6. Basic Network Design

1
6. Basic Network Design
2
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.

3
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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6
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

7
6. Basic Network Design

• 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

8
1. Shortest Path Problem (SPP)
1.1 Basic Concept of Shortest Path Problem

• SPP is perhaps the simplest of all network design
  problems.
• 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.

Data table of example network
i j cij
1 2 36
1 3 27
2 4 18
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
cij
i
j
9
1. Shortest Path Problem (SPP)

• 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

Data table of example network
i j cij
1 2 36
1 3 27
2 4 18
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
cij
i
j
10
1. Shortest Path Problem (SPP)

• 1.1 Basic Concept of Shortest Path Problem
• SPP can be formulated as follows

11
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
• 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 networks performance.

12
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.299333, 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.

13
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.
• 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.

14
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.

15
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.

• Path 1 1?2?4?8?10
• Objective function value z110
• Path 2 1?2?4?7?8?10
• Objective function value z109
• Path 3 1?3?5?4?7?8?10
• Objective function value z110

16
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
• 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 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.

17
1.4.1 Reviewing Encoding Methods

• a. 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.

18
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
• 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.

node ID 1 2 3 4 5 6 7
priority 2 1 6 4 5 3 7
2
5
s
t
1
1
1
4
7
path
3
6
1
4
3
7
19
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.

20
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
• 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.

locus 1 2 3 4
node ID 1 3 4 7
2
5
s
t
1
1
1
4
7
3
6
path
1
4
3
7
21
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.

22
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
• 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.

locus 1 2 3 4 5 6 7
node ID 3 1 4 7 2 4 6
2
5
s
t
1
1
1
4
7
3
6
path
3
1
4
7
23
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.

24
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 j1 to n // step 0 vk(j)? j for i1
to // step 1 repeat j?random1,
n l?random1, n until l?j
swap (vk(j), vk(l)) output the chromosome
vk // step 2 end
25
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
26
1.4.2 Priority-based Genetic Algorithm

• Illustration of Priority-based GA

Data table of example network
i j cij
1 2 36
1 3 27
2 4 18
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
1
1
Chromosome
Path 1?3?6?7?8?10 Objective function value
z106
27
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.

28
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
2 3 4
parent 1 1 7 2 3 4 6 5 8
3 5 7
parent 2 4 6 3 5 7 1 8 2
step 2 exchange substrings between
step 4 legalize offspring with mapping
relationship
parent 1 1 7 3 5 7 6 5 8
offspring 1 1 4 3 5 7 6 2 8
parent 2 4 6 2 3 4 1 8 2
offspring 2 7 6 2 3 4 1 8 5
29
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.
30
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.
31
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 .

32
1.4.3 Genetic Operators --- Crossover

• Weight Mapping Crossover (WMX)

33
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
parent 1
cut-point
1
4
3
7
6
3
5
4
7
1
2
parent 1
parent 2
1
4
2
7
5
4
1
5
6
2
7
3
parent 2
step 2 mapping the weight of the right segment
6
5
3
offspring 1
1
4
3
7
5
6
3
5
4
1
5
offspring 2
5
4
1
1
4
2
7
step 3 generate offspring with mapping
relationship
5
3
6
4
7
1
2
offspring 1
5
1
4
6
2
7
3
offspring 2
34
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

35
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.

36
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.

37
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.

38
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(t1) from P(t)
and C(t) by roulette wheel selection t ? t
1 end output best shortest path end
39
1.5 Numerical Examples

• Test Problems
• For examining the effect of different encoding
  methods, we applied Ahn et als 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 µ3
• Maximum generation maxGen 1000
• Terminating condition 100 generations with same
  fitness.
• Each solution is compared with Dijkstras
  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).

40
1.5 Numerical Examples

• The first numerical example, presented by Ahn et
  als 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.566-579, 2002.)

Fig.6.3 Example of the first numerical example
41
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.)

2.5
Dijkstras Algorithm Munemotos
Algorithm Inagakis Algorithm Ahns Algorithm
2.0
1.5
Objective Function Values
1.0
0.5
0
2
4
6
8
10
Generations
42
1.5 Numerical Examples

• Convergence property of Ahn et al.s algorithm
  and proposed algorithm for a Fixed Network With
  20 Nodes

et al.
Fig. 6.4 Convergence property of Ahn et al.s
algorithm and proposed algorithm.
43
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.)

Inagakis Algorithm Munemotos Algorithm Ahns
Algorithm
1200
1000
800
Objective Function Value
600
400
200
0
15
20
25
30
35
40
45
50
The Number of Nodes
44
1.5 Numerical Examples

• 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 (WMXInsertion 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 (WMXSwap 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.

Table 6.1 Performance comparisons with different
genetic operators
Test Problems ( of nodes/ of arcs) Optimal Solutions Best Solutions Best Solutions Best Solutions Best Solutions Best Solutions Best Solutions
Test Problems ( of nodes/ of arcs) Optimal 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 284.00 429.55 454.98 480.19 382.48 284.69 284.00
320/1845 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 FMXSwap Alg. 2 OXSwap Alg. 3 PXSwap Alg. 4 WMXSwap Alg. 5 WMXSwapImmigration(3) Alg. 6 WMXInsertionImmigration(3). Alg. 1 FMXSwap Alg. 2 OXSwap Alg. 3 PXSwap Alg. 4 WMXSwap Alg. 5 WMXSwapImmigration(3) Alg. 6 WMXInsertionImmigration(3). Alg. 1 FMXSwap Alg. 2 OXSwap Alg. 3 PXSwap Alg. 4 WMXSwap Alg. 5 WMXSwapImmigration(3) Alg. 6 WMXInsertionImmigration(3). Alg. 1 FMXSwap Alg. 2 OXSwap Alg. 3 PXSwap Alg. 4 WMXSwap Alg. 5 WMXSwapImmigration(3) Alg. 6 WMXInsertionImmigration(3). Alg. 1 FMXSwap Alg. 2 OXSwap Alg. 3 PXSwap Alg. 4 WMXSwap Alg. 5 WMXSwapImmigration(3) Alg. 6 WMXInsertionImmigration(3). Alg. 1 FMXSwap Alg. 2 OXSwap Alg. 3 PXSwap Alg. 4 WMXSwap Alg. 5 WMXSwapImmigration(3) Alg. 6 WMXInsertionImmigration(3). Alg. 1 FMXSwap Alg. 2 OXSwap Alg. 3 PXSwap Alg. 4 WMXSwap Alg. 5 WMXSwapImmigration(3) Alg. 6 WMXInsertionImmigration(3). Alg. 1 FMXSwap Alg. 2 OXSwap Alg. 3 PXSwap Alg. 4 WMXSwap Alg. 5 WMXSwapImmigration(3) Alg. 6 WMXInsertionImmigration(3).
45
1.5 Numerical Examples

• Comparison results of Ahns algorithm and
  Proposed algorithm

Table 6.2 Performance comparisons with Ahns
algorithm and Proposed algorithm.
Test Problems ( of nodes/ of arcs) Optimal Solutions Best Solutions Best Solutions CPU Times (ms) CPU Times (ms) Generation Num. of Obtained best result Generation Num. of Obtained best result
Test Problems ( of nodes/ of arcs) Optimal Solutions Ahns Alg. Prop. Alg. Ahns Alg. Prop. Alg. Ahns 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 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
46
1.5 Numerical Examples

• Different Parameter Settings

Table 6.3 Performance comparisons with different
parameter settings
Parameter Settings ( popSize / pC pM ) Test Problems ( of nodes/ of arcs) Optimal Solutions Best Solutions Best Solutions CPU Times ( ms ) CPU Times ( ms ) Generation Num. of Obtained best result Generation Num. of Obtained best result
Parameter Settings ( popSize / pC pM ) Test Problems ( of nodes/ of arcs) Optimal Solutions Ahns Alg. Prop. Alg. Ahns Alg. Prop. Alg. Ahns Alg. Prop. Alg.
10 / 0.3 0.1 20/49 142.00 156.20 142.00 10.42 8.37 38 27
10 / 0.3 0.1 80/120 389.00 389.00 389.00 32.80 31.10 5 1
10 / 0.3 0.1 80/632 291.00 313.20 291.00 29.40 34.40 43 16
10 / 0.3 0.1 160/2544 284.00 320.90 284.20 67.10 106.30 48 37
10 / 0.3 0.1 320/1845 394.00 478.70 394.00 120.30 250.20 68 18
10 / 0.3 0.1 320/10208 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
20 / 0.3 0.1 80/120 389.00 389.00 389.00 56.30 51.50 4 1
20 / 0.3 0.1 80/632 291.00 303.10 291.00 50.10 56.30 18 10
20 / 0.3 0.1 160/2544 284.00 298.70 284.20 122.10 181.20 44 35
20 / 0.3 0.1 320/1845 394.00 465.70 394.00 213.90 496.70 32 17
20 / 0.3 0.1 320/10208 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
20 / 0.7 0.5 80/120 389.00 389.00 389.00 118.50 96.80 1 1
20 / 0.7 0.5 80/632 291.00 291.00 291.00 109.50 118.50 19 10
20 / 0.7 0.5 160/2544 284.00 286.20 284.00 336.20 490.50 31 26
20 / 0.7 0.5 320/1845 394.00 403.40 394.00 779.80 1062.50 44 11
20 / 0.7 0.5 320/10208 288.00 288.90 288.00 1028.30 1498.50 38 26
47
1.5 Numerical Examples

• Different Parameter Settings with Ahns algorithm
  and Proposed algorithm

Parameter Settings ( popSize / pC pM ) Probability of obtaining the optimal solutions Probability of obtaining the optimal solutions
Parameter Settings ( popSize / pC pM ) Ahns 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
48
1.5 Numerical Examples

• Simulation ( of nodes 100, of arcs 859)

49
6. Basic Network Design

• Shortest Path Problem (SPP)
• 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
• Minimum Cost Flow (MCF) Problem
• Bicriteria Network Design Problem (BNP)
• Multi-criteria Network Design Problem

50
2. Maximum Flow (MXF) Problem
Data table of example network

• Online. Available http//www-b2.is.tokushima-u.
  ac.jp/
• ikeda/suuri/maxflow/Maxfl
  ow.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.

2.1 Basic Concept of Maximum Flow Problem
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 30
7 6 20
7 10 40
8 9 30
8 11 60
9 10 30
9 11 50
10 11 50
51
2. Maximum Flow (MXF) Problem
Data table of example network

• 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

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 30
7 6 20
7 10 40
8 9 30
8 11 60
9 10 30
9 11 50
10 11 50
uij
i
j
52
2. Maximum Flow (MXF) Problem

• 2.1 Basic Concept of Maximum Flow Problem
• MXF problem can be formulated as follows

53
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
• 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 that minimizes 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, the customers have specified precise
  dates (i.e., delivery dates) when the shipments
  must reach their destinations..

54
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.
  

55
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., 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.

56
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 j1 to n // step 0 vk(j)? j for i1
to // step 1 repeat j?random1,
n l?random1, n until l?j
swap (vk(j), vk(l)) output the chromosome
vk // step 2 end
57
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.

58
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 uij.
• 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.

59
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 an,
continue otherwise, perform the set of nodes Si
update as follows, return to step 1. step 4
calculate the flow flk of the path Plk. step 5
perform the flow capacity uij of each arc update.
Make a new flow capacity uij as follows step
6 if the flow capacity uij0, 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 .
60
2.4 Genetic Approach for solving MXF Problem
Data table of example network

• Illustration of Priority-based GA

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 30
7 6 20
7 10 40
8 9 30
8 11 60
9 10 30
9 11 50
10 11 50
f
f
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
61
2.4 Genetic Approach for solving MXF Problem

• Illustration of Priority-based GA

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
k i Si l Pk S1 fk
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 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
62
2.4 Genetic Approach for solving MXF Problem

• Illustration of Priority-based GA

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
k i Si l Pk S1 fk
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 1
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 2,4 70
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
63
2.4 Genetic Approach for solving MXF Problem

• Illustration of Priority-based GA

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
k i Si l Pk S1 fk
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 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
64
2.4 Genetic Approach for solving MXF Problem

• Illustration of Priority-based GA

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
k i Si l Pk S1 fk
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 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
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
65
2.4 Genetic Approach for solving MXF Problem
Data table of example network

• Illustration of Priority-based GA

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 30
7 6 20
7 10 40
8 9 30
8 11 60
9 10 30
9 11 50
10 11 50
40/40
60/60
5
8
2
60/60
60/60
30/30
20/20
30/30
20/30
s
t
160
160
60/60
50/50
6
9
1
3
11
50/50
40/40
20/30
10/30
20/20
50/50
40/60
40/40
30/40
7
10
4
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 z160
66
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
2 3 4
parent 1 1 7 2 3 4 6 5 8
3 5 7
parent 2 4 6 3 5 7 1 8 2
step 4 legalize offspring with mapping
relationship
step 2 exchange substrings between
parent 1 1 7 3 5 7 6 5 8
offspring 1 1 4 3 5 7 6 2 8
parent 2 4 6 2 3 4 1 8 2
offspring 2 7 6 2 3 4 1 8 5
67
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
• Selection The roulette wheel approach, a type of
  fitness-proportional selection, was adopted.

68
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(t1) from P(t) and C(t) by
roulette wheel selection t ? t
1 end output best maximum flow end
69
2.5 Numerical Examples

• Test Problems
• The numerical examples, presented by T. Munak