Chance Constrained Mixed Integer Program: Bilinear and Linear Formulations, and Benders Decomposition

In this paper, we study chance constrained mixed integer program with consideration of recourse decisions and their incurred cost, developed on a finite discrete scenario set. Through studying a non-traditional bilinear mixed integer formulation, we derive its linear counterparts and show that they could be stronger than existing linear formulations. We also develop a variant of Jensen's inequality that extends the one for stochastic program. To solve this challenging problem, we present a variant of Benders decomposition method in bilinear form, which actually provides an easy-to-use algorithm framework for further improvements, along with a few enhancement strategies based on structural properties or Jensen's inequality. Computational study shows that the presented Benders decomposition method, jointly with appropriate enhancement techniques, outperforms a commercial solver by an order of magnitude on solving chance constrained program or detecting its infeasibility

Discrete Optimization and Agent-Based Simulation for Regional Evacuation Network Design Problem

Natural disasters and extreme events are often characterized by their violence and unpredictability, resulting in consequences that in severe cases result in devastating physical and ecological damage as well as countless fatalities. In August 2005, Hurricane Katrina hit the Southern coast of the United States wielding serious weather and storm surges. The brunt of Katrina’s force was felt in Louisiana, where the hurricane has been estimated to total more than $108 billion in damage and over 1,800 casualties. Hurricane Rita followed Katrina in September 2005 and further contributed$12 billion in damage and 7 fatalities to the coastal communities of Louisiana and Texas. Prior to making landfall, residents of New Orleans received a voluntary, and then a mandatory, evacuation order in an attempt to encourage people to move themselves out of Hurricane Katrina’s predicted destructive path. Consistent with current practice in nearly all states, this evacuation order did not include or convey any information to individuals regarding route selection, shelter availability and assignment, or evacuation timing. This practice leaves the general population free to determine their own routes, destinations and evacuation times independently. Such freedom often results in inefficient and chaotic utilization of the roadways within an evacuation region, quickly creating bottlenecks along evacuation routes that can slow individual egress and lead to significant and potentially dangerous exposure of the evacuees to the impending storm. One way to assist the over-burdened and over-exposed population during extreme event evacuation is to provide an evacuation strategy that gives specific information on individual route selection, evacuation timing and shelter destination assignment derived from effective, strategic pre-planning. For this purpose, we present a mixed integer linear program to devise effective and controlled evacuation networks to be utilized during extreme event egress. To solve our proposed model, we develop a solution methodology based on Benders Decomposition and test its performance through an experimental design using the Central Texas region as our case study area. We show that our solution methods are efficient for large-scale instances of realistic size and that our methods surpass the size and computational limitations currently imposed by more traditional approaches such as branch-and-cut. To further test our model under conditions of uncertain individual choice/behavior, we create an agent-based simulation capable of modeling varying levels of evacuee compliance to the suggested optimal routes and varying degrees of communication between evacuees and between evacuees and the evacuation authority. By providing evacuees with information on when to evacuate, where to evacuate and how to get to their prescribed destination, we are able to observe significant cost and time increases for our case study evacuation scenarios while reducing the potential exposure of evacuees to the hurricane through more efficient network usage. We provide discussion on scenario performance and show the trade-offs and benefits of alternative batch-time evacuation strategies using global and individual effectiveness measures. Through these experiments and the developed methodology, we are able to further motivate the need for a more coordinated and informative approach to extreme event evacuation

Planning of integrated mobility-on-demand and urban transit networks

We envision a multimodal transportation system where Mobility-on-Demand (MoD) service is used to serve the first mile and last mile of transit trips. For this purpose, the current research formulates an optimization model for designing an integrated MoD and urban transit system. The proposed model is a mixed-integer non-linear programming model that captures the strategic behavior of passengers in a multimodal network through a passenger assignment model. It determines which transit routes to operate, the frequency of the operating routes, the fleet size of vehicles required in each transportation analysis zone to serve the demand, and the passenger flow on both road and transit networks. A Benders decomposition approach with several enhancements is proposed to solve the given optimization program. Computational experiments are presented for the Sioux Falls multimodal network. The results show a significant improvement in the congestion in the city center with the introduction and optimization of an integrated transportation system. The proposed design allocates more vehicles to the outskirt zones in the network (to serve the first mile and last mile of transit trips) and more frequency to the transit routes in the city center. The integrated system significantly improves the share of transit passengers and their level of service in comparison to the base optimized transit system. The sensitivity analysis of the bus and vehicle fleet shows that increasing the number of buses has more impact on improving the level of service of passengers compared to increasing the number of MoD vehicles. Finally, we provide managerial insights for deploying such multimodal service.Comment: 39 pages, 6 figure

Layout optimization and Sustainable development of waste water networks with the use of heuristic algorithms: The Luxemburgish case

Fresh water tends to increasingly comprise a scarcity today both in arid or demographically boosted regions of the world such as large and smaller cities. On this basis, research is directed towards minimization of fresh water supply into a Waste Water Network Topology (WWNT) and maximizing water re-use. This might be composed of a cluster of agents which have certain demands for fresh water as well as waste water dependent on their daily uses and living profiles. This work is divided into two parts. In the first part, different waste water flows within a reference building unit i.e. a typical household of four (4) occupants is simulated. This type of building represents a major part of the total building stock in Luxembourg. In its first part the present study attempts to examine the optimized fresh and waste water flow pathways between water using units of the building. Between water flows two domestic treatment units are adopted. The simulation of above mentioned system is attempted by adopting different algorithm methods such as the Sequential Quadratic Programming (SQP), the interior point and meta-heuristic optimization algorithms such as the Genetic Algorithms (GA’s).Suitable computational platform tools such as MATLAB and GAMS are incorporated. A comparison study on the most efficient approach is then realized on the single household unit by developing four (4) different mathematical model formulation versions. The second part of this study comprises simulation and development of the Waste Water Network Grid (WWNG) in the upscale level, such as the neighborhood level within or outside the urban context. This model encompasses all possible land uses and different kinds of buildings of different use envelopes thus demands. This range of units includes mainly building stock, agricultural and infrastructure of the tertiary sector. Integration of above mentioned model to the existing WWNG will enhance attempts to more closely reach the optimum points. The use of appropriate mathematical programming methods for the upscale level, will take place. Increased uncertainties within the built model will be attempted to be tackled by developing linear programming techniques and suitable assumptions without distorting initial condition largely. Assumptions are then drawn on the efficiency of the adopted method an additional essential task is the minimization of the overall infrastructure and network cost, which may in turn give rise to corresponding reduced waste effluents discharge off the proposed network. The case study comprises selected rural and semi-rural areas zone districts of similar living profiles outside the City of Luxembourg. Therefore a clustering of end users of similar demand will be attempted. Possible redesign of an optimized WWNG comprises a vital need within the context of large scale demographic growth of urban environments today.Open Acces

Interior Point Cutting Plane Methods in Integer Programming

This thesis presents novel approaches that use interior point concepts in solving mixed integer programs. Particularly, we use the analytic center cutting plane method to improve three of the main components of the branch-and-bound algorithm: cutting planes, heuristics, and branching. First, we present an interior point branch-and-cut algorithm for structured integer programs based on Benders decomposition. We explore using Benders decomposition in a branch-and-cut framework where the Benders cuts are generated using the analytic center cutting plane method. The algorithm is tested on two classes of problems: the capacitated facility location problem and the multicommodity capacitated fixed charge network design problem. For the capacitated facility location problem, the proposed approach was on average 2.5 times faster than Benders-branch-and-cut and 11 times faster than classical Benders decomposition. For the multicommodity capacitated fixed charge network design problem, the proposed approach was 4 times faster than Benders-branch-and-cut while classical Benders decomposition failed to solve the majority of the tested instances. Second, we present a heuristic algorithm for mixed integer programs based on interior points. As integer solutions are typically in the interior, we use the analytic center cutting plane method to search for integer feasible points within the interior of the feasible set. The algorithm searches along two line segments that connect the weighted analytic center and two extreme points of the linear programming relaxation. Candidate points are rounded and tested for feasibility. Cuts aimed to improve the objective function and restore feasibility are then added to displace the weighted analytic center until a feasible integer solution is found. The algorithm is composed of three phases. In the first, points along the two line segments are rounded gradually to find integer feasible solutions. Then in an attempt to improve the quality of the solutions, the cut related to the bound constraint is updated and a new weighted analytic center is found. Upon failing to find a feasible integer solution, a second phase is started where cuts related to the violated feasibility constraints are added. As a last resort, the algorithm solves a minimum distance problem in a third phase. For all the tested instances, the algorithm finds good quality feasible solutions in the first two phases and the third phase is never called. Finally, we present a new approach to generate good general branching constraints based on the shape of the polyhedron. Our approach is based on approximating the polyhedron using an inscribed ellipsoid. We use Dikin's ellipsoid which we calculate using the analytic center. We propose to use the disjunction that has a minimum width on the ellipsoid. We use the fact that the width of the ellipsoid in a given direction has a closed form solution in order to formulate a quadratic problem whose optimal solution is a thin direction of the ellipsoid. While solving a quadratic problem at each node of the branch-and-bound tree is impractical, we use a local search heuristic for its solution. Computational testing conducted on hard integer problems from MIPLIB and CORAL showed that the proposed approach outperforms classical branching

On High-Performance Benders-Decomposition-Based Exact Methods with Application to Mixed-Integer and Stochastic Problems

RÉSUMÉ : La programmation stochastique en nombres entiers (SIP) combine la difficulté de l’incertitude et de la non-convexité et constitue une catégorie de problèmes extrêmement difficiles à résoudre. La résolution efficace des problèmes SIP est d’une grande importance en raison de leur vaste applicabilité. Par conséquent, l’intérêt principal de cette dissertation porte sur les méthodes de résolution pour les SIP. Nous considérons les SIP en deux étapes et présentons plusieurs algorithmes de décomposition améliorés pour les résoudre. Notre objectif principal est de développer de nouveaux schémas de décomposition et plusieurs techniques pour améliorer les méthodes de décomposition classiques, pouvant conduire à résoudre optimalement divers problèmes SIP. Dans le premier essai de cette thèse, nous présentons une revue de littérature actualisée sur l’algorithme de décomposition de Benders. Nous fournissons une taxonomie des améliorations algorithmiques et des stratégies d’accélération de cet algorithme pour synthétiser la littérature et pour identifier les lacunes, les tendances et les directions de recherche potentielles. En outre, nous discutons de l’utilisation de la décomposition de Benders pour développer une (méta- )heuristique efficace, décrire les limites de l’algorithme classique et présenter des extensions permettant son application à un plus large éventail de problèmes. Ensuite, nous développons diverses techniques pour surmonter plusieurs des principaux inconvénients de l’algorithme de décomposition de Benders. Nous proposons l’utilisation de plans de coupe, de décomposition partielle, d’heuristiques, de coupes plus fortes, de réductions et de stratégies de démarrage à chaud pour pallier les difficultés numériques dues aux instabilités, aux inefficacités primales, aux faibles coupes d’optimalité ou de réalisabilité, et à la faible relaxation linéaire. Nous testons les stratégies proposées sur des instances de référence de problèmes de conception de réseau stochastique. Des expériences numériques illustrent l’efficacité des techniques proposées. Dans le troisième essai de cette thèse, nous proposons une nouvelle approche de décomposition appelée méthode de décomposition primale-duale. Le développement de cette méthode est fondé sur une reformulation spécifique des sous-problèmes de Benders, où des copies locales des variables maîtresses sont introduites, puis relâchées dans la fonction objective. Nous montrons que la méthode proposée atténue significativement les inefficacités primales et duales de la méthode de décomposition de Benders et qu’elle est étroitement liée à la méthode de décomposition duale lagrangienne. Les résultats de calcul sur divers problèmes SIP montrent la supériorité de cette méthode par rapport aux méthodes classiques de décomposition. Enfin, nous étudions la parallélisation de la méthode de décomposition de Benders pour étendre ses performances numériques à des instances plus larges des problèmes SIP. Les variantes parallèles disponibles de cette méthode appliquent une synchronisation rigide entre les processeurs maître et esclave. De ce fait, elles souffrent d’un important déséquilibre de charge lorsqu’elles sont appliquées aux problèmes SIP. Cela est dû à un problème maître difficile qui provoque un important déséquilibre entre processeur et charge de travail. Nous proposons une méthode Benders parallèle asynchrone dans un cadre de type branche-et-coupe. L’assouplissement des exigences de synchronisation entraine des problèmes de convergence et d’efficacité divers auxquels nous répondons en introduisant plusieurs techniques d’accélération et de recherche. Les résultats indiquent que notre algorithme atteint des taux d’accélération plus élevés que les méthodes synchronisées conventionnelles et qu’il est plus rapide de plusieurs ordres de grandeur que CPLEX 12.7.----------ABSTRACT : Stochastic integer programming (SIP) combines the difficulty of uncertainty and non-convexity, and constitutes a class of extremely challenging problems to solve. Efficiently solving SIP problems is of high importance due to their vast applicability. Therefore, the primary focus of this dissertation is on solution methods for SIPs. We consider two-stage SIPs and present several enhanced decomposition algorithms for solving them. Our main goal is to develop new decomposition schemes and several acceleration techniques to enhance the classical decomposition methods, which can lead to efficiently solving various SIP problems to optimality. In the first essay of this dissertation, we present a state-of-the-art survey of the Benders decomposition algorithm. We provide a taxonomy of the algorithmic enhancements and the acceleration strategies of this algorithm to synthesize the literature, and to identify shortcomings, trends and potential research directions. In addition, we discuss the use of Benders decomposition to develop efficient (meta-)heuristics, describe the limitations of the classical algorithm, and present extensions enabling its application to a broader range of problems. Next, we develop various techniques to overcome some of the main shortfalls of the Benders decomposition algorithm. We propose the use of cutting planes, partial decomposition, heuristics, stronger cuts, and warm-start strategies to alleviate the numerical challenges arising from instabilities, primal inefficiencies, weak optimality/feasibility cuts, and weak linear relaxation. We test the proposed strategies with benchmark instances from stochastic network design problems. Numerical experiments illustrate the computational efficiency of the proposed techniques. In the third essay of this dissertation, we propose a new and high-performance decomposition approach, called Benders dual decomposition method. The development of this method is based on a specific reformulation of the Benders subproblems, where local copies of the master variables are introduced and then priced out into the objective function. We show that the proposed method significantly alleviates the primal and dual shortfalls of the Benders decomposition method and it is closely related to the Lagrangian dual decomposition method. Computational results on various SIP problems show the superiority of this method compared to the classical decomposition methods as well as CPLEX 12.7. Finally, we study parallelization of the Benders decomposition method. The available parallel variants of this method implement a rigid synchronization among the master and slave processors. Thus, it suffers from significant load imbalance when applied to the SIP problems. This is mainly due to having a hard mixed-integer master problem that can take hours to be optimized. We thus propose an asynchronous parallel Benders method in a branchand- cut framework. However, relaxing the synchronization requirements entails convergence and various efficiency problems which we address them by introducing several acceleration techniques and search strategies. In particular, we propose the use of artificial subproblems, cut generation, cut aggregation, cut management, and cut propagation. The results indicate that our algorithm reaches higher speedup rates compared to the conventional synchronized methods and it is several orders of magnitude faster than CPLEX 12.7