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.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / David H. Ben-Arieh / Quest for continuous improvement among the organizations and issue of missing data for data analysis are never ending. This thesis brings these two topics under one roof, i.e., to evaluate the productivity of organizations with sparse data. This study focuses on Data Envelopment Analysis (DEA) to determine the efficiency of 41 member clinics of Kansas Association of Medically Underserved (KAMU) with missing data. The primary focus of this thesis is to develop new reliable methods to determine the missing values and to execute DEA. DEA is a linear programming methodology to evaluate relative technical efficiency of homogenous Decision Making Units, using multiple inputs and outputs. Effectiveness of DEA depends on the quality and quantity of data being used. DEA outcomes are susceptible to missing data, thus, creating a need to supplement sparse data in a reliable manner. Determining missing values more precisely improves the robustness of DEA methodology. Three methods to determine the missing values are proposed in this thesis based on three different platforms. First method named as Average Ratio Method (ARM) uses average value, of all the ratios between two variables. Second method is based on a modified Fuzzy C-Means Clustering algorithm, which can handle missing data. The issues associated with this clustering algorithm are resolved to improve its effectiveness. Third method is based on interval approach. Missing values are replaced by interval ranges estimated by experts. Crisp efficiency scores are identified in similar lines to how DEA determines efficiency scores using the best set of weights. There exists no unique way to evaluate the effectiveness of these methods. Effectiveness of these methods is tested by choosing a complete dataset and assuming varying levels of data as missing. Best set of recovered missing values, based on the above methods, serves as a source to execute DEA. Results show that the DEA efficiency scores generated with recovered values are close within close proximity to the actual efficiency scores that would be generated with the complete data. As a summary, this thesis provides an effective and practical approach for replacing missing values needed for DEA.

Doctor of Philosophy / Department of Industrial & Manufacturing Systems Engineering / Malgorzata J. Rys / Centerline rumble strips (CLRS) are effective in preventing cross-over crashes and are promoted in the United States (U.S.) as a low-cost safety measure. However, there may be negative issues and/or concerns that question their use under certain road conditions. This dissertation is the result of studying these issues and concerns to provide guidance to policy makers on future installations of CLRS, based on current good practices and on the results of specific investigations of exterior noise, safety effectiveness, economics, and drivers’ behavior, including their interaction with shoulders and shoulder rumble strips (SRS). From a survey conducted, good practices in the U.S. were summarized. From a before-and-after study of CLRS safety effectiveness, results showed that total correctable crashes were reduced by 29.21%. Crashes involving fatalities and injuries were reduced by 34.05%. Cross-over crashes were reduced by 67.19%, and run-off-the-road crashes were reduced by 19.19%. Both Naïve and Empirical Bayes methods were applied and showed statistically similar results. There was no statistical difference between football shaped and rectangular shaped CLRS. From the external noise study performed, it was found that external noise depends on vehicle speed, type of vehicle, and distance. Both football and rectangular CLRS substantially increased the levels of external noise at distances up to 45 m (150 ft). Therefore, before installing CLRS, the distance from houses or businesses should be considered. A distance of 60 m (200 ft) was recommended as the limit of the potential exterior noise problem area. From a study of drivers’ behavior, the analyzed configurations of rumble strips and shoulder width levels affected vehicular lateral position and speed levels, although speed deviations were not practically significant. The study of safety performance function models provided technical and economical recommendations for installation of CLRS. Overall, this study recommends the installation of CLRS on rural, two-lane, undivided rural roads in Kansas. Both patterns, rectangular and football, currently installed in Kansas have provided crash reductions, which have been reflected in economic benefits for society. Shoulder width and traffic volume should be considered as crash predictors for enhancement of the benefits. Guidelines were recommended for future better applications of CLRS.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd Easton / History has shown that epidemics can occur at random and without warning — devastating the populations which they impact. As a preventative measure, modern medicine has helped to reduce the number of diseases that can instigate such an event, nevertheless natural and man-made disease mutations place us continuously at risk of such an outbreak. As a second line of defense, extensive research has been conducted to better understand spread patterns and the efficacy of various containment and mitigation strategies. However, these simulation models have primarily focused on minimizing the impact to groups of people either from an economic or societal perspective and little study has been focused on determining the utility maximizing strategy for an individual. Therefore, this work explores the decisions of individuals to determine emergent behaviors and characteristics which lead to increased probability of survival during an epidemic. This is done by leveraging linear program optimization techniques and the concept of Agent Based Simulation, to more accurately capture the complexity inherent in most real-world systems via the interactions of individual entities. This research builds on 5 years of study focused on rural epidemic simulation, resulting in the development of a 4,000-line computer code simulation package. This adaptable simulation can accurately model the interactions of individuals to discern the impact of any general disease type, and can be implemented on the population of any contiguous counties within Kansas. Furthermore, a computational study performed on the 17 counties of northwestern Kansas provides game theoretical based insights as to what decisions increase the likelihood of survival. For example, statistically significant findings suggest that an individual is four times more likely to become infected if they rush stores for supplies after a government issued warning instead of remaining at home. This work serves as a meaningful step in understanding emergent phenomena during an epidemic which, subsequently, provides novel insight to an individual’s utility maximizing strategy. Understanding the main findings of this research could save your life.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Graph theory is a widely researched topic. A graph contains a set of nodes and a set of edges. The nodes often represent resources such as machines, employees, or plant locations. Each edge represents the relationship between a pair of nodes such as time, distance, or cost. Integer programs are frequently used to solve graphical problems. Unfortunately, IPs are NP-hard unless P = NP, which implies that it requires exponential effort to solve them. Much research has been focused on reducing the amount of time required to solve IPs through the use of valid inequalities or cutting planes. The theoretically strongest cutting planes are facet defining cutting planes. This research focuses on the node packing problem or independent set problem, which is a combinatorial optimization problem. The node packing problem involves coloring the maximum number of nodes such that no two nodes are adjacent. Node packings have been applied to airline traffic and radio frequencies. This thesis introduces a new class of graphical structures called suns. Suns produce previously undiscovered valid inequalities for the node packing polyhedron. Conditions are provided for when these valid inequalities are proven to be facet defining. Sun valid inequalities have the potential to more quickly solve node packing problems and could even be extended to general integer programs through conflict graphs.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Integer Programs (IP) are a class of discrete optimization that have been used commercially to improve various systems. IPs are often used to reach an optimal financial objective with constraints based upon resources, operations and other restrictions. While incredibly beneficial, IPs have been shown to be NP-complete with many IPs remaining unsolvable. Traditionally, Branch and Bound (BB) has been used to solve IPs. BB is an iterative algorithm that enumerates all potential integer solutions for a given IP. BB can guarantee an optimal solution, if it exists, in finite time. However, BB can require an exponential number of nodes to be evaluated before terminating. As a result, the memory of a computer using BB can be exceeded or it can take an excessively long time to find the solution. This thesis introduces a modified BB scheme called the Octanary Branching Algorithm (OBA). OBA introduces eight children in each iteration to more effectively partition the feasible region of the linear relaxation of the IP. OBA also introduces equality constraints in four of the children in order to reduce the dimension of the remaining nodes. OBA can guarantee an optimal solution, if it exists, in finite time. In addition, OBA has been shown to have some theoretical improvements over traditional BB. During computational tests, OBA was able to find the first, second and third integer solution with 64.8%, 27.9% and 29.3% fewer nodes evaluated, respectively, than CPLEX. These integers were 44.9%, 54.7% and 58.2% closer to the optimal solution, respectively, when compared to CPLEX. It is recommended that commercial solvers incorporate OBA in the initialization and random diving phases of BB.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd W. Easton / Integer programs (IPs) are mathematical models that can provide an optimal solution to a variety of different problems. They have been used to reduce costs and optimize organizations. Additionally, IPs are NP-complete resulting in many IPs that cannot be solved. Cutting planes or valid inequalities have been used to decrease the time required to solve IPs. Lifting is a technique that strengthens existing valid inequalities. Lifting inequalities can result in facet defining inequalities, which are the theoretically strongest valid inequalities. Because of these properties, lifting procedures are used in software to reduce the time required to solve an IP. The thesis introduces a new algorithm for exact synchronized simultaneous uplifting over an arbitrary initial inequality for knapsack problems. Synchronized Simultaneous Lifting (SSL) is a pseudopolynomial time algorithm requiring O(nb+n[superscript]3) effort to solve. It exactly uplifts two sets simultaneously into an initial arbitrary valid inequality and creates multiple inequalities of a particular form. This previously undiscovered class of inequalities generated by SSL can be facet defining. A small computational study shows that SSL is quick to execute, requiring on average less than a quarter of a second. Additionally, applying SSL inequalities to a knapsack problem enabled commercial software to solve problems that it could not solve without them.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Jessica L. Heier Stamm / The main goal of public health emergency preparedness efforts is to mitigate the impact of events on the health of the population. However, decision-makers must also remain conscientious of the costs associated with these efforts. Planning is further complicated by uncertainty about the location and volume of demand that will need to be met in an emergency, the speed with which demand must be met, and the potential scarcity of needed items once an emergency occurs. To address these challenges, public health emergency planners often keep inventory stockpiles that are distributed when an event happens. Managing these stockpiles is a difficult task, and inefficient stockpile location and equipment distribution strategies can be costly both in terms of cost and public health impact. This research is motivated by challenges faced by state public health departments in creating stockpile location and equipment distribution strategies. The primary emphasis is on facemasks and respirators used by health workers during an influenza pandemic, but the approach is generalizable to other scenarios. The model proposed here uses a two-stage approach to generate a holistic solution to the problem. The first stage uses a pull distribution strategy to make stockpile location decisions. Additionally, it determines how counties should be assigned to stockpiles to minimize both storage and distribution costs. The second stage adopts a push distribution strategy to determine optimal delivery routes based on the county assignments made in stage one. This stage offers guidance for public health planners who have made location-allocation decisions but who then face a different distribution scenario than what was anticipated in the original planning phase. Recourse methods for managing demand uncertainty are also proposed. A case study of the state of Kansas is conducted using the methods introduced in the thesis. The computational results yield several significant insights into the tradeoffs and costs of various facility location-allocation and vehicle routing decisions: • For the tested range of storage and distribution cost parameters, multiple stockpile locations are preferred over a single location. • In a pull distribution system, storage costs play a greater role in location-allocation decisions than distribution costs. • In the push distribution system, finding an optimal vehicle routing plan is computationally intensive for stockpiles with a large number of assigned counties. • Efficient heuristics perform well to design recourse routing plans when realized demand is greater than expected. • In the event that planners wish to specify routes well in advance, the results of this research suggest adopting a robust routing plan based on higher-than-expected demand levels. This thesis makes three important contributions. The first is an optimization approach that considers multiple distribution strategies. This is especially relevant when stockpiling for an influenza pandemic where stockpiles need to be located significantly before the material is needed, during which time the distribution strategy may change. Second, the case study demonstrates that the proposed methods are applicable to a large-scale problem arising in practice. Finally, this research illustrates for decision-makers the tradeoffs between different stockpile management strategies and between optimal and heuristic methods.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Integer Programming is used to solve numerous optimization problems. This class of mathematical models aims to maximize or minimize a cost function restricted to some constraints and the solution must be integer. One class of widely studied Integer Program (IP) is the Multiple Knapsack Problem (MKP). Unfortunately, both IPs and MKPs are NP-hard, potentially requiring an exponential time to solve these problems. Utilization of cutting planes is one common method to improve the solution time of IPs. A cutting plane is a valid inequality that cuts off a portion of the linear relaxation space. This thesis presents a new class of cutting planes referred to as merged knapsack cover inequalities (MKCI). These valid inequalities combine information from a cover inequality with a knapsack constraint to generate stronger inequalities. Merged knapsack cover inequalities are generated by the Merging Knapsack Cover Algorithm (MKCA), which runs in linear time. These inequalities may be improved by the Exact Improvement Through Dynamic Programming Algorithm (EITDPA) in order to make them stronger inequalities. Theoretical results have demonstrated that this new class of cutting planes may cut off some space of the linear relaxation region. A computational study was performed to determine whether implementation of merged knapsack cover inequalities is computationally effective. Results demonstrated that MKCIs decrease solution time an average of 8% and decrease the number of ticks in CPLEX, a commercial IP solver, approximately 4% when implemented in appropriate instances.

Doctor of Philosophy / Department of Industrial & Manufacturing Systems Engineering / David H. Ben-Arieh / Chih-Hang Wu / Patients who suffer from sepsis or septic shock are of great concern in the healthcare system. Recent data indicate that more than 900,000 severe sepsis or septic shock cases developed in the United States with mortality rates between 20% and 80%. In the United States alone, almost $17 billion is spent each year for the treatment of patients with sepsis. Clinical trials of treatments for sepsis have been extensively studied in the last 30 years, but there is no general agreement of the effectiveness of the proposed treatments for sepsis. Therefore, it is necessary to find accurate and effective tools that can help physicians predict the progression of disease in a patient-specific way, and then provide physicians recommendation on the treatment of sepsis to lower risk for patients dying from sepsis. The goal of this research is to develop a risk assessment tool and a risk management tool for sepsis. In order to achieve this goal, two system dynamic mathematical models (SDMMs) are initially developed to predict dynamic patterns of sepsis progression in innate immunity and adaptive immunity. The two SDMMs are able to identify key indicators and key processes of inflammatory responses to an infection, and a sepsis progression. Second, an integrated-mathematical-multi-agent-based model (IMMABM) is developed to capture the stochastic nature embedded in the development of inflammatory responses to a sepsis. Unlike existing agent-based models, this agent-based model is enhanced by incorporating developed SDMMs and extensive experimental data. With the risk assessment tools, a Markov decision process (MDP) is proposed, as a risk management tool, to apply to clinical decision-makings on sepsis. With extensive computational studies, the major contributions of this research are to firstly develop risk assessment tools to identify the risk of sepsis development during the immune system responding to an infection, and secondly propose a decision-making framework to manage the risk of infected individuals dying from sepsis. The methodology and modeling framework used in this dissertation can be expanded to other disease situations and treatment applications, and have a broad impact to the research area related to computational modeling, biology, medical decision-making, and industrial engineering.