**5. Sensitivity Analysis**

In this Section, we evaluate the sensitivity of the errors of the triangular approximates seen in Section 4 with respect to the parameter *λ*. The following assumptions are considered:


• The core of *λ* may be low (0.04), medium (0.5), or high (0.96).




Only results for *p*-(*N* ≥ 1) and *π*-4 are shown. Furthermore, in order to avoid very long calculations, we have performed them on a scale with five grades of possibility. However, it can be verified that for other transition and stationary probabilities, and for a greater scale of grades of possibility, the conclusions to be drawn are practically the same. Table 7 shows that:


Table 8 shows the mean asymptotic premiums for the cores of *λ* considered in Table 7 and the most uncertain scenario (left and right spread equal to 0.015). It can be checked that triangular approximates *b*∗ always reach a practically perfect match to *b*<sup>∗</sup>, i.e., errors (as defined for Table 6) are very close to 0.


**Table 7.** *α*-cuts of *p*-(*N* ≥ 1) and *π*-4, their triangular approximate and errors for different parameters *λ*.



**Table 7.** *Cont.*

Source: Own Elaboration.

**Table 8.** *α*-cuts of *b*<sup>∗</sup>, it's triangular approximate and errors for different parameters *λ*.


Source: Own Elaboration.

#### **6. Summary and Further Research**


BMSs are often modelled by means of MCs with crisp probabilities. In this paper, it is considered that transition probabilities of Markovian BMSs are not crisp but uncertain. This uncertainty is captured by using a FN, thus giving rise to the concept of FBMSs. FBMSs modeling is based on the concept of FMC by Buckley and Eslami in [12]. As a result, conventional BMSs can be understood as a particular case of our model where transition probabilities are singletons. The model in [11] represents the uncertainty by means of modal intervals. Since its results can be interpreted as the 0-cuts of ours, that model can also be seen as a particular case of our FBMS.


We assume, as it is often done in actuarial literature, that the number of claims in a period is a Poisson RV. Nonetheless, due to uncertainty, its parameter *λ* is not a real number but a TFN. So, to implement the model presented in the paper, it is necessary, firstly, to structure available information of the behavior of that RV. From this information, the Poisson parameter can be fitted by means of a TFN. Three alternatives to do so are proposed. Subsequently, by using *α*-cut arithmetic, transition probabilities, the stationary distribution function, and the mean asymptotic premium of the FBMS are obtained by means of their *α*-cuts. The lower and upper bounds of these *α*-cuts can be understood as the result of a sensitivity analysis of the BMS that evaluates two extreme scenarios with possibility *α*. That output can be very useful in actuarial decision-making processes since it provides a set of sensitivity analyses that is structured on the basis of their grade of reliability.

Although the mean number of claims, *λ*, is assumed to be a TFN, the outputs from our FBMS do not maintain that shape. However, in the numerical applications developed within the framework of the Irish BMS, we have verified that all the outputs obtained from a triangular *λ* are well approximated by a TFN that maintains the support and core of the original FN. This result is quite interesting. On the one hand, other more complex shapes of FNs can produce drawbacks in information modeling, such as problems with calculations in computer implementation. In this regard, we have observed that the number of optimizing problems to be solved in order to obtain transition probabilities, the stationary distribution, and the mean asymptotic premium is reduced drastically. Likewise, TFNs are very attractive from an insurance decision-making perspective since TFNs admit a very intuitive interpretation even without any knowledge of FST. At least, a TFN provides an estimate of the maximum, minimum, and most feasible values of a variable. Therefore, we feel that the triangular approximations introduced in this document would make it easier to use FMCs in the implementation of BMSs by the insurance industry.

Our methodologic approach can be extended, with the necessary adaptations, to other assumptions for the RV number of claims. Likewise, as far as we are concerned, there are several topics that may be the object of further research. Firstly, a wider investigation on how to apply a fuzzy Poisson regression in a BMS context must be carried out. Secondly, a more in-depth evaluation of the goodness of triangular approximations to BMS probabilities and the mean asymptotic premium is needed. In this respect, a wider range for the values of *λ*, a greater number of classes in the BMS, and other methods to fit triangular approximates must be tested. Thirdly, it is also needed to extend our model to the case in which fuzzy uncertainty in the BMS does not only appear in the number of claims but also their cost. Moreover, to model *λ*, instead of TFNs, other types of FNs, such as GFNs or IFNs, could be considered. We are aware that these tools allow capturing uncertainty with more nuances than FNs. However, their fitting has a greater cost than TFNs since it implies adjusting more parameters. Additionally, implementing computational operations with them is more expensive. This last issue is crucial in our context, especially in complex BMSs like, e.g., the German one. So, we feel that applying FNs suppose a balance between the simplicity of crisp or interval probabilities and more complex representations of uncertain quantities such as GFNs or IFNs. Finally, to evaluate the efficiency of a BMS, it is usually calculated the elasticity of the mean premium (4) with respect to the risk parameter *λ*. To do so, numerical simulations for point values of *λ* within the reference interval [0,1] are implemented (see, e.g., [2]). The use of fuzzy logic may be of interest in this concern. For example, that reference interval can be granulated into linguistic labels such as "low risk", "medium risk" and so on, similarly to that proposed by [27] and [57]. Therefore, elasticity evaluations may be made on the basis of linguistic labels instead of point values on [0,1]. Fuzzy linguistic Markov chains, presented by [58], may be the starting point for this.

**Author Contributions:** All authors have contributed equally to all sections and stages of this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding. The APC has been funded by the University of Barcelona.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** Authors acknowledge helpful suggestions of anonymous reviewers.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Appendix A. Pseudo-Codes of Numerical Applications 2 and 4**
