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7. Minimum Spanning Tree Problem

7. Minimum Spanning Tree Problem. 7. Minimum Spanning Tree Problem. Multicriteria Minimum Spanning Tree (mc-MST) 1.1 Basic Concept of mc-MST 1.2 Genetic Algorithms Approach 1.3 GA procedure for mc-MST 1. 4 Numerical Experiments 2. Degree-constrained Minimum Spanning Tree (dc-MST)

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7. Minimum Spanning Tree Problem

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  1. Graduate School of Information, Production and Systems, Waseda University 7. Minimum Spanning Tree Problem

  2. 7. Minimum Spanning Tree Problem • Multicriteria Minimum Spanning Tree (mc-MST) 1.1 Basic Concept of mc-MST 1.2 Genetic Algorithms Approach 1.3 GA procedure for mc-MST 1.4 Numerical Experiments 2. Degree-constrained Minimum Spanning Tree (dc-MST) 2.1 Basic Concept of dc-MST 2.2 Genetic Algorithms Approach 2.3 GA procedure for dc-MST 2.4 Numerical Experiments 3. Degree-based Permutation GA for dc-MST 3.1 Concept on Degree-based Permutation GA 3.2 Genetic Algorithms Approach 3.3 Degree-based Permutation GA for dc-MST 3.4 Numerical Experiments • Leaf-constrained Minimum Spanning Tree 4.1 Basic Concept of lc-MST 4.2 Genetic Algorithms Approach 4.3 GA procedure for lc-MST 4.4 Numerical Experiments WASEDA UNIVERSITY , IPS

  3. 7. Minimum Spanning Tree Problem • Introduction • The Minimum Spanning Tree (MST)problem was first formulated by Boruvka in 1926 when he developed a solution to finding the most economical layout of a power-line network. • Graham, R. &P. Hell: On the history of the minimum spanning tree problem, Annals of the History of Computing, vol. 7, pp.43-57, 1985. • Since then the minimum spanning tree formulation has been widely applied to many combinatorial optimization problems: • Transportation problems • Telecommunication network design • Distribution systems etc. • Kershenbaum, A.: Telecommunications Network Design Algorithms, McGrawHill, New York, 1993. WASEDA UNIVERSITY , IPS

  4. 7. Minimum Spanning Tree Problem 2 1 2 6 3 8 4 5 1 3 wij 7 6 3 2 4 7 9 i j 2 6 8 2 1 5 4 • Basic Concept of Minimum Spanning Tree Problem • Minimum Spanning Tree (MST) problem is one of the traditional optimization problems. • Given a finite connected graph, the problem is to find a least-weight subgraph connecting all vertices. Data table of example network Fig. 7.1 Example of network model Table for non-directed graph Tavakoly., B.:Gene Expression Data Clustering With Minimum Spanning Tree, Department of Information Systems and Computing, Brunel University, May 2003. WASEDA UNIVERSITY , IPS

  5. 7. Minimum Spanning Tree Problem • Notations • Indices i, j : the index of node, i, j =1, 2, …, n • Parameters n:the number of nodes in the network V:the finite set of nodes (vertices) representing terminals S: the subset of nodes wij: the weight of connecting node i to node j,i.e., the weight of link (i,j); the weight matrix (wij) is symmetric. • Decision Variables xij : the 0,1 decision variable; 1, if link (i,j) is selected, and 0, otherwise. WASEDA UNIVERSITY , IPS

  6. 7. Minimum Spanning Tree Problem Consider a connected graphG=(V, E), where V={v1, v2, …, vn} or {1, 2, …, n} is a finite set of nodes E={(i,j) | i,j∈V} is a finite set ofedgesrepresenting connections between these vertices. Each edge has a positive real number denoted by W={w12, w13, …, wn-1,n} representing distance or cost. 1 2 8 6 3 7 9 5 4 w12=2 ,x12=1 w56=6 ,x56=1 w18=3 ,x18=1 w23=4 ,x23=1 w78=1 ,x78=1 w67=5 ,x67=1 w79=4 ,x79=1 w49=8 ,x49=1 • Basic Concept of Minimum Spanning Tree Problem Fig. 7.2 Simple MST Model WASEDA UNIVERSITY , IPS

  7. 7. Minimum Spanning Tree Problem • In a real-life network optimization situation, the problem often requires satisfying additional constraints. • Capacitated MST • Kershenbaum, A.: Computing capacitated minimal spanning trees efficiently, Networks, vol.4, pp.299–310, 1974. • Degree-constrained MST • Narula, S.&C. Ho: Degree-constrained minimum spanning tree, Computers and Operations Research, vol.7, pp.239-249, 1980. • Stochastic MST • Ishii, H., H. Shiode,&T. Nishida: Stochastic spanning tree problem, Discrete Applied Mathematics, vol.3, pp.263-273,1981. • Quadratic MST • Xu, W.: Quadratic minimum spanning tree problems and related topics, Ph. D. dissertation, University of Maryland, 1984. • Probabilistic MST • Bertismas, D.: The probabilistic minimum spanning tree problem, Networks vol.20, pp.245-275, 1990. • Multicriteria MST • Zhou, G. & M. Gen: The genetic algorithms approach to the multicriteria minimum spanning tree problem, in Kim, J. H., X. Yao, & T. Furuhashi Ed.: Proc. of the First Asia-Pacific Conference on Simulated Evolution and Learning, pp.387-394, Taejon, 1996. • Leaf-constrained MST • Fernandes, L. M. & L. Gouveia: Minimal spanning trees with a constraint on the number of leaves, European J. of Operational Research, vol.104, pp.250-261, 1998. WASEDA UNIVERSITY , IPS

  8. 7. Minimum Spanning Tree Problem • Some researchers used genetic algorithms to deal with them. • Probabilistic MST • Abuali, F., R. Wainwright, & D. Schoenefeld: A new encoding scheme for spanning trees applied to the probabilistic minimum spanning tree problem, in Proc. of the 5th Inter. Conf. on Genetic Algorithms, San Francisco, pp.470-475, 1995. • Palmer, C. C.&A.Kershenbaum: An approach to a problem in network design using genetic algorithm, Networks, vol. 26, pp.151-163, 1995. • Degree-constrained MST • Zhou, G. &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. & M. Gen: A note on genetic algorithm approach to the degree-constrained spanning tree problems, Networks, vol.30, pp.105-109,1997. • Capacitated MST • Chandy, K. M. & T. Lo: The capacitated minimum spanning tree, Networks, vol. 3, pp. 173-182, 1973. • Zhou, G. & M. Gen: Approach to degree-constrained minimum spanning tree problem using genetic algorithm, in Gen, M. & R. Chen, Genetic algorithms & Engineering Design, John Wiley, New York, 2000. WASEDA UNIVERSITY , IPS

  9. 7. Minimum Spanning Tree Problem • Quadratic MST • Zhou, G. & M. Gen: An effective genetic algorithm approach to the quadratic minimum spanning tree problem, Computers and Operations Research, vol.25, no.3, pp.229-247,1998. • Knowles, J. & D.Corne:A new evolutionary approach to the degree-constrained minimum spanning tree problem, IEEE Transactions on Evolutionary Computation, vol.4, no.2, pp.125-134, 2000. • Multicriteria MST • Zhou, G. & M. Gen: Genetic Algorithm Approach on Multicriteria Minimum Spanning Tree Problem, European J. of Operational Research, vol.114, pp.141-151, 1999. • Gen, M., G. Zhou & M. Takayama: Matrix-based Genetic Algorithm Approach onBicriteria Minimum Spanning Tree Problem with Interval Coefficients, J. of Japan Society for Fuzzy Theory and Systems, vol. 10, no.6, pp.643-656, 2000. • Zhou, G., H. Min & M. Gen: Agenetic algorithm approach to the bi-criteria allocation of customers to warehouses, Int. J. of Production Economics, vol.86, pp.35-45, 2003. • Leaf-constrained MST • Zhou, G.& M. Gen: Leaf-constrained spanning tree problem with genetic algorithms approach, Beijing Mathematics, vol.7, no.2, pp.50-62, 1998. • Fernandes, L. M. & L.Gouveia: Minimal spanning trees with a constraint on the number of leaves, European J. of Operational Research, vol.104, pp.250-261, 1998. WASEDA UNIVERSITY , IPS

  10. 7. Minimum Spanning Tree Problem • Multicriteria Minimum Spanning Tree (mc-MST) 1.1 Basic Concept of mc-MST 1.2 Genetic Algorithms Approach 1.2.1 Reviewing Encoding Methods 1.2.2Genetic Representation 1.2.3 Genetic Operators 1.3 GA procedure for mc-MST 1.4 Numerical Experiments • Degree-constrained Minimum Spanning Tree 3. Degree-based Permutation GA for dc-MST 4. Leaf-Constrained Minimum Spanning Tree WASEDA UNIVERSITY , IPS

  11. 1. Multicriteria Minimum Spanning Tree • Multicriteria Minimum Spanning Tree (mc-MST) problem is of high importance in practical network optimization. • A new approach to solve the mc-MST problem by using the Genetic Algorithms (GAs) is presented. • A skillful encoding for trees, denoted as Prüfer Number, is adopted for the GAs operation. • Combined with some MultiCriteria Decision Making (MCDM) techniques, the proposed GAs approach can : (1) get the Paretooptimal solutions close to its ideal point as much as possible or; (2) maintain all Pareto optimal solutions along thePareto frontier. • Zhou, G. & M. Gen:Genetic algorithm approach on Multicriteria minimum spanning tree problem, European J. of Operation Research, vol. 114, pp. 141-152, 1999. • Chankong, V. & Y. Y. Haimes: Multiobjective Decision Making Theory and Methodology, North-Holland, New York, 1983. WASEDA UNIVERSITY , IPS

  12. 1.1 Basic Concept of mc-MST • mc-MST Formulation The Multicriteria Minimum Spanning Tree (mc-MST) problem which has q objective and n nodes can be formulated as follows: WASEDA UNIVERSITY , IPS

  13. 1.1 Basic Concept of mc-MST • Definition 1: Given a set of feasible solution , • solution is denoted as the Pareto optimal solution or • nondominated solution to the problem (1) if and only • if there is no any other solution , satisfying the • following conditions: , for some , for all • Definition 2:The point in objective • space of the problem (1) is denoted as the ideal point, • where A basic understanding of Multiplecriteria Decision Making (MCDM) can be illustrated by the following definition: WASEDA UNIVERSITY , IPS

  14. 1.1 Basic Concept of mc-MST • Definition 3: In optimizing each objective of the problem (1), • there are respectively q optimal solution • and their corresponding objective values: • Through these ppoints , there • is a hyperplane which satisfies: where and • are the solution of the following equations: • This hyperplane is denoted as adaptive objective evaluation • hyperplane to the problem (1) WASEDA UNIVERSITY , IPS

  15. 1.2 Genetic Algorithms Approach • Operated on the coding of solution set of the problem to be solved at hand, instead of the solutions themselves. • Operated on a set of candidate solutions of the problem instead of a single one. • Operated not wholly randomly but including both the stochastic search (Crossover and Mutation) and the directed search (Evaluation and Selection) in Optimization. • Operated with a survival of the fittest mechanism which can enforce the search got to the optimal solution. • Zhou, G. & M. Gen: Genetic Algorithm Approach on Multicriteria Minimum Spanning Tree Problem, European J. of Operational Research, vol.114, pp.141-151, 1999. WASEDA UNIVERSITY , IPS

  16. 1.2.1 Reviewing Encoding Methods • Edge Encoding • Edge Encoding: • Piggott,P.& F.Suraweera: Encoding graphs for genetic algorithms: An investigation using the minimum spanning tree problem, in X. Yao, ed., Progress in Evolutionary Computation, Springer-Verlag, Berlin, pp.305-314, 1995. • Vertex Encoding • Prüfer number Encoding: • Zhou, G. & M. Gen: Genetic algorithm approach on Multicriteria minimum spanning tree problem, European J. of Operational Research, vol.114, pp.141-152, 1999. • Edge and Vertex Encoding • Link and Node biased Encoding: • Palmer, C. C. &A.Kershenbaum: An approach to a problem in network design using genetic algorithms, Networks, vol.26, pp. 151-163, 1995. • Degree-based Permutation Encoding: • Zhou, G.&M. Gen: Approach to degree-constrained minimum spanning tree problem using genetic algorithm, in Gen, M. & R. Chen, Genetic algorithms & Engineering Design, John Wiley, New York, 2000. • Tree Encodings • For the MST problem, how to encode a tree is critical for GAs pproach. Typical encodings for representing a tree can be classified as follows: WASEDA UNIVERSITY , IPS

  17. 1.2.1 Reviewing Encoding Methods x23 2 3 x12 x27 x37 x34 x17 x47 1 7 4 x57 x67 x16 x45 x56 6 5 • Edge Encoding • The edge encoding is really an intuitive representation for a tree. • 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. • A bit string can represent a candidate solution by indicating which edges are used in a spanning tree as illustrated. x12 x16 x17 x23 x27 x34 x37 x45 x47 x56 x57 x67 0 0 1 1 1 0 0 1 0 1 1 0 Fig. 7.3 Illustration of Edge Encoding • Piggott, P.& F. Suraweera: Progress in Evolutionary Computation, Springer-Verlag ,1995 WASEDA UNIVERSITY , IPS

  18. 1.2.1 Reviewing Encoding Methods c11’ c12 ’ ... c1m ’ c21 ’ c22 ’ ... c2m ’ ... ... ... cm1 ’ cm2 ’ cmm ’ ... • Link and Node biased Encoding • In this encoding, It holds a bias value for each node and each edge. • Each node bias and each edge bias are an integer in the range from 0 to 255. • The spanning tree corresponding to the encoding is found by running Prim’s MST algorithm on a modified cost matrix C’=[cjk’] cjk ’= cjk + p1 bjk cmax + p2 (bj + bk) cmax genetic representation • Palmer, C.&A. Kershenbaum: Networks, vol. 26, 1995 WASEDA UNIVERSITY , IPS

  19. 1.2.1 Reviewing Encoding Methods 21 2 3 89 34 68 215 12 172 1 7 4 23 1 2 3 6 4 5 7 138 143 8 - 26 89 M M M 143 12 1 6 5 89 - 21 M M M 34 2 M 21 - 215 M M 68 3 M M 215 - 8 M 172 4 M M M 8 - 26 23 5 143 M M M 26 - 138 6 12 34 76 154 34 76 - 7 • Link and Node biased Encoding by Prim’s algorithm A chromosome represented by Link and Node Biased Encoding Fig. 7.4 Illustration of Link and Node Biased Encoding WASEDA UNIVERSITY , IPS

  20. 1.2.1 Reviewing Encoding Methods 1 3 w12=3 w35=6 8 2 5 w58=2 w25=7 7 w27=5 w68=4 6 4 w46=6 • Prüfer number Encoding • Prüfer number is adopted to represent all candidate solutions of the problem. • Any tree with n nodes, the encoding length is only n - 2 data set wij i j 1 2 3 4 5 6 Fig. 7.5 Illustration of Prüfer number Encoding • Zhou,G.&M.Gen: Networks, vol. 30, 1997 WASEDA UNIVERSITY , IPS

  21. 1.2.1 Reviewing Encoding Methods • Degree-based Permutation Encoding 1 Tree data set w12=5 w13=3 w14=6 2 3 4 w25=4 w26=3 w48=5 w47=3 5 6 7 8 Degree-based permutation 1 2 3 4 5 6 7 8 node ID j 1 2 5 6 3 4 7 8 degree yj at node ID j 3 3 1 1 1 3 1 1 Fig. 7.6 Illustration of Degree-based Permutation Encoding node ID is node number based on the depth first search (DFS), degree at node ID is the number of connecting nodes. • Zhou, G.&M. Gen: Genetic algorithms & Engineering Design, John Wiley, 2000 WASEDA UNIVERSITY , IPS

  22. 1.2.2 Genetic Representation • Procedure of Encoding for Prüfer number (Zhou & Gen, 1997) input: spanning tree data set T output: Prüfer number Pstep 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 jincident 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

  23. 1.2.2 Genetic Representation • Procedure of Decoding for Prüfer number (Zhou & Gen, 1997) input: Prüfer number P output: spanning tree data set 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

  24. 1.2.3 Genetic Operators • Crossover and Mutation • Prüfer number encoding can always represent a tree after any crossover or mutation operations, simply the uniform crossover operator is adopted. • Mutation is performed as random perturbation within the range from 1 to n. Fig. 7.7 Illustration of crossover operation (One-cut Point Crossover) Fig. 7.8 Illustration of mutation operation (Altering Mutation) WASEDA UNIVERSITY , IPS

  25. 1.2.3 Genetic Operators • Evaluation and Selection (Strategy I) input: chromosomes output: fitness values of each chromosome step 1: Decode all chromosomes and calculate their objective values in each objective function. step 2: Determine the fitness value eval(T) of all chromosomes according to the following formula: As to selection procedure, the -selection is adopted. WASEDA UNIVERSITY , IPS

  26. 1.2.3 Genetic Operators • Evaluation and Selection (Strategy II) input: chromosomes output: fitness values of each chromosome step 1: Determine all nondominated chromosomes and assign a large dummy fitness value to them. step 2: Calculate each chromosome’s niche count ; , where ; otherwise , if step 3: Calculate the shared fitness value of each chromosome by dividing its dummy fitness by its niche count. step 4: Ignore all sorted nondominated chromosomes, go to step 1 and continue the process. For selection operation roulette wheel is adopted. WASEDA UNIVERSITY , IPS

  27. 1.3GA procedure for mc-MST • GA procedure for Multicriteria Minimum Spanning Tree (mc-MST) procedure: GA for mc-MST input: network data (V, A, W), associated number set in each edge, GA parameters output: Pareto optimal solutions E(P,C) begin t ←0; initialize P(t) by Prüfer number encoding; objectives z1(P), z2(P),…, zq(P); create Pareto E(P) fitness eval(P) by Prüfer number decoding; while (not termination condition) do crossover P(t) to yield C(t) by uniform crossover operator; mutation P(t) to yield C(t) by random perturbation; objectives z1(C), z2(C) ,…, zq(C); update Pareto E(P,C) if (not preference condition) fitness eval(P,C) by strategy I; else fitness eval(P,C)by strategy II; select P(t+1) from P(t) and C(t) by roulette wheel selection; t ←t+1; end output Pareto optimal solutions E(P,C); end WASEDA UNIVERSITY , IPS

  28. 1.4 Numerical Experiments Table 7.1 Ideal points and extreme Pareto optimal solutions • Two networks with 10-vertex and 50-vertex are tested by • the proposed GAs approach. The weights defined on • two objectives are randomly generated and respectively • distributed over [0, 100] and [0, 50]. • The parameters for the GAs operation are set as follows: • population size popSize = 200 • crossover probability pC = 0.2 • mutation probability pM = 0.05 • maximum generation maxGen = 500 WASEDA UNIVERSITY , IPS

  29. 1.4 Numerical Experiments • Numerical Experiments - Result Fig. 7.9 Illustration of GAs approach on the mc-MST by Strategy I WASEDA UNIVERSITY , IPS

  30. 1.4 Numerical Experiments • Numerical Experiments - Result (a) 10-vertex network (b) 50-vertex network Fig. 7.10 Illustration of GAs approach on the mc-MST by Strategy II WASEDA UNIVERSITY , IPS

  31. 7. Minimum Spanning Tree Problem 1.Multicriteria Minimum Spanning Tree (mc-MST) • Degree-constrained Minimum Spanning Tree 2.1 Basic Concept of dc-MST 2.2 Genetic Algorithms Approach 2.3 GA procedure for dc-MST 2.4 Numerical Experiments 3.Degree-based Permutation GA for dc-MST 4. Leaf-constrained Minimum Spanning Tree WASEDA UNIVERSITY , IPS

  32. 2. Degree-constrained Minimum Spanning Tree • Description of Degree-constrained Minimum Spanning Tree • Degree-constrained Minimum Spanning Tree (dc-MST) problem is of high importance in practical network design. • The dc-MST problem is NP-hard, no effective algorithms exitto solve this problem. • A new approach to solve the dc-MST problem by using the Genetic Algorithms (GAs) is presented. • A skillful encoding for trees, denoted as Prüfer Number, is adopted for the GAs operation. • Numerical examples are suggested to test the effective-ness of the proposed GAs approach compared with the results by existing heuristics and their Lower Bound. WASEDA UNIVERSITY , IPS

  33. 2.1 Basic Concept of dc-MST yj：Degree value on vertex j • A spanning tree with degree dj = 3 y8=1≤ d8 y9=1≤ d9 y8=1≤ d8 y9=1≤ d9 8 9 8 9 y7=3≤ d7 y7=3≤ d7 7 7 y4=4>d4 y6=1≤ d6 y6=1≤ d6 y5=1≤ d5 y5=1≤ d5 y4=3≤d4 6 5 6 4 4 5 2 y7=3≤ d7 2 y7=3≤ d7 y1=1≤ d1 y1=1≤ d1 1 1 3 3 y3=1≤ d3 y3=1≤ d3 Number of degrees:y4= 4 case Number of degrees:y4= 3 case Fig. 7.11 A spanning tree with degree WASEDA UNIVERSITY , IPS

  34. 2.1 Basic Concept of dc-MST • Notations • Indices i, j : the index of node, i, j =1, 2, …, n • Parameters n:the number of nodes in the network V:the finite set of nodes (vertices) representing terminals dｊ :the constrained degree value on vertex ｊ S: the subset of nodes cij: the cost of connecting node i to node j,i.e., the cost of link (i,j); the cost matrix (cij) is symmetric. yj ：degree value on vertex • Decision Variable xij : the 0,1 decision variable; 1, if link (i,j) is selected, and 0, otherwise. WASEDA UNIVERSITY , IPS

  35. 2.1 Basic Concept of dc-MST • Mathematical Model of dc-MST Problem • If we assume that there is a degree constraint on each • vertex such that, at each vertex , the degree value • yj is at most a given value , then the problem can be • formulated as follows: y2=2≤d2 y1=1≤d1 w12 ,x12=1 … 2 1 … yn=1≤dn y3=1≤d3 w3n ,x3n=0 n 3 … … … wjn ,xjn=1 j … … … yj=3≤dj Fig. 7.12 Simple dc-MST Model WASEDA UNIVERSITY , IPS

  36. 2.2Genetic Algorithms Approach • Repairing illegal Chromosome • Sample: A spanning tree with degree di = 3 y5=1≤ d5 y7=1≤ d7 5 7 y6=4>d6 y8=2≤ d8 y4=1≤ d4 y6=3>d6 y8=2≤ d8 y4=1≤ d4 8 4 6 8 4 6 repair y2=3≤ d2 y3=1≤ d3 y1=2≤ d1 y2=3≤ d2 y3=1≤ d3 y1=2≤ d1 1 2 3 1 2 3 y9=1≤ d9 y7=1≤ d7 y9=1≤ d9 y5=2≤ d5 9 7 9 5 Case: Case: Number of degrees:y6= 3 Number of degrees:y6= 4 Fig. 7.13 A spanning tree with degree WASEDA UNIVERSITY , IPS

  37. 2.2Genetic Algorithms Approach parent 2 6 6 2 6 8 1 replace with a digit from offspring 2 6 6 2 3 8 1 • Repairing illegal Chromosome • Let be the set of vertices whose degree has not been checked and repaired in a chromosome. If a vertex violates the degree constraints(degree value d ),this means that the number of this vertex in the chromosome is more than d - 1, then decrease the number of the vertex as illustrated as follows: check the extra vertex Fig. 7.14 Illustration of degree modification on a chromosome WASEDA UNIVERSITY , IPS

  38. 2.2Genetic Algorithms Approach • Genetic Operations • One-cut point crossover and altering mutation are used • The selection operation consists of both - selection and roulette wheel selection. procedure:Selection input:P(t), C(t), , output:P(t+1) begin select best different chromosomes; ifthen select chromosomes by roulette wheel selection; output P(t+1); end WASEDA UNIVERSITY , IPS

  39. 2.2Genetic Algorithms Approach • Evaluation • Convert each chromosome into a tree in the form of edge set according to the Prüfer number decoding procedure. • Calculate the fitness value for each chromosome in theform of total cost of the tree according to the objective in the dc-MST Problem. WASEDA UNIVERSITY , IPS

  40. 2.3GA procedure for dc-MST procedure: GA for dc-MST input: network data (V, A, W), associated number set in each edge, GA parameters output: best MST solution begin t←0; initialize P(t) by Prüfer number encoding; checkP(t) degree constraint and repairingP(t); fitness eval(P) by Prüfer number decoding; while (not termination condition) do crossover P(t) to yield C(t) by one-cut point crossover; mutation P(t) to yield C(t) by altering mutation; checkC(t) degree constraint and repairingC(t); fitness eval(C) by Prüfer number decoding; select P(t+1) from P(t) and C(t) by roulette wheel and ( +)-selection; t ←t+1;end output best MST solution; end WASEDA UNIVERSITY , IPS

  41. 2.4 Numerical Experiments • The numerical example was given by Savelsbergh and Volgenant who solved it by using heuristic algorithm denoted as edge exchange. It is a 9-vertex completer undirected network and optimal solution is 2256. Table 7.2 The numerical example by Savelsbergh and Volgenant 1 - 224 224 361 671 300 539 800 943 2 - 200 200 447 283 400 728 762 3 - 400 566 447 600 922 949 4 - 400 200 200 539 583 5 - 600 447 781 510 6 - 283 500 707 7 - 361 424 8 - 500 9 - node i 1 2 3 4 5 6 7 8 9 • Savelsbergh, M. & T. Volgenant:Edge exchanges in the degree- constrained spanning tree problem, Computers & Operations Research, vol. 12, no. 4, pp. 341-348, 1985. WASEDA UNIVERSITY , IPS

  42. 2.4 Numerical Experiments • Numerical Experiments – Results Analysis The parameters for the GAs operation are set as follows: popSize = 300; crossover probability pC = 0.2; maxGen = 500; mutation probability pM = 0.2; Fig. 7.15 Solution distribution for the dc-MST using GAs WASEDA UNIVERSITY , IPS

  43. 2.4 Numerical Experiments • Numerical Experiments – Results Analysis Fig. 7.16 Solution comparison between two selection strategies WASEDA UNIVERSITY , IPS

  44. 2.4 Numerical Experiments • Numerical Experiments – Results Table 7.3 Comparison between results by the LB and GA LB: lower bound without degree constraint;GA: GA fordc-MST ; minVal: minimal value; percentage of results by GA to LB; WASEDA UNIVERSITY , IPS

  45. 7. Minimum Spanning Tree Problem • Multicriteria Minimum Spanning Tree 2. Degree-constrained Minimum Spanning Tree • Degree-based Permutation GA for dc-MST 3.1 Concept on Degree-based Permutation GA 3.2 Genetic Algorithms Approach 3.2.1 Genetic Representation 3.2.2 Genetic Operators 3.3 Degree-based Permutation GA for dc-MST 3.4 Numerical Experiments 4.Leaf-constrained Minimum Spanning Tree WASEDA UNIVERSITY , IPS

  46. 3. Degree-based Permutation GA for dc-MST • Spanning tree is a basic topology structure in network • design problems like transportation, telecommunication • and distribution system. • Degree-constrained minimum spanning tree (dc-MST) is a more realistic representation in practice but it is a • NP-hard problem. • A new encoding is developed to deal with this problem • by using Genetic Algorithms (GAs). WASEDA UNIVERSITY , IPS

  47. 3.1 Concept on Degree-based Permutation GA • Literature Review • The dc-MST problem was first proposed by Narula and Ho, regarded as NP-hard problem (1980). • Simple Heuristic Approach: • Narula and Ho (Computer & Operations Research, vol. 7, 1980) • Edge Exchange Heuristic Approach: • Savelsbergh and Volgenant (Computers & Operations Research, vol. 12, no. 4, 1985) • Langrangean Multipliers Approach: • Volgenant (European J. of Operational Research, vol. 39, 1989) • Genetic Algorithm Approach using Prüfer Number: • Zhou and Gen (Proceedings of 1996 IEEE Internat. Conference on System, Man and Cybernetics, 1996, and Networks, vol. 30, 1997) WASEDA UNIVERSITY , IPS

  48. 3.1 Concept on Degree-based Permutation GA • Any Genetic Algorithms approach designed for a particular problem should address two main factors: (1) genetic representation of solutions to the problem, usually termed as chromosome or individual. (2) genetic operators that would alter the genetic composition of chromosomes during the evolutionary process. WASEDA UNIVERSITY , IPS

  49. 3.2 Genetic Algorithms Approach • One of the classic theorems in graphical enumeration is Cayley'stheorem that there are n(n-2) distinct labeled trees on a complete network 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 as Prüfer 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

  50. 3.2 Genetic Algorithms Approach 5 5 1 2 3 3 4 4 1 2 6 6 • Shortcoming of the Prüfer number • The Prüfer number has the great possibility to lose fit structure evolved in the evolutionary process. Above figure shows that changing even one digit of a Prüfer number can change the tree dramatically. Fig. 7.17 Illustration of Prüfer number • Lo, C. C. & W. H. Chang: ”A Multiobjective Hybrid Genetic Algorithm for the Capacitated Multipoint Network Design Problem”, IEEE Transactions on Systems, Man, and Cybernetics vol.30, no.3, pp.461-470, 2000. WASEDA UNIVERSITY , IPS

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