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Published: 2021

Total Pages: 96

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In this work, we study decision making with personalized stochastic optimization models. The methods, we propose, develop custom-tailored stochastic optimization models for a specific decision maker, while preserving the robustness of an optimal decision as expressions of the decision maker's attitude towards ambiguity. We present an optimization model using a novel robust preference relationship -- reference-based almost stochastic dominance (RSD). We use decision maker's utility function as a reference to individualize constraints of stochastic dominance. The concept of RSD addresses the two problems in utility-based decision making: (i) ambiguity and inaccuracy in characterizing the decision maker's individual risk attitude, (ii) over-conservativeness of stochastic dominance representing general properties of risk aversion. The RSD rule reveals the maximum dominance level quantifying the robustness of the decision maker's preference between alternative choices. We develop an approximation model using Bernstein polynomials, show the asymptotic convergence of its optimal value and set of optimal solutions to the true counterparts as the degree of Bernstein polynomials increases, and analyze the convergence rate of its feasible region. We next develop a cut-generation algorithm to solve the approximation model. Finally, we further adapt this cut-generation algorithm to seek a valid option most robustly preferable to a random benchmark. The effectiveness and computational complexity of the model are illustrated using a portfolio optimization problem. We study the sensitivity of the personalized stochastic optimization models with regards to risk entangled with the decision maker's ambiguous preference itself. We present a bi-objective stochastic optimization model --expected utility and sensitivity-averse maximization (ESM), incorporating classical risk-aversion and sensitivity analysis with regards to decision maker's preference. Unlike classical sensitivity analysis approaches which are post-analyses after optimization, ESM incorporates sensitivity analysis in the optimization procedure in terms of the second objective function. It thus allows to produce solutions which are both risk-averse in the classical sense and sensitivity-averse with regards to ambiguity in the decision maker's preference. ESM adapts the sensitivity measure (SMU) from the general Bayesian sensitivity analysis to build connection between classical expected utility maximization and the sensitivity aversion. We develop two solution methods of ESM. A mixed-integer reformulation is given for a preference maximizer decision maker, while a linear programming reformulation for a risk-averse decision maker. The effect of ESM is illustrated using a homeland security budget allocation problem.