34
3 70 72 86 108
4 50 52 66 88
S, 5
T 1 101 24 (96 ) 8.00 AM 8.00 AM
. 1 8.00 8.00
5
.
F 1 102 (98) 8.00 8.30 = 24
+30
F 1 103 (112 ) 8.00 12.00 = 24
+4 =112
F 1 104 (38 )8.00 5.30 =
9 +30 = 38
T F 2 101 (9.00 8.00
) =23 = 92
F 2 102 (9.00 8.30 ) = 23
+30 = 94
F 2 103 (9.00 12.00 ) = 24
+3 = 108
F 2 104 (9.00 5.30 ) = 8
+30 = 34
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32
T F 3 101 (2.30 8.00
) =17 + 30 = 70
F 3 102 (2.30 8.30 ) = 18
= 72
F 3 103 (2.30 12.00 ) =
21 + 30 = 86
F 3 104 (2.30 5.30 ) =
24+3= 108
T F 4 101 (7.30 8.00
) =12 +30 = 50
F 4 102 (7.30 8.30 ) = 13
= 52
F 4 103 (7.30 12.00 ) =
16 + 30 = 66
F 4 104 (7.30 5.30 ) =
22= 88
T 2:
F 101 102 103 104
1 87 85 71 49
2 91 89 75 53
3 113 111 97 75
4 37 35 21 95
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A
S, 5
T 101 1 21 + 45 (87 ) 9.15 AM
7.00 AM 1 . 101 D 9.15 D
7.00 1 5
.
F 101 2 (91 ) 9.15 D D 8.00 = 22 +45
F 101 3 (113 ) 9.15 D D 1.30 = 28
+15 = 113 F 101 4 (38 ) 9.15 D D 6.30 = 9
+15 = 37
T F 102 1 (9.45 D 7.00
) =21 +15 = 85
F 102 2 (9.45 D D 8.00 ) = 22 +
15 = 89 F 102 3 (9.45 D D 1.30 ) = 27
+45 = 111
F 102 4 (9.45 D D 6.30 ) = 8
+45 = 35
F N. A() F N. D (D)
101 9.15 AM 1 7.00 AM
102 9.45 AM 2 8.00 AM
103 1.15 PM 3 1.30 PM
104 6.45 PM 4 6.30AM
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T F 103 1 (1.15 D 7.00
) =17 + 45 = 71
F 103 2 (1.15 D D 8.00 ) = 18 +
45 = 75
F 103 3 (1.15 D D 1.30 ) = 24
+15 = 97
F 103 4 (1.15 D D 6.30 ) = 5 +
15 = 21
T F 104 1 (6.45 D 7.00
) =12 + 15 = 49
F 104 2 (6.45 D D 8.00 ) = 13 +
15 = 53
F 104 3 (6.45 D D 1.30 ) =18
+45 = 75
F 104 4 (6.45 D D 6.30 ) = 23 + 45 = 95
S 3: T 1
T 2 * .
T 3:
F 101 102 103 104
1 87* 85* 71* 38
2 91* 89* 75 34
3 70 72 86 75
4 37* 35* 21* 88
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35
U H
0 0 0 0
12 8 8 0
0 0 28 50
4 0 0 100
T
F 1103
F 2104F 3101
F 4102
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36
I (
),
T .
I , .
T .
1:
A .
T ( ):
M
J
1 2 3 4 5
1 2.5 5.0 1.0 6 1.0
2 2.0 5.0 1.5 7 3.0
3 3.0 6.5 2.0 9 4.5
4 3.5 7.0 2.0 9 4.5
5 4.0 7.0 3.0 9 6.0
6 6.0 9.0 5.0 10 6.0
S .
:
T 65, . S
. J 6 . T
.
8/10/2019 Assignment Problem.pdf
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37
M
J
1 2 3 4 5 6
1 2.5 5.0 1.0 6 1.0 0
2 2.0 5.0 1.5 7 3.0 03 3.0 6.5 2.0 9 4.5 0
4 3.5 7.0 2.0 9 4.5 0
5 4.0 7.0 3.0 9 6.0 0
6 6.0 9.0 5.0 10 6.0 0
T, H .
R M
2.5 5.0 1.0 6 1.0 0
2.0 5.0 1.5 7 3.0 0
3.0 6.5 2.0 9 4.5 0
3.5 7.0 2.0 9 4.5 0
4.0 7.0 3.0 9 6.0 0
6.0 9.0 5.0 10 6.0 0
R
C
0.5 0 0 0 0 0
0 0 0.5 1 2.0 0
1.0 1.5 1.0 3 3.5 0
1.5 2.0 1.0 3 3.5 0
2.0 2.0 2.0 3 5.0 0
4.0 4.0 4.0 4 5.0 0
N=3, =55, N, 1.0
8/10/2019 Assignment Problem.pdf
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38
0.5 0 0 0 0 1
0 0 0.5 1 2.0 1
0 0.5 0 2 3.5 1
0.5 1.0 0 2 3.5 11.0 1.0 1.0 2 5.0 0
3.0 3.0 3.0 3 5.0 0
N=5, =66, N
T 1
.
0.5 0 0 0 0 2
0 0 0.5 1 2.0 2
0 0.5 0 2 3.5 2
0.5 1.0 0 2 3.5 2
0 0 0 1 5.0 0
2.0 2.0 2.0 2 5.0 0
N=5, =6, N, 1
.
0.5 0 0 0 0 2
0 0 0.5 0 1.0 2
0 0.5 0 1 2.5 2
0.5 1.0 0 1 2.5 2
0 0 0 0 4.0 0
2.0 2.0 2.0 1 4.0 0
8/10/2019 Assignment Problem.pdf
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39
T N=
.
0.5 0 0 0 0 2
0 0 0.5 0 1.0 2
0 0.5 0 1 2.5 2
0.5 1.0 0 1 2.5 2
0 0 0 0 4.0 0
2.0 2.0 2.0 1 4.0 0
M 1 J5 = 1
M 2 J2 = 5
M 3 J1 = 3
M 4 J3 = 2
M 5 J4 = 9
M 6 J6 = 6
26
8/10/2019 Assignment Problem.pdf
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2.
M T C 5 5
. T . A 5 .
T
L L
A B C D E
1 300 290 280 290 210
2 250 310 290 300 200
3 180 190 300 190 180
4 320 180 190 240 170
5 270 210 190 250 160
6 190 200 220 190 140
7 220 300 230 180 160
8 200 190 260 210 180
S;
T 85, , 3
.
T
L L
A B C D E F G H
1 300 290 280 290 210 0 0 0
2 250 310 290 300 200 0 0 0
3 180 190 300 190 180 0 0 0
4 320 180 190 240 170 0 0 0
5 270 210 190 250 160 0 0 0
6 190 200 220 190 140 0 0 0
7 220 300 230 180 160 0 0 0
8 200 190 260 210 180 0 0 0
8/10/2019 Assignment Problem.pdf
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41
O
.
R
300 290 280 290 210 0 0 0
250 310 290 300 200 0 0 0
180 190 300 190 180 0 0 0
320 180 190 240 170 0 0 0
270 210 190 250 160 0 0 0
190 200 220 190 140 0 0 0
220 300 230 180 160 0 0 0
200 190 260 210 180 0 0 0
S, (0) .
C
120 110 90 110 70 0 0 0
70 130 100 120 60 0 0 0
0 10 110 10 40 0 0 0
140 0 0 60 30 0 0 0
90 30 0 70 20 0 0 0
10 20 30 10 0 0 0 0
40 120 40 0 20 0 0 0
20 10 70 30 40 0 0 0
8/10/2019 Assignment Problem.pdf
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42
T1 G = 0
T2 F = 0
T3 A = 180
T4 B = 180
T5 C = 190T6 E = 140
T7 D = 180
T8 H = 0
870
120 110 90 110 70 0 0 0
70 130 100 120 60 0 0 0
0 10 110 10 40 0 0 0
140 0 0 60 30 0 0 0
90 30 0 70 20 0 0 0
10 20 30 10 0 0 0 0
40 120 40 0 20 0 0 0
20 10 70 30 40 0 0 0
8/10/2019 Assignment Problem.pdf
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43
()
T
.
S .
H ( ) ,
.. I .
H . P
(
).
S 9 ) I =1,
I , . T
. I , I I . T
=
.
A, . I
=1 I . ( )
1.
T
F
C12 C1
C21 C2
C1 C2
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44
1:
G ,
.
:
R
C
T
F
A1 A2 A3 A4 A5
A1 2 5 7 1
A2 6 3 8 2
A3 8 7 4 7
A4 12 4 6 5
A5 1 3 2 8
A1 A2 A3 A4 A5
A1 1 4 6 0
A2 4 1 6 0
A3 4 3 0 3
A4 8 0 2 1
A5 0 2 1 7
A1 A2 A3 A4 A5
A1 1 3 6 0
A2 4 0 6 0
A3 4 3 0 3
A4 8 0 1 1
A5 0 2 0 7
8/10/2019 Assignment Problem.pdf
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45
N=, 5=55
A1A5A1, A2A3A4A2
C = 1+3+4+4+1=13
A A1, A5
A1, A2,A3 A4
.
S 2:
(1) 1 12, 43, 45,
L 15 12 42 45
T A1A2A3A4A5A1
T 2+3+4+5+1 =15H R.2.
15 12 1 42 45
. T R 2.
1 3 6 0
4 0 6 0
4 3 0 3
8 0 1 1
0 2 0 7
1 3 6 0
4 0 6 0
4 3 0 3
8 0 1 1
0 2 0 7
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46
2
A 5 ,
. T
:
T I
A B C D E
A 4 7 3 4
B 4 6 3 4
C 7 6 7 5
D 3 3 7 7
E 4 4 5 7
?
:
U
R M
1 4 1 1
1 3 0 1
2 1 2 0
0 0 4 4
0 0 1 3
C
1 3 1 1
1 2 0 1
2 1 2 0
0 0 3 4
0 0 0 3
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47
D
1 3 1 1
1 2 0 1
2 1 2 00 0 3 4
0 0 0 3
N, 45, 1
0 2 1 1
0 1 0 1
1 0 2 0
0 0 3 5
0 0 0 4
N=
0 2 1 1
0 1 0 1
1 0 2 0
0 0 3 5
0 0 0 4
ABDA, CEC, THE COST =4+3+5+3+5=20
I 2
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48
S 12 15 35 32
0 2 1 1
0 1 0 1
1 0 2 0
0 0 3 5
0 0 0 4
AEBDA, CB, T
COST =4+3+6+3+5=21
N 32 31 41 42
0 2 1 1
0 1 0 1
1 0 2 0
0 0 3 5
0 0 0 4
AECA, BDB
COST= 4+3+7+3+5=22
NO E TR ITH SHIFITNG OF C15 TO C14 AND C24 TO C25
0 2 1 1
0 1 0 1
1 0 2 0
0 0 3 5
0 0 0 4
ADBECA COST=4+4+7+3+5=23
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49
3
S P
1 2 3 4 5
1 20 4 10
2 20 5 10
3 4 5 6 5
4 10 6 2
5 10 5 2
4.
S
C12 =20, C13 =4, C14 =10, C23 =5, C34 =6, C25 =10, C35 =6, C45 =20 = C= C
I .
:
S1 C = ,
=
A
R :
16 0 6
15 0 5
0 1 2 1
8 4 0
8 3 0
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C
15 0 6
15 0 5
0 0 2 18 4 0
7 3 0
D
15 0 6
15 0 5
0 0 2 1
8 4 0
7 3 0
N, 4=5 5
10 0 1
10 0 0
0 0 7 6
8 4 0
7 3 0
N , 45 7
3 0 1
3 0 0
0 0 14 13
1 4 0
0 3 0
N, 45 1
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2 0 0
2 0 0
0 0 2 2
0 4 0 0 4 0
N=,
U
2 0 0
2 0 0
0 0 2 2
0 4 0
0 4 0
145231
C = 10+5+4+20+10 = R. 49/
A 5 ABCD &E () 5
A B C D E
A 7 6 8 4
B 7 8 5 6
C 6 8 9 7
D 8 5 9 8
E 4 7 7 8
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52
R
3 2 4 0
4 3 0 1
0 2 3 13 0 4 3
0 3 3 1
C
3 0 4 0
4 1 0 1
0 2 3 1
3 0 2 3
0 3 1 1
N, 45, 1
3 0 4 0
4 0 0 0
0 2 3 0
3 0 1 2
0 3 0 1
N= 5=5
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T
3 0 4 0
4 0 0 0
0 2 3 0
3 0 1 2
0 3 0 1
ACEA, BDB COST=6+7+4+5+5=27
U
3
00+0=0
4 00+0=0
4 00+0=0
00+1=1
00+0=0
00+0=0
2 3 00+0=0
3 02+1=3
1 2
00+0=0
3 00+0=0
1
DB
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A C D E
A
00+0=0
4 00+0=0
B 4 0
0+0=0
0
0+0=0
C 00+0=0
3 00+0=0
E 00+0=0
00+0=0
1
, ,
1 .
A C D E
A
00+0=0
3 00+0=0
B 4 00+0=0
00+0=0
C 00+0=0
2 00+0=0
E 00+0=0
00+0=0
02+0=2
ED
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A C E
A
00+0=0
00+0=0
B 4 00+0=0
00+0=0
C 00+4=0
00+0=0
CA
C E
A
00+0=0
B 00+0=0
00+0=0
BC
AE
T O A
AEDBCA
COST=4+8+5+8+6=31
, , , ,
, ,
&
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