SIAM Undergraduate Research Online (SIURO)
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Undergraduate Student Research
Published electronically January 19, 2018
Authors: Vinit Ranjan, Junmo Ryang, and Kelly Zhang (Duke University)
Sponsor: David Kraines (Duke University)
Abstract: The goal of this paper was to assess the effect of self driving cars on traffic conditions in the Greater Seattle Area using data from the Consortium for Mathematics and Its Applications in the 2017 Mathematical Contest in Modeling. We built a computational, agent based, mathematical model by which we could vary parameters such as number of lanes and capacity of cars. After polling a sufficient sample space, we ran our model and used curve fitting techniques to create functions to model the system. We used the model to calculate average speeds for various highways in Seattle. After creating our model, we adapted the computational model to allow for self driving cars to show increase in average car speed in various conditions. Our results show the increase in average speed with an increasing percentage of self driving cars in terms of increased average speed. The advantages of our model are the agent based aspects, which allow us to observe and model the system's behavior. Using this data to interpolate surfaces allows for more analytic techniques as well. The computational model is also flexible enough to poll data for different traffic conditions in other cities.
Published electronically January 23, 2018
Author: Shankar Sivarajan (Indian Institute of Science)
Sponsor: Y. Narahari (Indian Institute of Science)
Abstract: In this paper, we propose a family of approval voting-schemes for electing committees based on the preferences of voters. In our schemes, we calculate the vector of distances of the possible committees from each of the ballots and, for a given p-norm, choose the one that minimizes the magnitude of the distance vector under that norm. The minisum and minimax methods suggested by previous authors and analyzed extensively in the literature naturally appear as special cases corresponding to p = 1 and p = 1; respectively. Supported by examples, we suggest that using a small value of p; such as 2 or 3, provides a good compromise between the minisum and minimax voting methods with regard to the weightage given to approvals and disapprovals. For large but finite p; our method reduces to finding the committee that covers the maximum number of voters, and this is far superior to the minimax method which is prone to ties. We also discuss extensions of our methods to ternary voting.
Published electronically January 24, 2018
Author: Kira Parker (University of Utah)
Sponsor: Braxton Osting (University of Utah)
Abstract: Ranking methods are used in all aspects of life, from Google searches to sports tournaments. Because all ranking methods necessarily have advantages and disadvantages, USA Climbing, the organizer of national climbing competitions in the United States, changed their ranking method three times between 2009 and 2016. The combined rank method employed in 2015 marked a drastic step away from the previous two in that it failed to meet the independence of irrelevant alternatives (IIA) criterion and was almost impossible for spectators to use to calculate ranks on their own. We compare this more recent rank aggregation method with older USA Climbing score aggregation methods as well as other methods from the literature. Three particularly important methods we consider are (i) the combined rank method, (ii) a combination of the previous two USA Climbing score aggregation methods (the merged method), and (iii) a linear programming (LP)-based rank aggregation method from the literature. Using data from the 2016 Bouldering Youth National Championships, we perform leave-one-out cross validation and the Friedman hypothesis test to conclude that at the 99% confidence level, the LP-based rank aggregation method has significantly more predictive power than the other two methods, while there was insufficient evidence to distinguish between the predictive power of the combined rank method and the merged method. However, due to the desirable properties, such as the IIA criterion, satisfied by the merged method, we recommend this method for use in competitive climbing.