In the present study, a robust modified Incompressible Smoothed Particle Hydrodynamics (ISPH) method is developed and applied to model a fluid mixer with a rotating stirrer and a tank body with free surface and horizontal oscillations. The mixer consists of a rectangular tank in which a stirrer rotates to mix the fluid. For the fluid in the tank, the free surface condition is considered. According to the Reynolds number, it is assumed that the mixing process is turbulent, and so a turbulent viscosity is defined. Although the smoothed particle hydrodynamics (SPH) method can result in some complexities, it is generally an easy and appropriate method for modeling mixing flow, free surface flow, and moving body problems, which will simultaneously be applied in the present study. The method is improved with kernel gradient corrective tensor, shifting particle algorithm, and turbulent viscosity, and it is validated against other well-known test cases and problems and is applied to model the mixing phenomenon. This study aims to increase the mixing rate and decrease the mixing time of the described mixer. To do so, the effect of the rotation of the stirrer on the mixing process is first investigated, after which the linear oscillation of the body is added and then the optimal shape of the tank is examined. The results show that the stirrer's rotating speed, the linear oscillation of the tank body, and streamlining all have major effects on the mixing rate.