Taxonomy Method - Wrocław Taxonomic Development Measure

Overview

d_i ∈ (−∞,1]. Higher d means closer to the ideal reference object. The Taxonomy method and Hellwig's method share the same algorithm; the key difference is historical context - Taxonomy originates in 1951 Wrocław School of Mathematics, Hellwig applied it to socio-economic development measurement in 1968.

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

   Z-score standardisation. Cost criteria are negated after standardisation so higher z always means better.

   $$z_{ij}=\frac{x_{ij}-{\bar{x}}_j}{s_j}\text{ (benefit)};z_{ij}=-\frac{x_{ij}-{\bar{x}}_j}{s_j}\text{ (cost)}$$

   Hellwig 1968, p.310 (extension of Wrocław taxonomy)

2. 2

   Define reference object (ideal) z⁺_j = max_i z_ij for all j (after negation of cost).

   $$z_j^{+}=\underset{i}{\max }{z}_{ij}\forall j$$

   Hellwig 1968, p.311

3. 3

   Compute Euclidean distance c_i from each alternative to the ideal.

   $$c_i=\sqrt{\sum _{j=1}^n(z_{ij}-z_j^{+}{)}^2}$$

   Hellwig 1968, p.311

4. 4

   Compute development measure d_i = 1 − c_i / c⁰ where c⁰ = c̄ + 2s_c. Rank descending.

   $$\bar{c}=\frac{1}{m}\sum _i{c}_i,s_c=\sqrt{\frac{1}{m}\sum _i(c_i-\bar{c}{)}^2},c^0=\bar{c}+2s_c;d_i=1-\frac{c_i}{c^0}$$

   Hellwig 1968, p.312

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

• Constant criterion: standard deviation = 0, z-score undefined - check E-2.

Works with

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