Matheuristics: using mathematics for heuristic design

Matheuristics are heuristic algorithms based on mathematical tools such as the ones provided by mathematical programming, that are structurally general enough to be applied to different problems with little adaptations to their abstract structure. The result can be metaheuristic hybrids having components derived from the mathematical model of the problems of interest, but the mathematical techniques themselves can define general heuristic solution frameworks. In this paper, we focus our attention on mathematical programming and its contributions to developing effective heuristics. We briefly describe the mathematical tools available and then some matheuristic approaches, reporting some representative examples from the literature. We also take the opportunity to provide some ideas for possible future development

Iterative restricted space search : a solving approach based on hybridization

Face à la complexité qui caractérise les problèmes d'optimisation de grande taille l'exploration complète de l'espace des solutions devient rapidement un objectif inaccessible. En effet, à mesure que la taille des problèmes augmente, des méthodes de solution de plus en plus sophistiquées sont exigées afin d'assurer un certain niveau d 'efficacité. Ceci a amené une grande partie de la communauté scientifique vers le développement d'outils spécifiques pour la résolution de problèmes de grande taille tels que les méthodes hybrides. Cependant, malgré les efforts consentis dans le développement d'approches hybrides, la majorité des travaux se sont concentrés sur l'adaptation de deux ou plusieurs méthodes spécifiques, en compensant les points faibles des unes par les points forts des autres ou bien en les adaptant afin de collaborer ensemble. Au meilleur de notre connaissance, aucun travail à date n'à été effectué pour développer un cadre conceptuel pour la résolution efficace de problèmes d'optimisation de grande taille, qui soit à la fois flexible, basé sur l'échange d'information et indépendant des méthodes qui le composent. L'objectif de cette thèse est d'explorer cette avenue de recherche en proposant un cadre conceptuel pour les méthodes hybrides, intitulé la recherche itérative de l'espace restreint, ±Iterative Restricted Space Search (IRSS)>>, dont, la principale idée est la définition et l'exploration successives de régions restreintes de l'espace de solutions. Ces régions, qui contiennent de bonnes solutions et qui sont assez petites pour être complètement explorées, sont appelées espaces restreints "Restricted Spaces (RS)". Ainsi, l'IRSS est une approche de solution générique, basée sur l'interaction de deux phases algorithmiques ayant des objectifs complémentaires. La première phase consiste à identifier une région restreinte intéressante et la deuxième phase consiste à l'explorer. Le schéma hybride de l'approche de solution permet d'alterner entre les deux phases pour un nombre fixe d'itérations ou jusqu'à l'atteinte d'une certaine limite de temps. Les concepts clés associées au développement de ce cadre conceptuel et leur validation seront introduits et validés graduellement dans cette thèse. Ils sont présentés de manière à permettre au lecteur de comprendre les problèmes que nous avons rencontrés en cours de développement et comment les solutions ont été conçues et implémentées. À cette fin, la thèse a été divisée en quatre parties. La première est consacrée à la synthèse de l'état de l'art dans le domaine de recherche sur les méthodes hybrides. Elle présente les principales approches hybrides développées et leurs applications. Une brève description des approches utilisant le concept de restriction d'espace est aussi présentée dans cette partie. La deuxième partie présente les concepts clés de ce cadre conceptuel. Il s'agit du processus d'identification des régions restreintes et des deux phases de recherche. Ces concepts sont mis en oeuvre dans un schéma hybride heuristique et méthode exacte. L'approche a été appliquée à un problème d'ordonnancement avec deux niveaux de décision, relié au contexte des pâtes et papier: "Pulp Production Scheduling Problem". La troisième partie a permit d'approfondir les concepts développés et ajuster les limitations identifiées dans la deuxième partie, en proposant une recherche itérative appliquée pour l'exploration de RS de grande taille et une structure en arbre binaire pour l'exploration de plusieurs RS. Cette structure a l'avantage d'éviter l'exploration d 'un espace déjà exploré précédemment tout en assurant une diversification naturelle à la méthode. Cette extension de la méthode a été testée sur un problème de localisation et d'allocation en utilisant un schéma d'hybridation heuristique-exact de manière itérative. La quatrième partie généralise les concepts préalablement développés et conçoit un cadre général qui est flexible, indépendant des méthodes utilisées et basé sur un échange d'informations entre les phases. Ce cadre a l'avantage d'être général et pourrait être appliqué à une large gamme de problèmes

Model-Based Heuristics for Combinatorial Optimization

Many problems arising in several and different areas of human knowledge share the characteristic of being intractable in real cases. The relevance of the solution of these problems, linked to their domain of action, has given birth to many frameworks of algorithms for solving them. Traditional solution paradigms are represented by exact and heuristic algorithms. In order to overcome limitations of both approaches and obtain better performances, tailored combinations of exact and heuristic methods have been studied, giving birth to a new paradigm for solving hard combinatorial optimization problems, constituted by model-based metaheuristics. In the present thesis, we deepen the issue of model-based metaheuristics, and present some methods, belonging to this class, applied to the solution of combinatorial optimization problems

Optimal Shipping Decisions in an Airfreight Forwarding Network

This thesis explores three consolidation problems derived from the daily operations of major international airfreight forwarders. First, we study the freight forwarder's unsplittable shipment planning problem in an airfreight forwarding network where a set of cargo shipments have to be transported to given destinations. We provide mixed integer programming formulations that use piecewise-linear cargo rates and account for volume and weight constraints, flight departure/arrival times, as well as shipment-ready times. After exploring the solution of such models using CPLEX, we devise two solution methodologies to handle large problem sizes. The first is based on Lagrangian relaxation, where the problems decompose into a set of knapsack problems and a set of network flow problems. The second is a local branching heuristic that combines branching ideas and local search. The two approaches show promising results in providing good quality heuristic solutions within reasonable computational times, for difficult and large shipment consolidation problems. Second, we further explore the freight forwarder's shipment planning problem with a different type of discount structure - the system-wide discount. The forwarder's cost associated with one flight depends not only on the quantity of freight assigned to that flight, but also on the total freight assigned to other flights operated by the same carrier. We propose a multi-commodity flow formulation that takes shipment volume and over-declaration into account, and solve it through a Lagrangian relaxation approach. We also model the "double-discount" scheme that incorporates both the common flight-leg discount (the one used in the unsplittable shipment problem) and the system-wide discount offered by cargo airlines. Finally, we focus on palletized loading using unit loading devices (ULDs) with pivots, which is different from what we assumed in the previous two research problems. In the international air cargo business, shipments are usually consolidated into containers; those are the ULDs. A ULD is charged depending on whether the total weight exceeds a certain threshold, called the pivot weight. Shipments are charged the under-pivot rate up to the pivot weight. Additional weight is charged at the over-pivot rate. This scheme is adopted for safety reasons to avoid the ULD overloading. We propose three solution methodologies for the air-cargo consolidation problem under the pivot-weight (ACPW), namely: an exact solution approach based on branch-and-price, a best fit decreasing loading heuristic, and an extended local branching. We found superior computational performance with a combination of the multi-level variables and a relaxation-induced neighborhood search for local branching

Essays On Perioperative Services Problems In Healthcare

One of the critical challenges in healthcare operations management is to efficiently utilize the expensive resources needed while maintaining the quality of care provided. Simulation and optimization methods can be effectively used to provide better healthcare services. This can be achieved by developing models to minimize patient waiting times, minimize healthcare supply chain and logistics costs, and maximize access. In this proposal, we study some of the important problems in healthcare operations management. More specifically, we focus on perioperative services and study scheduling of operating rooms (ORs) and management of necessary resources such as staff, equipment, and surgical instruments. We develop optimization and simulation methods to coordinate material handling decisions, inventory management, and OR scheduling. In Chapter 1 of this dissertation, we investigate material handling services to improve the flow of surgical materials in hospitals. The ORs require timely supply of surgical materials such as surgical instruments, linen, and other additional equipment required to perform the surgeries. The availability of surgical instruments at the right location is crucial to both patient safety and cost reduction in hospitals. Similarly, soiled material must also be disposed of appropriately and quickly. Hospitals use automated material handling systems to perform these daily tasks, minimize workforce requirements, reduce risk of contamination, and reduce workplace injuries. Most of the literature related to AGV systems focuses on improving their performance in manufacturing settings. In the last 20 years, several articles have addressed issues relevant to healthcare systems. This literature mainly focuses on improving the design and management of AGV systems to handle the specific challenges faced in hospitals, such as interactions with patients, staff, and elevators; adhering to safety standards and hygiene, etc. In Chapter 1, we focus on optimizing the delivery of surgical instrument case carts from material departments to ORs through automated guided vehicles (AGV). We propose a framework that integrates data analysis with system simulation and optimization. We test the performance of the proposed framework through a case study developed using data from a partnering hospital, Greenville Memorial Hospital (GMH) in South Carolina. Through an extensive set of simulation experiments, we investigate whether performance measures, such as travel time and task completion time, improve after a redesign of AGV pathways. We also study the impact of fleet size on these performance measures and use simulation-optimization to evaluate the performance of the system for different fleet sizes. A pilot study was conducted at GMH to validate the results of our analysis. We further evaluated different policies for scheduling the material handling activities to assess their impact on delays and the level of inventory required. Reducing the inventory level of an instrument may negatively impact the flexibility in scheduling surgeries, cause delays, and therefore, reduce the service level provided. On the other hand, increasing inventory levels may not necessarily eliminate the delays since some delays occur because of inefficiencies in the material handling processes. Hospitals tend to maintain large inventories to ensure that the required instruments are available for scheduled surgery. Typically, the inventory level of surgical instruments is determined by the total number of surgeries scheduled in a day, the daily schedule of surgeries that use the same instrument, the processing capacity of the central sterile storage division (CSSD), and the schedule of material handling activities. Using simulation-optimization tools, we demonstrate that integrating decisions of material handling activities with inventory management has the potential to reduce the cost of the system. In Chapter 2 we focus on coordinating OR scheduling decisions with efficient management of surgical instruments. Hospitals pay more attention to OR scheduling. This is because a large portion of hospitals\u27 income is due to surgical procedures. Inventory management of decisions follows the OR schedules. Previous work points to the cost savings and benefits of optimizing the OR scheduling process. However, based on our review of the literature, only a few articles discuss the inclusion of instrument inventory-related decisions in OR schedules. Surgical instruments are classified as (1) owned by the hospital and (2) borrowed from other hospitals or vendors. Borrowed instruments incur rental costs that can be up to 12-25\% of the listed price of the surgical instrument. A daily schedule of ORs determines how many rental instruments would be required to perform all surgeries in a timely manner. A simple strategy used in most hospitals is to first schedule the ORs, followed by determining the instrument assignments. However, such a strategy may result in low utilization of surgical instruments owned by hospitals. Furthermore, creating an OR schedule that efficiently uses available surgical instruments is a challenging problem. The problem becomes even more challenging in the presence of material handling delays, stochastic demand, and uncertain surgery duration. In this study, we propose an alternative scheduling strategy in which the OR scheduling and inventory management decisions are coordinated. More specifically, we propose a mixed-integer programming model that integrates instrument assignment decisions with OR scheduling to minimize costs. This model determines how many ORs to open, determines the schedule of ORs, and also identifies the instrument assignments for each surgery. If the level of instrument inventory cannot meet the surgical requirements, our model allows instruments to be rented at a higher cost. We introduce and evaluate the solution methods for this problem. We propose a Lagrangean decomposition-based heuristic, which is an iterative procedure. This heuristic separates the scheduling problem from the inventory assignment problem. These subproblems are computationally easier to solve and provide a lower bound on the optimal cost of the integrated OR scheduling problem. The solution of the scheduling subproblem is used to generate feasible solutions in every iteration. We propose two alternatives to find feasible solutions to our problem. These alternatives provide an upper bound on the cost of the integrated scheduling problem. We conducted a thorough sensitivity analysis to evaluate the impact of different parameters, such as the length of the scheduling horizon, the number of ORs that can be used in parallel, the number of surgeries, and various cost parameters on the running time and quality of the solution. Using a case study developed at GMH, we demonstrate that integrating OR scheduling decisions with inventory management has the potential to reduce the cost of the system. The objective of Chapter 3 is to develop quick and efficient algorithms to solve the integrated OR scheduling and inventory management problem, and generate optimal/near-optimal solutions that increase the efficiency of GMH operations. In Chapter 2, we introduced the integrated OR scheduling problem which is a combinatorial optimization problem. As such, the problem is challenging to solve. We faced these challenges when trying to solve the problem directly using the Gurobi solver. The solutions obtained via construction heuristics were much farther from optimality while the Lagrangean decomposition-based heuristics take several hours to find good solutions for large-sized problems. In addition, those methods are iterative procedures and computationally expensive. These challenges have motivated the development of metaheuristics to solve OR scheduling problems, which have been shown to be very effective in solving other combinatorial problems in general and scheduling problems in particular. In Chapter 3, we adopt a metaheuristic, Tabu search, which is a versatile heuristic that is used to solve many different types of scheduling problems. We propose an improved construction heuristic to generate an initial solution. This heuristic identifies the number if ORs to be used and then the assignment of surgeries to ORs. In the second step, this heuristic identifies instrument-surgery assignments based on a first-come, first-serve basis. The proposed Tabu search method improves upon this initial solution. To explore different areas of the feasible region, we propose three neighborhoods that are searched one after the other. For each neighborhood, we create a preferred attribute candidate list which contains solutions that have attributes of good solutions. The solutions on this list are evaluated first before examining other solutions in the neighborhood. The solutions obtained with Tabu search are compared with the lower and upper bounds obtained in Chapter \ref{Ch2}. Using a case study developed at GMH, we demonstrate that high-quality solutions can be obtained by using very little computational time

Scheduled service network design for integrated planning of rail freight transportation

Cette thèse étudie une approche intégrant la gestion de l’horaire et la conception de réseaux de services pour le transport ferroviaire de marchandises. Le transport par rail s’articule autour d’une structure à deux niveaux de consolidation où l’affectation des wagons aux blocs ainsi que des blocs aux services représentent des décisions qui complexifient grandement la gestion des opérations. Dans cette thèse, les deux processus de consolidation ainsi que l’horaire d’exploitation sont étudiés simultanément. La résolution de ce problème permet d’identifier un plan d’exploitation rentable comprenant les politiques de blocage, le routage et l’horaire des trains, de même que l’habillage ainsi que l’affectation du traffic. Afin de décrire les différentes activités ferroviaires au niveau tactique, nous étendons le réseau physique et construisons une structure de réseau espace-temps comprenant trois couches dans lequel la dimension liée au temps prend en considération les impacts temporels sur les opérations. De plus, les opérations relatives aux trains, blocs et wagons sont décrites par différentes couches. Sur la base de cette structure de réseau, nous modélisons ce problème de planification ferroviaire comme un problème de conception de réseaux de services. Le modèle proposé se formule comme un programme mathématique en variables mixtes. Ce dernie r s’avère très difficile à résoudre en raison de la grande taille des instances traitées et de sa complexité intrinsèque. Trois versions sont étudiées : le modèle simplifié (comprenant des services directs uniquement), le modèle complet (comprenant des services directs et multi-arrêts), ainsi qu’un modèle complet à très grande échelle. Plusieurs heuristiques sont développées afin d’obtenir de bonnes solutions en des temps de calcul raisonnables. Premièrement, un cas particulier avec services directs est analysé. En considérant une cara ctéristique spécifique du problème de conception de réseaux de services directs nous développons un nouvel algorithme de recherche avec tabous. Un voisinage par cycles est privilégié à cet effet. Celui-ci est basé sur la distribution du flot circulant sur les blocs selon les cycles issus du réseau résiduel. Un algorithme basé sur l’ajustement de pente est développé pour le modèle complet, et nous proposons une nouvelle méthode, appelée recherche ellipsoidale, permettant d’améliorer davantage la qualité de la solution. La recherche ellipsoidale combine les bonnes solutions admissibles générées par l’algorithme d’ajustement de pente, et regroupe les caractéristiques des bonnes solutions afin de créer un problème élite qui est résolu de facon exacte à l’aide d’un logiciel commercial. L’heuristique tire donc avantage de la vitesse de convergence de l’algorithme d’ajustement de pente et de la qualité de solution de la recherche ellipsoidale. Les tests numériques illustrent l’efficacité de l’heuristique proposée. En outre, l’algorithme représente une alternative intéressante afin de résoudre le problème simplifié. Enfin, nous étudions le modèle complet à très grande échelle. Une heuristique hybride est développée en intégrant les idées de l’algorithme précédemment décrit et la génération de colonnes. Nous proposons une nouvelle procédure d’ajustement de pente où, par rapport à l’ancienne, seule l’approximation des couts liés aux services est considérée. La nouvelle approche d’ajustement de pente sépare ainsi les décisions associées aux blocs et aux services afin de fournir une décomposition naturelle du problème. Les résultats numériques obtenus montrent que l’algorithme est en mesure d’identifier des solutions de qualité dans un contexte visant la résolution d’instances réelles.This thesis studies a scheduled service network design problem for rail freight transportation planning. Rails follow a special two level consolidation organization, and the car-to-block, block-to-service handling procedure complicates daily operations. In this research, the two consolidation processes as well as the operation schedule are considered simultaneously, and by solving this problem, we provide an overall cost-effective operating plan, including blocking policy, train routing, scheduling, make-up policy and traffic distribution. In order to describe various rail operations at the tactical level, we extend the physical network and construct a 3-layer time-space structure, in which the time dimension takes into consideration the temporal impacts on operations. Furthermore, operations on trains, blocks, and cars are described in different layers. Based on this network structure, we model the rail planning problem to a service network design formulation. The proposed model relies on a complex mixed-integer programming formulation. The problem is very hard to solve due to the computational difficulty as well as the tremendous size of the application instances. Three versions of the problem are studied, which are the simplified model (with only non-stop services), complete model (with both non-stop and multi-stop services) and very-large-scale complete model. Heuristic algorithms are developed to provide good feasible solutions in reasonable computing efforts. A special case with non-stop services is first studied. According to a specific characteristic of the direct service network design problem, we develop a tabu search algorithm. The tabu search moves in a cycle-based neighborhood, where flows on blocks are re-distributed according to the cycles in a conceptual residual network. A slope scaling based algorithm is developed for the complete model, and we propose a new method, called ellipsoidal search, to further improve the solution quality. Ellipsoidal search combines the good feasible solutions generated from the slope scaling, and collects the features of good solutions into an elite problem, and solves it with exact solvers. The algorithm thus takes advantage of the convergence speed of slope scaling and solution quality of ellipsoidal search, and is proven effective. The algorithm also presents an alternative for solving the simplified problem. Finally, we work on the very-large-size complete model. A hybrid heuristic is developed by integrating the ideas of previous research with column generation. We propose a new slope scaling scheme where, compared with the previous scheme, only approximate service costs instead of both service and block costs are considered. The new slope scaling scheme thus separates the block decisions and service decisions, and provide a natural decomposition of the problem. Experiments show the algorithm is good to solve real-life size instances