Optimal Path Problem with Possibilistic Weights - Geoinformatics for Intelligent Transportation - page 40

Fig. 1 Simple graph with Ni i nodes and directed arcs A j

fuzzy numbers for calculation of possibility and necessity of equality and/or
exceedance of fuzzy numbers.
The structure of paper is following: Sect. 2 offers basic preliminaries and notions
of graph theory, for further elaborations about this topic please see [ 12 ]. Section 2.1
brie

y summarizes informations about Fuzzy numbers, addition of Fuzzy numbers
and ranking of Fuzzy numbers in framework of possibility theory. The proposed
Algorithm is shown in Sect. 3 and a case study is presented in Sect. 4 . The
discussion and conclusions are presented in Sect. 5 .
Prior to explanation of the algorithm some basic de

fl

nitions need to be set up, the
full reference can be found in Bondy and Murty [ 12 ]. Through the work we consider
directed weighted network G(N, A) that consist of set of nodes N ={1, … ,n } and a set
of directed arcs A ={1, … ,m}. Each of these arcs is de ned by an ordered pair of

nodes (i, j) that i, j ∈ N and never i ≠ j. Since the arcs are directed then A(i, j)

≠ A(j, i).

Each of these arcs has a weight, that speci es cost for passing from start to nal

node. Usually this weight w k is speci ed as a crisp positive number and it is used to

identify the optimal path from starting node to the destination (Fig. 1 ).
The selection of optimal path is a process of selecting path from node Ni i to node
N j , where the sum of the weights is minimal. There are many algorithms that solve
the problem proposed by Bellman Ford et al. [ 7 ]. Between these algorithms the

Dijkstra algorithm [ 13 ] is undoubtedly one of the most commonly used, not only in
practical applications, but also in scienti

-

c studies [ 4 ].

2 Fuzzy Numbers and Possibility Theory

In case when there is need to model uncertainty that originates in indistinguish-
ability, vagues etc. it is not suitable to use statistical approaches and alternative
approaches is necessary [ 10 ]. Alternative framework for all essential operations can
be found in Fuzzy set theory and Possibility theory.

2.1 Fuzzy Numbers

Fuzzy numbers are special cases of convex, normal fuzzy sets de with at

least piecewise continuous membership function, that represent vague, imprecise or

ned on ℝ