This study proposes a strategy for optimal design of hollow fiber membrane networks for post combustion carbon capture from power plant multicomponent flue gas. A mathematical model describing multicomponent gas permeation through a separation membrane was customized into the flowsheet modeling package ASPEN PLUS. An N-stage membrane network superstructure was defined considering all possible flowsheeting configurations. An optimization formulation was then developed and solved using an objective function that minimizes the costs associated with operating and capital expenses. For a case study of flue gas feed flow rate of 298 m3/s with 13% CO2 and under defined economic parameters, the optimization resulted in the synthesis of a membrane network structure consisting of two stages in series. This optimal design was found while also considering feed and permeate pressures as well as recycle ratios between stages. The cost of carbon capture for this optimal membrane network is estimated to be $\$$28 per tonne of CO2 captured, considering a membrane permeance of 1000 GPU and membrane selectivity of 50. Following this approach, a reduction in capture cost to less than $\$$20 per tonne CO2 captured is possible if membranes with permeance of 2000 GPU and selectivity higher than 70 materialize.