Skip to main content
SpringerLink
Log in

Frank Choquet Bonferroni Mean Operators of Bipolar Neutrosophic Sets and Their Application to Multi-criteria Decision-Making Problems

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

In this study, comprehensive multi-criteria decision-making (MCDM) methods are investigated under bipolar neutrosophic environment. First, the operations of BNNs are redefined based on the Frank operations considering the extant operations of bipolar neutrosophic numbers (BNNs) lack flexibility and robustness. Subsequently, the Frank bipolar neutrosophic Choquet weighted Bonferroni mean operator and the Frank bipolar neutrosophic Choquet geometric Bonferroni mean operator are proposed based on the Frank operations of BNNs. The proposed operators simultaneously consider the interactions and interrelationships among the criteria by combining the Choquet integral operator and Bonferroni mean operators. Furthermore, MCDM methods are developed based on the proposed aggregation operators. A numerical example of plant location selection is conducted to explain the application of the proposed methods, and the influences of parameters are also discussed. Finally, the proposed methods are compared with several extant methods to verify their feasibility.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

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

    Article  MATH  Google Scholar 

  2. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)

    Article  MATH  Google Scholar 

  3. Atanassov, K., Gargov, G.: Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 343–349 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Smarandache, F.: A unifying field in logics: neutrosophic logic. Multiple-Valued Logic 8(3), 489–503 (1999)

    MathSciNet  Google Scholar 

  5. Wang, H., Smarandache, F., Zhang, Y., Sunderraman, R.: Single valued neutrosophic sets. Rev. Air Force Acad. 17, 10–14 (2010)

    MATH  Google Scholar 

  6. Wang, H., Madiraju, P., Zhang, Y., Sunderraman, R.: Interval neutrosophic sets. Mathematics 1, 274–277 (2004)

    Google Scholar 

  7. Gruber, J., Siegel, E.H., Purcell, A.L., Earls, H.A., Cooper, G., Barrett, L.F.: Unseen positive and negative affective information influences social perception in bipolar I disorder and healthy adults. J. Affect. Disord. 192, 191–198 (2015)

    Article  Google Scholar 

  8. Bistarelli, S., Pini, M.S., Rossi, F., Venable, K.B.: Modelling and solving bipolar preference problems. Ercim Workshop on Constraints Lisbona Giugno 23(4), 545–575 (2015)

    MATH  Google Scholar 

  9. Zhang, W.R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: The Workshop on Fuzzy Information Processing Society Bi Conference, pp. 305–309 (1994)

  10. Deli, I., Ali, M., Smarandache, F.: Bipolar neutrosophic sets and their application based on multi-criteria decision making problems. In: International Conference on Advanced Mechatronic Systems, pp. 249–254 (2015)

  11. Dey, P.P., Pramanik, S., Giri, B.C.: TOPSIS for solving multi-attribute decision making problems under bi-polar neutrosophic environment, Pons asbl, Brussels, European Union, pp. 65–77 (2016)

  12. Pramanik, S., Dey, P.P., Giri, B.C., Smarandache, F.: Bipolar neutrosophic projection based models for solving multi-attribute decision making problems. Neutrosophic Sets Syst. 15, 70–79 (2017)

    Google Scholar 

  13. Uluçay, V., Deli, I., Şahin, M.: Similarity measures of bipolar neutrosophic sets and their application to multiple criteria decision making. Neural Comput. Appl. (2016). doi:10.1007/s00521-016-2479-1

    Google Scholar 

  14. Li, Y.-Y., Zhang, H.-Y., Wang, J.-Q.: Linguistic neutrosophic sets and their application in multicriteria decision-making problems. Int. J. Uncertain. Quantif. 7(2), 135–154 (2017)

    Article  Google Scholar 

  15. Reimann, O., Schumacher, C., Vetschera, R.: How well does the OWA operator represent real preferences? Eur. J. Oper. Res. 258(3), 993–1003 (2017)

    Article  MathSciNet  Google Scholar 

  16. Merigó, J.M., Palacios-Marqués, D., Soto-Acosta, P.: Distance measures, weighted averages, OWA operators and Bonferroni means. Appl. Soft Comput. 50, 356–366 (2017)

    Article  Google Scholar 

  17. Grabisch, M., Murofushi, T., Sugeno, M.: Fuzzy measures and integrals: Theory and Applications, Heidelberg, PhysicaVerlag, Germany, pp. 314–319 (2000)

  18. Branke, J., Corrente, S., Greco, S., Słowiński, R., Zielniewicz, P.: Using Choquet integral as preference model in interactive evolutionary multiobjective optimization. Eur. J. Oper. Res. 250(3), 884–901 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Keikha, A., Nehi, H.M.: Fuzzified Choquet Integral and its applications in MADM: a review and a new method. Int. J. Fuzzy Syst. 17(2), 337–352 (2015)

    Article  Google Scholar 

  20. Martínez, G.E., Mendoza, D.O., Castro, J.R., Melin, P., Castillo, O.: Choquet integral and interval type-2 fuzzy Choquet integral for edge detection, vol 667, pp. 79–97 (2017)

  21. Wang, J.-Q., Cao, Y.-X., Zhang, H.-Y.: Multi-criteria decision-making method based on distance measure and choquet integral for linguistic Z-numbers. Cognit. Comput. (2017). doi:10.1007/s12559-017-9493-1

    Google Scholar 

  22. Sun, H.X., Yang, H.X., Wu, J.Z., Ouyang, Y.: Interval neutrosophic numbers Choquet integral operator for multi-criteria decision making. J. Intell. Fuzzy Syst. 28(6), 2443–2455 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Meng, F., Zhang, Q., Zhan, J.: The interval-valued intuitionistic fuzzy geometric choquet aggregation operator based on the generalized banzhaf index and 2-additive measure. Technol. Econ. Dev. Econ. 21(2), 186–215 (2015)

    Article  Google Scholar 

  24. Yuan, J., Li, C.: Intuitionistic trapezoidal fuzzy group decision-making based on prospect choquet integral operator and grey projection pursuit dynamic cluster. Math. Probl. Eng. (2017). doi:10.1155/2017/2902506

    MathSciNet  Google Scholar 

  25. Cheng, H., Tang, J.: Interval-valued intuitionistic fuzzy multi-criteria decision making based on the generalized Shapley geometric Choquet integral. J. Ind. Prod. Eng. 33(1), 1–16 (2016)

    Google Scholar 

  26. Bonferroni, C.: Sulle medie multiple di potenze. Bollettino dell’Unione Matematica Italiana 5(3–4), 267–270 (1950)

    MathSciNet  MATH  Google Scholar 

  27. Xu, Z., Yager, R.R.: Intuitionistic fuzzy Bonferroni means. IEEE Trans. Syst. Man Cybern. Part B Cybern. A Publ. IEEE Syst. Man Cybern. Soc. 41(2), 568–578 (2011)

    Article  Google Scholar 

  28. Zhou, W., He, J.M.: Intuitionistic fuzzy normalized weighted Bonferroni mean and its application in multicriteria decision making. J. Appl. Math. 2012(1110-757X), 1–16 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Sun, M.: Normalized geometric bonferroni operators of hesitant fuzzy sets and their application in multiple attribute decision making. J. Inf. Comput. Sci. 10(9), 2815–2822 (2013)

    Article  Google Scholar 

  30. Tian, Z.P., Wang, J., Wang, J.Q., Chen, X.H.: Multicriteria decision-making approach based on gray linguistic weighted Bonferroni mean operator. Int. Trans. Oper. Res. (2015). doi:10.1111/itor.12220

    Google Scholar 

  31. Liang, R.X., Wang, J.Q., Li, L.: Multi-criteria group decision-making method based on interdependent inputs of single-valued trapezoidal neutrosophic information. Neural Comput. Appl. (2016). doi:10.1007/s00521-016-2672-2

    Google Scholar 

  32. Tian, Z.P., Wang, J., Wang, J.Q., Zhang, H.Y.: An improved MULTIMOORA approach for multi-criteria decision-making based on interdependent inputs of simplified neutrosophic linguistic information. Neural Comput. Appl. (2016). doi:10.1007/s00521-016-2378-5

    Google Scholar 

  33. Liu, P., Wang, Y.: Multiple attribute decision-making method based on single-valued neutrosophic normalized weighted Bonferroni mean. Neural Comput. Appl. 25(7–8), 2001–2010 (2014)

    Article  Google Scholar 

  34. Tian, Z.P., Wang, J., Zhang, H.Y., Chen, X.H., Wang, J.Q.: Simplified neutrosophic linguistic normalized weighted Bonferroni mean operator and its application to multi-criteria decision-making problems. Filomat, 30(12), 3339–3360 (2016)

    Article  MATH  Google Scholar 

  35. Liu, P., Zhang, L., Liu, X., Wang, P.: Multi-valued neutrosophic number Bonferroni mean operators with their applications in multiple attribute group decision making. Int. J. Inf. Technol. Decis. Mak. 15(05), 1181–1210 (2016)

    Article  Google Scholar 

  36. Zhang, Z.: Interval-valued intuitionistic fuzzy Frank aggregation operators and their applications to multiple attribute group decision making. Neural Comput. Appl. (2016). doi:10.1007/s00521-015-2143-1

    Google Scholar 

  37. Wang, W.S., Hua-Can, H.E.: Research on flexible probability logic operator based on Frank T/S norms. Acta Electron. Sin. 37(5), 1141–1145 (2009)

    Google Scholar 

  38. Qin, J., Liu, X., Pedrycz, W.: Frank aggregation operators and their application to hesitant fuzzy multiple attribute decision making. Appl. Soft Comput. 41, 428–452 (2016)

    Article  Google Scholar 

  39. Tan, C., Yi, W., Chen, X.: Generalized intuitionistic fuzzy geometric aggregation operators and their application to multi-criteria decision making. J. Oper. Res. Soc. 66(11), 1919–1938 (2015)

    Article  Google Scholar 

  40. Peng, H.G., Wang, J.Q., Cheng, P.F.: A linguistic intuitionistic multi-criteria decision-making method based on the Frank Heronian mean operator and its application in evaluating coal mine safety. Int. J. Mach. Learn. Cybernet. (2017). doi:10.1007/s13042-016-0630-z

    Google Scholar 

  41. Ji, P., Wang, J.Q., Zhang, H.Y.: Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in selecting third-party logistics providers. Neural Comput. Appl. (2016). doi:10.1007/s00521-016-2660-6

    Google Scholar 

  42. Sugeno, M.: Theory of fuzzy integrals and its applications. Unpublished PhD. Tokyo, Japan: Tokyo Institute of Technology. (1974)

  43. Patrascu, V.: Multi-valued color representation based on Frank t-norm properties. In: Proceedings of the 12th Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2008), Malaga, Spain, pp. 1215–1222 (2008)

  44. Nancy, H.G.: Novel single-valued neutrosophic aggregated operators under frank norm operation and its application to decision-making process. Int. J. Uncertain. Quantif. 6(4), 361–375 (2016)

    Article  Google Scholar 

  45. Frank, M.J.: On the simultaneous associativity of F (x, y) and x + y − F (x, y). Aequ. Math. 19(1), 194–226 (1979)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the editors and anonymous reviewers for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 71571193).

Author information

Authors and Affiliations

  1. School of Business, Central South University, Changsha, 410083, People’s Republic of China

    Le Wang, Hong-yu Zhang & Jian-qiang Wang

Authors
  1. Le Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hong-yu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jian-qiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian-qiang Wang.

Appendix

Appendix

Proof of Theorem 3.

Proof

Theorem 3 will be proven by utilizing the mathematical induction of n as follows:

\(x_{i} = T_{i}^{ + } ,I_{i}^{ - } ,F_{i}^{ - }\) and \(y_{i} = I_{i}^{ + } ,F_{i}^{ + } ,T_{i}^{ - }\) are utilized to simplify the process. First, Eq. (22) must be proven.

$$\mathop { \oplus _{F} }\limits_{\begin{subarray}{l} i.j = 1 \\ i \ne j \end{subarray} }^{n} \frac{{w_{{\left( i \right)}} w_{{\left( j \right)}} }}{{1 - w_{{\left( i \right)}} }} \cdot _{F} \left( {\left( {\sigma _{{\left( i \right)}} } \right)^{{ \wedge _{F} s}} \otimes _{F} \left( {\sigma _{{\left( j \right)}} } \right)^{{ \wedge _{F} t}} } \right) = \left( {1 - \log _{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{\begin{subarray}{l} i,j = 1 \\ j \ne i \end{subarray} }^{n} {\left( {\frac{{\left( {\lambda - 1} \right)^{{s + t}} - \left( {\lambda ^{{x_{{\left( i \right)}} }} - 1} \right)^{s} \left( {\lambda ^{{x_{{\left( j \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{{s + t}} + \left( {\lambda - 1} \right)\left( {\lambda ^{{x_{{\left( i \right)}} }} - 1} \right)^{s} \left( {\lambda ^{{x_{{\left( j \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{{\left( i \right)}} w_{{\left( j \right)}} }}{{1 - w_{{\left( i \right)}} }}}} } } \right),\log _{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{\begin{subarray}{l} i,j = 1 \\ j \ne i \end{subarray} }^{n} {\left( {\frac{{\left( {\lambda - 1} \right)^{{s + t}} - \left( {\lambda ^{{1 - y_{{\left( i \right)}} }} - 1} \right)^{s} \left( {\lambda ^{{1 - y_{{\left( j \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{{s + t}} + \left( {\lambda - 1} \right)\left( {\lambda ^{{1 - y_{{\left( i \right)}} }} - 1} \right)^{s} \left( {\lambda ^{{1 - y_{{\left( j \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{{\left( i \right)}} w_{{\left( j \right)}} }}{{1 - w_{{\left( i \right)}} }}}} } } \right)} \right).$$
(22)

(a) For n = 2, the following equations can be easily calculated:

$$\begin{aligned} & \frac{{w_{\left( 1 \right)} w_{\left( 2 \right)} }}{{1 - w_{\left( 1 \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( 1 \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( 2 \right)} } \right)^{{ \wedge_{F} t}} } \right) \\ & \quad = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{\left( 1 \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( 2 \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{\left( 1 \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( 2 \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( 1 \right)} w_{\left( 2 \right)} }}{{1 - w_{\left( 1 \right)} }}}} } \right),} \right. \, \\ & \quad \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{\left( 1 \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( 2 \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{\left( 1 \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( 2 \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( 1 \right)} w_{\left( 2 \right)} }}{{1 - w_{\left( 1 \right)} }}}} } \right)} \right) \\ \end{aligned}$$
(23)

and

$$\begin{aligned} & \frac{{w_{\left( 2 \right)} w_{\left( 1 \right)} }}{{1 - w_{\left( 2 \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( 2 \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( 1 \right)} } \right)^{{ \wedge_{F} t}} } \right) \\ & \quad = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{\left( 2 \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( 1 \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{\left( 2 \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( 1 \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( 2 \right)} w_{\left( 1 \right)} }}{{1 - w_{\left( 2 \right)} }}}} } \right),} \right. \, \\ & \quad \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{\left( 2 \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( 1 \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{\left( 2 \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( 1 \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( 2 \right)} w_{\left( 1 \right)} }}{{1 - w_{\left( 2 \right)} }}}} } \right)} \right) .\\ \end{aligned}$$
(24)

Then,

$$\begin{aligned} \mathop { \oplus_{F} }\limits_{\begin{subarray}{l} i.j = 1 \\ i \ne j \end{subarray} }^{2} \frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( j \right)} } \right)^{{ \wedge_{F} t}} } \right) \hfill \\ = \frac{{w_{\left( 1 \right)} w_{\left( 2 \right)} }}{{1 - w_{\left( 1 \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( 1 \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( 2 \right)} } \right)^{{ \wedge_{F} t}} } \right) \oplus_{F} \frac{{w_{\left( 2 \right)} w_{\left( 1 \right)} }}{{1 - w_{\left( 2 \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( 2 \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( 1 \right)} } \right)^{{ \wedge_{F} t}} } \right) \hfill \\ = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{\begin{subarray}{l} i,j = 1 \\ j \ne i \end{subarray} }^{2} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( j \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( j \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }}}} } } \right),} \right. \hfill \\ \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{\begin{subarray}{l} i,j = 1 \\ j \ne i \end{subarray} }^{2} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( j \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( j \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }}}} } } \right)} \right). \hfill \\ \end{aligned}$$

That is, when n = 2, Eq. (22) is correct.

(b) Equation (22) is assumed to be correct when n = k. That is,

$$\begin{aligned} \mathop { \oplus_{F} }\limits_{\begin{subarray}{l} i.j = 1 \\ i \ne j \end{subarray} }^{k} \frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( j \right)} } \right)^{{ \wedge_{F} t}} } \right) \hfill \\ = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{\begin{subarray}{l} i,j = 1 \\ j \ne i \end{subarray} }^{k} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( j \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( j \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }}}} } } \right),} \right. \hfill \\ \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{\begin{subarray}{l} i,j = 1 \\ j \ne i \end{subarray} }^{k} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( j \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( j \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }}}} } } \right)} \right). \hfill \\ \end{aligned}$$
(25)

Thereafter, when n = k + 1, the following equation can be obtained:

$$\begin{aligned} & \mathop { \oplus_{F} }\limits_{\begin{subarray}{l} i,j = 1 \\ i \ne j \end{subarray} }^{k + 1} \frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( j \right)} } \right)^{{ \wedge_{F} t}} } \right) \\ & \quad = \mathop { \oplus_{F} }\limits_{\begin{subarray}{l} i,j = 1 \\ i \ne j \end{subarray} }^{k} \frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( j \right)} } \right)^{{ \wedge_{F} t}} } \right)\mathop { \oplus_{F} }\limits_{i = 1}^{k} \frac{{w_{\left( i \right)} w_{{\left( {k + 1} \right)}} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{{\left( {k + 1} \right)}} } \right)^{{ \wedge_{F} t}} } \right) \, \\ & \quad \mathop { \oplus_{F} }\limits_{j = 1}^{k} \frac{{w_{{\left( {k + 1} \right)}} w_{\left( j \right)} }}{{1 - w_{{\left( {k + 1} \right)}} }} \cdot_{F} \left( {\left( {\sigma_{{\left( {k + 1} \right)}} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( j \right)} } \right)^{{ \wedge_{F} t}} } \right). \\ \end{aligned}$$
(26)

Equations (27) and (28) must be proven to prove Eq. (26).

$$\begin{aligned} \mathop { \oplus_{F} }\limits_{i = 1}^{k} \frac{{w_{\left( i \right)} w_{{\left( {k + 1} \right)}} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{{\left( {k + 1} \right)}} } \right)^{{ \wedge_{F} t}} } \right) \hfill \\ = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{i = 1}^{k} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{{\left( {k + 1} \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{{\left( {k + 1} \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{{\left( {k + 1} \right)}} }}{{1 - w_{\left( i \right)} }}}} } } \right),} \right. \hfill \\ \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{i = 1}^{k} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{{\left( {k + 1} \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{{\left( {k + 1} \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{{\left( {k + 1} \right)}} }}{{1 - w_{\left( i \right)} }}}} } } \right)} \right). \hfill \\ \end{aligned}$$
(27)
$$\begin{aligned} \mathop { \oplus_{F} }\limits_{j = 1}^{k} \frac{{w_{{\left( {k + 1} \right)}} w_{\left( j \right)} }}{{1 - w_{{\left( {k + 1} \right)}} }} \cdot_{F} \left( {\left( {\sigma_{{\left( {k + 1} \right)}} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( j \right)} } \right)^{{ \wedge_{F} t}} } \right) \hfill \\ = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{j = 1}^{k} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{{\left( {k + 1} \right)}} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( j \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{{\left( {k + 1} \right)}} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( j \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{{\left( {k + 1} \right)}} w_{\left( j \right)} }}{{1 - w_{{\left( {k + 1} \right)}} }}}} } } \right),} \right. \hfill \\ \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{j = 1}^{k} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{{\left( {k + 1} \right)}} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( j \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{{\left( {k + 1} \right)}} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( j \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{{\left( {k + 1} \right)}} w_{\left( j \right)} }}{{1 - w_{{\left( {k + 1} \right)}} }}}} } } \right)} \right). \hfill \\ \end{aligned}$$
(28)

(1) Equation (27) can be proven by utilizing the mathematical induction of k as follows:

① For k = 2, the following equation can be obtained easily:

$$\begin{aligned} \mathop { \oplus_{F} }\limits_{i = 1}^{2} \frac{{w_{\left( i \right)} w_{\left( 3 \right)} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( 3 \right)} } \right)^{{ \wedge_{F} t}} } \right) = \frac{{w_{\left( 1 \right)} w_{\left( 3 \right)} }}{{1 - w_{\left( 1 \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( 1 \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( 3 \right)} } \right)^{{ \wedge_{F} t}} } \right) \oplus_{F} \frac{{w_{\left( 2 \right)} w_{\left( 3 \right)} }}{{1 - w_{\left( 2 \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( 2 \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( 3 \right)} } \right)^{{ \wedge_{F} t}} } \right) \hfill \\ = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{i = 1}^{2} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( 3 \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( 3 \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{\left( 3 \right)} }}{{1 - w_{\left( i \right)} }}}} } } \right),} \right. \hfill \\ \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{i = 1}^{2} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( 3 \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( 3 \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{\left( 3 \right)} }}{{1 - w_{\left( i \right)} }}}} } } \right)} \right). \hfill \\ \end{aligned}$$

② Equation (27) is assumed to be correct when k = l. That is,

$$\begin{aligned} & \mathop { \oplus_{F} }\limits_{i = 1}^{l} \frac{{w_{\left( i \right)} w_{{\left( {l + 1} \right)}} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{{\left( {l + 1} \right)}} } \right)^{{ \wedge_{F} t}} } \right) \\ & \quad = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{i = 1}^{l} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{{\left( {l + 1} \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{{\left( {l + 1} \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{{\left( {l + 1} \right)}} }}{{1 - w_{\left( i \right)} }}}} } } \right),} \right. \\ & \quad \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{i = 1}^{l} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{{\left( {l + 1} \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{{\left( {l + 1} \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{{\left( {l + 1} \right)}} }}{{1 - w_{\left( i \right)} }}}} } } \right)} \right). \\ \end{aligned}$$

Subsequently, when k = l + 1, the following equation can be obtained:

$$\begin{aligned} & \mathop { \oplus_{F} }\limits_{i = 1}^{l + 1} \frac{{w_{\left( i \right)} w_{{\left( {l + 2} \right)}} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{{\left( {l + 2} \right)}} } \right)^{{ \wedge_{F} t}} } \right) \\ & \quad = \mathop { \oplus_{F} }\limits_{i = 1}^{l} \frac{{w_{\left( i \right)} w_{{\left( {l + 2} \right)}} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{{\left( {l + 2} \right)}} } \right)^{{ \wedge_{F} t}} } \right) \oplus_{F} \frac{{w_{{\left( {l + 1} \right)}} w_{{\left( {l + 2} \right)}} }}{{1 - w_{{\left( {l + 1} \right)}} }} \cdot_{F} \left( {\left( {\sigma_{{\left( {l + 1} \right)}} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{{\left( {l + 2} \right)}} } \right)^{{ \wedge_{F} t}} } \right) \\ & \quad = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{i = 1}^{l} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{{\left( {l + 2} \right)}} }}{{1 - w_{\left( i \right)} }}}} } } \right),} \right. \\ & \quad \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{i = 1}^{l} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{{\left( {l + 2} \right)}} }}{{1 - w_{\left( i \right)} }}}} } } \right)} \right) \\ & \quad \oplus_{F} \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{{\left( {l + 1} \right)}} }} - 1} \right)^{s} \left( {\lambda^{{x_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{{\left( {l + 1} \right)}} }} - 1} \right)^{s} \left( {\lambda^{{x_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{{\left( {l + 1} \right)}} w_{{\left( {l + 2} \right)}} }}{{1 - w_{{\left( {l + 1} \right)}} }}}} } \right),} \right. \\ & \quad \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{{\left( {l + 1} \right)}} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{{\left( {l + 1} \right)}} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{{\left( {l + 1} \right)}} w_{{\left( {l + 2} \right)}} }}{{1 - w_{{\left( {l + 1} \right)}} }}}} } \right)} \right) \\ & \quad = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{i = 1}^{l + 1} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{{\left( {l + 2} \right)}} }}{{1 - w_{\left( i \right)} }}}} } } \right),} \right. \\ & \quad \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{i = 1}^{l + 1} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{{\left( {l + 2} \right)}} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{{\left( {l + 2} \right)}} }}{{1 - w_{\left( i \right)} }}}} } } \right)} \right). \\ \end{aligned}$$

Thereafter, when k = l + 1, Eq. (27) is correct. Therefore, Eq. (27) is correct for all k.

(2) Similarly, Eq. (28) can be proven.

Subsequently, when Eqs. (25), (27), and (28) are used, Eq. (26) can be converted into the following form:

$$\begin{aligned} \mathop { \oplus_{F} }\limits_{\begin{subarray}{l} i,j = 1 \\ i \ne j \end{subarray} }^{k + 1} \frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( j \right)} } \right)^{{ \wedge_{F} t}} } \right) \hfill \\ = \mathop { \oplus_{F} }\limits_{\begin{subarray}{l} i,j = 1 \\ i \ne j \end{subarray} }^{k} \frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( j \right)} } \right)^{{ \wedge_{F} t}} } \right)\mathop { \oplus_{F} }\limits_{i = 1}^{k} \frac{{w_{\left( i \right)} w_{{\left( {k + 1} \right)}} }}{{1 - w_{\left( i \right)} }} \cdot_{F} \left( {\left( {\sigma_{\left( i \right)} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{{\left( {k + 1} \right)}} } \right)^{{ \wedge_{F} t}} } \right) \hfill \\ \mathop { \oplus_{F} }\limits_{j = 1}^{k} \frac{{w_{{\left( {k + 1} \right)}} w_{\left( j \right)} }}{{1 - w_{{\left( {k + 1} \right)}} }} \cdot_{F} \left( {\left( {\sigma_{{\left( {k + 1} \right)}} } \right)^{{ \wedge_{F} s}} \otimes_{F} \left( {\sigma_{\left( j \right)} } \right)^{{ \wedge_{F} t}} } \right) \hfill \\ = \left( {1 - \log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{\begin{subarray}{l} i,j = 1 \\ j \ne i \end{subarray} }^{k + 1} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( j \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{x_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{x_{\left( j \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }}}} } } \right),} \right. \hfill \\ \left. {\log_{\lambda } \left( {1 + \left( {\lambda - 1} \right)\prod\limits_{\begin{subarray}{l} i,j = 1 \\ j \ne i \end{subarray} }^{k + 1} {\left( {\frac{{\left( {\lambda - 1} \right)^{s + t} - \left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( j \right)} }} - 1} \right)^{t} }}{{\left( {\lambda - 1} \right)^{s + t} + \left( {\lambda - 1} \right)\left( {\lambda^{{1 - y_{\left( i \right)} }} - 1} \right)^{s} \left( {\lambda^{{1 - y_{\left( j \right)} }} - 1} \right)^{t} }}} \right)^{{\frac{{w_{\left( i \right)} w_{\left( j \right)} }}{{1 - w_{\left( i \right)} }}}} } } \right)} \right). \hfill \\ \end{aligned}$$

Thus, when n = k + 1, Eq. (22) is correct. Then, Eq. (22) is correct for all n.

Therefore, Eq. (18) can be easily proven.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, L., Zhang, Hy. & Wang, Jq. Frank Choquet Bonferroni Mean Operators of Bipolar Neutrosophic Sets and Their Application to Multi-criteria Decision-Making Problems. Int. J. Fuzzy Syst. 20, 13–28 (2018). https://doi.org/10.1007/s40815-017-0373-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-017-0373-3

Keywords

Search

Navigation