Fuzzy set theory commonly represents imprecise quantities and parameters using fuzzy numbers (FNs) (Dubois & Prade, 1993).

Thus, the fuzzy number $\tilde{A}$ is interpreted as a value that is "approximately $A^{(c)}$ with the lower and upper extreme scenarios denoted as $A^{(l)}$ and $A^{(r)}$, respectively. For example, an economic prediction such as "the inflation next year will be approximately 3%, and we do not expect it to be below 2.5% or above 4%" can be represented as (2.5%, 3%, 4%).

An IFN is an imprecise quantity that is measured using a real number. If nonmembership is established as $v_A\left(x\right)=1-{\mu }_A\left(x\right) \forall x$, then $\tilde{A}$ is an FN.

Following Mitchell (2004), ${\mu }_{A^{*}}\left(x\right)=1-v_A\left(x\right)$ can be interpreted as the upper distribution function of the uncertain quantity $A$, and ${\mu }_A\left(x\right)$ can be interpreted as the lower distribution function. In this way, Dubois and Prade (2012) interpreted ${\mu }_{A^{*}}\left(x\right)$ and ${\mu }_A\left(x\right)$ as bipolar possibility distribution measurements in such a way that ${\mu }_{A^{*}}\left(x\right)$ accounts for the potential possibility and ${\mu }_A\left(x\right)$ accounts for the real possibility of $A$ being $x$.

Thus, TIFNs are an extension of TFNs, such that if $A^{\left(l\right)*}=A^{(l)}$ and $A^{\left(r\right)*}{=A}^{(r)}$, we deal with a conventional TFN (Kumar & Hussain, 2015). Triangular uncertain parameters are commonly considered in practical applications involving intuitionistic modelling (Mahapatra & Roy, 2013; Kahramam et al., 2015; Kumar & Hussain, 2015; Bhaumik et al., 2017; Rasheed et al., 2021). The argument put forward by Jiménez and Rivas (1998), based on applying the parsimony principle for the use of TFNs, can be extended to the use of TIFNs.

A TIFN adapts very naturally to the way humans make estimations by incorporating more nuances than are necessary to fit a TFN. Once again, we can see that $A^{(c)}$ represents the scenario with maximum reliability. Moreover, whereas $A^{(l)}$ and $A^{\left(l\right)*}$ are two extreme lower scenarios, $A^{(r)}$ and $A^{\left(r\right)*}$ are two extreme upper scenarios. $A^{(l)}$ is considerably lower than the central value, and $A^{\left(l\right)*}$ is exceptionally low compared to the possible values of the parameter. For example, in the context of a random variable, $A^{(l)}$ could be a reasonably small percentile (e.g., the 5th percentile), and $A^{\left(l\right)*}$ is an exceptionally extreme percentile (e.g., the 0.1 percentile). If $A^{\left(l\right)*}<x\le$ $A^{(l)}, {\mu }_A\left(x\right)=$0, meaning that we do not assign a likelihood to parameter $\tilde{A}$ taking the value $x$, but express some level of doubt about its nonmembership, $h_A\left(x\right)>0,$ because $v_A\left(x\right)<1$.

Similarly, $A^{(r)}$ can be described as a notably high realization of $\tilde{A}$ and can be assimilated to a relatively high percentile of a random variable (e.g., the 90th percentile). In contrast, $A^{\left(r\right)*}$ has a potentially extremely high value (e.g., 99.5th percentile).

Note that TIFNs allow modelling estimations of a parameter that, while its knowledge may also be vague and imprecise, contains more nuances than an FN. For instance, a statement such as "inflation is approximately 3%, ranging from 2.5% to 4%. However, it is sure that will not be lower than 2% or higher than 5%" could be quantified as 〈(2.5%,3%,4%)(2%,3%,5%)〉 using TIFNs.

In the introduction, we highlighted numerous applications of fuzzy subsets in finance and insurence pricing using fuzzy number inputs. In all these cases, the fundamental problem lies in evaluating actuarial functions whose inputs are given via fuzzy numbers. This requires the application of Zadeh's extension principle with max-min operators, typically implemented through functional analysis in alpha-level sets.

Shen and Chen (2012) and Bayeg and Mert (2021) generalize to IFN arithmetic the findings of Nguyen (1978), Deng and Shah (1987), and Buckley and Qu (1990) in their evaluation of functions with fuzzy estimates of variables through alpha cuts.

Let be a continuous and differentiable function $y=f\left(x_1,x_2,\dots ,x_n\right)$, such that the values of the input variables are given the means of IFNs ${\tilde{A}}_{(i)}, i=\mathrm{1,2},\dots ,n$. It generates IFN $\tilde{B}$, $\tilde{B}=f\left({\tilde{A}}_{(1)},{\tilde{A}}_{(2)},\dots ,{\tilde{A}}_{(n)}\right)$, whose characteristic functions ${\mu }_B\left(y\right)$ and $v_B\left(y\right)$ must be obtained by using a convolution of a τ-conorm with a τ-norm. For an extended exposition of the usual combinations of τ-conorms and τ-norms, see Atanassova (2007). Among these combinations, it is common to generalize Zadeh’s principle by using the min-norm and max-conorm as follows (Bayeg & Mert, 2021):

Therefore, if ${\tilde{A}}_{(i)}, i=\mathrm{1,2},\dots ,n$ are FNs, it is only necessary to obtain ${\mu }_B\left(y\right)$ using the usual max-min principle. Therefore, to obtain $B_{〈\alpha ,\beta 〉}$ from ${A_{(i)}}_{〈\alpha ,\beta 〉}, i=\mathrm{1,2},\dots ,n$, we must implement (Shen and Chen, 2012): and thus, given that $f$ is continuous, the $〈\alpha ,\beta 〉-$cuts of $\tilde{B}$ are: where: being,

Therefore, $B_{\alpha }$ and ${B^{*}}_{\beta }$ can be obtained analogously to the $\alpha$-cuts of the conventional fuzzy number functions. To obtain $B_{\alpha }$, given that the domain on which $f$ is evaluated is convex, as it is a rectangle in ${\mathbb{R}}^n$, the global optima in the domain $x\in A_{\alpha }$, ${\underset{\_}{B}}_{\alpha }$ (minimum of $f$), and ${\overline{B}}_{\alpha }$ (maximum f), well (Deng & Shah, 1987):

Similar considerations are made for the determination of ${B^{*}}_{\beta }$, which requires obtaining ${\underset{\_}{B^{*}}}_{\beta }$ and ${\overline{B^{*}}}_{\beta }$.

Following Buckley and Qu (1990), when $f$ monotonically increases with respect to $x_i, i=\mathrm{1,2},\dots m$ and monotonically decreases in $x_i, i=m+1,m+2,\dots ,n$,$m\le n$, $B_{\alpha }=\left[{\underset{\_}{B}}_{\alpha },{\overline{B}}_{\alpha }\right]$ is

By analogy, the β-cuts of ${B^{*}}_{\beta }$ are:

The sum and subtraction results of the two TIFNs are also TIFNs. Letting $\tilde{A}=〈\left(A^{(l)},A^{(c)},A^{(r)}\right), \left(A^{\left(l\right)*},A^{(c)},A^{\left(r\right)*}\right)〉$ and $\tilde{B}=〈\left(B^{(l)},B^{(c)},B^{(r)}\right), \left(B^{\left(l\right)*},B^{(c)},B^{\left(r\right)*}\right)〉$, we find that: The multiplication of a TIFN by a scalar is also a TIFN.

The evaluation of nonlinear functions using TIFNs does not produce a TIFN. Despite this limitation, Kreinovich et al. (2020) argue that linear shapes often offer an effective solution to practical issues in the majority of cases. In many instances, straightforward and intuitively clear methods have proven to be the most successful, combining both formulas and intuition. The alignment of the resulting membership function with the original fuzzy concept improved as the characteristic value approached the minimum value. Hence, it is reasonable to consider utilizing functions for alternative fuzzy modelling, where the characteristic value is minimized (Kreinovich et al., 2020).

Hence, the multiplication and division of two TIFNs does not result in TIFNs, as indicated by the exact membership function expressions in Mahapatra and Roy (2013). However, it is worth noting that they allow expressions similar to those presented for the product and division of TFNs in Kaufmann (1986), which are widely used in practical applications. Thus, the triangular approximation of the product of two TIFN $\tilde{A}$ and $\tilde{B}$ that are strictly nonnegative, that is, $A^{\left(l\right)*}\ge 0$ and $B^{\left(l\right)*}\ge 0$:Similarly, the triangular approximation of the division of two TIFN $\tilde{A}$ and $\tilde{B},$ when $A^{\left(l\right)*}\ge 0$ and $B^{\left(l\right)*}>0$ is

It is well known that there are many financial functions that, despite not being linear, when they are evaluated, the result is well approximated by a TFN that maintains the same support (the 0-cut) and core (the 1-cut). This encompasses the present value of a set of cash flows (Kaufmann, 1986), the final value of a pension plan (Jiménez & Rivas, 1998), or the internal rate of return (Terceño et al., 2003). In the actuarial field, this involves the estimation of claim reserves (Heberle & Thomas, 2014), asymptotic probabilities of the number of claims in a bonus-malus system (Villacorta et al., 2021), payment of an immediate annuity (Andrés-Sánchez et al., 2020), and price of LSs (Andrés-Sánchez and González-Vila, 2023). Following the same philosophy, Kumar and Bajaj (2014) postulate that the net present value function, when cash flows and the discount rate are estimated using TIFNs, can be approximated through a TIFN with the same <0,1>-cut and <1,0>-cut. Therefore, in this study, when the initial data are estimated by TIFNs ${\tilde{A}}_{(i)}, i=\mathrm{1,2},\dots ,n$, the approximate TIFN $\widetilde{B\text{'}}\approx \tilde{B}=f\left({\tilde{A}}_{\left(1\right)},{\tilde{A}}_{\left(2\right)},\dots ,{\tilde{A}}_{\left(n\right)}\right),$ is considered: in such a way that if $f$ is continuous and monotonically increasing with respect to the m first variables and monotonically decreasing with respect to the last n-m: By analogy with the error measurement in the triangular approximation of fuzzy numbers in Andrés-Sánchez and González-Vila (2023), the quality of the relative error measurement in the bounds of $B_{〈\alpha ,\beta 〉}$ calculated with (2a)-(2d) by those of its triangular approximation, (5a)-(5f), $B_{〈\alpha ,\beta 〉}\text{'}$ is, in $B_{\alpha }=\left[{\underset{\_}{B}}_{\alpha },{\overline{B}}_{\alpha }\right],$ and for ${B^{\mathrm{*}}}_{\beta }=\left[{\underset{\_}{B^{\mathrm{*}}}}_{\beta },{\overline{B^{\mathrm{*}}}}_{\beta }\right],$

Therefore, to measure the average relative deviations, we use the weights of (6a) and (6b). In the case of $B_{\alpha }=\left[{\underset{\_}{B}}_{\alpha },{\overline{B}}_{\alpha }\right]$ belonging to a greater $\alpha$-level, it implies a larger reliability. Therefore, we define the weighted average errors for the approximation of $B_{\alpha }$:

On the other hand, in ${B^{*}}_{\beta }=\left[{\underset{\_}{B^{*}}}_{\beta },{\overline{B^{*}}}_{\beta }\right]$, a greater nonmembership degree $\beta$ supposes lower reliability. Therefore, we define the weighted average errors for the approximation of ${B^{*}}_{\beta }$:

The consideration of heterogeneity in mortality involves obtaining death probabilities or instantaneous mortality rates appropriate for the particular for whom life contingencies are being priced (Olivieri, 2006). When the cause of substandard LE is common to a wide group of people, such as smoking, specific mortality tables can be developed. However, in many cases, this is not possible either because there are very specific causes of impairment (e.g., a rare disease) or because the cause of impairment is a combination of risk factors (Pitacco, 2019). Thus, a common alternative is to take as a reference standard death probability, symbolized as $q_x^S$ for a person aged $x$ to fit the actual probability $q_x.$ The probabilities $q_x^S$ consider common conditions affecting a large group, such as climate, pollution, healthcare system, gender, and smoking status.

Subsequently, $q_x^S$ must be transformed to obtain specific probabilities $q_x$, that is, $q_x=f\left(q_x^S\right)$, by introducing individual characteristics that shape the specificity of the evaluated person's LE, such as the presence of any preexisting disability (Olivieri, 2006).

There are numerous ways to obtain$q_x$ from $q_x^S$ in practice (Pitacco, 2019). One of the most common methodologies considered in this study involves setting a parameter $k>0$, the so-called mortality multiplier, such that: where if $k>1$, we have a substandard LE; if $k<1$, it is a preferred risk; and if $k=1$, it is a standard risk (Pitacco, 2019). It is clear that $\frac{\partial q_x}{\partial k},\frac{{\partial q}_x}{\partial q_x^S}\ge 0$.

Standard mortality probabilities can be derived from either a static or a dynamic survival table. In the latter scenario, as seen in the works of Koissi and Shapiro (2006), Andres-Sánchez and González-Vila (2019), and Szymański and Rossa (2021), future survival probabilities are estimated as FNs using fuzzy regression methods and the analytical groundwork of Lee-Carter (1992). The fuzziness of standard probabilities in the context of static mortality tables is used, for example, in Lemaire (1990). Thus, we suppose that the set of one-year death probabilities is given by IFNs, $\widetilde{q_x^S}$: