HF-MARCOS - Hesitant Fuzzy MARCOS

Overview

HF-MARCOS extends crisp MARCOS (Stević 2020) to hesitant fuzzy data. Each cell is an HFE - a finite set of membership degrees in [0,1] representing multiple expert assessments or hesitation. Group decisions are aggregated cell-wise via HFWA (Xia-Xu 2011) before the MARCOS pipeline. HFEs are then collapsed to crisp scores via mean (default) prior to AI/AAI construction. The compromise utility f(K_i) balances proximity to ideal and distance from anti-ideal; higher is better.

Output

utility, higher is better

Data

Hesitant, hfe matrix complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Hesitant fuzzy supplier selection (Li-Geng-Yuan 2023), Sustainability evaluation under epistemic uncertainty, Group decision making with linguistic / multi-valued expert ratings

How it works

1. 1

   Validate inputs: per-expert HFE decision matrices {H^(k)}_{k=1..K}, criterion weights w, criterion directions, expert weights λ.

   $$CheckE-\mathrm{1..}E-4;abortonerror.$$

   Li-Geng-Yuan 2023, §3.2

2. 2

   Aggregate K expert HFE matrices into a single HFE matrix H via HFWA: h_ij = HFWA_λ(h_ij^(1),…,h_ij^(K)) = ∪_{γ_k ∈ h_ij^(k)} { 1 − ∏_{k=1}^K (1 − γ_k)^{λ_k} }. (K=1 → identity)

   $$h_{i}j=HFW{A}_{l}ambda(h_i{j}^{(}1),...,h_i{j}^{(}K))=unio{n}_{gamm{a}_{k}in{h}_i{j}^{(}k)}1-pro{d}_{k=1}^K(1-gamm{a}_k{)}^{lambd{a}_k}$$

   Xia-Xu 2011 Def. 5; Li-Geng-Yuan 2023 §3.2 Eq.(7)

3. 3

   Compute HFE score x_ij = S(h_ij) = (1/|h_ij|) Σ_{γ ∈ h_ij} γ for every cell.

   $$x_{i}j=S(h_{i}j)=(1/|h_{i}j|)\ast su{m}_{gammain{h}_{i}j}gamma$$

   Xia-Xu 2011 Def. 6; Li-Geng-Yuan 2023 §3.2 Eq.(8)

4. 4

   Construct AI (ideal) and AAI (anti-ideal) per criterion: benefit C_j: AI_j = max_i x_ij, AAI_j = min_i x_ij; cost C_j: AI_j = min_i x_ij, AAI_j = max_i x_ij. Form extended matrix X_ext of shape (m+2)×n with rows [AAI; A_1; …; A_m; AI].

   $$A{I}_j=ma{x}_i{x}_{i}j(benefit)|mi{n}_i{x}_{i}j(cost);AA{I}_j=mi{n}_i{x}_{i}j(benefit)|ma{x}_i{x}_{i}j(cost)$$

   Stević 2020 Eq.(1); Li-Geng-Yuan 2023 §3.3 Eq.(9)

5. 5

   Normalise X_ext against AI: benefit C_j: n_ij = x_ij / AI_j; cost C_j: n_ij = AI_j / x_ij. Guard against division by zero via epsilon_zero.

   $$n_{i}j=x_{i}j/A{I}_j(benefit)|A{I}_j/x_{i}j(cost);replacexwithmax(x,epsilo{n}_{z}ero)forcostdenominators$$

   Stević 2020 Eq.(2); Li-Geng-Yuan 2023 §3.3 Eq.(10)

6. 6

   Weighted normalised matrix v_ij = w_j · n_ij; row-sum to get S_i = Σ_j v_ij for each row of X_ext (including AAI and AI rows).

   $$v_{i}j=w_j\ast n_{i}j;S_i=su{m}_j{v}_{i}j$$

   Stević 2020 Eqs.(3)-(4); Li-Geng-Yuan 2023 §3.3 Eqs.(11)-(12)

7. 7

   Utility ratios: K^+_i = S_i / S_AI; K^-_i = S_i / S_AAI. Utility functions: f(K^+_i) = K^-_i / (K^+_i + K^-_i); f(K^-_i) = K^+_i / (K^+_i + K^-_i). Final utility f(K_i) = (K^+_i + K^-_i) / (1 + (1 − f(K^+_i))/f(K^+_i) + (1 − f(K^-_i))/f(K^-_i)). Rank by f(K_i) descending.

   $$K{p}_i=S_i/S_{A}I;K{m}_i=S_i/S_{A}AI;f(K{p}_i)=K{m}_i/(K{p}_i+K{m}_i);f(K{m}_i)=K{p}_i/(K{p}_i+K{m}_i);f(K_i)=(K{p}_i+K{m}_i)/(1+(1-f(K{p}_i))/f(K{p}_i)+(1-f(K{m}_i))/f(K{m}_i))$$

   Stević 2020 Eqs.(5)-(9); Li-Geng-Yuan 2023 §3.3 Eqs.(13)-(17)

Fits when / Look elsewhere when

Assumptions to verify

• Each cell is a non-empty HFE with values in [0,1]
• Compensation across criteria is acceptable (MARCOS is fully compensatory)
• All experts share the same scale/interpretation of HFE membership degrees
• Criterion weights are externally sourced (HF-MARCOS does not produce weights)

Edge cases and pitfalls

• •zero via epsilon_zero.

• Empty HFE: at least one γ ∈ [0,1] required per cell.

• Confusing Stević 2020 (crisp MARCOS) with HF-MARCOS - Li-Geng-Yuan 2023 is the hesitant fuzzy extension.

• Defuzzifying HFEs before HFWA aggregation discards inter-expert hesitation information - aggregate first.

Works with

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