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The ion-acoustic (IA) shock waves and double layers (DLs) in an unmagnetized, dissipative, quantum electron-positron-ion (EPI) plasma (composed of a viscous heavy ion fluid, Fermi electrons and positrons) have been theoretically investigated. The higher-order Burgers and Gardner equations are derived by employing the reductive perturbation method. The basic features of the IA shock waves and the DLs are identified by analyzing the solutions of both the higher-order Burgers and Gardner equations. The ratio of the Fermi temperature of the positron to that of the electron, the Fermi pressure of electrons and positrons, the viscous force, the plasma particle number densities, etc. are found to change remarkably the basic features (viz. amplitude, width, phase speed, etc.) of the IA waves. The results of our investigation may be helpful in understanding the nonlinear features of localized IA waves propagating in quantum EPI plasmas which are ubiquitous in astrophysical, as well as laboratory, environments.