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Fuzzy Heat exchanger network synthesis problem (F-HEN) is formulated as a multi-objective mixed-integer nonlinear programming (F-MO-MTNLP) to handle the multiple objectives of minimum utility consumption, minimum total area, and even minimum number of matches. Heat-capacity flow rates are proposed as fuzzy variables. The nonlinear G-bell membership function has been employed to describe this variation of heat-capacity flow rates. The Fuzzy Analytic Hierarchy Process (FAHP) has been utilized to obtain the weights of objectives. The nonlinear G-bell membership function has also been employed to describe the pair wise Fuzzy Analytic Hierarchy Process. Two examples are presented to demonstrate the various aspects dealt in the work.