HF-DFT (Hesitant Fuzzy Decision Field Theory)

HF-DFT is a method for when you hold a decision table in which expert opinions are given as several plausible values (hesitant fuzzy numbers), such as "which route" or "which investment target," and you want the alternatives ranked through a dynamic simulation that models how the human mind actually reaches a decision. Most other ranking methods (TOPSIS, SAW and the like) produce a preference at a single instant, through a static calculation. HF-DFT is different: it rests on a model from psychology called decision field theory, and assumes that preference forms as small comparisons between alternatives accumulate over time. Its output is the preference value each alternative has accumulated by the end of the deliberation period, and the rank that value produces. The method was proposed by Song, Zhang, Xu, Hao and Wang in 2019, carrying classical decision field theory into a hesitant fuzzy setting.

HF-DFT's underlying idea is that real decision-making is not an instantaneous calculation but a process that unfolds over time. The human mind does not compare alternatives once and reach a final verdict; it shifts its attention from one criterion to another in turn, a small preference gap between alternatives forms with each shift, and these gaps accumulate over time. HF-DFT mimics this process mathematically: an instantaneous "advantage pressure" (valence) is computed for every alternative relative to the others, this pressure is added to what has accumulated before, the effect of competition and memory between alternatives is modelled through a feedback mechanism, and this cycle repeats until a set duration is reached or a threshold is crossed.

This idea carries a philosophical consequence: HF-DFT is a method of consistency and maturation. In static methods, a small gap between two alternatives' scores is reported in exactly the same way as a large one — "first, second." In HF-DFT, by contrast, the process implicitly carries information about how "robust" the gap is: even when the static scores are almost equal, the dynamic accumulation process can bring out a clear winner over time — or, conversely, if the process is cut short, no one may yet have come out ahead.

The method proceeds through eight steps.

First, combining expert opinions. Where several experts are involved, each one's hesitant fuzzy assessments are combined, using expert weights, into a single shared decision matrix.

Second, direction equalisation. "Lower is better" criteria, such as cost, are converted to "higher is better" by taking their complement, so that every subsequent step proceeds in the same direction.

Third, determining criterion weights. If weights are known, they are used directly; if partly known, they are optimised under linear constraints; if unknown, they are derived from the data (for instance, by entropy or by maximising the total score).

Fourth, distance between alternatives. How different every pair of alternatives is from one another is measured by a weighted distance between the hesitant fuzzy values, and a distance table is built.

Fifth, the feedback mechanism. From this distance table, a feedback table is built that sets the strength of competition between alternatives and how much memory is retained; competition is stronger between alternatives that closely resemble one another and weaker between alternatives that differ greatly.

Sixth, instantaneous advantage pressure. Each alternative's relative advantage over the others at that instant (valence) is computed from the weighted criterion scores.

Seventh, accumulation over time. In small time steps, each alternative's preference value is updated by adding the feedback effect and the new advantage pressure to its previous value. This cycle repeats until a set deliberation duration is reached, or until an alternative's preference value crosses a set threshold.

Eighth, the decision. At the end of the deliberation period, the alternative with the highest accumulated preference value is the best alternative; where values are tied, the score is examined first, and then the stability of that score.

The formulas behind each step are given on the DecisionMind method page; this card carries no formulas.

The accumulated preference value tells you how far ahead an alternative pulled by the end of the deliberation process; it says nothing more than that. A positive value shows that the alternative was ahead in the overall trend, a negative value that it trailed, but this is not a percentage or a probability. The magnitude of the value depends on the chosen deliberation duration and on the model's parameters (competition strength, memory effect); running the same decision matrix with a different duration or different parameters can change the magnitude of the values, though the ranking is generally more robust.

Thus instead of writing:

"HF-DFT found the best alternative"

the report should read:

"With this deliberation duration and these model parameters, the alternative that accumulated the most preference is this one; only the rank of the accumulated value should be read as reliable, not its magnitude"

HF-DFT works with hesitant fuzzy data: every cell holds not a single number or a single triple but a set made up of several plausible values. This is used to carry situations where several experts propose different scores on the same criterion, or a single expert remains undecided between several plausible values, without reducing them prematurely to one number. DecisionMind holds no extension of this method for other data types; it stands alone in its base form.

You need: alternatives in rows, criteria in columns, a hesitant fuzzy set made up of one or more plausible values in every cell; criterion weights (the method can derive them from the data if they are unknown); and model parameters (competition strength, memory effect, deliberation duration, time step). If how these parameters correspond to the real decision situation is not clear (how strong should "competition strength" be, for instance), working with default values affects the result; the choice of parameters must therefore be justified in the report.

HF-DFT is a sound choice if your assessments contain several plausible values, whether between experts or within a single expert's indecision, and you want the decision modelled not as a static calculation but as a process that matures over time. Its typical territory is strategic choices, such as route selection or investment-target selection, where a more realistic reflection of human behaviour and deliberation is wanted.

It should not be used in the following cases: if your data is already crisp and there is no need to model the decision as a dynamic process, a simpler method requiring fewer parameters (TOPSIS, SAW) is sufficient. If you cannot justify the model parameters (competition strength, memory effect, deliberation duration) against the real decision situation, using HF-DFT without knowing how much these parameters affect the result can be misleading.

Hesitation exists among experts, the decision process is to be modelled dynamically → HF-DFT

Data is crisp, no need for a dynamic process → static methods such as TOPSIS, SAW

No hesitation, but two degrees exist (support/rejection) → IF-DFT (intuitionistic fuzzy DFT)

Model parameters cannot be justified → HF-DFT should not be used without a parameter sensitivity analysis

HF-DFT's greatest strength is that it models the decision not with a static formula but with a dynamic process that mimics human deliberation; this draws on classical decision field theory's strong experimental grounding in psychology (Busemeyer and Townsend, 1993; Roe, Busemeyer and Townsend, 2001). Working with hesitant fuzzy data lets it carry disagreement among experts without reducing it prematurely to a single number. The method can bring out a clear winner over time even when the static scores sit very close to one another; this offers an additional perspective in situations of indecision.

Its limitations relate to the model's parameter burden and its process structure. First, the method requires several parameters (competition strength, memory effect, deliberation duration, time step); how these correspond to the real decision situation is often unclear and can remain subjective. Second, classical decision field theory itself has been tested comparatively against some alternative models in psychology, and is known not to explain every preference pattern with equal accuracy (Roe, Busemeyer and Townsend, 2001). Third, hesitant fuzzy sets can differ in length (two plausible values on one criterion, three on another), which requires a technical equalisation step; this equalisation, such as repeating the most pessimistic value, is itself an assumption. Fourth, the magnitude of the result depends on the chosen deliberation duration; if the duration is cut too short, the process may not yet have matured.

The most common mistake is reading the magnitude of the accumulated preference value as a percentage or a degree of certainty; the value only shows relative standing at the end of the process.

A second mistake is leaving the model parameters (competition strength, memory effect, deliberation duration) at their default values without justification; these parameters can substantially change the magnitude of the result. A third is ignoring the difference in length between hesitant fuzzy sets and failing to report how this equalisation affects the result. A fourth is setting the deliberation duration too short and deciding at a point where the process has not yet matured. A fifth is reading HF-DFT as if it were a static weighted-sum method, ignoring the dynamic process itself.

The governing principle is this:

An HF-DFT result is a summary of a preference process accumulated under the chosen model parameters and deliberation duration; if the parameters change, the magnitude of the accumulated value, and sometimes its rank, can change too.