Analytical solution of volumetric flexibility through symbolic computation

Abstract

Volumetric flexibility index is an indicator for describing the flexibility of a process design. However, there is no prior literature on how the theoretical value of the volumetric flexibility index can be obtained. In this work, we propose the first systematic method to accurately derive the analytical solution of the volumetric flexibility index through symbolic computation. First, based on the cylindrical algebraic decomposition method, the feasible space can be decomposed into finite valid cells with analytical expressions. Then, the integral operation is implemented to calculate the hypervolume for each cell. Finally, hypervolumes of all feasible cells are summed to determine the volumetric flexibility index. Case studies show that the proposed method can obtain the theoretical value of the volumetric flexibility index for nonconvex systems. In addition, the explicit information between the hypervolume and design variables can be derived, which can be further applied to hypervolume optimization problems under different designs.