Interval Number Grey Relational Analysis (Olson and Wu, 2008)

Four things change; the grey relational coefficient and the ranking logic do not.

Cells. In crisp GRA every cell is a single number. Here every cell is a lower and an upper bound: [a⁻, a⁺]. This is not uncertainty or a lack of information; it is known with certainty that the value lies between these two bounds. Weights can be crisp or interval numbers; DecisionMind supports a scalar (single-number) weight in this extension.

An interval must not be confused with a grey number. The two share the same shape, two bounds. The difference lies in context: the bound in Grey GRA is a "partial information" object that may narrow in future as more becomes known, and it is reduced to its midpoint (whitenisation) before entering the calculation. The bound here states that the value is exactly within that interval and is never reduced to a midpoint at any step; both bounds are carried separately right to the end.

Scale equalisation. Crisp GRA places every column into the 0–1 range according to its own smallest and largest value (min–max normalisation). This extension does not use min–max. First, direction is corrected on a cost criterion by reversing the sign of both bounds ([a⁻,a⁺] → [−a⁺,−a⁻]). Each column is then divided by the largest absolute value among that column's two bounds across all cells (the column's vector norm). This does not squeeze the column into 0–1; it merely scales its magnitude against the column's own extreme point.

The reference is built from both bounds. In crisp GRA the reference sequence is a fixed single normalised value, (1; 1; …; 1). Here the reference is an envelope built separately from the largest of each column's lower bounds and the largest of its upper bounds: U₀ = ([max a⁻], [max a⁺]). This is a reference that may not match any real alternative exactly, criterion by criterion assembling the best of each; the logic resembles the optimal-alternative envelope on the IV-ARAS card.

Distance. In crisp GRA, distance is the absolute difference of a single number. Here distance is whichever is larger of the absolute difference of the lower bounds and the absolute difference of the upper bounds (the worse of the two bounds is taken as the basis). This is not a midpoint difference; both bounds are compared separately and the larger deviation is chosen.

DecisionMind fixes, in this extension, vector-norm scaling, the two-bound envelope reference and the two-bound distance rule; the grey relational coefficient and degree are then computed with the same formula as crisp GRA, using the discrimination coefficient ρ.

The grey relational degree shows, as in crisp GRA, an alternative's relative closeness to the reference within this particular analysis; it cannot be compared with a different analysis.

The difference is here. The degree is computed not from the midpoint of the bounds but from the bounds themselves; however wide an alternative's interval is, it is neither penalised nor rewarded separately for that width. Width only affects the degree indirectly, through how far the lower and upper bounds deviate from the reference in the distance calculation. This differs from Grey GRA, which works from the midpoint and never sees the width at all; here width enters the calculation implicitly, because the two bounds are carried separately.

Thus instead of writing:

"According to interval-number GRA, A1 is more reliable because its interval is narrow"

the report should read:

"A1's grey relational degree of 0.714 is higher than A2's 0.667; this difference comes from A1's lower and upper bounds sitting closer to the reference on both criteria, not from the width of the interval itself"

Use this extension when a criterion value is known only through a lower and an upper bound, and these bounds are known to be exact, that is, not expected to narrow with future information. Examples: a minimum-maximum delivery time guaranteed in a contract, a device's operating range as stated by the manufacturer, an approved lower and upper limit for a budget item. Where the bound sits in a "partial information" context, that is, where it is expected to narrow to a more precise point in future, Grey GRA is the more suitable framework.

If the value is measured as a single number, remain with base GRA. If the table is mixed, DecisionMind requires a single data type; a crisp value is written as a zero-width interval in the form [a, a]. Base GRA's exit condition applies here too: if a criterion carries a threshold on which no compromise is ever acceptable, GRA's additive structure does not preserve it.

Allowing the lower bound to exceed the upper bound. Every interval must satisfy a⁻ ≤ a⁺; feeding this into the calculation without checking renders the scaling and reference steps meaningless.

Forgetting the cost conversion. If the sign reversal ([a⁻,a⁺] → [−a⁺,−a⁻]) is not applied on a "lower is better" criterion, the ranking silently reverses; this is the most common error point the manifest itself names.

Mistaking this for triangular fuzzy GRA. The input here is not a triangular fuzzy number (lower, most likely, upper) but a two-ended interval; there is no most-likely point. The arithmetic, too, runs not on the triangular fuzzy extension principle but on Olson and Wu's interval-product rules. Confusing the two produces a different normalised matrix.

Reducing to a midpoint and treating it as Grey GRA. This treats the interval as "partial information" and adds a whitenisation step that DecisionMind never performs in this extension; the result approaches Grey GRA's result but departs from this extension's own definition.

Pushing the discrimination coefficient (ρ) outside its bounds. ρ must lie in (0,1]; ρ far above 1 weakens discrimination, while ρ very close to zero makes every alternative other than the closest one resemble each other.

The governing principle is this:

In interval-number GRA both bounds are carried separately right to the end and are never reduced to a midpoint at any step. Skipping the cost conversion, confusing this with triangular fuzzy GRA, or reducing to a midpoint early and treating it as Grey GRA all break the method's principle of preserving the bounds as exact information.

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