Triangular fuzzy Bonferroni mean operators and their application to multiple attribute decision making

Abstract

In this paper, we investigate the multiple attribute decision making (MADM) problems with triangular fuzzy information. Motivated by the ideal of Bonferroni mean, we develop the aggregation techniques called the triangular fuzzy Bonferroni mean (TFBM) operator for aggregating the triangular fuzzy information. We study its properties and discuss its special cases. For the situations where the input arguments have different importance, we then define the triangular fuzzy weighted Bonferroni mean (TFWBM) operator, based on which we develop the procedure for multiple attribute decision making under the triangular fuzzy environments. Finally, a practical example for evaluating the modernization development of Chinese traditional medicine industry is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Multiple attribute decision making (MADM)triangular fuzzy numbers Bonferroni mean triangular fuzzy Bonferroni mean (TFBM) operator triangular fuzzy weighted Bonferroni mean (TFWBM) operator Chinese traditional medicine industry modernization development

1 Introduction

The information aggregation operators are an interesting research topic, which is receiving increasing attention. The fundamental aspect of the order weighted averaging (OWA) operator is a reordering step in which the input arguments are rearranged in descending order [1]. Since its appearance, it has been studied and applied in a wide range of problems [2 –11]. The ordered weighted geometric (OWG) operator is an aggregation operator that is based on the OWA operator and the geometric mean [12, 13]. In some situations, however, the input arguments take the form of fuzzy data rather than numerical ones because of time pressure, lack of knowledge, and the decision maker’s limited attention and information processing capabilities. Therefore, Xu [14] and Fan and Wang [15] developed the fuzzy ordered weighted averaging (FOWA) operator. Xu [16] introduced the fuzzy ordered weighted geometric (FOWG) operator. Xu and Wu [17] proposed the fuzzy induced ordered weighted averaging (FIOWA) operator. Xu and Da [18] developed the fuzzy induced ordered weighted geometric (FIOWG) operator. Xu [19] developed some fuzzy harmonic mean operators, such as fuzzy weighted harmonic mean (FWHM) operator, fuzzy ordered weighted harmonic mean (FOWHM) operator, fuzzy hybrid harmonic mean (FHHM) operator. Wei [20] proposed the fuzzy ordered weighted harmonic averaging(FOWHA) operator. Wei [21] developed the fuzzy induced ordered weighted harmonic mean (FIOWHM) operator and applied it to the group decision making. Wei [22] proposed the generalized triangular fuzzy correlated averaging operator and applied these operators to multiple attribute decision making. Merigo [23] presented the fuzzy probabilistic ordered weighted averaging (FPOWA) operator which is an aggregation operator that unifies the fuzzy probabilistic aggregation and the fuzzy OWA (FOWA) operator in the same formulation considering the degree of importance that each concept has in the analysis. Merigo and Casanovas [24] introduced several generalizations of the hybrid aggregation (HA) operator by using generalized and quasi-arithmetic means, fuzzy numbers and order inducing variables in the reordering step of the aggregation process. They presented the fuzzy generalized hybrid averaging (FGHA) operator, the fuzzy induced generalized hybrid averaging (FIGHA) operator, the Quasi-FHA operator and the Quasi-FIHA operator. The main advantage of these operators is that they generalize a wide range of fuzzy aggregation operators that can be used in a wide range of applications such as decision making problems. For example, they mentioned the fuzzy induced hybrid averaging (FIHA), the fuzzy weighted generalized mean (FWGM) and the fuzzy induced generalized OWA (FIGOWA). Merigo and Gil-Lafuente [25] proposed a wide range of fuzzy induced generalized aggregation operators such as the fuzzy induced generalized ordered weighted averaging (FIGOWA) and the fuzzy induced quasi-arithmetic OWA (Quasi-FIOWA) operator. They are aggregation operators that use the main characteristics of the fuzzy OWA (FOWA) operator, the induced OWA (IOWA) operator and the generalized (or quasi-arithmetic) OWA operator. Therefore, they use uncertain information represented in the form of fuzzy numbers, generalized (or quasi-arithmetic) means and order inducing variables. The main advantage of these operators is that they include a wide range of mean operators such as the FOWA, the IOWA, the induced Quasi-OWA, the fuzzy IOWA, the fuzzy generalized mean and the fuzzy weighted quasi-arithmetic average (Quasi-FWA). They further generalize this approach by using Choquet integrals, obtaining the fuzzy induced quasi-arithmetic Choquet integral aggregation (Quasi-FICIA) operator. Xu [26] considered situations with linguistic, interval or fuzzy preference information, and develop some fuzzy ordered distance measures, such as linguistic ordered weighted distance measure, uncertain ordered weighted distance measure, linguistic hybrid weighted distance measure, and uncertain hybrid weighted distance measure, etc. After that, based on hybrid weighted distance measures, they established a consensus reaching process of group decision making with linguistic, interval, triangular or trapezoidal fuzzy preferenceinformation.

The Bonferroni Mean (BM), originally introduced by Bonferroni [27], is one of the aggregation methods. Due to its capability to capture the interrelationship between input arguments, BM is very useful in various application fields and has attracted a lot of attentions from researchers. Yager [28] proposed some generalizations of the BM, that enchance its modeling capability, by replacing the simple averaging by other mean type operators, such as the ordered weighted averaging (OWA) operator [29] and Choquet integral [30]. Yager [31] and Beliakov et al. [32] proposed another generalized form of BM. Nevertheless, Zhu et al. [33] explored the geometric Bonferroni mean (GBM) considering both the BM and the geometric mean (GM). Up to now, there existed approaches developed for dealing with triangular fuzzy decision making problems which can’t consider the interrelationship between input triangular fuzzy arguments. Therefore, it is necessary to pay attention to this issue. The aim of this paper is to develop some approaches to triangular fuzzy decision making problems consider the interrelationship between input triangular fuzzy arguments. In order to do so, in this paper, we futher extend the Bonferroni mean [27, 33] to triangular fuzzy situations. We first develop the aggregation techniques called the triangular fuzzy Bonferroni mean (TFBM) operator for aggregating the triangular fuzzy information. We study its properties and discuss its special cases. For the situations where the input arguments have different importance, we then define the triangular fuzzy weighted Bonferroni mean (TFWBM) operator, based on which we develop the procedure for multiple attribute decision making under the triangular fuzzy environments. To do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to triangular fuzzy numbers and some operational laws of triangular fuzzy numbers and Bonferroni mean operators. In Section 3 we have developed some triangular fuzzy aggregation operators: triangular fuzzy Bonferroni mean (TFBM) operator and the triangular fuzzy weighted Bonferroni mean (TFWBM) operator and studied some desirable properties of the proposed operators. The prominent characteristic of these proposed operators is that they take into account interrelationship among the input arguments. In Section 4, we have applied these operators to develop the model for triangular fuzzy multiple attribute decision making problems with triangular fuzzy information. In Section 5, a practical example for evaluating the modernization development of chinese traditional medicine industry is given to verify the developed approach and to demonstrate its practicality and effectiveness. In Section 6, we conclude the paper and give some remarks.

2 Preliminaries

2.1 Triangular fuzzy numbers

In this section, we briefly describe some basic concepts and basic operational laws related to triangular fuzzy numbers.

Definition 1. [34] A triangular fuzzy numbers $\tilde{a}$ can be defined by a triplet (a^L, a^M, a^U). The membership function $μ_{\tilde{a}} (x)$ is defined as: $μ_{\tilde{a}} (x) = {\begin{matrix} 0, & x < a^{L}, \\ \frac{x - a^{L}}{a^{M} - a^{L}}, & a^{L} \leq x \leq a^{M}, \\ \frac{x - a^{U}}{a^{M} - a^{U}}, & a^{M} \leq x \leq a^{U}, \\ 0, & x \geq a^{U} . \end{matrix}$ (1) where 0 < a^L ≤ a^M ≤ a^U, a^L and a^U stand for the lower and upper values of the support of $\tilde{a}$ , respectively, and a^M for the modal value.

Definition 2. [34] Basic operational laws related to triangular fuzzy numbers: $\begin{matrix} \tilde{a} \oplus \tilde{b} & = & [a^{L}, a^{M}, a^{U}] \oplus [b^{L}, b^{M}, b^{U}] \\ = & [a^{L} + b^{L}, a^{M} + b^{M}, a^{U} + b^{U}] \end{matrix}$ $\begin{matrix} \tilde{a} \otimes \tilde{b} & = & [a^{L}, a^{M}, a^{U}] \otimes [b^{L}, b^{M}, b^{U}] \\ [a^{L} b^{L}, a^{M} b^{M}, a^{U} b^{U}] \\ λ \otimes \tilde{a} & = & λ \otimes [a^{L}, a^{M}, a^{U}] \\ = & [λ a^{L}, λ a^{M}, λ a^{U}], λ > 0 . \end{matrix}$

Definition 3. [19] Let $\tilde{b} = [b^{L}, b^{M}, b^{U}]$ and $\tilde{a} = [a^{L}, a^{M}, a^{U}]$ be two triangular fuzzy numbers, then the degree of possibility of a ≥ b is defined as

$\begin{matrix} p (a \geq b) \\ = λ max {1 - max [\frac{b^{M} - a^{L}}{a^{M} - a^{L} + b^{M} - b^{L}}, 0], 0} \\ + (1 - λ) max {1 - max [\frac{b^{U} - a^{M}}{a^{U} - a^{M} + b^{U} - b^{M}}, 0], 0} \end{matrix}$ (2) where the value λ is an index of rating attitude. It reflects the decision maker’s risk-bearing attitude. If λ > 0.5, the decision maker is risk lover. If λ = 0.5, the decision maker is neutral to risk. If λ < 0.5, the decision maker is risk avertor.

From Definition 3, we can easily get the following results easily:

0 ≤ p (a ≥ b) ≤ 1, 0 ≤ p (b ≥ a) ≤ 1;

p (a ≥ b) + p (b ≥ a) = 1 . Especially, p (a ≥ a) = p (b ≥ b) = 0.5.

2.2 Bonferroni mean

Bonferroni [27] originally introduced a mean type aggregation operator, called Bonferroni mean, which can provide for aggregation lying between the max, min operators and the logical “or” and “and” operators, which was defined as follows:

Definition 4. [27] Let p, q ≥ 0 and a_i (i = 1, 2, …, n) be a collection of non-negative real numbers. Then the aggregation functions:

$\begin{matrix} {BM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{n (n - 1)} \sum_{\binom{i, j = 1}{i \neq j}}^{n} a_{i}^{p} a_{j}^{q})}^{\frac{1}{p + q}} \end{matrix}$ (3) is called the Bonferroni mean (BM) operator.

3 Triangular fuzzy Bonferroni mean operators

The Bonferroni mean (BM) operator [27], however, have usually been used in situations where the input arguments are the non-negative real numbers. We shall extend the BM operators to accommodate the situations where the input arguments are triangular fuzzy numbers. In this section, we shall investigate the BM operator under triangular fuzzy environments. Based on Definition 4, we give the definition of the triangular fuzzy Bonferroni mean (TFBM) operator as follows:

Definition 5. Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, and let p, q > 0. If $\begin{matrix} {TFBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \end{matrix}$ $\begin{matrix} = {(\frac{1}{n (n - 1)} \sum_{\binom{i, j = 1}{i \neq j}}^{n} {\tilde{a}}_{i}^{p} {\tilde{a}}_{j}^{q})}^{\frac{1}{p + q}} \end{matrix}$

$\begin{matrix} = [{(\frac{1}{n (n - 1)} \sum_{\binom{i, j = 1}{i \neq j}}^{n} {(a_{i}^{L})}^{p} {(a_{j}^{L})}^{q})}^{\frac{1}{p + q}}, \\ {(\frac{1}{n (n - 1)} \sum_{\binom{i, j = 1}{i \neq j}}^{n} {(a_{i}^{M})}^{p} {(a_{j}^{M})}^{q})}^{\frac{1}{p + q}}, \\ {(\frac{1}{n (n - 1)} \sum_{\binom{i, j = 1}{i \neq j}}^{n} {(a_{i}^{U})}^{p} {(a_{j}^{U})}^{q})}^{\frac{1}{p + q}}] \end{matrix}$ (4) then TFBM^p,q is called the triangular fuzzy Bonferroni mean (TFBM) operator.

It can be easily proved that the TFBM operator has the following properties.

Theorem 1. (Idempotency) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}]$ (i = 1, 2, …, n) be a set of triangular fuzzy numbers. If all ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}])$ are equal, i.e. ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}]) = \tilde{a} (\tilde{a} = [a^{L}, a^{M}, a^{U}])$ for all j, then ${TFBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (5)

Theorem 2. (Boundedness) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}]$ (i = 1, 2, …, n) be a set of triangular fuzzy numbers, and let ${\tilde{a}}^{-} \underset{j}{= min} {\tilde{a}}_{j}, {\tilde{a}}^{+} \underset{j}{= min} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} \leq {TFBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ (6)

Theorem 3. (Monotonicity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}]$ (i = 1, 2, …, n) and … , n) be two sets of triangular fuzzy numbers, if ${\tilde{a}}_{j} \leq {\tilde{a}}_{j}^{'}$ , for all j, then $\begin{matrix} {TFBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \leq {TFBM}^{p, q} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (7)

Theorem 4. (Commutativity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}]$ (i = 1, 2, …, n) and … , n) be two sets of triangular fuzzy numbers, where is any permutation of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , then $\begin{matrix} {TFBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {TFBM}^{p, q} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (8)

Now we discuss some special cases of the TFBM with respect to the parameters p and q:

Case 1. If q → 0, then from the TFBM (4), it yields

$\begin{matrix} lim_{q \to 0} {TFBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = lim_{q \to 0} {(\frac{1}{n (n - 1)} \sum_{\binom{i, j = 1}{i \neq j}}^{n} {\tilde{a}}_{i}^{p} {\tilde{a}}_{j}^{q})}^{\frac{1}{p + q}} \\ = {(\frac{1}{n} \sum_{i = 1}^{n} {\tilde{a}}_{i}^{p})}^{\frac{1}{p}} \\ = {TFBM}^{p, 0} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \end{matrix}$ (9) which we call the triangular fuzzy generalized mean (TFGM) operator.

Case 2. If p = 2 and q → 0, then by the TFBM (4), we have ${TFBM}^{2, 0} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {(\frac{1}{n} \sum_{i = 1}^{n} {\tilde{a}}_{i}^{2})}^{\frac{1}{2}}$ (10) which we call the triangular fuzzy square mean (TFSM) operator.

Case 3. If p = 1 and q → 0, then the TFBM (4) reduces to triangular fuzzy mean (TFM) operator ${TFBM}^{1, 0} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \frac{1}{n} \sum_{i = 1}^{n} {\tilde{a}}_{i}$ (11) which we call the triangular fuzzy mean (TFM) operator.

Case 4. If p = 1 and q = 1, then by the TFBM (4), we have ${TFBM}^{1, 1} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {(\frac{1}{n (n - 1)} \sum_{\binom{i, j = 1}{i \neq j}}^{n} {\tilde{a}}_{i} {\tilde{a}}_{j})}^{\frac{1}{2}}$ (12) which we call the triangular fuzzy interrelated square mean (TFISM) operator.

Considering that the input arguments may have different importance, here we define the triangular fuzzy weighted Bonferroni mean (TFWBM) operator.

Definition 6. ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers and p, q > 0, w = (w₁, w₂, …, w_n) ^T is the weight vector of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , where w_i indicates the importance degree of ${\tilde{a}}_{i}$ , satisfying w_i > 0 (i = 1, 2, …, n), and $\sum_{i = 1}^{n} w_{i} = 1$ . If

$\begin{matrix} {TFWBM}_{w}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {(\frac{1}{n (n - 1)} \sum_{\binom{i, j = 1}{i \neq j}}^{n} {(w_{i} {\tilde{a}}_{i})}^{p} {(w_{j} {\tilde{a}}_{j})}^{q})}^{\frac{1}{p + q}} \end{matrix}$ (13) then ${TFWBM}_{w}^{p, q}$ is called the triangular fuzzy weighted Bonferroni mean (TFWBM) operator.

4 An approach to multiple attribute decision making with triangular fuzzy information

In this section, we shall utilize the triangular fuzzy weighted Bonferroni mean (TFWBM) operator to multiple attribute decision making for evaluating the modernization development of Chinese traditional medicine industry with triangular fuzzy information.

For a multiple attribute decision making problems with triangular fuzzy information, let A ={ A₁, A₂, …, A_m } be a discrete set of alternatives, G ={ G₁, G₂, …, G_n } be the set of attributes, whose weight vector is ω = (ω₁, ω₂, …, ω_n), with ω_j ≥ 0, j = 1, 2, …, n, $\sum_{j = 1}^{n} ω_{j} = 1$ . Suppose that $A = {({\tilde{a}}_{ij})}_{m \times n} = {[a_{ij}^{L}, a_{ij}^{M}, a_{ij}^{U}]}_{m \times n}$ is the decision making matrix, where ${\tilde{a}}_{ij}$ is the preference values, which take the form of triangular fuzzy numbers, given by the decision maker, for the alternative A_i ∈ A with respect to the attribute G_j ∈ G.

Then, we utilize the triangular fuzzy weighted Bonferroni mean (TFWBM) operator to develop an approach to multiple attribute decision making problems with triangular fuzzy information, which can be described as following:

Step 1. Normalize each attribute value ${\tilde{a}}_{ij}$ in the matrix A into a corresponding element in the matrix $R = {({\tilde{r}}_{ij})}_{m \times n} = {[r_{ij}^{L}, r_{ij}^{M}, r_{ij}^{U}]}_{m \times n}$ using the following formulas: $\begin{matrix} {\begin{matrix} r_{ij}^{L} = a_{ij}^{L} / \sum_{i = 1}^{m} a_{ij}^{U} \\ r_{ij}^{M} = a_{ij}^{M} / \sum_{i = 1}^{m} a_{ij}^{M} \\ r_{ij}^{U} = a_{ij}^{U} / \sum_{i = 1}^{m} a_{ij}^{L} \end{matrix} \\ for benefit attribute G_{j}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$ (14) $\begin{matrix} {\begin{matrix} r_{ij}^{L} = (1 / a_{ij}^{U}) / \sum_{i = 1}^{m} (1 / a_{ij}^{L}) \\ r_{ij}^{M} = (1 / a_{ij}^{M}) / \sum_{i = 1}^{m} (1 / a_{ij}^{M}) \\ r_{ij}^{U} = ((1 / a_{ij}^{L})) / \sum_{i = 1}^{m} (1 / a_{ij}^{U}) \end{matrix} \\ for cost attribute G_{j}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$ (15)

Step 2. Utilize the decision information given in matrix $\tilde{R}$ , and the TFWBM operator (in general, we can take p = q = 1)

$\begin{matrix} {\tilde{r}}_{i} = (r_{i}^{L}, r_{i}^{M}, r_{i}^{U}) \\ = {TFWBM}_{w}^{p, q} ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) \\ = {(\frac{1}{n (n - 1)} \sum_{\binom{k, l = 1}{k \neq l}}^{n} {(w_{k} {\tilde{r}}_{ik})}^{p} {(w_{l} {\tilde{r}}_{il})}^{q})}^{\frac{1}{p + q}} \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$ (16)

Step 3. To rank these collective overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ , we first compare each ${\tilde{r}}_{i}$ with all the ${\tilde{r}}_{j} (j = 1, 2, \dots, m)$ by using Eq. (2). For simplicity, we let $p_{ij} = p ({\tilde{r}}_{i} \geq {\tilde{r}}_{j})$ , then we develop a complementary matrix as P = (p_ij) _m×m, where p_ij ≥ 0, p_ij + p_ji = 1, p_ii = 0.5, i, j = 1, 2, …, n.

Summing all the elements in each line of matrix P, we have $p_{i} = \sum_{j = 1}^{m} p_{ij}, i = 1, 2, \dots, m .$ (17)

Then we rank the overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ in descending order in accordance with the values of p_i (i = 1, 2, …, m).

Step 4. Rank all the alternatives X_i (i = 1, 2, …, m) and select the best one(s) in accordance with the collective overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ .

Step 5. End.

5 Numerical example

In this section, we utilize a practical multiple attribute decision making problems for evaluating the modernization development of Chinese traditional medicine industry to illustrate the application of the developed approaches. The company employs some external professional organizations (or experts) to aid this decision-making. The expert team selects four attributes to evaluate the modernization development of Chinese traditional medicine industry of five cities: (1) G₁: overall efficiency, (2) G₂: technical advancement, (3) G₃: internationalization; (4) G₄: the rationalization of industrial organization. The five possible cities A_i (i = 1, 2, …, 5) are to be evaluated using the triangular fuzzy numbers by the decision makers under the above four attributes (whose weighting vector is ω = (0.2, 0.1, 0.3, 0.4), and construct the following matrix $A = {({\tilde{a}}_{ij})}_{5 \times 4}$ is shown in Table 1.

In the following, in order to select the most desirable cities, we utilize the TFWBM operator to develop an approach to multiple attribute decision making problems for evaluating the modernization development of Chinese traditional medicine industry with triangular fuzzy information, which can be described asfollowing:

Step 1. Calculate the normalized decision matrix $\tilde{R}$ . The result is shown in Table 2.

Step 2. Aggregate all triangular fuzzy preference value ${\tilde{r}}_{ij} (j = 1, 2, \dots, n)$ by using the TFWBM to derive the overall triangular fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, 3, 4, 5)$ of the city A_i. The aggregating results are shown in Table 3.

Step 3. According to the aggregating results shown in Table 3 and the formula of degree of possibility (2), the ordering of the cities are shown in Table 4. Note that “> ” means “preferred to”. As we can see, the best alternative is A₃.

From equation (16) and the aforementioned example, we can see that the value derived by the TFWBM operator depend on the choice of the parameters p and q, and these two parameters are not robust. In general, the larger the parameters p and q, the more the calculation effort needed, and in the special case where at least one of these two parameters takes the value of zero, the TFWBM operator can’t capture the interrelationship of the individual arguments. As a result, in practical applications, we generally take the values of the two parameters as p = q = 1, which is not only intuitive and simple but also the interrelationship of the individual arguments can be fully taken into account.

6 Conclusion

In this paper, we investigate the multiple attribute decision making problems for evaluating the modernization development of Chinese traditional medicine industry with triangular fuzzy information. Motivated by the ideal of Bonferroni mean, we develop the aggregation technique called the triangular fuzzy Bonferroni mean (TFBM) operator for aggregating the triangular fuzzy information. We study its properties and discuss its special cases. For the situations where the input arguments have different importance, we then define the triangular fuzzy weighted Bonferroni mean (TFWBM) operator, based on which we develop the procedure for multiple attribute decision making under the triangular fuzzy environments. Finally, a practical example for evaluating the modernization development of Chinese traditional medicine industry is given to verify the developed approach and to demonstrate its practicality and effectiveness. In the future, our results may be further generalized by using the well-known Choquet integral [4] and Dempster-Shafer belief structure [36, 37] which is an interesting issue remained to be studied.

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