Hesitant MOORA (Li, 2014)

Three things change; the logic of the vector norm and the ratio system does not.

Cells. In crisp MOORA every cell is a single number. Here every cell can be a set consisting of one or more possible values. Different cells can carry different numbers of values; one cell may hold a single agreed value, while a neighbouring cell holds three different expert opinions. Criterion weights are taken from outside as crisp numbers.

Defuzzification comes first. Every set is reduced to a single score at the very first step of the calculation. The default operator is the arithmetic mean of the set's elements. The founding paper (Li, 2014) also defines a power-average operator; this operator builds a weighting based on how close the elements are to one another. The two operators give the same number on a single-element set. On a set with more than one element they can give different numbers; for instance, on the set {0.20; 0.25; 0.90}, the mean operator gives 0.45, while the power-average operator gives 0.551. Which operator is used is a parameter, and it must be stated in the report.

Vector norm and ratio system. The defuzzified score matrix is normalised with crisp MOORA's vector norm (every value divided by the square root of the sum of the squared values in its column) and multiplied by the weights; the sum of the weighted shares in the cost criteria is subtracted from the sum of the weighted shares in the benefit criteria. This is exactly the same operation as crisp MOORA's ratio system.

DecisionMind fixes, in this extension, the default defuzzification operator (arithmetic mean), the power-average option, and, following that, crisp MOORA's vector norm and ratio system. Sets of different sizes create no problem here, because the comparison is made not between the sets themselves but between the scores derived from them; the length-equalisation rule found in hesitant methods such as TOPSIS or EDAS, which compute distances directly between cells, is not needed in this family.

The MOORA score is read as in crisp MOORA: it is an alternative's net superiority, its share on the beneficial criteria against its share on the harmful criteria, and it cannot be compared with a different analysis. The difference lies here: behind the score there is no longer the several plausible values the set once carried, because the cell was already reduced to a single score at the first step. Which defuzzification operator was used affects this score, and hence the result.

Thus instead of writing:

"Because Hesitant MOORA carries the uncertainty through to the end, the result is more reliable"

the report should read:

"The sets were reduced to a single score at the very first step, with the operator chosen (mean or power average); which operator was used affects the result and must be stated in the report"

Use this extension when several expert opinions, scenarios or repeated measurements are simultaneously accepted as plausible for a criterion-alternative cell, and one does not want to lose them to an early average. If there is a single reliable value, converting it into a set is not necessary; the rule on the hesitant data-type card applies here too. Crisp MOORA's exit condition applies unchanged: if no compromise is accepted on a criterion, this extension too is compensatory.

Failing to state the defuzzification operator in the report. Whether the mean or the power average was used produces a different score on sets with more than one element; this choice must not be left unreported.

Trying to equalise sets of different lengths. In this family, cells are first reduced to scores independently. Unlike distance-based hesitant methods, no length-equalisation rule is needed here; every set is scored directly from its own elements.

Expanding a single-element set with unjustified elements to "look sophisticated." The rule on the hesitant data-type card applies here too: every element must rest on its own source, an expert, a scenario or a measurement.

Violating the value range. Every element in a set must lie between 0 and 1.

The governing principle is this:

In Hesitant MOORA, uncertainty is reduced to a single score by the chosen operator at the very first step, where the matrix is built, before it ever enters the calculation. The report must not conceal this early defuzzification and the choice of operator.