This paper aims to study multi- objective assignment (NMOAS) problem with imprecise costs instead of its prices information. The NMOAS problem is considered by incorporating single valued trapezoidal neutrosophic numbers in the elements of cost matrices. After converting the NMOAS problem into the corresponding crisp multiobjective assignment (MOAS) problem based on the score function, an approach to find the most preferred neutrosophic solution is discussed. The approach is used through a weighting Tchebycheff problem which is applied by defining relative weights and ideal targets.  The advantage of this approach is more flexible than the standard multi- objective assignment problem, where it allows the decision maker (DM) to choose the targets he is willing. Finally, a numerical example is given to illustrate the utility, effectiveness and applicability of the approach.