We consider a class of routing optimization problems under uncertainty in which all decisions are made before the uncertainty is realized. The objective is to obtain optimal routing solutions that would, as much as possible, adhere to a set of specified requirements after the uncertainty is realized. These problems include finding an optimal routing solution to meet the soft time window requirements at a subset of nodes when the travel time is uncertain, and sending multiple capacitated vehicles to different nodes to meet the customers' uncertain demands. We introduce a precise mathematical framework for defining and solving such routing problems. In particular, we propose a new decision criterion, called the Requirements Violation (RV) Index, which quantifies the risk associated with the violation of requirements taking into account both the frequency of violations and their magnitudes whenever they occur. The criterion can handle instances when probability distributions are known, and ambiguity, when distributions are partially characterized through descriptive statistics such as moments information. We develop practically efficient algorithms involving Benders decomposition to find the exact optimal routing solution in which the RV Index criterion is minimized, and give numerical results from several computational studies that show the attractive performance of the solutions.

We consider a project selection problem where each project has an uncertain return with partially characterized probability distribution. The decision maker selects a feasible subset of projects so that the risk of the portfolio return not meeting a specified target is minimized. To model and evaluate this risk, we propose and justify a general performance measure, the underperformance riskiness index (URI). We define a special case of the URI, the entropic underperformance riskiness index (EURI), for the project selection problem. We minimize the EURI of the project portfolio, which is the reciprocal of the absolute risk aversion (ARA) of an ambiguity-averse individual with constant ARA who is indifferent between the target return with certainty and the uncertain portfolio return. The EURI extends the riskiness index of Aumann and Serrano (2008) by incorporating the target and distributional ambiguity, and controls the underperformance probability (UP) for any target level. Our model includes correlation and interaction effects such as synergies. Since the model is a discrete nonlinear optimization problem, we derive the optimal solution using Benders decomposition techniques. We show that computationally efficient solution of the model is possible. Furthermore, the project portfolios generated by minimizing the underperformance risk are more than competitive in achieving the target with those found by benchmark approaches, including maximization of expected return, minimization of UP, mean-variance analysis, and maximization of Roy's safetyfirst ratio (1952). When there is only a single constraint for the budget, we describe a heuristic which routinely provides project portfolios with near-optimal underperformance risk.

W e consider an appointment system where heterogeneous participants are sequenced and scheduled for service. Because service times are uncertain, the aims are to mitigate the unpleasantness experienced by the participants in the system when their waiting times or delays exceed acceptable thresholds and to address fairness in the balancing of service levels among participants. To evaluate uncertain delays, we propose the Delay Unpleasantness Measure, which takes into account the frequency and intensity of delays above a threshold, and introduce the concept of lexicographic min-max fairness to design appointment systems from the perspective of the worst-off participants. We focus our study in the healthcare industry in balancing physicians' overtime and patients' waiting times in which patients are distinguished by their service time characterizations. The model can be adapted in the robust setting when the underlying probability distribution is not fully available. To capture the correlation between uncertain service times, we suggest using the mean absolute deviations as the descriptive statistics in the distributional uncertainty set to preserve the linearity of the model. The optimal sequencing and scheduling decisions can be derived by solving a sequence of mixed-integer programming problems, and we report the insights from our computational studies.