Stability Radius of an Optimal Schedule: A Survey and Recent Developments

The usual assumption that the processing times of the operations are known in advance is the strictest one in deterministic scheduling theory and it essentially restricts its practical aspects. Indeed, this assumption is not valid for the most real-world processes. This survey is devoted to a stability analysis of an optimal schedule which may help to extend the significance of scheduling theory for some production scheduling problems. The terms ‘stability’, ‘sensitivity’ or ‘postoptimal analysis’ are generally used for the phase of an algorithm at which a solution (or solutions) of an optimization problem has already been found, and additional calculations are performed in order to investigate how this solution depends on the problem data. We survey some recent results in the calculation of the stability radius of an optimal schedule for a general shop scheduling problem which denotes the largest quantity of independent variations of the processing times of the operations such that this schedule remains optimal. We present formulas for the calculation of the stability radius, when the objective is to minimize mean or maximum flow time. The extreme values of the stability radius are of particular importance, and these cases are considered more in detail. Moreover, computational results on the calculation of the stability radius for randomly generated job shop scheduling problems are briefly discussed. We also show that the well-known test problem with 6 jobs and 6 machines has both stable and unstable optimal makespan schedules.