This thesis compares different mixed integer programming (MIP) formulations for sequencing of tasks in the context of avionics scheduling. Sequencing is a key concern in many discrete optimisation problems, and there are numerous ways of accomplishing sequencing with different MIP formulations. A scheduling tool for avionic systems has previously been developed in a collaboration between Saab and Linköping University. This tool includes a MIP formulation of the scheduling problem where one of the model components has the purpose to sequence tasks. In this thesis, this sequencing component is replaced with other MIP formulations in order to study whether the computational performance of the scheduling tool can be improved. Different scheduling instances and objective functions have been used when performing the tests aiming to evaluate the performances, with the computational times of the entire avionic scheduling model determining the success of the different MIP formulations for sequencing. The results show that the choice of MIP formulation makes a considerable impact on the computational performance and that a significant improvement can be achieved by choosing the most suitable one.

Background: In treatment planning for intensity-modulated proton therapy (IMPT), we aim to deliver the prescribed dose to the target yet minimize the dose to adjacent healthy tissue. Mixed-integer programming (MIP) has been applied in radiation therapy to generate treatment plans. However, MIP has not been used effectively for IMPT treatment planning with dose-volume constraints. In this study, we incorporated dose-volume constraints in an MIP model to generate treatment plans for IMPT. Methods: We created a new MIP model for IMPT with dose volume constraints. Two groups of IMPT treatment plans were generated for each of three patients by using MIP models for a total of six plans: one plan was derived with the Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method while the other plan was derived with our MIP model with dose-volume constraints. We then compared these two plans by dose-volume histogram (DVH) indices to evaluate the performance of the new MIP model with dose-volume constraints. In addition, we developed a model to more efficiently find the best balance between tumor coverage and normal tissue protection. Results: The MIP model with dose-volume constraints generates IMPT treatment plans with comparable target dose coverage, target dose homogeneity, and the maximum dose to organs at risk (OARs) compared to treatment plans from the conventional quadratic programming method without any tedious trial-and-error process. Some notable reduction in the mean doses of OARs is observed. Conclusions: The treatment plans from our MIP model with dose-volume constraints can meetall dose-volume constraints for OARs and targets without any tedious trial-and-error process. This model has the potential to automatically generate IMPT plans with consistent plan quality among different treatment planners and across institutions and better protection for important parallel OARs in an effective way.

This thesis proposes and verifies a number of optimization models for re-designing a master surgery schedule with minimized peak inpatient load at the ward. All models include limitations on Operating Rooms and surgeons availability. Surgeons` preference is included with regards to a consistent weekly schedule over a cycle. The uncertain in patients` length of stay was incorporated using discrete probability distributions unique to each surgeon. Furthermore, robust optimization was utilized to protect against the uncertainty in the number of inpatients a surgeon may send to the ward per block. Different scenarios were developed that explore the impact of varying the availability of operating rooms on each day of the week. The models were solved using Cplex and were verified by an Arena simulation model.

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The goal of this dissertation is to study optimization models that integrate location, production, inventory and transportation decisions for industrial products and apply the knowledge gained to develop supply chains for agricultural products (biomass). We estimate unit cost for the whole biomass-to-biofuels’ supply chain which is the per gallon cost for biofuels up till it reaches the markets. The unit cost estimated is the summation of location, production, inventory holding, and transportation costs. In this dissertation, we focus on building mathematical models for designing and managing the biomass-to-biofuels’ supply chains. The computational complexity of the developed models makes it advisable to use heuristic solution procedures. We develop a Lagrangean decomposition heuristic. In our heuristic, we divide the problem into two sub-problems, sub-problem 1 is a transportation problem and sub-problem 2 is a combination of a capacitated facility location and production planning problem. Subproblem 2 is further divided by commodities. The algorithm is tested for a number of different scenarios. We also develop a decision support system (DSS) for the biomass-to-biofuels’ supply chain. In our DSS, the main problem is divided into four easy-to-solve supply chain problems. These problems were determined based on our knowledge of supply chain and discussions with the experts from the biomass and biofuels’ sector. The DSS is coded using visual basic applications (VBA) for Excel and has a simple user interface which assists the user in running different types of supply chain problems and provides results in form of reports which are easy to understand.

Addressing fairness among users and risk mitigation in the context of resource allocation in transportation systems under uncertainties poses a crucial challenge yet to be satisfactorily resolved. This dissertation attempts to address this challenge, focusing on achieving a balance between system-wide efficiency and individual fairness in stochastic transportation resource allocation problems. To study complicated fair and risk-averse resource allocation problems - from public transit to urban air mobility and multi-stage infrastructure maintenance - we develop three models: DrFRAM, FairUAM, and FCMDP. Each of these models, despite being proven NP-hard even in a simplistic case, inspires us to develop efficient solution algorithms. We derive mixed-integer linear programming (MILP) formulations for these models, leveraging the unique properties of each model and linearizing non-linear terms. Additionally, we strengthen these models with valid inequalities. To efficiently solve these models, we design exact algorithms and approximation algorithms capable of obtaining near-optimal solutions. We numerically validate the effectiveness of our proposed models and demonstrate their capability to be applied to real-world case studies to adeptly address the uncertainties and risks arising from transportation systems. This dissertation provides a foundational platform for future inquiries of risk-averse resource allocation strategies under uncertainties for more efficient, equitable, and resilient decision-making. Our adaptable framework can address a variety of transportation-related challenges and can be extended beyond the transportation domain to tackle resource allocation problems in a broader setting. / Doctor of Philosophy / In transportation systems, decision-makers constantly strive to devise the optimal plan for the most beneficial outcomes when facing future uncertainties. When optimizing overall efficiency, individual fairness has often been overlooked. Besides, the uncertainties in the transportation systems raise serious questions about the adaptability of the allocation plan. In response to these issues, we introduce the concept of fair and risk-averse resource allocation under uncertainties in this dissertation. Our goal is to formulate the optimal allocation plan that is both fair and risk-averse amid uncertainties. To tackle the complexities of fair and risk-averse resource allocation problems, we propose innovative methods and practical algorithms, including creating novel formulations as well as deriving super-fast algorithms. These solution approaches are designed to accommodate the fairness, uncertainties, and risks typically in transportation systems. Beyond theoretical results, we apply our frameworks and algorithms to real-world case studies, thus demonstrating our approaches' adaptability to various transportation systems and ability to achieve various optimization goals. Ultimately, this dissertation aims to contribute to fairer, more efficient, and more robust transportation systems. We believe our research findings can help decision-makers with well-informed choices about resource allocation in transportation systems, which, in turn, lead to the development of more equitable and reliable systems, benefiting all the stakeholders.

In this thesis, we study the role of interdiction in the Vehicle Routing Problem (VRP), which naturally arises in humanitarian logistics and military applications. We assume that in a general network, each arc has a chance to be interdicted. When interdiction happens, the vehicle traveling on this arc is lost or blocked and thus unable to continue the trip. We model the occurrence of interdiction as a given probability and consider the multi-period expected delivery. Our objective is to minimize the total travel cost or to maximize the demand fulfillment, depending on the supply quantity. This problem is called the Vehicle Routing Problem with Interdiction (VRPI). We first prove that the proposed VRPI problems are NP-hard. Then we show some key analytical properties pertaining to the optimal solutions of these problems. Most importantly, we examine Dror and Trudeau's property applied to our problem setting. Finally, we present efficient heuristic algorithms to solve these problems and show the effectiveness through numerical studies. / Thesis / Master of Science (MSc)

The global energy requirement is increasing at a rapid pace and fossil fuels have been one of the major players in meeting this growing energy demand. However, the resources for fossil fuels are finite. Therefore, it is essential to develop renewable energy sources like biofuels to help address growing energy needs. A key aspect in the production of biofuel is the biomass logistics chain that constitutes a complex collection of activities, which must be judiciously executed for a cost-effective operation. In this thesis, we introduce a two-phase optimization-simulation approach to determine tactical biomass logistics-related decisions cost effectively in view of the uncertainties encountered in real-life. These decisions include number of trucks to haul biomass from storage locations to a bio-refinery, the number of unloading equipment sets required at storage locations, and the number of satellite storage locations required to serve as collection points for the biomass secured from the fields. Later, an operational-level decision support tool is introduced to aid the "feedstock manager" at the bio-refinery by recommending which satellite storage facilities to unload, how much biomass to ship, how to allocate existing resources (trucks and unloading equipment sets) during each time period, and how to route unloading equipment sets between storage facilities. Another problem studied is the "Bale Collection Problem" associated with the farmgate operation. It is essentially a capacitated vehicle routing problem with unit demand (CVRP-UD), and its solution defines a cost-effective sequence for collecting bales from the field after harvest. / Master of Science

We present works on theory and applications for Mixed Integer Quadratically Constrained Quadratic Programs (MIQCQP). We introduce new mixed integer programming (MIP)-based relaxation and approximation schemes for general Quadratically Constrained Quadratic Programs (QCQP's), and also study practical applications of QCQP's and Mixed-integer QCQP's (MIQCQP). We first address a challenging tank blending and scheduling problem regarding operations for a chemical plant. We model the problem as a discrete-time nonconvex MIQCP, then approximate this model as a MILP using a discretization-based approach. We combine a rolling horizon approach with the discretization of individual chemical property specifications to deal with long scheduling horizons, time-varying quality specifications, and multiple suppliers with discrete arrival times. Next, we study optimization methods applied to minimizing forces for poses and movements of chained Stewart platforms (SPs). These SPs are parallel mechanisms that are stiffer, and more precise, on average, than their serial counterparts at the cost of a smaller range of motion. The robot will be used in concert with several other types robots to perform complex assembly missions in space. We develop algorithms and optimization models that can efficiently decide on favorable poses and movements that reduce force loads on the robot, hence reducing wear on this machine, and allowing for a larger workspace and a greater overall payload capacity. In the third work, we present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions and formulate this approximation using mixed-integer programming (MIP). Combining this with a diagonal perturbation technique to convert a nonseparable quadratic function into a separable one, we present a mixed-integer convex quadratic relaxation for nonconvex quadratic optimization problems. We study the strength (or sharpness) of our formulation and the tightness of its approximation. We computationally demonstrate that our model outperforms existing MIP relaxations, and on hard instances can compete with state-of-the-art solvers. Finally, we study piecewise linear relaxations for solving quadratically constrained quadratic programs (QCQP's). We introduce new relaxation methods based on univariate reformulations of nonconvex variable products, leveraging the relaxation from the third work to model each univariate quadratic term. We also extend the NMDT approach (Castro, 2015) to leverage discretization for both variables in a bilinear term, squaring the resulting precision for the same number of binary variables. We then present various results related to the relative strength of the various formulations. / Doctor of Philosophy / First, we study a challenging long-horizon supply acquisition problem for a chemical plant. For this problem, constraints with products of variables are required to track raw material composition from supply carriers to storage tanks to the production feed. We apply a mixed-integer nonlinear program (MIP) approximation of the problem combined with a rolling planning scheme to obtain good solutions for industry problems within a reasonable time frame. Next, we study optimization methods applied to a robot designed as a stack of Stewart platforms (SPs), which will be used in concert with several other types robots to perform complex space missions. When chaining these SPs together, we obtain a robot that is generally stiffer more precise than a classic robot arm, enabling their potential use for a variety of purposes. Our methods can efficiently decide on favorable poses and movements for the robot that reduce force loads on the robot, hence reducing wear on this machine, and allowing for a larger usable range of motion and a greater overall payload capacity. Our final two works focus on MIP-based techniques for nonconvex QCQP's. In the first work, we break down the objective into an easy-to-handle term minus some squared terms. We then introduce an elegant new MIP-based approximation to handle these squared terms. We prove that this approximation has strong theoretical guarantees, then demonstrate that it is effective compared to other approximations. In the second, we directly model each variable product term using a MIP relaxation. We introduce two new formulations to do this that build on previous formulations, increasing the accuracy with the same number of integer variables. We then prove a variety of useful properties about the presented formulations, then compare them computationally on two families of problems.

Evacuation planning is an important disaster management tool. A large-scale evacuation of a region by automobile is a difficult task, especially as demand is often greater than supply. This is made more difficult as the imbalance of supply and demand actually reduces supply due to congestion. Currently, most of the emphasis in evacuation planning is on supply management. The purpose of this dissertation is to introduce and study sophisticated demand management tools, specifically, staging and routing of evacuees. These tools can be used to produce evacuation strategies that reduce or eliminate congestion. A strategic planning model is introduced that accounts for evacuation dynamics and the non-linearities in travel times associated with congestion, yet is tractable and can be applied to large-scale networks. Objective functions of potential interest in evacuation planning are introduced and studied in the context of this model. Insights into the use of staging and routing in evacuation management are delineated and solution techniques are developed. Two different strategic approaches are studied in the context of this model. The first strategic approach is to control the evacuation at a disaggregate level, where customized staging and routing plans are produced for each individual or family unit. The second strategic approach is to control the evacuation at a more aggregate level, where evacuation plans are developed for a larger group of evacuees, based on pre-defined geographic areas. In both approaches, shelter requirements and preferences can also be considered. Computational experience using these two strategic approaches, and their respective solution techniques, is provided using a real network pertaining to Virginia Beach, Virginia, in order to demonstrate the efficacy of the proposed methodologies. / Ph. D.