Multicriteria Adjustable Robustness

Multicriteria adjustable robust optimization (MARO) problems arise in a wide variety of practical settings, for example, in the design of a building's energy supply. However, no general approaches, neither for the characterization of solutions to this problem class, nor potential solution methods, are available in the literature so far. We give different definitions for efficient solutions to MARO problems and look at three computational concepts to deal with the problems. These computational concepts can also be understood as additional solution definitions. We assess the advantages and disadvantages of the different computational approaches and analyze their connections to our initial definitions of MARO-efficiency. We observe that an $\varepsilon$-constraint inspired first-scalarize-then-robustify computational 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 is only beneficial if the desired trade-off between objectives is already known because 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.