# Ranking of generalized fuzzy numbers based on accuracy of comparison

Document Type : Research Paper

Authors

1 Department of mathematics, Qaemshar Branch, Islamic Azad University, Qaemshahr, Iran

2 Department of Mathematics, Cotton University, Guwahati, Assam 781001, India

Abstract

Ranking generalized fuzzy numbers plays an important role in many applied models and, in particular, decision-making procedures. In ranking process of two generalized fuzzy numbers, it is natural to compare the sets of values in support of two generalised fuzzy numbers. Accordingly, the comparison of a real number and a generalised fuzzy number as well as two generalised fuzzy numbers have to be considered. On the other hand, it is seen that a definitive process of comparison of a real number and a generalised fuzzy number, as well as two generalised fuzzy numbers, is not possible. So in this study, a method for comparing a real number and a generalised fuzzy number with a degree of accuracy (between a zero and one) is defined and then the method is generalized to compare two generalised fuzzy numbers. In general, an index to rank a real number and generalised fuzzy number is constructed. Eventually, this index is extended to rank two generalised fuzzy numbers based on the concept of accuracy of comparison. The advantage of our method is that it can compare two generalised fuzzy numbers with an accuracy of comparison. Also, a definition is introduced to make a definitive comparison. Finally, the proposed method is illustrated by some numerical examples.

Keywords

#### References

[1] S. Abbasbandy, T. Hajjari, A new approach for ranking of trapezoidal fuzzy numbers, Computers and Mathematics with Applications, 57(3) (2009), 413-419.
[2] M. Adabitabar Firozja, B. Agheli, M. Hosseinzadeh, Ranking function of two LR-fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 26(3) (2014), 1137-1142.
[3] M. Akram, T. Allahviranloo, W. Pedrycz, M. Ali, Methods for solving LR-bipolar fuzzy linear systems, Soft Computing, 25(1) (2021), 85-108.
[4] T. Allahviranloo, Fuzzy fractional differential operators and equations, Studies in Fuzziness and Soft Computing Series, Springer Nature, 397 (2020), Doi: 10.1007/978-3-030-51272-9.
[5] T. Allahviranloo, S. Abbasbandy, R. Saneifard, A method for ranking of fuzzy numbers using new weighted distance, Mathematical and Computational Applications, 16(2) (2011), 359-369.
[6] T. Allahviranloo, M. A. Firozja, Ranking of fuzzy numbers by a new metric, Soft Computing, 14(7) (2010), 773-782.
[7] T. Allahviranloo, F. Hosseinzadeh Lotfi, M. Adabitabar Firozja, Efficiency in fuzzy production possibility set, Iranian Journal of Fuzzy Systems, 9(4) (2012), 17-30.
[8] T. Allahviranloo, R. Saneifard, Defuzzification method for ranking fuzzy numbers based on center of gravity, Iranian Journal of Fuzzy Systems, 9(6) (2012), 57-67.
[9] B. Asady, A. Zendehnam, Ranking fuzzy numbers by distance minimization, Applied Mathematical Modelling, 31(11) (2007), 2589-2598.
[10] Y. Barazandeh, B. Ghazanfari, A novel method for ranking generalized fuzzy numbers with two different heights and its application in fuzzy risk analysis, Iranian Journal of Fuzzy Systems, 18(2) (2021), 81-91.
[11] V. M. Cabral, L. C. Barros, On differential equations with interactive fuzzy parameter via t-norms, Fuzzy Sets and Systems, 358 (2019), 97-107.
[12] K. C. Chai, K. M. Tay, C. P. Lim, A new method to rank fuzzy numbers using Dempster-Shafer theory with fuzzy targets, Information Sciences, 346 (2016), 302-317.
[13] S. H. Chen, Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets and Systems, 17(2) (1985), 113-129.
[14] C. H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems, 95(3) (1998), 307-317.
[15] H. T. X. Chi, F. Y. Vincent, Ranking generalized fuzzy numbers based on centroid and rank index, Applied Soft Computing, 68 (2018), 283-292.
[16] T. C. Chu, H. T. Nguyen, Ranking alternatives with relative maximizing and minimizing sets in a fuzzy MCDM model, International Journal of Fuzzy Systems, 21(4) (2019), 1170-1186.
[17] T. C. Chu, C. T. Tsao, Ranking fuzzy numbers with an area between the centroid point and original point, Computers and Mathematics with Applications, 43(1-2) (2002), 111-117.
[18] R. Chutia, Ranking of Z-numbers based on value and ambiguity at levels of decision making, International Journal of Intelligent Systems, 36(1) (2021), 313-331.
[19] A. De, S. Das, S. Kar, Ranking of interval type 2 fuzzy numbers using correlation coefficient and Mellin transform, OPSEARCH, (2021), 1-31.
[20] M. De, B. Das, M. Maiti, EPL models with fuzzy imperfect production system including carbon emission: A fuzzy differential equation approach, Soft Computing, 24(2) (2020), 1293-1313.
[21] N. Deepa, K. Ganesan, Hybrid rough fuzzy soft classifier based multi-class classification model for agriculture crop selection, Soft Computing, 23(21) (2019), 10793-10809.
[22] Y. Deng, Z. Zhenfu, L. Qi, Ranking fuzzy numbers with an area method using radius of gyration, Computers and Mathematics with Applications, 51(6-7) (2006), 1127-1136.
[23] D. Dubois, H. Prade, Operations on fuzzy numbers, International Journal of Systems Science, 9(6) (1978), 613-626.
[24] S. A. Edalatpanah, Neutrosophic structured element, Expert Systems, 37(5) (2020), doi:10.1111/exsy.12542.
[25] M. Eshaghnezhad, F. Rahbarnia, S. Effati, A. Mansoori, An artificial neural network model to solve the fuzzy shortest path problem, Neural Processing Letters, 50(2) (2019), 1527-1548.
[26] S. Ezadi, T. Allahviranloo, Artificial neural network approach for solving fuzzy fractional order initial value problems under gH-differentiability, Mathematical Methods in the Applied Sciences, (2020), Doi:10.1002/mma.7287.
[27] B. Fathi Vajargah, Z. Hassanzadeh, Monte Carlo method for the real and complex fuzzy system of linear algebraic equations, Soft Computing, 24(2) (2020), 1255-1270.
[28] M. A. Firozja, F. R. Balf, S. Firouzian, Vague ranking of fuzzy numbers, Mathematical Sciences, 11(3) (2017), 189-193.
[29] R. Fuller, Neural fuzzy systems, Abo Akademi University, 1995.
[30] A. Gola, G. Kosowski, Development of computer-controlled material handling model by means of fuzzy logic and genetic algorithms, Neurocomputing, 338 (2019), 381-392.
[31] V. Gregori, J. J. Minana, D. Miravet, Contractive sequences in fuzzy metric spaces, Fuzzy Sets and Systems, 379 (2020), 125-133.
[32] Q. Gu, Z. Xuan, A new approach for ranking fuzzy numbers based on possibility theory, Journal of Computational and Applied Mathematics, 309 (2017), 674-682.
[33] G. Hesamian, M. G. Akbari, A preference index for ranking closed intervals and fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 25(05) (2017), 741-757.
[34] A. F. R. L. de Hierro, C. Roldan, F. Herrera, On a new methodology for ranking fuzzy numbers and its application to real economic data, Fuzzy Sets and Systems, 353 (2018), 86-110.
[35] W. Jiang, C. Xie, Y. Luo, Y. Tang, Ranking Z-numbers with an improved ranking method for generalized fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 32(3) (2017), 1931-1943.
[36] H. V. Long, M. Ali, M. Khan, D. N. Tu, A novel approach for fuzzy clustering based on neutrosophic association matrix, Computers and Industrial Engineering, 127 (2019), 687-697.
[37] S. Maurya, V. K. Jain, Fuzzy based energy efficient sensor network protocol for precision agriculture, Computers and Electronics in Agriculture, 130 (2016), 20-37.
[38] G. Medini, S. Bouamama, Application of genetic algorithms to distributed optimization problems under fuzzy constraints, Procedia Computer Science, 159 (2019), 1258-1266.
[39] F. Molinari, A new criterion of choice between generalized triangular fuzzy numbers, Fuzzy Sets and Systems, 296 (2016), 51-69.
[40] A. M. Nejad, M. Mashinchi, Ranking fuzzy numbers based on the areas on the left and the right sides of fuzzy number, Computers and Mathematics with Applications, 61(2) (2011), 431-442.
[41] A. T. Nguyen, T. Taniguchi, L. Eciolaza, V. Campos, R. Palhares, M. Sugeno, Fuzzy control systems: Past, present and future, IEEE Computational Intelligence Magazine, 14(1) (2019), 56-68.
[42] V. Pandiyaraju, R. Logambigai, S. Ganapathy, A. Kannan, An energy efficient routing algorithm for WSNs using intelligent fuzzy rules in precision agriculture, Wireless Personal Communications, (2020), 1-17.
[43] J. Qin, Y. Xi, W. Pedrycz, Failure mode and effects analysis (FMEA) for risk assessment based on interval type-2 fuzzy evidential reasoning method, Applied Soft Computing, (2020), Doi:10.1016/j.asoc.2020.106
134.
[44] K. Rashidi, K. Cullinane, A comparison of fuzzy DEA and fuzzy TOPSIS in sustainable supplier selection: Implications for sourcing strategy, Expert Systems with Applications, 121 (2019), 266-281.
[45] J. V. Riera, S. Massanet, H. Bustince, J. Fernandez, On admissible orders on the set of discrete fuzzy numbers for application in decision making problems, Mathematics, 9(1) (2021), 95.
[46] I. Saha, J. P. Sarkar, U. Maulik, Integrated rough fuzzy clustering for categorical data analysis, Fuzzy Sets and Systems, 361 (2019), 1-32.
[47] J. P. C. de Souza, A. L. M. Marcato, E. P. De Aguiar, M. A. Juca, A. M. Teixeira, Autonomous landing of UAV based on artificial neural network supervised by fuzzy logic, Journal of Control, Automation and Electrical Systems, 30(4) (2019), 522-531.
[48] B. Sun, W. Ma, X. Xiao, Three-way group decision making based on multigranulation fuzzy decision-theoretic rough set over two universes, International Journal of Approximate Reasoning, 81 (2017), 87-102.
[49] S. Suneela, S. Chakraverty, New ranking function for fuzzy linear programming problem and system of linear equations, Journal of Information and Optimization Sciences, 40(1) (2019), 141-156.
[50] M. Tavana, K. Khalili-Damghani, F. J. S. Arteaga, R. Mahmoudi, A. Hafezalkotob, Efficiency decomposition and measurement in two-stage fuzzy DEA models using a bargaining game approach, Computers and Industrial Engineering, 118 (2018), 394-408.
[51] J. F. Tian, M. H. Ha, D. Z. Tian, Tripled fuzzy metric spaces and fixed point theorem, Information Sciences, 518 (2020), 113-126.
[52] E. B. Tirkolaee, A. Mardani, Z. Dashtian, M. Soltani, G. W. Weber, A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design, Journal of Cleaner Production, 250 (2020), Doi: 10.1016/j.jclepro.2019.1195
17.
[53] S. C. Tong, Adaptive fuzzy control for uncertain nonlinear systems, Journal of Control and Decision, 6(1) (2019), 30-40.
[54] K. Venkatanareshbabu, S. Nisheel, R. Sakthivel, K. Muralitharan, Novel elegant fuzzy genetic algorithms in classification problems, Soft Computing, 23(14) (2019), 5583-5603.
[55] X. Wang, E. E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy Sets and Systems, 118(3) (2001), 375-385.
[56] Z. X. Wang, Y. J. Liu, Z. P. Fan, B. Feng, Ranking L-R fuzzy number based on deviation degree, Information Sciences, 179(13) (2009), 2070-2077.
[57] R. R. Yager, On a general class of fuzzy connectives, Fuzzy Sets and Systems, 4(3) (1980), 235-242.
[58] J. S. Yao, K. Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems, 116(2) (2000), 275-288.
[59] M. Yazdi, Hybrid probabilistic risk assessment using fuzzy FTA and fuzzy AHP in a process industry, Journal of Failure Analysis and Prevention, 17(4) (2017), 756-764.
[60] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338-353.
[61] F. Zhang, J. Ignatius, C. P. Lim, Y. Zhao, A new method for ranking fuzzy numbers and its application to group decision making, Applied Mathematical Modelling, 38(4) (2014), 1563-1582.
[62] M. Zhang, P. Shi, L. Ma, J. Cai, H. Su, Network-based fuzzy control for nonlinear Markov jump systems subject to quantization and dropout compensation, Fuzzy Sets and Systems, 371 (2019), 96-109.
[63] J. Zhou, Z. Lai, D. Miao, C. Gao, X. Yue, Multi granulation rough-fuzzy clustering based on shadowed sets, Information Sciences, 507 (2020), 553-573.