In this paper, an interactive approach for solving multi-level multi-objective fractional programming (ML-MOFP) problem under hybrid uncertainty is developed. The proposed interactive approach makes an extension work of Shi and Xia [22]. In the current model the left-hand- and right-hand-side variables in the constraints are influenced by hybrid uncertainty (i.e. both fuzziness and randomness); represented by fuzzy random variables (FRVs). In the first phase, we make the best use of the chance-constrained programming approach and the α -cut approach to obtain the equivalent deterministic model of the ML-MOFP problem with FRVs. Then, the linear model of the crisp ML-MOFP problem is formulated. In the second phase, the interactive approach simplifies the ML-MOLP model by changing it into isolated multi-objective decision-making (MODM) problems, to avoid non-convexity. Also, each separate MODM problem of the linear model is solved by the ε -constraint method and the concept of satisfactoriness. Finally, illustrative example and comparison with the existing techniques are provided to indicate the efficiency of the interactive approach. |
