### AIMS Mathematics

2022, Issue 9: 17327-17348. doi: 10.3934/math.2022954
Research article

# Fractional transportation problem under interval-valued Fermatean fuzzy sets

• Received: 15 June 2022 Revised: 09 July 2022 Accepted: 13 July 2022 Published: 25 July 2022
• MSC : 90C32, 90C70

• The concept of an interval-valued Fermatean fuzzy set (IVFFS), an extension of Fermatean fuzzy sets, is a more resilient and reliable tool for dealing with uncertain and incomplete data in practical applications. The purpose of this paper is to define a triangular interval-valued Fermatean fuzzy number (TIVFFN) and its arithmetic operations. Fractional transportation problems (FTPs) have important implications for cost reduction and service improvement in logistics and supply management. However, in practical problems, the parameters in the model are not precise due to some unpredictable factors, including diesel prices, road conditions, weather conditions and traffic conditions. Therefore, decision makers encounter uncertainty when estimating transportation costs and profits. To address these challenges, we consider a FTP with TIVFFN as its parameter and call it an interval-valued Fermatean fuzzy fractional transportation problem (IVFFFTP). A new method for solving this IVFFFTP is proposed without re-transforming the original problem into an equivalent crisp problem. Illustrative examples are discussed to evaluate the precision and accuracy of the proposed method. Finally, the results of the proposed method are compared with those of existing methods.

Citation: Muhammad Akram, Syed Muhammad Umer Shah, Mohammed M. Ali Al-Shamiri, S. A. Edalatpanah. Fractional transportation problem under interval-valued Fermatean fuzzy sets[J]. AIMS Mathematics, 2022, 7(9): 17327-17348. doi: 10.3934/math.2022954

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• The concept of an interval-valued Fermatean fuzzy set (IVFFS), an extension of Fermatean fuzzy sets, is a more resilient and reliable tool for dealing with uncertain and incomplete data in practical applications. The purpose of this paper is to define a triangular interval-valued Fermatean fuzzy number (TIVFFN) and its arithmetic operations. Fractional transportation problems (FTPs) have important implications for cost reduction and service improvement in logistics and supply management. However, in practical problems, the parameters in the model are not precise due to some unpredictable factors, including diesel prices, road conditions, weather conditions and traffic conditions. Therefore, decision makers encounter uncertainty when estimating transportation costs and profits. To address these challenges, we consider a FTP with TIVFFN as its parameter and call it an interval-valued Fermatean fuzzy fractional transportation problem (IVFFFTP). A new method for solving this IVFFFTP is proposed without re-transforming the original problem into an equivalent crisp problem. Illustrative examples are discussed to evaluate the precision and accuracy of the proposed method. Finally, the results of the proposed method are compared with those of existing methods.

 [1] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X [2] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3 [3] R. R. Yager, Pythagorean membership grades in multi-criteria decision making, IEEE T. Fuzzy Syst., 22 (2014), 958–965. https://doi.org/10.1109/TFUZZ.2013.2278989 doi: 10.1109/TFUZZ.2013.2278989 [4] R. R. Yager, Pythagorean fuzzy subsets, 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), 2013, 57–61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375 [5] X. Han, P. Rani, Evaluate the barriers of blockchain technology adoption in sustainable supply chain management in the manufacturing sector using a novel Pythagorean fuzzy-CRITIC-CoCoSo approach, Oper. Manag. Res., 2022. https://doi.org/10.1007/s12063-021-00245-5 [6] T. Senapati, R. R. Yager, Fermatean fuzzy sets, J. Ambient Intell. Human. Comput., 11 (2020), 663–674. https://doi.org/10.1007/s12652-019-01377-0 doi: 10.1007/s12652-019-01377-0 [7] T. Senapati, R. R. Yager, Fermatean fuzzy weighted averaging/geometric operators and its application in multi-criteria decision making methods, Eng. Appl. Artif. Intel., 85 (2019), 112–121. https://doi.org/10.1016/j.engappai.2019.05.012 doi: 10.1016/j.engappai.2019.05.012 [8] T. Senapati, R. R. Yager, Some new operations over Fermatean fuzzy numbers and application of Fermatean fuzzy WPM in multiple criteria decision making, Informatica, 30 (2019), 391–412. https://doi.org/10.15388/Informatica.2019.211 doi: 10.15388/Informatica.2019.211 [9] M. Akram, N. Ramzan, F. Feng, Extending COPRAS method with linguistic Fermatean fuzzy sets and hamy mean operators, J. Math., 2022 (2022), 8239263. https://doi.org/10.1155/2022/8239263 [10] M. Akram, R. Bibi M. A. Al-Shamiri, A decision-making framework based on 2-tuple linguistic Fermatean fuzzy hamy mean operators, Math. Probl. Eng., 2022 (2022), 1501880. https://doi.org/10.1155/2022/1501880 doi: 10.1155/2022/1501880 [11] M. Akram, G. Muhiuddin, G. Santos-Garcia, An enhanced VIKOR method for multi-criteria group decision-making with complex Fermatean fuzzy sets, Math. Biosci. Eng., 19 (2022), 7201–7231. https://doi.org/10.3934/mbe.2022340 doi: 10.3934/mbe.2022340 [12] M. Akram, G. Shahzadi, B. Davvaz, Decision-making model for internet finance soft power and sportswear brands based on sine-trigonometric Fermatean fuzzy information, Soft Comput., 2022. https://doi.org/10.1007/s00500-022-07060-5 [13] S. Jeevaraj, Ordering of interval-valued Fermatean fuzzy sets and its applications, Expert Syst. Appl., 185 (2021), 115613. https://doi.org/10.1016/j.eswa.2021.115613 doi: 10.1016/j.eswa.2021.115613 [14] P. Rani, A. R. Mishra, Interval-valued fermatean fuzzy sets with multi-criteria weighted aggregated sum product assessment-based decision analysis framework, Neural Comput. Appl., 34 (2022), 8051–8067. https://doi.org/10.1007/s00521-021-06782-1 doi: 10.1007/s00521-021-06782-1 [15] D. Sergi, I. U. Sari, T. Senapati, Extension of capital budgeting techniques using interval-valued Fermatean fuzzy sets, J. Intell. Fuzzy Syst., 42 (2022), 365–376, https://doi.org/10.3233/jifs-219196 doi: 10.3233/jifs-219196 [16] R. E. Bellman, L. A. Zadeh, Decision making in a fuzzy environment, Manage. Sci., 17 (1970), 141–164. https://doi.org/10.1287/mnsc.17.4.B141 doi: 10.1287/mnsc.17.4.B141 [17] H. J. Zimmerman, Fuzzy programming and linear programming with several objective functions, Fuzzy Set. Syst., 1 (1978), 45–55. https://doi.org/10.1016/0165-0114(78)90031-3 doi: 10.1016/0165-0114(78)90031-3 [18] S. K. Mahato, L. Sahoo, A. K. Bhunia, Effects of defuzzification methods in redundancy allocation problem with fuzzy valued reliabilities via genetic algorithm, Int. J. Inform. Comput. Secur., 2 (2013), 106–115. [19] L. Sahoo, Effect of defuzzification methods in solving fuzzy matrix games, J. New Theory, 8 (2015), 51–64. [20] A. A. H. Ahmadini, F. Ahmad, Solving intuitionistic fuzzy multiobjective linear programming problem under neutrosophic environment, AIMS Mathematics, 6 (2021), 4556–4580, https://doi.org/10.3934/math.2021269 doi: 10.3934/math.2021269 [21] J. Ahmed, M. G. Alharbi, M. Akram, S. Bashir, A new method to evaluate linear programming problem in bipolar single-valued neutrosophic environment, Comput. Model. Eng. Sci., 129 (2021), 881–906. https://doi.org/10.32604/cmes.2021.017222 doi: 10.32604/cmes.2021.017222 [22] M. Akram, I. Ullah, T. Allahviranloo, S. A. Edalatpanah, Fully Pythagorean fuzzy linear programming problems with equality constraints, Comput. Appl. Math., 40 (2021), 120. https://doi.org/10.1007/s40314-021-01503-9 doi: 10.1007/s40314-021-01503-9 [23] M. Akram, I. Ullah, T. Allahviranloo, S. A. Edalatpanah, $LR$-type fully Pythagorean fuzzy linear programming problems with equality constraints, J. Intell. Fuzzy Syst., 41 (2021), 1975–1992. https://doi.org/ 10.3233/JIFS-210655 doi: 10.3233/JIFS-210655 [24] M. Akram, G. Shahzadi, A. A. H. Ahmadini, Decision-making framework for an effective sanitizer to reduce COVID-19 under Fermatean fuzzy environment, J. Math., 2020 (2020), 3263407. https://doi.org/10.1155/2020/3263407 doi: 10.1155/2020/3263407 [25] M. Akram, I. Ullah, M. G. Alharbi, Methods for solving $LR$-type Pythagorean fuzzy linear programming problems with mixed constraints, Math. Probl. Eng., 2021 (2021), 4306058. https://doi.org/10.1155/2021/4306058 doi: 10.1155/2021/4306058 [26] M. A. Mehmood, M. Akram, M. G. Alharbi, S. Bashir, Solution of fully bipolar fuzzy linear programming models, Math. Probl. Eng., 2021 (2021), 9961891. https://doi.org/10.1155/2021/9961891 doi: 10.1155/2021/9961891 [27] M. A. Mehmood, M. Akram, M. G. Alharbi, S. Bashir, Optimization of $LR$-type fully bipolar fuzzy linear programming problems, Math. Probl. Eng., 2021 (2021), 1199336. https://doi.org/10.1155/2021/1199336 doi: 10.1155/2021/1199336 [28] F. L. Hitchcock, The distribution of product from several resources to numerous localities, J. Math. Phys., 20 (1941), 224–230. https://doi.org/10.1002/sapm1941201224 doi: 10.1002/sapm1941201224 [29] A. Charnes, W. W. Cooper, Programming with linear fractional functionals, Naval Res. Logist. Q., 9 (1962), 181–186. https://doi.org/10.1002/nav.3800090303 doi: 10.1002/nav.3800090303 [30] P. Pandey, A. P. Punnen, A simplex algorithm for piecewise-linear fractional programming problems, Eur. J. Oper. Res., 178 (2007), 343–358. https://doi.org/10.1016/j.ejor.2006.02.021 doi: 10.1016/j.ejor.2006.02.021 [31] K. Swarup, Transportation technique in linear fractional functional programming, J. Royal Naval Sci. Serv., 21 (1966), 256–260. [32] A. Gupta, S. Khanna, M. C. Puri, A paradox in linear fractional transportation problems with mixed constraints, Optimization, 27 (1993), 375–387, https://doi.org/10.1080/02331939308843896 doi: 10.1080/02331939308843896 [33] D. Monta, Some aspects on solving a linear fractional transportation problem, JAQM, 2 (2007), 343–348. [34] M. Sivri, I. Emiroglu, C. Guler, F. Tasci, A solution proposal to the transportation problem with the linear fractional objective function, 2011 Fourth International Conference on Modeling, Simulation and Applied Optimization, 2011, 1–9. https://doi.org/10.1109/ICMSAO.2011.5775530 [35] V. D. Joshi, N. Gupta, Linear fractional transportation problem with varying demand and supply, Le Mat., 66 (2011), 3–12. [36] A. Kumar, A. Kaur, Application of classical transportation methods to find the fuzzy optimal solution of fuzzy transportation problems, Fuzzy Inform. Eng., 3 (2011), 81–99. https://doi.org/10.1007/s12543-011-0068-7 [37] B. Kaushal, R. Arora, S. Arora, An aspect of bilevel fixed charge fractional transportation problem, Int. J. Appl. Comput. Math., 6 (2020), 14. https://doi.org/10.1007/s40819-019-0755-3 doi: 10.1007/s40819-019-0755-3 [38] N. Guzel, Y. Emiroglu, F. Tapci, C. Guler, M. Syvry, A solution proposal to the interval fractional transportation problem, Appl. Math. Inf. Sci., 6 (2012), 567–571. [39] A. Ebrahimnejad, An improved approach for solving fuzzy transportation problem with triangular fuzzy numbers, J. Intell. Fuzzy Syst., 29 (2015), 963–974. https://doi.org/ 10.3233/IFS-151625 doi: 10.3233/IFS-151625 [40] S. T. Liu, Fractional transportation problem with fuzzy parameters, Soft Comput., 20 (2016), 3629–3636. https://doi.org/ 10.1007/s00500-015-1722-5 doi: 10.1007/s00500-015-1722-5 [41] A. Ebrahimnejad, New method for solving fuzzy transportation problems with LR flat fuzzy numbers, Inform. Sci., 357 (2016), 108–124. https://doi.org/10.1016/j.ins.2016.04.008 doi: 10.1016/j.ins.2016.04.008 [42] S. Mohanaselvi, K. Ganesan, A new approach for solving linear fuzzy fractinal transportational problem, Int. J. Civil Eng. Technol., 8 (2017), 1123–1129. [43] M. R. Safi, S. M. Ghasemi, Uncertainty in linear fractional transportation problem, Int. J. Nonlinear Anal. Appl., 8 (2017), 81–93. http://doi.org/10.22075/ijnaa.2016.504 doi: 10.22075/ijnaa.2016.504 [44] S. K. Bharati, S. R. Singh, Transportation problem under interval-valued intuitionistic fuzzy environment, Int. J. Fuzzy Syst., 20 (2018), 1511–1522. https://doi.org/10.1007/s40815-018-0470-y doi: 10.1007/s40815-018-0470-y [45] A. Mahmoodirad, T. Allahviranloo, S. Niroomand, A new efective solution method for fully fuzzy transportation problem, Soft Comput., 23 (2019), 4521–4530. https://doi.org/10.1007/s00500-018-3115-z doi: 10.1007/s00500-018-3115-z [46] R. Kumar, S. A. Edalatpanah, S. Jha, R. Singh, A Pythagorean fuzzy approach to the transportation problem, Complex Intell. Syst., 5 (2019), 255–263. https://doi.org/10.1007/s40747-019-0108-1 doi: 10.1007/s40747-019-0108-1 [47] S. K. Bharati, Trapezoidal intuitionistic fuzzy fractional transportation problem, In: Soft computing for problem solving, Singapore: Springer, 2019,833–842. https://doi.org/10.1007/978-981-13-1595-4-66 [48] S. K. Bharati, Transportation problem with interval-valued intuitionistic fuzzy sets: Impact of a new ranking, Prog. Artif. Intell., 10 (2021), 129–145. https://doi.org/10.1007/s13748-020-00228-w doi: 10.1007/s13748-020-00228-w [49] J. Pratihar, R. Kumar, S. A. Edalatpanah, A. Dey, Modified Vogel's approximation method for transportation problem under uncertain environment, Complex Intell. Syst., 7 (2021), 29–40. https://doi.org/10.1007/s40747-020-00153-4 doi: 10.1007/s40747-020-00153-4 [50] C. Veeramani, S. A. Edalatpanah, S. Sharanya, Solving the multi-objective fractional transportation problem through the neutrosophic goal programming approach, Discrete Dyn. Nat. Soc., 2021 (2021), 7308042. https://doi.org/10.1155/2021/7308042 doi: 10.1155/2021/7308042 [51] L. Sahoo, A new score function based Fermatean fuzzy transportation problem, Res. Control Optim., 4 (2021), 100040. https://doi.org/10.1016/j.rico.2021.100040 doi: 10.1016/j.rico.2021.100040 [52] M. K. Sharma, Kamini, N. Dhiman, V. N. Mishra, H. G. Rosales, A. Dhaka, et al., A fuzzy optimization technique for multi-objective aspirational level fractional transportation problem, Symmetry, 13 (2021), 1465. https://doi.org/10.3390/sym13081465 doi: 10.3390/sym13081465 [53] M. A. El Sayed, M. A. Abo-Sinna, A novel approach for fully intuitionistic fuzzy multi-objective fractional transportation problem, Alex. Eng. J., 60 (2021), 1447–1463. https://doi.org/10.1016/j.aej.2020.10.063 doi: 10.1016/j.aej.2020.10.063 [54] S. A. Bas, H. G. Kocken, B. A. Ozkok, A novel iterative method to solve a linear fractional transportation problem, Pak. J. Stat. Oper. Res., 18 (2022), 151–166. https://doi.org/10.18187/pjsor.v18i1.3889 doi: 10.18187/pjsor.v18i1.3889 [55] A. Singh, R. Arora, S. Arora, Bilevel transportation problem in neutrosophic environment, Comput. Appl. Math., 41 (2022), 44. https://doi.org/10.1007/s40314-021-01711-3 doi: 10.1007/s40314-021-01711-3 [56] E. B. Bajalinov, Linear fractional programming theory methods applications and software, New York: Springer, 2003. https://doi.org/10.1007/978-1-4419-9174-4
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