Minimum Spanning Tree Problems with Possibilistic Arc Weights and their Single Most Vital Arcs

• Main Authors: Hui-Hsin Hsu, 許惠馨
• Other Authors: Kao-Cheng Lin
• Format: Others
• Language: zh-TW
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/59445356038175304316

碩士 === 南台科技大學 === 工業管理研究所 === 97 === When the arc weights are fuzzy numbers, the total weight of a spanning tree is also a fuzzy number. Since the number of spanning trees in a network is usually very large, how to select a spanning tree is an interesting problem. In this study, based on the possibility theory, two kinds of models for selecting a spanning tree are proposed. One is called the expected value model, and the other is called the possibilistic chance-constrained model. Finally, we also consider the problem of finding a single most vital arc in the network. In the expected value model, a ranking function is used in selecting a spanning tree. At first, some rules and criteria for choosing a ranking function for this model are proposed. Then, it is pointed out that when the ranking function is additive, the problem can be transformed into the classical minimum spanning tree problem by defuzzifying the arc weights using the ranking function. In particular, weighted averages of the upper bound and the lower bound of the probabilistic mean value or the possibilistic mean value are suggested, in which the weight can be used to represent the characteristic of the decision maker. Finally, we consider the problem of finding minimum spanning trees for all kinds of decision makers at the same time. In the possibilistic chance-constrained model, the arc weights used in planning are treated as decision variables and subject to a set of chance constraints. These constraints require the possibilities, that the arc weights used in planning are sufficient, are not less than specified levels. According to the characteristic of the decision maker, there are two kinds of chance-constrained model, the minimin model and the maximin model. It is shown that when the arc weights are non-interactive, these two models can be transformed into the classical minimum spanning tree problems easily. As to the single most vital arc problem, we point out that it should be divided into two situations. One situation is that the decision model of the network user is known in advance. Here, it is assumed that the user select a spanning tree using the expected value model. However, in practice, the characteristic of the user usually can’t be known in advance. Therefore, we propose a parametric analysis method for the single most vital arc in this situation. The other situation is that the decision model of the network user is not known in advance. For this situation, we use Lin and Chern’s approach to propose an algorithm for finding a single most vital arc.