dontesalgor 2bm angol - budapest university of technology ... · 2015.10.04. 1...

2015.10.04.

1

Algorithm for longest path

Generic shortest path algorithmSingle source shortest path algorithm

Bellman-Ford 1958

Construction Technology and Management, Mályusz Levente Decision Support models

Digraph

9

8

12

97

source

3

2

4

7

7

7

7

7

56

7

sink


Theorem of Graph

• Leonhard Euler 1707-1783• Königsbergi hidak 1736


2015.10.04.

2

Shortest Path problem• Algorithms

– Ford 1956– Bellman 1958– Dantzig 1958– Dijkstra 1959


Graph, network

• 1847 Kirchoff graph theory and application of electrical networks.1852 F. Guthrie: There are four colourproblem.1930 Kuratowski planar graph.1956-61 Ford-Fulkerson maximum flow, minimum cost flow1956 Polaris project, DuPont, RandCorporation, CPM, PERT, MPM, PDM


The longest path 1-4, the earliest and latest date policy

1 2 3 4

1 7 9

2 8 9

3 12

4

9

8

12

97

1

3

2

4


2015.10.04.

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From 1. node

1 2 3 4

1 7 9

2 8 9

3 12

4

9

8

12

97

1

3

2

4

0

7

9


From 2. node

1 2 3 4

1 7 9

2 8 9

3 12

4

9

8

12

97

1

3

2

4

0

7

9 15

16


From 3. node

1 2 3 4

1 7 9

2 8 9

3 12

4

9

8

12

97

1

3

2

4

0

7

9 15

16 27


2015.10.04.

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From 1., 2.,3., nodes

1 2 3 4

1 7 9

2 8 9

3 12

4

9

8

12

97

1

3

2

4

0 0

7 7

15 8 15

27 16 27


Backward from 4. node

1 2 3 4

1 7 9

2 8 9

3 12

4

9

8

12

97

1

3

2

4

0 0

7 7

15 8 15

27 16 27

9 12 0Construction Technology and Management, Mályusz Levente Decision Support models


1 2 3 4

1 7 9

2 8 9

3 12

4

9

8

12

97

1

3

2

4

0 0

7 7

15 8 15

27 16 27

21 9 20 12 0Construction Technology and Management, Mályusz Levente Decision Support models

2015.10.04.

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1 2 3 4

1 7 9

2 8 9

3 12

4

9

8

12

97

1

3

2

4

0 0

7 7

15 8 15

27 16 27

21 27 9 20 12 0Construction Technology and Management, Mályusz Levente Decision Support models

Latest date policy

1 2 3 4

1 7 9

2 8 9

3 12

4

9

8

12

97

1

3

2

4

0 0

7 7

15 8 15

27 16 27

21 27 9 20 12 0

0 7 15 27Construction Technology and Management, Mályusz Levente Decision Support models

Longest path 1-4 and lag times

1 2 3 4

1 7 9

2 8 9

3 -10 12

4

9

8

12

97

1

3

2

4-10


2015.10.04.

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From 1. node

1 2 3 4

1 7 9

2 8 9

3 -10 12

4

9

8

12

97

1

3

2

4-10

0

7

9


From 2. node

1 2 3 4

1 7 9

2 8 9

3 -10 12

4

9

8

12

97

1

3

2

4-10

0

7

9 15

16


From 3. node

1 2 3 4

1 7 9

2 8 9

3 -10 12

4

9

8

12

97

1

3

2

4-10

0

7

9 15

16 27


2015.10.04.

7

New iteration from 1,2,3. nodes

1 2 3 4

1 7 9

2 8 9

3 -10 12

4

9

8

12

97

1

3

2

4-10

0 0

7 7

15 8 15

27 16 27



1 2 3 4

1 7 9

2 8 9

3 -10 12

4

9

8

12

97

1

3

2

4-10

0 0

7 7

15 8 15

27 16 27

9 12 0



1 2 3 4

1 7 9

2 8 9

3 -10 12

4

9

8

12

97

1

3

2

4-10

0 0

7 7

15 8 15

27 16 27

21 9 20 12 0


2015.10.04.

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1 2 3 4

1 7 9

2 8 9

3 -10 12

4

9

8

12

97

1

3

2

4-10

0 0

7 7

15 8 15

27 16 27

21 27 9 20 12 0


Result: scheduling

1 2 3 4

1 7 9

2 8 9

3 -10 12

4

9

8

12

97

1

3

2

4-10

0 0

7 7

15 8 15

27 16 27

21 27 9 20 12 0


0

15

27

0

7

7

15

27

Longest path 1-4 and lag times

1 2 3 4

1 7 9

2 8 9

3 -7 12

4

9

8

12

97

1

3

2

4-7


2015.10.04.

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From 1. node

1 2 3 4

1 7 9

2 8 9

3 -7 12

4

9

8

12

97

1

3

2

4-7

0

7

9


From 2. node

1 2 3 4

1 7 9

2 8 9

3 -7 12

4

9

8

12

97

1

3

2

4-7

0

7

9 15

16


From 3. node

1 2 3 4

1 7 9

2 8 9

3 -7 12

4

9

8

12

97

1

3

2

4-7

0

7 8

9 15

16 27


2015.10.04.

10

New iteration from 1. node

1 2 3 4

1 7 9

2 8 9

3 -7 12

4

9

8

12

97

1

3

2

4-7

0 0

8 7 8

15 9 15

27 16 27



1 2 3 4

1 7 9

2 8 9

3 -7 12

4

9

8

12

97

1

3

2

4-7

0 0

8 7 8

15 16 9 15

27 16 27



1 2 3 4

1 7 9

2 8 9

3 -7 12

4

9

8

12

97

1

3

2

4-7

0 0

8 9 7 8

15 16 9 15

27 28 16 27


2015.10.04.

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3. Iteration from 1. node

1 2 3 4

1 7 9

2 8 9

3 -7 12

4

9

8

12

97

1

3

2

4-7

0 0 0

9 8 9 7 8

16 15 16 9 15

28 27 28 16 27



1 2 3 4

1 7 9

2 8 9

3 -7 12

4

9

8

12

97

1

3

2

4-7

0 0 0

9 8 9 7 8

16 17 15 16 9 15

28 27 28 16 27



1 2 3 4

1 7 9

2 8 9

3 -7 12

4

9

8

12

97

1

3

2

4-7

0 0 0

9 10 8 9 7 8

16 17 15 16 9 15

28 29 27 28 16 27


2015.10.04.

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Result: cycle, there is no solution

1 2 3 4

1 7 9

2 8 9

3 -7 12

4

8

12

97

1

3

2

4-7

0 0 0

9 10 8 9 7 8

16 17 15 16 9 15

28 29 27 28 16 27


9 9

Scheduling with calendar

55

6

M T W Th F Sat Su M T W

5 7 7 7 7 7 6 5 7 7


5 5 5 5 5 5 5 5 5 5


6 6 6 6 6 6 6 6 6 6

SS2

FS0

maxFF6


MPM network and the relationship with digraph

5 5K B

-5

MPM networkdigráfban


2015.10.04.

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Relationship between MPM and a digraph

5 5S F

-5

PDM networkdigraph

7

FS2

7S F

-7

2


Relationships of MPM and digraph

5 5S F

-5

PDM networkdigraph

7

SS3

7S F

-7

3maxSS4

3-4


Relationships of MPM and a digraph

5 5K B

-5

PDM networkdigraph

7

SS3

7K B

-7

3maxFF5

-5


2015.10.04.

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Normal critical activityactivity time increasing project

duration time increasing

FS0 FS0 FS0

FS0 FS0 FS0

0 3 3

3 3 6

6 3 9

0 3 3

3 2 5

5 3 8

0 3 3

3 4 7

7 3 10


activity time increasing project duration time decreasing

FF10 FF10 FF10

SS10 SS10 SS10

0 3 3

10 3 13

20 3 23

0 3 3

11 2 13

21 3 24

0 3 3

9 4 13

19 3 22


SS10 FF10 SS10 FF10 SS10 FF10

0 3 3

10 3 13

20 3 23

0 3 3

11 2 13

21 3 24

0 3 3

10 4 14

21 3 24

Bi- critical activityactivity time dec/increasing

project duration time increasing

SS10 FF10 SS10 FF10 SS10 FF10

2015.10.04.

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SS10 SS10 SS10

0 3 3

10 3 13

20 3 23

0 3 3

10 2 12

20 3 23

0 3 3

10 4 14

20 3 23

Neutral critical activityactivity time dec/increasing

project duration time does not change

SS10 SS10 SS10

dontesalgor 2bm angol - budapest university of technology ... · 2015.10.04. 1...

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