The Defense Language Institute (DLI) offers 23 beginning language courses and in 2004 began to provide a smaller class size for these courses. Restrictions on when classes can begin and a limited number of instructors prevent all students from being trained in a smaller class. This thesis develops integer linear programs (ILPs) that generate schedules for all student classes and maximize the number of smaller class starts for a given number of instructors. Secondary scheduling goals include avoiding weekly changes to instructor levels and scheduling preferences such as the number of classes to start simultaneously. The ILPs solve in less than one minute and offer a significant improvement in the number of students that may be trained in the smaller class size. Computational results using real data for the Arabic, Chinese-Mandarin, and Persian-Farsi courses verify the ILPs find feasible multiyear schedules that incorporate the DLI's scheduling preferences while exceeding the DLI's published schedule results. For example, the ILPs find schedules for Arabic that train 8%, 34% and 76% of students in the smaller class in 2006, 2007, and 2008, whereas DLI's manual schedules at best can train 8%, 7% and 64%.

The U.S. Army distributes its 51,000 competitive category officers among manning targets specified by location, rank and skill that change over time in response to changing requirements. The officer inventory also changes over time and does not exactly match the manning target requirements. The Army responds to imbalances by redistributing officers in order to provide each location with the minimum required officers while minimizing the number of unfilled targets and excess officers at each location. This thesis focuses on branch officers, branch targets and generalist targets with ranks from Branch Qualified Captain to Colonel. Using data provided by the Army, we formulate an integer programming model called DISTRIBUTOR. When DISTRIBUTOR allows all officers in the inventory to move, it finds only 340 unfilled targets but this requires 4,688 or 28% of the inventory to move. We reduce the number of moves by using DISTRIBUTOR in two sequential steps. The first step optimally distributes officers at each location and identifies the excess officers and unfilled targets at each location. The second step takes the excess officers and distributes them to unfilled targets at other locations. The two-step leaves only 346 targets unfilled (6 more) but requires only 1,373 or 8% of the inventory to move. By allowing rank substitution DISTRIBUTOR can reduce the unfilled targets to 70.

The thesis presents models for short term planning (24 hours) of a hyro dominated hydrothermal power system. The purpose of the models is to minimizae the system operation costs to provide a forecasted load and keep enough spinning reserve. / This thesis presents models for short term planning (24 hours) of a hydro dominated hydrothermal power system. The purpose of the models is to minimize the system operation cost to provide a forecasted load and keep enough spinning reserve.   The thesis focuses on two issues in hydro power modelling. The first issue is the relationship between water discharged and power generated. This relationship is a non-linear and non-convex function. If the plant has several units, the efficiency of the plant will have local maximums, so called local best-efficiency points. The second issue is to take into account the cost of start-ups of hydro units in the planning.   The hydro model is mixed-integer. Discharg􀁐s are allowed at zero flow, the local best-efficiency points and on the continuous part between the local best-efficiency point with the highest flow and the point with maximum flow. This last continuous part is modelled as a linear function. In order to get data for the start-up cost a survey among the largest power producers in Sweden has been made, where three questions about start-ups of hydro power units has been asked: What causes the costs in the start-up?, How much does a start-up cost? and How do start-ups effect the short-term scheduling strategies of power producers in Sweden? The results show that a fair estimate of the start-up cost is about $3/MW nominal output. For the thermal plants a standard model with polynomial operation cost, start-up costs and ramp-rate constraints has been used. The model also includes the possibilities of purchasing and selling power to forecasted prices.   The planning problem is formulated as a mathematical programming problem. The solution technique uses Lagrange relaxation to decompose the problem into subproblems. There will be one subproblem for each hydro and thermal plant. In order to find good feasible solutions a heuristic technique to change the integer variables in the hydro system has been developed. The Lagrange multipliers are updated with the subgradient method.   The models are tested in three different load situations; a winter day (heavy load), an autumn day (medium load) and a summer day (light load). The result shows that the method gives near optimal schedules in reasonable computation time in cases with a normal part of the thermal units committed. The assumed start-up cost results in that hydro units almost never are started or stopped for one hour only. / <p>QC 20161206</p>

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd W. Easton / The possibility of becoming a millionaire attracts over 200,000 daily fantasy sports (DFS) contest entries each Sunday of the NFL season. Millions of people play fantasy sports and the companies sponsoring daily fantasy sports are worth billions of dollars. This thesis develops optimization models for daily fantasy sports with an emphasis on tiered contests. A tiered contest has many different payout values, including the highly sought after million-dollar prize. The primary contribution of this thesis is the first model to optimize the expected payout of a tiered DFS contest. The stochastic integer program, MMIP, takes into account the possibility that selected athletes will earn a distribution of fantasy points, rather than a single predetermined value. The players are assumed to have a normal distribution and thus the team’s fantasy points is a normal distribution. The standard deviation of the team’s performance is approximated through a piecewise linear function, and the probabilities of earning cumulative payouts are calculated. MMIP solves quickly and easily fits the majority of daily fantasy sports contests. Additionally, daily fantasy sports have landed in a tense political climate due to contestants hopes of winning the million-dollar prize. Through two studies that compare the performance of randomly selected fantasy teams with teams chosen by strategy, this thesis conclusively determines that daily fantasy sports are not games of chance and should not be considered gambling. Besides creating the first optimization model for DFS tiered contests, this thesis also provides methods and techniques that can be applied to other stochastic integer programs. It is the author’s hope that this thesis not only opens the door for clever ways of modeling, but also inspires sports fans and teams to think more analytically about player selection.

Approved for public release; distribution is unlimited / The U.S. Army Wartime Ammunition Distribution System (WADS) will experience an unprecedented demand for ammunition under the operational concept of Airland Battle. To meet demand, proper storage facility location and an efficient flow through the distribution network will be required. Using information from Army Field Manuals, maps and simulation data for demand, both a mixed integer program (MIP) and a sequential, optimization-based heuristic are developed to model the WADS. The Generalized Algebraic Modelling System is used to implement both models. The sequential heuristic locates ammunition facilities with a binary integer program and then directs ammunition through those facilities utilizing a network flow model with side constraints. The MIP integrates location and flow decisions in the same model. For a general scenario, the sequential heuristic locates a 21 node, 30 arc network with ammunition flows over 30 time periods in 22 CPU seconds on an IBM 3033AP. For the same scenario the MIP obtains a solution for only a 3 time period problem in 87 CPU seconds. Keywords: Ammunition, Integer programming, Heuristic, Networks / http://archive.org/details/gamsbasedmodelof00cain / Captain, United States Army

A dissertation submitted to the Faculty of Science School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg. April 27, 2013. / In a number of situations the derivative of the objective function of an optimization problem is not available. This thesis presents a novel algorithm for solving mixed integer programs when this is the case. The algorithm is the first developed for problems of this type which uses a trust region methodology. Three implementations of the algorithm are developed and deterministic proofs of convergence to local minima are provided for two of the implementations. In the development of the algorithm several other contributions are made. The derivative free algorithm requires the solution of several mixed integer quadratic programming subproblems and novel methods for solving nonconvex instances of these problems are developed in this thesis. Additionally, it is shown that the current definitions of local minima for mixed integer programs are deficient and a rigorous approach to developing possible definitions is proposed. Using this approach we propose a new definition which improves on those currently used in the literature. Other components of this thesis are an overview of derivative based mixed integer non-linear programming, extensive reviews of mixed integer quadratic programming and deterministic derivative free optimization and extensive computational results illustrating the effectiveness of the contributions mentioned in the previous paragraphs.

Large integers are used in several key areas such as RSA (Rivest-Shamir-Adleman) public-key cryptographic system and elliptic curve public-key cryptographic system. To achieve higher levels of security requires larger key size and this becomes a limiting factor in prime finite field GF(p) arithmetic using large integers because operations on large integers suffer from the long carry propagation problem. Large integer representation has direct impact on the efficiency of the calculations and the hardware and software implementations. Attempts to use different representations such as residue number systems suffer from their own problems. In this dissertation, we propose a novel and efficient attribute-based large integer representation scheme capable of efficiently representing the large integers that are commonly used in cryptography such as the five NIST primes and the Pierpont primes used in supersingular isogeny Diffie-Hellman (SIDH) used in post-quantum cryptography. Moreover, we propose algorithms for this new representation to perform arithmetic operations such as conversions from and to binary representation, two’s complement, left-shift, numbers comparison, addition/subtraction, modular addition/subtraction, modular reduction, multiplication, and modular multiplication. Extensive numerical simulations and software implementations are done to verify the performance of the new number representation. Results show that the attribute-based large integer arithmetic operations are done faster in our proposed representation when compared with binary and residue number representations. This makes the proposed representation suitable for cryptographic applications on embedded systems and IoT devices with limited resources for better security level. / Graduate / 2020-07-04

Wang, Chongyu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 74-78). / Abstract also in Chinese. / Abstract --- p.1 / Abstract in Chinese --- p.3 / Acknowledgement --- p.4 / Contents --- p.5 / List of Tables --- p.7 / List of Figures --- p.8 / Chapter 1. --- Introduction --- p.9 / Chapter 2. --- Conventional Dynamic Programming --- p.15 / Chapter 2.1. --- Principle of optimality and decomposition --- p.15 / Chapter 2.2. --- Backward dynamic programming --- p.17 / Chapter 2.3. --- Forward dynamic programming --- p.20 / Chapter 2.4. --- Curse of dimensionality --- p.23 / Chapter 3. --- Surrogate Constraint Formulation --- p.26 / Chapter 3.1. --- Surrogate constraint formulation --- p.26 / Chapter 3.2. --- Singly constrained dynamic programming --- p.28 / Chapter 3.3. --- Surrogate dual search --- p.29 / Chapter 4. --- Distance Confined Path Algorithm --- p.34 / Chapter 4.1. --- Yen´ةs algorithm for the kth shortest path problem --- p.35 / Chapter 4.2. --- Application of Yen´ةs method to integer programming --- p.36 / Chapter 4.3. --- Distance confined path problem --- p.42 / Chapter 4.4. --- Application of distance confined path formulation to integer programming --- p.50 / Chapter 5. --- Convergent Surrogate Dual Search --- p.59 / Chapter 5.1. --- Algorithm for convergent surrogate dual search --- p.62 / Chapter 5.2. --- "Solution schemes for (Pμ{αk,αβ)) and f(x) = αk" --- p.63 / Chapter 5.3. --- Computational Results and Analysis --- p.68 / Chapter 6. --- Conclusions --- p.72 / Bibliography --- p.74

Generating employee rosters on a 24 hour, 7 day per week basis taking into account fluctuating demand for employees, employee skills, working conditions, training and employee preferences, while ensuring efficiency and equity between the employees is a very difficult task due to the very large number of possible rostering combinations available. The research done in this thesis sets to solve this exact problem for CS Energy's Swanbank Power Station located in Queensland, Australia. / thesis (PhDMathematics)--University of South Australia, 2002.

The acquisition and elimination of products and the resources needed to create them constitutes an important part of the business decision-making process. Activity-based costing (ABC) supports this process by providing a tool for evaluating the relative profitability of various products. It accomplishes this by allocating costs to products based on the activities, and in turn the resources demanded by those activities, required to produce them. In allocating indirect costs traditionally considered "fixed," such as equipment, administrative overhead, and support staff salaries, ABC treats all costs as variable in the long-run. However, many costs can only vary in discrete steps. For example, one usually cannot purchase a fractional piece of equipment; one chooses either to buy it or not to buy it. Also, in adding support staff, one will typically find that people demand full-time positions, so increments will come in discrete amounts. This stairstep semivariable nature of many costs runs counter to ABC's treatment of all costs as variable. In addition, different products often draw upon the same resources. This creates complex interactions, making it difficult to predict the ultimate consequences of adding or eliminating a particular product. Mixed integer programming (MIP) provides another tool for making these product/resource mix decisions. Unlike ABC, however, it can handle variables that take on integer values, and hence deal appropriately with stairstep semivariable costs. It also ensures that the decision recommended by the model will optimize profitability, given that a solution exists and the underlying assumptions hold true. In doing this, MIP automatically adjusts for all of the complex interactions that exist among the various products and resources. Using a simplified two product/two resource model, one can detail the mathematics behind ABC and MIP, and then link the two approaches through a common variable. This allows one to establish the conditions under which ABC and MIP will yield the same results, and those under which they will differ. Since MW produces an optimal solution, the fact that ABC yields a different result under specific circumstances underscores the danger of relying solely on the product margins generated by an ABC model. / Graduation date: 1998