Probabilistic hesitant fuzzy COPRAS (Song and Chen, 2021)

Three things change; the decision logic stays the same.

Cells. In plain hesitant COPRAS (HF-COPRAS), every cell is a set, that is, it carries more than one possible value, but how likely each of these values is remains unknown. Here every cell is a set of value-probability pairs: each possible degree comes with its own probability or frequency of occurrence, for example "the value 0.5 with 30 per cent probability, the value 0.7 with 70 per cent probability." The probabilities in a cell cannot sum to more than 1. Weights are given from outside, as single numbers.

Scale equalisation. Classical COPRAS turns every column into a share by dividing it by its own sum. There is no separate column normalisation here, because the values are already degrees between 0 and 1. Instead, DecisionMind first scales every cell by the criterion's weight; this scaling multiplies every degree in the set by the weight and leaves its probability unchanged. The scaled sets from benefit criteria are accumulated together, and those from cost criteria separately.

Distance, score and combination. The accumulated value-probability set is reduced to a single number by taking its expected value: every degree is multiplied by its own probability and the products summed. Taking the expected value this way makes no difference whether the sum is taken first and then the expected value, or the expected value of each cell is taken first and then summed; DecisionMind applies this step after the summing. The expected values of the benefit and cost sides are then combined with classical COPRAS's Q formula to build the relative significance value.

Result and defuzzification. Defuzzification sits in the middle of the calculation: the value-probability sets descend to an expected value immediately after the benefit and cost sums are built, neither at the very start nor at the very end. The output is a relative significance value in the same form as in classical COPRAS, together with a percentage derived from it.

DecisionMind fixes the expected-value function (a probability-weighted average) and its application after the summing step for this family.

The relative significance value and its percentage are read exactly as in classical COPRAS: a share relative to the best alternative, not an absolute percentage of success. The difference is this: this share now rests only on the probability-weighted average of the degrees in every cell. Two cells can share the same expected value while one comes from a narrow probability distribution (almost all probability concentrated on a single degree) and the other from a wide one (probability spread across several distant degrees). The relative significance value does not distinguish between these two.

Thus instead of writing:

"Probabilistic hesitant COPRAS shows A2 to be the best alternative"

the report should read:

"By expected value, A2 holds the highest relative significance (Q=0.508); this figure only reflects the probability-weighted average, and does not mean A2 is 'definitely the best' without separately examining how narrow or wide a probability distribution its degrees come from"

This extension is suitable when more than one plausible degree exists for the same criterion-alternative pair, and it is known how often each of these degrees was observed, or how probable each one is. Example: a supplier's past performance has been observed at several different levels, and how often each level occurred is on record. If the probabilities are unknown, and only which degrees are plausible is known, plain hesitant COPRAS (HF-COPRAS) is enough; inventing probabilities adds no information. Classical COPRAS's exit condition still applies here: if no compromise is acceptable on one criterion, this extension too is compensatory.

Letting the probabilities in a cell sum to more than 1. In that case the expected value is no longer a weighted average but a figure of unclear meaning.

Assuming the probabilities are equal. Giving every degree an equal probability (1/n) effectively reduces the cell to the plain average of a hesitant set, and hides the real contribution of the probability information.

Marking cost and benefit direction wrongly. If a cost criterion is written on the benefit side, a high expected value on that criterion is rewarded.

The "more sophisticated" fallacy. If the degrees have no real frequency of observation, adding arbitrary probabilities to a plain hesitant set adds no information; it only complicates the calculation.

The governing principle is this:

Probabilistic hesitant COPRAS exists to carry how often each possible degree is observed through to the expected value. If the probabilities do not come from a genuine observation or record, this extension adds nothing over plain hesitant COPRAS.