The pooling problem is an important optimization problem that is encountered in operation and scheduling of important industrial processes within petroleum refineries. The key objective of pooling is to mix various intermediate products to achieve desired properties and quantities of products. First, intermediate streams from various processing units are mixed and stored in intermediate tanks referred to as pools. The stored streams in pools are subsequently allowed to mix to meet varying market demands. While these pools enhance the operational flexibility of the process, they complicate the decisionmaking process needed for optimization. The problem to find the least costly mixing recipe from intermediate streams to pools and then from pools to sale products is referred to as the pooling problem. The research objective is to contribute an approach to solve this problem. The pooling problem can be formulated as an optimization program whose objective is to minimize cost or maximize profit while determining the optimal allocation of intermediate streams to pools and the blending of pools to final products. Because of the presence of bilinear terms, the resulting formulation is nonconvex which makes it very difficult to attain the global solution. Consequently, there is a need to develop computationally-efficient and easy-to-implement global-optimization techniques to solve the pooling problem. In this work, a new approach is introduced for the global optimization of pooling problems. The approach is based on three concepts: linearization by discretizing nonlinear variables, pre-processing using implicit enumeration of the discretization to form a convex-hull which limits the size of the search space, and application of integer cuts to ensure compatibility between the original problem and the discretized formulation. The continuous quality variables contributing to bilinear terms are first discretized. The discretized problem is a mixed integer linear program (MILP) and can be globally solved in a computationally effective manner using branch and bound method. The merits of the proposed approach are illustrated by solving test case studies from literature and comparison with published results.

The purpose of this thesis is to efficiently solve real life problems. We study LPs. We study an NLP and an MINLP based on what is known as the generalised pooling problem (GPP), and we study an MIP that we call the cattle mating problem. These problems are often very large or otherwise difficult to solve by direct methods, and are best solved by decomposition methods. During the thesis we introduce algorithms that exploit the structure of the problems to decompose them. We are able to solve row-linked, column-linked and general LPs efficiently by modifying the tableau simplex method, and suggest how this work could be applied to the revised simplex method. We modify an existing sequential linear programming solver that is currently used by Format International to solve GPPs, and show the modified solver takes less time and is at least as likely to find the global minimum as the old solver. We solve multifactory versions of the GPP by augmented Lagrangian decomposition, and show this is more efficient than solving the problems directly. We introduce a decomposition algorithm to solve a MINLP version of the GPP by decomposing it into NLP and ILP subproblems. This is able to solve large problems that could not be solved directly. We introduce an efficient decomposition algorithm to solve the MIP cattle mating problem, which has been adopted for use by the Irish Cattle Breeding Federation. Most of the solve methods we introduce are designed only to find local minima. However, for the multifactory version of the GPP we introduce two methods that give a good chance of finding the global minimum, both of which succeed in finding the global minimum on test problems.

Solution methodologies for mixed integer bilinear problems (MIBLP) are studied in this dissertation. This problem class is motivated using the pooling problem, a multicommodity network flow problem that typically arises in chemical engineering applications. Stronger than previously known results are provided to compare the strengths of polyhedral relaxations of the pooling problem. A novel single node flow relaxation, defined by a bilinear equality constraint and flow balance, is proposed for the pooling problem. Linear valid inequalities in the original space of variables are derived using a well-known technique called lifting. Mixed integer linear (MILP) formulations are proposed for generating feasible solutions to the pooling problem. Some of these MILP models arise from variable discretizations while others possess a network flow interpretation. The effectiveness of these MILP models is empirically validated on a library of medium and large-scale instances. General MIBLPs, not necessarily pooling problems, are solved using extended MILP reformulations. The reformulation is obtained by writing binary representation for each general integer variable. Facet-defining inequalities are provided for the reformulation of each bilinear term. New valid inequalities are also proposed for bilinear terms with a nontrivial upper bound. The proposed reformulation and cutting planes are compared against a global solver on five different classes of MIBLPs.

La dispersion spatiale de l'habitat et des activités de ces dernières décennies a fortement contribué à un allongement des distances et des temps de trajets domicile-travail. Cela a pour conséquence un accroissement de l'utilisation des voitures particulières, notamment au sein et aux abords des grandes agglomérations. Afin de réduire les impacts dus à l'augmentation du trafic routier, des services de covoiturage, où des usagers ayant la même destination se regroupent en équipage pour se déplacer, ont été mis en place partout dans le monde. Nous présentons ici nos travaux sur le problème de covoiturage régulier. Dans cette thèse, le problème de covoiturage régulier a été modélisé et plusieurs métaheuristiques de résolution ont été implémentées, testées et comparées. La thèse est organisée de la façon suivante: tout d'abord, nous commençons par présenter la définition et la description du problème ainsi que le modèle mathématique associé. Ensuite, plusieurs métaheuristiques pour résoudre le problème sont présentées. Ces approches sont au nombre de quatre: un algorithme de recherche locale à voisinage variable, un algorithme à base de colonies de fourmis, un algorithme génétique guidée et un système multi-agents génétiques auto-adaptatif. Des expériences ont été menées pour démontrer l'efficacité de nos approches. Nous continuons ensuite avec la présentation et la résolution d'une extension du problème de covoiturage occasionel comportant plusieurs destinations. Pour terminer, une plate-forme de test et d'analyse pour évaluer nos approches et une plate-forme de covoiturage sont présentées dans l'annexe. / Nowadays, the increased human mobility combined with high use of private cars increases the load on environment and raises issues about quality of life. The extensive use of private cars lends to high levels of air pollution, parking problem, traffic congestion and low transfer velocity. In order to ease these shortcomings, the car pooling program, where sets of car owners having the same travel destination share their vehicles, has emerged all around the world. We present here our research on the long-term car pooling problem. In this thesis, the long-term car pooling problem is modeled and metaheuristics for solving the problem are investigated. The thesis is organized as follows. First, the definition and description of the problem as well as its mathematical model are introduced. Then, several metaheuristics to effectively and efficiently solve the problem are presented. These approaches include a Variable Neighborhood Search Algorithm, a Clustering Ant Colony Algorithm, a Guided Genetic Algorithm and a Multi-agent Self-adaptive Genetic Algorithm. Experiments have been conducted to demonstrate the effectiveness of these approaches on solving the long-term car pooling problem. Afterwards, we extend our research to a multi-destination daily car pooling problem, which is introduced in detail manner along with its resolution method. At last, an algorithm test and analysis platform for evaluating the algorithms and a car pooling platform are presented in the appendix.