LINMAP: LINear programming technique for Multidimensional Analysis of Preference

Overview

LP-based ideal point from pairwise preference judgements. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: LP-based ideal point from pairwise preference judgements

Limitations

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

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)

When not to use

• •Criteria strongly correlated → consider DEMATEL/ANP for interdependence
• •Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)

Edge cases

• •See F.steps and D.parameters for LINMAP-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'LINMAP bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criteria preferences are independent (no synergistic interactions)
• •Hatalı: 'LINMAP bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Compensation is acceptable: high score on one criterion can offset low on another
• •Hatalı: 'LINMAP bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix is complete (no missing values)
• •Hatalı: LINMAP'yi 'Criteria strongly correlated → consider DEMATEL/ANP for interdependence' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: LINMAP'yi 'Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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. Formül: \min\sum_{(i,k)}s_{ik};\quad\text{s.t. }\sum_{j}w_{j}(x_{kj}-x_{j}^{*})^{2}-\sum_{j}w_{j}(x_{ij}-x_{j}^{*})^{2}\geq 1-s_{ik},\;s_{ik}\geq 0,\;w_{j}\geq 0 Anchor: Srinivasan & Shocker 1973, p.341 Eq.(2)
2. 2.Adım 2 (F2): Step 2: Compute weighted Euclidean distance D_i from each alternative to the inferred ideal point x*. Rank ascending (lower = better). Formül: D_{i} = \sqrt{\sum_{j=1}^{n}w_{j}(x_{ij}-x_{j}^{*})^{2}} Anchor: Srinivasan & Shocker 1973, p.341

Commonly paired with

• •AHP + LINMAP (high)
• •BWM + LINMAP (high)
• •ENTROPY + LINMAP (high)
• •CRITIC + LINMAP (high)
• •SWARA + LINMAP (high)

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