We consider an intuitionistic fuzzy shortest path problem (IFSPP) in a directed graph where the weights of the links are intuitionistic fuzzy numbers. We develop a method to search for an intuitionistic fuzzy shortest path from a source node to a destination node. We coin the concept of classical Dijkstra’s algorithm which is applicable to graphs with crisp weights and then extend this concept to graphs where the weights of the arcs are intuitionistic fuzzy numbers. It is claimed that the method may play a major role in many application areas of computer science, communication network, transportation systems, and so forth. in particular to those networks for which the link weights (costs) are ill defined.

Graphs [

One of the first studies on fuzzy shortest path problem (FSPP) in graphs was done by Dubois and Prade [

A graph

The intuitionistic fuzzy set theory of Atanassov [

In most of the real-life problems of networks, be it in a communication model or transportation model, the weights of the arcs are not always crisp but intuitionistic fuzzy numbers (or, at best fuzzy numbers). For example, Figure

But for such type of ill graph, there is no attempt made so far in the literature for searching an IF shortest path. In our method here, we solve this intuitionistic fuzzy shortest path problem (IFSPP) for graphs where we also use the notion of Dijkstra’s algorithm but with simple soft-computations without using any hybrid geometric operators, using only basics of Atanassov’s operators [

In this section we solve the IFSPP for graphs where we use the philosophy of Dijkstra’s algorithm but with simple soft-computations with IF data. Consider a directed graph

A graph

As shown in Figure

IF estimation procedure for

Diagram showing how the IF-RELAX algorithm works in a graph.

We extend the classical notion of relaxation to the case here with intuitionistic fuzzy number weights. We call it “IF relaxation.” For this, first of all we initialize the graph along with its starting vertex and IF shortest path estimate for each vertices of the graph

(1)

(2)

(3)

(4)

After the IF initialization, the process of IF relaxation of each arc begins, as shown in Figure

(1) IF

(2)

(3)

We now present our main algorithm to find single source IF shortest path in a graph. We name this “intuitionistic fuzzy shortest path algorithm,” that is, in short by the title IFSP algorithm. In this algorithm we use the previously designed above subalgorithms and also the subalgorithm EXTRACT-IF-MIN (

(1) IFISS (

(2)

(3)

(4)

(5) WHILE

(6)

(7)

(8) FOR each rn vertex

(9) DO IF

Consider the following directed graph

A graph

Our algorithm computes the following results:

with

There are many real-life problems in the networks of transportation, communication, circuit systems, and so forth which are initially modeled into graphs and hence solved. In many of these directed graphs, in reality, the weights of the arcs are not always crisp but fuzzy numbers. In this paper we develop a new method to solve the intuitionistic fuzzy shortest path problem (IFSPP) from a source vertex to a destination vertex in a directed graph. The importance of our method lies in its potential to give solution in intuitionistic fuzzy environment, unlike any of the existing algorithms of IFSPP. Obviously, our algorithm does also work in case few or all of the weights are fuzzy numbers or crisp numbers, as a special case of IF numbers.