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NETWORK MODELS
Short questions
1. What is a network problem?
2. What is unimodularity? Why are transportation and assignment problems
unimodular?
3. What is a spanning tree and minimum spanning tree?
4. Name two algorithms used to find the MST? Explain them?
5. Formulate the shortest path problem as a LP?
6. Is the shortest path problem unimodular?
7. How does the Dijkstra’s algorithm work when there are negative cost arcs?
8. How do we find the kth shortest path in a given network?
9. Which algorithm do we use when we have negative cost arcs in the shortest path
problem
10. How do you solve a constrained shortest path problem?
11. Define the maximum flow problem?
12. Formulate the maximum flow problem as a LP?
13. Write and explain the max flow min cut theorem?
14. Write the dual of the maximum flow problem and relate it to the cut problem?
15. What is a cut in the context of the maximum flow problem?
16. Show that every cut in the maximum flow problem can be represented as a
feasible solution to the dual of the maximum flow problem?
17. What is a flow augmenting path?
18. What is a backward arc and how do you explain the concept of flow using a
backward arc?
19. What is the significance of the shortest augmenting path algorithm?
20. What is a minimum cost flow problem?
21. Formulate the minimum cost flow problem as a LP?
22. How do you reduce the minimum cost flow problem to shortest path and
maximum flow problem?
23. What is a transhipment problem and how is it different from a transportation
problem?
24. How do you solve a transhipment problem as a transportation problem?
25. Explain the steps in the algorithm to solve the minimum cost flow problem?
Problems
1. You are given a network with six nodes and nine arcs. The data is shown in the
following Table
Arc
1-2
1-3
2-4
Length
14
6
12
Arc
1-4
1-5
3-5
length
8
11
9
Arc
4-6
3-6
5-6
Length
13
13
15
Find the length of the minimum spanning tree using Prim’s algorithm?
2. For the data given in problem 1 find length of the minimum spanning tree using
Kruskal’s algorithm?
3. You are given a network with eight nodes and fifteen arcs. The data is shown in
the following Table
Arc
1-2
1-3
1-4
1-6
2-3
Length
14
6
12
9
10
Arc
2-5
3-4
3-7
3-8
4-8
length
8
11
9
9
12
Arc
5-6
5-7
5-8
6-8
7-8
Length
13
13
15
7
3
Find the length of the minimum spanning tree using Prim’s algorithm?
4. For the data given in problem 3 find length of the minimum spanning tree using
Kruskal’s algorithm?
5. You are given a network with six nodes and nine arcs. The distance between the
points is given in the following Table
Arc
1-2
1-3
2-4
Distance
14
16
15
Arc
1-4
1-5
3-5
Distance
18
12
19
Arc
4-6
3-6
5-6
Distance
14
13
15
Find the length of the shortest path between nodes 1 and 6?
6. Consider the network given in problem 5. Find the shortest path and distance
between node 1 and all the nodes using the Dijkstra’s algorithm?
7. Consider the network given in problem 5. Find the shortest path and distance
between all nodes and 6 using Dijkstra’s algorithm?
8. Consider the network given in problem 5. Find the second and third shortest path
between nodes 1 and 6 using successive application of Dijkstra’s algorithm?
9. You are given a network with eight nodes and fifteen arcs. The distance between
the nodes is shown in the following Table
Arc
1-2
1-3
1-4
1-6
2-3
Distance
14
16
12
17
12
Arc
2-5
3-4
3-7
3-8
4-8
distance
12
11
9
10
11
Arc
5-6
5-7
5-8
6-8
7-8
distance
12
13
16
17
13
Find the shortest path and the distance between nodes 1 and 7?
10. Consider the network given in problem 9. Find the shortest path and distance
between node 1 and all the nodes using the Dijkstra’s algorithm?
11. Consider the network given in problem 9. Find the shortest path and distance
between all nodes and 8 using Dijkstra’s algorithm?
12. Consider the network given in problem 9. Find the second and third shortest path
between nodes 1 and 8 using successive application of Dijkstra’s algorithm?
13. You are given a network with six nodes and nine arcs. The capacity of the arcs is
given in the following Table
Arc
1-2
1-3
2-4
Capacity
24
30
25
Arc
1-4
1-5
3-5
Capacity
28
32
30
Arc
4-6
3-6
5-6
Capacity
34
30
25
Find the maximum flow between nodes 1 and 6? Start with the path 1-3-5-6 followed
by the path 1-2-4-6.
14. Consider the data given in problem 13. Find the minimum cut set corresponding
to the optimum solution in problem 14.
15. Consider the data given in problem 13. Find the maximum flow between 1 and 6
using the shortest augmenting path algorithm?
16. You are given a network with eight nodes and fifteen arcs. The data is shown in
the following Table
Arc
1-2
1-3
1-4
1-6
2-3
Capacity
34
26
32
17
16
Arc
2-5
3-4
3-7
3-8
4-8
capacity
25
15
29
19
17
Arc
5-6
5-7
5-8
6-8
7-8
Capacity
13
14
16
17
18
Find the maximum flow between nodes 1 and 8? Start with the path 1-2-5-7-8
followed by the path 1-6-5-2-3-7-8.
17. Consider the data given in problem 16. Find the minimum cut set corresponding
to the optimum solution in problem 16.
18. Consider the data given in problem 16. Find the maximum flow between 1 and 8
using the shortest augmenting path algorithm?
19. You are given a network with six nodes and nine arcs. The cost of transporting a
unit of product between the nodes is given in the following Table
Arc
1-2
1-3
2-4
Capacity
24
30
25
Arc
1-4
1-5
3-5
Capacity
28
32
30
Arc
4-6
3-6
5-6
Capacity
34
30
25
Assume that nodes 1, 2 and 3 supply 30, 20 and 25 units of the products while
nodes 4 and 6 are destination nodes with demand 40 and 35. Find the optimum
solution to the minimum cost flow problem?
20. Consider the data given in problem 19. Solve a transportation problem to obtain
the minimum cost flow?