Table 1. Shipping costs, Supply, and Demand
for Powerco Example
From
To
City 1 City 2
City 3 City 4
Supply
(Million kwh)
Plant 1
$8
$6
$10
$9
35
Plant 2
Plant 3
Demand
(Million kwh)
$9
$14
45
$12
$9
20
$13
$16
30
$7
$5
30
50
40
Transportation Tableau
1
.
Solution
1. Decision Variable:
Since we have to determine how much electricity
is sent from each plant to each city;
Xij = Amount of electricity produced at plant i
and sent to city j
X14 = Amount of electricity produced at plant 1
and sent to city 4
2
.
2. Objective function
Since we want to minimize the total cost of shipping
from plants to cities;
Minimize Z = 8X11+6X12+10X13+9X14
+9X21+12X22+13X23+7X24
+14X31+9X32+16X33+5X34
3
.
3. Supply Constraints
Since each supply point has a limited production
capacity;
X11+X12+X13+X14 <= 35
X21+X22+X23+X24 <= 50
X31+X32+X33+X34 <= 40
4
.
4. Demand Constraints
Since each supply point has a limited production
capacity;
X11+X21+X31 >= 45
X12+X22+X32 >= 20
X13+X23+X33 >= 30
X14+X24+X34 >= 30
5
.
5. Sign Constraints
Since a negative amount of electricity can not be
shipped all Xij’s must be non negative;
Xij >= 0 (i= 1,2,3; j= 1,2,3,4)
6
.
LP Formulation of Powerco’s Problem
Min Z = 8X11+6X12+10X13+9X14+9X21+12X22+13X23+7X24
+14X31+9X32+16X33+5X34
S.T.:
X11+X12+X13+X14 <= 35
(Supply Constraints)
X21+X22+X23+X24 <= 50
X31+X32+X33+X34 <= 40
X11+X21+X31 >= 45
(Demand Constraints)
X12+X22+X32 >= 20
X13+X23+X33 >= 30
X14+X24+X34 >= 30
Xij >= 0 (i= 1,2,3; j= 1,2,3,4)
7
.
General Description of a Transportation
Problem
1. A set of m supply points from which a good is
shipped. Supply point i can supply at most si
units.
2. A set of n demand points to which the good is
shipped. Demand point j must receive at least di
units of the shipped good.
3. Each unit produced at supply point i and shipped
to demand point j incurs a variable cost of cij.
8
.
Xij = number of units shipped from supply point i to
demand point j
i m j n
min
 c X
ij
ij
i 1 j 1
j n
s.t. Xij  si (i  1,2,..., m)
j 1
i m
X
ij
 dj ( j  1,2,..., n)
i 1
Xij  0(i  1,2,..., m; j  1,2,..., n)
9
.
Balanced Transportation Problem
If Total supply equals to total demand, the
problem is said to be a balanced
transportation problem:
j n
i m
s  d
i
i 1
j
j 1
10
.
Balancing a TP if total supply exceeds total
demand
If total supply exceeds total demand, we
can balance the problem by adding dummy
demand point. Since shipments to the
dummy demand point are not real, they are
assigned a cost of zero.
11
.
Balancing a transportation problem if total
supply is less than total demand
If a transportation problem has a total
supply that is strictly less than total
demand the problem has no feasible
solution. There is no doubt that in such a
case one or more of the demand will be left
unmet. Generally in such situations a
penalty cost is often associated with unmet
demand and as one can guess this time the
total penalty cost is desired to be minimum
12
.
Optimal solution is found as:
X12=10,
X13=25,
X21=45,
X23=5,
X32=10,
X34=30
cost 1020
13
.
Assignment Problems
Example: Machineco has four jobs to be completed.
Each machine must be assigned to complete one job.
The time required to setup each machine for completing
each job is shown in the table below. Machinco wants to
minimize the total setup time needed to complete the
four jobs.
14
.
Setup times
(Also called the cost matrix)
Time (Hours)
Job1
Job2
Job3
Job4
Machine 1
14
5
8
7
Machine 2
2
12
6
5
Machine 3
7
8
3
9
Machine 4
2
4
6
10
15
.
The Model
According to the setup table Machinco’s problem can be
formulated as follows (for i,j=1,2,3,4):
16
min Z  14 X 11  5 X 12  8 X 13  7 X 14  2 X 21  12 X 22  6 X 23  5 X 24
 7 X 31  8 X 32  3 X 33  9 X 34  2 X 41  X 42  6 X 43  10 X 44
s.t. X 11  X 12  X 13  X 14  1
X 21  X 22  X 23  X 24  1
X 31  X 32  X 33  X 34  1
X 41  X 42  X 43  X 44  1
X 11  X 21  X 31  X 41  1
X 12  X 22  X 32  X 42  1
X 13  X 23  X 33  X 43  1
X 14  X 24  X 34  X 44  1
Xij  0orXij  1
.
For the model on the previous page note that:
Xij=1 if machine i is assigned to meet the demands of
job j
Xij=0 if machine i is not assigned to meet the demands
of job j
In general an assignment problem is balanced
transportation problem in which all supplies and
demands are equal to 1.
17
.
The Assignment Problem
In general the LP formulation is given as
n
Minimize
n
 c
i 1 j 1
n
x
ij
xij
 1, i  1,
,n
Each supply is 1
 xij  1, j  1,
,n
Each demand is 1
j 1
ij
n
i 1
18
xij  0 or 1, ij
.
Solution in the example
•
•
•
•
X(1,2)=1
X(2,4)=1
X(3,3)=1
X(4,1)=1
19
.
Comments on the Assignment
Problem
• The Air Force has used this for assigning
thousands of people to jobs.
• This is a classical problem. Research on the
assignment problem predates research on LPs.
• Very efficient special purpose solution techniques
exist.
– 10 years ago, Yusin Lee and J. Orlin solved a problem
with 2 million nodes and 40 million arcs in ½ hour.
20
.
Transshipment Problems
A transportation problem allows only shipments that go
directly from supply points to demand points. In many
situations, shipments are allowed between supply points
or between demand points. Sometimes there may also
be points (called transshipment points) through which
goods can be transshipped on their journey from a
supply point to a demand point. Fortunately, the optimal
solution to a transshipment problem can be found by
solving a transportation problem.
21
.
Transshipment Example
• Example 5: Widgetco manufactures widgets at two
factories, one in Memphis and one in Denver. The
Memphis factory can produce as 150 widgets, and the
Denver factory can produce as many as 200 widgets
per day. Widgets are shipped by air to customers in
LA and Boston. The customers in each city require
130 widgets per day. Because of the deregulation of
airfares, Widgetco believes that it may be cheaper
first fly some widgets to NY or Chicago and then fly
them to their final destinations. The cost of flying a
widget are shown next. Widgetco wants to minimize
the total cost of shipping the required widgets to
customers.
22
.
Transportation Tableau Associated
with the Transshipment Example
•
•
•
•
•
•
•
•
•
•
NY Chicago LA
Boston Dummy Supply
Memphis $8
$13 $25 $28
$0
150
Denver $15
$12
$26 $25
$0
200
NY
$0
$6
$16 $17
$0
350
Chicago $6
$0
$14 $16
$0
350
Demand 350
350
130
130
90
Supply points: Memphis, Denver
Demand Points: LA Boston
Transshipment Points: NY, Chicago
The problem can be solved using the transportation simplex
method
23
.
Maximum Flow Problem
• Maximum flow problem description
– All flow through a directed and connected network
originates at one node (source) and terminates at one
another node (sink)
– All the remaining nodes are transshipment nodes
– Flow through an arc is allowed only 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
– The objective is to maximize the total amount of flow from
the source to the sink (measured as the amount leaving the
source or the amount entering the sink)
24
.
Maximum Flow Problem
• Typical applications
– Maximize the flow through a company’s
distribution network from its factories to its
customers
– Maximize the flow through a company’s supply
network from its vendors to its factories
– Maximize the flow of oil through a system of
pipelines
– Maximize the flow of water through a system of
aqueducts
– Maximize the flow of vehicles through a
transportation network
25
.
Maximum Flow Algorithm
• Some Terminology
– The residual network shows the remaining arc
capacities for assigning additional flows after some
flows have been assigned to the arcs
The residual capacity for assigning some flow from node B to node O
O
5
2
B
The residual capacity for flow from node O to Node B
26
.
Maximum Flow Algorithm
– An augmenting path is a directed path from the source to
the sink in the residual network such that every arc on this
path has strictly positive residual capacity
– The residual capacity of the augmenting path is the
minimum of these residual capacities (the amount of flow
that can feasibly be added to the entire path)
• Basic idea
– Repeatedly select some augmenting path and add a flow
equal to its residual capacity to that path in the original
network. This process continues until there are no more
augmenting paths, so that the flow from the source to the
sink cannot be increased further
27
.
Maximum Flow Algorithm
• The Augmenting Path Algorithm
– Assume that the arc capacities are either integers or rational
numbers
• 1. identify an augmenting path by finding some
directed path from the source to the sink in the
residual network such that every arc on this path has
strictly positive residual capacity. If no such path
exists, the net flows already assigned constitute an
optimal flow pattern
• 2. Identify the residual capacity c* of this augmenting
path by finding the minimum of the residual
capacities of the arcs on this path. Increase the flow in
this path by c*
28
.
Maximum Flow Example
• During the peak season the park management of the
Seervada park would like to determine how to route
the various tram trips from the park entrance (Station
O) to the scenic (Station T) to maximize the number
of trips per day. Each tram will return by the same
route it took on the outgoing trip so the analysis
focuses on outgoing trips only. To avoid unduly
disturbing the ecology and wildlife of the region,
strict upper limits have been imposed on the number
of outgoing trips allowed per day in the outbound
direction on each individual road. For each road the
direction of travel for outgoing trips is indicated by
an arrow in the next slide. The number at the base of
the arrow gives the upper limit on the number of
outgoing trips allowed per day.
29
.
Maximum Flow Example
• Consider the problem of sending as many units from node O to
node T for the following network (current flow, capacity):
(0,3)
A
(0,9)
D
(0,5)
(0,1)
(0,4)
T
(0,1)
(0,7)
(0,6)
B
O
(0,5)
(0,4)
(0,2)
(0,4)
C
30
.
E
Maximum Flow Example
•
Iteration 1: one of the several augmenting paths is OBET, which has a
residual capacity of min{7, 5, 6} = 5. By assigning the flow of 5 to this path, the
resulting network is shown above
(0,3)
A
(0,9)
D
(0,5)
(0,1)
(0,4)
T
(0,1)
(5,7)
(5,6)
B
O
(5,5)
(0,4)
(0,2)
(0,4)
C
31
.
E
Maximum Flow Example
• Iteration 2: Assign a flow of 3 to the augmenting path
OADT. The resulting residual network is
(3,3)
A
(3,9)
D
(3,5)
(0,1)
(0,4)
T
(0,1)
(5,7)
(5,6)
B
O
(5,5)
(0,4)
(0,2)
(0,4)
C
32
.
E
Maximum Flow Example
• Iteration 3: Assign a flow of 1 to the augmenting path
OABDT. The resulting residual network is
(3,3)
A
(4,9)
D
(4,5)
(1,1)
(1,4)
T
(0,1)
(5,7)
(5,6)
B
O
(5,5)
(0,4)
(0,2)
(0,4)
C
33
.
E
Maximum Flow Example
• Iteration 4: Assign a flow of 2 to the augmenting path OBDT. The
resulting residual network is
(3,3)
A
(6,9)
D
(4,5)
(1,1)
(3,4)
T
(0,1)
(7,7)
(5,6)
B
O
(5,5)
(0,4)
(0,2)
(0,4)
C
34
.
E
Maximum Flow Example
Iteration 5: Assign a flow of 1 to the augmenting path
OCEDT. The resulting residual network is
(3,3)
A
(7,9)
D
(4,5)
(1,1)
(3,4)
T
(1,1)
(7,7)
(5,6)
B
O
(5,5)
(1,4)
(0,2)
(1,4)
C
35
.
E
Maximum Flow Example
Iteration 6: Assign a flow of 1 to the augmenting path
OCET. The resulting residual network is
(3,3)
A
(7,9)
D
(4,5)
(1,1)
(3,4)
T
(1,1)
(7,7)
(6,6)
B
O
(5,5)
(2,4)
(0,2)
(2,4)
C
36
.
E
Maximum Flow Example
• There are no more flow augmenting paths, so the current
flow pattern is optimal
3
A
7
D
1
4
T
13
3
O
1
B
13
6
7
5
2
2
C
37
.
E
Minimum Spanning Tree Problems
Suppose that each arc (i,j) in a network has a length
associated with it and that arc (i,j) represents a way of
connecting node i to node j. For example, if each node
in a network represents a computer in a computer
network, arc(i,j) might represent an underground cable
that connects computer i to computer j. In many
applications, we want to determine the set of arcs in a
network that connect all nodes such that the sum of the
length of the arcs is minimized. Clearly, such a group of
arcs contain no loop.
38
.
Minimum Spanning Tree Problem
• An undirected and connected network is being
considered, where the given information includes
some measure of the positive length (distance, cost,
time, etc.) associated with each link
• Both the shortest path and minimum spanning tree
problems involve choosing a set of links that have the
shortest total length among all sets of links that
satisfy a certain property
– For the shortest-path problem this property is that the
chosen links must provide a path between the origin and
the destination
– For the minimum spanning tree problem, the required
property is that the chosen links must provide a path
between each pair of nodes
39
.
Some Applications
• Design of telecommunication networks (fiber-optic
networks, computer networks, leased-line telephone
networks, cable television networks, etc.)
• Design of lightly used transportation network to
minimize the total cost of providing the links (rail
lines, roads, etc.)
• Design of a network of high-voltage electrical
transmission lines
• Design of a network of wiring on electrical
equipment (e.g., a digital computer system) to
minimize the total length of the wire
• Design of a network of pipelines to connect a number
of locations
40
.
For a network with n nodes, a spanning tree is a group
of n-1 arcs that connects all nodes of the network and
contains no loops.
1
12
2
(1,2)-(2,3)-(3,1) is a loop
4
7
(1,3)-(2,3) is the minimum spanning tree
3
41
.
Minimum Spanning Tree Problem
Description
• You are given the nodes of the network but not the
links. Instead you are given the potential links and the
positive length for each if it is inserted into the
network (alternative measures for length of a link
include distance, cost, and time)
• You wish to design the network by inserting enough
links to satisfy the requirement that there be a path
between every pair of nodes
• The objective is to satisfy this requirement in a way
that minimizes the total length of links inserted into
the network
42
.
Minimum Spanning Tree Algorithm
• Greedy Algorithm
– 1. Select any node arbitrarily, and then connect (i.e., add a
link) to the nearest distinct node
– 2. Identify the unconnected node that is closest to a
connected node, and then connect these two nodes (i.e., add
a link between them). Repeat the step until all nodes have
been connected
– 3. Tie breaking: Ties for the nearest distinct node (step 1)
or the closest unconnected node (step 2) may be broken
arbitrarily, and the algorithm will still yield an optimal
solution.
• Fastest way of executing algorithm manually is the
graphical approach illustrated next
43
.
Example: The State University campus has five
computers. The distances between computers are given
in the figure below. What is the minimum length of
cable required to interconnect the computers? Note that
if two computers are not connected this is because of
underground rock formations.
1
1
2
2
6
5
4
4
44
3
2
4
3
5
.
2
Solution: We want to find the minimum spanning tree.
• Iteration 1: Following the MST algorithm discussed
before, we arbitrarily choose node 1 to begin. The
closest node is node 2. Now C={1,2}, Ć={3,4,5}, and
arc(1,2) will be in the minimum spanning tree.
1
1
2
2
6
5
4
4
45
3
2
4
3
5
.
2
• Iteration 2: Node 5 is closest to C. since node 5 is two
blocks from node 1 and node 2, we may include either
arc(2,5) or arc(1,5) in the minimum spanning tree. We
arbitrarily choose to include arc(2,5). Then C={1,2,5}
and Ć={3,4}.
1
1
2
2
6
5
4
4
46
3
2
4
3
5
.
2
• Iteration 3: Since node 3 is two blocks from node 5,
we may include arc(5,3) in the minimum spanning tree.
Now C={1,2,5,3} and Ć={4}.
1
1
2
2
6
5
4
4
47
3
2
4
3
5
.
2
• Iteration 4: Node 5 is the closest node to node 4. Thus,
we add arc(5,4) to the minimum spanning tree.
We now have a minimum spanning tree consisting of
arcs(1,2), (2,5), (5,3), and (5,4). The length of the
minimum spanning tree is 1+2+2+4=9 blocks.
1
1
2
2
6
5
4
4
48
3
2
4
3
5
.
2