Computing Shortest Path in a Single Valued Neutrosophic Hesitant Fuzzy Network

M. Saad, T. Mahmood, Kifayat Ullah, N. Jan


In engineering, computer sciences and many other applied sciences, finding shortest path in a network
is one of the famous problems. The aim of this manuscript is to develop a novel algorithm for finding
shortest path in a network where nodes and edges have some uncertainty. Firstly, the concept of singlevalued
neutrosophic hesitant fuzzy graph (SVNHFG) has been introduced with some related graph
theoretic results. Some examples are provided to understand the defined concepts. Then, the new
algorithm for solving shortest path problems (SPPs) has been introduced followed by a flowchart for a
stepwise description. A numerical example is provided in the environment of SVNHFGs to demonstrate
the proposed algorithm. The advantages of proposed method over the existing techniques are also

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R. Dial, F. Glover, D. Karney and D. Klingman, ‟A computational analysis of alternative algorithms and labeling techniques for finding shortest path trees”, Networks, vol. 9, no. 3, pp. 215-248, 1979.

G.Y. Handler and I. Zang, ‟A dual algorithm for the constrained shortest path problem”, Networks, vol. 10, no. 4, pp. 293-309, 1980.

Z. Teradata, ‟Selected multicriteria shortest path problems: An analysis of complexity, models and adaptation of standard algorithms”, Int. J Ap. Mat. Com-pol., vol. 17, no. 2, pp. 269-287, 2007.

S. Okada and S. Soper, ‟A shortest path problem on a network with fuzzy arc lengths”, Fuzzy Set Syst., vol. 109, no. 1, pp. 129-140, 2000.

K.C. Lin and M.S. Chern, ‟The fuzzy shortest path problem and its most vital arcs”, Fuzzy Set Syst., vol. 58, no. 3, pp. 343-353, 1993. [6] Y. Deng, Y. Chen, Y. Zhang and S. Mahadevan, ‟Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment”, Appl. Soft. Comput., vol. 12, no. 3, pp. 1231-1237, 2012.

S.N. Chuang and J.Y. Kung, ‟The fuzzy shortest path length and the corresponding shortest path in a network”, Comp. Oper. Res., vol. 32, no. 6, pp. 1409-1428, 2005. [8] S. Broumi, A. Bakal, M. Talea, F. Smarandache and L. Vladareanu. “Applying Dijkstra algorithm for solving neutrosophic shortest path problem”, Int. Conf. Adv. Mechatronic Syst., vol. 2, pp. 412-416, November 2016. [9] S. Broumi, A. Bakal, M. Talea, F. Smarandache and L. Vladareanu, “Computation of shortest path problem in a network with SV-trapezoidal neutrosophic numbersˮ, Int. Conf. Adv. Mechatronic Syst., vol. 2, pp. 417-422, November 2016. [10] S. Broumi, A. Bakal, M. Talea, F. Smarandache and M. Ali, ‟Shortest Path Problem under Bipolar Neutrosophic Setting”, Appl. Mech. Mater, vol. 859, pp. 59-66, 2017. [11] S. Broumi, A. Bakal, M. Talea, F. Smarandache and K.K. PK, ‟Shortest path problem on single valued neutrosophic graphs”, Infinite Study, vol. 2, no. 1, pp. 1-6, 2017. [12] S. Broumi, A. Bakal, M. Talea, F. Smarandache, K.K. Kishore and R. Sahin, ‟Shortest path problem under interval valued neutrosophic setting”, J. Fundam. Appl. Sci., vol. 10, no. 4, pp. 168-174, 2018. [13] S. Broumi, A. Bakal, M. Talea, F. Smarandache and K. Ullah, ‟Bipolar Neutrosophic Minimum Spanning Tree”, Infinite Study, vol. 1, pp. 2, 2018.

J. Ye, ‟Single-valued neutrosophic minimum spanning tree and its clustering method”, J Intell. Syst., vol. 23, no. 3, pp. 311-324, 2014.

L.A. Zadeh, ‟Fuzzy sets”, Inf. Control., vol. 8, no. 3, pp. 338-353, 1965.

K.S. Atanassov, ‟Intuitionistic fuzzy sets.” Fuzzy sets and Syst., vol. 20, no. 1, pp. 87-96, 1986. [17] Mukherjee, Anjan and S. Sarkar, "Several similarity measures of interval valued neutrosophic soft sets and their application in pattern recognition problems", Neutrosophic Sets Syst., vol. 6, pp. 55-61, 2014. [18] W. Haibin, F. Smarandache, Y. Zhang, and R. Sunderraman, ‟Single valued neutrosophic sets”, Infinite Study, vol. 17, no. 1, pp. 10, 2010.

B.C. Cuong, ‟Picture fuzzy sets”, J Comp. Sci. Cyberne., vol. 30, no. 4, pp. 409, 2014.

T. Mahmood, K. Ullah, Q. Khan and N. Jan, ‟An approach towards decision making and medical diagnosis problems using the concept of spherical fuzzy sets”, Neural Comput. Appl., vol. 31, no. 11, pp. 7041-7053, 2018.

Muhammad Saad et al. / The Nucleus 56, No. 3 (2019) 123-130

K. Ullah, T. Mahmood and N. Jan, ‟Similarity Measures for T-Spherical fuzzy sets with applications in pattern recognition”, Symmetry, vol. 10, no. 6, pp. 193, 2018.

V. Torra, ‟Hesitant fuzzy sets”, Int. J Intell. Syst., vol. 25, no. 6, pp. 529-539, 2010. [23] K. Ullah, Mahmood, N. Jan, S. Broumi S and Q. Khan, ‟On Bipolar-valued hesitant fuzzy sets and their applications in multi-attribute decision making”, The Nucleus, vol. 55, no. 2, pp. 93-101, 2018. [24] T. Mahmood, K. Ullah, Q. Khan and F. Smarandache, ‟Some aggregation operators for bipolar-valued hesitant fuzzy information”, J Fundam. Appl. Sci., vol. 10, no. 4, pp. 240-245, 2018. [25] T. Mahmood, J. Ye, and Q. Khan, ‟Vector similarity measures for simplified neutrosophic hesitant fuzzy set and their applications”, J Inequ. Spec. Func., vol. 7, no. 4. pp. 176-194, 2016.

T. Mahmood and M. Munir, ‟On bipolar fuzzy subgroups”, World Appl. Sci. J, vol. 27, no. 12. pp. 1806-1811, 2013.

Q. Khan, T. Mahmood and J. Ye, ‟Multiple attribute decision-making method under hesitant single valued neutrosophic uncertain linguistic environment”, Infinite Study, vol. 8, no. 2, pp. 1-17, 2017.

N. Jan, L. Zedam , T. Mahmood, K. Ullah, Z. Ali, ‟Multiple attribute decision making method under linguistic cubic information”, J Intell. Fuzzy Syst., vol. 36, no. 1, pp. 253-269, 2019.

A. Kaufmann, ‟Introduction à la théorie des sous-ensembles flous à l'usage des ingénieurs”, Élémentsthéoriques de base, vol. 1, pp. 41-189, 1973.

A. Rosenfeld, ‟Fuzzy graphs, in Fuzzy sets and their applications to cognitive and decision processes”, Elsevier, 1975.

R. Parvathi and M. Karunambigai, ‟Intuitionistic fuzzy graphs, in Computational Intelligence, Theory and Applications”, Springer, 2006.

R. Parvathi, M. Karunambigai and K. Atanassov. ‟Operations on intuitionistic fuzzy graphs”, Fuzzy Syst. IEEE Int. Conf. Fuzzy Syst., vol. 51, no. 5, pp. 1396-1409, 2009.

R. Parvathi, S. Thilagavathi and M. Karunambigai, ‟Intuitionistic fuzzy hypergraphs”, Cyberne. Info. Tech., vol. 9, no. 2, pp. 46-53, 2009. [34] R. Parvathi, S. Thilagavathi, G. Thamizhendhi and M.G. Karunambigai, ‟Index matrix representation of intuitionistic fuzzy graphs”, Notes Intuitionistic Fuzzy Sets, vol. 20, no. 2, pp. 100-108, 2014.

G. Pasi, R. Yager, and K. Atanassov. ‟Intuitionistic fuzzy graph interpretations of multi-person multi-criteria decision making”, 2nd Int. IEEE Conf. Gen Net Appro. Intell. Syst., vol. 2, pp. 434-439, 2004.

S.S. Dhavudh and R. Srinivasan, ‟Intuitionistic fuzzy graphs of second type”, Adv. Fuzzy Math., vol. 12, no. 2, pp. 197-204, 2017.

B. Davvaz, N. Jan, T. Mahmood and K. Ullah, ‟Intuitionistic fuzzy graphs of nth type with applications”, J Intell. Fuzzy Syst., vol. 36, no. 4, pp. 3923-3932, 2018. [38] Yaqoob, Naveed, M. Gulistan, S. Kadry and H.A. Wahab, ‟Complex intuitionistic fuzzy graphs with application in cellular network provider companies”, J Math., vol. 7, no. 1, pp. 35, 2019. [39] Hussain, S. Satham, R.J. Hussain, Y.B Jun and F. Smarandache, ‟Neutrosophic bipolar vague set and its application to neutrosophic bipolar vague graphs”, Neutrosophic Sets Syst., vol. 28, no. 1, pp. 8, 2019. [40] S. Broumi, M. Talea, A. Bakali and F. Smarandache, ‟Single valued neutrosophic graphs”, J New theory, vol. 10, pp. 86-101, 2015. [41] S. Broumi, M. Talea, A. Bakali and F. Smarandache, ‟Interval valued neutrosophic graphs”, Crit. Rev., vol. 10, pp. 5-33, 2016. [42] S. Broumi, M. Talea, A. Bakali and F. Smarandache, ‟On strong interval valued neutrosophic graphs”, Crit. Rev., vol. 12, pp. 49-71, 2016. [43] S. Broumi, K. Ullah, A. Bakali and M. Talea, ‟Novel system and method for telephone network planning based on neutrosophic graph”, Infinite Study, vol. 10, no. 4, pp. 403-434, 2018. [44] M. Gulistan, N. Yaqoob, Z. Rashid, F. Smarandache and H. A. Wahab, ‟A study on neutrosophic cubic graphs with real life applications in industries”, Symmetry, vol. 10, no. 6, pp. 203, 2018.

C. Zhang and D. Li, ‟Hesitant fuzzy graph and its application in multi-attribute decision making”, Int. J Patran. Recogn. vol. 30, no. 11, pp. 1012-1018, 2017. [46] Rashid, Sheikh, N. Yaqoob, M. Akram and M. Gulistan, ‟Cubic graphs with application”, Int. J Anal. Applications, vo. 16, no. 5, pp. 733-750, 2018.

J. Ye, ‟Multiple-attribute decision-making method under a single-valued neutrosophic hesitant fuzzy environment”, J Intell. Fuzzy. Syst., vol. 24, no. 1, pp. 23-36, 2015.


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