In this paper, we propose a novel solution strategy to explicitly describe the design space in which no recourse is considered for the realization of the parameters. First, to smooth the boundary of the design space, the Kreisselmeier-Steinhauser (KS) function is applied to aggregate all inequality constraints, and project them into the design space. Next, for creating a surrogate polynomial model of the KS function, we focus on finding the sampling points on the boundary of KS space. After testing the feasibility of Latin hypercube sampling points, two methods are presented to efficiently extend the set of boundary points. Finally, a symbolic computation method, cylindrical algebraic decomposition, is applied to transform the surrogate model into a series of explicit and triangular subsystems that can be further converted to describe the KS space. Two case studies are considered to show the efficiency of the proposed algorithm.

The existing methods of flexibility index are mainly based on mixed-integer linear or nonlinear programming methods, making it difficult to readily deal with complex mathematical models. In this article, a novel solution strategy is proposed for finding a reliable upper bound of the flexibility index where the process model is implemented in a black box that can be directly executed by a commercial simulator, and also avoiding the need for calculating derivatives. Then, the flexibility index problem is formulated as a sequence of univariate derivative-free optimization (DFO) models. An external DFO solver based on trust-region methods can be called to solve this model. Finally, after calculating the critical point of the model parameters, the vertex enumeration method and two gradient approximation methods are proposed to evaluate the impact of process parameters and to evaluate the flexibility index. A reaction model is studied to show the efficiency of the proposed algorithm.