Data types
Grey
This is the data structure that preserves the situation in which only a lower and an upper bound of a value are known, and no value between them is taken as more likely or more reasonable than another.
Example cell: 3, 5
What Is It?
A grey data structure expresses an alternative's standing on a criterion not as one number but as two bounds: at least this much, at most that much. "Maintenance cost between 30 and 50 thousand TL" is the grey number ⊗[30, 50]. No most-likely point, distribution or membership grade is defined within the bounds; the information consists of the bounds alone.
The name "grey" comes from grey systems theory: a situation where information is fully known is called "white", one where it is entirely unknown is "black", and one where it is partly known is "grey". A grey number is the simplest form of partial knowledge: where data are scarce, a record is short, or a measurement could not be completed in full, a lower and an upper bound are often the only reliable information left.
When to Use It
Grey data are used wherever nothing beyond the bounds is known about a value. They are valuable in problems with little data, in evaluations of a new product or a new market, in supplier quotes given as "somewhere between this and that", and in expert judgements that can state "at least, at most" but not "most likely".
Conversely, if a most-likely value exists within the bounds, a fuzzy structure suits better; if the bounds are the ends of a probability distribution, a stochastic structure does. Dropping to a grey structure while deliberately discarding information held within the bounds erases knowledge that is actually available.
Can Grey Data Be Built from Crisp Data?
Yes, but the bounds must have their own source. Two steps are needed.
The first is establishing what the value represents: a single measured quantity, or one not yet realised, or one that could not be fully observed? A measured quantity stays crisp. An unrealised quantity (a future cost, future demand) or a partly observed one (an incomplete record) may move to a grey structure.
The second is justifying the lower and upper bounds separately: the lowest and highest price in a contract, the best and worst value in a past record, or the two points an expert states as "certainly no lower than this, no higher than that". The bounds are read from a source, not guessed.
What must not be done is building a grey number by adding a fixed percentage either side of a precise figure. Writing 2,450 TL as ⊗[2,400, 2,500] has no source behind it.
A Grey Number Is Not the Same as a Fuzzy Number
A fuzzy number defines a most-likely value within its bounds and states that plausibility declines from that value towards the extremes. A grey number states nothing of the kind: every value within the bounds is equally possible, because nothing more is available.
For this reason:
"I don't know the most likely value, so I'll take the midpoint"
should give way to:
"If I don't know the most likely value, I keep the bounds and declare no midpoint"
Treating the midpoint of a grey number as its most likely value assumes knowledge that does not exist. Reducing a grey number to a single value at the results stage (whitenisation) may be needed; that is a calculation step, stated in the report, not something added to the data from the outset.
Strengths
The chief advantage of the grey structure is that it works honestly with little information. Where only bounds are available, the structure asks for exactly that much; it does not require inventing a distribution or a most-likely value that does not exist.
The assessment burden is also low. Asking an expert for "at least and at most" is easier than asking for "most likely" or a probability, and yields more consistent answers. In data-scarce problems, this is what gives the structure its practical value.
Limitations
A grey number carries no information about what lies within its bounds; where such information exists, the structure discards it. Comparing two grey numbers is not governed by a single rule: for grey numbers whose bounds overlap, the rule chosen can change the ranking.
Operations on grey numbers widen their range; across multi-step calculations, the resulting interval can be far wider than the inputs. There is more than one way to reduce a result to a single value (whitenisation), and the choice affects the outcome.
Interval Data and the Grey Number
The two are identical in form: two bounds. The difference lies in the context of use. An interval is a mathematical object stating that a value lies, with certainty, within those bounds. A grey number uses the same form in a "partial information" context, where knowledge is incomplete and may later narrow as new information arrives; grey-system methods work with the tools proper to that context (whitenisation, grey relational analysis).
In practice, both are entered as a lower and an upper bound. Which family of methods applies depends on whether the bounds express certainty or partial knowledge.
Common Mistakes
The most frequent mistake is reading the midpoint of a grey number as its most likely value. A grey number carries no such information.
The second is building a grey number by adding a percentage around a measured value; if the bounds have no source, there is no grey number either. The third is dropping to a grey structure while discarding a distribution or a most-likely value that is in fact already known; the information available is thereby erased.
The governing principle is this:
Both bounds of a grey number must be read from a source; no information that is not actually available — a most likely value, a distribution — should be added inside the bounds afterwards.
Examples
Each example opens with a familiar, single precise figure and shows the conditions under which, and the steps by which, that same figure moves into grey form.
1. Economics: This Year's Demand Was 48,000 Units
A precise figure. A product's sales this year came to 48,000 units. This has already happened and been counted; it does not become grey.
Step 1: What does the value represent? The decision is between three production-capacity options; the criterion is "next year's demand". This is a quantity that has not yet occurred.
Step 2: Derive the bounds from their own source. An industry report puts next year at "somewhere between an 8 per cent contraction and 12 per cent growth"; the annual change over the past five years has not strayed outside this band either. There is no view on a most likely value.
Grey form. The "next year's demand" criterion holds ⊗[44,200, 53,800]. No midpoint is declared; every value between the two bounds is equally possible.
Same figure, different situation. If the report instead says "most likely 50,000", the information exceeds the bounds and the structure moves to fuzzy: (44,200, 50,000, 53,800). If a demand distribution can be estimated from past years, a stochastic structure is used. The grey structure is the honest form for when only bounds are available.
2. Construction: The Last Project's Cost per Square Metre Was 18,500 TL
A precise figure. The realised cost per square metre of the last completed project was 18,500 TL, a precise figure from the accounts.
Step 1: What does the value represent? The decision is between three new projects; the criterion is "expected cost per square metre". Ground surveys are not yet complete and material prices are not yet fixed.
Step 2: Derive the bounds from their own source. The technical office states, based on ground results and price scenarios, that the cost "will not fall below 17,000 nor exceed 22,000", without being able to say which scenario is more likely.
Grey form. The "expected cost per square metre" criterion holds ⊗[17,000, 22,000].
Same figure, different situation. Once the ground survey is complete and one scenario stands out, the technical office might say "most likely 19,000", at which point a fuzzy structure, (17,000, 19,000, 22,000), carries more information. The completed project's 18,500 TL always stays crisp.
3. Agriculture: Last Year's Yield per Decare Was 420 kg
A precise figure. The current variety's realised yield last year was 420 kg per decare, a weighed, precise value.
Step 1: What does the value represent? The decision is between three new varieties; the criterion is "expected yield per decare". The new variety has been sown in only two trial plots in this region.
Step 2: Derive the bounds from their own source. The two plots yielded 380 and 470 kg. Deriving a mean or a distribution from two observations is not meaningful; only an observed lower and upper value are available.
Grey form. The "expected yield per decare" criterion holds ⊗[380, 470].
Same figure, different situation. If the same variety has been sown across eight plots over three years, a mean and a distribution can be calculated; the grey structure would then discard information, and a stochastic structure, or (where distinct conditions such as dry, normal and wet years are meaningful) a hesitant structure, would be more appropriate.
4. Contract: An Annual Maintenance Fee of Between 30 and 50 Thousand TL
A precise figure. A supplier's maintenance contract states "between 30 and 50 thousand TL a year, depending on intensity of use".
Step 1. The criterion is "annual maintenance cost"; the value that will actually occur depends on usage intensity, which is not yet known.
Step 2. The bounds are read from the contract: 30 at the lower end, 50 at the upper. The contract gives no most likely value, and there is no past record of usage intensity.
Grey form. ⊗[30, 50]. The bounds are entered in the same form for each of the three suppliers, and the system recommends a method suited to grey data.
5. What Not to Do
Building bounds by adding a percentage either side of a measured value such as 48,000 units, 18,500 TL or 420 kg: 420 → ⊗[400, 440]. The bounds have no source. Equally wrong is reading the midpoint of a grey number such as ⊗[30, 50], namely 40, as the "most likely cost"; a grey number carries no such information.
The numbers in the examples are fictional; they are not real data.
Short decision rule
A single, defensible figure → Classical (Crisp)
Only a lower and an upper bound, nothing more known within them → Grey
Bounds express certainty, a mathematical interval → Interval
Bounds plus a most likely value → Fuzzy
Bounds are the ends of a distribution → Stochastic
Bounds are derived from a group of experts' assessments → Rough set
Key sources
Deng, J. L. (1982). Control problems of grey systems. Systems & Control Letters, 1(5), 288–294. DOI: 10.1016/S0167-6911(82)80025-X
Deng, J. L. (1989). Introduction to grey system theory. The Journal of Grey System, 1(1), 1–24. (no DOI)
Liu, S., & Lin, Y. (2011). Grey Systems: Theory and Applications. Understanding Complex Systems. Springer. DOI: 10.1007/978-3-642-16158-2
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