Network Models
Chapters 6 and 7
McGraw-Hill/Irwin
7.1
© The McGraw-Hill Companies, Inc., 2003
Where network flows arise
• Transportation
– Transportation of goods over transportation networks
– Scheduling of fleets of airplanes: time/space networks
• Manufacturing
– Scheduling of goods for manufacturing
– Flow of manufactured items within inventory systems
• Communications
– Design and expansion of communication systems
– Flow of information across networks
• Personnel Assignment
– Assignment of crews to airline schedules
– Assignment of drivers to vehicles
McGraw-Hill/Irwin
7.2
© The McGraw-Hill Companies, Inc., 2003
Network Optimization Problem Types
Many optimization problems can be represented by a
graphical network representation.
Examples:
–
–
–
–
–
Distribution problems
Routing problems
Maximum flow problems
Designing computer / phone / road networks
Equipment replacement
McGraw-Hill/Irwin
7.3
© The McGraw-Hill Companies, Inc., 2003
Network examples
• Shortest path
• Maximum flow
• Transportation problem (Chapter 6)
• Assignment problem (Chapter 6)
• All are examples of a more general model type:
– The Minimum-Cost-Network Flow Model
McGraw-Hill/Irwin
7.4
© The McGraw-Hill Companies, Inc., 2003
Advantages of Network Models
• They can be solved very quickly with specialized algorithms.
• They have naturally integer solutions.
– By recognizing that a problem can be formulated as a network program, it
is possible to solve special types of integer programs without resorting to
the ineffective and time consuming integer programming algorithms.
• They are intuitive.
– Network models provide a language for talking about problems that is
much more intuitive than the “variables, objective, and constraints”
language of linear and integer programming.
• These advantages come with a drawback (of course):
– Network models cannot formulate the wide range of models that linear
and integer programs can.
– However, they occur often enough that they form an important tool for real
decision making.
McGraw-Hill/Irwin
7.5
© The McGraw-Hill Companies, Inc., 2003
Table of Contents
Chapter 7 (Network Optimization Problems)
Minimum-Cost Flow Problems (Section 7.1)
A Case Study: The BMZ Maximum Flow Problem (Section 7.2)
Maximum Flow Problems (Section 7.3)
Shortest Path Problems: Littletown Fire Department (Section 7.4)
Shortest Path Problems: General Characteristics (Section 7.4)
Shortest Path Problems: Minimizing Sarah’s Total Cost (Section 7.4)
Shortest Path Problems: Minimizing Quick’s Total Time (Section 7.4)
Minimum Spanning Trees: The Modern Corp. Problem (Section 7.5)
McGraw-Hill/Irwin
7.6
© The McGraw-Hill Companies, Inc., 2003
Distribution Unlimited Co. Problem
• The Distribution Unlimited Co. has two factories producing a product
that needs to be shipped to two warehouses
–
–
–
–
Factory 1 produces 80 units.
Factory 2 produces 70 units.
Warehouse 1 needs 60 units.
Warehouse 2 needs 90 units.
• There are rail links directly from Factory 1 to Warehouse 1 and Factory
2 to Warehouse 2.
• Independent truckers are available to ship up to 50 units from each
factory to the distribution center, and then 50 units from the distribution
center to each warehouse.
Question: How many units (truckloads) should be shipped along
each shipping lane?
McGraw-Hill/Irwin
7.7
© The McGraw-Hill Companies, Inc., 2003
The Distribution Network
80 units
produced
W1 60 units
needed
F1
DC
70 units
produced
McGraw-Hill/Irwin
W2
F2
7.8
90 units
needed
© The McGraw-Hill Companies, Inc., 2003
Data for Distribution Network
80 units
produced
$700/unit
F1
$300/unit
[50 unit s max.]
$200/unit
[50 unit s max.]
60 units
W1 needed
DC
$400/unit
[50 unit s max.]
70 units
produced
F2
$400/unit
[50 unit s max.]
$900/unit
W2
90 units
needed
Both transportation cost and arc capacity are considered.
McGraw-Hill/Irwin
7.9
© The McGraw-Hill Companies, Inc., 2003
A Network Model
[80]
[- 60]
$700
F1
$300
[50]
[0]
W1
$200
[50]
DC
$400
[50]
F2
$400
[50]
$900
[70]
McGraw-Hill/Irwin
W2
[- 90]
7.10
© The McGraw-Hill Companies, Inc., 2003
The Optimal Solution
[80]
[- 60]
(30)
F1
(50)
W1
(30)
[0]
DC
(30)
F2
(50)
(40)
[70]
McGraw-Hill/Irwin
W2
[- 90]
7.11
© The McGraw-Hill Companies, Inc., 2003
Terminology for Minimum-Cost Flow Problems
1. The model for any minimum-cost flow problem is represented by a
network with flow passing through it.
2. The circles in the network are called nodes.
3. Each node where the net amount of flow generated (outflow minus
inflow) is a fixed positive number is a supply node.
4. Each node where the net amount of flow generated is a fixed negative
number is a demand node.
5. Any node where the net amount of flow generated is fixed at zero is a
transshipment node. Having the amount of flow out of the node
equal the amount of flow into the node is referred to as conservation
of flow.
6. The arrows in the network are called arcs.
7. The maximum amount of flow allowed through an arc is referred to as
the capacity of that arc.
McGraw-Hill/Irwin
7.12
© The McGraw-Hill Companies, Inc., 2003
Assumptions of a Minimum-Cost Flow Problem
1. At least one of the nodes is a supply node.
2. At least one of the other nodes is a demand node.
3. All the remaining nodes are transshipment nodes.
4. Flow through an arc is only allowed in the direction indicated by the
arrowhead, where the maximum amount of flow is given by the capacity
of that arc. (If flow can occur in both directions, this would be
represented by a pair of arcs pointing in opposite directions.)
5. The network has enough arcs with sufficient capacity to enable all the
flow generated at the supply nodes to reach all the demand nodes.
6. The cost of the flow through each arc is proportional to the amount of
that flow, where the cost per unit flow is known.
7. The objective is to minimize the total cost of sending the available
supply through the network to satisfy the given demand. (An alternative
objective is to maximize the total profit from doing this.)
McGraw-Hill/Irwin
7.13
© The McGraw-Hill Companies, Inc., 2003
Properties of Minimum-Cost Flow Problems
• The Feasible Solutions Property: Under the previous
assumptions, a minimum-cost flow problem will have feasible
solutions if and only if the sum of the supplies from its supply
nodes equals the sum of the demands at its demand nodes.
• The Integer Solutions Property: As long as all the supplies,
demands, and arc capacities have integer values, any
minimum-cost flow problem with feasible solutions is
guaranteed to have an optimal solution with integer values for
all its flow quantities.
McGraw-Hill/Irwin
7.14
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Model
3
4
5
6
7
8
9
10
11
B
C
D
From
F1
F1
DC
DC
F2
F2
To
W1
DC
W1
W2
DC
W2
Ship
30
50
30
50
30
40
Total Cost
$110,000
E
F
G
Capacity
Unit Cost
$700
$300
$200
$400
$400
$900
<=
<=
<=
<=
[80]
50
50
50
50
H
[- 60]
$700
F1
$300
[50]
[0]
F2
3
4
5
6
7
8
W1
$200
[50]
[70]
McGraw-Hill/Irwin
$400
[50]
$900
J
Nodes Net Flow
F1
80
F2
70
DC
0
W1
-60
W2
-90
K
L
=
=
=
=
=
Supply/Demand
80
70
0
-60
-90
J
DC
$400
[50]
I
Net Flow
=SUMIF(From,I4,Ship)-SUMIF(To,I4,Ship)
=SUMIF(From,I5,Ship)-SUMIF(To,I5,Ship)
=SUMIF(From,I6,Ship)-SUMIF(To,I6,Ship)
=SUMIF(From,I7,Ship)-SUMIF(To,I7,Ship)
=SUMIF(From,I8,Ship)-SUMIF(To,I8,Ship)
W2
[- 90]
7.15
© The McGraw-Hill Companies, Inc., 2003
The SUMIF Function
• The SUMIF formula can be used to simplify the node flow constraints.
=SUMIF(Range A, x, Range B)
• For each quantity in (Range A) that equals x, SUMIF sums the
corresponding entries in (Range B).
• The net outflow (flow out – flow in) from node x is then
=SUMIF(“From labels”, x, “Flow”) – SUMIF(“To labels”, x, “Flow”)
McGraw-Hill/Irwin
7.16
© The McGraw-Hill Companies, Inc., 2003
Typical Applications of Minimum-Cost Flow
Problems
Kind of
Application
Supply
Nodes
Transshipment
Nodes
Demand
Nodes
Operation of a
distribution
network
Sources of goods
Intermediate
storage facilities
Customers
Solid waste
management
Sources of solid
waste
Processing facilities
Landfill locations
Operation of a
supply network
Vendors
Intermediate
warehouses
Processing facilities
Coordinating
product mixes at
plants
Plants
Production of a
specific product
Market for a
specific product
Cash flow
management
Sources of cash at a
specific time
Short-term
investment options
Needs for cash at a
specific time
McGraw-Hill/Irwin
7.17
© The McGraw-Hill Companies, Inc., 2003
The BMZ Maximum Flow Problem
• The BMZ Company is a European manufacturer of luxury
automobiles. Its exports to the United States are particularly important.
• BMZ cars are becoming especially popular in California, so it is
particularly important to keep the Los Angeles center well supplied
with replacement parts for repairing these cars.
• BMZ needs to execute a plan quickly for shipping as much as possible
from the main factory in Stuttgart, Germany to the distribution center
in Los Angeles over the next month.
• The limiting factor on how much can be shipped is the limited capacity
of the company’s distribution network.
Question: How many units should be sent through each shipping
lane to maximize the total units flowing from Stuttgart to Los
Angeles?
McGraw-Hill/Irwin
7.18
© The McGraw-Hill Companies, Inc., 2003
The BMZ Distribution Network
[60 unit s max.]
RO Rotte rdam
[50 unit s max.]
Ne w York NY
[80 unit s max.]
Ne w O rl e an s
LA
Los An gel e s
[70 unit s max]NO
McGraw-Hill/Irwin
{40 units max.]
ST S tu ttgart
[70 unit s max.]
Borde au x
[40 unit s max.]
BO
[50 unit s max.]
LI
Lis bon
[30 unit s max.]
7.19
© The McGraw-Hill Companies, Inc., 2003
A Network Model for BMZ
RO
[60]
[50]
NY
[80]
[40]
BO
LA
[70]
ST
[50]
[70]
NO
[40]
[30]
LI
McGraw-Hill/Irwin
7.20
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Model for BMZ
3
4
5
6
7
8
9
10
11
12
13
14
B
C
D
From
Stuttgart
Stuttgart
Stuttgart
Rotterdam
Bordeaux
Bordeaux
Lisbon
New Y ork
New Orleans
To
Rotterdam
Bordeaux
Lisbon
New Y ork
New Y ork
New Orleans
New Orleans
Los Angeles
Los Angeles
Ship
50
70
30
50
30
40
30
80
70
Maximum Flow
150
McGraw-Hill/Irwin
E
F
<=
<=
<=
<=
<=
<=
<=
<=
<=
Capacity
50
70
40
60
40
50
30
80
70
7.21
G
H
I
Nodes
Stuttgart
Rotterdam
Bordeaux
Lisbon
New Y ork
New Orleans
Los Angeles
Net Flow
150
0
0
0
0
0
-150
J
K
Supply/Demand
=
=
=
=
=
0
0
0
0
0
© The McGraw-Hill Companies, Inc., 2003
Assumptions of Maximum Flow Problems
1. All flow through the network originates at one node, called the
source, and terminates at one other node, called the sink. (The
source and sink in the BMZ problem are the factory and the
distribution center, respectively.)
2. All the remaining nodes are transshipment nodes.
3. Flow through an arc is only allowed in the direction indicated by
the arrowhead, where the maximum amount of flow is given by
the capacity of that arc. At the source, all arcs point away from
the node. At the sink, all arcs point into the node.
4. The objective is to maximize the total amount of flow from the
source to the sink. This amount is measured in either of two
equivalent ways, namely, either the amount leaving the source or
the amount entering the sink.
McGraw-Hill/Irwin
7.22
© The McGraw-Hill Companies, Inc., 2003
BMZ with Multiple Supply and Demand Points
• BMZ has a second, smaller factory in Berlin.
• The distribution center in Seattle has the capability of
supplying parts to the customers of the distribution center
in Los Angeles when shortages occur at the latter center.
Question: How many units should be sent through each
shipping lane to maximize the total units flowing from
Stuttgart and Berlin to Los Angeles and Seattle?
McGraw-Hill/Irwin
7.23
© The McGraw-Hill Companies, Inc., 2003
Network Model for the expanded BMZ
Problem
HA
[40]
[60]
BN
[30]
[20]
RO
SE
[40]
BE
[60]
[10]
LA
[20]
[50]
NY
[40]
[80]
BO
[70]
ST
[50]
[70]
NO
[40]
[30]
LI
McGraw-Hill/Irwin
7.24
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Model
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
B
C
D
From
Stuttgart
Stuttgart
Stuttgart
Berlin
Berlin
Rotterdam
Bordeaux
Bordeaux
Lisbon
Hamburg
Hamburg
New Orleans
New Y ork
New Y ork
Boston
Boston
To
Rotterdam
Bordeaux
Lisbon
Rotterdam
Hamburg
New Y ork
New Y ork
New Orleans
New Orleans
New Y ork
Boston
Los Angeles
Los Angeles
Seattle
Los Angeles
Seattle
Ship
40
70
30
20
60
60
30
40
30
30
30
70
80
40
10
20
Maximum Flow
220
McGraw-Hill/Irwin
E
F
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
Capacity
50
70
40
20
60
60
40
50
30
30
40
70
80
40
10
20
7.25
G
H
I
Nodes
Stuttgart
Berlin
Hamburg
Rotterdam
Bordeaux
Lisbon
Boston
New Y ork
New Orleans
Los Angeles
Seattle
Net Flow
140
80
0
0
0
0
0
0
0
-160
-60
J
K
Supply/Demand
=
=
=
=
=
=
=
0
0
0
0
0
0
0
© The McGraw-Hill Companies, Inc., 2003
Some Applications of Maximum Flow Problems
1. Maximize the flow through a distribution network, as for BMZ.
2. Maximize the flow through a company’s supply network from its
vendors to its processing facilities.
3. Maximize the flow of oil through a system of pipelines.
4. Maximize the flow of water through a system of aqueducts.
5. Maximize the flow of vehicles through a transportation network.
McGraw-Hill/Irwin
7.26
© The McGraw-Hill Companies, Inc., 2003
Littletown Fire Department
• Littletown is a small town in a rural area.
• Its fire department serves a relatively large geographical area that
includes many farming communities.
• Since there are numerous roads throughout the area, many
possible routes may be available for traveling to any given
farming community.
Question: Which route from the fire station to a certain farming
community minimizes the total number of miles?
McGraw-Hill/Irwin
7.27
© The McGraw-Hill Companies, Inc., 2003
The Littletown Road System
8
6
A
1
3
6
Fire
Station
4
D
B
McGraw-Hill/Irwin
7.28
Farming
Communit y
5
2
4
7
6
3
G
E
2
C
6
3
5
4
4
F
H
7
© The McGraw-Hill Companies, Inc., 2003
The Network Representation
A
3
(Origin)
O
1
6
4
B
6
4
5
2
8
D
5
E
4
7
C
McGraw-Hill/Irwin
3
6
3
F
G
2
4
6
T
(Destinat ion)
7
H
7.29
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Model
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
B
C
D
From
Fire St.
Fire St.
Fire St.
A
A
B
B
B
B
C
C
D
D
E
E
E
E
F
F
G
G
G
H
H
To
A
B
C
B
D
A
C
D
E
B
E
E
F
D
F
G
H
G
Farm Com.
F
H
Farm Com.
G
Farm Com.
On Route
1
0
0
1
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
Total Distance
19
McGraw-Hill/Irwin
E
F
G
Distance
3
6
4
1
6
1
2
4
5
2
7
3
8
3
6
5
4
3
4
3
2
6
2
7
7.30
H
I
Nodes
Fire St.
A
B
C
D
E
F
G
H
Farm Com.
Net Flow
1
0
0
0
0
0
0
0
0
-1
J
K
=
=
=
=
=
=
=
=
=
=
Supply/Demand
1
0
0
0
0
0
0
0
0
-1
© The McGraw-Hill Companies, Inc., 2003
Assumptions of a Shortest Path Problem
1. You need to choose a path through the network that starts at a
certain node, called the origin, and ends at another certain
node, called the destination.
2. The lines connecting certain pairs of nodes commonly are
links (which allow travel in either direction), although arcs
(which only permit travel in one direction) also are allowed.
3. Associated with each link (or arc) is a nonnegative number
called its length. (Be aware that the drawing of each link in
the network typically makes no effort to show its true length
other than giving the correct number next to the link.)
4. The objective is to find the shortest path (the path with the
minimum total length) from the origin to the destination.
McGraw-Hill/Irwin
7.31
© The McGraw-Hill Companies, Inc., 2003
Applications of Shortest Path Problems
1. Minimize the total distance traveled.
2. Minimize the total cost of a sequence of activities.
3. Minimize the total time of a sequence of activities.
McGraw-Hill/Irwin
7.32
© The McGraw-Hill Companies, Inc., 2003
Minimizing Total Cost: Sarah’s Car Fund
• Sarah has just graduated from high school.
• As a graduation present, her parents have given her a car fund of
$21,000 to help purchase and maintain a three-year-old used car
for college.
• Since operating and maintenance costs go up rapidly as the car
ages, Sarah may trade in her car on another three-year-old car one
or more times during the next three summers if it will minimize
her total net cost. (At the end of the four years of college, her
parents will trade in the current used car on a new car for Sarah.)
Question: When should Sarah trade in her car (if at all) during the
next three summers?
McGraw-Hill/Irwin
7.33
© The McGraw-Hill Companies, Inc., 2003
Sarah’s Cost Data
Operating and Maintenance Costs
for Ownership Year
Trade-in Value at End
of Ownership Year
Purchase
Price
1
2
3
4
1
2
3
4
$12,000
$2,000
$3,000
$4,500
$6,500
$8,500
$6,500
$4,500
$3,000
McGraw-Hill/Irwin
7.34
© The McGraw-Hill Companies, Inc., 2003
Shortest Path Formulation
25,000
17,000
10,500
10,500
(Origin) 0
5,500
1
5,500
2
5,500
3
5,500
4
(Destinat ion)
10,500
17,000
McGraw-Hill/Irwin
7.35
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Model
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Y ear
Y ear
Y ear
Y ear
1
2
3
4
From
Y ear 0
Y ear 0
Y ear 0
Y ear 0
Y ear 1
Y ear 1
Y ear 1
Y ear 2
Y ear 2
Y ear 3
C
D
E
Operating &
Maint. Cost
$2,000
$3,000
$4,500
$6,500
Trade-in Value
at End of Y ear
$8,500
$6,500
$4,500
$3,000
Purchase
Price
$12,000
On Route
0
1
0
0
0
0
0
0
1
0
Cost
$5,500
$10,500
$17,000
$25,000
$5,500
$10,500
$17,000
$5,500
$10,500
$5,500
To
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
1
2
3
4
2
3
4
3
4
4
Total Cost
McGraw-Hill/Irwin
F
G
H
Nodes Net Flow
Y ear 0
1
Y ear 1
0
Y ear 2
0
Y ear 3
0
Y ear 4
-1
I
J
=
=
=
=
=
Supply/Demand
1
0
0
0
-1
$21,000
7.36
© The McGraw-Hill Companies, Inc., 2003
Minimizing Total Time: Quick Company
• The Quick Company has learned that a competitor is planning to come out
with a new kind of product with great sales potential.
• Quick has been working on a similar product that had been scheduled to
come to market in 20 months.
• Quick’s management wishes to rush the product out to meet the
competition.
• Each of four remaining phases can be conducted at a normal pace, at a
priority pace, or at crash level to expedite completion. However, the
normal pace has been ruled out as too slow for the last three phases.
• $30 million is available for all four phases.
Question: At what pace should each of the four phases be
conducted?
McGraw-Hill/Irwin
7.37
© The McGraw-Hill Companies, Inc., 2003
Time and Cost of the Four Phases
Remaining
Research
Development
Design of
Mfg. System
Initiate Production
and Distribution
Normal
5 months
—
—
—
Priority
4 months
3 months
5 months
2 months
Crash
2 months
2 months
3 months
1 month
Level
Remaining
Research
Development
Design of
Mfg. System
Initiate Production
and Distribution
Normal
$3 million
—
—
—
Priority
6 million
$6 million
$9 million
$3 million
Crash
9 million
9 million
12 million
6 million
Level
McGraw-Hill/Irwin
7.38
© The McGraw-Hill Companies, Inc., 2003
Shortest Path Formulation
,
0
1 h)
as
r
(C
2
5
3 ty) 2, 21(Priorit y)3, 12(Priorit y)4, 9
o ri
i
(C 3
r
P
(
ras
1, 27 2
h)
( Cr
ash
2
5 al)
) 2, 18 5
4, 6
3,
9
m
)
(Priorit
y)
r
(Priorit
y)
ty
o
3
i
r
(C 3
(N
io
r as
(Origin) 0, 30 4
1, 24 (Pr
(Priorit y) (C 2
h)
(C 2
ra
2
s h 2, 15 5
ra s
4, 3
3,
6
)
h)
(Priorit
y)
((P
riority)
3 ri ty ) (C 3
ras
1, 21 ri o
h)
(Cr (P2
ash
5
3, 3 2
4, 0
) 2, 12(Priorit
y)
(Priorit y)
0
ra
1 h)
s
(C
0
T
(Destinat ion)
1 s h)
ra
(C
0
McGraw-Hill/Irwin
7.39
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Model
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
B
C
D
From
(0, 30)
(0, 30)
(0, 30)
(1, 27)
(1, 27)
(1, 24)
(1, 24)
(1, 21)
(1, 21)
(2, 21)
(2, 21)
(2, 18)
(2, 18)
(2, 15)
(2, 15)
(2, 12)
(3, 12)
(3, 12)
(3, 9)
(3, 9)
(3, 6)
(3, 6)
(3, 3)
(4, 9)
(4, 6)
(4, 3)
(4, 0)
To
(1, 27)
(1, 24)
(1, 21)
(2, 21)
(2, 18)
(2, 18)
(2, 15)
(2, 15)
(2, 12)
(3, 12)
(3, 9)
(3, 9)
(3, 6)
(3, 6)
(3, 3)
(3, 3)
(4, 9)
(4, 6)
(4, 6)
(4, 3)
(4, 3)
(4, 0)
(4, 0)
(T)
(T)
(T)
(T)
On Route
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
1
Total Time
10
McGraw-Hill/Irwin
E
F
G
Time
5
4
2
3
2
3
2
3
2
5
3
5
3
5
3
5
2
1
2
1
2
1
2
0
0
0
0
7.40
H
I
J
K
Nodes
(0, 30)
(1, 27)
(1, 24)
(1, 21)
(2, 21)
(2, 18)
(2, 15)
(2, 12)
(3, 12)
(3, 9)
(3, 6)
(3, 3)
(4, 9)
(4, 6)
(4, 3)
(4, 0)
(T)
Net Flow
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-1
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Supply/Demand
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-1
© The McGraw-Hill Companies, Inc., 2003
The Optimal Solution
Phase
Level
Time
Cost
Remaining research
Crash
2 months
$9 million
Priority
3 months
6 million
Crash
3 months
12 million
Priority
2 months
3 million
10 months
$30 million
Development
Design of manufacturing system
Initiate production and distribution
Total
McGraw-Hill/Irwin
7.41
© The McGraw-Hill Companies, Inc., 2003
Minimum Spanning Trees:
The Modern Corp. Problem
• Modern Corporation has decided to have a state-of-the-art fiberoptic network installed to provide high-speed communication
(data, voice, and video) between its major centers.
• Any pair of centers do not need to have a cable directly
connecting them in order to take advantage of the technology.
All that is necessary is to have a series of cables that connect the
centers.
Question: Which cables should be installed to provide highspeed communications between every pair of centers.
McGraw-Hill/Irwin
7.42
© The McGraw-Hill Companies, Inc., 2003
Modern Corporation’s Major Centers
B
2
7
2
G
5
5
A
4
C
E
7
4
1
D
McGraw-Hill/Irwin
3
1
F
4
7.43
© The McGraw-Hill Companies, Inc., 2003
The Optimal Solution
B
2
2
5
C
A
E
3
1
D
McGraw-Hill/Irwin
G
1
F
7.44
© The McGraw-Hill Companies, Inc., 2003
Assumptions of a Minimum-Spanning Tree
Problem
1. You are given the nodes of a network but not the links. Instead,
you are given the potential links and the positive cost (or a
similar measure) for each if it is inserted into the network.
2. You wish to design the network by inserting enough links to
satisfy the requirement that there be a path between every pair of
nodes.
3. The objective is to satisfy this requirement in a way that
minimizes the total cost of doing so.
McGraw-Hill/Irwin
7.45
© The McGraw-Hill Companies, Inc., 2003
Algorithm for a Minimum-Spanning-Tree
Problem
1. Choice of the first link: Select the cheapest potential link.
2. Choice of the next link: Select the cheapest potential link
between a node that already is touched by a link and a node that
does not yet have such a link.
3. Repeat step 2 over and over until every node is touched by a
link (perhaps more than one). At that point, an optimal solution
(a minimum spanning tree) has been obtained.
(Ties for the cheapest potential link at each step may be broken
arbitrarily.)
McGraw-Hill/Irwin
7.46
© The McGraw-Hill Companies, Inc., 2003
Application of Algorithm to Modern Corp.: First
Link
B
2
7
2
G
5
5
A
4
C
E
7
4
1
D
McGraw-Hill/Irwin
3
1
F
4
7.47
© The McGraw-Hill Companies, Inc., 2003
Application of Algorithm to Modern Corp.:
Second Link
B
2
7
2
G
5
5
A
4
C
E
7
4
1
D
McGraw-Hill/Irwin
3
1
F
4
7.48
© The McGraw-Hill Companies, Inc., 2003
Application of Algorithm to Modern Corp.:
Third Link
B
2
7
2
G
5
5
A
4
C
E
7
4
1
D
McGraw-Hill/Irwin
3
1
F
4
7.49
© The McGraw-Hill Companies, Inc., 2003
Application of Algorithm to Modern Corp.:
Fourth Link
B
2
7
2
G
5
5
A
4
C
E
7
4
D
McGraw-Hill/Irwin
3
1
1
F
4
7.50
© The McGraw-Hill Companies, Inc., 2003
Application of Algorithm to Modern Corp.: Fifth
Link
B
2
7
2
G
5
5
A
4
C
E
7
4
D
McGraw-Hill/Irwin
3
1
1
F
4
7.51
© The McGraw-Hill Companies, Inc., 2003
Application of Algorithm to Modern Corp.:
Final Link
B
2
7
2
G
5
5
A
4
C
E
7
4
D
McGraw-Hill/Irwin
3
1
1
F
4
7.52
© The McGraw-Hill Companies, Inc., 2003
Applications of Minimum-Spanning-Tree
Problems
1. Design of telecommunication networks (computer networks, leaseline telephone networks, cable television networks, etc.)
2. Design of a lightly-used transportation network to minimize the
total cost of providing the links (rail lines, roads, etc.)
3. Design of a network of high-voltage electrical power transmission
lines.
4. Design of a network of wiring on electrical equipment (e.g., a
digital computer system) to minimize the total length of the wire.
5. Design of a network of pipelines to connect a number of locations.
McGraw-Hill/Irwin
7.53
© The McGraw-Hill Companies, Inc., 2003
Network Optimization Problems
Many optimization problems can be represented by a
graphical network representation.
Examples:
–
–
–
–
–
Distribution problems
Routing problems
Maximum flow problems
Designing computer / phone / road networks
Equipment replacement
McGraw-Hill/Irwin
7.54
© The McGraw-Hill Companies, Inc., 2003
Components of a Minimum-Cost-Flow Model
• Nodes
– can represent a location, point in time, or state
– supply node (flow is generated)
– demand node (flow is consumed)
– transshipment node (flow in = flow out)
• Arcs
– can represent potential flow (e.g., a shipping lane) or a
transition from state to state.
– directional (one-way)
• if both ways, use two arcs
– cost (assumed proportional to flow)
– may have capacity limitations
McGraw-Hill/Irwin
7.55
© The McGraw-Hill Companies, Inc., 2003
Minimum-Cost-Flow Model
• Objective: Minimize the total cost of all flow, while sending supply,
subject to constraints, through the network to satisfy demand.
• Integer Solutions Property: If supplies, demands, and arc
capacities are integer, then the optimal flow will also be integer.
• Network Simplex Method: A streamlined version of the simplex
method.
– extremely efficient
– computer software may have graphical interface (with nodes and arcs)
• Excel uses the standard simplex method. However, the minimumcost-flow model is a useful tool for modeling a problem:
–
–
–
–
visual
intuitive
easy to set up
transforms easily to a spreadsheet model
McGraw-Hill/Irwin
7.56
© The McGraw-Hill Companies, Inc., 2003
Minimum-Cost-Flow Model
• Consider a directed network with n nodes. The decision
variables are xij, the flow through arc (i, j). The given
information includes:
– cij: cost per unit of flow from i to j (may be negative),
– uij: capacity (or upper bound) on flow from i to j,
– bi: net flow generated at i.
• This last value has a sign convention:
– bi > 0 if i is a supply node,
– bi < 0 if i is a demand node,
– bi = 0 if i is a transshipment node.
• The objective is to minimize the total cost of sending the supply
through the network to satisfy the demand.
McGraw-Hill/Irwin
7.57
© The McGraw-Hill Companies, Inc., 2003
Minimum-Cost-Flow Model
• Linear programming formulation for this model is…
McGraw-Hill/Irwin
7.58
© The McGraw-Hill Companies, Inc., 2003
Minimum-Cost-Flow Model
• Things you can do with this model…
– Lower bounds on arcs. If a variable xij has a lower bound of lij, upper bound
of uij, and cost of cij, change the problem as follows:
• Replace the upper bound with uij - lij,
• Replace the supply at i with bi - lij,
• Replace the supply at j with bi + lij,
– Now this is a minimum cost flow problem. Add cijlij to the objective after solving and
lij to the flow on arc (i, j) to obtain a solution of the original problem.
– Upper bounds on flow through a node. Replace the node i with nodes i' and i''.
Create an arc from i' to i'' with the appropriate capacity, and cost 0. Replace every arc
(j, i) with one from j to i' and every arc (i, j) with one from i'' to j. Lower bounds can
also be handled this way.
– Convex, piecewise linear costs on arc flows (for minimization). This is handled by
introducing multiple arcs between the nodes, one for each portion of the piecewise
linear function. The convexity will assure that costs are handled correctly in an optimal
solution.
McGraw-Hill/Irwin
7.59
© The McGraw-Hill Companies, Inc., 2003
Multi-Echelon Distribution
Consider a multi-echelon distribution problem. Product must be
distributed from a pair of factories to three warehouses. Product is then
shipped to five distribution centers. A private trucking fleet is used for all
shipping. Some shipping lanes are currently capacitated due to a limited
number of trucks.
$6.00
WH1
$4.00
[2000]
F1
[-800]
DC2
[-700]
DC3
[-1500]
DC4
[-900]
DC5
[-1100]
$6.75
[300]
[900]
DC1
$8.25
$3.75
$7.50
WH2
$2.50
[1200]
[3000]
[500]
F2
$6.50
$5.25
$8.75
WH3
$7.75
Question: How many units should be shipped along each
shipping lane?
McGraw-Hill/Irwin
7.60
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Model
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
C
D
From
F1
F1
F2
F2
WH1
WH1
WH2
WH2
WH2
WH3
WH3
To
WH1
WH2
WH2
WH3
DC1
DC2
DC2
DC3
DC4
DC4
DC5
Ship
900
1100
1200
1800
800
100
600
1500
200
700
1100
E
F
G
<=
Capacity
900
<=
1200
<=
300
<=
500
Unit Cost
$4.00
$3.75
$2.50
$5.25
$6.00
$6.75
$8.25
$7.50
$6.50
$8.75
$7.75
H
I
J
K
L
Nodes
F1
F2
WH1
WH2
WH3
DC1
DC2
DC3
DC4
DC5
Net Flow
2000
3000
0
0
0
-800
-700
-1500
-900
-1100
=
=
=
=
=
=
=
=
=
=
Supply/Demand
2000
3000
0
0
0
-800
-700
-1500
-900
-1100
Total Cost $57,800
J
3
4
5
6
7
8
9
10
11
12
13
McGraw-Hill/Irwin
7.61
Net Flow
=D4+D5
=D6+D7
=D8+D9-D4
=D10+D11+D12-D5-D6
=D13+D14-D7
=-D8
=-D9-D10
=-D11
=-D12-D13
=-D14
© The McGraw-Hill Companies, Inc., 2003
The SUMIF Function
• The SUMIF formula can be used to simplify the node flow constraints.
=SUMIF(Range A, x, Range B)
• For each quantity in (Range A) that equals x, SUMIF sums the
corresponding entries in (Range B).
• The net outflow (flow out – flow in) from node x is then
=SUMIF(“From labels”, x, “Flow”) – SUMIF(“To labels”, x, “Flow”)
McGraw-Hill/Irwin
7.62
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Model using SUMIF
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
C
D
From
F1
F1
F2
F2
WH1
WH1
WH2
WH2
WH2
WH3
WH3
To
WH1
WH2
WH2
WH3
DC1
DC2
DC2
DC3
DC4
DC4
DC5
Flow
900
1100
1200
1800
800
100
600
1500
200
700
1100
E
F
G
<=
Capacity
900
<=
1200
<=
300
<=
500
Unit Cost
$4.00
$3.75
$2.50
$5.25
$6.00
$6.75
$8.25
$7.50
$6.50
$8.75
$7.75
H
I
J
K
L
Nodes
F1
F2
WH1
WH2
WH3
DC1
DC2
DC3
DC4
DC5
Net Flow
2000
3000
0
0
0
-800
-700
-1500
-900
-1100
=
=
=
=
=
=
=
=
=
=
Supply/Demand
2000
3000
0
0
0
-800
-700
-1500
-900
-1100
Total Cost $57,800
J
3
4
5
6
7
8
9
10
11
12
13
McGraw-Hill/Irwin
Net Flow
=SUMIF($B$4:$B$14,I4,$D$4:$D$14)-SUMIF($C$4:$C$14,I4,$D$4:$D$14)
=SUMIF($B$4:$B$14,I5,$D$4:$D$14)-SUMIF($C$4:$C$14,I5,$D$4:$D$14)
=SUMIF($B$4:$B$14,I6,$D$4:$D$14)-SUMIF($C$4:$C$14,I6,$D$4:$D$14)
=SUMIF($B$4:$B$14,I7,$D$4:$D$14)-SUMIF($C$4:$C$14,I7,$D$4:$D$14)
=SUMIF($B$4:$B$14,I8,$D$4:$D$14)-SUMIF($C$4:$C$14,I8,$D$4:$D$14)
=SUMIF($B$4:$B$14,I9,$D$4:$D$14)-SUMIF($C$4:$C$14,I9,$D$4:$D$14)
=SUMIF($B$4:$B$14,I10,$D$4:$D$14)-SUMIF($C$4:$C$14,I10,$D$4:$D$14)
=SUMIF($B$4:$B$14,I11,$D$4:$D$14)-SUMIF($C$4:$C$14,I11,$D$4:$D$14)
=SUMIF($B$4:$B$14,I12,$D$4:$D$14)-SUMIF($C$4:$C$14,I12,$D$4:$D$14)
=SUMIF($B$4:$B$14,I13,$D$4:$D$14)-SUMIF($C$4:$C$14,I13,$D$4:$D$14)
7.63
© The McGraw-Hill Companies, Inc., 2003
The Minimum-Cost-Flow Model is an LP
• Any minimum cost flow model consists of a set of nodes:
– Supply node(s), with supply si
– Demand node(s), with demand di
– Transshipment nodes
• A set of arcs from node i to node j
– with cost cij
– some with limited capacity kij
• LP Formulation:
Let xij = flow from i to j
Minimize Cost = ∑ ij cij xij
subject to
Flow: ∑all j flowing out of i xij – ∑all j flowing into i xji = (si, di, or 0)
Capacity: xij ≤ kij
and xij ≥ 0.
McGraw-Hill/Irwin
7.64
© The McGraw-Hill Companies, Inc., 2003
Maximum Flow Problem
An oil company has the following pipeline network, where each pipeline
is labeled with its maximum flow rate (in thousands of gallons per hour).
8
D
B
7
3
10
A
6
1
F
4
10
2
C
12
G
2
8
E
Question: What is the maximum possible flow rate from A to G?
McGraw-Hill/Irwin
7.65
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Model
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
C
D
From
A
A
B
B
B
C
C
C
D
D
E
E
F
To
B
C
C
D
F
B
E
F
F
G
F
G
G
Ship
7
10
0
7
0
0
8
2
0
7
0
8
2
Maximum Flow
McGraw-Hill/Irwin
E
F
G
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<=
Capacity
10
10
1
8
6
1
12
4
3
7
2
8
2
H
I
Nodes
A
B
C
D
E
F
G
Net Flow
17
0
0
0
0
0
-17
J
K
Supply/Demand
=
=
=
=
=
0
0
0
0
0
17
7.66
© The McGraw-Hill Companies, Inc., 2003
Shortest Path Problem
The travel times along various routes in the Pacific Northwest is shown below.
Question: What is the quickest route from Seattle to Denver?
McGraw-Hill/Irwin
7.67
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Model
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
B
C
D
E
From
Seattle
Seattle
Seattle
Portland
Boise
Boise
Boise
Butte
Butte
Butte
Butte
Billings
Chey enne
Salt Lake City
Grand Junction
To
Portland
Boise
Butte
Boise
Butte
Chey enne
Salt Lake City
Billings
Chey enne
Salt Lake City
Boise
Chey enne
Denv er
Grand Junction
Denv er
On Route
0
0
1
0
0
0
0
1
0
0
0
1
1
0
0
Time (hr)
3
9
10
7
7
14
6
4
12
7
7
7
1
5
4
McGraw-Hill/Irwin
Total Time
F
G
H
Nodes
Seattle
Portland
Boise
Butte
Billings
Salt Lake City
Chey enne
Grand Junction
Denv er
Net Flow
1
0
0
0
0
0
0
0
-1
I
J
=
=
=
=
=
=
=
=
=
Supply/Demand
1
0
0
0
0
0
0
0
-1
22
7.68
© The McGraw-Hill Companies, Inc., 2003
Equipment Replacement
A production department needs to purchase a new machine. As the machine
ages, it requires additional maintenance and also has a higher defect rate. The
production department plans to replace the machine every few years. The
purchase price of a new machine is $10,000. The maintenance cost and cost of
defective product is given below. A used machine has no resale value.
Age of machine
Maintenance Cost
Cost of Defects
First Year
$3,000
$2,000
Second Year
$4,000
$4,000
Third Year
$6,000
$7,000
Fourth Year
$10,000
$11,000
Fifth Year
$20,000
$24,000
Question: What is the best replacement policy over the next
five years?
McGraw-Hill/Irwin
7.69
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet Solution
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Age of
Machine
First Y ear
Second Y ear
Third Y ear
Fourth Y ear
Fif th Y ear
C
D
Maintenance
Cost
$3,000
$4,000
$6,000
$10,000
$20,000
Cost of
Defects
$2,000
$4,000
$7,000
$11,000
$24,000
From
Y ear 0
Y ear 0
Y ear 0
Y ear 0
Y ear 0
Y ear 1
Y ear 1
Y ear 1
Y ear 1
Y ear 2
Y ear 2
Y ear 2
Y ear 3
Y ear 3
Y ear 4
McGraw-Hill/Irwin
To
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
Y ear
1
2
3
4
5
2
3
4
5
3
4
5
4
5
5
On Route
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
Total Cost
$59,000
E
F
G
H
I
J
Year
Y ear 0
Y ear 1
Y ear 2
Y ear 3
Y ear 4
Y ear 5
Net Flow
1
0
0
0
0
-1
=
=
=
=
=
=
Supply/Demand
1
0
0
0
0
-1
Purchase
Cost
$10,000
Cost
$15,000
$23,000
$36,000
$57,000
$101,000
$15,000
$23,000
$36,000
$57,000
$15,000
$23,000
$36,000
$15,000
$23,000
$15,000
7.70
© The McGraw-Hill Companies, Inc., 2003
Planning Vehicle Replacement at Phillips
Petroleum
• Phillips Petroleum had a fleet of 1,500 cars and 3,800 trucks.
• Modeled replacement strategy as shortest path model (20-year time
horizon)—solved model once for each class of vehicle.
• Could keep, purchase (replace), or lease, at 3-month intervals.
• Costs considered included:
– Maintenance and operating costs (fuel, oil, repair),
– Leasing cost for leased vehicles,
– Purchasing cost for purchased vehicles,
– State license fees and road taxes,
– Tax effects (investment tax credits, depreciation)
• First used to make lease-or-buy decision, then vehicle-replacement
strategy, and more recently for other equipment (non-vehicle).
For more details, see Waddell (1983) Jul-Aug Interfaces article, “A Model for Equipment
Replacement Decisions and Policies”, downloadable at www.mhhe.com/hillier2e/articles.
McGraw-Hill/Irwin
7.71
© The McGraw-Hill Companies, Inc., 2003