Probabilistic bicriteria models : sampling methodologies and solution strategies

Many complex systems involve simultaneous optimization of two or more criteria, with uncertainty of system parameters being a key driver in decision making. In this thesis, we consider probabilistic bicriteria models in which we seek to operate a system reliably, keeping operating costs low at the same time. High reliability translates into low risk of uncertain events that can adversely impact the system. In bicriteria decision making, a good solution must, at the very least, have the property that the criteria cannot both be improved relative to it. The problem of identifying a broad spectrum of such solutions can be highly involved with no analytical or robust numerical techniques readily available, particularly when the system involves nontrivial stochastics. This thesis serves as a step in the direction of addressing this issue. We show how to construct approximate solutions using Monte Carlo sampling, that are sufficiently close to optimal, easily calculable and subject to a low margin of error. Our approximations can be used in bicriteria decision making across several domains that involve significant risk such as finance, logistics and revenue management.

As a first approach, we place a premium on a low risk threshold, and examine the effects of a sampling technique that guarantees a prespecified upper bound on risk. Our model incorporates a novel construct in the form of an uncertain disrupting event whose time and magnitude of occurrence are both random. We show that stratifying the sample observations in an optimal way can yield savings of a high order. We also demonstrate the existence of generalized stratification techniques which enjoy this property, and which can be used without full distributional knowledge of the parameters that govern the time of disruption. Our work thus provides a computationally tractable approach for solving a wide range of bicriteria models via sampling with a probabilistic guarantee on risk. Improved proximity to the efficient frontier is illustrated in the context of a perishable inventory problem.

In contrast to this approach, we next aim to solve a bicriteria facility sizing model, in which risk is the probability the system fails to jointly satisfy a vector-valued random demand. Here, instead of seeking a probabilistic guarantee on risk, we instead seek to approximate well the efficient frontier for a range of risk levels of interest. Replacing the risk measure with an empirical measure induced by a random sample, we proceed to solve a family of parametric chance-constrained and cost-constrained models. These two sampling-based approximations differ substantially in terms of what is known regarding their asymptotic behavior, their computational tractability, and even their feasibility as compared to the underlying "true" family of models. We establish however, that in the bicriteria setting we have the freedom to employ either the chance-constrained or cost-constrained family of models, improving our ability to characterize the quality of the efficient frontiers arising from these sampling-based approximations, and improving our ability to solve the approximating model itself. Our computational results reinforce the need for such flexibility, and enable us to understand the behavior of confidence bounds for the efficient frontier.

As a final step, we further study the efficient frontier in the cost versus risk tradeoff for the facility sizing model in the special case in which the (cumulative) distribution function of the underlying demand vector is concave in a region defined by a highly-reliable system. In this case, the "true" efficient frontier is convex. We show that the convex hull of the efficient frontier of a sampling-based approximation: (i) can be computed in strongly polynomial time by relying on a reformulation as a max-flow problem via the well-studied selection problem; and, (ii) converges uniformly to the true efficient frontier, when the latter is convex. We conclude with numerical studies that demonstrate the aforementioned properties.