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Graduate School of Information, Production and Systems, Waseda University

6. Network Design Problems



Soft Computing Lab. WASEDA UNIVERSITY , IPS 2

6. Network Design Problems 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 solvesuch 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.




6. Network Design Problems 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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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






1.1 Basic Concept of Shortest Path Problem1.2 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 Algorithm1.4.3 Genetic Operators

1.5 Numerical Examples









SPP is perhaps the simplest of all network designproblems.

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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i j cij

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3 2 13

3 5 123 6 23

4 7 11

4 8 32

5 4 16

6 7 12

6 9 387 8 20

8 9 15

8 10 24

9 10 13

Data table of example network

i jcij

1.1 Basic Concept of Shortest Path Problem






Directed graph G =(V , A)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, if link i , j is includedinthepath

0, otherwise¿

xij

=¿ { ¿ ¿ ¿

¿

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i j cij

1 2 36

1 3 27

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3 2 13

3 5 12

3 6 23

4 7 11

4 8 32

5 4 16

6 7 12

6 9 38

7 8 20

8 9 15

8 10 24

9 10 13


i jcij






SPP can be formulated as follows:

),,2,1,(1or 0

)(1

)1,,3,2(0

)1(1

11

1 1

n ji x

ni

ni

i

x x

xc z

ij

n

k

ki

n

j

ij

n

i

n

j

ijij

==

=−

−=

=

=

−

=

∑∑

∑∑

==

= =

s.t.

min





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 How to determine the route road that have prohibitive weight restriction so that the

driver can reach the destination within the shortest possible time.

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 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’sperformance.






With the growth of the Internet, Internet Service Providers (ISPs) try to meetthe increasing traffic demand with new technology and improved utilizationof existing resources.

Routing of data packets can affect network utilization.

Packets are sent along network paths from source to destination followinga 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 verylarge 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.






OSPF uses a Shortest Path Routing (SPR) algorithm to compute routes inthe routing table.

The SPR algorithm computes the shortest (least cost) path between the router

and all the networks of the internetwork.

As the size of the link state database increases:

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.





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 graphin the form of adjacency lists and using a heap as priority queue toimplement 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 aweighted graph (where some of the edge weights may be negative).

Bellman-Ford is usually used only when there are negative edgeweights.

Floyd-Warshall Algorithm Floyd-Warshall algorithm is an algorithm to solve the all pairs shortest

path problem in a weighted, directed graph by multiplying anadjacency-matrix representation of the graph multiple times.





1.4 Genetic Approach for Solving SPP

How to encode a path in a network is critical for designing aGA.

Special difficulties:

a path contains variable number of nodes.

a random sequence of edges usually does not correspond to a

path.

Path 1 ： 1→2→4→8→10

Objective function value ： z =110

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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How to encode a solution of the problem into a chromosome is a key issuefor GAs.

For the nonstring coding approach, three critical issues emergedconcerning with the encoding and decoding between chromosomes andsolutions: The feasibility 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. The legality of a chromosome

Legality refers to the phenomenon of whether a chromosome represents asolution to a given problem.

The illegality of chromosomes originates from the nature of encodingtechniques.

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.




1.4.1 Reviewing Encoding Methodsa. Priority-based Chromosome (Cheng & Gen, 1997)

Cheng & Gen proposed a priority-based encoding method for solving resource-

constrained 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.





a. Priority-based Chromosome (Cheng & Gen, 1997)

Example: An example of generated chromosome and its decoded path

as follows:

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.

1 4

3 6

7

s t

2 5

1 1

node ID :1 2 3 4 5 6 7

priority :2 1 6 4 5 3 7

1 43 7 path :




1.4.1 Reviewing Encoding Methodsb. 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 theSizing 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.




1.4.1 Reviewing Encoding Methodsb. Variable-length Chromosome (Munemoto et al., 1998)

Example: An example of generated chromosome and its decoded path as follows:

Advantage:

The mapping from any chromosome to solution (decoding) belongs to 1-

to-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.

1 4

3 6

7

s t

2 5

1 1

1 43 7 path :

locus :1 2 3 4

node ID :1 3 4 7





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.





c. Fixed-length Chromosome (Inagaki et al., 1999)

Example: An example of generated chromosome and its decoded path as follows:

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 thespecial 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 thepriority-based encoding method.

In the special genetic operator phase, some offspring may generate newchromosomes that resemble the initial chromosomes in fitness, therebyretarding the process of evolution.

1 4

3 6

7

s t

2 5

1 1

1 43 7 path :

locus : 1 2 3 4 5 6 7

node ID :3 1 4 7 2 4 6





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.




1.4.2 Priority-based Genetic Algorithm

procedure 1: Priority-based Encoding

input: number of nodes n

output: chromosome v k

begin

for j=1 to n // step 0 v

k ( j )← j ;

for i=1 to // step 1

repeat

j ←random[1, n];

l ←random[1, n];

until l ≠ j

swap (v k ( j ), v

k (l ));

output the chromosome v k ; // step 2

end




begin

for j=1 to n // step 0 v

k ( j )← j ;


repeat

j ←random[1, n];

l ←random[1, n]; until l ≠ j

swap (v k ( j ), v

k (l ));


end

Priority-based Encoding Method

2/n





procedure 2: One Path Growthinput: number of nodes n, chromosome v

k ,

the set of nodes S i with all nodes adjacent to node i.

output: path P k

begin

initial source node i ←1, P k

←φ ; // step 0

while S i ≠φ do // step 1

select l from S i with the highest priority;

if v k (l )≠0 then

v k (l )=0;

P k

←P k

∪{ x il

};

i ←l ;

else S i ←S

i \{l }

end

output the complete path P k ; // step 2

end

Decoding Method





i j cij

1 2 36

1 3 27

2 4 18

3 2 13

3 5 123 6 23

4 7 11

4 8 32

5 4 16

6 7 12

6 9 387 8 20

8 9 15

8 10 24

9 10 13


Path: 1→3→6→7→8→10


priority: v ( j )

node ID: j

91108526437

10987654321

Chromosome:

Illustration of Priority-based GA

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It operates on two parents (chromosomes) at a time and generatesoffspring 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 salesmanproblem, Euro. J. of OR , vol.26, pp.187-210, 1986.

Heuristic crossover , and so on.

1.4.3 Genetic Operators --- Crossover




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 1 : select the substring at random

step 2 : exchange substrings between

step 3 : determine mapping relationship

step 4 : legalize offspring with mapping

relationship

parent 1: 1 7 2 3 4 6 5 8

parent 2: 4 6 3 5 7 1 8 2

substring selected

parent 1: 1 7 3 5 7 6 5 8

parent 2: 4 6 2 3 4 1 8 2

2 3 4

3 5 7 74

532

↔

↔↔

offspring 1: 1 4 3 5 7 6 2 8

offspring 2: 7 6 2 3 4 1 8 5





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

parent 2: 4 6 3 5 7 1 8 2

offspring : 6 5 2 3 4 7 1 8

substring selected

Fig. 6.1 Illustration of the OX operator.





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 thenodes are selected inconsecutively.

parent 1: 1 7 2 3 4 6 5 8

parent 2: 4 6 3 5 7 1 8 2

offspring : 3 7 5 1 4 6 2 8

Fig. 6.2 Illustration of the PX operator.





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 notpossible to succeed the character of the parents.

thereby retarding the process of evolution.

We proposed a new crossover operator, Weight MappingCrossover (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 .





Weight Mapping Crossover (WMX)

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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 isnumber 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, v

2; legalize offspring with mapping relationship.

6354712 parent 1 :

4156273 parent 2 :

cut-point

415

635653

541

5146273offspring 2 :

5364712offspring 1 :

step 1: select a cut-point

step 2: mapping the weight of the right segment

step 3: generate offspring with mapping relationship

1 43 7 parent 1 :

1 42 7 parent 2 :

offspring 1 :

1 42 7offspring 2 :

5

1 43 75




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 havebeen 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:

6354712 parent :

select a gene at random

6347152offspring :

insert it in a random position

1 43 7 parent :

offspring : 1 4 7




1.4.3 Genetic Operators --- Immigration

The trade-off between exploration and exploitation in

serial GAs for function optimization is a fundamentalissue.

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.




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.




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 inthe 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.

),( λ µ




1.4.4 Overall Procedure

procedure: Priority-based GA for Shortest Path Probleminput: 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

GA Procedure for Shortest Path Problem





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 testproblems: 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 areference 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).

1




1

1


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 Problemand the Sizing of Populations”, IEEE Trans. on Evol. Comput., Vol.6, No.6, pp.566-

579, 2002.)

Fig.6.3 Example of the first numerical example

1 5 N i l E l




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.)

Dijkstra’s AlgorithmMunemoto’s AlgorithmInagaki’s AlgorithmAhn’s Algorithm

2.5

2.0

1.5

1.0

0.5

0 2 4 6 8 10

Generations

Objective

Function

Values


1 5 N i l E l




240

260

240

260


Convergence property of Ahn et al.’s algorithm and proposed

algorithm for a Fixed Network With 20 Nodes

Fig. 6.4 Convergence property of Ahn et al.’s algorithm and proposed algorithm.

et al.

1 5 N i l E l




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

15 20 25 30 35 40 45 50

1200

1000

800

600

400

200

0

The Number of Nodes

Objective

FunctionValue


1 5 N i l E l





Discussion of the Results:

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 alternativesbecome to very large in test problems, the population be prematurely converged to a localoptimum of the function.

Table 6.1 Performance comparisons with different genetic operatorsTest Problems

(# of nodes/ # of arcs)

OptimalSolutions

Best Solutions

Alg. 1 Alg. 2 Alg. 3 Alg. 4 Alg. 5 Alg. 6

20/49 142.00 148.35 148.53 147.70 143.93 142.00 142.00

80/120 389.00 423.53 425.33 418.82 396.52 389.00 389.00

80/632 291.00 320.06 311.04 320.15 297.21 291.62 291.00

160/2544 284.00 429.55 454.98 480.19 382.48 284.69 284.00320/1845 394.00 754.94 786.08 906.18 629.81 395.01 394.00

320/10208 288.00 794.26 732.72 819.85 552.71 331.09 288.00

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).

1 5 N i l E l




Comparison results of Ahn’s algorithm and Proposed algorithm

Test Problems


OptimalSolutions

Best Solutions CPU Times (ms) Generation Num. of Obtained best result

Ahn’s Alg. Prop. Alg. Ahn’s Alg. Prop. Alg. Ahn’s Alg. Prop. Alg.

20/49 142.00 142.00 142.00 40.60 23.37 2 9

80/120 389.00 389.00 389.00 118.50 96.80 4 4

80/632 291.00 291.00 291.00 109.50 118.50 19 10

160/2544 284.00 286.20 284.00 336.20 490.50 31 26320/1845 394.00 403.40 394.00 779.80 1062.50 44 11

320/10208 288.00 288.90 288.00 1028.30 1498.50 38 26

20/49 80/120 80/630 160/2544 320/1845 320/10208

Bestsolutions

Problem size

Table 6.2 Performance comparisons with Ahn’s algorithm and Proposed algorithm.


1 5 N i l E l





Different Parameter Settings:

Parameter Settings

( popSize / pC : pM )

Test Problems


OptimalSolutions

Best Solutions CPU Times ( ms ) Generation Num. of Obtained best result

Ahn’s Alg. Prop. Alg. Ahn’s Alg. Prop. Alg. Ahn’s Alg. Prop. Alg.

10 / 0.3 : 0.1 20/49 142.00 156.20 142.00 10.42 8.37 38 27

80/120 389.00 389.00 389.00 32.80 31.10 5 1

80/632 291.00 313.20 291.00 29.40 34.40 43 16

160/2544 284.00 320.90 284.20 67.10 106.30 48 37

320/1845 394.00 478.70 394.00 120.30 250.20 68 18

320/10208 288.00 444.00 288.30 126.40 400.20 25 59

20 / 0.3 : 0.1 20/49 142.00 145.23 142.00 22.36 13.34 27 24

80/120 389.00 389.00 389.00 56.30 51.50 4 1

80/632 291.00 303.10 291.00 50.10 56.30 18 10

160/2544 284.00 298.70 284.20 122.10 181.20 44 35

320/1845 394.00 465.70 394.00 213.90 496.70 32 17

320/10208 288.00 373.10 288.60 311.00 631.10 61 35

20 / 0.7 : 0.5 20/49 142.00 142.00 142.00 40.60 23.37 6 9

80/120 389.00 389.00 389.00 118.50 96.80 1 1

80/632 291.00 291.00 291.00 109.50 118.50 19 10

160/2544 284.00 286.20 284.00 336.20 490.50 31 26

320/1845 394.00 403.40 394.00 779.80 1062.50 44 11

320/10208 288.00 288.90 288.00 1028.30 1498.50 38 26

Table 6.3 Performance comparisons with different parameter settings

1 5 N i l E l




Different Parameter Settings with Ahn’s algorithm and Proposed algorithm

Parameter Settings

( popSize / pC : pM )

Probability of obtaining the optimal solutions

Ahn’s Alg. Proposed Alg.

10 / 0.3 : 0.1 16.67% 66.67%

20 / 0.3 : 0.1 16.67% 66.67%

30 / 0.3 : 0.1 33.33% 83.33%

50 / 0.3 : 0.1 50.00% 100.00%

100 / 0.3 : 0.1 33.33% 100.00%

10 / 0.7 : 0.5 33.33% 83.33%20 / 0.7 : 0.5 50.00% 100.00%

30 / 0.7 : 0.5 50.00% 100.00%

50 / 0.7 : 0.5 83.33% 100.00%

100 / 0.7 : 0.5 83.33% 100.00%


1 5 N i l E l




1.5 Numerical Examples Simulation (# of nodes: 100, # of arcs: 859)

6 Network Design Problems

http://opt/scribd/conversion/tmp/scratch22662/C:/Documents%20and%20Settings/waseda/??????\GAohp-LastVer\Program\Chap6-NetDesign\run.bat






2. Maximum Flow (MXF) Problem2.1 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 Representation2.4.2 Genetic Operators





2 Maximum Flow (MXF) 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.

i j uij

1 2 60

1 3 601 4 60

2 3 30

2 5 40

2 6 30

3 4 30

3 6 50

3 7 30

4 7 40

5 8 60

6 5 20

6 8 30

6 9 40

6 10 30

7 6 20

7 10 40

8 9 30

8 11 60

9 10 30

9 11 50

10 11 50


i juij

2.1 Basic Concept of Maximum Flow Problem

60

1

2

3

4

5

6

7

8

9

10

11

s t60

60

30

30

20

20

30

30

60

50

50

40

30

50

30

40

60

30

40

30

40

f f








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


i juij

i j uij

1 2 60

1 3 601 4 60

2 3 30

2 5 40

2 6 30

3 4 30

3 6 50

3 7 30

4 7 40

5 8 60

6 5 20

6 8 30

6 9 40

6 10 30

7 6 20

7 10 40

8 9 30

8 11 60

9 10 30

9 11 50

10 11 50


60

1

2

3

4

5

6

7

8

9

10

11

s t60

60

30

30

20

20

30

30

60

50

50

40

30

50

30

40

60

30

40

30

40

f f







MXF problem can be formulated as follows:

f z =max

−

=−∑ ∑= =

f

f

x xn

j

n

k

kiij 0 t.s.1 1

)(

)1,,3,2(

)1(

ni

ni

i

=−=

=

A jiu x ijij ∈≤≤ ),(,0

0≥ f






2.2 Application of Maximum Flow Problem

This basic MXF model can be applied in many applicationssuch as:

Ahuj, R. K., T. L. Magnanti & J. B. Orlin: Network Flows. Prentice-Hall, Upper Saddle River, NJ, 1993.

Scheduling on Uniform Parallel Machines 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 Distributed computing on a two-processor computer concerns assigning

different modules (subroutines) of a program to two processors in a way thatminimizes the collective costs of interprocessor communication and

computation..

Tanker Scheduling Problem A steamship company has contracted to deliver perishable goods between

several different origin-destination pairs. Since the cargo is perishable, thecustomers have specified precise dates (i.e., delivery dates) when theshipments must reach their destinations..






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 inthe network.






Munakata, T. and D. J. Hashier : “A genetic algorithm applied to the maximum flowproblem,” 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.,

it has more "freedom" to choose.

In many other problems, selecting an edge may mean to simply add a fixed

distance.

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.


2 4 Genetic Approach for solving MXF Problem





2.4.1 Genetic Representation

procedure 1: Priority-based Encodinginput: number of nodes n


begin

for j=1 to n // step 0

v k ( j )← j ;


repeat

j ←random[1, n];

l ←random[1, n];

until l ≠ j

swap (v k ( j ), v

k (l ));


end

procedure 1: Priority-based Encodinginput: number of nodes n


begin


v k ( j )← j ;


repeat

j ←random[1, n];

l ←random[1, n];

until l ≠ j

swap (v k ( j ), v

k (l ));


end

2/n






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 Growthinput: number of nodes n, chromosome v

k ,

the set of nodes S i with all nodes adjacent to node i.

output: path P k

step 0: the source node i ←1, P k ←φ

step 1: if S i =φ , goto step 3; otherwise, continue.

step 2: select l from S i with the highest priority, and go back to step 1.

if v k (l )≠0 then

v k (l )=0;

P k ←P

k ∪{ x

il };

i ←l ;

else v k (l )=0

step 3: output the complete path P k ;

}...,,,,{ ,,,1 132211 mm l l l l l l l k x x x x P

−

=






The decoding procedure is a two-stage process.

Second stage: overall paths are generated by overall pathsgrowth procedure

For a given path, we can calculate its flow f k

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.






procedure 3: Overall-path Growth

input: network data (V, A, U ), chromosome v k , the set of nodes S

i with all nodes adjacent to node i

output: number of paths Lk , the flow f i k of each path, i ∈Lk

step 0: number of paths l ←0

step 1: if S 1=φ , go to step 7; otherwise, l ← l +1, continue.

step 2: the implementation of path P l k growth is based on procedure 2. Select the sink node a of path p

l k .

step 3: if the sink node a=n, continue; otherwise, perform the set of nodes S i update as follows, return to

step 1.

step 4: calculate the flow f l k of the path P

l k .

step 5: perform the flow capacity u ij of each arc update. Make a new flow capacity ũ ij as follows:

step 6: if the flow capacity ũ ij =0, perform the set of nodes S

i update which the node j adjacent to node i .

step 7: output number of paths Lk ← l -1, the flow f

i k of each path, i ∈L

k .

iaS S ii ∀−← },{

}),(|min{1 lk ij

k

l

k

l P jiu f f ∈+← −

}),(min{~ k l ijijij P jiuuu ∈−←

0~&),( ,}{ =∈−← ij

k

l ii u P ji j s s







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 j uij

1 2 60

1 3 601 4 60

2 3 30

2 5 40

2 6 30

3 4 30

3 6 50

3 7 30

4 7 40

5 8 60

6 5 20

6 8 30

6 9 40

6 10 30

7 6 20

7 10 40

8 9 30

8 11 60

9 10 30

9 11 50

10 11 50


60

1

2

3

4

5

6

7

8

9

10

11

s t60

60

30

30

20

20

30

30

60

50

50

40

30

50

30

40

60

30

40

30

40

Chromosome:

f f






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

Chromosome:

60

s

360

s

3

f

k i S i l P k S 1 f k

1 0 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 2, 3, 4 20

2 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 2, 3, 4 50

k : number of paths

i : start node

S i : the set of nodes

l : sink node

P k : the k th pathS 1 : the set of nodes with all nodes

adjacent to node 1

f k : maximum possible flow







Chromosome:

60

s

360

s

3

f


3 0 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 2, 4 60

4 0 11 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 2,4 70

k : number of paths

i : start node


l : sink node


adjacent to node 1


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







Chromosome:

60

s

360

s

3

f


5 0 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 2 100

6 0 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 2 110

k : number of paths

i : start node


l : sink node


adjacent to node 1


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







Chromosome:

60

s

360

s

3

f


7 0 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 2 140

8 0 11 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, 11 160

k : number of paths

i : start node


l : sink node


adjacent to node 1


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








Chromosome:

i j uij

1 2 60

1 3 60

1 4 60

2 3 30

2 5 40

2 6 30

3 4 30

3 6 50

3 7 30

4 7 40

5 8 60

6 5 20

6 8 30

6 9 40

6 10 307 6 20

7 10 40

8 9 30

8 11 60

9 10 30

9 11 50

10 11 50


60/60

1

2

3

4

5

6

7

8

9

10

11

s t60/60

40/60

20/20

20/20

30/30

20/30

60/60

50/50

50/50

40/40

20/30

50/50

10/3040/40

60/60

30/30

40/40

30/40

160 160

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 j xij / uij






Mutation: The swap mutation operator was used here, in which twopositions are selected at random and their contents are swapped

as follows:

Selection: The roulette wheel approach, a type of fitness-

proportional selection, was adopted.

8 parent : 5643271 8 parent : 5643271

8offspring : 5243671 8offspring : 5243671

exchanging points

2.4.2 Genetic Operators





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 themaximum flow problem,” Proc. of 5th Int. Conf. on Genetic Algorithms,pp. 488-493, 1993.






All the simulations were performed with Java onPentium 4 processor (1.5-GHz clock).





2.5 Numerical ExamplesTest Problem 1:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25 f

20

20

20

20

20

20

20

20

20

20

20

20

20

20

10

10

10

10

10

10

10

10

8

8

8

8

8

25

25

25

25

25

20

15

15

15

15

15

15

15

15

6

5

4

5

4

30

30

30

f

The first numerical example, presented by Munakata & Hashier , was

adopted. The problem comprises 25 nodes and 49 arcs. It is given asfollows:

i juij




2.5 Numerical Examples Process of Genetic Computing

f l o w




i juij

Objective function value: z=91 (optimal solution)

Generation Num. of Obtained best result: 67

Best Chromosome:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

2591 91

18/20

20/20

20/20

20/20

13/20

10/10

8/8

10/10

6/6

4/8

5/5

7/9

8/8

7/7

10/10

3/7

7/12

6/15

18/18

9/12

15/20

7/22

3/6

8/11

2/2

4/5

6/10

4/8

6/10

1/10

6/6

4/5

9/10

7/8

9/9

5/7

8/15

8/8

8/8

6/6

5/9

15/15

20/20

22/30

20/20

14/15

node ID : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

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





g




3.1 Basic Concept of MCF Problem

3.2 Application of MCF Problem

3.3 Methods for solving MCF Problem3.4 Genetic Approach for solving MCF Problem





5. Multicriteria Network Design Problem




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


i j cij

uij

1 2 18 60

1 3 19 601 4 17 60

2 3 13 30

2 5 16 40

2 6 14 30

3 4 15 30

3 6 16 50

3 7 17 304 7 19 40

5 8 19 60

6 5 15 20

6 8 16 30

6 9 15 40

6 10 18 30

7 6 15 20

7 10 13 40

8 9 17 30

8 11 18 60

9 10 14 30

9 11 19 50

10 11 17 50


i jcij , uij

18, 60

1

2

3

4

5

6

7

8

9

10

11

s t19, 60

17, 60

13, 30

15, 30

15, 20

15, 20

17, 30

14, 30

18, 60

17, 50

19, 50

16, 40

14, 30

16, 50

17, 30

19, 40

19, 60

16, 30

15, 40

18, 30

13, 40

q q

( )





( )

3.1 Basic Concept of MCF Problem

MCF problem can be formulated as follows:

∑∑= =

=n

i

n

j

ijij xc z 1 1

min

−

=−∑ ∑= =

q

q

x xn

i

n

k

kiij 0 t.s.1 1

)(

)1,,3,2(

)1(

ni

ni

i

=

−=

=

A jiu x ijij ∈≤≤ ),(,0

q: total flow valueq: total flow value





( )

3.2 Application of Minimum Cost Flow (MCF) Problem

This basic MCF model can be applied in many applicationssuch 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-unitcost on each arc. Each source has a supply of material, and eachdestination 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.





( )

3.3 Methods for solving MCF ProblemAhuj, 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.



3.4 Genetic Approach for solving MCF Problem




step 4: calculate the flow f l k and the cost c

l k of the path P

l k .

step 5: perform the flow capacity u ij

of each arc update. Make a new

flow capacity ũ ij as follows:

step 6: if the flow capacity ũ ij =0, perform the set of nodes S

i update

which the node j adjacent to node i .

step 7: output number of paths Lk ← l -1, the flow f

i k and the cost c

i k of

each path, i ∈Lk

.

k

l

n

i

n

j

k

l

k

l ij

k

l

k

l

lk ijk l k l

P ji f f ccc

P jiu f f

∈∀−+←

∈+←

∑∑= =

−−

−

),(),(

}),(|min{

1 1

11

1

}),(min{~ k

l ijijij P jiuuu ∈−←

0~&),( ,}{ =∈−← ijk

l ii u P ji j s s





Illustration of Priority-based GA i j cij

uij

1 2 18 60

1 3 19 601 4 17 60

2 3 13 30

2 5 16 40

2 6 14 30

3 4 15 30

3 6 16 50

3 7 17 304 7 19 40

5 8 19 60

6 5 15 20

6 8 16 30

6 9 15 40

6 10 18 30

7 6 15 20

7 10 13 40

8 9 17 30

8 11 18 60

9 10 14 30

9 11 19 50

10 11 17 50


i jcij , uij

18, 60

1

2

3

4

5

6

7

8

9

10

11

s t19, 60

17, 60

13, 30

15, 30

15, 20

15, 20

17, 30

14, 30

18, 60

17, 50

19, 50

16, 40

14, 30

16, 50

17, 30

19, 40

19, 60

16, 30

15, 40

18, 30

13, 40

q q

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

Chromosome:






k i S i l P k S 1 f l k cl

k

1 0 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 2, 3, 4 20 87

2 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 2, 3, 4 30 69

k : number of paths

i : start node


l : sink node

P k : the k th path

S 1 : the set of nodes with all nodes

adjacent to node 1

f l k : the total flow

cl k : minimum possible cost

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

Chromosome:






k i S i l P k S 1 f l k cl

k

3 0 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 2, 4 10 85

k : number of paths

i : start node


l : sink node

P k : the k th path


adjacent to node 1

f l k : the total flow

cl k : minimum possible cost

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

Chromosome:





step 1 : select the substring at random

step 2 : exchange substrings between

step 3 : determine mapping relationship

parent 1: 1 7 2 3 4 6 5 8

parent 2: 4 6 3 5 7 1 8 2

substring selected

parent 1: 1 7 3 5 7 6 5 8

parent 2: 4 6 2 3 4 1 8 2

2 3 4

3 5 7 74

532

↔

↔↔

offspring 1: 1 4 3 5 7 6 2 8

offspring 2: 7 6 2 3 4 1 8 5

• 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 4 : legalize offspring with mapping

relationship





Mutation: The swap mutation operator was used here, in which twopositions are selected at random and their contents are swapped

as follows:

Selection: The roulette wheel approach, a type of fitness-

proportional selection, was adopted.

8 parent : 5643271 8 parent : 5643271

8offspring : 5243671 8offspring : 5243671

exchanging points






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

GA Procedure for solving MCF Problem





Test Problems: The numerical examples, presented by Munakata &

Hashier , was adopted.

Munakata, T. and D. J. Hashier : “A genetic algorithm applied to themaximum flow problem,” Proc. of 5th Int. Conf. on Genetic Algorithms,pp. 488-493, 1993.






All the simulations were performed with Java onPentium 4 processor (1.5-GHz clock).





Test Problem 1:



1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25q

10, 20

13, 20

32, 20

135, 20

631, 20

8, 20

6, 20

7, 20

7, 20

7, 20

5, 20

14, 20

4, 20

14, 20

10, 10

35, 10

3, 10

33, 10

7, 10

7, 10

3, 10

10, 10

13, 8

15, 8

11, 8

5, 8

3, 8

11, 25

8, 25

35, 25

14, 25

12, 25

34, 20

10, 15

4, 15

9, 15

11, 15

12, 15

9, 15

14, 15

5, 15

10, 6

15, 533, 4

4, 5

13, 4

10, 30

2, 30

3, 30

q

i jcij , uij





Total flow value q = 70

Objective function value: z= 6969

Generation Num. of Obtained best result:863

Best Chromosome:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25q=70

10, 18/20

13, 14/20

32, 20/20

135, 14/20

631, 4/20

6, 10/20

7, 20/20

7, 18/20

14, 7/20

4, 15/20

10, 10/10

35, 10/10

3, 10/10

7, 4/10

3, 10/10

10, 10/10

13, 8/8

15, 8/8

3, 8/8

11, 10/25

8, 20/25

14, 20/25

10, 15/15

9, 15/15

11, 15/15

12, 7/15

9, 3/15

14, 15/15

10, 5/6

15, 5/5

33, 4/4

4, 2/5

13, 4/4

10, 30/30

2, 20/30

3, 20/30

q=70

node ID : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

priority: 1 16 11 9 6 5 7 8 15 10 3 12 13 21 4 22 14 18 20 24 17 25 23 2 19



3 10/10 29 10/11





Total flow value q = 72

Objective function value: z=5986

Generation Num. of Obtained best result:132

Best Chromosome:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

2, 18/20

10, 14/20

33, 20/20

126, 20/20

3, 10/10

14, 8/8

13, 6/6

32, 8/8

7, 3/5

12, 9/9

33, 8/8

4, 7/7

12, 10/10

30, 3/7

7, 8/18

6, 6/12

8, 15/20

3, 5/5

11,2 /22

6, 6/6

29, 10/11

30, 6/10

34,9/10

2, 3/9

35, 6/6

14, 2/5

8, 6/10

26, 8/9

12, 6/7

31, 3/15

8,4/ 4

30, 8/8

11, 4/6

2,10/15

10, 20/20

6, 16/30

7, 20/20

9, 6/15

node ID : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

priority: 10 25 15 24 11 4 7 8 12 6 5 9 13 14 3 16 17 18 1 20 22 19 23 21 2

q=72q=72





Process of Genetic Computing

5900

6000

6100

6200

6300

6400

6500

6600

6700

6800

6900

7000

0 50 100 150 200 250

generation

c o s t


http://opt/scribd/conversion/tmp/scratch22662/C:/Documents%20and%20Settings/lin/??????\Program\Chap6-NetDesign\run.bat




Simulation (# of nodes: 80, # of arcs: 857)







2. Maximum Flow (MXF) Problem3. Minimum Cost Flow (MCF) Problem


4.1 Introduction of BNP

4.2 BNP Formulation4.3 Genetic Approach for solving BNP


4.3.2 Decoding Method

4.3.3 Fitness Assignment







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 networkperformance. 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.

In a manufacturing system, the two criteria under consideration areminimizing cost and maximizing manufacturing. 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 theenhancement of customer services and asset utilization through a significantreduction 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.


http://techreports.isr.umd.edu/ARCHIVE/

http://techreports.isr.umd.edu/ARCHIVE/




The Bicriteria Network Design Problem (BNP) is known as NP-hard (Gareyand Johnson, 1979), it is not simply an extension from single objective totwo 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 theseobjectives usually conflict with each other in practice. The real solutions to theproblem are a set of Pareto optimal solutions (Chankong and Haimes, 1983). Chankong, V. and Y.Y. Haimes: Multiobjective Decision Making Theory and

Methodology. North-Holland, New York, 1983. For solving the BNP, the set of efficient paths may be very large and possibly

exponential in size. 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 toa decision maker.

Having to evaluate a large efficient set in order to select the best one poses aconsiderable cognitive burden on decision makers. Therefore, in such cases,obtaining the entire Pareto optimal set is of little interest to decision makers.




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.




In this study, we dominated BNP with more complexity cases as two criteriaproblem that maximum flow and minimum cost considered. Priority-based encoding method (Cheng and Gen, 1994) has been improved.

Cheng, R. and M. Gen: “Evolution program for resource constrained projectscheduling 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 canrepresent various efficient paths by each chromosome.

Considering the characteristic of priority-based encoding method, we proposeda new crossover operator called as Weight Mapping Crossover (WMX)

Insertion mutation operator and Immigration operator (Michael et al., 1991)was adopted. 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.

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 z 1 and minimum cost z

2 in the same time, from

a source to a sink.

0

),(, 0

)(

)1,,3,2(0

)1(

t.s.

min

max

1 1

1 1

2

1

≥

∈∀≤≤

=−

−=

=

=−

=

=

∑ ∑

∑∑

= =

= =

f

A jiu x

ni f

ni

i f

x x

xc z

f z

ijij

n

j

n

k

kiij

n

i

n

j

ijij





Priority-based encoding method




begin


v k ( j )← j ;for i=1 to // step 1

repeat

j ←random[1, n];

l ←random[1, n];

until l ≠ j

swap (v k ( j ), v

k (l ));


end




begin


v k ( j )← j ;for i=1 to // step 1

repeat

j ←random[1, n];

l ←random[1, n];

until l ≠ j

swap (v k ( j ), v

k (l ));


end

2/n



4.3.2 Decoding Method




procedure 3: Overall-path Growth

input: network data (V, A, C, U ), chromosome v k

, the set of nodes S i

with all nodes adjacent to node i

output: number of paths Lk

, the flow f i k and the cost c

i k of each path,

i ∈Lk

step 0: number of paths l ←0step 1: if S

1=φ , go to step 7; otherwise, l ← l +1, continue.

step 2: the implementation of path P l k growth is based on procedure 2.

Select the sink node a of path pl k .

step 3: if the sink node a=n, continue; otherwise, perform the set of

nodes S i update as follows, return to step 1.iaS S ii ∀−← },{



Illustration of Decoding Method




k i S i l P k S 1 z 1k z 2

k

7 0 11 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 2 140 12790

8 0 1

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, 11 160 14350

k : number of paths

i : start node


l : sink node

P k : the k th path


adjacent to node 1

z 1k : maximum possible flow

z 2k : minimum possible cost

18 6018 60

Chromosome:



4.3.3 Fitness Assignment




step 2: The adaptive weight for objective 1 and objective 2 are calculated by

the following equation:

step 3: Calculate the fitness value for each individual.

minmax

minmax

22

2

11

1

1

1

z z w

z z w

−=

−=

( ) popSizek

L

z cw z f wveval

k

L

i

k

i

k

i

k

k

∈∀+−−

=∑

= ,)()(

)( 1

min

22

min

11



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 parametersoutput: Pareto optimal solution E (t )

begin

t 0;

initialize P (t ) by priority-based encoding;

objectives z 1(P ), z 2(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 z 1(C ), z 2(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

GA Procedure for BNP

4.4 Numerical Examples Test Problems:



Soft Computing Lab.WASEDA UNIVERSITY , IPS

128

Test Problems: The numerical examples, presented by Munakata & Hashier , was

adopted.

Munakata, T. and D. J. Hashier: “A genetic algorithm applied to the maximum flowproblem,” 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



All the simulations were performed with Java on Pentium 4

processor (1.5-GHz clock).

4.4 Numerical ExamplesTest Problem 1:




129

Test Problem 1:



1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25 f

10, 20

13, 20

32, 20

135, 20

631, 20

8, 20

6, 20

7, 20

7, 20

7, 20

5, 20

14, 20

4, 20

14, 20

10, 10

35, 10

3, 10

33, 10

7, 10

7, 10

3, 10

10, 10

13, 8

15, 8

11, 8

5, 8

3, 8

11, 25

8, 25

35, 25

14, 25

12, 25

34, 20

10, 15

4, 15

9, 15

11, 15

12, 15

9, 15

14, 15

5, 15

10, 6

15, 533, 4

4, 5

13, 4

10, 30

2, 30

3, 30

f

i jcij , uij





130

z 1 z 2 z 1 z 2 z 1 z 2

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

Table 6.4 The Pareto optimal solutions of test problem 1

Gen, M., L. Lin & R. Cheng: “Bicriteria Network Optimization Problem using Priority-

based Genetic Algorithm,” IEEJ Trans. on Elect., Info. & Sys., Oct. 2004.

4.4 Numerical Examplesideal point: z 1=90, z 2=300




131Fig. 6.5 The Pareto optimal solutions of test problem 1

p 1 , 2

z 1=66, z 2=6575

0 20 40 60 80 100

flow

cost

0 -

-5000 -

-10000 -

-15000 -

-20000 -

- 25000 -




z 1 z 2 z 1 z 2 z 1 z 2 z 1 z 22 52 32 1633 54 3872 73 6944

8 248 34 1909 55 3990 74 7192

10 340 36 1937 58 4146 75 7402

15 495 38 2077 61 4671 76 7532

18 692 40 2485 63 5153 78 7847

19 1012 41 2581 65 5463 80 9228

20 1111 43 2731 66 5704 82 10395

21 1220 47 3080 67 6323 83 12508

25 1292 49 3302 68 6422 85 12610

26 1406 51 3551 71 6537 86 13151

27 1457 52 3739 72 6748 91 16752

28 1475

Gen, M., L. Lin & R. Cheng: “Bicriteria Network Optimization Problem using Priority-

based Genetic Algorithm,” IEEJ Trans. on Elect., Info. & Sys., Oct. 2004.

Table 6.5 The Pareto optimal solutions of test problem 2





Pareto optimal solutionideal oint

0 20 40 60 80 100

flow

cost

ideal point: z 1=91, z 2=52

z 1=61, z 2=4671

0 -

-2000 -

-4000 -

-6000 -

-8000 -

-10000 -

-12000 -

-14000 -

-16000 -

-18000 -

Fig. 6.6 The Pareto optimal solutions of test problem 2

4.4 Numerical Examples Simulation (# of nodes: 25, # of arcs: 56)






1 Shortest Path Problem (SPP)









5.1 Introduction of Multi-criteria Network Design Problem

5.2 Reviewing Solution Approaches for MNP


5. Multi-criteria Network Design Problem (MNP)

With the information superhighway fast becoming a reality




With the information superhighway fast becoming a reality,the problem of designing networks capable of

accommodating multimedia (both audio and video) traffic ina multicast (simultaneous transmission of data to multipledestinations) environment has come to assume paramountimportance 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 networkcomputers,” 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. 343-351,1983.

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.

5.1 Introduction of MNP

Network design problems where even one cost measure must




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 tominimize 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.


Multi-criteria network design problems, with separate cost functions for




g p , peach optimization criterion, also occur naturally in Information Retrievaland 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-boundedminimum Steiner tree construction,” Technical Report UCSC-CRL-94-39, UCSanta 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 ininterconnect delays thus making them a dominant factor in the timinganalysis of VLSI circuits.

Therefore VLSI circuit designers aim at finding minimum cost(spanning or Steiner) trees given delay bound constraints on source-sink connections.


For example, the problem of finding low-cost and low-transmission-delay




p , p g ymultimedia networks can be modeled as the (Diameter, Total cost,Spanning tree)-bicriteria problem:

given an undirected graph G = (V ,E ) with two weight functions c e and d e for each edge e∊E modeling construction and delay costs respectively, and abound D (on the total delay), find a minimum c -cost spanning tree such thatthe 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 easilyextended to the more general multicriteria optimization problems.

The applications set the stage for the formal definition of multicriterianetwork design problems. Marathe et al. explain this concept by giving aformal 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.


a AWA (Gen et al 1998)




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- (DebⅡ 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.


a AWA (Gen & Cheng 1998)




a. AWA (Gen & Cheng, 1998)

Gen & Cheng (1998) proposed an Adaptive Weight Approach (AWA)

which utilizes some useful information from the current population toreadjust 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 -)

where z k max and z k

min are the maximal and minimal values for the k th objective as

defined by the following equations:

][

][

minmin

2

min

1

maxmax

2

max

1

q

q

z z z

z z z

=

=

−

+

z

z

qk P f z

qk P f z

k k

k k

,,2,1},|)(min{

,,2,1},|)(max{

min

max

=∈=

=∈=

x x

x x

P : set of solution candidates.






a. AWA (Gen & Cheng, 1998) The weighted-sum objective function for a given chromosome x is given by the

following equation:

where w k is adaptive weight for objective k :

The equation driven above is a hyperplane defined by the following extreme points

in current solutions:

∑∑∑=== −

−=

−

−=−=

q

k k k

k k q

k k k

k k q

k

k k k z z

z f

z z

z z z z w z

1

minmax

min

1

minmax

min

1

min )()()(

x x

qk z z

wk k

k ,,2,1,1

minmax=

−=

][minminmin

2

max

1 qk z z z z

][minminmax

2

min

1 qk z z z z

][maxminmin

2

min

1 qk z z z z

][ minmaxmin

2

min

1 qk z z z z






Fig.6.7 Adaptive weights and adaptive hyperplane

+

z

1 z

2 z

min

2 z

max

1 z

max2 z

min

1 z

subspace

corresponding to

current solutions

adaptivemoving line

whole criteria space Z

positive ideal pointminimal rectangle containing

all current solutions

maximumextreme point

minimumextreme point

− z

),(max

2

min

1z z

),(min

2

max

1 z z

( & g, )

Adaptive moving line defined by the extreme points ( z 1

max, z 2

min) and ( z 1

max, z 2

min) are

shown as follows:




b. RWA (Murata et al., 1998)




( , )

For a problem to maximize q objective functions, weighted-sum objective

is given as the follows:

Random-weight wk is calculated by the following equation:

where r j 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:

where z min is the worst fitness value in the current population.

∑=

=q

k

k k f w z 1

)( x

qk

r

r w

q

j j

k k ,,2,1,

1

==

∑=

( )∑ =−

−= popSize

j j

ii

z z z z p

1 min

min


c SPEA (Zitzler et al 1999)




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 updatedpopulation.

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 withoutdestroying its characteristics.






The fitness assignment procedure is a two-stage process.

First, the individuals in the external nondominated set P ’ are ranked.

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 .

1+=

N

n si

f 1

f 2

3/8

5/8

3/8

Fig. 6.9 Two scenarios for a maximization problem with two objectives.






( , )

The fitness assignment procedure is a two-stage process.

Afterwards, the individuals in the population P are evaluated.

where the fitness of an individual j∈ P is calculated by summing the

strengths of all external nondominated solutions i∈ P’ that cover j.

f 1

f 2

3/8

5/8

3/8

Fig. 6.10 Two scenarios for a maximization problem with two objectives.

),1[,1,

N f s f j

jii

i j ∈+= ∑ where

11/8

16/8

19/8

13/813/8

16/8 11/8




d. NSGA- (DebⅡ et al., 2000)




( , )

The new population P t +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:

where, Ri is nondomination rank and Di is crowding distance

( ) ( ) ji

D D R R R R

n

ji ji ji then

andorif )()( >=<




Test Problems:




1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25 f

10, 20

13, 20

32, 20

135, 20

631, 20

8, 20

6, 20

7, 20

7, 20

7, 20

5, 20

14, 20

4, 20

14, 20

10, 10

35, 10

3, 10

33, 10

7, 10

7, 10

3, 10

10, 10

13, 8

15, 8

11, 8

5, 8

3, 8

11, 25

8, 25

35, 25

14, 25

12, 25

34, 20

10, 15

4, 15

9, 15

11, 15

12, 15

9, 15

14, 15

5, 15

10, 6

15, 5

33, 4

4, 5

13, 4

10, 30

2, 30

3, 30

f



i jcij , uij

5.3 Numerical Examples Test Problems:

The second numerical example presented by T Munakata & D J Hashier was adopted




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:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25 f f

2, 20

10, 20

33, 20

126, 20

634, 20

3, 10

14, 8

8, 10

13, 6

32, 8

7, 5

12, 9

33, 8

4, 7

12, 10

30, 7

11, 12

12, 15

3, 5

7, 18

12, 7

4, 6

6, 12

12, 8

9, 7

8, 20

12, 6

3, 5

11, 22

6, 6

29, 11

13, 2

11, 5

30, 10

6, 8

9, 10

34, 10

15, 10

2, 9

35, 6

14, 5

7, 8

8, 10

6, 8

26, 9

12, 7

31, 15

8, 4

30, 8

11, 6

35, 9

2,15

10, 20

6, 30

7, 20

9, 15

i jcij , uij


Reference solution set S* :




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: popSize =30


Mutation probability: pM =0.80 Immigration probability: μ=5

Stopping conditions: Evaluation of 100000 solutions.


Reference solution set S * to test problems:




p

0

5000

10000

15000

20000

0 20 40 60 80 100

flow

c o s

t

Fig. 6.11 The Reference solution set of Example 1 (|S *|=69)


Reference solution set S* to test problems:




Reference solution set S to test problems:

z 1 z 2 z 1 z 2 z 1 z 2 z 1 z 2

5 260 39 2380 62 5339 82 14088

6 318 40 2447 63 5743 83 14841

8 376 41 2531 64 5936 84 15561

10 510 42 2648 65 5962 85 16198

11 590 43 2696 66 6162 86 16842

12 644 45 2866 67 6382 87 17528

13 744 46 3042 68 6714 88 1833216 808 47 3049 69 6846 89 18988

18 918 48 3151 70 6952 90 19597

20 1030 49 3322 71 7182

23 1206 50 3393 72 7364

25 1320 51 3527 73 8043

28 1496 52 3530 74 8707

30 1650 53 3932 75 9375

32 1796 54 4066 76 1005733 1875 55 4072 77 10750

34 1955 56 4364 78 11438

35 2058 57 4479 79 12123

37 2162 58 4747 80 12735

38 2262 60 5010 81 13523

Table 6.6 The Reference solution set of Example 1 (|S *|=69)






0

5000

10000

15000

20000

0 20 40 60 80 100

p

flow

c o s

t

Fig.6.12 The Reference solution set of Example 2 (|S *|=77)






z 1 z 2 z 1 z 2 z 1 z 2 z 1 z 2 2 52 31 1408 55 3525 75 6815

5 115 32 1488 56 3610 76 7081

7 167 33 1595 57 3669 77 7227

8 248 34 1640 58 3732 78 7308

10 275 35 1753 59 3987 79 8002

11 306 36 1818 60 4189 80 8697

13 388 37 1892 61 4458 81 9319

15 495 38 1942 62 4551 82 9997

17 608 40 2171 63 4751 83 10667

18 653 42 2319 64 4867 84 11341

20 784 43 2405 65 5069 85 11949

22 872 44 2587 66 5341 86 12623

23 953 45 2653 67 5517 87 13455

24 965 47 2701 68 5583 88 14116

25 1035 48 2897 69 5809 89 14735

26 1168 49 2974 70 5941 90 15551

27 1217 50 3077 71 6128 91 16100

28 1275 51 3106 72 6319

29 1332 52 3198 73 6487

Table 6.7 The Reference solution set of Example 2 (|S *|=77)


Performance Measures:




We mainly use a performance measure based on: The number of obtained solutions |S j |

The ratio of nondominated solutions R NDS(S j )

The R NDS(S j ) measure can be written as follows:

The distance D1R

The D1Rmeasure can be written as follows:

where S* is a reference solution set for evaluation the solution set S j .

d xr is the distance between a solution x and a reference solution r .

[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.

[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.

{ }

j

j j

j NDS S

xr S r S xS S R

:*)(

∈∃∈−=

∑∈

∈=*

R }min{*

1D1

S r

jrx S xd S

( ) ( )222

2

11 )()()()( x f r f x f r f d rx −+−=





Table 6.8 Comparison with the four approaches using the |S j| measure.

Test Problems(# of nodes/ # of arcs)

|S j | CPU TimesRWA SPEA NSGA-Ⅱ AWA RWA SPEA NSGA-Ⅱ AWA

25/49 52 57 43 49 15122 17635 15693 14170

25/56 43 44 55 43 11918 16684 15981 14961

Table 6.9 Comparison with the four approaches using the RNDS (S j) measure.

Test Problems


RNDS (S j ) CPU Times

RWA SPEA NSGA-Ⅱ AWA RWA SPEA NSGA-Ⅱ AWA

25/49 0.57 0.54 0.39 0.61 15122 17635 15693 14170

25/56 0.41 0.34 0.36 0.53 11918 16684 15981 14961

Test Problems


D1R measure CPU Times

RWA SPEA NSGA-Ⅱ AWA RWA SPEA NSGA-Ⅱ AWA

25/49 191.21 315.61 228.65 143.58 15122 17635 15693 14170

25/56 203.96 224.40 185.89 141.43 11918 16684 15981 14961

Table 6.10 Comparison with the four approaches using the D1R measure.






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 |S j| measure.

Test Problems


|S j |

AWA RWA SPEA NSGA-Ⅱ

25/49 49 50 53 50

25/56 52 50 40 34

Table 6.12 Comparison with the four approaches using the RNDS (S j) measure.

Test Problems


RNDS (S j )


25/49 0.57 0.44 0.56 0.48

25/56 0.51 0.32 0.60 0.41

Test Problems


D1R measure


25/49 191.17 203.72 222.28 239.99

25/56 147.07 219.59 279.60 433.43







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 |S j | measure.

Test Problems


|S j |

RWA SPEA NSGA-Ⅱ AWA

25/49 50 53 50 49

25/56 50 40 34 52

Table 6.12 Comparison with the four approaches using the RNDS (S j ) measure.

Test Problems


RNDS (S j )


25/49 0.44 0.56 0.48 0.57

25/56 0.32 0.60 0.41 0.51

Test Problems


D1R measure


25/49 203.72 222.28 239.99 191.17

25/56 219.59 279.60 433.43 147.07




Conclusion In this study, we presented a GA approach used a priority-based chromosome

for solving the network design problems



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 alsocombines 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

ssp, mst, msf

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