Algorithms for Robust Multi-Objective Optimization
(Topic B.3)

– How can I find a variety of solutions for a problem with conflicting objectives today even though the objectives may change slightly until tomorrow? –

Whole life cycles of natural resources are affected by uncertainty. Growth and harvest of regenerative resources and hence their availability depend on weather and environmental circumstances.
Furthermore, quality characteristics, like tensile strength of wood beams, depend on the growth of resources. Hence, not only the availability of natural resources but also the quality of purchased resources has to be considered as uncertain.

As a consequence, uncertainty has to be considered in production planning for goods that are made from regenerative resources. One research area of mathematics that deals with uncertainty in optimization problems is robust optimization. In particular, robust optimization is suitable for handling uncertain optimization problems where no probability distribution of the uncertainty is known.
In robust optimization, we distinguish uncertainty in problem parameters, so-called parameter uncertainty, and uncertainty inside decision variables, so-called variable uncertainty. Examples for parameter uncertainty are availabilities of regenerative resources or the price that can finally be charged for a product. Variable uncertainty occurs whenever a computed solution can not be put into reality exactly. For instance, it is neither possible to cut logs exactly at a length of 3,3338 meters, nor is it practicable to fill a planter pot with exactly 312,55 grams of peat. However, small deviations of parameters inside a problem and small differences between solutions that were computed and solutions that were realized can have a significant impact on the performance of objectives.
When using regenerative resources inside production processes, producers might want to maximize different conflicting objectives. For instance, a company that processes regenerative resources might want to maximize profit and minimize negative environmental impacts at the same time. Optimization problems with more than one objective are called multi-objective. In general, uncertain multi-objective optimization problems are hard to solve.

Problems of this kind are treated in research area robust multi-objective optimization. Solutions to uncertain multi-objective problems are called robust efficient. Until now, research has been focused on uncertain multi-objective optimization problems that include only parameter uncertainty.
The main aim of this dissertation project is to investigate variable uncertainty in multi-objective optimization problems. Furthermore, we intend to develop solution techniques for multi-objective problems with variable uncertainty.
In addition, there exist no algorithms that are able to find all robust efficient solutions to multi-objective problems with parameter uncertainty until now. Therefore, a second aim of this dissertation project is to generate an algorithm that determines all robust efficient solutions of special classes of multi-objective problems with parameter uncertainty. Following the results of these two research questions, multi-objective problems that contain both, parameter uncertainty and variable uncertainty, will be investigated.
Finally, theoretical results such as potential algorithms and solution techniques will be applied to an uncertain multi-objective optimization problem that is a part of the dissertation project of Francesco Castellani. The respective multi-objective problem includes both parameter uncertainty and variable uncertainty.