A new approach to the bipolar Shilkret integral

Document Type : Original Article


Department of Applied Sciences, University of Technology, Baghdad, Iraq


Abstract: Capacity, also known as a non-additive measure, is an extension of the Lebesgue measure. In recent years, bi-capacity was presented as a generalization of capacity with several bipolar fuzzy integrals related to bi-capacity, one of them being the bipolar Shilkret integral. In this paper, we propose a new approach to calculating the bipolar Shilkret integral to be suitable for bipolar scales. Then, we give some main properties of this integral related to bi-capacity.


[1] J. Abbas, The Banzhaf interaction index for bi-cooperative games, International Journal of General Systems, 50(5) 2021, 486-500.
[2] J. Abbas, The Bipolar Choquet Integrals Based On Ternary-Element Sets, Journal of Artificial Intelligence and Soft Computing Research, 6(1) 2016, 13-21.
[3] J. Abbas, The Balancing Bipolar Choquet Integrals, International Journal Of Innovative Computing, Information And Control, 17(3) 2021, 949-957.
[4] J. Abbas, H. Jaferi, The alternative Representation of Bipolar Sugeno integral, Engineering and Technology Journal, 40(10) 2022, 1318-1324.
[5] J. Abbas, Bipolar Choquet integral of fuzzy events, In: IEEE Conference on Computational Intelligence in Multi-Criteria Decision-Making, Florida, USA, 2014, 116-123.
[6] J. Abbas, The 2-Additive Choquet Integral Of Bi-Capacities, In: Rutkowski L., Scherer R., Korytkowski M., Pedrycz W., Tadeusiewicz R., Zurada J. (eds) Artificial Intelligence and Soft Computing, ICAISC 2019, Lecture Notes in Computer Science, Vol. 11508, Springer, 2019, 287-295.
[7] J. Abbas, Shilkret Integral Based on Binary-Element Sets and its Application in the Area of Synthetic Evaluation, Engineering and Technology Journal, 33(B) 2015, 571–577.
[8] J.M. Bilbao, J.R. Fernandez, N. Jimenez, J.J. Lopez, A survey of bicooperative games, In: Pareto Optimality, Game Theory and Equilibrium, Springer New York, 2008, 187-216.
[9] A. Banerjee, P.K. Singh, R. Sarkar, Fuzzy integral based CNN classifier fusion for 3D skeleton action recognition, IEEE Trans. Circ. Syst. Video Technol, 31(6) 2020, 2206-2216.
[10] J. Bueno, C.A Dias, G.P. Dimuro, H.S. Santos, E.N. Borges, G. Lucca, H. Bustince, Aggregation functions based on the Choquet integral applied to image resizing, In Proceedings of the 11th Conference of the European Society for Fuzzy Logic and Technology, EUSFLAT 2019, Prague, Czech Republic, Atlantis Press: Dordrecht, The Netherlands, 2019, 460-466.
[11] D. Candeloro, R. Mesiar, A.R. Sambucini, A special class of fuzzy measures: Choquet integral and applications, Fuzzy Sets Syst., 355 2019, 83-99.
[12] G. Choquet, Theory of capacities, Ann. Inst. Fourier, 5 1953, 131-295.
[13] S. Dey, R. Bhattacharya, S. Malakar, S. Mirjalili, R. Sarkar, Choquet fuzzy integral-based classifier ensemble technique for COVID-19 detection, Computers in Biology and Medicine, 135 2021, 1-11.
[14] M. Grabisch, T. Murofushi, M. Sugeno, Fuzzy Measures and Integrals. Theory and Applications, Physica Verlag, Berlin Heidelberg, 2000.
[15] M. Grabisch, C. Labreuche, Bi-capacities, Part I: definition, Möbius transform and interaction, Fuzzy Sets Syst. 151 2005, 211-236.
[16] M. Grabisch, C. Labreuche, Bi-capacities II: The Choquet integral, Fuzzy Sets and Systems, 151 2005, 237-259.
[17] M. Grabisch, The symmetric Sugeno integral, Fuzzy Sets and Systems, 139 2003, 473-490.
[18] S. Greco, F. Rindone, Bipolar fuzzy integrals, Fuzzy Sets and Systems, 220 2013, 21- 33.
[19] S. Greco, M. Grabisch, M. Pirlot, Bipolar and bivariate models in multicriteria decision analysis: descriptive and constructive approaches, Int. J. Intell. Syst. 23 2008, 930-969.
[20] S. Greco, F. Rindone, The bipolar Choquet integral representation, Theory Decis 77 2014, 1-29.
[21] S. Greco, R. Mesiar, F. Rindone, Discrete bipolar universal integrals, Fuzzy Sets Syst. 252 2014, 55-65.
[22] M. Hesham, J. Abbas, Multi-criteria decision making on the optimal drug for rheumatoid arthritis disease, Iraqi Journal of Science, 62(5) 2021.
[23] P. Karczmarek, A. Kiersztyn, W. Pedrycz, Generalized Choquet integral for face recognition, Int. J. Fuzzy Syst. 20 2018, 1047-1055.
[24] F. Kareem, J. Abbas, A Generalization of the Concave Integral in Terms of Decomposition of the Integrated Function for Bipolar Scales, Journal of Applied Sciences and Nanotechnology, 1(4) 2021, 81–90.
[25] G. Lucca, J.A. Sanz, G.P. Dimuro, E.N Borges, H. Santos, H. Bustince, Analyzing the performance of different fuzzy measures with generalizations of the Choquet integral in classification problems, In Proceedings of the 2019 IEEE International Conference on Fuzzy Systems, New Orleans, LA, USA, 23-26 June 2019, 1-6.
[26] A. Naeem, J. Abbas, A Computational Model for Multi-Criteria Decision Making in Traffic Jam Problem, Journal of Automation, Mobile Robotics and Intelligent Systems, 15(2) 2022, 39-43.
[27] B. Mihailović, E. Pap, M. Štrboja, A. Simićević, A unified approach to the monotone integral-based premium principles under the CPT theory, Fuzzy Sets and Syst. 398 2020, 78-97.
[28] N. Shilkret, Maxitive measure and integration, Indag. Math., 33 1971, 109–116.
[29] M. Štrboja, E. Pap, B. Mihailović, Discrete bipolar pseudo-integrals, Inf. Sci. 468 2018, 72-88.
Volume 3, Issue 4
December 2022
Pages 46-54
  • Receive Date: 30 November 1999
  • Accept Date: 30 November 1999
  • First Publish Date: 01 December 2022