IV-PROJECTION (Interval-Valued Intuitionistic Fuzzy Projection Method)

IV-PROJECTION is a ranking method for when expert opinions in your decision table are given as interval-valued intuitionistic fuzzy numbers, where both the degree of support and the degree of rejection are expressed as an interval rather than a single figure, and you want to arrange alternatives (which supplier, which partner) into a single order. Its output is a projection score for every alternative together with the rank that score produces. It resembles TOPSIS's "closeness to the ideal" idea, but instead of distance it uses projection: it measures how far an alternative extends in the direction of the ideal alternative, taking into account not only closeness but also magnitude in that direction. Yue proposed the method in 2019; the paper's own application is the evaluation of inter-institutional knowledge-transfer effectiveness.

The idea behind IV-PROJECTION is to turn "what is the best alternative" into "who casts the longest projection in the direction of the ideal alternative." In geometry, the projection of one vector onto another jointly reflects how aligned the two vectors are in direction and how large they are in magnitude. IV-PROJECTION treats each alternative and the ideal alternative as vectors; if an alternative's projection onto the ideal is large, that alternative is both closely aligned with the ideal in direction and strong in value.

This idea carries a philosophical consequence: IV-PROJECTION is compensatory. A weakness on one criterion can be offset by strength on another, because the projection is a single figure into which every criterion's contribution is absorbed. The method recognises two forms of projection: the raw (unnormalised) projection mixes direction alignment and magnitude together, while the normalised projection divides this by the alternative's own magnitude to yield a pure measure of direction alignment. DecisionMind uses the normalised projection as its ranking score; this is philosophically the closest form to TOPSIS's closeness ratio.

The method proceeds through eight steps.

First, verifying the data. Every cell is an interval-valued intuitionistic fuzzy number: a support degree given as an interval ([lower, upper]) and a rejection degree also given as an interval. These four figures, support-lower, support-upper, rejection-lower, rejection-upper, are held together in a single cell.

Second, determining criterion weights. Every criterion is assigned a numerical weight; the weights must sum to 1.

Third, constructing the ideal alternative. For a "more is better" criterion, the ideal value is set to full support and zero rejection ([1,1] support, [0,0] rejection); for a "less is better" criterion, it is set to full rejection and zero support ([0,0] support, [1,1] rejection). This is a hypothetical alternative that is perfect on every criterion.

Fourth, the weighted inner product. A weighted inner product, using the criterion weights, is calculated between each real alternative and the ideal alternative; this is the first component measuring how closely two vectors point in the same direction.

Fifth, the weighted magnitudes. The weighted magnitude (norm) of each real alternative and of the ideal alternative is calculated.

Sixth, the raw projection. Each alternative's inner product with the ideal is divided by the ideal's magnitude to give the raw projection, showing how far the alternative "extends" in the ideal's direction, though it is not independent of the alternative's own magnitude.

Seventh, the normalised projection. The same inner product is divided instead by the product of the alternative's and the ideal's magnitudes. This gives a pure direction-alignment measure that also accounts for the alternative's own magnitude, and DecisionMind uses this as the ranking score.

Eighth, ranking. Alternatives are ranked from highest to lowest normalised projection score; the highest score belongs to the alternative most aligned with the ideal.

The formulas behind each step are given on the DecisionMind method page; this card carries no formulas.

The normalised projection score states how closely an alternative is aligned with the ideal alternative's direction, and nothing more. A score of 0.97 does not mean "97 per cent good," and the score has no fixed upper bound of exactly 1 as TOPSIS's closeness ratio does; the score reaches its highest value when the ideal alternative is compared with itself, but comparisons among real alternatives are valid only within this particular analysis's set of alternatives. It cannot be compared with a score from a different analysis, because the ideal alternative is rebuilt afresh in every analysis from that analysis's own criterion directions.

The raw projection and the normalised projection must not be confused with one another: the raw projection is divided only by the ideal's magnitude, while the normalised projection is divided by both the alternative's and the ideal's magnitude. DecisionMind uses the normalised projection for ranking.

Thus instead of writing:

"IV-PROJECTION found the best alternative"

the report should read:

"With these weights and this alternative set, the alternative most aligned with the ideal is this one; the ranking is sensitive to the criterion weights"

IV-PROJECTION works with interval-valued intuitionistic fuzzy data: each cell holds not a single figure but an interval for the degree of support and an interval for the degree of rejection, four figures in total. This data type is used to carry situations where even the expert's degrees of support and rejection are not given with full precision but as intervals: both "how much do I support this" and "how much do I reject this" are themselves uncertain. DecisionMind carries no extension of this method in other data types; it appears only in this base form.

You need alternatives in rows, criteria in columns, and in each cell an interval-valued intuitionistic fuzzy number (support-lower, support-upper, rejection-lower, rejection-upper); for every criterion, whether more is better or less is better; and criterion weights that sum to 1. A minimum of two alternatives and two criteria is required; three to ten criteria work comfortably, and the method has been tested with up to twenty-five alternatives. In every cell, the sum of the support upper bound and the rejection upper bound must not exceed 1; this is the basic constraint of an interval-valued intuitionistic fuzzy set, and data violating it is considered invalid.

If your expert opinions give the degree of support and the degree of rejection themselves as intervals, not just single figures, and you want to rank alternatives against an ideal reference, IV-PROJECTION is a suitable choice. Its typical territory is group decisions where multiple experts do not fully agree on the same judgement but the disagreement can reasonably be bounded by an interval: knowledge-transfer effectiveness evaluation, patent quality evaluation, and expert-driven evaluation under uncertainty.

It should not be used in the following cases. If your data consists of single figures only, crisp data, carrying this extra layer of uncertainty brings no benefit, and a simpler method such as TOPSIS suffices. If no compromise is possible on one criterion, IV-PROJECTION, being a compensatory method, is not appropriate. The ideal alternative cannot be constructed correctly without criterion directions (more/less is better) being marked correctly, so the method should not be run without this information.

Support/rejection degrees given as intervals, the goal is ranking → IV-PROJECTION

Data is crisp, no need for an extra layer of uncertainty → TOPSIS

No compromise allowed on one criterion, sub-threshold alternatives must be screened out → screening first, then ranking

Only two degrees (support/rejection) as single figures, no interval → single-valued intuitionistic fuzzy methods

IV-PROJECTION's greatest strength is that it extends TOPSIS's "closeness to the ideal" logic with a projection measure that carries both distance and magnitude together; this is a well-defined approach in the interval-valued intuitionistic fuzzy setting, with several variants developed in the literature (Tsao and Chen, 2016; Zheng, Xu and He, 2025). The interval-valued intuitionistic fuzzy data type lets an expert carry their support, their rejection, and the uncertainty in both as intervals, offering a richer representation of uncertainty than a single-valued intuitionistic fuzzy structure (Atanassov and Gargov, 1989). The method's computational burden is small and stays just as simple as the number of alternatives and criteria grows.

Its limitations stem from the structure of the projection measure itself and from its compensatory nature. First, a one-directional projection measures only alignment towards the ideal; it does not separately account for distance from the anti-ideal, the worst alternative, which can mean it carries less information than TOPSIS's dual-reference approach in some situations (Zheng, Xu and He, 2025, who propose a bidirectional projection to address this limitation). Second, a projection-based measure alone may not be sufficient for seeking a "compromise," which is why an alternative compromise-projection index has also been developed separately in the literature (Tsao and Chen, 2016). Third, there is the assumption of full compensation: a serious weakness on one criterion can be papered over by others. Fourth, the quality of the weights lies outside the method itself; a flawless calculation built on poor weights still produces a poor ranking.

The most common mistake is constructing the ideal alternative for a cost criterion incorrectly. For a "less is better" criterion, the ideal cell must be ⟨[0,0],[1,1]⟩ (full rejection, zero support); reverse this and the ideal alternative is built from the most expensive, slowest alternative, and the ranking is reversed.

A second mistake is confusing the raw projection with the normalised projection; DecisionMind uses the normalised projection for ranking, and the raw projection is a separate intermediate value. A third mistake is entering a cell as a single interval or a triangular fuzzy number instead of the four interval-valued intuitionistic fuzzy figures; this leads to a data-type mismatch. A fourth mistake is including, without noticing, an invalid cell whose support upper bound and rejection upper bound sum to more than 1. A fifth mistake is reading the projection score as a percentage and comparing scores from different analyses.

The governing principle is this:

An IV-PROJECTION result is an alignment summary, relative to the ideal alternative, of the criterion directions, weights and alternative set that you supplied; if any input is contested, the ranking is contested too.