Modeling Preferences
To be able to find the best solution for MCDA problems we need to know subjective preferences:
So given two alternatives $a$ and $b$ a decision maker can say if
- $a \ P \ b$ or $b \ P \ a$: $a$ is preferred to $b$ - the preference relation
- $a \ I \ b$ - the indifference relation
- $a \ J \ b$ - the incomparability relation, when you cannot compare things
Main properties:
- $P$ is asymmetric
- $a \ P \ b \equiv b \ \overline{P} \ a$
- $I$ is reflexive and symmetric:
- $a \ I \ a$ and $a \ I \ b \equiv b \ I \ a$
- $J$ is irreflexible and symmetric
- $a \ \overline{J} \ a$ and $a \ J \ b \equiv b \ J \ a$
Transitivity
We can show the preferences of a decision maker with a graph:
Preference Structures
How to build a mathematical model from statements of a decision maker?
There are several preference structures that can do that:
Preferential Independence
- Main Article: Preferential Independence
This is an important condition between preferences and criteria: the criteria should be preferentially independent.
Sources