8Inner Product of Fuzzy Vectors

A fuzzy interval in double-struck upper R is a fuzzy set in double-struck upper R such that its alpha‐level sets are bounded closed intervals. The purpose of this chapter is to study the inner product of vectors of fuzzy intervals using the extension principle and the expression in the decomposition theorem. Since the fuzzy linear optimization problems and fuzzy linear systems can be formulated in terms of inner product of fuzzy vectors, the results obtained in this chapter can be useful for studying the fuzzy linear optimization problems and fuzzy linear systems.

There are two types of inner product that will be studied in this chapter. The first type of inner product of fuzzy vectors is directly based on the inner product of vectors

bold x equals left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis and bold y equals left-parenthesis y 1 comma ellipsis comma y Subscript n Baseline right-parenthesis

given by the following expression

bold x bullet bold y equals x 1 y 1 plus midline-horizontal-ellipsis plus x Subscript n Baseline y Subscript n Baseline comma

where bold x and bold y are two vectors in ...

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