Extension card · Grey
Grey TODIM (Sen, Datta & Mahapatra, 2015)
Grey TODIM is the form of TODIM used when the values in the decision table are known not as single numbers but only by a lower and upper bound (a grey number). It runs the same loss-aversion logic on values reduced to the midpoint of the bounds.
Base method
TODIM →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Grey →
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?
Three things change; θ's mechanism for amplifying losses and the reference-criterion mechanism do not.
Cells. In crisp TODIM every cell is a single number. Here every cell is a lower and an upper bound: ⊗v = [v^L, v^U]. No most-likely value is declared within the bounds; when only two bounds are available, this is the structure. DecisionMind takes weights as crisp (single numbers) in this extension. Unlike the fuzzy or intuitionistic extensions, weights are not grey here; group decisions are not supported.
Whitenisation and scale equalisation. In crisp TODIM, scale equalisation works directly on the raw values. Here, every grey cell is first reduced to a single number by the average of its own lower and upper bound, that is, by whitenisation. Crisp TODIM's scale equalisation, dividing by the column maximum for a benefit criterion or entering the column minimum into the ratio for a cost criterion, is then applied to these whitenised numbers. Whitenisation removes the width within the bounds from the calculation; only the midpoint remains.
Loss-aversion coefficient. Unlike the fuzzy, intuitionistic and neutrosophic TODIM members, θ cannot be adjusted by the user in DecisionMind's Grey TODIM; θ is internally fixed at 1. The reason is this: in this member of the grey family, uncertainty has already been reduced to a bound at the whitenisation step. An additional behavioural tuning parameter would not add a contribution proportional to data quality. The gain–loss comparison itself, that is, the reference criterion determining relative weight and the amplification of the loss side, continues exactly as before.
Result. The global value is again a single number normalised between 0 and 1; the width of the bounds never returns to the result at any point, only the difference between the whitenised midpoints is carried forward.
DecisionMind holds whitenisation fixed as the midpoint (v^L+v^U)/2 in this extension, and fixes θ at 1.
How to Read the Output
The global value is read as in crisp TODIM: the lowest total dominance takes 0, the highest takes 1; it is not an absolute "good/bad" measure.
The difference is this. This value is derived only from the midpoint of the bounds; when reading the score gap between two alternatives, the width of the bounds should be asked about separately. A cell with a narrow bound (⊗[0.65; 0.75]) and one with the same midpoint but a wide bound (⊗[0.50; 0.90]) produce the same number after whitenisation, and the result does not show this difference.
Thus instead of writing:
"According to the Grey TODIM result, A2 is first, and uncertainty was accounted for"
the report should read:
"Ranking was done by reducing the bounds to their midpoint; A2 has the highest global value, but which cells entered with a wide bound (little information) must be stated separately, because this width is not reflected in the global value"
When to Prefer This over the Base Method
This extension is suitable when only a lower and upper bound are known about a criterion's value, and no value within the bounds is considered more likely than another. It is equally suitable when the intuition that the decision-maker is more sensitive to losses than to gains fits the nature of the decision.
If a most-likely value is known within the bounds, fuzzy TODIM, not grey, should be used; deliberately narrowing the bounds and descending to the grey structure erases available information. If the criteria are measured, crisp TODIM should be kept; adding bounds to a measured value afterwards is manufacturing uncertainty. If the table is mixed, DecisionMind requires a single data type; a measured criterion is written as a grey number whose lower and upper bound are the same figure. TODIM's exit condition applies in exactly the same way: if a criterion allows no compromise, elimination is applied first; if the loss-aversion assumption does not fit, a symmetrically compensatory method such as Grey TOPSIS or Grey VIKOR should be preferred.
Mistakes Specific to This Extension
Interpreting the whitenised midpoint as a "most-likely value." A grey number carries no most-likely-value information; the midpoint is only a reduction for computation, and the warning on the data-type card applies here too.
Assuming θ is adjustable and failing to report it. θ is fixed at 1 in this extension; if a different behavioural assumption is needed (such as modelling a decision-maker more sensitive to losses), this extension is not sufficient, and this limitation should be stated in the report.
Building a grey number without showing the source of the bounds. Adding a percentage around a measured value to build a bound (the most common error on the data-type card) applies here too, and runs TODIM's gain–loss logic over a fabricated width.
Not checking for a value-space violation. As the manifest warns, every cell's lower bound must be less than or equal to its upper bound (x̲ ≤ x̄); if this is not checked before the calculation, whitenisation produces a midpoint in the wrong direction.
The governing principle is this:
Grey TODIM carries the midpoint of the bounds, not their width; when interpreting a difference in the global value, which cells entered with a narrow bound and which with a wide one must be stated separately.
Cases
The first case is DecisionMind's verification example: a hand-traceable, three-alternative, three-criterion grey-interval table, built synthetically while staying faithful to the formulas, not taken from a paper or book page. The second case is an illustrative fiction.
1. Illustrative example: Three suppliers scored on three criteria with grey bounds
Three suppliers are assessed on three criteria; every cell consists of a lower and an upper bound, with no most-likely value declared within them. The first and second criteria are "more is better," the third "less is better." Weights are crisp numbers; the first criterion has the highest weight and is the reference criterion.
| Supplier | Criterion 1 | Criterion 2 | Criterion 3 |
|---|---|---|---|
| A1 | ⊗[0.65; 0.75] | ⊗[0.45; 0.55] | ⊗[0.55; 0.65] |
| A2 | ⊗[0.75; 0.85] | ⊗[0.55; 0.65] | ⊗[0.35; 0.45] |
| A3 | ⊗[0.55; 0.65] | ⊗[0.65; 0.75] | ⊗[0.45; 0.55] |
| Direction | more is better | more is better | less is better |
| Weight | 0.40 (reference) | 0.35 | 0.25 |
The method first whitenises every cell by the average of its lower and upper bound, then equalises the scale, dividing by the column maximum for a benefit criterion and entering the column minimum into the ratio for a cost criterion. It then compares each pair of suppliers in turn. It sums a positive contribution on the criteria where a supplier wins and a negative contribution amplified by θ=1 on the criteria where it loses, and scales the global value to the 0–1 range.
| Supplier | Global value | Rank |
|---|---|---|
| A2 | 1.000 | 1 |
| A3 | 0.509 | 2 |
| A1 | 0.000 | 3 |
The result reads as follows. A2 has the highest midpoint on the first (reference, heaviest) criterion, and this advantage offsets its weakness on the other criteria. A3 is best on no single criterion but comes second thanks to a balanced profile. A1 is third because it has the lowest midpoint on the first criterion. A global value of 0 does not mean "worthless" in an absolute sense; it only shows the lowest relative superiority among these three suppliers.
The board's hesitation is this: if A1's bound on the first criterion had been given with less certain information, for example if the upper bound were 0.90 instead of 0.75, that is, the same lower bound but a wider uncertainty, would the ranking change? Recomputed independently in Python, A3's global value falls from 0.509 to 0.434. But the A2–A3–A1 order does not break, because whitenisation only shifts the midpoint, and raising A1's midpoint also affects the other suppliers' relative position. If the weights of C1 and C3 were swapped, A3's value would fall to 0.455; the order would again be preserved.
In the report: "With the highest weight given to the first criterion, A2 is clearly ahead; even if A1's bound information on the same criterion were less certain (the upper bound rising to 0.90) or the weight order changed, the A2–A3–A1 order is preserved, though A3's relative distance from A2 is sensitive to these changes."
Source: This case is DecisionMind's verification example for the Grey TODIM engine; the matrix and weights were produced synthetically, faithful to the formulas, as a small example that can be checked by hand — not a table from a paper or book. It is an illustrative example. The figures for the bound-width and weight-swap scenarios were independently recomputed with the same algorithm by this card's author.
2. Agriculture: Ranking a new seed variety's suitability for a region
An agricultural cooperative will decide which of three new seed varieties to recommend. Criteria: expected yield per decare, drought resistance, and seed cost (the last one "less is better"). Because the varieties have been sown in only one or two trial plots in the region, deriving an average or a distribution for yield and resistance is not meaningful. Only the lowest and highest observed value is available; these bounds are entered as grey numbers. The cooperative gives the highest weight (reference criterion) to yield.
The method compares the three varieties pairwise. On each criterion, the difference between the whitenised midpoints is summed as a positive contribution on the winning side and as a negative contribution amplified by θ=1 on the losing side. Suppose the variety with the highest yield bound also has the most expensive seed cost, and it still comes first in global value, because yield is the reference criterion and carries the heaviest weight.
The cooperative's hesitation is this: the variety with the widest yield bound, that is, the one about which the least is known, may have a midpoint that came out high by chance. As the number of trial plots increases, this bound may narrow and the midpoint may change. The cooperative should consider waiting for an additional trial year to narrow the bound before choosing this variety; the global value does not show today's bound width.
In the report: "With the highest weight given to expected yield per decare, the first variety stands out clearly; however, this variety's yield bound is the widest (based on the least trial data), so the decision should not be finalised without verification from an additional trial year."
3. What Not to Do
Averaging the bounds in the table before whitenisation and interpreting the result as a "most-likely value" is an error: reporting the midpoint of ⊗[0.65; 0.75], namely 0.70, as "most-likely yield" assumes information a grey number does not carry. The second error is assuming θ is adjustable and trying to rerun the calculation with a different value; θ is fixed at 1 in this extension, and if a different behavioural assumption is needed, a different extension should be used. The third error is opening a measured cost (a finalised contract price) into a bounded grey number "to be cautious"; a measured value should be written as a grey number whose lower and upper bound are the same.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/grey-todim
Sen, D. K., Datta, S., & Mahapatra, S. S. (2015). Extension of TODIM combined with grey numbers: an integrated decision making module. Grey Systems: Theory and Application, 5(3), 367–391. DOI: 10.1108/gs-05-2015-0029
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)
Deng, J.-L. (1982). Control problems of grey systems. Systems & Control Letters, 1(5), 288–294. DOI: 10.1016/S0167-6911(82)80025-X
Liu, S., & Lin, Y. (2011). Grey Systems: Theory and Applications. Understanding Complex Systems. Springer. DOI: 10.1007/978-3-642-16158-2