CHAPTER 12
Network Models
TRUE/FALSE
12.1
The minimal-spanning tree technique finds the shortest route to a series of destinations.
12.2
In the minimal-spanning tree technique, it is necessary to start at the last node in the network.
12.3
The maximal-flow technique would be helpful to city planners in determining how freeways
should be expanded.
12.4
The shortest-route technique requires that each node in the network be connected to at least one
other node.
12.5
The shortest-route technique is the same as the minimal-spanning tree technique.
12.6
As illustrated in the text, the three network optimizing techniques are designed specifically to
solve primarily small, simple problems.
12.7
The Etah Dairy, New Delhi, India, has successfully used the maximal-flow technique to
improve milk volume.
12.8
Busy highways are often analyzed with the maximal-flow technique.
12.9
Transportation companies would definitely be interested in the shortest-route technique to
optimize travel.
376
Network Models  CHAPTER 12
12.10
Cable television companies would employ the shortest-route technique to lay out the cables
connecting individual houses.
12.11
The minimal-spanning tree problem will always have a unique optimal solution.
12.12
We may begin the maximal-flow technique by picking an arbitrary path through the network.
12.13
In the maximal-flow technique, pathways are indicated by nodes, and the connections between
them by lines.
12.14
The maximal-flow technique might be used by the U.S. Army Corps of Engineers to study
water run-off in an attempt to minimize the danger from floods.
12.15
The shortest-route technique might be used by someone planning a vacation in order to
minimize the required amount of driving.
12.16
The maximal-flow technique might be used by New York City traffic engineers to study the
impact of different stop light sequences on traffic moving through the city.
12.17
In the shortest-route technique, locations are indicated by nodes, and the pathways connecting
them by lines.
*12.18
A traveling salesman might use the shortest route technique to minimize the distance traveled to
reach one of his customers.
*12.19
A computer design engineer might use the minimal-spanning tree model to determine the data
paths among various elements in the computer.
377
Network Models  CHAPTER 12
*12.20
One might use the minimal-spanning model to minimize the time spent to complete a set of
tasks when the time to perform a task is dependent on the previous task accomplished.
*12.21
For a traveling salesman attempting to plan his/her route between customers, the shortest-route
model would be of greater assistance than the minimal spanning tree model.
*12.22
The maximal flow model might be of use to an engineer looking for spare capacity in an oil
pipeline system.
*12.23
The shortest route model assumes that one is trying to connect two end points in the shortest
manner possible, rather than attempting to connect all the nodes in the model.
*12.24
The maximal-flow model assumes that there are alternate arcs between each pair of nodes.
*12.25
The maximal-flow model assumes that there is a net flow from source to sink.
*12.26
All of the network models assume that each node is connected to every other node through at
most two arcs.
*12.27
A fundamental assumption of the minimal spanning tree model is that all nodes in the model
must be connected.
378
Network Models  CHAPTER 12
MULTIPLE CHOICE
12.28
If your goal was to construct a network in which all points were connected and the distance
between them was as short as possible, the technique that you would use is
(a)
(b)
(c)
(d)
12.29
The minimal-spanning technique would best be used
(a)
(b)
(c)
(d)
12.30
shortest-route.
maximal-flow.
minimal-spanning tree.
none of the above
to determine LAN network wiring within a building.
to minimize traffic flow on a busy highway.
by a trucking company making frequent but repeatable drops.
none of the above
Given the following distances between destination nodes, what is the minimum distance that
connects the nodes?
From
1
2
1
(a)
(b)
(c)
(d)
(e)
To
2
3
3
100
250
350
450
none of the above
379
Distance
100
150
200
Network Models  CHAPTER 12
12.31
Given the following distances between destination nodes, what is the minimum distance that
connects the nodes?
From
1
1
2
2
3
(a)
(b)
(c)
(d)
(e)
12.32
To
2
3
3
4
4
Distance
200
300
350
350
250
100
750
850
900
none of the above
Given the following distances between destination nodes, what is the minimum distance that
connects the nodes?
From
1
2
1
2
3
4
3
(a)
(b)
(c)
(d)
(e)
To
2
4
3
3
4
5
5
100
150
550
1225
none of the above
380
Distance
100
150
200
50
175
250
300
Network Models  CHAPTER 12
12.33
Given the following distances between destination nodes, what is the minimum distance that
connects the nodes?
From
1
1
2
2
1
3
3
4
(a)
(b)
(c)
(d)
(e)
12.34
300
525
675
1925
none of the above
to determine LAN network wiring within a building.
to maximize traffic flow on a busy highway.
by a trucking company making frequent but repeatable drops.
none of the above
A technique which allows a researcher to determine the greatest amount of material that can
move through a network is called
(a)
(b)
(c)
(d)
12.36
Distance
100
50
200
325
50
350
400
450
The maximal-flow technique would best be used
(a)
(b)
(c)
(d)
12.35
To
2
3
3
5
4
4
5
5
maximal-flow.
maximal-spanning.
shortest-route.
none of the above
The first step in the maximal-flow technique is to
(a)
(b)
(c)
(d)
(e)
pick the node with the maximum flow.
pick any path with some flow.
eliminate any node that has a zero flow.
add a dummy flow from the start to the finish.
none of the above
12.37 Pipeline fluid flows are indicated below. Determine the maximum flow from Node 1 to
Node 3.
From
Node
1
To
Node
3
381
Fluid
Flow
400
Network Models  CHAPTER 12
3
1
2
2
3
(a)
(b)
(c)
(d)
(e)
12.38
1
2
1
3
2
100
300
0
100
100
100
400
500
700
none of the above
Pipeline fluid flows are indicated below. Determine the maximum flow from Node 1 to Node
4.
From
Node
1
2
1
4
1
3
2
4
3
4
(a)
(b)
(c)
(d)
(e)
To
Node
2
1
4
1
3
1
4
2
4
3
200
300
600
700
none of the above
382
Fluid
Flow
400
0
200
200
200
0
200
200
300
300
Network Models  CHAPTER 12
12.39
(a)
(b)
(c)
(d)
12.40
determine the amount of LAN network wiring within a building.
minimize the amount of traffic flow on a busy highway.
determine the path for a truck making frequent but repeatable drops.
none of the above
When using the shortest-route technique, the first step is to
(a)
(b)
(c)
(d)
(e)
12.41
The shortest-route technique would best be used to
connect the nearest node that minimizes the total distance to the origin.
trace the path from the warehouse to the plant.
determine the average distance traveled from source to end.
find the nearest node to the origin and put a distance box by the node.
none of the above
The shortest route technique might be logically used for
(a)
(b)
(c)
(d)
finding the longest time to travel between two points.
finding the shortest travel distance between two points.
finding the most scenic route to allow travel to several places during a trip on spring break.
connecting all the points of a network together while minimizing the distance between
them.
(e) none of the above
12.42
Find the shortest route from Node 1 to Node 4 using the shortest-route technique.
From
Node
1
1
2
2
3
(a)
(b)
(c)
(d)
(e)
To
Node
2
3
3
4
4
250
400
450
600
none of the above
383
Distance
100
200
50
350
250
Network Models  CHAPTER 12
12.43
Find the shortest route from Node 1 to Node 4.
From
Node
1
1
1
2
2
3
(a)
(b)
(c)
(d)
(e)
12.44
Distance
250
150
400
50
100
200
300
350
400
450
none of the above
Find the shortest route from Node 1 to Node 6.
From
Node
1
1
2
2
4
3
3
5
(a)
(b)
(c)
(d)
(e)
12.45
To
Node
2
3
4
3
4
4
To
Node
2
3
4
3
6
4
5
6
Distance
150
200
200
50
100
300
350
100
300
450
550
650
none of the above
All the nodes must be connected in which of the following techniques?
(a)
(b)
(c)
(d)
shortest-route
maximal flow
minimal-spanning tree
all of the above
384
Network Models  CHAPTER 12
12.46
The minimal-spanning technique would best be used
(a)
(b)
(c)
(d)
(e)
12.47
The minimal-spanning technique would best be used
(a)
(b)
(c)
(d)
(e)
12.48
by a hospital trying to lay out a route for the collection of lab samples.
by a grocer trying to lay out a route for his delivery truck.
by an airline laying out flight routes.
none of the above
all of the above
by an architect to lay out corridors between offices in a new office building.
by a telephone company attempting to lay out wires in a new housing development.
by an airline laying out flight routes.
none of the above
all of the above
Given the following distances between destination nodes, what is the minimum distance that
connects the nodes?
From
1
2
1
2
3
4
(a)
(b)
(c)
(d)
(e)
To
2
3
3
4
5
5
290
310
620
460
none of the above
385
Distance
120
100
200
150
90
170
Network Models  CHAPTER 12
12.49
Given the following distances between destination nodes, what is the minimum distance that
connects the nodes?
From
1
1
1
2
2
3
3
4
4
5
(a)
(b)
(c)
(d)
(e)
12.50
To
2
3
5
3
4
4
5
5
6
6
Distance
200
300
400
300
400
200
200
100
300
400
1000
800
700
1100
none of the above
Given the following distances between destination nodes, what is the minimum distance that
connects the nodes?
From
1
1
2
2
2
3
3
4
4
5
(a)
(b)
(c)
(d)
(e)
To
2
3
3
4
5
4
5
5
6
6
900
650
400
1200
none of the above
386
Distance
100
200
100
150
200
150
300
250
200
100
Network Models  CHAPTER 12
12.51 Given the following distances between destination nodes, what is the minimum
distance that connects the nodes?
From
1
1
2
2
1
3
3
3
4
4
5
(a)
(b)
(c)
(d)
(e)
12.52
Distance
100
50
200
300
50
350
400
400
450
350
200
900
1200
1100
800
none of the above
The maximal-flow technique might be used
(a)
(b)
(c)
(d)
(e)
12.53
To
2
3
3
5
4
4
5
6
5
6
6
to minimize cabling in a local area network (LAN).
to help design an aqueduct system for bringing water to a city.
by a steamship company trying to maximize cargo carried.
all of the above
none of the above
The maximal-flow technique might be used
(a) to help design the moving sidewalks transporting passengers from one terminal to another
in a busy airport.
(b) by someone designing the traffic approaches to an airport.
(c) by someone attempting to design roads that would limit the flow of traffic through an area.
(d) all of the above
(e) none of the above
387
Network Models  CHAPTER 12
12.54
The second step in the maximal-flow technique is to
(a)
(b)
(c)
(d)
(e)
12.55
pick the node with the maximum flow.
pick any path with some flow.
increase the flow as much as possible.
add capacity to the path with minimum flow.
none of the above
Pipeline fluid flows are indicated below. Determine the maximum flow from Node 1 to Node
4.
From
Node
1
3
1
2
2
3
3
4
(a)
(b)
(c)
(d)
(e)
To
Node
3
1
2
1
3
2
4
3
100
150
200
50
none of the above
388
Fluid
Flow
200
0
150
50
100
100
150
50
Network Models  CHAPTER 12
12.56
Pipeline fluid flows are indicated below. Determine the maximum flow from Node 1 to Node
5.
From
Node
1
2
1
3
1
4
1
5
2
4
3
4
3
5
4
5
(a)
(b)
(c)
(d)
(e)
12.57
300
400
600
500
none of the above
plan the routes for a vacation driving tour.
plan the route for a school bus.
determine the path for a truck making frequent runs from a factory to a warehouse.
all of the above
none of the above
When using the shortest-route technique, the second step is to
(a)
(b)
(c)
(d)
(e)
12.59
Fluid
Flow
300
0
0
150
200
200
100
100
200
200
250
300
300
250
100
0
The shortest-route technique would best be used to
(a)
(b)
(c)
(d)
(e)
12.58
To
Node
2
1
3
1
4
1
5
1
4
2
4
3
5
3
5
4
find the next-nearest node to the origin and put the distance in a box by the node.
trace the path from the warehouse to the plant.
determine the average distance traveled from source to end.
find the nearest node to the origin and put a distance box by the node.
none of the above
The shortest route technique might be logically used for
(a) finding the longest distance to travel between two points.
389
Network Models  CHAPTER 12
(b)
(c)
(d)
(e)
12.60
finding the shortest time to travel between two points.
finding the most scenic route to allow travel to several places during a trip on spring break.
connecting all the points of a network together while minimizing the number of pathways.
none of the above
Find the shortest route from Node 1 to Node 5 using the shortest-route technique.
From
Node
1
1
2
2
3
3
4
(a)
(b)
(c)
(d)
(e)
To
Node
2
3
3
4
4
5
5
350
400
450
600
none of the above
390
Distance
200
150
50
300
250
200
150
Network Models  CHAPTER 12
12.61
Find the shortest route from Node 1 to Node 5.
From
Node
1
1
1
2
2
3
3
2
(a)
(b)
(c)
(d)
(e)
12.62
To
Node
2
3
4
3
4
4
5
5
Distance
250
150
200
50
150
150
100
150
200
350
250
450
none of the above
Find the shortest route from Node 1 to Node 6.
From
Node
1
1
2
2
2
3
3
4
5
(a)
(b)
(c)
(d)
(e)
To
Node
2
3
3
4
5
4
5
6
6
300
450
550
650
none of the above
391
Distance
150
200
100
200
50
350
300
100
100
Network Models  CHAPTER 12
12.63 Given the following traffic flows, in hundreds of cars per hour, what is the maximum
traffic flow from City 1 to City 7?
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
(a)
(b)
(c)
(d)
(e)
*12.64
From City
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
5
5
5
6
6
6
7
7
7
To City
2
3
5
1
4
5
1
5
6
2
5
7
1
2
3
4
6
7
3
5
7
4
5
6
Flow
4
8
5
0
3
3
0
3
1
3
3
4
1
0
2
0
1
5
1
4
1
2
1
0
1200
1400
900
800
none of the above
Which of the following models would likely be most useful in deciding where to run network
cables to wire the various rooms in your building?
(a)
(b)
(c)
(d)
(e)
shortest route
traveling salesman
minimal-spanning model
maximal flow model
none of the above
392
Network Models  CHAPTER 12
*12.65
Solve the minimal-spanning tree problem defined below:
Branch
1
2
3
4
5
6
(a)
(b)
(c)
(d)
(e)
*12.66
Start
Node
1
1
2
2
3
4
End
Node
3
2
4
5
4
6
Cost
5
1
3
4
6
2
total cost = 13
total cost = 15
total cost = 17
total cost = 11
none of the above
Find the shortest route from Node 1 to Node 6.
1
2
3
4
5
6
7
8
9
(a)
(b)
(c)
(d)
(e)
From
Node
1
1
2
2
2
3
4
4
5
To
Node
2
4
3
4
5
4
5
6
6
total distance = 350
total distance = 410
total distance = 270
total distance = 520
none of the above
393
Distance
100
215
70
200
110
320
200
200
200
Network Models  CHAPTER 12
*12.67
Given the following traffic flows, in hundreds of cars per hour, what is the maximum traffic
flow from Town 1 to Town 7?
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
(a)
(b)
(c)
(d)
(e)
From Town
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
5
5
5
6
6
6
7
7
7
max flow = 4 units
max flow = 6 units
max flow = 3 units
max flow = 9 units
none of the above
394
To Town
2
3
5
1
4
5
1
5
6
2
5
7
1
2
3
4
6
7
3
5
7
4
5
6
Flow
4
7
9
0
3
5
1
3
4
3
1
0
1
0
3
0
5
1
1
6
3
5
2
0
Network Models  CHAPTER 12
*12.68
Find the shortest route from Node 6 to Node 1.
1
2
3
4
5
6
7
8
9
(a)
(b)
(c)
(d)
(e)
*12.69
From
Node
1
1
2
2
2
3
3
4
5
To
Node
2
3
3
4
5
4
5
6
6
Distance
150
200
100
200
50
350
300
100
100
branches 9, 7, and 2
branches 8, 6, and 2
branches 8, 6, 7, and 1
branches 9, 5, and 1
none of the above
Given the pipeline fluid flows indicated below, determine the maximum flow from Node 1 to
Node 5.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
(a)
(b)
(c)
(d)
(e)
From
Node
1
2
1
3
1
4
1
5
2
4
3
4
3
5
4
5
To
Node
2
1
3
1
4
1
5
1
4
2
4
3
5
3
5
4
250
300
350
400
none of the above
395
Fluid
Flow
300
0
0
150
200
200
100
100
200
200
250
300
300
250
100
0
Network Models  CHAPTER 12
*12.70
Find the least amount of cable that will allow Jack’s Cable Company to connect the following
nodes (houses).
From
Node
1
1
1
2
2
3
(a)
(b)
(c)
(d)
(e)
*12.71
To
Node
2
3
4
3
4
4
Distance
250
150
400
50
100
200
250
400
350
300
none of the above
Given the following nodes and distances, determine the minimum length of cable necessary to
connect all six nodes.
1
2
3
4
5
6
7
8
9
(a)
(b)
(c)
(d)
(e)
From
Node
1
1
2
2
2
3
3
4
5
To
Node
2
3
3
4
5
4
5
6
6
200
300
400
500
none of the above
396
Distance
150
200
100
200
50
350
300
100
100
Network Models  CHAPTER 12
*12.72
Given the following list of nodes and distances, determine the minimal amount of cable needed
to connect all nodes.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
(a)
(b)
(c)
(d)
(e)
*12.73
Node
1
2
1
3
1
4
1
5
2
4
3
4
3
5
4
5
Node
2
1
3
1
4
1
5
1
4
2
4
3
5
3
5
4
Distance
300
0
0
150
200
200
100
100
200
200
250
300
300
250
100
0
400
300
200
100
none of the above
Given the following nodes and distances, determine the minimal length of cable necessary to
connect all nodes, using node 2 as the starting point.
1
2
3
4
5
6
7
8
9
10
(a)
(b)
(c)
(d)
(e)
From
1
1
1
2
2
3
3
4
4
5
To
2
3
5
3
4
4
5
5
6
6
Distance
200
300
400
300
400
200
200
100
300
400
1200
1100
900
700
none of the above
*12.74 What happens to the solution for the previous problem when we remove the possibility
of using link 9, from node 4 to node 6?
397
Network Models  CHAPTER 12
(a)
(b)
(c)
(d)
(e)
The minimum length increases to 1200.
The minimum length stays the same.
The minimum length decreases to 1000.
The minimum length increases to 1400.
none of the above
PROBLEMS
12.75
Find the shortest route from Node 1 to each of the other nodes in the transportation network
represented below.
Route from Node
1 to 2
1 to 3
2 to 3
2 to 4
3 to 4
3 to 5
4 to 5
4 to 6
5 to 6
398
Distance
50
100
75
60
80
70
65
200
150
Network Models  CHAPTER 12
12.76
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
1 to 2
1 to 4
2 to 3
2 to 4
3 to 5
3 to 6
4 to 5
4 to 7
5 to 7
6 to 7
12.77
Distances (100s ft)
4
3
2
4
1
5
3
3
2
6
Given a network with the following distances:
From Node
1
1
2
2
3
3
3
4
5
To Node
2
3
3
4
4
5
6
5
6
Distance
4
1
2
3
6
4
2
7
5
(a) Determine which nodes should be connected to get the minimum distance.
(b) Determine the minimum distance.
399
Network Models  CHAPTER 12
12.78 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?
From
1
2
3
4
To
1
1
2
-
2
2
2
-
3
2
-
4
4
3
5
-
5
3
2
3
5
-
2
2
1
-
.
12.79
Find the shortest route from Node 1 to each of the other nodes in the transportation network
represented below.
Route From Node
1 to 2
1 to 3
2 to 3
2 to 5
3 to 4
3 to 5
3 to 6
4 to 6
5 to 6
400
Distance
50
100
75
60
80
70
65
200
150
Network Models  CHAPTER 12
12.80
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
1 to 2
1 to 3
1 to 4
2 to 4
3 to 5
3 to 6
4 to 5
4 to 7
5 to 7
6 to 7
12.81
Distances (100s ft)
4
3
2
4
1
5
3
3
2
6
Brantley College has decided to "wire" their 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?
Distances in Hundreds of Feet
From
Building 1
Building 2
Building 3
Building 4
Building 5
Building 6
To
Building
1
Building
2
3
Building
3
7
5
401
Building
4
5
2
5
Building
5
5
6
4
5
Building
6
4
6
4
3
4
Network Models  CHAPTER 12
12.82
Given a network with the following distances:
From Node
1
1
2
2
3
3
3
4
4
5
5
6
To Node
2
4
3
4
4
5
6
5
7
6
7
7
Distance
4
1
2
3
6
4
2
7
5
5
8
4
(a) Determine which nodes should be connected to get the minimum distance.
(b) Determine the minimum distance.
402
Network Models  CHAPTER 12
12.83
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?
From
City 1
City 2
City 3
City 4
City 5
To
City 1
2
4
-
City 2
1
2
3
3
City 3
2
2
5
2
City 4
3
City 5
2
2
1
-
SHORT ANSWER/ESSAY
12.84
List three different network models that can be used to optimize a variety of problems.
12.85
Briefly describe the minimal-spanning technique.
12.86
Briefly describe the maximal-flow technique.
12.87
Briefly describe the minimal shortest-route technique.
403