The purpose of this thesis is to efficiently solve real life problems. We study LPs. We study an NLP and an MINLP based on what is known as the generalised pooling problem (GPP), and we study an MIP that we call the cattle mating problem. These problems are often very large or otherwise difficult to solve by direct methods, and are best solved by decomposition methods. During the thesis we introduce algorithms that exploit the structure of the problems to decompose them. We are able to solve row-linked, column-linked and general LPs efficiently by modifying the tableau simplex method, and suggest how this work could be applied to the revised simplex method. We modify an existing sequential linear programming solver that is currently used by Format International to solve GPPs, and show the modified solver takes less time and is at least as likely to find the global minimum as the old solver. We solve multifactory versions of the GPP by augmented Lagrangian decomposition, and show this is more efficient than solving the problems directly. We introduce a decomposition algorithm to solve a MINLP version of the GPP by decomposing it into NLP and ILP subproblems. This is able to solve large problems that could not be solved directly. We introduce an efficient decomposition algorithm to solve the MIP cattle mating problem, which has been adopted for use by the Irish Cattle Breeding Federation. Most of the solve methods we introduce are designed only to find local minima. However, for the multifactory version of the GPP we introduce two methods that give a good chance of finding the global minimum, both of which succeed in finding the global minimum on test problems.

We consider the problem of minimizing costs in the generation unit commitment problem, a cornerstone in electric power system operations, while enforcing an -- reliability criterion. This reliability criterion is a generalization of the well-known - criterion and dictates that at least fraction of the total system demand (for ) must be met following the failure of or fewer system components. We refer to this problem as the contingency-constrained unit commitment problem, or CCUC. We present a mixed-integer programming formulation of the CCUC that accounts for both transmission and generation element failures. We propose novel cutting plane algorithms that avoid the need to explicitly consider an exponential number of contingencies. Computational studies are performed on several IEEE test systems and a simplified model of the Western US interconnection network. These studies demonstrate the effectiveness of our proposed methods relative to current state-of-the-art.

This paper presents a method that uses designed experiments and statistical models to extract information about how solver parameter settings perform for classes of mixed integer programs. The use of experimental design facilitates fitting a model that describes the response surface across all combinations of parameter settings, even those not explicitly tested, allowing identification of both desirable and poor settings. Identifying parameter settings that give the best expected performance for a specific class of instances and a specific solver can be used to more efficiently solve a large set of similar instances, or to ensure solvers are being compared at their best.

Home schooling has steadily been increasing in the past decade. According to a survey in 2007, about 2.5 million children were being home schooled in the US. Typically, parents provide education at the convenience of their home and in some cases an instructor is appointed for the same. The Home School Timetabling problem (HSTP) deals with assigning subjects, timeslots and rooms to every student. In doing so, there are certain hard and specialty constraints that are to be satisfied. Integer programming (IP) has been used in solving the HSTP as it has the advantage of being able to provide information about the relative significance of each constraint with respect to the objective. A prototype in the form of a GUI has been built such that the parent can enter each student’s name, his/her subjects, duration, days and time for each subject, availability times of the parent etc. This data is then fed into the IP model so that it can generate a feasible timetable satisfying all of the constraints. When a solution is found it is formatted to provide the weekly timetable for each student, individually, as well as a complete timetable for all students each day.

Each year the Railway Applications Section (RAS) of the Institution for Operations Research and the Management Sciences (INFORMS) posits a research problem to the world in the form of a competition. For 2012, the contest involved solving the Train Dispatching Problem (TDP) on a realistic 85 edge network for three different sets of input data. This work is an independent attempt to match or improve upon the results of the top three finishers in the contest using mixed integer programming (MIP) techniques while minimizing the use of heuristics. The primary focus is to partition the network in a manner that reduces the number of binary variables in the formulation as much as possible without compromising the ability to satisfy any of the contest requirements. This resulted in the ability to optimally solve this model for RAS Data Set 1 in 29 seconds without any problem-specific heuristics, variable restrictions, or variable fixing. Applying some assumptions about train movements allowed the same Data Set 1 solution to be found in 5.4 seconds. After breaking the larger Data Sets 2 and 3 into smaller sub-problems, solutions for Data Sets 2 and 3 were 28% and 1% better, respectively, than those of the competition winner. The time to obtain solutions for Data Sets 2 and 3 was 90 and 318 seconds, respectively.

Effective aeromedical evacuation of casualties is one of the most important problems in military medical systems because high-priority casualties will not survive without timely medical care. The decision making process for aeromedical evacuation consists of the following components: (1) identifying which aeromedical evacuation asset (see figure 1) to dispatch to the casualty, (2) locating aeromedical evacuation assets strategically in anticipation of incoming demand, and (3) deciding which medical treatment facility to transport the casualty. These decisions are further complicated because prioritization of casualties is based on severity of injury while aeromedical evacuation assets and medical treatment facilities operate with varying capabilities. In this dissertation, discrete optimization models are developed to examine dispatch, delivery, and location logistics for the effective aeromedical evacuation of casualties in military medical systems.

Scheduling refinery operation promises a big cut in logistics cost, maximizes efficiency, organizes allocation of material and resources, and ensures that production meets targets set by planning team. Obtaining accurate and reliable schedules for execution in refinery plants under different scenarios has been a serious challenge. This research was undertaken with the aim to develop robust methodologies and solution procedures to address refinery scheduling problems with uncertainties in process parameters. The research goal was achieved by first developing a methodology for short-term crude oil unloading and transfer, as an extension to a scheduling model reported by Lee et al. (1996). The extended model considers real life technical issues not captured in the original model and has shown to be more reliable through case studies. Uncertainties due to disruptive events and low inventory at the end of scheduling horizon were addressed. With the extended model, crude oil scheduling problem was formulated under receding horizon control framework to address demand uncertainty. This work proposed a strategy called fixed end horizon whose efficiency in terms of performance was investigated and found out to be better in comparison with an existing approach. In the main refinery production area, a novel scheduling model was developed. A large scale refinery problem was used as a case study to test the model with scheduling horizon discretized into a number of time periods of variable length. An equivalent formulation with equal interval lengths was also presented and compared with the variable length formulation. The results obtained clearly show the advantage of using variable timing. A methodology under self-optimizing control (SOC) framework was then developed to address uncertainty in problems involving mixed integer formulation. Through case study and scenarios, the approach has proven to be efficient in dealing with uncertainty in crude oil composition.

Doctor of Philosophy / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Integer Programming (IP) problems are a common type of optimization problem used to solve numerous real world problems. IPs can require exponential computational effort to solve using the branch and bound technique. A popular method to improve solution times is to generate valid inequalities that serve as cutting planes. This dissertation introduces a new category of cutting planes for general IPs called inequality merging. The inequality merging technique combines two or more low dimensional inequalities, yielding valid inequalities of potentially higher dimension. The dissertation describes several theoretical results of merged inequalities. This research applies merging inequalities to a frequently used class of IPs called multiple knapsack (MK) problems. Theoretical results related to merging cover inequalities are presented. These results include: conditions for validity, conditions for facet defining inequalities, merging simultaneously over multiple cover inequalities, sequentially merging several cover inequalities on multiple variables, and algorithms that facilitate the development of merged inequalities. Examples demonstrate each of the theoretical discoveries. A computational study experiments with inequality merging techniques using benchmark MK instances. This computational study provides recommendations for implementing merged inequalities, which results in an average decrease of about 9% in computational time for both small and large MK instances. The research validates the effectiveness of using merged inequalities for MK problems and motivates substantial theoretical and computational extensions as future research.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Integer programming (IP) is a class of mathematical models useful for modeling and optimizing many theoretical and industrial problems. Unfortunately, IPs are NP-complete, and many integer programs cannot currently be solved. Valid inequalities and their respective cuts are commonly used to reduce the effort required to solve IPs. This thesis poses the questions, do valid equality cuts exist and can they be useful for solving IPs? Several theoretical results related to valid equalities are presented in this thesis. It is shown that equality cuts exist if and only if the convex hull is not full dimensional. Furthermore, the addition of an equality cut can arbitrarily reduce the dimension of the linear relaxation. In addition to the theory on equality cuts, the idea of infeasibility conditions are presented. Infeasibility conditions introduce a set of valid inequalities whose intersection is the empty set. infeasibility conditions can be used to rapidly terminate a branch and cut algorithm. Applying the idea of equality cuts to the multi-demand multidimensional knapsack problem resulted in a new class of cutting planes named anticover cover equality (ACE) cuts. A simple algorithm, FACEBT, is presented for finding ACE cuts in a branching tree with complexity O(m n log n). A brief computational study shows that using ACE cuts exist frequently in the MDMKP instances studied. Every instance had at least one equality cut, while one instance had over 500,000. Additionally, computationally challenging instances saw an 11% improvement in computational effort. Therefore, equality cuts are a new topic of research in IP that is beneficial for solving some IP instances.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Throughout an NFL season, 512 games are played in 17 weeks. For a given fan that follows one team, only 16 of those games usually matter, and the rest of the games carry little significance. The goal of this research is to provide substantial reasons for fans to watch other games. This research finds the easiest path to a division championship for each team. This easiest path requires winning the least number of games. Due to NFL’s complicated tiebreaker rules, games not involving the fan’s team can have major implications for that team. The research calls these games critical because if the wrong team wins, then the fan’s team must win additional games to become the division champion. To identify both the easiest path and the critical games, integer programming is used. Given the amount of two-team, three-team, and four-team division tie scenarios that can occur, 31 separate integer programs are solved for each team to identify the easiest path to the division championship. A new algorithm, Shortest Path of Remaining Teams (SPORT) is used to iteratively search through every game of the upcoming week to determine critical games. These integer programs and the SPORT algorithm were used with the data from the previous 2 NFL seasons. Throughout these 2 seasons, it was found that the earliest a team was eliminated from the possibility of winning a division championship was week 12, and occurred in 2012 and 2013. Also, throughout these 2 seasons, there was an average of 65 critical games per season, with more critical games occurring in the 2013-2014 season. Additionally, the 2012 season was used to compare flexed scheduled games with the critical games for those weeks and it was found that the NFL missed three weeks of potentially scheduling a critical game.