Extension card · Hesitant
Hesitant Fuzzy AHP (Zhu and Xu, 2014)
Hesitant Fuzzy AHP is the form of AHP for situations where a pairwise comparison rests not on a single ratio but on several defensible ratios at once. It converts these ratios directly into a weight vector through linear programming rather than an eigenvector.
Base method
AHP →
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 pairwise-comparison logic does not.
Cells. In crisp AHP every cell is a single number from Saaty's 1–9 scale. Here every cell carries more than one ratio from that same scale; the judgement "X1 is 2 or 3 times more important than X2", for instance, corresponds to the set {2; 3}. This set is different from the membership sets, between 0 and 1, seen on the Hesitant data-type card: here the values sit on a ratio, that is a multiplicative, scale running from 1/9 to 9. The set can also represent the separate ratios given by more than one expert; the method carries these ratios directly into the calculation without first reducing them to a single number. The reciprocal cell is built from the reciprocal of every element in the set.
Synthesis (in place of the eigenvector). Crisp AHP solves the matrix's principal eigenvector. No eigenvalue problem is set up here at all. Every pairwise comparison is turned into a constraint set of the form "the ratio of the weights is at least one of the values in this set". If the set holds more than one ratio, satisfying only one of these constraints is enough; all of them being satisfied at once is not required. DecisionMind makes this choice by trying every possible combination of the set's elements and keeping the combination that gives the highest overall consistency. The resulting weights are already crisp numbers; they are reached by a different route from crisp AHP's eigenvector, but they still arrive directly at a weight vector.
How consistency is handled here. The quantity corresponding to crisp AHP's Consistency Ratio (CR) is here the Group Satisfaction Index (GSI, symbol λ). GSI shows how far the chosen combination can satisfy all the comparisons at once; the closer it is to 1, the more the judgements agree with one another, and a negative value signals a strong contradiction. GSI is not on the same scale as CR and is not read against the same threshold (0.10); DecisionMind's default warning threshold is 0.5. Computing GSI takes a deviation parameter (t); this parameter sets how strictly the constraints are interpreted, and it must always be stated in the report.
Result and defuzzification. The output is directly a crisp weight vector that sums to 1. Unlike Fuzzy AHP, there is no intermediate step in which a fuzzy weight is produced first and defuzzified afterwards. Hesitancy is consumed in the combination search that decides which element is chosen; no uncertainty remains in the output itself, and GSI shows only how hard that search had to work.
DecisionMind holds fixed, in this classical form, the disjunctive search that tries combinations one by one, and the default GSI warning threshold (0.5). The deviation parameter t is chosen by the user.
How to Read the Output
The weight is read as in crisp AHP: it shows a criterion's relative importance within this comparison set alone, and sums to 1. GSI measures something different. Just as a low CR does not show that the judgements are "correct", a high GSI does not show that the decision is sound; it shows only how far the chosen ratio combination can satisfy the comparisons at once.
Thus instead of writing:
"Because GSI came out close to 1, these weights are correct"
the report should read:
"The chosen ratio combination satisfies the comparisons with a high degree of agreement (GSI=0.91); whether the weights are apt still depends on whether the decision-maker judged the criteria correctly, and GSI cannot be compared with crisp AHP's CR threshold (0.10)"
When to Prefer This over the Base Method
This extension is suitable when a comparison cannot be reduced to a single ratio, and this flexibility has a genuine source. Examples: an expert hesitating between two plausible ratios, such as "2 or 3 times"; several experts comparing the same pair with different but defensible ratios. The principle on the Hesitant data-type card applies here too: the set must rest on a genuine source; adding a second ratio to a single one arbitrarily adds no information.
If a comparison is already clear as a single ratio, crisp AHP is sufficient; turning a ratio into a set without justification only adds computational weight. Crisp AHP's exit condition applies here too: it works comfortably with two to twelve criteria, and the comparison burden grows quadratically as the number of criteria increases.
Mistakes Specific to This Extension
Not entering a set's elements in ascending order. The ratios within a set must be ordered from smallest to largest; if they are not, the search still returns a result, but the canonical interpretation of reciprocity (that a criterion's ratio to another, multiplied by the reverse ratio, equals 1) is broken.
Merging a set by averaging (GHMA/GHMWA) and then proceeding with a single ratio. This is a different method from the disjunctive search (HMPM) DecisionMind uses here, and it gives a different weight. The source paper explicitly separates these two routes; they should not be conflated.
Choosing too small a deviation parameter (t). In the source paper's own example, when t=0.05 is chosen, GSI for the same comparison set drops to −0.81; at t=1, the same set gives GSI=0.91. A small t can artificially push GSI negative even for a consistent set; the t value used must be stated in the report.
Reading GSI as though it were a CR threshold. GSI is good the closer it gets to 1, while CR is considered acceptable the lower it falls below 0.10; these are quantities on different scales, read in opposite directions. A sentence such as "consistent because GSI<0.10" is wrong here.
The governing principle is this:
Hesitant AHP weights are the linear-programming solution that gives the highest overall agreement (GSI) for a ratio combination chosen from the comparison sets; GSI must not be confused with crisp AHP's CR, and the deviation parameter used must be stated in the report.
Cases
The first case is a genuine literature case: Example 2 in Zhu and Xu's (2014) paper. The second case is an illustrative fiction.
1. Zhu and Xu's (2014) comparison of four objectives (Example 2, p. 797)
The source paper compares four objectives (X1–X4) in pairs; for some pairs the decision-maker gave more than one plausible ratio rather than a single one. The paper presents these four objectives as an abstract example, not tied to any particular sector, to demonstrate the method's linear-programming solution; this card carries the same abstraction across.
| Comparison | Ratio set |
|---|---|
| X1 / X2 | {2; 3} |
| X1 / X3 | {6; 7} |
| X1 / X4 | {2} |
| X2 / X3 | {2; 3} |
| X2 / X4 | {1/8; 1/7} |
| X3 / X4 | {1/3} |
The deviation parameter is taken as t=1. The method selects one element from each pair's ratio set, finds the combination that maximises the overall agreement of this selection, and produces a weight vector directly.
| Objective | Weight |
|---|---|
| X1 | 0.485 |
| X4 | 0.288 |
| X2 | 0.132 |
| X3 | 0.096 |
GSI comes out at λ=0.9096, clearly above DecisionMind's warning threshold (0.5).
The result reads as follows. X1 dominates the other three objectives (judged 2–3 times more important than X2, 6–7 times more important than X3, 2 times more important than X4). X4 is ranked second because it is judged far more important than X2 (7–8 times).
The decision-maker's hesitation is this: if the two weakest comparisons, X1–X2 and X2–X4, were softened substantially, would the ranking change? Verified with an independent Python computation, once the X2–X4 ratio is pulled from {7; 8} to {2; 3}, the weights become X1=0.496, X4=0.261, X2=0.157, X3=0.087, and GSI rises to 0.974; the ranking (X1≻X4≻X2≻X3) is unchanged. Once the X1–X2 ratio is pulled from {2; 3} to {1; 1.5}, the weights become X1=0.429, X4=0.288, X2=0.188, X3=0.096, and GSI falls to 0.853; the ranking again does not change. Even with these two comparisons substantially softened, the ranking stays solid; only GSI moves.
In the report: "X1 leads with the highest weight (0.485). Even when the two weakest comparisons (X1–X2, X2–X4) are substantially softened, the ranking (X1≻X4≻X2≻X3) does not change; t=1 was used, and GSI is 0.91, above the warning threshold."
Source: Zhu and Xu (2014), Example 2, p. 797 (the ratio sets are carried over verbatim from the paper). The weight vector and GSI were obtained by independently running DecisionMind's engine, and match the result in the paper's Table 2 (p. 797). The number of sensitivity scenarios was separately computed by this card's author with the same engine.
2. Care home: Weighting service-quality criteria
The board of a care-home chain will weight three criteria for use in a cross-branch service-quality audit: quality of medical care, variety of social activities, and comfort of the physical premises. As board members compared the criteria in pairs, some pairs were settled not on a single ratio but on a range they debated.
| Comparison | Ratio set |
|---|---|
| Medical care / Social activities | {2; 3} |
| Medical care / Physical comfort | {1.5; 2.5} |
| Social activities / Physical comfort | {1.2; 1.6} |
The method derives a weight vector directly from these three comparisons.
| Criterion | Weight |
|---|---|
| Quality of medical care | 0.527 |
| Variety of social activities | 0.261 |
| Comfort of physical premises | 0.213 |
GSI comes out at λ=0.9947; the board's judgements largely agree with one another.
The board's hesitation is this: the comparison between social activities and physical comfort put social activities slightly ahead in the first round. What happens if this comparison is revisited and pulled from {1.2; 1.6} to {0.8; 1}? This would mean social activities are now seen as slightly behind, or equal to, physical comfort. Verified with an independent Python computation, the weights become medical care 0.577, physical comfort 0.233, social activities 0.191, and the order of the last two criteria reverses. Medical care's first place does not change in either case.
In the report: "Quality of medical care leads with the highest weight (0.527), and this first place is solid. The order between social activities and physical comfort is sensitive to how that comparison is reviewed; in the second round this comparison was revisited, and physical comfort moved ahead."
3. What Not to Do
In the care-home example, fixing the comparison between social activities and physical comfort at {1.2; 1.6} based on the view of only one board member is wrong. If the other members' objections are not recorded, the genuine source of the hesitancy carried by the set is concealed. The second error is reporting GSI=0.9947 as "proof the criteria were correctly chosen"; GSI shows only how far the judgements agree with one another, not whether the criteria themselves are apt. The third error is sharing the weights without stating the deviation parameter t; a small value such as t=0.05 can give the same comparison set a much lower, even negative, GSI, and the reader would have no way of knowing why.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/hf-ahp
Zhu, B., & Xu, Z. (2014). Analytic hierarchy process-hesitant group decision making. European Journal of Operational Research, 239(3), 794–801. DOI: 10.1016/j.ejor.2014.06.019
Saaty, T. L. (1980). The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation. McGraw-Hill. ISBN: 978-0070543713. (no DOI)
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418
Xia, M., & Xu, Z. (2013). Managing hesitant information in GDM problems under conditions of fuzzy and multiplicative preference relations. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 21(6), 865–897. DOI: 10.1142/S0218488513500402
Mikhailov, L. (2000). A fuzzy programming method for deriving priorities in the analytic hierarchy process. Journal of the Operational Research Society, 51(3), 341–349. DOI: 10.1057/palgrave.jors.2600899