Minmax regret combinatorial optimization problems: an Algorithmic Perspective

Candia-Vejar, A (reprint author), Univ Talca, Modeling & Ind Management Dept, Curico, Chile.Uncertainty in optimization is not a new ingredient. Diverse models considering uncertainty have been developed over the last 40 years. In our paper we essentially discuss a particular uncertainty model associated with combinatorial optimization problems, developed in the 90's and broadly studied in the past years. This approach named minmax regret (in particular our emphasis is on the robust deviation criteria) is different from the classical approach for handling uncertainty, stochastic approach, where uncertainty is modeled by assumed probability distributions over the space of all possible scenarios and the objective is to find a solution with good probabilistic performance. In the minmax regret (MMR) approach, the set of all possible scenarios is described deterministically, and the search is for a solution that performs reasonably well for all scenarios, i.e., that has the best worst-case performance. In this paper we discuss the computational complexity of some classic combinatorial optimization problems using MMR. approach, analyze the design of several algorithms for these problems, suggest the study of some specific research problems in this attractive area, and also discuss some applications using this model

Mathematical optimization models for reallocating and sharing health equipment in pandemic situations

In this paper we provide a mathematical programming based decision tool to optimally reallocate and share equipment between different units to efficiently equip hospitals in pandemic emergency situations under lack of resources. The approach is motivated by the COVID-19 pandemic in which many Heath National Systems were not able to satisfy the demand of ventilators, sanitary individual protection equipment or different human resources. Our tool is based in two main principles: (1) Part of the stock of equipment at a unit that is not needed (in near future) could be shared to other units; and (2) extra stock to be shared among the units in a region can be efficiently distributed taking into account the demand of the units. The decisions are taken with the aim of minimizing certain measures of the non-covered demand in a region where units are structured in a given network. The mathematical programming models that we provide are stochastic and multiperiod with different robust objective functions. Since the proposed models are computationally hard to solve, we provide a divide-et-conquer math-heuristic approach. We report the results of applying our approach to the COVID-19 case in different regions of Spain, highlighting some interesting conclusions of our analysis, such as the great increase of treated patients if the proposed redistribution tool is applied.Spanish Ministerio de Ciencia e Innovacion, Agencia Estatal de Investigacion/FEDER PID2020-114594GB-C21Junta de Andalucia SEJ-584 FQM-331 P18-FR-1422 US-1256951 P18-FR-2369Spanish Government PEJ2018-002962-ADoctoral Program in Mathematics at the Universidad of GranadaProyect NetMeetData (Fundacion BBVA - Big Data)IMAG-Maria de Maeztu grant CEX2020-001105-M/AEICenter for Forestry Research & Experimentation (CIEF)European Commission CIGE/2021/16

A distributionally robust optimization approach for two-stage facility location problems

In this paper, we consider a facility location problem where customer demand constitutes considerable uncertainty, and where complete information on the distribution of the uncertainty is unavailable. We formulate the optimal decision problem as a two-stage stochastic mixed integer programming problem: an optimal selection of facility locations in the first stage and an optimal decision on the operation of each facility in the second stage. A distributionally robust optimization framework is proposed to hedge risks arising from incomplete information on the distribution of the uncertainty. Specifically, by exploiting the moment information, we construct a set of distributions which contains the true distribution and where the optimal decision is based on the worst distribution from the set. We then develop two numerical schemes for solving the distributionally robust facility location problem: a semi-infinite programming approach which exploits moments of certain reference random variables and a semi-definite programming approach which utilizes the mean and correlation of the underlying random variables describing the demand uncertainty. In the semi-infinite programming approach, we apply the well-known linear decision rule approach to the robust dual problem and then approximate the semi-infinite constraints through the conditional value at risk measure. We provide numerical tests to demonstrate the computation and properties of the robust solutions. © 2020, The Author(s)