A tutorial of integrating duality and branch and bound in earliness–tardiness scheduling with idle insertion time problems

We present a conceptual tutorial of the value of the dual representation to branching strategies of idle insertion time in a single-machine scheduling problem. We consider the problem where scheduling generally occurs in two phases: a sequence of activities is first determined, then idle time is inserted into the schedule to minimise the sum of early and tardy costs. Typically, the sequence of jobs is determined first using a heuristic method. The idle time insertion problem is solved via linear programming to establish starting and ending times of the activities. In this paper, we show how the dual of the idle time insertion problem can be used as a means of generating insights for earliness–tardiness sequencing algorithms. We provide an extended tutorial on a new way of branching based on dual information. We find that in our simple examples, the search space can be vastly reduced, improving the feasibility of branch-and-bound improvements to heuristic solutions. We suggest that the dual approach is a useful and intuitive way to improve scheduling solutions in short order.