A fuzzy multi-objective optimization model for sustainable location planning using volumetric fuzzy sets

The volumetric fuzzy sets (VFSs) are used to manage the uncertainties in two-dimensional variables. This study applies VFSs to find the optimal coordinates for a new metro station using an optimization model with multiple volumetric fuzzy goals. The proposed model considers two types of volumetric fuzzy goals: the type-1 fuzzy goals for closeness to the ideal point and the type-2 fuzzy goals for farness from the anti-ideal point. We consider the satisfaction of each alternative point to identify the metro station location based on its Euclidean distances from the ideal (anti-ideal) point. We first extract a satisfaction membership function for each goal based on the Euclidean distance between its ideal (anti-ideal) and the optimal points. We then design a volumetric fuzzy multi-objective model to obtain the optimal coordinates for a new metro station. The proposed model finds the optimal coordinates for the new metro station by maximizing the overall satisfaction realized from the fuzzy goals. To solve this model, we use the weighted max–min approach, where the weight of each goal represents its importance on different sustainability dimensions. These weights are derived from a trapezoidal fuzzy analytic hierarchy process. A case study is presented to demonstrate the applicability of the proposed model by solving a volumetric optimization model with four type-1 and two type-2 volumetric fuzzy goals.