Discrete location problems with flexible objectives

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Mathematical location theory is a dynamic field within Operations Research, notable for its intriguing structural properties and practical applications. Typically, these applications involve a discrete and finite set of potential locations, necessitating the use of discrete location models for effective decision-making. This work addresses fundamental mathematical location problems characterized by a discrete decision space, focusing on the ordered median function. This function allows for the integration of various decision criteria previously considered in location theory, such as pull, push, and push-pull objectives, into a cohesive framework. Additionally, it introduces mathematical models for new objectives, like balancing objectives, leading to what are termed ordered median problems. A key advantage of these problems is the feasibility of solution approaches for all special cases. The text presents several mixed-integer linear formulations for basic ordered median problems, both with and without capacity restrictions, and develops solution methodologies based on these formulations. The effectiveness of these approaches is evaluated through extensive computational studies. Furthermore, it introduces the ordered median extension of the classical transportation problem and demonstrates how the flexible ordered median function can be applied to various logistics challenges, including routing and scheduling.