Juan Valencia | Searcharxiv

arXiv2025

Limiting behavior of mixed coherent systems with L\'evy-frailty Marshall-Olkin failure times

In this paper we show a limit result for the reliability function of a system -- that is, the probability that the whole system is still operational after a certain given time -- when the number of components of the system grows to infinity. More specifically, we consider a sequence of mixed coherent systems whose components are homogeneous and non-repairable, with failure-times governed by a L\'evy-frailty Marshall-Olkin (LFMO) distribution -- a distribution that allows simultaneous component failures. We show that under integrability conditions the reliability function converges to the probability of a first-passage time of a L\'evy subordinator process. To the best of our knowledge, this is the first result to tackle the asymptotic behavior of the reliability function as the number of components of the system grows. To illustrate our approach, we give an example of a parametric family of reliability functions where the system failure time converges in distribution to an exponential random variable, and give computational experiments testing convergence.

math.PR↗

arXiv2024

Forecasting Outside the Box: Application-Driven Optimal Pointwise Forecasts for Stochastic Optimization

We study a class of two-stage stochastic programs, namely, those with fixed recourse matrix and fixed costs, and linear second stage. We show that, under mild assumptions, the problem can be solved with just one scenario, which we call an ``optimal scenario.'' Such a scenario does not have to be unique and may fall outside the support of the underlying distribution. Although finding an optimal scenario in general might be hard, we show that the result can be particularly useful in the case of stochastic optimization problems with contextual information, where the goal is to optimize the expected value of a certain function given some contextual information (e.g., previous demand, customer type, etc.) that accompany the main data of interest. The contextual information allows for a better estimation of the quantity of interest via machine learning methods. We focus on a class of learning methods -- sometimes called in the literature decision-focused learning -- that integrate the learning and optimization procedures by means of a bilevel optimization formulation, which determines the parameters for pointwise forecasts. By using the optimal scenario result, we prove that when such models are applied to the class of contextual two-stage problems considered in this paper, the pointwise forecasts computed from the bilevel optimization formulation actually yield asymptotically the best approximation of an optimal scenario within the modeler's pre-specified set of parameterized forecast functions. Numerical results conducted with inventory problems from the literature (with synthetic data) as well as a bike-sharing problem with real data demonstrate that the proposed approach performs well when compared to benchmark methods from the literature.

math.OC↗