Model for multiple attribute decision making with hesitant triangular fuzzy information and its applications

Abstract

We study on the multiple attribute decision making problems using the hesitant triangular fuzzy information. Afterwards, we have designed the hesitant triangular fuzzy Bonferroni Mean (HTFBM) operator and hesitant triangular fuzzy weighted Bonferroni Mean (HTFWBM) operator. We have utilized the HTFWBM operator to multiple attribute decision making for evaluating the risk of the e-commerce enterprises with hesitant triangular fuzzy information. In the end, an illustrative example for evaluating the risk of the e-commerce enterprises is proposed to demonstrate the effectiveness of our proposed method.

Keywords

Multiple attribute decision making hesitant triangular fuzzy Bonferroni mean (HTFBM) operator hesitant triangular fuzzy weighted Bonferroni mean (HTFWBM) operator risk evaluation e-commerce enterprises

1 Introduction

As a feature in the age of 21th century, E-commerce is becoming a management technique with global and strategic meaning, which brings new and great challenge and opportunities to the society. It has been worked as an important strategic policy in national economic and social development to advance its application, accelerate the technical reform and innovation of our traditional industries, drive the development of industry with information technology, utilize the advantage of mature technology and realize the rapid and span development in productivity. E-commerce indicates informatization and network of firm management process and activities. It has been researched widely, mainly including the law and feature of corporate management in network environment, the theory and technology ways of utilizing E-commerce, the requirement of law and society environment and the strategy and tactics of developing e-commerce. Rapid advances in information technologies have brought new opportunity to the economic increase. Internet and electronic-commerce technology have been becoming the new technologies applied by firms to reduce the consumption of resources and enhance the efficiency of business operations, they are greatly affecting and changing the economic and social life of human beings. The government of our country pays much attention on the development of information technologies and the Internet economy, and established the informationization strategy covering all domains of the modernization. Electronic-commerce has become the important guidance of industrial development in our country. Although the Internet-based electronic-commerce technology has many potential advantages and has begun to be applied by our firms, in the whole, the level or degree of its applications in our traditional manufacturing firms is still limited. There have been a lot of researches on the factors affecting the development and applications of electronic-commerce technology in the world, but few researches have conducted from the explanatory view tested empirically in the context of our firms. Supported by several research projects in electronic-commerce, they conducted an empirical study on the factors affecting the diffusion of electronic-commerce technology, according to the several theories of innovation diffusion and the data collected from the traditional manufacturing firm.

In 2010, Torra [1] proposed the hesitant fuzzy set which supports the membership with a set of possible values and then studied on the relationship between hesitant fuzzy set and intuitionistic fuzzy set, and demonstrated that the envelope of hesitant fuzzy set refers to an intuitionistic fuzzy set. Wei et al. [2] presented the concept of the hesitant triangular fuzzy set and developed some hesitant triangular fuzzy aggregation operators: the hesitant triangular fuzzy weighted averaging (HTFWA) operator, hesitant triangular fuzzy weighted geometric (HTFWG) operator, hesitant triangular fuzzy ordered weighted averaging (HTFOWA) operator, hesitant triangular fuzzy ordered weighted geometric (HTFOWG) operator, hesitant triangular fuzzy hybrid average (HTFHA) operator and hesitant triangular fuzzy hybrid geometric (HTFHG) operator. Zhao et al. [3] utilized Einstein operations to develop some hesitant triangular fuzzy aggregation operators: hesitant triangular fuzzy Einstein weighted average (HTFEWA) operator, hesitant triangular fuzzy Einstein weighted geometric (HTFEWG) operator, hesitant triangular fuzzy Einstein ordered weighted average (HTFEOWA) operator, hesitant triangular fuzzy Einstein ordered weighted geometric (HTFEOWG) operator, hesitant triangular fuzzy Einstein hybrid average (HTFEHA) operator and hesitant triangular fuzzy Einstein hybrid geometric (HTFEHG) operator.

The problems for evaluating the risk of the e-commerce enterprises is the multiple attribute decision making problems [4 –13]. In our research, the multiple attribute decision making problem for evaluating the risk of the e-commerce enterprises with the hesitant triangular fuzzy information is studied. Next, we proposed the hesitant triangular fuzzy Bonferroni Mean (HTFBM) operator and hesitant triangular fuzzy weighted Bonferroni Mean (HTFWBM) operator. We have applied the HTFWBM operator to multiple attribute decision making for evaluating the risk of the e-commerce enterprises with hesitant triangular fuzzy information. Finally an illustrative example for evaluating the risk of the e-commerce enterprises has been given to show the developed method.

2 Preliminaries

Wei et al. [2] presented hesitant triangular fuzzy set based on hesitant fuzzy set.

Definition 1. [2] Let X be a fixed set, a hesitant triangular fuzzy set (HTFS) on X is in terms of a function that when exploited to each x in X and returns a subset of values in [0, 1]. We describe the HTFS by the following equation: $E = {< x, {\tilde{h}}_{E (x)} > | x \in X}$ (1) where ${\tilde{h}}_{E (x)}$ refers to possible triangular fuzzy values in [0,1], which means possible membership degrees of the element x ∈ X to the set E. For convenience, we call ${\tilde{h}}_{E (x)} = \tilde{h} = (γ^{L}, γ^{M}, γ^{R})$ a hesitant triangular fuzzy element (HTFE) and $\tilde{H}$ the set of all HTFEs.

Definition 2. [2, 3] For an HTFE $\tilde{h}$ , $s (\tilde{h}) = \frac{1}{# \tilde{h}} \sum_{\tilde{γ} \in \tilde{h}} \tilde{γ}$ is named score function of $\tilde{h}$ , in which $# \tilde{h}$ denotes the number of the triangular fuzzy values in $\tilde{h}$ , and $s (\tilde{h})$ is an triangular fuzzy values belonging to [0, 1]. Supposing that there two HTFEs ${\tilde{h}}_{1}$ and ${\tilde{h}}_{2}$ , if $s ({\tilde{h}}_{1}) \geq s ({\tilde{h}}_{2})$ , then ${\tilde{h}}_{1} \geq {\tilde{h}}_{2}$ .

3 Hesitant triangular fuzzy Bonferroni mean operators

Bonferroni [14] firstly proposed a mean type aggregation operator, which is named Bonferroni mean, which can been utilized for aggregation lying between the max, min operators and the OR and AND operators, which was defined as follows:

Definition 3. [14] Let p, q ≥ 0 and a_i (i = 1, 2, …, n) be a collection of non-negative real numbers. Then the aggregaton functions: ${BM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) = {(\frac{1}{n (n - 1)} \sum_{i, j = 1 i \neq j}^{n} a_{i}^{p} a_{j}^{q})}^{\frac{1}{p + q}}$ (2) is called the Bonferroni mean (BM) operator.

Based on the above design, we design several hesitant triangular fuzzy Bonferroni Mean operator using the operations of HTFEs and Bonferroni Mean.

Definition 4. Assuming that ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ is a set of HTFEs, and p, q > 0, then we define the hesitant triangular fuzzy Bonferroni Mean (HTFBM) operator as follows: $\begin{array}{l} HTFBM ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = \cup_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}, {\tilde{γ}}_{2} \in {\tilde{h}}_{2}, \dots,} {\tilde{γ}}_{n} \in {\tilde{h}}_{n} \\ {{(\frac{1}{n (n - 1)} \oplus_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({\tilde{h}}_{i}^{p} \otimes {\tilde{h}}_{j}^{p}))}^{\frac{1}{p + q}}} \end{array}$ (3)

Afterwards, Theorem 1 can be obtained as follows.

Theorem 1. Assuming that ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ be a collection of HTFEs, and p, q > 0, then we define the hesitant triangular fuzzy Bonferroni Mean (HTF BM) operator as follows: $\begin{array}{l} HTFBM ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = \cup_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}, {\tilde{γ}}_{2} \in {\tilde{h}}_{2}, \dots,} {\tilde{γ}}_{n} \in {\tilde{h}}_{n} {(\frac{1}{n (n - 1)} \oplus_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({\tilde{h}}_{i}^{p} \otimes {\tilde{h}}_{j}^{p}))}^{\frac{1}{p + q}} \\ = \cup_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}, {\tilde{γ}}_{2} \in {\tilde{h}}_{2}, \dots,} {\tilde{γ}}_{n} \in {\tilde{h}}_{n} {{(\frac{1}{n (n - 1)} \oplus_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({(γ_{i}^{L})}^{p} \otimes {(γ_{j}^{L})}^{p}))}^{\frac{1}{p + q}}, \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({(γ_{i}^{M})}^{p} \otimes {(γ_{j}^{M})}^{p}))}^{\frac{1}{p + q}}, \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({(γ_{i}^{R})}^{p} \otimes {(γ_{j}^{R})}^{p}))}^{\frac{1}{p + q}}} \end{array}$ (4)

As the input arguments have various importance, we define hesitant triangular fuzzy weighted Bonferroni Mean (HTFWBM) operator as follows.

Definition 5. Let ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ be a set of HTFEs, and p, q > 0, thus, we define the hesitant triangular fuzzy weighted Bonferroni Mean (HTFWBM) operator as follows: $\begin{array}{l} HTFWBM ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = \cup_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}, {\tilde{γ}}_{2} \in {\tilde{h}}_{2}, \dots,} {\tilde{γ}}_{n} \in {\tilde{h}}_{n} \\ {{(\frac{1}{n (n - 1)} \oplus_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({(w_{i} {\tilde{h}}_{i})}^{p} \otimes {(w_{j} {\tilde{h}}_{j})}^{p}))}^{\frac{1}{p + q}}} \end{array}$ (5)

where w = (w₁, w₂, …, w_n) ^T refers to a weight vector of ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ , where w_j indicates the importance degree of ${\tilde{h}}_{j}$ , where w_j > 0 (j = 1, 2, …, n), and $\sum_{j = 1}^{n} w_{j} = 1$ .

Theorem 2. Let ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ be a collection of HTFEs, and p, q > 0, then we define the hesitant triangular fuzzy weighted Bonferroni Mean (HTF WBM) operator as follows:

$\begin{array}{l} HTFWBM ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = \cup_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}, {\tilde{γ}}_{2} \in {\tilde{h}}_{2}, \dots,} {\tilde{γ}}_{n} \in {\tilde{h}}_{n} {{(\frac{1}{n (n - 1)} \oplus_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} ({(w_{i} {\tilde{h}}_{i})}^{p} \otimes {(w_{j} {\tilde{h}}_{j})}^{p}))}^{\frac{1}{p + q}}} \\ = \cup_{{\tilde{γ}}_{1} \in {\tilde{h}}_{1}, {\tilde{γ}}_{2} \in {\tilde{h}}_{2}, \dots,} {\tilde{γ}}_{n} \in {\tilde{h}}_{n} {{(1 - \prod_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} {(1 - {(1 - {(1 - γ_{i}^{L})}^{w_{i}})}^{p} {(1 - {(1 - γ_{i}^{L})}^{w_{i}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}, \\ {(1 - \prod_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} {(1 - {(1 - {(1 - γ_{i}^{M})}^{w_{i}})}^{p} {(1 - {(1 - γ_{i}^{M})}^{w_{i}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}, \\ {(1 - \prod_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} {(1 - {(1 - {(1 - γ_{i}^{R})}^{w_{i}})}^{p} {(1 - {(1 - γ_{i}^{R})}^{w_{i}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}} \end{array}$ (6) where w = (w₁, w₂, …, w_n) ^T refers to a weight vector of ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ , where w_j indicates the importance degree of ${\tilde{h}}_{j}$ , satisfying w_j > 0 (j = 1, 2, …, n), and $\sum_{j = 1}^{n} w_{j} = 1$ .

4 Multiple attribute decision making with hesitant triangular fuzzy information

Let A ={ A₁, A₂, …, A_m } be a discrete set of alternatives, and G ={ G₁, G₂, …, G_n } be the set of attributes, ω = (ω₁, ω₂, …, ω_n) is the weighting vector of the attribute G_j (j = 1, 2, …, n), where ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ . When decision makers provide several triangular fuzzy values for the alternative A_i with attribute G_j, these values are regarded as a hesitant triangular fuzzy element ${\tilde{h}}_{ij}$ . Suppose that the decision matrix $H = {({\tilde{h}}_{ij})}_{m \times n}$ is the hesitant triangular fuzzy decision matrix, where ${\tilde{h}}_{ij} (i = 1, 2, \dots, m, j = 1, 2, \dots, n)$ meets the requirements of HTFEs.

Afterwards, we apply the hesitant triangular fuzzy weighted Bonferroni Mean (HTFWBM) operator to the MADM problems with hesitant triangular fuzzy information.

Step 1. We exploit the decision information given in matrix H, and the HTFWBM operator $\begin{matrix} {\tilde{h}}_{i} & = & HTFWBM ({\tilde{h}}_{i 1}, {\tilde{h}}_{i 2}, \dots, {\tilde{h}}_{in}) \\ = & \cup_{{\tilde{γ}}_{i 1} \in {\tilde{h}}_{i 1}, {\tilde{γ}}_{i 2} \in {\tilde{h}}_{i 2}, \dots, {\tilde{γ}}_{in} \in {\tilde{h}}_{in}} \\ {{(\frac{1}{n (n - 1)} \oplus_{k, j = 1 k \neq j}^{n} ({(w_{i} {\tilde{h}}_{ik})}^{p} \otimes {(w_{j} {\tilde{h}}_{ij})}^{p}))}^{\frac{1}{p + q}}} \\ = & \cup_{{\tilde{γ}}_{i 1} \in {\tilde{h}}_{i 1}, {\tilde{γ}}_{i 2} \in {\tilde{h}}_{i 2}, \dots, {\tilde{γ}}_{in} \in {\tilde{h}}_{in}} \\ {(1 - \prod_{k, j = 1 k \neq j}^{n} (1 - {(1 - {(1 - γ_{ik}^{L})}^{w_{i}})}^{p} \\ {{{(1 - {(1 - γ_{ij}^{L})}^{w_{i}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}, \\ (1 - \prod_{k, j = 1 k \neq j}^{n} (1 - {(1 - {(1 - γ_{ik}^{M})}^{w_{i}})}^{p} \end{matrix}$ (7)

to obtain the overall values ${\tilde{h}}_{i} (i = 1, 2, \dots, m)$ of the alternative A_i.

Step 2. Computing the scores s (h_i) (i = 1, 2, 3, 4, 5) of the overall hesitant triangular fuzzy values h_i (i = 1, 2, 3, 4, 5).

Step 3. In order to rank these overall hesitant triangular fuzzy values $S ({\tilde{h}}_{i}) (i = 1, 2, \dots, m)$ , then we develop a complementary matrix as P = (p_ij) _m×m, where [2] $p_{ij} \geq 0, p_{ij} + p_{ji} = 1, p_{ii} = 0.5 .$

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

Step 4. Rank all the alternatives A_i (i = 1, 2, …, m) and select the optimal elements in terms of p_i (i = 1, 2, …, m).

Step 5. End.

5 Numerical example

In recent years, the internet technology developed rapidly, online shopping is some branch of general IT life and have become a symbol of modern fashion life because of its convenience. It comes into people’s lives more quickly. The speed of the network development and the pace of life accelerated, what are making online shopping began to be accepted by the broad masses of the people. Of course, there are also many citizens don’t quite understand the process and steps of online shopping, some citizens still haven’t tried online shopping, and also some citizens who would like to open a shop on the net to sell goods, but feel that it is a bit complicated, even mysterious. At present, the electronic commerce system has a very beautiful interface, but a poor user experience, which does not accord with user routine operation habit. These facts haven’t only let consumers feel that online shopping is difficult, what lost the original intention that is convenient to customers, but also let the managers feel difficult to operate quickly, and then worry about their own items in the transaction process. Electronic commerce system is a platform which making webpage and operates the database. Because of the flexibility and convenience of the ASP language, the scalability of the system is quite large. We present an empirical case study of evaluating the e-commerce risk. The e-commerce risk of five possible e-commerce enterprises A_i (i = 1, 2, 3, 4, 5) is evaluated. After preliminary screening, five e-commerce enterprises A_i (i = 1, 2, …, 5) have been retained. Three decision makers construct a committee to act as decision makers. The company should make a decision in terms of to the following 4 attributes: G1 is the technological risk; G2 is the logistics risk; G3 is the market risk; G4 is the information transmission risk. The five possible e-commerce enterprises A_i (i = 1, 2, …, 5) are to be evaluated under the given 4 attributes in anonymity and the decision matrix $H = {({\tilde{h}}_{ij})}_{5 \times 4}$ is presented in Table 1, where ${\tilde{h}}_{ij} (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4)$ are in the form of HTFEs.

Next, we use the method which is developed for evaluating the risk of the e-commerce enterprises.

Step 1. We use the decision information given in matrix H, and the HTFWBM operator to compute the overall preference values ${\tilde{h}}_{i}$ of the e-commerce enterprises A_i (i = 1, 2, 3, 4, 5).

Step 3. Computing the scores $s ({\tilde{h}}_{i}) (i = 1, 2, 3, 4, 5)$ of the overall hesitant triangular fuzzy value h_i (i = 1, 2, 3, 4, 5), the score values for the information systems.

$\begin{matrix} h_{1} & = & (0.402, 0.534, 0.672) \\ h_{2} & = & (0.321, 0.426, 0.533) \\ h_{3} & = & (0.567, 0.654, 0.795) \\ h_{4} & = & (0.632, 0.698, 0.703) \\ h_{5} & = & (0.325, 0.487, 0.552) \end{matrix}$

Step 4. According to the aggregating results and the equation of degree of possibility, the rank list of the e-commerce enterprises is: A₃ > A₄ > A₁ > A₂ > A₅. Thus, the desirable e-commerce enterprise is A₃.

6 Conclusion

E-commerce is becoming the main trend of economic development with the rapid development of network technology and E-commerce. In the field ofE-commerce, information flow, logistics and capital flow are the three very important aspects, among which the capital flow is the most crucial part. Besides, the capital flow is also basic conditions for carrying out the transactions. The manifestation of capital flow in e-commerce is online payment, which transfers the information via electronic data in the opening system platform. Compared with those traditional payment methods, it has the advantages of low cost and high efficiency. Meanwhile, as a newborn payment method, the online payment faces with many difficulties and risks along with its rapid development. We focus on the multiple attribute decision making problems with the hesitant triangular fuzzy information. Then, we have developed the hesitant triangular fuzzy Bonferroni Mean (HTFBM) operator and hesitant triangular fuzzy weighted Bonferroni Mean (HTFWBM) operator. We have applied the HTFWBM operator to multiple attribute decision making for evaluating the risk of the e-commerce enterprises with hesitant triangular fuzzy information. In the end, an example for evaluating the risk of the e-commerce enterprises has been given to show the developed method.

Footnotes

Acknowledgments

The work was supported by the Scientific and technological innovation projects of the Education Department of Shanxi Province, based on the “Shanxi consumer network” to build the campus e-commerce business practice platform, the project number:20131115.

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