Introduction and review of the literature
A tool which represents the partnership or relationship function is called a Fuzzy Set (FS) and handles the real-world problems in which generally some type of uncertainty exists [
1]. This concept was generalized by Atanassov [
2] to intuitionistic fuzzy set (IFS) which is determined in terms of membership (MS) and non-membership (NMS) functions, the characteristic functions of the set. Beside this, several theories have been developed for uncertainties, including generalized orthopair FSs [
3], Pythagorean FSs [
4], picture FSs [
5], hesitant interval-based neutrosophic linguistic sets [
6], N-valued interval neutrosophic sets (NVINSs) [
7], generalized interval-valued triangular intuitionistic FSs [
8], interval-valued trapezoidal intuitionistic FSs [
9], interval-valued Pythagorean FSs [
10], interval-valued IFSs [
11], and interval type 2 FSs [
12].
In 1995, Smarandache [
13] premises the theme of neutrosophic sets (NS). The NS is to be a set of elements having a membership degree, indeterminate membership and also non-membership with the criterion less than or equal to 3. The neutrosophic number is an exceptional type of neutrosophic sets that extend the domain of numbers from those of real numbers to neutrosophic numbers. By generalizing SVNSs [
14], Wang et al. premised the idea of IVNS. The IVNS [
15] is a more general database to generalize the concept of different types of sets to express membership degrees’ truth, indeterminacy, and a false degree in terms of intervals. Thus, several papers are published in the field of fuzzy and neutrosophic sets [
46‐
62].
Harish [
16] proposed and analyzed an extension of the score function by incorporating hesitance. The authors presented an algorithm for the function including qualitative examples. Jun et al. [
17] discuss INSs in algebra of BCK/BCI. Mehmet [
18] put forward for analyzing the concept of the interval cut set (ICS) and strong ICS (
α,
β,
γ) of IVNSs with proof and examples. Also, there are other several extensions of NSs described in the literature including interval-valued bipolar neutrosophic sets [
19], hesitant interval neutrosophic linguistic set [
20], and interval neutrosophic hesitant fuzzy sets [
21]; for more details of neutrosophic set and their extensions, we refer the reader to [
22‐
28].
Among humanistic problems of computer science, finding the shortest path is one of the significant problems. Many of the algorithms existing for optimization assumed the edge weights as the absolute real numbers. Despite this, we need to deal inexplicit parameters such as scope, costs, time and requirements in real-world problems. For example, a substantial length of any road is permanent; still, traveling time along the road varies according to weather and traffic conditions. An uncertain fact of those cases directs us to adopt fuzzy logic, fuzzy numbers, intuitionistic fuzzy and so on. The SPP using fuzzy numbers is called fuzzy shortest path problem (FSPP). Several researchers are paying attention in fuzzy shortest path (FSP) and intuitionistic FSP algorithms.
Das and De [
29] employed Bellman dynamic programming problem for solving FSP based on value and ambiguity of trapezoidal intuitionistic fuzzy numbers. De and Bhincher [
30] have studied the FSP in a network under triangular fuzzy number (TFN) and trapezoidal fuzzy number (TpFN) using two approaches such as influential programming of Bellman and linear programming with multi-objective. Kumar et al. [
31] proposed a model to find the SP of the network under intuitionistic trapezoidal fuzzy number based on interval value. Meenakshi and Kaliraja [
32] formulated interval-valued FSPP for interval-valued type and developed a technique to solve SPP.
Elizabeth and Sujatha [
33] solved FSPP using interval-valued fuzzy matrices. Based on traditional Dijkstra algorithm, Enayattabar et al. [
34] solved SPP in the interval-valued pythagorean fuzzy setting. Dey et al. [
35] formulated fuzzy shortest path problem with interval type 2 fuzzy numbers. But, if the indeterminate information has appeared, all these kinds of shortest path problems failed. For this reason, some new approaches have been developed using neutrosophic numbers. Then neutrosophic shortest path was first developed by Broumi et al. [
36]. The authors in [
36] constructed an extension of Dijkstra algorithm to solve neutrosophic SPP. Then they used the extended version to treat the NSPP where the edge weight is characterized by IVNNs [
37].
Broumi et al. [
38‐
40] first introduced a technique of finding SP under SV-trapezoidal and triangular fuzzy neutrosophic environment. In [
41], the authors proposed another approach to solve SPP on a network using trapezoidal neutrosophic numbers. Broumi et al. [
42] developed a new algorithm to solve SPP using bipolar neutrosophic setting. In another paper, Broumi et al. [
43] discussed an algorithmic approach based on a score function defined in [
44] for solving NSPP on a network with IVNN as the edges. Liu and You proposed interval neutrosophic Muirhead mean operators and their applications in multiple-attribute group decision-making [
45]. Thus, several papers are published in the field of neutrosophic sets [
46‐
55]. Table
1 summarizes some contributions towards NSPP. Based on the idea of Bellman’s algorithm, SPP is solved for fuzzy network [
29‐
32]. This algorithm is not applied yet on neutrosophic network. Therefore, there is a need to establish a neutrosophic version of Bellman’s algorithm for neutrosophic shortest path problems.
Table 1
Authors’ contributions towards neutrosophic shortest path problem
| 2016 | Solved NSPP using Dijkstra algorithm |
| 2016 | Solved NSPP for interval-based data using Dijkstra algorithm |
| 2016 | Discovered the SP using SV-TpNNs |
| 2016 | Worked out SPP using single-valued neutrosophic graphs |
| 2017 | Solved SPP under neutrosophic setting as well as trapezoidal fuzzy |
| 2017 | Solved SPP under bipolar neutrosophic environment. |
| 2017 | Dealt SPP under interval-valued neutrosophic setting |
| 2018 | Proposed maximizing deviation method with partial weight in a decision-making problem under the neutrosophic environment |
This paper | – | Introduction of the neutrosophic version of a Bellman’s algorithm |
The main motivation of this study is to introduce an algorithmic approach for SPP in an uncertain environment which will be simple enough and effective in real-life problem. The main contributions of this paper are as follows.
-
We concentrate on a NSP on a neutrosophic graph in which an IVNN, instead of a real number/fuzzy number, is assigned to each arc length.
-
A modified Bellman’s algorithm is introduced to deal the shortest path problem in an uncertain environment.
-
Based on the idea discussed in [
15], we use an addition operation for adding the IVNNs corresponding to the edge weights present in the path. It is used to find the path length between source and destination nodes. We also use a ranking method to choose the shortest path associated with the lowest value of rank.
In this work, we are motivated to solve SPP by introducing a new version of BA where the edge weight is represented by IVNNs. The remaining part of the paper is presented as follows. The next section contains a few of the ideas and theories as overview of interval neutrosophic set followed by which the Bellman algorithm is discussed. In the subsequent section, an analytical illustration is presented, where our algorithm is applied. Then contingent study has been done with existing methods. Before the concluding section, advantages of the proposed algorithm are presented. Finally, conclusive observations are given.
Contingent study
In this section, the analysis of contingency for the proposed algorithm with existing approaches has been analyzed. A comparison of the results between the existing and new technique is shown in Table
4.
Table 4
Comparison of the sequence of nodes using neutrosophic shortest path and our proposed algorithm
Neutrosophic shortest path with interval-valued neutrosophic numbers [ 43] | 1 → 2 → 5 → 6 |
\( \left( {\left[ {0.35, 0.60} \right], \left[ {0.01, 0.04} \right], \left[ {0.008, 0.075} \right]} \right) \)
|
PA based on \( S_{\text{Ridvan }} \) | 1 → 2 → 5 → 6 |
\( 0.35 \)
|
PA based on \( S_{\text{Liu }} \) | 1 → 2 → 5 → 6 |
\( 4.65 \)
|
From the result, it is shown that the introduced algorithm contributes sequence of visited nodes which shown to be similar to neutrosophic shortest path presented in [
43].
The neutrosophic shortest path (NSP) remains the same, namely 1 → 2 → 5 → 6, but the neutrosophic shortest path length (NSPL) differs, namely \( \left( {\left[ {0.424, 0.608} \right], \left[ {0.012, 0.06} \right], \left[ {0.016, 0.125} \right]} \right), \) respectively. From here we come to the conclusion that there exists no unique method for comparing neutrosophic numbers and different methods may satisfy different desirable criteria.
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