Multicriteria adjustable robustness

Multicriteria adjustable robust optimization (MARO) problems are highly relevant for a wide variety of practical problems with a two-stage-decision, typically an initial purchase decision followed by the possibility to react during operation after uncertain parameters are revealed. However, no general approaches for the definition of efficient solutions to this problem class are available in the literature for the multicriteria case. The objective of this paper is to find a meaningful definition that in particular allows the computation of solutions. By combining well-known approaches from multicriteria optimization and robust optimization in a straightforward way, we give different definitions for efficient solutions to MARO problems and look at three computation-oriented approaches to deal with the problems. These computation-oriented approaches can also be understood as additional efficiency definitions. We assess the advantages and disadvantages of the different computation-oriented approaches and analyze their connections to our initial definitions of MARO-efficiency. We observe that an ɛ-constraint inspired first-scalarize-then-robustify approach is beneficial because it provides an efficient set that is easy to understand for decision makers and provides tight bounds on the worst-case evaluation for a particular efficient solution. In contrast, a weighted sum first-scalarize-then-robustify approach keeps the problem structure more simple but the efficient set might look ambiguous. Further, we demonstrate that a first-robustify procedure only gives bad bounds and can be too optimistic as well as too pessimistic.