Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

BaPCod - a generic branch-and-price code

This document presents a user guide for BaPCod version 0.63, a C++ library implementing a generic branch-cut-and-price solver. We give guidelines for installing BaPCod, using its modelling language, BaPCod parameterization, retrieving BaPCod statistics, and understanding BaP-Cod output. We also present the VRPSolver extension of BaPCod which allows one to model and efficiently solve a large number of vehicle routing and related problems

On the integration of Dantzig-Wolfe and Fenchel decompositions via directional normalizations

The strengthening of linear relaxations and bounds of mixed integer linear programs has been an active research topic for decades. Enumeration-based methods for integer programming like linear programming-based branch-and-bound exploit strong dual bounds to fathom unpromising regions of the feasible space. In this paper, we consider the strengthening of linear programs via a composite of Dantzig-Wolfe and Fenchel decompositions. We provide geometric interpretations of these two classical methods. Motivated by these geometric interpretations, we introduce a novel approach for solving Fenchel sub-problems and introduce a novel decomposition combining Dantzig-Wolfe and Fenchel decompositions in an original manner. We carry out an extensive computational campaign assessing the performance of the novel decomposition on the unsplittable flow problem. Very promising results are obtained when the new approach is compared to classical decomposition methods

Runway Operations Management: Models, Enhancements, and Decomposition Techniques

Air traffic loads have been on the rise over the last several decades and are expected to double, and possibly triple in some regions, over the coming decade. With the advent of larger aircraft and ever-increasing air traffic loads, aviation authorities are continually pressured to examine capacity expansions and to adopt better strategies for capacity utilization. However, this growth in air traffic volumes has not been accompanied by adequate capacity expansions in the air transport infrastructure. It is, therefore, predicted that flight delays costing multi-billion dollars will continue to negatively impact airline companies and consumers. In airport operations management, runways constitute a scarce resource and a key bottleneck that impacts system-wide capacity (Idris et al. 1999). Throughout the three essays that form this dissertation, enhanced optimization models and effective decomposition techniques are proposed for runway operations management, while taking into consideration safety and practical constraints that govern access to runways. Essay One proposes a three-faceted approach for runway capacity management, based on the runway configuration, a chosen aircraft assignment/sequencing policy, and an aircraft separation standard as typically enforced by aviation authorities. With the objective of minimizing a fuel burn cost function, we propose optimization-based heuristics that are grounded in a classical mixed-integer programming formulation. By slightly altering the FCFS sequence, the proposed optimization-based heuristics not only preserve fairness among aircraft, but also consistently produce excellent (optimal or near optimal) solutions. Using real data and alternative runway settings, our computational study examines the transition from the (Old) Doha International Airport to the New Doha International Airport in light of our proposed optimization methodology. Essay Two examines aircraft sequencing problems over multiple runways under mixed mode operations. To curtail the computational effort associated with classical mixed-integer formulations for aircraft sequencing problems, valid inequalities, pre-processing routines and symmetry-defeating hierarchical constraints are proposed. These enhancements yield computational savings over a base mixed-integer formulation when solved via branch-and-bound/cut techniques that are embedded in commercial optimization solvers such as CPLEX. To further enhance its computational tractability, the problem is alternatively reformulated as a set partitioning model (with a convexity constraint) that prompts the development of a specialized column generation approach. The latter is accelerated by incorporating several algorithmic features, including an interior point dual stabilization scheme (Rousseau et al. 2007), a complementary column generation routine (Ghoniem and Sherali, 2009), and a dynamic lower bounding feature. Empirical results using a set of computationally challenging simulated instances demonstrate the effectiveness and the relative merits of the strengthened mixed-integer formulation and the accelerated column generation approach. Essay Three presents an effective dynamic programming algorithm for solving Elementary Shortest Path Problems with Resource Constraints (ESPPRC). This is particularly beneficial, because the ESPPRC structure arises in the column generation pricing sub-problem which, in turn, causes computational challenges as noted in Essay Two. Extending the work by Feillet et al. (2004), the proposed algorithm dynamically constructs optimal aircraft schedules based on the shortest path between operations while enforcing time-window restrictions and consecutive as well as nonconsecutive minimum separation times between aircraft. Using the aircraft separation standard by the Federal Aviation Administration (FAA), our computational study reports very promising results, whereby the proposed dynamic programming approach greatly outperforms the solution of the sub-problem as a mixed-integer programming formulation using commercial solvers such as CPLEX and paves the way for developing effective branch-and-price algorithms for multiple-runway aircraft sequencing problems

A Branch-and-Cut based Pricer for the Capacitated Vehicle Routing Problem

openIl Capacitated Vehicle Routing Problem, abbreviato come CVRP, è un problema di ottimizzazione combinatoria d'instradamento nel quale, un insieme geograficamente sparso di clienti con richieste note deve essere servito da una flotta di veicoli stazionati in una struttura centrale. Negli ultimi due decenni, tecniche di Column generation incorporate all'interno di frameworks branch-price-and-cut sono state infatti l'approccio stato dell'arte dominante per la costruzione di algoritmi esatti per il CVRP. Il pricer, un componente critico nella column generation, deve risolvere il Pricing Problem (PP) che richiede la risoluzione di un Elementary Shortest Path Problem with Resource Constraints (ESPPRC) in una rete di costo ridotto. Pochi sforzi scientifici sono stati dedicati allo studio di approcci branch-and-cut per affrontare il PP. L'ESPPRC è stato tradizionalmente rilassato e risolto attraverso algoritmi di programmazione dinamica. Questo approccio, tuttavia, ha due principali svantaggi. Per cominciare, peggiora i dual bounds ottenuti. Inoltre, il tempo di esecuzione diminuisce all'aumentare della lunghezza dei percorsi generati. Per valutare la performance dei loro contributi, la comunità di ricerca operativa ha tradizionalmente utilizzato una serie d'istanze di test storiche e artificiali. Tuttavia, queste istanze di benchmark non catturano le caratteristiche chiave dei moderni problemi di distribuzione del mondo reale, che sono tipicamente caratterizzati da lunghi percorsi. In questa tesi sviluppiamo uno schema basato su un approccio branch-and-cut per risolvere il pricing problem. Studiamo il comportamento e l'efficacia della nostra implementazione nel produrre percorsi più lunghi comparandola con soluzioni all'avanguardia basate su programmazione dinamica. I nostri risultati suggeriscono che gli approcci branch-and-cut possono supplementare il tradizionale algoritmo di etichettatura, indicando che ulteriore ricerca in quest'area possa portare benefici ai risolutori CVRP.The Capacitated Vehicle Routing Problem, CVRP for short, is a combinatorial optimization routing problem in which, a geographically dispersed set of customers with known demands must be served by a fleet of vehicles stationed at a central facility. Column generation techniques embedded within branch-price-and-cut frameworks have been the de facto state-of-the-art dominant approach for building exact algorithms for the CVRP over the last two decades. The pricer, a critical component in column generation, must solve the Pricing Problem (PP), which asks for an Elementary Shortest Path Problem with Resource Constraints (ESPPRC) in a reduced-cost network. Little scientific efforts have been dedicated to studying branch-and-cut based approaches for tackling the PP. The ESPPRC has been traditionally relaxed and solved through dynamic programming algorithms. This approach, however, has two major drawbacks. For starters, it worsens the obtained dual bounds. Furthermore, the running time degrades as the length of the generated routes increases. To evaluate the performance of their contributions, the operations research community has traditionally used a set of historical and artificial test instances. However, these benchmark instances do not capture the key characteristics of modern real-world distribution problems, which are usually characterized by longer routes. In this thesis, we develop a scheme based on a branch-and-cut approach for solving the pricing problem. We study the behavior and effectiveness of our implementation in producing longer routes by comparing it with state-of-the-art solutions based on dynamic programming. Our results suggest that branch-and-cut approaches may supplement the traditional labeling algorithm, indicating that further research in this area may bring benefits to CVRP solvers

Does the Enlarged European Union Need a Better Fiscal Pact?

In this paper, we set out to examine an efficient fiscal policy framework for a monetary union. We find that a monetary union can survive with diverging fiscal policies and that the financial markets are efficient enough to separate between “good” and “bad” fiscal policies and punish the latter with higher costs of borrowing. Therefore, there is only limited spill over effect of “bad” fiscal policy within a monetary union if financial markets work efficiently. We argue, consequently, that fiscal rules in a monetary union are still important as they allow to overcome incentive incompatibility of national fiscal rules and as they may guide financial markets in assessing sustainability of national fiscal policies. Finally, we argue for adoption of an institutional rule, Fiscal Sustainability Council for enlarged European Union. The Council would periodically assess fiscal policies and set guidelines for annual deficits. We argue that in order to make the FSC relevant, governments would be obliged to deposit with the Council a substantial amount of bonds that would be regularly rolled over by the Council. By doing so, the Council would connect fiscal policy sustainability principle with financial markets and would guide financial markets evaluation of national fiscal policies.fiscal policy; European Union; sustainability

The Robust Bulk Ship Routing Problem with Batched Cargo Selection

Maritime transportation forms the backbone of the world merchandise trade. In this paper, we consider a problem that combines three interconnected subproblems in tramp shipping: the fleet adjustment problem, the cargo selection problem, and the ship routing problem. For cargo selection, we consider the decision behaviors under the setting of Contract of Affreightment (COA), in which cargoes should be rejected or accepted as a batch. In view of the uncertainties observed in maritime transportation, we formulate the problem in a robust way so that the solutions can protect the profitability of shipping companies against variations in voyage costs. We first provide compact mixed integer linear programming formulations for the problem and then convert them into a strengthened set covering model. A tailored branch-and-price-and-cut algorithm is developed to solve the set covering model. The algorithm is enhanced by a multi-cut generation technique aimed at tightening the lower bounds and a primal heuristic aimed at finding high-quality upper bounds. Extensive computational results show that our algorithm yields optimal or near-optimal solutions to realistic instances within short computing times and that the enhancement techniques significantly improve the efficiency of the algorithm.</p

Improvements on Column-Generation-Based Algorithms for Vehicle Routing and Other Combinatorial Problems

RÉSUMÉ : Plusieurs applications dans le contexte de la logistique et de la planification de la production peuvent être modélisées comme des problèmes d’optimisation combinatoire (POC). En particulier,l’un des problèmes les plus étudiés dans ce domaine est le problème de tournées de véhicules (PTV). Le PTV consiste à trouver des tournées de véhicules qui minimisent le coût total de transport pour visiter un ensemble de clients, de telle sorte que leur demande soit complètement satisfaite en une seule visite, et que la capacité des véhicules ne soit jamais dépassée. Présentement, la principale méthode de résolution exacte pour les PTVs est la génération de colonnes. Dans cette thèse, nous nous intéressons à l’étude des algorithmes de génération de colonnes et proposons de nouvelles idées pour améliorer leur efficacité. Dans le Chapitre 4, nous présentons une revue de littérature très exhaustive dans laquelle nous mettons en évidence les principales contributions algorithmiques et de modélisation apportées au cours des dernières années dans la cadre du développent des algorithmes de génération de colonnes et de plans coupants intégrés à des méthodes d’énumération implicite pour le PTV. Notre étude est divisée en deux parties principales. Dans la première partie, nous présentons des aspects qui peuvent s’appliquer à la plupart des variantes de PTV, à savoir : des algorithmes de résolution du sous-problème de la génération de colonnes, la séparation de plans coupants, les stratégies de branchement et la stabilisation des variables duales dans le problème-maître. La deuxième partie est dédiée à la résolution de problèmes spécifiques. Dans cette partie, nous discutons comment les spécificités de chaque problème peuvent êtres traitées lors du développement des algorithmes d’énumération implicite combinant génération de colonnes et plans coupants. On étude les attributs suivants : l’existence d’une flotte hétérogène et des dépôts multiples, la considération de fenêtres de temps souples chez les clients, la possibilité d’effectuer des livraisons fractionnées, les coûts dépendant du temps, la réalisation de cueillettes et livraisons, la présence d’incertitude dans les données et des aspects environnementaux. Dans le Chapitre 5, nous proposons un algorithme sélectif pour résoudre des sous-problèmes de la génération de colonnes afin de générer des routes relaxées de type arc-ng. Notre méthode considère une généralisation de la dominance par ensemble proposée par Bulhões et al. [1]. Les résultats numériques obtenus sur des instances du PTV avec fenêtres de temps montrent que le nouveau mécanisme aide à réduire le nombre d’étiquettes non-dominées dans l’algorithme d’étiquetage utilisé pour résoudre le sous-problème et, par conséquent, le temps de calcul. Enfin, dans le Chapitre 6, nous présentons une nouvelle méthode de stabilisation pour des POCs avec des structures qui favorisent l’parution de dégénérescence. Le nouvel algorithme de stabilisation, appelé dyn-SAR, est basé sur la séparation dynamique de contraintes agrégées, qui sont obtenues en additionnant des contraintes du problème maître de génération de colonnes. L’effet de stabilisation induit par dyn-SAR provient des fortes interactions qui surviennent entre les variables duales, ce qui n’est pas observé lors de la résolution explicite d’une formulation de partition d’ensemble (recouvrement / empaquetage). L’intérêt principal pour l’utilisation du dyn-SAR est dû à sa simplicité et généralité. Ce dernier aspect est confirmé dans nos expériences, où nous considérons des problèmes dont la fonction objectif et le sous-problème de génération de colonnes sont considérablement différents. Les résultats numériques montrent un avantage important du dyn-SAR par rapport à une méthode de génération de colonnes standard en termes de nombre d’itérations et de temps de calcul.----------ABSTRACT : Several applications arising in the context of logistics and production planning can be modeled as combinatorial optimization problems (COPs). In particular, one of the most studied problems in this field is the vehicle routing problem (VRP). The VRP is the problem of finding least-cost routes to visit a set of customers in such a way that their demand is completely satisfied in a single visit, and the capacity of vehicles is not exceeded. Nowadays, the leading exact method to cope with different classes of VRPs is column generation (CG). In this thesis, we are interested in studying CG algorithms and propose new ideas to enhance their efficiency. In Chapter 4, we present a methodological survey in which we highlight and discuss the main algorithmic and modeling contributions made over the years in the context of branch-priceand-cut methods for VRPs. Our study is divided into two main parts. In the first part, we discuss topics that may apply to most VRPs variants, namely: pricing algorithms, cut separation, branching strategies, and dual variable stabilization. The second part is more problem-oriented and describes how aspects such as heterogeneous fleet, multi-depots, soft time windows, split deliveries, time dependency, pickups and deliveries, uncertainty, and environmental aspects can be handled in devising branch-price-and-cut algorithms. In Chapter 5, we propose a selective pricing algorithm to solve pricing subproblems defined in terms of arc-ng-route relaxations. Our method extends the set-based dominance rule proposed by Bulhões et al. [1], making it more general and stronger. Computational experiments performed over instances of the VRP with time windows show that the proposed mechanism helps in reducing the number of non-dominated labels kept by the labeling algorithm and, as a consequence, the CPU time. Finally, in Chapter 6, we develop a new stabilization framework to tackle COPs with degenerate structures. The new stabilization method, called dyn-SAR, relies on the dynamic separation of aggregated constraints, which are obtained by summing up constraints from the CG master problem. The stabilization effect induced by dyn-SAR is due to strong interactions that arise from dual variables, which is not observed when solving explicitly a set-partitioning (covering/packing) formulation. The main interests in using the dyn-SAR method are its simplicity and generality. The latter aspect is confirmed in our experiments, where we solve instances from problems differing considerably in their objective function and pricing subproblem. Numerical results show a clear advantage of dyn-SAR over a standard CG method in terms of both the number of iterations and running time