FOCUS Some relationships between Losonczi’s based OWA generalizations and the Choquet–Stieltjes integral Vicenc ¸ Torra Æ Yasuo Narukawa Published online: 7 June 2009 Ó Springer-Verlag 2009 Abstract The number of aggregation operators existing nowadays is rather large. In this paper, we study some of these operators and establish some relationships between them. In particular, we focus on neat operators. We link some of these operators with the Losonczi’s mean. The results permit us to define a Losonczi’s OWA and a Losonczi’s WOWA. Keywords Aggregation operators  Losonczi’s mean  OWA operators  Choquet–Stieltjes integral 1 Introduction Aggregation operators (Calvo et al. 2002; Grabisch et al. 2000; Torra and Narukawa 2007a) are used to combine information to obtain a datum of better quality. In recent years, there is an increasing interest in these topics for their application in decision problems and artificial intelligence applications (Nettleton and Baeza-Yates 2008; Torra 2008). In this paper, we focus on the Losonczi’s mean, which was proposed in (Losonczi 1971) and which generalizes Bajraktarevic ´’s mean (Bajraktarevic ´ 1958, 1963). Our interest is to establish some relationships between these operators and some of the ones that have been defined more recently related to the ordered weighted averaging (OWA) operator. In particular, we will consider the OWA and the neat OWA. The structure of this paper is as follows. In Sect. 2, we review fuzzy measures and introduce a few results related to the Choquet–Stieltjes integral. In Sect. 3, we review the aggregation operators we need later on. Then, in Sect. 5, we establish the relationships between these operators. Section 6, uses Losonczi’s mean to introduce a Losonczi’s OWA (LoOWA) operator. The paper finishes with some conclusions. 2 Fuzzy measures and the Choquet–Stieltjes integral In this section, we define fuzzy measures, the Choquet integral and the Choquet Stieltjes integral, and show their basic properties. Let X be a locally compact Hausdorff space and B be a class of Borel sets, that is, the smallest r-algebra, which includes the class of all closed sets. We say that ðX; BÞ is a measurable space. Example 1 We consider two examples of Hausdorff spaces: 1. The set of all real numbers R is a locally compact Hausdorff space. If X ¼ R; B is the smallest r-algebra, which includes the class of all closed intervals. 2. Let X :¼f1; 2; ...; Ng: X is a compact Hausdorff space with a discrete topology. Then we have B¼ 2 X : Definition 1 Let ðX; BÞ be a measurable space. A fuzzy measure (or a non-additive measure) l is a real valued set function, l : B !½0; 1 with the following properties; 1. lð;Þ ¼ 0 V. Torra (&) IIIA-CSIC Institut d’Investigacio ´ en Intellige `ncia Artificial, Campus UAB s/n, 08193 Bellaterra, Catalonia, Spain e-mail: vtorra@iiia.csic.es Y. Narukawa Toho Gakuen, 3-1-10 Naka, Kunitachi, Tokyo 186-0004, Japan e-mail: narukawa@d4.dion.ne.jp 123 Soft Comput (2010) 14:465–472 DOI 10.1007/s00500-009-0454-9