-
1
IV. PROBLEM TRANSPORTA I DISTRIBUCIJE
1. Rijeite Kotzigovom metodom sljedei problem transporta:
ai
1 3 4
2 1 16
4 7 25
2 5 5
5 6 10
bj 20 40
Rjeenje:
1 3 -2 3 3
2 1 1 5 5
4 7 -3 1 2
2 5 -3 2 1
5 6 -1 4 4
0 4
0 16
20 5
0 5
0 10
3 4 1 16 4 20 7 5 5 5 6 10 228T = + + + + + =
0 4
0 16
15 10
5 0
0 10
3 4 1 16 4 15 7 10 2 5 6 10 228.T = + + + + + =
Napomena: Moe se krenuti i tako da se od drugog stupca oduzme
prvi s tim da se poslije
prvo popunjava drugi stupac.
2. Rijeite zadani problem transporta tako da poetno rjeenje
postavite metodom
sjeverozapadnog kornera, a testiranje optimalnosti provedite
metodom skakanja s kamena na
kamen:
ai
2 7 4 5
3 3 1 8
5 4 7 7
1 6 2 14
bj 7 9 18
-
2
Rjeenje:
5
2 6
3 4
14
5 0 0
2 6 02 5 3 2 3 6 4 3 7 4 2 14 102
0 3 4
0 0 14
NWX T
= = + + + + + =
12 12 2 3z c = 3+
13 13
7 5
2 3z c
=
= 3+
23 23
31 31
4 7 4 1
3 4 7 1 5
4 3
z c
z c
+ =
= + =
= 3+
41 41
5 1
2 7 4 3z c
=
= + 3+
42 42
1 2
2 7 4 6 7z c
=
= + =
5 -5 1
2 6 5
-1 3 4
-2 -7 14
23
32
22
33
0 4
3 4 7
6 4 2
4 4 0
x
x
x
x
= +
= + =
= =
= =
5
2 2 4
7
14
12 12 2 3z c = 3+
13 13
31 31
7 5
2 3 1 4 4
3
z c
z c
=
= + =
= 3
33 33
41 41
42 42
4 5 1
1 3 4 7 5
2 1 3 1 3
2 1 3 6 2
z c
z c
z c
+ =
= + =
= + =
= + =
5 -5 -4
2 2 4
-1 7 -5
3 -2 14
-
3
41
23
21
43
0 2 2
4 2 6
2 2 0
14 2 12
x
x
x
x
= + =
= + =
= =
= =
5
2 6
7
2 12
12 12
13 13
21 21
31 31
33 33
42 42
2 1 2 1 3 7 2
2 1 2 4 1
1 2 1 3 3
1 2 1 3 4 5 4
1 3 4 7 5
2 1 3 6 2
z c
z c
z c
z c
z c
z c
= + + =
= + =
= + =
= + + =
= + =
= + =
5 0 0
0 2 62 5 3 2 1 6 4 7 1 2 2 12 76.
0 7 0
2 0 12
X T
= = + + + + + =
3. Odredite optimalan program transporta na osnovu podataka iz
tablice:
ai
5 4 6 3 30
7 8 2 2 60
bj 10 40 20 30
Rjeenje:
ai
5 4 6 3 30
7 8 2 2 60
0 0 0 0 10
bj 10 40 20 30
Metoda NW kornera
10 20
20 20 20
10
10 20 0 0
0 20 20 20 5 10 4 20 8 20 2 20 2 20 370
0 0 0 10
NWX T
= = + + + + =
-
4
13 13
14 14
21 21
31 31
32 32
33 33
4 8 2 6 8
4 8 2 3 5
8 4 5 7 2
0 2 8 4 5 0 7
0 2 8 0 6
0 2
z c
z c
z c
z c
z c
z c
= + =
= + =
= + =
= + + =
= + =
= 2+ 0 0 =
10 20 -8 -5
2 20 20 20
7 6 0 10
31
24
12
34
22
11
0 10 10
20 10 30
20 10 30
10 10 0
20 10 10
10 10 0
x
x
x
x
x
x
= + =
= + =
= + =
= =
= =
= =
30
10 20 30
10
degeneracija!
ai
5 4 6 3 30+c
7 8 2 2 60+c
0 0 0 0 10+c
bj 10 40 20 30+3c
Metoda uzajamno preferiranih tokova
30+c
10-2c 20 30+3c
10 c
11 11
13 13
14 14
21 21
33 33
34 34
0 0 4 5 1
4 8 2 6 8
4 8 2 3 5
0 0 8 7 1
2 8 0 0 6
2 8 0 0 6
z c
z c
z c
z c
z c
z c
= + =
= + =
= + =
= + =
= + =
= + =
-1 30+c -8 -5
1 10-2c 20 30+3c
10 c -6 -6
-
5
( )
( )
21
32
31
22
0 10 2 10 2
10 2 10
10 10 2 2
10 2 10 2 0
x c c
x c c c
x c c
x c c
= + =
= + =
= =
= =
30+c
10-2c 20 30+3c
2c 10-c
11 11
13 13
14 14
22 22
33 33
34 34
0 0 4 5 1
4 0 0 7 2 6 7
4 0 0 7 2 3 4
7 0 0 8 1
2 7 0 0 5
2 7 0 0 5
z c
z c
z c
z c
z c
z c
= + =
= + + =
= + + =
= + =
= + =
= + =
0 30 0 0
10 0 20 30 4 30 7 10 2 20 2 30 290.
0 10 0 0
X T
= = + + + =
5. Rijeite zadani problem transporta tako da poetno rjeenje
postavite metodom uzajamno
preferiranih tokova.
ai
2 1 3 2 3
1 4 2 3 5
2 1 3 3 1
4 2 1 2 4
bj 1 4 6 1
Rjeenje:
ai
2 1 3 2 0 3
1 4 2 3 0 5
2 1 3 3 0 1
4 2 1 2 0 4
bj 1 4 6 1 1
Metoda uzajamno preferiranih tokova
3
1 1 2 1
1
4
degeneracija!
-
6
ai
2 1 3 2 0 3+c
1 4 2 3 0 5+c
2 1 3 3 0 1+c
4 2 1 2 0 4+c
bj 1 4 6 1 1+4c
Metoda uzajamno preferiranih tokova + MODI metoda
ai ui
-4 3+c -4 -2 -3 3+c 0
1 1-c 2-c 1 3c 5+c 3
-1 3 -1 0 1+c 1+c 3
-4 1 4+c 0 -1 4+c 2
bj 1 4 6 1 1+4c
vj -2 1 -1 0 -3
( )
( )
0
32
25
35
22
1 3 1 1 4 1 2 2 3 1 1 4 19
0 1 1
3 1 1 2
1 1 2
1 1 0
z
x c c
x c c c
x c c c
x c c
= + + + + + =
= + =
= + = +
= + =
= =
ai ui
-1 3+c -1 1 0 3+c 0
1 -3 2-c 1 1+2c 5+c 0
-1 1-c -1 0 2c 1+c 0
-4 -2 4+c 0 -1 4+c -1
bj 1 4 6 1 1+4c
vj 1 1 2 3 0
( ) ( )1 0 32 32
14
32
25
12
35
24
19 1 3 16
0 2 2
1 2 1
1 2 2 1 4
3 2 3
2 2 0
1 2
z z c z
x c c
x c c c
x c c c
x c c c
x c c
x c
= + = + =
= + =
= + = +
= + + = +
= + =
= =
=
ai ui
-2 3-c -2 2c -1 3+c 0
1 -2 2-c 1-2c 1+4c 5+c 1
-2 1+c -2 -1 -1 1+c 0
-4 -1 4+c 0 -1 4+c 0
bj 1 4 6 1 1+4c
vj 0 1 1 2 -1
-
7
( ) ( )
( )
( )
1
1 14 14
44
23
24
43
0 3 0 0 0
1 0 2 1 1
0 1 0 0 0
0 0 4 0 0
T 16 0 1 16
0 1 2 1 2
2 1 2 3 3
1 2 1 2 0
4 1 2 3 3
X
z c z
x c c
x c c c
x c c
x c c c
=
= + = + =
= + =
= + =
= =
= + = +
ai ui
-2 3-c -2 2c -1 3+c 0
1 -2 3-3c 0 1+4c 5+c 1
-2 1+c -2 -1 -1 1+c 0
-4 -1 3+3c 1-2c -1 4+c 0
bj 1 4 6 1 1+4c
vj 0 1 1 2 -1
( ) [ ]
2
1 2
0 3 0 0 0
1 0 3 0 1
0 1 0 0 0
0 0 3 1 0
1 , 0,1 .
X
X X X
=
= +
6. Rijeite zadani problem transporta tako da poetno rjeenje
postavite Vogelovom metodom.
ai
6 7 4 9 40
3 3 8 12 40
9 7 7 9 60
11 7 2 6 20
3 5 2 9 20
bj 90 60 20 50
Rjeenje:
ai
6 7 4 9 40+c
3 3 8 12 40+c
9 7 7 9 60+c
11 7 2 6 20+c
3 5 2 9 20+c
0 0 0 0 40+c
bj 90 60 20 50+6c
-
8
Vogelova metoda + MODI metoda
ai ui
40+c -3 -2 -3 40+c 0
2 40+c -7 -7 40+c -1
30-2c 20-c -2 10+4c 60+c 3
-5 -3 20 c 20+c 0
20+c -4 -3 -6 20+c -3
0 -2 -4 40+c 40+c -6
bj 90 60 20 50+6c
vj 6 4 2 6
( )
( )
0
21
32
22
31
6 40 3 40 9 30 7 20 9 10 2 20 3 20 960
0 30 2 30 2
20 30 2 50 3
40 30 2 10 3
30 2 30 2 0
z
x c c
x c c c
x c c c
x c c
= + + + + + + =
= + =
= + =
= + = +
= =
ai ui
40+c -1 0 -1 40+c 0
30-2c 10+3c -7 -7 40+c -3
-2 50-3c -2 10+4c 60+c 1
-7 -3 20 c 20+c -2
20+c -2 -1 -4 20+c -3
-2 -2 -4 40+c 40+c -8
bj 90 60 20 50+6c
vj 6 6 4 8
( ) ( )
1
0 21 21
13
21
32
40 0 0 0
30 10 0 0
0 50 0 10
0 0 20 0
20 0 0 0
0 0 0 40
T 960 30 2 900
0 10 3 10 3
30 2 10 3 40
50 3
X
z c z
x c c
x c c c
x c
=
= + = + =
= + + = +
= + + = +
= 10 3c+ +
( )
( )
( )
( )
44
11
22
34
43
60
10 3 10 4
40 10 3 30 2
10 3 10 3 0
10 4 10 3
20 10 3 10 3
x c c c
x c c c
x c c
x c c c
x c c
=
= + + = +
= + + =
= + + =
= + + =
= + =
-
9
ai ui
30-2c -1 10+3c -1 40+c 0
40+c 0 -7 -7 40+c -3
-2 60 -2 c 60+c 1
-7 -3 10-3c 10+4c 20+c -2
20+c -2 -1 -4 20+c -3
-2 -2 -4 40+c 40+c -8
bj 90 60 20 50+6c
vj 6 6 4 8
( ) [ ]
2
1 2
30 0 10 0
40 0 0 0
0 60 0 0
0 0 10 10
20 0 0 0
0 0 0 40
1 , 0,1 .
X
X X X
=
= +
7. Zadana je tablica jedininih trokova transporta:
ai
8 2 6 5 55
11 9 13 4 65
3 5 10 7 80
4 9 8 10 30
bj 110 60 20 40
Odredite optimalno rjeenje ako su kriterij optimalnosti
a) minimalni, b) maksimalni trokovi.
Rjeenje:
a)
Metoda uzajamno preferiranih tokova + MODI metoda
ai ui
-6 55+c 0 -8 55+c 0
-2 5-c 20-2c 40+4c 65+c 7
80+c -2 -3 -9 80+c 1
30-c -5 2c -11 30+c 2
bj 110 60 20 40+4c
vj 2 2 6 -3
1
0 55 0 0
0 5 20 40
80 0 0 0
30 0 0 0
X
=
-
10
( )
( )
min
13
22
12
23
T 2 55 9 5 13 20 4 40 3 80 4 30 935
0 20 2 20 2
5 20 2 25 3
55 20 2 35 3
20 2 20 2 0
x c c
x c c c
x c c c
x c c
= + + + + + =
= + =
= + =
= + = +
= =
ai ui
-6 35+3c 20-2c -8 55+c 0
-2 25-3c 0 40+4c 65+c 7
80+c -2 -3 -9 80+c 1
30-c -5 2c -11 30+c 2
bj 110 60 20 40+4c
vj 2 2 6 -3
( ) [ ]
2
1 2
0 35 20 0
0 25 0 40
80 0 0 0
30 0 0 0
1 , 0,1 ;
X
X X X
=
= +
b)
Metoda uzajamno preferiranih tokova + MODI metoda
ai ui
55 8 4 7 55 0
45 4 20 11 65 3
10 60 -5 10 80 -5
2 -1 0 30 30 -2
bj 110 60 20 40
vj 8 10 10 12
0
33
21
23
31
8 55 11 45 13 20 3 10 5 60 7 10 10 30 1895
0 10 10
45 10 55
20 10 10
10 10 0
z
x
x
x
x
= + + + + + + =
= + =
= + =
= =
= =
ai ui
55 3 4 2 55 0
55 -1 10 6 65 3
5 60 10 10 80 0
7 -1 5 30 30 3
bj 110 60 20 40
vj 8 5 10 7
-
11
( )1 0 33 33
22
33
23
32
1895 10 5 1945
0 10 10
10 10 20
10 10 0
60 10 50
z z c z
x
x
x
x
= + = + =
= + =
= + =
= =
= =
ai ui
55 4 5 3 55 0
55 10 1 7 65 3
4 50 20 10 80 -1
6 -1 5 30 30 2
bj 110 60 20 40
vj 8 6 11 8
( )2 1 22 22
42
34
32
44
1945 10 1 1955
0 30 30
10 30 40
50 30 20
30 30 0
z z c z
x
x
x
x
= + = + =
= + =
= + =
= =
= =
ai ui
55 4 5 3 55 0
55 10 1 7 65 3
4 20 20 40 80 -1
7 30 6 1 30 3
bj 110 60 20 40
vj 8 6 11 8
( )max 2 32 32
55 0 0 0
55 10 0 0, T 1955 30 1 1985.
0 20 20 40
0 30 0 0
X z c z
= = + = + =
8.
ai
3 5 2 8 1 20+c
4 2 9 7 6 80+c
5 3 3 1 2 100+c
0 0 0 0 0 10+c
bj 50 80 15 25 40+4c
-
12
Metoda uzajamno preferiranih tokova + MODI metoda
ai ui
1 -3 0 -8 20+c 20+c 0
c 80 -7 -7 -5 80+c 0
40-2c 0 15 25 20+3c 100+c 1
10+c -2 -2 -4 -3 10+c -4
bj 50 80 15 25 40+4c
vj 4 2 2 0 1
( )
( )
0
11
35
15
31
1 20 2 80 5 40 3 15 1 25 2 20 490
0 20 20
20 3 20 40 4
20 20 0
40 2 20 20 3
z
x c c
x c c c
x c c
x c c c
= + + + + + =
= + + = +
= + + + = +
= + + =
= + =
ai ui
20+c -4 -1 -9 -1 20+c 0
c 80 -7 -7 -5 80+c 1
20-3c 0 15 25 40+4c 100+c 2
10+c -2 -2 -4 -3 10+c -3
bj 50 80 15 25 40+4c
vj 3 1 1 -1 0
( ) ( )
( )
( )
1
0 11 11
32
21
22
31
20 0 0 0 0
0 80 0 0 0
20 0 15 25 40
10 0 0 0 0
T 490 20 1 470
0 20 3 20 3
20 3 20 2
80 20 3 60 3
20 3 20 3 0
X
z c z
x c c
x c c c
x c c
x c c
=
= + = + =
= + =
= + =
= = +
= =
ai ui
20+c -4 -1 -9 -1 20+c 0
20-2c 60+3c -7 -7 -5 80+c 1
0 20-3c 15 25 40+4c 100+c 2
10+c -2 -2 -4 -3 10+c -3
bj 50 80 15 25 40+4c
vj 3 1 1 -1 0
-
13
( ) [ ]
2
1 2
0
0
20 0 0 0 0
20 60 0 0 0
0 20 15 25 40
10 0 0 0 0
1 , 0,1 ;
T 3 10 1 10 2 60 9 15 7 5 5 40 3 10 1 20 2 30 640;
T T 640 470 170.
X
X X X
=
= +
= + + + + + + + + =
= =
9.
a) Odredite optimalno rjeenje problema transporta s minimalnim
trokovima (mogunost
degeneracije) ako je zadano:
ai
5 4 3 6 100
3 6 3 3 200
5 7 3 4 80
3 5 4 5 90
bj 150 50 220 80
b) Opiite znaenje koliine tereta u fiktivnom ishoditu.
c) Napiite funkciju cilja dualnog problema i odredite optimalne
vrijednosti dualnih varijabli.
Provjerite, na ovom primjeru, ispravnost fundamentalnog teorema
dualiteta.
Rjeenje:
a)
ai
5 4 3 6 100+c
3 6 3 3 200+c
5 7 3 4 80+c
3 5 4 5 90+c
0 0 0 0 30+c
bj 150 50 220 80+5c
Metoda uzajamno preferiranih tokova + MODI metoda
ai ui
-1 0 100+c -2 100+c 0
150 -3 -1 50+c 200+c -1
-1 -3 80+c 0 80+c 0
2 20-c 40-2c 30+4c 90+c 1
0 30+c -1 0 30+c -4
bj 150 50 220 80+5c
vj 4 4 3 4
-
14
( )
( )
0
41
24
21
44
3 100 3 150 3 50 3 80 5 20 4 40 5 30 1550
0 30 4 30 4
50 30 4 80 5
150 30 4 120 4
30 4 30 4 0
z
x c c
x c c c
x c c
x c c
= + + + + + + =
= + + = +
= + + + = +
= + =
= + + =
ai ui
-3 0 100+c -4 100+c 0
120-4c -1 1 80+5c 200+c 1
-3 -3 80+c -2 80+c 0
30+4c 20-c 40-2c -2 90+c 1
-2 30+c -1 -2 30+c -4
bj 150 50 220 80+5c
vj 2 4 3 2
( ) ( )
( )
( )
1 0 41 41
23
41
21
43
1550 30 2 1490
0 40 2 40 2
30 4 40 2 70 2
120 4 40 2 80 2
40 2 40 2 0
z z c z
x c c
x c c c
x c c c
x c c
= + = + =
= + =
= + + = +
= =
= =
ai ui
-2 1 100+c -3 100+c 0
80-2c -1 40-2c 80+5c 200+c 0
-2 -2 80+c -1 80+c 0
70+2c 20-c -1 -2 90+c 0
-2 30+c -2 -2 30+c -5
bj 150 50 220 80+5c
vj 3 5 3 3
( ) ( )
( )
( )
( )
2 1 23 23
12
41
23
13
21
42
1490 40 1 1450
0 20 20
70 2 20 90
40 2 20 60 3
100 20 80 2
80 2 20 60
20 20 0
z z c z
x c c
x c c c
x c c c
x c c c
x c c c
x c c
= + = + =
= + =
= + + = +
= + =
= + = +
= =
= =
-
15
ai ui
-2 20-c 80+2c -3 100+c 0
60-c -2 60-3c 80+5c 200+c 0
-2 -3 80+c -1 80+c 0
90+c -1 -1 -2 90+c 0
-1 30+c -1 -1 30+c -4
bj 150 50 220 80+5c
vj 3 4 3 3
( ) ( )2 12 12
0 20 80 0
60 0 60 80
0 0 80 0
90 0 0 0
0 30 0 0
T 1450 20 1 1430
X
z c z
=
= + = + =
b) Potranja drugog odredita nije u potpunosti zadovoljena (jer
je ostalo mjesta za 30
jedinica robe) zbog fiktivne ponude 5 30.a =
c) ( )1 2 3 4 5 1 2 3 4
1 1
1 2
1 3
1 4
2 1
2 2
2 3
2 4
3 1
3 2
3 3
3 4
4 1
4 2
4 3
4 4
5 1
5 2
5 3
5 4
max 100 200 80 90 30 150 50 220 80
5
4
3
6
3
6
3
3
5
7
3
4
3
5
4
5
0
0
0
0
u u u u u v v v v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
u v
+ + + + + + + +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-
16
( )
( )
1
2
3
4
5
1
2
3
4
max
0 0 0
0 0 0
0 0 0
0 0 0
4 0 4
max
3 4 1
4 4 0
3 4 1
3 4 1
i i i
j j j
u u u
u
u
u
u
u
v v v
v
v
v
v
=
= =
= =
= =
= =
= =
=
= =
= =
= =
= =
-
17
14.
Gradovi 1 2 3 4
1 110 100 200 100
2 110 60 190 60
3 100 60 80 60
4 200 190 80 80
40 0 0 20 360
1
360
Gradovi 1 2 3 4
1 10 100 80
2 10 100 110
3 100 100 800
4 80 110 800
Gradovi 1 2 3
1 10 0 0
2 10 0 0
4 80 110 80
0 10 0 90
1
360
32
( )3,4 ( )3,4
440 450
Gradovi 1 2 3 4
1 110 100 200 100
2 110 60 190 60
3 100 60 60
4 200 190 80 80
40 0 0 100 440
Gradovi 1 2 3 4
1 10 00 300
2 10 100 30
3 100 100
4 80 110 800
-
18
Gradovi 1 2 4
1 10 0 0
2 10 30 10
3 0 0 0
0 0 0 10
1
360
32
( )3,4 ( )3,4
440 450
4 520 5 450
( )4,3( )4, 3
Gradovi 1 2 3
1 200 00
2 10 100
4 300 20
Gradovi 2 3
1 0 0
2 0 0
0 0 0
1
360
32
( )3,4 ( )3,4
440 450
4 520 5 450
( )4,3( )4, 36 7 450480
( )4,1( )4,1
1 2 3 4 1, T 450 =
-
19
Gradovi 1 2 4
1 10 300
2 200 20
3 00 100
Gradovi 1 2
2 0 0
3 0 0
0 0 0
1
360
32
( )3,4 ( )3,4
440 450
4 520 5 450
( )4,3( )4, 36 7 450480
( )4,1( )4,1
8 480 9 450
( )1, 4( )1, 4
1 4 3 2 1, T 450. =
15.
Strojevi 1 2 3 4 5
1 20 35 18 18 18
2 20 26 24 25 20
3 35 26 21 23 21
4 17 30 18 21 17
5 18 25 23 21 18
0 2 1 0 0 97
1
97
-
20
Strojevi 1 2 3 4 5
1 30 16 00 20
2 40 5 4 5
3 14 3 20 2
4 00 11 40 4
5 30 5 4 3
Strojevi 2 3 4 5
1 16 0 0 0
3 3 0 2 0
4 11 0 4 0
5 5 4 3 3
2 0 0 0 5
1
97
32 101 102
( )2,1 ( )2,1
Strojevi 1 2 3 4 5
1 20 35 18 18 18
2 26 24 25 24
3 35 26 21 23 21
4 17 30 18 21 17
5 18 25 23 21 18
0 2 1 0 0 101
Strojevi 1 2 3 4 5
1 30 16 00 10
2 2 10 1
3 14 3 20 2
4 00 11 20 4
5 30 5 4 3
Strojevi 1 3 4 5
2 2 0 1 0
3 14 0 2 0
4 0 0 4 0
5 0 4 3 0
0 0 0 1 1
-
21
1
97
32 101
4 104 5 102
( )2,1 ( )2,1
( )1, 2 ( )1, 2
102
Strojevi 2 3 4 5
1 16 00 20
3 1 10 2
4 9 50 4
5 10 1 00
Strojevi 2 4 5
1 0 0 0
3 1 2 1
5 0 0 0
0 0 0 1
1
97
2 101
4 104 5 102
( )2,1 ( )2,1
( )1, 2 ( )1, 2
3 102
6 7 103107
( )4,3( )4,3
Strojevi 1 3 4 5
2 2 00 10
3 14 10 1
4 00 20 3
5 30 4 3
-
22
Strojevi 3 4 5
2 2 0 0
3 0 1 0
4 0 3 0
0 0 1 1
1
97
2 101
4 104 5 102
( )2,1 ( )2,1
( )1, 2 ( )1, 2
3 102
6 7 103107
( )4,3( )4,3
8 9 105103
( )5,1 ( )5,1
Strojevi 3 4 5
2 2 20
3 00 20
4 40 2
Strojevi 4 5
2 0 0
3 0 0
0 0 0
-
23
1
97
2 101
4 104 5 102
( )2,1 ( )2,1
( )1, 2 ( )1, 2
3 102
6 7 103107
( )4,3( )4,3
8 9 105103
( )5,1 ( )5,1
10 103 11 107
( )4,3 ( )4,3
1 2 4 3 5 1, t 103 =
Strojevi 2 4 5
1 00 10
3 10 1
5 00 00
Strojevi 2 4
3 0 0
5 0 0
0 0 0
-
24
1
97
2 101
4 104 5 102
( )2,1 ( )2,1
( )1, 2 ( )1, 2
3 102
6 7 103107
( )4,3( )4,3
8 9 105103
( )5,1 ( )5,1
10 103 11 107
( )4,3 ( )4,3
12 13 103104
( )1, 5( )1, 5
1 5 4 3 2 1, t 103. =
16.
Gradovi 1 2 3 4 5
1 5 8 7 8 5
2 1 2 4 5 1
3 5 6 1 2 1
4 3 4 5 6 3
5 7 8 1 2 1
0 0 0 0 1 12
1
12
Gradovi 1 2 3 4 5
1 30 3 2 2
2 10 1 3 3
3 4 5 10 20
4 10 1 2 2
5 6 7 20 1
-
25
Gradovi 1 3 4 5
2 1 3 3 1
3 4 0 0 0
4 0 2 2 0
5 6 0 1 0
0 0 0 0 1
1
12
2 13 3 15
( )1, 2 ( )1, 2
Gradovi 1 3 4 5
2 20 2 2
3 4 10 20
4 60 2 2
5 6 10 1
Gradovi 3 4 5
2 0 2 0
3 0 0 0
5 0 1 0
0 0 0 0
1
12
2 13
4 19 5 13
( )1, 2
( )4,1 ( )4,1
3 15
( )1, 2
Gradovi 3 4 5
2 20 2
3 10 20
5 10 1
-
26
Gradovi 4 5
3 0 0
5 1 1
0 0 1
1
12
2 13
4 19 5 13
( )1, 2
( )4,1 ( )4,1
3 15
( )1, 2
6 15 7 14
( )2,3( )2,3
Gradovi 4 5
3 0
5 0
1 2 3 5 4 1, T 14. =
17.
Gradovi 1 2 3 4 5
1 60 34 28 20 20
2 42 22 20 56 20
3 32 32 32 60 32
4 52 36 32 28 28
5 24 52 48 26 24
0 0 2 0 0 126
1
126
-
27
Gradovi 1 2 3 4 5
1 40 12 8 80
2 22 20 00 36
3 00 80 00 28
4 24 8 2 20
5 20 28 22 2
Gradovi 1 2 3 4
2 22 0 0 0
3 0 0 0 0
4 24 8 2 2
5 28 22 2 2
0 0 0 0 4
1
126
2 130 3 134
( )1, 5 ( )1, 5
Gradovi 1 2 3 4
2 22 00 00
3 220 60 00
4 22 6 60
5 26 20 200
Gradovi 2 3 4
2 0 0 0
4 6 0 0
5 26 0 0
6 0 0 6
-
28
1
126
2 130
4 152 5 136
3 134
( )1, 5 ( )1, 5
( )3,1( )3,1
Gradovi 1 2 3 4 5
1 60 34 28 28
2 42 22 20 56 20
3 32 32 32 60 32
4 52 36 32 28 28
5 24 52 48 26 24
0 0 2 0 0 134
Gradovi 1 2 3 4 5
1 32 4 40
2 22 20 00 36
3 00 80 00 28
4 24 8 2 300
5 20 28 22 2
Gradovi 1 2 3 4
1 32 4 0 0
2 22 0 0 0
3 0 0 0 0
5 0 28 22 0
0 0 0 0 0
-
29
1
126
2 130
4 152 5 136
3 134
( )1, 5 ( )1, 5
( )3,1( )3,16 7 134164
( )4, 5( )4, 5
Gradovi 1 2 3 4
1 32 4 40
2 22 40 00
3 00 280 00
5 220 28 22
Gradovi 1 3 4
1 4 0 0
2 22 0 0
5 0 22 0
0 4 0 4
1
126
2 130
4 152 5 136
3 134
( )1, 5 ( )1, 5
( )3,1( )3,16 7 134164
( )4, 5( )4, 5
8 9 162138
( )3,2 ( )3,2
Gradovi 2 3 4
2 00 00
4 200 00
5 20 200
-
30
Gradovi 3 4
2 0 0
5 0 0
0 0 0
1
126
2 130
4 152 5 136
3 134
( )1, 5 ( )1, 5
( )3,1( )3,16 7 134164
( )4, 5( )4, 5
8 9 162138
( )3,2 ( )3,2
10 156 11 136
( )4, 2 ( )4, 2
1 5 4 2 3 1, T 136. =
-
31
18.
Gradovi 1 2 3 4 5
1 26 36 28 38 26
2 26 42 34 46 26
3 36 42 50 40 36
4 28 34 50 30 28
5 38 46 40 30 30
0 0 10 0 2 158
1
158
Gradovi 1 2 3 4 5
1 60 00 2 10
2 60 6 8 18
3 20 6 14 2
4 00 6 12 20
5 8 16 00 20
Gradovi 1 3 4 5
2 6 8 18 6
3 0 14 2 0
4 0 12 0 0
5 8 0 0 0
0 0 0 0 6
1
158
2 164 3 164
( )1, 2 ( )1, 2
Gradovi 1 3 4 5
2 20 2 12
3 20 14 2
4 00 12 20
5 8 00 20
Gradovi 1 4 5
3 14 2 2
4 0 0 0
5 8 0 0
0 0 0 2
-
32
1
158
2 164 3 164
( )1, 2 ( )1, 2
4 166 5 166
( )2,3 ( )2,3
Gradovi 1 2 3 4 5
1 36 28 38 28
2 26 42 34 46 26
3 36 42 50 40 36
4 28 34 50 30 28
5 38 46 40 30 30
0 6 8 0 2 164
Gradovi 1 2 3 4 5
1 20 00 8
2 80 8 8 18
3 00 00 14 2
4 00 00 14 20
5 8 10 2 20
Gradovi 2 3 4 5
1 0 0 8 0
3 0 14 2 0
4 0 14 0 0
5 10 2 0 0
0 0 0 0 0
-
33
1
158
2 164
( )1, 2 ( )1, 2
4 166 5 166
( )2,3 ( )2,3
3 164
6 7 164172
( )2,1( )2,1
Gradovi 2 3 4 5
1 20 00 8
3 20 14 2
4 00 14 20
5 10 2 20
Gradovi 2 4 5
3 14 2 2
4 0 0 0
5 10 0 0
0 0 0 2
1
158
2 164
4 166 5 166
3 164
( )1, 2 ( )1, 2
( )2,3 ( )2, 36 7 164172
( )2,1( )2,1
8 166 9 166
( )1, 3 ( )1, 3
-
34
Gradovi 1 4 5
3 12 120
4 80 00
5 8 200
Gradovi 1 5
3 0 0
4 0 0
0 0 0
1 2 3 5 4 1, T 166 = preko vora 4;
Gradovi 2 4 5
3 12 120
4 100 00
5 10 220
Gradovi 2 5
3 0 0
4 0 0
0 0 0
1 3 5 4 2 1, T 166 = preko vora 8;
Gradovi 1 3 4 5
2 8 18 8
3 0 14 2 0
4 0 12 0 0
5 8 0 0 0
0 0 0 0 8
Gradovi 1 3 4 5
2 100 10
3 20 14 2
4 00 12 20
5 8 120 00
Gradovi 1 4 5
2 0 10 0
3 0 14 0
4 0 0 0
0 0 0 0
Gradovi 1 4 5
2 240 10
3 140 14
4 00 100
-
35
Gradovi 1 5
3 0 0
4 0 0
0 0 0
1 2 4 5 3 1, T 166 = preko vora 5;
Gradovi 2 3 4 5
1 0 8 0
3 0 14 2 0
4 0 14 0 0
5 10 2 0 0
0 2 0 0 2
Gradovi 2 3 4 5
1 80 8
3 20 14 2
4 00 12 20
5 10 120 00
Gradovi 2 4 5
1 0 8 0
3 0 14 0
4 0 0 0
0 0 0 0
Gradovi 2 4 5
1 220 8
3 140 14
4 00 80
Gradovi 2 5
3 0 0
4 0 0
0 0 0
1 4 5 3 2 1, T 166 = preko vora 9.
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