Batch Testing, Adaptive Algorithms, and Heuristic Applications for Stable Marriage Problems

In this dissertation we focus on different variations of the stable matching (marriage) problem, initially posed by Gale and Shapley in 1962. In this problem, preference lists are used to match n men with n women in such a way that no (man, woman) pair exists that would both prefer each other over their current partners. These two would be considered a blocking pair, preventing a matching from being considered stable. In our research, we study three different versions of this problem. First, we consider batch testing of stable marriage solutions. Gusfield and Irving presented an open problem in their 1989 book The Stable Marriage Problem: Structure and Algorithms\u3c\italic\u3e on whether, given a reasonable amount of preprocessing time, stable matching solutions could be verified in less than O(n^2) time. We answer this question affirmatively, showing an algorithm that will verify k different matchings in O((m + kn) log^2 n) time. Second, we show how the concept of an adaptive algorithm can be used to speed up running time in certain cases of the stable marriage problem where the disorder present in preference lists is limited. While a problem with identical lists can be solved in a trivial O(n) running time, we present an O(n+k) time algorithm where the women have identical preference lists, and the men have preference lists that differ in k positions from a set of identical lists. We also show a visualization program for better understanding the effects of changes in preference lists. Finally, we look at preference list based matching as a heuristic for cost based matching problems. In theory, this method can lead to arbitrarily bad solutions, but through empirical testing on different types of random sources of data, we show how to obtain reasonable results in practice using methods for generating preference lists “asymmetrically” that account for long-term ramifications of short-term decisions. We also discuss several ways to measure the stability of a solution and how this might be used for bicriteria optimization approaches based on both cost and stability

Uncertain Multi-Criteria Optimization Problems

Most real-world search and optimization problems naturally involve multiple criteria as objectives. Generally, symmetry, asymmetry, and anti-symmetry are basic characteristics of binary relationships used when modeling optimization problems. Moreover, the notion of symmetry has appeared in many articles about uncertainty theories that are employed in multi-criteria problems. Different solutions may produce trade-offs (conflicting scenarios) among different objectives. A better solution with respect to one objective may compromise other objectives. There are various factors that need to be considered to address the problems in multidisciplinary research, which is critical for the overall sustainability of human development and activity. In this regard, in recent decades, decision-making theory has been the subject of intense research activities due to its wide applications in different areas. The decision-making theory approach has become an important means to provide real-time solutions to uncertainty problems. Theories such as probability theory, fuzzy set theory, type-2 fuzzy set theory, rough set, and uncertainty theory, available in the existing literature, deal with such uncertainties. Nevertheless, the uncertain multi-criteria characteristics in such problems have not yet been explored in depth, and there is much left to be achieved in this direction. Hence, different mathematical models of real-life multi-criteria optimization problems can be developed in various uncertain frameworks with special emphasis on optimization problems

Fuzzy Techniques for Decision Making 2018

Zadeh's fuzzy set theory incorporates the impreciseness of data and evaluations, by imputting the degrees by which each object belongs to a set. Its success fostered theories that codify the subjectivity, uncertainty, imprecision, or roughness of the evaluations. Their rationale is to produce new flexible methodologies in order to model a variety of concrete decision problems more realistically. This Special Issue garners contributions addressing novel tools, techniques and methodologies for decision making (inclusive of both individual and group, single- or multi-criteria decision making) in the context of these theories. It contains 38 research articles that contribute to a variety of setups that combine fuzziness, hesitancy, roughness, covering sets, and linguistic approaches. Their ranges vary from fundamental or technical to applied approaches

Setting forest road maintenance and upgrade priorities based on environmental effects and expert judgment

The prioritization of road maintenance projects is an important forest engineering task subject to limited budgets and competing investment needs. Large investments are made each year to maintain and upgrade forest road networks to meet economic and environmental goals. Many models and guidelines are available for single-criterion analysis of forest roads, but guidelines for multi-criterion analysis are rare. Additionally, even single-criterion approaches often rely on expert judgment to inform models of user preferences and priorities. These preferences are used to make tradeoffs between alternatives that contain data that are physical and biological, quantitative and qualitative, and measured on many different scales. The Analytic Hierarchy Process (AHP) has the potential to provide a consistent approach to the ranking of forest road investments based on multiple criteria. AHP provides a consistent, quantifiable approach to problems involving multi-criterion analysis, but it has not been applied to road management. The road investment problem differs from traditional AHP applications in that a large number of alternatives are compared at one time. AHP methodology is discussed, including the foundations, assumptions, and potential for use in prioritizing forest road investments to meet economic and environmental goals. Using AHP, the problem of scheduling maintenance and upgrade of forest roads is presented as a hierarchy and pairwise comparisons are elicited from decision makers to determine the relative importance of road characteristics to management goals. Issues regarding the use of the AHP to prioritize forest road investments are discussed. Three metrics are proposed to assist in evaluating the quality of a solution and to revise preferences when necessary. Various models using mathematical programming are used to incorporate the results of an AHP analysis into the scheduling of forest road maintenance and upgrade activities involving non-monetary benefits. Solutions to two non-linear integer programming formulations are found using a threshold accepting heuristic

Kontextsensitivität für den Operationssaal der Zukunft

The operating room of the future is a topic of high interest. In this thesis, which is among the first in the recently defined field of Surgical Data Science, three major topics for automated context awareness in the OR of the future will be examined: improved surgical workflow analysis, the newly developed event impact factors, and as application combining these and other concepts the unified surgical display.Der Operationssaal der Zukunft ist ein Forschungsfeld von großer Bedeutung. In dieser Dissertation, die eine der ersten im kürzlich definierten Bereich „Surgical Data Science“ ist, werden drei Themen für die automatisierte Kontextsensitivität im OP der Zukunft untersucht: verbesserte chirurgische Worflowanalyse, die neuentwickelten „Event Impact Factors“ und als Anwendungsfall, der diese Konzepte mit anderen kombiniert, das vereinheitlichte chirurgische Display

Conflicting Objectives in Decisions

This book deals with quantitative approaches in making decisions when conflicting objectives are present. This problem is central to many applications of decision analysis, policy analysis, operational research, etc. in a wide range of fields, for example, business, economics, engineering, psychology, and planning. The book surveys different approaches to the same problem area and each approach is discussed in considerable detail so that the coverage of the book is both broad and deep. The problem of conflicting objectives is of paramount importance, both in planned and market economies, and this book represents a cross-cultural mixture of approaches from many countries to the same class of problem

Mathematical Knowledge for Teaching as Decision-Making for the University Mathematician Developing Coherence in Review of Discrete Mathematics

Mathematicians teaching at the university level have a deep understanding and appreciation for the mathematics that they teach. However, they rarely receive much formal training in teaching. Thus, university mathematicians must rely on their mathematical understandings and personally developed ideas of teaching to guide their decision-making. Relatively little research exists on mathematicians’ teaching practices. The purpose of this study was to examine the mathematical knowledge for teaching (MKT) of a university mathematician teaching discrete mathematics and how he leveraged his knowledge to make decisions and develop coherence among mathematical ideas during a semester review. An enactivist perspective examining a mathematician’s decision-making in planning, enacting, and reflecting upon their lessons in this study shed light on how this mathematician practically approached his teaching duties. By enacting four distinct coherence strategies, the mathematician in this case study revealed a personal standard for mathematical storytelling which guided his decision enactment. These strategies fostered rich connections among mathematical ideas and among topics from earlier in the semester meaningfully with a single culminating topic: the chromatic polynomial. Implications of this study for research include recognized advantages of graph theoretic visualizations for the analysis of teacher decisions and coherence, benefits of dual coding for the Knowledge Quartet MKT framework, and a stance on inactivism\u27s consideration of cognitive actions. For teaching, this research supports the benefits of mathematical storytelling and review units which feature a new context to reframe previously seen topics