Conference Agenda

The Data Envelopment Analysis (DEA) was initially developed for multidimensional output production processes wherein prices as valuation system are (partially) unavailable. Different DEA models were presented later, which also consider undesirable outputs (bads) of production. The most prevalent approach incorporating bads into these models is the assumption of weak disposability regarding these bads. One drawback of this approach is the potential violation of the monotonicity condition, which could lead to possible misclassifications of efficient and inefficient units and incorrect efficiency values. Unlike other constraints in DEA, weak disposability is represented by equations. As is generally known, the shadow prices in the dual program can now accept any sign and can be negative. If the bad increases, the eco-efficiency value will also increase, with negative associated shadow prices. This implausible phenomenon is examined, including the (economic) conditions that facilitate its occurrence, the data structure depending on the relations between inputs, outputs and bads, and its implications for the presentation of eco-DEA models and their outcomes. Furthermore, it is demonstrated that these models are related to the More-for-less paradox in linear programming in specific instances. Sufficient conditions for identifying units for which this phenomenon occurs are presented. Beyond that, it is demonstrated that under the weak disposability assumption the efficiency frontier changes with the orientation of the DEA model, in contrast to the standard DEA models; this result is also applicable to detection of such units.

Solving bi-level multi-criteria optimization problems is a complex endeavor. The lower-level optimization problem can create a complex feasible set for the upper-level problem. Even when bi-level problems are structured to allow for the independent consideration of one level, as is the case with lexicographic methods, the multi-criteria nature of these issues can still create difficulties.

Our focus is on hierarchical bi-level multi-criteria optimization problems, in particular those where the lower-level is independent, and the upper-level objectives do not depend on the lower-level variables. This problem class encompasses reoptimization problems where, subsequent to the initial optimization, secondary criteria need to be taken into account without compromising the quality of the initially optimized goals.

We demonstrate a method to transform hierarchical bi-level problems into equivalent single-level multi-criteria problems, by creating a fitting domination cone. The resulting single-level reduction has the same Pareto front as the bi-level problems’ optimistic solutions. This facilitates the application of standard multi-criteria optimization (MCO) methods, including the approximation of the entire Pareto front to a predefined accuracy and the navigation on it. Furthermore, we give an outlook on how the presented method can be iteratively applied to multi-level MCO problems, effectively simplifying them into single-level MCO problems.

Multiobjective branch and bound has gained quite some attention during the last few years. Despite this fact objective space methods are still the gold standard for general multiobjective integer programming problems, while multiobjective branch and bound algorithms are only competitive in specific situations. Multiobjective branch and bound algorithms run into difficulties as the size of the upper and the lower bound set can grow exponentially with the number of objective functions. Moreover, the higher the number the objective functions is, the weaker are the corresponding bounds in general. Therefore, a suitable branching and queuing strategy is of high importance. Thereby it is crucial not only to find efficient solutions in early stages of the algorithm but also to find a set of solutions whose images are close to and well distributed along the non-dominated frontier.

The adaptive use of objective space information can guide the search in promising directions to determine a good approximation of the Pareto front already in early stages of the algorithm. In particular we focus in this article on improving the branching and queuing of subproblems and the handling of lower bound sets. In our numerical test we evaluate the impact of the proposed methods in comparison to a standard implementation of multiobjective branch and bound on knapsack problems, generalized assignment problems and (un)capacitated facility location problems.