1 Affiliation : Department of Mathematics Lahore Garrison University, Av-4, Sec-C, DHA Phase VI, 54000 Lahore Pakistan
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2 Affiliation : Department of Mathematics, TED University, Ön Cebeci, Ziya Gökalp Cd. 48/A, 06420 Çankaya/Ankara, Turkey
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3 Affiliation : Department of Mathematics Lahore Garrison University, Av-4, Sec-C, DHA Phase VI, 54000 Lahore Pakistan
Email : firstname.lastname@example.org
4 Affiliation : Department of Mathematics Lahore Garrison University, Av-4, Sec-C, DHA Phase VI, 54000 Lahore Pakistan
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The concept of neutrosophic become really handy now a days and based on non-standard analysis to mention mathematical outcomes, uncertainty, non-completed situations, inconsistency, distinctness. The main concept of Neutrosophic set based on membership values of truth, indeterminacy and falsity, which are independent, and which play vital role in situations like uncertainty, incomplete and inconsistence. From triangular to octagonal Neutrosophic number. They play vital role in modeling problems, science, biology and many more. Hence it is clear that these are necessary and have real life applications, but some real-life problems have more edges and their triangular to octagonal fail to overcome this situation (mention in table 1). Hence, nonagonal neutrosophic numbers give a wide scope of utilizations while managing more variances in the decision-making condition with nine edges for membership values of truth, indeterminacy and falsity. In this current article we present compression between triangular to nonagonal neutrosophic number and their requirement, explore differential equations in Neutrosophic environment as Linear, symmetric and asymmetric types further, their ∂-cute and then we present a real-life problem and solved it with TOPSIS technique of MCDM.
Accuracy function , Neutrosophic number , Nonagonal Neutrosophic numbers (NNN) , MCDM , TOPSIS.
 Molodtsov, D. Soft set theory - First results, Computers and mathematics with applications. 37, pp.19- 31,1999.
 Zadeh, L.A. Fuzzy sets. Inf. Control , 8, pp.338–353, 1965.
 Chang, S.S.L.; Zadeh, L.A. On fuzzy mappings and control. IEEE Trans. Syst. Man Cybern, 2, pp.30–34,1972.
 Dubois, D.; Prade, H. Operations on fuzzy numbers. Int. J. Syst. Sci, 9, pp.613–626, 1978.
 Atanassov, K.T. Intuitionistic Fuzzy Sets; VII ITKR’s Session: Sofia, Bulgarian, 1983.
 Smarandache, F. A Unifying Field in Logics Neutrosophy: Neutrosophic Probability; American ResearchPress: Rehoboth, DE, USA, 1998.
 F. Smarandache, Neutrosophic set, a generalization of the intuitionistic fuzzy sets, Inter. J. Pure Appl. Math. Vol.24 , pp. 287–297, 2005.
 Chakraborty, A.; Mondal, S. P.; Ahmadian, A.; Senu, N.; Alam, S.; and Salahshour, S.; Different Forms of Triangular Neutrosophic Numbers, De-Neutrosophication Techniques, and their Applications, Symmetry, vol 10, 327, 2018. ; doi:10.3390/sym10080327.
 Chakraborty. A., Sankar P. Mondal, Shariful A. Ali A., Norazak S., Debashis De. and Soheil S., The Pentagonal Fuzzy Number:Its Different Representations, Properties, Ranking, Defuzzification and Application in Game Problems, Symmetry, vol 10, pp 248, 2018.
 Saqlain. M and Florentin S. Octagonal Neutrosophic Number: Its Different Representations, Properties, Graphs and De-Neutrosophication, International Journal of Neutrosophic Science, vol. 8, no. 1, pp. 19–33, 2020.
 Saqlain.M , A. Hamza, and S. Farooq, “Linear and Non-Linear Octagonal Neutrosophic Numbers: Its Representation, α-Cut and Applications,” International Journal of Neutrosophic Science, vol. 3, no. 1, pp. 29–43, 2020.
 Saqlain.M, A. Hamza, and M. Saeed, R. M. Zulqarnain, Aggregate, Arithmetic and Geometric Operators of Octagonal Neutrosophic Numbers and Its Application in Multi-Criteria Decision-Making Problems, Springer Book Series 2020.
 Dubois, D.; Prade, H.Operations on fuzzy numbers. Int. J. Syst. Sci. vol 9, pp. 613–626, 1978.
 Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy sets Syst., vol 20, pp. 87–96, 1986.
 Smarandache, F. A Unifying Field in Logics Neutrosophy: Neutrosophic Probability, Set and Logic, 3rd ed.; American Research Press: Washington, DC, USA, 2003.
 Wang, H.B.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single Valued Neutrosophic Sets. Tech. Sci. Appl. Math. 2010.
 Ye, J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Int. Fuzzy Syst., vol 26,pp. 2459–2466, 2014.
 Wang, J.Q.; Peng, J.J.; Zhang, H.Y.; Liu, T.; Chen, X.H. An uncertain linguistic multi-criteria group decisionmaking method based on a cloud model. Group Decis. Negot., vol 24, pp. 171–192, 2015.
 Peng, J.J.; Wang, J.Q.; Wu, X.H.; Zhang, H.Y.; Chen, X.H. The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their applicationin multi-criteria decision-making. Int. J. Syst. Sci. vol 46, pp. 2335– 2350, 2015.
 Peng, J.J.; Wang, J.Q.; Wang, J.; Zhang, H.Y.; Chen, X.H. Simplified neutrosophic sets and their applications in multi-criteria group decision making problems. Int. J. Syst. Sci., vol 47,pp. 2342–2358, 2016.
 Deli I, Broumi S. Neutrosophic soft matrices and NSM decision making, Journal of Intelligent and Fuzzy System vol 28, pp.2233–2241, 2015.
 Ma YX, Wang JQ, Wang J, Wu XH. An interval neutrosophic linguistic multi-criteria group decision– making the method and its application in selecting medical treatment options, Neural Computer Application. DOI:10.1007/s00521-016-2203-1. 2016.
 Abdel-Basset, M., Saleh, M., Gamal, A., & Smarandache, F. An approach of the TOPSIS technique for developing supplier selection with group decision making under type-2 neutrosophic number. Applied Soft Computing, vol77, pp. 438-452, 2019.
 Abdel-Baset, M., Chang, V., Gamal, A., & Smarandache, F. An integrated neutrosophic ANP and VIKOR method for achieving sustainable supplier selection: A case study in the importing field. Computers in Industry, vol 106, pp. 94-110, 2019.
 Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. A group decision-making framework based on the neutrosophic TOPSIS approach for smart medical device selection. Journal of medical systems, vol 43, issue 2, pp. 38-43, 2019.
 Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. A hybrid approach of neutrosophic sets and DEMATEL method for developing supplier selection criteria. Design Automation for Embedded Systems, pp.1-22, 2018.
 Nabeeh, N. A., Smarandache, F., Abdel-Basset, M., El-Ghareeb, H. A., & Aboelfetouh, A. An Integrated Neutrosophic-TOPSIS Approach and Its Application to Personnel Selection: A New Trend in Brain Processing and Analysis. IEEE Access, vol 7, pp. 29734-29744. 2019.
 Smarandache, F.” Neutrosophy. Neutrosophic probability, set, and logic, ProQuest Information & Learning,
Arbor, Michigan, USA, 1998.  Abdel-Baset, M., Chang, V., & Gamal, A. Evaluation of the green supply chain management practices: A novel neutrosophic approach. Computers in Industry, vol 108, pp. 210-220, 2019.
 Abdel-Basset, M., Saleh, M., Gamal, A., & Smarandache, F. An approach of TOPSIS technique for developing supplier selection with group decision making under type-2 neutrosophic number. Applied Soft Computing, vol 77, pp. 438-452, 2019.
 Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. A group decision making framework based on neutrosophic TOPSIS approach for smart medical device selection. Journal of medical systems, vol 43(2), pp. 38, 2019.
 Abdel-Basset, M., Atef, A., & Smarandache, F. A hybrid Neutrosophic multiple criteria group decision making approach for project selection. Cognitive Systems Research, vol 57, pp. 216-227, 2019.
 Abdel-Basset, Mohamed, Mumtaz Ali, and Asma Atef. "Resource levelling problem in construction projects under neutrosophic environment." The Journal of Supercomputing, pp.1-25, 2019.
 Saqlain M, Sana M, Jafar N, Saeed. M, Said. B, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure of Single valued Neutrosophic Hypersoft Sets, Neutrosophic Sets and Systems (NSS), vol 32, pp. 317-329, 2020.
 S. Pramanik, P. P. Dey and B. C. Giri, “TOPSIS for single valued neutrosophic soft expert set based multiattribute decision making problems,” Neutrosophic Sets and Systems, vol 10, pp. 88-95, 2015.
 Saqlain. M., Jafar. N. and Riffat. A., “Smart phone selection by consumers’ in Pakistan: FMCGDM fuzzy multiple criteria group decision making approach,” Gomal University Journal of Research, vol 34(1), pp. 27- 31, 2018.
 Saqlain. M, Jafar.N. M, and Muniba. K, “Change in The Layers of Earth in Term of Fractional Derivative:
A Study,” Gomal University Journal of Research, vol 34(2), pp. 1-13, 2018.
 Saqlain M, Jafar N, Hamid R, Shahzad A. “Prediction of Cricket World Cup 2019 by TOPSIS Technique of MCDM-A Mathematical Analysis,” International Journal of Scientific & Engineering Research, vol 10(2), pp. 789-792, 2019.
 Saqlain M, Saeed M, Ahmad M. R, Smarandache,. F. “Generalization of TOPSIS for Neutrosophic Hypersoft set using Accuracy Function and its Application,” Neutrosophic Sets and Systems (NSS), vol 27, pp. 131-137, 2019.
 Riaz.M., Saeed.M. Saqlain.M. and Jafar.N,”Impact of Water Hardness in Instinctive Laundry System based on Fuzzy Logic Controller,” Punjab University Journal of Mathematics, vol 51(4), pp. 73-84, 2018.
 Riaz. M., Saqlain. M. and Saeed. M., “Application of Generalized Fuzzy TOPSIS in Decision Making for Neutrosophic Soft set to Predict the Champion of FIFA 2018: A Mathematical Analysis,” Punjab University
Journal of Mathematics, vol 51(8), pp.111-126, 2019.
 I. Deli and S. Broumi, “Neutrosophic Soft Matrices and NSM-decision Making,” Journal of Intelligent and
Fuzzy Systems, vol 28(5), pp. 2233-2241 2015.
 T. Bera and N. K. Mahapatra, Introduction to neutrosophic soft groups, Neutrosophic Sets and Systems, vol
13, pp. 118-127, 2016. doi.org/10.5281/zenodo.570845.
 P. Biswas, S. Pramanik, and B. C. Giri. “A new methodology for neutrosophic multi-attribute decision
making with unknown weight information,” Neutrosophic Sets and Systems, vol 3, pp. 42-52, 2014.
 K. Mondal, and S. Pramanik. Neutrosophic decision making model of school choice. Neutrosophic Sets and
Systems, vol 7,pp. 62-68, 2015.
 Smarandache, F., Pramanik, S., “New Neutrosophic Sets via Neutrosophic Topological Spaces,” In
Neutrosophic Operational Research; Eds.; Pons Editions: Brussels, Belgium, vol I, pp. 189–209, 2017.
 Saqlain, M. Sana, M., Jafar. M. N., Saeed, M., Smarandache, F. “Aggregate Operators of Neutrosophic
Hypersoft Set,” Neutrosophic Sets and Systems, vol. 32, pp. 294-306, 2020. DOI: 10.5281/zenodo.3723155
 Saqlain, M., Jafar, M. N., Riaz, M. “A New Approach of Neutrosophic Soft Set with Generalized Fuzzy
TOPSIS in Application of Smart Phone Selection,” Neutrosophic Sets and Systems, vol. 32, pp. 307-316,
2020. DOI: 10.5281/zenodo.3723161.
 Saqlain, M., Jafar, M. N., Moin, S., Saeed, M. and Broumi, S. “Single and Multi-valued Neutrosophic
Hypersoft set and Tangent Similarity Measure of Single valued Neutrosophic Hypersoft Sets,” Neutrosophic
Sets and Systems, vol. 32, pp. 317-329, 2020. DOI: 10.5281/zenodo.3723165
 A. Chakraborty, S. Broumi, P.K Singh,”Some properties of Pentagonal Neutrosophic Numbers and its
Applications in Transportation Problem Environment,” Neutrosophic Sets and Systems, vol.28, pp.200- 215,
 A. Chakraborty, S. Mondal, S. Broumi, “De-Neutrosophication technique of pentagonal neutrosophic
number and application in minimal spanning tree,” Neutrosophic Sets and Systems, vol. 29, pp. 1-18, 2019.
 Edalatpanah, S. A., “A Direct Model for Triangular Neutrosophic Linear Programming,” International
Journal of Neutrosophic Science, Volume 1, Issue 1, pp. 19-28, 2020.
 Chakraborty, A. “A New Score Function of Pentagonal Neutrosophic Number and its Application in
Networking Problem,” International Journal of Neutrosophic Science, Volume 1, Issue 1, pp. 40-51, 2020.
 Parimala ,M,. Karthika, M, Florentin Smarandache , Said Broumi, “On αω-closed sets and its connectedness
in terms of neutrosophic topological spaces,” International Journal of Neutrosophic Science, Volume 2 ,
Issue 2, pp. 82-88, 2020.
 Pratihar, J.; Kumar, R.; Dey, A.; Broumi, S. Transportation problem in neutrosophic environment. In
Neutrosophic Graph Theory and Algorithms; IGI Global, pp.180-212, 2020.
 Pratihar, J.; Kumar, R.; Edalatpanah, S. A.; Dey, A. Modified Vogel_s approximation method for
transportation problem under uncertain environment. Complex & Intelligent Systems, pp.1-12, 2020.
 Mohapatra, H.; Panda, S.; Rath, A. K.; Edalatpanah, S. A.; Kumar, R. A tutorial on powershell pipeline and
its loopholes. International Journal of Emerging Trends in Engineering Research, 8, pp.975-982,2020.
 Kumar, R.; Edalatpanah, S. A.; Mohapatra, H. A note on ''Optimal path selection approach for fuzzy reliable
shortest path problem._. Journal of Intelligent & Fuzzy Systems, pp.1-4, 2020.
 Kumar, R.; Dey, A.; Broumi, S.; Smarandache, F. A study of neutrosophic shortest path problem. In
Neutrosophic Graph Theory and Algorithms; IGI Global, pp.148-179, 2020.
 Gayen, S.; Jha, S.; Singh, M.; Kumar, R. On a generalized notion of anti-fuzzy subgroup and some
characterizations. International Journal of Engineering and Advanced Technology 2019, 8, 385-390.
 Gayen, S.; Smarandache, F.; Jha, S.; Singh, M. K.; Broumi, S.; Kumar, R. Introduction to plithogenic
subgroup. In Neutrosophic Graph Theory and Algorithms; IGI Global, pp.213-259, 2020.
 Gayen, S.; Smarandache, F.; Jha, S.; Singh, M. K.; Broumi, S.; Kumar, R. Introduction to plithogenic
hypersoft subgroup. Neutrosophic Sets and Systems, 33, pp.208-233,2020.
 Gayen, S.; Smarandache, F.; Jha, S.; Kumar, R. Interval-valued neutrosophic subgroup based on intervalvalued
triple t-norm. In Neutrosophic sets in decision analysis and operations research; IGI Global,