This paper is devoted to finding integral solutions of algebraic equations. Only algebraic equations with integral coefficients are considered. The elementary properties of integers are assumed.

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Improving the explainability of results from machine learning methods has become an important research goal. In this thesis, we have studied the problem of making clusters more interpretable using a recent approach by Davidson et al., and Sambaturu et al., based on succinct representations of clusters. Given a set of objects S, a partition of S (into clusters), and a universe T of descriptors such that each element in S is associated with a subset of descriptors, the goal is to find a representative set of descriptors for each cluster such that those sets are pairwise-disjoint and the total size of all the representatives is at most a given budget. Since this problem is NP-hard in general, Sambaturu et al. have developed a suite of approximation algorithms for the problem. We also show applications to explain clusters of genomic sequences that represent different threat levels / Master of Science / Improving the explainability of results from machine learning methods has become an important research goal. Clustering is a commonly used Machine Learning technique which is performed on a variety of datasets. In this thesis, we have studied the problem of making clusters more interpretable; and have tried to answer whether it is possible to explain clusters using a set of attributes which were not used while generating these clusters.

SNCF is a large-sized railway transportation company that needs to be operated 365 days a year and 24 hours a day. In order to schedule a certain category of workers in train stations and selling points, rosters are designed to cover a cyclical demand. However, the highly combinatorial nature of the rostering problem makes it very difficult to solve manually, and experts spend a huge amount of time to derive implementable solutions that improve a number of preference criteria. This thesis presents two formulations based on mixed-integer programming to adress the cyclical rostering problem. The first one uses variables to express the nature of each day of the roster, whereas the second one uses patterns corresponding to feasible blocks of seven days and assigns them to each week of the roster. Different strategies relative to the management of some preference criteria are compared, some of them leading to significant reductions in computational times. Cuts are finally introduced to improve the bounds obtained by the linear relaxation of the mixed-integer programs. The impact of these cuts on computational times depends much on the problem. / Master of Science

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Approved for public release; distribution is unlimited / Surgical technicians at Naval hospitals provide a host of services related to surgical procedures that include handing instruments to surgeons, assisting operating room nurses, prepping and cleaning operating rooms, and administrative duties. At the Naval Medical Center San Diego (NMCSD), there are 83 surgical technicians that must be scheduled for these duties. The three military and one civilian hospital interviewed for this thesis manually schedule these duties. Weaknesses of these manual schedules exposed during interviews at these hospitals include assignment inequities and the time needed to create them. This thesis reports on an optimization based and spreadsheet implemented tool developed to schedule surgical technicians for both daily and weekly duties at a Naval hospital. We demonstrate the tool for the surgical technician department at NMCSD. The schedulers at NMCSD verify the utility of the developed tool and cite a drastic reduction in the time required to generate timely, equitable, and accurate schedules. The study also investigates historical operating room usage data and makes suggestions for improving scheduling practices based on these data. / Commander (Select), United States Navy

Logistics Operations Command and Control Capability Concept (LOCCC), developed by Jeff Grelson in 2000, introduces a new distribution principle to combat elements. This concept employs a supporting logistics unit in a general support role and controls it by a unique command center in order to minimize the footprint left by logistics, improve logistic and tactical responsiveness, and reduce the "iron mountain" on the battlefield. This thesis revisits the mathematical models and algorithms developed by Major Thomas Lenhardt to model LOCCC. We preprocess the network topology in order to convert it into an equivalent, simplified network that is computationally tractable with the existing optimization model by using exact and heuristic algorithms. We show that the simplifications and enhancements we propose help us to obtain much faster and better quality solutions than using the original, non-simplified networks. For example, in a ten-minute run, we can obtain a solution that is 98% better in some cases. We also apply the model to a Turkish Infantry Brigade to evaluate LOCCC with sustainment requirements and transportation assets of the Turkish Army.

With the drastic increase in the demand for air travel, taxiway aircraft traffic scheduling is becoming increasingly important in managing air traffic. In order to reduce traffic congestion on taxiways, this thesis proposes a tool for air traffic controllers to use in decision making: a taxiway air traffic model developed using Mixed Integer Programming (MIP) that can be applied to a rolling time horizon. The objective of this model is to minimize the total taxi time, and the output is a schedule and route for each aircraft. This MIP model assumes that only the origin and destination of each aircraft is fixed; due to some uncertain factors in the air arrival and departure process, it allows for the departure time and arrival time to vary within a certain time window. This MIP model features aircraft type, and also incorporates runway crossings and runway separations. The model is programmed using C++ and Solved in CPLEX 12.1. Runways 26R and 26L of George Bush International Airport are used to find solutions. The author presents a rolling horizon method by dividing the large scheduling issue into smaller time interval problems according to the scheduled times of departure or arrival. A bound is also proposed based on the discretized time interval problems. By using partial data from George Bush International Airport (IAH), solutions are obtained. The results are compared with the bound and show fairly high optimality. Compared with the previous research, this thesis presents a model with more flexibility by considering different operations. By using the rolling horizon method, the problem is broken into smaller units that can be solved efficiently without losing much optimality.