This work investigates the nonlinear propagation of dust ion-acoustic (DIA) shock waves (SWs) in a collisionless, four-component, unmagnetized, dusty multi-ion plasma. The four components are considered as the generalized \((\alpha ,q)\) -distributed electrons, positively charged Maxwell-Boltzmann distributed light ions, negatively charged inertial heavy ions, and negatively charged stationary dusts. To do so, several nonlinear evolution equations are formulated by implementing the reductive perturbation technique. The appropriate solutions to the obtained equations are provided. This work also reinvestigates the electrostatic DIA SWs in the dusty multi-ion nonextensive plasma because of the misleading information for the nonlinear coefficients and analytical solutions of higher-order Burgers equations (BEs) in Ref. [Eur Phys J Plus (2015) 130: 46, https://doi.org/10.1140/epjp/i2015-15046-0]. It is found from the electrostatic potential obtained by BE that the rarefactive electrostatic shocks are only supported in the presence of superthermal \((0.33< q < 1,\;\alpha = 0)\) , isothermal \((q = 1,\; \alpha = 0)\) , and nonthernal electrons \((0< \alpha < 0.35,\;q = 1)\) , but both compressive and rarefactive shocks are supported in the presence of subthermal \((q > 1,\; \alpha = 0)\) electrons. The effects of parameters on the DIA SWs are reported. It is also investigated the nature of electrostatic shocks and double layers around the critical values and at the critical values of any specific plasma parameters described by modified BE and mixed modified BE. The outcomes from this work would be helpful for better understanding the dynamics of DIA SWs in space environments and plasma laboratories.
