Extension card · Hesitant
Hesitant fuzzy linguistic CODAS (Yalçın and Pehlivan, 2019)
Hesitant fuzzy linguistic CODAS is the form of CODAS used when a criterion is scored not with a single verbal term but with a comparative verbal expression such as "at least good" or "between medium and good." It converts this expression into a fuzzy envelope and carries it as such through most of the calculation.
Base method
CODAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Hesitant →
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?
Four things change; the two-distance comparison logic stays the same.
Cells. In crisp CODAS every cell is a single number. Here every cell is a comparative verbal expression. A seven-term scale is used: N (none), VB (very bad), B (bad), M (medium), G (good), VG (very good), P (perfect). Criterion importance is likewise given verbally, on a separate seven-term scale: DL (very low), VL (low), L (somewhat low), M (medium), H (high), VH (very high), DH (highest). Every expression is first read as a set built from consecutive terms, then converted into a four-cornered envelope through Liu and Rodríguez's (2014) OWA operations. The method does not directly support a group decision; DecisionMind expects a single, already-aggregated expression matrix.
Scale equalisation. Crisp CODAS equalises every cell by dividing it by the column's largest value or by the ratio to its smallest. Division is also used here, but the divisor is the largest defuzzified (kappa) value in the column; all four corners of the envelope are divided by this single number. On a cost criterion, the envelope is reversed after this division: every corner is subtracted from one, and the corners' order is reversed.
Distance. In crisp CODAS, the negative-ideal is artificially constructed from each column's worst value. Here, the negative-ideal is the weighted envelope of the actual alternative that has the lowest defuzzified value on that criterion; it is not artificially assembled from extremes. Euclidean and city-block distances are calculated over the envelope's four corners. The inner corners, that is the most likely band, are given twice the weight of the outer corners.
Result and defuzzification. The assessment score is built with the same pairwise comparison rule as crisp CODAS (threshold θ, Euclidean first, city-block if that is not enough), and is already a single number. The envelope stays four-cornered until it enters the distance calculation; defuzzification is used only as an intermediate step, in scale equalisation and in choosing the negative-ideal.
For classical hesitant fuzzy linguistic CODAS, DecisionMind fixes the OWA smoothing operations and the threshold value (θ = 0.02).
How to Read the Output
The assessment score is read exactly as in crisp CODAS: it is a relative measure of standing, not a percentage, and it cannot be compared with a different analysis. The difference is this: the score comes from the four corners of envelopes built from comparative verbal expressions. When the verbal expression on a criterion narrows from "at least good" to "between medium and good," the envelope changes and the score shifts accordingly.
Thus instead of writing:
"The hesitant fuzzy linguistic CODAS score is an exact measure that directly reflects the expert's verbal expression"
the report should read:
"The score is the result of converting the verbal expression into a Liu-Rodríguez envelope and comparing this envelope against its kappa value; narrowing or widening the expression can change the score"
When to Prefer This over the Base Method
This extension is used when the expert gives a criterion not as a single verbal term but as a comparative expression such as "at least," "at most" or "between." Where a single verbal term is enough (just "good," say), a simpler linguistic extension may suffice instead. Measured criteria should not be carried into this extension. Where the matrix must hold a single data type, a measured value is written as the single nearest term on the scale. Base CODAS's exit condition applies here exactly as before.
Mistakes Specific to This Extension
Confusing a midpoint interval with the envelope. Taking an expression's lowest and highest term and treating the midpoint between them as the "envelope" is not the same as the genuine envelope built through Liu and Rodríguez's OWA operations. The two give different corner values.
Missing a cost criterion. In the illustrative example below, one criterion is of the cost type. Skipping the reversal on this criterion ties the negative-ideal to the wrong alternative, and swaps the two middle-ranked alternatives.
Using ready-made distance values from a published paper as input. Euclidean and city-block distances are this method's output, not its input. Injecting them from outside hides how the envelopes and weights actually combine.
Accepting a number in a printed table without question. Some of the source paper's own printed cells do not fully match the OWA formula the paper itself states. The engine stays faithful to the formula; where a difference comes from must be stated in the report.
The governing principle is this:
Hesitant fuzzy linguistic CODAS exists to convert the comparative verbal expression into a genuine envelope and carry that envelope through to the end. Using a crude midpoint instead of the envelope, or treating a published table's intermediate results as input, does not corrupt the calculation itself, only its apparent agreement with the source.
Cases
The first case is a genuine case from the literature: it is Yalçın and Pehlivan's (2019) personnel-selection example, and the verbal expressions are taken from the paper's own table (Table 3). The engine processes these expressions faithfully to the formula. Some of the paper's own printed intermediate-table cells (Tables 6-7) do not fully match its own OWA formula, whereas the engine produces a result consistent with that formula; this difference is stated explicitly below. The second case is an illustrative construction.
1. Recruitment: Selecting personnel among six candidates (Yalçın and Pehlivan, 2019)
An organisation will choose one of six candidates (A1-A6) against eleven competency criteria. All criteria except C5 are of the higher-is-better kind; C5 (a negative indicator such as a tendency towards inappropriate behaviour) is of the lower-is-better kind. Candidates are assessed on every criterion with a comparative verbal expression rather than a single term.
| Candidate | C1 | C2 | C3 | C4 | C5 (cost) | C6 |
|---|---|---|---|---|---|---|
| A1 | M | at least VG | at least M | at least G | between M and G | at least G |
| A2 | M | at least G | between M and VG | M | between B and M | at least G |
| A3 | between M and VG | at least G | M | M | between B and M | between M and VG |
| A4 | between M and VG | between G and VG | G | at least G | M | between M and VG |
| A5 | between M and VG | at least VG | M | between M and VG | between M and G | between M and VG |
| A6 | M | between B and G | M | between M and VG | M | between M and G |
| Candidate | C7 | C8 | C9 | C10 | C11 |
|---|---|---|---|---|---|
| A1 | at most M | at least G | between M and VG | M | at least G |
| A2 | between VB and M | at least G | between M and G | between M and G | at least G |
| A3 | between B and G | between G and VG | between M and G | between M and VG | at least G |
| A4 | between B and G | at least VG | between G and VG | between M and VG | between G and VG |
| A5 | between VB and G | at least G | between M and VG | M | at least G |
| A6 | between VB and M | G | M | between B and M | between M and G |
| Criterion importance | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Verbal expression | between L and H | DH | at least L | between M and VH | at most VH | at least M | between L and M | between H and VH | at least H | between L and VH | at least VH |
The method converts every verbal expression into a set built from consecutive terms, then into a four-cornered envelope through the Liu-Rodríguez OWA operations. It divides the envelopes by the largest defuzzified value, reverses C5, and multiplies corner by corner with the criterion-importance envelopes. It chooses the envelope of the candidate with the lowest defuzzified value as the negative-ideal, and calculates the Euclidean and city-block distances, giving more weight to the inner corners.
| Candidate | Assessment score | Rank |
|---|---|---|
| A4 | 8.3360 | 1 |
| A1 | 5.7160 | 2 |
| A2 | 1.8712 | 3 |
| A5 | 1.4335 | 4 |
| A3 | 0.2671 | 5 |
| A6 | -17.6237 | 6 |
The result reads as follows. A4 has the highest envelope on C2 (between G and VG) and strong expressions such as at least VG on C8. Both these criteria also carry high importance; DH, for C2, is the highest importance term on the scale. A6 finishes last, with medium or lower expressions on almost every criterion. Some of the source paper's own printed intermediate values in Tables 6-7 do not fully match the OWA formula the paper states. The engine has stayed faithful to the formula, so A4's score in particular comes out different from the number the paper prints. Both ends of the ranking, A4 first and A6 last, are preserved.
To see how robust the decision is, the criterion-importance envelopes were changed and the calculation rerun. C1's (communication) importance was raised to roughly 1.5 times its original value, and C9's (teamwork) importance was lowered to half in return. In that case, A2 in third place and A5 in fourth swap: A5 comes out at 1.619 and A2 at 1.536. A4's first place and A6's last place are not disturbed by this change.
In the report: "With the eleven criteria's stated verbal importance, A4 has the highest assessment score (8.34); the gap between third-placed A2 and fourth-placed A5 is sensitive to the relative importance of the communication and teamwork criteria. There is a small numerical difference from some of the source paper's intermediate table values; this difference comes from the engine staying faithful to the formula, and it is separately documented."
Source: Yalçın and Pehlivan (2019), Table 3 (verbal-expression matrix) and Tables 4-5 (envelopes). The assessment scores and the ranking were obtained by independently rerunning DecisionMind's hesitant fuzzy linguistic CODAS engine on these envelopes. There are small differences from the paper's own printed Tables 6-7 intermediate values; these arise because some of the paper's printed cells do not fully match its own stated OWA formula (details in the verification notes).
2. Care services: Choosing a maintenance-service provider for a nursing-home chain's new branch
A nursing-home chain will contract with one of three care-service providers for a branch it is about to open. The criteria are: the experience of the nursing staff, emergency-response time, and unit cost of service (the last of these is of the lower-is-better kind). The chain's management board has assessed the providers purely verbally, from previous references, using comparative expressions such as "at least good" or "between medium and good."
The method converts the three providers' expressions into envelopes, weights them with the criterion-importance envelopes, and chooses the envelope of the provider with the lowest defuzzified value as the negative-ideal. Suppose the provider with the strongest expression on experience also has the highest unit cost. It nonetheless comes out first, because the experience criterion carries higher importance than cost.
The management board's hesitation is this. Choosing the most expensive provider must be justified to the budget committee on grounds of cost. It should also be tested separately whether the ranking changes if a broad expression such as "at least good" is narrowed to a tighter one such as "between good and very good."
In the report: "Given the high importance placed on the experience criterion, the most experienced provider reaches the highest assessment score; this result holds despite the provider's cost disadvantage, and should be retested if the experience expression narrows."
3. What Not to Do
In the illustrative table, had C5 been mistakenly counted as a benefit criterion, that is, had the reversal step been skipped, third-placed A2 and fourth-placed A5 would swap (A5 at 2.929, A2 at -0.223); in the correct order, A2 is third and A5 fourth. A4's first place and A6's last place do not change under this error either, but the middle of the ranking is disturbed. The second error is using the paper's printed Euclidean and city-block distances directly as input; this skips how the envelopes and weights actually combine, and does not verify the engine's own calculation. The third error is mistaking the midpoint of an expression's lowest and highest terms for the envelope; this gives a different number from the genuine corner values produced by the Liu-Rodríguez OWA operation.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/hfl-codas
Yalçın, N., & Pehlivan, N. Y. (2019). Application of the Fuzzy CODAS Method Based on Fuzzy Envelopes for Hesitant Fuzzy Linguistic Term Sets: A Case Study on a Personnel Selection Problem. Symmetry, 11(4), 493. DOI: 10.3390/sym11040493
Keshavarz Ghorabaee, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2016). A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Economic Computation and Economic Cybernetics Studies and Research, 50(3), 25–44. (no DOI. This article is not registered in Crossref; see the Sources section of the base CODAS card.)
Liu, H., & Rodríguez, R. M. (2014). A fuzzy envelope for hesitant fuzzy linguistic term set and its application to multicriteria decision making. Information Sciences, 258, 220–238. DOI: 10.1016/j.ins.2013.07.027
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418