The propagation and collision of dust acoustic (DA) multi-solitons in a four component dusty plasma which consists of negatively and positively charged cold dust fluids, Boltzmann distributed electrons and ions have been studied in the presence of a polarization force acting on dust grains. Using the reductive perturbation technique (RPT), the Korteweged-de Vries (KdV) equation is derived. By using the Hirota bilinear method, the two-and three-soliton solutions of the obtained KdV equation have been successfully obtained. Phase shifts of the two-soliton and three-soliton have been deduced. It has been found that both compressive and rarefactive solitons exist in current dusty plasma model. The polarization force plays a significant role in the overtaking collision process of two-soliton and three-soliton, in such a way that the amplitude of the compressive (rarefactive) solitons decreases (increases) as polarization force increases. It was also found that the phase shifts are significantly affected by the dusty plasma parameters such as the dust charge number ratio, ion-to-electron temperature ratio, negative-to-positive dust mass ratio, the ratio of the densities of electrons to ions and the polarization force parameter.