This paper presents a flexible heat exchanger network (HEN) synthesis methodology to cope with (i) the fluctuations of uncertain process parameters and (ii) the gradually accumulated deposit, which have never been simultaneously considered yet. First, a called all-cycle flexibility analysis method is developed to perform the network evaluation throughout entire operation horizon. Then at Part (I) of the methodology, an optimization-based sequential framework is proposed for the design of flexible HENs. The three major steps are (i) initial HEN synthesis; (ii) all-cycle flexibility analysis; and (iii) network improvement toward qualified flexibility and cost expectation via a mixed-integer nonlinear programming (MINLP) model-based iteration between network optimization and flexibility analysis. Furthermore, the optimization of the cleaning schedule cooperates with that of the heat transfer areas in Part (II), striving for lower network redundancy and total cost. At last, two cases are studied to demonstrate the application of the methodology. Results indicate that the fouling in heat exchangers and the parametric fluctuations in operation act synergistically on the flexibility of HEN. And the flexible HEN solutions with and without periodic cleaning both can be obtained by employing the proposed framework with sufficient flexibility guaranteed.