Multi-attribute decision-making method based on a novel distance measure of linguistic intuitionistic fuzzy sets

Abstract

Linguistic intuitionistic fuzzy sets can qualitatively rather than quantitatively express data in the form of membership degree. But quantitative tools are required to handle qualitative information. Therefore, an improved linguistic scale function, which can more accurately manifest the subjective feelings of decision-makers, is employed to deal with linguistic intuitionistic information. Subsequently, due to some commonly used distance measures do not comprehensively evaluate the information of linguistic intuitionistic fuzzy sets, an improved distance measure of linguistic intuitionistic fuzzy sets is designed. It considers the cross-evaluation information to get more realistic reasoning results. In addition, a new similarity measure defined by nonlinear Gaussian diffusion model is proposed, which can provide different response scales for different information between various schemes. The properties of these measures are also studied in detail. On this basis, a method in linguistic intuitionistic fuzzy environment is developed to handle multi-attribute decision-making problems. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed method and the influence of the parameters is analyzed.

Keywords

Linguistic intuitionistic fuzzy set linguistic scale function distance measure similarity measure multi-attribute decision-making

1 Introduction

Multi-attribute decision-making (MADM) is a branch of decision-making which is considered as a human activity based on cognition. Before choosing the best scheme, a set of alternatives is assessed against multiple influential attributes. The problems of MADM are studied in various fields, and many practical problems are solved by many useful methods, such as artificial intelligence [1], web data [2], granular computing [3] and so on [4 –6].

Since the environment in reality is complicated, imprecise and vague, it is difficult for the decision-makers (DMs) to provide the correct decision in practical applications [7, 8]. Therefore, to handle the uncertainty in the data, Zadeh [9] proposed fuzzy sets (FSs), which is a powerful tool to address the vagueness [10]. After that, its extension has been studied, for example, intuitionistic fuzzy sets (IFSs) [11] has become more powerful tools to describe the MADM problems [12, 13]. Since then, the research on IFSs and its extension theory have been increased substantially. In 2018, various types of centroid transformations of intuitionistic fuzzy values are introduced and explored based on aggregation operators [14]. In 2019, a graphical ranking method of IFSs based on the uncertainty index and entropy is proposed [15]. In 2020, a new extension of the preference ranking organization method for enrichment evaluation is proposed by taking advantage of IFSs [16]. But the above fuzzy theory cannot handle qualitative information. Moreover, in practical problems, many qualitative attribute values cannot be numerically represented. For example, to measure one’s quality, DMs are prone to express it in terms of “excellent”, “good”, “bad”, etc. rather than in numerical terms [17]. Therefore, in the many complex and fuzzy decision-making problems, the expression of DMs’ opinions on decision objects is based on linguistic variables [18], which can enhance the flexibility and reliability of the classical decision-making model [19], and has been widely employed in various fields [20]. Based on some preliminary linguistic models, some extended linguistic concepts have been improved, such as hesitant fuzzy linguistic sets (HFLSs) [21], linguistic hesitant fuzzy sets (LHFSs) [22], etc. After that, combining linguistic models with IFS and considering linguistic membership and non-membership, the concept of linguistic intuitionistic fuzzy sets (LIFSs) is proposed [23]. The LIFSs can easily express the qualitative as well as the quantitative aspects.

When tackling decision-making problems with linguistic information, the DMs need to choose an effective calculation model to process linguistic information [24]. The 2-tuple linguistic computing model is one of the most commonly used computing models [25]. It not only improves the fuzzy linguistic method, but also increases the accuracy of linguistic computations [26]. Subsequently, a linguistic scale function (LSF) which can accurately manifest the DMs’ subjective feelings is proposed [27] based on the prospect theory and the numerical scale models of linguistic term sets (LTSs) [28, 29]. However, the loss-aversion attitude of DMs is not considered by the LSF in [27], which results in its inability to accurately reflect the DMs’ subjective feelings. To handle this problem, an improved LSF is proposed [30]. But the semantics corresponding to the linguistic terms in [30] cannot accurately reflect the DMs’ subjective cognition.

Distance measures and similarity measures are important tools for measuring the closeness of two patterns. They have influence on numerous research topics such as classification and decision-making. These measures could increase classification or recognition rate and refine the choice of appropriate solution for decision-making problems. In particular, when dealing with MADM problems, distance or similarity measures are usually used to compare all alternatives with positive and negative solutions to obtain the ranking of alternatives, such as the distance measures and similarity measures of IFSs in [31 –34]. Some basic properties of distance measure of LIFSs are introduced in [17]. And some distance measures of LIFSs are proposed [17 , 36] to tackle MADM problems. However, the distance measures of LIFSs as mentioned above have not thoroughly evaluated the linguistic intuitionistic fuzzy (LIF) information because the cross-evaluation between membership degrees and non-membership degrees are not considered. Hence, these measures are not really effective in the complex decision-making problems. For example, medical diagnosis depends not only on the current symptoms, but also on the medical history of a patient. In this case, if the distance measure uses the cross-evaluation, it is easy to evaluate the importance degrees between the membership and the non-membership degrees of a patient at present. It also can measure those degrees of a patient in the past time as well as the cross-time between the past and present. Therefore, in practical application, it will bring more information and accuracy of diagnosis for patients to evaluate LIF medical information fully through cross-evaluation. In addition, the most existing similarity measures are usually base on linear functions. That will bring difficulties to DMs. For example, in some practical problems, especially when the similarity degree of alternatives is high, the similarity measure based on linear function may not well distinguish these alternatives. So, it is necessary to construct a more flexible similarity measure.

To deal with the above drawbacks, in this paper, an improved LSF, which can more accurately express the DMs’ subjective feelings, is introduced based on the prospect theory. Then an improved distance measure of LIFSs is proposed, where the cross-evaluation is considered. Further, we study a similarity measure defined by nonlinear Gaussian diffusion model. It can provide different response scales for different information between various schemes. Finally, a method to address the decision-making problems in LIF environment is presented. The novelty of this paper in comparison with the relevant researches is highlighted as follows:

(1) To flexibly express semantics, an improved LSF which can more accurately reflect the DMs’ subjective feelings than [27, 30] is introduced based on the previous LSFs.

(2) By comparing with the common distance measures which have not thoroughly evaluated the information of LIFSs, a new distance measure of LIFSs which considers the cross-evaluation is designed to get more realistic reasoning results.

(3) Compared to the existing similarity measures based on linear functions [31, 32], a similarity measure of LIFSs which can more effectively reflect the response ability of different scales for the difference information between alternatives is proposed based on nonlinear Gaussian diffusion model [37].

(4) On these basis, a method for solving MADM problems is discussed, and the effectiveness of the method is illustrated by an example. Finally, the calculation results are compared with the existing methods to verify its effectiveness.

The rest of this paper is arranged as follows: Some basic concepts are reviewed in Section 2. In Section 3, a novel LSF is presented based on the prospect theory and LIFSs. After that, the improved distance measure and similarity measure of LIFSs is further defined in Section 4. Then, we give a method to cope with MADM problems under LIF environment in Section 5. For the sake of checking the validity of the proposed method, the comparative analysis and parameter analysis are carried out after a practical example in Section 6. Finally, the conclusion is given in Section 7.

2 Preliminaries

This section reviews and discusses some relevant concepts about IFSs, LTSs and LIFSs.

Definition 1 [11] Suppose that X = {x₁, x₂, ⋯, x_n} is a fixed set. An IFS A on X can be defined as: $A = {(x_{i}, μ_{A} (x_{i}), ν_{A} (x_{i})) | x_{i} \in X},$ where the function μ_A (x_i) ∈ [0, 1] and ν_A (x_i) ∈ [0, 1] represent the membership degree and non-membership degree, respectively. π_A (x_i) ∈ [0, 1] satisfying π_A (x_i) +μ_A (x_i) + ν_A (x_i) =1 means hesitation degree.

Definition 2. [38] Let S = {s_i|i = 0, 1, ⋯ , 2t} be a fixed LTS with odd cardinality, where s_i indicates a possible value for a linguistic variable and t is a nonnegative integer. The LTS S should satisfy the following conditions:

(1) The elements in S are ordered: s_i > s_j if and only if i > j;

(2) There exists a negation operator: neg (s_i) = s_2t-i.

Then the discrete term set S is extended to a continuous LTS $\tilde{S} = {s_{α} | s_{0} \leq s_{α} \leq s_{t}, α \in [0, l]}$ , in which s_α > s_β (α > β), and l is a sufficiently large positive integer [39]. The elements in $\tilde{S}$ fit all the features described above. s_α is called an original linguistic term if s_α ∈ S, and s_α is called a virtual linguistic term otherwise. A virtual linguistic term has no practical meaning but sorting alternatives.

Definition 3. [23] Let LIFS A in X = {x₁, x₂, ⋯ , x_m} be defined and $\tilde{S} = {s_{i} | 0 \leq i \leq 2 t}$ be a continuous LTS, where 2t is the maximal subscript of the linguistic term s_i in $\tilde{S}$ . $A = {〈 x_{i}, s_{u_{A}} (x_{i}), s_{v_{A}} (x_{i}) 〉 | x_{i} \in X},$ where $s_{u_{A}} (x_{i}), s_{v_{A}} (x_{i}) \in \tilde{S}$ indicate the linguistic membership and non-membership degrees of x_i to A, respectively. For any x_i ∈ X, 0 ≤ u_A (x_i) + v_A (x_i) ≤2t is always satisfied. The linguistic intuitionistic index of x_i to A is defined as s_{π
_A} (x_i) = s_{2t-u_A-v_A} (x_i). The pair (s_{u
_A} (x_i) , s_{v
_A} (x_i)) is called a linguistic intuitionistic fuzzy number (LIFN), and we simplified it as γ = (s_u, s_v). Moreover, if s_u, s_v ∈ S, then γ is an original LIFN; otherwise, γ is a virtual LIFN.

For example, a manufacturing company desires to search the best global supplier for one of its most critical parts used in assembling process. Alternative A₁ means the first supplier while attribute G₁ means quality of the product. Suppose that attribute G₁ of alternative A₁ is evaluated as γ = (s_u, s_v) = (s₄, s₂) according to the linguistic term set: S = {s₀ = extremelypoor, s₁ = verypoor, s₂ = poor, s₃ = fair, s₄ = good, s₅ = verygood, s₆ = extremelygood}. The linguistic membership s₄ indicates that the degree of alternative A₁ satisfies attribute G₁ is good, while linguistic non-membership s₂ indicates that the degree of alternative A₁ does not satisfy attribute G₁ is poor.

3 Linguistic scale functions

The LSFs make the qualitative and quantitative linguistic evaluation information both available for the DMs to make decision, which can provide more definite decision results based on semantics [27].

Definition 4. [27, 30] Let S = {s_i|i = 0, 1, 2, ⋯ , 2t} be a linguistic term set with odd cardinality, η_i ∈ R (i = 0, 1, ⋯ , 2t) be the linguistic preference values corresponding to the semantics of linguistic terms. Then the LSF f that performs the mapping from s_i to η_i is defined as $\begin{matrix} f : s_{i} ⟶ η_{i} (s_{i} \in S, η_{i} \in R, i = 0, 1, \dots, 2 t), \end{matrix}$ Clearly, the symbol η_i reflects the preference of DMs when they choose the linguistic term s_i ∈ S. Then the function f illustrates the semantics of s_i.

It has been argued that using only the subscripts of LTSs to cope with decision-making problems may result in information distortion[40]. To overcome this problem, on the basis of Definition 4, three common LSFs are constructed [27].

First, the linguistic information evaluation scale, which is simple and universal, but lacks reasonable theoretical basis is given as following [41]. $\begin{matrix} f_{1} (s_{θ}) = \frac{θ}{2 t}, θ = 0, 1, \dots, 2 t . \end{matrix}$ Decision criteria influence both positive and negative aspects on DMs’ mental stimulation. Therefore, a composite LSF is proposed: $f_{2} (s_{θ}) = {\begin{matrix} \frac{T^{t} - T^{t - θ}}{2 T^{t} - 2}, θ = 0, 1, \dots, t; \\ \frac{T^{t} + T^{θ - t} - 2}{2 T^{t} - 2}, θ = t + 1, \dots, 2 t . \end{matrix}$ In this case, the absolute deviation between adjacent linguistic subscripts θ increases with the expansion from the middle to both ends of LTSs. The parameter T can be used to adjust the absolute deviation between any two adjacent linguistic subscripts, which can be obtained by subjective methods [40].

The following LSF inspired by prospect theory is defined, as well as the DMs’ different sensitivities with respect to the absolute deviation between adjacent linguistic subscripts: $f_{3} (s_{θ}) = {\begin{matrix} \frac{t^{a} - (t - θ)^{a}}{2 t^{a}}, θ = 0, 1, \dots, t; \\ \frac{t^{b} + (θ - t)^{b}}{2 t^{b}}, θ = t + 1, \dots, 2 t; \end{matrix}$ where 0 < a, b ≤ 1. In this case, the absolute deviation between adjacent linguistic subscripts θ decreases when extending from the middle to both ends of the LTSs. If a=b=1, then $f_{3} (s_{θ}) = \frac{θ}{2 t} = f_{1} (s_{θ})$ . Several experiments have been done to illustrate that a = b = 0.8 is a fine default choice [42].

To facilitate the calculation without losing of information, the above function is expanded as $f_{i}^{*} : \tilde{S} \to R^{+}$ (R⁺ = {r|r ≥ 0, r ∈ R}) where i = 1, 2, 3, R is the set of all real number. The function $f_{i}^{*} (s_{θ}) = η_{θ}$ is a continuous and strictly monotonically increasing function. Thus, the inverse function of $f_{i}^{*}$ exists and is represented by ${f_{i}^{*}}^{- 1}$ .

In most practical decision process, people tend to make decision according to their risk preference attitudes as well as the way of thinking [30]. However, the LSFs listed above cannot accurately reflect the DMs’ subjective feelings, which include DMs’ risk preference attitude corresponding to the gains, losses and loss-aversion. To overcome this shortcoming, a LSF is proposed in [30].

Definition 5. [30] Let S = {s_θ|θ = 0, 1, ⋯ , 2t} be a LTS with odd cardinality, U (s_θ) ∈ R be the linguistic preference values corresponding to the semantics of linguistic terms. Thus, a function is constructed: $U (s_{θ}) = {\begin{matrix} t - λ (t - θ)^{a}, θ = 0, 1, \dots, t; \\ t + (θ - t)^{b}, θ = t + 1, \dots, 2 t; \end{matrix}$ where parameters a and b (0 ≤ a, b < 1) are the DMs’ risk preference attitude coefficients corresponding to the gains and losses, respectively. Parameter λ (λ > 1) is the loss-aversion coefficient which reflects the DMs’ loss-aversion attitudes by making the loss part of the S-shape curve steeper than the gain part [42, 43]. Similarly, the function U can be expanded to $U^{*} : \tilde{S} \to R$ which is continuous and strictly monotonically increasing.

Remark 1. When s_θ = s_t+1, the linguistic preference value U^* (s_t+1) = t + 1 is a fixed number for any b ∈ [0, 1), which makes it unable to accurately reflect the DMs’ subjective feelings.

Inspired by [30], a LSF which can more accurately reflect the DMs’ subjective feelings is proposed. This function can assign different linguistic preference values to the corresponding linguistic terms.

Definition 6. Assume S = {s_θ|θ = 0, 1, ⋯ , 2t} is a LTS with odd cardinality, V (s_θ) ∈ R be the linguistic preference values corresponding to the semantics of linguistic terms. Therefore, a new LSF can be showed as: $V (s_{θ}) = {\begin{matrix} \frac{t - λ (t^{\frac{1}{a}} - θ^{\frac{1}{a}})^{a}}{2 t}, θ = 0, 1, \dots, t; \\ \frac{t + (t^{\frac{1}{b}} - (2 t - θ)^{\frac{1}{b}})^{b}}{2 t}, θ = t + 1, \dots, 2 t; \end{matrix}$ (1) where parameters a and b (0 < a, b ≤ 1) can reflect the DMs’ risk preference attitude coefficients corresponding to the gains and losses, respectively. Parameter λ (λ > 1) is the loss-aversion coefficient which reflects the DMs’ loss-aversion attitudes and make the S-shape curve for the losses steeper than the gains [42, 43]. If a = b = 1, λ = 1, the $V (s_{θ}) = \frac{θ}{2 t}$ , which is the same as f₁ (s_θ). Similarity, expanding the function V into $V^{*} : \tilde{S} \to R$ , which satisfies V^* (s_θ) = η_θ and is a strictly monotonically increasing and continuous function. So, the inverse function of V^* exists and can be expressed by V^*^-1.

In addition, by Eq. (1), different values of risk preference parameters can be used to transform the semantics into their corresponding linguistic preference values and also can reflect different subjective feelings of DMs. For example, for a set of seven LTS S = {s₀ = none, s₁ = verylow, s₂ = low, s₃ = medium, s₄ = high, s₅ = veryhigh, s₆ = perfect}, by Eq. (1), the obtained linguistic preference values are shown as Fig. 1.

Fig. 1

A set of seven linguistic terms with their linguistic preference values.

From Fig. 1, it is noted that V is a strictly monotonically increasing and continuous function, where the absolute deviation of linguistic preference values between two adjacent subjective feelings gradually decreases. we can also see that the linguistic preference value $V^{*} (s_{t + 1}) = \frac{t + (t^{\frac{1}{b}} - (t - 1)^{\frac{1}{b}})^{b}}{2 t}$ is a variable with parameter b ∈ (0, 1], which can more accurately reflect the DMs’ subjective feelings than U^* (s_t+1).

It is noted that $f_{3}^{*}$ , U^* and V^* are strictly monotonically increasing and continuous LSFs where the absolute deviation of linguistic preference values between two adjacent subjective feelings gradually decreases. In order to more intuitively understand the difference between the above LSFs, we assume that λ = 1 and modify U appropriately into $\tilde{U}$ , $\tilde{U} (s_{θ}) = {\begin{matrix} \frac{t - (t - θ)^{a}}{2 t}, θ = 0, 1, \dots, t; \\ \frac{t + (θ - t)^{b}}{2 t}, θ = t + 1, \dots, 2 t . \end{matrix}$ Then we obtain a strictly monotonically increasing and continuous function ${\tilde{U}}^{*}$ , and draw the S-curved functions of $f_{3}^{*}$ , ${\tilde{U}}^{*}$ and V^*, as shown in Fig. 2.

Fig. 2

The curves of $f_{3}^{*}$ , V^* and ${\tilde{U}}^{*}$ with a = b = 0.8.

From Fig. 2, we can see that for LSFs $f_{3}^{*}$ , ${\tilde{U}}^{*}$ and V^*, when the values of parameters are the same, the change of V^* is more obvious than others. Therefore, V^* is more sensitive than $f_{3}^{*}$ , U^* and can better describe the subjective feelings of DMs.

Remark 2. The relationship between the sensitivity of the LSFs and the DMs’ subjective feelings is that, the more sensitive the LSF is, the more accurate the DMs’ subjective feeling is. The reason is demonstrated in the following. If a LSF is more sensitive, it will change with θ more obviously when the parameters are fixed. Then, the DMs’ subjective feeling reflected by the linguistic preference value is more reliable, which means the LSF is more distinguishable and can better describe the DMs’ subjective feeling.

4 Some novel information measures of LIFSs

In this section, a novel distance and similarity measures of LIFSs are constructed.

4.1 A novel distance measure of LIFSs

Inspired by [17, 33] and [34], some common distance measures between LIFSs can be listed as follows:

The Hamming distance measure between LIFSs A and B: $\begin{matrix} d_{Ham} (A, B) = & \frac{1}{2 m} \sum_{i = 1}^{m} (| f^{*} (s_{u_{A}} (x_{i})) - f^{*} (s_{u_{B}} (x_{i})) | \\ + | f^{*} (s_{v_{A}} (x_{i})) - f^{*} (s_{v_{B}} (x_{i})) | \\ + | f^{*} (s_{π_{A}} (x_{i})) - f^{*} (s_{π_{B}} (x_{i})) |) . \end{matrix}$ (2) The Euclidean distance measure between LIFSs A and B:

$\begin{matrix} d_{Euc} (A, B) = & (\frac{1}{2 m} \sum_{i = 1}^{m} ((f^{*} (s_{u_{A}} (x_{i})) - f^{*} (s_{u_{B}} (x_{i})))^{2} \\ + (f^{*} (s_{v_{A}} (x_{i})) - f^{*} (s_{v_{B}} (x_{i})))^{2} \\ {+ (f^{*} (s_{π_{A}} (x_{i})) - f^{*} (s_{π_{B}} (x_{i})))^{2})}^{\frac{1}{2}} . \end{matrix}$ (3)

The Hausdorff distance measure between LIFSs A and B: $\begin{matrix} d_{Hau} (A, B) = \frac{1}{m} \sum_{i = 1}^{m} max {| f^{*} (s_{u_{A}} (x_{i})) \\ - f^{*} (s_{u_{B}} (x_{i})) |, | f^{*} (s_{v_{A}} (x_{i})) - f^{*} (s_{v_{B}} (x_{i})) |}, \end{matrix}$ (4) where the f^* is a LSF that can be selected from Section 3. Moreover, the above distance measures Eq. (2)-(4) satisfies the three axioms of distance measure, but they have a shortcoming as demonstrated in the following.

Example 1. Let S = {s₀ = verypoor, s₁ = poor, s₂ = slightlypoor, s₃ = fair, s₄ = slightlygood, s₅ = good, s₆ = verygood} be a LTS for evaluation. Further, let A, B, C be three LIFSs on X = {x₁, x₂, ⋯ , x_m}, where A = {〈x_i, s₆, s₀, s₀〉}, B = {〈x_i, s₀, s₆, s₀〉}, C = {〈x_i, s₀, s₀, s₆〉}. Clearly, the difference between A and C is less than that between A and B. Nevertheless, as shown in Table 1, the values of d_Ham, d_Euc and d_Hau between A and C and that between A and B are equal, which reflects these distance measures can not well fit the reality.

Table 1

The values of distance measures d_Ham, d_Euc and d_Hau

	d _Ham	d _Euc	d _Hau
A, B	f^* (s₆) - f^* (s₀)	f^* (s₆) - f^* (s₀)	f^* (s₆) - f^* (s₀)
A, C	f^* (s₆) - f^* (s₀)	f^* (s₆) - f^* (s₀)	f^* (s₆) - f^* (s₀)
B, C	f^* (s₆) - f^* (s₀)	f^* (s₆) - f^* (s₀)	f^* (s₆) - f^* (s₀)

In view of the shortcoming of the above distance measures, a new distance measure between LIFSs is proposed.

Definition 7. Let $\tilde{S} = {s_{i} | 0 \leq i \leq 2 t}$ be a continuous LTS where 2t is the maximal subscript of the linguistic term s_i in $\tilde{S}$ . Suppose A and B are two LIFSs on the universe X = {x_i|i = 1, 2, ⋯ , m} which are defined by s_{u
_A}, s_{v
_A} and s_{u
_B}, s_{v
_B}, where s_{u
_A}, $s_{v_{A}} \in \tilde{S}$ and s_{u
_B}, $s_{v_{B}} \in \tilde{S}$ are their linguistic membership and linguistic non-membership degrees respectively. Then, a novel distance measure between LIFSs A and B is defined as follows. $\begin{matrix} d_{LIF} (A, B) = & \frac{1}{3 m} \sum_{i = 1}^{m} (| f^{*} (s_{u_{A}} (x_{i})) - f^{*} (s_{u_{B}} (x_{i})) | \\ + | f^{*} (s_{v_{A}} (x_{i})) - f^{*} (s_{v_{B}} (x_{i})) | \\ + | max {f^{*} (s_{u_{A}} (x_{i})), f^{*} (s_{v_{B}} (x_{i}))} \\ - max {f^{*} (s_{u_{B}} (x_{i})), f^{*} (s_{v_{A}} (x_{i}))} |), \end{matrix}$ (5) where | max {f^* (s_{u
_A} (x_i)) , f^* (s_{v
_B} (x_i))} - max {f^* (s_{u
_B} (x_i)) , f^* (s_{v
_A} (x_i))} | is called the cross-evaluation between A and B. And f^* is a LSF that can be selected from Section 3.

Theorem 1. Let $\tilde{S} = {s_{i} | 0 \leq i \leq 2 t}$ be a continuous LTS where 2t is the maximal subscript of the linguistic term s_i in $\tilde{S}$ and A, B, C be three LIFSs on universe X = {x₁, x₂, …, x_m}. The distance measure d_LIF : LIFS × LIFS → R⁺ satisfies the following axioms:

(i) 0 ≤ d_LIF (A, B) ≤1;

(ii) d_LIF (A, B) =0 if and only if A = B;

(iii) d_LIF (A, B) = d_LIF (B, A);

(iv) If A ⊆ B ⊆ C, then d_LIF (A, C) ≥ d_LIF (A, B) and d_LIF (A, C) ≥ d_LIF (B, C).

Proof. Let A, B, C be three LIFSs on the universe X = {x_i|i = 1, 2, ⋯ , m}. It is easy to prove the proposed distance measure satisfies the first three axioms. Then, the axiom (iv) is proved as follows: $\begin{matrix} d_{LIF} (A, B) = & \frac{1}{3 m} \sum_{i = 1}^{m} (| f^{*} (s_{u_{A}} (x_{i})) - f^{*} (s_{u_{B}} (x_{i})) | \\ + | f^{*} (s_{v_{A}} (x_{i})) - f^{*} (s_{v_{B}} (x_{i})) | \\ + | max {f^{*} (s_{u_{A}} (x_{i})), f^{*} (s_{v_{B}} (x_{i}))} \\ - max {f^{*} (s_{u_{B}} (x_{i})), f^{*} (s_{v_{A}} (x_{i}))} |); \\ d_{LIF} (A, C) = & \frac{1}{3 m} \sum_{i = 1}^{m} (| f^{*} (s_{u_{A}} (x_{i})) - f^{*} (s_{u_{C}} (x_{i})) | \\ + | f^{*} (s_{v_{A}} (x_{i})) - f^{*} (s_{v_{C}} (x_{i})) | \\ + | max {f^{*} (s_{u_{A}} (x_{i})), f^{*} (s_{v_{C}} (x_{i}))} \\ - max {f^{*} (s_{u_{C}} (x_{i})), f^{*} (s_{v_{A}} (x_{i}))} |) . \end{matrix}$ Since A ⊆ B ⊆ C, we can get that $\begin{matrix} s_{u_{A}} (x_{i}) \leq s_{u_{B}} (x_{i}) \leq s_{u_{C}} (x_{i}), \\ s_{v_{A}} (x_{i}) \geq s_{v_{B}} (x_{i}) \geq s_{v_{C}} (x_{i}), \forall x_{i} \in X . \end{matrix}$ Therefore, $\begin{matrix} | s_{u_{A}} (x_{i}) - s_{u_{B}} (x_{i}) | \leq | s_{u_{A}} (x_{i}) - s_{u_{C}} (x_{i}) |; \\ | s_{v_{A}} (x_{i}) - s_{v_{B}} (x_{i}) | \leq | s_{v_{A}} (x_{i}) - s_{v_{C}} (x_{i}) | . \end{matrix}$ Since the LSF f^* is a strictly monotone increasing function, we have $\begin{matrix} | f^{*} (s_{u_{A}} (x_{i})) - & f^{*} (s_{u_{B}} (x_{i})) | \\ \leq | f^{*} (s_{u_{A}} (x_{i})) - f^{*} (s_{u_{C}} (x_{i})) |; \\ | f^{*} (s_{v_{A}} (x_{i})) - & f^{*} (s_{v_{B}} (x_{i})) | \\ \leq | f^{*} (s_{v_{A}} (x_{i})) - f^{*} (s_{v_{C}} (x_{i})) | . \end{matrix}$ It is easy to obtain that $\begin{matrix} max {f^{*} (s_{u_{C}} (x_{i})), f^{*} (s_{v_{A}} (x_{i}))} \\ \geq max {f^{*} (s_{u_{B}} (x_{i})), f^{*} (s_{v_{A}} (x_{i}))} \\ \geq max {f^{*} (s_{u_{A}} (x_{i})), f^{*} (s_{v_{B}} (x_{i}))} \\ \geq max {f^{*} (s_{u_{A}} (x_{i})), f^{*} (s_{v_{C}} (x_{i}))} . \end{matrix}$ Therefore, we have $\begin{matrix} | max {f^{*} (s_{u_{A}} (x_{i})), f^{*} (s_{v_{B}} (x_{i}))} \\ - max {f^{*} (s_{u_{B}} (x_{i})), f^{*} (s_{v_{A}} (x_{i}))} | \\ \leq | max {f^{*} (s_{u_{A}} (x_{i})), f^{*} (s_{v_{C}} (x_{i}))} \\ - max {f^{*} (s_{u_{C}} (x_{i})), f^{*} (s_{v_{A}} (x_{i}))} | . \end{matrix}$ Hence d_LIF (A, B) ≤ d_LIF (A, C). By the same method, we can get d_LIF (B, C) ≤ d_LIF (A, C). The proof is completed.

Remark 3. Applying the constructed distance measure d_LIF to Example 1. It is easy to calculate that d_LIF (A, B) = f^* (s₆) - f^* (s₀) , d_LIF (A, C) = d_LIF (B, C) $= \frac{2}{3} (f^{*} (s_{6}) - f^{*} (s_{0}))$ . Therefore, we can see that d_LIF (A, B) is greater than d_LIF (A, C) or d_LIF (B, C). According to the above, the proposed distance measure d_LIF can overcome the shortcomings of the average typical distance measures, namely d_Ham, d_Euc and d_Hau in Eq. (2-4). We have both the difference between LIFSs A and C, and the difference between LIFSs B and C are less than that between LIFSs A and B.

After defining a distance measure d_LIF, the corresponding similarity measure of LIFSs will be discussed.

4.2 A similarity measure of LIFSs based on the proposed distance measure

Inspired by the distance measure d_LIF, a similarity measure of LIFSs is proposed after we give the definition of similarity measure.

Theorem 2. Let $\tilde{S} = {s_{i} | 0 \leq i \leq 2 t}$ be a continuous LTS, 2t is the maximal subscript of the linguistic term s_i in $\tilde{S}$ and A, B, C be three LIFSs on universe X = {x₁, x₂, …, x_m}. A mapping s_LIF : LIFS × LIFS → R⁺ is a similarity measure of LIFSs if it fulfils the following axioms:

(i) 0 ≤ s_LIF (A, B) ≤1;

(ii) s_LIF (A, B) =1 if and only if A = B;

(iii) s_LIF (A, B) = s_LIF (B, A);

(iv) If A ⊆ B ⊆ C, then s_LIF (A, C) ≤ s_LIF (A, B) and s_LIF (A, C) ≤ s_LIF (B, C).

Based on Theorem 1 and 2, it is easy to find that the distance measure and similarity measure can be regarded as two opposites of a problem. The similarity measures are basically based on Hamming distance, Euclidean distance measures and some others improved forms [33, 34], which can be generally used s = 1 - d (d and s represent distance measure and similarity measure, respectively). Based on the relationship between d and s, a more general expression of similarity measure can be given as follows: $s = F (d),$ where F (·) is an arbitrary strictly monotonically decreasing function.

The existing similarity measures usually base on linear functions [31, 32]. It will bring difficulties to DMs, especially when the similarity degree of alternatives is high, the similarity measure based on linear function may not have a good ability to distinguish alternatives. In this case, if the similarity measure can provide different scales of response capabilities for the different information between different alternatives, the difficulties faced by DMs will be greatly reduced. For example, for two alternatives with large difference, the similarity measure can be more easily identified, for two alternatives with small difference information, the similarity measure can appropriately enlarge the difference information and provide more distinct results. So, it is necessary to construct a different and more flexible similarity measure.

Based on Gaussian function, the diffusion process of pollutants can be reduced very well. Gaussian diffusion model is the most widely used diffusion model of pollutants [37]. The diffusion principle of the model is that the closer to the diffusion source, the higher the pollutant concentration, the faster the pollutant diffusion speed; similarly, the farther away from the pollution source, the lower the pollutant concentration, the slower the pollutant diffusion speed. The form of Gaussian diffusion model is as follows: $\begin{matrix} G (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}}, \end{matrix}$ (6) where parameter $\frac{1}{σ \sqrt{2 π}}$ determines the height of the diffusion source, μ is expectation which represents the initial position of the pollution source, σ is the variance information, x - μ represents the distance information from the pollution source and G (x) reflects the diffusion information from the pollution source to other points in the space. The principle of Gaussian diffusion model is similar to the relationship between similarity measure and distance measure. Inspired by the above, a similarity measure of LIFSs can be created as follow.

Definition 8. Let $\tilde{S} = {s_{i} | 0 \leq i \leq 2 t}$ be a continuous LTS, 2t is the maximal subscript of the linguistic term s_i in $\tilde{S}$ and A = {〈x_i, s_{u
_A} (x_i) , s_{v
_A} (x_i) 〉|x_i∈ X} , B = { 〈x_i, s_{u
_B} (x_i) , s_{v
_B} (x_i) 〉|x_i ∈ X } are two LIFSs on universe X = {x₁, x₂, …, x_m} where s_{u
_A} (x_i), s_{v
_A} (x_i), s_{u
_B} (x_i), $s_{v_{B}} (x_{i}) \in \tilde{S}$ . Then, a similarity measure between LIFSs A and B can be created as follow: $\begin{matrix} s_{LIF} (A, B) = c_{0} (ϕ) + c_{1} (ϕ) e^{- ϕ d_{LIF}^{2} (A, B)}, \end{matrix}$ (7) where $c_{1} (ϕ) = \frac{1}{1 - e^{- ϕ}}$ , c₀ (ϕ) =1 - c₁ (ϕ) and ϕ ≠ 0 is an adjustment parameter, which can provide different scale response capabilities according to different situations.

Property. The similarity measure (7) for LIFSs satisfies four axioms (i)-(iv) in Theorem 2.

Proof. Suppose A, B, C are three LIFSs on the universe X = {x₁, x₂, …, x_m}.

(i) We get that s_LIF (A, $B) = \frac{e^{- ϕ d_{LIF} (A, B)^{2}} - e^{- ϕ}}{1 - e^{- ϕ}}$ . When ϕ > 0, we can get $e^{- ϕ} \leq e^{- ϕ d_{LIF}^{2} (A, B)} \leq 1$ and 1 - e^-ϕ > 0, hence we have 0 ≤ s_LIF (A, B) ≤1; when ϕ < 0, we have 1 ≤ e^{-ϕd_LIF(A,B)²} ≤e^-ϕ and 1 - e^-ϕ < 0, we can get 0 ≤ s_LIF (A, B) ≤1.

(ii) It is easy to get that s_LIF (A, B) =1 if and only if d_LIF (A, B) =0. In addition, d_LIF (A, B) =0 if and only if A = B. Therefore, we have s_LIF (A, B) =1 if and only if A = B.

(iii) Based on that d_LIF (A, B) = d_LIF (B, A) is valid, s_LIF (A, B) = s_LIF (B, A) can be obtained.

(iv) If A ⊆ B ⊆ C, we have d_LIF (A, C) ≥ d_LIF (A, B) and d_LIF (A, C) ≥ d_LIF (B, C). In addition, s_LIF decreases as d_LIF increases. Therefore, we have s_LIF (A, C) ≤s_LIF (A, B) and s_LIF (A, C) ≤ s_LIF (B, C). The proof is completed.

In order to observe the change characteristics of similarity measure s_LIF (A, B) in Eq. (7) more intuitively, some similarity measure curves are drawn in Figs. 3-4, which are drawn with different values of parameter ϕ.

Fig. 3

s_LIF (A, B) changes with d_LIF (A, B) when ϕ > 0.

Fig. 4

s_LIF (A, B) changes with d_LIF (A, B) when ϕ < 0.

From Figs. 3-4, it can be seen that the similarity measure s_LIF (A, B) can show the response ability of different scales by adjusting parameter ϕ according to different difference information between alternatives. For example, in Fig. 3, for two alternatives with small difference information, we can appropriately adjust parameter ϕ to a small value, so that the similarity measure s_LIF (A, B) can appropriately enlarge the small difference information and get a more distinct result. The Fig. 4 shows that for two alternatives with large difference information, properly adjusting parameter ϕ to a large value can make the discrimination ability of the similarity measure s_LIF (A, B) better.

Remark 4. In this paper, the similarity measure s_LIF (A, B) based on Gaussian diffusion model Eq. (6) is constructed, which can show the response ability of different scales by adjusting the parameter ϕ according to the difference information between alternatives in actual situations. However, the similarity measure based on the linear function F (·) does not have this characteristic [31, 32].

For the sake of concrete explanation, it can display the response ability of different scales according to the difference information between alternatives. We give the following example:

Example 2. Suppose that the LTS S = {s₀ = extremelypoor, s₁ = verypoor, s₂ = poor, s₃ = slightly poor, s₄ = fair, s₅ = slightlygood, s₆ = good, s₇ = verygood, s₈ = extremelygood} is established for evaluation. The are four tour routes denoted by A_i (i = 1, 2, 3, 4). The attribute g_j (j = 1, 2, 3) stands for “economy”, “culture” and “security”, respectively. The four alternatives A_i is evaluated using LIFNs as listed in Table 2 and we set the parameters (a, b, λ) = (0.88, 0.88, 2.25) [44].

Table 2

The evaluation matrix M_LIF

	g ₁	g ₂	g ₃
A ₁	(s₄, s₂)	(s₄, s₂)	(s₄, s₂)
A ₂	(s₅, s₂)	(s₄, s₂)	(s₄, s₂)
A ₃	(s₅, s₂)	(s₅, s₂)	(s₄, s₂)
A ₄	(s₅, s₂)	(s₅, s₂)	(s₅, s₂)

According to the evaluation matrix M_LIF, by calculation, it is easy to get that d_LIF (A₁, A₂) =0.0361, d_LIF (A₁, A₃) =0.0722, d_LIF (A₁, A₄) =0.1083. Then, we have d_LIF (A₁, A₂) < d_LIF (A₁, A₃) < d_LIF (A₁, A₄). To demonstrate the characteristic of similarity measure s_LIF, the values of similarity measure s_LIF with different values of parameter ϕ and linear similarity measure s = 1 - d_LIF are given in Table 3. From Table 3, we get s_LIF (A₁, A₂) > s_LIF (A₁, A₃) > s_LIF (A₁, A₄). This means that A₁ and A₂ have the largest similarity, and A₁ and A₄ have the smallest similarity. According to the discussion of the difference for A_i, it can find that the conclusions obtained by using the distance measure and similarity measure are the same.

Table 3

The values of s_LIF (A₁, A_i) and s (A₁, A_i) with different values of parameter ϕ

	A ₂	A ₃	A ₄
s_LIF with ϕ = 50	0.9369	0.7704	0.5560
s_LIF with ϕ = 200	0.7704	0.3522	0.0956
s	0.9631	0.9278	0.8917

Note d_LIF (A₁, A₃) - d_LIF (A₁, A₂) = d_LIF (A₁, A₄) - d_LIF (A₁, A₃) =0.0361, which means the difference between d_LIF (A₁, A₃) and d_LIF (A₁, A₂) is equal to that between d_LIF (A₁, A₄) and d_LIF (A₁, A₃). We can get that s (A₁, A₂) =0.9631, s (A₁, A₃) =0.9278, s (A₁, A₄) =0.8917 are three numbers with little differences and close to 1, which makes it hard to distinguish them. In addition, the differences between distances are equal which leads to the differences of linear similarity measure s are equal. But we can see that s_LIF (A₁, A₂) - s_LIF (A₁, A₃) =0.1665 < d_LIF (A₁, A₃) - d_LIF (A₁, A₄) = 0.2144 when ϕ = 50; s_LIF (A₁, A₂) - s_LIF (A₁, A₃) =0.4181 > d_LIF (A₁, A₃) - d_LIF (A₁, A₄) =0.2567 when ϕ = 200. Comparing with the linear similarity measure s, the differences are quite large, and the difference between s_LIF (A₁, A₃) and s_LIF (A₁, A₂) is not equal to the difference between s_LIF (A₁, A₄) and s_LIF (A₁, A₃). Furthermore, the difference between s_LIF (A₁, A₃) and s_LIF (A₁, A₂) is smaller than that between s_LIF (A₁, A₄) and s_LIF (A₁, A₃) when ϕ = 50 while larger when ϕ = 200. The explanation to this phenomenon is that, when ϕ is a large number and d_LIF is small (close to 0), the similarity measure s_LIF is very sensitive to d_LIF. From Table 3 and Fig. 3, we can see that s_LIF is a very steep function when d_LIF ∈ [0, 0.1], which makes it more easy to distinct. For the situation described in Fig. 4, we can make a similar analysis.

5 A method to deal with problems of MADM under LIF circumstance

In most MADM methods based on distance measures, TOPSIS has been successfully applied [45, 46]. TOPSIS theorem uses the distance measure to calculate the closeness between each alternative and the positive or negative ideal alternatives, and then the optimal decision results are given [47]. Therefore, the accuracy and reliability of TOPSIS depend on the selection of positive and negative ideal alternatives and distance measures. The positive and negative ideal alternatives are defined in accordance with the best and worst alternatives of the given alternative set or obtained according to empirical knowledge. Therefore, the main factor determining the accuracy and reliability of TOPSIS is the distance measure. This section introduces a method based on distance measure d_LIF to deal with MADM problem under LIF circumstance.

Suppose that there exists an alternative set A = {A₁, A₂, ⋯ , A_n} which consists of n alternatives. DMs would like to choose the best alternative or rank the alternatives in preference order from the above set A according to the set of attributes G = {g₁, g₂, ⋯ , g_m} and the LTS S = {s_i|i = 0, 1, ⋯ , 2t} is established for evaluation. let M_LIF = (γ_ij) _n×m be a LIF matrix, where γ_ij = (s_{u
_ij}, s_{v
_ij}) is an evaluation value indicated by LIFN, and $Γ^{+} = (γ_{1}^{+}, \dots, γ_{m}^{+})$ and $Γ^{-} = (γ_{1}^{-}, \dots, γ_{m}^{-})$ , where $γ_{j}^{+} = (max_{i} s_{u_{ij}}, min_{i} s_{v_{ij}})$ , $γ_{j}^{-} = (min_{i} s_{u_{ij}}, max_{i} s_{v_{ij}})$ are the positive and negative ideal alternatives, respectively [47]. And a flow diagram (Fig. 5) is given to describe the algorithm intuitively.

Fig. 5

Flow diagram for the MADM method.

Step 1. Normalize the evaluation matrix.

To eliminate the influence of different dimensions on the operation process, all evaluation values should be normalized to the same magnitude grade, which can be expressed as follows [48]: $r_{ij} = {\begin{matrix} r_{ij} = (s_{u_{ij}}, s_{v_{ij}}), g_{j} is a gaining attribute; \\ {\bar{r}}_{ij} = (s_{v_{ij}}, s_{u_{ij}}), g_{j} is a cost attribute . \end{matrix}$

Step 2. By Eq. (5), we can calculate the distance between LIFSs A_i and Γ⁺ and the distance between LIFSs A_i and Γ^-, namely d_LIF (A_i, Γ⁺) and d_LIF (A_i, Γ^-).

Step 3. Using the distance d_LIF (A_i, Γ⁺) and d_LIF (A_i, Γ^-), the closeness χ_i for each alternative A_i can be obtained, namely, $χ_{i} = \frac{d_{LIF} (A_{i}, Γ^{+})}{d_{LIF} (A_{i}, Γ^{-}) + d_{LIF} (A_{i}, Γ^{+})} .$ (8)

Step 4. All alternatives are sorted according to the closeness coefficient χ_i. Since the corresponding A_i is better when χ_i is smaller, the optimal alternative is alternative which has the smallest closeness.

6 Application and simulation experiment

Next, to better illustrate the given method in Section 5, we apply it to a coal mine safety evaluation problem [17] as an example.

Example 3. Safety assessment is of great significance in coal mining. Therefore, a scientific and feasible method of coal mine safety evaluation is needed. Using the same LTS as in Example 2 to evaluate coal mine alternative A_i (i = 1, 2, 3, 4, 5). G = {g₁, g₂, g₃, g₄} is a set of attributes (Supposing all of the attributes are of the benefit type), where g_j (j = 1, 2, 3, 4) stands for “technological equipment”, “geological conditions”, “human diathesis”, “supplier’s profile” and “management quality”, respectively. Then, the LIF decision matrix ${\bar{M}}_{LIF} = (γ_{ij})_{5 \times 4}$ is constructed by DMs, as shown in Table 4.

Table 4
The evaluation matrix ${\bar{M}}_{LIF}$

g ₁ g ₂ g ₃ g ₄

A ₁ (s₅, s₁) (s₆, s₂) (s₇, s₁) (s₅, s₂)

A ₂ (s₆, s₂) (s₆, s₁) (s₅, s₂) (s₅, s₁)

A ₃ (s₇, s₁) (s₅, s₂) (s₄, s₃) (s₆, s₁)

A ₄ (s₆, s₂) (s₆, s₁) (s₅, s₂) (s₅, s₃)

A ₅ (s₆, s₁) (s₄, s₂) (s₆, s₂) (s₅, s₃)

	g ₁	g ₂	g ₃	g ₄
A ₁	(s₅, s₁)	(s₆, s₂)	(s₇, s₁)	(s₅, s₂)
A ₂	(s₆, s₂)	(s₆, s₁)	(s₅, s₂)	(s₅, s₁)
A ₃	(s₇, s₁)	(s₅, s₂)	(s₄, s₃)	(s₆, s₁)
A ₄	(s₆, s₂)	(s₆, s₁)	(s₅, s₂)	(s₅, s₃)
A ₅	(s₆, s₁)	(s₄, s₂)	(s₆, s₂)	(s₅, s₃)

Step 1. Since all attributes are benefit type, the normalization of the evaluations matrix can be omitted.

Step 2. From the information of matrix M_LIF, we can get Γ⁺ = ((s₇, s₁) , (s₆, s₁) , (s₇, s₁) , (s₆, s₁)), Γ^- = ((s₅, s₂) , (s₄, s₂) , (s₄, s₃) , (s₅, s₃)). And based on the proposed distance measure d_LIF from Eq. (5), the values of d_LIF (A_i, Γ⁺) and d_LIF (A_i, Γ^-) (i = 1, ⋯ , 5) are calculated, as shown in Table 5, where we set the parameters (a,b,λ)=(0.88,0.88,2.25), respectively [44].

Table 5

The values of d_LIF (A_i, Γ⁺) and d_LIF (A_i, Γ^-) (i = 1, ⋯ , 5)

	d_LIF (A_i, Γ⁺)	d_LIF (A_i, Γ^-)
A ₁	0.1056	0.2088
A ₂	0.1247	0.1897
A ₃	0.1572	0.1572
A ₄	0.1707	0.1437
A ₅	0.1978	0.1166

Step 3. According to Eq. (8), we can calculate the closeness coefficient of alternative A_i and the closeness coefficient χ_i can be obtained as χ₁ = 0.3359, χ₂ = 0.3967, χ₃ = 0.5000, χ₄ = 0.5429, χ₅ = 0.6291.

Step 4. The ranking result is A₁ > A₂ > A₃ > A₄ > A₅. It can be concluded that the best alternative A₁ is consistent with [17].

Comparative analysis: In order to verify the effectiveness of the proposed method, we apply other existing methods to Example 3 and make a comparative analysis.

From Table 6, it can be seen that A₁ is always the best alternative among the different existing methods mentioned above. Table 6 also illustrates that the ranking result given by the first method in [51] is the same with that given by the proposed method, the ranking result given by [49] is the same with the method in [50], the second method in [51] and the second method in [23]. On the basis of the above simulation results, there are some possible reasons for this difference in rankings.

(1) First, the present methods are defined by simply dealing with the subscript of the linguistic term [23] and the loss-aversion attitude of DMs is not considered [17], which makes it cannot accurately reflect the DMs’ subjective cognition. So, these strategies have some non-negligible shortcomings and leads to the loss and distortion of the original information. However, the proposed method in this paper is constructed based on an improved LSF and distance measure, which can accurately represent the semantics of linguistic terms and obtain the decision results that are consistent with the DMs’ cognition and can effectively negate the drawbacks in the existing methods.

(2) Second, the existing methods and the proposed method have different methods to determine the final ranking for alternatives. The linguistic score function and linguistic accuracy function are used to compare comprehensive evaluation values [23], and the possibility degree and complementary matrix are utilized to obtain final rankings in [49 –51]. In contrast, the proposed method depend on the improved distance measure of LIFSs to compare the comprehensive evaluation values for each alternative, and then the final ranking is determined.

Table 6

The ranking results using different existing methods

Methods	Ranking results
Method in this paper	A₁ > A₂ > A₃ > A₄ > A₅
Method in [17]	A₁ > A₃ > A₄ > A₅ > A₂
Method in [49]	A₁ > A₂ > A₄ > A₅ > A₃
Method in [50]	A₁ > A₂ > A₄ > A₅ > A₃
Method 1 in [51]	A₁ > A₂ > A₃ > A₄ > A₅
Method 2 in [51]	A₁ > A₂ > A₄ > A₅ > A₃
Method 1 in [23]	A₁ > A₃ > A₂ > A₅ > A₄
Method 2 in [23]	A₁ > A₂ > A₄ > A₅ > A₃

6.1 Discussion of the influence of parameters a, b and λ

The ranking result in Table 6 is obtained by the proposed method under the parameters (a, b, λ) = (0.88, 0.88, 2.25) [44]. When the values of parameters a, b and λ in the proposed LSF are different, it is necessary to discuss whether and how the ranking results change. Three sets of parameter values which can well reflect the DMs’ subjective feelings are (a, b, λ) = (0.88, 0.88, 2.25), (a, b, λ) = (0.88, 0.88, 1.25) and (a, b, λ) = (1.68, 0.88, 2.25) [44, 52]. Based on the above discussion, more ranking results are gave with different parameter values in Table 7.

Table 7
The ranking results using different values of parameters a, b and λ

(a, b, λ) Ranking results

(0.88, 0.88, 1.25) A₁ > A₂ > A₃ > A₄ > A₅

(0.88, 0.88, 2.25) A₁ > A₂ > A₃ > A₄ > A₅

(0.88, 1.68, 1.25) A₁ > A₃ > A₂ > A₄ > A₅

(0.88, 1.68, 2.25) A₁ > A₂ > A₃ > A₄ > A₅

(1.68, 0.88, 1.25) A₁ > A₂ > A₃ > A₄ > A₅

(1.68, 0.88, 2.25) A₁ > A₂ > A₃ > A₄ > A₅

(1.68, 1.68, 1.25) A₁ > A₃ > A₂ > A₄ > A₅

(1.68, 1.68, 2.25) A₁ > A₃ > A₂ > A₄ > A₅

(a, b, λ)	Ranking results
(0.88, 0.88, 1.25)	A₁ > A₂ > A₃ > A₄ > A₅
(0.88, 0.88, 2.25)	A₁ > A₂ > A₃ > A₄ > A₅
(0.88, 1.68, 1.25)	A₁ > A₃ > A₂ > A₄ > A₅
(0.88, 1.68, 2.25)	A₁ > A₂ > A₃ > A₄ > A₅
(1.68, 0.88, 1.25)	A₁ > A₂ > A₃ > A₄ > A₅
(1.68, 0.88, 2.25)	A₁ > A₂ > A₃ > A₄ > A₅
(1.68, 1.68, 1.25)	A₁ > A₃ > A₂ > A₄ > A₅
(1.68, 1.68, 2.25)	A₁ > A₃ > A₂ > A₄ > A₅

According to Table 7, it is noted that the ranking results change when the values of parameters a, b and λ change. First, we discuss the influence of λ by variable control. Table 7 shows that the rankings do not change with λ when a = 0.88, b = 0.88 or a = 1.68, b = 0.88 or a = 1.68, b = 1.68, while the ranking change with λ when a = 0.88, b = 1.68. This indicates that the ranking result is sensitive to λ. Second, how the ranking changes with parameter a is also analyzed. Table 7 shows that the rankings do not change with a when b = 0.88, λ = 1.25 or b = 0.88, λ = 2.25 or b = 1.68, λ = 1.25, while the ranking change with a when b = 1.68, λ = 2.25. This indicates the ranking result is sensitive to a. Third, we discuss how the ranking change with the value of parameter b. Table 7 shows that the rankings change with b except when a = 0.88, λ = 2.25. This indicates the ranking result is more sensitive to b than parameters a and λ in this coal mine safety evaluation problem.

In addition, whatever the value of the parameter, the best choice is always A₁. The consistency of the results fully prove the reliability and accuracy of the proposed method. The preceding discussion demonstrates that the values of parameters a, b and λ can influence the ranking results. Parameters a and b represent risk preference attitude coefficients of DMs towards gains and losses, respectively. The bigger the value of a or b is, the more strong DMs’ risk preference attitude toward gains or losses is. Parameter λ represents DMs’ attitudes toward loss-aversion. The bigger the value of λ, the more the DMs tend to avoid losses, vice versa.

Therefore, the method proposed in this paper has certain flexibility for coal mine safety evaluation. And DMs can choose appropriate values of parameters a, b and λ based on their subjective feelings to obtain desirable result.

7 Conclusion

In this paper, a novel LSF is proposed to convert the linguistic terms into the linguistic preference values. Considering the cross-evaluation, a new distance measure of LIFSs is proposed. Additionally, based on the principle of nonlinear Gaussian diffusion model, a similarity measure of LIFSs is further defined, which can provide different response capabilities according to the difference information between alternatives. On this basis, a method is developed to handle MADM problems, and the effectiveness of the method is illustrated by an example. Finally, the calculation results are compared with the existing methods to verify its effectiveness.

For future research, we can continue to improve LSFs to handle the information under the subscript-symmetric LTSs. Moreover, the method proposed in this paper can also be used to deal with MADM problems in supply chain management, medical diagnosis and other fields.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant Nos. 11671001 and 61876201), the Science and Technology Project of Chongqing Municipal Education Committee (Grants no. KJQN201800624) of China and the Project of Humanities and Social Sciences planning fund of Ministry of Education (18YJA630022) of China.

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Multi-attribute decision-making method based on a novel distance measure of linguistic intuitionistic fuzzy sets

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Linguistic scale functions

4.1 A novel distance measure of LIFSs

Table 4 The evaluation matrix M ¯ LIF g 1 g 2 g 3 g 4 A 1 (s5, s1) (s6, s2) (s7, s1) (s5, s2) A 2 (s6, s2) (s6, s1) (s5, s2) (s5, s1) A 3 (s7, s1) (s5, s2) (s4, s3) (s6, s1) A 4 (s6, s2) (s6, s1) (s5, s2) (s5, s3) A 5 (s6, s1) (s4, s2) (s6, s2) (s5, s3)

Footnotes

Acknowledgments

References