Making Robust Decisions in Discrete Optimization Problems as a Game against Nature

In this paper a discrete optimization problem under uncertainty is discussed. Solving such a problem can be seen as a game against nature. In order to choose a solution, the minmax and minmax regret criteria can be applied. In this paper an extension of the known minmax (regret) approach is proposed. It is shown how different types of uncertainty can be simultaneously taken into account. Some exact and approximation algorithms for choosing a best solution are constructed.Discrete optimization, minmax, minmax regret, game against nature

Energy Crop Supply in France: A Min-Max Regret Approach

This paper attempts to estimate energy crop supply using an LP model comprising hundreds of representative farms of the arable cropping sector in France. In order to enhance the predictive ability of such a model and to provide an analytical tool useful to policy makers, interval linear programming (ILP) is used to formalise bounded rationality conditions. In the presence of uncertainty related to yields and prices it is assumed that the farmer minimises the distance from optimality once uncertainty resolves introducing an alternative criterion to the classic profit maximisation rationale. Model validation based on observed activity levels suggests that about 40% of the farms adopt the min-max regret criterion. Then energy crop supply curves, generated by the min-max regret model, are proved to be upward sloped alike classic LP supply curves.interval linear programming, min-max regret, energy crops, France, Crop Production/Industries, Resource /Energy Economics and Policy, C61, D81, Q18,

Algorithm Engineering in Robust Optimization

Robust optimization is a young and emerging field of research having received a considerable increase of interest over the last decade. In this paper, we argue that the the algorithm engineering methodology fits very well to the field of robust optimization and yields a rewarding new perspective on both the current state of research and open research directions. To this end we go through the algorithm engineering cycle of design and analysis of concepts, development and implementation of algorithms, and theoretical and experimental evaluation. We show that many ideas of algorithm engineering have already been applied in publications on robust optimization. Most work on robust optimization is devoted to analysis of the concepts and the development of algorithms, some papers deal with the evaluation of a particular concept in case studies, and work on comparison of concepts just starts. What is still a drawback in many papers on robustness is the missing link to include the results of the experiments again in the design

Min-max regret versus gross margin maximization in arable sector modeling

"A sector model presented in this article, uses about 200 representative French cereal-oriented farms to estimate policy impacts by means of mathematical modeling. Usually, such models suppose that farmers intend to maximize expected gross margin. This rationality hypothesis however seems hardly justifiable, especially these days, when gross margin variability due to European Common Agricultural Policy changes may become significant. Increasing uncertainty introduces bounded rationality to the decision problem so that crop gross margins may be better approximated by interval rather than by expected (precise) values. The initial LP problem is specified as an “Interval Linear Programming (ILP)”. We assume that farmers tend to decide upon their surface allocation prudently in order to get through with minimum loss, which is precisely the rationale underlying the minimization of maximum regret decision criterion. Recent advances in operations research, namely Mausser and Laguna algorithms, are exploited to implement the min-max regret criterion to arable agriculture ILP. The validation against observed crop mix proved that as uncertainty increases about 40% of the farmers adopt the min-max regret decision rule instead of the gross margin maximization."Interval Linear Programming, Min-Max Regret, Common Agricultural Policy, Arable cropping, France

A linear programming based heuristic framework for min-max regret combinatorial optimization problems with interval costs

This work deals with a class of problems under interval data uncertainty, namely interval robust-hard problems, composed of interval data min-max regret generalizations of classical NP-hard combinatorial problems modeled as 0-1 integer linear programming problems. These problems are more challenging than other interval data min-max regret problems, as solely computing the cost of any feasible solution requires solving an instance of an NP-hard problem. The state-of-the-art exact algorithms in the literature are based on the generation of a possibly exponential number of cuts. As each cut separation involves the resolution of an NP-hard classical optimization problem, the size of the instances that can be solved efficiently is relatively small. To smooth this issue, we present a modeling technique for interval robust-hard problems in the context of a heuristic framework. The heuristic obtains feasible solutions by exploring dual information of a linearly relaxed model associated with the classical optimization problem counterpart. Computational experiments for interval data min-max regret versions of the restricted shortest path problem and the set covering problem show that our heuristic is able to find optimal or near-optimal solutions and also improves the primal bounds obtained by a state-of-the-art exact algorithm and a 2-approximation procedure for interval data min-max regret problems