1
Department of Mathematics, Lahore Garrison University, Lahore, Pakistan
(msaqlain@lgu.edu.pk)
2
Department of Mathematics, Lahore Garrison University, Lahore, Pakistan
(alifm2909@gmail.com )
3
Department of Mathematics, Lahore Garrison University, Lahore, Pakistan
(sarafarooq447@gmail.com)
Abstract :
In this paper, the primarily focus is to extend the concept of Octagonal Neutrosophic Numbers (ONN) since these numbers provide a wide range of applications while dealing with more fluctuations in the linguistic environment. Firstly, mathematical notions and definitions of Linear, Symmetric and Asymmetric types are proposed. Secondly, α-Cut is defined. Finally, a case study is done by using the TOPSIS technique of MCDM.
Keywords :
Accuracy Function , Neutrosophic Numbers , Octagonal Neutrosophic Numbers (ONN) , MCDM , TOPSIS.
References :
[1] F. Smarandache, Neutrosophic set, a generalization of the intuitionistic fuzzy sets, Inter. J. Pure Appl. Math. Vol.24 , pp. 287–297, 2005.
[2] 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.
[3] Chakraborty, A.; Mondal, S. P.;Mahata, A.; Alam, S.; Different linear and non-linear form of Trapezoidal Neutrosophic Numbers, De-Neutrosophication Techniques and its Application in time cost optimization technique, sequencing problem; RAIRO Operation Research, vol 11, 327, 2018. doi:10.1051/ro/2019090.
[4] 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.
[5] Dubois, D.; Prade, H.Operations on fuzzy numbers. Int. J. Syst. Sci. vol 9, pp. 613–626, 1978.
[6] Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy sets Syst., vol 20, pp. 87–96, 1986.
[7] Smarandache, F. A Unifying Field in Logics Neutrosophy: Neutrosophic Probability, Set and Logic, 3rd ed.; American Research Press: Washington, DC, USA, 2003.
[8] Wang, H.B.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single Valued Neutrosophic Sets. Tech. Sci. Appl. Math. 2010. Available online: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.309.9470&re p=rep1&type=pdf (accessed on 31 July 2018)
[9] Ye, J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Int. Fuzzy Syst., vol 26,pp. 2459–2466, 2014.
[10] 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.
[11] 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..
[12] Wang, J.Q.; Peng, J.J.; Zhang, H.Y.; Liu, T.; Chen, X.H. An uncertain linguistic multi-criteria group decision-making method based on a cloud model. Group Decis. Negot., vol 24, pp. 171–192, 2015.
[13] Deli I, Broumi S. Neutrosophic soft matrices and NSM decision making, Journal of Intelligent and Fuzzy System vol 28, pp.2233–2241, 2015.
[14] 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.
[15] 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.
[16] 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.
[17] 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.
[18] 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.
[19] 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.
[20] Smarandache, F.” Neutrosophy. Neutrosophic probability, set, and logic, ProQuest Information & Learning, Ann Arbor, Michigan, USA, 1998.
[21] 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..
[22] 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.
[23] 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.
[24] 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.
[25] 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.
[26] 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.
[27] 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.
[28] S. Pramanik, P. P. Dey and B. C. Giri, “TOPSIS for single valued neutrosophic soft expert set based multi-attribute decision making problems,” Neutrosophic Sets and Systems, vol 10, pp. 88-95, 2015.
[29] 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.
[30] 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.
[31] 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.
[32] 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.
[33] 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.
[34] 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.
[35] 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.
[36] 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.
[37] 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.
[38] K. Mondal, and S. Pramanik. Neutrosophic decision making model of school choice. Neutrosophic Sets and Systems, vol 7,pp. 62-68, 2015.
[39] 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.
[40] 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
[41] 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
[42] 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
[43] 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, 2019.
[44] 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. doi: 10.5281/zenodo.3514383.
[45] Edalatpanah, S. A., “A Direct Model for Triangular Neutrosophic Linear Programming,” International Journal of Neutrosophic Science, Volume 1, Issue 1, pp. 19-28, 2020
[46] 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
[47] 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
| Style | # |
|---|---|
| MLA | Muhammad Saqlain *, Ali Hamza , Sara Farooq. "Linear and Non-Linear Octagonal Neutrosophic Numbers: Its Representation, α-Cut and Applications." International Journal of Neutrosophic Science, Vol. 3, No. 1, 2020 ,PP. 29-43 (Doi : https://doi.org/10.54216/IJNS.030101) |
| APA | Muhammad Saqlain *, Ali Hamza , Sara Farooq. (2020). Linear and Non-Linear Octagonal Neutrosophic Numbers: Its Representation, α-Cut and Applications. Journal of International Journal of Neutrosophic Science, 3 ( 1 ), 29-43 (Doi : https://doi.org/10.54216/IJNS.030101) |
| Chicago | Muhammad Saqlain *, Ali Hamza , Sara Farooq. "Linear and Non-Linear Octagonal Neutrosophic Numbers: Its Representation, α-Cut and Applications." Journal of International Journal of Neutrosophic Science, 3 no. 1 (2020): 29-43 (Doi : https://doi.org/10.54216/IJNS.030101) |
| Harvard | Muhammad Saqlain *, Ali Hamza , Sara Farooq. (2020). Linear and Non-Linear Octagonal Neutrosophic Numbers: Its Representation, α-Cut and Applications. Journal of International Journal of Neutrosophic Science, 3 ( 1 ), 29-43 (Doi : https://doi.org/10.54216/IJNS.030101) |
| Vancouver | Muhammad Saqlain *, Ali Hamza , Sara Farooq. Linear and Non-Linear Octagonal Neutrosophic Numbers: Its Representation, α-Cut and Applications. Journal of International Journal of Neutrosophic Science, (2020); 3 ( 1 ): 29-43 (Doi : https://doi.org/10.54216/IJNS.030101) |
| IEEE | Muhammad Saqlain *, Ali Hamza, Sara Farooq, Linear and Non-Linear Octagonal Neutrosophic Numbers: Its Representation, α-Cut and Applications, Journal of International Journal of Neutrosophic Science, Vol. 3 , No. 1 , (2020) : 29-43 (Doi : https://doi.org/10.54216/IJNS.030101) |