As a result, a decision-making problem in strict uncertainty can be summarized in a decision table as shown in Table 1.

The decision-making literature proposes several criteria for solving decision problems from the information in a decision table. Table 2 summarizes the most relevant decision rules under strict uncertainty, including the underlying decision-making principle followed by each rule. The Laplace score function L i (Laplace, 1825) aggregates evaluations for each alternative over the whole range of scenarios and selects the alternative with the maximum aggregated value. The Wald maximin rule (Wald, 1950) assumes that the worst scenario will occur. Then, the decision-maker focuses only on the minimum evaluation for each alternative and selects the alternative with the maximum W i over the minimum evaluations. The Hurwicz criterion (Hurwicz, 1951) is characterized by parameter α that describes the attitude towards pessimism and optimism of a decision-maker who ultimately selects the alternative with maximum H i . The Savage minimax regret criterion (Savage, 1951) requires the consideration of minimum regret S i , defined as the difference between the best evaluation in the scenario and the particular evaluation of each alternative in this scenario. Finally, the Ballestero moderate pessimism rule (Ballestero, 2002) implies that the larger the range of evaluations for each scenario, the higher the distrust of the decision-maker towards the scenario. As a result, a set of weights inversely proportional to the range of evaluations for each scenario is used to adjust evaluations in score function B i .

Hayashi (2008) proposed the maximin joy criterion, as a reciprocal of the Savage minimax regret criterion. As an extension of the maximin joy criterion, Gaspars-Wieloch (2014) introduced the dominance joy criterion and the cumulative maximin joy criterion. More recently, Shestakevych and Volkov (2021) described the criterion of extreme optimism in which the maximum of maximum payoffs is considered to select the best alternative. Other criteria such as the Bayesian criterion, the Hermeyer criterion and the Hodge-Lehman criterion, as defined in Shestakevych and Volkov (2021), imply the assumption of probabilities for scenarios. This assumption is out of the scope of this paper because we focus on strict uncertainty, characterized by a complete absence of knowledge concerning the probabilities associated with future states.

Since the pioneering work by Zadeh (1965), the theory of fuzzy sets has been developed in several research fields. In this section, we provide a brief introduction to some basic concepts in fuzzy set theory.

Within the class of fuzzy numbers, the trapezoidal fuzzy number is most commonly used to quantify fuzzy evaluation in the decision-making process. The motivation behind their utilization comes from the simplicity of these membership functions (Delgado et al., 1998), and their characterization requires reasonably limited information. Therefore, the definitions of the triangular and trapezoidal fuzzy numbers and operational laws (Klir and Yuan, 1995; Pedrycz et al., 2011), are briefly provided below.

Let us denote the set of all trapezoidal fuzzy numbers on the real line R by F ( R ). The computation between the trapezoidal fuzzy numbers Tr 1 = ( a 1 , b 1 , c 1 , d 1 ) and Tr 2 = ( a 2 , b 2 , c 2 , d 2 ) from F ( R ) could be facilitated with the help of the following arithmetic operational laws:

Note that the set of fuzzy numbers does not possess a natural order. Therefore, we require a mechanism to order or rank fuzzy numbers. Several methods have been proposed in the literature to rank or generate an order for fuzzy numbers. These methods can be classified into three main categories (Wang and Kerre, 2001; Yatsalo and Martínez, 2018):

Though there are various methods to produce the ordering among the fuzzy numbers, most can generate only partial order. More precisely, these methods do not warrant the anti-symmetry property of the order relation. In this paper, we focus on the ordering mechanism that generates the total order of F ( R ).

In this regard, Zumelzu et al. (2022) introduced the total order of the fuzzy numbers based on the α-cut of the fuzzy numbers. It can overcome the drawback of defuzzification-based ranking methods. The key principle of this ordering mechanism is to compare the α-cuts of the fuzzy numbers, which are intervals. Therefore, it is necessary to introduce the concept of total ordering of intervals.

Let I ( R ) = { [ a , b ] : ( a , b ) ∈ R 2 , a ⩽ b } be all the close and bounded sub-intervals of R. Consider the usual lexicographic order on the R 2 given by ( a , b ) ≧ ( c , d ) ⇔ a ⩾ c ∧ b ⩾ d. The order relation ≧ is a partial order relation on R 2 . Further, the relation ≧ induces the partial order of I ( R ). We denote this partial order relation by ⩾ 2 and interpret it as [ a , b ] ⩾ 2 [ c , d ] ⇔ a ⩾ c ∧ b ⩾ d .

Bustince et al. (2013) introduced the notion of admissible order for intervals, which is linear and refined or encompasses a partial order.

Another concept that enables us to compare two fuzzy numbers in terms of the α-cut is the representation of fuzzy numbers using α-cuts, specifically in terms of an upper dense sequence of α-cuts. Let S = ( α i ) i ∈ N be a sequence in [ 0 , 1 ]. Then, S is said to be an upper dense sequence if, for every x ∈ [ 0 , 1 ] and any ϵ > 0, there exists i ∈ N such that α i ∈ [ x , x + ϵ [. It is noted that a fuzzy number can be represented via an upper dense sequence of α-cuts (Wang et al., 2005): Tr = ⋃ α i ∈ S Tr α i . Based on the admissible order relation of intervals and upper dense sequence of α-cuts representation of fuzzy numbers, we can compare whether all α-cuts of two fuzzy numbers are equal or there exists an α-cut that dominates based on admissible order relation.

It could be easily verified that the order relation ⪰ on F ( R ) induces a total order of the trapezoidal fuzzy numbers.

Note that although in Definition 6, we have mentioned that we need to consider an upper dense sequence of α-cuts to decide the order of any two trapezoidal fuzzy numbers. However, in practice, the trapezoidal fuzzy number could be fully characterized by its support, i.e. 0-cut, and core, i.e. 1-cut. Further, the two trapezoidal fuzzy numbers would be equal only when these two α-cuts are equal. Therefore, the condition of using an upper dense sequence of α-cuts for ordering the trapezoidal fuzzy numbers in Definition 6 could be reduced to the condition of using only two special α-cuts S ¯ = { 0 , 1 }.

Formally, two trapezoidal fuzzy numbers Tr 1 and Tr 2 are said to be equal Tr 1 = Tr 2 iff supp ( Tr 1 ) = supp ( Tr 2 ) and core ( Tr 1 ) = core ( Tr 2 ). Further, the condition for the order relation ⪰ on F ( R ) could be simplified as follows: Tr 1 ⪰ Tr 2 ⇔ ( Tr 1 = Tr 2 ) ∨ ( core ( Tr 1 ) ≻ core ( Tr 2 ) ) ∨ ( supp ( Tr 1 ) ≻ supp ( Tr 2 ) ) .

As the order relation ⪰ on F ( R ) induces a total order relation, any set S ⊂ F ( R ) would be total ordered.

Now, we utilize the ordering relation ⪰ to define the minimum and maximum operation on a set of fuzzy numbers.

Aggregating fuzzy numbers to reach a final decision is critical in any decision-making process. Typically, the aggregation operations combine several fuzzy numbers and produce a single representative fuzzy number (Klir and Yuan, 1995). Formally, an aggregation function over trapezoidal fuzzy numbers of dimension n can be represented as a function, ψ : F ( R ) n → F ( R ). Definition 10.

For a set of trapezoidal fuzzy numbers ( Tr 1 , … , Tr n ) ∈ F ( R ) n , the weighted fuzzy arithmetic mean operator WA ˜ : F ( R ) n → F ( R ) can be defined as: WA ˜ ( Tr 1 , … , Tr n ) = ⨁ i = 1 n ( w i ⊙ Tr i ), where w = ( w 1 , w 2 , … , w n ) ∈ [ 0 , 1 ] n such that w i ⩾ 0, i = 1 , … , n and ∑ i = 1 n w i = 1. Further, if Tr i = ( a i , b i , c i , d i ), i = 1 , … , n, then it can be computed by utilizing the operational laws of trapezoidal fuzzy numbers and given by WA ˜ ( Tr 1 , … , Tr n ) = ( ∑ i = 1 n w i a i , ∑ i = 1 n w i b i , ∑ i = 1 n w i c i , ∑ i = 1 n w i d i ).

As a consequence of Definition 11, the combination of alternative a i within scenario r j results in fuzzy evaluation V ˜ i j , following the same structure of Table 1. Again, for convenience, all fuzzy evaluations V ˜ are assumed to be of the type the more the better.

We next extend the decision rules described in Section 2.1 to a fuzzy environment. As a result of this extension, the Laplace (1825) criterion implies the use of fuzzy score function L ˜ i : (2) L ˜ i = ⨁ j = 1 n V ˜ i j .

The fuzzy maximin rule by Wald (1950) implies focusing on minimum evaluations for each alternative i over the n scenarios and selecting the alternative with maximum W ˜ i : (3) W ˜ i = min ˜ ( V ˜ i 1 , … , V ˜ i n ) .

The fuzzy Hurwicz (1951) criterion uses parameter α to express the attitude towards pessimism and optimism to select the alternative with maximum H ˜ i : (4) H ˜ i = α ⊙ min ˜ ( V ˜ i 1 , … , V ˜ i n ) ⊕ ( 1 − α ) ⊙ max ˜ ( V ˜ i 1 , … , V ˜ i n ) .

The minimax regret fuzzy criterion by Savage (1951) computes the fuzzy regret for each combination and scenario to select the alternative with minimum S ˜ i : (5) S ˜ i = max ˜ ( V ˜ i 1 , … , V ˜ i n ) ⊖ V ˜ i j .

The extension of the Ballestero (2002) moderate pessimism rule to a fuzzy context requires the computation of fuzzy weights w ˜ j : (6) w ˜ j = 1 max j ( V ˜ i j ) ⊖ min j ( V ˜ i j ) provided that supp ( max j ( V ˜ i j ) ⊖ min j ( V ˜ i j ) ) does not contain 0. However, this might be tedious to ensure in most of the cases, even when max j ( V ˜ i j ) is not equal to min j ( V ˜ i j ). To simplify this, we defuzzified the fuzzy values through the centre of gravity (COG) defuzzification method (Yager, 1978) and used these values to generate a weight proportional to the range of defuzzified values of the evaluations. Based on this, the weight could be computed as follows: (7) w j = 1 C O G ( max j ( V ˜ i j ) ) − C O G ( min j ( V ˜ i j ) ) where, C O G of a fuzzy number A with membership function μ A : X → [ 0 , 1 ] is computed as C O G ( A ) = ∫ X x μ A ( x ) d x / ∫ X μ A ( x ) d x. Finally, the decision maker aims to maximize score function B ˜ i : (8) B ˜ i = ⨁ j = 1 n w j ⊙ V ˜ i j .

Note that all decision rules with fuzzy payoffs produce an output. When the fuzzy payoffs are represented by trapezoidal or triangular fuzzy numbers, the quantities L ˜ i , W ˜ i , H ˜ i , S ˜ i and B ˜ i can be computed using the operational laws of fuzzy numbers described in Section 2. According to the previous decision rules, the decision-maker should choose the alternative with maximum L ˜ i , W ˜ i , H ˜ i , B ˜ i , and minimum S ˜ i . The maximum and minimum values of fuzzy numbers can be obtained according to the ranking principle described in Section 2. Therefore, we obtain the desired ranking of the fuzzy numbers.

Given a fuzzy decision table with fuzzy payoffs V ˜ i j , with alternatives indexed by i = { 1 , 2 , … , m } and scenarios indexed by j = { 1 , 2 , … , n }, a set of operational laws { ⊕ , ⊖ , ⊙ } defined for trapezoidal fuzzy numbers on F ( R ), a set of minimum and maximum operators { min ˜ , max ˜ }, and an aggregation operator WA ˜ defined on F ( R ) n , we derive the following properties:

We leave more involved properties for subsequent work.

The decision-making process under strict uncertainty is graphically summarized in Fig. 1. The process begins by gathering information about evaluating alternatives under different scenarios. The main assumption is that no information about the probability of occurrence of any scenario is available. According to Table 2, decision-makers can deploy different guiding principles by selecting one of the rules in the table. For instance, the Wald rule is considered appropriate for a pessimistic decision-maker because it is assumed that the worst scenario will occur, while the Ballestero rule fits a moderately pessimistic attitude. In addition, selecting any of the rules (Laplace, Wald, Hurwicz, Savage, Ballestero, and possibly others) implies a set of reasonable properties as described in Section 2.1.

By extending the concept of strict uncertainty to a fuzzy context, we also extend decision-making principles and properties to provide a more general decision-making framework. For instance, the moderate pessimism principle remains the same but requires adaptation through a fuzzy rule. Similarly, we can study new properties of fuzzy decision rules under strict uncertainty as proposed in Section 3.3. Finally, using a fuzzy decision rule implies using a ranking function to derive an admissible order of fuzzy numbers, pointing out which alternative is best. A wide range of ranking functions can provide the desired order of fuzzy numbers. However, decision-makers can also integrate their attitude by selecting a ranking function. For instance, decision-makers with risk aversion to low values in an interval fuzzy evaluation may select a ranking function focusing first on the left end-points of fuzzy numbers rather than modal values.

To summarize, our decision-making framework for fuzzy strict uncertainty combines the principles and properties derived from strict uncertainty. Uncertainty and imprecision are inherent to many decision-making problems in economics and finance. Indeed, forecasting the results (payoffs) of a set of alternatives under different future scenarios is crucial. Because of the difficulty of computing exact values for payoffs in economics and finance problems, we propose using fuzzy intervals as a suitable approximation method. Consider, for instance, the problem investors face when evaluating a set of investing alternatives under different future scenarios. They will probably have problems establishing a precise evaluation of future payoffs. A suitable way to solve this problem is by approximating these payoffs through interval fuzzy numbers. As a result, they must select a fuzzy decision rule adapted to the context of strict uncertainty. To this end, they can rely on the decision-making principle underlying each decision rule described in this paper to make a choice. In addition, their risk attitudes can also be integrated into the decision-making process by establishing the ranking function that ultimately produces an admissible order of fuzzy numbers.

A further detailed formalization of our methodology for decision-making under strict uncertainty is shown in Algorithm 1.

Portfolio selection is a critical task of financial decision-making that involves the allocation of a given budget to a set of assets to optimize returns while managing risks according to investors’ preferences. This process consists of constructing a diversified portfolio that balances profitability and risk. Investors typically seek to maximize returns while minimizing the inherent uncertainties associated with financial markets. As a result, the fundamental principle of portfolio selection is rooted in the trade-off between risk and return (Markowitz, 1952). The concept of diversification implies the selection of an optimal portfolio that balances the expected returns of individual assets with their corresponding risks.

Furthermore, the inherent uncertainty of expected returns poses a significant challenge in portfolio selection. Traditional models often rely on historical data and assumptions about future returns, which may not accurately capture the dynamic nature of financial markets. As a result, including fuzzy payoff decision-making methods (Wang and Zhu, 2002; Bilbao-Terol et al., 2006; Jalota et al., 2023), such as the one described in this paper, represents a reliable way to account for the uncertainty surrounding expected returns. This paper delves into portfolio selection through the strict uncertainty procedure proposed in Ballestero et al. (2007). The proposed methodology comprises two main steps:

Given the close price for a set of available assets included in the Dow Jones Industrial Average index from 2006-10-60 to 2020-12-31 as an input, we follow the next procedure to construct a decision table with fuzzy payoffs:

Once we have built a decision table with fuzzy payoffs summarized in Table 3, we can select the best portfolio according to the fuzzy strict uncertainty procedure described in Section 3.

To this end, we must select an admissible order described in Section 2.2. Investors are usually concerned about expected returns (represented here as the core of a triangular fuzzy number) and the volatility of returns (represented here as the support of a triangular fuzzy number). Moreover, a risky investor prioritizes returns over volatility (core over support), and a conservative investor prioritizes volatility over returns (support over core). Finally, investors are usually concerned with downside risk because they are more averse to returns below the mean value than to deviations above the mean value. As a result, given Tr 1 = ( a 1 , b 1 , c 1 ) and Tr 2 = ( a 2 , b 2 , c 2 ), and changing notation for economy of space ( C ( Tr 1 ) = core ( Tr 1 ) and S ( Tr 1 ) = supp ( Tr 1 )), we consider the following ranking functions.

Ranking 1. For risky investors. Given Tr 1 , Tr 2 ∈ R, where R is a set of fuzzy numbers, focus first on the core and second on the support of alternative fuzzy payoffs. Tr 1 ⪰ Tr 2 ⇔ ( C ( Tr 1 ) ≽ C ( Tr 2 ) ) ∨ ( C ( Tr 1 ) = C ( Tr 2 ) ) ∧ ( S ( Tr 1 ) ≽ S ( Tr 2 ) ) interpreted within the context of investing and assuming the interval order ≽ as ≽ L e x 1 the ordering conditions could be simplified as follows: Tr 1 ⪰ Tr 2 ⇔ ( b 1 > b 2 ) ∨ ( b 1 = b 2 ) ∧ ( a 1 > a 2 ) ∨ ( b 1 = b 2 ∧ a 1 = a 2 ) ∧ ( c 1 ⩾ c 2 ) .

Ranking 2. For conservative investors. Given Tr 1 , Tr 2 ∈ R, where R is a set of fuzzy numbers, focus first on the support and second on the core of alternative fuzzy payoffs. Tr 1 ⪰ Tr 2 ⇔ ( S ( Tr 1 ) ≽ S ( Tr 2 ) ) ∨ ( S ( Tr 1 ) = S ( Tr 2 ) ) ∧ ( C ( Tr 1 ) ≽ C ( Tr 2 ) ) interpreted within the context of investing and assuming the interval order ≽ as ≽ L e x 1 the ordering conditions could be simplified as follows: Tr 1 ⪰ Tr 2 ⇔ ( a 1 > a 2 ) ∨ ( a 1 = a 2 ) ∧ ( c 1 > c 2 ) ∨ ( a 1 = a 2 ∧ c 1 = c 2 ) ∧ ( b 1 ⩾ b 2 ) .

As a result, applying the Laplace rule with a fuzzy payoff as described in Section 3, we obtain the rankings summarized in Table 4 for risky (Ranking 1) and conservative (Ranking 2) investors. As expected, both rankings lead to an opposite evaluation of alternatives because the highest core of aggregated triangular payoffs comes in conjunction with the lowest endpoint of the support.

To apply the Wald rule with fuzzy payoffs, we need to find the minimum return for each PEP and scenario and select the PEP with the maximum return. For illustrative purposes and economy of space, we use Ranking 1 when applying the minimum operator from Definition 8 to find the minimum return, and then we use Rankings 1 and 2 when applying the maximum operator from Definition 9 to compare the ranking results for risky and conservative investors as shown in Table 4. The use of any other ranking operator is straightforward. In this case, we find ties when comparing the core of triangular payoffs, implying the use of the left and right support endpoints to obtain Ranking 1.

Similarly, using the minimum and maximum operators and setting α = 0.5 to represent a decision-maker characterized by neutrality concerning optimism and pessimism, we use the Hurwicz fuzzy decision rule under strict uncertainty to derive the rankings. In the case of the Savage fuzzy rule, we first build a new decision table with fuzzy regrets by using the maximum fuzzy operator and the initial decision table. Next, we follow the minimax criterion to select the minimum fuzzy regret among the maximum regrets for each PEP and scenario in an opposite way of the maximin criterion by Wald. Finally, to apply the fuzzy Ballestero rule, we find the minimum and maximum fuzzy payoffs for each scenario to compute their respective weights by computing the COG of a triangular fuzzy number. Then, we apply these weights to obtain the score function in equation (8) and derive rankings. The Savage, Ballestero and the extreme optimism criterion, described in Shestakevych and Volkov (2021), ranking results are summarized in Table 5. Furthermore, the results derived from the maximin joy criterion by Hayashi (2008), the dominance joy and the cumulative maximin joy criteria by Gaspars-Wieloch (2014) are shown in Table 6.