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The propagation of dust-acoustic (DA) shock waves (SWs) is studied in a four-component dusty plasma system containing viscous dust fluids of opposite polarity, Schamel distributed ions and Boltzmann distributed electrons. The reductive perturbation method is employed to derive a modified Burgers equation which gives rise to the DA shock waves with stronger nonlinearity. The viscous force acting in the dust fluids is identified as a source of dissipation, and is responsible for the formation of the DA shock waves. The basic characteristics (viz., speed, amplitude, width) of the DA shock waves are found to be significantly modified by the combined effects of opposite polarity dust fluids and trapped ions. The applications of this investigation in different space plasma environments and laboratory devices are pinpointed.