Extension card · Hesitant
Hesitant SAW
This is the form of SAW for situations where several plausible values on a criterion are held together. It carries these sets through the calculation and reduces the result to a single weighted score only at the last step.
Base method
SAW →
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?
Three things change; the logic of the weighted sum does not.
Cells. In crisp SAW every cell is a single number. Here every cell is a set; it holds several plausible values for the same criterion together, for instance {0.6; 0.7; 0.8}. Sets belonging to different alternatives in the same column can be of different length. In this case DecisionMind completes every column to the length of the longest set in that column; the missing place is filled with the set's own smallest value. This is a cautious completion: missing information is compensated for with the set's lowest end, not in its favour. Weights are taken from outside as crisp numbers.
Scale equalisation. Crisp SAW ratios every column against its own best value. There is no step of this kind here, because the values in a hesitant set are already degrees between 0 and 1. Instead, on a cost criterion, every element of the set is complemented one by one: 1−g is written in place of g. This is not the "swap between support and rejection" seen in intuitionistic or Pythagorean fuzzy sets; because a hesitant set carries only one type of degree, the complement is applied directly to every element.
Weighted sum and defuzzification. In crisp SAW every column is multiplied by its weight and summed. Here, every alternative's sets across the criteria are merged with the hesitant weighted-average (HFWA) operator. This merger processes, one by one, every possible combination (a Cartesian product) of the sets across the different criteria. Every combination produces a value according to its own weighted combination; these values form a new, generally larger set. This set grows rapidly as the number of criteria increases. Only at the last step is the arithmetic mean of this merged set's values taken, reducing it to a single score, and alternatives are ranked on this score.
DecisionMind fixes, in classical HF-SAW, the cautious length completion (with the smallest value), the HFWA merger and the arithmetic-mean score. Weights are taken from outside; the method does not generate weights.
How to Read the Output
The score, like the sum in crisp SAW, only ranks this set of alternatives; it is not a percentage or a probability, and it cannot be compared with a different analysis.
The difference is this. The score is the average of a set that has grown through the Cartesian product; this average conceals the set's breadth, that is, the genuine disagreement between the experts or the scenarios. Two alternatives can share the same average score while one came from a narrow set and the other from a wide one.
Thus instead of writing:
"Because the Hesitant SAW score comes out highest, this alternative is definitively the best"
the report should read:
"This score is the average of the weighted merger of the criterion sets; which alternative's merged set stayed wider, that is, more variable, must be reported separately"
When to Prefer This over the Base Method
Use this extension when there are genuinely more than one plausible and defensible value for the same criterion-alternative pair, for instance when several experts or several scenarios have each given an independent degree. The distinction on the data-type card applies here too: if there is a single measured value, a single number is written into the cell, and no set is fabricated. SAW's exit condition applies unchanged: if no compromise is accepted on a criterion, this extension too carries full compensatory logic and will not eliminate anything below a threshold.
Mistakes Specific to This Extension
Violating the value range. Every set must be a non-empty set consisting of values between 0 and 1.
Failing to state the length-completion rule. This kernel always fills the missing place with the set's smallest value; a different completion rule (for instance, with the largest value) gives a different score. The rule used must be stated in the report.
Failing to notice the size of the Cartesian product. The merged set grows rapidly as the number of criteria increases; even though a large set's average is a single number, the spread behind it must be examined separately.
The "more sophisticated" fallacy. When there is no genuine source supporting more than one value, opening a crisp number out into a few nearby numbers arbitrarily adds no information.
The governing principle is this:
A set must genuinely come from more than one defensible source for the same criterion-alternative pair. If the length-completion rule is left unjustified, the same data can produce a different score.
Cases
The first case is DecisionMind's validation example: the synthetic 3×3 table in the manifest is built faithfully to the formula chain and carries no literature page reference. The second case is an illustrative fiction.
1. Illustrative example (DecisionMind's validation example): Choosing an imaging device
A hospital will select one of three MRI-device bids. There are three criteria: image quality ("higher is better"), shortness of processing time ("higher is better") and annual maintenance expense ("lower is better"). Three specialists in the radiology unit each gave a degree, drawn from their own experience, for every device-criterion pair; the degrees were kept together without being reduced to an average.
| Device | Image quality | Shortness of processing time | Maintenance expense (lower is better) |
|---|---|---|---|
| M1 | {0.6; 0.7; 0.8} | {0.4; 0.5; 0.6} | {0.5; 0.6; 0.7} |
| M2 | {0.7; 0.8; 0.9} | {0.5; 0.6; 0.7} | {0.3; 0.4; 0.5} |
| M3 | {0.5; 0.6; 0.7} | {0.6; 0.7; 0.8} | {0.4; 0.5; 0.6} |
| Weight | 0.40 | 0.35 | 0.25 |
The method complements every degree in the maintenance-expense set, merges the sets from the three criteria with HFWA, and takes the average of the merged set.
| Device | Score | Rank |
|---|---|---|
| M2 | 0.7062 | 1 |
| M3 | 0.6238 | 2 |
| M1 | 0.5794 | 3 |
The result reads as follows. M2 carries the highest degrees on image quality ({0.7; 0.8; 0.9}), the most heavily weighted criterion, and it is also advantaged on maintenance expense (once complemented). Although M3 has the strongest set on processing time ({0.6; 0.7; 0.8}), this is not enough to offset M2's advantage on image quality.
The hospital's hesitation is this: what happens if the image-quality weight is lowered from 0.40 to 0.10 and processing time is given 0.75? Verified by running the kernel directly, when the weights are redistributed to (0.10; 0.75; 0.15), M3's score falls to 0.6702 and M2's to 0.6324; first place passes to M3, because M3's advantage on processing time now governs the most heavily weighted criterion.
In the report: "With the given weights (0.40; 0.35; 0.25), M2 has the highest score (0.7062). When the weight is markedly shifted towards processing time (0.10; 0.75; 0.15), first place passes to M3; the clinical priority among the criteria must therefore be justified separately."
Source: DecisionMind's HF-SAW validation example. The merging operator (HFWA) follows Xia and Xu's (2011) definition of hesitant fuzzy aggregation; the value space follows Torra's (2010) definition of the hesitant fuzzy set. No literature source is attached to this table in the manifest, so it is presented as an illustrative example; the scores and the weight-trade-off scenario were computed independently by this card's author by running the kernel directly. The manifest's own J.expected_primary field (M1 ≈ 0.5742; M2 ≈ 0.6986; M3 ≈ 0.6179) differs from the engine's current output by a small margin; the engine has flagged this difference as "O-2, a non-fatal gate deviation," and it does not change the ranking. This card uses the engine's current output, obtained by running it directly.
2. Theatre: A municipal theatre's choice of new-season play
A municipal theatre will stage one of three play scripts for its new season. There are three criteria: expected audience interest, cast suitability and stage/set cost ("lower is better"). The artistic direction and the dramaturge recorded their impressions from different reading sessions for every play as several plausible degrees, rather than reducing them to a single one.
The method complements the degrees on the cost criterion, merges the three criteria's sets with HFWA, and takes the average. Suppose the play with the highest expected audience interest also turned out to have a low set cost, and it took first place.
The theatre's hesitation is this: the breadth of the cast-suitability set, that is, how much the dramaturge and the director agreed on this point, does not show in the average score. For a play whose cast-suitability set turns out wide, the decision should not be finalised before the cast is settled.
In the report: "One play comes out ahead because of its expected audience interest and low set cost; the range of opinion in this play's cast-suitability evaluation is wide, and the decision should not be finalised before the cast is settled."
3. What Not to Do
In the illustrative example, had M1's maintenance-expense set {0.5; 0.6; 0.7} been arbitrarily narrowed to {0.55; 0.60; 0.65}, as if it were a single expert's hesitation, the set's genuine source (three separate expert evaluations) would be lost. The second mistake is writing the report without stating the length-completion rule; had completion with the largest value been chosen instead, the score would have come out differently. The third mistake is presenting the gap between M2's 0.7062 and M3's 0.6238 as a fixed superiority without ever stating the weight sensitivity; this gap can reverse with a single weighting decision.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/hf-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
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418
Xia, M., & Xu, Z. (2011). Hesitant fuzzy information aggregation in decision making. International Journal of Approximate Reasoning, 52(3), 395–407. DOI: 10.1016/j.ijar.2010.09.002