Uncertainty in Multi-Criteria Decision Making

Often, word uncertainty has been equated with random variability (Ferson and Ginzburg 1996). In decision making, the uncertainty is traditionally connected to the unknown or uncertain future states of nature: the consequences of alternative actions are not known, which makes choices risky.

Probability theory demands that probabilities should fulfil the Kolmogorov axioms. It means that the probability of each event has to be greater than or equal to zero (i.e. negative probabilities are not allowed), the probability of the entire set of events has to be one (i.e. some event in the whole set of events will occur with probability 1) and that the probability of a union of disjoint events is the sum of their individual probabilities (e.g. Williams 1991). It also means that the probability of a union of an event and its negation must add up to one.

Probabilities can be either objective or subjective. Objective probabilities can be calculated for events that can be repeated infinitely many times; for instance, the probabilities of obtaining heads or tails when tossing a coin. Subjective probabilities describe beliefs of persons; for instance, someone may believe that a certain sports team will win with 0.7 probability. The Bayesian probability theory may be the best known example of utilising subjective probabilities.

Additivity requirement in probability theory means that if the decision maker has no idea of the probability of some event, like whether there is life on Mars of not, both alternatives (there is and there is not) need to be assigned a 0.5 probability. Assigning so high a probability to either option may seem counterintuitive in the case of complete ignorance. There exist theories, in which also non-additive subjective beliefs can be dealt with, for example the possibility theory (Dubois and Prade 1988) and the evidence theory (Schafer 1976; Klir and Harmanec 1997). These theories were developed to deal with situations, where the classical or Bayesian probability theory was deemed too normative. They are argued to be suitable especially in cases where human opinions, judgement and decisions are involved (Zimmermann 1985; Dubois and Prade 1988).