A TODIM· SIR method for multiple attribute decision making with interval grey uncertain linguistic based on a new distance measure 1

Abstract

The multiple attribute decision making (MADM) with interval grey uncertain linguistic (IGUL) is a topic of current interest, and various methods have been developed. However, few approaches taking the behavioral characteristics of the decision maker into account. In this paper, an extension of TODIM (i.e., an acronym in Portuguese of interactive and multiple attribute decision making) method, in which three behavioral characteristics of the decision maker (i.e., risk aversion, reference dependence and loss aversion) are considered, is proposed. First, the δ-Hamming distance is defined to deal with the interval grey uncertain linguistic variables by considering the level of the decision maker’s risk aversion, and its distinguish ability is validated by compared to the classical Hamming distance. Then, the details of the TODIM· SIR method is demonstrated: (i) considering the reference dependence behaviour of the decision maker, the positive-ideal alternative (i.e., PIA) and the negative-ideal alternative (i.e., NIA) are defined, and the gain and loss degrees of each alternative relative to NIA and PIA are computed based on the δ-Hamming distance; (ii) taking the loss aversion behaviour of the decision maker into account, the perceived dominance degree of the decision maker for the gain and the loss is calculated; (iii) according to the idea of the Superiority and Inferiority Ranking method (i.e., SIR, an outranking method), the Gain-flow and the Loss-flow are defined, and the partial ranking orders and the complete ranking order are obtained. Finally, two numerical examples are given to illustrate the robustness and validity of the method, and a comparative analysis is also conducted to compare the TODIM· SIR method with both the classical TODIM method and the classical SIR method.

Keywords

TODIM SIR Multiple attribute decision making interval grey uncertain linguistic

1 Introduction

Multiple attribute decision making (MADM) which has been widely used in management, engineering, finance, military and many other fields, aims to support decision maker (DM) to rank alternatives based on multiple attributes [1 –3]. With the increasing complexity, vagueness and uncertainty of decision problems, the expression of decision-making information is becoming more and more diversified, such as crisp numbers [4], interval numbers [5], fuzzy numbers [6], grey numbers [7], interval-valued fuzzy-rough numbers [8], intuitionistic trapezoidal fuzzy numbers [9] or linguistic variables [10], etc. Since the linguistic variable is easier to express fuzzy information, it has become a common form of decision-making information. Moreover, considering the human limited cognition, the psychological behavior of DMs were taken into account when he or she gives his or her assessment information on attributes, and thus the hesitant fuzzy linguistic term sets [11, 12], linguistic intuitionistic fuzzy numbers [13] or intuitionistic hesitant linguistic sets [14] were defined and employed.

However, it is not reasonable just only consider the fuzziness in assessment information caused by uncertainties, but the greyness caused by incomplete and inadequate information should also be taken into account [15 –17]. From this point of view, the interval grey uncertain linguistic (IGUL) variables which have promising potential in expressing uncertain and incomplete information were proposed [16 –23], and a number of MADM techniques have been proposed. Wang and Wu [18] discussed the characteristics of the IGUL variables, on which an ordered weighted operator was proposed. Ma et al. [19] also discussed the IGUL correlated ordered arithmetic averaging operator and the induced IGUL correlated ordered arithmetic averaging operator based on the Choquet integral. Zhang and Xie [20] developed the IGUL weighted averaging operator, the IGUL ordered weighted averaging operator and the induced IGUL ordered weighted aggregation operator. Liu et al. [21] proposed the IGUL generalized hybrid averaging operator to deal with the MAGDM problems. Han et al. [22] defined the Hamming distance of the IGUL and also discussed the IGUL weighted geometric aggregation operator and the IGUL hybrid weighted geometric aggregation operator. Xu [23] defined the IGUL ordered weighted geometric operator which has been applied to evaluate the commercial banks financial risk.

On the other hand, some behavioral studies have shown that the DM who is bounded rational in decision process is more sensitive to losses than to absolutely commensurate gains [24 –27]. So, TODIM (an acronym in Portuguese of interactive and multiple attribute decision making) was early developed by Gomes and Lima to handle the problem effectively [4]. Since TODIM method was put forward in 1992, it has become one of the most valuable tool to deal with the MADM problem considering DM’s behaior [27]. Up to now, the applications range of the TODIM method have been expanded from initial handling crisp numbers [4] to processing interval numbers [28], interval type-2 fuzzy sets [29], grey numbers [30,31, 30,31] or linguistic variables [32], and some efforts have also been made to process heterogeneous data types [27 , 34]. Therefore, how to solve the MADM problem with IGUL variables considering DM’s psychological behavior is still a valuable research topic.

Motivated by the advantages of the TODIM method and IGUL variables, this paper proposes a novel MADM method by extending the TODIM method to IGUL variables. The contributions of the proposed method can be summarized briefly as follows:

A novel δ-Hamming distance is developed to deal with the IGUL variables considering the level of the DM’s risk aversion;

The classical TODIM method is extended to handle the multiple attribute decision making problems in an IGUL environment;

A framework of communication between the classical TODIM method and the classical Superiority and Inferiority Ranking method (i.e., SIR, an outranking methods) [35 –37] is established, which makes the applicability of the two methods has been further expanded.

The remainder of the paper is organized as follows: In Section 2, basic concepts of IGL, IGUL, interval grey number and the constant absolute risk aversion (CARA) utility function are explained. In Section 3, the δ-Hamming distance based on the CARA utility function is proposed. In Section 4, a TODIM· SIR method is proposed to handle MADM problems in an IGUL environment. In Section 5, two numerical examples are given to illustrate the validity and practicality of the proposed method. Finally, Section 6 presents the conclusion of this paper.

2 Preliminary concepts

Suppose that S = (s₀, s₁, . . . , s_l-1) is a finite and totally ordered discrete term set, where l is the odd number and usually equal to 3, 5, 7 or 9 in real situation. In this paper, l = 7, the set S could be given as follows: S = (s₀, s₁, s₂, s₃, s₄, s₅, s₆)={very poor, poor, slightly poor, fair, slightly good, good, very good}.

Definition 1. ([16]) Let $A_{\otimes}^{\sim} = (\tilde{A}, \underset{\otimes}{A})$ be the grey fuzzy number, if its fuzzy part $\tilde{A}$ is a linguistic variable s_α ∈ S, and its grey part $\underset{\otimes}{A}$ is a closed interval $[g_{A}^{L}, g_{A}^{U}]$ , then $A_{\otimes}^{\sim}$ is called the interval grey linguistic (IGL) variables. Let $A_{\otimes}^{\sim} = (s_{α}, [g_{A}^{L}, g_{A}^{U}])$ and $B_{\otimes}^{\sim} = (S_{β}, [g_{B}^{L}, g_{B}^{U}])$ be two interval grey linguistic variables, then the Hamming distance $d (A_{\otimes}^{\sim}, B_{\otimes}^{\sim})$ between $A_{\otimes}^{\sim} and B_{\otimes}^{\sim}$ is defined as follows:

$\begin{matrix} d (A_{\otimes}^{\sim}, B_{\otimes}^{\sim}) & = & \frac{1}{2 (l - 1)} (| α (1 - g_{A}^{L}) - β (1 - g_{B}^{L}) | \\ + | α (1 - g_{A}^{U}) - β (1 - g_{B}^{U}) |) \end{matrix}$ (1)

Definition 2. ([18, 21]) Let $A_{\otimes}^{\sim} = (\tilde{A}, \underset{\otimes}{A})$ be the grey fuzzy number, if its fuzzy part $\tilde{A}$ is an uncertain linguistic variable $\tilde{s} = [s_{α}, s_{β}]$ , s_α, s_β ∈ S, and its grey part $\underset{\otimes}{A}$ is a closed interval $[g_{A}^{L}, g_{A}^{U}] \subseteq [0, 1]$ , then $A_{\otimes}^{\sim}$ is called an interval grey uncertain linguistic (IGUL) variable.

Remark 1. ([16]) It is reasonable to utilize the linguistic variables $\tilde{A}$ to represent the fuzzy part, due to the linguistic variable is very suitable for describe fuzzy information. The grey part $\underset{\otimes}{A}$ which indicates the amount of information the DM obtained. The larger the greyness of $\underset{\otimes}{A}$ is, the lower the credibility of the obtained information is. In contrast, the smaller the greyness is, the higher the credibility and the higher the use value of the DM’s obtained information.

Definition 3. ([38]) Suppose $\underset{\otimes}{A} \in [g_{A}^{L}, g_{A}^{U}]$ , $g_{A}^{L} < g_{A}^{U}$ is an interval grey number, in the case of the distributing information of the values of grey number is lack, then $\overset{\land}{\otimes_{A}} = \frac{1}{2} (g_{A}^{L} + g_{A}^{U}) .$ (2)

Where ${\hat{\otimes}}_{A}$ is called the “kernel” of grey number $\underset{\otimes}{A}$ . The “kernel” of grey number $\underset{\otimes}{A}$ as the representation of grey numbers $\underset{\otimes}{A}$ . In fact, the “kernel” of grey number $\underset{\otimes}{A}$ , as a real number, can be completely operated by the operation of real numbers, such as plus, minus, multiplication, division, power, extract and so on.

Definition 4. ([39, 40]) The constant absolute risk aversion (CARA) utility function is defined as follows: $u (x) = 1 {- e}^{(δ - 1) x}$ (3)

Where δ is the level of relative risk aversion, 0< δ<1. The lager δ is, the greater degree of risk aversion of decision-maker is. x is attribute values in the field of decision making.

3 The δ-Hamming distance based on the CARA utility function

3.1 New distance measure between two IGUL variables

Definition 5. Suppose ${\hat{\otimes}}_{A}$ is the “kernel” of grey number $\underset{\otimes}{A}$ , then the DM’s perceived credibility degree function of grey number $\underset{\otimes}{A}$ is defined based on the Definition 4 as follows: $ρ (\overset{\land}{\otimes_{A}}) = {\begin{matrix} {1 - e}^{(δ - 1) e (1 - \overset{\land}{\otimes_{A}})}, \overset{\land}{0 < \otimes_{A} ⩽ 1;} \\ 1, \overset{\land}{\otimes_{A}} = 0 . \end{matrix}$ (4)

Where δ is the level of DM’s risk aversion.

Definition 6. Suppose $A_{\otimes}^{\approx} = ([s_{α 1}, s_{α 2}], [g_{A}^{L}, g_{A}^{U}])$ and $B_{\otimes}^{\approx} = ([s_{β 1}, s_{β 2}], [g_{B}^{L}, g_{B}^{U}])$ are two IGUL variables, and then taking the level of DM’s risk aversion into consideration, the δ-Hamming distance $d (A_{\otimes}^{\approx}, B_{\otimes}^{\approx})$ between $A_{\otimes}^{\approx}$ and $B_{\otimes}^{\approx}$ can be obtained: $d (A_{\otimes}^{\approx}, B_{\otimes}^{\approx}) = \frac{1}{2 (l - 1)} (\begin{matrix} | α 1 \times ρ (\overset{\land}{\otimes_{A}}) - β 1 \times ρ (\overset{\land}{\otimes_{B}}) | + \\ | α 2 \times ρ (\overset{\land}{\otimes_{A}}) - β 2 \times ρ (\overset{\land}{\otimes_{B}}) | \end{matrix})$ (5) where $\overset{\land}{\otimes_{A}}$ is the “kernel” of grey number $\underset{\otimes}{A}$ , $\overset{\land}{\otimes_{B}}$ is the “kernel” of grey number $\underset{\otimes}{B}$ .

$d (A_{\otimes}^{\approx}, B_{\otimes}^{\approx})$ satisfied the following formula: $d (A_{\otimes}^{\approx}, A_{\otimes}^{\approx}) = 0, 0 ⩽ d (A_{\otimes}^{\approx}, B_{\otimes}^{\approx}) ⩽ 1;$ (6) $d (A_{\otimes}^{\approx}, B_{\otimes}^{\approx}) = d (B_{\otimes}^{\approx}, A_{\otimes}^{\approx});$ (7) $d (A_{\otimes}^{\approx}, B_{\otimes}^{\approx}) + d (B_{\otimes}^{\approx}, C_{\otimes}^{\approx}) ⩾ d (A_{\otimes}^{\approx}, C_{\otimes}^{\approx}) .$ (8)

Formula (5) satisfied Equations (6)– (8) obviously.

3.2 The evaluation of δ-Hamming distance

To demonstrate the feasibility and effectiveness of the δ-Hamming distance, the distinguish ability which was defined by Li [41] is employed.

Definition 7. ([41]) (Distinguish ability) In a group of data, taking 20% of the maximum values (rounded to the nearest integer), denoted by H₁, H₂, . . . , H_N/5, subtract 20% of the minimum values (rounded to the nearest integer), denoted by L₁, L₁, . . . , L_N/5, then the distinguish ability η is defined: $η = \frac{\sum_{i = 1}^{N / 5} H_{i} - \sum_{i = N - N / 5 + 1}^{N} L_{i}}{N / 5} \times 100 %$ (9)

The larger the η is, the higher recognition rate of sample data.

We use the original data in articles [19 –22] to compare the classical Hamming distance [22] and the δ-Hamming distance, and the results are shown in Table 2. As can be seen in Table 2, the distinguish ability of the δ-Hamming distance is obviously higher than the classical, where the DM’s risk aversion coefficient δ=0.10. To analyze the relationship between δ and η, the sensitivity analysis is carried out on δ with original data in articles [21], and Table 1 shows the results. Known by Table 1, δ significant impact on η.

Table 1

The sensitivity analysis between δ and η using data in article [21]

δ	0.25	0.50	0.75	0.95
η	30.22%	23.76%	14.20%	3.36%

Table 2

4 groups of typical IGUL variables and the values of evaluation indexes

No.	Han[22]		Liu[21]		Ma[19]		Zhang[20]
	A	B	A	B	A	B	A	B
1	([4,6],[0.1,0.3])	([3,4],[0.1,0.2])	([4,5],[0.4,0.5])	([2,3],[0.2,0.2])	([4,5],[0.2,0.5])	([2,3],[0.2,0.3])	([4,5],[0.5,0.7])	([2,4],[0.5,0.6])
2	([4,5],[0.3,0.4])	([3,4],[0.3,0.5])	([4,5],[0.2,0.3])	([3,4],[0.2,0.4])	([3,5],[0.4,0.5])	([3,4],[0.2,0.4])	([2,3],[0.3,0.4])	([3,4],[0.3,0.4])
3	([2,3],[0.2,0.3])	([1,3],[0.3,0.4])	([3,4],[0.4,0.4])	([2,2],[0.4,0.4])	([3,6],[0.2,0.4])	([1,3],[0.3,0.4])	([4,5],[0.5,0.5])	([5,6],[0.7,0.8])
4	([5,6],[0.4,0.5])	([1,2],[0.2,0.4])	([3,3],[0.2,0.3])	([2,2],[0.3,0.4])	([4,6],[0.3,0.6])	([1,4],[0.3,0.5])	([0,2],[0.4,0.5])	([4,5],[0.6,0.9])
5	([1,2],[0.3,0.3])	([5,6],[0.3,0.4])	([5,6],[0.5,0.6])	([5,6],[0.4,0.5])	([3,4],[0.3,0.4])	([3,6],[0.4,0.5])	([1,2],[0.3,0.4])	([3,4],[0.6,0.7])
6	([5,6],[0.4,0.5])	([3,4],[0.1,0.4])	([2,2],[0.4,0.4])	([2,3],[0.2,0.3])	([4,5],[0.4,0.5])	([2,3],[0.2,0.4])	([5,6],[0.3,0.5])	([1,2],[0.1,0.2])
7	([4,5],[0.2,0.4])	([2,3],[0.3,0.3])	([4,5],[0.4,0.5])	([3,3],[0.3,0.3])	([3,4],[0.2,0.4])	([2,3],[0.3,0.4])	([2,3],[0.3,0.3])	([2,3],[0.6,0.7])
8	([2,3],[0.2,0.3])	([4,6],[0.1,0.5])	([4,4],[0.2,0.3])	([3,4],[0.2,0.4])	([2,5],[0.2,0.5])	([3,4],[0.2,0.5])	([5,6],[0.7,0.8])	([2,5],[0.3,0.4])
9	([4,5],[0.4,0.5])	([4,5],[0.2,0.4])	([2,2],[0.2,0.2])	([4,5],[0.2,0.4])	([4,5],[0.3,0.5])	([4,5],[0.2,0.4])	([5,6],[0.6,0.6])	([4,5],[0.6,0.7])
10	([2,4],[0.1,0.2])	([3,4],[0.3,0.3])	([4,5],[0.5,0.5])	([3,4],[0.3,0.3])	([2,4],[0.2,0.4])	([3,4],[0.3,0.4])	([4,6],[0.2,0.4])	([2,4],[0.4,0.6])
11	([3,5],[0.3,0.3])	([2,3],[0.1,0.4])	([3,3],[0.1,0.2])	([4,4],[0.2,0.3])	([3,4],[0.3,0.4])	([3,4],[0.2,0.3])	([0,3],[0.5,0.6])	([3,6],[0.2,0.3])
12	([1,2],[0.3,0.5])	([3,4],[0.2,0.3])	([4,4],[0.3,0.3])	([2,3],[0.2,0.3])	([2,4],[0.2,0.5])	([2,5],[0.2,0.3])	([1,2],[0.2,0.3])	([2,6],[0.2,0.3])
13	([3,4],[0.2,0.4])	([3,5],[0.2,0.2])	([2,3],[0.2,0.4])	([2,3],[0.3,0.3])	([2,5],[0.2,0.4])	([2,4],[0.3,0.3])	([1,2],[0.4,0.6])	([5,6],[0.3,0.4])
14	([2,4],[0.5,0.5])	([4,5],[0.3,0.4])	([2,3],[0.2,0.4])	([4,5],[0.3,0.4])	([3,4],[0.3,0.5])	([4,5],[0.3,0.4])	([2,3],[0.4,0.5])	([3,4],[0.3,0.5])
15	([4,5],[0.1,0.3])	([4,6],[0.2,0.5])	([3,4],[0.5,0.5])	([4,5],[0.3,0.4])	([4,5],[0.2,0.5])	([3,5],[0.3,0.4])	([5,6],[0.5,0.7])	([4,6],[0.5,0.6])
16	([3,5],[0.4,0.5])	([2,4],[0.2,0.3])	([4,5],[0.2,0.3])	([2,3],[0.1,0.3])	([2,4],[0.3,0.4])	([2,5],[0.1,0.3])	([2,3],[0.6,0.8])	([5,6],[0.3,0.5])
17	([3,5],[0.1,0.3])	([2,4],[0.4,0.5])	([2,3],[0.3,0.4])	([3,4],[0.4,0.5])	([3,5],[0.1,0.3])	([3,4],[0.4,0.5])	([1,3],[0.3,0.4])	([2,3],[0.3,0.6])
18	([5,6],[0.3,0.4])	([2,3],[0.1,0.2])	([3,4],[0.1,0.3])	([1,2],[0.1,0.2])	([4,5],[0.3,0.5])	([1,4],[0.1,0.4])	([4,6],[0.5,0.6])	([0,4],[0.6,0.7])
19	([3,4],[0.2,0.4])	([1,2],[0.2,0.4])	([4,5],[0.4,0.5])	([1,2],[0.1,0.2])	([3,4],[0.2,0.4])	([1,3],[0.1,0.5])
20	([4,5],[0.2,0.2])	([4,5],[0.4,0.5])	([4,4],[0.2,0.4])	([3,4],[0.3,0.4])	([4,5],[0.3,0.4])	([3,4],[0.3,0.4])
21	([2,3],[0.3,0.4])	([4,5],[0.2,0.3])	([4,5],[0.3,0.4])	([3,4],[0.2,0.3])	([2,4],[0.2,0.3])	([3,4],[0.2,0.3])
22	([2,4],[0.3,0.5])	([3,4],[0.2,0.4])	([2,3],[0.2,0.3])	([2,3],[0.1,0.2])	([2,5],[0.3,0.4])	([2,5],[0.1,0.3])
23	([4,5],[0.1,0.3])	([3,5],[0.1,0.3])	([2,3],[0.3,0.4])	([3,4],[0.2,0.3])	([3,4],[0.2,0.3])	([3,4],[0.2,0.3])
24	([4,5],[0.4,0.5])	([5,6],[0.4,0.5])	([4,4],[0.2,0.3])	([4,5],[0.4,0.5])	([3,5],[0.4,0.5])	([4,5],[0.1,0.4])
Classical Hamming^[22](η)	19.58%		19.92%		17.50%		28.96%
Novel Hamming distance (η)	33.03% (δ=0.10)		25.62 % (δ=0.10)		23.55% (δ=0.10)		41.26% (δ=0.10)

More specifically, when the degree of risk aversion of DM is relatively low, such as the risk aversion coefficient δ=0.25, the recognition rate of sample data is 30.22%, which higher than the classical Hamming distance’s (η=19.58%). However, when the risk aversion coefficient δ=0.75, the recognition rate of sample data is being dropped to 14.20%, which significantly lower than the classical Hamming distance’s η. It is known that when a DM’s risk aversion degree is higher, who would believe that the obtained information with larger greyness is lower credibility and lower usage value [16].

For example, if the greyness of the obtained information ${\hat{\otimes}}_{A}$ is 0.40, those DMs with different risk aversion attitudes would perceived different credibility degree of the obtained information (see Fig. 1).

Fig.1

The decision-maker’s perception utility function of the “kernel” of grey number.

To those DM with lower risk aversion (i.e., δ= 0.10) which indicates his or her tolerance for risk is higher, his or her perceived credibility degree of the obtained information may higher (i.e., 0.7796), and at this point he or she tends to believe that the obtained information has a higher reliability and value in the process of decision-making. But for those DMs with lager risk aversion (i.e., δ= 0.75), whose perceived credibility degree may lower (i.e., 0.3348), and he or she tends to believe that the obtained information has a lower reliability and value. In other words, it is difficult for DM to make a decision based on those obtained information with higher greyness. Thus, with the increasing degree of risk aversion of DMs, whose ability to distinguish information with higher greyness would continue to be reduced. In fact, the δ-Hamming distance could precisely describes the information processing of DMs which were mentioned above, and it is more flexible and practical.

4 The TODIM· SIR method

4.1 Problem statement

Let N = {1, 2, . . . , N} andM = {1, 2, . . . , M} be the subscripts sets of alternatives and attributes. Let P = {P₁, P₂, . . . , P_i, . . . , P_N} with N≥2 be a finite set of alternatives and C = {C₁, C₂, . . . , C_j, . . . , C_M} with M≥2 be a finite set of attributes. Let W = (w1, w2, . . . , w_M) ^T be an attribute weight vector, where w_j is the weight assigned to attribute C_j.

Let $D = {[A_{\otimes ij}^{\approx} = {([s_{α^{i}}, s_{β^{i}}], [g_{A}^{Li}, g_{A}^{Ui}])}_{j}]}_{N \times M}$ be a decision matrix, where IGUL variable $A_{\otimes ij}^{\approx} = {([s_{α^{i}}, s_{β^{i}}], [g_{A}^{Li}, g_{A}^{Ui}])}_{j}$ denotes the evaluation of alternative P_i for attribute C_j. The problem addressed in this study is how to rank alternatives according to decision matrix $D = {[A_{\otimes ij}^{\approx}]}_{N \times M}$ considering the DM’s behaviour. The resolution procedure of the TODIM· SIR method is shown in Fig. 2.

Fig.2

The resolution procedure of the TODIM· SIR method.

4.2 The proposed method

In this section, the TODIM· SIR method is presented considering the DM’s behaviour and the seven steps are described as follows:

Step 1. Normalize attribute values by Definition 3 and Definition 5. $\begin{matrix} A_{\otimes ij}^{\approx *} = {[α^{i} \times ρ (\overset{\land}{\otimes_{Ai}}), β^{i} \times ρ (\overset{\land}{\otimes_{Ai}})]}_{j}, \\ i \in N, j \in M . \end{matrix}$ (10)

Where $\overset{\land}{\otimes_{Ai}} = \frac{1}{2} {(g_{A}^{Li} + g_{A}^{Ui})}_{j},$ (11) $ρ (\overset{\land}{\otimes_{Ai}}) = {\begin{matrix} {1 - e}^{(δ - 1) e (1 - \overset{\land}{\otimes_{Ai}})}, \overset{\land}{0 < \otimes_{Ai} ⩽ 1;} \\ 1, \overset{\land}{\otimes_{Ai}} = 0 . \end{matrix}$ (12)

Then the positive-ideal alternative (i.e., PIA) and the negative-ideal alternative (i.e., NIA) can be defined. $\begin{matrix} P_{PIA} = {[max_{i \in N} {α^{i} \times ρ (\overset{\land}{\otimes_{Ai}})}, max_{i \in N} {β^{i} \times ρ (\overset{\land}{\otimes_{Ai}})}]}_{j}, \\ j \in M . \end{matrix}$ (13) $\begin{matrix} P_{NIA} = {[min_{i \in N} {α^{i} \times ρ (\overset{\land}{\otimes_{Ai}})}, min_{i \in N} {β^{i} \times ρ (\overset{\land}{\otimes_{Ai}})}]}_{j}, \\ j \in M . \end{matrix}$ (14)

Step 2. Define gains and losses.

Definition 8. The gain of alternative P_i relative to alternative P_NIA concerning attribute C_j, $G_{iNIA}^{j}$ , is given by

$\begin{matrix} G_{iNIA}^{j} \\ = \frac{1}{2 (l - 1)} (\begin{matrix} {| α^{i} \times ρ (\overset{\land}{\otimes_{Ai}}) - min_{i \in N} {α^{i} \times ρ (\overset{\land}{\otimes_{Ai}})} |}_{j} \\ + {| β^{i} \times ρ (\overset{\land}{\otimes_{Ai}}) - min_{i \in N} {β^{i} \times ρ (\overset{\land}{\otimes_{Ai}})} |}_{j} \end{matrix}) \end{matrix}$ (15)

Correspondingly, the loss of alternative P_k relative to alternative P_PIA concerning attribute C_j, $L_{iPIA}^{j}$ , is given by

$\begin{matrix} L_{iPIA}^{j} \\ = \frac{1}{2 (l - 1)} (\begin{matrix} {| α^{i} \times ρ (\overset{\land}{\otimes_{Ai}}) - max_{i \in N} {α^{i} \times ρ (\overset{\land}{\otimes_{Ai}})} |}_{j} \\ + {| β^{i} \times ρ (\overset{\land}{\otimes_{Ai}}) - max_{i \in N} {β^{i} \times ρ (\overset{\land}{\otimes_{Ai}})} |}_{j} \end{matrix}) \end{matrix}$ (16)

Step 3. Calculate the dominance degree for the gain $ϑ_{iINA}^{j (+)}$ and the loss $ϑ_{iPIA}^{j (-)}$ , respectively. ${\begin{matrix} ϑ_{iNIA}^{j (+)} = \sqrt{\frac{w_{j} G_{iNIA}^{j}}{w_{r} \sum_{j = 1}^{n} (w_{j} / w_{r})}}, i \in N, j \in M . \\ ϑ_{iPIA}^{j (-)} = \frac{- 1}{θ} \sqrt{\frac{w_{r} L_{iPIA}^{j}}{w_{j}} \sum_{j = 1}^{n} (w_{j} / w_{r})}, i \in N, j \in M . \end{matrix}$ (17)

Where w_r = max {w_j|j ∈ M} , θ is the attenuation factor of the loss and θ>0. When 0< θ<1, the influence of loss will increase [29].

Step 4. Define the Gain-flow and the Loss-flow as follows.

G-flow: $φ^{>} (P_{i}) = \sum_{j = 1}^{n} ϑ_{iNIA}^{j (+)}, j \in M .$ (18)

L-flow: $φ^{<} (P_{i}) = \sum_{j = 1}^{n} ϑ_{iPIA}^{j (-)}, j \in M .$ (19)

In accordance with the SIR method, the higher the G-flow φ^> (P_i) and the lower L-flow φ^> (P_i) is, the better the alternative P_i is.

Step 5. Determine the Gain ranking and Loss ranking.

Let P_i ∈ P and P_k ∈ P are two different alternatives, and G-flow and L-flow can be obtained by Step 1 to Step 4, respectively.

The Gain ranking (called G-ranking) can be obtained based on the descending order of φ^> (P_i) : $P_{i} P_{>} P_{k} if φ^{>} (P_{i}) > φ^{>} (P_{k});$ (20) $P_{i} I_{>} P_{k} if φ^{>} (P_{i}) = φ^{>} (P_{k}) .$ (21)

The Loss ranking (called L-ranking) can be obtained based on the descending order of φ^>(P_i): $P_{i} P_{<} P_{k} if φ^{<} (P_{i}) < φ^{<} (P_{k});$ (22) $P_{i} I_{<} P_{k} if φ^{<} (P_{i}) = φ^{<} (P_{k}) .$ (23)

In general, the G-ranking and L-ranking are two different complete rankings.

Step 6. Combine the G-ranking and L-ranking into a partial ranking.

The partial ranking can be obtained by employing the following intersection principle (Brans et al., 1986; Roy et al., 1992) [35]: ${\begin{matrix} P_{i} is indifferent to P_{k}, P_{i} I_{>} P_{k} and P_{i} I_{<} P_{k}; \\ P_{i} is preferred to P_{k}, {\begin{matrix} P_{i} P_{>} P_{k} and P_{i} P_{<} P_{k} \\ P_{i} P_{>} P_{k} and P_{i} I_{<} P_{k} \\ P_{i} I_{>} P_{k} and P_{i} P_{<} P_{k} \end{matrix}; \\ P_{i} is incomparable to P_{k}, {\begin{matrix} P_{i} P_{>} P_{k} and P_{k} P_{<} P_{i} \\ P_{k} P_{>} P_{i} and P_{i} P_{<} P_{k} \end{matrix} . \end{matrix}$ (24)

Step 7. Construct synthesizing flow.

When a complete ranking is requested, the net flow (N-flow) can be used. $φ_{n} (P_{i}) = φ^{>} (P_{i}) + φ^{<} (P_{i}) .$ (25)

5 Numerical example

5.1 Application

Example 1. Enterprises’ evaluation

In this section, the evaluation of enterprises’ technological innovation ability problem discussed by Liu [21] who also developed a decision making method using the IGUL variables is used as the benchmark. In this problem, there are four enterprises and four criteria (see Fig. 3), and the decision-making matrix D is shown as follow. w = (0.3, 0.2, 0.2, 0.3) is attribute weight vector. $D = [\begin{matrix} ([s_{3 . 68}, s_{4 . 68}], [0 . 20, 0 . 40]), ([s_{2 . 00}, s_{2 . 60}], [0 . 40, 0 . 40]), ([s_{3 . 08}, s_{4 . 08}], [0 . 50, 0 . 50]), ([s_{3 . 64}, s_{4 . 24}], [0 . 40, 0 . 50]) \\ ([s_{3 . 32}, s_{4 . 32}], [0 . 40, 0 . 50]), ([s_{3 . 36}, s_{4 . 36}], [0 . 40, 0 . 50]), ([s_{2 . 44}, s_{3 . 04}], [0 . 20, 0 . 40]), ([s_{2 . 40}, s_{3 . 40}], [0 . 50, 0 . 50]) \\ ([s_{3 . 60}, s_{3 . 60}], [0 . 20, 0 . 40]), ([s_{4 . 00}, s_{4 . 28}], [0 . 30, 0 . 40]), ([s_{2 . 52}, s_{2 . 80}], [0 . 40, 0 . 40]), ([s_{3 . 40}, s_{4 . 08}], [0 . 30, 0 . 30]) \\ ([s_{3 . 84}, s_{4 . 84}], [0 . 50, 0 . 60]), ([s_{2 . 32}, s_{2 . 92}], [0 . 40, 0 . 50]), ([s_{2 . 28}, s_{2 . 96}], [0 . 30, 0 . 40]), ([s_{2 . 88}, s_{3 . 88}], [0 . 40, 0 . 50]) \end{matrix}]$

Fig.3

The evaluation of enterprises’ technological innovation ability.

Considering the DM’s behaviour with regard to risk and loss aversion, the calculation process and results obtained using the TODIM· SIR method are represented below.

According to Equations (10)– (14), the standardized matrix $D^{*} = {[A_{\otimes ij}^{\approx *}]}_{N \times M}$ , the positive-ideal alternative (i.e., PIA) and negative-ideal alternative (i.e., NIA) are constructed as follows. Where the risk aversion coefficient of DM is δ=0.25.

$D^{*} = (\begin{matrix} [2 . 80, 3 . 56], [1 . 41, 1 . 83], [1 . 97, 2 . 61], [2 . 45, 2 . 86] \\ [2 . 24, 2 . 91], [2 . 27, 2 . 94], [1 . 85, 2 . 31], [1 . 53, 2 . 17] \\ [2 . 74, 2 . 74], [2 . 94, 3 . 14], [1 . 78, 1 . 98], [2 . 58, 3 . 10] \\ [2 . 31, 2 . 91], [1 . 56, 1 . 97], [1 . 67, 2 . 17], [1 . 94, 2 . 62] \end{matrix})$ $PIA = ([2 . 80, 3 . 56], [2 . 94, 3 . 14], [1 . 97, 2 . 61], [2 . 58, 3 . 10])$ $NIA = ([2 . 24, 2 . 74], [1 . 41, 1 . 83], [1 . 67, 1 . 98], [1 . 53, 2 . 17])$

The gain and loss of alternative P_i relative to alternative P_NIA and P_PIA concerning attribute C_j, $G_{iNIA}^{j}$ and $L_{iPIA}^{j}$ , can be obtained by Equations (15), (16), respectively. $G = (\begin{matrix} 0 . 1150, 0 . 0000, 0 . 0772, 0 . 1338 \\ 0 . 0147, 0 . 1632, 0 . 0429, 0 . 0000 \\ 0 . 0415, 0 . 2361, 0 . 0087, 0 . 1648 \\ 0 . 0198, 0 . 0238, 0 . 0164, 0 . 0708 \end{matrix}),$ $L = (\begin{matrix} 0 . 0000, 0 . 2361, 0 . 0000, 0 . 0310 \\ 0 . 1003, 0 . 0729, 0 . 0343, 0 . 1648 \\ 0 . 0735, 0 . 0000, 0 . 0685, 0 . 0000 \\ 0 . 0951, 0 . 2123, 0 . 0608, 0 . 0940 \end{matrix})$

The dominance degree for the gain and loss can be calculated by Equation (17), respectively. Generally, the attenuation factor of the loss θ is considered to be 1. $G^{D} = (\begin{matrix} 0 . 1857, 0 . 0000, 0 . 1243, 0 . 2003 \\ 0 . 0664, 0 . 1806, 0 . 0926, 0 . 0000 \\ 0 . 1116, 0 . 2173, 0 . 0417, 0 . 2223 \\ 0 . 0771, 0 . 0691, 0 . 0573, 0 . 1458 \end{matrix}),$ $L^{D} = (\begin{matrix} 0 . 0000, - 1 . 0865, 0 . 0000, - 0 . 3217 \\ - 0 . 5781, - 0 . 6039, - 0 . 4142, - 0 . 7412 \\ - 0 . 4949, 0 . 0000, - 0 . 5852, 0 . 0000 \\ - 0 . 5631, - 1 . 0302, - 0 . 5512, - 0 . 5597 \end{matrix})$

According to Equations (18)– (19) and Equation (25), the Gain-flow, the Loss-flow and the Net-flow can be defined as follows (see Table 3). Hence, by using Gain-flow and Loss-flow, the G-ranking and L-ranking are obtained:

Table 3

The Gain-flow, the Loss-flow and the Net-flow of four enterprises

P _i	Gain-flow		Loss-flow		Net-flow
	Superiority degree	Descending order	Inferiority degree	Descending order	Net degree	Descending order
P ₁	0.5103	2	– 1.4082	2	– 0.8979	2
P ₂	0.3397	4	– 2.3373	3	– 1.9976	3
$\begin{matrix} P_{3} \end{matrix}$	0.5929	1	– 1.0801	1	– 0.4872	1
$\begin{matrix} P_{4} \end{matrix}$	0.3493	3	– 2.7041	4	– 2.3548	4

Table 4

The interval grey uncertain linguistic decision making matrix

	Market risk:	Enterprise operation	Enterprise assets	Environmental risk: C₄
	C ₁	risk: C₂	structure risk: C₃
Commercial bank: P₁	(2,3],[0.20,0.40])	([5,6],[0.40,0.50])	([2,3],[0.30,0.40])	([2,4],[0.50,0.60])
Commercial bank: P₂	([3,4],[0.40,0.50])	([2,3],[0.30,0.40])	([4,5],[0.50,0.60])	([4,5],[0.60,0.70])
Commercial bank: $\begin{matrix} P_{3} \end{matrix}$	([4,5],[0.30,0.40])	([3,4],[0.50,0.60])	([4,5],[0.20,0.30])	([4,5],[0.20,0.30])

$\begin{matrix} P_{>} : P_{3} \to P_{1} \to P_{4} \to P_{2} \\ P_{<} : P_{3} \to P_{1} \to P_{2} \to P_{4} \end{matrix}$

Then, the result of partial ranking is constructed by combine the G-ranking with the L-ranking:

And the complete ranking order by Net-flow is: $P_{3} \to P_{1} \to P_{2} \to P_{4}$

Obviously, P₃ has the highest innovation ability, which is consistent with the conclusion of literature [21]. In accordance with the group decision-making matrix D, the alternative P₂ is relative better than P₄ on attributes C₂ and C₃, while not better than P₄ on attributes C₄. Moreover, it is hard to compare P₂ with P₄ on C₁. Therefore, it may be more reasonable to predicate the relationship between P₂ and P₄ is incomparable.

Example 2: the commercial banks financial risk evaluation

In this section, the evaluation of the commercial banks financial risk problem discussed by Xu [23] who developed the induced IGUL ordered weighted geometric (I-IGULOWG) operator to deal with the MADM problems is used as the benchmark. The attribute weight vector is w=(0.20, 0.30, 0.10, 0.40). According to Equations (10)– (14), the standardized matrix $D^{*} = {[A_{\otimes ij}^{\approx *}]}_{N \times M}$ , the positive-ideal alternative (i.e., PIA) and negative-ideal alternative (i.e., NIA) are constructed as follows. Where the risk aversion coefficient of DM is δ=0.25. $D^{*} = (\begin{matrix} [1 . 52, 2 . 28], [3 . 37, 4 . 04], [1 . 47, 2 . 20], [1 . 20, 2 . 40] \\ [2 . 02, 2 . 70], [1 . 47, 2 . 20], [2 . 40, 3 . 00], [2 . 04, 2 . 55] \\ [2 . 94, 3 . 67], [1 . 80, 2 . 40], [3 . 13, 3 . 92], [3 . 13, 3 . 92] \end{matrix}) .$ $PIA = ([2 . 94, 3 . 67], [3 . 37, 4 . 04], [3 . 13, 3 . 92], [3 . 13, 3 . 92]) .$ $NIA = ([1 . 52, 2 . 28], [1 . 47, 2 . 20], [1 . 47, 2 . 20], [1 . 20, 2 . 40]) .$

According to Equations (15)– (25), the Gain-flow, the Loss-flow and the Net-flow can be defined as follows (see Table 5). By using Gain-flow and Loss-flow, the G-ranking and L-ranking are obtained:

Table 5

The Gain-flow, the Loss-flow and the Net-flow of commercial banks

P_i	Gain-flow		Loss-flow		Net-flow
	Superiority degree	Descending order	Inferiority degree	Descending order	Net degree	Descending order
P ₁	0.3060	3	– 3.6069	2	– 3.3009	2
P ₂	0.4254	2	– 3.7936	3	– 3.3682	3
$\begin{matrix} P_{3} \end{matrix}$	0.8384	1	– 0.9446	1	– 0.1062	1

Table 6

The S-flow, the I-flow and the n-flow of four enterprises

P _i	S-flow		I-flow		n-flow
	Superiority flow	Descending order	Inferiority flow	Ascending order	Net degree	Descending order
P ₁	0.0052	2	0.0056	2	0.0109	2
P ₂	0.0029	3	0.0062	3	0.0091	3
$\begin{matrix} P_{3} \end{matrix}$	0.0098	1	0.0013	1	0.0111	1
$\begin{matrix} P_{4} \end{matrix}$	0.0009	4	0.0075	4	0.0084	4

$\begin{matrix} P_{>} : P_{3} \to P_{2} \to P_{1 .} \\ P_{<} :_{.} P_{3} \to P_{1} \to P_{2 .} \end{matrix}$

Then, the result of partial ranking can be also constructed by combine the G-ranking with the L-ranking:

And the complete ranking order by Net-flow is: $P_{3} \to P_{1} \to P_{2}$

Therefore, P₃ is the best commercial banks, which is also consistent with the conclusion of literature [23].

5.2 Comparisons and discussions

In this section, the example 1 in Section 5.1 is used to conduct the comparison analysis and sensitivity analysis.

5.2.1 Compared with the classical TODIM method

The first 4 steps in the classical TODIM method are in accordance with the TODIM· SIR method mentioned in Section 4.2, and the overall dominance degree of alternative P_i can be calculated by Equation (25). $\begin{matrix} σ_{P_{1}} & = & φ^{>} (P_{1}) + φ^{<} (P_{1}) = - 0 . 8979, \\ σ_{P_{2}} & = & φ^{>} (P_{2}) + φ^{<} (P_{2}) = - 1 . 9976, \\ σ_{P_{3}} & = & φ^{>} (P_{3}) + φ^{<} (P_{3}) = - 0 . 4872, \\ σ_{P_{4}} & = & φ^{>} (P_{4}) + φ^{<} (P_{4}) = - 2.3548 . \end{matrix}$

The overall prospect value of alternative P_i can be obtained by Equation (26). Then ɛ (P₁) = 0 .7801, ɛ (P₂) = 0 . 1913, ɛ (P₃) = 1 . 0000, ɛ (P₄) = 0.0000, and the ranking order of the four alternatives is determined, i.e.,P₃ > P₁ > P₂ > P₄. $ɛ (P_{i}) = \frac{σ_{P_{i}} - min_{i \in N} {\sum_{i = 1}^{N} σ_{P_{i}}}}{max_{i \in N} {\sum_{i = 1}^{N} σ_{P_{i}}} - min_{i \in N} {\sum_{i = 1}^{N} σ_{P_{i}}}}, i \in N .$ (26)

There is no doubt that the complete ranking order by Net-flow in the TODIM· SIR method is the same with the classical TODIM method. However, it is necessary to point out that the TODIM· SIR method could effectively identify the preferred/ indifferent/ incomparable relationship between alternatives (i.e., the relationship between P₂ and P₄ is incomparable), which due to one alternative is superior only if the value of it is greater than that of others [35]. In other words, the TODIM· SIR method provides two types of flows which could be used to rank the alternatives partially or completely according to the DM’s needs.

5.2.2 Compared with the classical SIR method

The first 2 steps in the classical SIR method are in accordance with the TODIM· SIR method mentioned in Section 4.2, and the Gaussian criterion which has been mostly selected by users for practical applications was employed to meet the DM’s preference attitude [35]. $\begin{matrix} f (G, L) = {\begin{matrix} 1 - exp (- G_{iNIA}^{2} (j) / 2 θ^{2}), \\ 1 - exp (- L_{iPIA}^{2} (j) / 2 θ^{2}) . \end{matrix} \\ i \in N, j \in M . \end{matrix}$ (27)

According to Equation (27), we can obtain the S-matrix and the I-matrix as follows, where θ=1: $S = (\begin{matrix} 0 . 0066, 0 . 0000, 0 . 0030, 0 . 0089 \\ 0 . 0001, 0 . 0132, 0 . 0009, 0 . 0000 \\ 0 . 0009, 0 . 0275, 0 . 0000, 0 . 0135 \\ 0 . 0002, 0 . 0003, 0 . 0001, 0 . 0025 \end{matrix}),$ $I = (\begin{matrix} 0 . 0000, 0 . 0275, 0 . 0000, 0 . 0005 \\ 0 . 0050, 0 . 0027, 0 . 0006, 0 . 0135 \\ 0 . 0027, 0 . 0000, 0 . 0023, 0 . 0000 \\ 0 . 0045, 0 . 0223, 0 . 0018, 0 . 0044 \end{matrix}) .$

Then the S-flows and the I-flows is calculated by the simple additive weighting (SAW) respectively, and the flows of SIR· SAW is obtained.

Hence, the complete or partial rankings are the same: $P_{3} \to P_{1} \to P_{2} \to P_{4}$

Which compared with the TODIM· SIR method’s results, the biggest difference is the partial rankings. One of the main reason is that we employ the prospect function instead of the Gaussian criterion in the TODIM· SIR method. As is known to all some studies about behavioral experiments have shown that the DM is more sensitive to losses [27]. Thus, in the situation of considering DM’s psychological behavior, it is more reasonable to use the prospect function. From this point of view, the TODIM· SIR method is the inheritance and development of the classical SIR method.

5.2.3 Discussions

In order to explore the influence of both the parameter δ and θ on the ranking order of alternatives, we modify the parameters and recalculate ranking orders as listed in Table 7. To save the space, the detailed calculation process is omitted.

Table 7
The S-flow and the I-flow of four enterprises

Different values of δ Different values of θ Ranking orders of alternatives

0.10 1.0 P_>: P₃ (0.6080)⟶ P₁ (0.5327)⟶ $\begin{matrix} P_{4} \end{matrix}$ (0.3817)⟶ P₂ (0.3616)_. <:. P₃ (– 1.4175)⟶ P₁ (-2.4199)⟶P₂(– 1.1459)⟶ $\begin{matrix} P_{4} \end{matrix}$ (-2.7725)_.

1.5 P_>: P₃ (0.6080)⟶ P₁ (0.5327)⟶P₄(0.3817)⟶P₂(0.3616)_.P_<:. P₃ (– 0.7640)⟶ P₁ (– 0.9450)⟶ P₂ (– 1.6133)⟶P₄(– 1.8484).

2.0 P_>: P₃ (0.6080)⟶ P₁ (0.5327)⟶P₄(0.3817)⟶P₂(0.3616)_.P_<:. P₃ (– 0.5730)⟶ P₁ (– 0.7087)⟶ P₂ (– 1.2100)⟶P₄(– 1.3863)_.

0.30 1.0 P_>: P₃ (0.5860)⟶ P₁ (0.5012)⟶P₄(0.3364)⟶ P₂ (0.3310)_.P_<:. P₃ (– 1.0547)⟶ P₁ (– 1.3997)⟶P₂(– 2.3006)⟶P₄(– 2.6718)_.

1.5 P_>: P₃ (0.5860)⟶ P₁ (0.5012)⟶P₄(0.3364)⟶P₂(0.3310)_.P_<:. P₃ (– 0.7031)⟶ P₁– 0.9331()⟶ P₂ (– 1.5338)⟶ $\begin{matrix} P_{4} \end{matrix}$ (– 1.7812)_.

2.0 P_>: P₃(0.5860)⟶ P₁ (0.5012)⟶P₄(0.3364)⟶ P₂ (0.3310)_.P_<:. P₃ (– 0.5273)⟶ P₁ (– 0.6999)⟶ P₂ (-1.1503)⟶P₄(-1.3359)_.

0.5 1.0 P_>: P₃(0.5445)⟶ P₁ (0.4534)⟶P₂(0.2863)⟶P₄(0.2616)_.P_<:. P₃ (– 0.4649)⟶P₁(– 0.6640)⟶P₂(-1.0437)⟶P₄(-1.2373)_.

1.5 P_>: P₃ (0.5445)⟶ P₁ (0.4534)⟶P₂(0.2863)⟶P₄(0.2616)_.P_<:. P₃ (– 0.6199)⟶ P₁ (– 0.8854)⟶ P₂ (– 1.3916)⟶ $\begin{matrix} P_{4} \end{matrix}$ (– 1.6498)_.

2.0 P_>: P₃ (0.5445)⟶ P₁ (0.4534)⟶ P₂ (0.2863)⟶P₄(0.2616)_.P_<:. P₃ (– 0.4649)⟶ P₁ (– 0.6640)⟶ P₂ (– 1.0437)⟶ $\begin{matrix} P_{4} \end{matrix}$ (– 1.2373)_.

Different values of δ	Different values of θ	Ranking orders of alternatives
0.10	1.0	P_>: P₃ (0.6080)⟶ P₁ (0.5327)⟶ $\begin{matrix} P_{4} \end{matrix}$ (0.3817)⟶ P₂ (0.3616)_. <:. P₃ (– 1.4175)⟶ P₁ (-2.4199)⟶P₂(– 1.1459)⟶ $\begin{matrix} P_{4} \end{matrix}$ (-2.7725)_.
	1.5	P_>: P₃ (0.6080)⟶ P₁ (0.5327)⟶P₄(0.3817)⟶P₂(0.3616)_.P_<:. P₃ (– 0.7640)⟶ P₁ (– 0.9450)⟶ P₂ (– 1.6133)⟶P₄(– 1.8484).
	2.0	P_>: P₃ (0.6080)⟶ P₁ (0.5327)⟶P₄(0.3817)⟶P₂(0.3616)_.P_<:. P₃ (– 0.5730)⟶ P₁ (– 0.7087)⟶ P₂ (– 1.2100)⟶P₄(– 1.3863)_.
0.30	1.0	P_>: P₃ (0.5860)⟶ P₁ (0.5012)⟶P₄(0.3364)⟶ P₂ (0.3310)_.P_<:. P₃ (– 1.0547)⟶ P₁ (– 1.3997)⟶P₂(– 2.3006)⟶P₄(– 2.6718)_.
	1.5	P_>: P₃ (0.5860)⟶ P₁ (0.5012)⟶P₄(0.3364)⟶P₂(0.3310)_.P_<:. P₃ (– 0.7031)⟶ P₁– 0.9331()⟶ P₂ (– 1.5338)⟶ $\begin{matrix} P_{4} \end{matrix}$ (– 1.7812)_.
	2.0	P_>: P₃(0.5860)⟶ P₁ (0.5012)⟶P₄(0.3364)⟶ P₂ (0.3310)_.P_<:. P₃ (– 0.5273)⟶ P₁ (– 0.6999)⟶ P₂ (-1.1503)⟶P₄(-1.3359)_.
0.5	1.0	P_>: P₃(0.5445)⟶ P₁ (0.4534)⟶P₂(0.2863)⟶P₄(0.2616)_.P_<:. P₃ (– 0.4649)⟶P₁(– 0.6640)⟶P₂(-1.0437)⟶P₄(-1.2373)_.
	1.5	P_>: P₃ (0.5445)⟶ P₁ (0.4534)⟶P₂(0.2863)⟶P₄(0.2616)_.P_<:. P₃ (– 0.6199)⟶ P₁ (– 0.8854)⟶ P₂ (– 1.3916)⟶ $\begin{matrix} P_{4} \end{matrix}$ (– 1.6498)_.
	2.0	P_>: P₃ (0.5445)⟶ P₁ (0.4534)⟶ P₂ (0.2863)⟶P₄(0.2616)_.P_<:. P₃ (– 0.4649)⟶ P₁ (– 0.6640)⟶ P₂ (– 1.0437)⟶ $\begin{matrix} P_{4} \end{matrix}$ (– 1.2373)_.

It can be seen from Table 7 that the ranking orders of alternatives are changed with different values of δ. More specifically, for δ= 0.1 or δ= 0.3, the relationship between P₂ and P₄ is incomparable. For δ= 0.5, P₂ is preferred to P₄. While the ranking orders of alternatives were not change when the value of θ changes. Although the ranking orders of alternatives were not change, but the DM’s perceive utility value changes with θ change. The reason of different results for the two parameters is that the characteristics of the original data determine its sensitivity for different parameters.

6 Conclusions

In this paper, a novel TODIM· SIR method is proposed to deal with the multiple attribute decision making problems under an IGUL environment, in which three behavioral characteristics of the DM (i.e., risk aversion, reference dependence and loss aversion) are considered. Due to existing distance measures of IGUL variables are deficient and cannot take into account the DM’s risk aversion attitudes during handle the grey part of IGUL variables, new δ-Hamming distance is put forward to address the existing problem. Then, considering both the reference dependence behaviour and the loss aversion behaviour of DM, the perceived dominance degree of the DM for the gain and the loss of alternatives are calculated, and the ranking steps of alternatives are presented. Finally, we use two examples to illustrate the practicality and the robustness of the proposed method. In comparison with the classical TODIM method and the classical SIR method, the significant characteristics of the TODIM· SIR method are that it has higher flexibility which enables DM to make rational decisions based on changes in the external environment and handle the IGUL information effectively. In future, we will study how to construct the novel TODIM· SIR method based on the Euclidean distance [42], which is another simple and effective distance measure method and widely used in the field of multiple attribute decision making.

Footnotes

Acknowledgments

The paper is partly supported by Fundamental Research Funds for the Central Universities (No. JBX170615, XJS16080), Natural Science Basic Research Plan in Shaanxi Province of China (No. 2017JQ7006).

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