Effect of the relative control phase as well as the squeezed phase parameter on the optical switching process of the optical bi-multistable behaviour are investigated in detail for a four-level Y-type atomic system; in the presence of a control field (coherent + dc fields) with squeezed vacuum field and quantum interference from spontaneous emission. The results indicated that (a) the quantum interference assists to decrease the rate of switching via the relative phase of the control field and makes the switching on/off points more rapid in squeezed vacuum compared to a normal vacuum case, and (b) the phase switching effects between the output field and the relative control phase increases the switching on/off processes in normal vacuum more than the squeezed vacuum case. In addition, we have demonstrated the possibility of displaying quantum beats with a novel type of bi-multistable behaviour in the cooperative switching diagram.