Composition by Joint Probabilities - Probabilistic Composition of Preferences, Theory and Applications - page 39

The term conservative, in this terminology, is associated to the idea of avoiding a
losses, of concern on avoiding the negative extremes. On the other hand, the term
progressive is associated to the desire to achieve the best gains.
With respect to optimism and pessimism, in the optimistic point of view it is
enough to reach satisfactory results on at least one criterion. All the criteria are
taken into account, but the composition uses the connective . What will be

computed are the probabilities of maximizing (or not minimizing, if the view on the
other axis is conservative) preference according to at least one among the multiple
criteria. From the pessimistic point of view, the preference will be measured by the
probability of maximizing (or not minimizing) the preference according to all the
criteria.
So, four kinds of composition without the use of weights may be considered: (1)
by the probability of being the best according to all the criteria considered, (2) by
the probability of being the best according to at least one of the criteria, (3) by the
probability of not being the worst by all the criteria considered, (4) the probability
of not being the worst by at least one of the criteria.
In the progressive and pessimistic composition is used the joint probability of the
intersection with respect to all the criteria of the events corresponding to the
alternative receiving the best evaluation.
In the progressive and optimistic composition, the global preference score is
given by the probability of the complement of the intersection of the events cor-
responding to not achieving the best evaluation.
In the conservative and pessimistic composition is used the joint probability
corresponding to the intersection of the events of not receiving the worst rating.
In the conservative and optimistic composition is used the probability of the
complement of the intersection of events corresponding to receiving the worst
ratings.
By dividing the criteria into groups, these four joint probabilities can be com-
puted for subsets of the set of criteria. After that, a choice between the optimistic
and the pessimistic approaches will be applied to determine how to combine the
groups to derive from the probabilities of preference inside each group a general
score of preference.
The criteria may be divided, for example, into a group related to attributes of
highest importance and another associated with secondary attributes, or into a group
of inputs and other of outputs. The composition of the criteria of high priority
would be taken from a pessimistic point of view, while secondary criteria would be
seen from an optimistic point of view. The outputs would be treated from a pro-
gressive perspective, while inputs suggest a conservative treatment. In both cases,
as the two groups cannot be thought as replaceable, a pessimistic approach must be
taken to combine them and the

“ or ”

final score will be given by the product of the
probabilities of preference according to each group.
For instance, if the criteria are ranked as in the Analytic Network Process (ANP)
of Saaty ( 1996 ), on criteria of bene

ts, opportunities, costs and risks, or of
strengths, opportunities, weaknesses and threats, of the SWOT analysis of

Humphrey ( 2005 ), for the first group, given its priority and positive feature, a