Shortest path problem using Bellman algorithm under neutrosophic environment
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Abstract
An elongation of the singlevalued neutrosophic set is an intervalvalued neutrosophic set. It has been demonstrated to deal indeterminacy in a decisionmaking problem. Realworld problems have some kind of uncertainty in nature and among them; one of the influential problems is solving the shortest path problem (SPP) in interconnections. In this contribution, we consider SPP through Bellman’s algorithm for a network using intervalvalued neutrosophic numbers (IVNNs). We proposed a novel algorithm to obtain the neutrosophic shortest path between each pair of nodes. Length of all the edges is accredited an IVNN. Moreover, for the validation of the proposed algorithm, a numerical example has been offered. Also, a comparative analysis has been done with the existing methods which exhibit the advantages of the new algorithm.
Keywords
Intervalvalued neutrosophic numbers Ranking methods Shortest path problem Bellman’s algorithm Directed graph networkIntroduction and review of the literature
A tool which represents the partnership or relationship function is called a Fuzzy Set (FS) and handles the realworld 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 nonmembership (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 intervalbased neutrosophic linguistic sets [6], Nvalued interval neutrosophic sets (NVINSs) [7], generalized intervalvalued triangular intuitionistic FSs [8], intervalvalued trapezoidal intuitionistic FSs [9], intervalvalued Pythagorean FSs [10], intervalvalued 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 nonmembership 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, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 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 intervalvalued 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, 23, 24, 25, 26, 27, 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 realworld 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 multiobjective. 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 intervalvalued FSPP for intervalvalued type and developed a technique to solve SPP.
Elizabeth and Sujatha [33] solved FSPP using intervalvalued fuzzy matrices. Based on traditional Dijkstra algorithm, Enayattabar et al. [34] solved SPP in the intervalvalued 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].
Authors’ contributions towards neutrosophic shortest path problem
Author and references  Year  Contribution 

Broumi et al. [36]  2016  Solved NSPP using Dijkstra algorithm 
Broumi et al. [37]  2016  Solved NSPP for intervalbased data using Dijkstra algorithm 
Broumi et al. [38]  2016  Discovered the SP using SVTpNNs 
Broumi et al. [40]  2016  Worked out SPP using singlevalued neutrosophic graphs 
Broumi et al. [41]  2017  Solved SPP under neutrosophic setting as well as trapezoidal fuzzy 
Broumi et al. [42]  2017  Solved SPP under bipolar neutrosophic environment. 
Broumi et al. [43]  2017  Dealt SPP under intervalvalued neutrosophic setting 
Broumi et al. [44]  2018  Proposed maximizing deviation method with partial weight in a decisionmaking problem under the neutrosophic environment 
This paper  –  Introduction of the neutrosophic version of a Bellman’s algorithm 

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.
Overview on intervalvalued neutrosophic set
In this part, we recall few primary notions pertaining to NSs, SVNSs, IVNSs and some existing ranking functions for IVNNs which are the background of this study and will help us to further research.
Definition 1 [13]
The values of the three MS functions are taken from ]^{−}0,1^{+}[. As we have difficulty of applying NSs to realtime issues, Wang et al. [14] put forward the approach of a SVNS, which is the simplification of a NS and can be applied to any realworld topic.
Definition 2 [14]
Definition 3 [15]
Now we consider a few mathematical operations on intervalvalued neutrosophic numbers (IVNNs)s.
Definition 4 [15]
Deneutrosophication formulas for intervalvalued neutrosophic numbers
 (i)
\( \dddot A_{1} < \dddot A_{2} \) if \( {\text{SF}}(\dddot A_{1} ) < {\text{SF}}(\dddot A_{2} ). \)
 (ii)
\( \dddot A_{1} > \dddot A_{2} \) if \( {\text{SF}}(\dddot A_{1} ) > {\text{SF}}(\dddot A_{2} ). \)
 (iii)
\( \dddot A_{1} = \dddot A_{2} \) if \( {\text{SF}}(\dddot A_{1} ) = {\text{SF}}(\dddot A_{2} ). \)
Computation of the shortest path based on neutrosophic numbers
In this section, the new algorithmic approach to solve IVNSP is provided. It is pretended that there are \( n \) nodes with the source node (SN), node 1 and destination node (DN), node n. The neutrosophic length between nodes \( i \) and \( j \) is denoted by \( d_{ij} \) and the set of all nodes having a connection with the node \( i \) is denoted by \( M_{N\left( i \right)} \).
Mathematical formulation of BELLMAN dynamic programming
In the posterior section, we present a simple illustration to show the brevity of our method.
Illustrative example
This part is based on a numerical problem adapted from [43] to show the potential application of the proposed algorithm.
Example 1
The details of edge information in terms of intervalvalued neutrosophic numbers
Edges  IVN distance  Edges  IVN distance 

1–2  \( \left( {\left[ {0.1,0.2} \right],\left[ {0.2,0.3} \right],\left[ {0.4,0.5} \right]} \right) \)  3–4  \( \left( {\left[ {0.2,0.3} \right],\left[ {0.2,0.5} \right],\left[ {0.4,0.5} \right]} \right) \) 
1–3  \( \left( {\left[ {0.2,0.4} \right],\left[ {0.3,0.5} \right],\left[ {0.1,0.2} \right]} \right) \)  3–5  \( \left( {\left[ {0.3,0.6} \right],\left[ {0.1,0.2} \right],\left[ {0.1,0.4} \right]} \right) \) 
2–3  \( \left( {\left[ {0.3,0.4} \right],\left[ {0.1,0.2} \right],\left[ {0.3,0.5} \right]} \right) \)  4–6  \( \left( {\left[ {0.4,0.6} \right],\left[ {0.2,0.4} \right],\left[ {0.1,0.3} \right]} \right) \) 
2–5  \( \left( {\left[ {0.1,0.3} \right],\left[ {0.3,0.4} \right],\left[ {0.2,0.3} \right]} \right) \)  5–6  \( \left( {\left[ {0.2,0.3} \right],\left[ {0.3,0.4} \right],\left[ {0.1,0.5} \right]} \right) \) 
The details of deneutrosophication value of edge (i, j)
Edges  \( S_{\text{Ridvan }} \)  \( S_{\text{Liu }} \) 

1–2  0.1  1.45 
1–3  0.175  1.75 
2–3  0.325  1.8 
2–5  0.125  1.6 
3–4  0.05  1.45 
3–5  0.45  2.05 
4–6  0.35  2 
5–6  0.125  1.6 
Therefore, the path P: 1 → 2 → 5 → 6 is recognized as the neutrosophic shortest path, and the crisp shortest path is 0.35.
Contingent study
Comparison of the sequence of nodes using neutrosophic shortest path and our proposed algorithm
Possible path  Sequence of nodes  Crisp shortest path length 

Neutrosophic shortest path with intervalvalued 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.
Advantages and limitations of the proposed algorithm
Advantages
 1.
Singlevalue neutrosophic numbers.
 2.
Bipolar neutrosophic numbers.
 3.
Trapezoidal neutrosophic numbers.
 4.
Cubic neutrosophic numbers.
 5.
Interval bipolar neutrosophic numbers.
 6.
Triangular neutrosophic numbers and so on.
Limitations
 1.
Slow response will be observed when there is a change in the network as this change will spread nodebynode.
 2.
If node failure occurs then routing loops may exist.
Conclusion
In this study, we describe the NSP, where edge weights are represented by IVNS. The advantage of using IVNSs in NSP is discussed in this paper. The classical Bellman’s algorithm is modified by incorporating the uncertainty using IVNSs for NPP between source and destination nodes. We use a numerical example to illustrate the efficiency of our proposed algorithm. The main goal of this work is to describe an algorithm for NSP in the neutrosophic environment using IVNS as edge weight. The proposed algorithm is very effective for reallife problem. In this paper, we have used a simple numerical example to illustrate our proposed algorithm. Therefore, as future work, we need to consider a largescale practical shortest path problem using our proposed algorithm and to compare our proposed algorithm with the existing algorithm in terms of strictness of optimality, efficiency, computational time, and other aspects.
Notes
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