早稻田大学遗传算法.ppt

Soft Computing Lab. 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) 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 7. Minimum Spanning Tree Problem 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. 7. 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. 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.