Optimality region for job permutation in single-machine scheduling with uncertain processing times

The problem of scheduling optimally a given set of jobs on a single machine is studied. The lower and upper bounds on the admissible duration of each job are known. The optimality criterion of the schedule is the minimum total completion time of a given set of jobs. Some properties of the optimality region for a job permutation are investigated. Polynomial algorithms for constructing the optimality region for a job permutation and also for calculating the volume of this region are developed. The existence conditions of an empty optimality region for a job permutation are determined. A criterion for the existence of a job permutation with the maximum possible volume of the optimality region is established.