Abstract
Multiobjective Programming (MOP) may be faced as the extension of classical single objective programming to the cases in which more than one objective function is explicitly considered in mathematical optimization models. However, if these functions are conflicting, a paradigm change is at stake. The concept of optimal solution no longer makes sense since, in general, there is no feasible solution that simultaneously optimizes all objective functions. Single objective programming follows the optimality paradigm, that is, there is a complete comparability between pairs of feasible alternatives and transitivity applies. This is a mathematically well-formulated problem, since we possess enough mathematical tools to solve the three fundamental questions of analysis: existence, unicity and construction of the solution. When more than one objective function is considered these properties are no longer valid.
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Notes
- 1.
Nothing would change, in substance, when considering minimization problems or cases where some objective functions should be maximized and others minimized. In this case the original problem is transformed into another problem where all objective functions are maximized (or minimized), by multiplying the minimizing objective functions by −1 (or the opposite).
- 2.
The mathematical definition of efficient solution can be done in several ways. For example, Yu (1974) presents it in terms of extreme point cones; Lin (1976) uses the notion of directional convexity and Payne et al. (1975) use a perturbation function similar to the one introduced by Geoffrion (1971) in another context.
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Antunes, C.H., Alves, M.J., Clímaco, J. (2016). Formulations and Definitions. In: Multiobjective Linear and Integer Programming. EURO Advanced Tutorials on Operational Research. Springer, Cham. https://doi.org/10.1007/978-3-319-28746-1_2
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DOI: https://doi.org/10.1007/978-3-319-28746-1_2
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