This thesis presents a period-by-period heuristic algorithm for a multi-product single-machine capacitated lotsizing problem where products sharing the same setup are grouped into the same family and backlogs are allowed. The family concept is employed to express the problem explicitly and to extend the existing heuristic methods for a multi-product single-machine capacitated lotsizing problem to more complex situations where grouping of similar products is required. The backlog allowance is another improvement over conventional period-by-period heuristics. In the proposed heuristic method, lotsizing decisions are made period by period. In each period, lotsizing indices and opportunity costs for production of each family are evaluated to obtain a feasible plan with low costs. Next, an improvement step is applied to lower the production costs by perturbing the obtained plan.