A theoretical framework and an efficient algorithm are presented to solve the problem of sequencing jobs on a single processor. The objective achieved is minimum total tardiness. Jobs must be independent, with deterministic processing times. A brief review of the literature concerning sequencing to achieve minimum total tardiness is presented. This review shows that Emmons’ algorithm generally results in a partial schedule, and an enumerative method, branch and bound method or dynamic programming method, was then applied to help obtain a complete sequence. Thus Emmons’ algorithm is applied as a precursor to several enumerative algorithms. Furthermore, to certain problems (i.e., LPSD: jobs which have the property of a longer processing time but a shorter due date), Emmons’ theorems would not apply before branching the problems. The algorithm presented in this paper effectively applies the partitioning method to eliminate the need for enumerative methods. A set of necessary conditions for an optimal sequence is presented with proofs in the theory section. This is followed by a statement of the algorithm. The algorithm is illustrated with an example problem taken from Ref. [7]. Computational results are then presented which show the efficiency of the algorithm relative to dynamic programming.