Robust optimization with incremental recourse

In this paper, we consider an adaptive approach to address optimization problems with uncertain cost parameters. Here, the decision maker selects an initial decision, observes the realization of the uncertain cost parameters, and then is permitted to modify the initial decision. We treat the uncertainty using the framework of robust optimization in which uncertain parameters lie within a given set. The decision maker optimizes so as to develop the best cost guarantee in terms of the worst-case analysis. The recourse decision is ``incremental"; that is, the decision maker is permitted to change the initial solution by a small fixed amount. We refer to the resulting problem as the robust incremental problem. We study robust incremental variants of several optimization problems. We show that the robust incremental counterpart of a linear program is itself a linear program if the uncertainty set is polyhedral. Hence, it is solvable in polynomial time. We establish the NP-hardness for robust incremental linear programming for the case of a discrete uncertainty set. We show that the robust incremental shortest path problem is NP-complete when costs are chosen from a polyhedral uncertainty set, even in the case that only one new arc may be added to the initial path. We also address the complexity of several special cases of the robust incremental shortest path problem and the robust incremental minimum spanning tree problem

On green routing and scheduling problem

The vehicle routing and scheduling problem has been studied with much interest within the last four decades. In this paper, some of the existing literature dealing with routing and scheduling problems with environmental issues is reviewed, and a description is provided of the problems that have been investigated and how they are treated using combinatorial optimization tools

The robust single machine scheduling problem with uncertain release and processing times

In this work, we study the single machine scheduling problem with uncertain release times and processing times of jobs. We adopt a robust scheduling approach, in which the measure of robustness to be minimized for a given sequence of jobs is the worst-case objective function value from the set of all possible realizations of release and processing times. The objective function value is the total flow time of all jobs. We discuss some important properties of robust schedules for zero and non-zero release times, and illustrate the added complexity in robust scheduling given non-zero release times. We propose heuristics based on variable neighborhood search and iterated local search to solve the problem and generate robust schedules. The algorithms are tested and their solution performance is compared with optimal solutions or lower bounds through numerical experiments based on synthetic data

Algorithm Engineering in Robust Optimization

Robust optimization is a young and emerging field of research having received a considerable increase of interest over the last decade. In this paper, we argue that the the algorithm engineering methodology fits very well to the field of robust optimization and yields a rewarding new perspective on both the current state of research and open research directions. To this end we go through the algorithm engineering cycle of design and analysis of concepts, development and implementation of algorithms, and theoretical and experimental evaluation. We show that many ideas of algorithm engineering have already been applied in publications on robust optimization. Most work on robust optimization is devoted to analysis of the concepts and the development of algorithms, some papers deal with the evaluation of a particular concept in case studies, and work on comparison of concepts just starts. What is still a drawback in many papers on robustness is the missing link to include the results of the experiments again in the design

Models and Algorithms for Addressing Travel Time Variability: Applications from Optimal Path Finding and Traffic Equilibrium Problems

An optimal path finding problem and a traffic equilibrium problem are two important, fundamental, and interrelated topics in the transportation research field. Under travel time variability, the road networks are considered as stochastic, where the link travel times are treated as random variables with known probability density functions. By considering the effect of travel time variability and corresponding risk-taking behavior of the travelers, this dissertation proposes models and algorithms for addressing travel time variability with applications from optimal path finding and traffic equilibrium problems. Specifically, two new optimal path finding models and two novel traffic equilibrium models are proposed in stochastic networks. To adaptively determine a reliable path with the minimum travel time budget required to meet the user-specified reliability threshold α, an adaptive α-reliable path finding model is proposed. It is formulated as a chance constrained model under a dynamic programming framework. Then, a discrete-time algorithm is developed based on the properties of the proposed model. In addition to accounting for the reliability aspect of travel time variability, the α-reliable mean-excess path finding model further concerns the unreliability aspect of the late trips beyond the travel time budget. It is formulated as a stochastic mixed-integer nonlinear program. To solve this difficult problem, a practical double relaxation procedure is developed. By recognizing travelers are not only interested in saving their travel time but also in reducing their risk of being late, a α-reliable mean-excess traffic equilibrium (METE) model is proposed. Furthermore, a stochastic α-reliable mean-excess traffic equilibrium (SMETE) model is developed by incorporating the travelers’ perception error, where the travelers’ route choice decisions are determined by the perceived distribution of the stochastic travel time. Both models explicitly examine the effects of both reliability and unreliability aspects of travel time variability in a network equilibrium framework. They are both formulated as a variational inequality (VI) problem and solved by a route-based algorithm based on the modified alternating direction method. In conclusion, this study explores the effects of the various aspects (reliability and unreliability) of travel time variability on travelers’ route choice decision process by considering their risk preferences. The proposed models provide novel views of the optimal path finding problem and the traffic equilibrium problem under an uncertain environment, and the proposed solution algorithms enable potential applicability for solving practical problems