1 Genetic algorithm approach on multi-criteria minimum spanning tree problem Kuo-Hsien Chuang 2009/01/06

2 Introduction  Minimum Spanning Tree to find a least cost spanning tree many efficient polynomial-time algorithms

3 Introduction  Multi-criteria Minimum Spanning Tree Multiple objectives Pareto optimal solutions

4 Problem description  G = (V, E) V = {v 1, v 2, … v n }, E = {e 1, e 2 … e m }  Each edge has p attributes Wi = {W 1i, W 2i … W pi } X = {x 1, x 2, … x m },

5 Problem description  Multi-objective

6 Multiple criteria decision making

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 Methods of objective weighting 9

Multiple criteria decision making  Method of Pareto optimal enumeration 10

GA approach  n vertices, n 的 (n-2) 次方種 tree  Chromosome representation Prufer number a permutation of n-2 digits 11

GA approach  Prufer number 12

GA approach  Crossover and mutation Uniform crossover 13

GA approach  mutation 14

GA approach  Evaluation and selection Evaluation for Strategy I Evaluation for Strategy II 15

GA approach  Evaluation for Strategy I  (μ+λ)selection in evolution strategy 16

GA approach  Evaluation for Strategy II 17

GA approach 18

GA approach  mc-MST genetic algorithm 19

GA approach 20

Experiment  tested on five numerical examples of the 10-vertex to 50-vertex 21

Experiment 22

Experiment 23

Experiment  According to the preference of decision maker, the proposed GA approach can obtain all Pareto optimal solutions close to the ideal point or produce a set of solutions distributed along the whole Pareto frontier.  Although this paper has only dealt with the classical MST problem with multi-criteria, it is easy to extend the proposed method to solve those degree-constrained MST, stochastic MST, probabilistic MSTand quadratic MST problems with multi-criteria. 24