Stochastic optimization of staffing for multiskill call centers

Dans cette thèse, nous étudions le problème d’optimisation des effectifs dans les centres d’appels, dans lequel nous visons à minimiser les coûts d’exploitation tout en offrant aux clients une qualité de service (QoS) élevée. Nous introduisons également l'utilisation de contraintes probabilistes qui exigent que la qualité de service soit satisfaite avec une probabilité donnée. Ces contraintes sont adéquates dans le cas où la performance est mesurée sur un court intervalle de temps, car les mesures de QoS sont des variables aléatoires sur une période donnée. Les problèmes de personnel proposés sont difficiles en raison de l'absence de forme analytique pour les contraintes probabilistes et doivent être approximées par simulation. En outre, les fonctions QoS sont généralement non linéaires et non convexes. Nous considérons les problèmes d’affectation personnel dans différents contextes et étudions les modèles proposés tant du point de vue théorique que pratique. Les méthodologies développées sont générales, en ce sens qu'elles peuvent être adaptées et appliquées à d'autres problèmes de décision dans les systèmes de files d'attente. La thèse comprend trois articles traitant de différents défis en matière de modélisation et de résolution de problèmes d'optimisation d’affectation personnel dans les centres d'appels à compétences multiples. Les premier et deuxième article concernent un problème d'optimisation d'affectation de personnel en deux étapes sous l'incertitude. Alors que dans le second, nous étudions un modèle général de programmation stochastique discrète en deux étapes pour fournir une garantie théorique de la consistance de l'approximation par moyenne échantillonnale (SAA) lorsque la taille des échantillons tend vers l'infini, le troisième applique l'approche du SAA pour résoudre le problème d’optimisation d'affectation de personnel en deux étapes avec les taux d’arrivée incertain. Les deux articles indiquent la viabilité de l'approche SAA dans notre contexte, tant du point de vue théorique que pratique. Pour être plus précis, dans le premier article, nous considérons un problème stochastique discret général en deux étapes avec des contraintes en espérance. Nous formulons un problème SAA avec échantillonnage imbriqué et nous montrons que, sous certaines hypothèses satisfaites dans les exemples de centres d'appels, il est possible d'obtenir les solutions optimales du problème initial en résolvant son SAA avec des échantillons suffisamment grands. De plus, nous montrons que la probabilité que la solution optimale du problème de l’échantillon soit une solution optimale du problème initial tend vers un de manière exponentielle au fur et à mesure que nous augmentons la taille des échantillons. Ces résultats théoriques sont importants, non seulement pour les applications de centre d'appels, mais également pour d'autres problèmes de prise de décision avec des variables de décision discrètes. Le deuxième article concerne les méthodes de résolution d'un problème d'affectation en personnel en deux étapes sous incertitude du taux d'arrivée. Le problème SAA étant coûteux à résoudre lorsque le nombre de scénarios est important. En effet, pour chaque scénario, il est nécessaire d'effectuer une simulation pour estimer les contraintes de QoS. Nous développons un algorithme combinant simulation, génération de coupes, renforcement de coupes et décomposition de Benders pour résoudre le problème SAA. Nous montrons l'efficacité de l'approche, en particulier lorsque le nombre de scénarios est grand. Dans le dernier article, nous examinons les problèmes de contraintes en probabilité sur les mesures de niveau de service. Notre méthodologie proposée dans cet article est motivée par le fait que les fonctions de QoS affichent généralement des courbes en S et peuvent être bien approximées par des fonctions sigmoïdes appropriées. Sur la base de cette idée, nous avons développé une nouvelle approche combinant la régression non linéaire, la simulation et la recherche locale par région de confiance pour résoudre efficacement les problèmes de personnel à grande échelle de manière viable. L’avantage principal de cette approche est que la procédure d’optimisation peut être formulée comme une séquence de simulations et de résolutions de problèmes de programmation linéaire. Les résultats numériques basés sur des exemples réels de centres d'appels montrent l'efficacité pratique de notre approche. Les méthodologies développées dans cette thèse peuvent être appliquées dans de nombreux autres contextes, par exemple les problèmes de personnel et de planification dans d'autres systèmes basés sur des files d'attente avec d'autres types de contraintes de QoS. Celles-ci soulèvent également plusieurs axes de recherche qu'il pourrait être intéressant d'étudier. Par exemple, une approche de regroupement de scénarios pour atténuer le coût des modèles d'affectation en deux étapes, ou une version d'optimisation robuste en distribution pour mieux gérer l'incertitude des données.In this thesis, we study the staffing optimization problem in multiskill call centers, in which we aim at minimizing the operating cost while delivering a high quality of service (QoS) to customers. We also introduce the use of chance constraints which require that the QoSs are met with a given probability. These constraints are adequate in the case when the performance is measured over a short time interval as QoS measures are random variables in a given time period. The proposed staffing problems are challenging in the sense that the stochastic constraints have no-closed forms and need to be approximated by simulation. In addition, the QoS functions are typically non-linear and non-convex. We consider staffing optimization problems in different settings and study the proposed models in both theoretical and practical aspects. The methodologies developed are general, in the sense that they can be adapted and applied to other staffing/scheduling problems in queuing-based systems. The thesis consists of three articles dealing with different challenges in modeling and solving staffing optimization problems in multiskill call centers. The first and second articles concern a two-stage staffing optimization problem under uncertainty. While in the first one, we study a general two-stage discrete stochastic programming model to provide a theoretical guarantee for the consistency of the sample average approximation (SAA) when the sample sizes go to infinity, the second one applies the SAA approach to solve the two-stage staffing optimization problem under arrival rate uncertainty. Both papers indicate the viability of the SAA approach in our context, in both theoretical and practical aspects. To be more precise, in the first article, we consider a general two-stage discrete stochastic problem with expected value constraints. We formulate its SAA with nested sampling. We show that under some assumptions that hold in call center examples, one can obtain the optimal solutions of the original problem by solving its SAA with large enough sample sizes. Moreover, we show that the probability that the optimal solution of the sample problem is an optimal solution of the original problem, approaches one exponentially fast as we increase the sample sizes. These theoretical findings are important, not only for call center applications, but also for other decision-making problems with discrete decision variables. The second article concerns solution methods to solve a two-stage staffing problem under arrival rate uncertainty. It is motivated by the fact that the SAA version of the two-stage staffing problem becomes expensive to solve with a large number of scenarios, as for each scenario, one needs to use simulation to approximate the QoS constraints. We develop an algorithm that combines simulation, cut generation, cut strengthening and Benders decomposition to solve the SAA problem. We show the efficiency of the approach, especially when the number of scenarios is large. In the last article, we consider problems with chance constraints on the service level measures. Our methodology proposed in this article is motivated by the fact that the QoS functions generally display ``S-shape'' curves and might be well approximated by appropriate sigmoid functions. Based on this idea, we develop a novel approach that combines non-linear regression, simulation and trust region local search to efficiently solve large-scale staffing problems in a viable way. The main advantage of the approach is that the optimization procedure can be formulated as a sequence of steps of performing simulation and solving linear programming models. Numerical results based on real-life call center examples show the practical viability of our approach. The methodologies developed in this thesis can be applied in many other settings, e.g., staffing and scheduling problems in other queuing-based systems with other types of QoS constraints. These also raise several research directions that might be interesting to investigate. For examples, a clustering approach to mitigate the expensiveness of the two-stage staffing models, or a distributionally robust optimization version to better deal with data uncertainty

Convex Nonlinear and Integer Programming Approaches for Distributionally Robust Optimization of Complex Systems

The primary focus of the dissertation is to develop distributionally robust optimization (DRO) models and related solution approaches for decision making in energy and healthcare service systems with uncertainties, which often involves nonlinear constraints and discrete decision variables. Without assuming specific distributions, DRO techniques solve for solutions against the worst-case distribution of system uncertainties. In the DRO framework, we consider both risk-neutral (e.g., expectation) and risk-averse (e.g., chance constraint and Conditional Value-at-Risk (CVaR)) measures. The aim is twofold: i) developing efficient solution algorithms for DRO models with integer and/or binary variables, sometimes nonlinear structures and ii) revealing managerial insights of DRO models for specific applications. We mainly focus on DRO models of power system operations, appointment scheduling, and resource allocation in healthcare. Specifically, we first study stochastic optimal power flow (OPF), where (uncertain) renewable integration and load control are implemented to balance supply and (uncertain) demand in power grids. We propose a chance-constrained OPF (CC-OPF) model and investigate its DRO variant which is reformulated as a semidefinite programming (SDP) problem. We compare the DRO model with two benchmark models, in the IEEE 9-bus, 39-bus, and 118-bus systems with different flow congestion levels. The DRO approach yields a higher probability of satisfying the chance constraints and shorter solution time. It also better utilizes reserves at both generators and loads when the system has congested flows. Then we consider appointment scheduling under random service durations with given (fixed) appointment arrival order. We propose a DRO formulation and derive a conservative SDP reformulation. Furthermore, we study a scheduling variant under random no-shows of appointments and derive tractable reformulations for certain beliefs of no-show patterns. One preceding problem of appointment scheduling in the healthcare service operations is the surgery block allocation problem that assigns surgeries to operating rooms. We derive an equivalent 0-1 SDP reformulation and a less conservative 0-1 second-order cone programming (SOCP) reformulation for its DRO model. Finally, we study distributionally robust chance-constrained binary programs (DCBP) for limiting the probability of undesirable events, under mean-covariance information. We reformulate DCBPs as equivalent 0-1 SOCP formulations under two moment-based ambiguity sets. We further exploit the submodularity of the 0-1 SOCP reformulations under diagonal and non-diagonal matrices. We derive extended polymatroid inequalities via submodularity and lifting, which are incorporated into a branch-and-cut algorithm incorporated for efficiently solving DCBPs. We demonstrate the computational efficacy and solution performance with diverse instances of a chance-constrained bin packing problem.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/149946/1/zyiling_1.pd

Service Region Design for Urban Electric Vehicle Sharing Systems

Emerging collaborative consumption business models have shown promise in terms of both generating business opportunities and enhancing the efficient use of resources. In the transportation domain, car sharing models are being adopted on a mass scale in major metropolitan areas worldwide. This mode of servicized mobility bridges the resource efficiency of public transit and the flexibility of personal transportation. Beyond the significant potential to reduce car ownership, car sharing shows promise in supporting the adoption of fuel- efficient vehicles, such as electric vehicles (EVs), due to these vehicles special cost structure with high purchase but low operating costs. Recently, key players in the car sharing business, such as Autolib, Car2Go and DriveNow, have begun to employ EVs in an operations model that accommodates one-way trips. On the one hand (and particularly in free-floating car sharing), the one-way model results in significant improvements in coverage of travel needs and therefore in adoption potential compared with the conventional round-trip-only model (advocated by ZipCar, for example). On the other hand, this model poses tremendous planning and operational challenges. In this work, we study the planning problem faced by service providers in designing a geographical service region in which to operate the service. This decision entails trade-offs between maximizing customer catchment by covering travel needs and controlling fleet operations costs. We develop a mathematical programming model that incorporates details of both customer adoption behavior and fleet management (including EV repositioning and charging) under imbalanced travel patterns. To address inherent planning uncertainty with regard to adoption patterns, we employ a distributionally robust optimization framework that informs robust decisions to overcome possible ambiguity (or lacking) of data. Mathematically, the problem can be approximated by a mixed integer second-order cone program, which is computationally tractable with practical scale data. Applying this approach to the case of Car2Go’s service with real operations data, we address a number of planning questions and suggest that there is potential for the future development of this service

On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling

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Addressing Nonlinearity and Uncertainty via Mixed Integer Programming Approaches

The main focus of the dissertation is to develop decision-making support tools that address nonlinearity and uncertainty appearing in real-world applications via mixed integer programming (MIP) approaches. When making decisions under uncertainty, knowing the accurate probability distributions of the uncertain parameters can help us predict their future realization, which in turn helps to make better decisions. In practice, however, it is oftentimes hard to estimate such a distribution precisely. As a consequence, if the estimated distribution is biased, the decisions thus made can end up with disappointing outcomes. To address this issue, we model uncertainty in the decision-making process through a distributionally robust optimization (DRO) approach, which aims to find a solution that hedges against the worst-case distribution within a pre-defined ambiguity set, i.e., a collection of probability distributions that share some distributional and/or statistical characteristics in common. The role of the ambiguity set is crucial as it affects both solution quality and computational tractability of the DRO model. In this dissertation, we tailor the ambiguity sets based on the available historical data in healthcare operations and energy systems, and derive efficient solution approaches for the DRO models via MIP approaches. Through extensive numerical studies, we show that the DRO solutions yield better out-of-sample performance, and the computational performance of the proposed MIP approaches is encouraging. Finally, we provide some managerial insights into the operations of healthcare and energy systems.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155030/1/msryu_1.pd

Optimisation du trafic aérien à l'arrivée dans la zone terminale et dans l'espace aérien étendu

Selon les prévisions à long terme du trafic aérien de l'Organisation de l'Aviation Civile Internationale (OACI) en 2018, le trafic mondial de passagers devrait augmenter de 4,2% par an de 2018 à 2038. Bien que l'épidémie de COVID-19 ait eu un impact énorme sur le transport aérien, il se rétablit progressivement. Dès lors, l'efficacité et la sécurité resteront les principales problématiques du trafic aérien, notamment au niveau de la piste qui est le principal goulot d'étranglement du système. Dans le domaine de la gestion du trafic aérien, la zone de manœuvre terminale (TMA) est l'une des zones les plus complexes à gérer. En conséquence, le développement d'outils d'aide à la décision pour gérer l'arrivée des avions est primordial. Dans cette thèse, nous proposons deux approaches d'optimisation qui visent à fournir des solutions de contrôle pour la gestion des arrivées dans la TMA et dans un horizon étendu intégrant la phase en route. Premièrement, nous abordons le problème d'ordonnancement des avions sous incertitude dans la TMA. La quantification et la propagation de l'incertitude le long des routes sont réalisées grâce à un modèle de trajectoire qui représente les informations temporelles sous forme de variables aléatoires. La détection et la résolution des conflits sont effectuées à des points de cheminement d'un réseau prédéfini sur la base des informations temporelles prédites à partir de ce modèle. En minimisant l'espérance du nombre de conflits, les vols peuvent être bien séparés. Outre le modèle proposé, deux autres modèles de la litérrature - un modèle déterministe et un modèle intégrant des marges de séparation - sont présentés comme références. Un recuit simulé (SA) combiné à une fenêtre glissante temporelle est proposé pour résoudre une étude de cas de l'aéroport de Paris Charles de Gaulle (CDG). De plus, un cadre de simulation basé sur l'approche Monte-Carlo est implémenté pour perturber aléatoirement les horaires optimisés des trois modèles afin d'évaluer leurs performances. Les résultats statistiques montrent que le modèle proposé présente des avantages absolus dans l'absorption des conflits en cas d'incertitude. Dans une deuxième partie, nous abordons un problème dynamique basé sur le concept de Gestion des Arrivées Étendue (E-AMAN). L'horizon E-AMAN est étendu jusqu'à 500 NM de l'aéroport de destination permettant ainsi une planification anticipée. Le caractère dynamique est traitée par la mise à jour périodique des informations de trajectoires réelles sur la base de l'approche par horizon glissant. Pour chaque horizon temporel, un sous-problème est établi avec pour objectif une somme pondérée de métriques de sécurité du segment en route et de la TMA. Une approche d'attribution dynamique des poids est proposée pour souligner le fait qu'à mesure qu'un aéronef se rapproche de la TMA, le poids de ses métriques associées à la TMA devrait augmenter. Une étude de cas est réalisée à partir des données réelles de l'aéroport de Paris CDG. Les résultats finaux montrent que grâce à cet ajustement anticipé, les heures d'arrivée des avions sont proches des heures prévues tout en assurant la sécurité et en réduisant les attentes. Dans la troisième partie de cette thèse, on propose un algorithme qui accélère le processus d'optimisation. Au lieu d'évaluer les performances de tous les aéronefs, les performances d'un seul aéronef sont concentrées dans la fonction objectif. Grâce à ce changement, le processus d'optimisation bénéficie d'une évaluation d'objectif rapide et d'une vitesse de convergence élevée. Afin de vérifier l'algorithme proposé, les résultats sont analysés en termes de temps d'exécution et de qualité des résultats par rapport à l'algorithme utilisé à l'origine.According to the long term air traffic forecasts done by International Civil Aviation Organization (ICAO) in 2018, global passenger traffic is expected to grow by 4.2% annually from 2018 to 2038 using the traffic data of 2018 as a baseline. Even though the outbreak of COVID-19 has caused a huge impact on the air transportation, it is gradually restoring. Considering the potential demand in future, air traffic efficiency and safety will remain critical issues to be considered. In the airspace system, the runway is the main bottleneck in the aviation chain. Moreover, in the domain of air traffic management, the Terminal Maneuvering Area (TMA) is one of the most complex areas with all arrivals converging to land. This motivates the development of suitable decision support tools for providing proper advisories for arrival management. In this thesis, we propose two optimization approaches that aim to provide suitable control solutions for arrival management in the TMA and in the extended horizon that includes the TMA and the enroute phase. In the first part of this thesis, we address the aircraft scheduling problem under uncertainty in the TMA. Uncertainty quantification and propagation along the routes are realized in a trajectory model that formulates the time information as random variables. Conflict detection and resolution are performed at waypoints of a predefined network based on the predicted time information from the trajectory model. By minimizing the expected number of conflicts, consecutively operated flights can be well separated. Apart from the proposed model, two other models - the deterministic model and the model that incorporates separation buffers - are presented as benchmarks. Simulated annealing (SA) combined with the time decomposition sliding window approach is used for solving a case study of the Paris Charles de Gaulle (CDG) airport. Further, a simulation framework based on the Monte-Carlo approach is implemented to randomly perturb the optimized schedules of the three models so as to evaluate their performances. Statistical results show that the proposed model has absolute advantages in conflict absorption when uncertainty arises. In the second part of this thesis, we address a dynamic/on-line problem based on the concept of Extended Arrival MANagement (E-AMAN). The E-AMAN horizon is extended up to 500NM from the destination airport so as to enhance the cooperation and situational awareness of the upstream sector control and the TMA control. The dynamic feature is addressed by periodically updating the real aircraft trajectory information based on the rolling horizon approach. For each time horizon, a sub-problem is established taking the weighted sum of safety metrics in the enroute segment and in the TMA as objective. A dynamic weights assignment approach is proposed to emphasize the fact that as an aircraft gets closer to the TMA, the weight for its metrics associated with the TMA should increase. A case study is carried out using the real arrival traffic data of the Paris CDG airport. Final results show that through early adjustment, the arrival time of the aircraft can meet the required schedule for entering the TMA, thus ensuring overall safety and reducing holding time. In the third part of this thesis, an algorithm that expedites the optimization process is proposed. Instead of evaluating the performance of all aircraft, single aircraft performance is focused and a corresponding objective function is created. Through this change, the optimization process benefits from fast evaluation of objective and high convergence speed. In order to verify the proposed algorithm, results are analyzed in terms of execution time and quality of result compared to the originally used algorithm

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Robust and stochastic approaches to network capacity design under demand uncertainty

This thesis considers the network capacity design problem with demand uncertainty using the stochastic, robust and distributionally robust stochastic optimization approaches (DRSO). Network modeling in itself has found wide areas of application in most fields of human endeavor. The network would normally consist of source (origin) and sink (destination) nodes connected by arcs that allow for flows of an entity from the origin to the destination nodes. In this thesis, a special type of the minimum cost flow problem is addressed, the multi-commodity network flow problem. Commodities are the flow types that are transported on a shared network. Offered demands are, for the most part, unknown or uncertain, hence a model that immune against this uncertainty becomes the focus as well as the practicability of such models in the industry. This problem falls under the two-stage optimization framework where a decision is delayed in time to adjust for the first decision earlier made. The first stage decision is called the "here and now", while the second stage traffic re-adjustment is the "wait and see" decision. In the literature, the decision-maker is often believed to know the shape of the uncertainty, hence we address this by considering a data-driven uncertainty set. The research also addressed the non-linearity of cost function despite the abundance of literature assuming linearity and models proposed for this. This thesis consist of four main chapters excluding the "Introduction" chapter and the "Approaches to Optimization under Uncertainty" chapter where the methodologies are reviewed. The first of these four, Chapter 3, proposes the two models for the Robust Network Capacity Expansion Problem (RNCEP) with cost non-linearity. These two are the RNCEP with fixed-charge cost and RNCEP with piecewise-linear cost. The next chapter, Chapter 4, compares the RNCEP models under two types of uncertainties in order to address the issue of usefulness in a real world setting. The resulting two robust models are also comapared with the stochastic optimization model with distribution mean. Chapter 5 re-examines the earlier problem using machine learning approaches to generate the two uncertainty sets while the last of these chapters, Chapter 6, investigates DRSO model to network capacity planning and proposes an efficient solution technique