Abstract
The field of aggregation theory addresses the mathematical formalization of aggregation processes. Historically, the developed mathematical framework has been largely confined to the aggregation of real numbers, while the aggregation of other types of structures, such as rankings, has been independently considered in different fields of application. However, one could lately perceive an increasing interest in the study and formalization of aggregation processes on new types of data. Mostly, this aggregation outside the framework of real numbers is based on the use of a penalty function measuring the disagreement with a consensus element. Unfortunately, there does not exist a comprehensive theoretical framework yet. In this paper, we propose a natural extension of the definition of a penalty function to a more general setting based on the compatibility with a given betweenness relation. In particular, we revisit one of the most common methods for the aggregation of rankings – the method of Kemeny – which will be positioned in the penalty-based aggregation framework.
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Notes
- 1.
Some other terms, such as ‘cost’, ‘disagreement’, ‘discrepancy’, ‘divergence’ or ‘error’, have been occasionally used for replacing the term ‘penalty’. However, in the field of aggregation theory, the term ‘penalty’ is nowadays considered the standard.
- 2.
We denote (local) penalty functions defined on \(\mathbb {R}\times \mathbb {R}\) by L and penalty functions defined on \(\mathbb {R}^{n}\times \mathbb {R}\) by P. For more details, we refer to [2].
- 3.
Note that we write the term ‘aggregation’ between quotation marks due to the fact that the penalty-based function associated with P is only assured to satisfy the boundary conditions (it is actually idempotent), while the property of increasingness might not hold. For more details, we refer to [2, 3].
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Pérez-Fernández, R., De Baets, B. (2018). Penalty-Based Aggregation Beyond the Current Confinement to Real Numbers: The Method of Kemeny Revisited. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT IWIFSGN 2017 2017. Advances in Intelligent Systems and Computing, vol 643. Springer, Cham. https://doi.org/10.1007/978-3-319-66827-7_16
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