nearest fuzzy number of type l-r to an arbitrary fuzzy number with applications to fuzzy linear system

The fuzzy operations on fuzzy numbers of type l-r are much easier than general fuzzy numbers. it would be interesting to approximate a fuzzy number by a fuzzy number of type l-r. in this paper, we state and prove two significant application inequalities in the monotonic functions set. these inequalities show that under a condition, the nearest fuzzy number of type l-r to an arbitrary fuzzy number exists and is unique. after that, the nearest fuzzy number of type l-r can be obtained by solving a linear system. note that the trapezoidal fuzzy numbers are a particular case of the fuzzy numbers of type l-r. the proposed method can represent the nearest trapezoidal fuzzy number to a given fuzzy number. finally, to approximate fuzzy solutions of a fuzzy linear system, we apply our idea to construct a framework to find solutions of crisp linear systems instead of the fuzzy linear system. the crisp linear systems give the nearest fuzzy numbers of type l-r to fuzzy solutions of a fuzzy linear system. the proposed method is illustrated with some examples.