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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
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
2015.10.04.
3
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
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
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
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
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
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
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
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
2015.10.04.
11
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
2015.10.04.
12
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
2015.10.04.
13
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
2015.10.04.
14
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
2015.10.04.
15
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
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