Optimal Configurations for Modular Systems

This work investigates how to determine optimal configurations for modular systems, a concept widely used in mechanical engineering to improve logistics and manufacturing processes. Although modular systems are common in industrial applications, systematically deriving an optimal configuration with methods of mathematical optimization remains an open challenge.

After introducing modular systems and reviewing relevant literature, we develop a mathematical optimization model that captures the structural relationships between components, variants, and the number of pieces in the modular system. Because the number of variants for the components is itself a decision variable, the initial formulation leads to a non implementable optimization model. To address this, we derive in a next step a well-defined mixed-integer optimization problem (MIP) that deals with the unknown number of variants. Although state-of-the-art MIP solvers such as Gurobi can solve MIPs with a large number of variables efficiently, numerical experiments reveal high computational effort for the modular system instances considered here, due to their strong combinatorial and logical structure.
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To exploit this structure, we propose decomposition-based solution methods and analyze their numerical behavior. Motivated by these results, we introduce discrete functions as an alternative representation and examine discrete convexity concepts, including one newly developed in this work. Building on this, we adapt several derivative-free optimization techniques, such as Steepest Descent and Coordinate Search, and an adaption of the Nelder–Mead method to discrete functions.