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61) Find the shortest route from Node 1 to Node 6.
From
Node To
Node
Distance
1 2 150
1 3 200
2 3 100
2 4 200
2 5 50
3 4 350
3 5 300
4 6 100
5 6 100
A) 300
B) 450
C) 550
D) 650
E) None of the above
62) Given the following traffic flows, in hundreds of cars per hour, what is the maximum traffic flow from City 1
to City 7?
From City To City Flow
1 1 2 4
2 1 3 8
3 1 5 5
4 2 1 0
5 2 4 3
6 2 5 3
7 3 1 0
8 3 5 3
9 3 6 1
10 4 2 3
11 4 5 3
12 4 7 4
13 5 1 1
14 5 2 0
15 5 3 2
16 5 4 0
17 5 6 1
18 5 7 5
19 6 3 1
20 6 5 4
21 6 7 1
22 7 4 2
23 7 5 1
24 7 6 0
A) 1200
B) 1400
C) 900
D) 800
E) None of the above
63) Solve the minimal-spanning tree problem defined below:
Branch Start Node End Node Cost
1 1 3 5
2 1 2 1
3 2 4 3
4 2 5 4
5 3 4 6
6 4 6 2
A) total cost = 13
B) total cost = 15
C) total cost = 17
D) total cost = 11
E) None of the above
64) Find the shortest route from Node 1 to Node 6.
From
Node To
Node
Distance
1 1 2 100
2 1 4 215
3 2 3 70
4 2 4 200
5 2 5 110
6 3 4 320
7 4 5 200
8 4 6 200
9 5 6 200
A) total distance = 350
B) total distance = 410
C) total distance = 270
D) total distance = 520
E) None of the above
65) Given the following traffic flows, in hundreds of cars per hour, what is the maximum traffic flow from Town 1
to Town 7?
From Town To Town Flow
1 1 2 4
2 1 3 7
3 1 5 9
4 2 1 0
5 2 4 3
6 2 5 5
7 3 1 1
8 3 5 3
9 3 6 4
10 4 2 3
11 4 5 1
12 4 7 0
13 5 1 1
14 5 2 0
15 5 3 3
16 5 4 0
17 5 6 5
18 5 7 1
19 6 3 1
20 6 5 6
21 6 7 3
22 7 4 5
23 7 5 2
24 7 6 0
A) max flow = 4 units
B) max flow = 6 units
C) max flow = 3 units
D) max flow = 9 units
E) None of the above
66) Find the shortest route from Node 6 to Node 1.
Branch From
Node To
Node
Distance
1 1 2 150
2 1 3 200
3 2 3 100
4 2 4 200
5 2 5 50
6 3 4 350
7 3 5 300
8 4 6 100
9 5 6 100
A) branches 9, 7, and 2
B) branches 8, 6, and 2
C) branches 8, 6, 7, and 1
D) branches 9, 5, and 1
E) None of the above
67) Given the pipeline fluid flows indicated below, determine the maximum flow from Node 1 to Node 5.
From
Node To
Node Fluid
Flow
1 1 2 300
2 2 1 0
3 1 3 0
4 3 1 150
5 1 4 200
6 4 1 200
7 1 5 100
8 5 1 100
9 2 4 200
10 4 2 200
11 3 4 250
12 4 3 300
13 3 5 300
14 5 3 250
15 4 5 100
16 5 4 0
A) 250
B) 300
C) 350
D) 450
E) None of the above
68) Find the least amount of cable that will allow Jack's Cable Company to connect the following nodes (houses).
From
Node To
Node
Distance
1 2 250
1 3 150
1 4 400
2 3 50
2 4 100
3 4 200
A) 250
B) 400
C) 350
D) 300
E) None of the above
69) Given the following nodes and distances, determine the minimum length of cable necessary to connect all six
nodes.
From
Node To
Node
Distance
1 1 2 150
2 1 3 200
3 2 3 100
4 2 4 200
5 2 5 50
6 3 4 350
7 3 5 300
8 4 6 100
9 5 6 100
A) 200
B) 300
C) 400
D) 500
E) None of the above
70) Given the following nodes and distances, determine the minimal length of cable necessary to connect all
nodes, using Node 2 as the starting point.
From To Distance
1 1 2 200
2 1 3 300
3 1 5 400
4 2 3 300
5 2 4 400
6 3 4 200
7 3 5 200
8 4 5 100
9 4 6 300
10 5 6 400
A) 1200
B) 1100
C) 900
D) 700
E) None of the above
71) Find the shortest route from Node 1 to each of the other nodes in the transportation network represented
below.
Route
from Node
Distance
1 to 2 50
1 to 3 100
2 to 3 75
2 to 4 65
3 to 4 80
3 to 5 70
4 to 5 65
4 to 6 200
5 to 6 130
72) As part of the planning for a major office development project, it is necessary to install telephone line to the
buildings. Information about the project is given below. The distances are provided in hundreds of feet. Which
offices should be connected so that total wiring costs (i.e., total distance) are minimized? What is the total length
of this?
Building Distances (100s ft)
1 to 2 4
1 to 4 3
2 to 3 2
2 to 4 4
3 to 5 1
3 to 6 5
4 to 5 3
4 to 7 3
5 to 7 2
6 to 7 6
73) A cable company must is to provide service for 7 houses in a particular neighborhood. They would like to
wire the neighborhood in a way to minimize the wiring costs (or distance). How should the cable company wire
the neighborhood and what would be the minimal length of the network?
House Distances (yards)
1 to 2 100
1 to 3 400
1 to 4 300
2 to 3 300
2 to 4 250
2 to 5 400
3 to 5 350
3 to 6 450
4 to 5 300
4 to 7 250
5 to 7 100
6 to 7 150
74) Given a network with the following distances:
From
Node To
Node
Distance
1 2 4
1 3 1
2 3 2
2 4 3
3 4 6
3 5 3
3 6 9
4 5 7
5 6 5
(a) Determine which nodes should be connected to get the minimum distance from Nodes 1 through 6.
(b) Determine the minimum distance.
75) The west-to-east air traffic system passing through the United States can handle aircraft flows with capacities
in hundreds of planes per hour as shown. What is the peak air traffic load (From City 1 to City 5) in aircraft per
hour that this system can handle?
76) Find the shortest route from Node 1 to each of the other nodes in the transportation network represented
below.
Route
From Node
Distance
1 to 2 50
1 to 3 100
2 to 3 75
2 to 5 60
3 to 4 80
3 to 5 70
3 to 6 65
4 to 6 200
5 to 6 150
77) A logistics company is determining the shortest route to get to a selected final destination. The information on
possible paths and distances is given below:
Route
From Node
Distance
1 to 2 50
1 to 3 100
1 to 4 125
2 to 4 75
2 to 5 180
3 to 4 100
3 to 5 125
4 to 5 170
4 to 6 200
5 to 6 125
5 to 7 100
6 to 7 75
78) As part of the planning for a major office development project, it is necessary to install telephone lines to the
buildings. Information about the project is given below. The distances are provided in hundreds of feet. Which
offices should be connected so that total wiring costs (i.e., total distance) are minimized? What is the total length
of this?
Buildings Distances (100s ft)
1 to 2 4
1 to 3 3
1 to 4 2
2 to 4 4
3 to 5 1
3 to 6 5
4 to 5 3
4 to 7 3
5 to 7 2
6 to 7 6
79) Brantley College has decided to "wire" its campus. The first stage in this effort is to install the "backbone," i.e.,
to connect all the buildings. The table below gives the distances between the various buildings on campus in
hundreds of feet. How should the buildings be connected to minimize the total length of cable? What length of
cable is required?
80) Given a network with the following distances:
From
Node To
Node
Distance
1 2 4
1 4 1
2 3 2
2 4 3
3 4 6
3 5 4
3 6 2
4 5 7
4 7 5
5 6 5
5 7 8
6 7 4
(a) Determine which nodes should be connected to get the minimum distance flowing from Node 1 through
Node 7.
(b) Determine the minimum distance.
81) The east-to-west (City 5 to City 1) air traffic system passing through the U.S. can handle aircraft flows with
capacities in hundreds of planes per hour as shown. What is the peak air traffic load in aircraft per hour from
City 5 to City 1 that this system can handle?
82) A water company is analyzing the flow of water through pipes in an office building. The flow capacities are
given in the table below. Flow is measured in 100 gallons/hour. What is the maximal flow of water from node 1
to node 5?
83) Describe the steps of the shortest-route technique.
84) Briefly describe the minimal-spanning technique.
85) Briefly describe the maximal-flow technique.
86) Briefly describe the minimal shortest-route technique.
