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A robust extension of VIKOR method for bipolar fuzzy sets using connection numbers of SPA theory based metric spaces

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The purpose of this study is to introduce an innovative multi-attribute group decision making (MAGDM) based on bipolar fuzzy set (BFS) by unifying“ VIseKriterijumska Optimizacija I Kompromisno Rasenje (VIKOR)” method. The VIKOR method is considered to be a useful MAGDM method, specifically in conditions where an expert is unable to determine his choice correctly at the initiation of designing a system. The method of VIKOR is suitable for problems containing conflicting attributes, with an assumption that compromising is admissible for conflict decision, the expert wishes a solution very near to the best, and the different alternatives or choices are processed according to all developed attributes. The theory of set pair analysis is a state-of-the-art uncertainty theory which consists of three factors, including “identity degree”, “discrepancy degree”, and “contrary degree” of connection numbers (CNs) and coincidence with many existing theories dealing with vagueness in the given information. Consequently, inspired by this, in the present study, we make an effort to improve the theory of data measurement by introducing some metric spaces using CNs of BFSs. In this research paper, we extend VIKOR method in the context of CNs based metrics, which are obtained form bipolar fuzzy numbers (BFNs). Firstly, we develop CNs of BFNs as well as metric spaces based on CNs. We also discuss some interesting properties of proposed metric spaces. Secondly, we develop VIKOR method using CNs based metrics to handle an MAGDM problem under bipolar fuzzy type information. The predominance of proposed metric spaces is also studied by the means of examples. Furthermore, we demonstrate the efficiency of the extended VIKOR method by solving a numerical example, sensitivity analysis and a detailed comparison with some existing approaches.

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References

  • Abdullah S, Aslam M, Ullah K (2014) Bipolar fuzzy soft sets and its applications in decision making problem. J Intell Fuzzy Syst 27(2):729–742

    MathSciNet  MATH  Google Scholar 

  • Akram M, Arshad M (2018) A novel trapezoidal bipolar fuzzy TOPSIS method for group decision-making. Group Decis Negot 28(3):565–584

    Google Scholar 

  • Akram M, Sumaiza Smarandache F (2018) Decision-making with bipolar neutrosophic TOPSIS and bipolar neutrosophic ELECTRE-I. Axioms 7(2):33

    Google Scholar 

  • Alghamdi MA, Alshehri NO, Akram M (2018) Multi-criteria decision-making methods in bipolar fuzzy environment. Int J Fuzzy Syst 20(6):20–57

    Google Scholar 

  • Atanassov KT (1999) Intuitionistic fuzzy sets: Theory and applications. In: Studies in fuzziness and soft computing, vol 35. Physica-Verlag, Heidelberg, pp 1–137

  • Atanassov KT, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fyzzy Sets and Systems 31(3):343–349

    MathSciNet  MATH  Google Scholar 

  • Benferhat S, Dubois D, Kaci S, Prade H (2006) Bipolar possibility theory in preference modeling:Representation, fusion and optimal solutions. Inf Fus 7(1):135–150

    Google Scholar 

  • Bera T, Mahapatra NK (2018) On neutrosophic soft metric space. Int J Adv Math 2018(1):180–200

    Google Scholar 

  • Bloch I (2009) Geometry of spatial bipolar fuzzy sets based on bipolar fuzzy numbers and mathematical morphology. In: International workshop on fuzzy logic and applications WILF 2009: fuzzy logic and applications, pp 237–245

  • Cao YX, Zhou H, Wang JQ (2018) An approach to interval-valued intuitionistic stochastic multi-criteria decision-making using set pair analysis. Int J Mach Learn Cybern 9(4):629–640

    Google Scholar 

  • Chang CL, Hsu CH (2009) Multi-criteria analysis via the VIKOR method for prioritizing land-use restraint strategies in the Tseng-Wen reservoir watershed. J Environ Manag 90:3226–3230

    Google Scholar 

  • Dubois D, Prade H (2009) An overview of the asymmetric bipolar representation of positive and negative information in possibility theory. Fuzzy Sets Syst 160(10):1355–1366

    MathSciNet  MATH  Google Scholar 

  • Dutta P, Doley D (2019) Medical diagnosis under uncertain environment through bipolar-valued fuzzy sets. In: Computer vision and machine intelligence in medical image analysis, pp. 127–135. https://doi.org/10.1007/978-981-13-8798-2-13

  • Fu S, Zhou H (2016) Triangular fuzzy number multi-attribute decision-making method based on set-pair analysis. J Softw Eng. https://doi.org/10.3923/jse.2016

    Article  Google Scholar 

  • Gao H, Wei GW, Huang YH (2018) Dual hesitant bipolar fuzzy Hamacher prioritized aggregation operators in multiple attribute decision making. IEEE Access 6(1):11508–11522

    Google Scholar 

  • Garg H, Arora R (2018) Dual hesitant fuzzy soft aggregation operators and their application in decision making. Cognit Comput 10(5):769–789

    Google Scholar 

  • Garg H, Kumar K (2018a) Group decision making approach based on possibility degree measures and the linguistic intuitionistic fuzzy aggregation operators using Einstein norm operations. J Mult Valued Log Soft Comput 31(1/2):175–209

    MathSciNet  MATH  Google Scholar 

  • Garg H, Kumar K (2018b) An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making. Soft Comput 22(15):4959–4970

    MATH  Google Scholar 

  • Garg H, Kumar K (2018c) Some aggregation operators for linguistic intuitionistic fuzzy set and its application to group decision-making process using the set pair analysis. Arab J Sci Eng 43(6):3213–3227

    MATH  Google Scholar 

  • Garg H, Kumar K (2018d) A novel exponential distance and its based TOPSIS method for interval-valued intuitionistic fuzzy sets using connection number of SPA theory. Artif Intell Rev. https://doi.org/10.1007/s10462-018-9668-5

    Article  Google Scholar 

  • Gul Z (2015) Some bipolar fuzzy aggregations operators and their applications in multi-criteria group decision making, M.Phil. Thesis

  • Han Y, Shi P, Chen S (2015) Bipolar-valued rough fuzzy set and its applications to decision information system. IEEE Trans Fuzzy Syst 23(6):2358–2370

    Google Scholar 

  • Han Y, Lu Z, Du Z, Luo Z, Chen S (2018) A YinYang bipolar fuzzy cognitive TOPSIS method to bipolar disorder diagnosis. Comput Methods Program Biomed 158:1–10

    Google Scholar 

  • Hwang CL, Yoon K (1981) Methods for multiple attribute decision-making. In: Fandel G, Trockel W (eds) Multiple attribute decision making. Springer, Berlin, pp 58–191

    Google Scholar 

  • Jana C, Pal M, Karaaslan F, Wang JQ (2018) Trapezoidal neutrosophic aggregation operators and its application in multiple attribute decision -making process. Sci Iran. https://doi.org/10.24200/sci.2018.51136.2024

    Article  Google Scholar 

  • Jana C, Pal M, Wang JQ (2019a) A robust aggregation operators for multi-criteria decision making method with bipolar fuzzy soft environment. Iran J Fuzzy Syst. https://doi.org/10.22111/ijfs.2019.4575

    Article  MATH  Google Scholar 

  • Jana C, Pal M, Wang JQ (2019b) Bipolar fuzzy Dombi aggregation operators and its application in multiple- attribute decision-making process. J Ambient Intell Humaniz Comput 10(9):3533–3549

    Google Scholar 

  • Jana C, Pal M, Wang JQ (2019c) Bipolar fuzzy Dombi prioritized aggregation operators and its application in multiple-attribute decision-making process. Soft Comput. https://doi.org/10.1007/s00500-019-04130-z

    Article  MATH  Google Scholar 

  • Jiang YL, Xu CF, Yao Y, Zhao KQ (2004) Systems information in set pair analysis and its applications. In: Proceedings of 2004 international conference on machine learning and cybernetics, vol. 3, pp. 1717–1722

  • Khullar D, Jha AK, Jena AB (2015) Reducing diagnostic errors—why now. N Engl J Med 373:2491–3

    Google Scholar 

  • Kumar K, Garg H (2017) Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making. Appl Intell 48(5):1–8

    Google Scholar 

  • Li PP (2016) The global implications of the indigenous epistemological system from the east how to apply Yin–Yang balancing to paradox management. Cross Cult Strateg Manag 23(1):42–47

    Google Scholar 

  • Lu M, Busemeyer JR (2014) Do traditional chinese theories of Yi–Jing Yin–Yang and Chinese medicine go beyond western concepts of mind and matter. Mind Matter 12(1):37–59

    Google Scholar 

  • Luo X, Wang X (2017) Extended VIKOR method for intuitionistic fuzzy multi-attribute decision-making based on a new distance measure. Math Problem Eng 2017:16 ID:4072486

    Google Scholar 

  • Mahmood T, Abdullah S, Bilal M, Rashid S (2016) Multiple criteria decision making based on bipolar valued fuzzy set. Ann Fuzzy Math Inf 11(6):1003–1009

    MathSciNet  MATH  Google Scholar 

  • Opricovic S (1998) Multi-criteria optimization of civil engineering systems. Faculty of Pennsylvania, Belgrade

    Google Scholar 

  • Opricovic S (2011) Fuzzy VIKOR with an application to water resources planning. Exp Syst Appl 38:12983–12990

    Google Scholar 

  • Patrascu V (2010) Cardinality and entropy for bifuzzy sets. In: International conference on information processing and management of uncertainty in knowledge-based systems IPMU 2010. theory and methods, pp. 656–665

  • Patrascu V (2015) Similarity Cardinality and entropy for bipolar fuzzy set in the framework of penta-valued representation. https://doi.org/10.13140/RG.2.1.4727.0566:Source

  • Pramanik S, Dalapati S, Alam S, Roy TK (2018) VIKOR based MAGDM strategy under bipolar neutro-sophic set environment. Neutrosophic Sets Syst 19:57–69

    Google Scholar 

  • Riaz M, Hashmi MR (2019) MAGDM for agribusiness in the environment of various cubic m-polar fuzzy averaging aggregation operators. J Intell Fuzzy Syst 37(3):3671–3691. https://doi.org/10.3233/JIFS-182809

    Article  Google Scholar 

  • Riaz M, Hashmi MR (2019) Linear diophantine fuzzy set and its applications towards multi-attribute decision making problems. J Intell Fuzzy Syst 37(4):5417–5439. https://doi.org/10.3233/JIFS-190550

    Article  Google Scholar 

  • Riaz M, Hashmi MR (2019) 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. https://doi.org/10.1007/s40314-019-0989-z

    Article  MathSciNet  MATH  Google Scholar 

  • Riaz M, Tehrim ST (2019) Cubic bipolar fuzzy ordered weighted geometric aggregation operators and their application using internal and external cubic bipolar fuzzy data. Comput Appl Math 38(2):1–25

    MathSciNet  MATH  Google Scholar 

  • Riaz M, Tehrim ST (2019) Multi-attribute group decision making based cubic bipolar fuzzy information using averaging aggregation operators. J Intell Fuzzy Syst 37(2):2473–2494

    Google Scholar 

  • Riaz M, Tehrim ST (2019) Bipolar fuzzy soft mappings with application to bipolar disorders. Int J Biomath. https://doi.org/10.1142/S1793524519500803

    Article  MathSciNet  MATH  Google Scholar 

  • Samanta S, Pal M (2014) Some more results on bipolar fuzzy sets and bipolar fuzzy intersection graphs. J Fuzzy Math 22(2):253–262

    MATH  Google Scholar 

  • Sarwar M, Akram M, Zafar F (2018) Decision making approach based on competition graphs and extended TOPSIS method under bipolar fuzzy environment. Math Comput Appl 23(4):68

    MathSciNet  Google Scholar 

  • Sayadi MK, Heydaria M, Shahanaghia K (2009) Extension of VIKOR method for decision making problem with interval numbers. Appl Math Modell 33:2257–2262

    MathSciNet  Google Scholar 

  • Shemshadi A, Shirazi H, Toreihi M, Torakh MJ (2011) A fuzzy VIKOR method for supplier selection based on entropy measure for objective weighting. Exp Syst Appl 38(10):12160–12167

    Google Scholar 

  • Singh PK, Kumar CA (2014) Bipolar fuzzy graph representation of concept lattice. Inf Sci 288:437–488

    MathSciNet  MATH  Google Scholar 

  • Tehrim ST, Riaz M (2019) A novel extension of TOPSIS to MCGDM with bipolar neutrosophic soft topology. J Intell Fuzzy Syst 37(4):5531–5549. https://doi.org/10.3233/JIFS-190668

    Article  Google Scholar 

  • Wang JQ, Gong L (2009) Interval probability stochastic multi-criteria decision-making approach based on set pair analysis. Control Decis 24:1877–1880

    Google Scholar 

  • Wei G, Alsaadi FE, Hayat T, Alsaedi A (2018) Bipolar fuzzy Hamacher aggregation operators in multiple attribute decision making. Int J Fuzzy Syst 20(1):1–12

    MathSciNet  MATH  Google Scholar 

  • Yang LH, Li SG, Yang WH, Lu Y (2013) Notes on bipolar fuzzy graphs. Inf Sci 242:113–121

    MathSciNet  MATH  Google Scholar 

  • Yoon KP, Kim WK (2017) The behavioral TOPSIS. Exp Syst Appl 89:266–272

    Google Scholar 

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

    MATH  Google Scholar 

  • Zhan J, Alcantud JCR (2018) A novel type of soft rough covering and its application to multi-criteria group decision-making. Artif Intell Rev. https://doi.org/10.1007/s10462-018-9617-3

    Article  Google Scholar 

  • Zhan J, Xu W (2018) Two types of coverings based multi-granulation rough fuzzy sets and applications to decision making. Artif Intell Rev. https://doi.org/10.1007/s10462-019-09729-0

    Article  Google Scholar 

  • Zhang WR (1994) Bipolar fuzzy sets and relations: a computational frame work for cognitive modeling and multi-agent decision analysis. In: Proceedings of IEEE conference, pp. 305–309

  • Zhang WR (1998) Bipolar fuzzy sets. In: Proceedings of FUZZY IEEE, pp. 835–840

  • Zhang WR (2013) Bipolar quantum logic gates and quantum cellular combinatorics a logical extension to quantum entanglement. J Quantum Inf Sci 3(2):93–105

    Google Scholar 

  • Zhang WR (2016) G-CPT symmetry of quantum emergence and submergence an information conservational multi-agent cellular automata unification of CPT symmetry and CP violation for equilibrium-based many world causal analysis of quantum coherence and decoherence. J Quantum Inf Sci 6(2):62–97

    Google Scholar 

  • Zhang WR, Zhang L (2004) Bipolar logic and bipolar fuzzy logic. Inf Sci 165(3–4):265–287

    MathSciNet  MATH  Google Scholar 

  • Zhang WR, Zhang HJ, Shi Y, Chen SS (2009) Bipolar linear algebra and YinYang N element cellular networks for equilibrium-based bio-system simulation and regulation. J Biol Syst 17(4):547–576

    MATH  Google Scholar 

  • Zhang WR, Pandurangi KA, Peace KE, Zhang Y, Zhao Z (2011) Mental squares: a generic bipolar support vector machine for psychiatric disorder classification, diagnostic analysis and neurobiological data mining. Int J Data Min Bio-inform 5(5):532–572

    Google Scholar 

  • Zhao K (1989) Set pair and set pair analysis-a new concept and systematic analysis method. In: Proceedings of the national conference on system theory and regional planning, pp 87–91

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  1. Department of Mathematics, University of the Punjab, Lahore, Pakistan

    Muhammad Riaz & Syeda Tayyba Tehrim

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  1. Muhammad Riaz
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  2. Syeda Tayyba Tehrim
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Correspondence to Muhammad Riaz.

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Riaz, M., Tehrim, S.T. A robust extension of VIKOR method for bipolar fuzzy sets using connection numbers of SPA theory based metric spaces. Artif Intell Rev 54, 561–591 (2021). https://doi.org/10.1007/s10462-020-09859-w

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