Fuzzy Interpolative Reasoning Based on Representative Values and Characteristic Points of Rough-Fuzzy Sets

In this paper, we propose a new fuzzy interpolative reasoning (FIR) method on the basis of representative values (RVs) and characteristic points (CPs) of rough-fuzzy sets (RFSs). Firstly, it calculates the Euclidean distance between the antecedent polygonal RFS
${\widetilde A_{ij}}$
and the observation polygonal RFS
${\widetilde A_j^\ast}$
based on the RVs
$Rep{(\tilde A_j^\ast)_x}$
and
$Rep{(\tilde A_j^\ast)_y}$
of antecedent polygonal RFS
$\widetilde A_j^\ast$
in the
${X}$
-axis and
${Y}$
-axis, respectively, and the RVs
$Rep{(\tilde A_j^\ast)_x}$
and
$Rep\,{(\tilde A_j^\ast)_y}$
of the observation polygonal RFS
$\widetilde A_j^\ast$
in the X-axis and Y-axis, respectively, where
${1 \leq i \leq q}$
and
${1 \leq j \leq m}$
. Then, it calculates the weight
$w_{ij}^{\prime}$
and the normalized weight
$w_{ij}$
of the
${j}$
th antecedent variable of fuzzy rule
${R_{i}}$
and calculates the weight
${w_{i}}$
of fuzzy rule
${R_{i}}$
, which are used to calculate the RVs
$Rep{(\underline {{{\widetilde B}^\ast}})_x}$
and
$Rep{(\overline {{{\tilde B}^\ast}})_x}$
of the lower approximate polygonal fuzzy set
$\underline {{{\tilde B}^\ast}}$
and the upper approximate polygonal fuzzy set (UAPFS)
${\overline {\tilde B}^\ast}$
of the interpolated consequence polygonal
${{\tilde B}^\ast}$
in the X-axis, respectively. Then, it calculates the CPs of the LAPFS
$\underline {{{\tilde B}^\ast}}$
and calculates the CPs of the
$\overline {{{\tilde B}^\ast}}$
of the interpolated consequence RFS
${{\tilde B}^\ast}$
based on the obtained RVs
$Rep{(\underline {{{\tilde B}^\ast}})_x}$
and
$Rep{(\overline {{{\tilde B}^\ast}})_x}$
of the LAPFS
$\underline {{{\tilde B}^\ast}}$
and the UAPFS
$\overline {{{\tilde B}^\ast}}$
of the interpolated consequence polygonal RFS
${{{\tilde B}^\ast}}$
in the
${X}$
-axis, respectively. Finally, it calculates the membership degree (MD)
$\underline {\mu _g^\ast}$
of the gth CP
$\underline {b_g^\ast}$
of the LAPFS
$\underline {{{\tilde B}^\ast}}$
and calculates the MD
$\overline {\mu _g^\ast}$
of the gth CP
$\overline {b_g^\ast}$
of the UAPFS
$\overline {{{\tilde B}^\ast}}$
of the interpolated consequence polygonal RFS
${{{\tilde B}^\ast}}$
, where
$0 \leq g \leq n - 1$
. The proposed FIR method offers us a very useful approach for FIR in rough-fuzzy environments.