Scale defines the set of values that can be assigned to an attribute.
Scales can be ordered or unordered, and ordered scales can be either ascending or descending. An unordered scale is just a set of values, whose relation with each other is unknown or undefined. In contrast, the values of an ordered scale are ordered preferentially, that is, according to their contribution to the quality of alternatives. The values of ascending scales are ordered from ‘bad’ to ‘good’ values, and the value of a descending scales are ordered from ‘good’ to ‘bad’ values. In both cases, ‘bad’ means something that is disadvantageous for the alternative and is not preferred by the decision maker. Analogously, ‘good’ represents an advantageous and preferred value. The ordering of scales plays an important role in the definition of aggregation functions, where it simplifies the definition of decision rules and allows checking of their consistency.
For emphasis and better visualization, ‘bad’ and ‘good’ values of ordered scales are printed in different fonts and colors. By default, ‘bad’ values appear in bold-red and ‘good’ values appear in italic-green.
A DEX qualitative scale consists of a list of values or categories. Generally, they are just words, such as ‘excellent’, ‘acceptable’, ‘inappropriate’, etc. Optionally, each may be associated with a textual description.
With preferentially ordered qualitative scales, each category can also be associated with a class, which is either ‘bad’, ‘neutral’ or ‘good’, For ascending scales, the default is that the lowest value is considered ‘bad’ and the highest ‘good’. The opposite order holds for descending scales. Value class can be assigned for each category individually, allowing to declare whole subsequences of categories as ‘bad’ or ‘good’.
Example qualitative scales
A DEX continuous scale represents all real (floating-point) numbers. Additionally, ordered continuous scales allow a definition of two thresholds, called bad and good threshold, that define value intervals that are considered ‘bad’ and ‘good’, respectively, on that scale. Optionally for continuous scales, it is possible to define:
measuremen unit, (a string of characters), and
number of decimals for displaying values on that scale (the default is -1, meaning as many decimals as needed).
continuous scales are represented using angular brackets.
<> represents a general continuous scale
<-5; 5> a scale with defined ‘bad’ (-5) and ‘good’ (+5) thresholds.
Continuous scales can only be assigned to basic attributes that are single descendants of some aggregate attribute. This configuration allows defining a discretization function and associating it with the aggregate attribute.
On size (number of values) of discrete scales:
For basic attributes: Use the least number of values that is still sufficient to distinguish between importantly different characteristics of alternatives. Usually, this means two to four values.
For aggregate attributes: The number of values should gradually increase from basic attributes towards the root(s). For example, three four-valued attributes might be aggregated into a five-valued attribute. Five-valued root attributes usually work well.
On scale ordering:
Use ordered scales whenever possible, they really help while defining functions.
Avoid descending scales. They are much less comprehensible than ascending scales. It is particularly confusing when both types are mixed together in a single function.