Accelerating TSP Solving by Using Cost-Based Solution Densities of Relaxations

RÉSUMÉ : Le problème du voyageur de commerce, ou problème du commis voyageur, est l’un des problèmes les plus importants dans le domaine de l’optimisation combinatoire. Il a fait l’objet d’innombrables travaux de recherche, à la fois théoriques et pratiques. Parmi les aspects de ce problème, nous nous intéressons particulièrement, dans le cadre de notre sujet, à certaines de ses relaxations, qui ont aussi été étudiées pour apporter de nouvelles approches à la résolution du problème. Les structures combinatoires de ces relaxations peuvent être encapsulées dans des contraintes globales existantes en programmation par contraintes (PPC), ce qui nous motive à tester une approche basée sur des travaux récents sur les heuristiques de dénombrement en PPC. L’objectif de ce projet est d’améliorer la résolution du problème du voyageur de commerce en appliquant les densités de solution aux relaxations du problème. On pose l’hypothèse qu’une arête a très peu de chance d’appartenir à la solution optimale du problème si plusieurs relaxations retournent de faibles densités de solution pour cette arête et qu’on peut donc l’éliminer pour nettoyer le graphe d’entrée du problème. On évalue donc chaque arête en fonction de leur densité de solution pour chaque relaxation et on élimine les arêtes évaluées comme "mauvaises" par toutes les relaxations. Pour l’expérimentation, cet algorithme de pré-traitement sera appliqué à plusieurs exemplaires de TSPLIB, une bibliothèque d’exemplaires du problème de voyageur de commerce. On évaluera d’abord le temps de calcul de notre méthode. Enfin, on résoudra nos exemplaires élagués avec différents solveurs (concorde, Gurobi et IBM CP Optimizer) et on comparera les résultats obtenus à la résolution des exemplaires originels. L’élagage est efficace si le temps de résolution gagné en pré-traitant les exemplaires compense le temps de pré-traitement.----------ABRACTS : The Traveling Salesman Problem is a combinatorial optimization problem, which, broadly speaking, consists of visiting a certain number n of cities, by passing through each city exactly once and by traveling the shortest possible distance. This problem is very prominent in research, as a representative of the NP-hard class of problems and as a problem with applications in various areas, including routing, networking and scheduling. Nowadays, integer programming methods dominate the landscape of TSP solvers, with the state-of-art solver concorde. As part of the efforts to solve the TSP, several of its relaxations have been studied, for computing lower bounds or domain filtering. Since these relaxations can provide insight on the combinatorial structure of the problem, we believe recent work in Constraint Programming concerning counting-based branching heuristics can bring new effective methods of using these relaxations. In this Master’s thesis, we present an approach to the traveling salesman problem which exploits cost-based solution densities from counting-based search. we propose a method for eliminating edges from the input graph of TSP instances in pre-processing, by using the solution densities from relaxations of the TSP to determine promising edges. Solution densities from different relaxations can also be combined for branching in a constraint programming solver. The efficiency and robustness of our pre-processing algorithm is evaluated by applying it to instances from TSPLIB and comparing the time to solve them with that of the original complete instances. We consider various solvers in our experimentation, namely the IBM CP Optimizer, concorde and Gurobi

09261 Abstracts Collection -- Models and Algorithms for Optimization in Logistics

From June 21 to June 26, 2009 the Dagstuhl Seminar Perspectives Workshop 09261 ``Models and Algorithms for Optimization in Logistics \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Polyhedral techniques in combinatorial optimization II: computations

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

Optimal Design and Operation of WHO-EPI Vaccine Distribution Chains

Vaccination has been proven to be the most effective method to prevent infectious diseases and in 1974 the World Health Organization (WHO) established the Expanded Programme on Immunization (EPI) to provide universal access to all important vaccines for all children, with a special focus on underserved low- and middle-income countries. However, there are still roughly 20 million infants worldwide who lack access to routine immunization services and remain at risk, and millions of additional deaths could be avoided if global vaccination coverage could improve. The broad goal of this research is to optimize the design and operation of the WHO-EPI vaccine distribution chain in these underserved low- and middle-income countries. We first present a network design problem for a general WHO-EPI vaccine distribution network by developing a mathematical model that formulates the network design problem as a mixed integer program (MIP). We then present three algorithms for typical problems that are too large to be solved using commercial MIP software. We test the algorithms using data derived from four different countries in sub-Saharan Africa and show that with our final algorithm, high-quality solutions are obtained for even the largest problems within a few minutes. We then discuss the problem of outreach to remote population centers when resources are limited and direct clinic service is unavailable. A set of these remote population centers is chosen, and over an appropriate planning period, teams of clinicians and support personnel are sent from a depot to set up mobile clinics at these locations to vaccinate people there and in the immediate surrounding area. We formulate the problem of designing outreach efforts as an MIP that is a combination of a set covering problem and a vehicle routing problem. We then incorporate uncertainty to study the robustness of the worst-case solutions and the related issue of the value of information. Finally, we study a variation of the outreach problem that combines Set Covering and the Traveling Salesmen Problem and provides an MIP formulation to solve the problem. Motivated by applications where the optimal policy needs to be updated on a regular basis and where repetitively solving this via MIP can be computationally expensive, we propose a machine learning approach to effectively deal with this problem by providing an opportunity to learn from historical optimal solutions that are derived from the MIP formulation. We also present a case study on outreach operations and provide numerical results. Our results show that while the novel machine learning based mechanism generates high quality solution repeatedly for problems that resemble instances in the training set, it does not generalize as well on a different set of optimization problems. These mixed results indicate that there are promising research opportunities to use machine learning to achieve tractability and scalability

Multi-shot Solution Prediction for Combinatorial Optimization

This paper aims to predict optimal solutions for combinatorial optimization problems (COPs) via machine learning (ML). To find high-quality solutions efficiently, existing methods use a ML model to predict the optimal solution and use the ML prediction to guide the search. Prediction of the optimal solution to sufficient accuracy is critical, however it is challenging due to the high complexity of COPs. Nevertheless, these existing methods are single-shot, i.e., predicting the optimal solution only for once. This paper proposes a framework that enables a ML model to predict the optimal solution in multiple shots, namely multi-shot solution prediction (MSSP), which can improve the quality of a ML prediction by harnessing feedback from search. Specifically, we employ a set of statistical measures as features, to extract useful information from feasible solutions found by the search method and inform the ML model as to which value a decision variable is likely to take in high-quality solutions. Our experiments on three NP-hard COPs show that MSSP improves the quality of a ML prediction substantially and achieves competitive results as compared with other search methods in terms of solution quality. Furthermore, we demonstrate that MSSP can be used as a pricing heuristic for column generation, to boost a branch-and-price algorithm for solving the graph coloring problem

Hybrid Vehicle-drone Routing Problem For Pick-up And Delivery Services Mathematical Formulation And Solution Methodology

The fast growth of online retail and associated increasing demand for same-day delivery have pushed online retail and delivery companies to develop new paradigms to provide faster, cheaper, and greener delivery services. Considering drones’ recent technological advancements over the past decade, they are increasingly ready to replace conventional truck-based delivery services, especially for the last mile of the trip. Drones have significantly improved in terms of their travel ranges, load-carrying capacity, positioning accuracy, durability, and battery charging rates. Substituting delivery vehicles with drones could result in $50M of annual cost savings for major U.S. service providers. The first objective of this research is to develop a mathematical formulation and efficient solution methodology for the hybrid vehicle-drone routing problem (HVDRP) for pick-up and delivery services. The problem is formulated as a mixed-integer program, which minimizes the vehicle and drone routing cost to serve all customers. The formulation captures the vehicle-drone routing interactions during the drone dispatching and collection processes and accounts for drone operation constraints related to flight range and load carrying capacity limitations. A novel solution methodology is developed which extends the classic Clarke and Wright algorithm to solve the HVDRP. The performance of the developed heuristic is benchmarked against two other heuristics, namely, the vehicle-driven routing heuristic and the drone-driven routing heuristic. Anticipating the potential risk of using drones for delivery services, aviation authorities in the U.S. and abroad have mandated necessary regulatory rules to ensure safe operations. The U.S. Federal Aviation Administration (FAA) is examining the feasibility of drone flights in restricted airspace for product delivery, requiring drones to fly at or below 400-feet and to stay within the pilot’s line of sight (LS). Therefore, a second objective of this research is considered to develop a modeling framework for the integrated vehicle-drone routing problem for pick-up and delivery services considering the regulatory rule requiring all drone flights to stay within the pilot’s line of sight (LS). A mixed integer program (MIP) and an efficient solution methodology were developed for the problem. The solution determines the optimal vehicle and drone routes to serve all customers without violating the LS rule such that the total routing cost of the integrated system is minimized. Two different heuristics are developed to solve the problem, which extends the Clarke and Wright Algorithm to cover the multimodality aspects of the problem and to satisfy the LS rule. The first heuristic implements a comprehensive multimodal cost saving search to construct the most efficient integrated vehicle-drone routes. The second heuristic is a light version of the first heuristic as it adopts a vehicle-driven cost saving search. Several experiments are conducted to examine the performance of the developed methodologies using hypothetical grid networks of different sizes. The capability of the developed model in answering a wide variety of questions related to the planning of the vehicle-drone delivery system is illustrated. In addition, a case study is presented in which the developed methodology is applied to provide pick-up and delivery services in the downtown area of the City of Dallas. The results show that mandating the LS rule could double the overall system operation cost especially in dense urban areas with LS obstructions

Modeling and Solving Large-scale Stochastic Mixed-Integer Problems in Transportation and Power Systems

In this dissertation, various optimization problems from the area of transportation and power systems will be respectively investigated and the uncertainty will be considered in each problem. Specifically, a long-term problem of electricity infrastructure investment is studied to address the planning for capacity expansion in electrical power systems with the integration of short-term operations. The future investment costs and real-time customer demands cannot be perfectly forecasted and thus are considered to be random. Another maintenance scheduling problem is studied for power systems, particularly for natural gas fueled power plants, taking into account gas contracting and the opportunity of purchasing and selling gas in the spot market as well as the maintenance scheduling considering the uncertainty of electricity and gas prices in the spot market. In addition, different vehicle routing problems are researched seeking the route for each vehicle so that the total traveling cost is minimized subject to the constraints and uncertain parameters in corresponding transportation systems. The investigation of each problem in this dissertation mainly consists of two parts, i.e., the formulation of its mathematical model and the development of solution algorithm for solving the model. The stochastic programming is applied as the framework to model each problem and address the uncertainty, while the approach of dealing with the randomness varies in terms of the relationships between the uncertain elements and objective functions or constraints. All the problems will be modeled as stochastic mixed-integer programs, and the huge numbers of involved decision variables and constraints make each problem large-scale and very difficult to manage. In this dissertation, efficient algorithms are developed for these problems in the context of advanced methodologies of optimization and operations research, such as branch and cut, benders decomposition, column generation and Lagrangian method. Computational experiments are implemented for each problem and the results will be present and discussed. The research carried out in this dissertation would be beneficial to both researchers and practitioners seeking to model and solve similar optimization problems in transportation and power systems when uncertainty is involved