ANALYTICAL EXPRESSIONS FOR THE ADDITION OF FUZZY INTERVALS

The addition of fuzzy quantities is undeniably the most important oper ation in fuzzy arithmetic. In this paper, we consider the general case of the addition of fuzzy intervals based on a continuous triangular n orm. We start our discussion with an extensive literature review in wh ich we recall explicit formulae for the addition based on the stronges t and the weakest triangular norm, for the addition based on the algeb raic product, and several recent results for the addition based on a c ontinuous Archimedean triangular norm, including specific theorems for strict and nilpotent triangular norms. Subsequently, the addition bas ed on an ordinal sum is studied, and it is shown how this addition can be transformed into a series of additions based on the summands of th is ordinal sum. This important observation implies that the addition b ased on an arbitrary continuous triangular norm can be practically per formed, provided the summands of the corresponding ordinal sum represe ntation and the fuzzy intervals involved fulfil the appropriate condit ions mentioned in the overview. This is illustrated by means of severa l examples. (C) 1997 Elsevier Science B.V.