Statement of Research

Sundeep Samson

Introduction: Decision making, as old as the human race itself, has always been an integral part of human existence. As the world evolved, decision making has became increasingly more complex as risky decisions are made in uncertain environments. The main emphasis of my ongoing research in multicriteria decision making with Dr. James A. Reneke, Dr. Margaret M. Wiecek and Dr. Georges M. Fadel is to quantify and optimize risk in order to make better decisions under uncertainty. My primary accomplishments pertain to decision making in the field of financial optimization, stochastic design optimization and currently in combining these two very different fields and are reported in my dissertation and journal publications. Many decision making problems involve minimizing risk, probability of something bad happening, under various criteria simultaneously. Because risk is usually quantified in terms of probability, decision making has been traditionally dependent on assumptions regarding basic underlying statistical distributions that model the unknown randomness in the system. Even though this helps the decision maker reduce the complexity of the overall decision problem, these assumptions could be very wrong and mislead the decision significantly. With the advent of advanced computing power and the drive to model decision problems as realistically as possible, assumptions are avoided and instead these unknowns are modeled using uncertainties. Adding uncertainty to the equation not only makes our problem complex and interesting, but more importantly it makes it very realistic. However, uncertainty defined in the literature is typically problem specific. My PhD research objective was to define and model non-problem specific uncertainty and risk and develop a performance based decision making methodology for problems with uncertainty. In the rest of this document, I briefly discuss the mathematical core behind my PhD research and present a couple of applications and my current and future research directions. Mathematical Core: I proposed an alternative modeling paradigm for uncertainty and risk based on the Knightian definition of uncertainty and risk and introduced our decision making methodology [1] by solving an illustrative problem presented by a group of decision makers from Sandia National Laboratories [2]. I was able to represent, aggregate and propagate the uncertainties and quantify risk which led to a preferred decision. Multicriteria decision problems for complex systems with interacting components require the algebra of operator representations and the separability of these representations to include the feedback of the interacting uncertainties. However, the sum of these separable representations need not be separable. Separable stochastic approximations of random fields and functional representations of risk are the major tools used in this methodology. More specifically, for a scalar random field defined on a two-dimensional rectangle, sufficient conditions on the field covariance kernel are given for the representation of discretization of the field based on a factorization of the discrete kernel. Fields satisfying the condition are said to be separable. The kernel of the discretized random field is represented by a matrix and Cholesky decomposition is used to factor the discrete kernel. Since the sum of two separable fields need not be separable, I proposed a separable approximation [3] for the sum and bounded the errors due to the approximation in a probabilistic sense. This approach reduces computational complexity for decision models while still remaining realistic. Ellsberg’s Paradox and Financial Optimization: I also ‘resolved’ [4] Ellsberg’s paradox [5] by applying my decision making methodology to the urn problems presented by Ellsberg. My preference rule which treats each criteria separately, yields consistent decision choices without violating Savage’s Sure-Thing principle [6]. Furthermore, I applied my methodology to a real portfolio selection problem [7] involving long term holdings of real estate investment trusts (REITs) with uncertain prime interest rates to construct a preferred portfolio.

Side-Impact Crashworthiness: In the Mechanical Engineering department, my research objectives were to introduce uncertainties in vehicular design models with respect to crashworthiness and engine packaging with deformable and/or heat radiating objects. I formulated a decision methodology with a two-level decision model, with the upper level decision model based on the second order statistics of the system performance and the lower level decision model based on a stochastic optimization problem. This two-level model [8] provides an opportunity for introducing uncertainty into the general stochastic optimization problem. In the engineering literature, random parameters with known distributions is classified as aleatory uncertainties and uncertain parameters without distributions as epistemic uncertainties. In the vehicle crashworthiness problem, unlike the existing research where only aleatory uncertainties are considered, I have identified and included epistemic uncertainties [9] to make the crashworthiness model more realistic I extended my PhD research to a real life engineering problem originally formulated at Ford Motor Company [10]. The original research, by ignoring the inherent epistemic uncertainties which model the hitting position and hitting height of the vehicle involved in a side and-impact accident, produced a design that could cost the driver heavily if the real accident were not similar to the crash-test conducted at the company. By including these epistemic uncertainties, I was able to propose a safer car design over a range of accident scenarios with a significantly small increase in cost [9]. Current and Future Research: Currently, I am working on a multidisciplinary, multicomponent design optimization model which combines the engineering stochastic design with the portfolio selection problem. While one component of the optimization problem deals with allocation of payload space under different scenarios with uncertainties that can be modeled as a portfolio problem, the other component is a structural design optimization problem with a possibly different set of uncertainties. The challenge is to consider any interacting component influence and develop a strategy to optimize the whole system simultaneously. As part of my future research, I plan to address some remaining questions from my previous work which includes developing a robust portfolio selection methodology and the vehicle underbody layout optimization problem. I also look forward to continue exploring new research areas by collaborating with other faculty and students. Decision problems as discussed here arise frequently in both industry and financial firms. These problems are best modeled with uncertainty and multiple criteria, creating a rich class of interesting research problems with practical relevance. While the two-level decision model discussed earlier is a direction of immediate interest to me, as a long term goal I would like to integrate my applied statistics background (from my undergraduate and master’s program in Madras Christian College) and operations research background to develop useful and powerful tools for optimal decision making. Network flows also fascinate me and I would like to spend time researching or initiating collaborative work in networks with uncertain demands and supply which is very relevant in today’s world. As all my interests have evolved over time, I am sure they will continue to change and grow.

References [1] S. Samson, J. A. Reneke, and M. M. Wiecek. A review of different perspectives on uncertainty and risk and an alternative modeling paradigm. Reliability Engineering & System Safety, 94 (2):558 – 567, 2009. [2] W. L. Oberkampf, J. C. Helton, C. A. Joslyn, S. F. Wojtkiewicz, and S. Ferson. Challenge problems: Uncertainty in system response given uncertain parameters. Reliability Engineering & System Safety, 85 (1):11–19, July-September 2004.

[3] J. A. Reneke and S. Samson. Models and risk analysis of uncertain complex systems. International Journal of Pure and Applied Mathematics, 44 (4):537–561, 2008. [4] S. Samson and J. A. Reneke. A multicriteria approach resolving Ellsbergs paradox with comments on criteria aggregation. Submitted to Journal of Nonlinear Studies, 2009. [5] D. Ellsberg. Risk, ambiguity, and Savage axioms. The Quarterly Journal of Economics, 75 (4):643–669, 1961. [6] L. J. Savage. The Foundation of Statistics. J. Wiley and Sons, New York and London, 1954. [7] S. Samson. Performance Based Decisions under Uncertainty and Risk. PhD thesis, Clemson University, Clemson, South Carolina, 2008. [8] J. A. Reneke, S. Samson, and M. M. Wiecek. Stochastic optimization and uncertainty: An example. International Journal of Pure and Applied Mathematics, 50 (2):181–186, 2009. [9] S. Samson, S. Thoomu, G. Fadel, and J. A. Reneke. Reliable design optimization under aleatory and epistemic uncertainty. In Proceeding of ASME Design Engineering Technical Conference and Design Automation Conference, 2009. [10] L. Gu, R. J. Yang, C. H. Tho, M. Makowski, O. Faruque, and Y. Li. Optimization and robustness for crashworthiness of side impact. International Journal of Vehicle Design (Special Issue), 26(4):348–360, 2001.

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