Abstract
Fuzzy sets were initially proposed to address ambiguities and uncertainties. However, in certain cases, the fuzzy sets show some degree of uncertainty and risk, when the available data are either obtained from unreliable sources or related to future events. To solve this problem, the R-numbers methodology has been recently developed as a powerful approach to model the risk of fuzzy sets and numbers due to risk factors. In R-numbers, only the variability of x values has been taken into account in risk modeling of the fuzzy sets, but not their membership function. Moreover, merely one source of risk factors related to fuzzy sets and numbers has been considered. Therefore, this article presents a new concept called R-sets, in which different risk cases of a membership function due to both future events and unreliable information sources are investigated, and the governing mathematical relations are presented. Subsequently, to overcome previous limitations of R-numbers, the R-sets are applied to develop a decision-making method, and it is tested by using a case study.
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