Exact models for selection problems: from clinical trials to network design

Discrete optimization is becoming an increasingly important tool for solving problems in the real world. The matching problem and the network design problem are two well studied selection problems in this area. They can be considered as modelling relations between nodes of a bi-graph or a graph, respectively. Using acute stroke trials as a context, the assignment algorithm is utilized to investigate a complex relationship between the overall degree of individual matching, the size of samples, and the quality of matching on variables. It is concluded that the post-hoc individual matching in parallel group randomized clinical trials cannot be recommended as a technique for treatment effect estimation. Based on the concept of the transshipment problem we proposed a mixed integer programming model to solve the asymmetric traveling salesman problems. The formulation is extendable to other transportation scheduling problems which are related to the traveling salesman problem (TSP) such as the Multiple TSP (m-TSP) and the Selective TSP (STSP). In addition to avoiding any cycles and being easy to implement, the model has a reasonable order of space complexity. It can be built on either a directed graph or an undirected graph. The reserve network design problem is a variation of the STSP which maximizes some utilities subject to various constraints. These constraints include a budget limitation and spatial attributes such as connectivity and compactness. The proposed model achieves the contiguity and to some extent compactness attributes. It does this without incurring the problem of sub-tours and requiring any regular shape assumptions. Furthermore, where full connectivity is not required, the model enables the trade-off between the number of contiguous areas and utility to be determined easily. The combinatorial structure of the reserve network design problem places it in the category of NP-hard problems which have exponential time complexity. We explored approaches to reduce the computational effort and introduced an approach with improved efficiency. Using this approach, the experimental results show the solution time significantly reduced on average

Approximation in Multiobjective Optimization with Applications

Over the last couple of decades, the field of multiobjective optimization has received much attention in solving real-life optimization problems in science, engineering, economics and other fields where optimal decisions need to be made in the presence of trade-offs between two or more conflicting objective functions. The conflicting nature of objective functions implies a solution set for a multiobjective optimization problem. Obtaining this set is difficult for many reasons, and a variety of approaches for approximating it either partially or entirely have been proposed. In response to the growing interest in approximation, this research investigates developing a theory and methodology for representing and approximating solution sets of multiobjective optimization problems. The concept of the tolerance function is proposed as a tool for modeling representation quality. Two types of subsets of the set being represented, covers and approximations, are defined, and their properties are examined. In addition, approximating the solution set of the multiobjective set covering problem (MOSCP), one of the challenging combinatorial optimization problems that has seen limited study, is investigated. Two algorithms are proposed for approximating the solution set of the MOSCP, and their approximation quality is derived. A heuristic algorithm is also proposed to approximate the solution set of the MOSCP. The performance of each algorithm is evaluated using test problems. Since the MOSCP has many real-life applications, and in particular designing reserve systems for ecological species is a common field for its applications, two optimization models are proposed in this dissertation for preserving reserve sites for species and their natural habitats