CODAS - Combinative Distance-Based Assessment

Overview

H_i can be any real number. Higher H means the alternative is further from the anti-ideal solution on both Euclidean and Taxicab metrics. The Taxicab distance acts as a secondary discriminator: it is used only when Euclidean distances are nearly equal (|E_i − E_k| < τ). τ=0.02 is the seminal paper default.

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 normalisation (max for benefit, min/x for cost).

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

   Keshavarz-Ghorabaee 2016, p.31 Eq.(2)

2. 2

   Weighted normalised matrix r_ij = w_j · n_ij.

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

   Keshavarz-Ghorabaee 2016, p.31 Eq.(3)

3. 3

   Negative-Ideal solution NI per criterion.

   $$N{I}_j=\underset{i}{\min }{r}_{ij}$$

   Keshavarz-Ghorabaee 2016, p.31 Eq.(4)

4. 4

   Euclidean E_i and Taxicab T_i distances from NI.

   $$E_i=\sqrt{\sum _{j=1}^n(r_{ij}-N{I}_j{)}^2},T_i=\sum _{j=1}^n|r_{ij}-N{I}_j|$$

   Keshavarz-Ghorabaee 2016, p.31 Eqs.(5)-(6)

5. 5

   Relative Assessment matrix h_ik with threshold function ψ.

   $$h_{ik}=(E_i-E_k)+\psi (E_i-E_k)\cdot (T_i-T_k),\psi (z)=\{\begin{array}{ll}1& |z|\ge \tau \\ 0& |z|<\tau \end{array}$$

   Keshavarz-Ghorabaee 2016, p.32 Eq.(7)

6. 6

   Assessment score H_i and descending ranking.

   $$H_i=\sum _{k=1}^m{h}_{ik}$$

   Keshavarz-Ghorabaee 2016, p.32 Eq.(8)

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

• τ sensitivity: a very small τ (near 0) causes T to dominate whenever E values differ at all; a very large τ (near 1) never invokes T - tune τ to the scale of your data.

Works with

How to cite