Decision SupportThe 2-rank additive model with axiomatic design in multiple attribute decision making
Introduction
Multiple attribute decision making (MADM) usually aims to obtain the ranking of alternatives or select the most preferred one(s) involving multiple attributes (Bana e Costa & Oliveira, 2012; Durbach & Stewart, 2012; Greco et al., 2016; Hwang & Yoon, 2012; Wallenius et al., 2008). The techniques of MADM have been widely used in many areas, including engineering, management, economy, and military (Geldermann et al., 2009; Hwang & Yoon, 2012; Ishizaka & Nemery, 2013).
However, in many real-world MADM problems, obtaining a complete ranking of alternatives can be very time-consuming and sometimes unnecessary. This is especially true with a large number of alternatives (Morente-Molinera et al., 2017; Morente-Molinera et al., 2018). Consequently, it is sometimes preferred to classify alternatives into several pre-defined and preference-ordered categories, leading to the multiple criteria/attribute sorting (MCS) problem (Greco et al., 1998; 2002; Pawlak & Słowiński, 1994; Roy, 1996; Zopounidis & Doumpos, 1999). To date, the MCS problem has become a frequently investigated topic (Zopounidis & Doumpos, 2002). Various MCS methods have been proposed in the existing studies. These methods can be classified into four main streams: the value function-based methods (Doumpos et al., 2001; Jacquet-Lagreze & Siskos, 1982; Keeney & Raiffa, 1976), the outranking relation-based methods (Corrente et al., 2013; Liu et al., 2015; Mousseau et al., 2000; Rocha & Dias, 2008), the distance-based methods (Chen et al., 2008; 2011; Vetschera et al., 2010) and the decision rule-based methods (Chakhar & Saad, 2012; Greco et al., 2001; Greco et al., 2010; Kadziński et al., 2014). Generally, thresholds are required in these MCS methods to identify the category that an alternative belongs to.
Although the MADM and the MCS problems have been intensively investigated, two gaps still remain unfilled.
- (1)
Typically, decision makers only need to divide alternatives into two ordered categories in practical MCS problems. For example, when evaluating scientific research projects, the science foundation committee has to divide scientific proposals into two categories: the supported and the rejected. Though of great importance, this special case of the MCS has not been sufficiently investigated based on the axiomatic design. To our knowledge, Bouyssou and Marchant (2007) used conjoint measurement to analyze the twofold partitions of alternatives that obtained using “non-compensatory sorting models’’. Also, the twofold MCS problem is related to the 2-rank problem which is presented by Hochbaum and Levin (2006), inspired by Kemeny and Snell's model for the group-ranking problem (Kemeny & Snell, 1962).
- (2)
Generally, the threshold of the existing classification models is provided exogenously by the decision maker, who either specifies a threshold in terms of evaluations of what means a “good” alternative, or in terms of cardinality of the set of “good” alternatives (e.g., what are the three best candidates in a job selection problem) (Jacquet-Lagreze & Siskos, 2001; Kadziński & Tervonen, 2013; Mousseau et al., 2001). However, in many actual problems, it is hard for decision makers to provide such thresholds. For instance, a company investigates consumer preference to make marketing plans. The company wants to know what products consumers prefer and what they do not. In this case, the company cannot present the threshold of consumers “preferred” and “not preferred” in advance.
In order to overcome these two shortcomings, we call the twofold MCS the 2-rank MADM in this paper, and use the axiomatic approach to study the 2-rank MADM problem in the framework of multiple attribute additive model, one of the classical MADM models (Keeney & Raiffa, 1976; Von Winterfeldt & Edwards, 1986). Furthermore, differently from the existing classification methods, in our model the threshold is determined endogenously by minimizing the weighted sum of violations. In a sense, we build the 2-rank model that minimizes the assignment conflicts, measured by the weighted sum of “advantages” of the alternatives in the bottom category with respect to the alternatives in the top category. This proposal is carried out by the following points:
- •
We present five axioms: Anonymity (ANO), Rankings Invariance (RI), Minimum Intensity of Reversed Rankings (MIRR), Rational Scale Invariance (RSI), and Inversion (INV) to fully characterize the desired properties of an “ideal” 2-rank MADM method. Based on these axioms, we propose an impossibility theorem, which states that all five axioms cannot be satisfied simultaneously due to the intrinsic conflict between RI and MIRR.
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We develop two simple 2-rank MADM methods: Permutation based 2-rank method (PRM) and ranking based 2-rank method (RRM). We prove that the PRM satisfies ANO, MIRR, RSI, and INV, but violates RI. In contrast, the RRM satisfies ANO, RI, RSI, and INV, but violates MIRR.
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We design simulation experiments to further explore the properties of the PRM and the RRM, and illustrate the two methods through a consumer preference analysis case study.
The rest of the paper is organized as follows. In Section 2, we briefly introduce the basic knowledge of MADM and define the 2-rank MADM problem. After that, we present five axioms with respect to the 2-rank MADM and the impossibility theorem in Section 3. In Section 4, we propose the PRM and the RRM, and examine the desired properties of them based on the established axioms. Afterwards, we present simulation experiments and a case study to further investigate the properties of the PRM and the RRM in Section 5. Finally, we end this paper with conclusions and future prospective researches in Section 6.
Section snippets
Preliminaries
In this section, we introduce some basic knowledge of MADM, and then define the 2-rank MADM problem.
Axioms in the 2-rank MADM
In this section, we present five axioms that identify an “ideal” 2-rank MADM method. After that, we propose an impossibility theorem, which states that no 2-rank MADM method satisfies all five axioms simultaneously.
Definition 3 (Anonymity (ANO)) Let be an MAEM. Let be a permutation on the set of alternatives and be a permutation on the set of attributes. In accordance with the permutation of attributes δ2, W is permuted as . Let be the MAEM obtained from
Two 2-rank MADM methods
In this section, we propose two simple 2-rank MADM methods: Permutation-based 2-rank method (PRM) and Ranking-based 2-rank method (RRM), and then discuss their properties based on the proposed axioms.
Simulation analysis and an illustrative example
In this section, we conduct two simulation experiments to explore the properties of the PRM and the RRM. After that, we use a consumer preference analysis case to illustrate these two methods.
Conclusions
We investigate the 2-rank problem with axiomatic design in the context of additive MADM. The main contributions are concluded as follows.
- (1)
We propose five axioms, ANO, RI, MIRR, RSI, and INV to thoroughly characterize an “ideal” 2-rank MADM method. Based on the axioms, we present an impossibility theorem, which states that no 2-rank MADM method satisfies all five axioms simultaneously due to the conflict between RI and MIRR.
- (2)
We present two simple 2-rank MADM methods: The PRM and the RRM. We prove
Acknowledgments
This work was supported by the grants (Nos. 71871149 and 71571125) from NSF of China, and the grants (Nos. sksyl201705, sksyl201821, and 2018hhs-58) from Sichuan University.
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