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Ordinal optimization through multi-objective reformulation
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DOI：10.1016/j.ejor.2023.04.042.png)
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Abstract
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We analyze combinatorial optimization problems with ordinal, i.e., non-additive, objective functions that assign categories (like good, medium and bad) rather than cost coefficients to the elements of feasible so-lutions. We review different optimality concepts for ordinal optimization problems and discuss their sim-ilarities and differences. We then focus on two prevalent optimality concepts that are shown to be equiv-alent. Our main focus lies on the investigation of a bijective linear transformation that transforms ordinal optimization problems to associated standard multi-objective optimization problems with binary cost co-efficients. Since this transformation preserves all properties of the underlying problem, problem-specific solution methods remain applicable. A prominent example is dynamic programming and Bellman's prin-ciple of optimality, that can be applied, e.g., to ordinal shortest path and ordinal knapsack problems. We investigate the interrelation between scalarization techniques and methods based on the hypervolume indicator when applied to the ordinal and the transformed problem, respectively. Furthermore, we ex-tend our results to multi-objective optimization problems that combine ordinal and real-valued objective functions.& COPY; 2023 Elsevier B.V. All rights reserved.
Keywords:
Multiple objective programming
Ordering cones
Ordinal objective functions
Combinatorial optimization
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