Abstract: With the application of the production possibility sets of intersection-form,this article presents the multivariate production function which is determined by the non-parametric method for the production department with multi-input factors.Furthermore,the necessary and sufficient conditions for distinguisking the future state of the returns to scale being increasing,co nstant,decreasing,saturated and congestion when the input scale being expanded are discussed in this paper.The results can be applied into scientific analysis that whether each decision making unit needs to expand the scale and how to expand it.

Key words: production possibility sets of intersection-form, non-parametric production function, returns to scale