QUALIFLEX - QUALItative FLEXible multiple criteria method

Overview

QUALIFLEX finds the permutation π* that maximises the weighted concordance index I(π). It is exact (exhaustive) for small m; for m > 7-8 use branch-and-bound or heuristic search. QUALIFLEX handles ordinal and cardinal weights equally well and naturally reveals near-optimal permutations (alternatives with similar I(π) values).

Output

rank position, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Complex multi-stakeholder decisions

How it works

1. 1

   Enumerate all m! permutations π of alternatives. For each permutation and each ordered pair (A_i, A_k) where π ranks A_i above A_k, compute concordance index c_jik = sign(x_ij − x_kj) adjusted for criterion direction.

   $$c_{jik}(\pi )=\{\begin{array}{ll}+1& \text{if }(x_{ij}>x_{kj}\text{ and benefit})\text{ or }(x_{ij}<x_{kj}\text{ and cost})\\ -1& \text{otherwise}\end{array}$$

   Paelinck 1978, p.194 Eq.(1)

2. 2

   Compute comprehensive concordance index I(π) = Σ_j Σ_{i≻k in π} w_j · c_jik(π). Select the permutation π* with maximum I(π) as the optimal ranking.

   $$I(\pi )=\sum _{j=1}^n\sum _{(i,k):\pi (i)<\pi (k)}{w}_j{c}_{jik}(\pi );{\pi }^{\ast }=\arg \underset{\pi }{\max }I(\pi )$$

   Paelinck 1978, p.194-195 Eq.(2)

Look elsewhere when

Assumptions to verify

• Decision-maker can specify preference (p), indifference (q), and veto (v) thresholds
• Non-compensatory preference structure

Edge cases and pitfalls

• Combinatorial explosion: m=8 gives 40320 permutations; m=10 gives 3,628,800 - use heuristics beyond m=7.

Works with

How to cite