The traditional methods for the solution of flexibility index and design problems are mainly developed by numerical calculation methods. In this work, a new solution approach is proposed for chemical systems described by polynomials to deduce explicit expressions of flexibility index on the value of the continuous design variables without solving any optimization problems. First, the flexibility index and design problems are reformulated as an existential quantifier model. Then, the cylindrical algebraic decomposition (CAD) method is introduced to project the solution space onto the dimension of design variables, flexibility index and uncertain parameters. Last, the analytical expressions between design variables and flexibility index can be deduced by the inscribed hyperrectangle checking rule. The case studies show that the proposed method is applicable to relatively small- or medium- scaled problems and the explicit relationship between the flexibility index and design variables can be deduced, regardless of linear or nonlinear systems.

解决柔性指标和设计问题的传统方法主要是通过数值计算方法来开发的。在这项工作中，为多项式描述的化学系统提出了一种新的解决方法，以在不解决任何优化问题的情况下，针对连续设计变量的值推导柔性指数的显式表达式。首先，将灵活性指标和设计问题重新表述为现有的量词模型。然后，引入了圆柱代数分解（CAD）方法，将求解空间投影到设计变量，柔性指数和不确定参数上。最后，可以通过内接的超矩形检查规则推导设计变量与柔性指标之间的解析表达式。