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PIF-CIMAS - Picture Fuzzy variant of CIMAS (Kara et al. 2024)

Picture Fuzzy criteria weighting via expert assessment (CIMAS) - PFWA aggregation

Kara, K., Yalçın, G. C., Kaygısız, E. G., Simic, V., Örnek, A. Ş., Pamucar, D.2024doi:10.1016/j.asoc.2024.111826 ↗

Overview

PIF-CIMAS extends CIMAS (Bošković et al. 2023) by replacing crisp expert/criterion ratings with Picture Fuzzy linguistic variables aggregated via score function Sc(W) = (1/3)(μ + (1−η) + (1−ν)). Reliability is verified via a second round.

Output: Weight, higher is better
Data: Picture Fuzzy, uncertainty tuples complete
Weights: Derived internally, no weight source needed
Size: 2+ alternatives, 3-10 criteria works best
Used for: Picture Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1
Steps 1-3 (paper Eqs 5-6): For each expert E_h providing a PFS importance assessment, compute crisp score Sc(E_h) = (μ_h + (1−η_h) + (1−ν_h))/3; then expert weights w_h = Sc(E_h) / Σ_k Sc(E_k).
$S c (E_{h}) = (m u_{h} + (1 - e t a_{h}) + (1 - n u_{h})) /3; w_{h} = S c (E_{h}) / s u m_{k} S c (E_{k})$
Kara et al. 2024, Eqs (5)-(6)
2
Step 4 (Eq 7): Build per-expert per-criterion PFS assessment matrix Z̃_{hj} (h=expert, j=criterion). Compute crisp Z_{hj} = Sc(Z̃_{hj}) = (μ_{hj} + (1−η_{hj}) + (1−ν_{hj}))/3.
$Z_{hj} = (m u_{hj} + (1 - e t a_{hj}) + (1 - n u_{hj})) /3$
Kara et al. 2024, Eq (7)
3
Step 6 (Eq 8): Column-sum normalisation across experts: Ỹ_{hj} = Z_{hj} / Σ_h Z_{hj}. Each column sums to 1.
$Y_{n} or m_{hj} = Z_{hj} / s u m_{h} Z_{hj}$
Kara et al. 2024, Eq (8)
4
Step 7 (Eq 9): Expert-weighted matrix L_{hj} = Ỹ_{hj} · w_h.
$L_{hj} = Y_{n} or m_{hj} * w_{h}$
Kara et al. 2024, Eq (9)
5
Steps 8-10 (Eqs 10-13): Per criterion j compute Y_j = max_h L_{hj}, X_j = min_h L_{hj}, range B_j = Y_j − X_j (paper's Γ_j), then w_j = B_j / Σ_j B_j.
$Y_{j} = ma x_{h} L_{hj}; X_{j} = mi n_{h} L_{hj}; B_{j} = Y_{j} - X_{j}; w_{j} = B_{j} / s u m_{j} B_{j}$
Kara et al. 2024, Eqs (10)-(13)
6
Step 11 (Eq 14): After a second elicitation round producing w_j^{(2r)}, compute Reliability Index RI = Σ_j |w_j·100 − w_j^{(2r)}·100| / 100. Require RI < 0.1 for acceptance. Second-round data is not part of the synthetic Block J fixture.
$R I = s u m_{j} ∣ w_{j} * 100 - w_{j}^{(2 r)} * 100∣/100 (r e q u i r esseco n d - r o u n d w_{j}^{(2 r)})$
Kara et al. 2024, Eq (14)

Fits when / Look elsewhere when

Fits when

•Preserves picture uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

•Crisp pairwise data available - use base CIMAS
•Experts disagree fundamentally on scale interpretation

Assumptions to verify

Experts can articulate preferences in Picture Fuzzy linguistic variables
Pairwise comparisons satisfy consistency requirements of underlying method

Edge cases and pitfalls

Applying PIF-CIMAS without verifying this assumption.
Requirement: Experts can articulate preferences in Picture Fuzzy linguistic variables
Applying PIF-CIMAS without verifying this assumption.
Requirement: Pairwise comparisons satisfy consistency requirements of underlying method
Using PIF-CIMAS when: Crisp pairwise data available.
An alternative method is recommended in this situation.
Using PIF-CIMAS when: Experts disagree fundamentally on scale interpretation.
An alternative method is recommended in this situation.

Works with

Its derived weights can feed

PIF-ARTASI PIF-CIMAS-ARTASI PIF-CODAS PIF-COPRAS PIF-EDAS PIF-MARCOS PIF-MOORA PIF-SAW PIF-TODIM PIF-TOPSIS PIF-VIKOR PIF-WASPAS

How to cite

Kara, K.; Yalçın, G. C.; Kaygısız, E. G.; Simic, V.; Örnek, A. Ş.; Pamucar, D. (2024). A picture fuzzy CIMAS-ARTASI model for website performance analysis in human resource management. Applied Soft Computing. https://doi.org/10.1016/j.asoc.2024.111826

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

PIF-CIMAS

Overview

Output

Weight, higher is better

Data

Picture Fuzzy, uncertainty tuples complete

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-10 criteria works best

Used for

Picture Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

Steps 1-3 (paper Eqs 5-6): For each expert E_h providing a PFS importance assessment, compute crisp score Sc(E_h) = (μ_h + (1−η_h) + (1−ν_h))/3; then expert weights w_h = Sc(E_h) / Σ_k Sc(E_k).

S c (E_{h}) = (m u_{h} + (1 - e t a_{h}) + (1 - n u_{h})) /3; w_{h} = S c (E_{h}) / s u m_{k} S c (E_{k})

Kara et al. 2024, Eqs (5)-(6)

Step 4 (Eq 7): Build per-expert per-criterion PFS assessment matrix Z̃_{hj} (h=expert, j=criterion). Compute crisp Z_{hj} = Sc(Z̃_{hj}) = (μ_{hj} + (1−η_{hj}) + (1−ν_{hj}))/3.

Z_{hj} = (m u_{hj} + (1 - e t a_{hj}) + (1 - n u_{hj})) /3

Kara et al. 2024, Eq (7)

Step 6 (Eq 8): Column-sum normalisation across experts: Ỹ_{hj} = Z_{hj} / Σ_h Z_{hj}. Each column sums to 1.

Y_{n} or m_{hj} = Z_{hj} / s u m_{h} Z_{hj}

Kara et al. 2024, Eq (8)

Step 7 (Eq 9): Expert-weighted matrix L_{hj} = Ỹ_{hj} · w_h.

L_{hj} = Y_{n} or m_{hj} * w_{h}

Kara et al. 2024, Eq (9)

Steps 8-10 (Eqs 10-13): Per criterion j compute Y_j = max_h L_{hj}, X_j = min_h L_{hj}, range B_j = Y_j − X_j (paper's Γ_j), then w_j = B_j / Σ_j B_j.

Y_{j} = ma x_{h} L_{hj}; X_{j} = mi n_{h} L_{hj}; B_{j} = Y_{j} - X_{j}; w_{j} = B_{j} / s u m_{j} B_{j}

Kara et al. 2024, Eqs (10)-(13)

Step 11 (Eq 14): After a second elicitation round producing w_j^{(2r)}, compute Reliability Index RI = Σ_j |w_j·100 − w_j^{(2r)}·100| / 100. Require RI < 0.1 for acceptance. Second-round data is not part of the synthetic Block J fixture.

R I = s u m_{j} ∣ w_{j} * 100 - w_{j}^{(2 r)} * 100∣/100 (r e q u i r esseco n d - r o u n d w_{j}^{(2 r)})

Kara et al. 2024, Eq (14)

Fits when / Look elsewhere when

Fits when

•Preserves picture uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

•Crisp pairwise data available - use base CIMAS
•Experts disagree fundamentally on scale interpretation

Edge cases and pitfalls

Applying PIF-CIMAS without verifying this assumption.

Requirement: Experts can articulate preferences in Picture Fuzzy linguistic variables

Applying PIF-CIMAS without verifying this assumption.

Requirement: Pairwise comparisons satisfy consistency requirements of underlying method

Using PIF-CIMAS when: Crisp pairwise data available.

An alternative method is recommended in this situation.

Using PIF-CIMAS when: Experts disagree fundamentally on scale interpretation.