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New similarity and divergence measures-based Pythagorean fuzzy MULTIMOORA approach for decision-making problems

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Abstract

Pythagorean fuzzy set (PFS), an extended version of intuitionistic fuzzy set (IFS), is verified to be fastest emerging research discipline and is a significant mode to tackle the imprecise and vague information. It provides a broader representation domain than the IFS. Similarity and divergence measures play paramount role in the field of IFSs’ doctrine as well as PFSs’ doctrine. These measures are integral aspects of utilizing generalized fuzzy sets in real-life problems. In this manuscript, novel similarity and divergence measures for PFSs are developed and compared with extant measures to illustrate their efficiency and reliability. Further, an extended version of multi-objective optimization based on the ratio analysis (MOORA) with the full multiplicative form (MULTIMOORA) approach is introduced by employing developed similarity and divergence measures on PFSs information. To reveal the feasibility and usefulness of the presented approach, a problem of medical equipment supplier selection is discussed from Pythagorean fuzzy perspective. Finally, a comparison with previously developed models is studied to certify the applicability and stability of the introduced MULTIMOORA technique. The outcomes of the work indicate that the present approach is more applicable and efficient than existing techniques under PFSs’ context.

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References

  • Ak MF, Gul M (2019) AHP–TOPSIS integration extended with Pythagorean fuzzy sets for information security risk analysis. Complex Intell Syst 5(2):113–126

    Article  Google Scholar 

  • Akram M, Ilyas F, Garg H (2020) Multi-criteria group decision making based on ELECTRE I method in Pythagorean fuzzy information. Soft Comput 24:3425–3453

    Article  Google Scholar 

  • Alipour M, Hafezi R, Rani P, Hafezi M, Mardani A (2021) A new Pythagorean fuzzy-based decision-making method through entropy measure for fuel cell and hydrogen components supplier selection. Energy 234:121208. https://doi.org/10.1016/j.energy.2021.121208

    Article  Google Scholar 

  • Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

    Article  MathSciNet  MATH  Google Scholar 

  • Bahadori M, Hosseini SM, Teymourzadeh E, Ravangard R, Raadabadi M, Alimohammadzadeh K (2017) A supplier selection model for hospitals using a combination of artificial neural network and fuzzy VIKOR. Int J Healthc Manag 5:1–9

    Google Scholar 

  • Baidya J, Garg H, Saha A, Mishra AR, Rani P, Dutta D (2021) Selection of third party reverses logistic providers: an approach of BCF-CRITIC-MULTIMOORA using Archimedean power aggregation operators. Complex Intell Syst. https://doi.org/10.1007/s40747-021-00413-x

    Article  Google Scholar 

  • Balezentis T, Zeng S (2013) Group multi-criteria decision making based upon interval-valued fuzzy numbers: an extension of the MULTIMOORA method. Expert Syst Appl 40:543–550

    Article  Google Scholar 

  • Biswas A, Sarkar B (2019) Pythagorean fuzzy TOPSIS for multicriteria group decision-making with unknown weight information through entropy measure. Int J Intell Syst 34(6):1108–1128

    Article  Google Scholar 

  • Bolturk E (2018) Pythagorean fuzzy CODAS and its application to supplier selection in a manufacturing firm. J Enterp Inf Manag 31(4):550–564

    Article  Google Scholar 

  • Brauers WKM, Zavadskas EK (2012) Robustness of MULTIMOORA: a method for multi-objective optimization. Informatica 23(1):1–25

    Article  MathSciNet  MATH  Google Scholar 

  • Brauers WKM, Zavadskas EK (2010) Project management by MULTIMOORA as an instrument for transition economies. Technol Econ Dev Econ 16(1):5–24

    Article  Google Scholar 

  • Brauers WKM, Baležentis A, Baležentis T (2011) MULTIMOORA for the EU member states updated with fuzzy number theory. Technol Econ Dev Econ 17(2):259–290

    Article  Google Scholar 

  • Chen TY (2019) Multiple criteria decision analysis under complex uncertainty: a Pearson-like correlation-based Pythagorean fuzzy compromise approach. Int J Intell Syst 34:114–151

    Article  Google Scholar 

  • Chen SM, Chang CH (2015) A novel similarity measure between Atanassov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition. Inf Sci 291:96–114

    Article  Google Scholar 

  • Giri BC, Molla MU, Biswas P (2022) Pythagorean fuzzy DEMATEL method for supplier selection in sustainable supply chain management. Expert Syst Appl. https://doi.org/10.1016/j.eswa.2021.116396

    Article  Google Scholar 

  • He J, Huang Z, Mishra AR, Alrasheedi M (2021) Developing a new framework for conceptualizing the emerging sustainable community-based tourism using an extended interval-valued Pythagorean fuzzy SWARA-MULTIMOORA. Technol Forecast Soc Change 171:120955. https://doi.org/10.1016/j.techfore.2021.120955

    Article  Google Scholar 

  • Hussain Z, Yang MS (2019) Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS. Int J Intell Syst 34(10):2633–2654

    Article  Google Scholar 

  • Jana C, Garg H, Pal M (2022) Multi-attribute decision making for power Dombi operators under Pythagorean fuzzy information with MABAC method. J Ambient Intell Humaniz Comput. https://doi.org/10.1007/s12652-022-04348-0

    Article  Google Scholar 

  • Jong JLD, Benton WC (2018) Dependence and power in healthcare equipment supply chains. Health Care Manag Sci 1:1–14

    Google Scholar 

  • Li DF, Cheng CT (2002) New similarity measures of intuitionistic fuzzy sets and its application to pattern recognitions. Pattern Recognit Lett 23:221–225

    Article  MATH  Google Scholar 

  • Liu HC, You JX, Lu C, Shan MM (2014) Application of interval 2-tuple linguistic MULTIMOORA method for health-care waste treatment technology evaluation and selection. Waste Manag 34(11):2355–2364

    Article  Google Scholar 

  • Liu P, Gao H, Fujita H (2021a) The new extension of the MULTIMOORA method for sustainable supplier selection with intuitionistic linguistic rough numbers. Appl Soft Comput. https://doi.org/10.1016/j.asoc.2020.106893

    Article  Google Scholar 

  • Liu P, Rani P, Mishra AR (2021b) A novel Pythagorean fuzzy combined compromise solution framework for the assessment of medical waste treatment technology. J Clean Prod. https://doi.org/10.1016/j.jclepro.2021.126047

    Article  Google Scholar 

  • Liu P, Wang P (2020) Multiple attribute group decision making method based on intuitionistic fuzzy Einstein interactive operations. Int J Fuzzy Syst 22:790–809

    Article  Google Scholar 

  • Liu Z, Chen J (2015) Study on supplier selection of medical equipment enterprises based on FAHP-TOPSIS. J Hunan Univ Technol 7(4):90–95

    Google Scholar 

  • Luo L, Zhang C, Liao H (2019) Distance-based intuitionistic multiplicative MULTIMOORA method integrating a novel weight-determining method for multiple criteria group decision making. Comput Ind Eng 131:82–98

    Article  Google Scholar 

  • Montes S, Couso I, Gil P, Bertoluzza C (2002) Divergence measure between fuzzy sets. Int J Approx Reason 30:91–105

    Article  MathSciNet  MATH  Google Scholar 

  • Mucong Z, Yan L (2019) Multiple-rules reasoning based on Triple I method on Atanassov’s intuitionistic fuzzy sets. Int J Approx Reason 113:196–206

    Article  MathSciNet  MATH  Google Scholar 

  • Peng X, Dai J (2017) Approaches to Pythagorean fuzzy stochastic multi-criteria decision making based on prospect theory and regret theory with new distance measure and score function. Int J Intell Syst 32(11):1187–1214. https://doi.org/10.1002/int.21896

    Article  Google Scholar 

  • Rani P, Mishra AR (2021) Fermatean fuzzy Einstein aggregation operators-based MULTIMOORA method for electric vehicle charging station selection. Expert Syst Appl 182:115267. https://doi.org/10.1016/j.eswa.2021.115267

    Article  Google Scholar 

  • Rani P, Mishra AR, Pardasani KR (2020a) A novel WASPAS approach for multi-criteria physician selection problem with intuitionistic fuzzy type-2 sets. Soft Comput 24:2355–2367. https://doi.org/10.1007/s00500-019-04065-5

    Article  Google Scholar 

  • Rani P, Mishra AR, Pardasani KR, Mardani A, Liao H, Streimikiene D (2019) A novel VIKOR approach based on entropy and divergence measures of Pythagorean fuzzy sets to evaluate renewable energy technologies in India. J Clean Prod 238:117936. https://doi.org/10.1016/j.jclepro.2019.117936

    Article  Google Scholar 

  • Rani P, Mishra AR, Razaei G, Liao H, Mardani A (2020b) Extended Pythagorean fuzzy TOPSIS method based on similarity measure for sustainable recycling partner selection. Int J Fuzzy Syst 22:735–747. https://doi.org/10.1007/s40815-019-00689-9

    Article  Google Scholar 

  • Rani P, Mishra AR, Mardani A (2020c) An extended Pythagorean fuzzy complex proportional assessment approach with new entropy and score function: Application in pharmacological therapy selection for type 2 diabetes. Appl Soft Comput. https://doi.org/10.1016/j.asoc.2020.106441

    Article  Google Scholar 

  • Rani P, Mishra AR, Saha A, Pamucar D (2021) Pythagorean fuzzy weighted discrimination-based approximation approach to the assessment of sustainable bioenergy technologies for agricultural residues. Int J Intell Syst. https://doi.org/10.1002/int.22408

    Article  Google Scholar 

  • Sarkar B, Biswas A (2021) Multicriteria decision making approach for strategy formulation using Pythagorean fuzzy MULTIMOORA. Expert Syst. https://doi.org/10.1111/exsy.12802

    Article  Google Scholar 

  • Shahzadi G, Akram M, Al-Kenani AN (2020) Decision-making approach under Pythagorean fuzzy Yager weighted operators. Symmetry 8:1–20

    Google Scholar 

  • Stanujkic D, Zavadskas EK, Brauers W, Karabasevic D (2015) An extension of the MULTIMOORA method for solving complex decision-making problems based on the use of interval-valued triangular fuzzy numbers. Transform Bus Econ 14(2B):355–375

    Google Scholar 

  • Tian ZP, Wang J, Wang JQ, Zhang HY (2019) An improved MULTIMOORA approach for multi-criteria decision-making based on interdependent inputs of simplified neutrosophic linguistic information. Neural Comput Appl 28:585–597

    Article  Google Scholar 

  • Vlachos IK, Sergiadis GD (2007) Intuitionistic fuzzy information—applications to pattern recognition. Pattern Recogn Lett 28:197–206

    Article  Google Scholar 

  • 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):634–652. https://doi.org/10.1002/int.21965

    Article  Google Scholar 

  • Wu X, Liao H, Xu Z, Hafezalkotob A, Herrera F (2018) Probabilistic linguistic MULTIMOORA: a multi-criteria decision making method based on the probabilistic linguistic expectation function and the improved Borda rule. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2018.2843330. (in press)

    Article  Google Scholar 

  • Xian S, Liu Z, Gou X, Wan W (2020) Interval 2-tuple Pythagorean fuzzy linguistic MULTIMOORA method with CIA and their application to MCGDM. Int J Intell Syst 35(4):650–681

    Article  Google Scholar 

  • Xiao F, Ding W (2019) Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis. Appl Soft Comput 79:254–267

    Article  Google Scholar 

  • Xu C (2022) An improved fuzzy multi-criteria algorithm for optimizing concentrated solar power (CSP) hybridized systems based on Pythagorean fuzzy set. Clean Eng Technol. https://doi.org/10.1016/j.clet.2022.100401

    Article  Google Scholar 

  • Xu ZS, Zhao N (2016) Information fusion for intuitionistic fuzzy decision making: an overview. Inf Fus 28:10–23

    Article  Google Scholar 

  • Yager RR (2014) Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst 22(4):958–965

    Article  Google Scholar 

  • Ye J (2011) Cosine similarity measures for intuitionistic fuzzy sets and their applications. Math Comput Model 53(1–2):91–97

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Zhang X (2016) A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making. Int J Intell Syst 31(6):593–611. https://doi.org/10.1002/int.21796

    Article  MathSciNet  Google Scholar 

  • Zhang X, Xu Z (2014) Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int J Intell Syst 29(12):1061–1078

    Article  MathSciNet  Google Scholar 

  • Zhou Q, Mo H, Deng Y (2020) A new divergence measure of Pythagorean fuzzy sets based on belief function and its application in medical diagnosis. Mathematics 8:1–20

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Mathematics, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Andhra Pradesh, 522302, India

    Pratibha Rani

  2. Department of Mathematics, Government College Raigaon, Satna, MP, India

    Arunodaya Raj Mishra

  3. School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan, 250014, Shandong, China

    Peide Liu

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  1. Pratibha Rani
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  2. Arunodaya Raj Mishra
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  3. Peide Liu
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Correspondence to Peide Liu.

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Communicated by Susana Montes.

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Rani, P., Mishra, A.R. & Liu, P. New similarity and divergence measures-based Pythagorean fuzzy MULTIMOORA approach for decision-making problems. Comp. Appl. Math. 42, 29 (2023). https://doi.org/10.1007/s40314-022-02150-4

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  • DOI: https://doi.org/10.1007/s40314-022-02150-4

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