A joint maximum likelihood estimation algorithm, based on the partial compensatory multidimensional logistic model (PCML) proposed by L. Zeng (1989), is presented. The algorithm simultaneously estimates item difficulty parameters, the strength of each dimension, and individuals' abilities on each of the dimensions involved in arriving at a correct response to the items. A maximum number of iterations was set for each estimation phase. In the model, iterations in a phase continue until either convergence is obtained or the maximum number of iterations is reached. In one particular phase, either item or person parameters are estimated, with the other parameter fixed. Estimation stops when the improvements are less than the prior chosen critical values. A parallel processing program, written in FORTRAN, was developed. This program and the algorithm were evaluated with several simulated item response data sets. Comparison of the estimates and the true simulated parameters suggested that the estimation was very accurate. Four tables present data from the simulations. (SLD)