PROBID (Preference Ranking on the Basis of Ideal-Average Distance)

PROBID is a ranking method for when you already hold a numerical decision table, placing the alternatives in a single order. Its output is a score for every alternative and a descending order based on that score. Rather than the two extreme points, ideal and anti-ideal, that TOPSIS builds, it constructs as many intermediate rank points as there are alternatives, and judges every alternative against the whole set of these rank points. It was proposed by Wang, Rangaiah and Wang in 2021 to reduce the sensitivity that arises, in chemical-engineering process-design selection, from TOPSIS's reliance on a single ideal point.

The idea behind TOPSIS is "find the alternative close to the best and far from the worst," done with two points. PROBID goes a step further: it orders every criterion column from largest to smallest (or smallest to largest for a "lower is better" criterion) and builds m hypothetical "rank points." The first rank point is made up of the best value on every criterion, the second of the second-best value on every criterion, and so on down to the last. In this way not only the extremes but the intermediate rungs too are taken into account. An alternative that sits close not only to the best point but also to the "good in the middle" points is rewarded for this in PROBID as well.

The philosophical consequence is that PROBID, like TOPSIS, is compensatory. A weakness on one criterion can be papered over by strength on another, because the outcome is still a single distance and a single score. What sets PROBID apart from TOPSIS is that this compensation is measured against the whole set of rank rungs, not just the two extreme points; the authors claim this structure disturbs the ranking less than TOPSIS does when the alternative set changes.

The method proceeds through five steps.

First, scale equalisation. Every criterion column is divided by its own magnitude: each value in the column is divided by the square root of the sum of the squared values in that column. Criterion direction is not applied at this step but at the third.

Second, weighting. Each equalised column is multiplied by its criterion's weight. The weights come from outside PROBID and must sum to 1.

Third, building the rank points. In the weighted table, every column is ordered smallest to largest for a "lower is better" criterion, and largest to smallest for a "higher is better" criterion. This produces as many hypothetical rank points as there are alternatives, m: the first is the "first-rank ideal," made up of the best value on every criterion, and the last is the "last-rank ideal," made up of the worst value on every criterion. The average of these m points is recorded separately as an "average ideal" point.

Fourth, distances. For every real alternative, the straight-line (Euclidean) distance to each rank point that has been built, and to the average ideal, is calculated.

Fifth, combination and ranking. The rank points are split into two halves: the upper half (rungs close to the best) and the lower half (rungs close to the worst). An alternative's distances to the upper half and its distances to the lower half are each weighted by the inverse of their rank number (full weight for the first rung, half weight for the second, and so on) and summed, giving what are called S+ and S− respectively. The ratio of these two is taken and compressed into the range 0 to 1 by an equation, to which the distance to the average ideal is then added. The alternative with the larger resulting score is the better one.

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 PROBID score tells you how well an alternative stands, relative to the other alternatives in this analysis, against both the best rung and the middle rungs; it says nothing more. The score is not a percentage and cannot be compared directly with TOPSIS's closeness score, because the two methods construct their score scale differently. Nor can the score be compared with a score from a different analysis; the rank points are built afresh, in every analysis, from that analysis's own alternatives.

The alternative ranked first is often not the outright best on any single criterion; it is the one whose closeness to both the upper and the middle rungs is balanced. A small score gap between two alternatives does not signal a robust ranking but one that is sensitive to the weights.

Thus instead of writing:

"PROBID found the best alternative"

the report should read:

"With these weights and this alternative set, the alternative with the highest PROBID score is this one; the ranking is sensitive to the weights and to the alternative set"

PROBID works with crisp data: one number per cell. You need alternatives in rows, criteria in columns, one number per cell with no empty cells; direction information for every criterion (higher or lower is better); and criterion weights summing to 1. PROBID does not produce weights, it asks for them from outside; you can derive weights from expert opinion (AHP, BWM, SWARA) or from the data itself (Entropy, CRITIC). DecisionMind currently holds no extension of this method; it works in its base, crisp form.

A minimum of two alternatives is required. As the number of alternatives grows, so does the computational burden, because PROBID calculates, for every alternative, as many distances as there are alternatives; beyond a hundred or so alternatives this grows substantially.

PROBID is a suitable choice if your criteria can be measured numerically, your table has no gaps, you accept compensation, and you are wary of TOPSIS's excessive sensitivity to a single ideal point. It is typically preferred in settings such as process design and engineering-alternative selection, where several reasonable intermediate solutions also carry information.

The case where it should not be used stems from the same root as TOPSIS: if you will not compromise on one criterion, PROBID is unsuitable, because it is compensatory. Where the number of alternatives is very large (in the hundreds), its computational burden grows faster than that of other ranking methods.

A numerical table, compensation accepted, wary of TOPSIS's sensitivity to a single extreme → PROBID

A very large number of alternatives, computation should stay light → simpler aggregation methods such as SAW, WPM

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

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

PROBID's greatest strength is that it bases the ranking not only on the two extreme points but on the intermediate rungs as well. This is intended to make the ranking less disturbed than TOPSIS's by small changes in the alternative set; the authors demonstrated this with both synthetic and real process-design examples (Wang, Rangaiah and Wang, 2021). Its computation stays close to TOPSIS in simplicity, and every step can be followed on the table.

PROBID, too, rests on the assumption of full compensation; this means a serious weakness on one criterion can be papered over by other criteria. Splitting the distance rungs into two halves and weighting them separately can make the ranking sensitive to small changes in the weights; this sensitivity appears to differing degrees across different datasets (Baydaş, Kavacık and Wang, 2024; Stević et al., 2024). Because it is a less widely known method, it does not yet have as broad a body of critique as AHP or TOPSIS; its fuzzy and linguistic extensions are still developing (Darko et al., 2023; Köseoğlu, 2025).

The most common mistake is marking criterion direction wrongly; this reverses the order in which the rank points are built and renders the result meaningless. A second mistake is assigning equal weights without justification. A third is comparing the PROBID score directly with TOPSIS's closeness score, or with a score from a different analysis; the two methods do not share the same score scale. A fourth is applying PROBID to very large numbers of alternatives (in the hundreds) without considering the computational burden.

The governing principle is this:

A PROBID result is a consistent summary of the directions, weights and alternative set you supplied; if any input is contested, the ranking is contested too, and the report must show this.