Application of Pentagonal Neutrosophic Number in Shortest Path Problem

: Real-human kind issues have distinct sort of ambiguity and among them; one of the critical troubles is solving the shortest path problem. In this contribution, we applied the developed score function and accuracy function of pentagonal neutrosophic number (PNN) into a shortage path selection problem. Further, a time dependent and heuristic cost function related shortest path algorithm is considered here in PNN area and solved it utilizing an influx of dissimilar rational & pioneer thinking. Lastly, estimation of total ideal time of the graph reflects the importance of this noble work.

how he/she will managed it and solve it? How we can relate PNN and crispificaton result ? From this aspect we actually try to develop this research article.

Novelties
Only some of the articles are published in PNN area till now. Although it can be applied in many fields and compute the results there. Our main focus is to apply the established PNN number in different areas. (i) Develop score and accuracy function.
(ii) Application of our established score function in shortest path problem.

Relationship between any two PNN:
Let us consider any two PNN defined as follows

Shortest Path Search Algorithm under PNN Environment:
Here we consider a problem in PNN environment to compute the shortest path in a very simple way. Shortest path Search Algorithm is one of the best and popular skills used in path finding and graph traversals. Many games and web-based graphs are used here to compute the shortest path very efficiently. This Algorithm finds the shortest path through the search space using the hemistich function. It uses a best first search graph algorithm and finds a least cost path from current node to destination node. Consider a weighted graph in PNN area [10] whose weights and heuristic cost function are given as a pentagonal neutrosophic number with multiple nodes and we want to reach the target node to starting node as quick as possible. It defined a heuristic cost function ( ) = ( ) + ℎ( ) where ( )= estimated cost of the cheapest solution, ( )= cost to reach node from the starting position, ℎ( ) = estimated heuristic cost. Time and cost these are hesitation factors in case of real life problem. Here we consider the functions ( ), ℎ( ) are both pentagonal neutrosophic number. This algorithm expands less search tree and computes the optimal result faster.

Algorithm
Step1: Convert all the PNN into crisp number using the established score value Section (3).

Step2: Placed the starting node to open list
Step 3: Check whether the open list is empty or not, if the list is empty then stop the process.
Step 4: choose the node from the open list, which has the least value of estimation function ( ), if node " " is target node then back to success and stop.
Step 5: Expand node "n" and produce all of its successes and put "n" in the closed list.
 For each successes "n", check whether "n" is already in the open or closed list.  If not then compute evaluation function for "n" and placed it into open list.
Step 6: Else if node "n" is already in open and closed then it should be attached to the back pointer which reflects the lowest ( ) value.

Connection
Edges Step-1 Network Diagram: Step-2 Crispification using the established Score function ( Step-3 Here, A is the starting node

Conclusion and future research scope
The concept of PNN has an adequate scope of utilization in various studies in different domain. In this research article, we strongly erect the perception of score and accuracy function from different aspects. Additionally, we consider a shortest path problem in PNN environment and resolve the problem applying the idea of score function.
Since, there may be no such articles is until now hooked up in PNN area, for this reason we cannot done comparison study of our work with the other established methods.
Further, researchers can immensely apply this idea of neutrosophic number in numerous flourishing research fields like engineering problem, mobile computing problems, diagnoses problem, realistic mathematical modeling, social media problem etc.