The three-dimensional BC model is simulated on a cellular automaton which improved from the Creutz cellular automaton for simple cubic lattice. The phase diagram characterizing phase transition of the model is obtained for a comparison with those obtained from other calculations. The simulations confirm the existence of a tricritical point at which the phase transition changes from second-order to first-order at D/J = 2.82. For the determined of the tricritical point, the thermodynamics quantities are computed using two different procedures which is called as the standard and the cooling algorithm for the anisotropy parameter values in the interval 3 >= D/J >= -8. The simulations indicates that the cooling algorithm is a suitable procedure for the calculations near the first-order phase transition region, and the cooling rate is an important parameter in the determining of the phase boundary. The estimated critical temperatures for D/J = 0 and 2.82 are compatible with the series expansion results.