Multicriteria Classification

In multiple criteria decision aiding (MCDA), multicriteria classification (or sorting) involves problems where a finite set of alternative actions should be assigned into a predefined set of preferentially ordered categories (classes). For example, credit analysts classify loan applications into risk categories (e.g., acceptable/unacceptable applicants), customers rate products and classify them into attractiveness groups, candidates for a job position are evaluated and their applications are approved or rejected, technical systems are prioritized for inspection on the basis of their failure risk, clinicians classify patients according to the extent to which they have a complex disease or not, etc.

Problem statement

In a multicriteria classification problem (MCP) a set

: $X=\$

of ''m'' alternative actions is available. Each alternative is evaluated over a set of ''n'' criteria. The scope of the analysis is to assign each alternative into a given set of categories (classes) ''C'' = .

The categories are defined in an ordinal way. Assuming (without loss of generality) an ascending order, this means that category ''c''₁ consists of the worst alternatives whereas ''c''_''k'' includes the best (most preferred) ones. The alternatives in each category cannot be assumed be equivalent in terms of their overall evaluation (the categories are not equivalence classes).

Furthermore, the categories are defined independently of the set of alternatives under consideration. In that regard, MCPs are based on an absolute evaluation scheme. For instance, a predefined specific set of categories is often used to classify industrial accidents (e.g., major, minor, etc.). These categories are not related to a specific event under consideration. Of course, in many cases the definition of the categories is adjusted over time to take into consideration the changes in the decision environment.

Relationship to pattern recognition

In comparison to statistical classification and pattern recognition in a machine learning sense, two main distinguishing features of MCPs can be identified:

# In MCPs the categories are defined in an ordinal way. This ordinal definition of the categories implicitly defines a preference structure. In contrast, machine learning is usually involved with nominal classification problems, where classes of observations are defined in a nominal way (i.e., collection of cases described by some common patterns), without any preferential implications.
# In MCPs, the alternatives are evaluated over a set of criteria. A criterion is an attribute that incorporates preferential information. Thus, the decision model should have some form of monotonic relationship with respect to the criteria. This kind of information is explicitly introduced (a priory) in multicriteria methods for MCPs.

Methods

The most popular modeling approach for MCPs are based on value function models, outranking relations, and decision rules:

* In a value function model, the classification rules can be expressed as follows: Alternative ''i'' is assigned to group ''c''_''r'' if and only if

:: t_

:where ''V'' is a value function (non-decreasing with respect to the criteria) and ''t''₁ > ''t''₂ > ... > ''t''_''k''−1 are thresholds defining the category limits.

:An important example of this approach is the use of the potentially all pairwise rankings of all possible alternatives (PAPRIKA) method to create models for classifying patients according to the extent to which they have a disease or not – e.g. Sjögren syndrome, gout, systemic sclerosis, etc.

* Examples of outranking techniques include the ÉLECTRE TRI method and its variants, models based on the PROMETHEE {{R from other capitalisation ...