La construction de modèles qui protègent contre les incertitudes dans les données, telles que la variabilité de l'information et l'imprécision est une des principales préoccupations en optimisation sous incertitude. L'incertitude peut affecter différentes domaines, comme le transport, les télécommunications, la finance, etc. ainsi que les différentes parts d'un problème d'optimisation, comme les coefficients de la fonction objectif et /ou les contraintes. De plus, l'ensemble des données incertaines peut être modélisé de différentes façons, comme sous ensembles compactes et convexes de l´espace réel de dimension n, polytopes, produits Cartésiens des intervalles, ellipsoïdes, etc.<p><p>Une des approches possibles pour résoudre des tels problèmes est de considérer les versions minimax regret, pour lesquelles résoudre un problème sous incertitude revient à trouver une solution qui s'écarte le moins possible de la valeur solution optimale dans tout les cas. <p><p>Dans le cas des incertitudes définies par intervalles, les versions minimax regret de nombreux problèmes combinatoires polynomiaux sont NP-difficiles, d'ou l'importance d'essayer de réduire l'espace des solutions. Dans ce contexte, savoir quand un élément du problème, représenté par une variable, fait toujours ou jamais partie d'une solution optimal pour toute réalisation des données (variables 1-persistentes et 0-persistentes respectivement), constitue une manière de réduire la taille du problème. Un des principaux objectifs de cette thèse est d'étudier ces questions pour quelques problèmes d'optimisation combinatoire sous incertitude.<p><p>Nous étudions les versions minimax regret du problème du choix de p éléments parmi m, de l'arbre couvrant minimum et des deux problèmes de plus court chemin. Pour de tels problèmes, dans le cas des incertitudes définis par intervalles, nous étudions le problème de trouver les variables 1- et 0-persistentes. Nous présentons une procédure de pre-traitement du problème, lequel réduit grandement la taille des formulations des versions de minimax regret.<p><p>Nous nous intéressons aussi à la version minimax regret du problème de programmation linéaire dans le cas où les coefficients de la fonction objectif sont incertains et l'ensemble des données incertaines est polyédral. Dans le cas où l'ensemble des incertitudes est défini par des intervalles, le problème de trouver le regret maximum est NP-difficile. Nous présentons des cas spéciaux ou les problèmes de maximum regret et de minimax regret sont polynomiaux. Dans le cas où l´ensemble des incertitudes est défini par un polytope, nous présentons un algorithme pour trouver une solution exacte au problème de minimax regret et nous discutons les résultats numériques obtenus dans un grand nombre d´instances générées aléatoirement.<p><p>Nous étudions les relations entre le problème de 1-centre continu et la version minimax regret du problème de programmation linéaire dans le cas où les coefficients de la fonction objectif sont évalués à l´aide des intervalles. En particulier, nous décrivons la géométrie de ce dernier problème, nous généralisons quelques résultats en théorie de localisation et nous donnons des conditions sous lesquelles certaines variables peuvet être éliminées du problème. Finalement, nous testons ces conditions dans un nombre d´instances générées aléatoirement et nous donnons les conclusions. / Doctorat en sciences, Orientation recherche opérationnelle / info:eu-repo/semantics/nonPublished

Collateral management has, during the most recent years, been an increasingly important part of a bank’s operation. The bank is facing an allocation problem of how to post collateral to all its counterparties in order to mitigate the credit risk and the number of transactions that requires collateralization is increasing. This master thesis has investigated if it is possible to effectively solve this allocation problem and hence reduce the cost of collateral management by using numerical optimization. Four mathematical linear optimization models of different structure and characteristics have been developed which aims to reflect the complex nature of the problem. These models have been solved with real-life data and it can be concluded that optimization can be used in order to reduce the cost of the collateral allocation and that the problem can be efficiently solved in an acceptable amount of time. The solution showed a reduced cost of almost 15% for the selected business day compared to the cost of current collateral allocations. / Säkerhetshantering har under de senaste åren varit en allt viktigare del av en banks verksamhet. Banken står inför ett allokeringsproblem kring vilka säkerheter som ska ställas till dess motparter för att minska den kreditrisk som uppkommer i samband med lånehandeln samt att antalet transaktioner som kräver ett säkerställande ökar. Denna uppsats har undersökt möjligheten till att effektivt lösa detta allokeringsproblem och därigenom minska kostnaden för säkerhetshantering genom att använda numerisk optimering. Fyra matematiska linjära optimeringsmodeller av olika strukturer och egenskaper har utvecklats som syftar till att återspegla problemets komplexa natur. Dessa modeller har lösts med verklig data och slutsatsen är att optimering kan användas för att minska kostnaden för en banks säkerställande och att problemet effektivt kan lösas inom en acceptabel tidsram. Lösningen visade en reducerad kostnad på nästan 15% för den valda arbetsdagen jämfört med kostnaden för nuvarande allokering av säkerheter.

Thesis: M.S., Massachusetts Institute of Technology, Department of Mathematics, 1979 / Bibliography: leaf 61. / by Paulo Arruda Villela. / M.S. / M.S. Massachusetts Institute of Technology, Department of Mathematics

Energy systems with a high portion of renewable energy from wind and solar power can suffer from fluctuations in production due to weak winds or cloudy weather, which may affect the electricity price. When producing heat and power in a combined heat and power plant, an additional heat storage tank can be used to store the heat surplus which is obtained when the power production is high, and the heat demand is low. To optimize heat and power production economically, short-term planning can be applied. Short-term planning covers the production in the near future of 1-3 days. The optimization in this degree project is based on the district heating production, which means that the heating demand always needs to be fulfilled. The district heating production is based on the weather. Therefore a suitable period for simulation is three days due to the accuracy of the weather forecasts are reasonable. The optimization is performed on the district heat system in Gävle, Sweden. The system comprises several different production units, such as combined heat and power plants, backup plants, and industrial waste heat recovery. Two different models are made, one using linear programming and one using mixed integer non-linear programming. The model stated as a linear programming problem is not as accurate as of the one stated as a mixed integer non-linear programming problem which uses binary variables. Historical input data from Bomhus Energi AB, a company owned together by the local heat and power supplier Gävle Energi AB and the pulp and paper manufacturer BillerudKorsnäs AB, was given to simulate different scenarios. The different scenarios have various average temperatures and in some scenarios are there some issues with the pulp and paper industry affecting the waste heat recovery. In all scenarios is the heat storage tank charged when the demand is low and then discharged when the demand increases to avoid starting some of the more expensive backup plants if possible. The simulation time varies a lot between the two approaches, from a couple of seconds to several hours. Particularly when observing scenarios with a rather high demand since the backup generators use binary variables which take a lot of time to solve.

Orientador: Fernando Antonio Campos Gomide / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-08T23:48:55Z (GMT). No. of bitstreams: 1 Amaral_WanessaMachadodo_M.pdf: 2424347 bytes, checksum: 390879b70ad2ca4dc593f415471fea5c (MD5) Previous issue date: 2007 / Resumo: A teoria de jogos é um ramo da teoria da decisão que modela e trata matematicamente situações de conflito de interesses entre entidades, onde o objetivo principal é escolher a melhor estratégia para cada uma delas, ou seja, aquela que se traduz em equilíbrio. Existem inúmeras áreas em que a teoria de jogos é utilizada. Uma das principais é a microeconomia, onde se aborda questões relativas ao comportamento de empresas e indústrias no mercado competitivo. A teoria de jogos é utilizada para encontrar a estratégia ótima para empresas com objetivos antagônicos, como exige o mercado. No entanto, os dados dos problemas reais nem sempre são precisos. A teoria de conjuntos nebulosos introduz flexibilidade na formulação desses problemas, pois permite a consideração de parâmetros imprecisos nos modelos. Esse trabalho aborda a teoria de jogos nebulosos. Estratégias de equilíbrio são analisadas e métodos computacionais desenvolvidos para a resolução dos modelos. É proposto um método baseado em computação evolutiva para obter soluções de equilíbrio de jogos nebulosos. Além disso propõe-se também um método baseado em a-cortes e no algoritmo de decomposição para a solução dos modelos bilineares associados a jogos nebulosos de soma não zero. Exemplos de aplicações são apresentados para ilustrar o potencial prático da teoria de jogos nebulosos / Abstract: Game theory is a branch of applied mathematics whose aim is to model and study decision making in conflicting situations. In these situations, the main goal is to choose the best strategy for all the players in the game, that is, to find the equilibrium solutions. Game theory can be defined as the study of how self-interested entities interact and make decisions. There are many applications of game theory in different areas. One of the main applications is in microeconomy, where situations of conflict between companies exist and there is a need to find the optimal strategies in that situation. In practice however, model parameters are imprecise. Fuzzy set theory allows modeling flexibility because imprecise data can be treated using fuzzy models. This work concerns Fuzzy Game Theory. Equilibrium strategies are studied and computational methods developed to solve fuzzy game problems. A new method to solve fuzzy games using evolutionary computation is introduced. A method based on a-cuts and on a decomposition algorithm to solve bilinear models also presented to solve fuzzy non zero-sum games. Algorithms were implemented and applications examples are discussed to illustrate the usefulness of fuzzy games in practice / Mestrado / Engenharia de Computação / Mestre em Engenharia Elétrica

Norrmejerier står inför förändringar vad gäller både mjölkkonsumtion och flytt av produktionen från Luleå mejeri till Umeå mejeri inom en snar framtid. Det har gett behov av ett verktyg för att snabbt kunna fatta beslut om systemet kan hantera en ökad mängd volym och antal produktkategorier. För att ta fram ett sådant verktyg skapades en matematisk optimeringsmodell uppbyggd i programvaran Python som gör det möjligt att köra programmet för olika scenarion. Modellen använder optimeringslösaren Pulp för att hitta en lösning på problemet. Den matematiska modellen baseras på Multi Commodity Flow Problem med tidsvariabel i kombination med Flow-shop scheduling och har modifierats efter systemet på Umeå mejeri. Det är en pessimistisk modell baserat på de antaganden som gjorts i rapporten. Programmet baseras på ett dygns produktion och avgör, genom att minimera den totala tiden det tar för flödet genom processen, om det finns kapacitet för en ökad produktion. Systemet i projektet är uppdelat i två subnätverk på grund av tidskomplexiteten och resultaten visar att implementering av en ytterligare produktkategori kan hanteras av båda subnätverken. En ökad volym med 10% av den befintliga kan endast hanteras av den första delen av nätverket. Det betyder att det finns tekniska begränsningar i det andra subnätverket. Genom tillägg av extra noder som kan användas till en viss straffkostnad kunde flaskhalsar identifieras och det visade sig att pastör 2P1 är en uppenbar flaskhals i systemet. Om man ökar produktionen ytterligare kan även silosarna behöva utökas för att hantera flödet. / Norrmejerier is facing changes in terms of both milk consumption and a move of the production from Luleå dairy to Umeå dairy in the near future. This has given rise to the need of a tool that quickly can make descisions about whether the system can handle an increased amount of volume and number of product categories. To produce such a tool a mathematical optimization model was created in Python which makes it possible to run the program for different scenarios. The model uses the optimization solver Pulp. The mathematical model is based on Multi Commodity Flow Problem with time variable combined with Flow-shop scheduling and has been modified according to the system at Umeå dairy. Based on the assumptions made in the report it is a pessimistic model. The program is based on one day's production and determines by minimizing the total time it takes for the flow to pass through the system, to see if there is enough capacity for increased production. The system in the project is divided into two subnetworks due to the time complexity and the results show that implementation of an additional product category can be handled by both subnetworks. An increased volume of 10% of the existing volume can only be handled by the first part of the network. This means that there are technical limitations in the second subnetwork. By adding extra nodes that can be used for a certain penalty cost, bottlenecks could be identified and it turned out that Pasteur 2P1 is an obvious bottleneck in the system. If the production increases further the silos may also need to be expanded to handle the flow in the system.

Agriculture in Cape Verde is severely constrained by a harsh physical environment, and large amounts of foreign aid are required to meet demand for food. Policy-makers believe that the development of irrigated farming offers the most potential for increasing food production, requiring a transition from the dominant irrigated crop, sugar cane, to food crops. Linear programming techniques are used to model a representative farm on the island of Santiago. Water constraints are varied parametrically, showing that revenues are extremely sensitive to frequency of irrigation, and that the dominance of low-profit crops is explained by unreliable and long watering intervals. The shift from cane to more profitable food crops will therefore require water reform aimed at increasing irrigation frequency and improving its reliability. Significant improvements in food production and farm incomes can be achieved even considering present supplies of water and land.

The current methods used by forensic experts to analyse blood spatter neglects the influence of gravitation and drag on the trajectory of the droplet. This research attempts to suggest a more accurate method to determine the trajectory of a blood droplet using multi-target tracking. The multi-target tracking problem can be rewritten as a linear programming problem and solved by means of optimization and numerical methods. A literature survey is presented on relevant articles on blood spatter analysis and multi-target tracking. In contrast to a more advanced approach that assumes a background in probability, mathematical modelling and forensic science, this dissertation aims to give a comprehensive mathematical exposition of particle tracking. The tracking of multi-targets, through multi-target tracking, is investigated. The dynamic programming methods to solve the multi-target tracking are coded in the MATLAB programming language. Results are obtained for different scenarios and option inputs. Research strategies include studying documents, articles, journal entries and books. / MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2014

Linear programming is widely used to solve various complex problems with many variables, subject to multiple constraints. Expert systems are created to provide expertise on complex problems through the application of inference procedures and advanced expert knowledge on facts relevant to the problem. The diet problem is well-known for its contribution to the development of linear programming. Over the years many variations and facets of the diet problem have been solved by means of linear programming techniques and expert systems respectively. In this study the feasibility of combining expert system technology and linear programming techniques to solve a diet problem topical to South Africa, is examined. A computer application is created that incorporates goal programming- and multi-objective linear programming models as the inference engine of an expert system. The program is successfully applied to test cases obtained through knowledge acquisition. The system delivers an eating-plan for an individual that conforms to the nutritional requirements of a healthy diet, includes the personal food preferences of that individual, and includes the food items that result in the lowest total cost. It further allows prioritization of the food preference and least cost factors through the use of weights. Based on the results, recommendations and contributions to the linear programming and expert system fields are presented. / MSc (Computer Science), North-West University, Potchefstroom Campus, 2014

Internal audit departments strive to control risk within an organization. To do this they choose specific audit areas to include in an audit plan. In order to select areas, they usually focus on those areas with the highest risk. Even though high risk areas are considered, there are various other restrictions such as resource constraints (in terms of funds, manpower and hours) that must also be considered. In some cases, management might also have special requirements. Traditionally this area selection process is conducted using manual processes and requires significant decision maker experience. This makes it difficult to take all possibilities into consideration while also catering for all resource constraints and special management requirements. In this study, mathematical techniques used in capital budgeting problems are explored to solve the IT audit area selection problem. A DSS is developed which implements some of these mathematical techniques such as a linear programming model, greedy heuristic, improved greedy heuristic and evolutionary heuristic. The DSS also implements extensions to the standard capital budgeting model to make provision for special management requirements. The performance of the mathematical techniques in the DSS is tested by applying different decision rules to each of the techniques and comparing those results. The DSS, empirical experiments and results are also presented in this research study. Results have shown that in most cases a binary 0-1 model outperformed the other techniques. Internal audit management should therefore consider this model to assist with the construction of an IT internal audit plan. / MSc (Computer Science), North-West University, Potchefstroom Campus, 2015