A new, unified theory and algorithms, based on multiparametric programming techniques, for the solution of flexibility analysis and design optimization problems in linear process systems are presented. They are used for the flexibility test and index problems in systems with deterministic parameters, and for the stochastic and expected stochastic flexibility evaluation problems in systems with stochastic parameters. They are computationally efficient and give the explicit dependence of various flexibility metrics on the values of the continous design variables. The latter feature enables the easy and efficient comparison of design alternatives. It also allows for the compact formulation of design optimization problems that can be solved parametrically to yield the exact algebraic form of the trade-off curve of economics against flexibility. Key features of the proposed approach are demonstrated through both mathematical and process examples.