COMET - Characteristic Objects METhod

Overview

P(x_i) ∈ [0,1]. Higher = better. COMET is provably rank-reversal free - adding/removing alternatives does not change relative rankings of existing alternatives. Tradeoff: requires expert elicitation for CO scores.

Output

utility, higher is better

Data

Crisp, crisp complete

Size

2+ alternatives, 2-4 criteria works best

Used for

Rank-reversal sensitive problems, expert-driven evaluation, MCDM with fuzzy rule base

How it works

1. 1

   Generate all Characteristic Objects (COs): Cartesian product of cvalues (Gray-code order for efficiency). If n criteria each with c_j characteristic values, total COs = Π_j c_j.

   $$CO=\{(c_{q_1}^{(1)},c_{q_2}^{(2)},\dots ,c_{q_n}^{(n)}):q_j\in \{1,\dots ,|cvalue{s}_j|\}\}$$

   Sałabun 2015 §3.1

2. 2

   Expert evaluation: apply expert_function to CO_matrix to obtain preference scores p(CO_k) ∈ [0,1] for each characteristic object. This is the knowledge elicitation step - captures DM preferences over the structured CO space.

   $$p(C{O}_k)\in [0,1],k=1,\dots ,\prod _j|cvalue{s}_j|$$

   Sałabun 2015 §3.2 - MEJ matrix

3. 3

   For each alternative x_i, compute membership degrees μ_ij for each criterion j relative to the characteristic values using Triangular Fuzzy Numbers (TFNs). Each x_ij triggers membership in adjacent characteristic levels.

   $$\mu (x,a,m,b)=\{\begin{array}{ll}0& x\le a\text{ or }x\ge b\\ (x-a)/(m-a)& a<x<m\\ (b-x)/(b-m)& m\le x<b\\ 1& x=m\end{array}$$

   Sałabun 2015 §3.3

4. 4

   Aggregate membership degrees across all COs to compute final preference P(x_i) = Σ_k [p(CO_k) · Π_j μ_j(x_ij, CO_k)]. Rank descending.

   $$P(x_i)=\sum _{k}p(C{O}_k)\cdot \prod _j{\mu }_j(x_{ij},C{O}_k),\text{rank descending}$$

   Sałabun 2015 §3.4

Look elsewhere when

Assumptions to verify

• Expert function is consistent with CO preference ordering
• cvalues span the range of actual alternative values
• Number of criteria ≤ 4-5 (exponential CO explosion otherwise)

Edge cases and pitfalls

• •If n criteria each with c_j characteristic values, total COs = Π_j c_j.

• Exponential complexity in number of criteria: 3 criteria × 3 cvalues each = 27 COs; 5 criteria × 5 cvalues = 3125 COs. Keep n small or use structured expert functions.

• Expert function must be consistent: p(CO_k) must respect ordinal preference ordering of COs.

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