The volumetric fuzzy set (VFS) deals with the uncertainties associated with two-dimensional variables. This study shows the application of VFS in finding the optimal coordinates for a new hospital by considering several fuzzy goals. For this purpose, we first define two types of volumetric fuzzy goals: type one following the proximity to an ideal point and type two following far from an anti-ideal point. Then, we extract the membership functions (MFs) for type-one and type-two volumetric fuzzy goals. These MFs determine the membership degrees of each point for volumetric fuzzy goals based on its distances from the ideal or anti-ideal points. We consider the Manhattan distance to extract these MFs because it better measures the real distance between two points in urban areas. Next, we design a new weighted volumetric fuzzy multi-objective decision-making (VFMODM) model considering type-one and type-two fuzzy goals. This model answers this question: what are the optimal coordinates for a new hospital to maximize overall satisfaction? Compared to existing fuzzy optimization models, the VFMODM model considers uncertainty for two-dimensional variables.
