Abstract
In this paper we show how the Belief-Function based Technique for Order Preference by Similarity to Ideal Solution (BF-TOPSIS) approach can be used for solving non-classical multi-criteria decision-making (MCDM) problems. We give simple examples to illustrate our presentation.
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Notes
- 1.
Analytic hierarchy process.
- 2.
Elimination and choice translating reality.
- 3.
Technique for order preference by similarity to ideal solution.
- 4.
In the MCDM context, a source of information consists in the list of scores values of alternatives related to a given criterion.
- 5.
The power set \(2^\mathcal {A}\) is the set of all subsets of \(\mathcal {A}\), empty set \(\emptyset \) and \(\mathcal {A}\) included.
- 6.
As proposed in Smets Transferable Belief Model for instance.
- 7.
Depending on the context of the MCDM problem, the score can be interpreted either as a cost/expense or as a reward/benefit. In the sequel, by convention and without loss of generality, we will interpret the score as a reward having monotonically increasing preference. Thus, the best alternative with respect to a given criterion will be the one providing the highest reward/benefit.
- 8.
Indeed, \(Bel_{ij}(X_i)\) and \(Bel_{ij}(\bar{X}_i)\) (where \(\bar{X}_i\) is the complement of \(X_i\) in the FoD \(\mathcal {A}\)) belong to [0, 1] and they are consistent because the equality \(Pl_{ij}(X_i)=1-Bel_{ij}(\bar{X}_i)\) holds. The proof is similar to the one given in [11].
- 9.
If \(X^j_{\max }=0\) then \(Bel_{ij}(X_i)=0\), and if \(X^j_{\min }=0\) then \(Pl_{ij}(X_i)=1\), so that \(Bel_{ij}(\bar{X}_i)=0\).
- 10.
When a score value is missing for some proposition \(X_i\) (i.e. if \(S_j(X_i)= \varnothing \)), then we take the vacuous BBA \(m_{ij}(X_i\cup \bar{X}_i)=1\).
- 11.
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Dezert, J., Han, D., Tacnet, JM., Carladous, S., Yin, H. (2016). The BF-TOPSIS Approach for Solving Non-classical MCDM Problems. In: Vejnarová, J., Kratochvíl, V. (eds) Belief Functions: Theory and Applications. BELIEF 2016. Lecture Notes in Computer Science(), vol 9861. Springer, Cham. https://doi.org/10.1007/978-3-319-45559-4_8
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