Cette thèse comporte deux parties majeures : la première partie est dédiée à l'étude du problème sparse CARP déterministe où nous avons développé une transformation du sparse CARP en un sparse CVRP. La seconde est consacrée au problème sparse CARP avec coûts sous incertitude. Nous avons donné une formulation mathématique du problème en min-max. Cette modélisation a permis d'identifier le pire scénario pour le problème robuste. Deux approches algorithmiques ont été proposées pour une résolution approchée. / This dissertation consists of two main parts : in the first part, we study the detreministic capacitated arc routing problem over sparse underlying graphs wher we have developed a new transformation techniquevof sparse CARP into sparse CVRP. The second part is consecrated about the sparse CARP with travel costs uncertainty. We have given a mathematical formulation of the probleme in min-max. A worst scenario for the robust problem is then identified, and two algorithmic approaches are proposed to determine a solution of the studied problem.

[pt] Com o aumento das questões energéticas e ambientais, os veículos elétricos (EVs) se tornarão um modo de transporte essencial na distribuição logística. Um cenário vital a ser considerado é a dependência do congestionamento do tráfego nos tempos de viagem dos veículos, como é comum nas áreas urbanas hoje. Esse recurso significa que a velocidade de um EV em cada rota pode ser distinta durante diferentes períodos. Como os EVs possuem autonomia limitada, vários trabalhos na literatura propuseram modelos de consumo de energia em função da velocidade e fatores aerodinâmicos. No entanto, sua aplicação permanece limitada e simplificada devido à sua dependência da velocidade e dos tempos de viagem. No caso da velocidade, os modelos da literatura trabalham sob uma velocidade média durante um determinado arco ou introduzem aproximações com métodos de linearização por partes. Em relação aos tempos de viagem, os atuais algoritmos de roteamento de veículos muitas vezes reformulam a rede viária em um gráfico completo onde cada arco representa o caminho mais rápido entre dois locais. Os resultados obtidos por esses métodos divergem da realidade, principalmente para problemas de roteamento de arco envolvendo serviços nos arcos de uma rede rodoviária. Por essas razões, definimos o Problema de Roteamento de Arco Capacitado Elétrico com tempos de viagem dependentes do tempo e taxa de consumo de energia dependente da velocidade. Ao longo de um horizonte de planejamento, cada arco está associado a uma função de velocidade passo a passo. O objetivo é atender um conjunto de arcos que demandam serviços por meio de uma frota de EVs com carga e capacidade de bateria limitadas, minimizando o tempo total de viagem. Além disso, a taxa de consumo de energia por unidade de tempo percorrido é considerada uma função não linear baseada na velocidade. Propomos um algoritmo de pré-processamento de consumo de energia de forma fechada sem aproximações. Nós o incorporamos em uma metaheurística Iterate Local Search e comparamos o impacto no projeto de rotas com os veículos convencionais. / [en] With energy and environmental issues rising, electric vehicles (EVs) will become an essential mode of transportation in logistics distribution. A vital scenario to consider is the dependence of traffic congestion on vehicle travel times, as it is common in urban areas today. This feature means that the speed of an EV on each route may be distinct during different periods. Because EVs have a limited driving range, various works in the literature have proposed energy consumption models as a function of speed and aerodynamic factors. However, their application remains limited and oversimplified due to their dependence on speed and travel times. In the case of speed, the models in the literature work under an average speed during a given arc or introduce approximations with piece-wise linearization methods. Regarding travel times, current vehicle routing algorithms often reformulate the road network into a complete graph where each arc represents the quickest path between two locations. The results obtained by these methods differ from reality, particularly for Arc Routing Problems involving services on the arcs of a road network. For these reasons, we define the Electric Capacitated Arc Routing Problem with Time-dependent Travel times, and Speed-dependent Energy Consumption Rate (E-TDCARP). Over a planning horizon, each arc is associated with a step-wise speed function. Based on this function, a vehicle s speed can change while traveling on a given arc. The objective is to serve a set of arcs that require services through a fleet of electric vehicles with limited load and battery capacity, minimizing the total travel time. Furthermore, the energy consumption rate per unit of time traveled (ECR) is considered a nonlinear function based on speed. We propose a closed-form energy consumption preprocessing algorithm without approximations. We embed it into an Iterate Local Search metaheuristic (ILS) for E-TDCARP and compare the impact on the design of routes between these alternative vehicles and conventional ones.

This thesis explores the use of a column generation method, a subgradient method, and a logic-based Benders decomposition method on a minimized makespan K-rural postman problem. The K-rural postman problem here describes a search and rescue mission using multiple identical unmanned aerial vehicles (UAVs) to cover an area, represented as a complete graph. Each decomposition method has a separate problem for each UAV. In the subgradient and column generation case, a heuristic is used to find an improved upper bound for the makespan. This upper bound can in turn be used to decrease the feasible regions of the subproblems. Moreover, because the subproblems are slow to solve, a maximum calculation time is used, resulting in a feasible solution and a lower bound for each subproblem. These two modifications to the decomposition methods result in a non-standard behaviour.  Multiple fictional problem instances of different sizes and numbers of UAVs were generated and used for evaluating the methods. A maximal time limit is used in these instances. We conclude that solving the original, non-decomposed, problem for smaller instances with a standard solver is faster and gives better results than the decomposition methods. For larger instances, solving the non-decomposed model led to memory issues on several occasions. However, the suggested subgradient and column generation methods can solve every problem. The logic-based Benders decomposition method performed best on instances with multiple UAVs, but had issues when fewer UAVs are utilized. / Den här masteruppsatsen utforskar användningen av en kolumngenereringsmetod, en subgradientmetod och en logikbaserad Benders dekompositionsmetod på en variant av lantbrevbärarproblemet. Vårat brevbärarprolem beskriver sök- och räddningsuppdrag där $K$ drönare används för att avsöka ett område med målfunktionen att minimera flygtiden för den långsammaste drönaren. Varje dekompositionsmetod använder sig av ett problem för varje drönare. I subgradient- och kolumngenereringsmetoden användes en heuristik för att hitta en bättre övre begränsning till drönarnas flygtid. Den förbättrade övre begränsningen kunde sedan användas för att minska det tillåtna området för de mindre problemen. Eftersom de mindre problem var svårlösta, användes en maximal beräkningstid vilket resulterade i att en tillåten lösning och undre gräns gavs för varje mindre problem. Dessa två modifikationer resulterade i icke typiska beteenden.  Metoderna utvärderades på flera fiktiva testinstanser av olika storlekar där antalet drönare varierar. En tidsbegränsning används på varje probleminstans. Slutsatserna från uppsatsen är de original brevbärare problemet ger bäst lösning och snabbast lösningstid i de mindre instanserna. Vid lösning av större probleminstanser, gav original problemet flerfaldiga gånger minnesproblem. Subgradient- och kolumngenereringsmetoden kunde däremot lösa varje probleminstans inom tidsbegränsningen, vilket gjorde de mer pålitliga. Logikbaserade Benders dekompositionsmetoden presterade bättre i instanser med flera drönare, men stötte på problem i instanser med färre drönare.

The multiple constant multiplication (MCM) operation is a fundamental operation in digital signal processing (DSP) and digital image processing (DIP). Examples of the MCM are in finite impulse response (FIR) and infinite impulse response (IIR) filters, matrix multiplication, and transforms. The aim of this work is minimizing the complexity of the MCM operation using common subexpression elimination (CSE) technique and redundant number representations. The CSE technique searches and eliminates common digit patterns (subexpressions) among MCM coefficients. More common subexpressions can be found by representing the MCM coefficients using redundant number representations. A CSE algorithm is proposed that works on a type of redundant numbers called the zero-dominant set (ZDS). The ZDS is an extension over the representations of minimum number of non-zero digits called minimum Hamming weight (MHW). Using the ZDS improves CSE algorithms' performance as compared with using the MHW representations. The disadvantage of using the ZDS is it increases the possibility of overlapping patterns (digit collisions). In this case, one or more digits are shared between a number of patterns. Eliminating a pattern results in losing other patterns because of eliminating the common digits. A pattern preservation algorithm (PPA) is developed to resolve the overlapping patterns in the representations. A tree and graph encoders are proposed to generate a larger space of number representations. The algorithms generate redundant representations of a value for a given digit set, radix, and wordlength. The tree encoder is modified to search for common subexpressions simultaneously with generating of the representation tree. A complexity measure is proposed to compare between the subexpressions at each node. The algorithm terminates generating the rest of the representation tree when it finds subexpressions with maximum sharing. This reduces the search space while minimizes the hardware complexity. A combinatoric model of the MCM problem is proposed in this work. The model is obtained by enumerating all the possible solutions of the MCM that resemble a graph called the demand graph. Arc routing on this graph gives the solutions of the MCM problem. A similar arc routing is found in the capacitated arc routing such as the winter salting problem. Ant colony optimization (ACO) meta-heuristics is proposed to traverse the demand graph. The ACO is simulated on a PC using Python programming language. This is to verify the model correctness and the work of the ACO. A parallel simulation of the ACO is carried out on a multi-core super computer using C++ boost graph library.