傳統的化工製程設計大多是以數學模型為基礎，並將經濟效益作為主要考量，但在實際操作過程中，數學模型中可能會有某些參數受到外部因素影響或是本身數值估算不精確，導致參數偏離原先程序設計時設定的數值，造成系統效率不如預期，甚至在極端情況下無法操作，因此在設計時除了考慮成本外，操作的可行性也需要重視。在過去的研究中已有學者針對參數不確定的動態系統定義動態彈性指標(dynamic flexibility index, FI_d)來評量系統的可操作性。然而在過往相關研究中所探討的系統大多以代數方程式或常微分方程式描述，探討以偏微分方程式為基礎的動態系統的研究較少，因此在本研究中我們同時探討以常為分方程式和偏微分方程式表示的動態系統。傳統的計算方法是使用窮舉頂點法，將所有頂點逐一代入限制條件，比較不同最適化結果後即可得到彈性指標值，但當限制式變多且複雜時，頂點數量會大幅增加，使得窮舉計算過於耗時，因此我們使用基因演算法(Genetic Algorithm, GA)輔助頂點法與活性約束法，希望能大幅減少計算量。本研究藉由整合MATLAB與GAMS撰寫程式，執行上述的多層最適化問題，以此分析數個動態系統案例，並比較其結果的精確度及收斂時間。

Traditional chemical process design is created mostly based on mathematical models, with economic benefit as the main evaluation criterion. However, in the actual plant operation, some model parameters may be affected by external factors or inaccurate estimates, causing the system efficiency to be lower than expected or even inoperable in extreme cases. Therefore, in addition to the operating and equipment costs, the operational feasibility also needs to be considered. In the past, the dynamic flexibility index (FId) was defined in various studies for use as a performance metric of any given dynamic system with uncertain parameters. However, the processes discussed in the past were described by either the algebraic equations or the ordinary differential equations (ODEs), and there have been essentially no studies concerning the dynamic systems characterized with partial differential equations (PDEs). Therefore, in this study, the dynamic systems expressed by both ODEs and PDEs are analyzed with the same basic approach. The traditional method for evaluating the dynamic flexibility index is to exhaustively search in all vertex directions, then maximize the above-mentioned performance measure along every direction one-at-a-time, and finally obtain FI_d by selecting the smallest metric among those determined with the exhaustive search. However, as the number of constraints becomes very large and the constraint formulations more complex, the vertex number should also increases drastically. This feature obviously makes the exhaustive search impractical.
Genetic algorithm (GA) assisted vertex method and the active set method have been used in this study to solve the dynamic programming models formulated with the ordinary or partial differential equations. The GA-based search approach was originally proposed by Ali et al. (2021) to relieve the computation load of this search procedure. The active set method was originally proposed by Grossmann and Floudas (1987), and Huang (2019) proposed additional constraints to reduce its convergence time. This research integrated MATLAB and GAMS to write a code to implement the above-mentioned multi-level optimization problem. Several examples are presented in this thesis to demonstrate the feasibility of the two proposed numerical methods. The computation accuracy and convergence time of their results have also been compared in detail. The advantage of the GA-assisted vertex method is that the time required for calculation is shorter, but it cannot guarantee that the result is the global optimal solution. The advantage of the active set method is that the result is more accurate, but the time required for the calculation increases significantly as Nz increases.

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