A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.Comment: 11 page

An Evolutionary Algorithm Using Duality-Base-Enumerating Scheme for Interval Linear Bilevel Programming Problems

Interval bilevel programming problem is hard to solve due to its hierarchical structure as well as the uncertainty of coefficients. This paper is focused on a class of interval linear bilevel programming problems, and an evolutionary algorithm based on duality bases is proposed. Firstly, the objective coefficients of the lower level and the right-hand-side vector are uniformly encoded as individuals, and the relative intervals are taken as the search space. Secondly, for each encoded individual, based on the duality theorem, the original problem is transformed into a single level program simply involving one nonlinear equality constraint. Further, by enumerating duality bases, this nonlinear equality is deleted, and the single level program is converted into several linear programs. Finally, each individual can be evaluated by solving these linear programs. The computational results of 7 examples show that the algorithm is feasible and robust

Fuzzy Bi-level Decision-Making Techniques: A Survey

© 2016 the authors. Bi-level decision-making techniques aim to deal with decentralized management problems that feature interactive decision entities distributed throughout a bi-level hierarchy. A challenge in handling bi-level decision problems is that various uncertainties naturally appear in decision-making process. Significant efforts have been devoted that fuzzy set techniques can be used to effectively deal with uncertain issues in bi-level decision-making, known as fuzzy bi-level decision-making techniques, and researchers have successfully gained experience in this area. It is thus vital that an instructive review of current trends in this area should be conducted, not only of the theoretical research but also the practical developments. This paper systematically reviews up-to-date fuzzy bi-level decisionmaking techniques, including models, approaches, algorithms and systems. It also clusters related technique developments into four main categories: basic fuzzy bi-level decision-making, fuzzy bi-level decision-making with multiple optima, fuzzy random bi-level decision-making, and the applications of bi-level decision-making techniques in different domains. By providing state-of-the-art knowledge, this survey paper will directly support researchers and practitioners in their understanding of developments in theoretical research results and applications in relation to fuzzy bi-level decision-making techniques

Solving a type of biobjective bilevel programming problem using NSGA-II

AbstractThis paper considers a type of biobjective bilevel programming problem, which is derived from a single objective bilevel programming problem via lifting the objective function at the lower level up to the upper level. The efficient solutions to such a model can be considered as candidates for the after optimization bargaining between the decision-makers at both levels who retain the original bilevel decision-making structure. We use a popular multiobjective evolutionary algorithm, NSGA-II, to solve this type of problem. The algorithm is tested on some small-dimensional benchmark problems from the literature. Computational results show that the NSGA-II algorithm is capable of solving the problems efficiently and effectively. Hence, it provides a promising visualization tool to help the decision-makers find the best trade-off in bargaining

Confident-DEA: A Unified Approach For Efficiency Analysis With Cardinal, Bounded And Ordinal Data

This paper proposes an extension to the existing literature in DEA, the authors call Confident-DEA approach. The proposed new approach involves a bi-level convex optimization model, and hence NP-hard, to which a solution method is suggested. Confident-DEA constitutes a generalization of DEA for dealing with imprecise data and hence a potential method for forecasting efficiency. Imprecision in data is defined as two forms, one is bounded data and the second is cardinal data. Complementing the methodology proposed by Cooper et al (1999) which provides single valued efficiency measures, Confident-DEA provides a range of values for the efficiency measures, e.g. an efficiency confidence interval, reflecting the imprecision in data. For the case of bounded data, a theorem defining the bounds of the efficiency confidence interval is provided. For the general case of imprecise data, that is a mixture of ordinal and cardinal data, a Genetic-Algorithm-based meta-heuristic is used to determine the upper and lower bounds defining the efficiency confidence interval. To the best knowledge of the authors, this is the first work combining Genetic algorithms with DEA. In both cases of imprecision, a Monte-Carlo type simulation is used to determine the distribution of the efficiency measures, taking into account the distribution of the bounded imprecise data over their corresponding intervals. Most of previous DEA works dealing with imprecise data implicitly assumed a uniform distribution. Confident-DEA, on the other hand, allows for any type of distribution and hence expands the scope of the analysis. The bounded data used in the illustrative examples are assumed to have truncated normal distributions. However, the methodology suggested here allows for any other distribution for the data