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9. Communication Network and LAN Design

9. Communication Network and LAN Design. 9. Communication Network and LAN Design. Centralized Network Design 1.1 Introduction of Centralized Network Design 1.2 Capacitated Multipoint Network Design Problem Backbone Network Design 2.1 Introduction of Backbone Network Design

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9. Communication Network and LAN Design

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  1. Graduate School of Information, Production and Systems, Waseda University 9. Communication Network and LAN Design

  2. 9. Communication Network and LAN Design • Centralized Network Design 1.1 Introduction of Centralized Network Design 1.2 Capacitated Multipoint Network Design Problem • Backbone Network Design 2.1 Introduction of Backbone Network Design 2.2 A Standard GA for Backbone Network Design 2.3 A Hybrid GA for Backbone Network Design • Bicriteria LAN Topological Design 3.1 Formulation of b-LAN Topology Design 3.2 B-LAN Topology Design based on GA 3.3 Numerical Example • Bi-level Hierarchical GA for Reliable Network Topological Design 4.1 Introduction 4.2 Problem Formulation 4.3 Genetic Approach 4.4 Numerical Example WASEDA UNIVERSITY , IPS

  3. 9. Communication Network and LAN Design • Computer Network Design Problems • The reason using computer networks: • To shareexpensivehardware/software resources • To provide accessto main system from distant locations • To improvethe reliabilityof computer systems • Network Design Problems: • Shortest Path Routing Problem • Network Design Problems Considering Reliability • Network Expansion Problems • LAN Topology Design Problems, etc. WASEDA UNIVERSITY , IPS

  4. vBNS Backbone Network Maphttp://www.mci.com/index.jsp WASEDA UNIVERSITY , IPS vBNS: very high speed Backbone Network Services

  5. NACSIS: National Center for Science Information Systemhttp://www.sinet.ad.jp/english/index.html WASEDA UNIVERSITY , IPS

  6. vBNS Logical Network Maphttp://www.mci.com/index.jsp WASEDA UNIVERSITY , IPS

  7. 9. Communication Network and LAN Design • Description of Computer Communication Network • Illustration of a network with spanning tree structure . . . Mux. Mux. Host Computer . . . Conc. Host Computer Host Computer Conc. Mux. Conc. Terminals Conc.: concentrator Mux.: multiplexer WASEDA UNIVERSITY , IPS

  8. 9. Communication Network and LAN Design • The use of communication networks has increased significantly in the last decade due to the dramatic growth in the use of internet for business and personal use. • As the society transforms itself to an information society the network becomes the primary source for information creation, storage, distribution and retrieval. • The design and development of a reliable network to support the primary resource of an information society becomes a very critical activity. • The reliability and service quality requirements of communication networks and the large investments in communication infrastructure have made it critical to design optimized networks that meet performance parameters. • These factors have promoted researcher to develop new models and methodologies for network design. WASEDA UNIVERSITY , IPS

  9. 9. Communication Network and LAN Design • A cost-effective structure for a large communication network is a multilevel hierarchical structure consisting of a backbone network (high level) and local access networks (low level). • Boorstyn, R. R. and H. Frank: Large-scale network topological optimization, IEEE Trans. on Communication, Vol. COM-25, No.1, pp. 29–47, 1977. • Figure shows the hierarchical structure of communication networks. • The backbone network, which connects local access networks to each other, is characterized by distributed traffic requirements and is generally implemented using packet switching techniques. Backbone Network Local Access Network : host Local Access Network : switching node : terminal WASEDA UNIVERSITY , IPS

  10. 9. Communication Network and LAN Design • In packet switching techniques, messages are broken into blocks of a certain size called packets; • The packets, when they contain the destination address, can follow different routes toward their destination. • The backbone network itself may be multilevel, incorporating, for example, terrestrial and satellite channels. • Local access networks have in general centralized traffic patterns (most traffic is to and from the gateway backbone node) and are implemented with centralized techniques such as multiplexing, concentrating, and polling to share data coming from several terminals having lower bit rates on a single high capacity link. • In special cases, the network may consist primarily of either centralized or distributed portions exclusively. • Next page show vBNS (very high speed Backbone Network System) backbone network and its logical network map funded in part by NSF. • vBNS is a research platform for advancement and development of high-speed scientific and engineering applications, data routing and data switching capabilities. WASEDA UNIVERSITY , IPS

  11. 9. Communication Network and LAN Design • The topological design problem for a large hierarchical network can be formulated as follows: • Terminal and host locations (terminal-to-host and host-to-host), • Traffic requirements (terminal-to-host and host-to-host) • Candidate sites for backbone nodes, and cost elements (line tariff structures, nodal processor costs, hardware costs, etc.) • Minimizing communication costs: Total communications costs = (backbone line costs) + (backbone node costs) + (local access line costs) + (local access hardware costs) WASEDA UNIVERSITY , IPS

  12. 9. Communication Network and LAN Design • Such that average delay and reliability requirements, which are index of quality of service, are met. • Average packet delay in a network can be defined as the mean time taken by a packet to travel from a source to a destination node. • Reliability is concerned with the ability of a network to be available the desired service to the end-users. • Reliability of a network can be measured using deterministic or probabilistic connectivity measures. • Colbourn, C. J.:The Combinatorics of Network Reliability, Oxford University Press, 1987. • Kershenbaum, A.:Telecommunications Network Design Algorithms. McGraw-Hill: New York, 1993. WASEDA UNIVERSITY , IPS

  13. 9. Communication Network and LAN Design • The global design problem consists of two subproblems: • The design of the backbone • The design of the local distribution networks. • The two subproblems interact with each other through the following parameters: • Backbone node numberandlocations • Terminalandhostassociation to backbone nodes • Delay requirementsfor backbone and local networks • Reliability requirementfor backbone and local networks WASEDA UNIVERSITY , IPS

  14. 9. Communication Network and LAN Design • Once the above variables are specified, the subproblems can be solved independently. • Boorstyn, R. R. and H. Frank: Large-scale network topological optimization, IEEE Transasctions on Communication, vol. COM-25, no 1, pp. 29–47, 1977. • Topological design of backbone and local access networks can be viewed as a search for topologies that minimize communication costs by taking into account delay and reliabilityconstraints. • This is NP hard problem which is usually solved by means of heuristic approaches. • For example, branch exchange, cut saturation algorithms, concave branch elimination are well known greedy heuristic approaches for backbone network design. • Gerla, M. and L. Kleinrock: “On the topological design of distributed computer networks”, IEEE Transactions on Communications, Vol. COM-25, No. 1, pp. 48–60, 1977. • Kershenbaum, A.:Telecommunications Network Design Algorithms. New York: McGraw-Hill, 1993. • Tanenbaum, A.S.:Computer Networks, Prentice Hall, New Jersey, 1981. • Easu and William’s algorithm, Sharma's algorithm and unified algorithm are known greedy heuristics for centralized network design. • Easu, L.R. and K.C. Williams: “On teleprocessing system design: a method for approximating the optimal network”, IBM System Journal, 5, pp 142-147, 1966. • Sharma, R. and M. El-Bardai:“Suboptimal communications networks synthesis”, Proceedings. of the IEEE International Conference on Communications, pp 19.11-19.16, 1970. • Kershenbaum, A. and W. Chou: “A unified algorithm for designing multidrop teleprocessing networks”, IEEE Transactions on Communications, Vol.22, pp. 1762-1772, 1974. WASEDA UNIVERSITY , IPS

  15. 9. Communication Network and LAN Design • Recently, genetic algorithms has been successfully applied to network design problems. The studies for centralized network design are given as: • Abuali, F. N., R. L. Wainwright, and D. A. Schoenefeld: “Determinant factorization: a new encoding scheme for spanning tree applied to the probability minimum spanning tree problem”, Proceedings of 6th International Conference on Genetic Algorithms, pp. 470–477, 1995. • Elbaum, R. and M. Sidi: “Topological design of local-area networks using genetic algorithms”, IEEE/ACM Transactions on Networking, Vol. 4, No.5, pp. 766–778, 1996. • Kim, J.R.:“Study on advanced genetic algorithms for reliable network design”, Ph.D. Disertation, Ashikaga Institute of Technology, 2000. • Kim, J.R. and M. Gen: “Genetic algorithm for solving bicriteria network topology design problem”, Proceedings. of the 1999 Congress on Evolutionary Computation, 2272-2279, 1999. • Kim, J. R. and Gen, M.:“A genetic algorithm for bicriteria communication network topology design”, Engineering Valuation and Cost Analysis, Vol. 3, pp. 351-363, 2000. • Lo, C.C. and W.H. Chang:“A multiobjective hybrid genetic algorithm for the capacitated multipoint network design problem”, IEEE Trans. on System, Man and Cybernetics-Part B, Vol. 30, No 3, pp 461-470, 2000. WASEDA UNIVERSITY , IPS

  16. 9. Communication Network and LAN Design • The studies for centralized network design are given as: • Palmer, C. C. and A. Kershenbaum:“An approach to a problem in network design using genetic algorithms”, Networks, Vol. 26, pp. 151-163, 1995. • Zhou, G. and M. Gen: “Approach to degree-constrained minimum spanning tree problem using genetic algorithm”, Engineering Design and Automation, Vol. 3, No 2, pp. 157-165, 1997. • Zhou, G. and M. Gen:“A note on genetic algorithm approach to the degree-constrained minimum spanning tree problem”, Networks,Vol. 30, pp. 105-109, 1997. • Zhou, G. and M. Gen:“A new tree encoding for the degree-constrained minimum spanning tree problem”, in Gen,M. & R. Cheng, Genetic Algorithms & Engineering Optimization, John Wiley & Sons, New York,2000. • Zhou, G. and M. Gen:“A genetic algorithm approach on tree-like telecommunication network design problem”, Journal of Operational Research Society, Vol. 54, No 3, pp. 248-254, 2003. WASEDA UNIVERSITY , IPS

  17. 9. Communication Network and LAN Design • The studies for backbone design are given as follows: • Davis, L. and S. Coombs: “Optimizing network link sizes with genetic algorithms”, Modeling and Simulation Methodology, Knowledge System’s Paradigms. Amsterdam, The Netherlands: Elsevier, 1989. • Kumar, A., R.M. Pathak, and Y.P. Gupta: “Genetic algorithm - based reliability optimization for computer network expansion”, IEEE Transactions on Reliability, Vol. 44, No 1, pp. 63 – 72, 1995. • Kumar, A., R.M. Pathak, Y.P. Gupta, and H.R. Parsei: “A genetic algorithm for distributed system topology design”, Computers andIndustrial Engineering, Vol. 28, No 3, pp. 659 – 670, 1995. • Ko, K.T., K.S. Tang, C.Y. Chan, K.F. Man and S. Kwong: “Using genetic algorithm to design mesh networks”, IEEE Computer, pp. 56-58, 1997. • Dengiz, B., F. Altiparmak, and A.E. Smith: “Efficient optimization of all-terminal reliable networks”, IEEE Transactions on Reliability, Vol. 41, No 1, pp. 18-26, 1997. • Dengiz, B., F. Altiparmak, and A.E. Smith: “Local search genetic algorithm for optimal design of reliable networks”, IEEE Transactions on Evolutionary Computation, Vol. 1, No 3, pp. 179-188, 1997. • Deeter, D.L., and A.E. Smith:“Economic design of reliable networks”, IIE Transactions, Vol. 30, pp. 1161-1174, 1998. WASEDA UNIVERSITY , IPS

  18. 9. Communication Network and LAN Design • The studies for backbone design are given as follows: • Pierre, S., and G. Legault: “A genetic algorithm for designing distributed computer network topologies”, IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics, Vol. 28, No 2, pp. 249-257, 1998. • Cheng, S.T.:“Topological optimization of a reliable communication network”, IEEE Transactions on Reliability, Vol. 47, No 3, pp. 225-233, 1998. • Konak, A., and A.E. Smith: “A hybrid genetic algorithm approach for backbone design of communication networks”, Proc. of the 1999 Congress on Evolutionary Computation, pp. 1817-1823, 1999. • Liu, B. and K. Iwamura: “Topological optimization models for communication network with multiple reliability goals”, Computers & Mathematics with Applications, Vol. 39, pp. 59-69, 2000. • Altiparmak, F., B. Dengiz, and A.E. Smith: “Optimal design of reliable computer networks: a comparison of metaheuristics”, Journal of Heuristics, Vol. 9, No 6, pp. 471-487, 2003. • Altiparmak, F., M. Gen, B. Dengiz, and A.E. Smith: “Topological optimization of communication networks with reliability constraint by an evolutionary approach”, Proc. of International Workshop on Reliability and Its Applications, pp. 183-188, 2003. WASEDA UNIVERSITY , IPS

  19. 9. Communication Network and LAN Design • Application of Shortest Path Routing Problem M.G.C. Resende: “A Genetic Algorithm with Optimized Crossover for Routing Crossover for Routing on the Internet,” INFORMS Annual Meeting, California, Nov., 2002. • OSPF (Open Shortest Path First) • OSPF is a commonly used intra-domain routing protocol (IGP). • Routers exchange routing information with all other routers in the autonomous system (AS). • Complete network topology knowledge is available to all routers, i.e. state of all routers and links in the AS. • OSPF routing • Assign an integer weight w∈ [1, wmax] to each link in AS. • In general, wmax= 65535=216−1. • Each router computes tree of shortest weight paths to all other routers in the AS, with itself as the root, using Dijkstra’s algorithm. WASEDA UNIVERSITY , IPS

  20. 9. Communication Network and LAN Design • OSPF weight setting • OSPF weights are assigned by network operator. • CISCO assigns, by default, a weight proportional to the inverse of the link bandwidth (Inv Cap). • If all weights are unit, the weight of a path is the number of hops in the path. • M.G.C. Resende propose a hybrid genetic algorithm to find good OSPF weights. • Memetic algorithm • Memetic Algorithms is a population-based approach for heuristic search in optimization problems. • It combine local search heuristics with crossover operators. For this reason, some researchers have viewed them as Hybrid Genetic Algorithms. • Genetic algorithm with optimized crossover WASEDA UNIVERSITY , IPS

  21. 9. Communication Network and LAN Design [ Resende, 2002 ] cost generation WASEDA UNIVERSITY , IPS

  22. 9. Communication Network and LAN Design [ Resende, 2002 ] cost generation WASEDA UNIVERSITY , IPS

  23. 9. Communication Network and LAN Design [ Resende, 2002 ] cost demand WASEDA UNIVERSITY , IPS

  24. 9. Communication Network and LAN Design • Centralized Network Design 1.1 Introduction of Centralized Network Design 1.2 Capacitated Multipoint Network Design Problem • Backbone Network Design • Bicriteria LAN Topological Design • Bi-level Hierarchical GA for Reliable Network Topological Design WASEDA UNIVERSITY , IPS

  25. 1.1 Introduction of Centralized Network Design • The problem of effectively transmitting data in a network involves the design of communication subnetworks, i.e.,Local access networks. • Local access networks are generally designed as centralized networks. • A centralized network is a network where all communication is to and from a central site (backbone node). • In such networks, terminals are connected directly to the central site. • Sometimes multipoint lines are used, where groups of terminals share a tree to the center and each multipoint line is linked to the central site by one link only. • This means that optimal topology for this problem corresponds to a tree in a graph G =(V, E) with all but one of nodes in V corresponding to the terminals. The remaining node refers to the central site, and edges in E correspond to the feasible communication wiring. • Each subtree rooted in the central site corresponds to a multipoint line. • Usually, the central site can handle, at most, a given fixed amount of information in communication. • This, in turn, corresponds to restricting the maximum amount of information flowing in any link adjacent to the central site to that fixed amount. WASEDA UNIVERSITY , IPS

  26. 1.1 Introduction of Centralized Network Design • In the combinatorial optimization literature, this problem is known as the constrainedMinimum Spanning Tree problem (c-MST). • The mathematical model of c-MST is as follows: WASEDA UNIVERSITY , IPS

  27. 1.1 Introduction of Centralized Network Design • Where • n:the number of nodes in the network • Tk: the kth multipoint link, and it may not exist for some k. • W: given weight • wij : the weight of the ith node to node j . • T: the spanning tree • cij: the cost of connecting node i to node j,i.e., the cost of link (i,j); the cost matrix (cij) is symmetric. • xij : the 0,1 decision variable; 1, if link (i,j) is selected, and 0, otherwise. • Constraint: (2) guarantees thatthe links chosen for the topologydo not includeany cycles. (3) guarantees thatenough linkswill be selectedto connect the network(of n nodes). (4) guarantees thatthe total weight of the terminalson each multipoint line does not exceedthe limit. WASEDA UNIVERSITY , IPS

  28. 1.1 Introduction of Centralized Network Design • The problem has been shown to be NP-hard by Papdimitriou. • Papadimitriou, C. H.:“The complexity of the capacitated tree problem”, Networks, Vol. 8, pp. 217-230, 1978. • Chandy and Lo, Kershenbaum and Chou, Elias and Ferguson had proposed heuristics approaches for this problem. • Chandy, K. M. and T. Lo: “The capacitated minimum spanning tree”, Networks, Vol. 3, pp. 173-182, 1973. • Kershenbaum, A. and W. Chou: “A unified algorithm for designing multidrop teleprocessing networks”, IEEE Transactions on Communications, Vol.22, pp. 1762-1772, 1974. • Elias, D. and M. Ferguson: “Topological design of multipoint teleprocessing networks”, IEEE Transactions on Communications, Vol. 22, pp. 1753-1762, 1974. • Gavish also studied a new formulation and its several relaxation procedures for the capacitated minimum directed tree problem. • Gavish, B.: “Topological design of centralized computer networks: formulation and algorithms”, Networks, Vol. 12, pp. 355-377, 1982. WASEDA UNIVERSITY , IPS

  29. 1.1 Introduction of Centralized Network Design • Recently, GA has been successfully applied to solve this problem and its variants. • Abuali, F. N., R. L. Wainwright, and D. A. Schoenefeld: “Determinant factorization: a new encoding scheme for spanning tree applied to the probability minimum spanning tree problem”, Proceedings of 6th International Conference on Genetic Algorithms, pp. 470–477, 1995. • Palmer, C. C. and A. Kershenbaum: “An approach to a problem in network design using genetic algorithms”, Networks, Vol. 26, pp. 151-163, 1995. • Zhou, G. and M. Gen: “Approach to degree-constrained minimum spanning tree problem using genetic algorithm”, Engineering Design and Automation, Vol. 3, No 2, pp. 157-165, 1997. • Zhou, G. and M. Gen: “A note on genetic algorithm approach to the degree-constrained minimum spanning tree problem”, Networks,Vol. 30, pp. 105-109, 1997. • Lo, C.C. and W.H. Chang: “A multiobjective hybrid genetic algorithm for the capacitated multipoint network design problem”, IEEE Trans. on System, Man and Cybernetics-Part B, Vol. 30, No 3, pp 461-470, 2000. • Zhou, G. and M. Gen: “A new tree encoding for the degree-constrained minimum spanning tree problem”, in Gen,M. & R. Cheng: Genetic Algorithms & Engineering Optimization, John Wiley & Sons, New York, 2000. • Zhou, G. and M. Gen: “A genetic algorithm approach on tree-like telecommunication network design problem”, Journal of Operational Research Society, Vol. 54, No 3, pp. 248-254, 2003. WASEDA UNIVERSITY , IPS

  30. 1.2 Capacitated Multipoint Network Design Problem • Many important real-world decision problem involves multiple and conflicts objectives that need to be tackled while respecting various constraints, leading to overwhelming problem complexity. • Lo and Chang propose a multiobjective hybrid GA (mo-hGA) for capacitated Multipoint Network Design (cMND)Problem to minimize total cost and delay. • Lo, C.C. and W.H. Chang:“A multiobjective hybrid genetic algorithm for the capacitated multipoint network design problem”, IEEE Trans. on System, Man and Cybernetics-Part B, Vol. 30, No 3, pp 461-470, 2000. • In their approach, the concept of subpopulation proposed by Schaffer had been used. • Schaffer, J. D.:“Multiple objective optimization with vector evaluated genetic algorithms”, Proceedings of 1st International Conference on Genetic Algorithms, pp. 93–100, 1985. • After four subpopulations generated using different mechanism, population of next generation was obtained by mixing these subpopulations. WASEDA UNIVERSITY , IPS

  31. 1.2 Capacitated Multipoint Network Design Problem • When two objectives are considered, the mathematical model given above for the cMND problem is reformulated as a c-MST problem is as follows: Where, dijis the average delay on link (i,j), the delay matrix [dij] is symmetric. WASEDA UNIVERSITY , IPS

  32. 1.2.1 Representation and Initialization • The genetic representation is a type of data structure that represents the candidate solutions of problems. • Usually, different problems have different data structures or genetic representations. • Gen, M. & R. Cheng: Genetic Algorithms and Engineering Optimization, John Wiley & Sons, New York, 2000. • For a c-MST problem, there are three ways of encoding tree: • Edge-based encoding, • Vertex-based encoding, • Edge and vertex-biased encoding. WASEDA UNIVERSITY , IPS

  33. 1.2.1 Representation and Initialization 1 2 3 4 5 6 1 2 3 4 5 6 • Edge-based Encoding: • A graph G = (V, E) is easily formed into a binary string which can be used as a chromosome for the GA. • Each element of the chromosome represents a possible edge in the graph so there are n(n-1)/2 edges. where n is the number of vertices. • The value of each element represents whether the specific edge connects with the pair of nodes or not. • Consider an example with 6 vertices shown as follows. The table shows connectivity matrix. Table 9.1 Connectivity matrix of links. 2 6 1 5 3 4 Fig. 9.2 Six node tree. WASEDA UNIVERSITY , IPS

  34. 1.2.1 Representation and Initialization k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 chromosome v 1 1 0 0 0 0 0 0 0 1 0 0 1 0 1 2 6 1 5 3 4 • Edge-based Encoding: • Note that there are (6x5)/2 = 15 possible edges for the example but only five are included. • The other 10 are not connected. • As it is seen that the index of chromosome is represented by following equation: • The chromosome representation is given as follows: WASEDA UNIVERSITY , IPS

  35. 1.2.1 Representation and Initialization • Edge-based Encoding: • This encoding method is really an intuitive representation for a tree. • However, for an n-node graph, a spanning tree should be a connected subgraph with n – 1 edges or non-loop subgraph with n – 1 edges. • But the edge based encoding can not preserve this property. • If the bit string contains other than n – 1 edges, it is not a tree. • Even if the bit string contains n – 1 edges, it unlikely represents a tree because there may exist loops. • Thus it is quite likely that none of them would be trees in the initial population or after crossover and mutation operators in GA approach. • Actually, the probability of obtaining a spanning tree with edge-based encoding is infinitesimally small as the number of vertices n increases. • There are n(n-2) spanning trees in a fully connected graph with n vertices. • While search space of the graph is 2(n(n-1)/2), the probability to randomly produce a spanning tree is only [n(n-2)]/ 2(n(n-1)/2)]. • Hence, this encoding is a poor encoding for spanning tree because of the extremely low probability of obtaining a tree. WASEDA UNIVERSITY , IPS

  36. 1.2.1 Representation and Initialization • Vertex-based Encoding: • One of the classic theorems in graphical enumeration is Cayley's theorem that there are n(n-2) distinct labeled treeson a complete graph with n vertices. • Prüfer provided a constructive proof of Cayley's theorem by establishing a one-to-one mapping between such trees and the set of all string of n-2 digits. • This means that it is possible to use only n-2 digits permutation to uniquely represent a tree where each digit is an integer between 1 and n inclusive. • This permutation is usually known asPrüfer number. • Prüfer, H.:“Neuer Beweis eines Satzes über Permutationen“, Archiv fuer Mathemtische und Physik, Vol. 27, pp. 742-744, 1918. • In this study,Prüfer numberis used torepresent a candidate tree and initial populationis generated using complete random method. WASEDA UNIVERSITY , IPS

  37. 1.2.1 Representation and Initialization • Encoding Procedure of Prüfer Number procedure: Encoding of Prüfer Number input: a tree T output: Prüfer number P step 1: Let node i be the smallest labeled leaf node in a labeled tree T. step 2: Let j be the first digit in the encoding, as the code j incident to i is uniquely determined. The encoding is built by appending digits from left to right. step 3: Remove node i and the link from i to j, thus there is a tree with k-1 nodes. step 4: Repeat the steps above until one link is left. P is obtained. WASEDA UNIVERSITY , IPS

  38. 1.2.1 Representation and Initialization T = {(1, 3), (2, 5), (3, 4), (4, 8), (5, 8), (5, 9), (6, 9), (7, 8), (8, 11), (10, 12), (11, 12)} 3 7 1 8 4 11 5 9 2 12 6 10 WASEDA UNIVERSITY , IPS

  39. 1.2.1 Representation and Initialization • Decoding Procedure of Prüfer Number procedure:Decoding of Prüfer Number input: Prüfer number P output: Tree T step 1: Let P be the original Prüfer number and let P' be the set of all nodes not included in P, which are designated as eligible nodes for consideration in building a tree T. step 2: Let i be the eligible node with the smallest label. Let j be the left most digit of P. Add the edge from i to j into the tree T. Remove i from P' and j from P. If j does not occur anywhere in P, put it into P'. Repeat the process until no digits are left in P. step 3: If no digits remain in P, there are exactly two nodes, r and s, still eligible for consideration. Add a link from r to s into tree and form a tree with k-1 links. WASEDA UNIVERSITY , IPS

  40. 1.2.1 Representation and Initialization 3 7 1 8 4 11 5 9 2 12 T = {(1, 3), (2, 5), (3, 4), (4, 8), (5, 8), (5, 9), (6, 9), (7, 8), (8, 11), (10, 12), (11, 12)} 6 10 WASEDA UNIVERSITY , IPS

  41. 1.2.1 Representation and Initialization • Edge and Vertex biased Encoding: • Palmer and Kershenbaumproposed theedge and vertex biased encodingmethod. Palmer, C. C. and A. Kershenbaum: “An approach to a problem in network design using genetic algorithms”, Networks, Vol. 26, pp. 151-163, 1995. • In this method, a tree is encoded using a modified cost matrix. • Based on the modified cost matrix, Prim algorithm is used to generate tree. • For a graph with n vertices, the chromosome of the representation has biases including vertex bias bi and edge bias bij, for the n vertices and each of the n(n-1)/2 edges, for a total of [n(n+1)/2] biases. • P1 and P2 are used as the multipliers of the maximum link costCmax. • The cost matrix (Cmax) is biased by bi, bij, P1, P2and Cmaxusing WASEDA UNIVERSITY , IPS

  42. 1.2.1 Representation and Initialization • Edge and Vertex biased Encoding: • Palmer and Kershenbaum claimed that this version of representation could encode any tree, given appropriate values of the bi,bjand bij. Palmer, C. C. and A. Kershenbaum: “An approach to a problem in network design using genetic algorithms”, Networks, Vol. 26, pp. 151-163, 1995. • However, as pointed out by Gen and Cheng, this encoding has three major disadvantages: Gen, M. and Cheng, R.: Genetic Algorithms & Engineering Optimization, John Wiley & Sons, New York, 2000. • It requires a very long encoding (memory cost); • It needs a conventional minimum spanning tree algorithm to generate a tree from its encoding (computation cost); • It containsno useful informationsuch as degree, connection, etc, about a tree. WASEDA UNIVERSITY , IPS

  43. 1.2.2 Subpopulation Generation Mechanisms • In the study, four different subpopulation mechanisms have been used [Lo-Cheng, 2000]: • Elitism reservation strategy • Stochastic universal sampling • Complete random method • Shifting Prüfer vector WASEDA UNIVERSITY , IPS

  44. 1.2.2 Subpopulation Generation Mechanisms a. Elitism Reservation Strategy: • In traditional GA’s, a chromosome in the current generation is selected into the next generation with certain probability. • The best chromosomes of the current generation may be lost due to mutation, crossover, or selection during the evolving process, and subsequently causes difficulty in reaching convergence. • In other word, it takes more generations; i.e., running time, to get quality solutions. • Tamaki et al.. proposed an elitism reservation strategy that permits chromosomes with the best fitness to survive and be carried into the next generation for multiobjective problems. • Tamaki, H., H. Kita and S. Kobayahi: “Multi-objective optimization by genetic algorithms: a review”, Proceeding of 3rd International Conference on Evolutionary Computation, pp. 517-522, 1996. WASEDA UNIVERSITY , IPS

  45. 1.2.2 Subpopulation Generation Mechanisms b. Shifting Prüfer Vector: • The shifting Prüfer vector is developed by Lo and Chang as a genetic operator. • The operator replaces the leftmost element of a Prüfer vector with a randomly selected nonleftmost element of the same vector. • The purpose is to maintain maximum locality and realize local search. • They had proofed that the new topology differs from the old one in at most two edges. • The figure illustrates the new tree after the shifting Prüfer vector is applied to the 7-node tree. It is seen from figures that the new tree and the old one differ in only one edge. 2 4 2 4 1 1 6 6 3 5 3 5 P = [1, 3, 5, 6] P’ = [3, 3, 5, 6] Fig. 9.3 New tree after applying the shifting Prüfer vector WASEDA UNIVERSITY , IPS

  46. 1.2.2 Subpopulation Generation Mechanisms c. Stochastic Universal Sampling: • A simple way to perform sampling is to spin a roulette wheel. • Unfortunately, this sampling method does not guarantee that any particular sample will actually be chosen in any given generation. • This is a well-known problem of the roulette wheel selection method. • Baker suggested the stochastic universal sampling method. • Baker, J.:“Adaptive selection methods for genetic algorithms”, Proceedings of 2th International Conference on Genetic Algorithms, pp. 100-111, 1987. • Baker’s algorithm completes the whole sampling in a single pass, and requires only one random number. • A wheel spin, whose size is equal to the population size, is divided into a number of equally spaced markers. A single spin is used to generate the random number. • The expected value ek for chromosome k is expressed as ek = popSizex pk, where popSize represents population size and pk represents selection probability. WASEDA UNIVERSITY , IPS

  47. 1.2.2 Subpopulation Generation Mechanisms d. Complete Random Method: • Population is generated according to random number and random position. • The major reason for using the complete random method is to maintain the diversity of the population. Mixing Method: • Firstly, four subpopulations are generated according to • The elitism reservation strategy • The shifting Prüfer vector • The stochastic universal sampling • The complete random method, • Then, mixing these four subpopulations produces the population of next generation. WASEDA UNIVERSITY , IPS

  48. 1.2.3 Mixing Method • Lo and Chang[IEEE-SMC, 2000] had explained the main idea on mixing method as follows: • There aretwo competing factorsin the selection procedure of a genetic search.They areselection pressureand population diversity. • To increase of selection pressure decreases the diversity of the population. • Michalewicz, Z.:Genetic Algorithms + Data Structure = Evolution Programs, 3nd ed., New York: Springer-Verlag, 1996. • The stochastic universal sampling method increases selection pressure. However, it may cause the premature convergence of a genetic search. • To decrease selection pressure, the complete random method can be used in conjunction with the stochastic universal sampling method. • Nevertheless, the best chromosomes of the current generation may be lost due to crossover and mutation. • In order to find globally optimal solutions, the shifting Prüfer vector is added to the selection procedure. WASEDA UNIVERSITY , IPS

  49. 1.2.4 mo-hGA procedure for Capacitated mo-NDP procedure: mo-hGA for Capacitated mo-NDP (Lo-Cheng 2000) input: network data (V, A, C, D, W), GA parameters output: Pareto optimal solutions E(P) begin t 0; initialize P(t) by Prüfer number encoding; objectives z1(P), z2(P) by Prüfer number decoding; create Pareto E(P); fitness eval(P); while (not termination condition) do generate subpopulation, C1(t) by elitism reservation strategy; generate subpopulation, C2(t) by shifting Prüfer vector; generate subpopulation, C3(t) by stochastic universal sampling; generate subpopulation, C4(t) by complete random method; objectives z1(Ci), z2(Ci), i=1, 2, 3, 4 by Prüfer number decoding; fitness eval(C) by Prüfer number decoding; select P(t+1) from Ci(t), i=1, 2, 3, 4 by mixing routine; t  t + 1; end output Pareto optimal solutions E(P); end WASEDA UNIVERSITY , IPS

  50. 1.2.5 Experimental Results • In order to evaluate the solutions of Capacitated mo-NDP obtained by mo-hGA, Lo and Chang examined a set of problems, with 7, 14, 28, and 56 nodes, respectively. • While they took population size as 100 for each problem, crossover and mutation probabilities and maximum number of generations had been changed according problem size. • mo-hGA was coded in the C++ language and run on an Intel Pentium-66 MHz PC with 64 MB RAM. Table 9.2 Parameters of GA WASEDA UNIVERSITY , IPS

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