Document Type : Research Paper


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

2 Department of Mathematical Sciences, University of South Africa, UNISA0003, South Africa


In the context of decision support systems, bi-capacities were introduced as an extension of classical capacities. Many bipolar fuzzy integrals related to the bi-capacities have been presented in recent years. One of these integrals is the Sugeno integral concerning aggregation on bipolar scales. The paper aims to build an equivalent representation of the bipolar Sugeno integral. Therefore, we first employ in this paper the framework based on a ternary-criterion set for proposing an alternative formula of the bipolar Sugeno integral to be suitable for bipolar scales.  Then, we discuss some basic properties and give an illustrative example of this representation. This representation is consistent as an extension of the representation concerning the classical capacities and aggregation on the Sugeno integral unipolar scales.


  • An alternative representation of the bipolar Sugeno integral was proposed to be suitable for bipolar scales.
  • Study the main properties of this integral.
  • This representation is consistent as a generalization of the expression concerning the classical Sugeno integral.


Main Subjects

[1] D. Candeloro, R.  Mesiar,A.R. Sambucini, “A special class of fuzzy measures: Choquet integral and applications”,. Fuzzy Sets Syst., 355(2019) 83–99.
[2] G. Lucca, J.A. Sanz, G.P. Dimuro, E.N.Borges, Santos, H.; Bustince, H. “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., (2019) 1–6 . doi: 10.1109/FUZZ-IEEE.2019.8858815
[3] M. Hesham ,  Abbas J., “Multi-criteria decision making on the optimal drug for rheumatoid arthritis disease”, Iraqi Journal of Science., 62(2021). doi: 10.24996/ijs.2021.62.5.28
[4] P. Karczmarek, Kiersztyn A., Pedrycz W., “Generalized Choquet integral for face recognition”, Int. J. Fuzzy Syst., 20(2018)1047-1055.
[5] A. Naeem, , Abbas, J. “A Computational Model for Multi-Criteria Decision Making in Traffic Jam Problem”, Journal of Automation, Mobile Robotics and Intelligent Systems., 15(2022) 39-43.doi: 10.14313/JAMRIS/2-2021/12
[6] M. Sugeno, “Theory of fuzzy integrals and its applications”, Ph.D. Thesis, Tokyo Institute of technology, 1974.
[7]  J.C.S. Bueno, Dias, C.A.; Dimuro, G.P.; Santos, H.S.; Borges, E.N.; Lucca, G.; Bustince, H. “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, 9–13 September 2019; Atlantis Press: Dordrecht, The Netherlands.,(2019). 460–466.
[8] Subhrajit Dey, Rajdeep Bhattacharya, Samir Malakar, Seyedali Mirjalili, Ram Sarkar, “Choquet fuzzy integral-based classifier ensemble technique for COVID-19 detection”, Computers in Biology and Medicine., 135(2021) 1-11. doi:10.1016/j.compbiomed.2021.104585
[9]  A. Banerjee, Singh P.K., Sarkar R., “Fuzzy integral based CNN classifier fusion for 3D skeleton action recognition”, IEEE Trans. Circ. Syst. Video Technol., 31 (2020) 2206 – 2216. doi10.1109/TCSVT.2020.3019293
[10] M. Grabisch, T. Murofushi,, Sugeno, M., Fuzzy Measures and Integrals. Theory and Applications, Physica Verlag, Berlin Heidelberg, 2000.
[11] G. Choquet G, “Theory of capacities”,  Ann. Inst. Fourier., 5(1953) 131-295.
[12] J. Abbas, Shilkret Integral Based on Binary-Element Sets and its Application in The Area of Synthetic Evaluation, Engineering & Technology Journal., 33 Part (B) 2015.
[13] I. Chitescu, “The Sugeno integral. A point of view”, Information Sciences., 582 (2022) 648-664.
[14] J. Abbas, Logical twofold integral”, Engineering & Technology Journal., 283 (2010).
[15] 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.
[16] Grabisch M., C. Labreuche: Bi-capacities, Part I: definition, Möbius transform and interaction, Fuzzy Sets Syst., 151 (2005) 211-236. 10.1016/j.fss.2004.08.012ff. ffhal-00187189f
[17]  M.  Grabisch, C. Labreuche, “Bi-capacities II: The Choquet integral”, Fuzzy Sets and Systems., 151(2005) 237-259.2005.
[18]   S. Greco, F.  Rindone, “Bipolar fuzzy integrals”, Fuzzy Sets and Systems.,  220(2013)  21-33 .
[19] F. Kareem, and Abbas J. “A Generalization of the Concave Integral in Terms of Decomposition of the Integrated Function for Bipolar Scales”, Journal of Applied Sciences and Nanotechnology., 1 (2021) 81–90. doi: 10.53293/jasn.2021.3985.1065
[20] S. Greco, Grabisch M., Pirlot M., Bipolar and bivariate models in multi-criteria decision analysis: descriptive and constructive approaches, Int. J. Intell. Syst., 23(2008)930-969.
[21] S. Greco, Rindone, F. “The bipolar Choquet integral representation”, Theory Decis., 77(2014) 1-29.
[22] J. Abbas, The Banzhaf interaction index for bi-cooperative games, International Journal of General Systems., 50(2021)486-500./
[23] M. Štrboja, Pap E., Mihailovi B., Discrete bipolar pseudo-integrals, Information Sciences., 468(2018) 72–88.
[24] J. Abbas, The Bipolar Choquet Integrals Based On Ternary-Element Sets. jaiscr., 6(2016)13-21. doi:
[25]  J. Abbas, A New Framework of bi-capacities and its Integrals, Journal of Al-Nahrain University Science., 18 (2015) 132-137.
[26]  J. Abbas .  The Balancing Bipolar Choquet Integrals, International Journal Of Innovative Computing, Information And Control, 17(2021)949–957. doi: 10.24507/ijicic.17.03.949
[27] J.M. Bilbao, Fern´andez, J.R., Jim´enez, N., L´opez, J.J. , A survey of bicooperative games, In: Pareto Optimality, Game Theory And Equilibria, Springer New York., (2008) 187-216 .
[28] Grabisch, M. ,“The symmetric Sugeno integral”, Fuzzy Sets and Systems., 139(2003) 473–490.