Extension card · Fuzzy
Scenario fuzzy MEREC
This is a form of MEREC for situations where criterion scores are given as triangular fuzzy numbers: it runs MEREC fully and separately on each triangle vertex (lower, middle, upper) and averages the three weight vectors. The output is an averaged criterion-weight vector; it does not produce a ranking.
Base method
MEREC →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Fuzzy →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Cells. In crisp MEREC every cell is a single number. Here every cell is a triangular fuzzy number: a lower (most pessimistic), a middle (most likely) and an upper (most optimistic) scenario value. MEREC does not need weights as input; here too it does not, since weight is the method's own output.
Scale equalisation. Unlike the Fuzzy MEREC card, this extension does not process the cell with a fuzzy normalisation formula and then defuzzify with a single graded average. Instead it treats the triangle's three vertices (lower, middle, upper) as three separate crisp decision tables. It runs the whole of crisp MEREC (direction-aware logarithmic normalisation, overall performance, criterion-removal effect, weight) independently on these three tables, three times over. Defuzzification comes only at the very end, by averaging the three weight vectors. This is the fundamental difference between Fuzzy MEREC's "single triangular normalisation first, then defuzzify" approach and "compute the three scenarios separately, then average."
Combination. The three scenarios' criterion-weight vectors (lower, middle, upper) are each produced separately so that they sum to 1, then averaged component by component and renormalised to sum to 1 again. DecisionMind also computes two diagnostic values: a dispersion measure showing how far the average weight differs from the middle scenario (the most likely value) alone, and a warning flag showing how widely the three scenarios' (lower, middle, upper) weight for a criterion spreads at most. If this spread is close to zero, it means the expert's triangles were built proportionally; the fuzziness has added no extra information to the weight.
DecisionMind fixes, in this internal derivation, the approach of solving the three scenarios separately with the full crisp MEREC and then averaging, along with these two diagnostic values.
How to Read the Output
The output, as in crisp MEREC, is a weight vector summing to 1, and it is read the same way: a high weight shows that overall performance is badly disturbed when that criterion is removed, not that the decision-maker considers it important. The difference is here: this weight has been averaged from three separate scenarios (lower, middle, upper). If all three scenarios point to the same criterion as most heavily weighted, the ranking is robust, though the magnitudes may vary by scenario. If one of the three scenarios points to a different criterion as most heavily weighted, reporting only the average conceals this disagreement.
Thus instead of writing:
"Fuzzy input was used, so the weight has been tested against uncertainty and is more robust"
the report should read:
"If the weights from the three scenarios (lower, middle, upper) point to the same criterion as most distinctive, the ranking is robust; the gap between the magnitudes reflects how wide the expert made the triangles, and should be shown separately in the report"
When to Prefer This over the Base Method
Where a pessimistic-likely-optimistic three-scenario estimate can be obtained for criterion values, instead of a single crisp number, and the goal is to see how much these three scenarios change the weight ranking, this extension can be used as an internal robustness check. It must be remembered, though, that this extension has no peer-reviewed source. An analysis in an academic publication that needs to cite something called "Fuzzy MEREC" should use Fuzzy MEREC (Saidin et al., 2023); this extension may be used only as a DecisionMind-internal sensitivity tool, with its source clearly stated. If criterion values are already crisp, crisp MEREC is sufficient.
Mistakes Specific to This Extension
Presenting this method as a published "Scenario Fuzzy MEREC" with a founding paper. This extension has no academic source; it has been derived from DecisionMind's own internal codebase. This must be stated clearly in the report.
Ignoring the warning flag. If the three scenarios' weight for a criterion is almost identical, this is usually a sign that the expert built the triangles proportionally and the fuzziness has added nothing to the weight; a spread that is not close to zero, on the other hand, shows the scenarios genuinely give different weights and should be reported.
Confusing this with Fuzzy MEREC (Saidin et al., 2023). The two are different algorithms. Fuzzy MEREC processes the cell with a single fuzzy normalisation and defuzzifies at the end with a graded average; this extension treats the three vertices as three separate crisp MEREC runs and averages at the end. Comparing their weights directly is wrong.
Leaving a zero or negative value in place. Crisp MEREC's general error applies here too: all three vertices must be positive, since logarithmic normalisation does not accept a zero or negative value.
The governing principle is this:
This extension is not a peer-reviewed literature method but a DecisionMind internal tool that solves the three scenarios (lower, middle, upper) separately with full crisp MEREC and then averages them. The weight it produces shows whether fuzziness has made a genuine contribution only once the three scenarios themselves are compared.
Cases
The first case is an internal record verified against DecisionMind's source code (a frozen code snapshot); it is evidence level C and does not rest on a peer-reviewed literature paper. The second case is an illustrative construction.
1. Illustrative example: Four alternatives, scenario fuzzy weighting of three criteria (DecisionMind's internal verification record)
A team has scored four alternatives with a pessimistic-likely-optimistic three-scenario estimate on three criteria; every cell is a triangular fuzzy number. C1 and C3 are "higher is better", C2 is "lower is better".
| Alternative | C1 (higher is better) | C2 (lower is better) | C3 (higher is better) |
|---|---|---|---|
| A1 | (1; 2; 3) | (4; 5; 6) | (7; 8; 9) |
| A2 | (2; 3; 5) | (7; 8; 10) | (3; 4; 6) |
| A3 | (5; 6; 7) | (2; 3; 4) | (8; 9; 10) |
| A4 | (4; 5; 6) | (6; 7; 8) | (1; 2; 3) |
(The MEREC family has no "Weight" row; here the weight is the method's own output.)
The method treats the triangle's lower, middle and upper vertices as three separate crisp tables, runs crisp MEREC independently on all three, then averages the three weight vectors.
| Criterion | Lower scenario | Middle scenario | Upper scenario | Average weight |
|---|---|---|---|---|
| C1 | 0.3267 | 0.3344 | 0.3720 | 0.3444 |
| C2 | 0.2399 | 0.2213 | 0.1795 | 0.2135 |
| C3 | 0.4334 | 0.4443 | 0.4485 | 0.4421 |
The result reads as follows. In all three scenarios, C3 receives the highest weight and C2 the lowest; the ranking (C3 > C1 > C2) is the same across all three scenarios. But the magnitudes vary by scenario: the gap between C1 and C3 is 0.1067 in the lower scenario, narrowing to 0.0765 in the upper scenario. This shows that the four alternatives' distinctiveness on C1 relatively increases in the optimistic scenario.
The team has one hesitation. When A3's C1 triangle is widened from (5; 6; 7) so that its upper vertex is pulled markedly wider, to (1; 6; 15), C1 (0.3983) overtakes C3 (0.3846) only in the upper scenario's weight vector; the lower and middle scenario still show C3 ahead. In the average weight, C3 is still ahead (0.4235 against 0.3698), because the rank change in a single scenario (the upper one) is diluted by being divided across three. This shows why it is risky to report only the average without checking whether there is a scenario-level disagreement.
In the report: "The weight of the three criteria has been derived by solving the pessimistic-likely-optimistic three scenarios separately with MEREC and averaging. C3 is the most distinctive criterion in all three scenarios (average weight 0.4421); its gap to C1 narrows in the optimistic scenario. In a sensitivity test where only the upper vertex was widened, the upper scenario alone puts C1 ahead of C3; this shows why a scenario-based check is necessary."
Source: an internal record verified against DecisionMind's source code (the analyse_asisance repository, a frozen code snapshot); it is evidence level C and does not rest on a peer-reviewed literature paper. The weight vectors, the scenario-level spread and the sensitivity test were recomputed by this card's author in Python (as an independent run of crisp MEREC on the three vertices, followed by averaging) and matched DecisionMind's own record to the decimal place.
2. Public transport: A municipality weighting bus-route performance criteria
A municipal transport department wants to derive from data the weights of four criteria to be used in its annual bus-route performance assessment: passenger occupancy rate, service-delay frequency ("lower is better"), fuel consumption ("lower is better") and passenger-satisfaction survey score. Traffic engineers have given three separate annual estimates, pessimistic, likely and optimistic, for each criterion; these estimates reflect seasonal fluctuation and the effect of roadworks.
The method splits the four criteria into three separate scenario tables, runs crisp MEREC independently on each table, and averages the three weight vectors. Suppose service-delay frequency receives the highest weight in all three scenarios, because this criterion showed the largest difference between routes. For fuel consumption, however, which criterion comes second changed between scenarios: passenger satisfaction came second in the pessimistic scenario, occupancy rate in the optimistic one.
The department's hesitation is this: service-delay frequency's weight is robust across all three scenarios and can be used confidently in prioritisation. But because the second-ranked criterion changes with the scenario, the priority order between occupancy rate and passenger satisfaction should not be settled by a single average weight; both criteria should be included in the short-term improvement plan.
In the report: "The weight of the four criteria has been derived by solving the three scenarios separately with MEREC and averaging. Service-delay frequency receives the highest weight in all three scenarios. The second-place ranking between occupancy rate and passenger satisfaction changes with the scenario, so both have been included in the priority plan."
3. What Not to Do
Had C2 (a cost-type criterion similar to service-delay frequency) been mistakenly marked "higher is better" in the illustrative example, the removal effect would be computed in reverse in all three scenarios. A second error is presenting this extension in a paper with a single founding citation such as "Fuzzy MEREC (Saidin et al., 2023)"; this extension has a separate, non-peer-reviewed source, and this must be stated clearly. A third error is ignoring that the C1-C3 gap narrows across scenarios and reporting only the average weight as the "definitive result"; as the sensitivity test above shows, even a single scenario's widening can change the ranking within that scenario alone.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/scenario-fuzzy-merec
Keshavarz-Ghorabaee, M., Amiri, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2021). Determination of Objective Weights Using a New Method Based on the Removal Effects of Criteria (MEREC). Symmetry, 13(4), 525. DOI: 10.3390/sym13040525
Saidin, M. S., Lee, L. S., Marjugi, S. M., Ahmad, M. Z., & Seow, H.-V. (2023). Fuzzy Method Based on the Removal Effects of Criteria (MEREC) for Determining Objective Weights in Multi-Criteria Decision-Making Problems. Mathematics, 11(6), 1544. DOI: 10.3390/math11061544
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI: 10.1016/S0019-9958(65)90241-X