A bicriteria approach to robust optimization

The classic approach in robust optimization is to optimize the solution with respect to the worst case scenario. This pessimistic approach yields solutions that perform best if the worst scenario happens, but also usually perform bad for an average case scenario. On the other hand, a solution that optimizes the performance of this average case scenario may lack in the worst-case performance guarantee. In practice it is important to find a good compromise between these two solutions. We propose to deal with this problem by considering it from a bicriteria perspective. The Pareto curve of the bicriteria problem visualizes exactly how costly it is to ensure robustness and helps to choose the solution with the best balance between expected and guaranteed performance. In this paper we consider linear programming problems with uncertain cost functions. Building upon a theoretical observation on the structure of Pareto solutions for these problems, we present a column generation approach that requires no direct solution of the computationally expensive worst-case problem. In computational experiments we demonstrate the effectiveness of both the proposed algorithm, and the bicriteria perspective in general

Robustness of solutions to the capacitated facility location problem with uncertain demand

We investigate the properties of robust solutions of the Capacitated Facility Location Problem with uncertain demand. We show that the monotonic behavior of the price of robustness is not guaranteed, and therefore that one cannot discriminate among alternative robust solutions by simply relying on the trade-off price-vs-robustness. Furthermore, we report a computational study on benchmark instances from the literature and on instances derived from a real-world application, which demonstrates the validity in practice of our findings

An Experimental Comparison of Uncertainty Sets for Robust Shortest Path Problems

Through the development of efficient algorithms, data structures and preprocessing techniques, real-world shortest path problems in street networks are now very fast to solve. But in reality, the exact travel times along each arc in the network may not be known. This led to the development of robust shortest path problems, where all possible arc travel times are contained in a so-called uncertainty set of possible outcomes. Research in robust shortest path problems typically assumes this set to be given, and provides complexity results as well as algorithms depending on its shape. However, what can actually be observed in real-world problems are only discrete raw data points. The shape of the uncertainty is already a modelling assumption. In this paper we test several of the most widely used assumptions on the uncertainty set using real-world traffic measurements provided by the City of Chicago. We calculate the resulting different robust solutions, and evaluate which uncertainty approach is actually reasonable for our data. This anchors theoretical research in a real-world application and gives an indicator which robust models should be the future focus of algorithmic development

Multi-objective network optimization: models, methods, and applications

There can be an array of planning objectives to consider when identifying alternatives for using, modifying, or restoring natural or built environments. In this respect, multi-objective network optimization models can provide decision support to both managers and users of the system. While there can be an infinite number of feasible solutions to any multi-objective optimization problem in large networks (e.g., urban transportation systems), the efficient ones are usually more desirable in the decision-making process. However, identification of efficient solutions can be challenging in practical applications. To address this issue, this dissertation details mathematical formulations and solution algorithms for a range of real-world planning problems in the context of intelligent transportation systems, vehicle routing problem, natural conservation and landscape connectivity. While the combination of objectives being optimized is unique for each application, the underlying phenomena involves modeling movement between origins and destinations of a networked system. To demonstrate the type of insights that can be achieved using these modeling approaches, the location and number of times solutions appear in different realizations of system and given different solution approaches (e.g., exact and approximate methods) are visualized on network using a commercial geographic information system

A Bicriteria Approach to Robust Optimization

The classic approach in robust optimization is to optimize the solution with respect to the worst case scenario. This pessimistic approach yields solutions that perform best if the worst scenario happens, but also usually perform bad on average. A solution that optimizes the average performance on the other hand lacks in worst-case performance guarantee. In practice it is important to find a good compromise between these two solutions. We propose to deal with this problem by considering it from a bicriteria perspective. The Pareto curve of the bicriteria problem visualizes exactly how costly it is to ensure robustness and helps to choose the solution with the best balance between expected and guaranteed performance. Building upon a theoretical observation on the structure of Pareto solutions for problems with polyhedral feasible sets, we present a column generation approach that requires no direct solution of the computationally expensive worst-case problem. In computational experiments we demonstrate the effectivity of both the proposed algorithm, and the bicriteria perspective in general

A Bicriteria Approach to Robust Optimization

The classic approach in robust optimization is to optimize the solution with respect to the worst case scenario. This pessimistic approach yields solutions that perform best if the worst scenario happens, but also usually perform bad on average. A solution that optimizes the average performance on the other hand lacks in worst-case performance guarantee. In practice it is important to find a good compromise between these two solutions. We propose to deal with this problem by considering it from a bicriteria perspective. The Pareto curve of the bicriteria problem visualizes exactly how costly it is to ensure robustness and helps to choose the solution with the best balance between expected and guaranteed performance. Building upon a theoretical observation on the structure of Pareto solutions for problems with polyhedral feasible sets, we present a column generation approach that requires no direct solution of the computationally expensive worst-case problem. In computational experiments we demonstrate the effectivity of both the proposed algorithm, and the bicriteria perspective in general