A Method for Ranking Non-Linear Qualitative Decision Preferences using Copulas

A Method for Ranking Non-Linear Qualitative Decision Preferences using Copulas

Biljana Mileva-Boshkoska (Jožef Stefan International Postgraduate School, Slovenia) and Marko Bohanec (Jožef Stefan Institute and University of Nova Gorica, Slovenia)
Copyright: © 2012 |Pages: 17
DOI: 10.4018/jdsst.2012040103
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This paper addresses the problem of option ranking in qualitative evaluation models. Current approaches make the assumptions that when qualitative data are suitably mapped into discrete quantitative ones, they form monotone or closely linear tabular value functions. Although the power of using monotone and linear functions to model decision maker’s preferences is impressive, there are many cases when they fail to successfully model non-linear decision preferences. Therefore, the authors propose a new method for ranking discrete non-linear decision maker preferences based on copula functions. Copulas are functions that capture the non-linear dependences among random variables. Hence each attribute is considered as a random variable. The variables are nested into hierarchical copula structures to determine the non-linear dependences among all attributes at hand. The obtained copula structure is used for obtaining regression function and consequently for option ranking. The application of the method is presented on two examples.
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Solving decision problems described with multiple attributes is part of our daily life. The main difficulties occur when the decision problems become too complex. The complexity of these problems increases in cases when:

  • The number of attributes increase,

  • The set of possible values that the attributes may receive increases (in general people may consistently distinguish five levels, such as five grades at school, while experts may consistently distinguish up to seven levels (Saaty, 2003)),

  • Several options belong to the same qualitative class of preferable options, therefore it is difficult to identify the best one (when one has to choose from many varieties, the decision making becomes a difficult task (“The tyranny of choice,” 2010)), and

  • The problem itself is ill-structured and it is impossible to examine it through single attribute, criteria or point of view that would lead to the optimum decision (Zopounidis & Doumpos, 2006).

Addressing these issues in different decision problems evolved into development of different quantitative and qualitative multi-attribute decision analysis (MADM) methods and tools (Adam & Humphreys, 2008; Figueira et al., 2005; Bouyssou et al., 2006). Quantitative methods represent attributes with quantitative (numerical) values. The group of classical quantitative methods is large and includes, among others, outranking methods (ELECTRE and its variants, PROMETEE) (Figueira et al., 2005), methods based on Multiple Attribute Utility Theory (MAUT) (Jacquet-Lagreze & Siskos, 1982) and Analytical Hierarchy Process (AHP) (Saaty, 2008). Qualitative methods, on the other hand, represent attributes with qualitative (symbolic) values. Methods that belong to this group are ZAPROS (Moshkovich & Larichev, 1995), which is based on verbal decision theory, Rough Sets (Greco et al., 2001) and Doctus (Baracskai & Dörfler, 2003). This paper builds on the DEX method (Bohanec & Rajkovič, 1990), which is a member of the latter group. DEX has been successfully used in a wide range of applications, such as environmental (Bohanec et al., 2008), agricultural (Pavlovič et al., 2011), and in medicine and healthcare (Bohanec, Zupan, & Rajkovič, 2000). DEX is implemented in the computer program called DEXi (Bohanec, 2011).

In DEX, the aggregation of discrete qualitative attributes is specified with a table whose rows are interpreted as if-then rules. Specifically, the decision maker’s preferences over the available options are defined using an attribute that are called a qualitative class. Options that are almost equally preferred belong to the same qualitative class. Consequently, a partial ordering of options is obtained.

Qualitative evaluation of options suffers from two problems: it provides only partial ranking of options instead of full ranking, and is insensitive to small differences among options. One possible way to overcome these problems is to combine qualitative and quantitative evaluation. In addition to qualitative evaluation, which ranks options into classes, we wish to numerically rank options within classes. In this paper, we propose an approach that constructs such a quantitative evaluation model automatically from decision maker’s qualitatively specified preferences.

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