Grey Projection

Grey Projection is a method for ranking alternatives when the data you hold is not a single exact number but a lower bound and an upper bound (a grey number). It carries Grey Systems Theory's (Deng, 1982) principle that "a decision can still be made with little data and incomplete information" into multi-criteria ranking. The method is an extension of Grey Relational Analysis (GRA); in place of GRA's distance measurement, it uses projection onto ideal and anti-ideal points. The output is a closeness coefficient for every alternative and the rank that coefficient produces; the larger the coefficient, the closer the alternative sits to the ideal.

The starting point of Grey Systems Theory is that a grey region lies between the fully known and the entirely unknown, and that many real decisions are made within this grey region. Consider a weather forecast. Saying "18 to 22 degrees tomorrow" rather than "20 degrees tomorrow" honestly admits incomplete information while still allowing a decision to be made. Grey Projection accepts criterion values as such intervals in the same way, and compares the direction each alternative forms from these intervals against the best and worst reference directions. Using projection in place of TOPSIS's distance measurement produces a comparison that takes direction into account as well as distance alone; of two alternatives equally distant from the ideal, the one more closely aligned with the ideal direction comes out ahead.

The method proceeds through four steps.

First, defuzzification and scale equalisation. Each cell's grey number is first defuzzified: the lower and upper bounds are averaged down to a single number. Every column is then normalised against its own magnitude: benefit criteria are scaled against the largest upper bound in the column, cost criteria against the smallest lower bound. This step makes criteria in different units comparable.

Second and third steps, finding the ideal and anti-ideal directions and computing the projections. In the normalised table, the best value on each criterion builds the ideal direction, and the worst value builds the anti-ideal direction. The method projects each alternative's weighted, normalised vector onto both the ideal direction and the anti-ideal direction; this projection is a number measuring how closely the alternative aligns with that direction.

Fourth, computing the closeness coefficient and ranking. Grey Projection divides the projection onto the ideal by the sum of the projections onto the ideal and the anti-ideal. For an alternative strongly aligned with the ideal, this ratio approaches 1. Alternatives are ranked by this coefficient from the highest to the lowest.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The closeness coefficient shows how closely an alternative, given this interval data, aligns with the ideal direction; it says nothing more than that. A coefficient around 0.50 does not mean "an average decision," but a roughly equal projection onto the ideal and anti-ideal directions. It cannot be compared with a coefficient from a different analysis, because the ideal and anti-ideal directions are built afresh in every analysis from that analysis's own data.

In methods that use grey data, the gap between coefficients can come out smaller than in methods using crisp data, because the defuzzification step erases some of the information about the width of the interval. A small gap between coefficients points to a ranking sensitive to the width of the intervals; the report should show this sensitivity.

Thus instead of writing:

"Grey Projection found the best alternative"

the report should read:

"With this interval data and these weights, the alternative most closely aligned with the ideal direction is this one; if the gap between the coefficients is small, the width of the intervals should be separately reported"

Grey Projection works with grey numbers: every cell consists of a lower bound and an upper bound, not a single exact number. As of this writing, DecisionMind holds no extension alongside the base method; if your data is crisp, exact methods such as GRA or TOPSIS can be used directly without converting it into grey numbers.

You need: a lower bound and an upper bound for every criterion of every alternative (the lower bound must not exceed the upper bound), a direction for every criterion, and criterion weights that sum to 1. A minimum of two alternatives and two criteria is required; the recommended size is two to twenty-five alternatives and three to ten criteria.

Grey Projection is a sound choice if your data is expressed as an interval because of incomplete measurement, a short observation period or expert estimation, and you do not want to ignore this uncertainty. Its typical territory is engineering and planning problems decided with a small sample or incomplete information.

It should not be used where your data is already fully known as single exact numbers; converting it into grey numbers adds needless uncertainty. If your intervals are very wide, the defuzzification step can oversimplify the information; in that case the interval itself should be shown separately in the report.

Data given as intervals, information is incomplete → Grey Projection

Data is a single exact number → exact ranking methods such as GRA, TOPSIS

Uncertainty comes from expert language (low, medium, high) → fuzzy or intuitionistic methods

Not a ranking but weights are needed → AHP, BWM, SWARA (subjective), Entropy, CRITIC (objective)

The method's greatest strength is that it accepts incomplete or uncertain data as an interval rather than forcing it into a single number. It can work with a small number of observations, drawing on Grey Systems Theory's search for solutions to small-sample problems. Its use of projection, which accounts for alignment with the ideal direction as well as distance alone, offers an alternative to TOPSIS's distance-based approach.

Its limitations stem from this same flexibility. First, the defuzzification step partly loses information about the width of the interval; a very wide interval and a narrow one sharing the same average become indistinguishable. Second, there is no single, universally accepted founding article under the name "grey projection"; the method is one of several projection-based adaptations of Grey Relational Analysis and can vary slightly from application to application. Third, it carries the same limitations as GRA: criteria are treated as independent, and the quality of the weights lies outside the method itself.

The most common mistake is entering a lower bound greater than the upper bound; this breaks the definition of the interval and invalidates the calculation. A second mistake is reading the closeness coefficient as a percentage or a probability; the coefficient only shows relative alignment with the ideal direction. A third is setting the interval very wide and then interpreting the defuzzified average as if it were an exact number; the width of the interval should be separately stated in the report. A fourth is converting crisp data into grey numbers without justification, needlessly complicating the method.

The governing principle is this:

A Grey Projection result depends on the width of the intervals you supplied and on the weights; if the intervals are very wide, a single defuzzified coefficient conceals that width, and the report must show it separately.