Abstract
Uncertainty is excessively a common, inevitable, and conspicuopus aspect of any decision-making process, including transportation problems. Since its inception, a plethora of uncertainty representation methods has been put forward to deal with uncertainty by various researchers. Among those, fuzzy set and Intuitionistic fuzzy set is remarkably effective representation methods of uncertainty modeling. However, the existing uncertainty modeling methods have some severe limitations. Consequently, here we adopt the concept of the Pythagorean fuzzy set, an extension of the intuitionistic fuzzy set for its extensive flexibility characteristic and advantages. On the other hand, the similarity measure plays a crucial role in transportation problems under uncertainty. Therefore, we strive to introduce an advanced similarity measure of Pythagorean fuzzy sets. The proposed similarity measure is constructed based on the distances of the degree of membership, non-membership, and hesitancy of Pythagorean fuzzy sets. The present similarity measure also holds the general axioms of the similarity measure. Furthermore, we adopt some numerical examples to showcase the superiority of the proposed similarity measure and apply it to solve transportation problems. The core motive for transportation problems is minimizing transportation costs, and hence, we modified Monalisa’s method of Pythagorean fuzzy sets with the help of the proposed similarity measure. The proposed method has been demonstrated with an example and compared its output with the other pre-existing methods available in the literature. At length, statistical tests and result analysis are drawn to judge the significance of the proposed method.
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Agheli B, Firozja MA, Garg H (2022) Similarity measure for Pythagorean fuzzy sets and application on multiple criteria decision making. J Stat Manag Syst 25(4):749–769. https://doi.org/10.1080/09720510.2021.1891699
Alves IC, Yamakami A, Gomide F (1995). An approach for fuzzy linear multicommodity transportation problems and its application. In: Proceedings of 1995 IEEE international conference on fuzzy systems, vol 2, pp 773–780. IEEE
Arsham H, Kahn AB (1989) A simplex-type algorithm for generaltransportation problems: an alternative to stepping-stone. J Oper Res Soc 40(6):581–590
Atanassov KT (1999) Intuitionistic fuzzy sets. In: Intuitionistic fuzzy sets Physica, Heidelberg, pp 1–137
Bharathi SD, Kanmani G (2022) Solving Pythagorean transportation problem using ArithmeticC Mean and Harmoni mean. Int J Mech Eng 7
Bharati SK (2021) Transportation problem with interval-valued intuitionistic fuzzy sets: impact of a new ranking. Progr Artif Intell 10(2):129–145
Boora R, Tomar VP (2022) Two trigonometric intuitionistic fuzzy similarity measures. Int J Decis Support Syst Technol (IJDSST) 14(1):1–23
Boran FE, Akay D (2014) A biparametric similarity measure on intuitionistic fuzzy sets with applications to pattern recognition. Inf Sci 255:45–57
Chen SM, Chang CH (2015) A novel similarity measure between Atanssov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition. Inf Sci 291:96–114
Chen SM (1997) Similarity measures between vague sets and between elements. IEEE Trans Syst Man Cyber 27:153–158
Chhibber D, Srivastava PK, Bisht DC (2022) Optimization of a transportation problem under Pythagorean fuzzy environment. J MESA 13(3):769–776
Dafermos SC (1972) The traffic assignment problem for multiclass-user transportation networks. Transp Sci 6(1):73–87
Ejegwa PA (2020) Distance and similarity measures for Pythagorean fuzzy sets. Granular Comput 5(2):225–238
Evans JR (1979) Aggregation in the generalized transportation problem. Comput Oper Res 6(4):199–204
Flood MM (1956) The traveling-salesman problem. Oper Res 4(1):61–75
Geetha SS, Selvakumari K (2020) A new method for solving Pythagorean fuzzy transportation problem. PalArch’s J Archaeol Egypt/Egyptol 17(7):4825–4834
Hong DH, Kim C (1999) A note on similarity measures between vague sets and between elements. Inf Sci 115:83–96
Hung WL, Yang MS (2004) Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance. Pattern Recognit Lett 25:1603–1611
Hussain RJ, Kumar PS (2012) Algorithmic approach for solving intuitionistic fuzzy transportation problem. Appl Math Sci 6(80):3981–3989
Jeyalakshmi K, Chitra L, Veeramalai G, Krishna Prabha S, Sangeetha S (2021) Pythagorean fuzzy transportation problem via Monalisha technique. Ann RSCB 25(3):2078–2086
Krishna PS, Sangeetha S, Hema P, Basheerd M, Veeramalae G (2021) Geometric mean with Pythagorean fuzzy transportation problem. Turk J Comput Math Educ 12(7):1171–1176
Kumar R, Edalatpanah SA, Jha S et al (2019) A Pythagorean fuzzy approach to the transportation problem. Complex Intell Syst 5:255–263. https://doi.org/10.1007/s40747-019-0108-1
Kundu P, Kar S, Maiti M (2014) Fixed charge transportation problem with type-2 fuzzy variables. Inf Sci 255:170–186
Li DF, Cheng CT (2002) New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recognit Lett 23:221–225
Li F, Xu ZY (2001) Measures of similarity between vague sets. J Softw 12:922–927
Li Y, Olson DL, Qin Z (2007) Similarity measures between intuitionistic fuzzy (vague) sets: a comparative analysis. Pattern Recognit Lett 28:278–285
Liang ZZ, Shi PF (2003) Similarity measures on intuitionistic fuzzy sets. Pattern Recognit Lett 24:2687–2693
Mao Y, Zhong H, Xiao X, Li X (2017) A segment-based trajectory similarity measure in the urban transportation systems. Sensors 17(3):524
Mitchell HB (2003) On theDengfeng-Chuntian similarity measure and its application to pattern recognition. Pattern Recognit Lett 24:3101–3104
Molodtsov D (1999) Soft set theory-First results. Comput Math Appl 37(4):19–31. https://doi.org/10.1016/S0898-1221(99)00056-5
Nagar P, Srivastava P. K, Srivastava A (2021). A new dynamic score function approach to optimize a special class of Pythagorean fuzzy transportation problem. Int J Syst Assur Eng Manag 1–10
Ngastiti PTB, Surarso B (2018) Zero point and zero suffix methods with robust ranking for solving fully fuzzy transportation problems. J Phys 1022(1):012005
Pawlak Z (1982) Rough set. Int J Comput Inform Sci 11(5):341–356. https://doi.org/10.1007/BF01001956
Peng X (2019) New similarity measure and distance measure for Pythagorean fuzzy set. Complex Intell Syst 5:101–111
Peng X, Yuan H, Yang Y (2017) Pythagorean fuzzy information measures and their applications. Int J Intell Syst 32:991–1029
Pratihar J, Kumar R, Edalatpanah SA (2021) Modified Vogel’s approximation method for transportation problem under uncertain environment. Complex Intell Syst 7:29–40. https://doi.org/10.1007/s40747-020-00153-4
Rani D, Gulati TR (2014) Fuzzy optimal solution of interval-valued fuzzy transportation problems. In: Proceedings of the third international conference on soft computing for problem solving, pp 881–888. Springer, New Delhi
Ranjan K, Sripati J, Ramayan S (2017) Shortest path problem in network with type-2 triangular Fuzzy arc length. J Appl Res Ind Eng 4(1):1–7
Ropke S, Pisinger D (2006) An adaptive large neighborhood search heuristic for the pickup and delivery problem with time windows. Transp Sci 40(4):455–472
Sahoo L (2021) A new score function based Fermatean fuzzy transportation problem. Results Control Optim 4:100040
Sakawa M, Nishizaki I, Uemura Y (2001) Fuzzy programming and profit and cost allocation for a production and transportation problem. Eur J Oper Res 131(1):1–15
Singh SK, Yadav SP (2016) A new approach for solving intuitionistic fuzzy transportation problem of type-2. Ann Oper Res 243(1):349–363
Tada M, Ishii H (1996) An integer fuzzy transportation problem. Comput Math Appl 31(9):71–87
Umamageswari RM, Uthra G (2020) A Pythagorean fuzzy approach to solve transportation problem. Adalya J 9(1):1301–1308
Wei G, Wei Y (2018) Similarity measures of Pythagorean fuzzy sets based on the cosine function and their applications. Int J Intell Syst 33(3):634–653
Yager RR (2013). Pythagorean fuzzy subsets. The 9th joint world congress on fuzzy systems and NAFIPS annual meeting. In: IFSA/NAFIPS 2013, pp 57–61. Edmonton, Canada
Ye J (2011) Cosine similarity measures for intuitionistic fuzzy sets and their applications. Math Comput Model 53:91–97
Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–356. https://doi.org/10.1016/S0019-9958(65)90241-X
Zeng W, Li D, Yin Q (2018) Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making. Int J Intell Syst 33(11):2236–2254
Zhang X (2016) A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making. Int J Intell Syst 31:593–611
Zhang Q, Hu J, Feng J, Liu A, Li Y (2019) New similarity measures of Pythagorean fuzzy sets and their applications. IEEE Access 7:138192–138202
Zhang Q, Hu J, Feng J, Liu A (2020) Multiple criteria decision making method based on the new similarity measures of Pythagorean fuzzy set. J Intell Fuzzy Syst 39(1):809–820
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Saikia, B., Dutta, P. & Talukdar, P. An advanced similarity measure for Pythagorean fuzzy sets and its applications in transportation problem. Artif Intell Rev 56, 12689–12724 (2023). https://doi.org/10.1007/s10462-023-10421-7
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DOI: https://doi.org/10.1007/s10462-023-10421-7