Graduate School of Information, Production and Systems, Waseda University
6. Basic Network Design
6. Basic Network Design
 Genetic Algorithms (GAs) are one of the most powerful and broadly applicable
stochastic search and optimization techniques based on principles from
evolution theory (Holland, 1976):
 Michalewicz, Z. : Genetic Algorithm + Data Structure = Evolution Programs, 2nd ed.,
Springer-Verlag, New York, 1994
 Gen, M. & R. Cheng: Genetic Algorithms & Engineering Design, John Wiley & Sons,
New York, 1997.
 Recent advances in evolutionary computation have made it possible to solve
such practical network optimization problems:
 Ali, M. & F. Kamoun： “Neural Networks for Shortest Path Computation and Routing in
Computer Networks”, IEEE Trans. on Neural Networks, vol.4, pp.941-954, 1993.
 Perfetti, R. : “Optimization Neural Network for Solving Flow Problems”, IEEE Trans.
on Neural Network, Vol.6, No.5, pp.1287-1291, 1995.
 Gen, M. & K. Ida: Neural Networks and Optimization with Mathematica, Kyoritsu
Shuppan, 1998 in Japanese.
 Ahn, C. W., R. Ramakrishna, C. Kang & I. Choi: “Shortest Path Routing Algorithm
using Hopfield Neural Network”, Electronic Letter, Vol.37, No.19, pp.1176-1178, 2001.
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6. Basic Network Design
 In the past few years, the genetic algorithms community has turned much of its
attention toward the optimization of network design problems:
 Munakata, T. & D. J. Hashier: “A genetic algorithm applied to the maximum flow problem”, Proc. of
the 5th Inter. Conf. on Genetic Algorithms, San Francisco, pp.488-493, 1993.
 Gen, M. & R. Cheng: Genetic Algorithms and Engineering Design, John Wiley & Sons, New York,
1997.
 Munetomo, M., Y. Takai & Y. Sato: “A migration Scheme for the Genetic Adaptive routing Algorithm”,
Proc. of IEEE Int. Conf. Systems, Man, and Cybernetics, pp.2774-2779, 1998.
 Inagaki, J., M. Haseyama & H. Kitajima: “A Genetic Algorithm for Determining Multiple Routes and
Its Applications”, Proc. of IEEE Int. Symp. Circuits and Systems, pp.137-140, 1999.
 Gen, M. & R. Cheng: Genetic Algorithms and Engineering Optimization, John Wiley & Sons, New
York, 2000.
 Gen, M., R. Cheng & S.S. Oren: "Network design techniques using adapted genetic algorithms",
Advances in Engineering Software, Vol.32, pp.731-744, 2001.
 Ahn, C.W. & R. Ramakrishna: “A Genetic Algorithm for Shortest Path Routing Problem and the
Sizing of Populations”, IEEE Trans. on Evol. Comput., Vol.6, No.6, pp.566-579, 2002.
 Zhou, G. & M. Gen: “A Genetic Algorithm Approach on Tree-like Telecommunication Network
Design Problem”, J. of Operational Research Society, Vol. 54, No. 3, pp.248-254, 2003.
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vBNS Backbone Network Map
http://www.mci.com/index.jsp
Soft Computing Lab.
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WASEDA UNIVERSITY
IPS high speed Backbone Network Services
vBNS: ,very
vBNS Logical Network Map
http://www.mci.com/index.jsp
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6. Basic Network Design
1. Shortest Path Problem (SPP)
2. Maximum Flow (MXF) Problem
3. Minimum Cost Flow (MCF) Problem
4. Bicriteria Network Design Problem (BNP)
5. Multi-criteria Network Design Problem
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6. Basic Network Design
1.
Shortest Path Problem (SPP)
1.1 Basic Concept of Shortest Path Problem
1.2 Application of Shortest Path Problem
1.3 Methods for solving SPP
1.4 Genetic Approach for solving SPP
1.4.1 Reviewing Encoding Methods
1.4.2 Priority-based Genetic Algorithm
1.4.3 Genetic Operators
1.5 Numerical Examples
2.
Maximum Flow (MXF) Problem
3.
Minimum Cost Flow (MCF) Problem
4.
Bicriteria Network Design Problem (BNP)
5.
Multi-criteria Network Design Problem
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1. Shortest Path Problem (SPP)
1.1 Basic Concept of Shortest Path Problem
 SPP is perhaps the simplest of all network design
problems.
Data table of example network
 For this problem, the object is to find a path of
minimum cost (or length) from a specified source
node s to another specified sink node t, assuming
that each arc (i, j)∈A has an associated cost (or
length) cij.
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1. Shortest Path Problem (SPP)
1.1 Basic Concept of Shortest Path Problem

Directed graph G=(V, A)
Data table of example network
i
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where V is a set of nodes, A is a set of links.

cij is a cost associated with each arc(i, j)

Source node: node 1

Destination node: node n

Indicator variable:
 1,
xij  
 0,
if link (i , j ) is included in the path
otherwise
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1. Shortest Path Problem (SPP)
1.1 Basic Concept of Shortest Path Problem
 SPP can be formulated as follows:
n
n
min z   cij xij
i 1 j 1
n
s.t.
n
x x
j 1
ij
k 1
ki
 1　(i  1)

  0 (i  2,3,  , n  1)
 1　(i  n)

xij  0 or 1　(i, j  1, 2,  , n)
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1. Shortest Path Problem (SPP)
1.2 Application of Shortest Path Problem

This basic model can be applied in many applications such as:
Evans, J. R. and E. Minieka: Optimization Algorithms for Networks and Graphs.
New York: Marcel-Dkker, 1992.

Transportation Planning


Salesperson Routing


Suppose that a sales person want to go to Los Angeles from Boston and
stop over in several city to get some commission. How can she determine
the route?
Investment Planning


How to determine the route road that have prohibitive weight restriction so
that the driver can reach the destination within the shortest possible time.
How to determine the invest strategy to get an optimal investment plan.
Message routing in communication systems


The Routing algorithm computes the shortest (least cost) path between the
router and all the networks of the internetwork.
It is one of the most important issues that has a significant impact on the
network’s performance.
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1. Shortest Path Problem (SPP)
1.2 Application of Shortest Path Problem


With the growth of the Internet, Internet Service Providers (ISPs) try to meet
the increasing traffic demand with new technology and improved utilization
of existing resources.
Routing of data packets can affect network utilization.
 Packets are sent along network paths from source to destination following
a protocol.
 Open Shortest Path First (OSPF) is the most commonly used protocol.


Ericsson, M., M.G.C. Resende & P.M. Pardalos: “A Genetic Algorithm for the
Weight Setting Problem in OSPF Routing”, J. of Combinatorial Optimization,
No.6, pp.299–333, 2002.
OSPF is designed for exchanging routing information within a large or very
large internetwork.
 The biggest advantage of OSPF is that it is efficient.


OSPF requires very little network overhead even in very large internetworks.
The biggest disadvantage of OSPF is its complexity.

OSPF requires proper planning and is more difficult to configure and administer.
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1. Shortest Path Problem (SPP)
1.2 Application of Shortest Path Problem

OSPF uses a Shortest Path Routing (SPR) algorithm to compute routes in
the routing table.


As the size of the link state database increases:


The SPR algorithm computes the shortest (least cost) path between the router
and all the networks of the internetwork.
Memory requirements and route computation times increase.
Genetic Algorithm (GA) approaches to the SPR problem in OSPF.
Ahn, C.W. & R. Ramakrishna: “A Genetic Algorithm for Shortest Path Routing
Problem and the Sizing of Populations”, IEEE Trans. on Evol. Comput., Vol.6,
No.6, pp.566-579, 2002.
Lin, L., M. Gen & R. Cheng: “Priority-based Genetic Algorithm for Shortest Path
Routing Problem in OSPF”, Proc. of 3rd Inter. Conf. on Information and
Management Sciences, Dunhuang, China, June 5-10, 2004.

The objective of this research considers the quality of solution (path optimality)
within the shortest route computation times.
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1. Shortest Path Problem (SPP)
1.3 Methods for Solving SPP
 Dijkstra Shortest Path Algorithm
 Dijkstra, E. W.: "A Note on Two Problems in Connection with Graphs",
Numerische Math., No.1, pp.269-271, 1959.
 Dijkstra's algorithm can be implemented efficiently by storing the graph
in the form of adjacency lists and using a heap as priority queue to
implement the Extract-Min function.
 Computes shortest paths in a graph with non-negative edge weights.
 Bellman-Ford Algorithm
 Bellman-Ford algorithm computes single-source shortest paths in a
weighted graph (where some of the edge weights may be negative).
 Bellman-Ford is usually used only when there are negative edge
weights.
 Floyd-Warshall Algorithm
 Floyd-Warshall algorithm is an algorithm to solve the all pairs shortest
path problem in a weighted, directed graph by multiplying an
adjacency-matrix representation of the graph multiple times.
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1. Shortest Path Problem (SPP)
1.4 Genetic Approach for Solving SPP
 How to encode a path in a network is critical for designing a
GA.
 Special difficulties:
 a path contains variable number of nodes.
 a random sequence of edges usually does not correspond to a
path.
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 Path 1： 1→2→4→8→10
Objective function value： z=110
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 Path 2： 1→2→4→7→8→10
Objective function value： z=109
 Path 3： 1→3→5→4→7→8→10
Objective function value： z=110
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1. Shortest Path Problem (SPP)
1.4.1 Reviewing Encoding Methods


How to encode a solution of the problem into a chromosome is a key issue
for GAs.
For the nonstring coding approach, three critical issues emerged
concerning with the encoding and decoding between chromosomes and
solutions:

The feasibility of a chromosome


The legality of a chromosome




Feasibility refers to the phenomenon of whether a solution decoded from a
chromosome lies in the feasible region of a given problem.
Legality refers to the phenomenon of whether a chromosome represents a
solution to a given problem.
The illegality of chromosomes originates from the nature of encoding
techniques.
Repairing techniques are usually adopted to convert an illegal chromosome to
a legal one.
The uniqueness of mapping



The mapping from chromosomes to solutions (decoding) may belong to one of
the following three cases: (a) 1-to-1 mapping; (b) n-to-1 mapping; (c) 1-to-n
mapping.
The 1-to-1 mapping is the best one among three cases
And 1-to-n mapping is the most undesired one.
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1.4.1 Reviewing Encoding Methods
a. Priority-based Chromosome (Cheng & Gen, 1997)
 Cheng & Gen proposed a priority-based encoding method for solving resourceconstrained project scheduling problem (rcPSP) first. And also adopted this method for
solving SPP in 1997.

Cheng, R. & M. Gen: “Resource Constrained Project Scheduling Problem using Genetic Algorithm”, Inter. J. of
Intelligent Auto. and Soft Comput., Vol.3, pp.273-286, 1997.

Gen, M., R. Cheng & D. Wang: “Genetic Algorithms for Solving Shortest Path Problems”, Proc. of IEEE Int. Conf. on
Evol. Comput., Indianapolis, Indiana, pp.401-406, 1997.
 They adopted an indirect approach:
 The path is generated by sequential node appending procedure with beginning from
the specified node 1 and terminating at the specified node n.
 At each step, there are usually several nodes available for consideration.
 They gave each node a priority with a random mechanism and add the one with the
highest priority into path.


As we know, a gene in a chromosome is characterized by two factors:

locus, i.e., the position of gene located within the structure of chromosome,

allele, i.e., the value which the gene takes.
In the priority-based encoding method, the position of a gene is used to represent node ID and its value is used to
represent the priority of the node for constructing a path among candidates. A path can be uniquely determined from this
encoding.
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1.4.1 Reviewing Encoding Methods
a. Priority-based Chromosome (Cheng & Gen, 1997)
 Example: An example of generated chromosome and its decoded path
as follows:
2
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1
5
t 1
7
4
3
6
node ID :
1
2
3
4
5
6
7
priority :
2
1
6
4
5
3
7
path :
1
3
4
7
 Advantage:
 Any permutation of the encoding corresponds to a path (legality).
 Most existing genetic operators can be easily applied to the encoding.
 Any path has a corresponding encoding (completeness); any point in
solution space is accessible for genetic search.
 Disadvantage:
 At some case, n-to-1 mapping may occur for the encoding.
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1.4.1 Reviewing Encoding Methods
b. Variable-length Chromosome (Munemoto et al., 1998)
 Munemoto et. al. (1998) proposed a variable-length encoding method for
network routing problems in a wired or wireless environment. Ahn et. al.
(2002) also used the encoding method for solving the shortest path routing
(SPR) problem.
 Munetomo, M., Y. Takai & Y. Sato: “A migration Scheme for the Genetic Adaptive routing
Algorithm”, Proc. of IEEE Int. Conf. Systems, Man, and Cybernetics, pp.2774-2779, 1998.
 Ahn, C.W. & R. Ramakrishna: “A Genetic Algorithm for Shortest Path Routing Problem and the
Sizing of Populations”, IEEE Trans. on Evol. Comput., Vol.6, No.6, pp.566-579, 2002.
 The proposed encoding method consists of sequences of positive integers
that represent the IDs of nodes through which a path passes.
 Each locus of the chromosome represents an order of a node (indicated by the gene of the
locus) in a path.
 The length of the chromosome is variable, but is should not exceed the maximum length n,
where n is the total number of nodes in the network, since it never needs more than n number of
nodes to form a path.
 The gene of the first locus encodes the source node, and the gene of second locus is randomly
or heuristically selected from the nodes connected with the source node.
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1.4.1 Reviewing Encoding Methods
b. Variable-length Chromosome (Munemoto et al., 1998)

Example: An example of generated chromosome and its decoded path as follows:
2
1 s
1
t 1
7
4
3
locus :
1
2
3
4
node ID : 1
3
4
7
5
6
path :
1
3
4
7
 Advantage:
 The mapping from any chromosome to solution (decoding) belongs to 1to-1 mapping (uniqueness).
 Theoretically, convergence performance is better than the priority-based
encoding method.
 Disadvantage:
 In general, the genetic operators may generate infeasible chromosomes
(illegality) that violate the constraints, generating loops in the paths.
 Repairing techniques are usually adopted to convert an illegal
chromosome to a legal one.
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1.4.1 Reviewing Encoding Methods
c. Fixed-length Chromosome (Inagaki et al., 1999)
 Inagaki et al. (1999) proposed a fixed-length encoding method
determining multiple routes in routing applications.
 Inagaki, J., M. Haseyama & H. Kitajima: “A Genetic Algorithm for Determining
Multiple Routes and Its Applications”, Proc. of IEEE Int. Symp. Circuits and
Systems, pp.137-140, 1999.
 The proposed method are sequences of integers and each gene
represents the node ID through which it passes.
 To encode a route from node 1 to node n, put i in the jth locus of the
chromosome.
 This process is reiterated from the specified node 1 and terminating at
the specified node n.
 If the route does not pass through a node x, select one node randomly
from the set of nodes which are connected with node x, and put it in
the xth locus.
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1.4.1 Reviewing Encoding Methods
c. Fixed-length Chromosome (Inagaki et al., 1999)

Example: An example of generated chromosome and its decoded path as follows:
2
1 s
1
5
t 1
7
4
3
6
locus :
1
2
3
4
5
6
7
node ID :
3
1
4
7
2
4
6
path :
1
3
4
7
 Advantage:
 Any path has a corresponding encoding (completeness).
 Any point in solution space is accessible for genetic search.
 Any permutation of the encoding corresponds to a path (legality) using the
special genetic operators.
 Disadvantage:
 At some case, n-to-1 mapping may occur for the encoding.
 Furthermore the probability of occurrence of n-to-1 mapping is higher than
the priority-based encoding method.
 In the special genetic operator phase, some offspring may generate new
chromosomes that resemble the initial chromosomes in fitness, thereby
retarding the process of evolution.
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1.4.1 Reviewing Encoding Methods
 Compared with the Performance of Different Encoding Methods:
 Variable-length encoding method
 Convergence performance is best than others.
 However, the genetic operators may generate infeasible chromosomes
(illegality).
 Repairing techniques have to be adopted to convert an illegal
chromosome to a legal one. For the computation times, variable-length
encoding method may be slow in several large network design problems.
 Fixed-length encoding method
 n-to-1 mapping may occur for the encoding.
 The special genetic operators have to been adopted; thereby some
offspring may generate new chromosomes that resemble the initial
chromosomes in fitness.
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1.4.2 Priority-based Genetic Algorithm
 Priority-based Encoding Method
procedure 1: Priority-based Encoding
input: number of nodes n
output: chromosome vk
begin
for j=1 to n
// step 0
vk(j) j;
for i=1 to n / 2
// step 1
repeat
jrandom[1, n];
lrandom[1, n];
until l≠j
swap (vk(j), vk(l));
output the chromosome vk;
// step 2
end
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1.4.2 Priority-based Genetic Algorithm
 Decoding Method
procedure 2: One Path Growth
input: number of nodes n, chromosome vk ,
the set of nodes Si with all nodes adjacent to node i.
output: path Pk
begin
initial source node i1, Pk ;
// step 0
while Si ≠ do
// step 1
select l from Si with the highest priority;
if vk(l)≠0 then
vk(l)=0;
Pk  Pk{xil};
il;
else Si  Si \{l}
end
output the complete path Pk ;
// step 2
end
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1.4.2 Priority-based Genetic Algorithm
 Illustration of Priority-based GA
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Data table of example network
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node ID: j
1
2
3
4
5
6
7
9
10
priority: v(j)
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Path:
8
1→3→6→7→8→10
Objective function value： z=106
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1.4.3 Genetic Operators --- Crossover
 It operates on two parents (chromosomes) at a time and generates
offspring by combining both chromosomes’ features.

In network design problem, crossover plays the role of exchanging each partial route of two chosen
parents in such a manner that the offspring produced by the crossover represents.
 In this study, the nature of the priority-based encoding is a kind of permutation
representation.

Generally, this representation will yield illegal offspring by one-point crossover or other simple
crossover operators.
 During the past decade, several crossover operators have been proposed for
permutation representation, such as:


Partial-mapped crossover (PMX)
Goldberg, D. & R. Lingle, Alleles: “loci and the traveling salesman problem”, Proc. of the 1st Inter.
Conf. on GA, pp.154-159, 1985.
Order crossover (OX):
Davis, L. : “Applying adaptive algorithms to domains”, Proc. of the Inter. Joint Conf. on
Artificial Intelligence, pp.1162-164, 1985.

Position-based crossover (PX)
Davis, L. : “Applying adaptive algorithms to domains”, Proc. of the Inter. Joint Conf. on
Artificial Intelligence, pp.1162-164, 1985.

Cycle crossover (CX)
Oliver, I. & J. Holland: “A study of permutation crossover operators on the traveling salesman
problem, Euro. J. of OR, vol.26, pp.187-210, 1986.
 Heuristic crossover, and so on.
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1.4.3 Genetic Operators --- Crossover
 Partial-Mapped Crossover (PMX)
 PMX was proposed by Goldberg and Lingle.
Goldberg, D. & R. Lingle, Alleles: “loci and the traveling salesman problem”, Proc.
of the 1st Inter. Conf. on GA, pp.154-159, 1985.
 PMX can be viewed as an extension of two-point crossover for binary
string to permutation representation.
 It uses a special repairing procedure to resolve the illegitimacy caused
by the simple two-point crossover.
step 3 : determine mapping relationship
step 1 : select the substring at random
substring selected
parent 1:
1
7
2
3
4
6
5
8
parent 2:
4
6
3
5
7
1
8
2
step 2 : exchange substrings between
3
4
3
5
7
235
47
step 4 : legalize offspring with mapping
relationship
parent 1:
1
7
3
5
7
6
5
8
parent 2:
4
6
2
3
4
1
8
2
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offspring 2:
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1.4.3 Genetic Operators --- Crossover
 Order Crossover (OX)
 OX was proposed by Davis.
Davis, L. : “Applying adaptive algorithms to domains”, Proc. of the Inter.
Joint Conf. on Artificial Intelligence, pp.1162-164, 1985.
 It can be viewed as a kind of variation of PMX with a different
repairing procedure.
parent 1:
1
7
2
3
4
6
5
8
substring selected
offspring:
6
5
2
3
4
7
1
8
parent 2:
4
6
3
5
7
1
8
2
Fig. 6.1 Illustration of the OX operator.
Soft Computing Lab.
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29
1.4.3 Genetic Operators --- Crossover
 Position-based Crossover (PX)
 PX was proposed by Syswerda.
Davis, L. : “Applying adaptive algorithms to domains”, Proc. of the Inter.
Joint Conf. on Artificial Intelligence, pp.1162-164, 1985.
 It is essentially a kind of uniform crossover for permutation
representation together with a repairing procedure.
 It also can be viewed as a kind of variation of OX in which the
nodes are selected inconsecutively.
parent 1:
1
7
2
3
4
6
5
8
offspring:
3
7
5
1
4
6
2
8
parent 2:
4
6
3
5
7
1
8
2
Fig. 6.2 Illustration of the PX operator.
Soft Computing Lab.
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30
1.4.3 Genetic Operators --- Crossover
 However, in all of above approaches:
 the mechanism of the crossover is not the same as that of the
conventional one-point crossover.
 Some offspring may generate new chromosomes that are not possible
to succeed the character of the parents.
 thereby retarding the process of evolution.
 We proposed a new crossover operator, Weight Mapping
Crossover (WMX).
 WMX can be viewed as an extension of one-point crossover for
permutation representation.
 As one-point crossover:
 Two chromosomes (parents) would be to choose a random cut-point.
 Generate the offspring by using segment of own parent to the left of
the one-cut point
 Then remapping the right segment that base on the weight of other
parent of right segment .
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31
1.4.3 Genetic Operators --- Crossover
 Weight Mapping Crossover (WMX)
procedure : Weight Mapping Crossover
input : v1, v2 , n
output : v1', v2'
begin
p  random [1, n];
l  n  p;
for i  1 to l
for j  1 to l
if v1'[ p  i ]  s2 [ j ] then
for i  1 to l
v1'[ p  i ]  s1[ j ];
s1[i ]  v1[ p  i ];
for j  1 to l
if v2'[ p  i ]  s1[ j ] then
s2 [i ]  v2 [ p  i ];
s1[]  sorting ( s1[]);
v2'[ p  i ]  s2 [ j ];
s2 []  sorting ( s2 []);
v1'  v1[1 : p ] // v2 [ p  1 : n];
v2'  v2 [1 : p ] // v1[ p  1 : n];
Soft Computing Lab.
output v1' , v2' ;
end
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32
1.4.3 Genetic Operators --- Crossover
 Weight Mapping Crossover (WMX)
 As shown Fig., first we choose a random cut-point p.
 calculate l that is the length of right segments of chromosomes, where n is
number of nodes in the network.
 Then get mapping relationship by sorting the weight of the right segments s1[∙]
and s2[∙].
 As one-point crossover, generate the offspring v1’, v2’ by exchange substrings
between parents v1, v2; legalize offspring with mapping relationship.
step 1: select a cut-point
cut-point
parent 1 :
2
1
7
4
5
3
6
parent 2 :
3
7
2
6
5
1
4
parent 1 : 1
3
4
7
parent 2 : 1
2
4
5
7
7
step 2: mapping the weight of the right segment
5
3
6
5
1
4
3
5
6
offspring 1 : 1
3
4
5
1
4
5
offspring 2 : 1
2
4
7
step 3: generate offspring with mapping relationship
offspring 1 :
2
1
7
4
6
3
5
offspring 2 :
3
7
2
6
4
1
5
Soft Computing Lab.
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33
1.4.3 Genetic Operators --- Mutation
 It is relatively easy to produce some mutation operators for
permutation representation.
 During the past decade, several mutation operators have
been proposed for permutation representation, such as:




Inversion
Insertion
Displacement
Swap mutation.
 Insertion Mutation
 Selects a gene at random and inserts it in a random position as
follows:
select a gene at random
parent :
2
1
7
4
5
3
6
parent : 1
3
4
offspring : 1
4
7
insert it in a random position
offspring :
Soft Computing Lab.
2
5
1
7
4
3
6
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7
34
1.4.3 Genetic Operators --- Immigration
 The trade-off between exploration and exploitation in
serial GAs for function optimization is a fundamental
issue.
 If a GA is biased towards exploitation:
 highly fit members are repeatedly selected for recombination.
 Although this quickly promotes better members, the population
can prematurely converge to a local optimum of the function.
 If a GA is biased towards exploration:
 Large numbers of schemata are sampled which tends to inhibit
premature convergence.
 Unfortunately, excessive exploration results in a large number
of function evaluations, and defaults to random search in the
worst case.
Soft Computing Lab.
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35
1.4.3 Genetic Operators --- Immigration
 To search effectively and efficiently, a GA must maintain a balance
between these two opposing forces.
Michael, C.M., C.V. Stewart & R.B. Kelly: “Reducing the Search Time of A Steady
State Genetic Algorithm using the Immigration Operator”, Proc. of IEEE Int. Conf.
on Tools for AI San Jose, CA, pp.500-501, 1991.
 Michael et. al. (1991) proposed an immigration operator which, for certain types
of functions, allows increased exploration while maintaining nearly the same level
of exploitation for the given population size.
 Immigration operator
step 1: The algorithm is modified to include immigration, with each
generation generated.
step 2: Evaluate μ random members (μ, called the immigration rate).
step 3: Replace the μ worst members of the population with the μ random
members.
 This study experimentally examines the immigration operator, and
present the effectiveness of this approach for solving network design
problems in next section.
Soft Computing Lab.
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1.4.3 Genetic Operators --- Selection
 Selection operators: two basic types of selection scheme
used commonly in current practice.
 Proportionate selection: picks out chromosomes based on their
fitness values relative to the fitness of the other chromosomes in
the population.
 Roulette wheel selection
 Stochastic remainder selection
 Stochastic universal selection
 Ordinal-based selection: upon their rank within the population.
The chromosomes are ranked according to their fitness values.




Tournament selection
(  ,  ) selection
Truncation selection
Linear ranking selection
 In this study, the roulette wheel selection, a type of
Proportionate selection, is adopted.
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37
1.4.4 Overall Procedure
 GA Procedure for Shortest Path Problem
procedure: Priority-based GA for Shortest Path Problem
input: network data (V, A, C), GA parameters
output: best shortest path
begin
t  0;
initialize P(t) by priority-based encoding;
fitness eval(P);
while (not termination condition) do
crossover P(t) to yield C(t) by weight mapping crossover;
mutation P(t) to yield C(t) by insertion mutation;
immigration operation to yield C(t)
fitness eval(C);
select P(t+1) from P(t) and C(t) by roulette wheel selection;
t  t + 1;
end
output best shortest path;
end Lab.
Soft Computing
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38
1.5 Numerical Examples
 Test Problems:
 For examining the effect of different encoding methods, we applied Ahn
et al’s method and priority-based encoding method to the 6 test
problems:
 Ahn, C.W. & R. Ramakrishna: “A Genetic Algorithm for Shortest Path Routing Problem
and the Sizing of Populations.” IEEE Trans. Evol. Comput., Vol.6, No.6, pp.566-579, 2002.
 OR-Notes. [Online]. Available: http://mscmga.ms.ic.ac.uk/jeb/or/orweb.html
 Using the following parameter specifications.






Population size:
popSize =20
Crossover probability: pC =0.70
Mutation probability: pM =0.50
Immigration rate:
μ=3
Maximum generation: maxGen =1000
Terminating condition: 100 generations with same fitness.
 Each solution is compared with Dijkstra’s algorithm that provides a
reference point (optimal solution).
 Each algorithm was applied to each test problem 20 times (i.e., 20 runs)
using different initial populations.
 All the simulations were performed with Java on Pentium 4 processor
(1.5-GHz clock).
Soft Computing Lab.
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1.5 Numerical Examples
 The first numerical example, presented by Ahn et al’s was adopted.
The problem comprises 20 nodes and 49 arcs. It is given as follows:
(Ahn, C.W. & R. Ramakrishna: “A Genetic Algorithm for Shortest Path Routing Problem
and the Sizing of Populations”, IEEE Trans. on Evol. Comput., Vol.6, No.6, pp.566579, 2002.)
1
1
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UNIVERSITY
, IPS
Fig.6.3 Example
of the
first numerical
example
40
1.5 Numerical Examples
 Convergence property of each algorithm for a Fixed Network
With 20 Nodes
(Ahn, C.W. & R. Ramakrishna: “A Genetic Algorithm for Shortest Path Routing Problem and
the Sizing of Populations”, IEEE Trans. on Evol. Comput., Vol.6, No.6, pp.566-579, 2002.)
Objective Function Values
2.5
Dijkstra’s Algorithm
Munemoto’s Algorithm
Inagaki’s Algorithm
Ahn’s Algorithm
2.0
1.5
1.0
0.5
0
Soft Computing Lab.
2
4
6
Generations
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8
10
41
1.5 Numerical Examples
 Convergence property of Ahn et al.’s algorithm and proposed
algorithm for a Fixed Network With 20 Nodes
260
Ahn et
et al.
al. 's Algorithm
Objective Function Values
240
Proposed Algorithm
220
200
180
160
140
120
0
50
100
150
200
250
300
350
400
450
500
Generations
Fig. 6.4 Convergence property of Ahn et al.’s algorithm and proposed algorithm.
Soft Computing Lab.
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42
1.5 Numerical Examples
 Comparison with results
(Ahn, C.W. & R. Ramakrishna: “A Genetic Algorithm for Shortest Path Routing Problem and
the Sizing of Populations”, IEEE Trans. on Evol. Comput., Vol.6, No.6, pp.566-579, 2002.)
Inagaki’s Algorithm
Munemoto’s Algorithm
Ahn’s Algorithm
Objective Function Value
1200
1000
800
600
400
200
0
15
Soft Computing Lab.
20
25
30
35
40
The Number of Nodes
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45
50
43
1.5 Numerical Examples
 Discussion of the Results:
Table 6.1 Performance comparisons with different genetic operators
Test Problems
(# of nodes/ # of arcs)
20/49
80/120
80/632
160/2544
320/1845
320/10208
Optimal
Solutions
142.00
389.00
291.00
284.00
394.00
288.00
Best Solutions
Alg. 1
148.35
423.53
320.06
429.55
754.94
794.26
Alg. 2
148.53
425.33
311.04
454.98
786.08
732.72
Alg. 3
147.70
418.82
320.15
480.19
906.18
819.85
Alg. 4
143.93
396.52
297.21
382.48
629.81
552.71
Alg. 5
Alg. 6
142.00
389.00
291.62
284.69
395.01
331.09
142.00
389.00
291.00
284.00
394.00
288.00
Alg. 1: FMX+Swap; Alg. 2: OX+Swap; Alg. 3: PX+Swap; Alg. 4: WMX+Swap;
Alg. 5: WMX+Swap+Immigration(3); Alg. 6: WMX+Insertion+Immigration(3).




The quality of solution with different genetic operators is investigated in Table 1.
The path optimality is defined in all test problems, by Alg.6 (WMX+Insertion+
Immigration) that the GA finds the global optimum (i.e., the shortest path).
The path optimality is defined in #1, #2 test problems, by Alg.5 (WMX+Swap+
Immigration), The near optimal result is defined in other test problems.
By Alg.1 ~ Alg.4, the path optimality is not defined. Since the number of possible
alternatives become to very large in test problems, the population be prematurely
converged to a local optimum of the function.
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1.5 Numerical Examples
 Comparison results of Ahn’s algorithm and Proposed algorithm
Table 6.2 Performance comparisons with Ahn’s algorithm and Proposed algorithm.
Test Problems
(# of nodes/ # of arcs)
142.00
389.00
291.00
284.00
394.00
288.00
Best Solutions
Ahn’s Alg.
142.00
389.00
291.00
286.20
403.40
288.90
Prop. Alg.
142.00
389.00
291.00
284.00
394.00
288.00
CPU Times (ms)
Ahn’s Alg.
40.60
118.50
109.50
336.20
779.80
1028.30
Prop. Alg.
23.37
96.80
118.50
490.50
1062.50
1498.50
Generation Num. of
Obtained best result
Ahn’s Alg.
2
4
19
31
44
38
Prop. Alg.
9
4
10
26
11
26
Best solutions
20/49
80/120
80/632
160/2544
320/1845
320/10208
Optimal
Solutions
20/49
Soft Computing Lab.
80/120
80/630
160/2544
320/1845
WASEDA
UNIVERSITY
, IPS
Problem
size
320/10208
45
1.5 Numerical Examples
 Different Parameter Settings:
Table 6.3 Performance comparisons with different parameter settings
Parameter Settings
( popSize / pC : pM )
10 / 0.3 : 0.1
20 / 0.3 : 0.1
20 / 0.7 : 0.5
Best Solutions
CPU Times ( ms )
Generation Num. of
Obtained best result
Test Problems
(# of nodes/ # of arcs)
Optimal
Solutions
20/49
80/120
80/632
160/2544
142.00
389.00
291.00
284.00
156.20
389.00
313.20
320.90
142.00
389.00
291.00
284.20
10.42
32.80
29.40
67.10
8.37
31.10
34.40
106.30
38
5
43
48
27
1
16
37
320/1845
394.00
478.70
394.00
120.30
250.20
68
18
320/10208
20/49
80/120
80/632
160/2544
320/1845
320/10208
20/49
80/120
80/632
160/2544
320/1845
320/10208
288.00
142.00
389.00
291.00
284.00
394.00
288.00
142.00
389.00
291.00
284.00
394.00
288.00
444.00
145.23
389.00
303.10
298.70
465.70
373.10
142.00
389.00
291.00
286.20
403.40
288.90
288.30
142.00
389.00
291.00
284.20
394.00
288.60
142.00
389.00
291.00
284.00
394.00
288.00
126.40
22.36
56.30
50.10
122.10
213.90
311.00
40.60
118.50
109.50
336.20
779.80
1028.30
400.20
13.34
51.50
56.30
181.20
496.70
631.10
23.37
96.80
118.50
490.50
1062.50
1498.50
25
27
4
18
44
32
61
6
1
19
31
44
38
59
24
1
10
35
17
35
9
1
10
26
11
26
Soft Computing Lab.
Ahn’s Alg.
Prop. Alg.
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Ahn’s Alg.
Prop. Alg.
Ahn’s Alg.
Prop. Alg.
46
1.5 Numerical Examples
 Different Parameter Settings with Ahn’s algorithm and Proposed algorithm
Parameter Settings
( popSize / pC : pM )
10 / 0.3 : 0.1
20 / 0.3 : 0.1
30 / 0.3 : 0.1
50 / 0.3 : 0.1
100 / 0.3 : 0.1
10 / 0.7 : 0.5
20 / 0.7 : 0.5
30 / 0.7 : 0.5
50 / 0.7 : 0.5
100 / 0.7 : 0.5
Probability of obtaining the optimal solutions
Ahn’s Alg.
Proposed Alg.
16.67%
66.67%
16.67%
66.67%
33.33%
83.33%
50.00%
100.00%
33.33%
100.00%
33.33%
83.33%
50.00%
100.00%
50.00%
100.00%
83.33%
100.00%
83.33%
100.00%
Probability of obtaining
the optimal solutions
100%
80%
60%
40%
20%
0%
10/ 0.3:0.1
20/ 0.3:0.1
Soft Computing Lab.
30/ 0.3:0.1
50/ 0.3:0.1
100/ 0.3:0.1
10/ 0.7:0.5
20/ 0.7:0.5
Parameter
Settings , IPS
WASEDA
UNIVERSITY
30/ 0.7:0.5
50/ 0.7:0.5
100/ 0.7:0.5
47
1.5 Numerical Examples
 Simulation (# of nodes: 100, # of arcs: 859)
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48
6. Basic Network Design
1.
Shortest Path Problem (SPP)
2.
Maximum Flow (MXF) Problem
2.1 Basic Concept of Maximum Flow Problem
2.2 Application of Maximum Flow Problem
2.3 Methods for solving MXF Problem
2.4 Genetic Approach for solving MXF Problem
2.4.1 Genetic Representation
2.4.2 Genetic Operators
2.5 Numerical Examples
3.
Minimum Cost Flow (MCF) Problem
4.
Bicriteria Network Design Problem (BNP)
5.
Multi-criteria Network Design Problem
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2. Maximum Flow (MXF) Problem
Data table of example network
2.1 Basic Concept of Maximum Flow Problem
 [Online]. Available: http://www-b2.is.tokushima-u.ac.jp/
~ikeda/suuri/maxflow/Maxflow.shtml.en
 MXF is in a sense a complementary model to SPP.
 MXF seeks a feasible solution that sends the maximum
amount of flow from a specified source node s to another
specified sink node t.
 If we interpret uij as the maximum flow rate of arc (i, j),
MXF identifies the maximum steady-state flow that the
network can send from node s to node t per unit time.
2
60
f
40
30
30
1
60
3
30
4
Soft Computing Lab.
50
30
40
60
30
20
s
60
5
6
20
7
40
30
40
8
60
30
t
9
30
10
50
11
f
50
i
WASEDA UNIVERSITY , IPS
uij
j
i
1
1
1
2
2
2
3
3
3
4
5
6
6
6
6
7
7
8
8
9
9
10
j
2
3
4
3
5
6
4
6
7
7
8
5
8
9
10
6
10
9
11
10
11
11
uij
60
60
60
30
40
30
30
50
30
40
60
20
30
40
30
20
40
30
60
30
50
50
50
2. Maximum Flow (MXF) Problem
2.1 Basic Concept of Maximum Flow Problem

Directed graph G=(V, A)
where V is a set of nodes, A is a set of links.

uij is a capacity associated with each link(i, j)

Source node: node 1

Destination node: node n
2
60
f
40
30
30
1
60
3
30
4
50
30
40
60
30
20
s
60
5
6
20
7
40
30
40
8
60
30
t
9
50
30
50
10
i
Soft Computing Lab.
f
11
uij
j
WASEDA UNIVERSITY , IPS
Data table of example network
i
1
1
1
2
2
2
3
3
3
4
5
6
6
6
6
7
7
8
8
9
9
10
j
2
3
4
3
5
6
4
6
7
7
8
5
8
9
10
6
10
9
11
10
11
11
uij
60
60
60
30
40
30
30
50
30
40
60
20
30
40
30
20
40
30
60
30
50
50
51
2. Maximum Flow (MXF) Problem
2.1 Basic Concept of Maximum Flow Problem
 MXF problem can be formulated as follows:
max z  f
 f (i  1)

s. t.  xij   xki   0 (i  2,3, , n  1)
j 1
k 1
 f (i  n)

0  xij  uij , (i, j )  A
n
n
f 0
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52
2. Maximum Flow (MXF) Problem
2.2 Application of Maximum Flow Problem

This basic MXF model can be applied in many applications
such as:
Ahuj, R. K., T. L. Magnanti & J. B. Orlin: Network Flows. Prentice-Hall, Upper
Saddle River, NJ, 1993.

Scheduling on Uniform Parallel Machines


Distributed Computing on a Two-Processor Computer


The feasible scheduling problem, described in the preceding paragraph, is a
fundamental problem in this situation and can be used as a subroutine for
more general scheduling problems, such as the maximum lateness problem,
the (weighted) minimum completion time problem, and the (weighted)
maximum utilization problem.
Distributed computing on a two-processor computer concerns assigning
different modules (subroutines) of a program to two processors in a way that
minimizes the collective costs of interprocessor communication and
computation..
Tanker Scheduling Problem

Soft Computing Lab.
A steamship company has contracted to deliver perishable goods between
several different origin-destination pairs. Since the cargo is perishable, the
customers have specified precise dates (i.e., delivery dates) when the
shipments must reach their destinations..
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2. Maximum Flow (MXF) Problem
2.3 Methods for solving MXF Problem
 Ford-Fulkerson Algorithm
 It works by finding a flow augmenting path in the graph. By adding the
flow augmenting path to the flow already established in the graph, the
maximum flow will be reached when no more flow augmenting paths
can be found in the graph.
 Maximum Flow Algorithm
 An incremental algorithm for max-flow problem that tries to find the
max-flow in the network as an edge is deleted or inserted in the
network, is presented.
 It has also been shown that other cases of a unit change can be
considered as a special case of insertion and deletion of an edge in
the network.
Soft Computing Lab.
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54
2. Maximum Flow (MXF) Problem
2.4 Genetic Approach for solving MXF Problem
Munakata, T. and D. J. Hashier: “A genetic algorithm applied to the maximum flow
problem,” Proc. of 5th Int. Conf. on Genetic Algorithms, pp. 488-493, 1993.
The maximum flow problem appears to be more challenging in applying GAs
than many other common graph problems (e.g., shortest path, minimum
spanning tree)


Its unique characteristic:

A flow at each edge can be anywhere between zero and its flow capacity, i.e.,
has more "freedom" to choose.

In many other problems, selecting an edge may mean to simply add a fixed
distance.
it
In the maximum flow problem, two conditions must be satisfied:

The flow at each edge must be between zero and its flow capacity.

At each vertex, the incoming flow and outgoing flow must balance.
Soft Computing Lab.
WASEDA UNIVERSITY , IPS
55
2.4 Genetic Approach for solving MXF Problem
2.4.1 Genetic Representation
procedure 1: Priority-based Encoding
input: number of nodes n
output: chromosome vk
begin
for j=1 to n
// step 0
vk(j) j;
for i=1 to n / 2
// step 1
repeat
jrandom[1, n];
lrandom[1, n];
until l≠j
swap (vk(j), vk(l));
output the chromosome vk;
// step 2
end
Soft Computing Lab.
WASEDA UNIVERSITY , IPS
56
2.4 Genetic Approach for solving MXF Problem
 The decoding procedure is a two-stage process.
 First stage: the path is generated by one-path growth procedure
 It is given in procedure 2
 With beginning from the specified node 1 and terminating at the specified
node n. At each step, add the one with the highest priority into path.
procedure 2: One-path Growth
input: number of nodes n, chromosome vk ,
the set of nodes Si with all nodes adjacent to node i.
output: path Pk
step 0: the source node i1, Pk 
step 1: if Si=, goto step 3; otherwise, continue.
step 2: select l from Si with the highest priority, and go back to step 1.
if vk(l)≠0 then
vk(l)=0;
Pk  Pk{xil};
il;
else vk(l)=0
step 3: output the complete path Pk ;
Pk  {x1l1 , xl1 ,l2 , xl2 ,l3 , ..., xlm 1 ,lm }
Soft Computing Lab.
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57
2.4 Genetic Approach for solving MXF Problem
 The decoding procedure is a two-stage process.
 Second stage: overall paths are generated by overall paths
growth procedure
 For a given path, we can calculate its flow fk
 By removing the used capacity from uij of each arc, we have a
new network with the new flow capacity ũij.
 With the one-path growth procedure (procedure 2), we can obtain
the second path.
 By repeating this procedure we can obtain the maximum flow for
the given chromosome till no new network can be defined in this
way.
 It is given in procedure 3.
Soft Computing Lab.
WASEDA UNIVERSITY , IPS
58
2.4 Genetic Approach for solving MXF Problem
procedure 3: Overall-path Growth
input: network data (V, A, U), chromosome vk , the set of nodes Si with all nodes adjacent to node i
output: number of paths Lk , the flow fik of each path, iLk
step 0: number of paths l0
step 1: if S1=, go to step 7; otherwise, l l +1, continue.
step 2: the implementation of path Plk growth is based on procedure 2. Select the sink node a of path plk.
step 3: if the sink node a=n, continue; otherwise, perform the set of nodes Si update as follows, return to
step 1.
S i  S i  {a}, i
step 4: calculate the flow flk of the path Plk.
f l k  f l k1  min{ uij | (i, j )  Plk }
step 5: perform the flow capacity uij of each arc update. Make a new flow capacity ũij as follows:
u~ij  u ij  min{ u ij (i, j )  Pl k }
step 6: if the flow capacity ũij=0, perform the set of nodes Si update which the node j adjacent to node i.
step 7: output number of paths Lk  l -1, the flow fik of each path, iLk .
si  si  { j} , (i, j )  Pl k & u~ij  0
Soft Computing Lab.
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59
2.4 Genetic Approach for solving MXF Problem
 Illustration of Priority-based GA
40
2
60
f
30
30
1
60
30
20
s
60
60
5
50
3
30
40
7
t
50
9
30
20
40
4
60
40
6
30
30
8
30
f
11
50
10
Chromosome:
node ID :
1
2
3
4
5
6
7
8
9
10
11
priority :
2
1
6
4
11
9
8
10
5
3
7
Soft Computing Lab.
WASEDA UNIVERSITY , IPS
Data table of example network
i
1
1
1
2
2
2
3
3
3
4
5
6
6
6
6
7
7
8
8
9
9
10
j
2
3
4
3
5
6
4
6
7
7
8
5
8
9
10
6
10
9
11
10
11
11
uij
60
60
60
30
40
30
30
50
30
40
60
20
30
40
30
20
40
30
60
30
50
50
60
2.4 Genetic Approach for solving MXF Problem
 Illustration of Priority-based GA
2
60
f
60
Chromosome:
60
node ID :
1
2
3
4
5
6
7
8
9
10
11
priority :
2
1
6
4
11
9
8
10
5
3
7
i
1
0
2
Si
l
Pk
fk
2, 3, 4
20
2, 3, 4
50
1
1
2, 3, 4
3
1, 3
2
4, 6, 7
6
1, 3, 6
3
5, 8, 9, 10
5
1, 3, 6, 5
4
8
8
1, 3, 6, 5, 8
5
9, 11
11
1, 3, 6, 5, 8, 11
0
1
1
2, 3, 4
3
1, 3
2
4, 6, 7
6
1, 3, 6
3
8, 9, 10
8
1, 3, 6, 8
4
9, 11
11
1, 3, 6, 8, 11
Soft Computing Lab.
60
30
20
8
60
30
t
3
30
4
S1
5
30
30
s
1
k
40
50
30
40
6
20
7
40
30
40
9
30
50
f
11
50
10
k : number of paths
i : start node
Si : the set of nodes
l : sink node
Pk : the kth path
S1 : the set of nodes with all nodes
adjacent to node 1
fk : maximum possible flow
WASEDA UNIVERSITY , IPS
61
2.4 Genetic Approach for solving MXF Problem
 Illustration of Priority-based GA
2
60
1
Chromosome:
node ID :
1
2
3
4
5
6
7
8
9
10
11
priority :
2
1
6
4
11
9
8
10
5
3
7
4
60
i
0
Si
l
Pk
1
1
2, 3, 4
3
1, 3
2
4, 7
7
1, 3, 7
3
6, 10
6
1, 3, 7, 6
4
9, 10
9
1, 3, 7, 6, 9
5
10, 11
11
1, 3, 7, 6, 9, 11
0
S1
60
2, 4
4
1, 4
2
7
7
1, 4, 7
3
6, 10
6
1, 4, 7, 6
4
9, 10
9
1, 4, 7, 6, 9
5
10, 11
11
1, 4, 7, 6, 9, 11
Soft Computing Lab.
60
30
20
8
60
30
t
3
30
4
50
30
40
6
20
7
40
30
40
9
30
50
f
11
50
10
fk
2, 4
60
2,4
70
1
1
5
30
30
s
f
k
3
40
k : number of paths
i : start node
Si : the set of nodes
l : sink node
Pk : the kth path
S1 : the set of nodes with all nodes
adjacent to node 1
fk : maximum possible flow
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62
2.4 Genetic Approach for solving MXF Problem
 Illustration of Priority-based GA
2
60
1
Chromosome:
node ID :
1
2
3
4
5
6
7
8
9
10
11
priority :
2
1
6
4
11
9
8
10
5
3
7
6
60
i
0
Si
l
Pk
1
1
2, 4
4
1, 4
2
7
7
1, 4, 7
3
10
10
1, 4, 7, 10
4
11
11
1, 4, 7, 10, 11
0
60
fk
2
100
2
110
1
1
2
2
1, 2
2
3, 5, 6
5
1, 2, 5
3
8
8
1, 2, 5, 8
4
9, 11
11
1, 2, 5, 8, 11
Soft Computing Lab.
60
30
20
8
60
30
t
3
30
4
S1
5
30
30
s
f
k
5
40
50
30
40
6
20
7
40
30
40
9
30
50
f
11
50
10
k : number of paths
i : start node
Si : the set of nodes
l : sink node
Pk : the kth path
S1 : the set of nodes with all nodes
adjacent to node 1
fk : maximum possible flow
WASEDA UNIVERSITY , IPS
63
2.4 Genetic Approach for solving MXF Problem
 Illustration of Priority-based GA
2
60
1
Chromosome:
node ID :
1
2
3
4
5
6
7
8
9
10
11
priority :
2
1
6
4
11
9
8
10
5
3
7
8
i
0
Si
l
Pk
1
1
2
2
1, 2
2
3, 5, 6
5
1, 2, 5
3
8
8
1, 2, 5, 8
4
9
9
1, 2, 5, 8, 9
5
10, 11
11
1, 2, 5, 8, 9, 11
0
1
2
3
4
5
60
S1
2
60
Soft Computing Lab.
2
6
9
10
11
1, 2
1, 2, 6
1, 2, 6, 9
1, 2, 6, 9, 10
1, 2, 6, 9, 10, 11
60
30
20
8
60
30
t
3
30
4
50
30
40
6
20
7
40
30
40
9
30
50
f
11
50
10
fk
140
1
2
3, 6
9, 10
10
11
5
30
30
s
f
k
7
40
k : number of paths
i : start node
Si : the set of nodes
l : sink node
Pk : the kth path
S1 : the set of nodes with all nodes
adjacent to node 1
fk : maximum possible flow
160
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64
2.4 Genetic Approach for solving MXF Problem
Data table of example network
 Illustration of Priority-based GA
40/40
2
60/60
160
60/60
5
8
60/60
20/30 20/20 30/30 30/30
s
60/60
1
3
6
50/50
40/40
40/40
4
30/40
7
9
20/30
10/30 20/20
40/60
50/50
t
11
160
50/50
10
Chromosome:
node ID :
1
2
3
4
5
6
7
8
9
10
11
priority :
2
1
6
4
11
9
8
10
5
3
7
Objective function value： z=160
i
Soft Computing Lab.
xij / uij
j
WASEDA UNIVERSITY , IPS
i
1
1
1
2
2
2
3
3
3
4
5
6
6
6
6
7
7
8
8
9
9
10
j
2
3
4
3
5
6
4
6
7
7
8
5
8
9
10
6
10
9
11
10
11
11
uij
60
60
60
30
40
30
30
50
30
40
60
20
30
40
30
20
40
30
60
30
50
50
65
2.4 Genetic Approach for solving MXF Problem
2.4.2 Genetic Operators
• Here the position-based crossover operator proposed by PMX (Partial
Mapped Crossover) (Gen-Cheng97, pp.119-125) was adopted.
• It uses a special repairing procedure to resolve the illegitimacy caused by
the simple two-point crossover as follows:
step 3 : determine mapping relationship
step 1 : select the substring at random
substring selected
parent 1:
parent 2:
1
4
7
6
2
3
3
5
4
7
6
1
5
8
3
4
3
5
7
235
47
8
2
step 4 : legalize offspring with mapping
relationship
step 2 : exchange substrings between
parent 1:
1
7
3
5
7
6
5
8
parent 2:
4
6
2
3
4
1
8
2
Soft Computing Lab.
2
offspring 1:
1
4
3
5
7
6
2
8
offspring 2:
7
6
2
3
4
1
8
5
WASEDA UNIVERSITY , IPS
66
2.4 Genetic Approach for solving MXF Problem
2.4.2 Genetic Operators
 Mutation: The swap mutation operator was used here, in which
two positions are selected at random and their contents are
swapped as follows:
exchanging points
parent:
1
7
2
3
4
6
5
8
offspring:
1
7
6
3
4
2
5
8
 Selection: The roulette wheel approach, a type of fitnessproportional selection, was adopted.
Soft Computing Lab.
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67
2.4 Genetic Approach for solving MXF Problem
 GA Procedure for Maximum Flow Problem
procedure: Priority-based GA for Maximum Flow Problem
input: network data (V, A, U), GA parameters
output: best maximum flow
begin
t  0;
initialize P(t) by priority-based encoding;
fitness eval(P);
while (not termination condition) do
crossover P(t) to yield C(t) by partial mapped crossover;
mutation P(t) to yield C(t) by swap mutation;
fitness eval(C);
select P(t+1) from P(t) and C(t) by roulette wheel selection;
t  t + 1;
end
output best maximum flow;
end
Soft Computing Lab.
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2.5 Numerical Examples
Test Problems:
 The numerical examples, presented by T. Munakata &
D.J. Hashier, was adopted.
Munakata, T. and D. J. Hashier: “A genetic algorithm applied to the
maximum flow problem,” Proc. of 5th Int. Conf. on Genetic Algorithms,
pp. 488-493, 1993.
 Using the following parameter specifications.





Population size:
popSize =10
Crossover probability:
pC =0.50
Mutation probability:
pM =0.50
Maximum generation:
maxGen =1000
Terminating condition: 100 generations with same fitness.
 All the simulations were performed with Java on
Pentium 4 processor (1.5-GHz clock).
Soft Computing Lab.
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2.5 Numerical Examples
 Test Problem 1:
The first numerical example, presented by Munakata & Hashier, was
adopted. The problem comprises 25 nodes and 49 arcs. It is given as
follows:
10
2
7
15
8
6
20
f
20
1
5
4
8
8
10
9
15
13
20
20
14
20
20
15
4
5
10
10
6
15
11
22
15
20
16
20
30
8
19
8
20
10
25
23
25
f
30
10
25
30
24
25
15
10
25
18
20
10
Soft Computing Lab.
20
8
20
20
10
17
25
15
5
4
15
20
20
15
3
20
12
i
uij
j
21
WASEDA UNIVERSITY , IPS
70
2.5 Numerical Examples
10/10
2
7
12
14/15
8/8
5/25
9/20
4/6
18/20
5/5
4 /4
3
8
13/20
1
20/20
8/8
10/10
4
15/15
9
7/15
10
19
5/10
23 30/30
6/8 10/10
25
90
30/30
20 25/25
20/20
15
24
9/15
5/10
9/20
10/10
15/15
10/10
6
30/30
5/25
15/15
4/4
5
22
4/8
10/20
14
2/20
20/20
25/25
18
15/20
20/20
19/20
10/10
4/8
20/20
13
15/15
2/5
90
17
11
16
i
21
xij / uij
j
Objective function value: z=90 (optimal solution)
Generation Num. of Obtained best result: 34
Best Chromosome:
node ID :
priority:
1
2
3
4
5
6
8 25 2 12 15 20
Soft Computing Lab.
7
8
9
10
11
1 16 21 14 7
12
13
14
15
16
6 18 23 3 13
WASEDA UNIVERSITY , IPS
17
18
19
20
21
22
23
24
25
4 17 5 11 9 24 19 10 22
71
2.5 Numerical Examples
flow
 Process of Genetic Computing
Soft Computing Lab.
WASEDA UNIVERSITY , IPS
72
2.5 Numerical Examples
 Test Problem 2:
The first numerical example, presented by Munakata & Hashier, was
adopted. The problem comprises 25 nodes and 56 arcs. It is given as
follows:
10
2
5
8
20
18
6
3
20
f
1
20
9
4
7
12
Soft Computing Lab.
6
6
15
30
25
f
20
8
23
6
22
10
22
8
6
5
14
5
15
18
9
6
10
7
20
20
9
5
10
7
20
20
17
10
13
8
7
10
8
5
15
21
8
12
8
9
10
12
6
8
16
5
7
20
8
2
7
10
11
11
19
8
10
15
i
uij
j
9
15 UNIVERSITY , IPS
WASEDA
24
73
2.5 Numerical Examples
10/10
2
8/8
18/18
7
6/6
3
5/7
20/20
17
8/15
5/5
91
20/20
1
7/9
4
6/10
13
1/10
8/8
20/20
15/20
9
7/7
10/10
5
13/20
7/22
10
7/12
23
6/6
14/15
i
19
uij
j
5/9
9/10
15
25 91
8/8
3/6
6/15
6
22/30
20/20
6/6
4/5
3/7
22
8/8
18
14
15/15
21
4/8
9/12
8
9/9
6/10
12
4/8
20/20
16
4/5
20
7/8
2/2
10/10
18/20
8/11
11
24
Objective function value: z=91 (optimal solution)
Generation Num. of Obtained best result: 67
Best Chromosome:
node ID :
1
13
14
15
priority: 20 10 5 22 25 23 11 6 18 1 16 12 3
2
7 14 15 19 4 13 17 8 21 24 9
Soft Computing Lab.
2
3
4
5
6
7
8
9
10
11
12
16
WASEDA UNIVERSITY , IPS
17
18
19
20
21
22
23
24
25
74
2.5 Numerical Examples
flow
 Process of Genetic Computing
Soft Computing Lab.
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2.5 Numerical Examples
 Simulation (# of nodes: 80, # of arcs: 826)
Soft Computing Lab.
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76
6. Basic Network Design
1.
Shortest Path Problem (SPP)
2.
Maximum Flow (MXF) Problem
3.
Minimum Cost Flow (MCF) Problem
3.1 Basic Concept of MCF Problem
3.2 Application of MCF Problem
3.3 Methods for solving MCF Problem
3.4 Genetic Approach for solving MCF Problem
3.4.1 Genetic Representation
3.4.2 Genetic Operators
3.5 Numerical Examples
4.
Bicriteria Network Design Problem (BNP)
5.
Multicriteria Network Design Problem
Soft Computing Lab.
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3. Minimum Cost Flow (MCF) Problem
Data table of example network
3.1 Basic Concept of MCF Problem
 MCF model is the most fundamental of all network design
problems.
 The problem is to determine the minimum cost plan for sending
flow through the network to satisfy supply and demand
requirements.
 The problem is defined by a given set of arcs and a given set of
nodes, where each are has a known capacity uij and unit cost cij
and each node has a fixed external flow.
 SPP model is a special case of MCF when the flow q =1.
2
18, 60
q
16, 40
5
19, 60
8
14, 30
13, 30
15, 20
s
18, 60
16, 30 17, 30
t
1
19, 60
3
16, 50
6
17, 30
15, 30
18, 30
15, 20
17, 60
4
15, 40
19, 40
7
9
19, 50
11
14, 30
17, 50
13, 40
10
i
Soft Computing Lab.
q
cij, uij
WASEDA UNIVERSITY , IPS
j
i
1
1
1
2
2
2
3
3
3
4
5
6
6
6
6
7
7
8
8
9
9
10
j
2
3
4
3
5
6
4
6
7
7
8
5
8
9
10
6
10
9
11
10
11
11
cij
18
19
17
13
16
14
15
16
17
19
19
15
16
15
18
15
13
17
18
14
19
17
uij
60
60
60
30
40
30
30
50
30
40
60
20
30
40
30
20
40
30
60
30
50
50
78
3. Minimum Cost Flow (MCF) Problem
Data table of example network
3.1 Basic Concept of MCF Problem





Directed graph G=(V, A)
where V is a set of nodes, A is a set of links.
uij is a capacity associated with each link(i, j)
cij is unit cost associated with each link(i, j)
Source node: node 1
Destination node: node n
2
18, 60
q
16, 40
5
19, 60
8
14, 30
13, 30
15, 20
s
18, 60
16, 30 17, 30
t
1
19, 60
3
16, 50
6
17, 30
15, 30
18, 30
15, 20
17, 60
4
15, 40
19, 40
7
19, 50
9
14, 30
17, 50
13, 40
10
i
Soft Computing Lab.
q
11
cij, uij
j
WASEDA UNIVERSITY , IPS
i
1
1
1
2
2
2
3
3
3
4
5
6
6
6
6
7
7
8
8
9
9
10
j
2
3
4
3
5
6
4
6
7
7
8
5
8
9
10
6
10
9
11
10
11
11
cij
18
19
17
13
16
14
15
16
17
19
19
15
16
15
18
15
13
17
18
14
19
17
uij
60
60
60
30
40
30
30
50
30
40
60
20
30
40
30
20
40
30
60
30
50
50
79
3. Minimum Cost Flow (MCF) Problem
3.1 Basic Concept of MCF Problem
 MCF problem can be formulated as follows:
n
n
min z   cij xij
i 1 j 1
 q (i  1)

s. t.  xij   xki   0 (i  2,3, , n  1)
i 1
k 1
 q (i  n)

n
n
0  xij  uij , (i, j )  A
q: total flow value
Soft Computing Lab.
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3. Minimum Cost Flow (MCF) Problem
3.2 Application of Minimum Cost Flow (MCF) Problem

This basic MCF model can be applied in many applications
such as:
Ahuj, R. K., T. L. Magnanti & J. B. Orlin: Network Flows, Prentice-Hall, Upper
Saddle River, NJ, 1993.

Transportation Problem
 There are a set of nodes called sources, and a set of nodes called
destinations. All arcs go from a source to a destination. There is a per-unit
cost on each arc. Each source has a supply of material, and each
destination has a demand.
 It can be solved by applying Min-cost Flow Algorithm

Distribution Problem
 The distribution of a product from manufacturing plants to warehouses, or
from warehouses to retailers
 The flow of raw material and intermediate goods through the various
machining stations in a production line

The routing of automobiles through an urban street network

The routing of calls through the telephone system.
Soft Computing Lab.
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3. Minimum Cost Flow (MCF) Problem
3.3 Methods for solving MCF Problem
Ahuj, R. K., T. L. Magnanti & J. B. Orlin: Network Flows, Prentice-Hall, Upper
Saddle River, NJ, 1993.
 Successive Shortest Path Algorithm

The successive shortest path algorithm maintains optimality of the solution at
every step and strives to attain feasibility.
 Primal-dual Algorithm


The primal-dual algorithm for the minimum cost flow problem is similar to the
successive shortest path algorithm in the sense that it also maintains a
pseudoflow that satisfies the reduced cost optimality conditions and gradually
converts it into a flow by augmenting flows along shortest paths.
Out-of-Kilter Algorithm

The out-of-kilter algorithm, which satisfies only the mass balance constraints,
so intermediate solutions might violate both the optimality conditions and the
flow bound restrictions.
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3.4 Genetic Approach for solving MCF Problem
3.4.1 Genetic Representation
procedure 1: priority-based encoding
input: number of nodes n
output: chromosome vk
begin
for j=1 to n
// step 0
vk(j) j;
for i=1 to n / 2
// step 1
repeat
jrandom[1, n];
lrandom[1, n];
until l≠j
swap (vk(j), vk(l));
output the chromosome vk;
// step 2
end
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3.4 Genetic Approach for solving MCF Problem
 The decoding procedure is a two-stage process.
 First stage: the path is generated by One-path growth procedure
 It is given in procedure 2
 With beginning from the specified node 1 and terminating at the specified
node n. At each step, add the one with the highest priority into path.
procedure 2: One-path Growth
input: number of nodes n, chromosome vk ,
the set of nodes Si with all nodes adjacent to node i.
output: path Pk
step 0: the source node i1, Pk 
step 1: if Si=, goto step 3; otherwise, continue.
step 2: select l from Si with the highest priority, and go back to step 1.
if vk(l)≠0 then
vk(l)=0;
Pk  Pk{xil};
il;
else vk(l)=0
step 3: output the complete path Pk .
Pk  {x1l1 , xl1 ,l2 , xl2 ,l3 , ..., xlm 1 ,lm }
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3.4 Genetic Approach for solving MCF Problem
 The decoding procedure is a two-stage process.
 Second stage: overall paths are generated by overall paths growth
procedure
 For a given path, we can calculate its flow fk and the cost ck.
 By removing the used capacity from uij of each arc, we have a new
network with the new flow capacity ũij.
 With the one-path growth procedure (procedure 2), we can obtain
the second path.
 By repeating this procedure we can obtain the maximum flow for
the given chromosome till the flow fk is larger than total flow value q.
 It is given in procedure 3.
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3.4 Genetic Approach for solving MCF Problem
procedure 3: Overall-path Growth
input: network data (V, A, C, U), chromosome vk , the set of nodes Si
with all nodes adjacent to node i
output: number of paths Lk , the flow fik and the cost cik of each path,
iLk
step 0: number of paths l0
step 1: if flow fk >q, go to step 7; otherwise, l l +1, continue.
step 2: the implementation of path Plk growth is based on procedure 2.
Select the sink node a of path plk.
step 3: if the sink node a=n, continue; otherwise, perform the set of
nodes Si update as follows, return to step 1.
S i  S i  {a}, i
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3.4 Genetic Approach for solving MCF Problem
step 4: calculate the flow flk and the cost clk of the path Plk.
f l k  f l k1  min{ uij | (i, j )  Plk }
c c
k
l
k
l 1
n
n
  cij ( f l k  f l k1 ), (i, j )  Pl k
i 1 j 1
step 5: perform the flow capacity uij of each arc update. Make a new
flow capacity ũij as follows:
u~ij  u ij  min{ u ij (i, j )  Pl k }
step 6: if the flow capacity ũij=0, perform the set of nodes Si update
which the node j adjacent to node i.
si  si  { j} , (i, j )  Pl k & u~ij  0
step 7: output number of paths Lk  l -1, the flow fik and the cost cik of
each path, iLk .
Soft Computing Lab.
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3.4 Genetic Approach for solving MCF Problem
Data table of example network
 Illustration of Priority-based GA
16, 40
2
18, 60
q
19, 60
5
8
14, 30
13, 30
15, 20
s
18, 60
16, 30 17, 30
t
1
19, 60
16, 50
3
15, 40
6
17, 30
15, 30
9
18, 30
19, 40
4
11
q
14, 30
15, 20
17, 60
19, 50
17, 50
13, 40
7
10
cij, uij
i
j
Chromosome:
node ID :
1
2
3
4
5
6
7
8
9
10
11
priority :
2
1
6
4
11
9
8
10
5
3
7
Soft Computing Lab.
WASEDA UNIVERSITY , IPS
i
1
1
1
2
2
2
3
3
3
4
5
6
6
6
6
7
7
8
8
9
9
10
j
2
3
4
3
5
6
4
6
7
7
8
5
8
9
10
6
10
9
11
10
11
11
cij
18
19
17
13
16
14
15
16
17
19
19
15
16
15
18
15
13
17
18
14
19
17
uij
60
60
60
30
40
30
30
50
30
40
60
20
30
40
30
20
40
30
60
30
50
50
88
3.4 Genetic Approach for solving MCF Problem
 Illustration of Priority-based GA
2
18, 60
16, 40
13, 30
15, 20
19, 60
3
16, 50
6
17, 30
15, 30
1
2
3
4
5
6
7
8
9
10
11
priority :
2
1
6
4
11
9
8
10
5
3
7
18, 60
16, 30 17, 30
4
15, 40
18, 30
15, 20
17, 60
node ID :
8
t
1
Chromosome:
19, 60
14, 30
s
q
5
19, 40
7
9
19, 50
q
11
14, 30
17, 50
13, 40
10
q = 60
k
i
1
0
2
1
2
3
4
5
0
1
2
3
4
Si
l
Pk
S1
flk
clk
20
87
30
69
1
2, 3, 4
4, 6, 7
5, 8, 9, 10
8
9, 11
3
6
5
8
11
2, 3, 4
4, 6, 7
8, 9, 10
9, 11
3
6
8
11
Soft Computing Lab.
1, 3
1, 3, 6
1, 3, 6, 5
1, 3, 6, 5, 8
1, 3, 6, 5, 8, 11 2, 3, 4
1
1, 3
1, 3, 6
1, 3, 6, 8
1, 3, 6, 8, 11 2, 3, 4
WASEDA UNIVERSITY , IPS
k : number of paths
i : start node
Si : the set of nodes
l : sink node
Pk : the kth path
S1 : the set of nodes with all nodes
adjacent to node 1
flk : the total flow
clk : minimum possible cost
89
3.4 Genetic Approach for solving MCF Problem
 Illustration of Priority-based GA
2
18, 60
16, 40
13, 30
15, 20
3
16, 50
6
17, 30
15, 30
4
1
2
3
4
5
6
7
8
9
10
11
priority :
2
1
6
4
11
9
8
10
5
3
7
q = 60
k
i
3
0
1
2
3
4
5
Si
l
Pk
S1
flk
clk
2, 4
10
85
1
2, 3, 4
4, 7
6, 10
9, 10
10, 11
Soft Computing Lab.
3
7
6
9
11
1, 3
1, 3, 7
1, 3, 7, 6
1, 3, 7, 6, 9
1, 3, 7, 6, 9, 11
18, 60
16, 30 17, 30
WASEDA UNIVERSITY , IPS
15, 40
18, 30
15, 20
17, 60
node ID :
8
t
19, 60
1
Chromosome:
19, 60
14, 30
s
q
5
19, 40
7
9
19, 50
q
11
14, 30
17, 50
13, 40
10
k : number of paths
i : start node
Si : the set of nodes
l : sink node
Pk : the kth path
S1 : the set of nodes with all nodes
adjacent to node 1
flk : the total flow
clk : minimum possible cost
90
3.4 Genetic Approach for solving MCF Problem
 Illustration of Prioritybased GA
i
19, 20/60
5
2
Data table of example network
j
8
16, 30/30
18, 50/60
15, 20/20
s
60
cij, xij /uij
t
19, 60/60
1
16, 50/50
3
15, 10/40
6
9
19, 10/50
11
17, 10/30
15, 10/20
7
4
10
Chromosome:
node ID :
1
2
3
4
5
6
7
8
9
10
11
priority :
2
1
6
4
11
9
8
10
5
3
7
Objective function value： z=20*87+30*69+10*85
=4660
Soft Computing Lab.
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i
1
1
1
2
2
2
3
3
3
4
5
6
6
6
6
7
7
8
8
9
9
10
j
2
3
4
3
5
6
4
6
7
7
8
5
8
9
10
6
10
9
11
10
11
11
cij
18
19
17
13
16
14
15
16
17
19
19
15
16
15
18
15
13
17
18
14
19
17
uij
60
60
60
30
40
30
30
50
30
40
60
20
30
40
30
20
40
30
60
30
50
5091
3.4 Genetic Approach for solving MCF Problem
3.4.2 Genetic Operators
• Here the position-based crossover operator proposed by PMX (Partial
Mapped Crossover) (Gen-Cheng97, pp.119-125) was adopted.
• It uses a special repairing procedure to resolve the illegitimacy caused by
the simple two-point crossover as follows:
step 3 : determine mapping relationship
step 1 : select the substring at random
substring selected
parent 1:
parent 2:
1
4
7
6
2
3
3
5
4
7
6
1
5
8
3
4
3
5
7
235
47
8
2
step 4 : legalize offspring with mapping
relationship
step 2 : exchange substrings between
parent 1:
1
7
3
5
7
6
5
8
parent 2:
4
6
2
3
4
1
8
2
Soft Computing Lab.
2
offspring 1:
1
4
3
5
7
6
2
8
offspring 2:
7
6
2
3
4
1
8
5
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3.4 Genetic Approach for solving MCF Problem
3.4.2 Genetic Operators
 Mutation: The swap mutation operator was used here, in which
two positions are selected at random and their contents are
swapped as follows:
exchanging points
parent:
1
7
2
3
4
6
5
8
offspring:
1
7
6
3
4
2
5
8
 Selection: The roulette wheel approach, a type of fitnessproportional selection, was adopted.
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3.4 Genetic Approach for solving MCF Problem
 GA Procedure for solving MCF Problem
procedure: Priority-based GA for Solving MCF Problem
input: network data (V, A, C, U), GA parameters
output: best minimum cost
begin
t  0;
initialize P(t) by priority-based encoding;
fitness eval(P);
while (not termination condition) do
crossover P(t) to yield C(t) by partial mapped crossover;
mutation P(t) to yield C(t) by swap mutation;
fitness eval(C);
select P(t+1) from P(t) and C(t) by roulette wheel selection;
t  t + 1;
end
output best minimum cost;
end
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3.5 Numerical Examples
Test Problems:
 The numerical examples, presented by Munakata &
Hashier, was adopted.
Munakata, T. and D. J. Hashier: “A genetic algorithm applied to the
maximum flow problem,” Proc. of 5th Int. Conf. on Genetic Algorithms,
pp. 488-493, 1993.
 Using the following parameter specifications.





Population size:
popSize =10
Crossover probability:
pC =0.50
Mutation probability:
pM =0.50
Maximum generation:
maxGen =1000
Terminating condition: 100 generations with same fitness.
 All the simulations were performed with Java on
Pentium 4 processor (1.5-GHz clock).
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3.5 Numerical Examples
 Test Problem 1:
The first numerical example, presented by Munakata & Hashier, was
adopted. The problem comprises 25 nodes and 49 arcs. It is given as
follows:
2
10, 10
7
10, 15
13, 8
10, 6
10, 20
13, 20
q
1
32, 20
4
8
9, 15
15, 8
35, 10
9 12, 15
13
631, 20
7, 20
14
10
6
7, 10
14, 15
18
15
19
22
14, 20
10, 10
35, 25
20 14, 25
4, 20
16
10, 30
23 2, 30
3, 8 3, 10
14, 20
21
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q
3, 30
24
12, 25
5, 15
11
8, 25
5, 8
7, 20
9, 15
33, 10
7, 10
Soft Computing Lab.
7, 20
5, 20
5
11, 8
34, 20
135, 20
3, 10
13, 4
17
6, 20
11, 15
4, 5
8, 20
11, 25
4, 15
15, 5
33, 4
3
12
i
cij , uij
j
96
3.5 Numerical Examples
10, 10/10
2
7
10, 15/15
13, 8/8
12
11, 10/25
6, 10/20
10, 5/6
10, 18/20
3
17
15, 5/5
33, 4/4
8 9, 15/15
13
18 8, 20/25 22
10, 30/30
13, 14/20
11, 15/15
4, 2/5
q=70
32, 20/20
15, 8/8
1
4
9 12, 7/15
35, 10/10
135, 14/20
3, 10/10
9, 3/15
13, 4/4
5
10
631, 4/20
14, 15/15
14
15
7, 18/20
19
10, 10/10
q=70
23 2, 20/30 25
3, 8/8
3, 20/30
3, 10/10
14, 7/20 2014, 20/25 24
4, 15/20
7, 4/10
6
7, 20/20
11
16
21
Total flow value q = 70
Objective function value: z= 6969
Generation Num. of Obtained best result:863
Best Chromosome:
node ID :
priority:
1
2
3
1 16 11
Soft Computing Lab.
4
5
6
7
8
9
10
11
12
13
14
15
16
9
6
5
7
8 15 10 3 12 13 21 4 22 14 18 20 24 17 25 23 2 19
WASEDA UNIVERSITY , IPS
17
18
19
20
21
22
23
24
25
97
3.5 Numerical Examples
 Process of Genetic Computing
cost
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3.5 Numerical Examples
 Test Problem 2:
The first numerical example, presented by Munakata & Hashier, was
adopted. The problem comprises 25 nodes and 56 arcs. It is given as
follows:
2
3, 10
8, 10
2, 20
3
3, 5
14, 8
7
8
7, 5
q
1
33, 20
4
13
33, 8
5
4, 7
12, 10
11, 12
6
12, 15
8, 20
12, 6
10
17
18
15
31, 15
22
6, 30
q
25
7, 20
30, 8
23
11, 6
19
6, 6
10, 20
8, 4
35, 6
14, 5
7, 8
2,15
21
12, 7
15, 10
11, 22
20
26, 9
34, 10
2, 9
14
3, 5
30, 7
634, 20
Soft Computing Lab.
9, 7
9
6, 8
9, 10
12, 8
16
30, 10
6, 12
12, 9
126, 20
11, 5
12
4, 6
6, 8
13, 2
12, 7
32, 8
29, 11
7, 18
13, 6
10, 20
11
9, 15
35, 9
8, 10
WASEDA UNIVERSITY , IPS
i
24
cij , uij
j
99
3.5 Numerical Examples
3, 10/10
2
20
14, 8/8
7, 8/18
7
2, 18/20
29, 10/11
11
13, 6/6
3
16
30, 6/10
12
32, 8/8
26, 8/9
12, 6/7
10, 14/20
6, 6/12
8
10, 20/20
17
31, 3/15
7, 3/5
q=72
1
33, 20/20
12, 9/9
4
34,9/10
13
8, 15/20
9
5
3, 5/5
30, 3/7
35, 6/6
14, 2/5
14
11,2 /22
10
6, 16/30
25
q=72
7, 20/20
18
2, 3/9
4, 7/7
12, 10/10
22
8,4/ 4
33, 8/8
126, 20/20
2,10/15
21
30, 8/8
23
11, 4/6
9, 6/15
19
6, 6/6
8, 6/10
15
6
24
Total flow value q = 72
Objective function value: z=5986
Generation Num. of Obtained best result:132
Best Chromosome:
node ID :
1
6
7
priority: 10 25 15 24 11 4
7
Soft Computing Lab.
2
3
4
5
8
9
10
11
12
13
14
15
8 12 6
5
9 13 14 3 16 17 18 1 20 22 19 23 21 2
WASEDA UNIVERSITY , IPS
16
17
18
19
20
21
22
23
24
25
100
3.5 Numerical Examples
 Process of Genetic Computing
7000
6900
6800
6700
cost
6600
6500
6400
6300
6200
6100
6000
5900
0
50
100
150
200
250
generation
Soft Computing Lab.
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3.5 Numerical Examples
 Simulation (# of nodes: 80, # of arcs: 857)
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6. Basic Network Design
1.
Shortest Path Problem (SPP)
2.
Maximum Flow (MXF) Problem
3.
Minimum Cost Flow (MCF) Problem
4.
Bicriteria Network Design Problem (BNP)
4.1 Introduction of BNP
4.2 BNP Formulation
4.3 Genetic Approach for solving BNP
4.3.1 Genetic Representation
4.3.2 Decoding Method
4.3.3 Fitness Assignment
4.3.4 Genetic Operators
4.4 Numerical Examples
5. Multi-criteria Network Design Problem
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4. Bicriteria Network Design Problem (BNP)

In real-life applications, network design is often the case that the network to be
built is required to optimize multicriteria simultaneously.

Marathe, M. V., R. Ravi, R. Sundaram, S. S. Ravi, D. J. Rosenkrantz, and H. B. Hunt: “Bicriteria
network design problems,” J. Algorithms, vol. 28, no. 1, pp. 142-171, Jul. 1998.

Lo, C. and W. Chang: “A multiobjective hybrid genetic algorithm for the capacitated multipoint network
design problem,” IEEE Trans. Syst., Man, Cybern. B, vol. 30, no. 3, pp. 461-470, Jun. 2000.

Kim, J. R. and M. Gen: “A genetic algorithm for bicriteria communication network topology design,”
Eng. Val. Cost Analysis, vol. 3, pp. 351-363, 2000.

Raghavan, S., M. O. Ball, and V. S. Trichur: “Bicriteria product design optimization,” Institute for
Systems Research, Tech. Rep. TR 2001-8, 2001.
[Online]. Available: http://techreports.isr.umd.edu/ARCHIVE/

Kumar, R., P. P. Parida, and M. Gupta: “Topological design of communication networks using
multiobjective genetic optimization,” Proc. Cong. Evol. Comput., May. 2002, vol. 1, pp. 425-430.

Yuan, D.: “A bicriteria optimization approach for robust OSPF routing,” Proc. IPOM, 2003, pp. 91-98.

Medaglia, A. L. and S. Fang: “A genetic-based framework for solving (multi-criteria) weighted matching
problems,” Eur. J. Oper. Res., vol. 149, pp.77-101, Jan. 2003.

Yang, H., M. Maier, M. Reisslein, and W. M. Carlyle: “A genetic algorithm-based methodology for
optimizing multiservice convergence in a metro WDM network,” J. Lightwave Technol., vol. 21, no. 5,
pp. 1114-1133, May. 2003.

Zhou, G., H. Min, and M. Gen: “A genetic algorithm approach to the bi-criteria allocation of customers
to warehouses,” Int. J. Production Economics, vol. 86, pp. 35-45, Oct. 2003.
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4.1 Introduction of BNP
 The problems may arise when designing:

In a communication network, find a set of links which consider the low cost
(or delay) and the high throughput (or reliability) for increasing the network
performance.




In a manufacturing system, the two criteria under consideration are
minimizing cost and maximizing manufacturing.



Marathe, M. V., R. Ravi, R. Sundaram, S. S. Ravi, D. J. Rosenkrantz, and H. B. Hunt:
“Bicriteria network design problems,” J. Algorithms, vol. 28, no. 1, pp. 142-171, Jul. 1998.
Yuan, D.: “A bicriteria optimization approach for robust OSPF routing,” Proc. IPOM, 2003,
pp. 91-98.
Yang, H., M. Maier, M. Reisslein, and W. M. Carlyle: “A genetic algorithm-based
methodology for optimizing multiservice convergence in a metro WDM network,” J.
Lightwave Technol., vol. 21, no. 5, pp. 1114-1133, May. 2003.
Raghavan, S., M. O. Ball, and V. S. Trichur: “Bicriteria product design optimization,”
Institute for Systems Research, Tech. Rep. TR 2001-8, 2001.
[Online]. Available: http://techreports.isr.umd.edu/ARCHIVE/
In a logistic system, the main drive to improve logistics productivity is the
enhancement of customer services and asset utilization through a significant
reduction in order cycle time (lead time) and logistics costs.

Zhou, G. , H. Min, and M. Gen: “A genetic algorithm approach to the bi-criteria allocation of
customers to warehouses,” Int. J. Production Economics, vol. 86, pp. 35-45, Oct. 2003.
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4.1 Introduction of BNP

The Bicriteria Network Design Problem (BNP) is known as NP-hard
(Garey and Johnson, 1979), it is not simply an extension from single
objective to two objectives.
Garey, M. and D. Johnson: Computers and Intractability: A Guide to the Theory of
NP-Completeness, W.H. Freeman, New York, 1979.
 In generally, we can not get the optimal solution of the problem because these
objectives usually conflict with each other in practice. The real solutions to the
problem are a set of Pareto optimal solutions (Chankong and Haimes, 1983).


For solving the BNP, the set of efficient paths may be very large and possibly
exponential in size.



Chankong, V. and Y.Y. Haimes: Multiobjective Decision Making Theory and
Methodology. North-Holland, New York, 1983.
Thus the computational effort required to solve it can increase exponentially
with the problem size in the worst case.
While the tractability of the problem is of importance when solving large scale
problems, the issue concerning with the size of the efficient set is important to
a decision maker.
Having to evaluate a large efficient set in order to select the best one poses a
considerable cognitive burden on decision makers. Therefore, in such cases,
obtaining the entire Pareto optimal set is of little interest to decision makers.
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4.1 Introduction of BNP

Recently, GAs have received considerable attention regarding their
potential as a novel approach to multiobjective optimization problems,
known as evolutionary or genetic multiobjective optimization.
Deb, K. : Multiobjective Optimization Using Evolutionary Algorithms, Wiley, Chichester,
UK, 2001.

The basic feature of GAs is the multiple directional and global search by
maintaining a population of potential solutions from generation to generation.

The population-to-population approach is hopeful to explore Pareto optimal or
nondominated solutions.

GAs do not have much mathematical requirements about the problems and can
handle any kind of objective functions and constraints.

Due to their evolutionary nature, GAs can search for solutions without regard to the
specific inner workings of the problem.

Therefore, GAs is possibly well suited to the multiobjective optimization problems.
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4.1 Introduction of BNP
 The bicriteria shortest path problem is one of BNPs, which of finding a
diameter-constrained shortest path from a specified source node s to
another specified sink node t.

This problem, termed the multi-objective shortest path problem
(MOSP) in the literature is NP-complete.
 Warburton (1987) presented the first fully polynomial approximation
scheme (FPAS) for it.
Warburto, A.: “Approximation of Pareto optima in multiple-objective, shortest
path problems,” Operations Research, vol. 35, no. 1, pp. 70-79, 1987.
 Hassin (1992) provided a strongly polynomial FPAS for the problem which
improved the running time of Warburton.
Hassin, R.: “Approximation schemes for the restricted shortest path problem,”
Math. Of Operations Research, vol. 17, no. 1, pp. 36-42, Feb. 1992.
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4.1 Introduction of BNP

The general classes of BNPs with minimum two objectives (under different
cost functions) are defined and extended to the more multi-criteria network
design problems.

Ravi et al. (1994) presented an approximation algorithm for finding good
broadcast networks.
Ravi, R.: "Rapid rumor ramification: approximating the minimum broadcast time,”
Proc. 35th Annual IEEE Foundations of Comput. Sci., pp. 202-213, 1994.

Ganley et al. (1995) consider a more general problem with more than two
objective functions.
Ganley, J. L., M. J. Golin, and J. S. Salowe: “The multi-weighted spanning tree
problem,” Proc. 1st COCOON, pp. 141-150, Springer-Verlag, LNCS, 1995.

Marathe et al. (1998) consider three different criteria of network and presented
the first polynomial-time approximation algorithms for a large class of BNP.
Marathe, M. V., R. Ravi, R. Sundaram, S. S. Ravi, D. J. Rosenkrantz, and H. B.
Hunt: “Bicriteria network design problems,” J. Algorithms, vol. 28, no. 1, pp.
142-171, 1998.
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4.1 Introduction of BNP

In this study, we dominated BNP with more complexity cases as two criteria
problem that maximum flow and minimum cost considered.

Priority-based encoding method (Cheng and Gen, 1994) has been improved.




Considering the characteristic of priority-based encoding method, we proposed
a new crossover operator called as Weight Mapping Crossover (WMX)
Insertion mutation operator and Immigration operator (Michael et al., 1991)
was adopted.



Cheng, R. and M. Gen: “Evolution program for resource constrained project
scheduling problem,” Proc. of Int. Conf. Evol. Comput., pp.736-741, 1994.
For maximizing flow, different form other genetic representation methods, such
as path oriented encoding method, priority-based encoding method can
represent various efficient paths by each chromosome.
Michael, C.M., C.V. Stewart and R. B. Kelly: “Reducing the search time of a
steady state genetic algorithm using the immigration operator”, Proc. IEEE Int.
Conf. Tools for AI, San Jose, CA, pp.500-501, 1991.
These methods provide a search capability that results in improved quality of
solution and enhanced rate of convergence.
For ensure the population diversity in MOGA, Adaptive Weight Approach
(AWA) which is one of weighted-sum approach, was adopted.


Gen, M. and R. Cheng: Genetic Algorithms and Engineering Optimization, John
Wiley & Sons, New York, 2000.
Their elements represent that weights are adjusted adaptively based on the
current generation to obtain search pressure toward the positive ideal point.
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4.2 BNP Formulation
4.2 BNP Formulation

In this study, we present a mathematical programming formulation of the bicriteria
network design model including MXF model and MCF model.

Different from the generic BNP, the problem’s efficient set of paths may be very large,
possibly exponential in size. Thus the computational effort required to solve it can
increase exponentially with the problem size in the worst case.
 In a network with flow capacities and costs on the arcs, BNP is to determine
both the maximum possible flow z1 and minimum cost z2 in the same time, from
a source to a sink.
max z1  f
n
min
n
z 2   cij xij
i 1 j 1
 f (i  1)

s. t.  xij   xki   0 (i  2,3,  , n  1)
j 1
k 1
 f (i  n)

0  xij  uij , (i, j )  A
n
n
f 0
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4.3.1 Genetic Representation
 Priority-based encoding method
procedure 1: Priority-based Encoding
input: number of nodes n
output: chromosome vk
begin
for j=1 to n
// step 0
vk(j) j;
for i=1 to n / 2
// step 1
repeat
jrandom[1, n];
lrandom[1, n];
until l≠j
swap (vk(j), vk(l));
output the chromosome vk;
// step 2
end
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4.3.2 Decoding Method
 The decoding procedure is a two-stage process.
 First stage: the path is generated by One-path growth procedure
 It is given in procedure 2
 With beginning from the specified node 1 and terminating at the specified
node n. At each step, add the one with the highest priority into path.
procedure 2: One-path Growth
input: number of nodes n, chromosome vk ,
the set of nodes Si with all nodes adjacent to node i.
output: path Pk
step 0: the source node i1, Pk 
step 1: if Si=, goto step 3; otherwise, continue.
step 2: select l from Si with the highest priority, and go back to step 1.
if vk(l)≠0 then
vk(l)=0;
Pk  Pk{xil};
il;
else vk(l)=0
step 3: output the complete path Pk ;
Pk  {x1l1 , xl1 ,l2 , xl2 ,l3 , ..., xlm 1 ,lm }
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4.3.2 Decoding Method
 The decoding procedure is a two-stage process.
 Second stage: overall paths are generated by overall paths
growth procedure
 For a given path, we can calculate its flow fk and the cost ck.
 By removing the used capacity from uij of each arc, we have a
new network with the new flow capacity ũij.
 With the one-path growth procedure (procedure 2), we can obtain
the second path.
 By repeating this procedure we can obtain the maximum flow for
the given chromosome till no new network can be defined in this
way.
 It is given in procedure 3.
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4.3.2 Decoding Method
procedure 3: Overall-path Growth
input: network data (V, A, C, U), chromosome vk , the set of nodes Si
with all nodes adjacent to node i
output: number of paths Lk , the flow fik and the cost cik of each path,
iLk
step 0: number of paths l0
step 1: if S1=, go to step 7; otherwise, l l +1, continue.
step 2: the implementation of path Plk growth is based on procedure 2.
Select the sink node a of path plk.
step 3: if the sink node a=n, continue; otherwise, perform the set of
nodes Si update as follows, return to step 1.
S i  S i  {a}, i
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4.3.2 Decoding Method
step 4: calculate the flow flk and the cost clk of the path Plk.
f l k  f l k1  min{ uij | (i, j )  Plk }
c c
k
l
k
l 1
n
n
  cij ( f l k  f l k1 ), (i, j )  Pl k
i 1 j 1
step 5: perform the flow capacity uij of each arc update. Make a new
flow capacity ũij as follows:
u~ij  u ij  min{ u ij (i, j )  Pl k }
step 6: if the flow capacity ũij=0, perform the set of nodes Si update
which the node j adjacent to node i.
si  si  { j} , (i, j )  Pl k & u~ij  0
step 7: output number of paths Lk  l -1, the flow fik and the cost cik of
each path, iLk .
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Illustration of Decoding Method
Chromosome:
node ID :
1
priority :
2
2
3
1
6
4
4
5
11
6
9
7
8
8
9
10
5
10
11
3
7
18, 60
f
16, 40
2
19, 60
5
14, 30
13, 30
15, 20
s
19, 60
16, 50
3
6
15, 40
17, 30
15, 30
4
1
0
2
Si
l
Pk
S1
z1k
z2k
1
2, 3, 4
3
1, 3
2
4, 6, 7
6
1, 3, 6
3
5, 8, 9, 10
5
1, 3, 6, 5
4
8
8
1, 3, 6, 5, 8
5
9, 11
11
7
f
11
14, 30
17, 50
13, 40
10
k : number of paths
Si : the set of nodes
l : sink node
1, 3, 6, 5, 8, 11 2, 3, 4
20
1380
1
Pk : the kth path
S1 : the set of nodes with all nodes
adjacent to node 1
z1k : maximum possible flow
z2k : minimum possible cost
1
2, 3, 4
3
1, 3
2
4, 6, 7
6
1, 3, 6
3
8, 9, 10
8
1, 3, 6, 8
4
9, 11
11
1, 3, 6, 8, 11
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19, 40
9
19, 50
i : start node
1
0
18, 30
15, 20
17, 60
i
18, 60
16, 30 17, 30
t
1
k
8
2, 3, 4
50
5730
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Illustration of Decoding Method
Chromosome:
node ID :
1
priority :
2
2
3
1
6
4
5
4
11
6
9
7
8
8
9
10
10
5
11
3
16, 40
2
7
18, 60
13, 30
15, 20
19, 60
16, 50
3
6
15, 40
17, 30
15, 30
3
0
4
Si
l
Pk
S1
z1k
z2k
1
2, 3, 4
3
1, 3
2
4, 7
7
1, 3, 7
3
6, 10
6
1, 3, 7, 6
4
9, 10
9
1, 3, 7, 6, 9
5
10, 11
11
1, 3, 7, 6, 9, 11
7
f
11
14, 30
17, 50
13, 40
10
i : start node
Si : the set of nodes
l : sink node
2, 4
60
6580
1
Pk : the kth path
S1 : the set of nodes with all nodes
1
2, 4
4
1, 4
2
7
7
1, 4, 7
3
6, 10
6
1, 4, 7, 6
4
9, 10
9
1, 4, 7, 6, 9
5
10, 11
11
1, 4, 7, 6, 9, 11
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19, 40
9
19, 50
k : number of paths
1
0
4
18, 30
15, 20
17, 60
i
18, 60
16, 30 17, 30
t
1
k
8
14, 30
s
f
19, 60
5
adjacent to node 1
z1k : maximum possible flow
z2k : minimum possible cost
2,4
70
7430
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Illustration of Decoding Method
Chromosome:
node ID :
1
priority :
2
2
3
1
6
4
4
5
11
6
9
7
8
8
10
9
5
10
11
3
7
18, 60
f
16, 40
2
19, 60
5
14, 30
13, 30
15, 20
s
16, 50
3
6
15, 40
17, 30
15, 30
4
5
0
6
Si
l
Pk
S1
z1k
z2k
1
1
2, 4
4
1, 4
2
7
7
1, 4, 7
3
10
10
1, 4, 7, 10
4
11
11
1, 4, 7, 10, 11
0
2
2
1, 2
2
3, 5, 6
5
1, 2, 5
3
8
8
1, 2, 5, 8
4
9, 11
11
1, 2, 5, 8, 11
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19, 40
7
9
19, 50
f
11
14, 30
17, 50
13, 40
10
k : number of paths
i : start node
Si : the set of nodes
l : sink node
2
100
9410
1
1
18, 30
15, 20
17, 60
i
18, 60
16, 30 17, 30
t
19, 60
1
k
8
Pk : the kth path
S1 : the set of nodes with all nodes
2
110
10120
adjacent to node 1
z1k : maximum possible flow
z2k : minimum possible cost
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Illustration of Decoding Method
Chromosome:
node ID :
priority :
1
2
2
1
3
6
4
4
5
11
6
9
7
8
8
9
10
10
5
11
3
7
18, 60
f
16, 40
2
19, 60
5
14, 30
13, 30
15, 20
s
16, 50
3
6
15, 40
17, 30
15, 30
7
0
8
Si
l
Pk
S1
z1k
z2k
4
18, 30
15, 20
17, 60
i
18, 60
16, 30 17, 30
t
19, 60
1
k
8
19, 40
7
9
19, 50
f
11
14, 30
17, 50
13, 40
10
1
k : number of paths
1
2
2
1, 2
2
3
4
3, 5, 6
8
9
5
8
9
1, 2, 5
1, 2, 5, 8
1, 2, 5, 8, 9
5
10, 11
11
1, 2, 5, 8, 9, 11
0
i : start node
Si : the set of nodes
l : sink node
2
140
12790
1
S1 : the set of nodes with all nodes
1
2
2
1, 2
2
3, 6
6
1, 2, 6
3
9, 10
9
1, 2, 6, 9
4
10
10
1, 2, 6, 9, 10
5
11
11
1, 2, 6, 9, 10, 11
Soft Computing Lab.
Pk : the kth path
adjacent to node 1
z1k : maximum possible flow
z2k : minimum possible cost
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Illustration of Decoding Method
Data table of example network
 Illustration of Priority-based GA
- cost
-0
40, 1380
-2000
-4000
50, 5730
60, 6580
70, 7430
-6000
-8000
100, 9410
110, 10120
-10000
-12000
140, 12790
-14000
160, 14350
-16000
flow
0
20
40
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100
120
140
160
180
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i
1
1
1
2
2
2
3
3
3
4
5
6
6
6
6
7
7
8
8
9
9
10
j
2
3
4
3
5
6
4
6
7
7
8
5
8
9
10
6
10
9
11
10
11
11
cij
18
19
17
13
16
14
15
16
17
19
19
15
16
15
18
15
13
17
18
14
19
17
uij
60
60
60
30
40
30
30
50
30
40
60
20
30
40
30
20
40
30
60
30
50
50
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4.3.3 Fitness Assignment
 In this study, evaluate the fitness of each individual in the GA
approach for BNP, we design an adaptive evaluation function based
on the AWA as procedure 4.
procedure 4: Adaptive Weight Approach
input: chromosome vk , kpopSize, number of paths Lk , the flow fik and the cost cik of each
path, iLk
output: fitness value eval(vk), kpopSize
step 1: define two extreme points: the maximum extreme point z+ and the minimum extreme
point z- in criteria space as follows:
z   { z1max , z2max }
z   { z1min , z2min }
where z1max, z２max, z1min and z2min are the maximal value and minimal value for
objective 1 and objective 2 in the current population. They are defined as follows:
z1max  max{ f i k | i  Lk , k  popSize}
z 2max  max{ cik | i  Lk , k  popSize}
z1min  min{ f i k | i  Lk , k  popSize}
z 2min  min{ cik | i  Lk , k  popSize}
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4.3.3 Fitness Assignment
step 2: The adaptive weight for objective 1 and objective 2 are calculated by
the following equation:
1
w1  max
z1  z1min
w2 
1
z 2max  z 2min
step 3: Calculate the fitness value for each individual.
 w ( f  z )  w (c  z )
Lk
eval (vk ) 
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k
1
i
min
1
2
k
i
min
2
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4.3.4 Genetic Operators
 Crossover Operator: Weight Mapping Crossover (WMX)
 We proposed a new crossover operator: WMX.
 WMX can be viewed as an extension of one-point crossover for
permutation representation.
step 1: select a cut-point
cut-point
parent 1 :
2
1
7
4
5
3
6
parent 1 : 1
3
4
7
parent 2 :
3
7
2
6
5
1
4
parent 2 : 1
2
4
5
7
offspring 1 : 1
3
4
5
7
offspring 2 : 1
2
4
7
step 2: mapping the weight of the right segment
5
3
6
5
1
4
3
1
5
6
4
5
step 3: generate offspring with mapping relationship
offspring 1 :
2
1
7
4
6
3
5
offspring 2 :
3
7
2
6
4
1
5
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4.3.4 Genetic Operators
 Mutation Operator: Insertion Mutation
 Selects a gene at random and inserts it in a random position
as follows:
select a gene at random
parent :
2
1
7
4
5
3
6
insert it in a random position
offspring :
2
5
1
7
4
3
parent : 1
3
4
offspring : 1
4
7
7
6
 Selection:
 the roulette wheel selection, a type of Proportionate selection,
is adopted.
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4.3.4 Genetic Operators
 Immigration Operator:

Moed et. al. (1990) proposed an immigration operator which, for
certain types of functions:
 Allows increased exploration
 Maintaining nearly the same level of exploitation for the given
population size.

Immigration operator procedure:
step 1: The algorithm is modified to include immigration, with each
generation generated.
step 2: Evaluate μ random members (μ, called the immigration rate).
step 3: Replace the μ worst members of the population with the μ
random members.
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4.3.5 GA Procedure for BNP
 GA Procedure for BNP
procedure: Priority-based GA for BNP
input: network data (V, A, C, U), GA parameters
output: Pareto optimal solution E(t)
begin
t  0;
initialize P(t) by priority-based encoding;
objectives z1(P), z2(P);
create Pareto E(P);
fitness eval(P) by adaptive weight approach;
while (not termination condition) do
crossover P(t) to yield C(t) by weight mapping crossover;
mutation P(t) to yield C(t) by insertion mutation;
immigration operation to yield C(t) ;
objectives z1(C), z2(C);
update Pareto E(P, C);
fitness eval(P, C) by adaptive weight approach;
select P(t+1) from P(t) and C(t) by roulette wheel selection;
t  t + 1;
end
output Pareto optimal solution E(t);
end
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4.4 Numerical Examples
 Test Problems:
 The numerical examples, presented by Munakata & Hashier, was
adopted.
Munakata, T. and D. J. Hashier: “A genetic algorithm applied to the maximum flow
problem,” Proc. of 5th Int. Conf. on Genetic Algorithms, pp. 488-493 , 1993.
 Using the following parameter specifications.





Population size:
popSize =20
Crossover probability:
pC =0.40
Mutation probability:
pM =0.60
Maximum generation:
maxGen =1000
Terminating condition: 100 generations with same fitness.
 All the simulations were performed with Java on Pentium 4
processor (1.5-GHz clock).
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4.4 Numerical Examples
 Test Problem 1:
The first numerical example, presented by Munakata & Hashier, was
adopted. The problem comprises 25 nodes and 49 arcs. It is given as
follows:
2
10, 10
7
10, 15
13, 8
10, 6
10, 20
13, 20
f
1
32, 20
4
8
9, 15
15, 8
35, 10
9 12, 15
13
631, 20
7, 20
14
10
6
7, 10
14, 15
18
15
19
22
14, 20
10, 10
35, 25
20 14, 25
4, 20
16
10, 30
23 2, 30
3, 8 3, 10
14, 20
21
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f
3, 30
24
12, 25
5, 15
11
8, 25
5, 8
7, 20
9, 15
33, 10
7, 10
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7, 20
5, 20
5
11, 8
34, 20
135, 20
3, 10
13, 4
17
6, 20
11, 15
4, 5
8, 20
11, 25
4, 15
15, 5
33, 4
3
12
i
cij , uij
j
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4.4 Numerical Examples
Table 6.4 The Pareto optimal solutions of test problem 1
z1
z2
z1
z2
z1
z2
4
300
30
2470
72
7703
5
345
33
2786
73
8382
8
600
38
2926
75
9762
10
696
40
3046
78
11799
12
993
43
3274
80
13147
15
1001
47
3674
82
14531
18
1226
52
4074
85
17115
20
1568
56
4830
87
17941
21
1629
59
5406
88
19254
23
1833
66
6575
89
19333
28
2178
69
7145
90
20007
 Gen, M., L. Lin & R. Cheng: “Bicriteria Network Optimization Problem using Prioritybased Genetic Algorithm,” IEEJ Trans. on Elect., Info. & Sys., Oct. 2004.
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4.4 Numerical Examples
ideal point: z1=90, z2=300
0-
-5000 -
z1=66, z2=6575
-10000 cost
-15000 -
-20000 -
- 25000 0
20
40
60
80
100
flow
Fig. 6.5 The Pareto optimal solutions of test problem 1
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4.4 Numerical Examples
 Test Problem 2:
The first numerical example, presented by Munakata & Hashier, was
adopted. The problem comprises 25 nodes and 56 arcs. It is given as
follows:
2
3, 10
8, 10
2, 20
3
3, 5
14, 8
7
8
7, 5
f
1
33, 20
4
13
33, 8
5
4, 7
12, 10
11, 12
6
12, 15
8, 20
12, 6
10
17
18
15
31, 15
22
6, 30
f
25
7, 20
30, 8
23
11, 6
19
6, 6
10, 20
8, 4
35, 6
14, 5
7, 8
2,15
21
12, 7
15, 10
11, 22
20
26, 9
34, 10
2, 9
14
3, 5
30, 7
634, 20
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9, 7
9
6, 8
9, 10
12, 8
16
30, 10
6, 12
12, 9
126, 20
11, 5
12
4, 6
6, 8
13, 2
12, 7
32, 8
29, 11
7, 18
13, 6
10, 20
11
9, 15
35, 9
8, 10
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cij , uij
j
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4.4 Numerical Examples
Table 6.5 The Pareto optimal solutions of test problem 2
z1
z2
z1
z2
z1
z2
z1
z2
2
8
10
15
18
19
20
21
25
26
27
52
248
340
495
692
1012
1111
1220
1292
1406
1457
32
34
36
38
40
41
43
47
49
51
52
1633
1909
1937
2077
2485
2581
2731
3080
3302
3551
3739
54
55
58
61
63
65
66
67
68
71
72
3872
3990
4146
4671
5153
5463
5704
6323
6422
6537
6748
73
74
75
76
78
80
82
83
85
86
91
6944
7192
7402
7532
7847
9228
10395
12508
12610
13151
16752
28
1475
 Gen, M., L. Lin & R. Cheng: “Bicriteria Network Optimization Problem using Prioritybased Genetic Algorithm,” IEEJ Trans. on Elect., Info. & Sys., Oct. 2004.
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4.4 Numerical Examples
ideal point: z1=91, z2=52
0 -2000 -4000 -
z1=61, z2=4671
-6000 -8000 cost -10000 -12000 -14000 -16000 -
Pareto optim alsolution
idealpoint
-18000 0
20
60
40
80
100
flow
Fig. 6.6 The Pareto optimal solutions of test problem 2
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4.4 Numerical Examples
 Simulation (# of nodes: 25, # of arcs: 56)
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6. Basic Network Design
1. Shortest Path Problem (SPP)
2. Maximum Flow (MXF) Problem
3. Minimum Cost Flow (MCF) Problem
4. Bicriteria Network Design Problem (BNP)
5. Multi-criteria Network Design Problem
5.1 Introduction of Multi-criteria Network Design Problem
5.2 Reviewing Solution Approaches for MNP
5.3 Numerical Examples
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5. Multi-criteria Network Design Problem (MNP)
 With the information superhighway fast becoming a reality,
the problem of designing networks capable of
accommodating multimedia (both audio and video) traffic in
a multicast (simultaneous transmission of data to multiple
destinations) environment has come to assume paramount
importance

Chow, C.-H.: “On multicast path finding algorithms,” Proceedings of IEEE
INFOCOM, pp.1274-1283, 1991.

Frank, A., L. Wittie, and A. Bernstein: “Multicast communication in network
computers,” IEEE Software, Vol. 2, No. 3, pp. 49-61,1985.
Kadaba, B. and J. Jaffe: “Routing to multiple destinations in computer
networks,” IEEE Transactions on Communications, Vol. COM-31, pp. 343351,1983.
Kompella, V.P., J.C. Pasquale and G.C. Polyzos: “Multicasting for
multimedia applications,” Proceedings of IEEE INFOCOM, 1992.
Kompella, V.P., J.C. Pasquale and G.C. Polyzos: “Multicast routing for
multimedia communication,” IEEE/ACM Transactions on Networking, pp.
286-292, 1993.



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5.1 Introduction of MNP
 Network design problems where even one cost measure must
be minimized, are often NP-hard. But, in real-life applications, it
is often the case that the network to be built is required to
minimize multiple cost measures simultaneously, with different
cost functions for each measure.
 For example, in the problem of finding good multicast trees,
each edge has associated with it two edge costs:
 The construction cost: It is typically a measure of the amount of
buffer space or channel bandwidth used
 The delay cost: It is a combination of the propagation, transmission
and queuing delays.
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5.1 Introduction of MNP
 Multi-criteria network design problems, with separate cost functions for
each optimization criterion, also occur naturally in Information Retrieval
and VLSI designs.


Bookstein, A. & S.T. Klein: “Construction of Optimal Graphs for Bit-Vector
Compression,” Proc. 13th ACM-SIGIR, vol. 16, pp. 387-400, 1990.
Zhu, Q., M. Parsa & W.W.M. Dai: “An iterative approach for delay-bounded
minimum Steiner tree construction,” Technical Report UCSC-CRL-94-39,
UC Santa Cruz, 1994.
 With the advent of deep micron VLSI designs, the feature size has
shrunk to sizes of 0.5 microns and less.
 As a result, the interconnect resistance, being proportional to the
square of the scaling factor, has increased significantly.
 An increase in interconnect resistance has led to an increase in
interconnect delays thus making them a dominant factor in the
timing analysis of VLSI circuits.
 Therefore VLSI circuit designers aim at finding minimum cost
(spanning or Steiner) trees given delay bound constraints on
source-sink connections.
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5.1 Introduction of MNP

For example, the problem of finding low-cost and low-transmission-delay
multimedia networks can be modeled as the (Diameter, Total cost,
Spanning tree)-bicriteria problem:



given an undirected graph G = (V,E) with two weight functions ce and de for
each edge e∊E modeling construction and delay costs respectively, and a
bound D (on the total delay), find a minimum c-cost spanning tree such that
the diameter of the tree under the d-costs is at most D.
It is easy to see that the notion of bicriteria optimization problems can be easily
extended to the more general multicriteria optimization problems.
The applications set the stage for the formal definition of multicriteria
network design problems. Marathe et al. explain this concept by giving a
formal definition of a bicriteria network design problem.


Marathe, M. V., R. Ravi, R. Sundaram, S. S. Ravi, D. J. Rosenkrantz, and H. B.
Hunt: “Bicriteria network design problems,” J. Algorithms, vol. 28, no. 1, pp.
142-171, 1998.
Marathe et al. study the complexity and approximability of a number of
bicriteria network design problems. The three objectives considered:



total cost
diameter
degree of the network.
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5.2 Reviewing Solution Approaches for MNP
a. AWA (Gen et al., 1998)
 Gen, M. & R. Cheng: Genetic Algorithms and Engineering
Optimization, John Wiley & Sons, New York, 2000.
b. RWA (Murata et al., 1998)
 Gen, M. & R. Cheng: Genetic Algorithms and Engineering
Optimization, John Wiley & Sons, New York, 2000.
c. SPEA (Zitzler et al., 1999)
 Zitzler, E. & L. Thiele: “Multiobjective Evolutionary Algorithms: A
Comparative Case Study and the Strength Pareto Approach”, IEEE
Trans. on Evol. Comput., Vol.3, No.4, pp.257-271, 1999.
d. NSGA-Ⅱ(Deb et al., 2000)
 Deb, K., A. Pratap, S. Agarwal and T. Meyarivan: “A Fast and Elitist
Multiobjective Genetic Algorithm: NSGA-Ⅱ”, IEEE Trans. on Evol.
Comput., Vol.6, No.2, 182-197, 2002.
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5.2 Reviewing Solution Approaches for MNP
a. AWA (Gen & Cheng, 1998)
 Gen & Cheng (1998) proposed an Adaptive Weight Approach (AWA)
which utilizes some useful information from the current population to
readjust weights to obtain a search pressure toward a positive ideal point.
 Gen, M. & R. Cheng: Genetic Algorithms and Engineering Optimization, John Wiley & Sons,
New York, 2000.
 For the examined solutions at each generation, they define two extreme points
(maximum: z+, minimum: z-)
z   [ z1max z2max  zqmax ]
z   [ z1min z2min  zqmin ]
where zkmax and zkmin are the maximal and minimal values for the kth objective as
defined by the following equations:
zkmax  max{ f k ( x ) | x  P},
k  1, 2, , q
zkmin  min{ f k ( x ) | x  P},
k  1, 2, , q
P: set of solution candidates.
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5.2 Reviewing Solution Approaches for MNP
a. AWA (Gen & Cheng, 1998)

The weighted-sum objective function for a given chromosome x is given by the
following equation:
zk  zkmin
z ( x )   wk ( zk  z )   max min 
 zk
k 1
k 1 z k
q
q
min
k


k 1
f k ( x )  zkmin
zkmax  zkmin
where wk is adaptive weight for objective k :
wk 

q
z
max
k
1
,
min
 zk
k  1, 2, , q
The equation driven above is a hyperplane defined by the following extreme points
in current solutions:
[ z1max
[ z1min

[ z1min

[ z1min
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z2min  zkmin  zqmin ]
z2max  zkmin  zqmin ]
z2min  zkmax  zqmin ]
z2min  zkmin  zqmax ]
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5.2 Reviewing Solution Approaches for MNP
a. AWA (Gen & Cheng, 1998)
 Adaptive moving line defined by the extreme points (z1max, z2min) and (z1max, z2min)
are shown as follows:
z2
minimal rectangle containing
all current solutions
positive ideal point
whole criteria space Z
z
min
1
(z
max
2
z 2min
z
,z
max
2
)
z  maximum
extreme point
( z1max , z2min )

minimum
adaptive
moving line
subspace
extreme point corresponding to
current solutions
z1min
z1max
z1
Fig.6.7 Adaptive weights and adaptive hyperplane
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5.2 Reviewing Solution Approaches for MNP
b. RWA (Murata et al., 1998)
 Murata, Ishibuchi & Tanaka (1998) proposed a Random-Weight Approach
(RWA) to obtaining a variable search direction toward the Pareto frontier.
 Gen, M. & R. Cheng: Genetic Algorithms and Engineering Optimization, John Wiley & Sons,
New York, 2000.
 Ishibuchi, H., T. Yoshida and T. Murata: “Balance Between Genetic Search and Local Search in
Memetic Algorithms for Multiobjective Permutation Flowshop Scheduling”, IEEE Trans. on Evol.
Comput., Vol.7, No.2, pp.204-223, 2003.
 Fixed-weight approach gives the GAs a tendency to sample the area toward a fixed
point in the criterion space.
 Random-weight approach gives the GAs a tendency to demonstrate a variable
search direction, therefore, the ability to sample the area uniformly over the entire
frontier.
f1
f1
fixed search
direction
multiple search
direction
f2
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Fig. 6.8
Search UNIVERSITY
on a fixed direction
in criterion space
f2
145
5.2 Reviewing Solution Approaches for MNP
b. RWA (Murata et al., 1998)
 For a problem to maximize q objective functions, weighted-sum objective
is given as the follows:
q
z   wk f k ( x )
k 1
 Random-weight wk is calculated by the following equation:
wk 
rk

q
,
k  1, 2, , q
r
j 1 j
where rj are non-negative random number between [0, 1].
 Before selecting a pair of parents for crossover operation, a new set of
random weights is specified. The selection probability pi for individual i is
then defined by the following linear scaling function:
pi 

zi  zmin
popSize
j 1
z
j
 z min 
where zmin is the worst fitness value in the current population.
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5.2 Reviewing Solution Approaches for MNP
c. SPEA (Zitzler et al., 1999)
 Zitzler & Thiele (1999) proposed a new evolutionary approach to multicriteria
optimization, the Strength Pareto Evolutionary Algorithm (SPEA), that
combines several features of previous multiobjective EA’s in a unique manner.
 Zitzler, E. & L. Thiele: “Multiobjective Evolutionary Algorithms: A Comparative Case Study and the
Strength Pareto Approach”, IEEE Trans. on Evol. Comput., Vol.3, No.4, pp.257-271, 1999.
 It is characterized by:
 Storing nondominated solutions externally in a second, continuously updated
population.
 Evaluating an individual’s fitness dependent on the number of external
nondominated points that dominate it.
 Preserving population diversity using the Pareto dominance relationship.
 Incorporating a clustering procedure in order to reduce the nondominated set without
destroying its characteristics.
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5.2 Reviewing Solution Approaches for MNP
c. SPEA (Zitzler et al., 1999)
 The fitness assignment procedure is a two-stage process.
 First, the individuals in the external nondominated set P’ are ranked.
n
si 
N 1
where si is proportional to the number of population members j∈P for
which i ≻ j. n is the number of individuals in P that are covered
by i and N is the size of P.
f1
3/8
5/8
3/8
f2
Fig. 6.9 Two scenarios for a maximization problem with two objectives.
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5.2 Reviewing Solution Approaches for MNP
c. SPEA (Zitzler et al., 1999)
 The fitness assignment procedure is a two-stage process.
 Afterwards, the individuals in the population P are evaluated.
f j  1
s ,
i
where f j [1, N )
i, i j
where the fitness of an individual j∈P is calculated by summing the
strengths of all external nondominated solutions i∈P’ that cover j.
f1
3/8
11/8
5/8
16/8
13/8
13/8
3/8
19/8
16/8
11/8
f2
Fig. 6.10 Two scenarios for a maximization problem with two objectives.
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5.2 Reviewing Solution Approaches for MNP
d. NSGA-Ⅱ(Deb et al., 2000)
 Deb, Pratap, Agarwal & Meyarivan (2000) suggest a nondominated sorting-based
Multiobjective Evolutionary Algorithm (MOEA), called Nondominated Sorting Genetic
AlgorithmⅡ(NSGA-Ⅱ), which alleviates the three difficulties:
 Deb, K., A. Pratap, S. Agarwal and T. Meyarivan: “A Fast and Elitist Multiobjective Genetic
Algorithm: NSGA-Ⅱ”, IEEE Trans. on Evol. Comput., Vol.6, No.2, 182-197, 2002.
 Computational complexity
 Nonelitism approach
 The need for specifying a sharing parameter
 They presented the proposed NSGA-Ⅱ approach:
 Fast nondominated sort
 They first sorting a population into different nondomination levels.
 Nondomination rank (Ri);
 Crowding distance assignment
 The crowding-distance computation requires sorting the population according to
each objective function value in ascending order of magnitude.
 Crowding distance (Di)
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5.2 Reviewing Solution Approaches for MNP
d. NSGA-Ⅱ(Deb et al., 2000)
 The new population Pt+1 is now used for selection, crossover, and mutation to
create a new population Qt+1.
 It is important to note that they used a binary tournament selection operator,
but the selection criterion is based on the crowded-comparison operator ≺n .
 Crowded-comparison operator is defined as follows:
if Ri  R j  or ( Ri  R j ) and ( Di  D j ) 
then
i n j
where, Ri is nondomination rank and Di is crowding distance
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5.3 Numerical Examples
Test Problems:
 For examining the effect of different solution approaches of
multiobjective GAs, we applied SPEA, NSGAⅡ, RWA and AWA
to the 2 test problems.
 Munakata, T. & D. J. Hashier: “A genetic algorithm applied to the
maximum flow problem.” Proc. of the 5th Inter. Conf. on Genetic
Algorithms, San Francisco, pp.488-493, 1993.
 Using the following parameter specifications.





Population size:
Crossover probability:
Mutation probability:
Immigration rate:
Stopping conditions:
popSize =20
pC =0.70
pM =0.50
μ=3
Evaluation of 5000 solutions.
 Each algorithm was applied to each test problem 20 times (i.e.,
20 runs) using different initial populations.
 All the simulations were performed with Java on Pentium 4
processor (1.5-GHz clock).
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5.3 Numerical Examples
 Test Problems:
The first numerical example, presented by Munakata & Hashier, was
adopted. The problem comprises 25 nodes and 49 arcs. It is given as
follows:
2
10, 10
7
10, 15
13, 8
10, 6
10, 20
13, 20
f
1
32, 20
4
8
9, 15
15, 8
35, 10
9 12, 15
13
631, 20
7, 20
14
10
6
7, 10
14, 15
18
15
19
22
14, 20
10, 10
35, 25
20 14, 25
23 2, 30
14, 20
21
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f
3, 30
24
12, 25
5, 15
16
10, 30
3, 8 3, 10
4, 20
11
8, 25
5, 8
7, 20
9, 15
33, 10
7, 10
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7, 20
5, 20
5
11, 8
34, 20
135, 20
3, 10
13, 4
17
6, 20
11, 15
4, 5
8, 20
11, 25
4, 15
15, 5
33, 4
3
12
i
cij , uij
j
153
5.3 Numerical Examples
 Test Problems:
The second numerical example, presented by T. Munakata & D.J. Hashier, was adopted.
The problem comprises 25 nodes and 56 arcs. It is given as follows:
2
3, 10
3, 5
14, 8
7
8, 10
2, 20
3
33, 20
4
8
9, 7
9
4, 7
12, 10
10
11, 12
6
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12, 15
12, 7
10, 20
17
31, 15
8, 4
35, 6
14, 5
6, 30
f
25
30, 8
23
11, 6
7, 8
15
22
7, 20
18
19
6, 6
2,15
21
15, 10
11, 22
20
26, 9
34, 10
2, 9
14
3, 5
30, 7
634, 20
8, 20
12, 6
16
6, 8
9, 10
12, 8
13
126, 20
6, 8
30, 10
6, 12
12, 9
33, 8
5
11, 5
12
4, 6
7, 5
1
7, 18
13, 6
32, 8
29, 11
13, 2
12, 7
10, 20
f
11
8, 10
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9, 15
35, 9
i
cij , uij
j
24
154
5.3 Numerical Examples
 Reference solution set S* :
 The reference solution set S* of each test problem was
found using the SPEA, NSGA-2, RWA, and AWA. Each
algorithm was applied to each test problem with much
longer computation time and larger memory storage than
the other computational experiments in this study.
 More specifically, we used the following parameter
specifications in all the three algorithms for finding the
reference solution set of each test problem.





Population size:
Crossover probability:
Mutation probability:
Immigration probability:
Stopping conditions:
Soft Computing Lab.
popSize =30
pC =0.70
pM =0.80
μ=5
Evaluation of 100000 solutions.
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5.3 Numerical Examples
 Reference solution set S* to test problems:
0
cost
5000
10000
15000
20000
0
20
40
60
80
100
flow
Fig. 6.11 The Reference solution set of Example 1 (|S*|=69)
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5.3 Numerical Examples
 Reference solution set S* to test problems:
Table 6.6 The Reference solution set of Example 1 (|S*|=69)
z1
5
6
8
10
11
12
13
16
18
20
23
25
28
30
32
33
34
35
37
38
Soft Computing Lab.
z2
260
318
376
510
590
644
744
808
918
1030
1206
1320
1496
1650
1796
1875
1955
2058
2162
2262
z1
39
40
41
42
43
45
46
47
48
49
50
51
52
53
54
55
56
57
58
60
z2
2380
2447
2531
2648
2696
2866
3042
3049
3151
3322
3393
3527
3530
3932
4066
4072
4364
4479
4747
5010
z1
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
z2
5339
5743
5936
5962
6162
6382
6714
6846
6952
7182
7364
8043
8707
9375
10057
10750
11438
12123
12735
13523
WASEDA UNIVERSITY , IPS
z1
82
83
84
85
86
87
88
89
90
z2
14088
14841
15561
16198
16842
17528
18332
18988
19597
157
5.3 Numerical Examples
 Reference solution set S* to test problems:
0
cost
5000
10000
15000
20000
0
20
40
60
80
100
flow
Fig.6.12 The Reference solution set of Example 2 (|S*|=77)
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5.3 Numerical Examples
 Reference solution set S* to test problems:
Table 6.7 The Reference solution set of Example 2 (|S*|=77)
z1
2
5
7
8
10
11
13
15
17
18
20
22
23
24
25
26
27
28
29
30
Soft Computing Lab.
z2
52
115
167
248
275
306
388
495
608
653
784
872
953
965
1035
1168
1217
1275
1332
1344
z1
31
32
33
34
35
36
37
38
40
42
43
44
45
47
48
49
50
51
52
54
z2
1408
1488
1595
1640
1753
1818
1892
1942
2171
2319
2405
2587
2653
2701
2897
2974
3077
3106
3198
3446
z1
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
z2
3525
3610
3669
3732
3987
4189
4458
4551
4751
4867
5069
5341
5517
5583
5809
5941
6128
6319
6487
6652
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z1
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
z2
6815
7081
7227
7308
8002
8697
9319
9997
10667
11341
11949
12623
13455
14116
14735
15551
16100
159
5.3 Numerical Examples
Performance Measures:
 We mainly use a performance measure based on:
 The number of obtained solutions |Sj|
 The ratio of nondominated solutions RNDS(Sj)
 The RNDS(Sj) measure can be written as follows:
RNDS ( S j ) 
S j  x  S j r  S * : r  x
Sj
 The distance D1R
 The D1R measure can be written as follows:
D1R 
1
min{ d rx x  S j }

S * rS *
where S* is a reference solution set for evaluation the solution set Sj.
dxr is the distance between a solution x and a reference solution r.
d rx 
f
1
(r )  f1 ( x)    f 2 (r )  f 2 ( x) 
2
2
[Ref.] Ishibuchi, H., T. Yoshida & T. Murata: “Balance Between Genetic Search and Local Search in Memetic
Algorithms for Multiobjective Permutation Flowshop Scheduling”. IEEE Trans. On Evol. Comp., Vol.
7, No. 2, pp. 204-223, 2003.
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5.3 Numerical Examples
 Discussion of the Results:
 Comparison with different approaches using the |Sj|
Table 6.8 Comparison with the four approaches using the |Sj| measure.
Test Problems
(# of nodes/ # of arcs)
25/49
25/56
|Sj|
RWA
52
43
CPU Times
SPEA NSGA-Ⅱ
57
44
43
55
AWA
RWA
49
43
SPEA NSGA-Ⅱ
15122 17635
11918 16684
AWA
15693
15981
14170
14961
 Comparison with different approaches using the RNDS(Sj)
Table 6.9 Comparison with the four approaches using the RNDS(Sj) measure.
Test Problems
(# of nodes/ # of arcs)
25/49
25/56
RNDS(Sj)
CPU Times
RWA
SPEA NSGA-Ⅱ
AWA
0.57
0.41
0.54
0.34
0.61
0.53
0.39
0.36
RWA
SPEA NSGA-Ⅱ
15122 17635
11918 16684
15693
15981
AWA
14170
14961
 Comparison with different approaches using the D1R
Table 6.10 Comparison with the four approaches using the D1R measure.
Test Problems
(# of nodes/ # of arcs)
25/49
25/56
Soft Computing Lab.
D1R measure
CPU Times
RWA
SPEA NSGA-Ⅱ
AWA
191.21
203.96
315.61
224.40
143.58
141.43
228.65
185.89
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RWA
SPEA NSGA-Ⅱ
15122 17635
11918 16684
15693
15981
AWA
14170
14961
161
5.3 Numerical Examples
 Different Parameter Settings:

Comparison with different approaches using the stopping conditions: under the
same computation time: 10,000 ms.
Table 6.11 Comparison with the four approaches using the |Sj| measure.
|Sj|
Test Problems
(# of nodes/ # of arcs)
25/49
25/56
AWA
RWA
49
52
50
50
SPEA NSGA-Ⅱ
53
40
50
34
Table 6.12 Comparison with the four approaches using the RNDS(Sj) measure.
Test Problems
(# of nodes/ # of arcs)
25/49
25/56
RNDS(Sj)
AWA
RWA
SPEA NSGA-Ⅱ
0.57
0.51
0.44
0.32
0.56
0.60
0.48
0.41
Table 6.13 Comparison with the four approaches using the D1R measure.
Test Problems
(# of nodes/ # of arcs)
25/49
25/56
Soft Computing Lab.
D1R measure
AWA
RWA
SPEA NSGA-Ⅱ
191.17
147.07
203.72
219.59
222.28
279.60
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239.99
433.43
162
5.3 Numerical Examples
 Different Parameter Settings:

Comparison with different approaches using the stopping conditions: under the
same computation time: 10,000 ms.
Table 6.11 Comparison with the four approaches using the |Sj| measure.
|Sj|
Test Problems
(# of nodes/ # of arcs)
RWA
25/49
25/56
SPEA NSGA-Ⅱ
50
50
53
40
AWA
50
34
49
52
Table 6.12 Comparison with the four approaches using the RNDS(Sj) measure.
Test Problems
(# of nodes/ # of arcs)
25/49
25/56
RNDS(Sj)
RWA
SPEA NSGA-Ⅱ
AWA
0.44
0.32
0.56
0.60
0.57
0.51
0.48
0.41
Table 6.13 Comparison with the four approaches using the D1R measure.
Test Problems
(# of nodes/ # of arcs)
25/49
25/56
Soft Computing Lab.
D1R measure
RWA
SPEA NSGA-Ⅱ
AWA
203.72
219.59
222.28
279.60
191.17
147.07
239.99
433.43
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5.3 Numerical Examples
 Discussion of the Results:
1
1
D1R
Sj
R NDS ( S j )
D1R
Sj
R NDS ( S j )
Test problem 1 (25/56)
Test problem 1 (25/49)
 D1R: distance
 RNDS(Sj): ratio of nondominated solutions
 |Sj|: No. of obtained solutions
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Conclusion



In this study, we presented a GA approach used a priority-based chromosome
for solving the network design problems.
 It is easy to verify that any permutation of the encoding corresponds to the
paths.
 So that most existing genetic operators can easily be applied to the
encoding.
 Also, any path has a corresponding encoding.
 Therefore, any point in solution space is accessible for genetic search.
For solving the MXF/MCF, and Multi-criteria Network Design Problem, we also
combines an adaptive evaluation function based on the AWA.
 The fitness values of all individuals are calculated according to this
adaptive evaluation function.
 In each generation, the set of Pareto solutions is updated by deleting all
dominated solutions and adding all newly generated Pareto solutions.
Computer simulations show the several numerical experiments by using
several network optimization problems, and show the effectiveness of the
proposed method.
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