A risk attitudinal ranking approach of triangular intuitionistic fuzzy numbers and their application to MADM problems

Abstract

In this work, a new ranking approach of triangular intuitionistic fuzzy numbers (TIFNs) is proposed and applied to solve multi-attribute decision making (MADM) problems. Firstly, I use a weighted average operator to introduce the risk preference of the decision maker (DM), and construct the score and accuracy expected functions of TIFN. Then, based on these concepts, a ranking method for TIFNs is developed considering the DM’s risk preference. Subsequently, a decision procedure is described to deal with the MADM under a triangular intuitionistic fuzzy environment. Moreover, a novel distance measure for TIFNs is defined and utilized to determine the attribute weights. Finally, a trusted cloud service selection example verifies the effectiveness and practicality of the developed method.

Keywords

Multi-attribute decision making triangular intuitionistic fuzzy numbers score expected function accuracy expected function distance measure

1 Introduction

In a fuzzy environment, ranking approaches of fuzzy numbers are used as an important means in multi-attribute decision making (MADM). As a fuzzy number may be considered to be a representation for an ill-defined quantity, it is not possible to order them. Therefore, how to compare or rank them becomes an interesting and important research topic. To handle this issue, roughly speaking, a fuzzy quantity is changed into a crisp value and compared by crisp value. Since the concept of maximizing a set was first introduced to order fuzzy numbers by Jain [17], various ranking methods of fuzzy numbers are proposed by a large amount of literature [1 , 23].

The MADM problem is a fairly common activity in practical life such as cloud service selection, supply chain management and economic society analysis. The MADM methods usually involve two steps: (1) aggregate the different attribute values of each alternative into an overall value, then (2) rank the overall values. Due to the uncertainty of the socioeconomic situation, it is necessary to study the ranking methods with fuzzy numbers for the characteristics of candidate alternatives.

As an extension of the concept of Zadeh’s fuzzy set [12] whose element is only a membership function, the Intuitionistic fuzzy set (IFS) developed by Atanassov [9] has become a very important branch of the fuzzy field. However, it is difficult for IFS to give crisp values for its membership and non-membership degree because of the complexity of the practical decision environment. Thus, in [10], IFS is further extended to the interval-valued intuitionistic fuzzy sets (IVIFS) by applying intervals to represent the membership and non-membership degree.Correspondingly, the ranking methods of the intuitionistic fuzzy numbers (IFNs) and interval-valued intuitionistic fuzzy numbers (IVIFNs) have received more and more attention. In particular, the score and accuracy system are designated to compare IFNs and IVIFNs. Xu and Yager in [27] deal with ranking of IFNs by utilizing the score function [20] and the accuracy function [4]. Lin [11] proposed an improved score function of IFN, which includes hesitation.Ye [7] and Chen et al. [13] developed a new score function and accuracy function of IVIFN, respectively, by incorporating the hesitation. Later, Wu and Chiclana [6] investigated a risk attitudinal ranking method for IVIFNs on the basis of the corresponding expected score values and expected accuracy values.

However, IFS and IVIFS only use discrete domains. To extend discrete sets to consecutive sets, Shu et al. [14] defined trapezoidal intuitionistic fuzzy numbers (TrIFNs) and triangular intuitionistic fuzzy numbers (TIFNs) whose domains are continuous sets. Their prominent characteristics are that their non-membership and membership degrees are trapezoidal fuzzy numbers or triangular fuzzy numbers rather than crisp values and intervals. Thus, TrIFNs and TIFNs may express and describe uncertain or imprecise evaluation information more abundantly and flexibly than IFNs and IVIFNs. For instance, the product quality of a seller in cloud service selection may be rated as TIFN ((3,4,5);0.6,0.1), which indicates that the minimum and maximum ratings of the seller with product quality are 3 and 5 respectively, the most possible ratings are about 4 with the minimum dissatisfaction 0.1 and maximum satisfaction 0.6. Likewise, the ranking of TIFNs and TrIFNs needs to be addressed in MADM problems. For example, in [5] the ranking of TrIFNs is transformed into the ranking of interval numbers. Li et al. [3] utilized the value-index represented central value of TIFNs, and the ambiguity-index reflected the global spread of TIFNs to rank TIFNs. Zhang et al. [15] defined the distance measure of TIFNs and extended idea of TOPSIS to rank TIFNs. Wan and Dong [22] presented an expectant score for TrIFNs based on the geometric idea. Ye [18] introduced a definition of the expected value for TrIFN, which was later improved by Liu and Liu [16] with an additional accuracy function. Ju et al. [25] defined the expected function and score function of TrIFN from the membership degree and non-membership degree functions. Wan [21] introduced the concepts of the weighted possibility mean for TIFNs, and a new ranking approach for TIFNs was proposed according to these concepts. Recently, a ratio method for ranking ITFNs was developed by Li [2]. This ratio ranking method is computed as the value index out of the ambiguity index and applied to build an extended MADM method. However, in contrast to the above methods [3 , 25], the ratio ranking method incorporating preference information of decision maker (DM) can improve the quality of ranking. Although this ranking method [2] is subsequently useful and valuable, there are some limitations (see the cases discussed in subsection 4.2 in detail): (1) Li’s ratio method [2] cannot differentiate some kinds of TIFNs; (2) the risk attitude parameter of Li’s ratio method [2] may be ineffective on adjusting value-index and ambiguity-index; (3) The attribute weights in Li’s ratio method [2] are provided in advance. In addition, there is little research on the ranking of TIFNs based on their score and accuracy expected function.

In this work, a novel ranking method of TIFNs by combining the score expected function with an accuracy expected function is proposed. The novelty of the proposed method is that these functions fully consider the DM’s risk attitude in comparison to TIFNs, and solve MADM problems with TIFNs. The paper is organized as follows. In Section 2, we introduce several basic concepts and definitions of TIFN.Section 3 gives an approach to ranking TIFNs based on mean-index and variance-index. Section 4 constructs MADM problems with TIFNs and presents the corresponding decision procedure. A trusted cloud service selection example and a comparison analysis are provided in Section 5. Finally, we conclude inSection 6.

2 Preliminaries

2.1 The definition and distance of TIFNs

The basic definition and distance measure of TIFNs are introduced as follows.

Definition 1 [3]. A triangular intuitionistic fuzzy number $\tilde{a} = ((a, b, c); u_{\tilde{a}}, v_{\tilde{a}})$ is a special intuitionistic fuzzy set on a real number set R, whose membership function and non-membership function are defined as follows: $μ_{\tilde{a}} (x) = {\begin{matrix} (a - x) u_{\tilde{a}} / - (a - b), & if a \leq x < b \\ u_{\tilde{a}}, & if x = b \\ (x - c) u_{\tilde{a}} / - (b - c), & if b < x \leq c \\ 0, & otherwise \end{matrix}$ (1)

$υ_{\tilde{a}} (x) = {\begin{matrix} [(x - a) v_{\tilde{a}} + b - x] / - (b - a), & if a \leq x < b \\ ν_{\tilde{a}}, & if x = b \\ [(c - x) v_{\tilde{a}} + x - b] / - (c - b), & if b < x \leq c \\ 1, & otherwise \end{matrix}$ (2)

where $u_{\tilde{a}}$ denotes the membership degree maximum and $ν_{\tilde{a}}$ denotes the non-membership degree minimum, such that they satisfy the conditions: $0 \leq u_{\tilde{a}} \leq 1$ , $0 \leq ν_{\tilde{a}} \leq 1$ , and $0 \leq u_{\tilde{a}} + ν_{\tilde{a}} \leq 1$ . Moreover, the IF index of an element x in $\tilde{a}$ can be denoted as $π_{\tilde{a}} (x) = 1 - μ_{\tilde{a}} (x) - υ_{\tilde{a}} (x)$ which gives the indeterminacy membership degree of an element x in $\tilde{a}$ . For more details regarding the arithmetical operations on TIFNs, refer to [2].

Definition 2. For two given TIFNs ${\tilde{t}}_{1} = ((a_{1}, b_{1},$ $c_{1}); u_{{\tilde{a}}_{1}}, ν_{{\tilde{a}}_{1}})$ and ${\tilde{t}}_{2} = ((a_{2}, b_{2}, c_{2}); u_{{\tilde{a}}_{2}}, ν_{{\tilde{a}}_{2}})$ , the distance between ${\tilde{t}}_{1}$ and ${\tilde{t}}_{2}$ is defined as follows:

$\begin{matrix} d ({\tilde{t}}_{1}, {\tilde{t}}_{2}) & = & \frac{1}{12} [| (| 1 + u_{{\tilde{a}}_{1}} - v_{{\tilde{a}}_{1}} |) a_{1} \\ - (| 1 + u_{{\tilde{a}}_{2}} - v_{{\tilde{a}}_{2}} |) a_{2} | \\ + (| 1 + u_{{\tilde{a}}_{1}} - v_{{\tilde{a}}_{1}} |) b_{1} \\ - (| 1 + u_{{\tilde{a}}_{2}} - v_{{\tilde{a}}_{2}} |) b_{2} | \\ + | (| 1 + u_{{\tilde{a}}_{1}} - v_{{\tilde{a}}_{1}} |) c_{1} \\ - (| 1 + u_{{\tilde{a}}_{2}} - v_{{\tilde{a}}_{2}} |) c_{2} |] \\ + \frac{1}{2} max (| u_{{\tilde{a}}_{1}} - u_{{\tilde{a}}_{2}} |, | v_{{\tilde{a}}_{1}} - v_{{\tilde{a}}_{2}} |) \end{matrix}$ (3)

When $u_{{\tilde{a}}_{i}} = 1$ and $v_{{\tilde{a}}_{i}} = 0$ , Equation (3) reduces to the Hamming distance of the triangular fuzzy number.

Theorem 1. The distance of TIFNs satisfies the following properties:

$d ({\tilde{t}}_{1}, {\tilde{t}}_{2}) \in [0, 1]$ ;

$d ({\tilde{t}}_{1}, {\tilde{t}}_{2})$ = 0 if and only if ${\tilde{t}}_{1} = {\tilde{t}}_{2}$ ;

$d ({\tilde{t}}_{1}, {\tilde{t}}_{2}) = d ({\tilde{t}}_{2}, {\tilde{t}}_{1})$

If ${\tilde{t}}_{3} = ((a_{3}, b_{3}, c_{3}); u_{{\tilde{a}}_{3}}, ν_{{\tilde{a}}_{3}})$ is any TIFN, then $d ({\tilde{t}}_{1}, {\tilde{t}}_{3}) \leq d ({\tilde{t}}_{1}, {\tilde{t}}_{2}) + d ({\tilde{t}}_{2}, {\tilde{t}}_{3})$ .

It can be proved from simple analysis that Equation (3) meets the above properties.

2.2 Cut sets of TIFNs

Definition 3 [3]. A α - cut set of a TIFN $\tilde{a} = ((a, b, c); u_{\tilde{a}}, ν_{\tilde{a}})$ is defined as ${\tilde{a}}_{α} = {x | μ_{\tilde{a}} (x) \geq α}$ , which is calculated by: $\begin{matrix} {\tilde{a}}_{α} = [L_{\tilde{a}} (α), R_{\tilde{a}} (α)] \\ = [a - \frac{(a - b) α}{u_{\tilde{a}}}, c + \frac{(b - c) α}{u_{\tilde{a}}}], \end{matrix}$ (4)

where $α \in [0, u_{\tilde{a}}]$ .

Definition 4 [3]. A β - cut set of a TIFN $\tilde{a} = ((a, b, c); u_{\tilde{a}}, ν_{\tilde{a}})$ is defined as ${\tilde{a}}_{β} = {x | υ_{\tilde{a}} (x) \leq β}$ , which is calculated by: $\begin{matrix} {\tilde{a}}_{β} = [L_{\tilde{a}} (β), R_{\tilde{a}} (β)] \\ = [\frac{(a - b) β + b - a ν_{\tilde{a}}}{1 - ν_{\tilde{a}}}, \frac{(c - b) β + b - c ν_{\tilde{a}}}{1 - ν_{\tilde{a}}}], \end{matrix}$ (5)

where $β \in [ν_{\tilde{a}}, 1]$ .

2.3 The ranking method of TIFNs

Yager [18] pointed out that any interval number into a single value could be aggregated by using the continuous ordered weighted average (COWA) operator. Thus, ${\tilde{a}}_{α} = [L_{\tilde{a}} (α), R_{\tilde{a}} (α)]$ can be expressed in the following form: $G ({\tilde{a}}_{α}, λ) = (1 - λ) L_{\tilde{a}} (α) + λ R_{\tilde{a}} (α)$ (6)

where λ ∈ [0, 1] is a weight that represents the DM’s intentions. λ ∈ [0, 0.5) shows that the DM’s attitude prefers pessimistic; λ ∈ (0.5, 1] shows that the DM’s attitude inclined to be optimistic; λ = 0.5 shows that the DM’s attitude maintains neutrality.

Definition 5. Let ${\tilde{a}}_{α}$ be a α - cut set of $\tilde{a}$ and $E_{μ} (\tilde{a})$ be the membership expected function of $μ_{\tilde{a}} (x)$ , i.e., $E_{μ} (\tilde{a}) = \int_{0}^{u_{\tilde{α}}} G ({\tilde{a}}_{α}, λ) d α .$ (7)

According to Equations. (4) and (6), the membership expected function of $\tilde{a}$ can be calculated by: $E_{μ} (\tilde{a}) = \frac{u_{\tilde{a}}}{2} [(1 - λ) (a + b) + λ (b + c)] .$ (8)

Definition 6. Let ${\tilde{a}}_{β}$ be a β - cut set of $\tilde{a}$ and $E_{υ} (\tilde{a})$ be the non-membership expected function of $\tilde{a}$ , i.e., $\begin{matrix} E_{υ} (\tilde{a}) = \int_{v_{\tilde{α}}}^{1} G ({\tilde{a}}_{β}, λ) d α \\ = \frac{1}{2} [λ (3 v_{\tilde{α}} - 1) (a - c) + b + a + v_{\tilde{α}} (b - 3 a)] . \end{matrix}$ (9)

Inspired by the score function and accurate function introduced by Xu [27], the score expected function and accurate expected function of TIFN can be defined as follows:

Definition 7. Let $\tilde{a}$ be a TIFN; the score expected function and accurate expected function of $\tilde{a}$ are defined by $S (\tilde{a}) = E_{μ} (\tilde{a}) - E_{υ} (\tilde{a}),$ (10) $A (\tilde{a}) = E_{μ} (\tilde{a}) + E_{υ} (\tilde{a}) .$ (11)

Definition 8. Let $\tilde{a}$ and ${\tilde{a}}_{2}$ be two TIFNs. According to their score expected function and accurate expected function, we introduce a new lexicographic ranking method for TIFNs, which can be performed as follows:

If $S ({\tilde{a}}_{1}) < S ({\tilde{a}}_{2})$ for the same given λ, then ${\tilde{a}}_{1} < {\tilde{a}}_{2}$ ;

If $S ({\tilde{a}}_{1}) > S ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ ;

If $S ({\tilde{a}}_{1}) = S ({\tilde{a}}_{2})$ , then

(i) $A ({\tilde{a}}_{1}) > A ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ ;

(ii) $A ({\tilde{a}}_{1}) < A ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} < {\tilde{a}}_{2}$ ;

(iii) $A ({\tilde{a}}_{1}) = A ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} = {\tilde{a}}_{2}$ ;

Remark 1. The proposed ranking method shows that: (1) it fully considers the risk attitude of DM, thus the decision results in dealing with MADM problems are more persuasive; (2) compared to method [2], the score expected function $S (\tilde{a})$ and accurate expected function $A (\tilde{a})$ are linear functions, resulting in more stability and simplicity.

Remark 2. Considering that (1) the proposed ranking method is based on the cut sets of TIFNs which are closed intervals, and (2) Li and Chen [24] proved that the cut sets of TrIFNs are also closed intervals, we can easily extend the proposed ranking method to compare TrIFNs.

3 A method for MADM with TIFNs

MADM is the procedure to find the best alternative among a set of feasible alternatives that are characterized by multiple, usually conflicting, attributes. We suppose that (1) S = {S₁, S₂, …, S_m} is a discrete alternative set, (2) A = {a₁, a₂, …, a_n} represents a attribute set, and (3) W = {w₁, w₂, …, w_n} indicates the weight vector of the attributes a_j (j = 1, 2, …, n), where w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ . Let the rating of alternative S_i ∈ S (i = 1, 2, …, m) on each attribute a_j ∈ A (j = 1, 2, …, n) be a TIFN ${\tilde{a}}_{ij}^{'} = ((a_{ij}^{'}, b_{ij}^{'}, c_{ij}^{'}); u_{{\tilde{a}}_{ij}^{'}}, ν_{{\tilde{a}}_{ij}^{'}})$ . Thus, the evaluation information matrix $\tilde{A} = ({\tilde{a}}_{ij}^{'})_{m \times n}$ can be used to concisely describe the MADM problem with TIFNs.

Based on the proposed ranking method distance measure of TIFNs, the decision procedure for the MADM problems can be followed in detail asfollows:

Step 1. Normalize the TIFN decision matrix. Since different physical dimensions may be measured in various of means, it is necessary to normalize the matrix $\tilde{A} = ({\tilde{a}}_{ij}^{'})_{m \times n}$ into the matrix $\tilde{R} = ({\tilde{r}}_{ij})_{m \times n}$ , where ${\tilde{r}}_{ij} = ((a_{ij}, b_{ij}, c_{ij}); u_{{\tilde{a}}_{ij}}, ν_{{\tilde{a}}_{ij}})$ . In this paper, we employ the following normalized transformation: ${\tilde{r}}_{ij} = ((\frac{a_{ij}^{'}}{c_{j}^{+}}, \frac{b_{ij}^{'}}{c_{j}^{+}}, \frac{c_{ij}^{'}}{c_{j}^{+}}); u_{{\tilde{a}}_{ij}}, v_{{\tilde{a}}_{ij}}) (j \in B)$ (12)

and ${\tilde{r}}_{ij} = ((\frac{a_{j}^{-}}{c_{ij}^{'}}, \frac{a_{j}^{-}}{b_{ij}^{'}}, \frac{a_{j}^{-}}{a_{ij}^{'}}); u_{{\tilde{a}}_{ij}}, v_{{\tilde{a}}_{ij}}) (j \in C),$ (13)

where $c_{j}^{+} = max {c_{ij}^{'} | i = 1, 2, \dots, m}$ , $a_{j}^{-} = min {a_{ij}^{'} | i = 1, 2, \dots, m}$ , $u_{{\tilde{a}}_{ij}} = u_{{\tilde{a}}_{ij}^{'}}$ , $v_{{\tilde{a}}_{ij}} = ν_{{\tilde{a}}_{ij}^{'}}$ , and B, C are the benefit and cost attribute sets, respectively.

Step 2. Determine the attribute weights. Chen and Yang [26] indicated that as the attribute value comes closer to the mean value, the greater the effect and the larger the weight. Motivated by this idea, the attribute weights can be derived as follows: $w_{j} = 1 - d (r_{j}, \bar{r}) / - \sum_{j = 1}^{n} 1 - d (r_{j}, \bar{r})$ (14)

where $\bar{r}$ is the mean vector of all attribute columns, r _j is the column vector of attribute a_j, and $d (r_{j}, \bar{r})$ is the distance between r _j and $\bar{r}$ .

Step 3. Calculate the overall evaluation values of alternatives. According to Definition 2 in [2], the overall evaluation values ${\tilde{O}}_{I}$ of alternatives S_I can be derived by the following expression $\begin{matrix} {\tilde{O}}_{i} = \sum_{j = 1}^{n} w_{j} {\tilde{r}}_{ij} = ((\sum_{j = 1}^{n} w_{j} a_{ij}, \sum_{j = 1}^{n} w_{j} b_{ij}, \sum_{j = 1}^{n} w_{j} c_{ij}); \\ min_{j} u_{{\tilde{a}}_{ij}}, max_{j} v_{{\tilde{a}}_{ij}}) \end{matrix}$ (15)

which are TIFNs.

Step 4. Compute the score expected value and accurate expected value of ${\tilde{E}}_{I}$ by Equations. (10)and (11).

Step 5. Rank the alternatives S_i using Definition 8 and perform a sensitivity analysis with the decision maker’s intentions λ.

4 Illustrative example

4.1 Trusted cloud service selection example

Cloud service trading usually takes place between parties who are autonomous, in an environment where the buyer often does not have enough information about the users and services. Therefore, it is very important that users can identify the most trusted cloud service. Suppose that a user wants to establish a trusted cloud service. After initial screening, the three possible cloud service providers (S₁, S₂ and S₃) can be evaluated under five related credential factors: (1) a₁ is the accessibility of the cloud service; (2) a₂ is the cloud service attitude; (3) a₃ is the technology security of the cloud environment; (4) a₄ is the cloud service dependability; (5) a₅ is the cloud resource ability. The rate values are real numbers in [0,1]. Using statistical methods, the decision matrix is shown in Table 1. In order to create a new document, do the following:

Decision contains the following steps:

Step 1. Used Equations. (12), (13), we have acquired the normalized TIFN evaluation matrix $\tilde{R}$ _, which is listed in Table 2.

Step 2. By Equation (14), we acquire the attribute weights w₁ = 0.2059, w₂ = 0.1874, w₃ = 0.2137, w₄ = 0.1930, and w₅ = 0.1999.

Step 3. Using Equation (15), the overall attribute values of alternatives S_i can be calculated asfollows: $\begin{matrix} {\tilde{O}}_{1} & = & ((0.4206, 0.5801, 0.6801); 0.5, 0.4), \\ {\tilde{O}}_{2} & = & ((0.4374, 0.5374, 0.8197); 0.4, 0.4), \\ {\tilde{O}}_{3} & = & ((0.4575, 0.5969, 0.9373); 0.4, 0.4) . \end{matrix}$ Step 4. According to Equation (10), the score expected value of ${\tilde{O}}_{I}$ can be computed as follows: $\begin{matrix} S ({\tilde{O}}_{1}) & = & - 0.1138 - 0.0908 λ, \\ S ({\tilde{O}}_{2}) & = & - 0.1375 + 0.1147 λ, \\ S ({\tilde{O}}_{3}) & = & - 0.1612 + 0.1439 λ . \end{matrix}$

Step 5. We perform a sensitivity analysis with λ, described as in Fig. 1.

From Fig. 1, if λ ∈ [0, 0.812], it can be seen that $S ({\tilde{O}}_{1}) > S ({\tilde{O}}_{2}) > S ({\tilde{O}}_{3})$ . Therefore, according to Definition 8, the ranking order of the three trusted cloud services is S₁ ≻ S₂ ≻ S₃, and the most trusted candidate is S₁. Otherwise, if λ ∈ (0.812, 0.892], we obtain the ranking order S₂ ≻ S₁ ≻ S₃, and the most trusted candidate is S₂; if λ ∈ (0.892, 0.990], we have the ranking order S₂ ≻ S₃ ≻ S₁, and the most trusted candidate is S₂; if λ ∈ (0.990, 1], we get the ranking order S₃ ≻ S₂ ≻ S₁, and the most trusted candidate is S₃. Obviously, when the decision maker’s risk preference value is different, the corresponding ranking results of alternatives are also not quite identical. Thus, the proposed method is reasonable.

4.2 Comparison analysis with existing methods

In the following, we compare the ranking results obtained by method [2] and our proposed method.

The method [2] and our proposed method can be used to solve MADM problems in which the attribute values are expressed by TIFNs. Li et al. defined the values-index and ambiguity-index, and applied these to MADM problems with TIFNs. We apply this method [2] to deal with the above trusted cloud service selection example. The ranking orders of S_I are obtained as follows: S₂ ≻ S₃ ≻ S₁ for λ ∈ [0, 0.685], S₃ ≻ S₂ ≻ S₁ for λ ∈ (0.685, 0.816], and S₃ ≻ S₁ ≻ S₂ for λ ∈ (0.816, 1]. It is observed that the above ranking orders are a subset of the ranking results obtained by our method. This shows that the proposed method is more effective.

The proposed method develops a new distance for TIFNs and utilizes the distance between attribute vectors and the attribute mean vector to determine the attribute weights, which is more reasonable and objective, whereas those in the method [2] need to be provided in advance, which is highly subjective.

In the situation of $u_{\tilde{a}} + ν_{\tilde{a}} = 1$ for a TIFN $\tilde{a} = ((a, b, c); u_{\tilde{a}}, ν_{\tilde{a}})$ , the risk attitude parameter is ineffective when adjusting value-index and ambiguity-index. Consider a TIFN ${\tilde{S}}_{1} = ((0.592, 0.774, 0.910); 0.6, 0.4)$ . By the method [2], it is clear that the ambiguity-index and value-index are constant for different risk attitude preferences such as $V_{λ} ({\tilde{S}}_{1}) = 0.461$ and $A_{λ} ({\tilde{S}}_{1}) = 0.019$ , which are even incorrect according to intuition. By using the score expected function, we obtain $S ({\tilde{S}}_{1}) = 0.0728 - 0.1272 λ$ . It is observed that the ranking index with different risk attitudes may generate different results. However, this effect of ranking method cannot be reflected in Li’s method [2]. Thus, the proposed method is more reasonable.

Consider two TIFNs ${\tilde{a}}_{1} = ((0.3, 0.45, 0.9);$ 0.9, 0.1) and ${\tilde{a}}_{2} = ((0.2, 0.5, 0.8); 0.9, 0.1)$ . Intuitively, the ranking order is not equal. However, by the method [2], we have $V_{μ} ({\tilde{a}}_{1}) = V_{υ} ({\tilde{a}}_{1}) = 0.405$ , $V_{μ} ({\tilde{a}}_{2}) = V_{υ} ({\tilde{a}}_{2}) = 0.405$ , $A_{λ} ({\tilde{a}}_{1}) = 0.162$ and $A_{λ} ({\tilde{a}}_{2}) = 0.162$ ; therefore, the ranking order is ${\tilde{a}}_{1} = {\tilde{a}}_{2}$ . Therefore, method [2] cannot distinguish between these fuzzy numbers. By the score expected function in Definition 7, we have $S ({\tilde{a}}_{1}) = - 0.015 + 0.06 λ$ and $S ({\tilde{a}}_{1}) = - 0.03 + 0.06 λ$ . Clearly, ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ , which shows the validation of our method.

5 Conclusion

This paper presents a new method for ranking TIFNs. In this approach, the score and accuracy expected function of TIFNs are defined. Then, the ranking index value is obtained based on the score and accuracy expected function, then used to solve MADM problems with TIFNs. The trusted cloud service selection example given here illustrates that the proposed method is feasible and valid. The proposed MADM method has the following main advantages: (1) it develops a new distance for TIFNs and applies it to determine the attribute weights; (2) it proposes a new effective risk attitudinal ranking method for TIFNs; and (3) it presents MADM decision steps that consider the DM’s attitudinal character. Additionally, the proposed method can provide decision makers with a new idea for ranking TIFNs. It enriches the theories and approaches for ranking TIFNs. However, the proposed method is only suitable for the problem of a single DM, and the risk attitude parameter λ in score expected and accurate expected functions need to be given by DM in advance. Therefore, future work plans to improve the score expected and accurate expected functions as well as the ranking method. Also, this work will be extended to solve multi-attribute group decision-making problems.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Nos. 61602219 and 71661010).

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