Modern Methodologies for Practical Discrete Optimization

Modern discrete optimization problems, especially those motivated by practice, continue to grow in complexity and scale. The development of modern methodologies
to address such problems is of paramount importance, and even more so given that advanced analytical tools become increasingly widespread among businesses. This
dissertation consists of three projects, commonly aligned in the pursuit of solving and analyzing larger, more complex optimization problems involving discrete variables. In the first project (Chapter 2), we consider the home healthcare problem, inspired by a real-world hospice care firm's operational need to perform weekly updates to a
rolling visitation schedule of patients by home health aides. The need to jointly perform assignment and routing of aides combined with scheduling constraints makes this problem especially challenging. We propose an exact method based on logic-based Benders decomposition (LBBD) that allows the assignment and routing aspects of the problem to be treated independently and show that it is superior to a monolithic MILP model for all but a few instances. We also find that a variation of the standard LBBD method, called Branch-and-Check (B&C), performs better for the instances
tested. Thus, the contributions of this chapter are
 an exact method for solving the home healthcare problem based on logic-based Benders decomposition (LBBD),
 experimental verification that the method can find provably optimal solutions to realistic size instances of the problem within one hour, and a direct computational comparison of standard LBBD and Branch-and-Check on a class of problems derived from real-world data. In the second project (Chapter 3), we propose a generic framework for robust scheduling problems where the uncertainty set has a \combinatorial" structure that can be efficiently queried. This contrasts with many robust scheduling problems where
the uncertainty set is assumed to be polyhedral, or more generally, convex. The idea is to avoid including the whole uncertainty set in the model by dynamically generating
the scenarios to only include those that correspond to the \worst-case", which are often fewer in number. We apply our framework to the robust job shop scheduling
problem with machine breakdowns and the robust parallel machine scheduling problem with machine breakdowns and show that we can indeed compute a robust optimal solution by only considering a fraction of the entire uncertainty set. Thus, the contributions of this chapter are a generic algorithm for solving robust discrete optimization problems with combinatorial uncertainty sets, and experimental verification that, for certain robust machine scheduling problems, we can guarantee robustness to the entire uncertainty set by only considering a small percentage of the scenarios. In the third project (Chapter 4), we consider post-optimality analysis of mixed integer linear programming (MILP) problems, where the goal is to provide a systematic
method of analyzing the set of near-optimal solutions. Following a line of work initiated by [HH06] and extended by [HS17], we use the notion of sound reduction and sound decision diagrams as a compact and transparent representation of the set of near-optimal solutions. We also propose a novel method for handling continuous
variables with decision diagrams and report preliminary results on some MIPLIB instances to investigate the practicality of our procedure.