Speeding up Martins' algorithm for multiple objective shortest path problems

The latest transportation systems require the best routes in a large network with respect to multiple objectives simultaneously to be calculated in a very short time. The label setting algorithm of Martins efficiently finds this set of Pareto optimal paths, but sometimes tends to be slow, especially for large networks such as transportation networks. In this article we investigate a number of speedup measures, resulting in new algorithms. It is shown that the calculation time to find the Pareto optimal set can be reduced considerably. Moreover, it is mathematically proven that these algorithms still produce the Pareto optimal set of paths

Labeling Methods for Partially Ordered Paths

The landscape of applications and subroutines relying on shortest path computations continues to grow steadily. This growth is driven by the undeniable success of shortest path algorithms in theory and practice. It also introduces new challenges as the models and assessing the optimality of paths become more complicated. Hence, multiple recent publications in the field adapt existing labeling methods in an ad-hoc fashion to their specific problem variant without considering the underlying general structure: they always deal with multi-criteria scenarios and those criteria define different partial orders on the paths. In this paper, we introduce the partial order shortest path problem (POSP), a generalization of the multi-objective shortest path problem (MOSP) and in turn also of the classical shortest path problem. POSP captures the particular structure of many shortest path applications as special cases. In this generality, we study optimality conditions or the lack of them, depending on the objective functions' properties. Our final contribution is a big lookup table summarizing our findings and providing the reader an easy way to choose among the most recent multicriteria shortest path algorithms depending on their problem's weight structure. Examples range from time-dependent shortest path and bottleneck path problems to the fuzzy shortest path problem and complex financial weight functions studied in the public transportation community. Our results hold for general digraphs and therefore surpass previous generalizations that were limited to acyclic graphs

Multi-Criterion Dynamic Traffic Assignment Models and Algorithms for Road Pricing Applications with Heterogeneous Users

This study develops a simulation-based dynamic traffic assignment, or dynamic user equilibrium (DUE), model for dynamic road pricing applications. This proposed model is considered as the bi-criterion DUE (BDUE) model, because it explicitly considers heterogeneous users with different values of time (VOT) choose paths that minimize the two path attributes: travel time and out-of-pocket cost. This study assumed trip-makers would select their respective least generalized cost paths, the generalized cost being the sum of travel cost and travel time weighted by the trip-maker's VOT. The VOT is modeled as a continuous random variable distributed across all users in a network. The BDUE problem is formulated as an infinite dimensional variational inequality (VI), and solved by a column generation-based algorithmic framework which embeds (i) a parametric analysis (PAM) to obtain the VOT breakpoints which determine multiple user classes, and find the set of extreme non-dominated paths, (ii) a simulator to determine experienced travel times, and (iii) a multi-class path flow equilibrating scheme to update path assignments. The idea of finding and assigning heterogeneous trips to the set of extreme non-dominated paths is based on the assumption that in the disutility minimization path choice model with convex utility functions, all trips would choose only among the set of extreme non-dominated paths. Moreover, to circumvent the difficulty of storing the grand path set and assignment results for large-scale network applications, a vehicle-based implementation technique is proposed. This BDUE model is generalized to the multi-criterion DUE (MDUE) model, in which heterogeneous users with different VOT and values of reliability (VOR) make path choices so as to minimize their path travel cost, travel time, and travel time variability. Another important extension of the BDUE model is the multi-criterion simultaneous route and departure time user equilibrium (MSRDUE) model, which considers heterogeneous trip-makers with different VOT and values of schedule delay (VOSD) making simultaneous route and departure time choices so as to minimize their respective trip costs, defined as the sum of travel cost, travel time weighted by VOT, and schedule delay weighted by VOSD. The MSRDUE problem is also solved by the column generation-based algorithmic framework. The Sequential Parametric Analysis Method (SPAM) is developed to find the VOT and VOSD breakpoints that define multiple user classes, and determine the least trip cost alternative (a combination of departure time and path) for each user class

On the Development of a New Class of Covering Path Models

The basis of this dissertation work stems from the fact that if one examines system route maps for many bus transit systems in U.S. cities, an interesting pattern emerges. Routes often utilize embedded loops to increase accessibility coverage of a system at the expense of adding a marginal amount of length to the overall path. Further, such routes frequently share a common corridor with respect to traveling in opposing directions, but they may depart spatially from each other in terms of direction. These departures in direction represent embedded loops that are traversed in only one direction. However, the literature has not explored this issue, and in fact, often discourages or outright prevents any loops from occurring whether they are loops that are traversed in both directions or traversed in only one direction. Furthermore, past research on covering path models has not accounted for travel in opposing directions, even when attempting to model transit lines. This is due in part to the roots of the covering path literature.This dissertation presents an analysis of past work and from that defines several new problems that are ‘loop agnostic’ – that is, they neither prevent nor encourage the formation of loops in an optimal route, essentially a new class of covering path problems. Although several loop agnostic models are developed in this dissertation to better represent the maximal covering shortest path problem, these models only capture one aspect of loop use. In the classic Maximal Covering Shortest Path problem, it is assumed that its use in transit will be traversed in both directions. Further, the classic formulation prevents most loops from occurring. A new form of this model is developed that allows loops to be part of a solution, whenever such loops provide an improvement in the objective function value. This model is called “loop agnostic” as the model neither prevents nor requires loops to be used in a solution. This means that a loop can be present as part of the path, as an out-and-back path or a more complex loop which visits several other nodes before returning to a previously visited node, or even as a ‘lollipop’ shaped route attached to the origin node or the destination node. If one assumes that the covering path can be traversed both in the outbound and inbound directions (which past work has done), any loops that are present will be traversed in both directions and is what we refer to as a bi-directional loop. When addressing the question of bi-directionality in real world systems it is possible that a loop is traversed in only one direction. Such “uni-directional” loops are formed whenever inbound/outbound paths diverge and can be observed in many transit system maps, like those of Bozeman, MT; Eau Claire, WI; and San Luis Obispo, CA. This dissertation also proposes a new problem, the Bi-Directional MCSP, and formulates two new models that account for travel based upon inbound and outbound path directions which allows for the use of shared arcs and uni-directional loops as well as bi-directional loops.This dissertation also presents results from the application of these new models as well as a new heuristic to a hypothetical test network as well as a real world network from Richardson/Garland, Texas. Results demonstrate that loops are present in many optimal solutions and that the route designs that utilize loop structures such as a ‘lollipop,’ ‘barbell,’ and ‘figure eight’ may well be superior to route designs that do not incorporate loops. This gives credence to the designs of virtually all transit systems in small and medium sized cities in the United States

Wielokryterialne, mrowiskowe algorytmy optymalizacji w nawigacji samochodowej

Rozwiązywanie złożonych problemów optymalizacji dyskretnej znajduje zastosowania praktyczne w wielu dziedzinach aktywności człowieka. Przykładem możne być wyszukiwanie optymalnej drogi między dwoma punktami na mapie drogowej w nawigacji samochodowej z zastosowaniem wielu kryteriów oceny. Problem jest znany w literaturze jako bardziej ogólny, wielokryterialny problem najkrótszej drogi w grafie (ang. multi-objective shortest path problem – MOSP). Na początku rozprawy zostały omówione różne, klasyczne podejścia do rozwiązywania problemów optymalizacji wielokryterialnej, ze szczególnym uwzględnieniem podejścia zaproponowanego przez włoskiego ekonomistę Vilfredo Pareto. W metodzie tej zakłada się, że w procesie optymalizacji wielokryterialnej bardzo rzadko możliwe jest wyznaczenie jednego rozwiązania, które jest optymalne z punktu widzenia każdego kryterium oceny równocześnie. W związku z tym wynikiem optymalizacji wielokryterialnej metoda Pareto jest najczęściej zbiór rozwiązań niezdominowanych. Każde rozwiązanie, które należy do tego zbioru charakteryzuje się tym, że nie da się już polepszyć żadnego z kryterium oceny bez pogorszenia pozostałych. W rozprawie zdefiniowano wielokryterialny problem wyszukiwania najkrótszej drogi w grafie oraz dokonano przeglądu różnych metod rozwiązywania tego problemu. Charakteryzują się one dużą złożonością obliczeniową, co zachęca do stosowania algorytmów wyznaczających rozwiązania przybliżone. Są wśród nich algorytmy optymalizacji mrowiskowej, które są skuteczną metodą rozwiązywania złożonych problemów optymalizacji dyskretnej. Optymalizacja mrowiskowa (ang. ant colony optimization – ACO) jest paradygmatem związanym z tworzeniem algorytmów heurystycznych dla rozwiązywania problemów optymalizacji dyskretnej, które należą do licznego grona algorytmów inspirowanych przez naturę. Jest on oparty na kolonii sztucznych mrówek, które współpracują i komunikują się za pośrednictwem sztucznych śladów feromonowych. Pierwszym algorytmem w tej klasie był system mrówkowy, który wywodzi się z badań w dziedzinie systemów naśladujących rzeczywiste zachowania mrówek i został zaproponowany w 1991 r. przez M. Dorigo, V. Maniezzo i A. Colorniego jako algorytm rozwiązujący problem komiwojażera. Mrówki podróżują w przestrzeni rozwiązań, która zwykle ma strukturę grafową. Następny punkt swojej drogi mrówki wybierają z prawdopodobieństwem zależącym od dwóch rodzajów informacji związanych z krawędzią: statycznej informacji heurystycznej, np. odległości między węzłami oraz śladu feromonowego, który zmienia się w trakcie obliczeń i jest środkiem „porozumiewania się” mrówek. Algorytmy mrowiskowe stały się podstawą zaproponowanych w rozprawie oryginalnych wielokryterialnych algorytmów optymalizacji w nawigacji samochodowej. Na podstawie algorytmu AVN, znanego z literatury, zaprezentowano nowe, udoskonalone algorytmy NAVN i MultiNAVN będące głównymi efektami prac nad niniejszą rozprawa. Zaproponowano dwie wersje algorytmu MultiNAVN: z kryterium zastępczym (MultiNAVNZ) oraz losowym wyborem kryterium (MultiNAVN-L), które zostały poddane eksperymentom na danych rzeczywistych. Wykorzystano cztery kryteria oceny rozwiązań: długość drogi, szerokość drogi, liczba skrzyżowań oraz bezpieczeństwo. Dane dotyczące trzech pierwszych kryteriów pozyskano z bazy danych systemu OpenStreetMap, a jako kryterium bezpieczeństwa wykorzystano informacje o wypadkach i kolizjach z systemu Policji o nazwie SEWiK. Oba algorytmy wyznaczają przybliżone zbiory rozwiązań niezdominowanych, których jakość może być mierzona odległością tych zbiorów (np. metryka Hausdorffa) od pełnych zbiorów rozwiązań wyznaczonych algorytmem deterministycznym. Za pomocą wielu eksperymentów wykazano, że zaproponowane algorytmy z powodzeniem wyznaczają drogi dla rzeczywistych map drogowych, przy czym wyniki są porównywalne, a nierzadko lepsze od rezultatów uzyskiwanych za pomocą innych algorytmów

Special Topics in Information Technology

This open access book presents thirteen outstanding doctoral dissertations in Information Technology from the Department of Electronics, Information and Bioengineering, Politecnico di Milano, Italy. Information Technology has always been highly interdisciplinary, as many aspects have to be considered in IT systems. The doctoral studies program in IT at Politecnico di Milano emphasizes this interdisciplinary nature, which is becoming more and more important in recent technological advances, in collaborative projects, and in the education of young researchers. Accordingly, the focus of advanced research is on pursuing a rigorous approach to specific research topics starting from a broad background in various areas of Information Technology, especially Computer Science and Engineering, Electronics, Systems and Control, and Telecommunications. Each year, more than 50 PhDs graduate from the program. This book gathers the outcomes of the thirteen best theses defended in 2019-20 and selected for the IT PhD Award. Each of the authors provides a chapter summarizing his/her findings, including an introduction, description of methods, main achievements and future work on the topic. Hence, the book provides a cutting-edge overview of the latest research trends in Information Technology at Politecnico di Milano, presented in an easy-to-read format that will also appeal to non-specialists