CILOS: Criterion Impact LOSs objective weighting method

Overview

Relative criterion-loss matrix weighting (Mirkin theorem based). Output typically weight (higher value = preferred).

Strengths

• •Method-specific: Relative criterion-loss matrix weighting (Mirkin theorem based)

Limitations

• •Assumes: Decision matrix exists with measurable criteria
• •Assumes: Sufficient inter-alternative variation per criterion

Assumptions to verify

• •Decision matrix exists with measurable criteria
• •Sufficient inter-alternative variation per criterion

When not to use

• •No data variation (constant criterion) → weight degenerates
• •Expert judgment is the actual driver → use subjective weighting

Edge cases

• •zero.
• •when all off-diagonal p_ij > 0), F has rank m-1 and the solution is unique up to a multiplicative constant.

Common pitfalls

• •Hatalı: 'CILOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix exists with measurable criteria
• •Hatalı: 'CILOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Sufficient inter-alternative variation per criterion
• •Hatalı: CILOS'yi 'No data variation (constant criterion) → weight degenerates' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: CILOS'yi 'Expert judgment is the actual driver → use subjective weighting' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Convert minimized criteria to maximized via Eq.(8): r̃_ij = min_i r_ij / r_ij. Max-criteria left as-is. Formül: \tilde{r}_{ij} = \frac{\min_{i} r_{ij}}{r_{ij}}\ \text{for}\ j \in J^{-};\quad \tilde{r}_{ij} = r_{ij}\ \text{for}\ j \in J^{+} Anchor: Zavadskas & Podvezko 2016, §3, Eq.(8)
2. 2.Adım 2 (F2): Step 2: Column-sum normalisation per Eq.(4): r̃_ij ← r̃_ij / Σ_i r̃_ij so each column sums to 1. Formül: \tilde{r}_{ij} \leftarrow \dfrac{\tilde{r}_{ij}}{\sum_{i=1}^{n} \tilde{r}_{ij}} Anchor: Zavadskas & Podvezko 2016, §2, Eq.(4)
3. 3.Adım 3 (F3): Step 3: For each criterion j, find x_j = max_i x_ij = x_{k_j,j}; collect row k_j. Stack rows so that the principal diagonal of m×m matrix A contains the column-max values: a_ii = x_i, a_ij = x_{k_j,i}. Formül: x_{j} = \max_{i} x_{ij} = x_{k_{j}, j};\quad A = \|a_{ij}\|, a_{ii} = x_{i}, a_{ij} = x_{k_{j}, i} Anchor: Zavadskas & Podvezko 2016, §3, paragraph after Eq.(8)
4. 4.Adım 4 (F4): Step 4: Relative loss matrix P via Eq.(9): p_ij = (x_j - a_ij)/x_j = (a_jj - a_ij)/a_jj; diagonal p_ii=0. Formül: p_{ij} = \dfrac{x_{j} - a_{ij}}{x_{j}} = \dfrac{a_{jj} - a_{ij}}{a_{jj}},\quad p_{ii} = 0 Anchor: Zavadskas & Podvezko 2016, §3, Eq.(9)
5. 5.Adım 5 (F5): Step 5: Weight-system matrix F via Eqs.(13)-(15): F_ii = -Σ_i p_ij (negative diagonal = column-sum of losses); F_ij (off-diag) = p_ji (TRANSPOSE of P's off-diagonal). The result is an m×m matrix whose row sums are zero. Formül: F_{ii} = -\sum_{k=1}^{m} p_{ki};\quad F_{ij} = p_{ij}\ (i \ne j) Anchor: Zavadskas & Podvezko 2016, §3, Eqs.(13)-(15)
6. 6.Adım 6 (F6): Step 6: Solve homogeneous system F·q^T = 0. By Mirkin's theorem (when all off-diagonal p_ij > 0), F has rank m-1 and the solution is unique up to a multiplicative constant. Formül: F\, \mathbf{q}^{T} = \mathbf{0} Anchor: Zavadskas & Podvezko 2016, §3, Eq.(14)
7. 7.Adım 7 (F7): Step 7: Normalise q so that Σ q_i = 1; these are the CILOS weights. Formül: q_{j} \leftarrow \dfrac{q_{j}}{\sum_{k=1}^{m} q_{k}} Anchor: Zavadskas & Podvezko 2016, §3, paragraph after Eq.(15)

Commonly paired with

• •CILOS + TOPSIS (high)
• •CILOS + VIKOR (high)
• •CILOS + EDAS (high)
• •CILOS + WASPAS (high)
• •CILOS + MARCOS (high)

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