Entropy Weighting (Shannon Entropy Weighting)

The entropy method is not a ranking method; it does not rank alternatives, it produces the weights of the criteria. Given a numerical decision table of alternatives and criteria, it does not ask anyone "which criterion is more important". Looking at the distribution within the table, it assigns every criterion a weight, the whole summing to 1. These weights then serve as input to a ranking method such as TOPSIS, VIKOR or SAW.

The method rests on the concept of entropy that Shannon defined in 1948 for communication theory; its adaptation to decision analysis was popularised by Hwang and Yoon's 1981 book. In the Turkish literature it is the most widely used of the objective weighting methods.

The idea behind entropy is this: if a criterion takes almost the same value across every alternative, it does not distinguish the alternatives from one another and contributes nothing to the decision. Conversely, if the alternatives take widely differing values on a criterion, that criterion is the one determining the decision. Entropy measures how "evenly distributed" the values on a criterion are: an even distribution gives high entropy and little information; a distribution bunched together or split towards the extremes gives low entropy and a great deal of information. Weight is proportional to the amount of information.

A consequence of this philosophy is that the method draws no distinction between "importance" and "discriminating power". A criterion an expert regards as highly important, but that turns out similar across every alternative, is unimportant to the entropy method. This is not a flaw but a design choice: the method answers not "which criterion is valuable" but "which criterion is this table speaking through". If this distinction is acceptable, the method fits; if the decision-maker's own preference must show in the outcome, a subjective method, or a combination of the two, is needed.

The method proceeds through four steps.

Step zero, converting cost criteria. Entropy works on a "larger value is better" logic. "Lower is better" criteria such as price are converted first. The classical option is to divide the smallest value in the column into every value; the cheapest alternative then becomes 1, and more expensive ones fall below 1. Other forms of conversion exist (taking the reciprocal of the value, subtracting from the sum of the largest and smallest) and can give different weights. DecisionMind uses division by the smallest value as its default and states this choice in the report.

First step, conversion to probabilities. The values in each column are divided by the column total, giving each alternative's "share" for that criterion; the shares sum to 1. This step brings criteria in different units (currency, days, points) onto the same footing.

Second step, entropy. For each criterion, a dispersion measure is calculated from the shares and scaled between 0 and 1 according to the number of alternatives. If the shares are equal to one another, entropy is 1 (the criterion says nothing); if the shares are concentrated on one alternative, entropy approaches 0.

Third step, divergence. Entropy is subtracted from 1: 1 minus entropy is the amount of information the criterion carries.

Fourth step, weight. Each criterion's amount of information is divided by the total amount of information across all criteria. The result is a weight vector summing to 1.

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

An entropy weight measures not how important a criterion is in the decision-maker's eyes, but how much it discriminates between alternatives in the table at hand. A weight of 0.49 does not mean "this criterion is half the decision"; it means "roughly half of the difference between alternatives in this table comes from this criterion". The same criterion can take a completely different weight in a different set of alternatives, because the weight is a property of the table, not of the criterion.

For this reason, entropy weights cannot be carried over to another study and cannot be generalised as "the criterion's importance is such-and-such". If a criterion is identical across every alternative, its weight comes out at zero; this says the criterion is not useful for this decision, not that it is unimportant. If the number of alternatives is small (three or four), a single extreme value can drag a criterion's weight; weights should then be read with caution.

Thus instead of writing:

"The entropy analysis showed that cost was the most important criterion"

the report should read:

"In this set of alternatives, the criterion that separates the alternatives most is cost; the entropy weight of 0.49 reflects this discriminating power, not the decision-maker's order of priority"

Classical entropy works with crisp data: one number per cell. Values must be positive; zero and negative values cause problems at the conversion-to-probability step and must first be shifted or rescaled. DecisionMind holds fuzzy and intuitionistic fuzzy extensions alongside the base method (three members in total); if the data is an approximate expert judgement, these extensions should be considered.

You need alternatives in rows, criteria in columns, a positive number in every cell, no empty cells. You also need, for every criterion, whether higher or lower is better, since cost criteria will be converted. Weights are not entered; the method produces them and hands them to a ranking method. At least two alternatives and two criteria are required; the more alternatives there are, the more reliable the weights become, because the distribution then depends less on a single value.

Entropy suits cases with no expert opinion, no expert agreement, or where the decision-maker's priorities should not influence the result. It works best on large tables of genuine measurements (country, company or regional comparisons; financial ratios; performance indicators), because the weights are then derived from a real distribution. It is also combined with a subjective weighting method (AHP, BWM), so that part of the weight comes from preference and part from data.

The situations where it should not be used follow from its philosophy. If the decision-maker clearly regards one criterion as a priority, entropy cannot see this. If the number of alternatives is very small, a single extreme value drags the weights. If the table consists of expert scores confined to a narrow range (all between 6 and 8), discriminating power is already low, and entropy weights become sensitive to random differences. If a criterion is constant, its weight comes out at zero and it should be removed from the table.

No expert opinion available or wanted, table is a genuine measurement → Entropy

The relationship between criteria should also be taken into account → CRITIC

The decision-maker's priority should be visible in the result → AHP, BWM, SWARA (subjective)

Both preference and data matter → a combination of subjective and objective weights

Few alternatives, narrow-range expert scores → read entropy weights with caution or switch to a subjective method

The entropy method's most important strength is its objectivity: the same table gives everyone the same weight, regardless of who asks. There is no need to gather experts, run a survey, or hold pairwise comparisons; this makes it practical for large indicator sets and repeated annual assessments. Its calculation is transparent and traceable on the table itself, and it automatically suppresses non-discriminating criteria, helping the criteria that matter to the decision stand out among many.

Its limitations arise from the same objectivity. First, "discriminating power" is not the same as "importance"; the method cannot see the decision-maker's values and can render an important but little-varying criterion insignificant. Second, the weights depend on the set of alternatives: when an alternative is added or removed, every weight changes, so weights cannot be carried over to another study. Third, the way cost criteria are converted changes the weight; different conversions give different weight to the same table, and the choice must be stated in the report.

Fourth, it is sensitive to extreme values; in small tables, a single outlier can drag a criterion's weight. Fifth, it cannot work directly with zero or negative values. For comparative assessments of objective weighting methods, see Zavadskas and Podvezko (2016); for the limits of entropy with uncertain data, see Lotfi and Fallahnejad (2010).

The most common mistake is reporting an entropy weight as "importance". The sentence "the data showed that cost was the most important criterion" is wrong; the data shows that cost is the criterion that discriminates most.

A second mistake is feeding cost criteria into the calculation without converting them; the most expensive alternative then takes the highest share and the weight is built in the wrong direction. A third is failing to state the conversion method in the report, or comparing it with another study's different conversion method. A fourth is placing excessive trust in a weight derived from a table of only two or three alternatives. A fifth is leaving a constant criterion in the table and mistaking its zero weight for "a method error". A sixth is computing entropy weights once and reusing them unchanged in the following years' tables; every table produces its own weights.

The governing principle is this:

An entropy weight is a property of the table, not of the criterion; the weight changes when the set of alternatives changes, and the report must call this "discriminating power", not "importance".