The Minimum Convex-Cost Network Flow Problems (MC-CNFP) is a class of network flow problem, with nonlinear (convex) cost function, has an important role in operations research. The convexity of the cost (or objective) function arise in various ways such as strictly convex, mixed linear and strictly convex, piece-wise linear convex, pseudo-convex etc. A number of solution methods developed in the last few decades on the view of different type convexity of cost function. In this dissertation we have present a 'Proposed Solution Algorithm' for MC-CNFP, which can deals with convex cost function. This solution algorithm is constructed on the basis of 'Network Simplex Method' for minimum cost network flow problem with linear cost function, 'Zangwill's Convex Simplex Method' (Zangwill I. W., 1967) and it's extension for quadratic programming problem (Swarup K., 1977) and the decomposition of the convex simplex method for large scale convex programming problems (Hsia W. S., 1973, 1974, 1975) and Non-Linear Transportation Problems (Terefe K., 2007).