Extension card · Neutrosophic
Interval neutrosophic MULTIMOORA (Stanujkić et al., 2021)
Interval neutrosophic MULTIMOORA is the form of MULTIMOORA for situations where each component of a criterion assessment's truth-indeterminacy-falsity triple is given not as a single number but as an interval. The ratio system, the reference point and the full multiplicative form are each computed separately over these intervals; the result is then combined into a single ranking by the average-rank rule.
Base method
MULTIMOORA →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Neutrosophic →
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?
Four things change; the general flow is preserved.
Cells. In crisp MULTIMOORA every cell is a single number. Here every cell consists of six numbers: the lower and upper bound of truth, the lower and upper bound of indeterminacy, the lower and upper bound of falsity. The expert gives each of T, I and F as an interval rather than a single value. Weights are crisp numbers. The method's literature (Stanujkić et al., 2021) also defines a step that combines several decision-makers' scores into a single group matrix using a border–median technique. DecisionMind does not run this combination inside the engine; it asks the user for an already-combined, single interval neutrosophic decision matrix.
Scale equalisation. Crisp MULTIMOORA equalises every column by dividing by the square root of the sum of the squares. This family has no such division step. Instead, the benefit criteria in each alternative's row are combined directly into a SINGLE interval neutrosophic number using a weighted neutrosophic aggregation operator (IVNWA); the cost criteria are combined separately in the same way. Because this operator already carries the weights within itself and keeps the result within the [0,1] bound of the value space, no separate division step, as in crisp MULTIMOORA, is needed.
Distance / score / combination. The ratio system reduces the benefit side's combined value to a single number with a score function, does the same for the cost side, and takes the weighted difference of the two scores (the cost-side score, multiplied by the sum of the cost weights, is subtracted from the benefit-side score multiplied by the sum of the benefit weights). The reference point sets an ideal point for each criterion according to its direction: on a benefit criterion, the highest score, the lowest indeterminacy, the lowest falsity; on a cost criterion, the lowest score, the lowest indeterminacy, the highest falsity. Because the indeterminacy component is considered undesirable in both directions, it does not switch direction; only truth and falsity swap places. The largest component-based distance of each alternative to this reference is taken. The full multiplicative form combines the benefit and cost sides separately with a geometric operator (IVNWG) and takes the RATIO of the two sides' scores; crisp MULTIMOORA's multiplication/division logic corresponds here to a division at the score level.
Result and defuzzification. The output is a single number. When combining the three sub-rankings, this extension does NOT use Neutrosophic MULTIMOORA's (N-MULTIMOORA) "dominance in at least two sub-methods" rule. Instead it computes an average-rank measure equivalent to crisp MULTIMOORA's own Borda rule: each alternative's positions across the three sub-rankings are averaged, and the smallest average takes first place in the final ranking. This shows that a different extension in the same family (N-MULTIMOORA) uses a different combination rule; the two do not automatically behave the same way.
DecisionMind fixes, in this extension, the IVNWA/IVNWG operators, the score function, the direction-branching definition of the reference point (including the fact that indeterminacy does not switch direction), and the average-rank combination rule.
How to Read the Output
The final ranking is a combined summary of the three viewpoints (average performance, distance from the worst case, multiplicative balance); reading it is essentially the same as in crisp MULTIMOORA. The difference lies in the combination rule: the total/average-rank rule applies here, not N-MULTIMOORA's "dominance in two sub-methods" rule. The interval's width (upper bound minus lower bound) also dissolves into the score once it is reduced to a single number; two alternatives coming out with a similar score does not mean their intervals are equally wide.
Thus instead of writing:
"Interval neutrosophic MULTIMOORA carries uncertainty as an interval, so the result is more reliable"
the report should read:
"Each component's interval is preserved up to the point it enters the score function; but the final ranking is built with crisp MULTIMOORA's total/average-rank rule, not N-MULTIMOORA's dominance-in-two-of-three rule"
When to Prefer This over the Base Method
This extension is used when experts can only give the degrees of truth, indeterminacy and falsity of a judgement as an interval rather than a single number, for instance a statement such as "truth is between 0.60 and 0.80." If an interval is not needed, that is, if every component is given as a single number, single-valued neutrosophic MULTIMOORA (N-MULTIMOORA) suffices and demands less data collection; an interval is an assumption here, not automatically a "richer" model.
A measured criterion should not be written directly into the T interval while leaving the remaining components incomplete; this is the mistake the neutrosophic data-type card warns against. Crisp MULTIMOORA's exit condition applies here too: all values must stay within the valid interval so that the full multiplicative form does not become undefined, and this method should not be used if no compromise can ever be made on one criterion.
Mistakes Specific to This Extension
Reducing the interval to a single number too early. Taking the average of the lower and upper bound and treating it like single-valued neutrosophic data removes this extension's one contribution: carrying the interval through to the score function.
Reversing the indeterminacy component on a cost criterion. The reference point swaps the positions of truth and falsity on a cost criterion but does NOT change indeterminacy; indeterminacy is considered undesirable in both directions. Reversing indeterminacy as well means applying the criterion's direction twice and builds the reference point incorrectly.
Carrying the dominance-in-two-of-three rule over to this extension. This extension uses the total/average-rank rule; treating the same family's N-MULTIMOORA rule ("dominance in at least two sub-methods") as valid here is wrong and can produce a different final ranking.
Assuming the group-combination step runs in the engine. The border–median group combination in the literature does not run in this engine; the user enters a single, already-combined matrix. If more than one expert's scores exist, the combination must be done outside the card, at the data-preparation stage.
The governing principle is this:
In interval neutrosophic MULTIMOORA, each component's interval is preserved up to the score function; the final ranking is built with the total/average-rank rule, not N-MULTIMOORA's dominance-in-two-of-three rule, and the indeterminacy component is not changed when the cost direction is applied.
Cases
The first case is a case from the literature: the cloud-service selection example from Stanujkić and colleagues' (2021) book chapter; the figures are taken from that chapter's table, and the provider names have been numbered here so as not to carry a real institution's name. The second case is an illustrative construction.
1. IT: An organisation's choice of cloud infrastructure provider (Stanujkić et al., 2021, Table 18)
An organisation will choose among four cloud infrastructure providers (Provider 1–4). All six criteria are higher-is-better and equally weighted (each 1/6 ≈ 0.167): cost suitability, availability, storage, processing power, performance, security. Every cell is the interval form of the truth-indeterminacy-falsity triple.
Truth (T) intervals:
| Provider | Cost suitability | Availability | Storage | Processing power | Performance | Security |
|---|---|---|---|---|---|---|
| 1 | 0.90–1.00 | 1.00–1.00 | 0.90–1.00 | 0.70–0.70 | 0.90–1.00 | 0.90–0.90 |
| 2 | 1.00–1.00 | 1.00–1.00 | 1.00–1.00 | 0.60–0.60 | 1.00–1.00 | 0.90–1.00 |
| 3 | 0.70–0.80 | 0.90–0.90 | 0.60–0.70 | 0.50–0.50 | 0.90–0.90 | 0.70–0.70 |
| 4 | 0.70–0.80 | 0.80–0.90 | 0.70–0.80 | 0.50–0.60 | 0.80–0.90 | 0.70–0.80 |
Indeterminacy (I) intervals:
| Provider | Cost suitability | Availability | Storage | Processing power | Performance | Security |
|---|---|---|---|---|---|---|
| 1 | 0.10–0.60 | 0.00–0.10 | 0.00–0.10 | 0.30–0.30 | 0.00–0.10 | 0.10–0.10 |
| 2 | 0.00–0.00 | 0.00–0.00 | 0.00–0.00 | 0.00–0.00 | 0.00–0.00 | 0.10–0.10 |
| 3 | 0.20–0.30 | 0.00–0.00 | 0.20–0.20 | 0.00–0.10 | 0.00–0.10 | 0.10–1.00 |
| 4 | 0.20–0.30 | 0.10–0.20 | 0.10–0.20 | 0.20–0.30 | 0.10–0.20 | 0.10–0.20 |
Falsity (F) intervals:
| Provider | Cost suitability | Availability | Storage | Processing power | Performance | Security |
|---|---|---|---|---|---|---|
| 1 | 0.10–0.60 | 0.00–0.10 | 0.00–0.10 | 0.10–0.20 | 0.00–0.10 | 0.10–0.10 |
| 2 | 0.00–0.00 | 0.00–0.00 | 0.00–0.00 | 0.20–0.20 | 0.00–0.00 | 0.10–0.10 |
| 3 | 0.10–0.20 | 0.00–0.00 | 0.00–0.00 | 0.00–0.10 | 0.00–0.10 | 0.10–1.00 |
| 4 | 0.10–0.20 | 0.10–0.20 | 0.10–0.20 | 0.20–0.30 | 0.10–0.20 | 0.10–0.20 |
The method combines each provider's six criteria into a single interval neutrosophic number using the weighted neutrosophic aggregation operator. In this example every criterion is on the benefit side; the cost side is empty. The reference point is built from each criterion's ideal cell (highest score, lowest indeterminacy-falsity), and the full multiplicative form is computed with the geometric operator. These three computations were produced by running DecisionMind's engine (method_runner.py IVN-MULTIMOORA) and matched the manifest's recorded values exactly.
| Provider | Ratio system | Rank | Reference point (distance) | Rank | Full multiplicative | Rank |
|---|---|---|---|---|---|---|
| 1 | 0.8812 | 3 | 0.0833 | 3 | 0.7128 | 2 |
| 2 | 1.0000 | 1 | 0.0278 | 1 | 0.9112 | 1 |
| 3 | 0.8877 | 2 | 0.1278 | 4 | 0.0568 | 4 |
| 4 | 0.6251 | 4 | 0.0722 | 2 | 0.6027 | 3 |
Provider 2 is first on all three sub-methods; this is a strong first place. Provider 1 and Provider 3, by contrast, are disputed between the sub-methods. The ratio system places Provider 3 second, whereas the reference point and the full multiplicative form put Provider 1 ahead. Provider 3's full multiplicative score of 0.0568 is last, because its indeterminacy and falsity interval on the security criterion is far wider than on its other criteria.
| Provider | Average rank | Final rank |
|---|---|---|
| 2 | 1.00 | 1 |
| 1 | 2.67 | 2 |
| 4 | 3.00 | 3 |
| 3 | 3.33 | 4 |
The organisation's hesitation is this: Provider 3's fall to last place stems largely from a single criterion, security. The upper bound of the indeterminacy and falsity interval on this criterion is markedly wider than for the other providers. Before eliminating Provider 3, the organisation should separately verify the source of this one criterion's assessment, that is, which expert or document it came from.
In the report: "Under the average-rank rule, Provider 2 is clearly first in the final ranking; it comes first on all three sub-methods. Provider 1 is second, Provider 4 third, Provider 3 fourth; Provider 3's fall to the back stems largely from the wide indeterminacy-falsity interval on the security criterion, and this criterion should be separately verified."
Source: Stanujkić, D., Zavadskas, E. K., Smarandache, F., Brauers, W. K. M., & Karabašević, D.'s (2021) cloud-computing-technology-selection example (Table 18); the provider names have been numbered in this card so that no real institution's name appears. DecisionMind's engine in this family has been flagged in the direction test of the suspicious-method sweep; the detail is in the sign-off note.
2. Care home: A foundation's choice of care-service provider
A foundation will grant service authority to one of three care-home operators. The criteria are compliance with medical care quality, compliance with staff competence, and compliance with physical-premises standards; all three are higher-is-better. The inspection team expresses, for each operator, how much support it gives these three criteria as an interval, because different observers gave slightly different scores across different inspection visits; the width of the interval reflects this disagreement among observers.
The method combines each operator's interval neutrosophic triples on the three criteria using the weighted aggregation operator, builds the reference point, and computes the full multiplicative form. Suppose the operator with the highest truth interval on medical care quality comes first on the ratio system, but this operator's interval on the staff-competence criterion is very wide, that is, observers disagreed considerably.
The foundation's hesitation is this: a wide interval, even where the average value is high, points to serious disagreement among observers. The foundation should look not only at the average rank but also at which operator has the narrowest, least disputed, interval on which criterion; if the disagreement is large, an additional inspection visit can be requested.
In the report: "Under the average-rank rule, the operator with the highest truth interval is ahead. The width of the interval on the staff-competence criterion shows disagreement among observers, and an additional inspection visit is recommended before a decision is made."
3. What Not to Do
The first error is taking the average of the lower and upper bound of every cell in the illustrative example (for instance, giving Provider 1's cost suitability T = 0.95, I = 0.35, F = 0.35) and running single-valued neutrosophic MULTIMOORA (N-MULTIMOORA); this erases the uncertainty information the interval carries and conflates two methods. The second error is also reversing the indeterminacy component when building the reference point for a hypothetical cost criterion; indeterminacy is considered undesirable in every direction and is not reversed. The third error is recomputing the final ranking with N-MULTIMOORA's dominance-in-two-of-three rule; this extension uses the total/average-rank rule, and the two rules can produce a different final ranking.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/ivn-multimoora
Stanujkić, D., Zavadskas, E. K., Smarandache, F., Brauers, W. K. M., & Karabašević, D. (2021). Cloud computing technology selection using a novel neutrosophic extension of the MULTIMOORA method. In F. Smarandache & M. Abdel-Basset (Eds.), Neutrosophic Operational Research: Methods and Applications (pp. 367–395). Springer. DOI: 10.1007/978-3-030-57197-9_18
Stanujkic, D., Zavadskas, E. K., Smarandache, F., Brauers, W. K. M., & Karabasevic, D. (2017). A neutrosophic extension of the MULTIMOORA method. Informatica, 28(1), 181–192. DOI: 10.15388/Informatica.2017.125
Brauers, W. K. M., & Zavadskas, E. K. (2010). Project management by MULTIMOORA as an instrument for transition economies. Technological and Economic Development of Economy, 16(1), 5–24. DOI: 10.3846/tede.2010.01
Wang, H., Smarandache, F., Zhang, Y., & Sunderraman, R. (2010). Single valued neutrosophic sets. Multispace and Multistructure, 4, 410–413. (no DOI)
Smarandache, F. (1998). Neutrosophy: Neutrosophic probability, set, and logic. American Research Press, Rehoboth. (no DOI)