HF-TRADEOFF-PORT (Hesitant Fuzzy Score-Deviation Trade-off Portfolio Selection)

HF-TRADEOFF-PORT is a portfolio selection method for when you hold several investment or project options and must decide how to split a resource (capital, budget) among them. Unlike TOPSIS or VIKOR, it does not answer "which one comes first"; it answers "how much share should each option receive." Its output is an allocation (weight) vector summing to one, together with the highest expected return score attainable under the chosen risk limit. Zhou and Xu proposed it in 2018 for situations where return, according to expert opinion or across different scenarios, can take more than one value (hesitant); it is the classical Markowitz portfolio model carried into a hesitant fuzzy setting.

Markowitz's classical 1952 idea can be summarised as "put your eggs in different baskets, and weigh return against risk together." HF-TRADEOFF-PORT carries the same idea, but here each basket's (option's) return is not a single number. Return is a set of several possible values, because experts disagree or different scenarios yield different outcomes. The method extracts two things from this set: a score (the set's mean, the expected return) and a deviation (the inconsistency within the set itself, that is, risk). It then sets a deviation limit according to the investor's risk type and searches for the highest score while staying under that limit.

This idea carries a philosophical consequence: HF-TRADEOFF-PORT builds a bounded trade-off. Return is not traded against risk without limit; the question "how much risk is acceptable" is answered first, and only then is the best return sought within that bound. This differs from the logic of "deliver the highest return whatever the cost"; the investor bounds their own risk first, and the method works within that bound.

The method proceeds through four steps.

First, gathering hesitant values. For every option, the hesitant scores across the criteria (sets containing more than one possible value) are combined into a single hesitant return set for that option.

Second, finding the score and deviation bounds. The method solves two separate optimisation problems that find the lowest and highest risk (deviation) and the lowest and highest expected return (score) attainable across every possible allocation. This step yields the range "with this option set, risk can be at least this much and at most that much."

Third, dividing the risk range into three. The risk range found is split into thirds: a value close to the upper bound for a risk-seeking investor, a value close to the lower bound for a risk-averse investor, and the midpoint of the two for a neutral investor.

Fourth, finding the best allocation. The allocation (the share assigned to each option) that maximises expected return while staying under the chosen risk limit is computed. The same problem can also be set up the other way round: a return target can be fixed and the allocation that minimises risk sought; the two formulations can give different results for the same investor type.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The output is an allocation vector summing to one; each figure shows the share of the resource assigned to that option. An option's share coming out at zero does not mean "this option is bad"; it means that, within the chosen risk limit and option set, other options offer a better score-deviation balance. The accompanying score is the highest expected return attainable under that risk limit; it cannot be compared with a score obtained under a different risk limit or a different option set.

Thus instead of writing:

"HF-TRADEOFF-PORT found the best portfolio"

the report should read:

"With this risk limit and this option set, the allocation giving the highest expected return is as follows; the allocation changes when the risk limit changes"

HF-TRADEOFF-PORT works with hesitant data: each option's return is not a single number but a set of several possible values (for instance, if three experts gave different return estimates, or three separate scenarios produce three different outcomes). DecisionMind holds no extension of this method; the base method works on its own. The method does not ask for criterion weights from outside; the allocation it produces reflects the distribution of resources among options, not criterion importance. You need at least two options and a return set with more than one value for each option, and the investor's risk type (seeking, neutral, averse) must be specified.

HF-TRADEOFF-PORT is appropriate when you are splitting a resource (budget, capital, time) among several options and each option's return is expressed as a set of uncertain values rather than a single exact figure. If returns are known as exact numbers, the classical mean-variance portfolio model is sufficient; using this method when there is no hesitation adds unnecessary complexity. If your goal is not allocation but choosing a single option (an all-or-nothing decision), this method is the wrong tool.

Splitting a resource across options, return hesitant → HF-TRADEOFF-PORT

Splitting a resource, return an exact number → the classical mean-variance (Markowitz) portfolio model

Choosing a single option, not an allocation → ranking methods such as TOPSIS, VIKOR

Efficiency comparison via an input-output ratio → DEA methods such as HFEA, HFPE

The method's greatest strength is that, instead of collapsing risk into one number and imposing it on the investor, it offers three distinct limits for three risk types; every limit and the best return under it can be computed transparently. It does not collapse the hesitation in return (the several possible values) into a single mean at an early stage and lose it, but carries it through to the end of the computation (Zhou and Xu, 2018). By carrying the classical Markowitz logic into settings where expert opinion is inherently uncertain, it makes it possible to work with a richer data type.

The three-way split of the risk range is a mathematical convenience; the investor's actual risk tolerance may not coincide with any of these three points, in which case the investor's own risk value should be supplied as the limit instead. The two models formulated as max-score and min-deviation can give different portfolios for the same risk type; one does not substitute for the other. A single-valued (non-hesitant) option's deviation comes out at zero; whether this stems from the option genuinely being risk-free or merely from only one estimate having been collected for it must be questioned separately. Because the method was only proposed in 2018, a broad independent body of critique and application, of the kind that exists for TOPSIS or AHP, has not yet formed.

The most common mistake is to interpret an option whose deviation comes out at zero (usually because it is single-valued) as "risk-free" and direct most of the resource there; zero deviation sometimes stems from data scarcity, not genuine absence of risk. A second mistake is to generalise from two risk limits giving very similar scores to "the risk limit does not matter"; this closeness may be specific to that particular option set. A third mistake is to assume the max-score model's result will match the min-deviation model's result and run only one without ever testing the other. The governing principle is this:

An HF-TRADEOFF-PORT result is a reflection of the chosen risk limit and option set; if the risk limit was chosen without justification, the allocation is contestable too.