Decision SupportSubjective and objective information in linguistic multi-criteria group decision making
Introduction
Decision making problems are very common in the literature (Figueira, Greco, & Ehrgott, 2005). Most of the decisions are taken from an intuitive point of view or only with some very basic information. However, in real life, the problems are often not so easy and it is necessary to analyse the information in more detail. Moreover, sometimes the available information is vague and imprecise and it cannot be assessed with the usual exact numbers. Therefore, a better approach could be the use of linguistic information. Linguistic information was introduced by Zadeh (1975) as a tool for representing imprecise expressions such as low, high and very high by using fuzzy sets (Zadeh, 1965). Since then, other authors have studied alternative representations of the linguistic information, including the 2-tuple linguistic approach (Herrera & Martinez, 2000), the continuous linguistic framework (Xu, 2004a, Xu, 2004b), uncertain linguistic variables (Xu, 2004c), linguistic unbalanced information (Herrera, Herrera-Viedma, & Martínez, 2008), uncertain linguistic unbalanced information (Xu, 2009), linguistic cloud information (Wang, Peng, Zhang, & Chen, 2014), and hesitant fuzzy linguistic term sets (Liao, Xu, Zeng, & Merigó, 2015).
A further interesting issue when dealing with decision making problems is the necessity of aggregating the information. In the literature, there are a wide range of aggregation operators (Beliakov, Pradera and Calvo, 2007, Grabisch, Marichal, Mesiar and Pap, 2011). Some very well-known ones are the weighted average, the probabilistic aggregation and the ordered weighted average (Yager, 1988, Yager, Kacprzyk and Beliakov, 2011). These operators have also been studied under linguistic information forming the linguistic weighted average, the linguistic probabilistic aggregation and the linguistic OWA (LOWA) operator (Herrera, Herrera-Viedma, & Verdegay, 1995). Since its introduction, many other authors have developed new linguistic aggregation operators. Merigó and Casanovas (2010) studied the use of distance measures with linguistic information and Zeng and Su (2012) generalized it by using induced generalized aggregation operators. Zhou, Wu, and Chen (2014) extended the linguistic distances with continuous aggregation operators. Merigó, Casanovas, and Martínez (2010) developed a new approach for linguistic decision making with Dempster–Shafer belief structure. Xu, Merigó, and Wang (2012) and Zhou and Chen (2012) analysed a new framework with power averages. Merigó & Gil-Lafuente (2012) studied the use of induced and generalized aggregation operators. Liu and Jin (2012) and Liu, Jin, Zhang, Su, and Wang (2011) developed new approaches by using interval numbers in the linguistic variables. Wei and Zhao (2012) studied the use of dependent aggregation operators and Yu, Xu, Liu, and Chen (2012) suggested a new approach with 2-dimension linguistic information.
Recently, Merigó (2012a) has suggested the probabilistic weighted averaging (PWA) operator. It is an aggregation operator that unifies the probability and the weighted average (WA) in the same formulation and considering the degree of importance that each concept has in the analysis. In decision making, it is very useful for representing subjective and objective information in the same representation. It is worth noting that this approach follows a methodology similar to other methods suggested for the unification of the probability and the OWA operator (Merigó, 2012b). Furthermore, note that several authors have already suggested some approaches for the integration of the probability and the weighted average with the OWA operator such as the immediate probabilities (Engemann, Filev and Yager, 1996, Yager, Engemann and Filev, 1995), the weighted OWA (WOWA) operator (Torra, 1997) and the hybrid average (Xu & Da, 2003).
The objective of this paper is to present a new approach for dealing with linguistic information and decision making problems. In so doing, introduce the linguistic probabilistic weighted average (LPWA). Its main advantage is that it unifies the probability and the weighted average considering the degree of importance that each concept has in the analysis. Additionally, in an uncertain environment where the information is uncertain, it can be assessed with linguistic variables. We study some of its main properties and particular cases including the linguistic average (LA), the linguistic weighted average (LWA), the linguistic probabilistic aggregation (LPA), the arithmetic – LWA (A-LWA) and the arithmetic LPA (A-LPA) operator.
Further generalizations are presented by using quasi-arithmetic means forming the quasi-arithmetic LPWA (Quasi-LPWA) operator. It generalizes a wide range of aggregation functions including quadratic and geometric aggregations and the LPWA operator. Another interesting extension suggested for the LPWA operator is the use of moving averages obtaining the linguistic probabilistic weighted moving average (LPWMA). Its main advantage is that it can assess dynamic information in an appropriate way such as in time series forecasting. Moreover, it is able to deal with imprecise environments that can be represented with linguistic information. A further extension of the LPWMA operator is introduced by using quasi/arithmetic means forming the Quasi/LPWMA operator.
The applicability of the LPWA operator and all its extensions is very broad. All of the studies that use the probability or the weighted average in a linguistic context can be revised and extended with this approach. The main reason for this is that the LPWA operator can always be reduced to the LPA or LWA operator. However, by using this approach it is possible to represent the information in a more complete way because in uncertain or imprecise problems it is very common to have some part of the information given in a subjective way and another part in an objective way. This paper focuses on a linguistic multi-person multi-criteria decision making problem under subjective and objective risk. By using multi-criteria and multi-person information it is possible to obtain more robust information because real world problems depend on several aspects. Usually, the knowledge given by one expert is less than that from several experts. This decision making process produces a new aggregation operator that is called the multi-person multi-criteria LPWA (MPMC-LPWA) operator.
An illustrative example regarding the use of this new decision system is also studied. The example presents a judicial decision making problem regarding the creation of a new regulation in the European Union (EU). EU law implies the establishment of a lot of decisions in tax law, business law and so on, because different political parties may have different interests and it is necessary to reach an agreement. Under uncertain and imprecise environments, it is not easy to form a single decision because the future may produce different events that may completely change the potential results and therefore, the decisions. Thus, this new approach tries to represent all these possible scenarios and select the alternative that seems to be in closest accordance to the interests of the decision maker.
This paper is organized as follows. Section 2 briefly reviews some basic concepts to be used throughout the paper. Section 3 introduces the LPWA operator. Section 4 presents a generalization by using quasi-arithmetic means and Section 5 an extension with moving averages. Section 6 develops an application in linguistic multi-person multi-criteria decision making and Section 7 an illustrative example in judicial decision making. Section 8 ends the paper summarizing the main conclusions.
Section snippets
Preliminaries
In this Section, we briefly describe the linguistic approach to be used throughout the paper, the weighted and the probabilistic aggregation operators and the linguistic aggregation operators.
Linguistic probabilistic weighted aggregation operators
Linguistic probabilistic and weighted aggregation operators are those functions that use probabilities and weighted averages in the same formulation and in an uncertain environment that can be assessed with linguistic information. A fundamental aggregation operator under this context is the linguistic PWA (LPWA) operator. It unifies the probability and the weighted average in the same formulation and considering the degree of importance that each concept has in the aggregation. Moreover, it is
Generalized aggregation operators in the LPWA operator
A practical generalization of the LPWA operator can be developed by using generalized and quasi-arithmetic means (Fodor, Marichal and Roubens, 1995, Merigó and Gil-Lafuente, 2013). Since the quasi-arithmetic mean includes the generalized mean as a particular case, let us look into it. In the LPWA operator, the use of the quasi-arithmetic mean forms the quasi-arithmetic LPWA (Quasi-LPWA) operator. Its main advantage is that it provides a robust formulation that includes a wide range of
Moving averages in the LPWA operator
Moving averages are aggregation operators that deal with dynamic information. This is especially useful when dealing with different periods of time as happens in time series forecasting (Merigó and Yager, 2013, Yager, 2008). Following the methodology explained by Merigó (2012a) regarding the use of the moving average in the PWA operator, it is possible to extend this approach to the LPWA framework forming the linguistic probabilistic weighted moving average (LPWMA). The main advantage of this
Multi-person multi-criteria decision-making with the LPWA operator
Decision making problems are very common in the literature (Zavadskas and Turskis, 2011, Zeng, 2013) because the real world is plenty of decisions, some of them being of extreme relevance that may condition the future of a country or even the World. A very remarkable example is the European Union (EU) as it is still in the construction process and many issues that involve decision and agreement processes have to be considered (Merigó, Lobato-Carral, & Carrilero-Castillo, 2012). Some important
Numerical example
In this Section, let us present a simple numerical example of the new approach in a judicial linguistic multi-criteria multi-person decision making problem. The example is focused on a judicial problem regarding the creation of a new regulation in the EU. Note that the example is based on the use of the methodology explained in Section 6. However, in real world problems, many other issues may appear since different political parties may have different interests and the agreement should be
Conclusions
A new linguistic multi-criteria group decision making method has been introduced. It deals with uncertain environments that can be assessed with linguistic information. We have used linguistic probabilistic and weighted aggregation operators, such as the LPWA operator, in the analysis. It is a new aggregation operator that unifies the probability and the weighted average considering the degree of importance that each concept has in the analysis and in an uncertain environment assessed with
Acknowledgment
We would like to thank the reviewers for valuable comments that have improved the quality of the paper. Support from the European Commission through the project PIEF-GA-2011-300062 is gratefully acknowledged.
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