Rough MOORA

Cells. In crisp MOORA every cell is a single number. Here every cell is, as described on the rough data-type card, a single lower and upper bound [L, U] calculated from a group of experts' crisp scores. This differs from rough MABAC: there, a cell is a four-number hybrid structure carrying both an intuitionistic fuzzy judgement and group disagreement; here, a cell is a plain rough number with only two numbers. Criterion weights, too, can be entered as rough numbers, but when given as a single crisp number, its lower and upper bound are taken as equal ([w, w]).

Scale equalisation. Crisp MOORA divides every column by the square root of the sum of squares of its own values. Rough MOORA applies the same operation to the lower and upper bounds separately, but with a cross-division rule. Every cell's lower bound is divided by the square root of the sum of squares of the column's upper bounds, and its upper bound by the square root of the sum of squares of the column's lower bounds. This cross-division comes from the division rule of rough-number arithmetic and ensures the normalised interval does not exceed its true magnitude.

Ratio system. The weighted lower and upper bounds are summed separately for the "higher is better" criteria, and separately again for the "lower is better" criteria. The difference between these two summed intervals is taken with the rough-number subtraction rule: the result's lower bound is found by subtracting the harmful sum's upper bound from the beneficial sum's lower bound, and the upper bound by subtracting the harmful sum's lower bound from the beneficial sum's upper bound. This is the interval form of crisp MOORA's plain subtraction, and it carries the harmful side's uncertainty along with the beneficial side's.

Result and defuzzification. The subtraction gives an interval [y_L, y_U]. DecisionMind reduces this interval to a single number by its midpoint ([y_L + y_U] / 2); this is the last step, not the first. If the interval is wide, that is, if the expert group disagrees on this alternative, the midpoint conceals this width.

DecisionMind fixes this cross-normalisation, the rough-number subtraction rule and the midpoint defuzzification in classical rough MOORA. This extension has no single founding literature source; the rules of rough-number arithmetic (Zhai, Khoo & Zhong, 2008) have been applied within DecisionMind to MOORA's ratio-system steps. The reference-point and full-multiplicative forms are not implemented in this family; only the ratio system exists.

The output is a net score and a ranking, as in crisp MOORA, and it is read the same way. The difference is this. Beneath this score lies information about how much the expert group agreed on each criterion, and midpoint defuzzification conceals this information. If two alternatives' scores are close, this gap can depend on which criterion the group failed to agree on.

Thus instead of writing:

"Rough MOORA accounts for group disagreement, so the result is more reliable"

the report should read:

"Disagreement within the group has been converted into an interval for every cell, the ratio system has been run over these intervals, and the result has been reduced to a midpoint in the last step; wherever a criterion shows a wide border region, the report should show this separately"

This extension is suitable when a group of experts assesses every alternative on every criterion with crisp scores, and the disagreement between the experts itself is meant to feed into the decision. Reducing the scores to an average and running crisp MOORA erases this disagreement. The exit condition on the rough data-type card applies here too. With a single expert, or where experts give very similar scores, the lower and upper bound narrow to a tight interval, and the extension says nothing different from the crisp method.

Where experts give their scores as an intuitionistic fuzzy judgement (with a membership and non-membership degree), this extension does not suffice; that situation needs a hybrid structure, as in rough MABAC, and DecisionMind does not implement this hybrid structure for MOORA. The exit condition is the same as for crisp MOORA: if no compromise is acceptable on one criterion, this extension too is compensatory and will not eliminate anything below a threshold.

Entering the bounds by hand. A rough number's lower and upper bound are calculated from the group's crisp scores; a hand-entered interval is not a rough number but a grey number, and should not enter this extension.

Confusing cross-normalisation with plain normalisation. The lower bound is divided by the column's upper bounds, and the upper bound by the column's lower bounds. Skipping this cross rule and dividing each bound by its own type's magnitude makes the normalised interval look narrower than its true size.

Confusing rough-number subtraction with plain subtraction. The difference between the beneficial and harmful summed intervals is found by subtracting the upper bound from the lower bound, and the lower bound from the upper bound. Subtracting only the lower bounds, or only the upper bounds, of the two intervals from one another gives a wrong interval.

Ignoring a wide border region. Even if the midpoint distinguishes two alternatives, one interval being much wider than the other is itself a finding. It means there is genuine disagreement within the group for that alternative and should be stated separately in the report.

The governing principle is this:

A rough MOORA score carries the group's disagreement as an interval all the way to the end of the ratio system, and reduces it to a single number only at the last step. Any application that invents the bounds, or ignores a wide interval, destroys this contribution.