On Distance Function among Finite Set of Points

In practical purposes for some geometrical problems in computer science we have as information the coordinates of some finite points in surface instead of the whole body of a surface. The problem arised here is:”How to define a distance function in a finite space?”as we will show the appropriate function for this purpose is not a metric function. Here we try to define this distance function in order to apply it in further proposes, specially in the field setting of transportation theory and vehicle routing problem.More precisely in this paper we consider VRP problem for two dimensional manifolds in R.