Description

Optimization is of central concern to a number of discip- lines. Operations Research and Decision Theory are often consi- dered to be identical with optimizationo But also in other areas such as engineering design, regional policy, logistics and many others, the search for optimal solutions is one of the prime goals. The methods and models which have been used over the last decades in these areas have primarily been "hard" or "crisp", i. e. the solutions were considered to be either fea- sible or unfeasible, either above a certain aspiration level or below. This dichotomous structure of methods very often forced the modeller to approximate real problem situations of the more-or-less type by yes-or-no-type models, the solutions of which might turn out not to be the solutions to the real prob- lems. This is particularly true if the problem under considera- tion includes vaguely defined relationships, human evaluations, uncertainty due to inconsistent or incomplete evidence, if na- tural language has to be modelled or if state variables can only be described approximately. Until recently, everything which was not known with cer- tainty, i. e. which was not known to be either true or false or which was not known to either happen with certainty or to be impossible to occur, was modelled by means of probabilitieso This holds in particular for uncertainties concerning the oc- currence of events.