Risk Minimization, Regret Minimization and Progressive Hedging Algorithms

This paper begins with a study on the dual representations of risk and regret measures and their impact on modeling multistage decision making under uncertainty. A relationship between risk envelopes and regret envelopes is established by using the Lagrangian duality theory. Such a relationship opens a door to a decomposition scheme, called progressive hedging, for solving multistage risk minimization and regret minimization problems. In particular, the classical progressive hedging algorithm is modified in order to handle a new class of linkage constraints that arises from reformulations and other applications of risk and regret minimization problems. Numerical results are provided to show the efficiency of the progressive hedging algorithms.Comment: 21 pages, 2 figure

Re-Solving Stochastic Programming Models for Airline Revenue Management

We study some mathematical programming formulations for the origin-destination model in airline revenue management. In particular, we focus on the traditional probabilistic model proposed in the literature. The approach we study consists of solving a sequence of two-stage stochastic programs with simple recourse, which can be viewed as an approximation to a multi-stage stochastic programming formulation to the seat allocation problem. Our theoretical results show that the proposed approximation is robust, in the sense that solving more successive two-stage programs can never worsen the expected revenue obtained with the corresponding allocation policy. Although intuitive, such a property is known not to hold for the traditional deterministic linear programming model found in the literature. We also show that this property does not hold for some bid-price policies. In addition, we propose a heuristic method to choose the re-solving points, rather than re-solving at equally spaced times as customary. Numerical results are presented to illustrate the effectiveness of the proposed approach

Improved genetic algorithms by means of fuzzy crossover operators for revenue management in airlines

Abstract: Revenue Management is an economic policy that increases the earned profit by adjusting the service demand and inventory. Revenue Management in airlines correlates with inventory control and price levels in different fare classes. We focus on pricing and seat allocation problems in airlines by introducing a constrained optimization problem in Binary Integer Programming (BIP) formulation. Two BIP problems are represented. Moreover, some improved Genetic Algorithms (GAs) approaches are used to solve these problems. We introduce new crossover operators that assign a Fuzzy Membership Function to each parent in GAs. We achieve better outputs with new methods that take lower calculation times and earn higher profits. Three different test problems in different scales are selected to evaluate the effectiveness of each algorithm. This paper defines new crossover operators that help to reach better solutions that take lower calculation times and more earned profits

Stochastic programming approaches to air traffic flow management under the uncertainty of weather

As air traffic congestion grows, air traffic flow management (ATFM) is becoming a great concern. ATFM deals with air traffic and the efficient utilization of the airport and airspace. Air traffic efficiency is heavily influenced by unanticipated factors, or uncertainties, which can come from several sources such as mechanical breakdown; however, weather is the main unavoidable cause of uncertainty. Because weather is unpredictable, it poses a critical challenge for ATFM in current airport and airspace operations. Convective weather results in congestion at airports as well as in airspace sectors. During times of congestion, the decision as how and when to send aircraft toward an airspace sector in the presence of weather is difficult. To approach this problem, we first propose a two-stage stochastic integer program by emphasizing a given single sector. By considering ground delay, cancellation, and cruise speed for each flight on the ground in the first stage, as well as air holding and diversion recourse actions for each flight in the air in the second stage, our model determines how aircraft are sent toward a sector under the uncertainty of weather. However, due to the large number of weather scenarios, the model is intractable in practice. To overcome the intractability, we suggest a rolling horizon method to solve the problem to near optimal. Lagrangian relaxation and subgradient method are used to justify the rolling horizon method. Since the rolling horizon method can be solved in real time, we can apply it to actual aircraft schedules to reduce the costs incurred on the ground as well as in airspace. We then extend our two-stage model to a multistage stochastic program, which increases the number of possible weather realizations and results a more efficient schedule in terms of costs. The rolling horizon method as well as Lagrangian relaxation and subgradient method are applied to this multistage model. An overall comparison among the previously described methodologies are presented.Ph.D.Committee Chair: Johnson, Ellis; Committee Co-Chair: Clarke, John-Paul; Committee Member: Ahmed, Shabbir; Committee Member: Sokol, Joel; Committee Member: Solak, Sena

TOWARDS AN EFFICIENT DECISION POLICY FOR CLOUD SERVICE PROVIDERS

Cloud service providers may face the problem of how to price infrastructure services and how this pricing may impact the resource utilization. One aspect of this problem is how Cloud service providers would decide to accept or reject requests for services when the resources for offering these services become scarce. A decision support policy called Customized Bid-Price Policy (CBPP) is proposed in this paper to decide efficiently, when a large number of services or complex services can be offered over a finite time horizon. This heuristic outperforms well-known policies, if bid prices cannot be updated frequently during incoming requests and an automated update of bid prices is required to achieve more accurate decisions. Since CBPP approximates the revenue offline before the requests occur, it has a low runtime compared to other approaches during the online phase. The performance is examined via simulation and the pre-eminence of CBPP is statistically proven

Large scale multistage stochastic integer programming with applications in electric power systems

Multistage stochastic integer programming (MSIP) is a framework for sequential decision making under uncertainty, where the uncertainty is modeled by a general stochastic process, and the decision space involves integer variables and complicated constraints. Many power system applications, such as generation capacity planning and scheduling under uncertainty stemming from renewable generation, demand variability and price volatility, can be naturally formulated as MSIP problems. In this thesis, we develop general purpose solution methods for large-scale MSIP problems and demonstrate their effectiveness on various power systems applications. In the first part of this thesis, we consider an MSIP approach for electrical power generation capacity expansion problems under demand and fuel price uncertainty. We propose a partially adaptive stochastic mixed integer optimization model in which the capacity expansion plan is fully adaptive to the uncertainty evolution up to a certain period, and is static thereafter. Any solution to the partially adaptive model is feasible to the multistage model and we provide analytical bounds on the quality of such a solution. We propose an algorithm that solves a sequence of partially adaptive models, to recursively construct an approximate solution to the multistage problem. We apply the proposed approach to a realistic generation expansion case study. In the second part of this thesis, we develop decomposition algorithms for general MSIP problems with binary state variables. By exploiting the binary nature of the state variables, we extend the nested Benders decomposition algorithm to this problem class. Key to our developments are new families of cuts that guarantee finite convergence of the proposed algorithm. We also propose a stochastic variant of the nested Benders decomposition algorithm, called Stochastic Dual Dynamic integer Programming (SDDiP), and give a rigorous proof of its finite convergence with probability one to an optimal policy. We provide extensive computational results using the SDDiP approach for generation capacity planning, portfolio optimization, and airline revenue management problems. The final part of this thesis focuses on adapting the SDDiP approach to solve the multistage stochastic unit commitment (MSUC) problem. Unit commitment is a key operational problem in power systems used to determine the optimal generation schedule over the next day or week. Incorporating uncertainty in this already difficult optimization problem imparts severe challenges. We reformulate the MSUC problem such that each stage problem only depends on information from the previous stage and the uncertainty realization. This new formulation is amenable to our SDDiP approach. We propose a variety of computational enhancements to adapt the method to MSUC. Through extensive computational results, we demonstrate the effectiveness of our approach in solving realistic scale MSUC problems.Ph.D