Extension card · Rough
Rough TODIM (Tiwari, Khanna & Tandon, 2024)
Rough TODIM is the form of TODIM for situations where every cell in the decision matrix is given as a lower and upper bound derived from disagreement within a group. It carries out the pairwise gain-loss comparison over these intervals.
Base method
TODIM →
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 reference-criterion logic continues, but in this extension the reference criterion is always the highest-weighted criterion, and the loss-aversion coefficient θ is fixed at 1; neither can be changed from the interface.
Cells. In crisp TODIM 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's crisp scores. This is a plain rough number; it is not, as in rough MABAC, a four-number hybrid structure combined with an intuitionistic fuzzy judgement, but a stand-alone interval, as in rough MOORA. Criterion weights are also entered as decimal numbers, but they need not sum to 1; DecisionMind converts them to relative weights by relating them to the reference criterion (the highest-weighted criterion).
Scale equalisation. Crisp TODIM divides every column by its own column sum. Rough TODIM follows a different path here: it divides by a single global largest upper bound across the whole matrix, not column by column. This is the source paper's own choice; the paper assumes that all the customer preference scores come from the same scale (the same survey scale), and uses a single common divisor.
Gain-loss comparison. In crisp TODIM the difference between two values is a plain subtraction, and this difference carries both magnitude and direction. With rough numbers these two separate. The distance is the sum of the difference between the two intervals' lower bounds and the difference between their upper bounds. Which alternative "wins" is determined by looking at the sum of the interval's lower and upper bound. The source paper does not explicitly define this comparison rule; DecisionMind has chosen the sum (lower + upper) rule, but a different rule (looking only at the lower bound, for instance) could give a different result.
Result. The global value is again a single number normalised between 0 and 1. Beneath this value lies information about how wide a border region the group's disagreement has spread across, and this information becomes invisible at the last step.
DecisionMind fixes global-maximum normalisation and the sum (lower + upper) comparison rule in this extension; θ is fixed at 1, and the reference criterion is always the highest-weighted criterion.
How to Read the Output
The global value is read exactly as in crisp TODIM: the lowest total dominance takes 0, the highest takes 1, and this holds only for this alternative set. The difference is this. Beneath this value lies group disagreement, and this disagreement is carried as interval width, but the rule that determines the winning-losing direction is not fully defined in the source paper. For this reason the same data can give a different ranking under a different comparison rule; the rule DecisionMind has chosen should be stated in the report.
Thus instead of writing:
"The rough TODIM result is exact, because the calculation is TODIM itself"
the report should read:
"The rule by which the intervals are compared (here, the sum of the lower and upper bound) affects the result; because this rule is not fully defined in the source paper, a different piece of software could produce a different ranking"
When to Prefer This over the Base Method
This extension is suitable when a group of assessors evaluates every alternative with crisp scores and the disagreement within the group itself is meant to feed into the decision. The exit condition on the rough data-type card applies here too: with a single expert, or where experts give very similar scores, the rough number narrows to a tight interval and says nothing different from crisp TODIM.
TODIM's exit condition applies exactly: if no compromise is acceptable on one criterion, this extension too is compensatory and will not eliminate anything below a threshold. Where experts give their scores as an intuitionistic fuzzy judgement, this extension does not suffice; DecisionMind does not implement this hybrid structure for TODIM.
Mistakes Specific to This Extension
Forgetting that the comparison rule is undefined in the source paper. The source paper does not explicitly write down the rule that determines which criterion two rough numbers "win" on. DecisionMind uses the sum of the lower and upper bound. This is the most important extension-specific mistake: comparing the results one-to-one with another piece of software, or with the paper's own table, is misleading.
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.
Trying to change θ or the reference criterion. Neither can be changed from the interface in this extension; θ is fixed at 1, and the reference criterion is always the highest-weighted criterion.
Expecting the weights to sum to 1. In this family, weights can be entered relative to the reference criterion (Wrc) and need not sum to 1; DecisionMind also accepts normalised weights and relates them to the reference internally.
The governing principle is this:
Rough TODIM carries group disagreement as an interval, but the rule that determines which alternative wins on a criterion is not fully defined in the source paper. The rule DecisionMind has chosen (the sum of the lower and upper bound) must be stated explicitly in the report.
Cases
The first case is taken from Tiwari, Khanna and Tandon's (2024) published paper: an evaluation of four weightlifting-bench design concepts using customer preference scores. The second case is illustrative fiction.
1. Design: Comparing four bench design concepts by customer preference (Tiwari, Khanna & Tandon, 2024)
A product-development team is comparing four weightlifting-bench design concepts on four criteria. The criteria are preference scores derived from customer and designer opinions (all read in the "higher is better" direction), and every cell is a [lower, upper] rough number computed from a group assessment. The second criterion is the reference criterion (the highest relative weight).
| Concept | C1 | C2 (reference) | C3 | C4 |
|---|---|---|---|---|
| A1 | [4.50; 5.83] | [5.50; 6.50] | [1.13; 1.88] | [3.00; 6.04] |
| A2 | [3.50; 6.33] | [5.50; 6.50] | [2.29; 4.79] | [1.54; 3.50] |
| A3 | [3.22; 4.75] | [2.13; 5.75] | [5.13; 5.88] | [5.50; 6.50] |
| A4 | [1.25; 2.75] | [1.13; 1.88] | [4.50; 4.70] | [2.17; 3.83] |
| Relative weight (Wrc) | 0.857 | 1.000 | 0.667 | 0.333 |
The method divides every column by a single global largest upper bound across the whole matrix (6.50), relates the relative weights to C2, compares every pair of concepts by the sum of their lower and upper bound, and scales the global value to the 0-1 range.
| Concept | Global value | Rank |
|---|---|---|
| A3 | 1.000 | 1 |
| A1 | 0.753 | 2 |
| A2 | 0.557 | 3 |
| A4 | 0.000 | 4 |
The result reads as follows. A3 has the widest upper bound on the third and fourth criteria, and carries a relatively narrow but not low interval on the reference criterion, C2; in total it comes first. A4 has the lowest intervals on three of the four criteria and finishes last.
The team's hesitation is this: the source paper's own Table 9 reports the order A2 > A1 > A4 > A3 for this data, that is, it places A2 first and A3 last. DecisionMind's engine produces the order A3 > A1 > A2 > A4 with the same data. The difference does not come from a fabricated figure, but from the fact that the paper does not explicitly write down which rule it uses when comparing two rough intervals (only the lower bound, only the upper bound, or their sums); DecisionMind has chosen the sum (lower + upper) rule, and the paper's own intermediate values in Table 8 cannot be reproduced with this rule. When the weights are changed (setting C4's relative weight equal to the reference, or making C1 the new reference), the ranking does not change for these four concepts; the real uncertainty lies not in the weights but in the comparison rule.
In the report: "With DecisionMind's sum (lower + upper) comparison rule, A3 is clearly ahead. The source paper's own table reports a different order (A2 first); this difference stems from the paper not fully defining its comparison rule, and the report must note this uncertainty explicitly."
Source: Tiwari, V., Khanna, P., & Tandon, P. (2024). Capturing Design Intent During Concept Evaluation Using Rough Numbers and TODIM Method. Computer-Aided Design & Applications, 21(2), 215–228 (open access). The matrix, weights and normalisation are taken from the paper's Table 7b and Equations 4-5. The global values were verified for this card by independently running DecisionMind's ROUGH-TODIM engine (method_runner.py ROUGH-TODIM) and re-running the kernel functions directly in Python; the mismatch with the paper's own Table 9 order is stated explicitly above.
2. Care services: A care home's choice of outsourced cleaning and support firm
A care-home operator will choose among three outsourced cleaning and patient-support firms. Three criteria apply: service quality, staff turnover rate (converted into a preference score so that "lower is better" is meant), and price suitability. Five different assessors (the care-home manager, the head of nursing, and three family representatives) score each firm on these criteria with crisp scores from 1 to 9; the scores come out distinctly different from one another.
The method derives rough number bounds from each assessor group's scores, divides by the global maximum, builds relative weights, and compares the three firms pairwise. Suppose the firm with the highest interval on service quality also has a good interval on price suitability, and comes out first on the global value.
The operator's hesitation is this: this firm's interval on the staff-turnover criterion is wide, meaning the assessors have not agreed on this point. The global value does not show this width. The operator should separately clarify the staff-turnover commitment before signing, and should note how the comparison rule (sum of lower and upper bound) affects this decision.
In the report: "The firm that stands out on service quality and price suitability comes first on the global value; there is wide disagreement among the assessors on the staff-turnover criterion, and this should be addressed separately before the contract is signed."
3. What Not to Do
Treating DecisionMind's A3-first result in the illustrative example as merely a different phrasing of the same calculation as the source paper's A2-first result in Table 9: both come from the same data but use a different comparison rule, so they cannot be compared directly. The second error is averaging the five assessors' scores first and then writing a [lower, upper] interval by hand around that average; this is not a rough number but a fabricated grey number. The third error is trying to change θ or the reference criterion and reporting "the same result would come out under a different scenario"; in this extension both are fixed from the interface.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/rough-todim
Tiwari, V., Khanna, P., & Tandon, P. (2024). Capturing Design Intent During Concept Evaluation Using Rough Numbers and TODIM Method. Computer-Aided Design & Applications, 21(2), 215–228. DOI: 10.14733/cadaps.2024.215-228
Gomes, L. F. A. M., & Lima, M. M. P. P. (1992). TODIM: Basics and application to multicriteria ranking of projects with environmental impacts. Foundations of Computing and Decision Sciences, 16, 113–127. (no DOI)
Pawlak, Z. (1982). Rough sets. International Journal of Computer & Information Sciences, 11(5), 341–356. DOI: 10.1007/BF01001956
Zhai, L.-Y., Khoo, L.-P., & Zhong, Z.-W. (2008). A rough set enhanced fuzzy approach to quality function deployment. The International Journal of Advanced Manufacturing Technology, 37(5-6), 613–624. DOI: 10.1007/s00170-007-0989-9