Calculation of the fuzzy importance of attributes based on the correlation coefficient,applied to the quality of hotel services

Abstract

A successful indirect method to calculate the importance of the attributes in services is that of correlation between the performance of attributes and the overall satisfaction such as they are perceived by customers. Often, in practice, the input data must be considered as fuzzy numbers, therefore it is completely justified our aim in the present paper, to give an indirect method of calculation of the importance of attributes in the fuzzy case, based on the correlation coefficient. The main benefits are related especially with the importance-performance analysis and multicriteria decision making methods. The effective computation is based on the Zadeh’s extension principle. The expected value, as a very simple and with suitable properties characteristic associated to a fuzzy number, is used for defuzzification and an immediate interpretation of the results. A practical method, algorithms and illustrative examples are given. The results of a survey with respect to the quality of hotel services in Oradea (Romania) are processed in accordance with our method.

Keywords

Fuzzy number trapezoidal fuzzy number fuzzy importance correlation coefficient expected value

1 Introduction

Fuzzy numbers or generalizations of fuzzy numbers are successful used for solving real life problems (see, e.g., [2 , 54]). Due to the subjectivity and fuzziness of human perceptions, fuzzy numbers are considered more suitable than crisp numbers to represent the customer’s opinions related with the performance and importance of service attributes (see, e.g., [16 , 53]).

The determination of the importance of attributes is an essential step in importance-performance analysis and multicriteria decision making methods. A multicriteria decision making method is a procedure for evaluating/ranking some alternatives, under a number of attributes, by a committee of decision-makers, a set of customers, visitors, etc. (see, e.g., [7 , 53]). Importance-performance analysis is a simple and effective technique which can assist practitioners in identifying improvement priorities and direct quality-based marketing strategies (see, e.g., [1 , 33]). The measurement of the importance of attributes can be obtained by direct or indirect methods. Even if the direct methods (by surveys or conjoint analysis) are still widely used, they have some disadvantages: they become unfeasible when more than few attributes are implied, the scores have a very small inter-itemic variation - with scores uniformly high, they increase the dimension of the survey or are influenced by the punctual performance of the products or services, etc. (see, e.g., [1 , 20]). Over the past few decades, many authors consider that the importance of attributes can be inferred through mathematical methods (see [32]). The most appropriate method to obtain derived importance of attributes is still subject to continuous debates (see [28]).

The calculation of the importance of attributes as the correlation coefficient between performance of attributes and the overall customer satisfaction is between the most popular (see [22 , 41–43]). The method is shortly described in Section 3. The correlation coefficient of fuzzy numbers as a fuzzy number was introduced in [40]. In the present paper we benefit from this contribution and we propose an indirect method for computing the fuzzy importance of attributes following the idea in the crisp case (Section 4). The effective calculation is based on the Zadeh’s extension principle, recalled in Section 2. The numerical results are obtained by working on different α-cuts, taking into account that an analytical solution is difficult to be given. Even so, the amount of calculation is not very short, so that the practical method described in Section 5 is preferable when the number of customers and/or attributes is large. The practical method has another important advantage: furnish us trapezoidal fuzzy numbers which can be easily compared, interpreted and handled in subsequent processing of data. For an easy implementation of the proposed method two algorithms are given in Section 6. They also make possible an immediate interpretation of the results, based on the expected value. In addition to the examples given in the previous sections, in Section 7 a complete application of our method is presented. It is based on a survey regarding the quality of hotel services, applied to four 4-stars hotels from Oradea, Romania.

2 Fuzzy numbers and linguistic variables

We begin by recalling some basic notions and notations used in this paper, especially related with fuzzy numbers.

Definition 1. [59] A fuzzy set A (fuzzy subset of X) is defined as a mapping A : X → [0, 1], where A (x) is the membership degree of x to the fuzzy set A.

The fuzzy numbers generalize the real numbers. They are fuzzy subsets of the real line with some additional properties.

Definition 2. (see [24] or [25]) A fuzzy number A is a fuzzy subset of the real line, $A : ℝ \to [0, 1]$ , satisfying the following properties:

A is normal (i. e. there exists $x_{0} \in ℝ$ such that A (x₀) = 1);

A is fuzzy convex (i. e. A (λx₁ + (1 - λ) x₂) ≥ min(A (x₁) , A (x₂)) , for every $x_{1}, x_{2} \in ℝ$ and λ ∈ [0, 1]);

A is upper semicontinuous on $ℝ$ (i. e. ∀ɛ > 0, ∃ δ > 0 such that A (x) - A (x₀) < ɛ, whenever);

$cl {x \in ℝ : A (x) > 0}$ is compact, where, for a set M, cl (M) denotes the closure of that set.

The α-cut, α ∈ (0, 1], of a fuzzy number A is a crisp set defined as $A_{α} = {x \in ℝ : A (x) \geq α} .$ The support suppA and the 0-cut A₀ of a fuzzy number A are defined as $supp A = {x \in ℝ : A (x) > 0}$ and $A_{0} = cl {x \in ℝ : A (x) > 0} .$ Every α-cut, α ∈ [0, 1] , of a fuzzy number A is a closed interval $A_{α} = [A_{L} (α), A_{U} (α)],$ where $A_{L} (α) = inf {x \in ℝ : A (x) \geq α},$ (1) $A_{U} (α) = sup {x \in ℝ : A (x) \geq α},$ (2)for any α ∈ (0, 1] and [A_L (0) , A_U (0)] = A₀. The pair of functions (A_L, A_U) gives a parametric representation of a fuzzy number A.

Fuzzy numbers with simple membership functions are preferred in practice. The most often used are so-called trapezoidal fuzzy numbers, that is fuzzy numbers with α-cuts given by $T_{α} = [t_{1} + (t_{2} - t_{1}) α, t_{4} - (t_{4} - t_{3}) α], α \in [0, 1],$ where $t_{1}, t_{2}, t_{3}, t_{4} \in ℝ, t_{1} \leq t_{2} \leq t_{3} \leq t_{4}$ . We denote by T = (t₁, t₂, t₃, t₄) a such fuzzy number. When t₂ = t₃ we obtain a triangular fuzzy number. When t₁ = t₂ = t₃ = t₄ = r we obtain the real number r.

Throughout this paper we denote by $F (ℝ)$ the set of all fuzzy numbers.

The Zadeh extension principle ([59, 60], see also the recent book [12]) allows to extend real functions, particularly the basic operations, for fuzzy numbers. We have the following definition of the extension principle (see, e.g., [34], p. 41):

Definition 3. Let X₁, . . . , X_n, Z be non-empty sets and let us consider the function F : X → Z, where X is the product space X = X₁ × X₂ × . . . × X_n . Furthermore, we consider the fuzzy sets A₁, . . . , A_n such that A_i : X_i → [0, 1] , i ∈ { 1, . . . , n } . Taking use of the function F we can define the fuzzy set F (A₁, A₂, . . . , A_n) : Z → [0, 1] by $\begin{matrix} F (A_{1}, . . ., A_{n}) (z) \\ = {\begin{matrix} sup_{(x_{1}, . . ., x_{n}) \in F^{- 1} (z)} min {A_{1} (x_{1}), . . ., A_{n} (x_{n})}, \\ if z \in F (X), \\ 0, otherwise . \end{matrix} \end{matrix}$ (3)

From practical point of view, the following result, which gives sufficient conditions for closed operations on fuzzy numbers, is very important.

Theorem 4. ([45], Proposition 5.1) Let us consider the continuous function $f : ℝ^{n} \to ℝ$ and suppose that A₁, . . . , A_n are fuzzy numbers. Then F (A₁, . . . , A_n) obtained by the extension principle (Definition 3) is a fuzzy number given by ${(F (A_{1}, . . ., A_{n}))}_{α} = f ({(A_{1})}_{α}, . . ., {(A_{n})}_{α}),$ (4)for α ∈ [0, 1].

Taking into account Theorem 4, the addition, scalar multiplication, difference, product and division of fuzzy numbers can be introduced starting from addition, difference, product and division of real numbers. The expected value EV (A) of a fuzzy number A is a very important characteristic especially in the ranking of fuzzy numbers (see [8]). It is given by [26, 35] $EV (A) = \frac{1}{2} \int_{0}^{1} (A_{L} (α) + A_{U} (α)) d α .$ (5)

We immediately obtain $EV (t_{1}, t_{2}, t_{3}, t_{4}) = \frac{1}{4} (t_{1} + t_{2} + t_{3} + t_{4}) .$ (6)

The linguistic variables are useful in the description of the situations where the classical quantitative expressions are inadequate. Most often, the answers to questions in a survey are expressed by linguistic variables. For example, if we opt for a 7-level scale in a survey then we may consider the linguistic variables for the rating by customers of the performance of the given attributes in the set {very poor, poor, medium poor, medium, medium good, good, very good} and their representations as trapezoidal fuzzy numbers in Table 1 (see [23]).

Of course, other linguistic variables and/or fuzzy numbers can be subjectively chosen too (see, e.g., [9, 54]). We point out here that in [21] the fuzzy numbers representing the linguistic variables are decided by customers, taking into account their responses regarding the range of each linguistic variable. Some tentatives to assign fuzzy numbers to linguistic variables in an objective way are presented in [39 , 58].

3 Calculation of the importance of attributes by correlation method

A successful indirect method to calculate the importance of the attributes is that of correlation between the performance perceived for each attribute and the overall customer satisfaction (see, e.g., [22 , 43]). The results obtained in this way are fructified especially related with the importance-performance analysis ([41], see also [21 , 43]). The approach based on the correlation coefficient as an indirect method of computation of the importance of attributes has some advantages which are pointed out in the literature [10 , 30].

In statistical analysis, the strength of the relationship between two variables is measured by the correlation coefficient. Given X = (X₁, . . . , X_p) and Y = (Y₁, . . . , Y_p), the correlation coefficient between X and Y, denoted by r_X,Y, is introduced as (see, e.g., [48]) $r_{X, Y} = \frac{\sum_{i = 1}^{p} (X_{i} - X^{M}) (Y_{i} - Y^{M})}{\sqrt{\sum_{i = 1}^{p} {(X_{i} - X^{M})}^{2} \sum_{i = 1}^{p} {(Y_{i} - Y^{M})}^{2}}},$ (7)where $X^{M} = \frac{1}{p} \sum_{i = 1}^{p} X_{i}$ and $Y^{M} = \frac{1}{p} \sum_{i = 1}^{p} Y_{i}$ . Let us consider n attributes of a service, C₁, . . . , C_n, and m customers, consumers of that service, D₁, . . . , D_m. We denote by X_ij the performance of the attribute C_j in the opinion of the customer D_i, i∈ { 1, . . . , m } , j ∈ { 1, . . . , n } and by X_i the overall level of satisfaction of the customer D_i, i∈ { 1, . . . , m }. Starting from the correlation coefficient defined in (7), the importance of the attribute C_j, j∈ { 1, . . . , n }, denoted by w_j, is given as the correlation coefficient between (X_1j, . . . , X_mj) and (X₁, . . . , X_m), therefore $w_{j} = \frac{\sum_{i = 1}^{m} (X_{ij} - X_{j}^{M}) (X_{i} - X^{M})}{\sqrt{\sum_{i = 1}^{m} {(X_{ij} - X_{j}^{M})}^{2} \sum_{i = 1}^{m} {(X_{i} - X^{M})}^{2}}},$ (8)where $X_{j}^{M} = \frac{1}{m} \sum_{i = 1}^{m} X_{ij}$ and $X^{M} = \frac{1}{m} \sum_{i = 1}^{m} X_{i}$ .

It is well-known that r_X,Y ∈ [- 1, 1], therefore w_j ∈ [- 1, 1]. Negative values of the importance of attributes are obtained by applying other indirect methods (multiple regression, partial least squares, principal components regression, etc.) too (see [28 , 37]). The researchers do not discriminate the negative values of importance and continue their study (see [10 , 38]), or put the negative values of importance equal to 0, that is ignore the corresponding attributes (see [51]), or consider that the negative values of importance do not make sense and/or they are difficult to interpret (see [30, 46]), or explain them as a consequence of complex interrelationship among attributes that are examined concurrently [1 , 18], or they do not obtain negative values in their empirical studies and avoid any discussion on this possibility [21 , 41–43]. Even if sometimes other methods are preferred because they avoid negative values, the regression analysis and correlation coefficient are considered the most suitable methods for deriving importance measures. The negativity of the importance measures is not a problem when the results have subsequent employments, between these the best example being the importance-performance analysis.

4 Calculation of the fuzzy importance of attributes by the correlationmethod

Following the idea in the crisp case (Equation (8)), in this section we propose an indirect method for calculating the importance of attributes when the input data are expressed by fuzzy numbers.

In many papers ([15, 56]) the correlation coefficient of fuzzy data is considered to be crisp, contradicting our intuition. When the observations are fuzzy, the correlation coefficient should be fuzzy as well. For the first time, in [40], the correlation coefficient for a sample of p pairs of observations $(\tilde{X_{i}}, \tilde{greaterthan Y_{i}}), i \in {1, . . ., p}$ , given as fuzzy numbers, is introduced such that the result is a fuzzy number too. It is given by $\tilde{r_{X, Y}} = \frac{\sum_{i = 1}^{p} (\tilde{X_{i}} - \tilde{X^{M}}) (\tilde{greaterthan Y_{i}} - \tilde{Y^{M}})}{\sqrt{\sum_{i = 1}^{p} {(\tilde{X_{i}} - \tilde{X^{M}})}^{2} \sum_{i = 1}^{p} {(\tilde{greaterthan Y_{i}} - \tilde{Y^{M}})}^{2}}},$ (9)where $\tilde{X^{M}} = \frac{1}{p} \cdot \sum_{i = 1}^{p} \tilde{X_{i}}$ and $\tilde{Y^{M}} = \frac{1}{p} \cdot \sum_{i = 1}^{p} \tilde{greaterthan Y_{i}}$ . Because it is almost impossible to derive the membership function of $\tilde{r_{X, Y}}$ from Equation (9) directly, the effective calculation is based on Zadeh’s extension principle (see [40]).

All the premises to extend Equation (8) to fuzzy numbers are fulfilled. We introduce $\tilde{w_{j}} = \frac{\sum_{i = 1}^{m} (\tilde{X_{ij}} - \tilde{X_{j}^{M}}) \cdot (\tilde{X_{i}} - \tilde{X^{M}})}{\sqrt{\sum_{i = 1}^{m} {(\tilde{X_{ij}} - \tilde{X_{j}^{M}})}^{2} \cdot \sum_{i = 1}^{m} {(\tilde{X_{i}} - \tilde{X^{M}})}^{2}}}$ (10)as the fuzzy importance of the attribute C_j, j∈ { 1, . . . , n }, where $\tilde{X_{ij}}$ denotes the performance of the attribute C_j, j∈ { 1, . . . , n } in the opinion of the customer D_i, i∈ { 1, . . . , m }, expressed as a fuzzy number, $\tilde{X_{i}}$ denotes the overall satisfaction in the opinion of the customer D_i, i∈ { 1, . . . , m }, expressed as a fuzzy number, and, in addition, we denote $\tilde{X_{j}^{M}} = \frac{1}{m} \cdot \sum_{i = 1}^{m} \tilde{X_{ij}} and \tilde{X^{M}} = \frac{1}{m} \cdot \sum_{i = 1}^{m} \tilde{X_{i}} .$

As in the definition of the fuzzy correlation coefficient introduced in [40], formula (10) is rather formal, the effective calculus of the fuzzy number $\tilde{w_{j}}$ is based on the Zadeh’s extension principle, in fact on the result in Theorem 4. According with Theorem 4 (see also [40]), we find ${(\tilde{w_{j}})}_{L} (α)$ and ${(\tilde{w_{j}})}_{U} (α)$ , for every α ∈ [0, 1], by solving the following pair of crisp mathematical programs: ${(\tilde{w_{j}})}_{L} (α) = min f_{j} (x_{1 j}, . . ., x_{mj}, x_{1}, . . ., x_{m})$ (11)such that ${(\tilde{X_{ij}})}_{L} (α) \leq x_{ij} \leq {(\tilde{X_{ij}})}_{U} (α),$ (12) ${(\tilde{X_{i}})}_{L} (α) \leq x_{i} \leq {(\tilde{X_{i}})}_{U} (α),$ (13)for every i∈ { 1, . . . , m }, and ${(\tilde{w_{j}})}_{U} (α) = max f_{j} (x_{1 j}, . . ., x_{mj}, x_{1}, . . ., x_{m})$ (14)such that ${(\tilde{X_{ij}})}_{L} (α) \leq x_{ij} \leq {(\tilde{X_{ij}})}_{U} (α),$ (15) ${(\tilde{X_{i}})}_{L} (α) \leq x_{i} \leq {(\tilde{X_{i}})}_{U} (α),$ (16)for every i∈ { 1, . . . , m }, where

$\begin{matrix} f_{j} (x_{1 j}, . . ., x_{mj}, x_{1}, . . ., x_{m}) \\ = \frac{\sum_{i = 1}^{m} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij}) (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i})}{\sqrt{\sum_{i = 1}^{m} {(x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})}^{2} \sum_{i = 1}^{m} {(x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i})}^{2}}} \end{matrix}$ (17)

for every j∈ { 1, . . . , n }. We can reconstitute the fuzzy number $\tilde{w_{j}}$ from the Negoiţă-Ralescu characterization theorem [44].

Remark 5. Since the real numbers can be considered as fuzzy numbers, in the crisp case the problems in Equations (11)–(12) and Equations (14)–(16) become identical and trivial because ${(\tilde{X_{ij}})}_{L} (α) = {(\tilde{X_{ij}})}_{U} (α),$ for every α∈ [0, 1] , i ∈ { 1, . . . , m } , j ∈ { 1, . . . , n }. The calculation of the importance of attributes reduces to Equation (8) in the crisp case.

Remark 6. The positive (negative) support of the importance of an attribute $\tilde{w_{j}}$ , indicates a positive (negative) correlation between the opinion of the customers with respect to the performance of attribute C_j and the overall customers satisfaction. With other words, from a mathematical point of view, an improvement of the performance with respect to attribute C_j will increase (decrease) the overall customers satisfaction. In fact, as we already explained in details at the end of Section 3, this situation is not completely clarified in the crisp case and, all the more, in the fuzzy case. Nevertheless, the calculation of the importance of attributes by correlation coefficient is very useful in subsequent developments, the most illustrative example being the fuzzy importance-performance analysis.

It is difficult to give analytical solutions of the systems given by Equations (11)–(13) and Equations (14)–(16) even if the constrained variable method and reduced gradient method could be useful. Nevertheless, numerical solutions can be easily obtained by finding a finite set of α-cuts, α∈ { α₀ = 0 < α₁ < . . . < α_r-1 < α_r = 1 }, for every $\tilde{w_{j}}, j \in {1, . . ., n}$ . The idea was launched in [40] related with the calculation of the fuzzy correlation coefficient. Of course, we obtain a good solution by choosing small differences α_k+1 - α_k for any k∈ { 0, . . . , r - 1 } and a large r, but we pay a better quality by an increasing volume of computation.

In the sequel we illustrate the above theoretical development by the following example.

Example 7. We consider four customers (D₁, D₂, D₃, D₄) and five attributes (C₁, C₂, C₃, C₄, C₅) to evaluate the quality of a certain service. The customers use the linguistic variables in Table 1 to evaluate the subjective attributes C₁, C₂, C₃, C₄, C₅ and the overall satisfaction related with the service under study. The performance matrix $\tilde{X_{ij}}, i \in {1, 2, 3, 4}, j \in {1, 2, 3, 4, 5}$ and the overall customers satisfaction vector $\tilde{X_{i}}, i \in {1, 2, 3, 4}$ are given in Tables 2 and 3, respectively.

As a basis for the demonstration of the feasibility of the proposed method, we consider $α_{k} = \frac{k}{10}, k \in {0, 1, . . ., 10}$ . Let us denote (see Equation (17))

$\begin{matrix} f_{1} (x_{11}, x_{21}, x_{31}, x_{41}, x_{1}, x_{2}, x_{3}, x_{4}) \\ = \frac{\sum_{i = 1}^{4} (x_{i 1} - \frac{1}{4} \sum_{i = 1}^{4} x_{i 1}) (x_{i} - \frac{1}{4} \sum_{i = 1}^{4} x_{i})}{\sqrt{\sum_{i = 1}^{4} {(x_{i 1} - \frac{1}{4} \sum_{i = 1}^{4} x_{i 1})}^{2} \sum_{i = 1}^{4} {(x_{i} - \frac{1}{4} \sum_{i = 1}^{4} x_{i})}^{2}}} \end{matrix}$ (18)

for finding the α_k-cuts of $\tilde{w_{1}}$ , the importance of the attribute C₁. Taking into account Equations (12), (13) and the representations as trapezoidal fuzzy numbersof the linguistic variables in Table 1, we get $(\tilde{w_{1}})_{L} (\frac{k}{10}), k \in {0, 1, . . ., 10}$ as the minimum of the function f₁ in (18) under conditions $4 + \frac{k}{10} \leq x_{11} \leq 6 - \frac{k}{10}$ (19) $7 + \frac{k}{10} \leq x_{21} \leq 9 - \frac{k}{10}$ (20) $8 + \frac{k}{10} \leq x_{31} \leq 10$ (21) $7 + \frac{k}{10} \leq x_{41} \leq 9 - \frac{k}{10}$ (22) $7 + \frac{k}{10} \leq x_{1} \leq 9 - \frac{k}{10}$ (23) $4 + \frac{k}{10} \leq x_{2} \leq 6 - \frac{k}{10}$ (24) $7 + \frac{k}{10} \leq x_{3} \leq 9 - \frac{k}{10}$ (25) $4 + \frac{k}{10} \leq x_{4} \leq 6 - \frac{k}{10} .$ (26)

Then, taking into account Equations (15) and (16) we get ${(\tilde{w_{1}})}_{U} (\frac{k}{10}), k \in {0, 1, . . ., 10}$ as the maximum of the same function f₁ in (18) under the same conditions (19)–(26). We use a program that, for each fixed k ∈ {0, 1, …, 10} stores the values on the left side and right side of Equations (19)–(26), determines the elements of the corresponding Cartesian product (eight sets in this example) ranged by a particular step (0.1 in our example), using the backtracking method, calculates the value of the function f₁ in these points of the Cartesian product and holds the minimum value and, respectively, the maximum value. We obtain the values in the first and second line of Table 4. We continue the procedure for the remaining attributes C_j, j∈ { 2, 3, 4, 5 }. The output data, that is the importance of attributes, given by their α_k-cuts, k∈ { 0, . . . , 10 } are listed in Table 4 and they are plotted in Figs. 1–5.

We obtain that the most important attribute is C₃ and the least important is C₂. On the other hand, it seems that the performance of C₁, C₂ and C₅ has a negative correlation with the overall customers satisfaction. From the mathematical point of view this means that an improvement of the performance with respect to C₁, C₂ and C₅ do not grow up the overall satisfaction (see also the comments at the end of Section 3).

In Example 7 the importance of attributes is rather negative (at least for C₁, C₂ and C₅). This is not at all surprising because the input data are theoretical. The relationship between the performance of attributes and the overall customer satisfaction is not of the same kind as in a real situation. As we prove in the below example, if we increase the relationship between the performance of attributes and the overall satisfaction, by our method we obtain positive values for the importance of attributes, in concordance with our intuition.

Example 8. We modify the attribute performance matrix $\tilde{X_{ij}}, i \in {1, 2, 3, 4}, j \in {1, 2, 3, 4, 5}$ in Example 7 according with the data given in Table 5 and we keep the overall customers satisfaction in Example 7 (see Table 3).

We apply again our method and the program of computation of the fuzzy importance of an attribute as the correlation coefficient between the performance of the attribute and the overall satisfaction. We obtain the results in Table 6.

The most important attribute seems to be C₄ and the least important seems to be C₅. The support of each $\tilde{w_{j}}, j \in {1, . . ., 5}$ is rather positive. As an immediate conclusion, by a natural defuzzification of the fuzzy numbers $\tilde{w_{j}}, j \in {1, . . ., n}$ , we obtain positive real numbers, our intuition being confirmed in this way. In Section 6 we will return to the discussion on the defuzzification and an easy interpretation of the results obtained by our method.

5 Practical calculation of the fuzzy importance of attributes

By analyzing the method described in the previous section we observe that an analytical solution cannot be given for the problems in Equations (11)–(13) and Equations (14)–(16). An approximate solution based on α-cuts (see Examples 7 and 8) is difficult to be obtained in the case of many attributes and/or customers because the time of work increases too much. In addition, any solution which is not a fuzzy number with a simple form leads to sophisticated subsequent calculations. An idea to overcome this situation could be based on the recent proposed approximations of fuzzy numbers (see, e.g., [5 , 55]). Nevertheless, the best idea seems to be based on the naive method of approximation of fuzzy numbers (see e.g. [29]). Namely, to keep the fuzzy framework, the essence of information and for an easy subsequent handling, the trapezoidal fuzzy number (A_L (0) , A_L (1) , A_U (1) , A_U (0)) as the approximation of $A \in F (ℝ), A_{α} = [A_{L} (α), A_{U} (α)], α \in [0, 1]$ , is a good choice. With the same notations as in Section 4, instead to solve the parametric mathematical programs given in Equations (11)–(13) and Equations (14)–(16), it is enough to solve the following four problems to find the fuzzy importance $\tilde{w_{j}} = (p_{j}, q_{j}, r_{j}, s_{j}) :$ $p_{j} = min f_{j} (x_{1 j}, . . ., x_{mj}, x_{1}, . . ., x_{m})$ such that ${(\tilde{X_{ij}})}_{L} (0) \leq x_{ij} \leq {(\tilde{X_{ij}})}_{U} (0), \forall i \in {1, . . ., m},$ ${(\tilde{X_{i}})}_{L} (0) \leq x_{i} \leq {(\tilde{X_{i}})}_{U} (0), \forall i \in {1, . . ., m},$ $q_{j} = min f_{j} (x_{1 j}, . . ., x_{mj}, x_{1}, . . ., x_{m})$ such that ${(\tilde{X_{ij}})}_{L} (1) \leq x_{ij} \leq {(\tilde{X_{ij}})}_{U} (1), \forall i \in {1, . . ., m},$ ${(\tilde{X_{i}})}_{L} (1) \leq x_{i} \leq {(\tilde{X_{i}})}_{U} (1), \forall i \in {1, . . ., m},$ $r_{j} = max f_{j} (x_{1 j}, . . ., x_{mj}, x_{1}, . . ., x_{m})$ such that ${(\tilde{X_{ij}})}_{L} (1) \leq x_{ij} \leq {(\tilde{X_{ij}})}_{U} (1), \forall i \in {1, . . ., m},$ ${(\tilde{X_{i}})}_{L} (1) \leq x_{i} \leq {(\tilde{X_{i}})}_{U} (1), \forall i \in {1, . . ., m}$ and $s_{j} = max f_{j} (x_{1 j}, . . ., x_{mj}, x_{1}, . . ., x_{m})$ such that ${(\tilde{X_{ij}})}_{L} (0) \leq x_{ij} \leq {(\tilde{X_{ij}})}_{U} (0), \forall i \in {1, . . ., m},$ ${(\tilde{X_{i}})}_{L} (0) \leq x_{i} \leq {(\tilde{X_{i}})}_{U} (0), \forall i \in {1, . . ., m},$ where $\begin{matrix} f_{j} (x_{1 j}, . . ., x_{mj}, x_{1}, . . ., x_{m}) \\ = \frac{\sum_{i = 1}^{m} (x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij}) (x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i})}{\sqrt{\sum_{i = 1}^{m} {(x_{ij} - \frac{1}{m} \sum_{i = 1}^{m} x_{ij})}^{2} \sum_{i = 1}^{m} {(x_{i} - \frac{1}{m} \sum_{i = 1}^{m} x_{i})}^{2}}} . \end{matrix}$

The benefits of the practical method are illustrated by the following example.

Example 9. We consider the data from Example 7. By applying the practical method, we find $\tilde{w_{1}} = (p_{1}, q_{1}, r_{1}, s_{1})$ as follows: $\begin{matrix} p_{1} & = min f_{1} (x_{11}, x_{21}, x_{31}, x_{41}, x_{1}, x_{2}, x_{3}, x_{4}), \\ s_{1} & = max f_{1} (x_{11}, x_{21}, x_{31}, x_{41}, x_{1}, x_{2}, x_{3}, x_{4}), \end{matrix}$ where 4 ≤ x₁₁ ≤ 6, 7 ≤ x₂₁ ≤ 9, 8 ≤ x₃₁ ≤ 10, 7 ≤ x₄₁ ≤ 9, 7 ≤ x₁ ≤ 9, 4 ≤ x₂ ≤ 6, 7 ≤ x₃ ≤ 9 and 4_leqx4 ≤ 6 and $\begin{matrix} q_{1} & = min f_{1} (x_{11}, x_{21}, x_{31}, x_{41}, x_{1}, x_{2}, x_{3}, x_{4}), \\ r_{1} & = max f_{1} (x_{11}, x_{21}, x_{31}, x_{41}, x_{1}, x_{2}, x_{3}, x_{4}), \end{matrix}$ where x₁₁ = x₂ = x₄ = 5, x₂₁ = x₄₁ = x₁ = x₃ = 8 and 9 ≤ x₃₁ ≤ 10 . We obtain $\tilde{w_{1}} = (- 0.979, - 0.333, - 0.140, 0.693) .$ We proceed similarly for $\tilde{w_{2}}, \tilde{w_{3}}, \tilde{w_{4}}$ and $\tilde{w_{5}}$ . We have $\begin{matrix} \tilde{w_{2}} & = (- 0.997, - 0.577, - 0.436, 0.474), \\ \tilde{w_{3}} & = (- 1.000, 0.905, 1.000, 1.000), \\ \tilde{w_{4}} & = (- 0.822, 0.192, 0.408, 0.999), \\ \tilde{w_{5}} & = (- 0.992, - 0.485, - 0.302, 0.677) . \end{matrix}$ The results are plotted in Figs. 6–10. We get similar conclusions as in Example 7.

On the same idea as in Example 8, we modify the input data to see the relevance of the relationship between the performance of the attributes and the overall satisfaction in the calculation of the fuzzy importance of attributes.

Example 10. We consider the data furnished by Example 8 and apply the practical method. We get $\begin{matrix} \tilde{w_{1}} & = (- 0.384, 0.816, 0.962, 1.000), \\ \tilde{w_{2}} & = (- 0.357, 0.816, 0.962, 1.000), \\ \tilde{w_{3}} & = (- 0.121, 0.832, 0.949, 1.000), \\ \tilde{w_{4}} & = (0.377, 0.928, 0.970, 1.000), \\ \tilde{w_{5}} & = (- 0.812, 0.447, 0.894, 1.000) \end{matrix}$ and similar conclusions as in Example 8.

6 Algorithms and interpretation of the results

The results obtained by fuzzy methods can be easily interpreted after defuzzification. There are many possibilities to defuzzify a fuzzy number, based on the center of area, mean of maxima, expected value, etc. Because the expected value is very simple and has suitable properties [35], it is often used in applications (see [7 , 49], etc.). On the other hand, sometimes it is more important to obtain an ordering of the fuzzy numbers, in our case a hierarchy of the importance of attributes. It is proved (see [8]) that a simple and effective ranking index is given by the expected value (see Equation (5)), that is, for $A, B \in F (ℝ),$ $\begin{matrix} A & ≺ B if and only if EV (A) < EV (B), \\ A & \sim B if and only if EV (A) = EV (B) \end{matrix}$ and $A ⪯ B if and only if EV (A) \leq EV (B) .$ According with the methods given in Sections 4 and 5, for every attribute C_j, j∈ { 1, . . . , n } we have determined the corresponding fuzzy importance ${\tilde{w}}_{j}, j \in {1, . . ., n}$ given on α_k-levels, k∈ { 0, . . . , r }, that is we know the values ${(\tilde{w_{j}})}_{L} (α_{k})$ and ${(\tilde{w_{j}})}_{U} (α_{k}), k \in {0, . . ., r}$ . In the sequel we elaborate a ranking method, based on the expected value and adjusted to our structure of data.

Let $A \in F (ℝ)$ and α_k ∈ [0, 1] , α₀ = 0 < α₁ < . . . < α_r-1 < α_r = 1, r ≥ 2. Because $\int_{0}^{1} A_{L} (α) d α$ can be approximated by $\sum_{k = 1}^{r} \frac{α_{k} - α_{k - 1}}{2} (A_{L} (α_{k}) + A_{L} (α_{k - 1}))$ and $\int_{0}^{1} A_{U} (α) d α$ by $\sum_{k = 1}^{r} \frac{α_{k} - α_{k - 1}}{2} (A_{U} (α_{k}) + A_{U} (α_{k - 1})),$ we obtain that the expected value of $A \in F (ℝ)$ can be approximated by the value EV^~ (A), where (see Equation (5)) ${EV}^{~} (A) = \sum_{k = 1}^{r} \frac{α_{k} - α_{k - 1}}{4}$ $\times (A_{L} (α_{k}) + A_{U} (α_{k}) + A_{L} (α_{k - 1}) + A_{U} (α_{k - 1})) .$ In the particular case $α_{k} = \frac{k}{r}, k \in {0, . . ., r}$ we get $\begin{matrix} {EV}^{~} (A) & = & \frac{A_{L} (0) + A_{U} (0) + A_{L} (1) + A_{U} (1)}{4 r} \\ + \frac{1}{2 r} \sum_{k = 1}^{r - 1} A_{L} (\frac{k}{r}) + \frac{1}{2 r} \sum_{k = 1}^{r - 1} A_{U} (\frac{k}{r}) . \end{matrix}$ (27)

In the remaining case r = 1, corresponding to the practical method in Section 5, we obtain the expected value of the naive trapezoidal approximation (A_L (0) , A_L (1) , A_U (1) , A_U (0)) of $A \in F (ℝ)$ , that is (see Equation (6)) ${EV}^{~} (A) = \frac{A_{L} (0) + A_{U} (0) + A_{L} (1) + A_{U} (1)}{4} .$ (28)

Example 11. We consider again Example 7. By using Equation (27) and the results in Table 4 we obtain $\begin{matrix} {EV}^{~} (\tilde{w_{1}}) & = - 0.215, \\ {EV}^{~} (\tilde{w_{2}}) & = - 0.416, \\ {EV}^{~} (\tilde{w_{3}}) & = 0.461, \\ {EV}^{~} (\tilde{w_{4}}) & = 0.209, \\ {EV}^{~} (\tilde{w_{5}}) & = - 0.291, \end{matrix}$ that is the most important attribute is C₃, the least important is C₂ and C₄, C₁ and C₅ have an intermediate importance in the decreasing ordering. The results are represented in Figs. 1–5.

If $\tilde{w_{j}}, j \in {1, . . ., 5}$ are obtained by the practical method described in Section 5 then, by considering the results in Example 9 and Equation (28), we obtain $\begin{matrix} {EV}^{~} (\tilde{w_{1}}) & = - 0.190, \\ {EV}^{~} (\tilde{w_{2}}) & = - 0.384, \\ {EV}^{~} (\tilde{w_{3}}) & = 0.476, \\ {EV}^{~} (\tilde{w_{4}}) & = 0.194, \\ {EV}^{~} (\tilde{w_{5}}) & = - 0.275 \end{matrix}$ and the same ordering between the importance of the attributes as above. The results are represented inFigs. 6–10.

Example 12. By reconsidering Example 8, using Equation (27) and the results in Table 6 we get $\begin{matrix} {EV}^{~} (\tilde{w_{1}}) & = 0.657, \\ {EV}^{~} (\tilde{w_{2}}) & = 0.655, \\ {EV}^{~} (\tilde{w_{3}}) & = 0.710, \\ {EV}^{~} (\tilde{w_{4}}) & = 0.838, \\ {EV}^{~} (\tilde{w_{5}}) & = 0.393, \end{matrix}$ which confirm the positivity of the importance of attributes in the case of a raised relationship between the performance of attributes and the overall satisfaction. By the practical method, with the results obtained in Example 10 and applying Equation (28) we get $\begin{matrix} {EV}^{~} (\tilde{w_{1}}) & = 0.599, \\ {EV}^{~} (\tilde{w_{2}}) & = 0.605, \\ {EV}^{~} (\tilde{w_{3}}) & = 0.665, \\ {EV}^{~} (\tilde{w_{4}}) & = 0.819, \\ {EV}^{~} (\tilde{w_{5}}) & = 0.382 . \end{matrix}$ Even if there exist some small differences between the results, the hierarchy of the importance of attributes is preserved.

Summarizing the theoretical development in the present paper, the following algorithms can be applied to give a numerical solution for the fuzzy importance $\tilde{w_{j}}$ of the attribute C_j, j∈ { 1, . . . , n }, with $\tilde{X_{ij}}$ asthe performance of the attribute C_j, j∈ { 1, . . . , n } in theopinion of the customer D_i, i∈ { 1, . . . , m } and $\tilde{X_{i}}$ the overall satisfaction in the opinion of the customer D_i, i∈ { 1, . . . , m }. An ordering of the importance of attributes is obtained too.

The first algorithm is dedicated to the case when we find out for the α_k-cuts of the fuzzy importance of attributes, $α_{k} = \frac{k}{r}, k \in {0, . . ., r}, r \geq 2 .$

Algorithm 1.

let r ≥ 2

for $j = \bar{1, n}$

for $k = \bar{0, r}$

for $i = \bar{1, m}$

compute ${(\tilde{X_{ij}})}_{L} (\frac{k}{r})$

compute ${(\tilde{X_{ij}})}_{U} (\frac{k}{r})$

compute ${(\tilde{X_{i}})}_{L} (\frac{k}{r})$

compute ${(\tilde{X_{i}})}_{U} (\frac{k}{r})$

find ${(\tilde{w_{j}})}_{L} (\frac{k}{r})$ by solving Equations (11)–(13) for

$α : = \frac{k}{r}$

find ${(\tilde{w_{j}})}_{U} (\frac{k}{r})$ by solving Equations (14)–(16) for

$α : = \frac{k}{r}$

end for

compute

$\begin{matrix} {EV}^{~} (\tilde{w_{j}}) = \frac{{(\tilde{w_{j}})}_{L} (0) + {(\tilde{w_{j}})}_{U} (0)}{4 r} \\ + \frac{{(\tilde{w_{j}})}_{L} (1) + {(\tilde{w_{j}})}_{U} (1)}{4 r} \\ + \frac{1}{2 r} \sum_{k = 1}^{r - 1} {(\tilde{w_{j}})}_{L} (\frac{k}{r}) + \frac{1}{2 r} \sum_{k = 1}^{r - 1} {(\tilde{w_{j}})}_{U} (\frac{k}{r}) . \end{matrix}$ (29)end for

order $\tilde{w_{j}}$ , $j = \bar{1, n}$ using the increasing sequence of the values ${EV}^{~} (\tilde{w_{j}})$ , $j = \bar{1, n}$ .

If r = 1 and we consider Equation (28) instead of Equation (27) (that is we keep only the first and the second term in the sum in Equation (29)) then we obtain the following algorithm corresponding to the practical method described in Section 5.

Algorithm 2.

for $j = \bar{1, n}$

for $i = \bar{1, m}$

compute ${(\tilde{X_{ij}})}_{L} (0)$

compute ${(\tilde{X_{ij}})}_{U} (0)$

compute ${(\tilde{X_{i}})}_{L} (0)$

compute ${(\tilde{X_{i}})}_{U} (0)$

compute ${(\tilde{X_{ij}})}_{L} (1)$

compute ${(\tilde{X_{ij}})}_{U} (1)$

compute ${(\tilde{X_{i}})}_{L} (1)$

compute ${(\tilde{X_{i}})}_{U} (1)$

end for

find ${(\tilde{w_{j}})}_{L} (0)$ by solving Equations (11)–(13) for α : =0

find ${(\tilde{w_{j}})}_{U} (0)$ by solving Equations (11)–(13) for α : =0

find ${(\tilde{w_{j}})}_{L} (1)$ by solving Equations (14)–(16) for α : =1

find ${(\tilde{w_{j}})}_{U} (1)$ by solving Equations (11)–(13) for α : =1

compute

$\begin{matrix} {EV}^{~} (\tilde{w_{j}}) = \frac{{(\tilde{w_{j}})}_{L} (0) + {(\tilde{w_{j}})}_{U} (0)}{4} \\ + \frac{{(\tilde{w_{j}})}_{L} (1) + {(\tilde{w_{j}})}_{U} (1)}{4} . \end{matrix}$ (30)end for

order $\tilde{w_{j}}$ , $j = \bar{1, n}$ using the increasing sequence of the values ${EV}^{~} (\tilde{w_{j}})$ , $j = \bar{1, n}$ .

7 Application: Derived importance of attributes for the quality of hotel services

In this section an example case is presented to prove the implementation of the proposed method.

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 (see Table 7), the SERVQUAL scale was considered. The Internet access was added as a specific attribute for urban tourism. The complete list of attributes was included in Table 7.

The validity of the questionnaire was verified with the α-Cronbach coefficient, the value obtained, 0.827, being a satisfactory one (see [11]). The data regarding the performance of attributes and the overall customer satisfaction (OCS) are summarized in the first columns of Table 9. They are measured on a five Likert scale, the possible answers being in the set {Very poor (VP), Poor (P), Medium (M), Good (G), Very good (VG)}. The corresponding trapezoidal fuzzy numbers to linguistic variables are indicated in Table 8.

We apply Algorithm 2 to calculate the importance of attributes in the fuzzy case and their crisp values obtained by defuzzification with the expected value. The results are shown in the last two columns of Table 9. The decreasing ordering of the importance of attributes is the following: 4 > 9 >8 > 2 >6 > 10 > 5 >14 > 3 >16 > 17 > 18 > 1 >11 > 13 > 21 > 15 > 7 >12 > 19 > 20.

The results on the importance of attributes in Table 9 may be useful in many subsequent studies, the most important being related with crisp or fuzzy multicriteria decision making methods and importance-performance analysis.

On the other hand, let us point out that the choosing of a suitable set of attributes is a very important step in any analysis related with different areas in the services industries (see, e.g., [38]). In our empirical study, the importance of eleven of the attributes (2 - 6, 8 - 10, 14, 16, 17) is greater than average and the importance of six of the attributes (2, 4, 6, 8 - 10) is significantly greater than average. This means that the number of attributes may be reduced to eleven, or, more drastic, to six, for any subsequent study implying the 4-stars hotels in Oradea, with obviousbenefits.

8 Conclusions

The correct determination of the importance of attributes is an essential step in decision theory. In the present paper we propose an indirect method of calculation of the importance of attributes in a fuzzy environment. It is based, as in the crisp case, on the calculation of the correlation coefficient between the performance of attributes, from the point of view of the customers, and the overall customers satisfaction. The benefits are important (they are pointed out in thepaper) even if the using of the Zadeh extension principle generates some handling problems. In fact, for large dimensional problems (many attributes and/or customers) and for an easy interpretation, the practical method given in Section 5 is more suitable to be applied. The paper is a part of the development of an importance-performance analysis in a fuzzy environment, the existing papers (see [14, 21]) assuming a premature defuzzification of data.

AcknowledgmentS

The work of the first author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0861.

References

Abalo

, Varela

and Manzano

, Importance values for importance-performance analysis: A formula for spreading out values derived from preference rankings, Journal of Business Research 60 (2007), 115–121.

Akyar

, Akyar

and Düzce

S.A.

, Fuzzy risk analysis based on a geometric ranking method for generalized trapezoidal fuzzy numbers, Journal of Intelligent and Fuzzy Systems 25 (2013), 209–217.

Azzopardi

and Nash

, A critical evaluation of importance-performance analysis, Tourism Management 35 (2013), 222–233.

Bacon

D.R.

, A comparison of approaches to importance-performance analysis, International Journal of Marketing Research 45 (2003), 55–71.

Ban

, Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval, Fuzzy Sets and Systems 159 (2008), 1327–1344.

Ban

, Brândaş

, Coroianu

, Negruţiu

and Nica

, Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value, Computers and Mathematics with Applications 61 (2011), 1379–1401.

Ban

and Ban

, Optimization and extensions of a fuzzy multicriteria decision making method and applications to selection of touristic destinations, Expert Systems with Applications 39 (2012), 7216–7225.

Ban

A.I.

and Coroianu

, Simplifying the search for effective ranking of fuzzy numbers, IEEE Transactions on Fuzzy Systems 23 (2015), 327–339.

Ban

O.I.

, (2011)Fuzzy multicriteria decision making method applied to selection of the best touristic destinations, International Journal of Mathematical Models and Methods in Applied Science 5 (2014), 264–271.

10.

Ban

O.I.

and Bogdan

, Using partial correlation coefficient to assess the importance of quality attributes, Study case for tourism Romanian consumers, International Journal of Social Science and Humanity 1 (2013), 21–24.

11.

Ban

O.I.

and Meşter

I.T

, Using Kano two dimensional service quality classification and characteristic analysis from the perspective of hotels’ clients of Oradea, Journal of Tourism-Studies and Research in Tourism 18 (2014), 30–36.

12.

Bede

, Mathematics of fuzzy sets and fuzzy logic, Springer, Studies in Fuzziness and Soft Computing, Heidelberg, New York, New York, 2013.

13.

Chen

S.M.

, (1996)Evaluating weapon systems using fuzzy arithmetic operations, Fuzzy Sets and Systems 77 , 265–276.

14.

Cheng

, Guo

J.J.

and Ling

, Fuzzy importance-performance analysis of visitor satisfaction for theme park: The case of Fantawild Adventure in Taiwan, China, Current Issues in Tourism, 2013 in press. doi:10.1080/13683500.2013.777399

15.

Chiang

D.A.

and Lin

N.P.

, Correlation of fuzzy sets, Fuzzy Sets and Systems 102 (1999), 221–226.

16.

Chien

C.J.

and Tsai

H.H.

, Using fuzzy numbers to evaluate perceived service quality, Fuzzy Sets and Systems 116 (2000), 289–300.

17.

Chiou

H.K.

, Tzeng

G.H.

and Cheng

D.C.

, Evaluating sustainable fishing development strategies using fuzzy MCDM approach, Omega 33 (2005), 223–234.

18.

Chu

, Stated-importance versus derived-importance customer satisfaction measurement, Journal of Services Marketing 16 (2002), 285–301.

19.

Chu

T-C

and Lin

, An extension to fuzzy MCDM, Computers and Mathematics with Applications 57 (2009), 445–454.

20.

Deng

W.J.

and Pei

, Fuzzy neural based importanceperformance analysis for determining critical service attributes, Expert Systems with Applications 36 (2009), 3774–3784.

21.

Deng

W-J

, Fuzzy importance-performance analysis for determining critical service attributes, International Journal of Service Industry Management 19 (2008), 252–270.

22.

Deng

, Using a revised importance-performance analysis approach: The case of Taiwanese hot springs tourism, Tourism Management 28 (2007), 1274–1284.

23.

Deng

, Chan

F.T.S.

, Wu

and Wang

, A new linguistic MCDM method based on multiple-criterion data fusion, Expert Systems with Applications 38 (2011), 6985–6993.

24.

Diamond

and Kloeden

, Metric Spaces of Fuzzy Sets. Theory and Applications, World Scientific, Singapore, 1994.

25.

Dubois

and Prade

, Operations on fuzzy numbers, International Journal of Systems Science 9 (1978), 613–626.

26.

Dubois

and Prade

, The mean value of a fuzzy number, Fuzzy Sets and Systems 24 (1987), 279–300.

27.

Ebrahimnejad

and Verdagay

J.L.

, A novel approach for sensitivity analysis in linear programs with trapezoidal fuzzy numbers, Journal of Intelligent and Fuzzy Systems 27 (2014), 173–185.

28.

Feng

, Mangan

, Wong

, Xu

and Lalwani

, Investigating the different approaches to importance-performance analysis, The Service Industries Journal 34 (2014), 1021–1041.

29.

Grzegorzewski

and Mrówka

, Trapezoidal approximations of fuzzy numbers, Fuzzy Sets and Systems 153 (2005), 115–135.

30.

Gustafsson

and Johnson

M.D.

, Determining attribute importance in a service satisfaction model, Journal of Service Research 7 (2004), 124–141.

31.

Hair

J.F.

, Anderson

R.E.

, Tatham

R.L.

and Black

W.C.

, Multivariate Data Analysis, Pretince-Hall, Upper Saddle River, New Jersey, 1995.

32.

Hancock

G.R.

and Klockars

A.J.

, The effect of scale manipulations on validity: Targeting frequency rating scales for anticipated performance levels, Applied Ergonomics 22 (1991), 147–154.

33.

Hansen

and Bush

R.J.

, Understanding customer quality requirements: Model and application, Industrial Marketing Management 28 (1999), 119–130.

34.

Hanss

, Applied Fuzzy Arithmetic, Springer, 2005.

35.

Heilpern

, The expected value of a fuzzy number, Fuzzy Sets and Systems 47 (1992), 81–86.

36.

Hsieh

T.Y.

, Lu

S.T.

and Tzeng

G.H.

, Fuzzy MCDM approach for planning and design tenders selection in public office buildings, International Journal of Project Management 22 (2004), 573–584.

37.

H-Y

, Lee

Y-C

, Yen

T-M

and Tsai

C-H

, Using BPNN and DEMATEL to modify importance-performance analysis model-A study of the computer industry, Expert Systems with Applications 36 (2009), 9969–9979.

38.

Lai

I.K.W.

and Hitchcock

, Importance-performance analysis in tourism: A framework for researchers, Tourism Management 48 (2015), 242–267.

39.

, A novel Likert scale based on fuzzy sets theory, Expert Systems with Applications 40 (2013), 1609–1618.

40.

Liu

S-T

and Kao

, Fuzzy measures for correlation coefficient of fuzzy numbers, Fuzzy Sets and Systems 128 (2002), 267–275.

41.

Matzler

, Sauerwein

and Heischmidt

K.A.

, Importanceperformance analysis revisited: The role of the factor structure of customer satisfaction, The Service Industries Journal 23 (2003), 112–129.

42.

Mount

D.J.

and Sciarini

M.P.

, IPA and DSI: Enhancing the usefulness of student evaluation of teaching data, Journal of Hospitality & Tourism Education 10 (1999), 8–14.

43.

Mount

D.J.

, An empirical application of quantitative derived importance-performance analysis (QDIPA) for employee satisfaction, Journal of Quality Assurance in Hospitality & Tourism 6 (2005), 65–76.

44.

Negoiţă

and Ralescu

, Application of Fuzzy Sets to System Analysis, Wiley, New York, 1975.

45.

Nguyen

H.T.

, (1978)A note on the extension principle for fuzzy sets, Journal of Mathematical Analysis and Applications 64 , 369–380.

46.

Ryan

M.J.

, Rayner

and Morrison

, Diagnosing customer loyalty drivers: Partial least squares vs regression, Marketing Research 11(Summer) (1999), 19–29.

47.

Shahsavari-Pour

, Tavakkoli-Moghaddam

and Basiri

M-A

, A new method for trapezoidal fuzzy numbers ranking based on the Shadow length and its application to manager’s risk taking, Journal of Intelligent and Fuzzy Systems 26 (2014), 77–89.

48.

Snedecor

G.W.

and Cochran

W.G.

, Statistical Methods Iowa State University Press, Iowa, 1967.

49.

Tian

K.Y.

, An empirical study of factors that affect tourist satisfaction in scenic areas based on Fuzzy-IPA, Tourism Tribune 25 (2010), 61–65.

50.

Usha Madhuri

, Babu

S. Suresh

and Shankar

N. Ravi

, Fuzzy risk analysis based on the novel fuzzy ranking with new arithmetic operations of linguistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems 26 (2014), 2391–2401.

51.

Van Ryzin

G.G.

, Importance-performance analysis of citizen satisfaction surveys, Public Administration 85 (2007), 215–226.

52.

Wan

S-P

and Li

D-F

, Possibility mean and variance based method for multi-attribute decision making with triangular intuitionsitic fuzzy numbers, Journal of Intelligent and Fuzzy Systems 24 (2013), 743–754.

53.

W.Y.

, Hsiao

S.W.

and Kuo

H.P.

, Fuzzy set theory based decision model for determining market position and developing strategy for hospital service quality, Total Quality Management 15 (2004), 439–456.

54.

Yeh

C-H

and Kuo

Y-L

, Evaluating passenger services of Asia-Pacific international airports, Transportation Research Part E 39 (2003), 35–48.

55.

Yeh

C-T

, On improving trapezoidal and triangular approximations of fuzzy numbers, International Journal of Approximate Reasoning 48 (2008), 297–313.

56.

, Correlation of fuzzy numbers, Fuzzy Sets and Systems 55 (1993), 303–307.

57.

S.C.

and Yu

M.N.

, Fuzzy partial credit scaling: A valid approach for scoring the BECK depression inventory, Social Behavior and Personality: An International Journal 35 (2007), 1163–1172.

58.

S.C.

and Yu

, Fuzzy item response model: A new approach to generate membership function to score psychological measurement, Quality and Quantity 43 (2009), 381–390.

59.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.

60.

Zadeh

L.A.

, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978), 3–28.