Computing Conference 2017 18-20 July2017 | London, UK 1 | Page Shortest Path Problem Under Trapezoidal Neutrosophic Information Said Broumi Laboratory of Information processing, Faculty of Science Ben M’Sik, University Hassan II, B.P 7955, Sidi Othman, Casablanca, Morocco broumisaid78@gmail.com Mohamed Talea Laboratory of Information processing, Faculty of Science Ben M’Sik, University Hassan II, B.P 7955, Sidi Othman, Casablanca, Morocco taleamohamed@yahoo.fr Assia Bakali Ecole Royale Navale, Boulevard Sour Jdid, B.P 16303 Casablanca, Morocco. assiabakali@yahoo.fr Florentin Smarandache Department of Mathematics, University of New Mexico,705 Gurley Avenue, Gallup, NM 87301, USA fsmarandache@gmail.com Abstract— In this study, we propose an approach to determine the shortest path length between a pair of specified nodes s and t on a network whose edge weights are represented by trapezoidal neutrosophic numbers. Finally, an illustrative example is provided to show the applicability and effectiveness of the proposed approach. Keywords—trapezoidal fuzzy neutrosophic sets; score function; Shortest path problem I. INTRODUCTION In 1995, Smarandache introduced the concept of Neutrosophy. It is a branch of philosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Smarandache [1] introduced the concept of neutrosophic set (NS) and neutrosophic logic as generalization of the concepts of fuzzy sets [3], intuitionistic fuzzy sets [4]. Neutrosophic set has the ability to deal with certain type of uncertain information such as incomplete, indeterminate and inconsistent information, which exist in real world, cannot be dealt with fuzzy sets as well as intuitionistic fuzzy sets. The concept of neutrosophic set is characterized by three independent membership degrees namely truth-membership degree (T), indeterminacy- membership degree (I), and falsity-membership degree (F). In order to practice NSs in real –life applications conveniently. Wang et al. [5] introduced,, a subclass of the neutrosophic sets, called single-valued neutrosophic sets in which the values of the three membership function T, I, F belongs to the unit interval [0, 1]. SVNS was studied deeply by many researchers. The concept of single valued neutrosophic sets has caught attension to the researcher on various topics such as to be such as the decision making problem, medical diagnosis and so on. Additional literature on single valued neutrosophic can be found in [6-14]. Also Later on, Smarandache extended the neutrosophic set to neutrosophic overset, underset, and offset [15]. However, in uncertain and complex situations, the truth-membership, the indeterminacy- membership and the falsity-membership independently of SVNS cannot be represented with exact real numbers or interval numbers Moreover, triangular fuzzy number can handle effectively fuzzy data rather than interval number. For this purpose, Biswas et al. [16] proposed the concept of triangular fuzzy number neutrosophic sets (TFNNS) by combining triangular fuzzy numbers (TFNs) and single valued neutrosophic set (SVNS) and define some of its operational rules and developed triangular fuzzy neutrosophic number weighted arithmetic averaging and triangular fuzzy neutrosophic number weighted arithmetic geometric averaging operators to solve multi-attribute decision making problem . In TFNNS the truth, indeterminacy and the falsity-membership functions are expressed with triangular fuzzy numbers instead of real numbers. In addition, Ye [17] presented the concept of trapezoidal fuzzy neutrosophic set and developed trapezoidal fuzzy neutrosophic number weighted arithmetic averaging and trapezoidal fuzzy neutrosophic number weighted arithmetic geometric averaging operators to solve multi-attribute decision making problem. Very Recently, Broumi et al. [18-26] presented the concept of neutrosophic graphs, interval valued neutrosophic graphs and bipolar single valued neutrosophic graphs. Smarandache and Kandasamy [27-29] proposed another variant of neutrosophic graphs based on literal indeterminacy. The shortest path problem is one of the most fundamental problems in graph theory which has many applications diversified field such operation research, computer science, communication network and so on. In a network, the shortest path problem concentrate at finding the path from one source to destination node with minimum weight, where some weight is attached to each edge connecting a pair of nodes. In the literature, many shortest path problems [30-39] that have been studied with different types of input data, including fuzzy set, intuitionistic fuzzy sets, trapezoidal intuitionistic fuzzy sets vague set. Till now, few research papers deal with shortest path in neutrosophic environment. Broumi et al. [40] proposed an algorithm for solving neutrosophic shortest path problem based on score