Data types
Neutrosophic
This is the data structure that preserves the degrees of truth, indeterminacy and falsity in an assessment as three independent values.
Example cell: 0.7, 0.2, 0.1
What Is It?
A neutrosophic data structure expresses an alternative's standing on a criterion not as one value but through three separate degrees: how far the assessment is true (T), how far it remains indeterminate (I), and how far it is false (F). Each degree lies between 0 and 1 and, in the single-valued form used in decision analysis, the three are independent of one another; their sum may be anything between 0 and 3.
What distinguishes this structure is that indeterminacy is kept as a separate, independent component. In an intuitionistic fuzzy structure, the hesitancy share is what remains once membership and non-membership are subtracted; in a neutrosophic structure, indeterminacy is assessed directly and is not derived from truth or falsity. This makes it possible to represent, separately, both a case with strong supporting and strong opposing evidence at once, and a case with a large unknown share alongside either.
When to Use It
This structure suits problems where the information available on a criterion is incomplete, inconsistent or contradictory, and where that condition matters to the decision itself. It is valuable for assessing alternatives with no track record, expert opinions resting on conflicting sources, partly observable systems, and analyses where the unknown share must be explicitly reported.
Conversely, if the assessment rests on a reliable measurement or experts express no share of uncertainty, converting the data to a neutrosophic structure merely to use a more elaborate model is unnecessary.
Can Neutrosophic Data Be Built from Crisp Data?
Yes, but a conversion is needed first: a neutrosophic triple expresses the degrees of truth, indeterminacy and falsity of a judgement, not of a number. "Gold is priced at 2,450 TL" is a measurement; it is neither true nor false. "Gold will deliver price stability over the next six months" is a judgement, and for this judgement the evidence for, the evidence against, and the unknown share can each be assessed separately.
The first step in moving a crisp value into a neutrosophic structure is therefore converting the criterion into a judgement: the proposition "the alternative meets this criterion". The second step is deriving each of the three components from its own source of information: T from evidence in favour, F from evidence against, and I from a separate measure such as missing data, conflicting forecasts, or an unreliable source.
If the value itself is known with certainty, and other criteria in the same decision matrix are neutrosophic so this one must be written in the same form, the honest embedding is: if the judgement's degree of fulfilment is t, then (t, 0, 1 − t). This triple states "no indeterminacy" and adds no new information; it is for consistency with the structure, not a justification for neutrosophic analysis.
What must not be done is writing components around a precise figure by feel: expressing 2,450 TL as (0.7, 0.1, 0.2) has no information source behind it.
Indeterminacy Is Not the Same as Hesitation
The I component in a neutrosophic structure is not the numerical equivalent of "I am undecided". An expert torn between two values is the domain of the hesitant structure; the I component instead measures how far the information behind the assessment is missing or inconsistent.
For this reason, it is also wrong to expect the three components to sum to 1. If the sum is 1, the assessment is in fact a dependent triple of "yes / abstain / no" proportions, for which a picture fuzzy structure may be more suitable. A neutrosophic structure is called for precisely where the components do not constrain one another.
For this reason:
"Neutrosophic is used because the expert is undecided"
should give way to:
"This is used because the information behind the assessment is missing or contradictory, and that condition is preserved as a separate component"
Strengths
The chief advantage of the neutrosophic structure is that it does not conflate the unknown with negative evidence. "There is no information on this" and "there is negative evidence on this" are held in separate components, so a gap in the data does not unfairly penalise an alternative.
It can also represent contradictory evidence. Where strong evidence exists both for and against an alternative, other structures must resolve the two against each other; a neutrosophic structure carries both at once and makes the contradiction itself visible to the decision process.
Limitations
Asking for three independent components is not intuitive for experts and raises the assessment burden. In practice, experts may fail to distinguish the indeterminacy component consistently, and true independence between the components may not actually hold.
Ranking requires reducing the three components to a single value, and more than one score function has been proposed for this reduction; which one is chosen, and in particular how the indeterminacy component is weighted, can affect the result. Nor should every kind of uncertainty be represented with a neutrosophic structure: measurement error, randomness, interval uncertainty and expert hesitation are distinct concepts, and the data structure should be chosen according to the true source of the uncertainty.
When Are Interval-Valued and Bipolar Neutrosophic Data Used?
In the single-valued structure, each component is one figure. Where experts can only give the components as intervals ("truth between 0.6 and 0.8"), the interval neutrosophic structure is more suitable.
Where an assessment needs positive and negative effects expressed separately, with different signs, for instance where a supplier's positive contribution and negative risk are assessed as independent components, the bipolar neutrosophic structure may be used.
These distinctions matter particularly in problems where the certainty of expert assessment is low, such as early-stage technology selection, entry into a new market, and the evaluation of a strategic partner.
Common Mistakes
The most frequent mistake is forcing the three components to sum to 1. This reduces the structure to a dependent triple and removes the justification for using a neutrosophic structure at all.
The second is calculating the indeterminacy component as 1 − T − F, in which case I carries no independent information. Interpreting T as a probability, adding components around a precise value without justification, or converting verbal expressions ("good", "medium") into triples without a defined rule cause similar methodological problems.
The governing principle is this:
Every component of a neutrosophic triple must rest on its own source of information for the same assessment, and must not be derived from the other two.
Examples
Each example opens with a familiar, single precise figure and shows the conditions under which, and the steps by which, that same figure moves into neutrosophic form.
1. Business: A Supplier's Reference Score Is 8/10
A precise figure. A supplier new to the sector scored 8 out of 10 on a reference survey. This is a measurement; it does not become neutrosophic on its own.
Step 1: Convert the criterion into a judgement. The decision is between three suppliers; the criterion is "reliability". The judgement: "This supplier meets its delivery and quality commitments."
Step 2: Derive the three components from three sources.
- •T (for): two reference letters are positive and the sample batch was accepted → 0.60
- •F (against): one trade report mentions late delivery, and a trial order arrived two days late → 0.20
- •I (unknown): there is no past delivery record and financial statements were not shared → 0.50
Neutrosophic form. On "reliability": (0.60, 0.50, 0.20). The sum exceeds 1, which is normal, since the three components come from separate sources. The score of 8/10 alone carries no information about a half-unknown share.
Same figure, different situation. If the same 8/10 score belonged to a decade-old supplier with a complete record, the unknown share would shrink: (0.85, 0.05, 0.15). The figure is the same; confidence in the information behind it differs, and the structure keeps that difference in a separate component.
2. Medicine: Treatment Efficacy Is 65 Per Cent
A precise figure. A review summarises a treatment's efficacy in a given patient group as 65 per cent, a measured ratio.
Step 1: Convert the criterion into a judgement. The decision is between three treatment options; the criterion is "efficacy". The judgement: "This treatment is effective in this patient group."
Step 2: Derive the three components from three sources.
- •T (for): three of five studies report a significant benefit; the pooled effect is 65 per cent → 0.65
- •F (against): two studies found no benefit and one reported an adverse effect → 0.40
- •I (unknown): the studies have small samples, wide confidence intervals, and two come from the same centre → 0.40
Neutrosophic form. On "efficacy": (0.65, 0.40, 0.40). Both T and F are high; the contradiction stays visible rather than being buried in an average. The 65 per cent figure alone carries no information about conflicting evidence.
Same figure, different situation. If a large, multi-centre study is published confirming the 65 per cent figure, the unknown share shrinks and the opposing evidence weakens: (0.70, 0.10, 0.20).
3. Economics: An Incentive's Employment Effect Is 12 Per Cent
A precise figure. An evaluation report states that a regional investment incentive raised employment by 12 per cent, a precise figure in the report.
Step 1: Convert the criterion into a judgement. The decision is between three incentive designs; the criterion is "employment effect". The judgement: "This incentive creates lasting employment in the region."
Step 2: Derive the three components from three sources.
- •T (for): four of seven studies find a positive effect → 4/7 = 0.57
- •F (against): three studies find no effect or report a displacement effect → 3/7 = 0.43
- •I (unknown): regional data arrive with a two-year lag and informal employment is not measured → 0.50
Neutrosophic form. On "employment effect": (0.57, 0.50, 0.43). The three components answer three separate questions: how far the evidence favours it, how far it opposes it, and how incomplete the data are.
Same figure, different situation. If the same 12 per cent comes from current administrative data that also covers informal employment, the I component falls: (0.60, 0.15, 0.35). The reported figure is unchanged; the unknown share changes.
4. Engineering: Equipment Success Rate Is 0.78
A precise figure. A maintenance unit records a piece of equipment's successful cycle rate as 0.78.
Step 1: Convert the criterion into a judgement. The decision is between three pieces of equipment; the criterion is "performance". The judgement: "This equipment meets its mission profile."
Step 2: Derive the three components from three sources. T comes from the success rate, I from the rate of missing records, and F from the failure rate, each held in its own record:
- •T = 0.78, I: 30 per cent of records are missing → 0.30, F: failure rate → 0.10
Neutrosophic form. (0.78, 0.30, 0.10). Each component has its own measurement. Generating (0.78, 0, 0.22) from the success rate alone would not be neutrosophic data.
Same figure, interval form. If an expert can only give the components as intervals (T between 0.7 and 0.85, I between 0.2 and 0.4, F between 0.05 and 0.15), an interval-valued neutrosophic structure is used instead of reducing them to single values.
5. What Not to Do
Converting a reference score of 8/10, an efficacy of 65 per cent, or an employment effect of 12 per cent directly into a triple: 65 per cent → (0.65, 0, 0.35). This triple states "no indeterminacy" and empties the structure of its only contribution; 8/10 → (0.8, 0.1, 0.1) is a guess in which no component rests on any source of information. The figure must first be converted into a judgement, and only then can each component be derived from its own source.
The numbers in the examples are fictional; they are not real data.
Short decision rule
A single, reliable value → Crisp
Membership and non-membership, sum at most 1 → Intuitionistic fuzzy
Yes / abstain / no, sum at most 1 → Picture fuzzy
True / indeterminate / false, independent of one another → Neutrosophic
Components given as intervals → Interval-valued neutrosophic
Positive and negative effect with separate signs → Bipolar neutrosophic
Key sources
Smarandache, F. (1998). Neutrosophy: Neutrosophic Probability, Set, and Logic. American Research Press, Rehoboth. (no DOI)
Wang, H., Smarandache, F., Zhang, Y., & Sunderraman, R. (2010). Single valued neutrosophic sets. Multispace and Multistructure, 4, 410–413. (no DOI)
Ye, J. (2013). Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. International Journal of General Systems, 42(4), 386–394. DOI: 10.1080/03081079.2012.761609
Ye, J. (2014). A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. Journal of Intelligent & Fuzzy Systems, 26(5), 2459–2466. DOI: 10.3233/IFS-130916
Peng, J.-J., Wang, J.-Q., Wang, J., Zhang, H.-Y., & Chen, X.-H. (2016). Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. International Journal of Systems Science, 47(10), 2342–2358. DOI: 10.1080/00207721.2014.994050
Biswas, P., Pramanik, S., & Giri, B. C. (2016). TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Computing and Applications, 27(3), 727–737. DOI: 10.1007/s00521-015-1891-2