The paper deals with a flow distribution problem with a piecewise-linear cost function. The problem is formulated as a piecewise-linear programming problem which is not separable with respect to separate variable group. The method for solving this problem is based on the extension of the idea of the simplex method to the class of non-separable piecewise-linear problems. It secures finding of a local solution to the problem after a finite number of iterations. The method uses the peculiarities of the problem constraints that make it possible to decompose the matrix of constraints to smaller ones and thus to diminish the volume of calculations.