Abstract
This chapter presents the mathematical models of best matching in a progressive manner, from simple one-to-one matching to more advanced matching instances, following the 3 + 1 dimensions of the PRISM taxonomy ; i.e., matching processes with different set characteristics, conditions, and criteria, in static or dynamic environments. The purpose is to demonstrate the mathematical formulation procedures, and the evolution of best matching models as the characteristics of the 3 + 1 dimensions of matching processes change and become more complex.
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Notes
- 1.
These assumptions are merely made for simplicity of mathematical representations, and can be modified/lifted in different cases, as required.
- 2.
Note that a resource-constrained matching problem is essentially a generalized matching problem.
- 3.
The one-to-many and many-to-many matching instances can be formulated in a similar manner.
- 4.
Note that a matching problem with resource sharing is essentially a resource-constrained matching problem.
- 5.
The cost parameters must be normalized in order for the first and the second terms of the objective function to be consistent with each other in terms of units (e.g., revenue/benefit; time saved/time added).
- 6.
Other cases can be formulated in a similar manner to one-to-one and many-to-one matching processes.
- 7.
Note that if i and j are neutral about all pairs of \( p \in I \) and \( q \in J \), their IP is equivalent to their initial preference.
- 8.
See Chap. 2, Sect. 2.1.3, for detailed description of each criteria function.
- 9.
As discussed earlier, such preferences may be representatives of various criteria such as cost, time, distance, or geometric fitness. The matching processes with higher dimensions can be formulated in a similar manner.
- 10.
Note that all criteria must be normalized or have the same unit to ensure consistency in the model.
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Moghaddam, M., Nof, S.Y. (2017). Mathematical Models of Best Matching. In: Best Matching Theory & Applications. Automation, Collaboration, & E-Services, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-46070-3_3
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