The spin-1 Ising (BEG) model with the nearest-neighbor bilinear and biquadratic interactions and single-ion anisotropy is simulated on a cellular automaton which is improved from the Creutz cellular automaton (CCA) for a simple cubic lattice. The simulations have been made for several sets of parameters K/J and D/J in the -3 < D/J <= 0 and -1 <= K/J <= 0 parameter regions. The re-entrant and double re-entrant phase transitions of the BEG model are determined from the temperature variations of the thermodynamic quantities (M, Q and chi). The phase diagrams characterizing phase transitions are compared with those obtained from other methods. (c) 2006 Elsevier B.V. All rights reserved.