PROBID - Preference Ranking on the Basis of Ideal-Average Distance

Overview

Higher p_i = better. PROBID considers all m ideal solutions (ranked alternatives) and weights closer ideals more heavily. Complexity O(m²n) - suitable for moderate-size problems.

Data

Crisp

Weights

Needs a weight source

How it works

1. 1

   Apply vector normalization F_ij=f_ij/√Σf_kj² to every criterion; criterion direction is handled while constructing the ordered ideal solutions.

   $$F_{ij}=f_{ij}/\sqrt{\sum _k{f}_{kj}^2}$$

   Wang et al. 2021 Eq. (2.1)

2. 2

   Weighted normalized matrix v_ij = w_j · r_ij.

   $$v_{ij}=w_j\cdot r_{ij}$$

   Wang et al. 2021 §2

3. 3

   Construct A(k) from the k-th largest benefit value and k-th smallest cost value; compute the average solution.

   Wang et al. 2021 Eqs. (2.3)-(2.5)

4. 4

   Euclidean distances: S_i(k) = distance from alt i to k-th ideal A(k); S_i(avg) = distance from alt i to average ideal Ā.

   $$S_{i(k)}=\sqrt{\sum _j(v_{ij}-A_j^{(k)}{)}^2};S_{i(avg)}=\sqrt{\sum _j(v_{ij}-{\bar{A}}_j{)}^2}$$

   Wang et al. 2021 §2

5. 5

   Positive ideals use k=1..⌈m/2⌉; negative ideals use k=⌊m/2⌋+1..m, so odd m shares the middle ideal while even m does not. Then R_i=S_pos/S_neg and P_i=1/(1+R_i²)+S_i(avg).

   Wang et al. 2021 Eqs. (2.8)-(2.11)

Edge cases and pitfalls

• O(m²) distance matrix - for large m (>100) becomes expensive.

How to cite