Balance Operator: A New Vision on Aggregation Operators

We introduce a new way to visualize the behavior of aggregation operators based on the analogy with a balance. The use of an analogy permits an intuitive representation of the operator. This metaphor is enough general to include operators of the most common aggregation families. So we obtain a general framework, that allows us to compare the different existing operators. In this paper we will show at first how we establish the analogy, then we will remind which operators are concerned. Subsequently, using the metaphor we will show how to construct the visualization of the behavior. And finally we will discover that the general visualization can be used for particular aggregation analysis. Introduction The problem of aggregating fuzzy sets in a meaningful way has been of a central interest since the late 1970s. In most cases, the aggregation operators are defined on a pure axiomatic basis forgetting usually to give a global intuitive vision of the comportment. That is why, in this paper we present a new way, based on a metaphor, to visualize aggregation operators. The use of a metaphor offers the possibility to illustrate the mathematical and axiomatic choices, providing in this way an intuitive vision of the behavior of the operator. On this article we will begin with the construction of the analogy, starting from a physical model of a balance and finishing with an aggregation operator: the balance aggregator. After this, we will note that our metaphor is very ample: the balance aggregator is enough general to include operators of the most common aggregation families: t-norms, t-conorm, arithmetic mean, OWA, MICA and Uninorm. That means, not only that we can use it with a lot of classical operators, but also that we obtain a general framework for discussion and comparison. Then we will, employing the analogy, construct the visualization of the behavior of an aggregation operator. We will at first explain how to construct the visualization of the general comportment and then how to add the visualization of the sensibility. We will finally see that we can exploit the last visualizations for the analysis of particular aggregations. The Balance Framework The idea of the metaphor consists in establishing a strength relationship between a real world object, on which we have a lot of natural intuition, with an abstract mathematical formula. We will start with a physical model of a real balance and then we will state that the mathematical formula, which computes the total weight, is actually a general form of aggregation operators. The physical model The physical model of balance we use is shown in the Figure 1. Figure 1: The balance model m1g m2g d1 d2