Method of ideal solution in fuzzy set theory and multicriteria decision making

• Authors: Gleb Beliakov
• Venue: PRICAI'00 Proceedings of the 6th Pacific Rim international conference on Artificial intelligence
• Year: 2000

Multicriteria decision-making (MCDM) refers to making selections among some courses of action in the presence of multiple, usually conflicting criteria. In a large class of methods of MCDM, one can represent the preference relation x on a set of alternatives X with a single-valued function u(x) on X, called utility, such that for any x,yeX xyy ⇔ u(x)u(y). Utility is defined up to an increasing monotone transform, and consequendy, it can always be normalised to the unit interval. In multiattribute utility theory the multicriteria problem is essentially substituted with a vector maximisation problem, maximise u(X)=u(u1,(x2), u2(x2),...,un. Utility is equivalent to membership functions employed in Fuzzy Set Theory (FST) [3] and methods of combining individual utility values into the overall utility u(X) correspond to aggregation operators in FST. A general aggregation operator is a monotone function f: [0,1]n→[0,1],f(0)=0,f(1)=1. Triangular norms, averaging operators and ordered weighted aggregation are examples of aggregation operators. Method of Ideal solution, in which the alternatives are ranked according to their distances (similarities) to the Ideal or nadir points, is also well known in the literature [4]. L-p metrics are commonly used in this context, and they correspond to the traditional methods in MCDM (additive and multiplicative utility, maximin) and FST (averaging operators, max, min). There is a semantic equivalence between the three concepts: utility, similarity and aggregation [3]. What was missing, however, is the equivalence at the syntactical level, which means that every method of utility combination has its mathematical counterparts among aggregation operators and similarity relations. This work establishes this syntactical equivalence and demonstrates that every aggregation operator can be put into correspondence to a monotone pseudometric, in which the distance to the Ideal (nadir) is measured. It provides methods to build new aggregation operators based on metrics, and to incorporate criteria importance into the metric. It extends the results to fuzzy MCDM via aggregation of fuzzy sets of type II. Various examples are provided, but for more detailed discussion the reader is referred to [1]. Paper [2] provides details of construction of aggregation operators based on empirical data. 1. Beliakov, G.: Definition of general aggregation operators through similarity relations. Fuzzy Sets Systems 114 (2000) 437-453.