Balaji Nagarajan

Robust Scheduling Approaches to Hedge against Processing Time Uncertainty

Introduction Robust scheduling is one of the approaches adopted to deal with uncertainties in stochastic scheduling. Since these uncertainties, arising as they do from variability in the scheduling parameters, significantly affect the performance of a system, a viable approach has been to determine a robust schedule that is least susceptible to disturbances or variations in the input parameters and that has minimum variability in its output performance measure. For instance, when processing times are uncertain, a robust schedule would have the least variance of the output performance measure, although it might not necessarily have the least expected value. As also mentioned in Chapter 1, of the various parameters involved in scheduling, we intend to focus on the uncertainty factor associated with job processing times. The following sections detail different robust scheduling methodologies that are available to model certain scheduling environments and the underlying principles behind those various approaches. Our primary focus is to elicit and delve in detail on the modeling approaches and not to expostulate the solution methodologies. However, we do briefly discuss the various solution approaches that are used to solve these robust scheduling problems and also state relevant results (as propositions) that are used in developing these solution methodologies. We intend to capture the core techniques that have been used in modeling robustness in scheduling and would advise interested readers to refer to the appropriate literature for a detailed and complete exposition.

Single-Machine Models

Introduction In this chapter we consider the problem of sequencing a given number of jobs on a single machine and devise methodologies and develop closed-form expressions (wherever possible) to compute the expectation and variance of various performance measures. Numerical illustrations indicating the significance and applicability of our work are also presented through example problems. Besides being of significance in their own right, single-machine problems also constitute a good starting point for analyzing more complex scheduling environments of flow shop and job shop. The different performance measures that we consider for the single-machine case can be classified into two categories: Completion-time-based , which includes total completion time (total flow time), total weighted completion time, and total weighted discounted completion time Due-date-based , which includes total tardiness, total weighted tardiness, total number of tardy jobs, total weighted number of tardy jobs, mean lateness, and maximum lateness. The processing time of a job is assumed to be a random variable that follows an arbitrary probability distribution. First, we present the notation that is used in this chapter. Some of this notation has been introduced already in earlier chapters. However, we include it here for the sake of completeness.