DNMA (Double Normalization-based Multiple Aggregation)

DNMA is a ranking method that arranges alternatives into a single order once you hold a decision table filled with numbers. Its output is a single score for each alternative: this score is built from the combination of three different aggregation logics, and alternatives are ranked by this score. Liao and Wu proposed the method in 2020. It arose from a concern: in multi-expert, multi-criteria decision problems, a decision-maker's choice of a single aggregation method (weighted sum alone, or weighted product alone) can change the result. DNMA produces no weights; it takes them from outside.

Multi-criteria ranking methods share another known weakness. Whichever aggregation logic is chosen, weighted sum, weighted product, or closeness to the ideal, different alternatives can come out ahead. The reason is this: a weighted sum readily compensates for a weakness on one criterion, whereas a weighted product is less tolerant of an extremely low value. DNMA's philosophical move is this: rather than giving a single answer to "which aggregation is correct", the method calculates all three at once and trusts none of them fully. DNMA calculates three separate measures for every alternative: a weighted sum (WSM, fully compensatory), a weighted product (WPM, less tolerant of low values), and a third measure based on closeness-to-the-ideal logic.

The second philosophical move is seen in how DNMA combines these three measures. DNMA does not simply average the three scores; it extracts both a score and a rank from each aggregation. It builds the combined score to reward alternatives that are "scored highly AND consistently ranked well across all three methods". If an alternative ranks well under only one aggregation method but poorly under the other two, this inconsistency lowers the combined score. The philosophical consequence is this: DNMA is a ranking approach that hands over trust in any single aggregation model to the mutual confirmation of several models, a kind of "cross-method consensus".

The method proceeds through four steps.

First, two separate normalisations. DNMA equalises every column of the decision table in two different ways. The first is a vector normalisation that takes direction into account; for cost criteria this operation is reversed. The second is a linear-maximum normalisation, also accounting for direction (relative to the column's largest value for a benefit criterion, or relative to the ratio of the column's smallest value for a cost criterion). These two differently normalised tables form the basis for the three aggregations in the next step.

Second, three sub-utility calculations. From the normalised tables, DNMA calculates three separate aggregations: a fully compensatory weighted sum (WSM-like, the "complete" model), a weighted product less tolerant of compensation (WPM-like, the "incomplete" model), and a third measure based on closeness-to-the-ideal logic (TOPSIS-like). Every alternative receives a separate score under each of these three models.

Third, ranks and deviation measures. DNMA extracts each alternative's rank within each of the three models from that model's own scores, and also calculates how much the scores within each model deviate from one another. DNMA thus holds both three scores and three ranks together for every alternative.

Fourth, the combined score. DNMA builds a single combined score by subtracting the weighted sum of the normalised squared ranks across the three models from the weighted sum of the normalised squared scores across the three models. This score jointly rewards a high score AND low (good) rank consistency. DNMA ranks alternatives by this combined score, from highest to lowest.

You can find the formulas for each step, the intermediate tables and the citation formats on the DecisionMind method page; this card carries no formulas.

DNMA's combined score is a figure that jointly summarises the magnitude of an alternative's scores under three different aggregation logics and the consistency of its ranks; a higher score is better. Unlike TOPSIS's closeness score, this figure does not arise from a single distance measure, but from how much three separate models confirm one another. If an alternative's score is high, this does not just mean "it came out well under one model"; it means "it was consistently ranked well under more than one aggregation logic". This is DNMA's central claim to reliability.

The score itself is neither a percentage nor a probability, and it can come out negative. If an alternative's rank inconsistency is penalised more heavily than the magnitude of its scores, the score comes out negative. A negative score does not mean "failure"; it only shows the relatively weakest position among these particular alternatives. The score cannot be compared directly with a score calculated with a different set of alternatives or different weights.

Thus instead of writing:

"By the DNMA score, A2's performance is 1.0, meaning perfect"

it is correct to write:

"A2 is the alternative that all three aggregation models consistently place at the top; this combined score only shows an order among these three alternatives, not an absolute percentage of success"

DNMA works with crisp data: a single number per cell. You need: alternatives in rows, criteria in columns, one number per cell; direction information for each criterion; weights that sum to 1. Optionally, additional coefficients can be defined specifying how much weight is given to the score and rank components of the three aggregation models; if unspecified, DecisionMind gives all three equal weight. DNMA produces no weights, taking them from outside; you can obtain these weights from methods such as AHP, BWM, Entropy or CRITIC. DecisionMind holds four DNMA members alongside the base method (including fuzzy, negative and positive extensions). A minimum of two alternatives and two criteria is required; three to twelve criteria is recommended. If every value in a criterion column is equal, that is, the largest and smallest values coincide, linear normalisation becomes undefined. Such constant columns should be avoided.

If your criteria can be measured numerically, DNMA is a suitable choice. It is especially useful if, instead of relying on a single aggregation logic (sum alone, or product alone, or closeness to the ideal alone), you want a result confirmed by all three together. It typically finds use in multi-expert, multi-criteria evaluations and in supplier or alternative selection.

There are some situations where you should not use DNMA. Do not use it where simplicity and the traceability of a single calculation logic matter; combining three separate models is a longer method to explain than a single one. Do not use DNMA if a criterion column is constant, that is, shows no variation, and linear normalisation would be undefined. Nor is DNMA suitable if no compromise is accepted on one criterion, that is, if veto logic is required.

If you do not want to rely on a single aggregation logic and want the consistency of more than one model → DNMA

If a simple, single-aggregation (sum only) ranking suffices → SAW, TOPSIS

If explanatory simplicity matters (a logic explainable in one step is wanted) → SAW

If a criterion column is constant (the same value across all alternatives) → remove this criterion first or choose a different form of normalisation

If no compromise is accepted on one criterion → the ELECTRE family

DNMA's greatest strength is this: it does not depend on a single aggregation logic (sum alone or product alone). It uses three different logics together and looks for them to confirm one another. This approach provides a safeguard against the arbitrariness of choosing a single method (Liao and Wu, 2020). Second, DNMA looks not only at the magnitude of the score but also at rank consistency. This means an alternative cannot dominate the result on the strength of a single model where it happened to do well by chance. Third, researchers have extended the method to uncertain data types such as fuzzy, intuitionistic fuzzy and probabilistic linguistic since its publication (Liao et al., 2019; Liao et al., 2024). This shows the design is flexible. Fourth, Liao and Wu designed DNMA specifically for multi-expert evaluation settings. The method directly answers the need to combine multiple viewpoints in that scenario.

Its limitations come from the same design. First, explaining three separate aggregation models and the rank-consistency penalty together requires a longer explanation than single-model methods such as SAW or TOPSIS. Answering the decision-maker's question of "why did this order come out" takes more steps. Second, the relative weights of the combined score's squared-score and squared-rank terms lie in the analyst's hands. These weights can be distributed equally or unequally across the three models, and this choice can affect the result. Third, if a criterion column is constant, linear normalisation remains undefined; this is a constraint that must be checked separately during data preparation. Fourth, because it includes a fully compensatory (WSM) component, DNMA is ultimately a partly compensatory method as well. It has no veto logic; a very poor value on one criterion does not entirely prevent an alternative from looking good on the other measures.

The most common mistake is this: the user ignores the fact that DNMA calculates three separate aggregation models and interprets the result as a single "closeness-to-ideal score" (as in TOPSIS). Yet DNMA's score arises from the combination of three models; the analyst must state this explicitly in the report. A second mistake is this: the user, not knowing the combined score can come out negative, mistakes a negative value for a calculation error. Yet a negative score only shows a relatively weak position. A third mistake is this: the user marks a criterion's direction (benefit/cost) incorrectly. This error affects both the vector and the linear normalisation, and hence all three models. A fourth mistake is this: the user includes a criterion column in the analysis without noticing it is constant. In that case, linear normalisation remains undefined. A fifth mistake is this: the user presents the result without reporting the weights given to the three models' score and rank components (equal or unequal). Yet these weights are genuine inputs that affect the result.

The governing principle is this:

DNMA's combined score is a summary of the score and rank consistency of three different aggregation models; if an alternative looks good under only one model, this is exactly the kind of fragility DNMA is designed to catch, and the report must show it.