The research on lot sizing is extensive; however, no author in the literature reviewed to date provides an optimal solution algorithm to a prevalent problem which is found in manufacturing. A multi-level, general product-structure, variable-cost model is presented which follows the procedure of a closed-loop material requirements planning (MRP) system, and incorporates many conditions that production and material managers find in practice. A branch and bound (B&B7) algorithm is developed. The efficiency of B&B is derived from effective lower bounds and solution procedures which are determined on the basis of the space-time structure of the MRP lot-sizing problem and its non-convex total-cost function. This path-dependent lower bound is computationally efficient and guarantees an optimal solution. The B&B algorithm is tested on problems and compared to heuristics in the literature.