Extension card · Rough
Rough ARAS (Daoud Ben Amor, Moalla Frikha & Martínez López, 2021)
This is the form of ARAS that works with rough numbers for situations where criterion scores come from a group assessment by several experts and the disagreement between them needs to be preserved. It carries uncertainty as a lower and upper bound all the way to the final step, and still ranks alternatives by a single degree of utility.
Base method
ARAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Rough →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Four things change; the degree-of-utility logic does not.
Cells. In crisp ARAS every cell is a single number. Here every cell is a rough number: a lower bound and an upper bound. These bounds are not entered from outside; they are calculated from the group's crisp scores, and how that calculation works is explained on the data-type card. Weights can be given as crisp numbers or as rough intervals; in the engine's validation example the weights are already reduced to crisp numbers.
Optimal alternative. As in crisp ARAS, a hypothetical optimal alternative is built here too, but now it is an interval: for a "higher is better" criterion, the highest observed upper bound and the highest observed lower bound are taken; for a "lower is better" criterion, the lowest lower bound and the lowest upper bound. This is a reference built from the rough numbers' own extremes.
Scale equalisation. Crisp ARAS reverses the cost criterion and divides every column by its own column sum. Rough ARAS applies the same logic through interval arithmetic: for a cost criterion, the lower and upper bounds swap places to reverse it, after which each alternative's lower bound is divided by the column's sum of upper bounds, and its upper bound by the column's sum of lower bounds. This cross-division ensures the normalised lower bound always stays less than or equal to the upper bound.
Result and defuzzification. The weighted lower and upper bounds are summed among themselves, giving an optimality interval for every alternative. The degree of utility is likewise built as an interval: the alternative's lower bound is divided by the optimal alternative's upper bound, and the alternative's upper bound by the optimal alternative's lower bound. The average of these two ratios gives a single degree of utility (K). DecisionMind holds this average, the midpoint, fixed; no adjustable risk parameter, of the kind seen in rough TOPSIS, is used here.
DecisionMind fixes cross-normalisation and midpoint defuzzification in this method. Weights are taken from outside; the method does not generate weights.
How to Read the Output
The degree of utility K is read as in crisp ARAS: a ratio relative to the optimal alternative, meaningful only for this alternative set and these weights.
The difference is here: beneath this ratio now lies the group's own disagreement. Because K is computed from a midpoint, an alternative with narrow bounds and one with wide bounds can reach a similar K value; K does not distinguish between the two. If two alternatives' K values are close, it is also worth checking which one has the narrower bound interval, that is, which one is less contested within the group.
Thus instead of writing:
"According to rough ARAS, A3 is the best alternative"
the report should read:
"Relative to the optimal alternative derived from this alternative set, A3 has the highest degree of utility; this ratio is computed from the midpoint of the group's score intervals and does not by itself show which alternative carries the wider group disagreement"
When to Prefer This over the Base Method
Use this extension when several experts score the same alternative and the disagreement between the experts itself matters to the decision. It also suits cases where crisp ARAS's "percentage relative to the optimal" output is wanted under this kind of group uncertainty. With a single expert, or experts giving similar scores, the rough number narrows to a tight interval and says nothing different from crisp ARAS; in that case crisp ARAS is sufficient.
The exit condition is the same as for crisp ARAS. The matrix must be of a single data type; if no compromise is acceptable on one criterion, this extension too is compensatory.
Mistakes Specific to This Extension
Writing the bounds by hand. A rough number's lower and upper bound are calculated from the group's crisp scores; writing an interval by hand "to represent uncertainty" has nothing to do with rough-set logic, and produces a grey or interval number instead.
Skipping the reversal on a cost criterion. Dividing without swapping the lower and upper bounds sets up both the optimal alternative and the normalised bounds in the wrong direction, corrupting every K value.
Comparing K directly with crisp ARAS as though it were a single exact number. K is a midpoint; without also reporting the width of the interval beneath it, the two methods' results cannot be compared directly.
Building a rough number from too few experts. With only two or three experts, the bounds can shift substantially if a single score changes; a wide boundary region in that case may come from a small sample rather than genuine disagreement.
The governing principle is this:
The degree of utility is a ratio computed from the midpoint of the group's score intervals; unless the width of that interval is separately shown in the report, K alone does not say how large the group's disagreement is.
Cases
The first case is based on an investment-firm selection example from Daoud Ben Amor, Moalla Frikha and Martínez López's (2021) paper, using the interval values that emerge from the paper's linguistic-hierarchy aggregation. DecisionMind's engine applies its own simplified ARAS steps to these intervals and does not give the same ranking as the paper's full method; this difference is shown explicitly below. The second case is illustrative fiction.
1. Investment: Choosing among four sectors for investment (Daoud Ben Amor et al., 2021)
An investment firm will invest in one of four sectors (defence equipment manufacturing, food, computer hardware, automotive). Four decision-makers scored each sector on four criteria (pollution cost, potential customer volume, sector stability, financial profitability) using a linguistic hierarchy; these assessments have been reduced to a rough number interval ([lower, upper]). Pollution cost is "lower is better"; the other three criteria are "higher is better".
| Sector | Pollution cost | Customer volume | Stability | Profitability |
|---|---|---|---|---|
| A1 (defence equipment) | [0.390; 0.782] | [0.314; 0.754] | [0.381; 0.680] | [0.342; 0.748] |
| A2 (food) | [0.628; 0.863] | [0.426; 0.714] | [0.363; 0.655] | [0.598; 0.920] |
| A3 (computer) | [0.267; 0.703] | [0.579; 0.765] | [0.258; 0.464] | [0.503; 0.800] |
| A4 (automotive) | [0.411; 0.848] | [0.474; 0.763] | [0.348; 0.589] | [0.488; 0.852] |
| Direction | lower is better | higher is better | higher is better | higher is better |
| Weight | 0.257 | 0.235 | 0.240 | 0.268 |
The method first reverses pollution cost, cross-divides every sector's lower and upper bound by the column sums, multiplies by the weights, and sums across the four criteria. Finally, it divides each sector's optimality by that of the hypothetical optimal sector and takes the midpoint.
| Sector | Degree of utility (K) | Rank |
|---|---|---|
| A3 (computer) | 1.577 | 1 |
| A1 (defence equipment) | 1.462 | 2 |
| A4 (automotive) | 1.457 | 3 |
| A2 (food) | 1.379 | 4 |
The result reads as follows. A3 comes first because it is strong on profitability, the heaviest criterion, and also carries the lowest pollution cost. A2 finishes last despite having the highest value on profitability, because its pollution cost is the highest of the four sectors, and since this criterion is "lower is better", it is this that drags A2 down.
An important clarification is needed here. The paper itself combines the decision-makers' linguistic assessments with a much richer aggregation method (the IVL2TWA operator, an extended linguistic hierarchy) and arrives at the order A3 ≻ A4 ≻ A2 ≻ A1. DecisionMind's engine takes only the outer bounds (the lowest and highest ends) of the aggregated intervals in the paper's Table III and applies the standard rough ARAS steps to them; this simplification gives A3 ≻ A1 ≻ A4 ≻ A2. The two rankings agree only on the first sector (A3); the relative order of A1, A4 and A2 differs between the paper's rich aggregation and DecisionMind's simplified interval arithmetic. This demonstrates the engine's own internal consistency; it does not reproduce the paper's full numerical result.
The firm's hesitation: if the pollution-cost weight is lowered from 0.257 to 0.10 and the financial-profitability weight is raised from 0.268 to 0.70 (recomputed independently in Python), the ranking changes completely: A2 (food) rises to first with 1.439, and A1 (defence equipment) falls to last with 1.274. The reason is that A2 has the clearly highest value on profitability while A1 has the lowest; as weight shifts towards profitability, this difference comes to dominate the decision.
In the report: "With the given weights, the computer sector (A3) has the highest degree of utility (K=1.577). This ranking agrees with the paper's own rich linguistic aggregation (A3≻A4≻A2≻A1) only on the first sector; DecisionMind applies a simplified calculation on the outer bounds of the intervals. When the weight is shifted towards profitability (0.70), the food sector (A2) moves to first place; the weight distribution should therefore be separately justified by the investment board."
Source: Daoud Ben Amor, W., Moalla Frikha, H., & Martínez López, L. (2021). The Interval Rough Number of the Extended ARAS Method for Solving Multi-Criteria Group Decision Making. 2021 International Conference on Decision Aid Sciences and Application (DASA), Table III (interval values) and Table IV (the paper's own degrees of utility). DecisionMind's K values and the weight-change scenario were obtained by running the engine's own rough ARAS steps independently; they are not transcribed from the paper.
2. Construction: A contractor's choice of site-equipment rental quotation
A construction firm will choose one of three equipment rental quotations for a large site. Three criteria have been set: equipment suitability, delivery time (lower is better) and rental cost (lower is better). The firm's four site engineers scored each quotation separately on these three criteria on a 1-9 scale; the scores came out distinctly different from one another, and this disagreement has been converted into rough number intervals.
The method compares the three quotations against the degree of utility relative to the optimal quotation built from the best end on each criterion. Suppose the quotation with the widest bound interval on equipment suitability, that is, the one with the greatest disagreement among engineers, nonetheless received the highest degree of utility, because its midpoint was higher than the other quotations'.
The firm's hesitation: this quotation's wide bound interval shows that the engineers genuinely disagree on how suitable the equipment is for the site. The degree of utility conceals this disagreement and carries only the midpoint. The firm should ask separately why the engineers disagree before approving the quotation with the highest degree of utility.
In the report: "The quotation with the highest degree of utility has the widest disagreement interval among engineers on the equipment-suitability criterion; the reason for this disagreement should be separately confirmed with the site team before approval."
3. What Not to Do
Had pollution cost mistakenly been marked "higher is better" in the investment example, the optimal sector would have been built from the most polluting alternative, and A2's disadvantage would have turned into an advantage. The second error is averaging the four decision-makers' linguistic scores first and only then running crisp ARAS; this entirely erases the interval width in Table III, that is, the disagreement between decision-makers. The third error is presenting DecisionMind's order A3≻A1≻A4≻A2 as though it were the same as the paper's A3≻A4≻A2≻A1; the two agree only on the first sector, and this difference must not be hidden in the report.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/rough-aras
Daoud Ben Amor, W., Moalla Frikha, H., & Martínez López, L. (2021). The Interval Rough Number of the Extended ARAS Method for Solving Multi-Criteria Group Decision Making. 2021 International Conference on Decision Aid Sciences and Application (DASA). DOI: 10.1109/DASA53625.2021.9681928
Pawlak, Z. (1982). Rough sets. International Journal of Computer & Information Sciences, 11(5), 341–356. DOI: 10.1007/BF01001956
Zavadskas, E. K., & Turskis, Z. (2010). A new additive ratio assessment (ARAS) method in multicriteria decision-making. Technological and Economic Development of Economy, 16(2), 159–172. DOI: 10.3846/tede.2010.10
Song, W., Ming, X., & Wu, Z. (2013). An integrated rough number-based approach to design concept evaluation under subjective environments. Journal of Engineering Design, 24(5), 320–341. DOI: 10.1080/09544828.2012.732994