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Genetic algorithm approach on multi-criteria minimum spanning tree problem

Kuo-Hsien Chuang 2009/01/06

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Introduction

Minimum Spanning Tree to find a least cost spanning tree many efficient polynomial-time

algorithms

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Introduction

Multi-criteria Minimum Spanning Tree Multiple objectives Pareto optimal solutions

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Problem description

G = (V, E) V = {v1, v2, …vn}, E = {e1, e2…em}

Each edge has p attributes Wi = {W1i, W2i … Wpi} X = {x1, x2, …xm},

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Problem description

Multi-objective

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Multiple criteria decision making

Multiple criteria decision making

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Multiple criteria decision making

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Multiple criteria decision making

Methods of objective weighting

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Multiple criteria decision making Method of Pareto optimal enumeration

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GA approach

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

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GA approach

Prufer number

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GA approach

Crossover and mutation Uniform crossover

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GA approach

mutation

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GA approach

Evaluation and selection Evaluation for Strategy I Evaluation for Strategy II

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GA approach

Evaluation for Strategy I

(μ+λ)selection in evolution strategy

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GA approach

Evaluation for Strategy II

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GA approach

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GA approach

mc-MST genetic algorithm

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GA approach

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Experiment

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

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Experiment

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Experiment

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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.

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