NAIADE (Novel Approach to Imprecise Assessment and Decision Environments. Munda, 1995) is an outranking multi-criteria method that manages mix information; criterion scores may be expressed in crisp, stochastic, fuzzy numbers or linguistic expressions.
NAIADE is a discrete multi-criteria method (i.e. the set of alternative is finite) that does a pair-wise comparisons between alternatives in order to generate a ranking of options.
a) Definition of alternatives
The alternatives are defined by clicking on the first row of the Criteria/Alternatives or
Groups/Alternatives matrix. The name and
description of the alternatives proposed are simply defined (See Figure 8).
b) Definition of criteria
Clicking on the first column of the
Criteria/Alternatives matrix (), it activates the window where the type of criterion (fuzzy, stochastic, qualitative or quantitative) and its
characteristic parameters, such as the measurement units, the optimisation criterion (maximisation or minimisation) and the crossover points of its preference relations can be defined (See Figure 9).
The crossover points of the preference relations represent the preference and indifference thresholds. They can be defined either by directly assigning it a numerical value or by moving graphically the vertical blue line. The vertical blue line represents the crossover
21 This text is based upon the NAIADE Manual – version 1.0.ENG. Joint Research Centre of the European Commission. Ispra site. Italy. 1996.
Figure 8: Dialogue box for the definition of alternatives
Figure 9: Dialogue box for the definition of criteria
value or the point where the preference function intersects the crossover line (i.e. when it reaches the value of 0.5).
Four preference relations have to be defined by means of establishing two preference and two indifference thresholds (crossover points).
The Much Better (red) and Better (yellow) thresholds are the minimum difference between the performances of two alternatives that makes one option preferred instead of the other.
The Almost Equal (blue) and Equal (green) thresholds are the maximum difference between the criterion scores of two alternatives that makes almost no difference and no difference between them respectively.
The number at the end of the X axis represents the base scale value, which must be higher than the Much Better threshold.
c) Filling the matrices
The matrices are simply filled by clicking on the cell to be defined. In the Impact matrix the types of input data depend upon the type of criteria chosen for the analysis whilst in the Equity matrix the data are always of linguistic type.
For fuzzy criteria a window is popped up from which the fuzzy set and its
parameters can be defined when choosing from four possible types (see Figure 10).
For stochastic criteria a window appears where the type of probability density function and its parameters can be
defined (see Figure 11). Finally, when the criterion is qualitative a window appears where the user simply selects the variable (see Figure 12). The user must assign a name to the linguistic variables which will appear in the matrix. If the criterion is numerical the user can simply type the number in the corresponding box.
Figure 11: Dialogue box for the definition of stochastic variables
Figure 10: Dialogue box for the definition of fuzzy numbers
d) Running the aggregation
i. Semantic distance
In order to compare the criteria values for the
alternatives, it is necessary to introduce the concept of distance. In the case of numeric evaluation, the
distance is simply defined as the difference between the two numbers. In the case of fuzzy or stochastic evaluation, the concept of semantic distance is used.
Semantic distance measures the distance between two functions: it takes into account the position and also the shape of the two functions (either for fuzzy
membership functions or probability density functions).
ii. Preference relations and pairwise comparisons of alternatives
Preference relationships are defined by means of six functions that allows to express (depending on the distance between alternatives), for each criteria, an index of credibility of the statements that an
alternative is much better, better, approximately equal, equal, worse and much worse than another. The
credibility index goes from 0 (definitely non-credible) to 1 (definitely credible) increasing monotonically within this range.
Figure 9 shows the functions that defines the preference relations Much Better (red) and Better (yellow), which are symmetrical respectively to the Much Worse and Worse preference relations (See Box 3).
iii. Criteria aggregation
The pairwise comparison of alternatives generate a set of indexes of credibility. Then, through an aggregation algorithm of the credibility indexes, NAIADE calculates a preference intensity index of one alternative with respect to another.
In particular the parameter is used to expressα the minimum requirements on the credibility indexes. Only those criteria whose indexes are above the threshold will be counted positivelyα in the aggregation (See below).
In order to use information on the diversity among the assessment of the single fuzzy relations, according to each criterion, the entropy concept is useful. Entropy is calculated as an index varying from 0 to 1 that gives an indication of the variance of the credibility indexes that are above the threshold, and around the crossover value 0.5 (maximum fuzziness).
Figure 12: Dialogue box for the definition of linguistic variables
Box 3: Example of the determination of the index of credibility
Consider the preference relations presented in Figure 9. The red curve expresses the index of credibility that alternative A is Much Better than alternative B.
The threshold for the Much Better preference relation is set up in 10.000 €.
If the performances of alternatives A and B are 45.000 and 30.000 Euros respectively, then the the index of credibility that A is Much Better than B is around 0,7.
An entropy value of 0 means that all criteria give an exact indication (either definitely credible or definitely non-credible), whereas an entropy value of 1 means that all criteria give an indication biased by the maximum fuzziness (0.5).
Figure 13 shows a graphic representation of the indexes of credibility (in yellow), and of the intensity of preference index and entropy (in pink).
In this case, the horizontal red line represents the value defined by the parameter. Theα indexes of credibility that are above this value will be counted positively in the aggregation.
iv. Rankings
The intensity of preference for each preference relationship (Much Better, Better, Almost Equal, Equal, Worse and Much worse) is calculated by means of aggregating the respective indexes of credibility.
Then, the ranking of alternatives is obtained from the intersection of two separate rankings. One of the rankings is based on the aggregation of: the intensity of preference indexes and the entropy of the Better and Much better preference relations. The other ranking is based on the aggregation of the intensity of preference indexes and the entropy of the Worse and Much worse preference relations.
The final ranking is a result of a sort of each against all competence, and mutual preference independence does not hold. Therefore, positions in the ranking may be contradictory with binary relationships between alternatives. For example, A can rank better than B in the final ranking, but B can be better than A according to most of the criteria.
In this regard, NAIADE uses the information provided by the preference intensity index and correspondent entropies in order to build the degrees of truth ( ) of the followingτ
statements:
“according to most of the criteria”:
a is better than b a and b are indifferent a is worse than b
Figure 13: Indexes of credibility, intensity of preference index and entropy