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Intelligent Network Design with Genetic Algorithms

Explore the application of genetic algorithms in solving network design problems such as shortest path, maximum flow, minimum cost flow, and multi-criteria design problems. Learn the basic concepts and methods for solving these problems through numerical examples.

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Intelligent Network Design with Genetic Algorithms

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  1. 智能控制与系统专题 主讲老师：龚涛 信息科学与技术学院 2018年10月 主教材：《基于正常模型的人工免疫系统及其应用》 参考教材：《人工智能及其应用》(第3版，研究生用书) 网络课程： http://isc.ytxxchina.com http://taogong.ytxxchina.com 进化算法的网络设计应用

  2. 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. 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. Basic Network Design • Shortest Path Problem (SPP) • Maximum Flow (MXF) Problem • Minimum Cost Flow (MCF) Problem • Bicriteria Network Design Problem (BNP) • Multi-criteria Network Design Problem

  7. 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 sto another specified sink nodet, assuming that each arc (i, j)∈A has an associated cost (or length) cij. Data table of example network 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, Ais a set of links. • cijis a cost associated with each arc(i, j) • Source node: node 1 • Destination node: node n • Indicator variable: Data table of example network 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 network’s 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.299–333, 2002. • OSPF is designed for exchanging routing information within a large or very large internetwork. • The biggest advantage of OSPF is that it is efficient. • OSPF requires very little network overhead even in very large internetworks. • The biggest disadvantage of OSPF is its complexity. • OSPF requires proper planning and is more difficult to configure and administer.

  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：　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

  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. 2 5 s t 1 1 1 4 7 path : 3 6 3 1 4 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. 2 5 s t 1 1 1 4 7 3 6 path : 3 1 4 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 iin 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. 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 j=1 ton // step 0 vk(j) j; for i=1 to // step 1 repeat jrandom[1, n]; lrandom[1, n]; until l≠j swap (vk(j), vk(l)); output the chromosome vk; // step 2 end

  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 initialsource 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 18 32 2 4 8 24 36 20 16 11 s t 13 5 10 1 7 15 27 12 12 13 38 23 3 6 9 node ID: j 1 2 3 4 5 6 7 8 9 10 priority: v(j) 7 3 4 6 2 5 8 10 1 9 • Illustration of Priority-based GA Data table of example network 1 1 Chromosome: Path: 1→3→6→7→8→10 Objective function value： z=106

  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 step 2 : exchange substrings between step 4 : legalize offspring with mapping relationship

  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. substring selected 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. 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 3 1 4 7 parent 1 : 2 1 7 4 5 3 6 parent 2 : 2 1 4 5 7 parent 2 : 3 7 2 6 5 1 4 step 2: mapping the weight of the right segment 3 5 6 offspring 1 : 3 1 4 5 7 5 3 6 5 1 4 offspring 2 : 2 1 4 7 1 4 5 step 3: generate offspring with mapping relationship offspring 1 : 2 1 7 4 6 3 5 offspring 2 : 3 7 2 6 4 1 5

  34. 1.4.3 Genetic Operators --- Mutation select a gene at random parent : 2 1 7 4 5 3 6 parent : 3 1 4 7 insert it in a random position offspring : 1 4 7 offspring : 2 5 1 7 4 3 6 • 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(t+1) 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 al’s method and priority-based encoding method to the 6 test problems: • Ahn, C.W. & R. Ramakrishna: “A Genetic Algorithm for Shortest Path Routing Problem and the Sizing of Populations.” IEEE Trans. Evol. Comput., Vol.6, No.6, pp.566-579, 2002. • OR-Notes. [Online]. Available:http://mscmga.ms.ic.ac.uk/jeb/or/orweb.html • Using the following parameter specifications. • Population size: popSize =20 • Crossover probability: pC =0.70 • Mutation probability: pM =0.50 • Immigration rate: μ=3 • Maximum generation: maxGen =1000 • Terminating condition: 100 generations with same fitness. • Each solution is compared with Dijkstra’s algorithm that provides a reference point (optimal solution). • Each algorithm was applied to each test problem 20 times (i.e., 20 runs) using different initial populations. • All the simulations were performed with Java on Pentium 4 processor (1.5-GHz clock).

  40. 1.5 Numerical Examples • The first numerical example, presented by Ahn et al’s was adopted. The problem comprises 20 nodes and 49 arcs. It is given as follows: (Ahn,C.W. & R. Ramakrishna:“A Genetic Algorithm for Shortest Path Routing Problem and the Sizing of Populations”, IEEE Trans. on Evol. Comput., Vol.6, No.6, pp.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 Dijkstra’s Algorithm Munemoto’s Algorithm Inagaki’s Algorithm Ahn’s Algorithm 2.0 1.5 Objective Function Values 1.0 0.5 2 0 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.) Inagaki’s Algorithm Munemoto’s Algorithm Ahn’s 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 (WMX+Insertion+ Immigration) that the GA finds the global optimum (i.e., the shortest path). • The path optimality is defined in #1, #2 test problems, by Alg.5 (WMX+Swap+ Immigration), The near optimal result is defined in other test problems. • By Alg.1 ~ Alg.4, the path optimality is not defined. Since the number of possible alternatives become to very large in test problems, the population be prematurely converged to a local optimum of the function. Table 6.1 Performance comparisons with different genetic operators

  45. 1.5 Numerical Examples Best solutions 20/49 80/120 80/630 160/2544 320/1845 320/10208 Problem size • Comparison results of Ahn’s algorithm and Proposed algorithm Table 6.2 Performance comparisons with Ahn’s algorithm and Proposed algorithm.

  46. 1.5 Numerical Examples • Different Parameter Settings: Table 6.3 Performance comparisons with different parameter settings

  47. 1.5 Numerical Examples • Different Parameter Settings with Ahn’s algorithm and Proposed algorithm

  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 40 60 5 8 2 60 30 30 60 30 20 30 s t f f 60 50 50 40 6 9 1 3 11 30 30 30 20 30 60 uij 50 i j 40 40 7 10 4 Data table of example network 2.1 Basic Concept of Maximum Flow Problem • [Online]. Available: http://www-b2.is.tokushima-u.ac.jp/ ~ikeda/suuri/maxflow/Maxflow.shtml.en • MXF is in a sense a complementary model to SPP. • MXF seeks a feasible solution that sends the maximum amount of flow from a specified source node s to another specified sink node t. • If we interpret uij as the maximum flow rate of arc (i, j), MXF identifies the maximum steady-state flow that the network can send from node sto node t per unit time.

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