Extension card · Grey
Grey SAW
Grey SAW is the form of SAW that works with grey numbers when criteria are known only by a lower and upper bound. It whitenises every cell, then applies crisp SAW's weighted sum.
Base method
SAW →
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?
Two things change; the decision logic does not.
Cells and scale equalisation. In crisp SAW every cell is a single number, and every column is scaled against its own best value. Here every cell is a grey number: only a lower (⊗v^L) and an upper (⊗v^U) bound; there is no most-likely value inside. DecisionMind's engine merges scale equalisation and whitenisation, that is, the reduction of the grey number to a single number, into one step. Every column is scaled against the column's largest upper bound if it is "more is better," or against the column's smallest lower bound if it is "less is better." The two ratios coming from the lower and upper bound are then averaged to give a single equalised, whitenised value. Crisp SAW has a single scaling step; here this one step both equalises the scale and reduces the lower and upper bound to a single number.
Weighted sum. The whitenised and equalised column is multiplied by the criterion weight and summed. Everything after this is exactly the same as crisp SAW.
DecisionMind holds this whitenisation rule (the average of the ratios from the lower and upper bound) fixed in classic Grey SAW; a different whitenisation rule (the midpoint alone, the upper bound alone) can give a different ranking.
How to Read the Output
The total score is a weighted average, as in crisp SAW, and is read the same way: it is not a percentage, it cannot be compared with a different analysis, and full compensation applies.
The difference is this. The score has been produced by reducing a quantity known only by its bounds to a single number. Since it is not known which value within the bounds is more likely, this reduction is an assumption (the equally weighted average of the lower and upper bound); a different whitenisation rule can give a different score, even a different ranking. This is why the report should state not only the score but also the width of the bounds and the score's sensitivity to the whitenisation rule.
Thus instead of writing:
"This alternative is certainly the best because its Grey SAW score is highest"
the report should read:
"This score is a value known only by its lower and upper bound, reduced to a single number by a particular whitenisation rule; for criteria where the bounds are wide, this reduction is a cruder assumption"
When to Prefer This over the Base Method
Use this when only a lower and upper bound are known about a criterion, and no value in between is considered more likely than another. Opening a single, measured and defensible value into a grey bound, for instance turning 40 into ⊗[35,45], is not modelling uncertainty, it is manufacturing it. DecisionMind requires a single data type; if the table is mixed, a measured value is also written as a grey number with the lower and upper bound equal.
SAW's exit condition applies here in exactly the same way: if a criterion allows no compromise, this extension too carries a full-compensation logic and will not eliminate anything below a threshold.
Mistakes Specific to This Extension
The lower bound being greater than the upper bound. In every cell the lower bound must be less than or equal to the upper bound (the GIN constraint); the reverse invalidates the calculation.
Assuming the midpoint of a grey number is a most-likely value. A grey number does not carry this information; whitenisation is an assumption, not a value read from the data.
Changing the whitenisation rule and expecting the same result. Using only the upper bound or only the lower bound instead of the average of the lower and upper bound produces a different score; which rule was used should be stated in the report.
Discarding a known distribution or most-likely value within the bounds and descending to grey. If more information than this is actually available (a most-likely value or a distribution), reducing it to a grey bound erases the available information; a fuzzy or probabilistic structure would be more accurate.
The governing principle is this:
Grey SAW exists to honestly carry a quantity known only by its bounds; whitenisation is a computational step, and which rule was used must be clearly stated in the report.
Cases
The first case is DecisionMind's verification example: the manifest attaches no page-level literature source to this 3×3 grey-interval table; it was built as a fiction faithful to the formulas and traceable by hand. The second case is an illustrative fiction.
1. Illustrative example: Three energy-storage technology candidates compared on three criteria (DecisionMind verification example)
A firm will choose a new energy-storage technology; the technologies have so far only been tested in a limited number of pilot deployments, and only the lowest and highest value observed is known for each criterion. There are three criteria: efficiency (C1, more is better), flexibility (C2, more is better), maintenance cost (C3, less is better).
| Candidate | C1 | C2 | C3 |
|---|---|---|---|
| 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 scales and whitenises each column (against the column's largest upper bound for C1 and C2, against the column's smallest lower bound for C3), then multiplies by the weight and sums.
| Candidate | Score | Rank |
|---|---|---|
| A2 | 0.8787 | 1 |
| A3 | 0.7858 | 2 |
| A1 | 0.7096 | 3 |
The result reads as follows. A2 holds the widest upper bound on both efficiency and flexibility, and the lowest maintenance cost; being superior to the others on every criterion makes it clearly first. A3, although best on flexibility, falls behind A2 on efficiency, which puts it second.
The firm's hesitation is this: the weights are debated within the committee; what if flexibility (C2) were considered more important than efficiency (C1)? If the weights were redistributed to C1=0.30 and C2=0.45, A2 would still hold first place at 0.8646; this has been verified by an independent Python computation. A2's lead is solid against a weight swap of this size. Conversely, if C2 were given an excessively high weight of 0.65 or above, A3 would move ahead; this lies outside a realistic weight range.
In the report: "A2 is first with a score of 0.8787, holding the widest upper bound on efficiency and flexibility and the lowest maintenance cost; this ranking is robust against redistributing the weights within a reasonable range."
Source: DecisionMind Grey SAW manifest, verification example; the bounds and weights were built synthetically, and the calculation steps follow Deng's (1982) grey-system framework and Fishburn's (1967) weighted-sum definition. Since the manifest gives no page-level literature source for this table, it is presented as an illustrative example.
2. Sport: Assessing a new contract candidate by performance criteria
A sports club will make an offer to one of three player candidates before the transfer window closes. Criteria: seasonal goal/assist contribution, injury history (less is better), and team-fit potential. Two of the candidates come from other leagues, so their statistics are available only as a "worst season" to "best season" range; the club does not want to give a single "expected" figure.
The method scales and whitenises the three candidates against their own column bounds, and computes the weighted sum. Suppose the candidate with the widest contribution range, but also the widest injury-history range, comes first, and the candidate with a narrower contribution but almost no injury history comes second, with a small gap between them.
The club's hesitation is this: if the upper end of the first candidate's injury-history bound, that is, the worst-case scenario, materialises, this cost item could strain the budget. Grey SAW only processes the given bounds and weights; how the injury risk is reflected in the contract terms is a separate decision and is not included in the method's score.
In the report: "The ranking rests on the three candidates' observed performance and injury-history bounds; if the wide upper end of the first candidate's injury history materialises, an additional budget risk arises that is not reflected in the score."
3. What Not to Do
The first error is interpreting the midpoint of A2's efficiency bound, 0.80, as "most-likely efficiency" and adding a separate comment based on it; a grey number does not carry this information, and the midpoint has no special meaning. The second error is scaling the C3 column (maintenance cost, less is better) against the largest upper bound as though it were "more is better"; this makes the most expensive candidate look best and reverses the ranking. The third error is averaging the bounds first (for instance turning A1's C1 into 0.70) and running crisp SAW; this silently erases the information carried in the width of the bounds, for example the fact that A1's C1 is noticeably narrower than A2's.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/grey-saw
Fishburn, P. C. (1967). Additive utilities with incomplete product sets: Application to priorities and assignments. Operations Research, 15(3), 537–542. DOI: 10.1287/opre.15.3.537
Deng, J. (1982). Control problems of grey systems. Systems and Control Letters, 1(5), 288–294. DOI: 10.1016/S0167-6911(82)80025-X