**1. Introduction**

Since the 80s and 90s, tourism has become a global economic phenomenon, which is a situation that has encouraged the search for competitive models that reveal what makes one destination more interesting than another [1]. Tourism is regarded as a sound alternative to achieve the economic development and social well-being of nations, but especially for developing or less developed ones [2]. However, tourism market trends and the life cycle of destinations bring about the need to undertake renovation processes [3]. Furthermore, the resources that position a destination as the most competitive today may not have any significance in the future [4].

The study of competitiveness became a topic of interest in tourism sector research in the 1990s, with the first researcher to conduct studies on tourism competitiveness being Poon [5]. Subsequently, to understand the role that competitiveness plays in tourism, researchers such as Ritchie and Crouch [6] defined a theoretical and conceptual framework to reveal how a tourist destination manages its competitiveness. The concept progressed from

**Citation:** Flores-Romero, M.B.; Pérez-Romero, M.E.; Álvarez-García, J.; del Río-Rama, M.d.l.C. Fuzzy Techniques Applied to the Analysis of the Causes and Effects of Tourism Competitiveness. *Mathematics* **2021**, *9*, 777. https://doi.org/10.3390/ math9070777

Academic Editor: Jorge de Andres Sanchez

Received: 22 February 2021 Accepted: 29 March 2021 Published: 2 April 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

a perspective focused on tourist attractiveness to the strategic promotion of the tourism industry in a more holistic way, which considers different advantages of competitiveness [7].

Therefore, tourism competitiveness is presented as a key instrument to turn tourism into a factor of economic development [8–10]; it provides countries with the opportunity to maintain their position as leaders in the tourism activity [11] or to obtain a favorable position [12]. The continual interest of the tourism sector in competitiveness is due to the diversification that occurred in this activity; there are destinations competing to have more tourist arrivals or more tourism expenditure, which are indicators that reflect the economic prosperity of their residents [1].

Tourism competitiveness is defined as the ability of a destination to intensify tourism expenditure and gain more visitors while offering satisfying and memorable experiences in a profitable way, as well as improving the well-being of local residents and preserving the natural capital of the destination for future generations [6]. The concept proposed by Ritchie and Crouch [6] was adapted as the primary definition of tourism competitiveness [13], from which others arose such as the one proposed by Acerenza [14], who in a simple way defines tourism competitiveness as the ability of a destination to attract tourists. On the other hand, Dupeyras and MacCallum [15] conceptualize destination competitiveness as the ability of a place to take advantage of its attractiveness and offer quality and innovative tourist services, as well as to gain national and global market shares, while ensuring that the available resources that support tourism are used efficiently and sustainably.

In practice, tourism competitiveness is a construct in which various tangible and intangible factors participate, although it is only in a few, which are known as critical factors, where the greatest options for success or failure lie [16]. Tourism competitiveness brings about collective improvement, both in organizations and institutions, in favor of strengthening the tourism sector and focusing on increasing tourist flow and jobs [17]. It can also be said that destination competitiveness is the ability of the destination to conceive, integrate and provide tourist experiences, which include value-added goods and services that tourists consider substantial [18] and is characterized as a crucial element for the success of tourist destinations [19].

As a line of research, tourism competitiveness has had an interesting development in recent years, with one of its fields being the identification of the factors that affect it [2] and its relationship with variables such as tourism performance. Research papers in which this subject was addressed include those by Imali [20], Milicevic et al. [21], Hanafiah and Zulkifly [22]; Armenski et al. [23], Andradres and Dimanche [24], Amaya et al. [25], Cucculelli and Goffi [26], García and Siles [1], Decasper [27], Castellanos et al. [28], Leung and Baloglu [12], Goffi [19], Gandara et al. [17], Bolaky [29], Rodrigues and Carrasqueira [30] and Pascarella and Fontes [4], to name a few. Models have also been proposed to explain this phenomenon, including those developed by Poon [5], Hassan [31], Health [32], Ritchie and Crouch [6], Dwyer and Kim [33], Acerenza [14], Wei-Chiang [18], Alonso [16] and Jiménez and Aquino [34], among others.

Based on the research carried out to date on tourism competitiveness, it is possible to determine the causes that generate it and its effects on a tourist destination. However, a question arises: do decision makers in the tourism sector share the same perspective as academics/researchers regarding the relationship between the causes and effects of tourism competitiveness?

In this context, the aim of this paper is to compare the opinions held by two groups of experts, decision makers versus academics/researchers, both from the tourism sector, regarding the relationship between the causes and effects of tourism competitiveness. The methodology used to achieve these objectives is the experton and forgotten effects theories, and the Hamming distance.

This manuscript is structured in four sections. The first section establishes the bases of tourism competitiveness and its causes and effects, and sets out the objective of the research. The second section describes the materials and methods; the experton theory, the forgotten effects theory and the Hamming distance are discussed. In the third section, the results obtained are presented and discussed. In the fourth section, the conclusions of the paper are discussed.

## **2. Materials and Methods**

Firstly, the causes and effects of tourism competitiveness were identified by reviewing the models proposed by Poon [5], Hassan [31], Health [32], Ritchie and Crouch [6], Dwyer and Kim [33], Acerenza [14], Wei-Chiang [18], Alonso [16], and Jiménez and Aquino [34]. The causes and effects found are shown in Tables 1 and 2.

At the same time, 8 experts were selected according to their academic and professional background; these people are directly immersed in the tourism sector, either offering a service or conducting research, as applicable. According to the purpose of this work, the governmen<sup>t</sup> sector has been excluded. Two groups of experts were formed: the first group was made up of academics/researchers on the subject of tourism and competitiveness; the second group of experts was made up of people who work in the tourism sector. Next, both groups evaluated the cause–effect, cause–cause and effect–effect relationships based on the endecadary scale shown in Table 3, proposed by Kaufmann and Gil-Aluja [46]. It is worth mentioning that all the experts had the same weight in the evaluation.

> **Table 1.** Causes of tourist competitiveness.


Source: Reproduced from [39], Journal of Intelligent & Fuzzy Systems: 2021.


**Table 2.** Effects of tourist competitiveness.

Source: Reproduced from [39], Journal of Intelligent & Fuzzy Systems: 2021.

**Table 3.** Endecadary scale.


Source: Kaufmann and Gil-Aluja [46].

Based on the evaluations conducted by the experts, six expert tables were constructed, with three for each group. To construct each experton, first the absolute frequencies were calculated (number of experts suggesting each result), then the data were normalized through relative frequencies (division of the absolute frequencies by the total number of experts) and finally, the accumulated relative frequencies were obtained [47–49]. This is done level by level for the 11-point endecadary scale [49] (Table 3).

The forgotten effects theory was used to identify the variables and relationships that remain hidden or generate an indirect impact. This was done for each group of experts, for which the constructed expertons *M* (cause–effect relationship), *A* (cause–cause relationship) and *B* (effect–effect relationship) were used, which correspond to the direct incidence matrices. The matrices *M*, *A* and *B* were convoluted in the following way: *A* ◦ *M* = *AM*

and *AM* ◦ *B* = *M*∗ (*M*∗ represents the accumulated effects matrix), then the forgotten effects matrix was calculated with the formula *O* = *M*∗ − *M*.

Finally, the Hamming distance between expertons was used to compare the opinions that both groups of experts have regarding tourism competitiveness, particularly its causes, effects and the relationship between them. This last step follows a similar process to the Hamming distance between fuzzy subsets in the discrete domain. In this case, all expert evaluations are considered, except for the level α = 0, and the result is normalized by dividing by the number of *n* evaluations considered. The formulation of the Hamming distance between experts is as follows:

$$Distance\ to\ the\ left=d\_I(A,B) = \frac{1}{2n} \sum\_{j=1}^{n} |a\_1(a\_j) - b\_1(a\_j)|\tag{1}$$

$$Distance \text{ to the right} = d\_D(A, B) = \frac{1}{2n} \sum\_{j=1}^{n} |a\_2(a\_j) - b\_2(a\_j)| \tag{2}$$

$$\text{Total distance} = d(A, B) = d\_I(A, B) + d\_D(A, B) \tag{3}$$

It should be noted that only one of the distances was used (left or right), since the information was in individual data and not in intervals.

## *2.1. Theory of Expertons*

There are phenomena in nature that humans evaluate through a subjective opinion and can hardly be classified according to whether or not a property is fulfilled [50]. In this regard, the theory of expertons suggests that to obtain realistic data of phenomena that cannot be measured directly, it is useful to have an aggregate set of the assessments given by experts [46,51].

The theory of expertons extends the probabilistic set concept [52] to uncertain environments that can be evaluated with interval numbers and allows for the analysis of group information considering all individual opinions, producing a single final result. Thus, it makes the information more robust because it is evaluated by several experts and the use of several experts in the analysis generally leads to better decisions [47].

An experton is defined as a generalization of a probabilistic set when the accumulated probabilities are replaced by intervals that decrease monotonically [53]. It arises as a result of a procedure of aggregation of different expert opinions regarding the degree of veracity of a statement and provides the percentage of experts who agree that the veracity of the statement is at least the given value [49].

The theory of expertons, which was developed by Kaufmann [51], is defined as follows:

**Definition 1.** *Let E be a referential set, finite or not; where r experts are required to express their subjective opinion about each element of E in the form of a confidence interval:*

$$\forall \mathbf{x} \in E: \left[ a\_\*^{j}(\mathbf{x}), \ a\_j^\*(\mathbf{x}) \right] \subset \left[ 0, \ \mathbf{1} \right] \tag{4}$$

*where* ⊂ *is an inclusion set and j is the expert. A cumulative complementary law <sup>F</sup>*∗(*<sup>a</sup>*, *x*) *can be established for all <sup>a</sup>j*∗(*x*) *and <sup>F</sup>*<sup>∗</sup>(*<sup>a</sup>*, *x*) *for all a*∗*j* (*x*)*, which is due to a statistic that indicates that for each x* ∈ *E, the lower limits are one way and the upper limits another way. By substituting this approach in Equation (4), the following is obtained:*

$$\forall \mathbf{x} \in E, \; \forall \mathbf{x} \in [0, \ 1]: \tilde{A}(\mathbf{x}) = [F\_\*(a, \mathbf{x}), \; F^\*(a, \mathbf{x})] \tag{5}$$

The symbol *A* that appears in Equation (5) recalls the nature of the concept.


So, the referential set *E* is the following experton:

$$\forall \mathbf{x} \in E, \forall \mathbf{x} \in [0, 1]: \left[F\_\*(\mathbf{a}, \mathbf{x}), \ F^\*(\mathbf{a}, \mathbf{x})\right] = 1\tag{6}$$

Additionally, an empty experton is given by:

$$\forall \mathbf{x} \in E: \left[ F\_\*\left(\mathbf{a}, \mathbf{x}\right), F^\*\left(\mathbf{a}, \mathbf{x}\right) \right] = \begin{cases} 1, & \mathbf{a} = 0 \\ 0, & \mathbf{a} \neq 0 \end{cases} \tag{7}$$

And it has the following properties:

$$\begin{array}{ccl} \forall \mathbf{x} \in E, \forall a, a' & \in \ [0, 1]: (a < a')\\ & \implies \left( \left[ F\_\*(a', \mathbf{x}), F^\*(a', \mathbf{x}) \right] \subset\_i \left[ F\_\*(a, \mathbf{x}), F^\*(a', \mathbf{x}) \right] \right) \end{array} \tag{8}$$

The expression ⊂*i* that appears in Equation (8) is known as the inclusion interval. It can be expressed as follows:

$$\left(\left(\mathfrak{a} < \mathfrak{a}'\right) \implies \left(\left(F\_\*\left(\mathfrak{a}', \mathfrak{x}\right) \le F\_\*\left(\mathfrak{a}, \mathfrak{x}\right)\right) \text{ and } \left(F^\*\left(\mathfrak{a}', \mathfrak{x}\right) \le F^\*\left(\mathfrak{a}, \mathfrak{x}\right)\right)\right) \tag{9}$$

When a final consideration or interpretation of the phenomena is required, the experton can be reduced to a single representative value by reducing the entropy of the results. This can be obtained by calculating the mathematical expectation of the probabilistic set [53].

## *2.2. Forgotten Effects Theory*

Any activity is subject to cause–effect incidents [54] and to the possibility of forgetting some causal relationships that are not explicit, obvious or visible [55]. In situations of uncertainty and volatility, there are variables that are not immediately detectable because they are hidden as a result of an accumulation of causes [46]. To identify the incidents that are not so evident between variables, but are fundamental for decision making, the theory of forgotten effects has proven to be an effective approach when seeking to maximize the information retrieved from the complex relationships between variables and to minimize errors that can occur in these processes [56].

The theory of forgotten effects is an extension of fuzzy logic applications [57]. This theory allows for an approach to the objective of globalizing the direct and indirect incidences between a set of causes and effects [46], since it suggests that all events, phenomena and facts that surround people are based on some type of system or subsystem. Therefore, they are subject to some type of cause–effect relationship, with the possibility that voluntarily or involuntarily some relationships are not directly perceived [58].

The theory of forgotten effects developed by Kaufmann and Gil-Aluja [46] is defined as follows:

**Definition 2.** *The existence of two sets, A* = *aii* = 1, 2, . . . , *n and B* = *bjj* = 1, 2, . . . , *m, is assumed. It is conjectured that an impact of ai on bj prevails if the value of the characteristic membership function of ai*, *bj is evaluated in a* [0, 1] *range, that is:*

$$\forall \ (a\_i, b\_j) \Rightarrow \mu \left(a\_i, b\_j\right) \in \ [0, 1] \tag{10}$$

The set of pairs of elements evaluated is the direct incidence matrix (*M*), which shows the cause–effect relationship in different degrees caused by the corresponding set *A* (causes) and set *B* (effects), as shown below:


The next step is to proceed to detecting the forgotten effects. For this, it is assumed that there is a third set of elements, called *C*, expressed in the following way *C* = *Ck k* = 1, 2, . . . , *k* . This set consists of elements that are effects of set *B* and has an incidence matrix which is expressed as follows:


So far, there are two incidence matrices, *M* and *N*, and they both have element *B* in common. This relationship is expressed as:

$$M \subset A \\ x \\ B \text{ y } N \subset B \\ x \mathbb{C} \tag{11}$$

Next, the max–min operator (convolution) is used to detect the relationship between sets *A* and *C* using *B*. As a result, a new incidence matrix is generated, which is expressed by:

$$\begin{array}{c} M \circ N = P\\ P \subset A \ge \text{C} \end{array} \tag{12}$$

This new relationship is formulated in the following way:

$$\forall (a\_i, c\_z \in A \ge \text{C}) \tag{13}$$

$$
\mu(a\_{\!i}, c\_{\!z})M \diamond N = \forall\_{\!bj} \left( \mu\_M(a\_{\!i}, b\_{\!j}) \land \mu\_N(b\_{\!j}, c\_{\!z}) \right) \tag{14}
$$


The matrix that results from the max–min operation is:

Matrix *P* defines the causal relationships between the elements of the *A* and *C* sets, at the intensity or degree that *B* allows for.

#### *2.3. The Hamming Distance between Fuzzy Subsets*

The distance measurement plays a vital role in pattern recognition, information fusion, decision making and other fields [59]. It is an important issue in fuzzy sets and their extensions [60,61]. There are several distance measures that have been introduced by researchers to solve problems in various fields; among these distances, the Euclid distance and the Hamming distance are widely used [62]. Specifically, the Hamming distance was developed by Richard Wesley Hamming in 1950 and is defined as follows [63]:

**Definition 3.** *The distance <sup>D</sup>*(*<sup>x</sup>*, *y*) *between two x and y points is defined as the number of coordinates for which x and y are different. This function fulfils the three usual conditions for a metrics:*

> *<sup>D</sup>*(*<sup>x</sup>*, *y*) = 0 if and only if *x* = *y* (15)

$$D(\mathbf{x}, \mathbf{y}) = D(\mathbf{y}, \mathbf{x}) > 0 \text{ if } \mathbf{x} \neq \mathbf{y} \tag{16}$$

$$D(z, y) \, + D(y, z) \, \ge D(x, z) \text{ (triangular inequality)} \tag{17}$$

The Hamming distance, like the Theory of Expertons, is considered a fuzzy numerical model [64]; it is known for its ability to calculate the difference between two sets or elements and is identified as one of the multi-criteria decision-making approaches. Also, when counting the number of specific differences between two permutations, it is a natural choice to measure the distance between assignments or pairings [65]. This approach helps to solve many problems related to biology, science and technology due to its ability to construct some related distance measures, in particular, similarity and proximity, which usually become a norm in several problems [66].

Before defining the Hamming distance between fuzzy subsets in the discrete domain,it is necessary to understand the concepts of fuzzy sets and subsets, which in the words of Zadeh [67] are defined below:

**Definition 4.** *Let X be a universe of analysis, then a fuzzy set A in X is defined as a set of pairs established in the following way:*

$$A = \{ \langle \mathbf{x}, \mu\_A(\mathbf{x}) \rangle : \mathbf{x} \in X \} \tag{18}$$

*where μA* : *X* → [0, 1] *is the membership function that characterizes the universe of analysis A and μA*(*x*) *is the degree of membership of x in A.*

**Definition 5.** *A fuzzy subset A of a universe of analysis X is characterized by a membership function μA* : *X* → [0, 1] *which associates to each element x of X a number μA*(*x*) *in an interval* [0, 1]*; thus, μA*(*x*) *represents the degree of membership of x in A.*

Regarding the degree of membership *μA*(*x*) shown in the two previous definitions, it can be interpreted as the degree of compatibility of *x* with the concept represented by *A* or as the degree of possibility of *x* given *A*. In these cases, the function *μA* : *X* → [0, 1] can be referred to as a compatibility function. It should be noted that the meaning attributed to a particular numerical value of the membership function is purely subjective in nature [57]. From the above, it is possible to state that the degree of non-membership of *x* in *A* is equal to 1 − *μA*(*x*) [68].

Therefore, the Hamming distance of two fuzzy subsets is defined as follows [69]:

**Definition 6.** *Given two fuzzy subsets A and B with a reference set X = { x*1 *, x*2*,* ... *, xn} and membership functions μA and μB, the Hamming distance is defined as:*

$$d(A, B) = \sum\_{j=1}^{n} |\mu\_A(\mathbf{x}\_j) - \mu\_B(\mathbf{x}\_j)| \tag{19}$$

*where μA and μB* ∈ [0, 1].

In this case, the Hamming distance measures the relationship between variables in a study of facts and how they fit a profile. Finally, it calculates the distance between the extremes of the intervals [64].
