Extension card · Grey
Grey COPRAS (Zavadskas, Kaklauskas, Turskis and Tamošaitienė, 2009)
Grey COPRAS is the form of COPRAS used when criterion values are known only by a lower and an upper bound, that is, when they are grey. DecisionMind reduces every interval to its midpoint, that is, whitens it, and then runs COPRAS's benefit/cost ratio calculation on these single numbers exactly as usual.
Base method
COPRAS →
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 benefit/cost ratio logic does not.
Cells. In crisp COPRAS every cell is a single number. Here every cell is an interval: a lower bound and an upper bound (⊗[lower, upper]). Criterion weights are entered as crisp, single numbers; only the decision matrix is grey, not the weights.
Scale equalisation. DecisionMind first reduces every grey cell to its midpoint. Whitening, the average of the lower and upper bound, is the very first step of the calculation. The interval collapses to a single number before normalisation and weighting take place. Crisp COPRAS's division-by-column-sum normalisation then runs on these whitened numbers exactly as it stands.
Distance / score / aggregation. The whitened, weighted values are separated into a benefit sum and a cost sum, as in crisp COPRAS, and the relative significance value (Q) is built by the same ratio. After this step the calculation is entirely independent of the interval's width.
Result and defuzzification. No separate defuzzification step is needed, because defuzzification, whitening, has already been done in the very first step. The output is, directly, a single degree of utility in exactly the same form as crisp COPRAS (best alternative = 100).
In this family, DecisionMind fixes whitening as the midpoint, applied once, at the very start of the calculation. No path is followed in which the interval's lower and upper bound are carried separately and merged at the end. This may conflict with the expectation that "a grey number is carried as an interval throughout the calculation." In DecisionMind the interval collapses to a single point before normalisation.
How to Read the Output
The output is a degree of utility and a ranking, in exactly the same form as crisp COPRAS, and it is read the same way. The best alternative is assigned 100, and the others receive a percentage relative to it.
This score rests on the interval's midpoint and says nothing about the interval's width. A narrow gap between two alternatives should be read with different confidence depending on whether it comes from wide-interval data or narrow-interval data. Because DecisionMind does not carry this width into the score, the report itself must separately state how wide the intervals are.
Thus instead of writing:
"Grey COPRAS gives a more reliable result because it accounts for the interval"
the report should read:
"Because the bounds are the only information available, their midpoints have been taken, and the width of the interval is not reflected in the degree of utility; A2 leads at 100.00, and this advantage holds until the weight is shifted markedly in C2's favour"
When to Prefer This over the Base Method
Use this method when only a lower and an upper bound are known about the criteria, and no "most likely" value in between can be claimed. There is no need to open a single measured value into an interval by adding a margin on either side; because DecisionMind requires a single data type, a measured criterion in the same matrix is written as an interval with the lower bound equal to the upper bound, giving it zero width and adding no information.
Crisp COPRAS's exit condition applies here too. This extension is also compensatory and will not eliminate anything below a threshold when no compromise is acceptable on one criterion. If a most-likely value is known within the bounds, a fuzzy structure is more suitable than grey; if the bounds are the ends of a distribution, a probabilistic structure is more suitable.
Mistakes Specific to This Extension
Violating the value range. Every cell's lower bound must be less than or equal to its upper bound; if this order is broken, the calculation becomes meaningless.
Forgetting that the whitening method is a hidden assumption. DecisionMind uses the midpoint, (lower+upper)/2; this is the canonical choice but not the only one. A whitening that takes, instead of the interval's midpoint, only the upper bound (optimistic) or only the lower bound (pessimistic) produces a different ranking and different scores. The rule used should be stated in the report.
Not knowing that whitening happens at the very start. DecisionMind does not carry the interval through to the end of the calculation; it reduces it to the midpoint in the first step. Interpreting intermediate results under the assumption that "the grey method preserves the interval to the end" is incorrect.
Mistaking the interval's midpoint for the most likely value. The whitened number is a computational step; it does not mean "the middle value is more likely." A grey number, by definition, treats no point within the bounds as more likely than another.
The governing principle is this:
Grey COPRAS exists to stay honest when nothing beyond the bounds is known; whatever the whitening rule, it must be stated explicitly in the report, and the interval's midpoint must never be presented as though it were the most likely value.
Cases
The first case is DecisionMind's validation example: there is no single, commonly accepted, page-traceable literature example for the grey COPRAS family, so the manifest uses a synthetic, three-alternative, three-criterion table that can be traced by hand. The second case is illustrative fiction.
1. Illustrative example: Three alternatives, three criteria (DecisionMind validation example)
Three alternatives are evaluated on three criteria; the first two criteria are "higher is better," and the third is "lower is better" (cost). Scores are given as grey numbers, ⊗[lower, upper].
| 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 reduces every cell to its midpoint. For A1-C1, for example, (0.65+0.75)/2 = 0.70. It then normalises against the column sum and multiplies by the weight. It gathers the benefit and cost sums, computes the relative significance value (Q), and divides by the highest to convert it into a percentage.
| Alternative | Degree of utility | Rank |
|---|---|---|
| A2 | 100.00 | 1 |
| A3 | 92.75 | 2 |
| A1 | 85.23 | 3 |
The result reads as follows. A2 has the highest midpoint on C1, the most heavily weighted criterion (0.80), and the lowest cost midpoint on C3 (0.40); its lagging behind A3 on C2 (0.60 against 0.70) is not enough to close these two advantages. A1 comes last, being lowest on C1 and most expensive on C3.
The decision's hesitation: if C1's weight is lowered from 0.40 to 0.30 and C2's raised from 0.35 to 0.45 (with C3 held at 0.25), A2 still comes first at 100.00, but A3 rises to 98.18. The gap narrows to 1.82 points. If the weight is shifted one step further in C2's favour, to C1=0.20 and C2=0.55, A3 moves ahead at 100.00 and A2 falls to 96.33. The ranking reverses. This shows how much the relative weight of C1 and C2 determines the ranking.
In the report: "With the given weights (C1=0.40, C2=0.35, C3=0.25), A2 has the highest degree of utility (100.00); its gap over A3 (92.75) closes, and the ranking reverses, if C2's weight is raised enough to overtake C1's (C1=0.20, C2=0.55)."
Source: DecisionMind's Grey COPRAS manifest, validation example; the steps follow the grey-number COPRAS definition of Zavadskas, Kaklauskas, Turskis and Tamošaitienė (2009). The degrees of utility and the sensitivity-scenario figures were obtained by independently re-running the DecisionMind engine's steps in this manifest.
2. Energy: Choosing a new solar panel supplier
An energy company must choose among three panel suppliers for a rooftop power-plant project. The criteria are: expected annual energy yield, maintenance cost over the panel's lifetime (the last one "lower is better"), and delivery time (also "lower is better"). Because the company has not yet run a field trial, it cannot give a "most likely" value for any supplier; only a lower and an upper bound can be read from each supplier's technical datasheet and from two pilot installations.
The method reduces each supplier's interval on every criterion to its midpoint, normalises against the column sum, and multiplies by the weights. It gathers the weighted yield value into the benefit sum, and maintenance and delivery time into the cost sum. It then collapses this to a single degree of utility. Suppose the supplier with the highest yield interval also has the widest maintenance-cost interval, and still comes out first, because the weight on yield has been set higher than that on maintenance.
The company's hesitation: for the supplier with a wide maintenance-cost interval, the midpoint may be an optimistic estimate. If the upper end of the interval is realised, does the overall cost advantage reverse? This question cannot be answered by looking at the degree of utility alone. The company should not proceed to a contract without also running a check calculation using the upper bound instead of the midpoint.
In the report: "With the high weight given to expected yield, the most efficient panel reaches the highest degree of utility; because the maintenance-cost interval is wide, this advantage rests only on the midpoint scenario and should be separately tested against the upper-bound scenario."
3. What Not to Do
Had C3 (cost) been marked as "higher is better" in the illustrative example, the most expensive alternative would have been included in the benefit sum; the ranking would not change, but the scores would cluster together, something like 100.00 / 99.41 / 98.46, concealing A2's real advantage (100.00 / 92.75 / 85.23). The second error is using only the upper bound (optimistic) instead of the interval's midpoint as though it were the "true" value, and fixing it as such. In this example A1's degree of utility falls from 85.23 to 81.93, and A3's from 92.75 to 90.28; the ranking does not change, but the result becomes dependent on which bound was chosen, and this choice must not be concealed in the report. The third error is entering a lower bound greater than the upper bound in a cell, or fabricating bounds with no basis at all, for example by adding a percentage margin around a measured number. In that case the division-by-column-sum step produces a meaningless result.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/grey-copras
Zavadskas, E. K., Kaklauskas, A., Turskis, Z., & Tamošaitienė, J. (2009). Multi-attribute decision-making model by applying grey numbers. Informatica, 20(2), 305–320. DOI: 10.15388/informatica.2009.252
Zavadskas, E. K., & Kaklauskas, A. (1996). Determination of an efficient contractor by using the new method of multicriteria assessment. In International Symposium for the Organization and Management of Construction: Shaping Theory and Practice, Vol. 2, 94–104. (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