DEX Values

DEX values are used throughout DEX models to provide input values and carry results of evaluations of decision alternatives. A value that can be assigned to an attribute is always bound to the context provided by that attribute’s scale.

A special DEX value undefined can be assigned to any attribute, including those whose scales are undefined. In principle, any operation (aggregation, discretization) involving undefined gives an undefined result. However, for qualitative attributes, DEXi Suite allows an evaluation mode in which all undefined values are expanded to the full set of the corresponding attribute’s values prior to evaluation. In this way, undefined is interpreted as unknown.

DEX values assigned to continuous attributes can only be real (floating-point) numbers. No intervals or other representations of uncertainty are possible.

DEX value types assigned to qualitative attributes are richer and allow expressing uncertainty and imprecision. There are four main qualitative value types: single value, interval, set and value distribution. An additional offset value type is used in qualitative-quantitative evaluation.

For illustrations below, let us take an ordered qualitative scale, consisting of four values: bad, acc, good, excellent.

Single Values

A single DEX value consists of a single element of the corresponding scale.

Example: acc

Interval Values

An interval value is defined by two bounds, lower and upper, and consists of all scale elements between these bounds, including the bounds. An interval is usually denoted lower:upper.

Example: acc:excellent: includes acc, good and excellent

Notational shortcuts are used for preferentially ordered scales:

• <=acc: worse than or equal to acc, equivalent to bad:acc

• >=acc: better than or equal to acc, at least acc, equivalent to acc:excellent

Sets

A value set consists of a set of elements of the corresponding scale. Sets are usually denoted as lists of ;-separated values. All intervals can be interpreted as sets, while the converse is not always true.

Example: bad; acc; excellent

Special notation * is used to denote the full set of values.

Example: * = bad:excellent = bad; acc; good; excellent

Value Distributions

A value distribution associates each qualitative scale value with some number, generally denoted p and normally expected to be in the [0,1] interval. Depending on the context and used evaluation method, p can be interpreted as probability or fuzzy set membership. Distributions are usually displayed in terms of a ;-separated list of pairs value/p. p = 0 is assumed for elements not appearing in the list.

Example: bad/0.5; good/0.2; excellent/0.3 (omitted: acc/0)

Offset Values

Offset values are used in qualitative-quantitative evaluation to facilitate ordering of alternatives within qualitative classes. An offset value is composed of a single qualitative value and a numeric offset on the interval [-0.5, +0.5]. For instance, acc-0.25 denotes the qualitative value acc having the offset -0.25. Similarly, acc+0.11 is the same qualitative value with the offset +0.11.

Offset values are meaningful only with ordered value scales. Numeric offsets represent evaluations within the associated qualitative values, providing means to preferentially compare and order these values. In this way, acc-0.25 is considered worse than acc+0.11. Decreasing negative offsets denote increasingly worse acc values, and increasing positive offsets denote increasingly better acc values. The extreme cases are acc-0.5 and acc+0.5.