'A Bayesian Optimisation Algorithm for the Nurse Scheduling Problem'

Abstract- A Bayesian optimization algorithm for the nurse scheduling problem is presented, which involves choosing a suitable scheduling rule from a set for each nurse's assignment. Unlike our previous work that used GAs to implement implicit learning, the learning in the proposed algorithm is explicit, i.e. eventually, we will be able to identify and mix building blocks directly. The Bayesian optimization algorithm is applied to implement such explicit learning by building a Bayesian network of the joint distribution of solutions. The conditional probability of each variable in the network is computed according to an initial set of promising solutions. Subsequently, each new instance for each variable is generated by using the corresponding conditional probabilities, until all variables have been generated, i.e. in our case, a new rule string has been obtained. Another set of rule strings will be generated in this way, some of which will replace previous strings based on fitness selection. If stopping conditions are not met, the conditional probabilities for all nodes in the Bayesian network are updated again using the current set of promising rule strings. Computational results from 52 real data instances demonstrate the success of this approach. It is also suggested that the learning mechanism in the proposed approach might be suitable for other scheduling problems.

'Bayesian Optimisation for Nurse Scheduling'

A Bayesian optimisation algorithm for a nurse scheduling problem is presented, which involves choosing a suitable scheduling rule from a set for each nurse's assignment. When a human scheduler works, he normally builds a schedule systematically following a set of rules. After much practice, the scheduler gradually masters the knowledge of which solution parts go well with others. He can identify good parts and is aware of the solution quality even if the scheduling process is not yet completed, thus having the ability to finish a schedule by using flexible, rather than fixed, rules. In this paper, we design a more human-like scheduling algorithm, by using a Bayesian optimisation algorithm to implement explicit learning from past solutions. A nurse scheduling problem from a UK hospital is used for testing. Unlike our previous work that used Genetic Algorithms to implement implicit learning [1], the learning in the proposed algorithm is explicit, i.e. we identify and mix building blocks directly. The Bayesian optimisation algorithm is applied to implement such explicit learning by building a Bayesian network of the joint distribution of solutions. The conditional probability of each variable in the network is computed according to an initial set of promising solutions. Subsequently, each new instance for each variable is generated by using the corresponding conditional probabilities, until all variables have been generated, i.e. in our case, new rule strings have been obtained. Sets of rule strings are generated in this way, some of which will replace previous strings based on fitness. If stopping conditions are not met, the conditional probabilities for all nodes in the Bayesian network are updated again using the current set of promising rule strings. For clarity, consider the following toy example of scheduling five nurses with two rules (1: random allocation, 2: allocate nurse to low-cost shifts). In the beginning of the search, the probabilities of choosing rule 1 or 2 for each nurse is equal, i.e. 50%. After a few iterations, due to the selection pressure and reinforcement learning, we experience two solution pathways: Because pure low-cost or random allocation produces low quality solutions, either rule 1 is used for the first 2-3 nurses and rule 2 on remainder or vice versa. In essence, Bayesian network learns 'use rule 2 after 2-3x using rule 1' or vice versa. It should be noted that for our and most other scheduling problems, the structure of the network model is known and all variables are fully observed. In this case, the goal of learning is to find the rule values that maximize the likelihood of the training data. Thus, learning can amount to 'counting' in the case of multinomial distributions. For our problem, we use our rules: Random, Cheapest Cost, Best Cover and Balance of Cost and Cover. In more detail, the steps of our Bayesian optimisation algorithm for nurse scheduling are: 1. Set t = 0, and generate an initial population P(0) at random; 2. Use roulette-wheel selection to choose a set of promising rule strings S(t) from P(t); 3. Compute conditional probabilities of each node according to this set of promising solutions; 4. Assign each nurse using roulette-wheel selection based on the rules' conditional probabilities. A set of new rule strings O(t) will be generated in this way; 5. Create a new population P(t+1) by replacing some rule strings from P(t) with O(t), and set t = t+1; 6. If the termination conditions are not met (we use 2000 generations), go to step 2. Computational results from 52 real data instances demonstrate the success of this approach. They also suggest that the learning mechanism in the proposed approach might be suitable for other scheduling problems. Another direction for further research is to see if there is a good constructing sequence for individual data instances, given a fixed nurse scheduling order. If so, the good patterns could be recognized and then extracted as new domain knowledge. Thus, by using this extracted knowledge, we can assign specific rules to the corresponding nurses beforehand, and only schedule the remaining nurses with all available rules, making it possible to reduce the solution space. Acknowledgements The work was funded by the UK Government's major funding agency, Engineering and Physical Sciences Research Council (EPSRC), under grand GR/R92899/01. References [1] Aickelin U, "An Indirect Genetic Algorithm for Set Covering Problems", Journal of the Operational Research Society, 53(10): 1118-1126,

Scheduling of track inspection and maintenance activities in railroad networks

The U.S. railroad companies spend billions of dollars every year on railroad track maintenance in order to ensure safety and operational efficiency of their railroad networks. Besides maintenance costs, other costs such as train accident costs, train and shipment delay costs and rolling stock maintenance costs are also closely related to track maintenance activities. Optimizing the track maintenance process on the extensive railroad networks is a very complex problem with major cost implications. Currently, the decision making process for track maintenance planning is largely manual and primarily relies on the knowledge and judgment of experts. There is considerable potential to improve the process by using operations research techniques to develop solutions to the optimization problems on track maintenance. In this dissertation study, we propose a range of mathematical models and solution algorithms for three network-level scheduling problems on track maintenance: track inspection scheduling problem (TISP), production team scheduling problem (PTSP) and job-to-project clustering problem (JTPCP). TISP involves a set of inspection teams which travel over the railroad network to identify track defects. It is a large-scale routing and scheduling problem where thousands of tasks are to be scheduled subject to many difficult side constraints such as periodicity constraints and discrete working time constraints. A vehicle routing problem formulation was proposed for TISP, and a customized heuristic algorithm was developed to solve the model. The algorithm iteratively applies a constructive heuristic and a local search algorithm in an incremental scheduling horizon framework. The proposed model and algorithm have been adopted by a Class I railroad in its decision making process. Real-world case studies show the proposed approach outperforms the manual approach in short-term scheduling and can be used to conduct long-term what-if analyses to yield managerial insights. PTSP schedules capital track maintenance projects, which are the largest track maintenance activities and account for the majority of railroad capital spending. A time-space network model was proposed to formulate PTSP. More than ten types of side constraints were considered in the model, including very complex constraints such as mutual exclusion constraints and consecution constraints. A multiple neighborhood search algorithm, including a decomposition and restriction search and a block-interchange search, was developed to solve the model. Various performance enhancement techniques, such as data reduction, augmented cost function and subproblem prioritization, were developed to improve the algorithm. The proposed approach has been adopted by a Class I railroad for two years. Our numerical results show the model solutions are able to satisfy all hard constraints and most soft constraints. Compared with the existing manual procedure, the proposed approach is able to bring significant cost savings and operational efficiency improvement. JTPCP is an intermediate problem between TISP and PTSP. It focuses on clustering thousands of capital track maintenance jobs (based on the defects identified in track inspection) into projects so that the projects can be scheduled in PTSP. A vehicle routing problem based model and a multiple-step heuristic algorithm were developed to solve this problem. Various side constraints such as mutual exclusion constraints and rounding constraints were considered. The proposed approach has been applied in practice and has shown good performance in both solution quality and efficiency.

Data-Aware Scheduling in Datacenters

Datacenters have emerged as the dominant form of computing infrastructure over the last two decades. The tremendous increase in the requirements of data analysis has led to a proportional increase in power consumption and datacenters are now one of the fastest growing electricity consumers in the United States. Another rising concern is the loss of throughput due to network congestion. Scheduling models that do not explicitly account for data placement may lead to a transfer of large amounts of data over the network causing unacceptable delays. In this dissertation, we study different scheduling models that are inspired by the dual objectives of minimizing energy costs and network congestion in a datacenter. As datacenters are equipped to handle peak workloads, the average server utilization in most datacenters is very low. As a result, one can achieve huge energy savings by selectively shutting down machines when demand is low. In this dissertation, we introduce the network-aware machine activation problem to find a schedule that simultaneously minimizes the number of machines necessary and the congestion incurred in the network. Our model significantly generalizes well-studied combinatorial optimization problems such as hard-capacitated hypergraph covering and is thus strongly NP-hard. As a result, we focus on finding good approximation algorithms. Data-parallel computation frameworks such as MapReduce have popularized the design of applications that require a large amount of communication between different machines. Efficient scheduling of these communication demands is essential to guarantee efficient execution of the different applications. In the second part of the thesis, we study the approximability of the co-flow scheduling problem that has been recently introduced to capture these application-level demands. Finally, we also study the question, "In what order should one process jobs?'' Often, precedence constraints specify a partial order over the set of jobs and the objective is to find suitable schedules that satisfy the partial order. However, in the presence of hard deadline constraints, it may be impossible to find a schedule that satisfies all precedence constraints. In this thesis we formalize different variants of job scheduling with soft precedence constraints and conduct the first systematic study of these problems.

Joint Optimization for Social Content Delivery in Wireless Networks

Over the last decade, success of social networks has significantly reshaped how people consume information. Recommendation of contents based on user profiles is well-received. However, as users become dominantly mobile, little is done to consider the impacts of the wireless environment, especially the capacity constraints and changing channel. In this dissertation, we investigate a centralized wireless content delivery system, aiming to optimize overall user experience given the capacity constraints of the wireless networks, by deciding what contents to deliver, when and how. We propose a scheduling framework that incorporates content-based reward and deliverability. Our approach utilizes the broadcast nature of wireless communication and social nature of content, by multicasting and precaching. Results indicate this novel joint optimization approach outperforms existing layered systems that separate recommendation and delivery, especially when the wireless network is operating at maximum capacity. Utilizing limited number of transmission modes, we significantly reduce the complexity of the optimization. We also introduce the design of a hybrid system to handle transmissions for both system recommended contents ('push') and active user requests ('pull'). Further, we extend the joint optimization framework to the wireless infrastructure with multiple base stations. The problem becomes much harder in that there are many more system configurations, including but not limited to power allocation and how resources are shared among the base stations ('out-of-band' in which base stations transmit with dedicated spectrum resources, thus no interference; and 'in-band' in which they share the spectrum and need to mitigate interference). We propose a scalable two-phase scheduling framework: 1) each base station obtains delivery decisions and resource allocation individually; 2) the system consolidates the decisions and allocations, reducing redundant transmissions. Additionally, if the social network applications could provide the predictions of how the social contents disseminate, the wireless networks could schedule the transmissions accordingly and significantly improve the dissemination performance by reducing the delivery delay. We propose a novel method utilizing: 1) hybrid systems to handle active disseminating requests; and 2) predictions of dissemination dynamics from the social network applications. This method could mitigate the performance degradation for content dissemination due to wireless delivery delay. Results indicate that our proposed system design is both efficient and easy to implement.

'Bayesian Optimisation for Nurse Scheduling'

A Bayesian optimisation algorithm for a nurse scheduling problem is presented, which involves choosing a suitable scheduling rule from a set for each nurse's assignment. When a human scheduler works, he normally builds a schedule systematically following a set of rules. After much practice, the scheduler gradually masters the knowledge of which solution parts go well with others. He can identify good parts and is aware of the solution quality even if the scheduling process is not yet completed, thus having the ability to finish a schedule by using flexible, rather than fixed, rules. In this paper, we design a more human-like scheduling algorithm, by using a Bayesian optimisation algorithm to implement explicit learning from past solutions. A nurse scheduling problem from a UK hospital is used for testing. Unlike our previous work that used Genetic Algorithms to implement implicit learning [1], the learning in the proposed algorithm is explicit, i.e. we identify and mix building blocks directly. The Bayesian optimisation algorithm is applied to implement such explicit learning by building a Bayesian network of the joint distribution of solutions. The conditional probability of each variable in the network is computed according to an initial set of promising solutions. Subsequently, each new instance for each variable is generated by using the corresponding conditional probabilities, until all variables have been generated, i.e. in our case, new rule strings have been obtained. Sets of rule strings are generated in this way, some of which will replace previous strings based on fitness. If stopping conditions are not met, the conditional probabilities for all nodes in the Bayesian network are updated again using the current set of promising rule strings. For clarity, consider the following toy example of scheduling five nurses with two rules (1: random allocation, 2: allocate nurse to low-cost shifts). In the beginning of the search, the probabilities of choosing rule 1 or 2 for each nurse is equal, i.e. 50%. After a few iterations, due to the selection pressure and reinforcement learning, we experience two solution pathways: Because pure low-cost or random allocation produces low quality solutions, either rule 1 is used for the first 2-3 nurses and rule 2 on remainder or vice versa. In essence, Bayesian network learns 'use rule 2 after 2-3x using rule 1' or vice versa. It should be noted that for our and most other scheduling problems, the structure of the network model is known and all variables are fully observed. In this case, the goal of learning is to find the rule values that maximize the likelihood of the training data. Thus, learning can amount to 'counting' in the case of multinomial distributions. For our problem, we use our rules: Random, Cheapest Cost, Best Cover and Balance of Cost and Cover. In more detail, the steps of our Bayesian optimisation algorithm for nurse scheduling are: 1. Set t = 0, and generate an initial population P(0) at random; 2. Use roulette-wheel selection to choose a set of promising rule strings S(t) from P(t); 3. Compute conditional probabilities of each node according to this set of promising solutions; 4. Assign each nurse using roulette-wheel selection based on the rules' conditional probabilities. A set of new rule strings O(t) will be generated in this way; 5. Create a new population P(t+1) by replacing some rule strings from P(t) with O(t), and set t = t+1; 6. If the termination conditions are not met (we use 2000 generations), go to step 2. Computational results from 52 real data instances demonstrate the success of this approach. They also suggest that the learning mechanism in the proposed approach might be suitable for other scheduling problems. Another direction for further research is to see if there is a good constructing sequence for individual data instances, given a fixed nurse scheduling order. If so, the good patterns could be recognized and then extracted as new domain knowledge. Thus, by using this extracted knowledge, we can assign specific rules to the corresponding nurses beforehand, and only schedule the remaining nurses with all available rules, making it possible to reduce the solution space. Acknowledgements The work was funded by the UK Government's major funding agency, Engineering and Physical Sciences Research Council (EPSRC), under grand GR/R92899/01. References [1] Aickelin U, "An Indirect Genetic Algorithm for Set Covering Problems", Journal of the Operational Research Society, 53(10): 1118-1126,

'A Bayesian Optimisation Algorithm for the Nurse Scheduling Problem'

Abstract- A Bayesian optimization algorithm for the nurse scheduling problem is presented, which involves choosing a suitable scheduling rule from a set for each nurse's assignment. Unlike our previous work that used GAs to implement implicit learning, the learning in the proposed algorithm is explicit, i.e. eventually, we will be able to identify and mix building blocks directly. The Bayesian optimization algorithm is applied to implement such explicit learning by building a Bayesian network of the joint distribution of solutions. The conditional probability of each variable in the network is computed according to an initial set of promising solutions. Subsequently, each new instance for each variable is generated by using the corresponding conditional probabilities, until all variables have been generated, i.e. in our case, a new rule string has been obtained. Another set of rule strings will be generated in this way, some of which will replace previous strings based on fitness selection. If stopping conditions are not met, the conditional probabilities for all nodes in the Bayesian network are updated again using the current set of promising rule strings. Computational results from 52 real data instances demonstrate the success of this approach. It is also suggested that the learning mechanism in the proposed approach might be suitable for other scheduling problems.

'Explicit Learning: an Effort towards Human Scheduling Algorithms'

Abstract Scheduling problems are generally NP-hard combinatorial problems, and a lot of research has been done to solve these problems heuristically. However, most of the previous approaches are problem-specific and research into the development of a general scheduling algorithm is still in its infancy. Mimicking the natural evolutionary process of the survival of the fittest, Genetic Algorithms (GAs) have attracted much attention in solving difficult scheduling problems in recent years. Some obstacles exist when using GAs: there is no canonical mechanism to deal with constraints, which are commonly met in most real-world scheduling problems, and small changes to a solution are difficult. To overcome both difficulties, indirect approaches have been presented (in [1] and [2]) for nurse scheduling and driver scheduling, where GAs are used by mapping the solution space, and separate decoding routines then build solutions to the original problem. In our previous indirect GAs, learning is implicit and is restricted to the efficient adjustment of weights for a set of rules that are used to construct schedules. The major limitation of those approaches is that they learn in a non-human way: like most existing construction algorithms, once the best weight combination is found, the rules used in the construction process are fixed at each iteration. However, normally a long sequence of moves is needed to construct a schedule and using fixed rules at each move is thus unreasonable and not coherent with human learning processes. When a human scheduler is working, he normally builds a schedule step by step following a set of rules. After much practice, the scheduler gradually masters the knowledge of which solution parts go well with others. He can identify good parts and is aware of the solution quality even if the scheduling process is not completed yet, thus having the ability to finish a schedule by using flexible, rather than fixed, rules. In this research we intend to design more human-like scheduling algorithms, by using ideas derived from Bayesian Optimization Algorithms (BOA) and Learning Classifier Systems (LCS) to implement explicit learning from past solutions. BOA can be applied to learn to identify good partial solutions and to complete them by building a Bayesian network of the joint distribution of solutions [3]. A Bayesian network is a directed acyclic graph with each node corresponding to one variable, and each variable corresponding to individual rule by which a schedule will be constructed step by step. The conditional probabilities are computed according to an initial set of promising solutions. Subsequently, each new instance for each node is generated by using the corresponding conditional probabilities, until values for all nodes have been generated. Another set of rule strings will be generated in this way, some of which will replace previous strings based on fitness selection. If stopping conditions are not met, the Bayesian network is updated again using the current set of good rule strings. The algorithm thereby tries to explicitly identify and mix promising building blocks. It should be noted that for most scheduling problems the structure of the network model is known and all the variables are fully observed. In this case, the goal of learning is to find the rule values that maximize the likelihood of the training data. Thus learning can amount to 'counting' in the case of multinomial distributions. In the LCS approach, each rule has its strength showing its current usefulness in the system, and this strength is constantly assessed [4]. To implement sophisticated learning based on previous solutions, an improved LCS-based algorithm is designed, which consists of the following three steps. The initialization step is to assign each rule at each stage a constant initial strength. Then rules are selected by using the Roulette Wheel strategy. The next step is to reinforce the strengths of the rules used in the previous solution, keeping the strength of unused rules unchanged. The selection step is to select fitter rules for the next generation. It is envisaged that the LCS part of the algorithm will be used as a hill climber to the BOA algorithm. This is exciting and ambitious research, which might provide the stepping-stone for a new class of scheduling algorithms. Data sets from nurse scheduling and mall problems will be used as test-beds. It is envisaged that once the concept has been proven successful, it will be implemented into general scheduling algorithms. It is also hoped that this research will give some preliminary answers about how to include human-like learning into scheduling algorithms and may therefore be of interest to researchers and practitioners in areas of scheduling and evolutionary computation. References 1. Aickelin, U. and Dowsland, K. (2003) 'Indirect Genetic Algorithm for a Nurse Scheduling Problem', Computer & Operational Research (in print). 2. Li, J. and Kwan, R.S.K. (2003), 'Fuzzy Genetic Algorithm for Driver Scheduling', European Journal of Operational Research 147(2): 334-344. 3. Pelikan, M., Goldberg, D. and Cantu-Paz, E. (1999) 'BOA: The Bayesian Optimization Algorithm', IlliGAL Report No 99003, University of Illinois. 4. Wilson, S. (1994) 'ZCS: A Zeroth-level Classifier System', Evolutionary Computation 2(1), pp 1-18.

BOA for nurse scheduling

Our research has shown that schedules can be built mimicking a human scheduler by using a set of rules that involve domain knowledge. This chapter presents a Bayesian Optimization Algorithm (BOA)for the nurse scheduling problem that chooses such suitable scheduling rules from a set for each nurse’s assignment. Based on the idea of using probabilistic models, the BOA builds a Bayesian network for the set of promising solutions and samples these networks to generate new candidate solutions. Computational results from 52 real data instances demonstrate the success of this approach. It is also suggested that the learning mechanism in the proposed algorithm may be suitable for other scheduling problems.

Train scheduling in high speed railways: considering competitive effects

The railway planning problem is usually studied from two different points of view: macroscopic and microscopic. We propose a macroscopic approach for the high-speed rail scheduling problem where competitive effects are introduced. We study train frequency planning, timetable planning and rolling stock assignment problems and model the problem as a multi-commodity network flow problem considering competitive transport markets. The aim of the presented model is to maximize the total operator profit. We solve the optimization model using realistic probleminstances obtained from the network of the Spanish railwa operator RENFE, including other transport modes in Spain

Coordinated Scheduling of Wind-Thermal Gencos in Day-Ahead Markets

This paper presents a stochastic mixed-integer linear programming approach for solving the self-scheduling problem of a price-taker thermal and wind power producer taking part in a pool-based electricity market. Uncertainty on electricity price and wind power is considered through a set of scenarios. Thermal units are modeled by variable costs, start-up costs and technical operating constraints, such as: ramp up/down limits and minimum up/down time limits. An efficient mixed-integer linear program is presented to develop the offering strategies of the coordinated production of thermal and wind energy generation, aiming to maximize the expected profit. A case study with data from the Iberian Electricity Market is presented and results are discussed to show the effectiveness of the proposed approach.

Methods for integrating machine learning and constrained optimization

In the framework of industrial problems, the application of Constrained Optimization is known to have overall very good modeling capability and performance and stands as one of the most powerful, explored, and exploited tool to address prescriptive tasks. The number of applications is huge, ranging from logistics to transportation, packing, production, telecommunication, scheduling, and much more. The main reason behind this success is to be found in the remarkable effort put in the last decades by the OR community to develop realistic models and devise exact or approximate methods to solve the largest variety of constrained or combinatorial optimization problems, together with the spread of computational power and easily accessible OR software and resources. On the other hand, the technological advancements lead to a data wealth never seen before and increasingly push towards methods able to extract useful knowledge from them; among the data-driven methods, Machine Learning techniques appear to be one of the most promising, thanks to its successes in domains like Image Recognition, Natural Language Processes and playing games, but also the amount of research involved. The purpose of the present research is to study how Machine Learning and Constrained Optimization can be used together to achieve systems able to leverage the strengths of both methods: this would open the way to exploiting decades of research on resolution techniques for COPs and constructing models able to adapt and learn from available data. In the first part of this work, we survey the existing techniques and classify them according to the type, method, or scope of the integration; subsequently, we introduce a novel and general algorithm devised to inject knowledge into learning models through constraints, Moving Target. In the last part of the thesis, two applications stemming from real-world projects and done in collaboration with Optit will be presented.

Adjustable robust optimization with nonlinear recourses

Over the last century, mathematical optimization has become a prominent tool for decision making. Its systematic application in practical fields such as economics, logistics or defense led to the development of algorithmic methods with ever increasing efficiency. Indeed, for a variety of real-world problems, finding an optimal decision among a set of (implicitly or explicitly) predefined alternatives has become conceivable in reasonable time. In the last decades, however, the research community raised more and more attention to the role of uncertainty in the optimization process. In particular, one may question the notion of optimality, and even feasibility, when studying decision problems with unknown or imprecise input parameters. This concern is even more critical in a world becoming more and more complex —by which we intend, interconnected —where each individual variation inside a system inevitably causes other variations in the system itself. In this dissertation, we study a class of optimization problems which suffer from imprecise input data and feature a two-stage decision process, i.e., where decisions are made in a sequential order —called stages —and where unknown parameters are revealed throughout the stages. The applications of such problems are plethora in practical fields such as, e.g., facility location problems with uncertain demands, transportation problems with uncertain costs or scheduling under uncertain processing times. The uncertainty is dealt with a robust optimization (RO) viewpoint (also known as "worst-case perspective") and we present original contributions to the RO literature on both the theoretical and practical side.

User scheduling for low earth orbit multi-user multiple-input-multiple-output non-terrestrial network systems

In rural and isolated areas without cellular coverage, Satellite Communication (SatCom) is the best candidate to complement terrestrial coverage. However, the main challenge for future generations of wireless networks will be to meet the growing demand for new services while dealing with the scarcity of frequency spectrum. As a result, it is critical to investigate more efficient methods of utilizing the limited bandwidth; and resource sharing is likely the only choice. The research community’s focus has recently shifted towards the interference management and exploitation paradigm to meet the increasing data traffic demands. In the Downlink (DL) and Feedspace (FS), LEO satellites with an on-board antenna array can offer service to numerous User Terminals (UTs) (VSAT or Handhelds) on-ground in FFR schemes by using cutting-edge digital beamforming techniques. Considering this setup, the adoption of an effective user scheduling approach is a critical aspect given the unusually high density of User terminals on the ground as compared to the on-board available satellite antennas. In this context, one possibility is that of exploiting clustering algorithms for scheduling in LEO MU-MIMO systems in which several users within the same group are simultaneously served by the satellite via Space Division Multiplexing (SDM), and then these different user groups are served in different time slots via Time Division Multiplexing (TDM). This thesis addresses this problem by defining a user scheduling problem as an optimization problem and discusses several algorithms to solve it. In particular, focusing on the FS and user service link (i.e., DL) of a single MB-LEO satellite operating below 6 GHz, the user scheduling problem in the Frequency Division Duplex (FDD) mode is addressed. The proposed State-of-the-Art scheduling approaches are based on graph theory. The proposed solution offers high performance in terms of per-user capacity, Sum-rate capacity, SINR, and Spectral Efficiency.

Models and algorithms for real-world optimization problems

This thesis deals with efficient solution of optimization problems of practical interest. The first part of the thesis deals with bin packing problems. The bin packing problem (BPP) is one of the oldest and most fundamental combinatorial optimiza- tion problems. The bin packing problem and its generalizations arise often in real-world ap- plications, from manufacturing industry, logistics and transportation of goods, and scheduling. After an introductory chapter, I will present two applications of two of the most natural extensions of the bin packing: Chapter 2 will be dedicated to an application of bin packing in two dimension to a problem of scheduling a set of computational tasks on a computer cluster, while Chapter 3 deals with the generalization of BPP in three dimensions that arise frequently in logistic and transportation, often com- plemented with additional constraints on the placement of items and characteristics of the solution, like, for example, guarantees on the stability of the items, to avoid potential damage to the transported goods, on the distribution of the total weight of the bins, and on compatibility with loading and unloading operations. The second part of the thesis, and in particular Chapter 4 considers the Trans- mission Expansion Problem (TEP), where an electrical transmission grid must be expanded so as to satisfy future energy demand at the minimum cost, while main- taining some guarantees of robustness to potential line failures. These problems are gaining importance in a world where a shift towards renewable energy can impose a significant geographical reallocation of generation capacities, resulting in the ne- cessity of expanding current power transmission grids.