PIV (Proximity Indexed Value)

PIV is a ranking method for when you already hold a decision table filled with numbers, placing the alternatives in a single order. Its output is a deviation score for every alternative: the smaller the score, the closer the alternative sits to the ideal. It was proposed by Mufazzal and Muzakkir in 2018 to reduce the rank reversal seen in methods such as TOPSIS, which look at both the ideal and the anti-ideal.

PIV's starting point is an objection to TOPSIS's philosophy. TOPSIS judges alternatives by both their closeness to the ideal and their distance from the anti-ideal; but because the anti-ideal point is built from the worst values in the alternative set, it shifts whenever that set changes, and this shift contributes to rank reversal. PIV dispenses with this second reference point entirely and focuses solely on the question "how close am I to the ideal"; distance itself is measured by the sum of absolute differences (L1 distance) rather than by squaring and summing.

The philosophical consequence is that PIV remains compensatory but now rests on a single reference point. A weakness on one criterion can still be offset by strength on another, because the deviations combine into a single sum; but that sum only asks "how far short of the best did I fall," never "how far did I get from the worst."

The method proceeds through five steps.

First, scale equalisation. As in TOPSIS, every column is divided by its own magnitude: each value in a column is divided by the square root of the sum of the squared values in that column (vector normalisation). This makes criteria expressed in different units comparable with one another.

Second, weighting. Each equalised column is multiplied by its criterion's weight.

Third, a single reference point. In the weighted table, only the best value on each criterion (largest for a benefit, smallest for a cost) is identified; unlike TOPSIS, no anti-ideal (worst) point is built.

Fourth, absolute deviation. For every alternative, the difference between its value on each criterion and that criterion's ideal value is calculated regardless of sign, as an absolute value.

Fifth, total deviation and ranking. These absolute differences are summed across criteria. PIV ranks alternatives from the smallest sum to the largest; the smallest total deviation marks the best alternative.

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 PIV score is a distance, and, unlike TOPSIS's closeness score, a small value is good. This distinction matters: in TOPSIS a high score is good, in PIV a low score is good. The score is not a percentage and cannot be set against a PIV score from a different analysis, because the ideal point is built afresh, in every analysis, from that analysis's own alternatives.

Because PIV looks only at distance to the ideal, it has no answer to the question "how far has this alternative got from the worst." This gives a simpler frame of reference than TOPSIS, but in situations where distance from the anti-ideal also carries meaningful information, that information is lost.

Thus instead of writing:

"PIV found the best alternative outright"

the report should read:

"With these weights and this alternative set, the alternative closest to the ideal (with the smallest deviation) is this one; distance from the anti-ideal is not included in this score"

PIV works with crisp data. DecisionMind holds no extension of this base method.

You need alternatives in rows, criteria in columns, one number per cell with no empty cells; direction information for every criterion; and criterion weights summing to 1. PIV does not produce weights, it asks for them. Benefit and cost criteria can sit in the same table, but each must be normalised separately according to its own direction; skipping this distinction produces a wrong result. A minimum of two alternatives and two criteria is required.

PIV is a suitable choice if your criteria can be measured numerically, your table has no gaps, and you want a TOPSIS-like compensatory ranking, but you are working in a setting where alternatives are frequently added to or removed from the set and rank-reversal risk is therefore of particular concern.

It should not be used where distance from the anti-ideal also carries meaningful information for the decision; if the question "how far has the field got from the worst" matters, TOPSIS gives this information more directly than PIV does. If you will not compromise on one criterion, PIV is unsuitable for the same reason as TOPSIS, because deviations are summed and so compensated for.

A numerical table, compensation accepted, rank-reversal concern is prominent → PIV

Distance from both the ideal and the anti-ideal is wanted → TOPSIS

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)

PIV's greatest strength is its conceptual simplicity. Using a single reference point simplifies both the calculation and its interpretation; by removing the anti-ideal's tendency to shift with the alternative set, it is expected to be more resistant to rank reversal than TOPSIS (Mufazzal and Muzakkir, 2018). The computational burden is light, and its steps are familiar to anyone used to TOPSIS.

Its limitations still stem from its compensatory structure and from its young literature. Although PIV was proposed with a claim to reduce rank reversal, this immunity is not complete; a later study showed that under certain conditions PIV also experiences rank reversal, and proposed an improved version (Saluja, Mathew and Singh, 2023). Second, because the method was proposed in 2018, it has not yet accumulated the breadth of application and critique that TOPSIS has. Third, using a single reference point is a simplification; the assumption that distance from the anti-ideal is genuinely unimportant does not hold in every decision problem.

The most common mistake is reading the PIV score as if it were a TOPSIS closeness score and assuming a high score is good; in PIV a low score is good, because the score is a distance, not a closeness.

A second mistake is treating benefit and cost criteria as if they shared one direction, normalising them together; each criterion must be normalised separately according to its own direction. A third mistake is presenting the PIV score as a percentage or an absolute measure of performance; the score is meaningful only for this particular alternative set and these particular weights.

The governing principle is this:

A PIV result is only a summary of distance to the ideal; a low score is good, and it says nothing about distance from the anti-ideal.