Abstract
Making beneficial decisions is a vital part of running an efficacious business. Factors affecting decision-making in business contain alignment with important themes, internal and external information. The multi-attribute decision-making tool is the sub-part of the strategic decision-making technique used for evaluating awkward and unreliable problems that occurred in genuine life troubles. The major contribution of this analysis is to analyze the latest principle of complex Pythagorean fuzzy soft setting and their algebraic operations. Additionally, interaction aggregation operators are a massive part of the operators to use for aggregating the group of attributes into a singleton set. Moreover, by using the interaction aggregation operators and complex Pythagorean fuzzy soft sets, we initiated the principle of complex Pythagorean fuzzy soft interaction weighted averaging operator, complex Pythagorean fuzzy soft interaction weighted geometric operator and their well-known properties. To investigate the supremacy of the elaborated operators, a multi-attribute decision-making procedure is illustrated under the complex Pythagorean fuzzy soft information. Finally, we discussed the advantages and sensitive analysis of the presented works and drew their geometrical expressions to explore the consistency and dominancy of the initiated work.
Similar content being viewed by others
References
Agarwal M, Biswas KK, Hanmandlu M (2013) Generalized intuitionistic fuzzy soft sets with applications in decision-making. Appl Soft Comput 13(8):3552–3566
Akram M, Naz S (2019) A novel decision-making approach under a complex Pythagorean fuzzy environment. Math Comput Appl 24(3):73–93
Akram M, Garg H, Zahid K (2020) Extensions of ELECTRE-I and TOPSIS methods for group decision-making under complex Pythagorean fuzzy environment. Iran J Fuzzy Syst 17(5):147–164
Akram M, Khan A, Borumand Saeid A (2021) Complex Pythagorean Dombi fuzzy operators using aggregation operators and their decision-making. Expert Syst 38(2):12626–12642
Ali Z, Mahmood T (2020) Maclaurin symmetric mean operators and their applications in the environment of complex q-rung orthopair fuzzy sets. Comput Appl Math 39:1–27
Ali Z, Mahmood T, Yang MS (2020) TOPSIS method based on complex spherical fuzzy sets with Bonferroni mean operators. Mathematics 8(10):1739–1757
Ali Z, Mahmood T, Aslam M, Chinram R (2021a) Another view of complex intuitionistic fuzzy soft sets based on prioritized aggregation operators and their applications to multiattribute decision making. Mathematics 9(16):19221–19942
Ali Z, Mahmood T, Ullah K, Khan Q (2021b) Einstein geometric aggregation operators using a novel complex interval-valued pythagorean fuzzy setting with application in green supplier chain management. Rep Mech Eng 2(1):105–134
Alkouri AMDJS, Salleh AR (2012) Complex intuitionistic fuzzy sets. AIP Conf Proc Am Inst Phys 1482(1):464–470
Arora R, Garg H (2018a) Prioritized averaging/geometric aggregation operators under the intuitionistic fuzzy soft set environment. Sci Iran 25(1):466–482
Arora R, Garg H (2018b) Robust aggregation operators for multi-criteria decision-making with the intuitionistic fuzzy soft set environment. Sci Iran 25(2):931–942
Ashraf S, Abdullah S, Aslam M, Qiyas M, Kutbi MA (2019) Spherical fuzzy sets and its representation of spherical fuzzy t-norms and t-conorms. J Intell Fuzzy Syst 36(6):6089–6102
Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96
Babitha KV, Sunil JJ (2011) Generalized intuitionistic fuzzy soft sets and their applications. Gen Math Notes (ISSN 2219-7184)
Bas E, Yolcu U, Egrioglu E (2021) Intuitionistic fuzzy time series functions approach for time series forecasting. Granul Comput 6(3):619–629
Chen Z, Aghakhani S, Man J, Dick S (2010) ANCFIS: a neuro-fuzzy architecture employing complex fuzzy sets. IEEE Trans Fuzzy Syst 19(2):305–322
Deng H, Sun X, Liu M, Ye C, Zhou X (2016) Image enhancement based on intuitionistic fuzzy sets theory. IET Image Proc 10(10):701–709
Ejegwa PA, Jana C, Pal M (2021) Medical diagnostic process based on modified composite relation on pythagorean fuzzy multi-sets. Granul Comput 3(1):1–9
Feng F, Fujita H, Ali MI, Yager RR, Liu X (2018) Another view is on generalized intuitionistic fuzzy soft sets and related multiattribute decision-making methods. IEEE Trans Fuzzy Syst 27(3):474–488
Feng F, Xu Z, Fujita H, Liang M (2020) Enhancing PROMETHEE method with intuitionistic fuzzy soft sets. Int J Intell Syst 35(7):1071–1104
Gao J, Guo F, Ma Z, Huang X (2021) A multi-criteria decision-making framework for large-scale rooftop photovoltaic project site selection based on intuitionistic fuzzy sets. Appl Soft Comput 102:107098–107122
Garg H, Arora R (2018a) Generalized and group-based generalized intuitionistic fuzzy soft sets with applications in decision-making. Appl Intell 48(2):343–356
Garg H, Arora R (2018b) Bonferroni means aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making. J Oper Res Soc 69(11):1711–1724
Garg H, Rani D (2019a) Some results on information measures for complex intuitionistic fuzzy sets. Int J Intell Syst 34(10):2319–2363
Garg H, Rani D (2019b) Some generalized complex intuitionistic fuzzy aggregation operators and their application to the multicriteria decision-making process. Arab J Sci Eng 44(3):2679–2698
Garg H, Rani D (2021) Novel similarity measure based on the transformed right-angled triangles between intuitionistic fuzzy sets and their applications. Cogn Comput 13(2):447–465
Garg H, Gandomi AH, Ali Z, Mahmood T (2022) Neutrality aggregation operators based on complex q-rung orthopair fuzzy sets and their applications in multiattribute decision-making problems. Int J Intell Syst. https://doi.org/10.1002/int.22657
Hassaballah M, Ghareeb A (2017) A framework for objective image quality measures based on intuitionistic fuzzy sets. Appl Soft Comput 57:48–59
Hayat K, Ali MI, Cao BY, Karaaslan F, Yang XP (2018) Another view of aggregation operators on group-based generalized intuitionistic fuzzy soft sets: multi-attribute decision-making methods. Symmetry 10(12):753–789
Jan N, Nasir A, Alhilal MS, Khan SU, Pamucar D, Alothaim A (2021) Investigation of cyber-security and cyber-crimes in oil and gas sectors using the innovative structures of complex intuitionistic fuzzy relations. Entropy 23(9):1112
Jana C, Senapati T, Pal M (2019a) Pythagorean fuzzy Dombi aggregation operators and its applications in multiple attribute decision-making. Int J Intell Syst 34(9):2019–2038
Jana C, Muhiuddin G, Pal M (2019b) Some Dombi aggregation of Q-rung orthopair fuzzy numbers in multiple-attribute decision making. Int J Intell Syst 34(12):3220–3240
Jana C, Muhiuddin G, Pal M (2020) Multiple-attribute decision making problems based on SVTNH methods. J Ambient Intell Humaniz Comput 11(9):3717–3733
Jiang Y, Tang Y, Chen Q, Liu H, Tang J (2010) Interval-valued intuitionistic fuzzy soft sets and their properties. Comput Math Appl 60(3):906–918
Jiang Y, Tang Y, Chen Q (2011) An adjustable approach to intuitionistic fuzzy soft sets based on decision making. Appl Math Model 35(2):824–836
Jin Y, Ashraf S, Abdullah S (2019) Spherical fuzzy logarithmic aggregation operators based on entropy and their application in decision support systems. Entropy 21(7):628–645
Karaaslan F (2016) Intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets with applications in decision making. Ann Fuzzy Math Inform 11(4):607–619
Khan MJ, Kumam P, Liu P, Kumam W, Ashraf S (2019) A novel approach to generalized intuitionistic fuzzy soft sets and its application in decision support systems. Mathematics 7(8):742–758
Liang D, Xu Z (2017) The new extension of the TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets. Appl Soft Comput 60:167–179
Liu P, Ali Z, Mahmood T (2020) The distance measures and cross-entropy are based on complex fuzzy sets and their application in decision-making. J Intell Fuzzy Syst 39(3):3351–3374
Liu S, Yu W, Chan FT, Niu B (2021) A variable weight-based hybrid approach for multi-attribute group decision making under interval-valued intuitionistic fuzzy sets. Int J Intell Syst 36(2):1015–1052
Mahmood T (2020) A novel approach towards bipolar soft sets and their applications. J Math 2020:Article ID 4690808
Mahmood T, ur Rehman U (2022) A novel approach towards bipolar complex fuzzy sets and their applications in generalized similarity measures. Int J Intell Syst. https://doi.org/10.1002/int.22639
Mahmood T, Ullah K, Khan Q, Jan N (2019) An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets. Neural Comput Appl 31(11):7041–7053
Maji PK, Biswas R, Roy AR (2001a) Fuzzy soft sets. J Fuzzy Math 9(3):589–602
Maji PK, Biswas R, Roy AR (2001b) Intuitionistic fuzzy soft sets. J Fuzzy Math 9(3):677–692
Mao J, Yao D, Wang C (2013) Group decision-making methods based on intuitionistic fuzzy soft matrices. Appl Math Model 37(9):6425–6436
Molodtsov D (1999) Soft set theory—first results. Comput Math Appl 37(5):19–31
Munir M, Kalsoom H, Ullah K, Mahmood T, Chu YM (2020) T-spherical fuzzy Einstein hybrid aggregation operators and their applications in multi-attribute decision making problems. Symmetry 12(3):365–378
Muthukumar P, Krishnan GSS (2016) A similarity measure of intuitionistic fuzzy soft sets and their application in medical diagnosis. Appl Soft Comput 41:148–156
Nasir A, Jan N, Gumaei A, Khan SU, Albogamy FR (2021) Cybersecurity against the loopholes in industrial control systems using interval-valued complex intuitionistic fuzzy relations. Appl Sci 11(16):7668–7689
Ngan RT, Ali M, Tamir DE, Rishe ND, Kandel A (2020) Representing complex intuitionistic fuzzy set by quaternion numbers and applications to decision making. Appl Soft Comput 87:105961–106018
Quek SG, Selvachandran G, Davvaz B, Pal M (2019) The algebraic structures of complex intuitionistic fuzzy soft sets associated with groups and subgroups. Sci Iran 26(3):1898–1912
Ramot D, Milo R, Friedman M, Kandel A (2002) Complex fuzzy sets. IEEE Trans Fuzzy Syst 10(2):171–186
Ramot D, Friedman M, Langholz G, Kandel A (2003) Complex fuzzy logic. IEEE Trans Fuzzy Syst 11(4):450–461
Riaz M, Hashmi MR (2020) Soft rough Pythagorean m-polar fuzzy sets and Pythagorean m-polar fuzzy soft rough sets with application to decision-making. Comput Appl Math 39(1):1–36
Riaz M, Naeem K, Afzal D (2020a) A similarity measure under Pythagorean fuzzy soft environment with applications. Comput Appl Math 39(4):1–17
Riaz M, Hashmi MR, Kalsoom H, Pamucar D, Chu YM (2020b) Linear diophantine fuzzy soft rough sets for the selection of sustainable material handling equipment. Symmetry 12(8):1215–1237
Sarkar B, Biswas A (2021) Pythagorean fuzzy AHP-TOPSIS integrated approach for transportation management through a new distance measure. Soft Comput 25(5):4073–4089
Shahzadi G, Muhiuddin G, Arif Butt M, Ashraf A (2021) Hamacher interactive hybrid weighted averaging operators under Fermatean fuzzy numbers. J Math 2021
Szmidt E, Kacprzyk J, Bujnowski P (2014) How to measure the amount of knowledge conveyed by Atanassov’s intuitionistic fuzzy sets. Inf Sci 257:276–285
Ullah K, Hassan N, Mahmood T, Jan N, Hassan M (2019) Evaluation of investment policy based on multi-attribute decision-making using interval valued T-spherical fuzzy aggregation operators. Symmetry 11(3):357–389
Ullah K, Mahmood T, Ali Z, Jan N (2020a) On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition. Complex Intell Syst 6(1):15–27
Ullah K, Mahmood T, Jan N, Ahmad Z (2020b) Policy decision making based on some averaging aggregation operators of t-spherical fuzzy sets; a multi-attribute decision making approach. Ann Optim Theory Pract 3(3):69–92
Wei G (2017) Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making. J Intell Fuzzy Syst 33(4):2119–2132
Wu X, Song Y, Wang Y (2021) Distance-based knowledge measure for intuitionistic fuzzy sets with its application in decision making. Entropy 23(9):1119–1137
Xue Y, Deng Y, Garg H (2021) Uncertain database retrieval with measure—based belief function attribute values under intuitionistic fuzzy set. Inf Sci 546:436–447
Yager RR (2009) On generalized Bonferroni mean operators for multi-criteria aggregation. Int J Approx Reason 50(8):1279–1286
Yager RR (2013) Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst 22(4):958–965
Yang J, Yao Y (2021) A three-way decision-based construction of shadowed sets from Atanassov intuitionistic fuzzy sets. Inf Sci 577:1–21
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Zhang G, Dillon TS, Cai KY, Ma J, Lu J (2009) Operation properties and δ-equalities of complex fuzzy sets. Int J Approx Reason 50(8):1227–1249
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Anibal Tavares de Azevedo.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mahmood, T., Ali, Z. A method to multiattribute decision making problems under interaction aggregation operators based on complex Pythagorean fuzzy soft settings and their applications. Comp. Appl. Math. 41, 227 (2022). https://doi.org/10.1007/s40314-022-01888-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-022-01888-1
Keywords
- Interaction aggregation operators
- Complex Pythagorean fuzzy soft sets
- Strategic decision-making techniques