Survey and Review

Abstract

The worst traffic jam known to man was probably the one that occurred last August in China. Dubbed the “Mother of All Traffic Jams” by the press, vehicles crawled in stop-and-go traffic for more than 9 days over a stretch of the Beijing Tibet Expressway. It was anything but “express”—the queues of vehicles stretched to over 60 miles. What is even more amazing was the quick appearance, and disappearance, of the congestion. It was as if someone had flipped a switch. This cautionary tale should convince you that modeling traffic is important. It is important to understand what causes traffic jams, and how to best deal with them when they occur. The first article in this issue's Survey and Review section discusses in detail efforts to model traffic. In fact the article goes beyond traffic models, which are basically one-dimensional, to talk about modeling the movement of crowds and swarms, which are two- and three-dimensional. The authors describe three main approaches to modeling traffic and crowds. Since the players in a traffic flow are vehicles, one could consider writing down equations that govern the interactions of these individual “points.” On the other hand, it might be more efficient to view traffic as a “blob” and treat it as a continuum. This second approach uses concepts from gas dynamics. Yet another approach is to incorporate the probabilistic nature of traffic flow at a macroscopic scale and apply ideas from kinetic theory. The paper is up-to-date and self-contained. The authors, Nicola Bellomo and Christian Dogbe, make the material highly accessible. They also provide critical reviews of the different modeling efforts. The second article in this issue concerns robust optimization. This is an important and rapidly developing research area in optimization. The basic premise of robust optimization is to find the best solution to an optimization problem when you are uncertain about the parameters describing the problem. Such problems occur in many applications. For example, when designing a bridge, the uncertainties could be the loads from the vehicles going over it, the wind conditions, the added loads from heavy snow, and the rare occurrence of earthquakes. One is interested in maximizing the strength of the bridge while minimizing the cost under these uncertain conditions. The article, by Dimitris Bertsimas, David Brown, and Constantine Caramanis, offers a gentle introduction to the subject of robust optimization. It leads the reader into the key ideas of the area through an example from financial portfolio optimization, where uncertainty abounds. From there, the paper goes deeper into the mathematical structure of robust optimization problems. Its relation to stochastic programming, where statistics of the uncertainties are available and used, is also explored. The authors also provide an excellent exposition to robust adaptable optimization where the problem involves multistage decisions. The paper contains a fine portfolio of applications of robust optimization. The list of examples is long, and the domains in which they arise very diverse. The paper ends by offering a survey of current research directions. Both papers represent valuable resources to the mathematical sciences. They can be read in depth to gain a full understanding of the intricacies and mathematical richness of these two research areas. The casual reader can also quickly get a view of the research landscape of these fields.