Workforce scheduling is widely used within most industries. A well-outlined and efficient schedule gives cost savings, such as reduced number of overtime hours, increases overall utilization, and facilitates meeting demands. A large and complex schedule, for example, scheduling of a health care workforce, needs to consider many parameters when constructed; it is essential to account for all critical constraints regarding who can dispense a particular medicine, laws restricting the health care system, etcetera. This thesis evaluates two different methods for implementing a workforce scheduling system for one of Linköping’s most well-known restaurants and bars for students, using mixed integer programming and heuristics. Flamman Pub & Disco recruits new employees prior to every semester. Usually, the workforce consists of around 100 employees, and the vast majority of them work either in the bar or in the kitchen. Historically, the scheduling process has been handled manually using Excel. This does, however, take up much time for the operations manager, something considered frowned upon. Therefore, this thesis suggests an automated scheme for future scheduling processes. Because Flamman is a student organization, they do not hold the capital to invest in expensive licensed optimization software. However, literature studies have shown that heuristics such as large neighborhood search can generate sufficient performance, and therefore the investigation of free-of-charge software using a heuristic approach is conducted. The constructed framework uses a mixed integer programming model, which also lays the cornerstone for the two heuristics: a reverse constructive heuristic and a large neighborhood search. The results retrieved from the analysis prove that a heuristic can be a helpful tool for upcoming recruitment periods. There are, however, recommended areas for improvement regarding the current state of the heuristic.

<p>Sustainability is an important consideration in municipalities, and decision-makers are often faced with making trade-offs in decisions on cost, location selection, and allocation. These decisions are often riddled with complexities and uncertainties, and a heuristic approach can become costly and inefficient. Integer programming is a branch of mathematical programming that can be applied to problems such as these and can be a helpful tool in the decision making process.</p> <p>This thesis presents three papers that develop and apply integer programming-based methodologies and their applications to problems faced by many communities: urban noise, school location-allocation and municipal solid waste collection.</p> <p>An interval-integer approach is applied to an urban noise problem to address the uncertainties in noise, variations in acceptable noise levels. The model determines the most optimal combination of common noise-control techniques to reduce noise in communities to acceptable levels based on the type of community.</p> <p>Changing demographics and urban sprawl have resulted in a decrease in school enrolment in parts of many municipalities, resulting in schools being closed and students driving longer distances to get to school. A mixed-integer linear programming approach is applied to select the best school option based on minimizing vehicle distances.</p> <p>Curbside solid waste collection, which is commonly used in North American municipalities, is a costly, labour intensive process. With fuel prices rising, it is becoming more expensive for municipalities to provide these services. This paper proposes a waste dropoff depot system for solid waste collection and provides a GIS-based integer programming model for site selection and bin sizing.</p> / Master of Applied Science (MASc)

<p> Solving the integer least squares problem min ||Hs- x|| 2 , where the unknown vector s is comprised of integers, the coefficient matrix H and given vector x are comprised of real numbers arises in many applications and is equivalent to find the closest lattice point to a given one known as NP-hard. In multiple antenna systems, the received signal represented by vector xis not arbitrary, but an lattice point perturbed by an additive noise vector whose statistical properties are known. It has been shown the Sphere Decoding, in which the lattice points inside a hyper-sphere are generated and the closest lattice point to the received signal is determined, together with Maximum Likelihood (ML) method often yields a near-optimal performance on average (cubic) while the worst case complexity is still exponential. By using lattice basis reduction as pre-processing step in the sub-optimum decoding algorithms, we can show that the lattice reduction aided sphere decoding (LRSD) achieves a better performance than the maximum likelihood sphere decoding (MLSD) in terms of symbol error rate (SER) and average algorithm running time. In the FIR (Finite Impulse Response) MIMO channel, the channel matrix is Toeplitz and thus gives us the leverage to use the fact that all its column vectors all linearly independent and the matrix itself is often well-conditioned. </p> <p> In this thesis, we will develop a lattice reduction added sphere decoding algorithm along with an improved LLL algorithm, and provide the simulations to show that this new algorithm achieves a better performance than the maximum likelihood sphere decoding. </p> <p> In chapter 1, we define our system model and establish the foundations for understanding of mathematical model - namly the integer least squares problem, and thus the choice of the simulation data. In chapter 2, we explain the integer least squares problems and exploit serveral ways for solving the problems, then we introduce the sphere decoding and maximum likelihood at the end. In chapter 3, we explore the famous LLL reduction algorithm named after Lenstra, Lenstra and Lovasz in details and show an example how to break Merkle-Hellman code using the LLL reduction algorithm. Finally, in chapter 4 we give the LLL reduction aided sphere decoding algorithm and the experiment setup as well as the simulation results against the MLSD and conclusions, further research directions. </p> / Thesis / Master of Science (MSc)

This thesis presents frameworks for the optimal strategic production planning of petroleum refineries operating in competition in multiple markets. The game theoretic concept of the Cournot oligopoly is used as the basic competitive model, and the Nash equilibrium as the solution concept for the formulated problems, which are reformulated into potential games. Nonlinear programming potential game frameworks are developed for static and dynamic production planning problems, as well for mixed integer nonlinear expansion planning problems in which refiners have access to potential upgrades increasing their competitiveness. This latter model represents a novel problem in game theory as it contains both integer and continuous variables and thus must satisfy both discrete and continuous mathematical definitions of the Nash equilibrium. The concept of the mixed-integer game is introduced to explore this problem and the theoretical properties of the new class of games, for which conditions are identified defining when a class of two-player games will possess Nash equilibria in pure strategies, and conjectures offered regarding the properties of larger problems and the class as a whole. In all examples, petroleum refinery problems are solved to optimality (equilibrium) to illustrate the competitive utility of the mathematical frameworks. The primary benefit of such frameworks is the incorporation of the influence of market supply and demand on refinery profits, resulting in rational driving forces in the underlying production planning problems. These results are used to justify the development of frameworks for enterprise optimization as a means of decision making in competitive industries. / Thesis / Doctor of Philosophy (PhD) / This thesis presents a mathematical framework in which refinery production planning problems are solved to optimal solutions in competing scenarios. Concepts from game theory are used to formulate these competitive problems into mathematical programs under single objective functions which coordinate the interests of the competing refiners. Several different cases are considered presenting refinery planning problems as static and dynamic programs in which decisions are time independent or dependent, respectively. A theoretical development is also presented in the concept of the mixed integer game, a game theoretic problem containing both continuous and discrete valued variables and which must satisfy both continuous and discrete definitions of Nash equilibrium. This latter development is used to examine refinery problems in which individual refiners have access to numerous unit upgrades which can potentially improve performance. The results are used to justify a game theoretic approach to enterprise optimization.

Hundreds of steel products need to be scheduled on a single or parallel machine in a steel plant every week. A good feasible schedule may save the company millions of dollars compared to a bad one. Single and parallel machine scheduling are also encountered often in many other industries, making it a crucial research topic for both the process system engineering and operations research communities. Single or parallel machine scheduling can be a challenging combinatorial optimization problem when a large number of jobs are to be scheduled. Each job has unique job characteristics, resulting in different setup times/costs depending on the processing sequence. They also have specific release dates to follow and due dates to meet. This work presents both an exact method using mixed-integer quadratic programming, and an approximate method with metaheuristics to solve real-world large-scale single/parallel machine scheduling problems faced in a steel plant. More than 1000 or 350 jobs are to be scheduled within a one-hour time limit in the single or parallel machine problem, respectively. The objective of the single machine scheduling is to minimize a combined total changeover, total earliness, and total tardiness cost, whereas the objective of the parallel machine scheduling is to minimize an objective function comprising the gaps between jobs before a critical time in a schedule, the total changeover cost, and the total tardiness cost. The exact method is developed to benchmark computation time for a small-scale single machine problem, but is not practical for solving the actual large-scale problem. A metaheuristic algorithm centered on variable neighborhood descent is developed to address the large-scale single machine scheduling with a sliding-window decomposition strategy. The algorithm is extended and modified to solve the large-scale parallel machine problem. Statistical tests, including Student's t-test and ANOVA, are conducted to determine efficient solution strategies and good parameters to be used in the metaheuristics. / Thesis / Master of Applied Science (MASc)

For manufacturing companies, the supply chain operations can be very large. Both supplies and delivery to the end customer need to be strategically planned and executed. Inthis thesis, we have looked closer at one of the largest heavy vehicle brands in the world,Scania, and zoomed into a certain part of their supply chain. All parts that are in atruck have their origin, and from this origin, the parts are sent over and over again tothe production facilities of Scania as trucks and buses are being produced. To make surethat the flow of parts to the production units is as efficient as possible, Scania providesits own packaging to the suppliers, and that is what this thesis analyses.We investigate how Scania can make sure that their empty packaging is delivered to thesuppliers in the most cost and CO2 efficient way possible. We begin by describing thecurrent state of the packaging logistics network and what the transport flows look liketoday. From here, we describe the circumstances that are important for the supply chainoperations. Further on we describe theory related to the subject, such as various locationmodels, graphs, networks, and sustainability-related topics among other things. Withhelp from the presented theory and by data preprocessing, we are able to translate theproblem into a mixed integer linear program which tries to minimise the total transportation costs related to the distribution network of packaging. We present our findings anddiscuss the relevance of the results obtained. Finally, we give our recommendations andprovide suggestions for further studies within the area.

In formulating discrete optimization problems, it is not only important to have a correct mathematical model, but to have a well structured model that can be solved effectively. Two important characteristics of a general integer or mixed-integer program are its size (the number of constraints and variables in the problem), and its strength or tightness (a measure of how well it approximates the convex hull of feasible solutions). In designing model formulations, it is critical to ensure a proper balance between compactness of the representation and the tightness of its linear relaxation, in order to enhance its solvability. In this dissertation, we consider these issues pertaining to the modeling of mixed-integer 0-1 programming problems in general, as well as in the context of several specific real-world applications, including a telecommunications network design problem and an airspace management problem. We first consider the Reformulation-Linearization Technique (RLT) of Sherali and Adams and explore the generation of reduced first-level representations for mixed-integer 0-1 programs that tend to retain the strength of the full first-level linear programming relaxation. The motivation for this study is provided by the computational success of the first-level RLT representation (in full or partial form) experienced by several researchers working on various classes of problems. We show that there exists a first-level representation having only about half the RLT constraints that yields the same lower bound value via its relaxation. Accordingly, we attempt to a priori predict the form of this representation and identify many special cases for which this prediction is accurate. However, using various counter-examples, we show that this prediction as well as several variants of it are not accurate in general, even for the case of a single binary variable. Since the full first-level relaxation produces the convex hull representation for the case of a single binary variable, we investigate whether this is the case with respect to the reduced first-level relaxation as well, and show similarly that it holds true only for some special cases. Empirical results on the prediction capability of the reduced, versus the full, first-level representation demonstrate a high level of prediction accuracy on a set of random as well as practical, standard test problems. Next, we focus on a useful modeling concept that is frequently ignored while formulating discrete optimization problems. Very often, there exists a natural symmetry inherent in the problem itself that, if propagated to the model, can hopelessly mire a branch-and-bound solver by burdening it to explore and eliminate such alternative symmetric solutions. We discuss three applications where such a symmetry arises. For each case, we identify the indistinguishable objects in the model which create the problem symmetry, and show how imposing certain decision hierarchies within the model significantly enhances its solvability. These hierarchies render an otherwise virtually intractable formulation computationally viable using commercial software. For the first problem, we consider a problem of minimizing the maximum dosage of noise to which workers are exposed while working on a set of machines. We next examine a problem of minimizing the cost of acquiring and utilizing machines designed to cool large facilities or buildings, subject to minimum operational requirements. For each of these applications, we generate realistic test beds of problems. The decision hierarchies allow all previously intractable problems to be solved relatively quickly, and dramatically decrease the required computational time for all other problems. For the third problem, we investigate a network design problem arising in the context of deploying synchronous optical networks (SONET) using a unidirectional path switched ring architecture, a standard of transmission using optical fiber technology. Given several rings of this type, the problem is to find a placement of nodes to possibly multiple rings, and to determine what portion of demand traffic between node pairs spanned by each ring should be allocated to that ring. The constraints require that the demand traffic between each node pair should be satisfiable given the ring capacities, and that no more than a specified maximum number of nodes should be assigned to each ring. The objective function is to minimize the total number of node-to-ring assignments, and hence, the capital investment in add-drop multiplexer equipments. We formulate the problem as a mixed-integer programming model, and propose several alternative modeling techniques designed to improve the mathematical representation of this problem. We then develop various classes of valid inequalities for the problem along with suitable separation procedures for tightening the representation of the model, and accordingly, prescribe an algorithmic approach that coordinates tailored routines with a commercial solver (CPLEX). We also propose a heuristic procedure which enhances the solvability of the problem and provides bounds within 5-13% of the optimal solution. Promising computational results that exhibit the viability of the overall approach and that lend insights into various modeling and algorithmic constructs are presented. Following this we turn our attention to the modeling and analysis of several issues related to airspace management. Currently, commercial aircraft are routed along certain defined airspace corridors, where safe minimum separation distances between aircraft may be routinely enforced. However, this mode of operation does not fully utilize the available airspace resources, and may prove to be inadequate under future National Airspace (NAS) scenarios involving new concepts such as Free-Flight. This mode of operation is further compounded by the projected significant increase in commercial air traffic. (Free-Flight is a paradigm of aircraft operations which permits the selection of more cost-effective routes for flights rather than simple traversals between designated way-points, from various origins to different destinations.) We begin our study of Air Traffic Management (ATM) by first developing an Airspace Sector Occupancy Model (AOM) that identifies the occupancies of flights within three dimensional (possibly nonconvex) regions of space called sectors. The proposed iterative procedure effectively traces each flight's progress through nonconvex sector modules which comprise the sectors. Next, we develop an Aircraft Encounter Model (AEM), which uses the information obtained from AOM to efficiently characterize the number and nature of blind-conflicts (i.e., conflicts under no avoidance or resolution maneuvers) resulting from a selected mix of flight-plans. Besides identifying the existence of a conflict, AEM also provides useful information on the severity of the conflict, and its geometry, such as the faces across which an intruder enters and exits the protective shell or envelope of another aircraft, the duration of intrusion, its relative heading, and the point of closest approach. For purposes of evaluation and assessment, we also develop an aggregate metric that provides an overall assessment of the conflicts in terms of their individual severity and resolution difficulty. We apply these models to real data provided by the Federal Aviation Administration (FAA) for evaluating several Free-Flight scenarios under wind-optimized and cruise-climb conditions. We digress at this point to consider a more general collision detection problem that frequently arises in the field of robotics. Given a set of bodies with their initial positions and trajectories, we wish to identify the first collision that occurs between any two bodies, or to determine that none exists. For the case of bodies having linear trajectories, we construct a convex hull representation of the integer programming model of Selim and Almohamad, and exhibit the relative effectiveness of solving this problem via the resultant linear program. We also extend this analysis to model a situation in which bodies move along piecewise linear trajectories, possibly rotating at the end of each linear translation. For this case, we again compare an integer programming approach with its linear programming convex hull representation, and exhibit the relative effectiveness of solving a sequence of problems based on applying the latter construct to each time segment. Returning to Air Traffic Management, another future difficulty in airspace resource utilization stems from a projected increase in commercial space traffic, due to the advent of Reusable Launch Vehicle (RLV) technology. Currently, each shuttle launch cordons off a large region of Special Use Airspace (SUA) in which no commercial aircraft are permitted to enter for the specified duration. Of concern to airspace planners is the expense of routinely disrupting air traffic, resulting in circuitous diversions and delays, while enforcing such SUA restrictions. To provide a tool for tactical and planning purposes in such a context within the framework of a coordinated decision making process between the FAA and commercial airlines, we develop an Airspace Planning Model (APM). Given a set of flights for a particular time horizon, along with (possibly several) alternative flight-plans for each flight that are based on delays and diversions due to special-use airspace (SUA) restrictions prompted by launches at spaceports or weather considerations, this model prescribes a set of flight-plans to be implemented. The model formulation seeks to minimize a delay and fuel cost based objective function, subject to the constraints that each flight is assigned one of the designated flight-plans, and that the resulting set of flight-plans satisfies certain specified workload, safety, and equity criteria. These requirements ensure that the workload for air-traffic controllers in each sector is held under a permissible limit, that any potential conflicts which may occur are routinely resolvable, and that the various airlines involved derive equitable levels of benefits from the overall implemented schedule. In order to solve the resulting 0-1 mixed-integer programming problem more effectively using commercial software (CPLEX-MIP), we explore the use of various facetial cutting planes and reformulation techniques designed to more closely approximate the convex hull of feasible solutions to the problem. We also prescribe a heuristic procedure which is demonstrated to provide solutions to the problem that are either optimal or are within 0.01% of optimality. Computational results are reported on several scenarios based on actual flight data obtained from the Federal Aviation Administration (FAA) in order to demonstrate the efficacy of the proposed approach for air traffic management (ATM) purposes. In addition to the evaluation of these various models, we exhibit the usefulness of this airspace planning model as a strategic planning tool for the FAA by exploring the sensitivity of the solution provided by the model to changes both in the radius of the SUA formulated around the spaceport, and in the duration of the launch-window during which the SUA is activated. / Ph. D.

As exploration in space increases through the use of larger telescopes, more sophisticated structures, and physical exploration, the use of autonomous robots will become instrumental to build and maintain the infrastructures required for this exploration. These systems must be autonomous to deal with the infeasibility of teleoperation due signal delay and task complexity. The reality of using robots in the real world without direct human input will require the autonomous systems to have the capability of responding to errors that occur in an assembly scenario on their own. As such, a system must be in place to allow for the sequencing and allocation of tasks to the robotic workforce autonomously, giving the ability to re-plan in real world stochastic environments. This work presents four contributions towards a system allowing for the autonomous sequencing and allocation of tasks for in-space assembly problems. The first contribution is the development of the Stochastic Assembly Problem Definition (SAPD) to articulate all of the features in an assembly problem that are applicable to the task sequencing and allocation. The second contribution is the formulation of a mixed integer program to solve for assembly schedules that are optimal or a quantifiable measurement from optimal. This contribution is expanded through the development of a genetic algorithm formulation to utilize the stochastic information present in the assembly problem. This formulation extends the state-of-the-art techniques in genetic algorithms to allow for the inclusion of new constraints required for the in-space assembly domain. The third contribution addresses how to estimate a robot's ability to complete a task if the robot must be assigned to a task it was previously not expected to work on. This is accomplished through the development of four metrics and analyzed through the use of screw theory kinematics. The final contribution focuses on a set of metrics to guide the selection of a good scheduling method for different assembly situations. The experiments in this work demonstrate how the developed theory can be utilized and shows the scheduling systems producing the best or close to the best schedules for assemblies. It also shows how the metrics used to quantify and estimate robot ability are applied. The theory developed in this work provides another step towards autonomous systems that are capable of assembling structures in-space without the need for human input. / Doctor of Philosophy / As space exploration continues to progress, autonomous robots are needed to allow for the necessary structures to be built in-space, on Mars, and on the Lunar surface. Since it is not possible to plan for every possible thing that could go wrong or break, the robots must be able to figure out how to build and repair structures without human input. The work presented here develops a framework that allows this in-space assembly problem to be framed in a way the robots can process. It then provides a method for generating assembly schedules that describe very good, if not the best way to complete the assembly quickly while still taking into account randomness that may be present. Additionally, this work develops a way to quantify and estimate how good robots will be at a task they have not attempted before. Finally, a set of considerations are proposed to aid in determining what scheduling method will work best for different assembly scenarios. The experiments in this work demonstrate how the developed theory can be used and shows the scheduling systems producing the best or close to the best schedules for assemblies. It also shows how the methods used to define robot ability are applied. The work developed here provides another step towards autonomous systems that are capable of assembling structures in-space without the need for human input.

Massive infrastructure networks such as electric power, natural gas, or water systems play a pivotal role in everyday human lives. Development and operation of these networks is extremely capital-intensive. Moreover, security and reliability of these networks is critical. This work identifies and addresses a diverse class of computationally challenging and time-critical problems pertaining to these networks. This dissertation extends the state of the art on three fronts. First, general proofs of uniqueness for network flow problems are presented, thus addressing open problems. Efficient network flow solvers based on energy function minimizations, convex relaxations, and mixed-integer programming are proposed with performance guarantees. Second, a novel approach is developed for sample-efficient training of deep neural networks (DNN) aimed at solving optimal network dispatch problems. The novel feature here is that the DNNs are trained to match not only the minimizers, but also their sensitivities with respect to the optimization problem parameters. Third, control mechanisms are designed that ensure resilient and stable network operation. These novel solutions are bolstered by mathematical guarantees and extensive simulations on benchmark power, water, and natural gas networks. / Doctor of Philosophy / Massive infrastructure networks play a pivotal role in everyday human lives. A minor service disruption occurring locally in electric power, natural gas, or water networks is considered a significant loss. Uncertain demands, equipment failures, regulatory stipulations, and most importantly complicated physical laws render managing these networks an arduous task. Oftentimes, the first principle mathematical models for these networks are well known. Nevertheless, the computations needed in real-time to make spontaneous decisions frequently surpass the available resources. Explicitly identifying such problems, this dissertation extends the state of the art on three fronts: First, efficient models enabling the operators to tractably solve some routinely encountered problems are developed using fundamental and diverse mathematical tools; Second, quickly trainable machine learning based solutions are developed that enable spontaneous decision making while learning offline from sophisticated mathematical programs; and Third, control mechanisms are designed that ensure a safe and autonomous network operation without human intervention. These novel solutions are bolstered by mathematical guarantees and extensive simulations on benchmark power, water, and natural gas networks.

The effective maintenance scheduling of power system generators is very important to a power utility for the economical and reliable operation of a power system. Many mathematical methods have been implemented for generator maintenance scheduling (GMS). However, these methods have many limitations and require many approximations. Here a Genetic Algorithm is proposed for GMS problems in order to overcome some of the limitations of the conventional methods. This paper formulates a general GMS problem using a reliability criterion as an integer programming problem, and demonstrates the use of GAs with three different problem encodings: binary, binary for integer and integer. The GA performances for each of these representations are analysed and compared for a test problem based on a practical power system scenario. The effects of different GA parameters are also studied. The results show that the integer GA is a very effective method for GMS problems.