Research on hotel supply chain risk assessment with dual generalized triangular fuzzy Bonferroni mean operators

Abstract

In this paper, we investigate the multiple attribute decision making (MADM) problems with triangular fuzzy information. Motivated by the ideal of dual generalized Bonferroni mean and dual generalized geometric Bonferroni mean, we develop two aggregation techniques called the dual generalized triangular fuzzy Bonferroni mean (DGTFBM) operator and the dual generalized triangular fuzzy geometric Bonferroni mean (DGTFGBM) 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 dual generalized triangular fuzzy weighted Bonferroni mean (DGTFWBM) operator and the dual generalized triangular fuzzy weighted geometric Bonferroni mean (DGTFWGBM) operator, based on which we develop two procedure for multiple attribute decision making under the triangular fuzzy environments. Finally, a practical example for hotel supply chain risk assessment is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Multiple attribute decision making (MADM)triangular fuzzy numbers dual generalized Bonferroni mean dual generalized geometric Bonferroni mean dual generalized triangular fuzzy Bonferroni mean (DGTFBM) operator hotel supply chain risk assessment

1. Introduction

The information aggregation operators are an interesting research topic, which is receiving increasing attention. The fundamental aspect of the 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 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 preference information.

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 is no approach developed for dealing with triangular fuzzy decision making problems 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 dual generalized Bonferroni mean [27]and dual generalized geometric Bonferroni mean [33] to triangular fuzzy situations. We first develop two aggregation techniques called the dual generalized triangular fuzzy Bonferroni mean (DGTFBM) operator and the dual generalized triangular fuzzy geometric Bonferroni mean (DGTFGBM) 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 dual generalized triangular fuzzy weighted Bonferroni mean (DGTFWBM) operator and the dual generalized triangular fuzzy weighted geometric Bonferroni mean (DGTFWGBM) operator, based on which we develop two 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: dual generalized triangular fuzzy Bonferroni mean (DGTFBM) operator, the dual generalized triangular fuzzy geometric Bonferroni mean (DGTFGBM) operator, the dual generalized triangular fuzzy weighted Bonferroni mean (DGTFWBM) operator and the dual generalized triangular fuzzy weighted geometric Bonferroni mean (DGTFWGBM) 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 some models for triangular fuzzy multiple attribute decision making (MADM) problems with triangular fuzzy information. In Section 5, a practical example for hotel supply chain risk assessment 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: $μ_{\tilde{a}} (x) = {\begin{matrix} 0, x < a^{L}, \\ \frac{x - a^{L}}{a^{M} - a^{L}}, a^{L} ⩽ x ⩽ a^{M}, \\ \frac{x - a^{U}}{a^{M} - a^{U}}, a^{M} ⩽ x ⩽ a^{U}, \\ 0, x ⩾ a^{U} . \end{matrix}$ (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: $\begin{matrix} \tilde{a} \oplus \tilde{b} & = & [a^{L}, a^{M}, a^{U}] \oplus [b^{L}, b^{M}, b^{U}] \\ = & [a^{L} + b^{L}, a^{M} + b^{M}, a^{U} + b^{U}] \\ \tilde{a} \otimes \tilde{b} & = & [a^{L}, a^{M}, a^{U}] \otimes [b^{L}, b^{M}, b^{U}] \\ = & [a^{L} b^{L}, a^{M} b^{M}, a^{U} b^{U}] λ \otimes \tilde{a} \\ = & λ \otimes [a^{L}, a^{M}, a^{U}] \\ = & [λ a^{L}, λ a^{M}, λ a^{U}], λ > 0 \end{matrix}$

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 $\begin{matrix} p (a ⩾ b) \\ = λ max {1 - max [\frac{b^{M} - a^{L}}{a^{M} - a^{L} + b^{M} - b^{L}}, 0], 0} \\ + (1 - λ) max {1 - max [\frac{b^{U} - a^{M}}{a^{U} - a^{M} + b^{U} - b^{M}}, 0], 0} \end{matrix}$ (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. Dual generalized 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 b_i (i = 1, 2, ⋯ , n) be a collection of nonnegative crisp numbers. The dual generalized Bonferroni mean (DGBM) is defined as follows: $\begin{array}{l} {DGBM}^{R} (b_{1}, b_{2}, \dots, b_{n}) \\ = (\frac{1}{n!} \sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} b^{r j}_{i j}))^{^{^{^{^{_{^{^{^{_{^{1 / \sum_{j = 1}^{n} r j}}}}}}}}}}} \end{array}$ (3)

Where R = (r₁, r₂, …, r_n) ^T is the parameter vector with r_i ⩾ 0 (i = 1, 2, …, n) .

3. Dual generalized triangular fuzzy Bonferroni mean operators

3.1. DGTFBM and DGTFWBM operators

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

Definition 5. 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 r_i ⩾ 0 (i = 1, 2, …, n). If $\begin{array}{l} D G T F B M^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = (\frac{1}{n!} \sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} {\tilde{a}}_{i_{j}}^{r j}))^{1 / \sum_{j = 1}^{n} r_{j}} \\ = [(\frac{1}{n!} {\sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} {(a_{_{j}^{i}}^{L})}^{r j}))}^{1 / \sum_{j = 1}^{n} r_{j}}, (\frac{1}{n!} {\sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} {(a_{i_{j}^{}}^{M})}^{r j}))}^{1 / \sum_{j = 1}^{n} r_{j}}, (\frac{1}{n!} {\sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} {(a_{i_{j}^{}}^{U})}^{r j}))}^{1 / \sum_{j = 1}^{n} r_{j}}] \end{array}$ (4) then DGTFBM^R is called the dual generalized triangular fuzzy Bonferroni mean (DGTFBM) operator.

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

Property 1. (Idempotency) 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. 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 ${DGTFBM}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (5)

Property 2. (Boundedness) 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 ${\tilde{a}}^{-} \underset{j}{= min} {\tilde{a}}_{j}, {\tilde{a}}^{+} \underset{j}{= min} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} ⩽ {DGTFBM}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) ⩽ {\tilde{a}}^{+}$ (6)

Property 3. (Monotonicity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, if ${\tilde{a}}_{j} ⩽ {\tilde{a}}_{j}^{'}$ , for all j, then

$\begin{matrix} {DGTFBM}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ ⩽ {DGTFBM}^{R} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (7)

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

Now we discuss some special cases of the DGTFBM with respect to the parameters r_i ⩾ 0 (i = 1, 2, …, n):

If R = (λ, 0, 0, …, 0) , then we obtain $D G T F B M^{(λ, 0, 0, \dots, 0)} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {(\frac{1}{n} \sum_{i = 1}^{n} {\tilde{a}}_{i}^{λ})}^{1 / λ}$ (9)

Which is the generalized averaging operator.

If R = (s, t, 0, 0, …, 0) , then we obtain

$D G T F B M^{(s, t, 0, 0, \dots, 0)} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {(\frac{1}{n (n - 1)} \sum_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} {\tilde{a}}_{i}^{s} {\tilde{a}}_{j}^{t})}^{1 / (s + t)}$ (10)

Which is the BM.

If R = (s, t, r, 0, 0, …, 0) , then we obtain $D G T F B M^{(s, t, r, 0, 0, \dots, 0)} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {(\frac{1}{n (n - 1) (n - 2)} \sum_{\begin{array}{l} i, j, k = 1 \\ i \neq j \neq k \end{array}}^{n} {\tilde{a}}_{i}^{s} {\tilde{a}}_{j}^{t} {\tilde{a}}_{k}^{r})}^{1 / (s + t + k)}$ (11)

Which is the GBM.

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

Definition 6. 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 r_i ⩾ 0 (i = 1, 2, …, n), w = (w₁, w₂, ⋯ , w_n) ^T is the weight vector of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , wherew_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

$\begin{array}{l} D G T F W B M_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {(\frac{1}{n!} \sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} w_{i_{j}} {\tilde{a}}^{r_{j}}_{i_{j}}))}^{1 / \sum_{j = 1}^{n} r_{j}} \\ = [{(\frac{1}{n!} \sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} w_{i_{j}} {(a_{i_{j}}^{L})}^{r_{j}}))}^{1 / \sum_{j = 1}^{n} r_{j}}, {(\frac{1}{n!} \sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} w_{i_{j}} {(a_{i_{j}}^{M})}^{r_{j}}))}^{1 / \sum_{j = 1}^{n} r_{j}}, {(\frac{1}{n!} \sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} w_{i_{j}} {(a_{i_{j}}^{U})}^{r_{j}}))}^{1 / \sum_{j = 1}^{n} r_{j}}] \end{array}$ (12) then ${DGTFWBM}_{w}^{R}$ is called the dual generalized triangular fuzzy weighted Bonferroni mean (DGTFWBM) operator.

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

Property 5. (Idempotency) 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. 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 ${DGTFWBM}_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (13)

Property 6. (Boundedness) 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 ${\tilde{a}}^{-} \underset{j}{= min} {\tilde{a}}_{j}, {\tilde{a}}^{+} \underset{j}{= min} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} ⩽ {DGTFWBM}_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) ⩽ {\tilde{a}}^{+}$ (14)

Property 7. (Monotonicity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, if ${\tilde{a}}_{j} ⩽ {\tilde{a}}_{j}^{'}$ , for all j, then $\begin{matrix} {DGTFWBM}_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ ⩽ {DGTFWBM}_{w}^{R} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (15)

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

$\begin{matrix} {DGTFWBM}_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {DGTFWBM}_{w}^{R} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (16)

3.2. DGTFGBM and DGTFWGBM operators

In the following, Zhu et al. [33] explored the geometric Bonferroni mean (GBM) considering both the BM and the geometric mean (GM).

Definition 7 [33]. Letr_i (i = 1, 2, ⋯ , n) ⩾ 0 and b_i (i = 1, 2, ⋯ , n) be a collection of non-negative real numbers. Then the aggregation functions: ${DGGBM}^{R} (b_{1}, b_{2}, \dots, b_{n}) = \frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\sum_{j = 1}^{n} r_{j} b_{i_{j}}))}^{\frac{1}{n!}}$ (17) is called the dual generalized geometric Bonferroni mean (DGGBM) operator.

The dual generalized geometric Bonferroni mean (DGGBM) operator [10], however, has usually been used in situations where the input arguments are the non-negative real numbers. We shall extend the DGGBM operators to accommodate the situations where the input arguments are triangular fuzzy information. In this section, we shall investigate the DGGBM operator under triangular fuzzy environments. Based on Definition 7, we give the definition of the dual generalized triangular fuzzy geometric Bonferroni mean (DGTFGBM) operator as follows:

Definition 8. 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 r_i (i = 1, 2, ⋯ , n) ⩾ 0. If $D G T F G B M^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\sum_{j = 1}^{n} r_{j} {\tilde{a}}_{i_{j}}))}^{\frac{1}{n!}}$

$\begin{matrix} = [\frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{matrix} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{matrix}}^{n} (\sum_{j = 1}^{n} r_{j} a_{i_{j}}^{L}))}^{\frac{1}{n!}}, \frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{matrix} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{matrix}}^{n} (\sum_{j = 1}^{n} r_{j} a_{i_{j}}^{M}))}^{\frac{1}{n!}}, \\ \frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{matrix} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{matrix}}^{n} (\sum_{j = 1}^{n} r_{j} a_{i_{j}}^{U}))}^{\frac{1}{n!}}] (18) \end{matrix}$ then DGTFGBM^R is called the dual generalized triangular fuzzy geometric Bonferroni mean (DGTFGBM) operator.

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

Property 9. (Idempotency) 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. 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 ${DGTFGBM}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (19)

Property 10. (Boundedness) 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 ${\tilde{a}}^{-} \underset{j}{= min} {\tilde{a}}_{j}, {\tilde{a}}^{+} \underset{j}{= min} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} ⩽ {DGTFGBM}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) ⩽ {\tilde{a}}^{+}$ (20)

Property 11. (Monotonicity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, if ${\tilde{a}}_{j} ⩽ {\tilde{a}}_{j}^{'}$ , for all j, then $\begin{matrix} {DGTFGBM}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) ⩽ \\ {DGTFGBM}^{R} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (21)

Property 12. (Commutativity) Let Let ${\tilde{a}}_{i} = [a_{i}^{L},$ $a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, where ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ is any permutation of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , then $\begin{matrix} {DGTFGBM}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {DGTFGBM}^{R} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (22)

Now we discuss some special cases of the DGTFGBM with respect to the parameters r_i (i = 1, 2, ⋯ , n):

(1) If R = (λ, 0, 0, …, 0) , then we obtain

$\begin{matrix} {DGTFGBM}^{(λ, 0, 0, \dots, 0)} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \frac{1}{λ} {(\prod_{i = 1}^{n} λ {\tilde{a}}_{i})}^{\frac{1}{n}} \end{matrix}$ (23)

Which is the generalized geometric averaging operator.

(2) If R = (s, t, 0, 0, …, 0) , then we obtain $\begin{matrix} {DGTFGBM}^{(s, t, 0, 0, \dots, 0)} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \frac{1}{(s + t)} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (s {\tilde{a}}_{i} + t {\tilde{a}}_{j}))}^{\frac{1}{n (n - 1)}} \end{matrix}$ (24)

Which is the GBM.

(3) If R = (s, t, r, 0, 0, …, 0) , then we obtain $\begin{matrix} {DGTFGBM}^{(s, t, r, 0, 0, \dots, 0)} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \frac{1}{(s + t + k)} {(\prod_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} (s {\tilde{a}}_{i} + t {\tilde{a}}_{j} + r {\tilde{a}}_{k}))}^{\frac{1}{n (n - 1) (n - 2)}} \end{matrix}$ (25)

Which is the GGBM.

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

Definition 9. 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 r_i (i = 1, 2, ⋯ , n) ⩾ 0, w = (w₁, w₂, ⋯ , w_n) ^T is the weight vector of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , wherew_i indicates the importance degree of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}]$ , satisfying w_i > 0 (i = 1, 2, ⋯ , n), and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} {DGTFWGBM}_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{matrix} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{matrix}}^{n} (\sum_{j = 1}^{n} r_{j} {({\tilde{a}}_{i_{j}})}^{w_{i_{j}}}))}^{\frac{1}{n!}} \\ = [\frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{matrix} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{matrix}}^{n} (\sum_{j = 1}^{n} r_{j} {(a_{i_{j}}^{L})}^{w_{i_{j}}}))}^{\frac{1}{n!}}, \frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{matrix} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{matrix}}^{n} (\sum_{j = 1}^{n} r_{j} {(a_{i_{j}}^{M})}^{w_{i_{j}}}))}^{\frac{1}{n!}}, \\ \frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{matrix} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{matrix}}^{n} (\sum_{j = 1}^{n} r_{j} {(a_{i_{j}}^{U})}^{w_{i_{j}}}))}^{\frac{1}{n!}}] \end{matrix}$ (26) then ${DGTFWGBM}_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n})$ is called the dual generalized triangular fuzzy weighted geometric Bonferroni mean (DGTFWGBM) operator.

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

Property 13. (Idempotency) 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. 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 ${DGTFWGBM}_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (27)

Property 14. (Boundedness) 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 ${\tilde{a}}^{-} \underset{j}{= min} {\tilde{a}}_{j}, {\tilde{a}}^{+} \underset{j}{= min} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} ⩽ {DGTFWGBM}_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) ⩽ {\tilde{a}}^{+}$ (28)

Property 15. (Monotonicity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M},$ $a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, if ${\tilde{a}}_{j} ⩽ {\tilde{a}}_{j}^{'}$ , for all j, then

$\begin{matrix} {DGTFWGBM}_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) ⩽ \\ {DGTFWGBM}_{w}^{R} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (29)

Property 16. (Commutativity) Let Let ${\tilde{a}}_{i} = [a_{i}^{L},$ $a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, where ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ is any permutation of ${\tilde{a}}_{i} = [a_{i}^{L},$ $a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , then $\begin{matrix} {DGTFWGBM}_{w}^{R} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {DGTFWGBM}_{w}^{R} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (30)

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

In this section, we shall utilize the dual generalized triangular fuzzy weighted Bonferroni mean (DGTFWBM) operator(dual generalized triangular fuzzy weighted geometric Bonferroni mean (DGTFWGBM) operator) to multiple attribute decision making 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 dual generalized triangular fuzzy weighted Bonferroni mean (DGTFWBM) operator (dual generalized triangular fuzzy weighted geometric Bonferroni mean (DGTFWGBM) 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 $X = {({\tilde{x}}_{ij})}_{m \times n} = {[x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{U}]}_{m \times n}$ using the following formulas: ${\begin{matrix} x_{ij}^{L} = a_{ij}^{L} / \sum_{i = 1}^{m} a_{ij}^{U} \\ x_{ij}^{M} = a_{ij}^{M} / \sum_{i = 1}^{m} a_{ij}^{M} \\ x_{ij}^{U} = a_{ij}^{U} / \sum_{i = 1}^{m} a_{ij}^{L} \end{matrix},$ (31) forbenefitattributeG_j, i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n, k = 1, 2, ⋯ , t

$\begin{matrix} {\begin{matrix} x_{ij}^{L} = (1 / a_{ij}^{U}) / \sum_{i = 1}^{m} (1 / a_{ij}^{L}) \\ x_{ij}^{M} = (1 / a_{ij}^{M}) / \sum_{i = 1}^{m} (1 / a_{ij}^{M}) \\ x_{ij}^{U} = ((1 / a_{ij}^{L})) / \sum_{i = 1}^{m} (1 / a_{ij}^{U}) \end{matrix}, \\ for cost attribute G_{j}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n, k = 1, 2, \dots, t \end{matrix}$ (32)

Step 2. Utilize the decision information given in matrix X, and the DGTFWBM operator(in general, we can take R = (1, 1, …, 1)) ${\tilde{x}}_{i} = (x_{i}^{L}, x_{i}^{M}, x_{i}^{U}) = D G T F W B M_{w}^{R} ({\tilde{x}}_{i 1}, {\tilde{x}}_{i 2}, \dots, {\tilde{x}}_{i n}) = ({(\frac{1}{n!} \sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} w_{i_{j}} {(x_{i_{j}}^{L})}^{r_{j}}))}^{1 / \sum_{j = 1}^{n} r_{j}}, {(\frac{1}{n!} \sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} w_{i_{j}} {(x_{i_{j}}^{M})}^{r_{j}}))}^{1 / \sum_{j = 1}^{n} r_{j}}, {(\frac{1}{n!} \sum_{\begin{array}{l} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{array}}^{n} (\prod_{j = 1}^{n} w_{i_{j}} {(x_{i_{j}}^{U})}^{r_{j}}))}^{1 / \sum_{j = 1}^{n} r_{j}}) i = 1, 2, \dots, m, j = 1, 2, \dots, n$ (33)

Or the DGTFWGBM(in general, we can take R = (1, 1, …, 1)) $\begin{matrix} {\tilde{x}}_{i} = (x_{i}^{L}, x_{i}^{M}, x_{i}^{U}) = {DGTFWGBM}_{w}^{R} ({\tilde{x}}_{i 1}, {\tilde{x}}_{i 2}, \dots, {\tilde{x}}_{in}) \\ = [\frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{matrix} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{matrix}}^{n} (\sum_{j = 1}^{n} r_{j} {(x_{i_{j}}^{L})}^{w_{i_{j}}}))}^{\frac{1}{n!}}, \frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{matrix} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{matrix}}^{n} (\sum_{j = 1}^{n} r_{j} {(x_{i_{j}}^{M})}^{w_{i_{j}}}))}^{\frac{1}{n!}}, \\ \frac{1}{\sum_{j = 1}^{n} r_{j}} {(\prod_{\begin{matrix} i_{1}, i_{2}, \dots, i_{n} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{n} \end{matrix}}^{n} (\sum_{j = 1}^{n} r_{j} {(x_{i_{j}}^{U})}^{w_{i_{j}}}))}^{\frac{1}{n!}}] \end{matrix} i = 1, 2, \dots, m, j = 1, 2, \dots, n$ (34) to derive the overall preference values ${\tilde{x}}_{i} (i = 1, 2, \dots, m)$ of the alternative A_i.

Step 3. To rank these collective overall preference values ${\tilde{x}}_{i} (i = 1, 2, \dots, m)$ , we first compare each ${\tilde{x}}_{i}$ with all the ${\tilde{x}}_{j} (j = 1, 2, \dots, m)$ by using Equation.(2). For simplicity, we let $p_{ij} = p ({\tilde{x}}_{i} ⩾ {\tilde{x}}_{j})$ , 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 $p_{i} = \sum_{j = 1}^{m} p_{ij}, i = 1, 2, \dots, m$ (35)

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

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

Step 5. End.

5. Numerical example

In this section, we utilize a practical multiple attribute decision making problems for hotel supply chain risk assessment to illustrate the application of the developed approaches. Suppose an organization plans to implement ERP system. The first step is to form a project team that consists of CIO and two senior representatives from user departments. By collecting all possible information about ERP vendors and systems, project term choose five potential hotels A_i (i = 1, 2, ⋯ , 5) as candidates. The company employs some external professional organizations (or experts) to aid this decision-making. The Project team selects four attributes to evaluate the alternatives: (1) function and technology G₁, (2) strategic fitness G₂, (3) vendor’s ability G₃; (4) vendor’s reputation G₄. The five possible alternativesA_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 isω = (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.

Table 1
Decision matrix A

G ₁ G ₂ G ₃ G ₄

A ₁ (0.72,0.76,0.80) (0.65,0.70,0.75) (0.64,0.66,0.69) (0.80,0.85,0.90)

A ₂ (0.67,0.77,0.83) (0.73,0.75,0.78) (0.33,0.40,0.43) (0.88,0.90,0.93)

A ₃ (0.90,0.93,0.95) (0.80,0.82,0.83) (0.50,0.52,0.55) (0.95,0.97,0.98)

A ₄ (0.97,0.98,1.00) (0.67,0.70,0.73) (0.71,0.72,0.73) (0.82,0.85,0.88)

A ₅ (0.78,0.79,0.81) (0.63,0.64,0.66) (0.56,0.58,0.61) (0.78,0.79,0.81)

	G ₁	G ₂	G ₃	G ₄
A ₁	(0.72,0.76,0.80)	(0.65,0.70,0.75)	(0.64,0.66,0.69)	(0.80,0.85,0.90)
A ₂	(0.67,0.77,0.83)	(0.73,0.75,0.78)	(0.33,0.40,0.43)	(0.88,0.90,0.93)
A ₃	(0.90,0.93,0.95)	(0.80,0.82,0.83)	(0.50,0.52,0.55)	(0.95,0.97,0.98)
A ₄	(0.97,0.98,1.00)	(0.67,0.70,0.73)	(0.71,0.72,0.73)	(0.82,0.85,0.88)
A ₅	(0.78,0.79,0.81)	(0.63,0.64,0.66)	(0.56,0.58,0.61)	(0.78,0.79,0.81)

In the following, in order to select the most desirable candidate, we utilize the DGTFWBM (or DGTFWGBM) operator to develop an approach to multiple attribute decision making problems with triangular fuzzy information, which can be described as following:

Step 1. Calculate the normalized decision matrix X. The result is shown in Table 2.

Table 2

Decision matrix X

	G ₁	G ₂	G ₃	G ₄
A ₁	(0.164,0.180,0.198)	(0.173,0.194,0.216)	(0.148,0.167,0.182)	(0.178,0.195,0.213)
A ₂	(0.153,0.182,0.205)	(0.195,0.208,0.224)	(0.238,0.276,0.352)	(0.196,0.206,0.220)
A ₃	(0.205,0.220,0.235)	(0.213,0.227,0.239)	(0.186,0.212,0.232)	(0.211,0.222,0.232)
A ₄	(0.221,0.232,0.248)	(0.179,0.194,0.210)	(0.140,0.153,0.164)	(0.182,0.195,0.208)
A ₅	(0.178,0.187,0.200)	(0.168,0.177,0.190)	(0.168,0.190,0.208)	(0.173,0.181,0.191)

Step 2. Aggregate all triangular fuzzy preference value ${\tilde{x}}_{ij} (j = 1, 2, \dots, n)$ by using the DGTFWBM (or DGTFWGBM) to derive the overall triangular fuzzy preference values ${\tilde{x}}_{i} (i = 1, 2, 3, 4, 5)$ of the hotels 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 alternatives are shown in Table 4. Note that > means “preferred to”. As we can see, depending on the aggregation operators used, the ordering of the alternatives is same and the best hotels is A₄.

Table 3

The overall preference values of the alternatives

	DGTFWBM	DGTFWGBM
A ₁	(0.040,0.044,0.048)	(0.666,0.680,0.695)
A ₂	(0.035,0.040,0.044)	(0.643,0.663,0.677)
A ₃	(0.044,0.047,0.050)	(0.676,0.687,0.697)
A ₄	(0.045,0.048,0.051)	(0.684,0.694,0.704)
A ₅	(0.039,0.041,0.044)	(0.661,0.670,0.681)

Table 4

Ordering of the hotels

	Ordering
DGTFWBM	A₄ > A₃ > A₁ > A₅ > A₂
DGTFWGBM	A₄ > A₃ > A₁ > A₅ > A₂

In this section, we have proposed two approaches to solve the multiple attribute decision making problems with triangular fuzzy information. From the above analysis, we can see that the main advantages of the proposed operators and approaches over the traditional triangular fuzzy operators and approaches are not only due to the fact that our operators accommodate the triangular fuzzy environment but also due to the consideration of the interrelationship among the input arguments, which makes it more feasible and practical.

6. Conclusion

Footnotes

Acknowledgments

The work was supported by the National Natural Science Foundation of China under No. 71373174 and Key social science program of Tianjin Municipal Education Commission under No. 2016JWZD35.

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