CRADIS - Compromise Ranking of Alternatives from Distance to Ideal Solution

Overview

Q_i ≥ 0. Higher Q means closer to ideal and farther from anti-ideal simultaneously. CRADIS combines D⁺ (ideal gap) and D⁻ (anti-ideal gap) through an arithmetic mean of reciprocal ratios, making it similar in spirit to TOPSIS but using linear sum normalisation and weighted linear distances.

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

1. 1

   Linear max normalisation per criterion direction.

   $$n_{ij}=\{\begin{array}{ll}{x}_{ij}/{\max }_i{x}_{ij}& j\in J^{+}\\ {\min }_i{x}_{ij}/x_{ij}& j\in J^{-}\end{array}$$

   Puška 2022, p.4 Eq.(1)

2. 2

   Weighted normalised matrix v_ij = w_j n_ij.

   $$v_{ij}=w_j{n}_{ij}$$

   Puška 2022, p.4 Eq.(2)

3. 3

   Deviations from ideal v_j^I = max v_ij and anti-ideal v_j^A = min v_ij.

   $$d_i^{+}=\sum _{j=1}^n(v_j^I-v_{ij}),d_i^{-}=\sum _{j=1}^n(v_{ij}-v_j^A)$$

   Puška 2022, p.4 Eqs.(3)-(4)

4. 4

   Utility ratios: s^+_i = min_k(d^+_k) / d^+_i (closeness to ideal, best alternative gets 1) and s^-_i = d^-_i / max_k(d^-_k) (closeness to anti-ideal, best alternative gets 1).

   $$s_i^{+}=\frac{{\min }_k{d}_k^{+}}{d_i^{+}},s_i^{-}=\frac{d_i^{-}}{{\max }_k{d}_k^{-}}$$

   Puška et al. 2022 (DOI 10.1007/s10668-021-01902-2), Eqs.(5)-(6): K_o = T_o^{opt}/T_o^i (= min/d^+), K_a = T_a^i/T_a^{opt} (= d^-/max)

5. 5

   CRADIS final score Q_i = (s^+_i + s^−_i)/2 and descending ranking.

   $$Q_i=\frac{s_i^{+}+s_i^{-}}{2}$$

   Puška 2022, p.5 Eq.(7)

Look elsewhere when

Assumptions to verify

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Edge cases and pitfalls

• D⁺_i = 0 for any alternative (it is already at the ideal on all criteria): Q_i formula divides by D⁺ - handle separately as perfect score.

Works with

How to cite