It is shown that the methods and algorithms, developed in (A. Capani et al., Computing minimal finite free resolutions, {\it Journal of Pure and Applied Algebra}, (117& 118)(1997), 105 -- 117; M. Kreuzer and L. Robbiano, {\it Computational Commutative Algebra 2}, Springer, 2005.) for computing minimal homogeneous generating sets of graded submodules and graded quotient modules of free modules over a commutative polynomial algebra, can be adapted for computing minimal homogeneous generating sets of graded submodules and graded quotient modules of free modules over a weighted $\mathbb{N}$-graded solvable polynomial algebra, where solvable polynomial algebras are in the sense of (A. Kandri-Rody and V. Weispfenning, Non-commutative Gr��bner bases in algebras of solvable type. {\it J. Symbolic Comput.}, 9(1990), 1--26). Consequently, algorithmic procedures for computing minimal finite graded free resolutions over weighted $\mathbb{N}$-graded solvable polynomial algebras are achieved.

25 pages. Introduction Section 1, Algorithm 2, Algorithm 3, and the last paragraph of the proof of Theorem 5.4 are revised; several new references are added. arXiv admin note: substantial text overlap with arXiv:1401.5464