Extension card · Grey
Grey CODAS
Grey CODAS is the form of CODAS that works with grey numbers when criterion values are known only by a lower and an upper bound. It computes two different distance measures over the bounds, and ranks the result, again, with a single assessment score.
Base method
CODAS →
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; the two-distance comparison logic does not.
Cells. In crisp CODAS every cell is a single number. Here every cell is two numbers: a lower bound and an upper bound; no "most likely" value is defined in between. Criterion weights here remain crisp numbers. This DecisionMind form of Grey CODAS requires weights as crisp values, not as intervals. This design is for situations where uncertainty is carried only in the criterion values, not in the weights.
Scale equalisation (direction-sensitive, including bound swap). Crisp CODAS divides every cell by the column's largest value (benefit) or takes the ratio to its smallest value (cost). With grey numbers this division is direction-sensitive and swaps bounds: for a benefit criterion, [lower/Upper_max, upper/Upper_max] is used (Upper_max being the largest upper value in the column); for a cost criterion, [Lower_min/upper, Lower_min/lower] is used (Lower_min being the smallest lower value in the column). Here the position of the lower and upper bound is deliberately swapped (bound swap). The reason is that on a cost criterion a small raw value must correspond to a large normalised value, and this is preserved only by reversing the bounds.
Whitening and distance. Every normalised, weighted grey interval is reduced to a single number by the average of its lower and upper bound (whitening); the negative-ideal is built from the smallest of these whitened values. Euclidean and Taxicab distances are computed, as in crisp CODAS, between these whitened numbers.
The assessment score and the pairwise comparison rule (threshold τ, Euclidean first, Taxicab if the gap does not exceed it) are exactly the same as in crisp CODAS. In classical Grey CODAS, DecisionMind fixes this whitening rule ((lower+upper)/2) and the threshold value (τ=0.02).
How to Read the Output
The assessment score is read the same way as in crisp CODAS: it is a relative-position measure, not a percentage, and it is rebuilt when the alternative set changes. The difference is this: only bound information stands behind the score. When the score gap between two alternatives is small, it must additionally be asked whether this gap is sensitive to the midpoint of the bounds, that is, the whitened value, or to the actual width of the interval. The wider the interval, the weaker the midpoint's power to represent the "true" value.
Thus instead of writing:
"Grey CODAS gave a decisive ranking even with little data, so it is reliable"
the report should read:
"The midpoints of the bounds have been ranked against the negative-ideal; this ranking is recorded together with the fact that only a lower and upper bound are available, and its reliability decreases as the bounds widen"
When to Prefer This over the Base Method
Use this method when only a lower and an upper bound are known about criterion values, that is, there is no "most likely" point or distribution, and the need to distinguish close rivals with two different distance measures, CODAS's own rationale, still applies. Measured criteria should not be carried into this extension; if the matrix must be of a single type, a measured value is written as a grey number with the lower bound equal to the upper bound. Base CODAS's exit condition applies here unchanged: a compensatory method is unsuitable when no compromise is acceptable on one criterion.
Mistakes Specific to This Extension
Whitening first and then running crisp CODAS. Reducing the bounds to their average from the outset and running crisp CODAS is not Grey CODAS; the direction-sensitive, bound-swapped normalisation is skipped. In the illustrative example below, this path gives A1 as -0.4115, A2 as 0.4068 and A3 as 0.0047; the correct Grey CODAS gives -0.3785, 0.3755 and 0.0030. Because the bounds are symmetric in this fixture, the ranking does not change, but the bound-swap step in normalisation has been skipped, and the ranking can change with asymmetric bounds.
Forgetting the bound swap on a cost criterion. If the benefit criterion's normalisation formula is applied to a cost criterion without reversing the bounds, the low-cost alternative is penalised and the ranking is reversed; this is the most common implementation error.
Reading the midpoint of a grey number as a "most likely" value. Whitening is only a computational step and adds no new information about the data; the midpoint cannot be claimed to be the point closest to the "true" value.
Giving weights as grey too. This DecisionMind extension requires crisp weights; if weights are also to be given as grey, a separate extension (grey-weighted grey CODAS) is required, and this card does not cover that form.
The governing principle is this:
Grey CODAS is for working honestly with bound information alone; any implementation that averages the bounds from the outset, or skips the bound swap on a cost criterion, breaks the principle of working honestly with little data.
Cases
The first case is DecisionMind's validation example: a small three-alternative, three-criterion grey-interval table generated by the engine itself, faithful to the formulas; it is not a page or table from the literature. The second case is illustrative fiction.
1. Illustrative example: Three alternatives, three criteria (DecisionMind validation example)
This example is not a literature case; it is a small table built to make GREY-CODAS's formulas traceable by hand. Three alternatives are evaluated on three criteria with grey intervals; the first two criteria are "higher is better," and the third is "lower is better" (cost). The threshold value is τ=0.02.
| Alternative | C1 | C2 | C3 (cost) |
|---|---|---|---|
| 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 | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method scales every bound in the benefit criteria against the column's largest upper value, and in the cost criterion against the column's smallest lower value with the bounds swapped; it multiplies by the weights, whitens every interval by its average, builds the negative-ideal from the smallest whitened value, and computes Euclidean and Taxicab distances. Because all pairwise Euclidean gaps among the three alternatives exceed the threshold (0.02), the Taxicab distance does not come into play in any comparison in this example.
| Alternative | Assessment score | Rank |
|---|---|---|
| A2 | 0.3755 | 1 |
| A3 | 0.0030 | 2 |
| A1 | -0.3785 | 3 |
The result reads as follows. A2 has the highest bounds (⊗[0.75; 0.85]) on C1, the most heavily weighted criterion, and the lowest, that is best, bounds on the cost criterion C3; these two advantages more than compensate for its relative weakness on C2. The gap between A3 and A1 is small (0.38).
There is a hesitation in this decision. When C2's weight is raised from 0.35 to 0.54 (with C1 and C3 reduced proportionally, to C1≈0.28 and C3≈0.18) and the calculation redone, A3 moves ahead (0.2512) and A2 drops to second (0.2299). A2's lead is broken once C2's weight is increased enough to overtake C1's; this is a moderately robust result.
In the report: "With the given weights (C1=0.40, C2=0.35, C3=0.25) and the τ=0.02 threshold, A2 has the highest assessment score (0.3755); when C2's weight is raised above 0.54, A3 moves ahead (0.2512 / 0.2299), so A2's lead is moderately sensitive to C2's weight."
Source: DecisionMind's validation fixture for the Grey CODAS engine; the steps follow, of the two published grey-CODAS variants defined by Tadić et al. (2020) and Badi et al. (2023), the bound-swapped form using threshold τ=0.02 (Badi et al., 2023). All figures have been independently recomputed by this card's author with the same algorithm and confirmed against the engine's output.
2. Sport: A club choosing a youth player to promote to the first team
A sports club's technical staff must promote one of three youth-academy players to the first-team squad. The criteria are: sprint speed, decision-making score under ball control, and injury history (the last one "lower is better"). Because the players have so far played in only a small number of official matches, each criterion is recorded not as a single number but as the observed lowest and highest performance bound; the scouting reports give no "most likely" value.
The method normalises the three players' bounds direction-sensitively, multiplies by the weights, whitens the result, and computes the two distances to the negative-ideal. Suppose the player with the highest sprint-speed bound also has the widest injury-history interval. He still comes out first, because the weight on sprint speed exceeds that on injury history. The Euclidean gap between the second and third player falls below the threshold, so the ranking between them is decided by the Taxicab distance.
The staff also has a hesitation. The wide bounds built from a small number of match observations carry uncertainty about how well they represent the player's true level; as the season progresses the bounds will narrow, so the decision could be postponed a few weeks and recalculated. It should also be separately noted in the report that the ranking between the second and third player comes from the Taxicab distance, and that these two could swap places if the threshold value were changed.
In the report: "With the high weight given to sprint speed, the fastest player reaches the highest assessment score; the ranking between the second and third player is sensitive to the threshold value (τ), and it should be separately stated that the bounds come from a small number of match observations and may narrow as the season progresses."
3. What Not to Do
The first error is reducing the illustrative example's grey intervals to their average from the outset and running crisp CODAS. In this fixture the ranking comes out the same (A2, A3, A1), but the direction-sensitive, bound-swapped normalisation step has been skipped, and the ranking can change with asymmetric bounds. The second error is forgetting the bound swap on a cost criterion (such as injury history in the sport example) and applying the benefit criterion's normalisation formula directly; this penalises the low value and reverses the ranking. The third error is presenting a wide grey interval built from a small number of observations as though it were the player's "true" level, without ever stating the width of the bounds in the report.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/grey-codas
Keshavarz Ghorabaee, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2016). A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Economic Computation and Economic Cybernetics Studies and Research, 50(3), 25–44. (no DOI; not registered in Crossref. Because the two published grey CODAS extensions differ from each other in how they handle the cost criterion, this card's "base method" attribution stays with crisp CODAS; see the review notes.)
Deng, J. (1982). Control problems of grey systems. Systems & Control Letters, 1(5), 288–294. DOI: 10.1016/S0167-6911(82)80025-X
Tadić, S., Krstić, M., Roso, V., & Brnjac, N. (2020). Dry Port Terminal Location Selection by Applying the Hybrid Grey MCDM Model. Sustainability, 12(17), 6983. DOI: 10.3390/su12176983
Badi, I., Alosta, A., Elmansouri, O., Abdulshahed, A., & Elsharief, S. (2023). An application of a novel grey-CODAS method to the selection of hub airport in North Africa. Decision Making: Applications in Management and Engineering, 6(1), 18–33. DOI: 10.31181/dmame0313052022i