Title: Robust and minimum spanning tree in fuzzy environment

Authors: Arindam Dey; Sahanur Mondal; Tandra Pal

Addresses: Department of Computer Science and Engineering, Saroj Mohan Institute of Technology, Hooghly, India ' Department of Computer Science and Engineering, National Institute of Technology, Durgapur, India ' Department of Computer Science and Engineering, National Institute of Technology, Durgapur, India

Abstract: This paper proposes an algorithm to find the fuzzy minimum spanning tree (FMST) of an undirected weighted fuzzy graph, in which mixed fuzzy numbers, either triangular or trapezoidal, are used to represent the lengths/costs of the arcs. In the proposed algorithm, we incorporate the uncertainty in Kruskal's algorithm for MST using fuzzy number as arc length. The concept of possibility programming is used to compare between the fuzzy number (i.e., costs of arcs) and addition operation of fuzzy numbers is used to find the cost of the spanning tree. We also investigate the robust version of the FMST problem. We define two measures for robustness of an FMST: absolute robustness and relative robustness. We characterize the fuzzy worst case scenarios for a given fuzzy spanning tree for both the measures. The corresponding fuzzy robust spanning trees are respectively defined as absolute robust fuzzy spanning tree (ARFST) and relative robust fuzzy spanning tree (RRFST). We extend Kruskal's algorithm to compute the ARFST and RRFST in fuzzy environment. An example of fuzzy graph is used to illustrate the effectiveness of the proposed methods.

Keywords: possibility programming; robust spanning tree; Kruskal's algorithm; minimum spanning tree; triangular fuzzy number; trapezoidal fuzzy number; fuzzy graph; fuzzy minimum spanning tree; absolute fuzzy robust spanning tree; relative fuzzy robust spanning tree.

DOI: 10.1504/IJCSM.2019.103679

International Journal of Computing Science and Mathematics, 2019 Vol.10 No.5, pp.513 - 524

Received: 05 Feb 2017
Accepted: 19 Jul 2017
Published online: 22 Nov 2019 *

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