Grey relational analysis (interval grey)

Three things change; the logic of the reference sequence and the grey relational coefficient does not.

Cells. In crisp GRA every cell is a single number. Here every cell is a lower and an upper bound: ⊗[x̲, x̄]. These bounds carry no most-likely value; only "at least this much, at most that much" information is available. Weights remain crisp numbers; GRA does not generate weights, it takes them from outside.

Scale equalisation. There is an important difference here. Crisp GRA places every column into the 0–1 range according to its own smallest and largest value, and builds the reference sequence as a constant (1; 1; …; 1). In Grey GRA, DecisionMind does not apply this min–max normalisation.

Instead, every cell is first whitenised, that is, the midpoint of the lower and upper bound is taken: v̂ = (x̲+x̄)/2. The reference sequence is then built from the best of these whitenised values within its own column: from the largest for a more-is-better criterion, from the smallest for a less-is-better criterion. This reference is not a constant 1. Criteria must be given on a mutually comparable scale, for instance all as a 0–1 ratio or an index. If columns in different units (currency, count, days) are entered as they are, the criterion with the widest numerical range dominates the deviation (Δ), and the effect of the other criteria becomes invisible.

Distance and grey relational grade. The absolute difference (Δ) between the whitenised value and the reference is computed. This deviation is converted into a grey relational coefficient according to the smallest and largest deviation in the data set and the distinguishing coefficient (ζ, fixed at 0.5 in DecisionMind). The coefficients are multiplied by the weight and summed. This step works by the same logic as crisp GRA, only the inputs are the whitenised grey values.

DecisionMind holds the distinguishing coefficient (ζ = 0.5) and the whitenisation rule (taking the midpoint) fixed in Grey GRA; there is no separate defuzzification step at the result stage, because whitenisation has already been done at the start.

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

The difference is here. The grade is computed from the midpoint of the bounds, and the width of the bounds, that is, the size of the uncertainty, does not visibly enter the score; only the midpoint enters. If two alternatives' midpoints are close but one comes from a very narrow interval (⊗[48,52]) and the other from a very wide one (⊗[20,80]), the grey relational grades can come out almost the same. Yet the second alternative's information is much weaker. This is why the report should state not only the grade but also the width of the bounds.

Thus instead of writing:

"According to Grey GRA, A2 gave a more reliable result because it accounted for uncertainty"

the report should read:

"A2's midpoint is closer to the reference (grey relational grade 0.825); this result rests on the midpoint of the bounds, and A2's bound width should also be reported separately"

This extension is used when only a lower and upper bound are known about criterion values, and no point in between can be considered more likely than another. Examples: new supplier bids evaluated with little data, new product or market assessments with a short track record, expert judgements able to say "at least this much, at most that much" but not "most likely."

If the value is measured by a single figure, or if a known most-likely point exists within the bounds, the base method should be kept. In that case the fuzzy extension carries more information; discarding the bounds and descending to the grey structure erases information. If the table is mixed, that is, some criteria are crisp and others grey, DecisionMind requires a single data type. A crisp value is written into the grey structure as ⊗[x, x], with zero bound width. The base GRA's exit condition applies here in exactly the same way: if a criterion has a threshold on which no compromise can ever be made, GRA's additive structure does not preserve it.

Violating GIN. In every cell, the lower bound cannot be greater than the upper bound (x̲ ≤ x̄); the reverse entry cannot be computed and must be corrected first.

Leaving criteria in their own unit without scale equalisation. As explained above, Grey GRA does not place columns into 0–1. Putting a cost column in currency into the same table as a score column already on a 0–1 scale hands the deviation calculation over to the absolute magnitude of the cost column. Criteria must be brought to a comparable scale (index, ratio) before the analysis.

Assuming whitenisation is done at the end of the analysis rather than the start. Taking the midpoint (whitenisation) is the first step here, not a defuzzification done at the end as in crisp GRA; the reference and the deviation are built without the bound width ever entering. A use that does not additionally add the width to the report as a separate measure of uncertainty has failed to account for the summing of the bounds at all.

Never questioning the distinguishing coefficient. ζ = 0.5 is DecisionMind's fixed value; as in crisp GRA, this choice can influence the result more or less here too, especially in tables where the deviations (Δ) are close to one another.

Marking a criterion's direction wrongly. If a less-is-better criterion is marked as more-is-better, the reference sequence (the best of the whitenised values) is built from the wrong end of the column and the ranking reverses; this is the grey counterpart of the same error in crisp GRA.

The governing principle is this:

Grey GRA is a GRA that runs from the midpoint of the bounds; if the columns have not first been brought to a comparable scale and the bound width is not reported separately, the honesty promised by the name "grey" remains only on paper.

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