determining malmquist productivity index in dea and dea-r based on value ‎efficiency

malmquist productivity index (mpi) is a numeric index that is of great importance in measuring productivity and its changes. in recent years, tools like dea have been utilized for determining mpi. in the present paper, some models are recommended for calculating mpi when there are just ratio data available. then, using dea and dea-r, some models are proposed under the constant returns to scale (crs) technology and based on value efficiency (ve) in order to calculate mpi when there is just a ratio of the output to the input data (and vice versa). finally, in an applied study on 30 welfare service companies under crs technology, the progress and/or regression of companies are determined in dea and dea-‎r.‎