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“On the Linguistic Weighted Average”
by Dongrui Wu and Jerry M. Mendel
June 2006
The focus of this report is the linguistic weighted average (LWA), which is a generalization of the fuzzy weighted average (FWA) that is obtained by replacing the type-1 fuzzy inputs in the FWA by interval type-2 fuzzy sets. Consequently, the output of the LWA is an interval type-2 fuzzy set. This report proposes algorithms to compute this output. First, the LWA is located in a taxonomy of weighted averages and a motivating example is provided. Then, the problem of the LWA is defined. Because the FWA is employed to compute the LWA, the main results of the FWA are also reviewed. Next, several theorems are introduced and proved for the LWA. It is shown that finding the LWA can be decomposed to finding two FWAs. The detailed algorithms for the underlying FWAs are described, and -cuts selection schemes which significantly reduce the computational cost are also proposed, followed by some examples to illustrate the LWA. Several special cases in which the interval type-2 inputs degrade to crisp numbers or type-1 fuzzy sets are provided, to verify the LWA algorithms. Results show that the LWA in the degraded cases coincides with the weighted average. An application, where the LWA is employed in a paper review process, is included to demonstrate how the LWA can be used in distributed and hierarchical decision-making.
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