Parametric rough bi-level multi-objective fractional programming problems

<p>The parametric rough bi-level multi-objective fractional programming problem (PRBL-MOFPP) is investigated in this article. In the right-hand aspect of the rough set of constraints, the suggested PRBL-MOFPP has a scalar parameter. The PRBL-MOFPP is converted into two issues congruent to the upper and lower approximation models (UAM and LAM) in the first phase. Both UAM and LAM are solved using the fuzzy goal programming (FGP) technique. The parametric UAM and LAM were formulated in the second phase, and the Lagrangian function for UAM and LAM was derived. Furthermore, both models were subjected to Karush-Kuhn-Tucker (KKT) optimality conditions. Finally, the surely and possibly stable set of the first kind (SSFK) are studied. An algorithm for determining the SSFK for PRBL-MOFPP, as well as numerical examples, are exhibited.<strong></strong></p><p> </p>