Hellwig's Method - Development Pattern

Overview

m_i ∈ (−∞,1]. Higher m means closer to the ideal pattern. m_i close to 1 = excellent development level. m_i < 0 means the alternative is farther from ideal than the critical distance d⁰. Hellwig's method is weight-free by design - all criteria are treated equally. It is widely used in regional/country development studies.

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

   Standardise each criterion: z_ij = (x_ij − μ_j)/σ_j.

   $$z_{ij}=\frac{x_{ij}-{\mu }_j}{{\sigma }_j}$$

   Hellwig 1968, p.310 Eq.(1)

2. 2

   Reference pattern z_0j = max z_ij (benefit) or min (cost).

   $$z_{0j}=\underset{i}{\max }{z}_{ij}(J^{+})\text{or}\underset{i}{\min }{z}_{ij}(J^{-})$$

   Hellwig 1968, p.310 Eq.(2)

3. 3

   Euclidean distance from reference: d_i = √Σ (z_ij − z_0j)².

   $$d_{i0}=\sqrt{\sum _{j=1}^n(z_{ij}-z_{0j}{)}^2}$$

   Hellwig 1968, p.310 Eq.(3)

4. 4

   Hellwig measure h_i = 1 − d_i/d_0; d_0 = μ_d + 2σ_d; descending ranking.

   $$h_i=1-\frac{d_{i0}}{d_0},d_0=\bar{d}+2{\sigma }_d$$

   Hellwig 1968, p.311 Eq.(4)

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 column: standard deviation = 0, z-score undefined - check E-2.

Works with

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