Abstract
The inner product of vectors of non-normal fuzzy intervals will be studied in this paper by using extension principle and expression in decomposition theorem. The membership functions of inner product will be different with respect to these two different methodologies. Since the non-normal fuzzy interval is more general than the normal fuzzy interval, the corresponding membership functions will become more complicated. Therefore, we shall establish their relationship including the equivalence and fuzziness based on the \(\alpha \)-level sets. The potential application of inner product of fuzzy vectors is to study the fuzzy linear programming problems.
Highlights
A fuzzy interval in R is a fuzzy set in R such that its α-level sets are bounded and closed intervals.The purpose of this paper is to study the inner product of vectors of fuzzy intervals using two different methodologies called the extension principle and the form of decomposition theorem
Since the fuzzy linear optimization problems can be formulated as the form of inner product of fuzzy vectors, the results obtained in this paper can be useful for studying the fuzzy linear optimization problems
The first type of inner product of fuzzy vectors is directly based on the inner product of vectors x and y given by the following expression x
Summary
Version of Record: A version of this preprint was published at Soft Computing on November 16th, 2021.
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