Journal
IRANIAN JOURNAL OF FUZZY SYSTEMS
Volume 18, Issue 4, Pages 1-17Publisher
UNIV SISTAN & BALUCHESTAN
Keywords
Aggregation operator; fuzzy numbers; normalized fuzzy weights; fuzzy probabilities; fuzzy weighted aver-age; fuzzy weighted average
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Funding
- Palacky University in Olomouc [IGA PrF 2020 015]
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The paper investigates whether the properties of the weighted average operator can be observed in its fuzzy extension, with results showing strict monotonicity in case of positive fuzzy weights, and symmetry in case of equal fuzzy weights, but not coinciding with the fuzzy arithmetic mean operator.
Weighted average with normalized weights is a widely used aggregation operator that takes into account the varying degrees of importance of the numbers in a data set. It possesses some important properties, like monotonicity, continuity, additivity, etc., that play an important role in practical applications. The inputs of the aggregation as well as the normalized weights are usually not known precisely. In such a case, their values can be expressed by fuzzy numbers, and the fuzzy weighted average of fuzzy numbers with normalized fuzzy weights needs to be employed in the model. The aim of the paper is to reveal whether and in which way the properties of the weighted average operator can be observed also for its fuzzy extension. It is shown that it possesses three conditions characteristic for aggregation operators - identity, monotonicity and boundary conditions, and besides that, also compensation, idempotency, stability for linear transformation, 1-lipschitzianity, and continuity. Furthermore, it is proved that it preserves strict monotonicity in case of positive fuzzy weights, and symmetry in case of equal fuzzy weights, although it does not coincide with the fuzzy arithmetic mean operator in such a case. One of the most valuable result of the study is the fact that in contrast to the crisp weighted average operator, it is not additive. The importance of the obtained results is discussed and illustrated by several illustrative examples.
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