LINMAP - LINear programming technique for Multidimensional Analysis of Preference

Overview

LINMAP infers both criterion weights and an ideal point from pairwise preference judgements. It requires no explicit weight elicitation. The closer an alternative is to the inferred ideal (lower D), the better. If the LP is inconsistent (Σ s > 0), some judgements are violated - examine which pairs are most inconsistent.

Output

utility, lower 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

   Formulate LP: find non-negative weights w_j and ideal point x*_j minimising inconsistency with pairwise judgements. For each (A_i ≻ A_k): d²(A_k, x*) − d²(A_i, x*) ≥ 1 − s_ik where s_ik ≥ 0 is slack. Minimise Σ s_ik.

   $$\min \sum _{(i,k)}{s}_{ik};\text{s.t. }\sum _j{w}_j(x_{kj}-x_j^{\ast }{)}^2-\sum _j{w}_j(x_{ij}-x_j^{\ast }{)}^2\ge 1-s_{ik},s_{ik}\ge 0,w_j\ge 0$$

   Srinivasan & Shocker 1973, p.341 Eq.(2)

2. 2

   Compute weighted Euclidean distance D_i from each alternative to the inferred ideal point x*. Rank ascending (lower = better).

   $$D_i=\sqrt{\sum _{j=1}^n{w}_j(x_{ij}-x_j^{\ast }{)}^2}$$

   Srinivasan & Shocker 1973, p.341

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

• Inconsistent judgements: if Σ s_ik > 0, the LP found the best-fit solution but some preference relations are violated.

Works with

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