Linguistic Decision Making

The complexity and uncertainty of objective thing and the fuzziness of human thought result in decision making with linguistic information in a wide variety of practical problems, such as personnel evaluation, military system performance evaluation, online auctions, supply chain management, venture capital, and medical diagnostics. In such problems, a realistic approach may be to use linguistic assessments instead of numerical values by means of linguistic variables, that is, variables whose values are not numbers but words or sentences in a natural or artificial language (Fan and Wang, 2003; 2004; Fan et al., 2002; Herrera and Herrera-Viedma, 2003; 2000a; 2000b; 1997; Herrera and Martínez, 2001a; 2001b; 2000a; 2000b; Herrera and Verdegay, 1993; Herrera et al., 2005; 2003; 2001a; 2001b; 2000; 1997; 1996a; 1996b; 1995; Herrera-Viedma, 2001; Herrera-Viedma and Peis, 2003; Herrera-Viedma et al., 2005; 2004; 2003; Wang and Chuu, 2004; Xu, 2010; 2009a; 2009b; 2009c; 2008; 2007a; 2007b; 2007c; 2007d; 2007e; 2006a; 2006b; 2006c; 2006d; 2006e; 2006f; 2006g; 2006h; 2006i; 2005a; 2005b; 2005c; 2005d; 2004a; 2004b; 2004c; 2004d; 2004e; 2004f; Xu and Da, 2003; 2002; Zadeh and Kacprzyk, 1999a; 1999b). This may arise for different reasons (Chen and Hwang, 1992): ① the information may be unquantifiable due to its nature; and ② the precise quantitative information may not be stated because either it is unavailable or the cost of its computation is too high and an “approximate value” may be tolerated. For example, when evaluating the “comfort” or “design” of a car, linguistic labels like “good”, “fair” and “poor” are usually used, and evaluating a car’s speed, linguistic labels like “very fast”, “fast” and “slow” can be used (Bordogna et al., 1997; Levrat et al., 1997).

When a problem is solved using linguistic information, it implies the need for computing with words (Zadeh and Kacprzyk, 1999a; 1999b). Thus, how to fuse the input linguistic information is an interesting and important research topic. Linguistic aggregation operators are a powerful tool to deal with linguistic information. Over the last decades, many scholars have focused their investigation on linguistic aggregation techniques and various linguistic aggregation operators have been proposed (Xu, 2007b), including the linguistic max and min operators (Xu and Da 2003b; Yager, 1998; 1996; 1995; 1992), linguistic median operator (Yager, 1998; 1996), Linguistic weighted median operator (Yager, 1998; 1996), linguistic max-min weighted averaging operator (Yager, 1995), extension principle-based linguistic aggregation operator (Bonissone and Decker, 1986; Bordogna and Passi, 1993; Chang and Chen, 1994; Chen, 1997; Degani and Bortolan, 1988; Delgado et al., 1998; Lee, 1996), symbol-based linguistic aggregation operator (Delgado et al., 1993a), 2-tuple arithmetic mean operator (Herrera and Martínez, 2000a), 2-tuple weighted averaging operator (Herrera and Martínez, 2000a), 2-tuple OWA (orded weighted averaging) operator (Herrera and Martínez, 2000a), linguistic weighted OWA operator (Torra, 1997), linguistic averaging operator (Xu, 2006b; 2005b; 2004d; 2004f), linguistic weighted disjunction operator (Herrera and Herrera-Viedma, 2000b; 1997), linguistic weighted conjunction operator (Herrera and Herrera-Viedma, 2000b; 1997), linguistic weighted averaging operator (Herrera and Herrera-Viedma, 1997; Xu, 2006b; 2005b; 2004d), ordinal OWA operator (Bordogna et al., 1997; Yager, 1995; 1992), ordinal linguistic aggregation operator (Delgado et al., 1993b), ordinal hybrid aggregation operator (Xu, 2004d), linguistic OWA operator (Delgado et al., 1998; Herrera and Herrera-Viedma, 2000a; Herrera et al., 2001b; 1997; 1996b; 1995; Xu, 2006a; 2006b; 2004b; 2004e; Xu and Da, 2003b), inverse-linguistic OWA operator (Herrera and Herrera-Viedma, 1997; Herrera et al., 2001b), linguistic hybrid aggregation operator (Xu, 2006a), induced linguistic OWA operator (Xu, 2006b), uncertain linguistic averaging operator (Xu, 2006b; 2004a), uncertain linguistic weighted averaging operator (Xu, 2006b), uncertain linguistic OWA operator (Xu, 2006b; 2004a), induced uncertain linguistic OWA operator (Xu, 2006b; 2006c), uncertain linguistic hybrid aggregation operator (Xu, 2004a), dynamic linguistic weighted averaging operator (Xu, 2007d), dynamic linguistic weighted geometric operator (Xu, 2009a), linguistic correlated averaging operator and linguistic correlated geometric operator (Xu, 2009b), etc. These linguistic aggregation operators have been studied and applied in a wide variety of areas, such as engineering (Levrat et al., 1997; Xu, 2004d), decision making (Bordogna et al., 1997; Chen and Hwang, 1992; Delgado et al., 2002; 1998; 1994; 1993a; 1993b; Herrera and Martínez, 2001a; Herrera et al., 2000; 1997; 1996a; 1996b; 1995; Herrera and Herrera-Viedma, 2003; 2000a; 2000b; 1997; Herrera and Martínez, 2001a; 2000b; Huynh and Nakamori, 2005; Xu, 2007a; 2006a; 2006c; 2006d; 2006e; 2006f; 2006g; 2006h; 2006i; 2005a; 2005b; 2005c; 2004a; 2004b; 2004c; 2004d; 2004e; 2004f; Tong and Bonissone, 1980; Yager, 1995; Yager and Kacprzyk, 1997), information retrieval (Bordogna and Passi, 1993; Delgado et al., 2002; Herrera-Viedma, 2001; Herrera-Viedma et al., 2003; Herrera-Viedma and Peis, 2003; Kostek, 2003), marketing (Herrera et al., 2002; Yager et al., 1994), scheduling (Adamopoulos and Pappis, 1996), biotechnology (Chang and Chen, 1994), materials selection (Chen, 1997), software system (Lee, 1996), personnel management (Herrera et al., 2001b), educational grading system (Law, 1996), medical diagnosis (Becker, 2001; Degani and Bortolan, 1988), artificial intelligence (Zadeh and Kacprzyk, 1999a; 1999b), supply chain management and maintenance service (Xu, 2004d), etc. Xu (2007b) provided a comprehensive survey of the existing main linguistic aggregation operators.

Preference relations (or called pairwise comparison matrices, judgment matrices) are very useful in expressing decision maker’s preference information over objects by comparing each pair of them in decision making problems of various fields, including politics, social psychology, engineering, management, business and economics, etc. During the past years, the use of preference relations is receiving increasing attention, and a number of studies have focused on this issue. In some situations, such as personnel appraisal, performance evaluation of weapon equipments, the partner selection of supply chain management, etc., a decision maker usually provides his/her preference information by using linguistic labels and constructs linguistic preference relations (Chen and Fan, 2005; Delgado et al., 1998; Fan and Xiao, 2002; Herrera and Herrera-Viedma, 2003; 2000; 1997; Herrera et al., 2005; 1997; 1996a; 1996b; Xu, 2008; 2007b; 2006a; 2006b; 2006c; 2005b; 2004a; 2004b; Xu and Wu, 2004). In this chapter, we shall introduce the concept of linguistic preference relation, uncertain linguistic preference relation, incomplete linguistic preference relation, consistent linguistic preference relation, and acceptable linguistic preference relation, etc., and their desirable properties. We also introduce in detail a series of approaches for decision making based on these linguistic preference relations.

A variety types of multi-attribute decision making (MADM) problems occur in our daily life, such as investment decision making, project appraisal, maintenance and repair services, weapon system performance evaluation, plant siting, tendering and bidding, and comprehensive evaluation of economic benefits, etc. (Wang and Fu, 1993).