In problems of graphs involving uncertainties, the fuzzy shortest path problem is one of the most studied topics, since it has a wide range of applications in different areas and therefore deserves special attention. In this paper, algorithms are proposed for the fuzzy shortest path
problem, where the arc length of the network takes imprecise numbers, instead of real numbers, namely, level

Many researchers have focused on fuzzy shortest path problem in a network, since it is important to many applications such as communications, routing, and transportation. 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. The fuzzy set theory, proposed by Zadeh [

The fuzzy shortest path problem was first analysed by Dubois and Prade [

This paper is organized as follows. In Section

A digraph is a graph each of whose edges are directed. Hence, an acyclic digraph is a directed graph without cycle.

Level

Level

Let

Let

In this paper, we introduce the following definitions.

Let the level

Consider Figure

Let

For

Level

Let the level

the level

the level

Generalizing Hamming distance and Euclidean distance results in Minkowski distance. It becomes the Hamming distance (HD) for

Let the level

the level

the level

A fifth type of graph fuzziness occurs when the graph has known vertices and edges, but unknown weights (or capacities) on the edges. Thus, only the weights are fuzzy.

To plan the quickest automobile route from one city to another. Unfortunately, the map gives distances, not travel times, so it is not known exactly how long it takes to travel any particular road segment.

For each

For the sake of verification, Definition

Nayeem and Pal extended the acceptability index originally proposed by Sengupta and Pal [

If

To find the fuzzy shortest path, the above acceptability index was slightly modified in [

The acceptability index defined in [

A fuzzy number

Consider Figure

if

Triangular LR acceptability index.

In many practical situations, we often need to employ a measurement tool to distinguish between two similar sets or groups. For this purpose, several similarity measures had been presented in [

In this paper, we introduce a new method called intersection index which acts as the measurement tool between

digraph and the arc length takes the level

Construct a network with 6 vertices and 7 edges as cited in Figure

let

Assume the arc lengths as

The possible paths and the corresponding path lengths are as follows:

See Table

Path

Results of the network based on level

Paths | Ranking | |
---|---|---|

0.11 | 3 | |

0.54 | 2 | |

0.7 | 1 |

The classical network.

See Tables

Results of the network based on level

Paths | Ranking | |
---|---|---|

63.5 | 3 | |

60 | 2 | |

58 | 1 |

Path

Results of the network based on level

Paths | Ranking | |||

43 | 25.5 | 21.82 | 3 | |

15 | 9 | 7.78 | 2 | |

7 | 7 | 7 | 1 |

Path

using Definition

identified as the shortest path and the corresponding path length is the shortest path length.

They are the same as in Example

See Table

Path

Results of the network based on level

Paths | Ranking | ||
---|---|---|---|

71.5 | 70.67 | 3 | |

61 | 61.33 | 2 | |

59 | 58.67 | 1 |

One way to verify the solution obtained is to make an exhaustive comparison, now comparing the result obtained in this paper with the existing results to generalize our proposed approach. See Tables

Results of the network based on signed distance ranking index.

Paths | Ranking | |
---|---|---|

70.25 | 3 | |

61.5 | 2 | |

58.5 | 1 |

Path

Results of the network based on acceptability index.

Paths | Ranking | ||

0.59 | 0.85 | 3 | |

0.16 | 0.22 | 2 | |

0 | 0 | 1 |

Path

Hence, we find that the solution obtained for FSPP in this paper coincides with the solution of the existing methods.

Fuzzy shortest path length and the shortest path are the useful information for the decision makers in a fuzzy shortest path problem. Due to its practical application, many researchers have focused on the fuzzy shortest path problem, and some algorithms were developed for the same. Hence, in this paper, we have defined few indices and developed the new algorithms based on them and verified with the existing methods. The ranking given to the paths is helpful for the decision makers as they make decision in choosing the best of all the possible path alternatives. Hence, we conclude that the algorithms developed in the current research are the simplest and the alternative method for getting the shortest path in fuzzy environment.

The authors are grateful to the referees for their suggestions to improve the presentation of the paper.