Evaluation of the intuitionistic fuzzy importance of attributes based on the correlation coefficient under weakest triangular norm and application to the hotel services

Abstract

The correlation between the performance of attributes and the overall satisfaction perceived by the customers is a successful indirect approach to evaluate the importance of the attributes in services. Recently, the fuzzy importance of attributes was discussed as the fuzzy correlation between the performance of attributes and the overall satisfaction. In this paper, the concept of fuzzy set is extended to the idea of intuitionistic fuzzy set and a novel simple approach is presented to evaluate the importance of attributes as the intuitionistic fuzzy correlation between the performance of attributes and the overall satisfaction. The qualitative input data such as customer’s opinions are available and expressed as linguistic terms. Each of these linguistic terms is mathematically represented by membership and non-membership functions of intuitionistic fuzzy number instead of a fuzzy number or a crisp number. The calculation is based on the weakest triangular norm (t-norm) arithmetic operations on triangular intuitionistic fuzzy numbers. The proposed approach has been applied to a survey with respect to the quality of hotel services in Oradea (Romania) and then compared with other existing approaches to calculating the importance of hotel quality attribute.

Keywords

Correlation coefficient intuitionistic fuzzy number weakest t-norm arithmetic triangular intuitionistic fuzzy number intuitionistic fuzzy importance

1. Introduction

Due to subjective and fuzziness of human perceptions, fuzzy numbers or generalizations of fuzzy numbers are often used for modeling and solving many real life problems having uncertain and/or incomplete information in different areas including decision making, engineering, science, economy, social sciences and other areas [1 , 48]. The fuzzy numbers are more suitable than crisp numbers to represent the customer’s opinions related to the performance and importance of service attributes (see, e.g., [5 , 51]).

The traditional likert scale is the widely used psychometric scale that measures the customer’s responses in a survey research. Each customer indicates their level of agreement with a declarative statement (see, e.g., [46]). For example, for a 5-point Likert scale, each scale point could be labeled according to its agreement level: 1 = strongly disagree (SD), 2 = disagree (D), 3 = neither disagree nor agree (NN), 4 = agree (A), and 5 = strongly agree (SA).

However, traditional Likert scale has some weaknesses. Li [42] addressed that the traditional Likert scale leads to loss of some information during measurement. The other weakness of this scaling comes from its closed response format [10]. To improve the traditional Likert scale, many researchers have applied fuzzy logic [34, 42]. Since the human thinking is subjective and ambiguous, therefore it would be more appropriate to model customer’s responses as fuzzy numbers instead of crisp numbers (see, e.g., [1 , 49]).

Fuzzy set theory has been shown to be a useful tool to handle vague situations by attributing a degree to which a certain object belongs to a set (see [25, 26]). In real life, a person may assume that an object belongs to a set to a certain degree, but it is possible that he is not so sure about it. In other words, there may be a hesitation or uncertainty about the membership degree of x in A. In fuzzy set theory, there is no means to incorporate that hesitation in the membership degrees. A possible solution was to use intuitionistic fuzzy set (IFS), defined by Atanassov [21, 22]. The concept of an IFS can be viewed as an alternative approach to define a fuzzy set in cases where available information is not sufficient for the definition of an imprecise concept by means of a conventional fuzzy set. Presently, intuitionistic fuzzy sets (IFSs) are being used in different research fields [28 , 52]. Burillo [39] proposed definition of intuitionistic fuzzy number (IFN), studied perturbations of intuitionistic fuzzy numbers (IFNs) and the first properties of the correlation between these numbers.

Correlation is a statistical technique that measures the strength and the direction of the relationship between two variables. The Pearson’s correlation coefficient has been used to analyze the similarity between lists of ratings in recommender systems [13]. Batyrshin et al. [17] presented that Pearson’s correlation coefficient can mislead the analysis of relationships between profiles of ratings in recommender systems for bipolar rating scales. Batyrshin et al. [12, 17] proposed a new correlation measures for measuring similarity and association of bipolar rating profiles. The correlation coefficient between the performance of attributes and the overall customer satisfaction has also been preferred by many researchers to calculate the importance of attributes (see [8, and 50]).

The determination of the importance of attributes is an essential step in several methods related to the decision theory. The measurement of the importance of attributes can be obtained by direct or indirect methods. The direct methods (by surveys or conjoint analysis) are still widely used, but they have significant disadvantages pointed out in many papers (e.g., [9, and 48]). Over the past few decades, many authors consider that the importance of attributes can be inferred through mathematical methods (see [16]). The most appropriate method to obtain the derived importance of attributes is still subject to continuous debates (see [27]). The calculation of the importance of attributes as the correlation coefficient between the performance of attributes and the overall customer satisfaction is often preferred (see [8, and 50]). The correlation coefficient of fuzzy numbers as a fuzzy number was introduced in [47]. When the input data are considered as fuzzy numbers, many methods [2, 3] have been proposed to determine the fuzzy importance of attributes based on the fuzzy correlation coefficient between the performance of attributes and the overall customer satisfaction.

In this paper, the concept of fuzzy set is extended to the concept of IFS and a novel indirect method is proposed to compute the intuitionistic fuzzy importance of attributes based on the correlation coefficient of IFNs under the weakest t-norm arithmetic operations (see [6 , 32]). The customer’s opinions or responses are measured on 5-point Likert scale and expressed as linguistic terms. The customer’s opinions are mathematically represented by membership and non-membership functions of IFNs instead of fuzzy numbers or crisp numbers. The main obvious advantages of using weakest t-norm arithmetic operations on triangular IFNs (TIFNs) are that the calculation is simplified, more exact results are obtained with smaller fuzziness (see, e.g. [22 , 41]).

The rest part of this paper is organized as follows. In Section 2, we give the review of basic concepts related to IFSs. Evaluation of the importance of attributes by the correlation coefficient method is presented in section 3. In Section 4, a novel algorithm is proposed for evaluating the intuitionistic fuzzy importance as correlation coefficient using weakest t-norm based arithmetic operations. In section 5, the proposed method is exemplified to the study of the quality of hotel services. A survey applied to the customers of four 4-stars hotels from Oradea, Romania gives us the input data. The final section makes conclusions.

2. Preliminaries

2.1. Intuitionistic fuzzy set

Let X be a universe of discourse. Then an IFS $\tilde{A}$ on X is a set of ordered triples given by $\tilde{A} = {< x, μ_{\tilde{A}} (x), ν_{\tilde{A}} (x) > : x \in X},$ (1) where $μ_{\tilde{A}} : X \to [0, 1]$ and $ν_{\tilde{A}} : X \to [0, 1]$ are functions such that $0 \leq μ_{\tilde{A}} (x) + ν_{\tilde{A}} (x) \leq 1, \forall x \in X .$ For each x, the numbers $μ_{\tilde{A}} (x)$ and $ν_{\tilde{A}} (x)$ represent respectively the degree of membership and degree of non-membership of the element x to $\tilde{A} \subseteq X$ . For each x, the intuitionistic fuzzy index of x in $\tilde{A}$ is defined as $π_{\tilde{A}} (x) = 1 - μ_{\tilde{A}} (x) - ν_{\tilde{A}} (x)$ .

2.2. Intuitionistic fuzzy number

An IFS $\tilde{A} = {〈 x, μ_{\tilde{A}} (x), ν_{\tilde{A}} (x) 〉 : x \in R}$ on real line R is called an IFN [22 , 40] if

$\tilde{A}$ is IF-normal (i.e. there exist at least two points x₁, x₂ ∈ X such that $μ_{\tilde{A}} (x_{1}) = 1$ and $ν_{\tilde{A}} (x_{2}) = 1$ .

$\tilde{A}$ is IF-convex (i.e. its membership function μ is fuzzy convex and its non-membership function is fuzzy concave).

$μ_{\tilde{A}} (x)$ is upper semicontinuous and $ν_{\tilde{A}} (x)$ is lower semicontinuous.

The support $S (\tilde{A}) = {x \in X : ν_{\tilde{A}} (x) < 1}$ of IFN $\tilde{A}$ is bounded.

We can describe an IFN $\tilde{A}$ with membership function $μ_{\tilde{A}} (x)$ in the form $μ_{\tilde{A}} (x) = {\begin{matrix} 0, & if x < a_{1} \\ f_{A} (x), & if a_{1} \leq x < a_{2} \\ 1, & if a_{2} \leq x \leq a_{3} \\ g_{A} (x), & if a_{3} < x \leq a_{4} \\ 0, & if x > a_{4} \end{matrix}$ (2) and non-membership function $ν_{\tilde{A}} (x)$ as $ν_{\tilde{A}} (x) = {\begin{matrix} 1, & if x < a_{1}^{'} \\ h_{A} (x), & if a_{1}^{'} \leq x < a_{2}^{'} \\ 0, & if a_{2}^{'} \leq x \leq a_{3}^{'} \\ k_{A} (x), & if a_{3}^{'} < x \leq a_{4}^{'} \\ 1, & if x > a_{4}^{'} \end{matrix}$ (3) where $a_{1}^{'}, a_{1}, a_{2}^{'}, a_{2}, a_{3}, a_{3}^{'}, a_{4}, a_{4}^{'} \in R$ such that $a_{1}^{'} \leq a_{1} \leq a_{2}^{'} \leq a_{2} \leq a_{3} \leq a_{3}^{'} \leq a_{4} \leq a_{4}^{'}$ . The functions f_A, g_A, h_A and k_A are called the sides of IFN, where functions f_A, k_A are non-decreasing and functions g_A, h_A are non-increasing.

The expected interval $EI (\tilde{A})$ of an IFN $\tilde{A}$ is a crisp interval given by (see [40])

$EI (\tilde{A}) = [E_{*} (\tilde{A}), E^{*} (\tilde{A})]$ (4) where $E_{*} (\tilde{A}) = \frac{a_{1}^{'} + a_{2}}{2} + \frac{1}{2} \int_{a_{1}^{'}}^{a_{2}^{'}} h_{A} (x) dx - \frac{1}{2} \int_{a_{1}}^{a_{2}} f_{A} (x) dx$ $E^{*} (\tilde{A}) = \frac{a_{3} + a_{4}^{'}}{2} + \frac{1}{2} \int_{a_{3}}^{a_{4}} g_{A} (x) dx - \frac{1}{2} \int_{a_{3}^{'}}^{a_{4}^{'}} k_{A} (x) dx$

The expected value $EV (\tilde{A})$ of an IFN $\tilde{A}$ is the center of the expected interval of $\tilde{A}$ i.e. $EV (\tilde{A}) = \frac{E_{*} (\tilde{A}) + E^{*} (\tilde{A})}{2}$ (5)

The IFNs with simple membership and non-membership functions are preferred in practice. A triangular IFN denoted by $\tilde{A} = (m, α, β, γ, δ)$ can also be defined by membership function $μ_{\tilde{A}}$ and non-membership function $ν_{\tilde{A}}$ as $μ_{\tilde{A}} (x) = {\begin{matrix} \frac{x - (m - α)}{α}, & if m - α \leq x \leq m, \\ \frac{(m + β) - x}{β} & if m \leq x \leq m + β, \\ 0, & otherwise \end{matrix}$ (6) $ν_{\tilde{A}} (x) = {\begin{matrix} \frac{m - x}{γ} & if m - γ \leq x \leq m, \\ \frac{x - m}{δ}, & if m \leq x \leq m + δ, \\ 0, & otherwise \end{matrix}$ (7)

For triangular IFN (TIFN) $\tilde{A} = (m, α, β, γ, δ)$ $EV (\tilde{A}) = m + \frac{(β + δ) - (α + γ)}{8}$ (8)

2.3. Triangular norm

A t-norm T is an associative, commutative, non-decreasing binary function on [0,1] i.e. T : [0, 1] ² → [0, 1] such that T (x, 1) = x for each x ∈ [0, 1]. Mathematically, the most important t- norms (see [15]) are

Algebraic product: T_P (x, y) = xy

Standard intersection: T_M (x, y) = min(x, y)

Bounded difference: T_L (x, y) = max(0, x + y - 1)

$\begin{matrix} Weakest t - norm (T_{w}) : T_{W} (x, y) \\ = {\begin{matrix} x, & if y = 1, \\ y, & if x = 1, \\ 0, & otherwise . \end{matrix} \end{matrix}$

The present research applies T_w based arithmetic operations on TIFNs due to its shape preserving and fuzziness reducing characteristics within an uncertain environment [3 , 19 and 20].

2.4. T_w-based arithmetic operations on TIFNs

The T_w-based arithmetic operations have some obvious advantages: the calculation is drastically simplified, smaller fuzzy spreads are obtained in results, and the obtained results are more exact. The T_w-based addition and multiplication preserve the shape of IFNs, in particular they preserve the TIFNs [31, 32].

Let $\tilde{A} = (a, α_{A}, β_{A}, γ_{A}, δ_{A})$ and $\tilde{B} = (b, α_{B},$ β_B, γ_B, δ_B) be two TIFNs. Then T_w-based arithmetic operations on $\tilde{A} and \tilde{B}$ are summarized as follows: $\tilde{A} \oplus \tilde{B} = (a, α_{A}, β_{A}, γ_{A}, δ_{A}) \oplus (b, α_{B}, β_{B}, γ_{B}, δ_{B}) = (a + b, max (α_{A}, α_{B}), max (β_{A}, β_{B}), max (γ_{A}, γ_{B}), max (δ_{A}, δ_{B}))$ (9) $\tilde{A} ⊖ \tilde{B} = (a, α_{A}, β_{A}, γ_{A}, δ_{A}) ⊖ (b, α_{B}, β_{B}, γ_{B}, δ_{B}) = (a - b, max (α_{A}, β_{B}), max (β_{A}, α_{B}), max (γ_{A}, δ_{B}), max (δ_{A}, γ_{B}))$ (10) $λ \tilde{A} = (λ a, λ α_{A}, λ β_{A}, λ γ_{A}, λ δ_{A}), λ \in R, λ > 0$ (11) $\sqrt{\tilde{A}} = \sqrt{(a, α_{A}, β_{A}, γ_{A}, δ_{A})} = (\sqrt{a}, \frac{α_{A}}{\sqrt{a}}, \frac{β_{A}}{\sqrt{a}}, \frac{γ_{A}}{\sqrt{a}}, \frac{δ_{A}}{\sqrt{a}}) for a > 0$ (12) $\tilde{A} \otimes \tilde{B} = (a, α_{A}, β_{A}, γ_{A}, δ_{A}) \otimes (b, α_{B}, β_{B}, γ_{B}, δ_{B}) = {\begin{matrix} (ab, max (α_{A} b, α_{B} a), max (β_{A} b, β_{B} a) \\ max (γ_{A} b, γ_{B} a), max (δ_{A} b, δ_{B} a)), & if a, b > 0 \\ (ab, max (- β_{A} b, - β_{B} a), max (- α_{A} b, - α_{B} a), \\ max (- δ_{A} b, - δ_{B} a), max (- γ_{A} b, - γ_{B} a)), & if a, b < 0 \\ (ab, max (α_{A} b, - β_{B} a), max (β_{A} b, - α_{B} a), \\ max (γ_{A} b, - δ_{B} a), max (δ_{A} b, - γ_{B} a)), & if a < 0, b > 0 \\ (ab, max (- β_{A} b, α_{B} a), max (- α_{A} b, β_{B} a), \\ max (- δ_{A} b, γ_{B} a), max (- γ_{A} b, δ_{B} a)), & if a > 0, b < 0 \\ (0, - β_{A} b, - α_{A} b, - δ_{A} b, - γ_{A} b), & if a = 0, b < 0 \\ (0, 0, 0) & if a = 0, b = 0 \end{matrix}$ (13) $\begin{matrix} \tilde{A} \emptyset \tilde{B} = (a, α_{A}, β_{A}, γ_{A}, δ_{A}) \emptyset (b, α_{B}, β_{B}, γ_{B}, δ_{B}) \\ = {\begin{matrix} (\frac{a}{b}, max (\frac{α_{A}}{b}, \frac{a β_{B}}{b (b + β_{B})}), max (\frac{β_{A}}{b}, \frac{a α_{B}}{b (b - α_{B})}), \\ max (\frac{γ_{A}}{b}, \frac{a δ_{B}}{b (b + δ_{B})}), max (\frac{δ_{A}}{b}, \frac{a γ_{B}}{b (b - γ_{B})})), if a, b > 0, b - α_{B} > 0, b - γ_{B} > 0 \\ (\frac{a}{b}, max (\frac{- β_{A}}{b}, - \frac{a α_{B}}{b (b - α_{B})}), max (\frac{α_{A}}{b}, \frac{a β_{B}}{b (b + β_{B})}), \\ max (\frac{- δ_{A}}{b}, - \frac{a γ_{B}}{b (b - γ_{B})}), max (\frac{γ_{A}}{b}, \frac{a δ_{B}}{b (b + δ_{B})})), if a, b < 0, b + β_{B} < 0, b + δ_{B} < 0 \\ (\frac{a}{b}, max (\frac{α_{A}}{b}, - \frac{a α_{B}}{b (b - α_{B})}), max (\frac{β_{A}}{b}, - \frac{a β_{B}}{b (b + β_{B})}), \\ max (\frac{γ_{A}}{b}, - \frac{a γ_{B}}{b (b - γ_{B})}), max (\frac{δ_{A}}{b}, - \frac{a δ_{B}}{b (b + δ_{B})})), if a < 0, b > 0, b - α_{B} > 0, b - γ_{B} > 0 \\ (\frac{a}{b}, max (- \frac{β_{A}}{b}, \frac{a β_{B}}{b (b + β_{B})}), max (- \frac{α_{A}}{b}, \frac{a α_{B}}{b (b - α_{B})}), \\ max (- \frac{δ_{A}}{b}, \frac{a δ_{B}}{b (b + δ_{B})}), max (- \frac{γ_{A}}{b}, \frac{a γ_{B}}{b (b - γ_{B})})), if a > 0, b < 0, b + β_{B} < 0, b + δ_{B} < 0 \\ (0, - \frac{β_{A}}{b}, - \frac{α_{A}}{b}, - \frac{δ_{A}}{b}, - \frac{γ_{A}}{b}), if a = 0, b < 0 (15) \end{matrix} \end{matrix}$

2.5. Linguistic terms

When the classical quantitative data are unavailable or inadequate, then qualitative data such as expert’s opinions can be used in the description of the situations. Most often, experts are more comfortable answering to questions using linguistic terms. An adequate representation of the linguistic terms is by trapezoidal or triangular shaped fuzzy numbers or IFNs. For example, if we opt for a 7-level scale in a survey then we may consider the linguistic terms in the set {very poor, poor, medium poor, medium, medium good, good, very good} for rating the performance of the given attributes by the customers. Of course, other linguistic terms and/or fuzzy numbers or extensions of fuzzy numbers can be subjectively chosen too (see, e.g., [37]). We point out here that in [49] the fuzzy numbers representing the linguistic terms are decided by customers, taking into account their responses regarding the range of each linguistic term. Some tentative to assign fuzzy numbers to linguistic variables in an objective way are presented in [42, 44]. In the present study, linguistic terms are mathematically represented by TIFNs instead of fuzzy numbers.

3. Evaluation of the importance of attributes by correlation method

Let us consider n attributes A₁, A₂,…, A_n of a service and m customers C₁, C₂,… C_m, consumers of that service. Let X_ij be the performance of the attribute A_j, j = 1, 2,…, n in the opinion of customer C_i, i = 1, 2,…, m, X_i be the overall level of satisfaction of the customer C_i and W_ij be the importance of the attribute A_j in the opinion of C_i. The importance W_j of the attribute A_j can be calculated by a direct method, aggregating the values W_ij. Direct methods are still widely used but they have some important disadvantages, pointed out in [2 , 48].

The correlation coefficient between the performance perceived for each attribute and the overall satisfaction is a successful indirect method to determine the importance of the attributes in the crisp case (see [8, 50]). Based on the classical definition of the correlation coefficient of two variables [14], the importance of the attribute A_j is given as the correlation coefficient between {X_1j, X_2j,…, X_mj} and {X₁, X₂,…, X_m}:

$W_{j} = \frac{\sum_{i = 1}^{m} (X_{ij} - X_{j}^{M}) (X_{i} - X^{M})}{\sqrt{\sum_{i = 1}^{m} {(X_{ij} - X_{j}^{M})}^{2} \sum_{i = 1}^{m} {(X_{i} - X^{M})}^{2}}}$ (14)

$where X_{j}^{M} = \frac{1}{m} \sum_{i = 1}^{m} X_{ij} and X^{M} = \frac{1}{m} \sum_{i = 1}^{m} X_{i}$ (15)

Liu and Kao [47] introduced the correlation coefficient for two variables expressed by fuzzy numbers. Following the idea in the crisp case, Ban et al. [2, 3] presented methods to compute the fuzzy importance of attributes as the correlation coefficient between the performance of attributes and the overall satisfaction when the input data were considered as fuzzy numbers. In fuzzy set theory, there is no means to incorporate the hesitation occurring in the membership degrees. To overcome this problem, in this paper, the concept of fuzzy number is extended by the idea of IFN and a novel approach is proposed to evaluate the importance of attributes as the intuitionistic fuzzy correlation coefficient between the performance of attributes and the overall satisfaction when qualitative input data is mathematically represented by TIFNs. It is well known that T_w - based addition and multiplication preserves the shape of fuzzy numbers and more exact results are obtained with smaller fuzzy spreads. Therefore, T_w - based arithmetic operations on TIFNs are used in the calculation.

If the IFN ${\tilde{X}}_{ij}$ denotes the performance of the attribute A_j in the opinion of the customer C_i and the IFN ${\tilde{X}}_{i}$ denotes the overall satisfaction in the opinion of the customer C_i, then the intuitionistic fuzzy importance of the attribute A_j can be formally defined as ${\tilde{W}}_{j} = \frac{\sum_{i = 1}^{m} ({\tilde{X}}_{ij} - {\tilde{X}}_{j}^{M}) ({\tilde{X}}_{i} - {\tilde{X}}^{M})}{\sqrt{\sum_{i = 1}^{m} {({\tilde{X}}_{ij} - {\tilde{X}}_{j}^{M})}^{2} \sum_{i = 1}^{m} {({\tilde{X}}_{i} - {\tilde{X}}^{M})}^{2}}}$ (16) $where {\tilde{X}}_{j}^{M} = \frac{1}{m} \sum_{i = 1}^{m} {\tilde{X}}_{ij} and {\tilde{X}}^{M} = \frac{1}{m} \sum_{i = 1}^{m} {\tilde{X}}_{i} .$ (17)

4. Evaluation of the intuitionistic fuzzy importance of attributes by correlation method under T_w - based arithmetic operations on TIFNs

In this section, a T_w -based algorithm is described to calculate the intuitionistic fuzzy importance of the attributes when known qualitative input data are mathematically represented as TIFNs.

Let ${\tilde{X}}_{ij} = (x_{ij}, α_{ij}, β_{ij}, γ_{ij}, δ_{ij})$ be the performance of the attribute A_j, j = 1, 2,…, n in the opinion of the customer C_i, i = 1, 2,…, m and ${\tilde{X}}_{i} = (x_{i}, α_{i}^{'}, β_{i}^{'}, γ_{i}^{'}, δ_{i}^{'})$ be the overall satisfaction in the opinion of the customer C_i, i = 1, 2,…, m. The steps of proposed algorithm to evaluate the intuitionistic fuzzy importance $\tilde{W_{j}}$ of the attribute A_j, j = 1, 2,…, n are given below.

Step 1. Compute the mean of ${\tilde{X}}_{ij}$ and ${\tilde{X}}_{i}$ (use (10) and (12)) $\begin{matrix} {\tilde{X}}^{M} = \frac{1}{m} \sum_{i = 1}^{m} {\tilde{X}}_{i} = (\frac{1}{m} \sum_{i = 1}^{m} x_{i}, \frac{1}{m} max_{1 \leq i \leq m} α_{i}^{'}, \\ \frac{1}{m} max_{1 \leq i \leq m} β_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} γ_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} δ_{i}^{'}) \end{matrix}$ $\begin{matrix} {\tilde{X}}_{j}^{M} = \frac{1}{m} \sum_{i = 1}^{m} {\tilde{X}}_{ij} = (\frac{1}{m} \sum_{i = 1}^{m} x_{ij}, \frac{1}{m} max_{1 \leq i \leq m} α_{ij}, \\ \frac{1}{m} max_{1 \leq i \leq m} β_{ij}, \frac{1}{m} max_{1 \leq i \leq m} γ_{ij}, \frac{1}{m} max_{1 \leq i \leq m} δ_{ij}) \end{matrix}$

Step 2. Compute (use (11)) $\begin{matrix} {\tilde{X}}_{i} - {\tilde{X}}^{M} = \\ (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}, max {α_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} β_{i}^{'}}, \\ max {β_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} α_{i}^{'}}, max {γ_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} δ_{i}^{'}}, \\ max {δ_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} γ_{i}^{'}}) \end{matrix}$ $\begin{matrix} {\tilde{X}}_{ij} - {\tilde{X}}_{j}^{M} = \\ (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij}, max {α_{ij}, \frac{1}{m} max_{1 \leq i \leq m} β_{ij}}, \end{matrix}$

$\begin{matrix} max {β_{ij}, \frac{1}{m} max_{1 \leq i \leq m} α_{ij}}, max {γ_{ij}, \frac{1}{m} max_{1 \leq i \leq m} δ_{ij}}, \\ max {δ_{ij}, \frac{1}{m} max_{1 \leq i \leq m} γ_{ij}}) \end{matrix}$

Step 3. Compute (use (14)) ${\tilde{R}}_{i} = {({\tilde{X}}_{i} - {\tilde{X}}_{M})}^{2} = (c_{i}^{'}, φ_{i}^{'}, ψ_{i}^{'}, ρ_{i}^{'}, σ_{i}^{'}), where$ $\begin{matrix} c_{i}^{'} = {(x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i})}^{2}, \\ If x_{i} \geq \frac{1}{m} \sum_{i = 1}^{m} x_{i} then φ_{i}^{'} \\ = max {α_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} β_{i}^{'}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}) \\ ψ_{i}^{'} = max {β_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} α_{i}^{'}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}) \\ ρ_{i}^{'} = max {γ_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} δ_{i}^{'}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}) \\ σ_{i}^{'} = max {δ_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} γ_{i}^{'}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}) \end{matrix}$

$\begin{matrix} If x_{i} \leq \frac{1}{m} \sum_{i = 1}^{m} x_{i} then φ_{i}^{'} \\ = - max {β_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} α_{ii}^{'}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}) \\ ψ_{i}^{'} = - max {α_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} β_{i}^{'}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}) \\ ρ_{i}^{'} = - max {δ_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} γ_{i}^{'}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}) \\ σ_{i}^{'} = - max {γ_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} δ_{i}^{'}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}) \end{matrix}$

Step 4. Compute ${\tilde{S}}_{ij} = {({\tilde{X}}_{ij} - {\tilde{X}}_{j}^{M})}^{2} = (c_{ij}, φ_{ij},$ }, ψ_ij ρ_ij, σ_ij), $where c_{ij} = {(x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})}^{2},$ (use (14)) $\geq \frac{1}{m} \sum_{i = 1}^{m} x_{ij} then$ $φ_{ij} = max {α_{ij}, \frac{1}{m} max_{1 \leq i \leq m} β_{ij}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})$ $ψ_{ij} = max {β_{ij}, \frac{1}{m} max_{1 \leq i \leq m} α_{ij}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})$ $ρ_{ij} = max {γ_{ij}, \frac{1}{m} max_{1 \leq i \leq m} δ_{ij}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})$ $σ_{ij} = max {δ_{ij}, \frac{1}{m} max_{1 \leq i \leq m} γ_{ij}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})$ $x_{ij} \leq \frac{1}{m} \sum_{i = 1}^{m} x_{ij} then$ $φ_{ij} = - max {β_{ij}, \frac{1}{m} max_{1 \leq i \leq m} α_{ij}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})$ $ψ_{ij} = - max {α_{ij}, \frac{1}{m} max_{1 \leq i \leq m} β_{ij}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})$ $ρ_{ij} = - max {δ_{ij}, \frac{1}{m} max_{1 \leq i \leq m} γ_{ij}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})$ $σ_{ij} = - max {γ_{ij}, \frac{1}{m} max_{1 \leq i \leq m} δ_{ij}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})$

Step 5. Compute $\tilde{R} = \sum_{i = 1}^{m} {\tilde{R}}_{i} = \sum_{i = 1}^{m} {({\tilde{X}}_{i} - {\tilde{X}}_{M})}^{2}$ $= (c^{'}, φ^{'}, ψ^{'}, ρ^{'}, σ^{'}), (use (10))$ $\begin{matrix} where c^{'} = \sum_{i = 1}^{n} c_{i}^{'}, φ^{'} = max_{1 \leq i \leq m} φ_{i}^{'}, ψ^{'} = max_{1 \leq i \leq m} ψ_{i}^{'}, \\ ρ^{'} = max_{1 \leq i \leq m} ρ_{i}^{'}, σ^{'} = max_{1 \leq i \leq m} σ_{i}^{'} \end{matrix}$

Step 6. Compute ${\tilde{S}}_{j} = \sum_{i = 1}^{m} {\tilde{S}}_{ij} = \sum_{i = 1}^{m} {({\tilde{X}}_{ij} - {\tilde{X}}_{j}^{M})}^{2}$ $= (c_{j}, φ_{j}, ψ_{j}, ρ_{j}, σ_{j}) (use (10))$ $\begin{matrix} where c_{j} = \sum_{i = 1}^{m} c_{ij}, φ_{j} = max_{1 \leq i \leq m} φ_{ij}, ψ_{j} = max_{1 \leq i \leq m} ψ_{ij} \\ ρ_{j} = max_{1 \leq i \leq m} ρ_{ij} and σ_{j} = max_{1 \leq i \leq m} σ_{ij} \end{matrix}$

Step 7. Compute $r_{ij}^{1} = max {α_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} β_{i}^{'}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij}),$ $r_{ij}^{2} = max {α_{ij}, \frac{1}{m} max_{1 \leq i \leq m} β_{ij}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}),$ $r_{ij}^{3} = max {β_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} α_{i}^{'}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij}),$ $r_{ij}^{4} = max {β_{ij}, \frac{1}{m} max_{1 \leq i \leq m} α_{ij}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}),$ $\begin{matrix} r_{ij}^{5} = max {γ_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} δ_{i}^{'}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij}), \\ r_{ij}^{6} = max {γ_{ij}, \frac{1}{m} max_{1 \leq i \leq m} δ_{ij}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}), \end{matrix}$ $\begin{matrix} r_{ij}^{7} = max {δ_{i}^{'}, \frac{1}{m} max_{1 \leq i \leq m} γ_{i}^{'}} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij}), \\ r_{ij}^{8} = max {δ_{ij}, \frac{1}{m} max_{1 \leq i \leq m} γ_{ij}} (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}) . \end{matrix}$

Step 8. Compute ${\tilde{Q}}_{ij} = ({\tilde{X}}_{i} - {\tilde{X}}^{M}) ({\tilde{X}}_{ij} - {\tilde{X}}_{j}^{M})$ $= (a_{ij}, θ_{ij}, ω_{ij}, λ_{ij}, π_{ij}) (use (14))$ $\begin{matrix} where a_{i} = (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i}) (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij}), \\ If x_{i} > \frac{1}{m} \sum_{i = 1}^{m} x_{i} and x_{ij} > \frac{1}{m} \sum_{i = 1}^{m} x_{ij} then \end{matrix}$ $\begin{matrix} θ_{ij} = max {r_{ij}^{1}, r_{ij}^{2}}, ω_{ij} = max {r_{ij}^{3}, r_{ij}^{4}} \\ λ_{ij} = max {r_{ij}^{5}, r_{ij}^{6}}, π_{ij} = max {r_{ij}^{7}, r_{ij}^{8}} \end{matrix}$ $If x_{i} < \frac{1}{m} \sum_{i = 1}^{m} x_{i} and x_{ij} < \frac{1}{m} \sum_{i = 1}^{m} x_{ij} then$ $\begin{matrix} θ_{ij} = max {- r_{ij}^{3}, - r_{ij}^{4}}, ω_{i} = max {- r_{ij}^{1}, - r_{i}^{2}} \\ λ_{ij} = max {- r_{ij}^{7}, - r_{ij}^{8}}, π_{i} = max {- r_{ij}^{5}, - r_{ij}^{6}} \end{matrix}$ $If x_{i} < \frac{1}{m} \sum_{i = 1}^{m} x_{i} and x_{ij} > \frac{1}{m} \sum_{i = 1}^{m} x_{ij} then$ $θ_{ij} = max {r_{ij}^{1}, - r_{ij}^{4}}, ω_{ij} = max {r_{ij}^{3}, - r_{ij}^{2}}$ $λ_{ij} = max {r_{ij}^{5}, - r_{ij}^{8}}, π_{ij} = max {r_{ij}^{7}, - r_{ij}^{6}}$ $If x_{i} > \frac{1}{m} \sum_{i = 1}^{m} x_{i} and x_{ij} < \frac{1}{n} \sum_{i = 1}^{m} x_{ij} then$ $θ_{ij} = max {- r_{ij}^{3}, r_{ij}^{2}}, ω_{ij} = max {- r_{ij}^{1}, r_{ij}^{4}}$ $λ_{ij} = max {- r_{ij}^{7}, r_{ij}^{6}}, π_{ij} = max {- r_{ij}^{5}, r_{ij}^{8}}$

Step 9. Compute ${\tilde{Q}}_{j} = \sum_{i = 1}^{m} {\tilde{Q}}_{ij} = (a_{j}, θ_{j},$ ω_j, λ_j, π_j), $\begin{matrix} where a_{j} = \sum_{i = 1}^{n} a_{ij}, θ_{j} = max_{1 \leq i \leq m} θ_{ij}, \\ ω_{j} = max_{1 \leq i \leq m} ω_{ij}, λ_{j} = max_{1 \leq i \leq m} λ_{ij}, π_{j} = max_{1 \leq i \leq m} π_{ij} \end{matrix}$

Step 10. Compute (use (14)) $\begin{matrix} {\tilde{P}}_{j} = \tilde{R} \times {\tilde{S}}_{j} = (\sum_{i = 1}^{m} {\tilde{R}}_{i}) \times (\sum_{i = 1}^{m} {\tilde{S}}_{ij}) \\ = (c^{'}, φ^{'}, ψ^{'}, ρ^{'}, σ^{'}) \times (c_{j}, φ_{j}, ψ_{j}, ρ_{j}, σ_{j}) \\ = (b_{j}, p_{j}, q_{j}, r_{j}, s_{j}) \end{matrix}$ $\begin{matrix} where b_{j} = c^{'} c_{j}, \\ p_{j} = max (c^{'} φ_{j}, c_{j} φ^{'}), q_{j} = max (c_{j} ψ^{'}, c^{'} ψ_{j}), \\ r_{j} = max (c_{j} ρ^{'}, c^{'} ρ_{j}), s_{j} = max (c_{j} σ^{'}, c^{'} σ_{j}) \end{matrix}$

Step 11. Compute (use (13)) ${\tilde{T}}_{j} = \sqrt{{\tilde{P}}_{j}} = \sqrt{\tilde{R} \times {\tilde{S}}_{j}} = (d_{j}, t_{j}, u_{j}, v_{j}, w_{j})$ $\begin{matrix} where d_{j} = \sqrt{b_{j}}, t_{j} = \frac{p_{j}}{\sqrt{b_{j}}}, u_{j} = \frac{q_{j}}{\sqrt{b_{j}}}, \\ v_{j} = \frac{r_{j}}{\sqrt{b_{j}}}, w_{j} = \frac{s_{j}}{\sqrt{b_{j}}}, for b_{j} > 0 \end{matrix}$

Step 12. Compute (use (15)) $\begin{matrix} \tilde{W_{j}} = \frac{\sum_{i = 1}^{m} ({\tilde{X}}_{ij} - {\tilde{X}}_{j}^{M}) ({\tilde{X}}_{i} - {\tilde{X}}^{M})}{\sqrt{\sum_{i = 1}^{m} {({\tilde{X}}_{ij} - {\tilde{X}}_{j}^{M})}^{2} \sum_{i = 1}^{m} {({\tilde{X}}_{i} - {\tilde{X}}^{M})}^{2}}} \\ = \frac{{\tilde{Q}}_{j}}{{\tilde{T}}_{j}} = \frac{(a_{j}, θ_{j}, ω_{j}, λ_{j}, π_{j})}{(d_{j}, t_{j}, u_{j}, v_{j}, w_{j})} = (f_{j}, k_{j}, l_{j}, m_{j}, n_{j}) \end{matrix}$ $If a_{j} > 0 then f_{j} = \frac{a_{j}}{d_{j}}$ $\begin{matrix} k_{j} = max {\frac{φ_{j}}{d_{j}}, \frac{a_{j} u_{j}}{d_{j} (d_{j} + u_{j})}}, \\ l_{j} = max {\frac{w_{j}}{d_{j}}, \frac{a_{j} t_{j}}{d_{j} (d_{j} - t_{j})}} \end{matrix}$ $\begin{matrix} m_{j} = max {\frac{λ_{j}}{d_{j}}, \frac{a_{j} w_{j}}{d_{j} (d_{j} + w_{j})}}, \\ n_{j} = max {\frac{π_{j}}{d_{j}}, \frac{a_{j} v_{j}}{d_{j} (d_{j} - v_{j})}} \\ If a_{j} = 0 then f_{j} = 0 \end{matrix}$ $k_{j} = \frac{φ_{j}}{d_{j}}, l_{j} = \frac{w_{j}}{d_{j}}, m_{j} = \frac{λ_{j}}{d_{j}}, n_{j} = \frac{π_{j}}{d_{j}}$ $\begin{matrix} If a_{j} < 0 then f_{j} = \frac{a_{j}}{d_{j}} \\ k_{j} = max {\frac{φ_{j}}{d_{j}}, - \frac{a_{j} t_{j}}{d_{j} (d_{j} - t_{j})}}, \\ l_{j} = max {\frac{w_{j}}{d_{j}}, - \frac{a_{j} u_{j}}{d_{j} (d_{j} + u_{j})}} \\ m_{j} = max {\frac{λ_{j}}{d_{j}}, - \frac{a_{j} v_{j}}{d_{j} (d_{j} - v_{j})}}, \\ n_{j} = max {\frac{π_{j}}{d_{j}}, - \frac{a_{j} w_{j}}{d_{j} (d_{j} + w_{j})}} \end{matrix}$

Based on the above described algorithm, the intuitionistic fuzzy importance $\tilde{W_{j}}$ of the attribute A_j, j = 1, 2,…, n can be evaluated.

4.1. Ordering of intuitionistic fuzzy importance of attributes

Sometimes, an ordering of the importance of attributes is more important than evaluations of these. The obtained results can be easily interpreted after IF-defuzzification. There are several methods to defuzzify an IFN. The expected value based method is very simple and has suitable properties [43].

A simple and effective ranking index on IFNs, particularly on triangular shaped, is given by the expected value. For two IFNs $\tilde{A}$ and $\tilde{B}$ , we introduce $\tilde{A} ≺ \tilde{B}$ if and only if $EV (\tilde{A}) < EV (\tilde{B})$ $\tilde{A} \sim \tilde{B}$ if and only if $EV (\tilde{A}) = EV (\tilde{B})$ $\tilde{A} \underset{0.3 em -}{≺} \tilde{B}$ if and only if $EV (\tilde{A}) \leq EV (\tilde{B})$ The computed intuitionistic fuzzy importance of attributes can be IF-defuzzified to obtain a crisp value/importance and simultaneously a hierarchy of them. In section 4, we have determined the intuitionistic fuzzy importance $\tilde{W_{j}}, j = 1, 2, . . ., n$ for each attribute A_j, j = 1, 2,…, n by applying proposed algorithm. A hierarchy of importance of attributes A_j, j = 1, 2,…, n can also be obtained by using expected value.

5. Application: Study of the hotel services by the proposed method

The linguistic terms or variables are very useful and widely used in different areas of research for describing situations where the classical quantitative expressions are unavailable or inadequate. For example, in surveys, it is more suitable to use linguistic terms than real numbers to express the answers to questions. Later on, these linguistic terms can be quantified by using membership functions and non-membership functions of IFNs.

The proposed method is exemplified by the study of the quality of hotel services and the same survey as in [2] is considered. During two weeks in June 2012, a number of 125 questionnaires was applied to customers of four 4-stars hotels from Oradea, Romania. For the establishment of the attributes, the SERVQUAL scale was considered. The complete list of attributes is included in Table 1. The value of α-Cronbach coefficient (0.827) was satisfactory for the verification of validity of the questionnaire (see [34]). The performance of attributes and the overall customer satisfaction (OCS) are measured on a five Likert scale {Very poor (VP), Poor (P), Medium (M), Good (G), Very good (VG)} and summarized in Table 2. The linguistic terms are quantified as TIFNs and indicated in Table 2.

Table 1
List of attributes

No. Attribute

1 The room facilities are appropriate

2 The room is clean enough

3 The hotel has sufficient restaurant facilities

4 The staff has an appropriate and professional look

5 The location of the hotel is suitable

6 The staff provide correct information to guests

7 The staff is able to offer services in a short period of time

8 The staff is able to resolve guests’ problems

9 The staff is able to provide information in a short time

10 The availability of staff

11 Clients complaints are resolved quickly

12 Different payment facilities are available

13 The safety of the installations in the hotel

14 Service professionalism

15 Service customization

16 Friendliness of staff

17 Proper opening hours of hotel’s facilities

18 The hotel has entertaining facilities

19 Big variety and proper quality of meals

20 Internet connection is available

21 Aesthetics of rooms and of the hotel

No.	Attribute
1	The room facilities are appropriate
2	The room is clean enough
3	The hotel has sufficient restaurant facilities
4	The staff has an appropriate and professional look
5	The location of the hotel is suitable
6	The staff provide correct information to guests
7	The staff is able to offer services in a short period of time
8	The staff is able to resolve guests’ problems
9	The staff is able to provide information in a short time
10	The availability of staff
11	Clients complaints are resolved quickly
12	Different payment facilities are available
13	The safety of the installations in the hotel
14	Service professionalism
15	Service customization
16	Friendliness of staff
17	Proper opening hours of hotel’s facilities
18	The hotel has entertaining facilities
19	Big variety and proper quality of meals
20	Internet connection is available
21	Aesthetics of rooms and of the hotel

Table 2

Linguistic terms with corresponding IFNs

Performance of attributes/Overall satisfaction	Rating of importance	Triangular IFN
Very poor (VP)	Unimportant (U)	1, 1, 1, 1, 1)
Poor (P)	Less important (L)	(3, 2, 1, 3, 2)
Medium (M)	Medium important(M)	(5, 2, 2, 3, 3)
Good (G)	Important (I)	(7, 1, 2, 2, 3)
Very Good (VG)	Very important (VI)	(9, 1, 1, 2, 2)

To compute the importance of attributes in the intuitionistic fuzzy case, the proposed algorithm has applied to the input data in Table 3 and the obtained results are shown in Table 4. The crisp importance of attributes is evaluated by IF-defuzzification of obtained intuitionistic fuzzy importance and shown in Table 4. A hierarchy of the evaluated crisp importance of the attributes is also obtained and shown in Table 4.

Table 3

Performance of attributes

No	Performance
	VP	P	M	G	VG
1	0	2	8	43	72
2	0	1	9	33	82
3	0	1	10	49	65
4	0	2	5	46	72
5	0	3	15	49	58
6	0	2	11	35	77
7	0	0	13	29	83
8	0	2	13	41	69
9	0	2	13	53	57
10	0	2	10	42	71
11	0	3	10	46	66
12	0	2	13	35	75
13	0	1	9	43	72
14	0	2	10	48	65
15	0	1	16	57	51
16	0	2	14	42	67
17	0	2	12	49	62
18	0	4	13	42	66
19	0	2	10	61	52
20	0	2	15	39	69
21	0	0	10	29	86
OCS	0	0	9	45	71

Table 4

The derived intuitionistic fuzzy importance and crisp importance of attributes

No.	Intuitionistic fuzzy importance	Crisp importance	Ordering
1	(0.967, 0.046, 0.046, 0.069, 0.069)	0.967	2
2	(0.882, 0.049, 0.049, 0.074, 0.074)	0.882	15
3	(0.923, 0.046, 0.046, 0.069, 0.069)	0.923	7
4	(0.952, 0.049, 0.049, 0.073, 0.073)	0.952	4
5	(0.855, 0.038, 0.038, 0.057, 0.057)	0.855	17
6	(0.904, 0.044, 0.044, 0.066, 0.066)	0.904	13
7	(0.856, 0.029, 0.029, 0.044, 0.044)	0.856	16
8	(0.934, 0.042, 0.042, 0.062, 0.062)	0.934	6
9	(0.850, 0.041, 0.041, 0.061, 0.061)	0.850	18
10	(0.965, 0.044, 0.044, 0.066, 0.066)	0.965	3
11	(0.919, 0.041, 0.041, 0.062, 0.062)	0.919	8
12	(0.913, 0.042, 0.042, 0.063, 0.063)	0.913	11
13	(0.973, 0.048, 0.048, 0.072, 0.072)	0.973	1
14	(0.916, 0.044, 0.044, 0.065, 0.065)	0.916	9
15	(0.806, 0.041, 0.041, 0.061, 0.061)	0.806	20
16	(0.916, 0.041, 0.041, 0.061, 0.061)	0.916	10
17	(0.886, 0.042, 0.042, 0.063, 0.063)	0.886	14
18	(0.910, 0.038, 0.038, 0.057, 0.057)	0.910	12
19	(0.823, 0.043, 0.043, 0.064, 0.064)	0.823	19
20	(0.937, 0.040, 0.040, 0.060, 0.060)	0.937	5
21	(0.799, 0.028, 0.028, 0.042, 0.042)	0.799	21

5.1. Comparisons with other existing methods

The proposed method is compared with the method given in [2, 3] for calculating the importance of attributes. Ban et al. [2] proposed an indirect method for computing the fuzzy importance of attributes by using the correlation coefficient. The α-cut based calculation was performed. In 2015, Ban et al. [3] presented an indirect method to compute the derived fuzzy importance of the attributes in the case of input data given by triangular fuzzy numbers. The effective calculation was based on the T_w-extension principle.

The fuzzy approach enables us to draw a useful conclusion regarding the importance of attributes in the absence of quantitative input data. It is worth mentioning that the IFS theory is the generalization of fuzzy set theory to allow describing scenarios when exact knowledge about the fuzziness of quantitative data is not expressible with a certain level of confidence. The proposed algorithm is based on T_w-based arithmetic operations on IFNs. Therefore the calculation is simplified and more exact results are obtained with less uncertainty. For this reason, the IFS theory based approach presented in this paper provides more flexibility to the analysts than the fuzzy set theory based approach in terms of expressing input data as membership and non-membership functions.

6. Conclusion

It is well-known that the determination of the importance of attributes is an essential step in multi-criteria decision making methods and importance-performance analysis. In the present paper, an indirect method has been proposed for evaluating of the intuitionistic fuzzy importance of attributes as the correlation between the intuitionistic fuzzy performance of attributes and intuitionistic fuzzy overall level of satisfaction. It is well known that T_w-based addition and multiplication preserves the shape of fuzzy numbers or IFNs and more exact results are obtained with less uncertainty. The T_w-based arithmetic operations on TIFNs are used in calculation. The major advantage of using IFSs over classical fuzzy sets is that IFSs separate the positive and negative evidence for membership of an element in the set. Unlike [2, 3], where input data are represented by triangular membership functions to calculate the fuzzy importance of attributes, but in the proposed method, the qualitative input data, which is given as linguistic terms, has been mathematically represented by both membership and non-membership functions to calculate the intuitionistic fuzzy importance of attributes. The proposed method has been demonstrated via application: study of hotel services. The results show that the proposed approach offers a useful way to compute the importance of attributes when quantitative input data are unavailable or insufficient and experts cannot express the fuzziness in the data under conformable confidence.

In the future, we hope to extend this work by looking at how the importance of attributes estimated by the proposed method can be affected by the different choices of membership functions and non-membership functions, expert opinions etc.

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Evaluation of the intuitionistic fuzzy importance of attributes based on the correlation coefficient under weakest triangular norm and application to the hotel services

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1. Intuitionistic fuzzy set

2.4. Tw-based arithmetic operations on TIFNs

3. Evaluation of the importance of attributes by correlation method

4.1. Ordering of intuitionistic fuzzy importance of attributes

5. Application: Study of the hotel services by the proposed method

6. Conclusion

References

2.4. T_w-based arithmetic operations on TIFNs