We examine a one-machine scheduling problem to minimize the sum of weighted completion times subject to deadline constraints. Since this problem was proved to be NP-complete, most current researches have been focused on dominant lower bounds and effective precedence conditions that can be used in the framework of a branch and bound approach. In this thesis, the dominance relations among several lower and upper bounds are represented clearly by schedule graphs. An existing decomposition rule is computationally experimented. Based on the observation, blockwise heuristics are devised to improve the efficiency of branch and bound algorithm rithms for large problems. Finally, we show that a branch and bound algorithm may be improved by excluding some of the popular precedence conditions and incorporating a dominance property between nodes.