CILOS - Criterion Impact LOSs objective weighting method

Overview

CILOS assigns higher weight to criteria where choosing the optimal alternative for that criterion causes larger losses on other criteria. It is fully data-driven. CILOS weights are often combined with ENTROPY weights to produce IDOCRIW integrated objective weights.

Output

Weight, higher is better

Data

Crisp, complete numeric matrix

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Any (objective weighting)

How it works

1. 1

   Convert minimized criteria to maximized via Eq.(8): r̃_ij = min_i r_ij / r_ij. Max-criteria left as-is.

   $${\tilde{r}}_{ij}=\frac{{\min }_i{r}_{ij}}{r_{ij}}\text{for}j\in J^{-};{\tilde{r}}_{ij}=r_{ij}\text{for}j\in J^{+}$$

   Zavadskas & Podvezko 2016, §3, Eq.(8)

2. 2

   Column-sum normalisation per Eq.(4): r̃_ij ← r̃_ij / Σ_i r̃_ij so each column sums to 1.

   $${\tilde{r}}_{ij}\leftarrow \frac{{\tilde{r}}_{ij}}{{\sum }_{i=1}^n{\tilde{r}}_{ij}}$$

   Zavadskas & Podvezko 2016, §2, Eq.(4)

3. 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}.

   $$x_j=\underset{i}{\max }{x}_{ij}=x_{k_j,j};A=\Vert a_{ij}\Vert ,a_{ii}=x_i,a_{ij}=x_{k_j,i}$$

   Zavadskas & Podvezko 2016, §3, paragraph after Eq.(8)

4. 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.

   $$p_{ij}=\frac{x_j-a_{ij}}{x_j}=\frac{a_{jj}-a_{ij}}{a_{jj}},p_{ii}=0$$

   Zavadskas & Podvezko 2016, §3, Eq.(9)

5. 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.

   $$F_{ii}=-\sum _{k=1}^m{p}_{ki};F_{ij}=p_{ij}(i\ne j)$$

   Zavadskas & Podvezko 2016, §3, Eqs.(13)-(15)

6. 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.

   $$F{\mathbf{q}}^T=0$$

   Zavadskas & Podvezko 2016, §3, Eq.(14)

7. 7

   Normalise q so that Σ q_i = 1; these are the CILOS weights.

   $$q_j\leftarrow \frac{q_j}{{\sum }_{k=1}^m{q}_k}$$

   Zavadskas & Podvezko 2016, §3, paragraph after Eq.(15)

Look elsewhere when

Assumptions to verify

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

Edge cases and pitfalls

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

• If multiple alternatives share the maximum p_ij for a criterion, choose the first (or break by alternative_id) - the loss matrix row is still well-defined.

• Homogeneous criterion (all alternatives equal): per paper §3, entropy weight → 0, CILOS weight → undefined (P column becomes zero, F singular). Engine should fall back gracefully.

• Eq.(8) division by zero if min_i r_ij = 0 for a cost criterion - preprocess to add small epsilon or reject input.

Works with

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