Extension card · Linguistic
2-tuple linguistic SAW (Cid-López et al., 2018)
This is the form of SAW for situations where expert scores are chosen from a pre-declared term set, and where the aggregation result is preserved with its shift rather than rounded to a term.
Base method
SAW →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Linguistic →
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?
Cells. In crisp SAW every cell is a single number. Here every cell consists of a term and a shift. An expert score is chosen from the term set; if the aggregation result falls between two terms, it is not rounded to the nearest one but kept with its shift. Criterion weights are crisp numbers; the method does not generate weights.
Scale equalisation. Crisp SAW scales every column against its own best value, so that the best alternative scores exactly 1 on every criterion. 2-tuple linguistic SAW does not carry out this scaling. Every term, together with its shift, is converted directly into a number, and no further best-value scaling is run on this number; criterion direction is applied by reversing the number's sign on cost criteria before summation. This is a point of difference from both crisp SAW and 2-tuple linguistic TOPSIS and VIKOR: those two methods have at least an ideal/anti-ideal or a best/worst scaling, whereas this method has none.
Weighted sum. The method signs each term's numerical equivalent according to the criterion's direction, multiplies it by the criterion's weight, and sums this across alternatives. This sum is the direct counterpart of crisp SAW's weighted sum; the only difference is that the input has moved from a term to a number.
Result and defuzzification. The total score here is again a single number, but it does not sit between 0 and 1 as it does in crisp SAW, because there is no best-value scaling step; the score is directly the weighted sum of the terms. The transformation that converts a term into a number is one-to-one and reversible; a result that falls between two terms is kept together with its shift.
DecisionMind fixes, for classical 2-tuple linguistic SAW, a nine-level term set, the absence of a separate scale-equalisation step, and the term-to-number conversion. Weights are taken from outside as crisp numbers.
How to Read the Output
The total score gives an alternative's weighted average; unlike crisp SAW, it is not a percentage or a probability. The difference is here: because there is no best-value scaling step, the size of the score depends less on the alternative set than it does in crisp SAW, but the assumption of full compensation applies just as it does there. A serious weakness on one criterion can be directly offset by strength on another criterion whose weight is large enough.
Thus instead of writing:
"2-tuple linguistic SAW found the best alternative"
the report should read:
"Under these weights and the assumption of full compensation, this is the alternative with the highest weighted sum; the ranking is sensitive to a one-level change in a certain criterion"
When to Prefer This over the Base Method
Use this extension when experts assess a criterion with a word chosen from a term set, when the aggregation result needs to be preserved without rounding to a term, and when a fast, hand-traceable ranking is sufficient.
It should not be confused with the other member of the same family. If an expert gives not a single term but several terms with probabilities, the probabilistic linguistic structure is used; the probabilistic linguistic member of SAW in this family is a separate extension. If the expert has an "approximately this number" in mind, fuzzy SAW is appropriate.
The case for staying with the base method is unchanged: a measured criterion stays measured. SAW's assumption of full compensation is unsuitable for decisions where no compromise is acceptable on one criterion, and this does not change for linguistic data either.
Mistakes Specific to This Extension
Dropping the shift and rounding to the nearest term. Dropping the shift before the weighted sum defeats the method's property of not losing information.
Carrying crisp SAW's best-value scaling step over here. 2-tuple linguistic SAW sums the terms directly; adding a best-value scaling step afterwards defines a different method and must be stated separately in the report.
Forgetting the sign on a cost criterion. On a "lower is better" criterion, the term's numerical equivalent enters the sum with a negative sign; if the sign is forgotten, the high-cost alternative appears advantaged.
Letting the term set differ from expert to expert. The number of levels and their order must remain the same for every expert and every alternative.
The governing principle is this:
2-tuple linguistic SAW exists to carry the shift produced when terms are aggregated without losing information; its simplicity is a virtue, but any application that forgets the assumption of full compensation or the cost-criterion sign produces a wrong ranking.
Cases
The first case is DecisionMind's validation example. In the manifest, this 3x3 table is recorded as a synthetic fixture with no page reference; the figures have been independently recomputed with the Δ and Δ⁻¹ formulas. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Choosing among three course providers
An organisation will choose among three course providers for staff training. Criteria: the instructor's subject knowledge, how current the content is, and the course fee; the last is a "lower is better" criterion. Human-resources staff score each provider with a word chosen from the term set; the term set has nine levels: very low, low, quite low, slightly low, medium, slightly high, quite high, high, very high.
| Provider | Subject knowledge | Content currency | Course fee |
|---|---|---|---|
| A1 | quite high, slight downward shift | medium | slightly high, slight downward shift |
| A2 | quite high, slight upward shift | slightly high, slight downward shift | slightly low, slight upward shift |
| A3 | slightly high, slight downward shift | quite high, slight downward shift | medium |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method converts every term, together with its shift, into a number, multiplies it by the weight as is for benefit criteria and with a reversed sign for the cost criterion, and sums.
| Provider | Total score | Rank |
|---|---|---|
| A2 | 3.44 | 1 |
| A3 | 2.88 | 2 |
| A1 | 2.44 | 3 |
The result reads as follows. A2 sits at the top level on subject knowledge, the heaviest criterion, and also at the lowest level on course fee; these two together carry it ahead of the others. A1 sits at the same level as A2 on subject knowledge but remains at the lowest level on content currency, and this places it last.
The organisation's hesitation is this: if A1's course fee fell by one level, that is, from slightly high to quite low, A1's score would rise from 2.44 to 2.69. A3 would still stay ahead, but the gap would fall from 0.44 to 0.19. The report should state that the order of these two providers is sensitive to a single one-level change in the fee bid.
In the report: "Expert scores were chosen from a nine-level term set, and the aggregation result was preserved with its shift. A2 is ahead with a score of 3.44. The gap between A3 and A1 is sensitive to a single one-level change in the course fee."
Source: DecisionMind's L2T-SAW validation example. The term set and the training scenario were constructed for this card; the figures were taken from the manifest's synthetic fixture and independently recomputed with the Δ and Δ⁻¹ formulas.
2. Waste management: Choosing among three recycling-facility operating bids
A municipality will choose among three bids to operate a recycling facility. Criteria: the facility's processing capacity, sorting quality and operating cost; the last is a "lower is better" criterion. A municipal council committee assesses each bid with a word chosen from the term set.
The method converts every term into a number, multiplies benefit criteria as is and the cost criterion with a reversed sign by the weight, and sums. Say the bid with the highest processing capacity also turned out to be the most expensive; because capacity is the heaviest criterion here, its advantage there outweighed its cost disadvantage in the sum and moved this bid into first place.
The committee's hesitation is this: if there is a budget ceiling, the method does not respect it on its own, because cost is summed directly with the other criteria. Any bid exceeding the budget should be set aside before the analysis, and the remaining bids ranked with this method.
In the report: "Bid scores were chosen from the term set and summed. The bid with the highest capacity is ranked first. This result should not be used directly unless a budget ceiling is applied as a separate pre-screening step."
3. What Not to Do
Taking A1's course-fee term in the illustrative example and scaling it against the best value as in crisp SAW before summing: this is not the method that 2-tuple linguistic SAW defines, and it produces a different score. The second error is forgetting the negative sign when summing the course-fee criterion; in that case the most expensive provider wrongly appears advantaged. The third error is allowing one expert to use a nine-level term set and another a five-level one; the number of levels must be the same for every expert.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/l2t-saw
Cid-López, A., Hornos, M. J., Carrasco-González, R. A., & Herrera-Viedma, E. (2018). Prioritization of the launch of ICT products and services through linguistic multi-criteria decision-making. Technological and Economic Development of Economy, 24(3), 1231–1257. DOI: 10.3846/tede.2018.1423
Herrera, F., & Martínez, L. (2000). A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8(6), 746–752. DOI: 10.1109/91.890332
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
Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning—I. Information Sciences, 8(3), 199–249. DOI: 10.1016/0020-0255(75)90036-5