Two Combinatorial Optimization Problems at the Interface of Computer Science and Operations Research

Solving large combinatorial optimization problems is a ubiquitous task across multiple disciplines. Developing efficient procedures for solving these problems has been of great interest to both researchers and practitioners. Over the last half century, vast amounts of research have been devoted to studying various methods in tackling these problems. These methods can be divided into two categories, heuristic methods and exact algorithms. Heuristic methods can often lead to near optimal solutions in a relatively time efficient manner, but provide no guarantees on optimality. Exact algorithms guarantee optimality, but are often very time consuming. This dissertation focuses on designing efficient exact algorithms that can solve larger problem instances with faster computational time. A general framework for an exact algorithm, called the Branch, Bound, and Remember algorithm, is proposed in this dissertation. Three variations of single machine scheduling problems are presented and used to evaluate the efficiency of the Branch, Bound, and Remember algorithm. The computational results show that the Branch, Bound, and Remember algorithms outperforms the best known algorithms in the literature. While the Branch, Bound, and Remember algorithm can be used for solving combinatorial optimization problems, it does not address the subject of post-optimality selection after the combinatorial optimization problem is solved. Post-optimality selection is a common problem in multi-objective combinatorial optimization problems where there exists a set of optimal solutions called Pareto optimal (non-dominated) solutions. Post-optimality selection is the process of selecting the best solutions within the Pareto optimal solution set. In many real-world applications, a Pareto solution set (either optimal or near-optimal) can be extremely large, and can be very challenging for a decision maker to evaluate and select the best solution. To address the post-optimality selection problem, this dissertation also proposes a new discrete optimization problem to help the decision-maker to obtain an optimal preferred subset of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. To solve this problem, exact algorithms and heuristic methods are presented. Different multi-objective problems with various numbers of objectives and constraints are used to compare the performances of the proposed algorithms and heuristics

Integrated Scheduling Problems in Healthcare and Logistics

Scheduling is one of the important components of operation management in different services. The goal of scheduling is to allocate limited available resources over time for performing a set of activities such that one or more objectives are optimized. In this thesis, we study several interesting applications of scheduling in health care and logistics. We present several formulations and algorithms to efficiently solve the scheduling problems that arise in these areas. We first study static and dynamic variants of a multi-appointment, multi-stage outpatient scheduling problem that arises in oncology clinics offering chemotherapy treatments. We present two integer programming formulations that integrate numerous scheduling decisions, features, and objectives of a major outpatient cancer treatment clinic in Canada. We also develop integrated and sequential scheduling strategies for the dynamic case in which arriving requests are processed at specific points of time. The results of computational experiments show that the proposed scheduling strategies can achieve significant improvements with respect to the several performance measures compared to the current scheduling procedure used at the clinic. We next present a daily outpatient appointment scheduling problem that simultaneously determines the start times of consultation and chemotherapy treatment appointments for different types of patients in an oncology clinic under uncertain treatment times. We formulate this stochastic problem using two two-stage stochastic programming models. We also propose a sample average approximation algorithm to obtain high quality feasible solutions. We use an efficient specialized algorithm that quickly evaluates any given first-stage solution for a large number of scenarios. We perform several computational experiments to compare the performance of proposed two-stage stochastic programming models. In the next part of the experiments, we show that the quality of the first-stage solutions obtained by the sample average approximation is significantly higher than those of the expected value problem, and the value of stochastic solution is extremely high specially for higher degrees of uncertainty. Finally, we address two variants of a cross-dock scheduling problem with handling times that simultaneously determines dock-door assignments and the scheduling of the trucks. In the general variant of the problem we assume that unit-load transfer times are door dependent, whereas in the specific case variant, unit-load transfer times are considered to be identical for all pairs of doors. We present constraint programming formulations for both variants of the problem, and we compare the performance of these models with mixed integer programming models from the literature. For the specific case, we propose several families of valid inequalities that are then used within a branch-and-cut framework to improve the performance of a time-index model. To solve the general problem efficiently, we also develop an approximate algorithm that first solves the specific case problem with the developed branch-and-cut algorithm to obtain a valid lower-bound, and then applies a matheuristic to obtain a valid upper-bound for the general problem and to compute the optimality gap. According to the computational experiments, we show that the proposed formulations and algorithms are able to solve the studied problems efficiently, and they outperform other models and heuristics that were previously developed for the problem in the literature

An ant colony algorithm for the sequential testing problem under precedence constraints.

We consider the problem of minimum cost sequential testing of a series (parallel) system under precedence constraints that can be modeled as a nonlinear integer program. We develop and implement an ant colony algorithm for the problem. We demonstrate the performance of this algorithm for special type of instances for which the optimal solutions can be found in polynomial time. In addition, we compare the performance of the algorithm with a special branch and bound algorithm for general instances. The ant colony algorithm is shown to be particularly effective for larger instances of the problem