A unified model between the weighted average and the induced OWA operator José M. Merigó ⇑ Department of Business Administration, University of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain article info Keywords: Weighted average OWA operator Aggregation operators Multi-person decision-making abstract We present a new model that uses the weighted average (WA) and the induced ordered weighted aver- aging (IOWA) operator in the same formulation. We call it the induced ordered weighted averaging– weighted average (IOWAWA) operator. We study some of its main properties and we see that it has a lot of particular cases such as the WA and the OWA operator. The main advantage of the IOWAWA oper- ator is that it unifies the IOWA operator with the WA in the same formulation considering the degree of importance that each concept has in the aggregation. We analyze the applicability of this new approach and we see that it is very broad because it can be applied in a wide range of fields such as statistics, eco- nomics, decision theory and engineering. Theoretically, we could state that all the previous models and applications based on the WA and the IOWA can be revised and improved with this new approach because they will be included in this framework as a particular case. We focus on an application in a multi-person decision-making in political management. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction In the literature, we find a wide range of methods for aggregat- ing the information (Beliakov, Pradera, & Calvo, 2007; Merigó, 2008; Torra & Narukawa, 2007; Yager & Kacprzyk, 1997). The weighted average (WA) is one of the most common aggregation operators found in the literature. It can be used in a wide range of different problems including statistics, economics and engineer- ing. Another interesting aggregation operator that has not been used so much in the literature is the ordered weighted averaging (OWA) operator (Yager, 1988). The OWA operator provides a parameterized family of aggregation operators that range from the maximum to the minimum. Since its introduction, the OWA operator has been studied in a wide range of applications (Ahn, 2009; Alonso, Cabrerizo, Chiclana, Herrera, & Herrera-Viedma, 2009; Chang & Wen, 2010; Cheng, Wang, & Wu, 2009; Kacprzyk & Zadrozny, 2009; Liu, Cheng, Chen, & Chen, 2010; Merigó & Gil-Lafuente, 2008, 2010; Xu, 2009; Yager, 1993, 1996a, 2006, 2007, 2009a, 2009b; Yager & Kacprzyk, 1997; Zhao, Xu, Ni, & Liu, 2010). A very practical extension of the OWA is the induced OWA (IOWA) operator (Yager & Filev, 1999). It is an extension of the OWA operator that uses order-inducing variables in the reordering of the arguments. Its main advantage is that it can represent more complex situations because it can include a wide range of factors in the reordering process rather than simply consider the values of the arguments. Recently, several authors have developed different extensions and applications of the IOWA operator. For example, Merigó and Gil-Lafuente (2009) generalized it by using generalized and quasi-arithmetic means. Merigó, López-Jurado, Gracia, and Casanovas (2009) provided a more general formulation by using hybrid aggregations. Wu, Li, Li, and Duan (2009) developed a con- tinuous geometric version. Tan and Chen (2010) suggested an extension by using the Choquet integral. Herrera-Viedma, Chiclana, Herrera, and Alonso (2007) developed an extension for group decision-making problems. Merigó and Casanovas (2009) introduced an application in decision-making with Dempster– Shafer theory. For further reading, see for example Chen and Chen (2003), Merigó (2008), Merigó & Casanovas (2010a, 2011a, 2011c, 2011d), Merigó, Gil-Lafuente, and Barcellos (2010), Wei (2010) and Yager (2003). Recently, some authors (Torra, 1997; Torra & Narukawa, 2007, 2010; Wei, 2009; Xu, 2010; Xu & Da, 2003; Yager, 1998b) have tried to unify the OWA operator with the WA in the same formu- lation. It is worth mentioning the work developed by Torra (1997) with the introduction of the weighted OWA (WOWA) oper- ator and the work of Xu and Da (2003) about the hybrid averaging (HA) operator. Both models arrived to a unification between the OWA and the WA because both concepts were included in the for- mulation as particular cases. However, as it has been studied by Merigó (2008), these models seem to be a partial unification but not a real one because they can unify them but they cannot con- sider how relevant these concepts are in the specific problem con- sidered. For example, in some problems we may prefer to give more importance to the OWA operator because we believe that it 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.03.034 ⇑ Tel.: +34 93 402 19 62; fax: +34 93 403 98 82. E-mail address: jmerigo@ub.edu Expert Systems with Applications 38 (2011) 11560–11572 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa