HF-MaxScore-Portfolio (Hesitant Fuzzy Maximum-Score Portfolio Selection)

HF-MaxScore-Portfolio is a portfolio-selection method. It is used when you hold several investment alternatives (a stock, a project, a programme) and, instead of numerical historical data on these alternatives, you hold qualitative, hesitant fuzzy assessments of them, such as "this stock's growth potential could be anywhere between 0.3 and 0.9." It decides how a limited budget should be split across the alternatives. Its output is not a ranking but a vector of investment shares that sum to one. It was proposed by Zhou and Xu in 2018, in a study that carried Markowitz's classical mean-variance portfolio theory into a hesitant fuzzy setting.

HF-MaxScore-Portfolio's underlying idea is to model situations where the decision question is not "which one is best" but "how much should I put where." Markowitz's classical portfolio theory, proposed in 1952, finds this split from historical return and risk data. HF-MaxScore-Portfolio works on the same question with assessments made up of several plausible values (hesitant fuzzy), given by expert or investor judgement in place of historical numerical data.

This idea carries a philosophical consequence. When combining alternatives, the method does not use a plain weighted average; it combines the hesitation sets with a probabilistic "either one or the other" logic. Because of this, even where two alternatives' standalone scores sit close to one another, splitting the budget between both can yield a higher total score than putting the whole budget on one alone. The method can thus justify diversification, not putting all your eggs in one basket, directly with numbers.

The method proceeds through four steps.

First, gathering criterion-level assessments. For every alternative, assessments on several qualitative criteria are gathered, each one a hesitant fuzzy set.

Second, combining at the level of the alternative. An alternative's assessments across its different criteria are combined into a single overall hesitant fuzzy assessment, giving one assessment set per alternative.

Third, finding the distribution that gives the highest score. Subject to the investment shares summing to one and none being negative, the distribution is found that maximises the total score obtained by combining the alternatives' overall assessments with these shares. This combination is not a plain weighted average; it is an operation that accounts for every plausible-value combination in the alternatives' hesitation sets. Even where the model would allow less than the full budget to be used, the optimal solution always uses the whole budget; it leaves nothing unspent.

Fourth, reporting the distribution. The share allocated to each alternative is reported, either as a percentage or as an amount.

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

The output is not a ranking but a vector of shares that sum to one. If an alternative's share is high, a large part of the budget should go to it; if its share is zero, the model has found that allocating budget to this alternative does not raise the total score. A zero share does not mean "this alternative is bad," only that "allocating a share to this alternative does not raise the total score."

Thus instead of writing:

"HF-MaxScore-Portfolio chose the best stock"

the report should read:

"With these assessments, the distribution below maximises the total score; a low share for an alternative does not mean it is bad, only that its contribution to the distribution is limited"

HF-MaxScore-Portfolio works with hesitant fuzzy data: every cell holds not a single number but a set made up of several plausible values. DecisionMind holds no separate data-type extension of this method; it stands alone in its base form.

You need: a table of alternatives (stocks, projects) by criteria, with a hesitant fuzzy assessment set in every cell; and, where more than one criterion is used, criterion weights. This method contains no term separately penalising investment risk or how scattered the assessments are; it only maximises the total score. A decision-maker who also wants risk or scatter taken into account should look at DecisionMind's sibling methods, such as HF-TradeOff-Portfolio or HF-EHVaR.

HF-MaxScore-Portfolio is a sound choice if your decision is not "which one should I pick" but "how should I split the budget," and the data you hold is qualitative, hesitant expert assessment rather than historical numerical return and risk data.

It should not be used in the following cases: if you hold historical numerical return and risk data, classical Markowitz portfolio theory, or its forms that work with measurable data, may be more suitable. If your decision really is choosing a single winner rather than splitting a budget, a ranking method (TOPSIS, SAW) gives a more direct answer. If you want risk or the scatter of the assessments built into the model, a portfolio method that carries a risk term (HF-TradeOff-Portfolio) is needed.

Splitting a budget, qualitative hesitant assessment → HF-MaxScore-Portfolio

Splitting a budget, risk/scatter must also be accounted for → HF-TradeOff-Portfolio, HF-EHVaR

Splitting a budget, historical numerical return/risk data exists → classical Markowitz portfolio theory

A single winner is to be chosen, not a split → TOPSIS, SAW and other ranking methods

HF-MaxScore-Portfolio's greatest strength is that it puts a budget-splitting decision on a quantitative footing using only qualitative expert judgement, without historical numerical data. Working with hesitant fuzzy data lets it carry the uncertainty in an assessment without reducing it prematurely to a single number. Using a probabilistic combination, rather than a plain average, when combining alternatives' hesitation sets can show directly, through the numbers, when diversification is advantageous.

Its limitations stem from the model carrying no risk term. First, the model contains no penalty for risk or scatter; on some data patterns it can therefore put the entire budget on a single alternative (a corner solution), which is mathematically correct but can be an economically undesirable outcome for a cautious investor. Second, the probabilistic combination of hesitation sets makes the computational burden grow quickly as the number of alternatives and criteria increases. Third, the method's result depends directly on how the criterion assessments were gathered (which expert held how much hesitation about which criterion); if these assessments change, the distribution changes with them. Fourth, the method measures only central tendency (the score); on its own, it is not sufficient for an investor sensitive to tail risk, the possibility of a rare but large loss.

The most common mistake is interpreting an alternative that receives a zero share as "rejected by the model, or found to be bad"; a zero share only shows that allocating a share to that alternative does not raise the total score.

A second mistake is accepting, without question, a corner solution in which the model puts the entire budget on a single alternative; in that case, whether the assessments genuinely reflect such a sharp difference should be checked. A third mistake is reading HF-MaxScore-Portfolio as if it were a ranking method and presenting the shares as "scores"; the shares are a budget split, not a score. A fourth is forgetting that risk or scatter is never included in the model, and presenting the result as if it were a cautious investment decision.

The governing principle is this:

An HF-MaxScore-Portfolio result is the budget split that maximises the total score given the qualitative assessments supplied; unless risk or scatter is separately assessed, this split should not on its own be treated as a cautious investment decision.