Το πρόβλημα εύρεσης συντομότερης διαδρομής είναι ένα από τα πιο θεμελιώδη προβλήματα μονοκριτηριακής βελτιστοποίησης σε δίκτυα. Σε πολλές εφαρμογές ωστόσο, μας ενδιαφέρουν περισσότερα από ένα κριτήρια προς βελτιστοποίηση. Για παράδειγμα, στην δρομολόγηση σε ένα οδικό δίκτυο με διόδια, μας ενδιαφέρει ταυτόχρονα η ελαχιστοποίηση του χρόνου και του κόστους σε χρήματα. Παρόμοια παραδείγματα βρίσκουμε και στον χώρο των δικτύων τηλεπικοινωνιών, όπου εξετάζονται κριτήρια όπως η καθυστέρηση, η πιθανότητα λάθους, ο αριθμός συνδέσμων και άλλα. Σε αυτές οι περιπτώσεις η ``καλύτερη\\\\ / The shortest path problem is perhaps the most fundamental single objective optimization problem in networks. In many applications however we are interested in more than two objectives. For example, when routing in a network with tolls, we are interested in minimizing both the time and the cost. Similar examples can be also found in communication networks where the criteria under investigation are the delay, the fault probability, the number of hops and other. In such cases the \\\\

In this thesis, we study robust combinatorial problems with interval data. We introduce several new measures of robustness in response to the drawbacks of existing measures of robustness. The idea of these new measures is to ensure that the solutions are satisfactory for the decision maker in all scenarios, including the worst case scenario. Therefore, we have introduced a threshold over the worst case costs, in which above this threshold, solutions are no longer satisfactory for the decision maker. It is, however, important to consider other criteria than just the worst case. Therefore, in each of these new measures, a second criteria is used to evaluate the performance of the solution in other scenarios such as the best case one. We also study the robust deviation p-elements problem. In fact, we study when this solution is equal to the optimal solution in the scenario where the cost of each element is the midpoint of its corresponding interval. Then, we finally formulate the robust combinatorial problem with interval data as a bicriteria problem. We also integrate the decision maker's preferences over certain types of solutions into the model. We propose a method that uses these preferences to find the set of solutions that are never preferred by any other solution. We call this set the final set. We study the properties of the final sets from a coherence point of view and from a robust point of view. From a coherence point of view, we study necessary and sufficient conditions for the final set to be monotonic, for the corresponding preferences to be without cycles, and for the set to be stable. Those that do not satisfy these properties are eliminated since we believe these properties to be essential. We also study other properties such as the transitivity of the preference and indifference relations and more. We note that many of our final sets are included in one another and some are even intersections of other final sets. From a robust point of view, we compare our final sets with different measures of robustness and with the first- and second-degree stochastic dominance. We show which sets contain all of these solutions and which only contain these types of solutions. Therefore, when the decision maker chooses his preferences to find the final set, he knows what types of solutions may or may not be in the set. Lastly, we implement this method and apply it to the Robust Shortest Path Problem. We look at how this method performs using different types of randomly generated instances.

In 1985, Moffat-Takaoka (MT) algorithm was developed to solve the all pairs shortest path (APSP) problem. This algorithm manages to get time complexity of O(n² log n) expected time when the end-point independent model of probabilistic assumption is used. However, the use of a critical point introduced in this algorithm has made the implementation of this algorithm quite complicated and the running time of this algorithm is difficult to analyze. Therefore, this study introduces a new deterministic algorithm for the APSP that provides an alternative to the existing MT algorithm. The major advantages of this approach compared to the MT algorithm are its simplicity, intuitive appeal and ease of analysis. Moreover, the algorithm was shown to be efficient as the expected running time is the same O(n² log n). Performance of a good algorithm depends on the data structure used to speed up the operations needed by the algorithm such as insert, delete-min and decrease-key operations. In this study, two new data structures have been implemented, namely quaternary and dimensional heaps. In the experiment carried out, the quaternary heap that employed similar concept with the trinomial heap with a special insertion cache function performed better than the trinomial heap when the number of n vertices was small. Likewise, the dimensional heap data structure executed the decrease-key operation efficiently by maintaining the thinnest structure possible through the use of thin and thick edges, far surpassing the existing binary, Fibonacci and 2-3 heaps data structures when a special acyclic graph was used. Taken together all these promising findings, a new improved algorithm running on a good data structure can be implemented to enhance the computing accuracy and speed of todays computing machines.

This dissertation discusses two independent topics: a resource-constrained shortest-path problem (RCSP) and a literature review on scheduling problems involving sequence-dependent setup (SDS) times (costs). RCSP is often used as a subproblem in column generation because it can be used to solve many practical problems. This dissertation studies RCSP with multiple resource constraints on an acyclic graph, because many applications involve this configuration, especially in column genetation formulations. In particular, this research focuses on a dynamic RCSP since, as a subproblem in column generation, objective function coefficients are updated using new values of dual variables at each iteration. This dissertation proposes a pseudo-polynomial solution method for solving the dynamic RCSP by exploiting the special structure of an acyclic graph with the goal of effectively reoptimizing RCSP in the context of column generation. This method uses a one-time “preliminary” phase to transform RCSP into an unconstrained shortest path problem (SPP) and then solves the resulting SPP after new values of dual variables are used to update objective function coefficients (i.e., reduced costs) at each iteration. Network reduction techniques are considered to remove some nodes and/or arcs permanently in the preliminary phase. Specified techniques are explored to reoptimize when only several coefficients change and for dealing with forbidden and prescribed arcs in the context of a column generation/branch-and-bound approach. As a benchmark method, a label-setting algorithm is also proposed. Computational tests are designed to show the effectiveness of the proposed algorithms and procedures. This dissertation also gives a literature review related to the class of scheduling problems that involve SDS times (costs), an important consideration in many practical applications. It focuses on papers published within the last decade, addressing a variety of machine configurations - single machine, parallel machine, flow shop, and job shop - reviewing both optimizing and heuristic solution methods in each category. Since lot-sizing is so intimately related to scheduling, this dissertation reviews work that integrates these issues in relationship to each configuration. This dissertation provides a perspective of this line of research, gives conclusions, and discusses fertile research opportunities posed by this class of scheduling problems. since, as a subproblem in column generation, objective function coefficients are updated using new values of dual variables at each iteration. This dissertation proposes a pseudo-polynomial solution method for solving the dynamic RCSP by exploiting the special structure of an acyclic graph with the goal of effectively reoptimizing RCSP in the context of column generation. This method uses a one-time

A stochastic shortest path problem is an undiscounted infinite-horizon Markov decision process with an absorbing and costree target state, where the objective is to reach the target state while optimizing total expected cost. In almost all cases, the objective in solving a stochastic shortest path problem is to minimize the total expected cost to reach the target state. But in probabilistic model checking, it is also useful to solve a problem where the objective is to maximize the expected cost to reach the target state. This thesis considers the maximum-time stochastic shortest path problem, which is a special case of the maximum-cost stochastic shortest path problem where actions have unit cost. The contribution is an efficient approach to computing high-quality bounds on the optimal solution for this problem. The bounds are useful in themselves, but can also be used by other algorithms to accelerate search for an optimal solution.

This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path.

Το πρόβλημα εύρεσης συντομότερων διαδρομών είναι ένα από τα πιο θεμελιώδη προβλήματα μονοκριτηριακής βελτιστοποίησης σε δίκτυα. Σε αυτό το πρόβλημα αναζητείται η συντομότερη διαδρομή μεταξύ δύο δεδομένων σημείων ελαχιστοποιώντας ένα κριτήριο κόστους. Σε πολλές εφαρμογές ωστόσο, μας ενδιαφέρουν περισσότερα από ένα κριτήρια προς βελτιστοποίηση. Για παράδειγμα, στην εύρεση διαδρομών σε ένα οδικό δίκτυο με διόδια, μας ενδιαφέρει εκτός από την διανυμένη απόσταση και η ελαχιστοποίηση του χρόνου και του κόστους. Παρόμοια παραδείγματα βρίσκουμε και στον χώρο των δικτύων τηλεπικοινωνιών, όπου εξετάζονται κριτήρια όπως η καθυστέρηση, η πιθανότητα λάθους, ο αριθμός συνδέσμων και άλλα. Σε αυτές τις περιπτώσεις η καλύτερη λύση δεν μπορεί να οριστεί με μονοσήμαντο τρόπο, και συνεπώς καταφεύγουμε σε αντισταθμίσεις μεταξύ των παραγόντων, που είναι γνωστές ως σύνολο λύσεων κατά Pareto. Παρόλο που για το πρόβλημα μονοκριτηριακής εύρεσης συντομότερων διαδρομών υπάρχουν πολλοί αποδοτικοί αλγόριθμοι για την επίλυση του προβλήματος, το αντίστοιχο πολυκριτηριακό πρόβλημα είναι πολύ πιο σύνθετο. Μέχρι τώρα, αυτό το πρόβλημα έχει αποδειχθεί ότι είναι NP-πλήρες. Επιπλέον, έχει αποδειχθεί ότι το πλήθος των λύσεων σε αυτό το πρόβλημα αυξάνεται εκθετικά σε σχέση με το μέγεθος της εισόδου. Υπάρχουν δύο βασικές προσεγγίσεις επίλυσης τέτοιων προβλημάτων, όπου εξετάζονται πολλαπλά κριτήρια. α) Η πρώτη μέθοδος βρίσκει προσεγγιστικές λύσης κατά έναν ορισμένο παράγοντα. Οι προσεγγιστικές μέθοδοι δεν βρίσκουν απαραίτητα ακριβείς λύσεις, αλλά είναι σχετικά γρήγορες και προσφέρουν εγγύηση για το ποσοστό απόκλισης από την βέλτιστη λύση. β) Η δεύτερη μέθοδος χρησιμοποιεί ευρετικές βελτιώσεις για να επιταχύνει τους ήδη υπάρχοντες αλγορίθμους. Τέτοιες τεχνικές βρίσκουν ακριβείς λύσεις, και το ζητούμενο είναι να επιτευχθεί μια πολύ καλή χρονική απόδοση. Στην παρούσα διπλωματική εργασία επικεντρωνόμαστε στην δεύτερη μέθοδο, υποκινούμενοι από την μεγάλη ζήτηση πρακτικών εφαρμογών για εύρεση αποτελεσματικής και ακριβούς λύσης του προβλήματος συντομότερων διαδρομών υπό πολλαπλά κριτήρια. Πιο συγκεκριμένα, στην εργασία αυτή παρουσιάζουμε ένα ενοποιημένο πλαίσιο για την αποδοτική επίλυση αυτών των προβλημάτων. Προτείνουμε νέες μεθόδους ή βελτιώσεις των υπαρχόντων. Υλοποιήσαμε τις μεθόδους που παρουσιάζουμε συνοδεύοντάς τις με μια εκτενή πειραματική μελέτη πάνω σε δίκτυα ευρείας κλίμακας. / We present new implementations of heuristic algorithms for the solution of the multiobjective shortest path problem, using a new graph structure specifically suited for large scale road networks. We enhance the heuristics with further optimizations and experimentally evaluate the performance of our enhanced implementation on real world road networks achieving 10 times better performance with respect to the best previous study.

A stochastic shortest path (SSP) problem is an undiscounted Markov decision process with an absorbing and zero-cost target state, where the objective is to reach the target state with minimum expected cost. This problem provides a foundation for algorithms for decision-theoretic planning and probabilistic model checking, among other applications. This thesis describes an implementation and evaluation of recently developed error bounds for SSP problems. The bounds can be used in a test for convergence of iterative dynamic programming algorithms for solving SSP problems, as well as in action elimination procedures that can accelerate convergence by excluding provably suboptimal actions that do not need to be re-evaluated each iteration. The techniques are shown to be effective for both decision-theoretic planning and probabilistic model checking.

This thesis deals with shortest paths problem in graphs. Shortest paths problem is the basic issue of graph theory with many pracitcal applications. We can divide this problem into two following generalizations: single-source shortest path problem and all-pairs shortest paths problem. This text introduces principles and algorithms for generalizations. We describe both classical and new more efficient methods. It contains information about how some of these algorithms were implemented and offers an experimental comparison of these algorithms.

Le besoin de se déplacer est un besoin fondamental dans la vie de tous les jours. Avec l’extension continue des zones urbaines, l’augmentation de la population et l’amélioration du niveau de vie des citoyens, le nombre de voitures ne cesse d’augmenter. Ceci étant, la plupart des transports publics proposés aujourd’hui obéissent à des règles qui manquent de souplesse et qui incluent rarement le caractère dynamique, en temps et en espace, de la demande. Cela réduit ainsi l’attractivité de ces services et les rendant même parfois difficilement supportables. De ce fait, la majorité des usagers utilisent encore leur propre véhicule. Ce grand nombre de véhicules, qui est en augmentation continue sur les réseaux routiers, provoque de nombreux phénomènes de congestion induisant une surconsommation de carburant, des émissions inutiles de gaz à effet de serre et une perte de temps importante. Pour y remédier, nous proposons dans cette thèse de nouveaux systèmes de déplacement des usagers avec différents modèles d’optimisation pour la mobilité partagée (covoiturage et taxis-partagés) ainsi que la combinaison de la mobilité partagée avec les transports publics. Les expérimentations sont réalisées sur de vrais réseaux routiers ainsi que sur des données réelles. Ces nouveaux systèmes améliorent considérablement la qualité de service des systèmes classiques existants en termes de coût et de flexibilité tout en ayant un temps de calcul raisonnable. / The travelling is a fundamental part of everyday life. The continuous expansion of urban areas combined with the population increasing and the improvement of life standards increases the need of mobility and the use of private cars. Furthermore, the majority of public transportations are subject to rules lacking of flexibility and rarely taking into account the dynamic context. The attractiveness of public transportation is therefore reduced and, as a consequence, its financial support, resulting in a further deterioration of the public services quality and flexibility. Therefore, the majority of users still use their own vehicles. The number of vehicles is continuously increasing on road networks causing important phenomena of congestion, high fuel consumption and emissions of greenhouse gases, time loss. This unpleasant situation forces communities to consider alternative solutions for the mobility such as ride-sharing, an interesting alternative to solo car use. The overall objective of this thesis is to propose new travel systems for users through the introduction of optimization models for shared mobility (ride-sharing and taxi-sharing) and the combination of shared mobility and public transportation. The computational experiments are carried out on real road networks and real data. Our numerical results show the effectiveness of our approach, which improves the quality of service compared to the traditional systems in terms of cost and flexibility. The running time remains reasonable to allow using our framework in real-time transportation applications.