Extension card · Grey
Grey MARCOS (Badi & Pamučar, 2020)
This is the form of MARCOS that works with grey numbers when criterion values are known only by a lower and upper bound. The bounds are first whitenised to a single midpoint, and the remaining calculation then follows the same ideal/anti-ideal ratio as crisp MARCOS.
Base method
MARCOS →
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?
One thing changes, and it happens right at the start: the shape of the cells and the moment at which this shape is reduced to a single number. The remaining ideal/anti-ideal ratio logic is the same.
Cells. In crisp MARCOS every cell is a single number. In Grey MARCOS every cell is two bounds: ⊗[lower, upper]. Criterion weights remain crisp; the method does not generate weights, it requires them from outside. The founding paper (Badi & Pamučar, 2020) proposed the method for a steel-producing firm's supplier selection. DecisionMind's Grey MARCOS does not combine the scores of multiple decision-makers.
Scale equalisation. Crisp MARCOS builds its ideal and anti-ideal references from the observed best and worst value of the columns. Grey MARCOS first whitenises every grey interval to a single number: the average of the lower and upper bound. This is so that MARCOS's ideal/anti-ideal reference logic can work on a single number rather than an interval. After whitenisation, the ideal and anti-ideal are built from these single numbers just as in crisp MARCOS, and normalisation runs by the same rule (dividing by the ideal for a benefit criterion, dividing the ideal by the cell for a cost criterion).
Distance / score / aggregation. After the whitenisation step, the weighted sum, the utility ratios and the final utility function run with exactly the same formulas as crisp MARCOS's steps. In the utility ratios, the ideal's total is always normalised to 1, and the anti-ideal's total is scaled relative to it. The difference lies only in the whitenisation at the first step.
Result and defuzzification. Defuzzification (whitenisation) is done at the first step, BEFORE the ideal and anti-ideal are built. This is the exact reverse order of Fuzzy MARCOS, where uncertainty is carried through to the fifth step and only defuzzified there; here it is defuzzified at the first step, and the remaining steps run like crisp MARCOS.
What DecisionMind holds fixed in Grey MARCOS is this: the whitenisation rule, that is, (lower + upper) / 2, and the normalisation that fixes the ideal's total score at 1 (F1, F3).
How to Read the Output
The output is a final utility degree and a ranking, as in crisp MARCOS, and it is read the same way. It cannot be compared with the degree of a different analysis.
The difference is here: because whitenisation is done at the first step, the width of the bounds, that is, the size of the uncertainty, never reaches the result. If a narrow interval ([48, 52]) and a wide interval ([20, 80]) share the same average (50), Grey MARCOS treats them identically.
Thus instead of writing:
"Grey MARCOS carries the uncertainty within the bounds into the result"
the report should read:
"Grey MARCOS reduces the bounds to a single midpoint and runs crisp MARCOS on these points; the width of the bounds does not reach the result and must be recalled separately when interpreting input quality. A series with a narrow uncertain range and a series with a wide uncertain range can give the same result in this method"
When to Prefer This over the Base Method
Use this when only a lower and upper bound are known about a criterion value, and there is no most-likely value or distribution within it (the criterion set on the grey data-type card). The principle of not adding an unsourced margin to a measured value applies here too; the bounds must have their own source.
The exit condition is the same as crisp MARCOS. If a criterion allows no compromise, this extension too is compensatory. If the alternative set is very small, the ideal and anti-ideal references remain overly sensitive to the alternatives themselves.
Mistakes Specific to This Extension
Violating the bounds' x̲ ≤ x̄ rule. If the lower bound is written greater than the upper bound, the ideal/anti-ideal references are corrupted too.
Changing the whitenisation rule without noticing. (Lower + upper)/2 is the canonical choice but not the only option; a different whitenisation rule (such as taking only the upper bound) can produce a different ranking. In the illustrative example below, giving crisp MARCOS only the optimistic (upper) end of the bound completely reverses the order: with correct whitenisation A1 is first and A2 third, whereas with only the upper bound A2 is first and A1 third.
Assuming the midpoint of the bounds is a "most-likely value." A grey number does not carry this information; the average is only a computational step for whitenisation, not information added to the input as "most likely."
The governing principle is this:
Grey MARCOS reduces the bounds to a single midpoint and runs crisp MARCOS on this point; the whitenisation rule should be stated in the report, and the width of the bounds should be interpreted separately, as input quality. The result itself does not carry this width.
Cases
The first case is DecisionMind's verification example: a small three-alternative, three-criterion grey-interval table with the same weight and direction structure as crisp MARCOS, built not from a book or paper page but so the formulas can be traced by hand. The second case is an illustrative fiction.
1. Illustrative example: Three alternatives, three criteria (DecisionMind verification example)
| Alternative | K1 | K2 | K3 (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 | more is better | more is better | less is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method first whitenises every bound by averaging, for instance (0.65+0.75)/2 = 0.70 for A1-K1. It then takes the best value of the whitenised columns as ideal and the worst as anti-ideal; on K1 and K2 the highest value is ideal, on K3 the lowest value is ideal. It divides each value by the ideal, weights and sums, and builds the utility ratios.
| Alternative | Final utility degree | Rank |
|---|---|---|
| A1 | 0.6750 | 1 |
| A3 | 0.6663 | 2 |
| A2 | 0.6577 | 3 |
The result reads as follows. In the whitenised table, A1 is neither best nor worst on K1, at 0.70. But on K3, the cost criterion, it carries the lowest value at 0.60, which means it is closest to its ideal there. Staying balanced on the two most heavily weighted criteria (K1=0.40, K3=0.25) carries it to first place. The three alternatives' degrees are very close to one another (between 0.658 and 0.675); this comes from the small gap between the whitenised averages.
The decision may hesitate here. If K1 and K2's weights are swapped (K1=0.35, K2=0.40, K3 stays at 0.25), the ranking itself changes: A3 moves to first at 0.6756, and A1 drops to second at 0.6700. The gap is only 0.0056. In this three-alternative example, the K1–K2 weight balance directly determines the ranking.
In the report: "With the given weights (K1=0.40, K2=0.35, K3=0.25), A1 has the highest final utility degree (0.6750); if K1 and K2's weights are swapped, A3 moves ahead and the A1–A3 gap is only 0.0056. The ranking is fragile against this weight balance."
Source: DecisionMind Grey MARCOS manifest, verification example; the steps follow the Badi & Pamučar (2020) definition. The figures for the weight-swap scenario were independently recomputed with the same algorithm by this card's author.
2. Mining: Choosing loading equipment for a new mine site
A mining operation will choose one of three models of loading equipment for a newly opened site. Three criteria: fuel consumption (less is better), loading capacity, maintenance interval. Because the site's ground conditions are not yet fully known, only a lower and upper bound can be obtained from manufacturer data and a few trial runs at similar pits; there is no most-likely value.
The method builds the ideal and anti-ideal from each criterion's whitenised (averaged) value, normalises, weights, and computes the utility ratios. Suppose the model with the lowest fuel-consumption bound also has the narrowest maintenance-interval bound, meaning it requires frequent maintenance. It nonetheless stays in second place, because fuel consumption is given a higher weight than the maintenance interval.
The operation may hesitate here. The maintenance-interval bounds come from only two trial sites. If a third site's data is added, the bounds may narrow or widen, and the whitenised midpoint will change too. The operation should not deploy this equipment at scale before gathering additional site data.
In the report: "With the high weight given to fuel consumption, the most efficient model remains in second place; because the maintenance-interval bounds are derived from only two trial sites, the whitenised values may change as further data arrives."
3. What Not to Do
The first error, in the illustrative example, is taking only the optimistic (upper) end of the bound and running crisp MARCOS. In that case A1 drops from 0.675 to 0.597 and falls to third place, while A2 rises from 0.658 to 0.727 and moves to first. The order is completely reversed. Changing the whitenisation rule, that is, using the optimistic end instead of the average, is a hidden way of making the input look better than it really is. The second error is interpreting the midpoint of the interval ⊗[0.55; 0.65], namely 0.60, as a "most-likely value." A grey number does not carry this information; 0.60 is only an intermediate step of the whitenisation calculation. The third error is adding the width of the bounds separately to the final degree as a kind of "risk score." The width has already been erased in whitenisation and cannot be added back afterwards; it should be modelled as a separate criterion.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/grey-marcos
Badi, I., & Pamučar, D. (2020). Supplier selection for steelmaking company by using combined Grey-MARCOS methods. Decision Making: Applications in Management and Engineering, 3(2), 37–48. DOI: 10.31181/dmame2003037b
Stević, Ž., Pamučar, D., Puška, A., & Chatterjee, P. (2020). Sustainable supplier selection in healthcare industries using a new MCDM method: Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS). Computers & Industrial Engineering, 140, 106231. DOI: 10.1016/j.cie.2019.106231
Deng, J. L. (1982). Control problems of grey systems. Systems & Control Letters, 1(5), 288–294. DOI: 10.1016/S0167-6911(82)80025-X
Demir, G., Chatterjee, P., Kadry, S., Abdelhadi, A., & Pamučar, D. (2024). Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS) Method: A Comprehensive Bibliometric Analysis. Decision Making: Applications in Management and Engineering, 7(2), 313–336. DOI: 10.31181/dmame7220241137