![Page 1: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/1.jpg)
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
Construction Technology and Management, Mályusz Levente Decision Support models
Theorem of Graph
• Leonhard Euler 1707-1783• Königsbergi hidak 1736
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 2: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/2.jpg)
2015.10.04.
2
Shortest Path problem• Algorithms
– Ford 1956– Bellman 1958– Dantzig 1958– Dijkstra 1959
Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 3: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/3.jpg)
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 4: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/4.jpg)
2015.10.04.
4
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
Backward 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 0
7 7
15 8 15
27 16 27
21 9 20 12 0Construction Technology and Management, Mályusz Levente Decision Support models
![Page 5: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/5.jpg)
2015.10.04.
5
Backward 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 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
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 6: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/6.jpg)
2015.10.04.
6
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 7: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/7.jpg)
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
Construction Technology and Management, Mályusz Levente Decision Support models
Backward from 4. 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 0
7 7
15 8 15
27 16 27
9 12 0
Construction Technology and Management, Mályusz Levente Decision Support models
Backward 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 0
7 7
15 8 15
27 16 27
21 9 20 12 0
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 8: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/8.jpg)
2015.10.04.
8
Backward 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 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
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 7 15 27Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 9: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/9.jpg)
2015.10.04.
9
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 10: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/10.jpg)
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
Construction Technology and Management, Mályusz Levente Decision Support models
New iteration 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 0
8 7 8
15 16 9 15
27 16 27
Construction Technology and Management, Mályusz Levente Decision Support models
New iteration 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 0
8 9 7 8
15 16 9 15
27 28 16 27
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 11: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/11.jpg)
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
Construction Technology and Management, Mályusz Levente Decision Support models
3. Iteration 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 0 0
9 8 9 7 8
16 17 15 16 9 15
28 27 28 16 27
Construction Technology and Management, Mályusz Levente Decision Support models
3. Iteration 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 0 0
9 10 8 9 7 8
16 17 15 16 9 15
28 29 27 28 16 27
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 12: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/12.jpg)
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
M T W Th F Sat Su M T W
5 5 5 5 5 5 5 5 5 5
M T W Th F Sat Su M T W
6 6 6 6 6 6 6 6 6 6
SS2
FS0
maxFF6
Construction Technology and Management, Mályusz Levente Decision Support models
MPM network and the relationship with digraph
5 5K B
-5
MPM networkdigráfban
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 13: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/13.jpg)
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
Construction Technology and Management, Mályusz Levente Decision Support models
Relationships of MPM and digraph
5 5S F
-5
PDM networkdigraph
7
SS3
7S F
-7
3maxSS4
3-4
Construction Technology and Management, Mályusz Levente Decision Support models
Relationships of MPM and a digraph
5 5K B
-5
PDM networkdigraph
7
SS3
7K B
-7
3maxFF5
-5
Construction Technology and Management, Mályusz Levente Decision Support models
![Page 14: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/14.jpg)
2015.10.04.
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Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
Construction Technology and Management, Mályusz Levente Decision Support models
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
![Page 15: dontesalgor 2bm angol - Budapest University of Technology ... · 2015.10.04. 1 Algorithmforlongestpath Genericshortestpathalgorithm Singlesourceshortestpathalgorithm Bellman-Ford](https://reader031.vdocument.in/reader031/viewer/2022041302/5e1295cc0936493e9c1a129d/html5/thumbnails/15.jpg)
2015.10.04.
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Construction Technology and Management, Mályusz Levente Decision Support models
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