A New Score Function of Pentagonal Neutrosophic Number and its Application in Networking Problem

: Pentagonal neutrosophic number is an extended version of single typed neutrosophic number. Real-humankind problems have different sort of ambiguity in nature and amongst them; one of the important problems is solving the networking problem. In this contribution, the conception of pentagonal neutrosophic number has been focused in a distinct framework of reference. Here, we develop of a new score function and its estimation have been formulated in different perspectives. Further, a time computing based networking problem is considered here in pentagonal neutrosophic arena and solved it using an influx of dissimilar logical & innovative thinking. Lastly, computation of total completion time of the problem reflects the impotency of this noble work


Introduction:
Researchers though have various fields to work on but hesitant theory is one of the vital topics in today's world to deal with. Professor Zadeh [1] was first to familiarize with the fuzzy set theory (in 1965) to handle hesitant idea. The theory of fuzziness has a leading feature to solve clear soundly engineering and statistical problem. Applying the uncertainty theory, plentiful varieties of realistic problem can be solved, networking problem, decision making problem, influence on social science, etc. Pertaining the concept of Zadeh's research paper, Atanassov [2] created phenomenally the intuitionistic fuzzy set where he meticulously elucidates the concept of membership and nonmembership function. With the going researches triangular [3], trapezoidal [4], pentagonal [5], hexagonal [6] fuzzy numbers are constructed in indefinite environment. The notion of triangular intuitionistic fuzzy set was put forth by Liu & Yuan [7]. The elementary idea of trapezoidal intuitionistic fuzzy set in research arena was constructed by Ye [8]. Unsurprisingly a basic question ascends onto our mind that how can mathematical model deal with the idea of vagueness? Different sorts of methodologies have been devised by the researchers to describe intricately the conceptions of some new uncertain parameters and to handle these complicated problems, the decision makers put forth their various ideas in disjunctive areas. F. Smarandache [9] in 1998 germinated the notion of having neutrosphic set holding three different fundamental element (i) truth, (ii) indeterminate, and (iii) falsity. Each and every attributes of the neutrosophic sets are very relevant factors to our real life models. Afterwards, Wang et al. [10] progressed with single typed neutrosophic set which serves the solution to any sort of complicated problem in a very efficient way. Later on, Chakraborty et al. [11,12] abstracted the concept of triangular and trapezoidal neutrosophic numbers and functionalized it in diversified real life problem fruitfully. Also, Maity et al. [13] constructed ranking and defuzzification using totally dissimilar sort of attributes. Bosc and Pivert [14] fostered the concept of bipolarity to deal with human decision making problem on the base of positive and negative sides. Lee [15] continued the expound the theory of bipolar fuzzy set into their research article. Broadening the hypothesis into groups and semi-groups structure field was done by Kang and Kang [16]. With ongoing researches Deli et al. [17] build up with the concept of bipolar neutrosophic number and applied it in the field of decision-making associated problem. Broumi et al. [18] put forward the concept of bipolar neutrosophic graph theory and successively Ali and Smarandache [19] proposed the perception of the indeterminate complex neutrosophic set. Chakraborty [20] acquainted us with bipolar number in distinctive aspects. Sequentially, the notion of use of operators in bipolar neutrosophic set was put forward by Wang et al. [21]. He applied in decision related problem. Researchers dealing with the evaluation of any scientific decision, multi-criteria decision making (MCDM) problem is at the utmost concerned. Nowadays utilization of group of criteria is more likely agreeable. Application of MCDM has a wider aspect in disjunctive fields under numerous skepticism frameworks. Researchers show their enthusiasm against problems relating to multi-criteria group decision making (MCGDM) problem. Several applications and progressions on neutrosophic theory could be found under multi-criteria decision making problem, moving with literature surveys showed in [22][23][24][25], graph theory [26][27][28][29][30], optimization systems [31][32][33] etc. On the recent times, Abdel [34] structured the view point of plithogenic set which had a vast significant in uncertain field in research area. Correspondingly, Chakraborty [35,36] established the outset of cylindrical neutrosophic number and applied it in networking planning problem, MCDM problem and in minimal spanning tree problem. Recently, the concept of pentagonal fuzzy number was first incubated by R. Helen [37]. Later on, utilizing this concept, Christi [38] established pentagonal intuitionistic number and solved transportation problem very proficiently. Chakraborty [39,40] set forth the idea of pentagonal neutrosophic number and its application in transportation problem and graphical research area. Also, Ye [41] manifested the idea of Single valued neutrosophic minimal spanning tree and clustering method & Mandal & Basu [42] focused on similarity measure based spanning tree problems in neutrosophic arena. Mullai et.al [43] ignited the minimum spanning tree problem & Broumi et.al [44] introduced shortest path problem in neutrosophic graphs. Also, Broumi et.al [45] manifested neutrosophic shortest path to solve Dijkstra algorithm & some published articles [46][47] are addressed here related with neutrosophic domain which plays an essential role in research arena. This paper deals with the conception of pentagonal neutrosophic number in different aspect. Nowadays researchers are very much interested in doing networking problem in neutrosophic domain. In this article, we consider a networking based PERT problem in pentagonal neutrosophic where we utilize the idea of our developed score function for solving the problem.

Motivation
The idea of vagueness plays an essential role in construction of mathematical modeling, economic problem and social real life problem etc. Now there will be a vital point that if someone considers pentagonal neutrosophic number in networking domain then what will be the final solution and the critical path? How should we convert a pentagonal neutrosophic number into crisp number? From this aspect we actually try to develop this research article.

Novelties
Till date lots of research works are already published under neutrosophic environment. Under numerous fields researches have established formulas to work on. Although many fields are unknown and works are still going on. Our job is to give a try on developing new ideas on unfamiliar points. (i) To develop score and accuracy function.
(ii) Usage of our function in networking problem. ̃ is called neut-convex indicating that ̃ is a subset of a real line by meeting the resulting conditions:

Preliminaries
i.

Proposed Score Function:
Score function utterly relies upon the value of exact membership indicator degree, inexact membership indicator degree and hesitancy membership indicator degree for a pentagonal neutrosophic number. The fundamental use of score function is to drag the judgment of conversion of pentagonal neutrosophic number to real number. A score function is developed for any Pentagonal Single typed Neutrosophic Number (PSNN).

Relationship between any two pentagonal neutrosophic fuzzy numbers:
Let us consider any two pentagonal neutrosophic fuzzy number defined as follows

PERT in Pentagonal Neutrosophic Environment and the Proposed Model
PERT system or Project Evaluation and Review Technique are a project managing scheme which is used to plan, arrange, systemize and equalize tasks amongst a project. These techniques basically examine the minimum time required in finishing the total task and also calculate the time required in completion of each task for the given project.
PERT arrangement entails the specified steps: 1. Identification of specified activities and milestones.
2. In determination of accurate sequence of the activities.
3. In construction of a network map.
4. Evaluation of time needed for ac task.
5. Determination of the critical path.
6. Updating the PERT chart on progression with the project.
The chief purpose of PERT chart is to simplify and to decrease both time and cost of completion of any decision forming project. This method is proposed for wide-ranging, one-time, non-routine difficult projects having high degree of dependence. In projects, where series of tasks are present, some are always executed successively while rest are accomplished matching with other activities. In case of new projects having huge uncertainty in technology and networking system PERT is essentially used. For handling the uncertainties, triangular neutrosophic setting For PERT activity duration has been introduced.
The three time estimations for activity duration are: Optimistic Time(̌): In general, the optimistic time requires minimum time for completing the activities and it is considered with three standards deviations from mean and approximately there is 1% chance for the activity to complete within time.  Draw the project network and find the probability that the project is completed in 5.6 days?
Step-1 Step-3 Probability that the project will be finished within 5.6 days is ( ≤

Conclusion and future research scope
The idea of pentagonal neutrosophic number is intriguing, competent and has an ample scope of utilization in various research domains. In this research article, we vigorously erect the perception of pentagonal neutrosophic number from different aspects. We introduced a score function here in pentagonal neutrosophic domain. Additionally, we consider a networking problem in neutrosophic environment and solve the problem utilizing the idea of score function. Since, there is no such articles is till now established in pentagonal networking neutrosophic arena, thus we cannot compare our work with other 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.