Solving multiobjective mixed integer convex optimization problems

Multiobjective mixed integer convex optimization refers to mathematical programming problems where more than one convex objective function needs to be optimized simultaneously and some of the variables are constrained to take integer values. We present a branch-and-bound method based on the use of properly defined lower bounds. We do not simply rely on convex relaxations, but we built linear outer approximations of the image set in an adaptive way. We are able to guarantee correctness in terms of detecting both the efficient and the nondominated set of multiobjective mixed integer convex problems according to a prescribed precision. As far as we know, the procedure we present is the first deterministic algorithm devised to handle this class of problems. Our numerical experiments show results on biobjective and triobjective mixed integer convex instances

The multi-objective Steiner pollution-routing problem on congested urban road networks

This paper introduces the Steiner Pollution-Routing Problem (SPRP) as a realistic variant of the PRP that can take into account the real operating conditions of urban freight distribution. The SPRP is a multi-objective, time and load dependent, fleet size and mix PRP, with time windows, flexible departure times, and multi-trips on congested urban road networks, that aims at minimising three objective functions pertaining to (i) vehicle hiring cost, (ii) total amount of fuel consumed, and (iii) total makespan (duration) of the routes. The paper focuses on a key complication arising from emissions minimisation in a time and load dependent setting, corresponding to the identification of the full set of the eligible road-paths between consecutive truck visits a priori, and to tackle the issue proposes new combinatorial results leading to the development of an exact Path Elimination Procedure (PEP). A PEP-based Mixed Integer Programming model is further developed for the SPRP and embedded within an efficient mathematical programming technique to generate the full set of the non-dominated points on the Pareto frontier of the SPRP. The proposed model considers truck instantaneous Acceleration/Deceleration (A/D) rates in the fuel consumption estimation, and to address the possible lack of such data at the planning stage, a new model for the construction of reliable synthetic spatiotemporal driving cycles from available macroscopic traffic speed data is introduced. Several analyses are conducted to: (i) demonstrate the added value of the proposed approach, (ii) exhibit the trade-off between the business and environmental objectives on the Pareto front of the SPRP, (iii) show the benefits of using multiple trips, and (iv) verify the reliability of the proposed model for the generation of driving cycles. A real road network based on the Chicago's arterial streets is also used for further experimentation with the proposed PEP algorithm. © 2019 Elsevier Lt

Multi-objective combinatorial optimization problems in transportation and defense systems

Multi-objective Optimization problems arise in many applications; hence, solving them efficiently is important for decision makers. A common procedure to solve such problems is to generate the exact set of Pareto efficient solutions. However, if the problem is combinatorial, generating the exact set of Pareto efficient solutions can be challenging. This dissertation is dedicated to Multi-objective Combinatorial Optimization problems and their applications in system of systems architecting and railroad track inspection scheduling. In particular, multi-objective system of systems architecting problems with system flexibility and performance improvement funds have been investigated. Efficient solution methods are proposed and evaluated for not only the system of systems architecting problems, but also a generic multi-objective set covering problem. Additionally, a bi-objective track inspection scheduling problem is introduced for an automated ultrasonic inspection vehicle. Exact and approximation methods are discussed for this bi-objective track inspection scheduling problem --Abstract, page iii

Nonconvex and mixed integer multiobjective optimization with an application to decision uncertainty

Multiobjective optimization problems commonly arise in different fields like economics or engineering. In general, when dealing with several conflicting objective functions, there is an infinite number of optimal solutions which cannot usually be determined analytically. This thesis presents new branch-and-bound-based approaches for computing the globally optimal solutions of multiobjective optimization problems of various types. New algorithms are proposed for smooth multiobjective nonconvex optimization problems with convex constraints as well as for multiobjective mixed integer convex optimization problems. Both algorithms guarantee a certain accuracy of the computed solutions, and belong to the first deterministic algorithms within their class of optimization problems. Additionally, a new approach to compute a covering of the optimal solution set of multiobjective optimization problems with decision uncertainty is presented. The three new algorithms are tested numerically. The results are evaluated in this thesis as well. The branch-and-bound based algorithms deal with box partitions and use selection rules, discarding tests and termination criteria. The discarding tests are the most important aspect, as they give criteria whether a box can be discarded as it does not contain any optimal solution. We present discarding tests which combine techniques from global single objective optimization with outer approximation techniques from multiobjective convex optimization and with the concept of local upper bounds from multiobjective combinatorial optimization. The new discarding tests aim to find appropriate lower bounds of subsets of the image set in order to compare them with known upper bounds numerically.Multikriterielle Optimierungprobleme sind in diversen Anwendungsgebieten wie beispielsweise in den Wirtschafts- oder Ingenieurwissenschaften zu finden. Da hierbei mehrere konkurrierende Zielfunktionen auftreten, ist die Lösungsmenge eines derartigen Optimierungsproblems im Allgemeinen unendlich groß und kann meist nicht in analytischer Form berechnet werden. In dieser Dissertation werden neue Branch-and-Bound basierte Algorithmen zur Lösung verschiedener Klassen von multikriteriellen Optimierungsproblemen entwickelt und vorgestellt. Der Branch-and-Bound Ansatz ist eine typische Methode der globalen Optimierung. Einer der neuen Algorithmen löst glatte multikriterielle nichtkonvexe Optimierungsprobleme mit konvexen Nebenbedingungen, während ein zweiter zur Lösung multikriterieller gemischt-ganzzahliger konvexer Optimierungsprobleme dient. Beide Algorithmen garantieren eine gewisse Genauigkeit der berechneten Lösungen und gehören damit zu den ersten deterministischen Algorithmen ihrer Art. Zusätzlich wird ein Algorithmus zur Berechnung einer Überdeckung der Lösungsmenge multikriterieller Optimierungsprobleme mit Entscheidungsunsicherheit vorgestellt. Alle drei Algorithmen wurden numerisch getestet. Die Ergebnisse werden ebenfalls in dieser Arbeit ausgewertet. Die neuen Algorithmen arbeiten alle mit Boxunterteilungen und nutzen Auswahlregeln, sowie Verwerfungs- und Terminierungskriterien. Dabei spielen gute Verwerfungskriterien eine zentrale Rolle. Diese entscheiden, ob eine Box verworfen werden kann, da diese sicher keine Optimallösung enthält. Die neuen Verwerfungskriterien nutzen Methoden aus der globalen skalarwertigen Optimierung, Approximationstechniken aus der multikriteriellen konvexen Optimierung sowie ein Konzept aus der kombinatorischen Optimierung. Dabei werden stets untere Schranken der Bildmengen konstruiert, die mit bisher berechneten oberen Schranken numerisch verglichen werden können

Mathematical models and solution algorithms for the vehicle routing problem with environmental considerations

Urban freight distribution is essential for the functioning of urban economies. However, it is contributing significantly to problems such as traffic congestion and environmental pollution. The main goal of this research is to contribute to greening urban freight distribution by developing new mathematical models and solution algorithms pertaining to the two major steams in Vehicle Routing Problems (VRPs) with environmental considerations: (i) VRPs with an explicit fuel consumption estimation component as a proxy for emissions, and (ii) VRPs with vehicles in the fleet that run on a cleaner alternative fuel such as electricity. In the first stream, this thesis develops and solves a new realistic multi-objective variant of the pollution-routing problem, referred to as the Steiner Pollution-Routing Problem (SPRP), that is studied directly on the original urban roadway network. The proposed variant is capable of incorporating the real operating conditions of urban freight distribution, and striking a balance between traditional business and environmental objectives, while integrating all factors that have a major impact on fuel consumption, including the time-varying congestion speed, vehicle load, vehicle’s physical and mechanical characteristics, and acceleration and deceleration rates. The thesis develops new combinatorial results that facilitate problem solution on the original roadway network and also introduces new mathematical models for synthesizing the expected second-by-second driving cycle of a vehicle over a given road-link at a given time of the day. New efficient multi-objective optimisation heuristics are also developed for addressing realistic instances of the SPRP. On the other hand, in the latter stream discussed above, to tackle the significantly impeding problem of range anxiety in the face of goods distribution using Electric Commercial Vehicles (ECVs), a paradigm shift in the routing of ECVs is proposed by introducing the Electric Vehicle Routing Problem with Synchronised Ambulant Battery Swapping/Recharging (EVRP-SABS). The proposed problem exploits new technological developments corresponding to the possibility of mobile battery swapping (or recharging) of ECVs using a Battery Swapping Van (BSV). In the EVRP-SABS, routing takes place in two levels for the ECVs that carry out delivery tasks, and for the BSVs that provide the running ECVs with fully charged batteries on their route. There is, therefore, a need to establish temporal and spatial synchronisations between the vehicles in the two levels and to do so new combinatorial results and a new solution algorithm is proposed

Modelling and Solving the Single-Airport Slot Allocation Problem

Currently, there are about 200 overly congested airports where airport capacity does not suffice to accommodate airline demand. These airports play a critical role in the global air transport system since they concern 40% of global passenger demand and act as a bottleneck for the entire air transport system. This imbalance between airport capacity and airline demand leads to excessive delays, as well as multi-billion economic, and huge environmental and societal costs. Concurrently, the implementation of airport capacity expansion projects requires time, space and is subject to significant resistance from local communities. As a short to medium-term response, Airport Slot Allocation (ASA) has been used as the main demand management mechanism. The main goal of this thesis is to improve ASA decision-making through the proposition of models and algorithms that provide enhanced ASA decision support. In doing so, this thesis is organised into three distinct chapters that shed light on the following questions (I–V), which remain untapped by the existing literature. In parentheses, we identify the chapters of this thesis that relate to each research question. I. How to improve the modelling of airline demand flexibility and the utility that each airline assigns to each available airport slot? (Chapters 2 and 4) II. How can one model the dynamic and endogenous adaptation of the airport’s landside and airside infrastructure to the characteristics of airline demand? (Chapter 2) III. How to consider operational delays in strategic ASA decision-making? (Chapter 3) IV. How to involve the pertinent stakeholders into the ASA decision-making process to select a commonly agreed schedule; and how can one reduce the inherent decision-complexity without compromising the quality and diversity of the schedules presented to the decision-makers? (Chapter 3) V. Given that the ASA process involves airlines (submitting requests for slots) and coordinators (assigning slots to requests based on a set of rules and priorities), how can one jointly consider the interactions between these two sides to improve ASA decision-making? (Chapter 4) With regards to research questions (I) and (II), the thesis proposes a Mixed Integer Programming (MIP) model that considers airlines’ timing flexibility (research question I) and constraints that enable the dynamic and endogenous allocation of the airport’s resources (research question II). The proposed modelling variant addresses several additional problem characteristics and policy rules, and considers multiple efficiency objectives, while integrating all constraints that may affect airport slot scheduling decisions, including the asynchronous use of the different airport resources (runway, aprons, passenger terminal) and the endogenous consideration of the capabilities of the airport’s infrastructure to adapt to the airline demand’s characteristics and the aircraft/flight type associated with each request. The proposed model is integrated into a two-stage solution approach that considers all primary and several secondary policy rules of ASA. New combinatorial results and valid tightening inequalities that facilitate the solution of the problem are proposed and implemented. An extension of the above MIP model that considers the trade-offs among schedule displacement, maximum displacement, and the number of displaced requests, is integrated into a multi-objective solution framework. The proposed framework holistically considers the preferences of all ASA stakeholder groups (research question IV) concerning multiple performance metrics and models the operational delays associated with each airport schedule (research question III). The delays of each schedule/solution are macroscopically estimated, and a subtractive clustering algorithm and a parameter tuning routine reduce the inherent decision complexity by pruning non-dominated solutions without compromising the representativeness of the alternatives offered to the decision-makers (research question IV). Following the determination of the representative set, the expected delay estimates of each schedule are further refined by considering the whole airfield’s operations, the landside, and the airside infrastructure. The representative schedules are ranked based on the preferences of all ASA stakeholder groups concerning each schedule’s displacement-related and operational-delay performance. Finally, in considering the interactions between airlines’ timing flexibility and utility, and the policy-based priorities assigned by the coordinator to each request (research question V), the thesis models the ASA problem as a two-sided matching game and provides guarantees on the stability of the proposed schedules. A Stable Airport Slot Allocation Model (SASAM) capitalises on the flexibility considerations introduced for addressing research question (I) through the exploitation of data submitted by the airlines during the ASA process and provides functions that proxy each request’s value considering both the airlines’ timing flexibility for each submitted request and the requests’ prioritisation by the coordinators when considering the policy rules defining the ASA process. The thesis argues on the compliance of the proposed functions with the primary regulatory requirements of the ASA process and demonstrates their applicability for different types of slot requests. SASAM guarantees stability through sets of inequalities that prune allocations blocking the formation of stable schedules. A multi-objective Deferred-Acceptance (DA) algorithm guaranteeing the stability of each generated schedule is developed. The algorithm can generate all stable non-dominated points by considering the trade-off between the spilled airline and passenger demand and maximum displacement. The work conducted in this thesis addresses several problem characteristics and sheds light on their implications for ASA decision-making, hence having the potential to improve ASA decision-making. Our findings suggest that the consideration of airlines’ timing flexibility (research question I) results in improved capacity utilisation and scheduling efficiency. The endogenous consideration of the ability of the airport’s infrastructure to adapt to the characteristics of airline demand (research question II) enables a more efficient representation of airport declared capacity that results in the scheduling of additional requests. The concurrent consideration of airlines’ timing flexibility and the endogenous adaptation of airport resources to airline demand achieves an improved alignment between the airport infrastructure and the characteristics of airline demand, ergo proposing schedules of improved efficiency. The modelling and evaluation of the peak operational delays associated with the different airport schedules (research question III) provides allows the study of the implications of strategic ASA decision-making for operations and quantifies the impact of the airport’s declared capacity on each schedule’s operational performance. In considering the preferences of the relevant ASA stakeholders (airlines, coordinators, airport, and air traffic authorities) concerning multiple operational and strategic ASA efficiency metrics (research question IV) the thesis assesses the impact of alternative preference considerations and indicates a commonly preferred schedule that balances the stakeholders’ preferences. The proposition of representative subsets of alternative schedules reduces decision-complexity without significantly compromising the quality of the alternatives offered to the decision-making process (research question IV). The modelling of the ASA as a two-sided matching game (research question V), results in stable schedules consisting of request-to-slot assignments that provide no incentive to airlines and coordinators to reject or alter the proposed timings. Furthermore, the proposition of stable schedules results in more intensive use of airport capacity, while simultaneously improving scheduling efficiency. The models and algorithms developed as part of this thesis are tested using airline requests and airport capacity data from coordinated airports. Computational results that are relevant to the context of the considered airport instances provide evidence on the potential improvements for the current ASA process and facilitate data-driven policy and decision-making. In particular, with regards to the alignment of airline demand with the capabilities of the airport’s infrastructure (questions I and II), computational results report improved slot allocation efficiency and airport capacity utilisation, which for the considered airport instance translate to improvements ranging between 5-24% for various schedule performance metrics. In reducing the difficulty associated with the assessment of multiple ASA solutions by the stakeholders (question IV), instance-specific results suggest reductions to the number of alternative schedules by 87%, while maintaining the quality of the solutions presented to the stakeholders above 70% (expressed in relation to the initially considered set of schedules). Meanwhile, computational results suggest that the concurrent consideration of ASA stakeholders’ preferences (research question IV) with regards to both operational (research question III) and strategic performance metrics leads to alternative airport slot scheduling solutions that inform on the trade-offs between the schedules’ operational and strategic performance and the stakeholders’ preferences. Concerning research question (V), the application of SASAM and the DA algorithm suggest improvements to the number of unaccommodated flights and passengers (13 and 40% improvements) at the expense of requests concerning fewer passengers and days of operations (increasing the number of rejected requests by 1.2% in relation to the total number of submitted requests). The research conducted in this thesis aids in the identification of limitations that should be addressed by future studies to further improve ASA decision-making. First, the thesis focuses on exact solution approaches that consider the landside and airside infrastructure of the airport and generate multiple schedules. The proposition of pre-processing techniques that identify the bottleneck of the airport’s capacity, i.e., landside and/or airside, can be used to reduce the size of the proposed formulations and improve the required computational times. Meanwhile, the development of multi-objective heuristic algorithms that consider several problem characteristics and generate multiple efficient schedules in reasonable computational times, could extend the capabilities of the models propositioned in this thesis and provide decision support for some of the world’s most congested airports. Furthermore, the thesis models and evaluates the operational implications of strategic airport slot scheduling decisions. The explicit consideration of operational delays as an objective in ASA optimisation models and algorithms is an issue that merits investigation since it may further improve the operational performance of the generated schedules. In accordance with current practice, the models proposed in this work have considered deterministic capacity parameters. Perhaps, future research could propose formulations that consider stochastic representations of airport declared capacity and improve strategic ASA decision-making through the anticipation of operational uncertainty and weather-induced capacity reductions. Finally, in modelling airlines’ utility for each submitted request and available time slot the thesis proposes time-dependent functions that utilise available data to approximate airlines’ scheduling preferences. Future studies wishing to improve the accuracy of the proposed functions could utilise commercial data sources that provide route-specific information; or in cases that such data is unavailable, employ data mining and machine learning methodologies to extract airlines’ time-dependent utility and preferences