Nilay Noyan

Risk-averse two-stage stochastic programming with an application to disaster management

Traditional two-stage stochastic programming is risk-neutral; that is, it considers the expectation as the preference criterion while comparing the random variables (e.g., total cost) to identify the best decisions. However, in the presence of variability risk measures should be incorporated into decision making problems in order to model its effects. In this study, we consider a risk-averse two-stage stochastic programming model, where we specify the conditional-value-at-risk (CVaR) as the risk measure. We construct two decomposition algorithms based on the generic Benders-decomposition approach to solve such problems. Both single-cut and multicut versions of the proposed decomposition algorithms are presented. We adapt the concepts of the value of perfect information (VPI) and the value of the stochastic solution (VSS) for the proposed risk-averse two-stage stochastic programming framework and define two stochastic measures on the VPI and VSS. We apply the proposed model to disaster management, which is one of the research fields that can significantly benefit from risk-averse two-stage stochastic programming models. In particular, we consider the problem of determining the response facility locations and the inventory levels of the relief supplies at each facility in the presence of uncertainty in demand and the damage level of the disaster network. We present numerical results to discuss how incorporating a risk measure affects the optimal solutions and demonstrate the computational effectiveness of the proposed methods.

Alternate risk measures for emergency medical service system design

The stochastic nature of emergency service requests and the unavailability of emergency vehicles when requested to serve demands are critical issues in constructing valid models representing real life emergency medical service (EMS) systems. We consider an EMS system design problem with stochastic demand and locate the emergency response facilities and vehicles in order to ensure target levels of coverage, which are quantified using risk measures on random unmet demand. The target service levels for each demand site and also for the entire service area are specified. In order to increase the possibility of representing a wider range of risk preferences we develop two types of stochastic optimization models involving alternate risk measures. The first type of the model includes integrated chance constraints (ICCs ), whereas the second type incorporates ICCs  and a stochastic dominance constraint. We develop solution methods for the proposed single-stage stochastic optimization problems and present extensive numerical results demonstrating their computational effectiveness.

Relaxations of linear programming problems with first order stochastic dominance constraints

Linear stochastic programming problems with first order stochastic dominance (FSD) constraints are non-convex. For their mixed 0-1 linear programming formulation we present two convex relaxations based on second order stochastic dominance (SSD). We develop necessary and sufficient conditions for FSD, used to obtain a disjunctive programming formulation and to strengthen one of the SSD-based relaxations.

Stochastic network design for disaster preparedness

This article introduces a risk-averse stochastic modeling approach for a pre-disaster relief network design problem under uncertain demand and transportation capacities. The sizes and locations of the response facilities and the inventory levels of relief supplies at each facility are determined while guaranteeing a certain level of network reliability. A probabilistic constraint on the existence of a feasible flow is introduced to ensure that the demand for relief supplies across the network is satisfied with a specified high probability. Responsiveness is also accounted for by defining multiple regions in the network and introducing local probabilistic constraints on satisfying demand within each region. These local constraints ensure that each region is self-sufficient in terms of providing for its own needs with a large probability. In particular, the Gale–Hoffman inequalities are used to represent the conditions on the existence of a feasible network flow. The solution method rests on two pillars. A preprocessing algorithm is used to eliminate redundant Gale–Hoffman inequalities and then proposed models are formulated as computationally efficient mixed-integer linear programs by utilizing a method based on combinatorial patterns. Computational results for a case study and randomly generated problem instances demonstrate the effectiveness of the models and the solution method.

Single-Leg Airline Revenue Management with Overbooking

Airline revenue management is concerned with identifying the maximum revenue seat allocation policies. Because a major loss in revenue results from cancellations and no-shows, overbooking has received significant attention in the literature over the years. In this study, we propose new static and dynamic single-leg overbooking models. In the static case we introduce two models: the first one aims to determine the overbooking limit and the second one is about finding the overbooking limit and the booking limits to allocate the virtual capacity among multiple fare classes. Because the second static model is hard to solve, we also introduce computationally tractable models that give upper and lower bounds on its optimal expected net revenue. In the dynamic case, we propose a dynamic programming model, which is based on two streams of events. The first stream corresponds to the arrival of booking requests and the second one corresponds to the cancellations. We conduct simulation experiments to illustrate the effectiveness of the proposed models.

A Stochastic Optimization Model for Designing Last Mile Relief Networks

In this study, we introduce a distribution network design problem that determines the locations and capacities of the relief distribution points in the last mile network, while considering demand-and network-related uncertainties in the post-disaster environment. The problem addresses the critical concerns of relief organizations in designing last mile networks, which are providing accessible and equitable service to beneficiaries. We focus on two types of supply allocation policies and propose a hybrid version considering their different implications on equity and accessibility. Then, we develop a two-stage stochastic programming model that incorporates the hybrid allocation policy and achieves high levels of accessibility and equity simultaneously. We devise a branch-and-cut algorithm based on Benders decomposition to solve large problem instances in reasonable times and conduct a numerical study to demonstrate the computational effectiveness of the solution method. We also illustrate the application of our model on a case study based on real-world data from the 2011 Van earthquake in Turkey.

Mathematical programming approaches for generating p-efficient points

Probabilistically constrained problems, in which the random variables are finitely distributed, are non-convex in general and hard to solve. The p-efficiency concept has been widely used to develop efficient methods to solve such problems. Those methods require the generation of p-efficient points (pLEPs) and use an enumeration scheme to identify pLEPs. In this paper, we consider a random vector characterized by a finite set of scenarios and generate pLEPs by solving a mixed-integer programming (MIP) problem. We solve this computationally challenging MIP problem with a new mathematical programming framework. It involves solving a series of increasingly tighter outer approximations and employs, as algorithmic techniques, a bundle preprocessing method, strengthening valid inequalities, and a fixing strategy. The method is exact (resp., heuristic) and ensures the generation of pLEPs (resp., quasi pLEPs) if the fixing strategy is not (resp., is) employed, and it can be used to generate multiple pLEPs. To the best of our knowledge, generating a set of pLEPs using an optimization-based approach and developing effective methods for the application of the p-efficiency concept to the random variables described by a finite set of scenarios are novel. We present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques.

Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints

Stochastic dominance relations are well studied in statistics, decision theory and economics. Recently, there has been significant interest in introducing dominance relations into stochastic optimization problems as constraints. In the discrete case, stochastic optimization models involving second order stochastic dominance constraints can be solved by linear programming. However, problems involving first order stochastic dominance constraints are potentially hard due to the non-convexity of the associated feasible regions. In this paper we consider a mixed 0–1 linear programming formulation of a discrete first order constrained optimization model and present a relaxation based on second order constraints. We derive some valid inequalities and restrictions by employing the probabilistic structure of the problem. We also generate cuts that are valid inequalities for the disjunctive relaxations arising from the underlying combinatorial structure of the problem by applying the lift-and-project procedure. We describe three heuristic algorithms to construct feasible solutions, based on conditional second order constraints, variable fixing, and conditional value at risk. Finally, we present numerical results for several instances of a real world portfolio optimization problem.

Arrival Rate Approximation by Nonnegative Cubic Splines

We describe an optimization method to approximate the arrival rate of data such as e-mail messages, Web site visits, changes to databases, and changes to Web sites mirrored by other servers. We model these arrival rates as non-homogeneous Poisson process based on observed arrival data. We estimate the arrival function by cubic splines using the maximum likelihood principle. A critical feature of the model is that the splines are constrained to be everywhere nonnegative. We formulate this constraint using a characterization of nonnegative polynomials by positive semidefinite matrices. We also describe versions of our model that allow for periodic arrival rate functions and input data of limited precision. We formulate the estimation problem as a convex program related to semidefinite programming and solve it with a standard nonlinear optimization package called KNITRO. We present numerical results using both an actual record of e-mail arrivals over a period of sixty weeks, and artificially generated data sets. We also present a cross-validation procedure for determining an appropriate number of spline knots to model a set of arrival observations

Bilinear optimality constraints for the cone of positive polynomials

For a proper cone