Single machine earliness and tardiness scheduling

We examine the problem of scheduling a given set of jobs on a single machine to minimize total early and tardy costs without considering machine idle time. We decompose the problem into two subproblems with a simpler structure. Then the lower bound of the problem is the sum of the lower bounds of two subproblems. A lower bound of each subproblem is obtained by Lagrangian relaxation. Rather than using the well-known subgradient optimization approach, we develop two efficient multiplier adjustment procedures with complexity O(nlog n) to solve two Lagrangian dual subproblems. A branch-and-bound algorithm based on the two efficient procedures is presented, and is used to solve problems with up to 50 jobs, hence doubling the size of problems that can be solved by existing branch-and-bound algorithms. We also propose a heuristic procedure based on the neighborhood search approach. The computational results for problems with up to 3000 jobs show that the heuristic procedure performs much better than known heuristics for this problem in terms of both solution efficiency and quality. In addition, the results establish the effectiveness of the heuristic procedure in solving realistic problems to optimality or near optimality.