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Improved digraph and matrix assessment model using bipolar fuzzy numbers

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Abstract

Most of the real-world decision making problems are based on the bipolar behavior of information evaluations. Bipolar fuzzy numbers as a generalization of fuzzy numbers are an efficient mathematical tool to cope with bipolar vagueness and uncertainty in data that frequently arise in human decision making problems. This research study focuses on the novel decision making techniqye by integrating the notion of trapezoidal bipolar fuzzy numbers with digraph and matrix approach. Firstly, the initial data in converted into trapezoidal bipolar fuzzy numbers to construct the interralation criteria matrices. Secondly, the bipolar fuzzy criterion matrix for each alternative is formed by interchanging the main diagonal entries of the interrelation matrix by the aggregated bipolar fuzzy numbers. Thirdly, the criterion priority index is obtained by defuzzifying the bipolar fuzzy criterion matrix using a score function for the assessment of the ordering of alternatives. The significance of the proposed approach is illustrated by an application example of agricultural farming for the assessment of greenhouse organic and inorganic farming systems. The effectiveness of the obtained results for the given decision-making problem is analyzed by providing a comparative study with the existing techniques.

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References

  1. Agrawal, S., Singh, R.K., Murtaza, Q.: Disposition decisions in reverse logistics: Graph theory and matrix approach. J. Clean. Prod. 137, 93–104 (2016)

    Google Scholar 

  2. Akram, M., Amjad, U., Davvaz, B.: Decision-making analysis based on bipolar fuzzy N-soft information. Comput. Appl. Math. 40(6), 182 (2021)

    MathSciNet  Google Scholar 

  3. Akram, M., Arshad, M.: A novel trapezoidal bipolar fuzzy TOPSIS method for group decision-making. Group Decis. Negot. 28(3), 565–584 (2019)

    Google Scholar 

  4. Akram, M., Dar, J.M., Naz, S.: Certain graphs under Pythagorean fuzzy environment. Compl. Intell. Syst. 5(2), 127–144 (2019)

    Google Scholar 

  5. Akram, M., Dudek, W.A., Ilyas, F.: Group decision-making based on Pythagorean fuzzy TOPSIS method. Int. J. Intell. Syst. 34(7), 1455–1475 (2019)

    Google Scholar 

  6. Akram, M., Habib, A., Alcantud, J.C.R.: An optimization study based on Dijkstra algorithm for a network with trapezoidal picture fuzzy numbers. Neural Comput. Appl. 33(4), 1329–1342 (2021)

    Google Scholar 

  7. Akram, M., Habib, A., Ilyas, F., Dar, J.: Specific types of Pythagorean fuzzy graphs and application to decision-making. Mathematical and Computational Applications 23(3), 42 (2018)

    MathSciNet  Google Scholar 

  8. Akram, M., Luqman, A.: Granulation of ecological networks under fuzzy soft environment. Soft. Comput. 24(16), 11867–11892 (2020)

    Google Scholar 

  9. Akram, M., Sarwar, M., Dudek, W.A.: Graphs for the analysis of bipolar fuzzy information, p. 452. Springer, Berlin (2021)

    Google Scholar 

  10. Akram, M., Shahzadi, S., Rasool, A., Sarwar, M.: Decision-making methods based on fuzzy soft competition hypergraphs. Compl. Intell. Syst. 8(3), 2325–2348 (2022)

    Google Scholar 

  11. Akram, M., Shumaiza Ahmad, N.A.: MCDM for selection of green suppliers under bipolar fuzzy PROMETHEE process. Symmetry 12(1), 77 (2020). https://doi.org/10.3390/sym12010077

    Article  Google Scholar 

  12. Akram, M., Shumaiza, Arshad, M.: Bipolar fuzzy TOPSIS and bipolar fuzzy ELECTRE-I methods to diagnosis. J. Comput. Appl. Math.39, Article No. 7 (2019c)

  13. Akram, M., Bibi, R.: Multi-criteria group decision-making based on an integrated PROMETHEE approach with 2-tuple linguistic Fermatean fuzzy sets. Granular Comput. 8(5), 917–941 (2023)

    Google Scholar 

  14. Alghamdi, M.A., Alshehri, N.O., Akram, M.: Multi-criteria decision-making methods in bipolar fuzzy environment. Int. J. Fuzzy Syst. 208(6), 2057–2064 (2018)

    Google Scholar 

  15. Ali, G., Ansari, M.N.: Multiattribute decision-making under Fermatean fuzzy bipolar soft framework. Granular Comput. 7(2), 337–352 (2022)

    Google Scholar 

  16. Aruldoss, M., Lakshmi, M.T., Venkatesan, V.P.: A survey on multi criteria decision making methods and its applications. Am. J. Inform. Syst. 1(1), 31–43 (2013)

    Google Scholar 

  17. Baykasoglu, A.: A review and analysis of graph theoretical-matrix permanent approach to decision making with example applications. Artif. Intell. Rev. 42(4), 573–605 (2014)

    Google Scholar 

  18. Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Manage. Sci. 17(4), 141–164 (1970)

    MathSciNet  Google Scholar 

  19. Brans, J.P., Mareschal, B., Vincke, P.: PROMETHEE: a new family of outranking methods in multicriteria analysis. Oper. Res. Int. J. 3, 477–490 (1984)

    Google Scholar 

  20. Chiou, C.Y., Hsu C.W., Hwang, W.Y.: Comparative investigation on green supplier selection of the American, Japanese and Taiwanese electronics industry in China. In 2008 IEEE International Conference on Industrial Engineering and Engineering Management, 1909–1914 (2008)

  21. Feng, F., Akram, M., Davvaz, B., Fotea, V.L.: Attribute analysis of information systems based on elementary soft implications. Knowl.-Based Syst. 70, 281–292 (2014)

    Google Scholar 

  22. Feng, F., Wan, Z., Alcantud, J.C.R., Garg, H.: Three-way decision based on canonical soft sets of hesitant fuzzy sets. AIMS Math. 7(2), 2061–2083 (2022)

    MathSciNet  Google Scholar 

  23. Gandhi, O.P., Agrawal, V.P.: FMEA-A digraph and matrix approach. J. Reliabil. Eng. Syst. Safety 3(5), 147–158 (1992)

    Google Scholar 

  24. Geng, X.L., Liu, Q.M.: A hybrid service supplier selection approach based on variable precision rough set and VIKOR for developing product service system. Int. J. Comput. Integr. Manuf. 28(10), 1063–1076 (2015)

    Google Scholar 

  25. Jana, C.: Multiple attribute group decision-making method based on extended bipolar fuzzy MABAC approach. Comput. Appl. Math. 40(6), 1–17 (2021)

    MathSciNet  Google Scholar 

  26. Hwang, C.L., Yoon, K.: Multiple attribute decision-making methods and applications. Springer, Berlin (1981)

    Google Scholar 

  27. Jana, C., Pal, M.: Extended bipolar fuzzy EDAS approach for multi-criteria group decision-making process. Comput. Appl. Math. 40(1), 1–15 (2021)

    MathSciNet  Google Scholar 

  28. Jana, C., Pal, M., Wang, J.: A robust aggregation operator for multi-criteria decision-making method with bipolar fuzzy soft environment. Iran. J. Fuzzy Syst. 16(6), 1–16 (2019)

    MathSciNet  Google Scholar 

  29. Jing, S., Tang, Y., Yan, J.: The Application of fuzzy VIKOR for the design scheme selection in lean management. Math. Probl. Eng. 2018, 15 (2018)

    Google Scholar 

  30. Joshi, R.: Multi-criteria decision making based on novel fuzzy knowledge measures. Granular Comput. 8(2), 253–270 (2023)

    Google Scholar 

  31. Kannan, D., de Sousa Jabbour, A.B.L., Jabbour, C.J.C.: Selecting green suppliers based on GSCM practices: Using fuzzy TOPSIS applied to a Brazilian electronics company. Eur. J. Operat. Res. 233(2), 432–447 (2014)

    MathSciNet  Google Scholar 

  32. Latinopoulos, D.: Multicriteria decision-making for efficient water and land resources allocation in irrigated agriculture. Environ. Dev. Sustain. 11(2), 329–343 (2009)

    Google Scholar 

  33. Liu, H.C., Chen, Y.Z., You, J.X., Li, H.: Risk evaluation in failure mode and effects analysis using fuzzy digraph and matrix approach. J. Intell. Manuf. 27(4), 805–816 (2014)

    Google Scholar 

  34. Luqman, A., Akram, M., Alcantud, J.C.R.: Digraph and matrix approach for risk evaluations under Pythagorean fuzzy information. Expert Syst. Appl. 170, 114518 (2021)

    Google Scholar 

  35. Poulik, S., Ghorai, G.: Applications of graph’s complete degree with bipolar fuzzy information. Complex Intell. Syst. 8(2), 1115–1127 (2022)

    Google Scholar 

  36. Opricovic, S., Tzeng, G.H.: Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS. Eur. J. Oper. Res. 156(2), 445–455 (2004)

    Google Scholar 

  37. Rao, R.V., Gandhi, O.P.: Failure cause analysis of machine tools using digraph and matrix methods. Int. J. Mach. Tools Manuf. 42(4), 521–528 (2002)

    Google Scholar 

  38. Rao, R.V., Padmanabhan, K.K.: Rapid prototyping process selection using graph theory and matrix approach. J. Mater. Process. Technol. 194(1–3), 81–88 (2007)

    Google Scholar 

  39. Sanayei, A., Mousavi, S.F., Yazdankhah, A.: Group decision-making process for supplier selection with VIKOR under fuzzy environment. Expert Syst. Appl. 37, 24–30 (2010)

    Google Scholar 

  40. Sarwar, M.: Decision-making approaches based on color spectrum and D-TOPSIS method under rough environment. Comput. Appl. Math. (2020). https://doi.org/10.1007/s40314-020-01284-7

    Article  MathSciNet  Google Scholar 

  41. Sarwar, M., Akram, M., Shahzadi, S.: Bipolar fuzzy soft information applied to hypergraphs. Soft. Comput. 25(5), 3417–3439 (2021)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  43. Sarwar, M., Zafar, F., Akram, M.: Novel group decision making approach based on the rough soft approximations of graphs and hypergraphs. J. Appl. Math. Comput. 69(3), 2795–2830 (2023)

    MathSciNet  Google Scholar 

  44. Sharma, H., Shanker, S., Barve, A., Muduli, K., Kumar, A., Luthra, S.: Interval-valued intuitionistic fuzzy digraph-matrix approach with PERMAN algorithm for measuring COVID-19 impact on perishable food supply chain. Environ. Develop. Sustain. (2022). https://doi.org/10.1007/s10668-022-02487-0

    Article  Google Scholar 

  45. Shumaiza Akram, M., Al-Kenani, A.N.: Multiple-attribute decision making ELECTRE II method under bipolar fuzzy model. Algorithms 12(11), 226 (2019)

    MathSciNet  Google Scholar 

  46. Tola, K.A., Repalle, V.N.S.R., Ashebo, M.A.: Interval-valued bipolar fuzzy line graphs. BMC. Res. Notes 16(1), 118 (2023)

    Google Scholar 

  47. Wang, L., Garg, H., Li, N.: Pythagorean fuzzy interactive hamacher power aggregation operators for assessment of express service quality with entropyweight. Soft. Comput. 25(2), 973–993 (2021)

    Google Scholar 

  48. Zadeh, L.A.: Fuzzy set. Inf. Control 8(3), 338–353 (1965)

    Google Scholar 

  49. Zhang, W.R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: Proceedings of the IEEE conference fuzzy information processing society biannual conference, San Antonio, TX, USA, Vol. 18–21, pp. 305–309 (1994)

  50. Zhang, W.R.: (Yin) (Yang) bipolar fuzzy sets. In: Proceedings of the 1998 IEEE international conference on fuzzy systems proceedings, IEEE world congress on computational intelligence (Cat. No.98CH36228), Anchorage, AK, USA, 4-9 May 1998; pp. 835–840

  51. Zhu, G.N., Ma, J., Hu, J.: A fuzzy rough number extended AHP and VIKOR for failure mode and effects analysis under uncertainty. Adv. Eng. Inform. 51, 101454 (2022)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Okara, Okara, 56300, Pakistan

    Fariha Zafar, Iqra Abdul Majeed & Soha Javed

  2. Department of Mathematics, GC Women University, Sialkot, 51310, Pakistan

    Musavarah Sarwar

  3. Department of Computer Science, University of Gujrat, Gujrat, Pakistan

    Nauman Riaz Chaudary

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  1. Fariha Zafar
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  2. Musavarah Sarwar
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  3. Iqra Abdul Majeed
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Contributions

Fariha Zafar and Musavarah Sarwar proposed the idea, design, analysis of the manuscript, made modifications to produce final manuscript. Iqra Abdul Majeed and Soha Javed analyzed the results and wrote initial manuscript. Nauman Riaz Chaudary made corrections and proof read the article.

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Correspondence to Musavarah Sarwar.

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Zafar, F., Sarwar, M., Majeed, I.A. et al. Improved digraph and matrix assessment model using bipolar fuzzy numbers. J. Appl. Math. Comput. 70, 4157–4188 (2024). https://doi.org/10.1007/s12190-024-02125-0

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