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Abstract: In decision analysis, and especially in multiple criteria decision analysis, several non additive integrals have been introduced in the last years. These include the Choquet integral, the Shilkret integral and the Sugeno integral, among others. In the context of multiple criteria decision analysis, these integrals are used to aggregate the evaluations of possible choice alternatives, with respect to several criteria, into a single overall evaluation. The use of mentioned integrals in the aggregation process requests the starting evaluations to be expressed in terms of exact evaluations. In this paper we present the robust Choquet, Shilkret and Sugeno integrals, computed with respect to an interval-capacity. These are quite natural generalizations of the Choquet, Shilkret and Sugeno integrals, useful to aggregate interval-evaluations of choice alternatives into a single overall evaluation. We show that, when the interval-evaluations collapse into exact evaluations, our definitions of robust integrals collapse into the previous definitions. We also provide an axiomatic characterization of the robust Choquet integral. [Copyright &y& Elsevier]

Abstract: In decision analysis, and especially in multiple-criteria decision analysis, several non-additive integrals have been introduced in the last 60 years. These include the Choquet integral, the Shilkret integral and the Sugeno integral, among others. Recently, a bipolar Choquet integral was proposed for cases in which the underlying scale is bipolar. In this paper we propose a bipolar Shilkret integral and a bipolar Sugeno integral. Moreover, we provide an axiomatic characterization of all three bipolar fuzzy integrals. [Copyright &y& Elsevier]

We propose the concepts of superadditive and of subadditive transformations of aggregation functions acting on non-negative reals, in particular of integrals with respect to monotone measures. We discuss special properties of the proposed transforms and links between some distinguished integrals. Superadditive transformation of the Choquet integral, as well as of the Shilkret integral, is shown to coincide with the corresponding concave integral recently introduced by Lehrer. Similarly the transformation of the Sugeno integral is studied. Moreover, subadditive transformation of distinguished integrals is also discussed. [ABSTRACT FROM AUTHOR]

Extending the idea of Even and Lehrer (2014) [3] , we discuss a general approach to integration based on a given decomposition system equipped with a weighting function, and a decomposition of the integrated function. We distinguish two type of decompositions: sub-decomposition based integrals (in economics linked with optimization problems to maximize the possible profit) and super-decomposition based integrals (linked with costs minimization). We provide several examples (both theoretical and realistic) to stress that our approach generalizes that of Even and Lehrer (2014) [3] and also covers problems of linear programming and combinatorial optimization. Finally, we introduce some new types of integrals related to optimization tasks. [ABSTRACT FROM AUTHOR]

The concept of semicopula plays a fundamental role in the aggregation theory on interval [ 0 , 1 ] . Semicopulas are applied, for example, in the definition of universal integrals. We present an extension of the notion of semicopula to the case of symmetric bipolar interval [ − 1 , 1 ] . We call this extension bipolar semicopula. The last definition can be used to obtain a simplified definition of the bipolar universal integral. Moreover, bipolar semicopulas allow for an extension of theory of quasi-copulas to the interval [ − 1 , 1 ] . [ABSTRACT FROM AUTHOR]

The concept of universal integral, recently proposed, generalizes the Choquet, Shilkret and Sugeno integrals. Those integrals admit a discrete bipolar formulation, useful in those situations where the underlying scale is bipolar. In this paper we propose the concept of discrete bipolar universal integral, in order to provide a common framework for bipolar discrete integrals, including as special cases the discrete Choquet, Shilkret and Sugeno bipolar integrals. Moreover we provide two different axiomatic characterizations of the proposed discrete bipolar universal integral. [ABSTRACT FROM AUTHOR]