Design program was developed to determine the external shape of the supersonic axisymmetric inlet by combining conical flow solver and approximation technique of conical shock with gradient-based optimization algorithm. Inlet designs were carried out under various operation conditions through optimization with respectively two object functions which consist of pressure recovery and cowl drag and with constraints about shock position, cowl shape, and minimum throat area. New object function consisting of pressure recovery and drag of the external cowl was proposed and the optimized shapes from new object function were compared to the ones from the old object function which maximize only the pressure recovery. Through computations of inviscid and turbulent flow, was tested performance of the design program and performance estimated in design program agreed well with computation results for inlets designed under various flight conditions.