Order Relation between Intervals and Its Application to Shortest Path Problem

Z-Dijkstra’s Algorithm to solve Shortest Path Problem in a Z-Graph

“…The Fuzzy Shortest Path Problem (FSPP) 18,23,25,[33][34][35][36][37][39][40][41][42][43][44][45][46][47][48] is a generalization of the classical SPP for applications in ill-defined environment and has been found important to many applications such as Communication or Transportation Network, Computational Geometry, Graph Algorithms, Geographical Information Systems (GIS), Network Optimization, etc. In traditional shortest path problems, the arc length of the network takes precise numbers, but in the real-world problem, the arc length may represent transportation time or cost which can be known only approximately due to vagueness of information, and hence it can be considered a fuzzy number or an intuitionistic fuzzy number.…”

“…Lin and Chen 36 found the fuzzy shortest path length in a network by means of a fuzzy linear programming approach. Okada and Soper [39][40][41][42][43][44][45] proposed a fuzzy algorithm, which was based on multiple-labelling methods to offer non-dominated paths to a decision maker. Chuang and Kung 20 proposed a fuzzy shortest path length procedure that can find a fuzzy shortest path length among all possible paths in a network.…”

Solving the Shortest Path Problem with Interval Arcs

“…The only earlier work on this problem is that of Okada and Gen (1994). However, the algorithm which they proposed to solve the shortest path problem tends to generate a set of incomparable intervals as non-dominated solutions of the problem.…”

“…However, the algorithm which they proposed to solve the shortest path problem tends to generate a set of incomparable intervals as non-dominated solutions of the problem. Okada and Gen (1994) first tried to resolve the issue of incomparability of intervals using a ranking strategy of their own but their method seems to work in an ad hoc manner and so the result is not always unique and self-explanatory when used in an algorithm for solving the shortest path problem. In the following sub-section, first we give a brief comparison of different order relations (viz., Moore (1979), Ishibuchi and Tanaka (1990), Kundu (1997)) and then we propose a methodology that considers fuzzy preference ordering (Sengupta and Pal (2000)) of intervals, which gives a more clear and convincing decision than that of Okada and Gen (1994).…”

Critical Path for a Grey Interval Project Network

“…The whitening value of a grey number ± x is defined as a deterministic number with its value lying between the upper and lower bounds of The operations of grey numbers were shown in (Gou et al, 1995;Okada and Gen, 1994;Wu and Huan, 2004;Liu and Chen, 1991;Xiao et al, 2005;Chans and Zielinski, 2002). ; the dissimilarity is the network activity times are grey num-…”