Quantification of process flexibility via space projection

Abstract

Flexibility index is an effective indicator for characterizing the operational flexibility of a process design model. In this work, the concept of δ‐space projection is proposed and defined from the view of solution space, and the flexibility index can be quantified without solving any optimization problems. First, the flexibility index problem is reformulated as an existential quantifier model. Then, the projection operator in the cylindrical algebraic decomposition (CAD) method is introduced to project the solution space onto the one‐dimensional feasible space of the flexibility index. Subsequently, all the candidate values of the flexibility index, which may refer to the locations of the critical points, that is, vertexes and tangency points, are extracted. Last, two δ‐checking rules based on the traditional and improved CAD methods are proposed for finding the flexibility index. The case studies show that the proposed method can exactly find the flexibility index, regardless of convex or nonconvex systems.

SyPSE: A Symbolic Computation Toolbox for Process Systems Engineering Part II─Design for PSE Applications

“…FI = 1 implies that the design has exactly the flexibility to satisfy the constraints; FI < 1 indicates the fractional deviation that can actually be handled; FI > 1 implies that the design can tolerate a wider range of uncertainties. Zhao et al proposed a method to exactly evaluate the maximum flexibility index by the projection operation instead of solving multi-level optimization problems, which is integrated in the solution strategy of SyPSE. First, eq can be reformulated into a constraint set as eq . …”

“…Thus, the space of δ can be decomposed into a series of cells, which is the candidate set of the flexibility index. Similarly, by lifting these cells of δ, the first cell satisfying the δ-checking rule proposed by Zhao et al is the maximum flexibility index. Through binary search for the candidates, the efficiency of locating the flexibility index can be dramatically increased.…”

“…With model triangularization as a key technique, Zhao and Chen first represented the process model as an existential quantifier formula and applied the CAD method to accurately describe the design space and explicitly express the relationships of uncertain parameters. Then, Zhao et al proposed a space projection method to deal with flexibility index problems. Zheng et al further studied the volumetric flexibility index to evaluate the design models.…”

Design space description through adaptive sampling and symbolic computation

“…The method is suitable for convex and nonconvex systems described by polynomials. Zhao et al 23 proposed a space projection method based on the CAD method to deal with flexibility index problems. Due to the heavy computational burden of the CAD method, the above methods are only applicable to relatively small‐scale problems.…”

Flexibility index of black‐box models with parameter uncertainty through derivative‐free optimization

“…Flexibility has been studied over three decades and as an important property that must be incorporated into a design, and it has been applied in several industrially inspired case studies, 2‐5 such as process synthesis of air separation, heat exchanger network design, and supply chain networks. For reviews of previous work on flexibility analysis, see references 6‐9 …”