Multi-Criteria Decision-Making Method Based on Distance Measure and Choquet Integral for Linguistic Z-Numbers

Abstract

Z-numbers are a new concept considering both the description of cognitive information and the reliability of information. Linguistic terms are useful tools to adequately and effectively model real-life cognitive information, as well as to characterize the randomness of events. However, a form of Z-numbers, in which their two components are in the form of linguistic terms, is rarely studied, although it is common in decision-making problems. In terms of Z-numbers and linguistic term sets, we provided the definition of linguistic Z-numbers as a form of Z-numbers or a subclass of Z-numbers. Then, we defined some operations of linguistic Z-numbers and proposed a comparison method based on the score and accuracy functions of linguistic Z-numbers. We also presented the distance measure of linguistic Z-numbers. Next, we developed an extended TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method based on the Choquet integral for multi-criteria decision-making (MCDM) problems with linguistic Z-numbers. Finally, we provided an example concerning the selection of medical inquiry applications to demonstrate the feasibility of our proposed approach. We then verified the applicability and superiority of our approach through comparative analyses with other existing methods. Illustrative and comparative analyses indicated that the proposed approach was valid and feasible for different decision-makers and cognitive environments. Furthermore, the final ranking results of the proposed approach were closer to real decision-making processes. Linguistic Z-numbers can flexibly characterize real cognitive information as well as describe the reliability of information. This method not only is a more comprehensive reflection of the decision-makers’ cognition but also is more in line with expression habits. The proposed method inherited the merits of the classical TODIM method and considers the interactivity of criteria; therefore, the proposed method was effective for dealing with real-life MCDM problems. Consideration about bounded rational and the interactivity of criteria made final outcomes convincing and consistent with real decision-making.

Introduction

To make computer systems study, think, and make correct decisions like a human brain, significant research based on cognitive information has been conducted in many fields [1,2,3,4]. Oliva et al. [5] proposed a cognitive model of the acquisition of verbal morphology that matched different kinds of errors children make. Peng et al. [6] proposed a human-like cognitive representation of Chinese concepts that overcame the scarceness of available resources and improved the state of the art. Guo et al. [7] developed a novel framework for a discriminative extreme learning machine for pattern classification. These studies not only enrich the field of cognitive computing but also provide many effective algorithms to solve practical problems. Decision-making is a human activity based on cognitive information. Human beings inevitably are faced with various decision-making problems, which involve multiple fields such as artificial intelligence [8], green product development [9], and tour recommendation [10].

With increasing complexity in the decision-making environment, decision-makers (DMs) no longer are satisfied with using numerical values to represent their cognition for alternatives. Therefore, scholars have tried to express this fuzziness in human cognition using fuzzy sets (FSs) [11], intuitionistic fuzzy sets (IFSs) [12], and hesitant fuzzy sets (HFSs) [13]. On this basis, many scholars have studied how DMs make their decisions based on cognitive information [14,15,16,17]. These studies enrich the field of cognitive computing and also provide effective algorithms for solving decision-making problems. Although the FSs, IFSs, and HFSs have made great progress in the expression of human cognition, they are inefficient at expressing the reliability of relevant cognitive information. The reliability of different people’s cognitive information will be different as a result of age, experience, knowledge, and other factors. Therefore, this paper introduced Z-numbers [18], a new concept that arose in 2011, to address these shortcomings.

Zadeh [18] first introduced the Z-number concept to describe uncertain information, which is a generalized notion. A Z-number is an ordered pair of fuzzy numbers, Z = (A, B). The first component A is a restriction on the values, in which a real-valued uncertain variable X is allowed to take. The second component B is a measure of reliability (certainty) of the first component A. In daily decision-making situations, human cognition usually is presented in the form of Z-numbers. For example, the phrase “Usually, it takes about 1 h for Robert to get home from work” can be represented by a Z-number, Z = (about 1 h,   usually). The information can be formalized as a Z-number-based evaluation “X is Z = (A, B).” In this example, the variable X represents the time for Robert to get home from work, “A = about 1 h” is a fuzzy number used to describe the time restriction, and “B = usually” is a fuzzy number to describe a soft constraint on a partial reliability of A.

Recently, Z-numbers have been a subject of great interest to researchers. In general, current research on Z-numbers can be roughly divided into two areas. The first area focuses on the fundamental research on Z-numbers, including operations [19, 20], converting methods [21], and extension studies [22,22,23,24,25,26,27]. Aliev et al. [19] and Bhanu [20] presented some operations for Z-numbers. These operations were still too complex, although they did suggest a few simplifications in these papers. Considering the complexity in directly computing with Z-numbers, Kang et al. [21] proposed a method of converting a Z-number to a classical fuzzy number, but this method may lead to the loss of original information. Therefore, fundamental research on Z-numbers, especially their operations, must be further studied.

The second area of research on Z-numbers focuses mainly on decision-making methods [28,27,28,29,32]. For example, Kang et al. [29, 30] and Yaakob and Gegov [31] proposed methods for solving multi-criteria decision-making (MCDM) problems with Z-numbers, but they based these methods on the converting method described by Kang et al. [21] and computed the evaluations of alternatives as real numbers. These methods imply significant loss of information contained in the original Z-numbers [18]. Other studies have extended Zadeh’s basic Z-numbers into a tool for computing with words (CWWs) [33,32,33,36]. The Z-number [18] is a powerful general formal work, generalizing the concept of classic sets because the two components A and B in a Z-number can be composed of different fuzzy numbers. Therefore, it also is necessary and worthwhile to conduct further analysis and research on some specific types and subsets of Z-numbers.

Because of the fuzziness and uncertainty of decision-making problems and the inherent vagueness of human cognition, it is more appropriate for DMs to use natural language to represent real-world information. For example, when evaluating the comfort or design of a car, linguistic terms like “good,” “fair,” and “poor” can be used. Natural language usually involves ambiguity and uncertainty. Among the uncertainties involved in natural language, randomness and fuzziness are the most important aspects [37, 38]. The fuzziness of a concept mainly refers to uncertainty regarding the range of extension of that concept [37, 38]. For example, it is difficult to obtain the exact intensions or establish precise boundaries of extensions of the linguistic term “good.” In some semantic situations, “good” represents the meaning of “very good,” whereas sometimes it means “fair.” The randomness of a concept means that any concept is related to the external world in various ways and is not an isolated fact [37,38,39]. For example, linguistic terms such as “usually,” “often,” and “sometimes” embody randomness. Most often, the fuzziness and randomness of concepts are tightly related and inseparable, and both terms can be used to describe the uncertainty of natural languages [37, 38].

The fuzziness and randomness inherent in linguistic terms exactly correspond to the restriction and probability measure of Z-numbers. On one hand, the fuzziness of linguistic terms can help DMs better describe qualitative information in the real world (i.e., as the restriction on the values that a real-valued uncertain variable is allowed to take). On the other hand, the randomness of linguistic terms can be used to characterize the probability measure because some linguistic terms, such as “usually,” can describe the probabilities of events occurring. Therefore, in terms of Z-numbers and linguistic term sets, in this paper, we proposed linguistic Z-numbers as a subclass of Z-numbers. A linguistic Z-number is an extension of a Z-number that represents the two components of the Z-number with linguistic terms. A linguistic Z-number may use linguistic terms such as “fair,” “good,” and “very good” to represent the fuzzy restriction and may use linguistic terms like “seldom,” “often,” and “usually,” or “uncertain,” “certain,” and “sure” as a measure of reliability. Linguistic Z-numbers can represent most decision-making information in real life. For example, a linguistic Z-number, (very  good, sure), can be used to evaluate the profitability of a company, and another linguistic Z-number, (fast, usually), can be used to describe the waiting time of a bus. Compared with classic FSs, linguistic Z-numbers can be more flexible, comprehensive, and accurate when representing cognitive information, and thus reduce the loss of decision-making information.

In real-life decision-making processes, people encounter situations in which criteria or preferences are interactive or influenced by each other. Sugeno [40] introduced the concept of a fuzzy measure that makes only a monotonicity instead of an additivity property. Later, Murofushi and Sugeno [41] introduced the Choquet integral as an extension of the Lebesgue integral; it considered the importance of criteria represented by fuzzy measures. Moreover, current research about Z-numbers usually assumes that DMs can make rational choices when facing complicated and various information. DMs’ behaviors, however, usually is influenced by many factors like character, risk preference, knowledge level, and psychological state, as well as external environmental factors. As such, DMs usually are bounded rational rather than absolutely rational. To overcome this shortcoming, Gomes and Lima [42, 43] proposed the TODIM (an acronym in Portuguese of interactive multi-criteria decision-making) method based on prospect theory [44,45,46]. The TODIM method reflects some behavioral characteristic of DMs, such as reference dependence and loss aversion, and it has been extended to different decision environments [47,48,49,50]. These extended TODIM methods are unable to handle linguistic Z-number MCDM problems with interactive criteria. Therefore, in this paper, we combined the classical TODIM method with the Choquet integral to handle linguistic Z-number MCDM problems.

The primary aims of this study follow:

• (1) DMs’ cognitions are only partly reliable because of different ages, backgrounds, and so on. Therefore, this paper gave the definition of linguistic Z-numbers as a subclass of Z-numbers, to describe cognitive information and measure the reliability of information.

• (2) The operations of Z-numbers defined in previous works are too complex, and the converting method for Z-numbers has the limitation of losing original information. Therefore, we modified and defined some new operations of linguistic Z-numbers.

• (3) DMs are bounded rational because they are influenced easily many factors such as character, and cognition preference. Most actual decision-making problems are MCDM problems with interactive criteria. Therefore, we aimed to combine the TODIM method with the Choquet integral to make decision results closer to actual situations.

This paper is organized as follows: the “Preliminaries” section briefly reviews some basic concepts of linguistic term sets, linguistic scale functions (LSFs), fuzzy measures and the Choquet integral, and Z-numbers. The “Linguistic Z-numbers and Their Operations” section defines linguistic Z-numbers and their operations and proposes a method for comparing two linguistic Z-numbers. In addition, the distance measure for linguistic Z-numbers is given. The “Extended Linguistic Z-numbers TODIM Approach Based on the Choquet Integral” section develops an extended TODIM approach based on the Choquet integral for solving MCDM problems with linguistic Z-numbers. The “Illustrative Example” section provides an illustrative example, a sensitivity analysis, and a comparative analysis. Conclusions are drawn in the “Conclusions” section.

Preliminaries

This section briefly reviews some definitions and basic operations of linguistic term sets, LFSs, fuzzy measures and the Choquet integral, and Z-numbers.

Linguistic Term Sets and Their Extension

Let S = {s _α |α = 0, 1,  … , 2t} be a finite and totally ordered discrete term set with odd cardinality, where s _α represents a possible value of` a linguistic variable, and usually S should satisfy the following characteristics [51]:

• (1) The set S is ordered: s _α  > s _β if and only if α > β.

• (2) There is a negation operator: neg(s _α ) = s _{2t − α} .

In the process of information aggregation, the aggregated results often do not match the elements in the language assessment scale. To preserve all provided information, Xu [52, 53] extended the discrete linguistic term set S to a continuous one: $\overline{S}=\left\{{s}_{\alpha}\left|\alpha \in \left[0,\kern0.3em l\right]\right.\right\}$, in which s _α  > s _β if α > β, and l(l > 2t) is a sufficiently large positive integer. If s _α  ∈ S, then s _α is called an original linguistic term; otherwise, s _α is called a virtual linguistic term. In general, DMs use original linguistic terms to evaluate alternatives. Virtual linguistic terms appear in operations only to avoid information loss and generally to enhance the decision-making process. Virtual linguistic terms have no practical meaning, and their main role is ranking alternatives.

Linguistic Numerical Scale Models

To use data more efficiently and express semantics more flexibly, LFSs assign different semantics to linguistic terms under different situations [54]. For the linguistic term s _α in a linguistic term set S, where S = {s _α |α = 0, 1,  … , 2t}, the relationship between element s _α and its subscript α is strictly monotonically increasing [52].

• Definition 1. [54]. If θ _i  ∈ R ⁺(R ⁺ = {r|r > 0, r ∈ R}), is a numeric value, then the LFS F that conducts the mapping from s _i to θ _i (i = 0, 1,  … , 2t) is defined as follows:

$$ F:{s}_i\to {\theta}_i\kern0.5em \left(i=0,1,\dots, 2t\right), $$

(1)

where 0 ≤ θ ₀ ≤ θ ₁ ≤  …  ≤ θ _2t .

Obviously, function F is a strictly monotonically increasing function with regard to label i, and the symbol θ _i (i = 0, 1,  … , 2t) reflects the preferences of the DMs when they are using linguistic items s _i  ∈ S (i = 0, 1,  … , 2t). Therefore, the function or value in fact denotes the semantics of the linguistic term.

For example, the following functions are possible choices for LFSs:

$$ {F}_1\left({\theta}_i\right)={\theta}_i=\frac{i}{2t}\left(0\le i\le 2t\right). $$

(2)

In Formula (2), the evaluation scale of the linguistic information is averaged, as follows:

$$ {F}_2\left({\theta}_i\right)={\theta}_i={\left(\frac{i}{2t}\right)}^t\left(0\le i\le 2t\right). $$

(3)

In Formula (3), as linguistic label i increases, the absolute deviation between adjacent linguistic subscripts first increases and then decreases, as follows:

$$ {F}_3\left({\theta}_i\right)={\theta}_i={\left(\frac{i}{2t}\right)}^{\frac{1}{t}}\left(0\le i\le 2t\right) $$

(4)

In Formula (4), as linguistic label i increases, the absolute deviation between adjacent linguistic subscripts first decreases and then increases, as follows:

$$ {F}_4\left({\theta}_i\right)={\theta}_i=\left\{\begin{array}{l}\frac{a^t-{a}^{t-i}}{2{a}^t-2}\kern3.199999em \left(0\le i\le t\right)\\ {}\frac{a^t+{a}^{i-t}-2}{2{a}^t-2}\kern1.5em \left(t+1\le i\le 2t\right)\end{array}\right.. $$

(5)

In Formula (5), with the extension from the middle of the given linguistic term set to both ends, the absolute deviation between adjacent linguistic subscripts also increases. The value of a can be obtained through experiments or subjective methods. Bao et al. [55] stated that a most likely will be obtained in the interval of [1.36, 1.4] according to experimental research. In addition, a can be determined thorough a subjective method, namely, assuming that the indicator A is far more important that indicator B. If the importance ratio is m, then a ^k = m (k represents the scale level); thus, $a=\sqrt[k]{m}$. At present, most scholars believe that m = 9 is the upper limit of the importance ratio; therefore, with respect to the scale level of 7, $a=\sqrt[7]{9}\approx 1.37$ can be calculated.

Assuming t = 3 and a = 1.4, the features of Formulas (2), (3), (4), and (5) can be depicted graphically as shown in Fig. 1.

Fig. 1

Graphical demonstration of Formulas (2), (3), (4), and (5)

Fuzzy Measures and the Choquet Integral

This section introduces concepts about fuzzy measures and the Choquet integral.

• Definition 2. [56]. Let X = {x ₁, x ₂,  … , x _n } be a fixed set, and P(X) be the power set of X. A fuzzy measure on X is a set function μ : P(X) → [0, 1], satisfying the following conditions:

• (1) μ(∅) = 0, μ(X) = 1.

• (2) When A , B ∈ P(X) and A ⊆ B, μ(A) < μ(B).

• Definition 3. [40]. Let X = {x ₁, x ₂,  … , x _n } be a fixed set. A fuzzy measure g on X is called λ-fuzzy measure if it satisfies the following conditions:

$$ g\left(A\cup B\right)=g(A)+g(B)+\lambda g(A)g(B), $$

(6)

where λ ∈ (−1, ∞) for ∀A , B ∈ P(X) and A ∩ B = ∅.

If X is a finite set, ${\cup}_{i=1}^n{x}_i=X$, the λ-fuzzy measure g satisfies the following equation:

$$ g\left({\cup}_{i=1}^n{x}_i\right)=\left\{\begin{array}{l}\frac{1}{\lambda}\left({\prod}_{i=1}^n\left(1+\lambda g\left({x}_i\right)\right)-1\right)\kern0.9000001em \lambda \ne 0,\\ {}{\sum}_{i=1}^ng\left({x}_i\right)\kern6.999996em \lambda =0,\end{array}\right. $$

(7)

where x _i  ∩ x _j  = ∅ for all i , j = 1 , 2 ,  …  , n, and i ≠ j; g(x _i ) for a subset with a single element x _i is called a fuzzy density, denoted as g _i  = g(x _i ), which is the subjective weight of criteria c _i in MCDM. For interactive criteria, generally, ${\sum}_{i=1}^ng\left({x}_i\right)\ne 1$.

Especially for every subset A ∈ P(X), we have

$$ g(A)=\left\{\begin{array}{l}\frac{1}{\lambda}\left(\prod_{i\in A}\left(1+\lambda g\left({x}_i\right)\right)-1\right)\kern1.5em \lambda \ne 0,\\ {}\sum_{i\in A}g(i)\kern8.499996em \lambda =0.\end{array}\right. $$

(8)

The value λ can be determined uniquely based on Eq. (6) from g(X) = 1 by solving the equation

$$ \lambda +1={\prod}_{i=1}^n\left(1+\lambda {g}_i\right) $$

(9)

Note that λ also can be determined uniquely by g(X) = 1.

• Definition 4. [57]. Let f be a real-valued function on X, and μ be a fuzzy measure on X. The discrete generalized Choquet integral of f with respect to μ is defined by

$$ {C}_{\mu }(f)={\sum}_{i=1}^p\left(\mu \left({B}_{\delta (i)}\right)-\mu \left({B}_{\delta \left(i-1\right)}\right)\right)\cdot f\left({x}_{\delta (i)}\right)+{\sum}_{i=p+1}^n\left(\mu \left({A}_{\delta (i)}\right)-\mu \left({A}_{\delta \left(i-1\right)}\right)\right)\cdot f\left({x}_{\delta (i)}\right), $$

(10)

where the subscript {δ(1), δ(2),  … , δ(n)} is a permutation on X such thatf(x _δ(1)) ≤ f(x _δ(2)) ≤  …  ≤ f(x _δ(p)) ≤ 0 ≤ f(x _{δ(p + 1)}) ≤  …  ≤ f(x _δ(n)) and B _σ(l) = {c _σ(1), c _σ(2), …c _σ(l)}, andc _σ(0) = ∅, A _σ(l) = {c _σ(l), c _{σ(l + 1)},  … , c _σ(n)}, and c _{σ(n + 1)} = ∅.

Z-Numbers

DMs have remarkable capability to make rational decisions based on information that is uncertain, imprecise, or incomplete. Thus, Zadeh [18] introduced the concept of Z-numbers to better describe real-life decision-making information.

• Definition 5. [18]. A Z-number is an ordered pair of fuzzy numbers, (A, B), which is associated with a real-valued uncertain variable X, where A is a fuzzy restriction on the values that the variable X is allowed to take, and B is a measure of the reliability of the first component. Typically, A and B are described in a natural language.

The form of Z-numbers may look similar to the rough membership function in rough set theory [58], but they are different concepts. The rough set theory, proposed by Pawlak, is an extension of the classical fuzzy set theory. The main idea of the rough set theory is to induce the decision-making problem or rules of classification by knowledge reduction. A rough membership function is used to measure the degree with which any object with given criterion values belongs to a given set [59]. Conceptually, a rough membership degree denotes the degree of an object belonging to a set and represents the objective fact. A Z-number is a description of an object and considers the reliability of the description, and simultaneously expresses the fuzziness and randomness of the object.

• Example 1. The phase “I’m very sure that China has a population of more than 1.3 billion” can be expressed as a Z-number, (more than 1.3 billion, very  sure). According to Definition 5, the uncertain variable X refers to the population of China, and “more than 1.3 billion” is the component A of the Z-number. This represents the restriction that X is allowed to take. “Very sure” can be interpreted as a response to the question: How sure are you that X is A?

Zadeh [18] outlined the procedures for the operations on Z-numbers, but these operations have some limitations. First, too many unknown variables must be given in advance. In particular, it is difficult for DMs who are not experts in fuzzy decision-making to determine the membership functions of A and B, respectively. In particular, the operations for Z-numbers [18] are complex and include several variational problems [19]. Considering these limitations, Kang [21] provided a method to transform Z-numbers into real numbers to avoid directly calculating with Z-numbers. This process of transformation may cause loss and distortion of information.

Linguistic Z-Numbers and Their Operations

This section proposes linguistic Z-numbers in terms of Z-numbers and linguistic term sets. Furthermore, the operations, comparison method, and distance of linguistic Z-numbers are provided.

• Definition 6. Let X be a universe of discourse, S ₁ = {s ₀, s ₁,  … , s _2l } and ${S}_2=\left\{{s}_0^{\prime },{s}_1^{\prime },\dots, {s}_{2k}^{\prime}\right\}$ be two finite and totally ordered discrete linguistic term sets, where l and k are nonnegative integers. Furthermore, let A _ϕ(x) ∈ S ₁ and B _φ(x) ∈ S ₂. A linguistic Z-number set Z in X is an object having the following form:

$$ Z=\left\{\left(x,{A}_{\phi (x)},{B}_{\varphi (x)}\right)\left|x\in X\right.\right\}, $$

(11)

which is characterized by two linguistic terms A _ϕ(x) and B _φ(x), where A _ϕ(x) is a fuzzy restriction on the values that the uncertain variable is allowed to take, and B _φ(x) is a measure of reliability of the first component. Usually, the linguistic term sets S ₁ and S ₂ are different and represent different preference information, respectively.

When X has only one element, the linguistic Z-number set is reduced to (A _ϕ(α), B _φ(α)). For convenience, z _α  = (A _ϕ(α), B _φ(α)) is called a linguistic Z-number, where A _ϕ(α) ∈ S ₁ and B _φ(α) ∈ S ₂ are two linguistic terms.

• Example 2. Assume l = 3 and k = 2. Let A = {A ₀, A ₁, A ₂, A ₃, A ₄, A ₅, A ₆}={very poor,   poor ,  slightly poor ,  fair ,  slightly good ,  good ,  very good} be a set of linguistic terms used to represent evaluation information and B = {B ₀, B ₁, B ₂, B ₃, B ₄} = {seldom ,  ocassionally ,  frequently ,  regularly ,  usually} be a linguistic term set used to express the reliability measure of linguistic Z-numbers. Let X = {x ₁, x ₂, x ₃} be the universe of discourse, a linguistic Z-number set Z = {(x, A _ϕ(x), B _φ(x))|x ∈ X} = {(x ₁, A ₂, B ₂), (x ₂, A ₄, B ₃), (x ₃, A ₃, B ₄)}. Then (A ₂, B ₂), (A ₄, B ₃), and (A ₃, B ₄) are linguistic Z-numbers that can be denoted by (slightly  poor, frequently),(slightly  good, regularly), and (fair, usually), respectively.

In a decision-making example, a patient is going to conduct a general evaluation of the service of a hospital. The person easily might use (good, usually) to express his impressions of the hospital. In this case, the linguistic Z-number ( good,  usually) shows the service of the hospital and also describes the probability that “the hospital’s service is good” is true. Compared with linguistic term sets, linguistic Z-numbers also measure the reliability of the event when describing qualitative information. This is more precise and comprehensive than a result represented as a single linguistic term. Linguistic Z-numbers are a subclass of Z-numbers and not only inherit the advantages of Z-numbers but also employ linguistic terms to represent the two components of Z-numbers.

Linguistic Z-Number Operations

As discussed in the “Fuzzy Measures and the Choquet Integral” section, the operations of Z-numbers [18]are too complex to be applied in MCDM methods, and they are not applicable to linguistic Z-numbers. For linguistic Z-numbers, the restriction of the variable and the probability measure of restriction are represented in the form of linguistic terms. We usually assume the membership degrees of the two components of linguistic Z-numbers are equal to 1, which is not suitable for processing by the extension principle. Therefore, we must develop some new operations, considering both the flexibility of linguistic term sets and the reliability measure of Z-numbers. The operations of linguistic Z-numbers must meet different semantic environments, and the evaluation and reliability measures of linguistic Z-numbers should not be processed separately. Motivated by this, we defined the following operations for linguistic Z-numbers.

• Definition 7. Let z _i  = (A _ϕ(i), B _φ(i)) and z _j  = (A _ϕ(j), B _φ(j)) be two linguistic Z-numbers; and f ^∗ and g ^∗ functions be the possible functions of F ₁(θ _i ), F ₂(θ _i ), F ₃(θ _i ), and F ₄(θ _i ). Some operators of linguistic Z-numbers are defined as follows:

• (1) neg(z _i ) = (f ^∗−1(f ^∗(A _2l ) − f ^∗(A _ϕ(i))), g ^∗−1(g ^∗(B _2k ) − g ^∗(B _φ(i))));

• (2)
  $$ \begin{array}{l}{z}_i\oplus {z}_j=\left({f}^{\ast -1}\left({f}^{\ast}\left({A}_{\phi (i)}\right)+{f}^{\ast}\left({A}_{\phi (j)}\right)\right),\right.\hfill \\ {}\left.{g}^{\ast -1}\left(\frac{f^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)+{f}^{\ast}\left({A}_{\phi (j)}\right)\times {g}^{\ast}\left({B}_{\varphi (j)}\right)}{f^{\ast}\left({A}_{\phi (i)}\right)+{f}^{\ast}\left({A}_{\phi (j)}\right)}\right)\right);\hfill \end{array} $$

• (3) λz _i  = (f ^{∗ − 1}(λf ^∗(A _ϕ(i))), B _φ(i)), where λ ≥ 0;

• (4) z _i  ⊗ z _j  = (f ^{∗ − 1}(f ^∗(A _ϕ(i))f ^∗(A _ϕ(j))), g ^{∗ − 1}(g ^∗(B _φ(i))g ^∗(B _φ(j)))); and

• (5) z _i ^λ = (f ^{∗ − 1}(f ^∗(A _ϕ(i))^λ), g ^{∗ − 1}(g ^∗(B _φ(i))^λ)), where λ ≥ 0.

Compared with the operations outlined by Zadeh [18], the operations in Definition 7 are relatively simple and applicable to different semantic environments, and the two components of linguistic Z-numbers are not processed separately. Furthermore, combing the randomness of linguistic terms, the reliability measure of linguistic Z-numbers also can be considered. Compared with the method converting the Z-numbers into crisp values [21], the defined operations effectively keep the consistency and integrity of the information and ensure the rationality of the MCDM methods based on the operations.

• Example 3. Using the linguistic term sets A and B in Example 2, reconsider the hospital evaluation problem under two parameters. The two parameters are service (c ₁) and professional capability (c ₂), and they are equally important: z _α  = (A ₅, B ₁) indicates the statement that the service of the hospital is occasionally good, and z _β  = (A ₁, B ₄) represents that the professional capability of the hospital is usually poor. Assume that λ = 2, a = 1.4, f ^∗(θ _i ) = F ₄(θ _i ), and g ^∗(θ _i ) = F ₁(θ _i ), then neg(z _α ) = (A ₁, B ₃), and the comprehensive value can be calculated as z = 0.5z _α  ⊕ 0.5z _β  = (A ₃, B _1.6744). If the two components in the linguistic Z-number with reliability are processed separately, however, the result is ${z}^{\prime }=\left({F}_4^{-1}\left(0.5{F}_4\left({A}_1\right)+0.5{F}_4\left({A}_5\right)\right),{F}_1^{-1}\left(0.5{F}_1\left({B}_1\right)+0.5{F}_1\left({B}_4\right)\right)\right)=\left({A}_3,{B}_{2.5}\right)$.

Obviously, the reliability measure yielded by the first method is smaller than that obtained by the second. Although the two components in linguistic Z-numbers play different roles in the evaluation process, they must work together to depict evaluation information. It would be more reasonable to gather the two components to derive a comprehensive reliability measure. The evaluation value A ₅ in terms of service (c ₁) is larger than the evaluation value A ₁ in terms of professional capability (c ₂). The reliability measure B ₁ of A ₅ is smaller than the reliability measure B ₄ of A ₁. The evaluation value of service (c ₁) will be salient in the comprehensive evaluation value. Thus, the reliability measure of the comprehensive value yielded by the processed operation rules is close to B ₁ but not to B ₄.

Theorem 1 Let z _i  = (A _ϕ(i), B _φ(i)), z _j  = (A _ϕ(j), B _φ(j)), and z _k  = (A _ϕ(k), B _φ(k)) be three linguistic Z-numbers, and f ^∗ and g ^∗ be LFSs. Then, the following properties are true:

• (1) z _i  ⊕ z _j  = z _j  ⊕ z _i ;

• (2) z _i  ⊗ z _j  = z _j  ⊗ z _i ;

• (3) λ(z _i  ⊕ z _j ) = λz _i  ⊕ λz _j  , λ > 0;

• (4) (z _i  ⊗ z _j )^λ = z _i ^λ ⊗ z _j ^λ;

• (5) λ ₁ z _i  ⊕ λ ₂ z _i  = (λ ₁ + λ ₂)z _i  , λ ₁ ≥ 0 , λ ₂ ≥ 0;

• (6) ${z_i}^{\lambda_1}\otimes {z_i}^{\lambda_2}={z_i}^{\left({\lambda}_1+{\lambda}_2\right)},{\lambda}_1\ge 0,{\lambda}_2\ge 0$;

• (7) (z _i  ⊕ z _j ) ⊕ z _k  = z _i  ⊕ (z _j  ⊕ z _k ); and

• (8) (z _i  ⊗ z _j ) ⊗ z _k  = z _i  ⊗ (z _j  ⊗ z _k ).

Properties (1), (2), (7), and (8) can be obtained easily. In the following, we will prove Property (3) of Theorem 1.

Proof .

$$ \begin{array}{l}\mathrm{Since}\ \lambda >0,\hfill \\ {}\lambda \left({z}_i\oplus {z}_j\right)=\lambda \left({f}^{\ast -1}\left({f}^{\ast}\left({A}_{\phi (i)}\right)+{f}^{\ast}\left({A}_{\phi (j)}\right)\right),\right.\left.{g}^{\ast -1}\left(\frac{f^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)+{f}^{\ast}\left({A}_{\phi (j)}\right)\times {g}^{\ast}\left({B}_{\varphi (j)}\right)}{f^{\ast}\left({A}_{\phi (i)}\right)+{f}^{\ast}\left({A}_{\phi (j)}\right)}\right)\right)\hfill \\ {}=\left({f}^{\ast -1}\left(\lambda {f}^{\ast}\left({A}_{\phi (i)}\right)+\lambda {f}^{\ast}\left({A}_{\phi (j)}\right)\right),\right.\left.{g}^{\ast -1}\left(\frac{f^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)+{f}^{\ast}\left({A}_{\phi (j)}\right)\times {g}^{\ast}\left({B}_{\varphi (j)}\right)}{f^{\ast}\left({A}_{\phi (i)}\right)+{f}^{\ast}\left({A}_{\phi (j)}\right)}\right)\right),\hfill \end{array} $$

and

$$ \begin{array}{l}\lambda {z}_i\oplus \lambda {z}_j\hfill \\ {}=\left({f}^{\ast -1}\left(\lambda {f}^{\ast}\left({A}_{\phi (i)}\right)\right),{B}_{\varphi (i)}\right)\oplus \left({f}^{\ast -1}\left(\lambda {f}^{\ast}\left({A}_{\phi (j)}\right)\right),{B}_{\varphi (j)}\right)\hfill \\ {}=\left({f}^{\ast -1}\left(\lambda {f}^{\ast}\left({A}_{\phi (i)}\right)+\lambda {f}^{\ast}\left({A}_{\phi (j)}\right)\right),\right.\left.{g}^{\ast -1}\left(\frac{\lambda {f}^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)+\lambda {f}^{\ast}\left({A}_{\phi (j)}\right)\times {g}^{\ast}\left({B}_{\varphi (j)}\right)}{\lambda {f}^{\ast}\left({A}_{\phi (i)}\right)+\lambda {f}^{\ast}\left({A}_{\phi (j)}\right)}\right)\right)\hfill \\ {}=\left({f}^{\ast -1}\left(\lambda {f}^{\ast}\left({A}_{\phi (i)}\right)+\lambda {f}^{\ast}\left({A}_{\phi (j)}\right)\right),\right.\left.{g}^{\ast -1}\left(\frac{f^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)+{f}^{\ast}\left({A}_{\phi (j)}\right)\times {g}^{\ast}\left({B}_{\varphi (j)}\right)}{f^{\ast}\left({A}_{\phi (i)}\right)+{f}^{\ast}\left({A}_{\phi (j)}\right)}\right)\right)\hfill \end{array} $$

Then, λ(z _i  ⊕ z _j ) = λz _i  ⊕ λz _j can be obtained.

Similarly, Properties (4), (5), and (6) can be proved.

Comparison Method for the Linguistic Z-Numbers

The score and accuracy functions are significant and effective tools for comparing fuzzy numbers. Inspired by the score and accuracy functions of intuitionistic linguistic sets [41, 60, 61], this subsection defines the score and accuracy functions of linguistic Z-numbers and develops a comparison method.

• Definition 8. Let ${z}_{\alpha }=\left({A}_{\phi_{\left(\alpha \right)}},{B}_{\varphi_{\left(\alpha \right)}}\right)$ be a linguistic Z-number. Then, the score function S(z _α ) of z _α can be defined as follows:

$$ S\left({z}_{\alpha}\right)={f}^{\ast}\left({A}_{\phi \left(\alpha \right)}\right)\times {g}^{\ast}\left({B}_{\varphi \left(\alpha \right)}\right) $$

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The accuracy function of z _α can be defined as follows:

$$ A\left({z}_{\alpha}\right)={f}^{\ast}\left({A}_{\phi \left(\alpha \right)}\right)\times \left(1-{g}^{\ast}\left({B}_{\varphi \left(\alpha \right)}\right)\right) $$

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• Definition 9. Let z _i  = (A _ϕ(i), B _φ(i)) and z _j  = (A _ϕ(j), B _φ(j)) be two linguistic Z-numbers, and the comparison method can be defined as follows:

• (1) When A _ϕ(i) > A _ϕ(j) and B _φ(i) > B _φ(j), z _i is strictly great than z _i , denoted by z _i  > z _j ;

• (2) When S(z _i ) > S(z _j ) or S(z _i ) = S(z _j ) and A(z _i ) > A(z _j ), z _i is greater than z _i , denoted by z _i  ≻ z _j ;

• (3) When S(z _i ) = S(z _j ) and A(z _i ) = A(z _j ), z _i equals z _j , denoted by z _i  ∼ z _j ; and

• (4) When S(z _i ) = S(z _j ) and A(z _i ) < A(z _j ) or S(z _i ) < S(z _j ), z _i is less than z _j , denoted by z _i  ≺ z _j .

• Example 4. Let z ₁ and z ₂ be two linguistic Z-numbers. According to Definitions 8 and 9, the following results can be obtained:

If a = 1.4, f ^∗(θ _i ) = F ₄(θ _i ), and g ^∗(θ _i ) = F ₁(θ _i ), then

• (1) If z ₁ = (A ₄, B ₄) and z ₂ = (A ₂, B ₃) are two linguistic Z-numbers, then A ₄ > A ₂ and B ₄ > B ₃, and therefore z ₁ > z ₂;

• (2) If Z ₁ = (A ₆, B ₂) and Z ₂ = (A ₄, B ₃) are two linguistic Z-numbers, then S(z ₁) = 0.5 and S(z ₂) = 0.461, and S(z ₁) > S(z ₂), and therefore z ₁ ≻ z ₂.

Distance Between the Linguistic Z-Numbers

• Definition 10. Let ${z}_i=\left({A}_{\phi_{(i)}},{B}_{\varphi_{(i)}}\right)$, ${z}_j=\left({A}_{\phi_{(j)}},{B}_{\varphi_{(j)}}\right)$, and ${z}_k=\left({A}_{\phi_{(k)}},{B}_{\varphi_{(k)}}\right)$ be three linguistic Z-numbers, and f ^∗ and g ^∗ be two LFSs. Let Ω be the set of linguistic Z-numbers, R be the set of real numbers, and d be mapping from Ω × Ω to R. Then, d(z _i , z _j ) is the distance between linguistic Z-numbers z _i and z _j , if d(z _i , z _j ) satisfies the following properties:

• (1) d(z _i , z _j ) ≥ 0, if and only if z _i  = z _j , d(z _i , z _j ) = 0;

• (2) d(z _i , z _j ) = d(z _j , z _i ); and

• (3) If ${A}_{\phi_{(i)}}\le {A}_{\phi_{(j)}}\le {A}_{\phi_{(k)}}$ and ${B}_{\varphi_{(i)}}\le {B}_{\varphi_{(j)}}\le {B}_{\varphi_{(k)}}$, then d(z _i , z _j ) ≤ d(z _i , z _k ) and d(z _j , z _k ) ≤ d(z _i , z _k ).

• Definition 11. Let ${z}_i=\left({A}_{\phi_{(i)}},{B}_{\varphi_{(i)}}\right)$ and ${z}_j=\left({A}_{\phi_{(j)}},{B}_{\varphi_{(j)}}\right)$ be two linguistic Z-numbers, and f ^∗ and g ^∗ be two LFSs. Then, the distance between linguistic Z-number z _i and z _j can be defined as follows:

$$ \begin{array}{l}d\left({z}_i,{z}_j\right)=\frac{1}{2}\left(\left|{f}^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)-{f}^{\ast}\left({A}_{\phi (j)}\right)\times {g}^{\ast}\left({B}_{\varphi (j)}\right)\right|\right.\\ {}\kern4.099998em \left.+ \max \left\{\left|{f}^{\ast}\left({A}_{\phi (i)}\right)-{f}^{\ast}\left({A}_{\phi (j)}\right)\right|,\left|{g}^{\ast}\left({B}_{\varphi (i)}\right)-{g}^{\ast}\left({B}_{\varphi (j)}\right)\right|\right\}\right)\\ {}\kern4.099998em \end{array} $$

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Proof Obviously, Eq. (14) satisfies Properties (1) and (2) in Definition 10. We will prove that Eq. (14) satisfies Property (3) in the following.

$$ \begin{array}{l}\mathrm{Since}\ {A}_{\phi_{(i)}}\le {A}_{\phi_{(j)}}\le {A}_{\phi_{(k)}},\mathrm{and}\kern0.5em {f}^{\ast }\ \mathrm{is}\ \mathrm{strictly}\ \mathrm{a}\ \mathrm{monotonically}\ \mathrm{increasing}\ \mathrm{a}\mathrm{nd}\ \mathrm{continuous}\ \mathrm{function},\hfill \\ {}{f}^{\ast}\left({A}_{\phi_{(i)}}\right)\le {f}^{\ast}\left({A}_{\phi_{(j)}}\right)\le {f}^{\ast}\left({A}_{\phi_{(k)}}\right).\mathrm{Similarly},\mathrm{we}\ \mathrm{have}\ {g}^{\ast}\left({B}_{\varphi_{(i)}}\right)\le {g}^{\ast}\left({B}_{\varphi_{(j)}}\right)\le {g}^{\ast}\left({B}_{\varphi_{(k)}}\right).\mathrm{Then}\ \mathrm{we}\ \mathrm{have}\hfill \\ {}{f}^{\ast}\left({A}_{\phi (i)}\right)\le {f}^{\ast}\left({A}_{\phi (j)}\right)\le {f}^{\ast}\left({A}_{\phi (k)}\right)\Rightarrow \left|{f}^{\ast}\left({A}_{\phi (i)}\right)-{f}^{\ast}\left({A}_{\phi (j)}\right)\right|\le \left|{f}^{\ast}\left({A}_{\phi (i)}\right)-{f}^{\ast}\left({A}_{\phi (k)}\right)\right|,\hfill \\ {}{g}^{\ast}\left({B}_{\varphi_{(i)}}\right)\le {g}^{\ast}\left({B}_{\varphi_{(j)}}\right)\le {g}^{\ast}\left({B}_{\varphi_{(k)}}\right)\Rightarrow \left|{g}^{\ast}\left({B}_{\varphi_{(i)}}\right)-{g}^{\ast}\left({B}_{\varphi_{(j)}}\right)\right|\le \left|{g}^{\ast}\left({B}_{\varphi_{(i)}}\right)-{g}^{\ast}\left({B}_{\varphi_{(k)}}\right)\right|,\mathrm{and}\hfill \\ {}{f}^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)\le {f}^{\ast}\left({A}_{\phi (j)}\right)\times {g}^{\ast}\left({B}_{\varphi (j)}\right)\le {f}^{\ast}\left({A}_{\phi (k)}\right)\times {g}^{\ast}\left({B}_{\varphi (k)}\right)\Rightarrow \hfill \\ {}\left|{f}^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)\right.\left.-{f}^{\ast}\left({A}_{\phi (j)}\right)\times {g}^{\ast}\left({B}_{\varphi (j)}\right)\right|\le \left.\left|{f}^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)\right.-{f}^{\ast}\left({A}_{\phi (k)}\right)\times {g}^{\ast}\left({B}_{\varphi (k)}\right)\right|.\hfill \end{array} $$

Thus, we obtain

$$ \begin{array}{l}\kern0.8000001em \left(\frac{1}{2}\right.\left(\left|{f}^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)-{f}^{\ast}\left({A}_{\phi (j)}\right)\times {g}^{\ast}\left({B}_{\varphi (j)}\right)\right|\right.+ \max \left\{\left|{f}^{\ast}\left({A}_{\phi (i)}\right)-{f}^{\ast}\left({A}_{\phi (j)}\right)\right|\right.,\left.\left.\left.\left|{g}^{\ast}\left({B}_{\varphi (i)}\right)-{g}^{\ast}\left({B}_{\varphi (j)}\right)\right|\right\}\right)\right)\\ {}\le \left(\frac{1}{2}\right.\left(\left|{f}^{\ast}\left({A}_{\phi (i)}\right)\times {g}^{\ast}\left({B}_{\varphi (i)}\right)-{f}^{\ast}\left({A}_{\phi (k)}\right)\times {g}^{\ast}\left({B}_{\varphi (k)}\right)\right|\right.\kern0.3em +\left.\left. \max \left\{\left|{f}^{\ast}\left({A}_{\phi (i)}\right)-{f}^{\ast}\left({A}_{\phi (k)}\right)\right|,\left|{g}^{\ast}\left({B}_{\varphi (i)}\right)-{g}^{\ast}\left({B}_{\varphi (k)}\right)\right|\right\}\right)\right)\end{array} $$

According to this proof, Eq. (14) satisfies the properties defined in Definition 10.

Extended Linguistic Z-Numbers TODIM Approach Based on the Choquet Integral

This section develops an extended TODIM approach based on the Choquet integral for MCDM problems with linguistic Z-numbers.

For an MCDM problem with linguistic Z-numbers, let A = {a ₁, a ₂,  … , a _m } be a discrete set of alternatives and let C = {c ₁, c ₂,  … , c _n } be a collection of criteria. Let $D={\left[{z}_{ij}\right]}_{m\times n}={\left[\left({A}_{\phi_{ij}},{B}_{\varphi_{ij}}\right)\right]}_{m\times n}$ be the decision-making matrix. The linguistic Z-numbers are $\left({A}_{\phi_{ij}},{B}_{\varphi_{ij}}\right)$, which represent the evaluation information of alternative a _i  (i = 1, 2,  ⋯ , m) under criteria c _j  (j = 1, 2,  ⋯ , n). The goal is to determine the rank of alternatives.

Preprocessing of Decision-Making Information

Because of evaluation information, especially qualitative evaluation information, decision-making in real-life situations is perplexing and diverse. Usually, this evaluation information is not in the form of Z-numbers. Thus, real-life evaluation information must be transformed into linguistic Z-numbers. This subsection introduces two methods for obtaining linguistic Z-numbers.

• 1. Consider the situation that the decision-making information is given directly by DMs. In this situation, DMs easily give evaluation information in the form of linguistic Z-numbers. A DM can choose a linguistic term as his evaluation for something from a specified linguistic term set and then can choose a linguistic term from another linguistic term set to represent his confidence in the evaluation.

• Example 5. A questionnaire about the iPhone 7 has seven options about the phone’s appearance: {very poor,   poor ,  slightly poor ,  fair ,  slightly good ,  good ,  very good}. Another five options, {uncertain , slightly  uncertain , medium , slightly  sure , sure} represent the respondents’ degree of confidence in choosing one of the first seven options. Tom is one of the respondents, and he chose the options “very good” and “sure” because he is a fan of iPhones. Tom’s evaluation about the iPhone 7’s appearance can be represented in the form of the linguistic Z-number (very  good, sure).

  • 2. Consider the situation that the decision-making information is based on a large number of known evaluation information. The specific steps for obtaining linguistic Z-numbers are as follows:

• (a) Determine the linguistic term set S ₁ = {s ₀, s ₁,  … , s _2l } according to known evaluation information, and determine the linguistic term set ${S}_2=\left\{{s}_0^{\prime },{s}_1^{\prime },\dots, {s}_{2k}^{\prime}\right\}$ to represent frequency information.

• (b) Calculate the frequency,f _i , of the linguistic term s _i  , s _i  ∈ S ₁ for alternative a _i with respect to criteria c _j .

• (c) Choose an appropriate linguistic term, ${s}_j^{\prime },{s}_j^{\prime}\in {S}_2$, as the reliability of the evaluation information, according to the correspondence between the linguistic term ${s}_j^{\prime }$ and frequency f _i .

• Example 6. Using the linguistic term sets A and B in Example 2, assume a website has 10,000 reviews about the service of a hotel, 8000 of which say the service of the hotel is “good.” Assume that if the frequency of an event occurring is between 80 and 90%, then we can use “usually” to describe the occurrence of the event. Obviously, the frequency of “good” is 0.8. Therefore, we can use the linguistic Z-number (good, usually) to evaluate the service of the hotel.

Procedures of the Linguistic Z-Numbers MCDM Method

• Step 1. Convert the decision-making information into linguistic Z-numbers and normalize the decision information.

Use the preprocessing method described in “Preprocessing of Decision-Making Information” section to obtain the decision information in the form of linguistic Z-numbers. Then, standardize the decision-making information. The types of criteria in MCDM can be divided into maximizing and minimizing criteria. For minimizing criteria, the negation operator in Definition 7 is used to normalize linguistic Z-numbers.

• Step 2. Confirm the fuzzy measure of criteria C.

For real decision-making problems, the fuzzy density g(x _i ) can be given by the DM. According to Eq. (9), the parameter λ can be determined. According to Eq. (7), the fuzzy measure g of criteria can be obtained.

• Step 3. Determine criteria weights.

Using the fuzzy measure obtained in Step 2, criteria weights can be obtained according to the following equation:

$$ {\omega}_{\sigma (j)}=\mu \left({A}_{\sigma (j)}\right)-\mu \left({A}_{\sigma \left(j+1\right)}\right),j=1,2,\dots, n, $$

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where A _σ(j) = {c _σ(j), c _{σ(j + 1)},  … , c _σ(n)},c _σ(n) = ∅, and (σ(1), σ(2),  … , σ(n)) is a permutation on (1, 2,  … , n); w _rj  = w _σ(j)/w _σ(r)(j = 1, 2,  … , n) is the reference weights of criteria c _σ(j) with respect to reference criteria; and w _σ(r) = max {w _j , j = 1, 2,  … , n}.

• Step 4. Calculate the dominance degree of alternative a _i over alternative a _k with respect to criteria C _j .

$$ {\phi}_{c_j}\left({a}_i,{a}_k\right)=\left\{\begin{array}{l}\sqrt{\frac{w_{rj}\left(d\left({z}_{ij},{z}_{kj}\right)\right)}{\sum_{j=1}^n{w}_{rj}}}\kern6.399996em s\left({z}_{ij}\right)>s\left({z}_{kj}\right),\\ {}0\kern13.00001em s\left({z}_{ij}\right)=s\left({z}_{kj}\right),\\ {}-\frac{1}{\theta}\sqrt{\frac{\sum_{j=1}^n{w}_{rj}\left(d\left({z}_{ij},{z}_{kj}\right)\right)}{w_{rj}}}\kern2.3em s\left({z}_{ij}\right)<s\left({z}_{kj}\right).\end{array}\right. $$

(16)

Obviously, Eq. (16) presents three scenarios: (1) if s(z _ij ) > s(z _kj ), then ${\phi}_{c_j}\left({a}_i,{a}_k\right)$ represents a gain; (2) if s(z _ij ) = s(z _kj ), then ${\phi}_{c_j}\left({a}_i,{a}_k\right)$ represents a nil; and (3) if s(z _ij ) < s(z _kj ), then ${\phi}_{c_j}\left({a}_i,{a}_k\right)$ represents a loss. The shape of the function ${\phi}_{c_j}\left({a}_i,{a}_k\right)$ is shown in Fig. 2. Above the horizontal axis, a concave curve represents gains; below the horizontal axis, a convex curve represents losses. The shape of the function ${\phi}_{c_j}\left({a}_i,{a}_k\right)$ is the same as the gain-loss function of prospect theory [44].

Fig. 2

Prospect value function in the TODIM method [42]

Parameter θ is the attenuation factor of the loss and denotes the degree of loss aversion of DMs. The greater the θ, the lower loss aversion [62]. Different choices of θ lead to different shapes of the prospect value function in TODIM, as illustrated in Fig. 2. The loss aversion coefficient usually is determined by designing experiments [63] to obtain the optimal value.

• Step 5. Calculate the overall dominance degree of alternative a _i over alternative a _k .

$$ \delta \left({a}_i,{a}_k\right)={\sum}_{l=1}^p{\phi}_{(l)}\left({a}_i,{a}_k\right)\left[\mu \left({B}_{\sigma (l)}-{B}_{\sigma \left(l-1\right)}\right)\right]+{\sum}_{l=p+1}^n{\phi}_{(l)}\left({a}_i,{a}_k\right)\left[\mu \left({A}_{\sigma (l)}-{A}_{\sigma \left(l+1\right)}\right)\right] $$

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where (ϕ ₍₁₎(a _i , a _k ), ϕ ₍₂₎(a _i , a _k ),  … , ϕ _(n)(a _i , a _k )) is a permutation such thatϕ ₍₁₎(a _i , a _k ) ≤ ϕ ₍₂₎(a _i , a _k ) ≤  …  ≤ ϕ _(p)(a _i , a _k ) ≤ 0 ≤ ϕ _(p + 1)(a _i , a _k ) ≤  …  ≤ ϕ _(n)(a _i , a _k ), andB _σ(l) = {c _σ(1), c _σ(2), …c _σ(l)}, c _σ(0) = ∅, A _σ(l) = {c _σ(l), c _{σ(l + 1)},  … , c _σ(n)}, and c _{σ(n + 1)} = ∅.

• Step 6. Calculate the comprehensive evaluation value of each alternative.

Calculate the comprehensive evaluation value of the alterative according the following expression:

$$ \xi \left({a}_i\right)=\frac{\sum_{k=1}^m\delta \left({x}_i,{x}_k\right)-\underset{i}{ \min }{\sum}_{k=1}^m\delta \left({x}_i,{x}_k\right)}{\underset{i}{ \max }{\sum}_{k=1}^m\delta \left({x}_i,{x}_k\right)-\underset{i}{ \min }{\sum}_{k=1}^m\delta \left({x}_i,{x}_k\right)} $$

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• Step 7. Rank the alternatives.

The ranking of alternatives can be determined according to the comprehensive evaluation value. The bigger ξ(a _i ), the better alternative A _i .

Illustrative Example

This section provides a practical example concerning the selection of medical inquiry applications (apps) to highlight the feasibility of our proposed approach. It provides a sensitivity analysis and a comparative analysis with existing methods to confirm the suitability and superiority of the proposed method.

Background

With the rapid development of mobile communication information technology and the popularization of smartphones, the use of numerous apps is becoming part of daily living for entertainment and convenience [64]. It is no exaggeration to say that some people believe they cannot live without the use of their apps. Compared with traditional personal computer software, the biggest feature of these apps is their universality and mobility. According to statistics, more than 2 million apps are available in the App Store (iOS) and more than 1 million apps are available in the Google Play Store (Android). Although a general classification system has formed and stabilized in the app market, since 2013, the number of the same types of apps have surged, and the homogenization phenomenon is serious. Therefore, understanding how to evaluate and choose an app is particularly important for consumers and developers.

With the continuous development of artificial intelligence technology research and application, some artificial intelligence apps, such as medical inquiry apps, bring convenience. Currently, an app evaluation website intends to evaluate five medical inquiry apps in current market. The five apps are a ₁ , a ₂ , a ₃ , a ₄ , a ₅. Many factors are involved in app evaluation, and four main factors are considered: application platform (c ₁: developers, attention, download, critic rating, the number of system platforms); user experience (c ₂: activity, frequency of use, total time of use, recognition, application rating, ease of operation, functional navigation, timeliness); visual foreground (c ₃: interface design, content attributes, basic performance, hardware properties); and network background (c ₄: network support, network services, security).

According to the first method of the preprocessing of decision-making information described in “Preprocessing of Decision-Making Information” section, several DMs are gathered to determine the evaluation information regarding the five medical apps with respect to these four criteria. We used the linguistic term sets A = {A ₀, A ₁, A ₂, A ₃, A ₄, A ₅, A ₆}={very  poor, poor, slightly  poor, fair , slightly  good , good , very  good} and B = {B ₀ , B ₁ , B ₂ , B ₃ , B ₄} = {uncertain , slightly  uncertain , medium , slightly  sure , sure}. The DMs can use the linguistic terms in A to evaluate the five apps with respect to the criteria and can use the linguistic terms in B to express their confidence in the evaluation information. For example, after discussion, the DMs are sure that the user experience (c ₂) of the third app (a ₃) is fair, that is, (fair, sure), that is, (A ₃, B ₄). Following preliminary treatment, the evaluation information can be represented in the form of linguistic Z-numbers (Table 1).

Table 1 Evaluation information given by DMs

In the following, we use the proposed approach to evaluate the five apps and choose the best one.

Illustration of the Proposed Method

To obtain the optimal alternatives, assume f ^∗(θ _i ) = F ₄(θ _i ) and g ^∗(θ _i ) = F ₁(θ _i ), and apply the following steps:

• Step 1. Convert the decision-making information into linguistic Z-numbers, and normalize the decision information.

The decision-making information is listed in Table 1. Because criteria in this example are maximized, we do not need to normalize the decision matrix in this step.

• Step 2. Determine the fuzzy measure of criteria C.

Assume g(c ₁) = 0.4, g(c ₂) = 0.27, g(c ₃) = 0.35, and g(c ₄) = 0.3. Calculate λ =  − 0.58 according to Eq. (9). Then, according to Eq. (7), the fuzzy density of criteria can be calculated as follows:

g(c ₁, c ₃) = 0.6688, g(c ₁, c ₄) = 0.6304, g(c ₂, c ₃) = 0.5652, g(c ₂, c ₄) = 0.523,g(c ₃, c ₄) = 0.5891, g(c ₁, c ₂, c ₃) = 0.834, g(c ₁, c ₂, c ₄) = 0.8017, g(c ₁, c ₃, c ₄) = 0.8524 , g(c ₂, c ₃, c ₄) = 0.7768, and g(c ₁, c ₂, c ₃, c ₄) = 1.

• Step 3. Determine criteria weights.

Criteria weights based on the fuzzy measure can be determined according to Eq. (15). The results are as follows: w ₁ = 0.17, w ₂ = 0.17, w ₃ = 0.32, and w ₃ = 0.35.

• Step 4. Calculate the dominance degree of alternative a _i over alternative a _k with respect to criteria C _j .

In the TODIM method, the value of θ usually is equal to 1 [48, 65]. To prevent the loss of generality, assume θ = 1. According to Eq. (16), the dominance degree of alternative A _i over alternative A _k with respect to criteria C _j can be obtained as follows:

$$ \begin{array}{c}\hfill \begin{array}{c}\hfill {\phi}_{c_1}\left({a}_i,{a}_k\right)=\left(\begin{array}{rrrrr}\hfill 0& \hfill 0.2030& \hfill 0.2607& \hfill -1.0699& \hfill 0.1544\\ {}\hfill -1.2061& \hfill 0& \hfill 0.1635& \hfill -1.0554& \hfill -0.7831\\ {}\hfill -1.5486& \hfill -0.9714& \hfill 0& \hfill -1.4344& \hfill -1.2477\\ {}\hfill 0.1801& \hfill 0.1776& \hfill 0.2414& \hfill 0& \hfill 0.1819\\ {}\hfill -0.9173& \hfill 0.1318& \hfill 0.2100& \hfill -1.0810& \hfill 0\end{array}\right),\hfill \\ {}\hfill {\phi}_{c_2}\left({a}_i,{a}_k\right)=\left(\begin{array}{rrrrr}\hfill 0& \hfill -0.9801& \hfill -1.3488& \hfill -1.4696& \hfill -1.1482\\ {}\hfill 0.1650& \hfill 0& \hfill -0.9265& \hfill -1.0951& \hfill -0.5981\\ {}\hfill 0.2270& \hfill 0.1560& \hfill 0& \hfill -0.5837& \hfill -0.9922\\ {}\hfill 0.2474& \hfill 0.1843& \hfill 0.0982& \hfill 0& \hfill 0.1544\\ {}\hfill 0.1933& \hfill 0.1007& \hfill 0.1670& \hfill -0.9173& \hfill 0\end{array}\right),\hfill \end{array}\hfill \\ {}\hfill {\phi}_{c_3}\left({a}_i,{a}_k\right)=\left(\begin{array}{rrrrr}\hfill 0& \hfill 0.4347& \hfill 0.3254& \hfill 0.4347& \hfill 0.3980\\ {}\hfill -1.3719& \hfill 0& \hfill -0.9094& \hfill 0& \hfill -1.0455\\ {}\hfill -1.0272& \hfill 0.2881& \hfill 0& \hfill 0.2881& \hfill 0.2377\\ {}\hfill -1.3719& \hfill 0& \hfill -0.9094& \hfill 0& \hfill -1.0455\\ {}\hfill -1.2562& \hfill 0.3313& \hfill -0.7502& \hfill 0.3313& \hfill 0\end{array}\right),\mathrm{and}\hfill \\ {}\hfill {\phi}_{c_4}\left({a}_i,{a}_k\right)=\left(\begin{array}{rrrrr}\hfill 0& \hfill 0.2206& \hfill 0.2396& \hfill 0.2486& \hfill 0.2206\\ {}\hfill -0.6367& \hfill 0& \hfill -0.6457& \hfill -0.6457& \hfill 0\\ {}\hfill -0.6915& \hfill 0.2238& \hfill 0& \hfill -0.8494& \hfill 0.2711\\ {}\hfill -0.7174& \hfill 0.2238& \hfill -0.8494& \hfill 0& \hfill 0.2238\\ {}\hfill -0.6367& \hfill 0& \hfill -0.7824& \hfill -0.6457& \hfill 0\end{array}\right).\hfill \end{array} $$

• Step 5. Calculate the overall dominance degree of alternative a _i over alternative a _k .

According to Eq. (17), the overall dominance degree of each alternative A _i over alternative A _k can be determined as follows:

$$ \delta \left({a}_i,{a}_k\right)=\left(\begin{array}{rrrrr}\hfill 0& \hfill -0.0063& \hfill -0.1232& \hfill -0.8061& \hfill -0.0773\\ {}\hfill -0.9370& \hfill 0& \hfill -0.5899& \hfill -0.7772& \hfill -0.7144\\ {}\hfill -0.9612& \hfill -0.2049& \hfill 0& \hfill -0.7686& \hfill -0.5548\\ {}\hfill -0.4914& \hfill 0.1577& \hfill -0.4044& \hfill 0& \hfill -0.2122\\ {}\hfill -0.7968& \hfill 0.1746& \hfill -0.3330& \hfill -0.6109& \hfill 0\end{array}\right). $$

• Step 6. Calculate the comprehensive evaluation value of each alternative.

Following on Step 5, the comprehensive evaluation value ξ(a _i ) of alternative a _i (i = 1, 2,  … , 5) can be obtained by Eq. (18) as follows:

ξ(a ₁) = 0.9698, ξ(a ₂) = 0, ξ(a ₃) = 0.2558, ξ(a ₄) = 1, ξ(a ₅) = 0.7023.

• Step 7. Rank the alternatives.

According to the comprehensive evaluation values, the final ranking of alternatives is a ₄ ≻ a ₁ ≻ a ₅ ≻ a ₃ ≻ a ₂, where a ₄ is the best alternative.

Sensitivity Analysis

To illustrate the influence of the LFSs and parameter θ on the decision-making process in this example, different LFSs and θ were used to rank the alternatives, and the ranking results are shown in Tables 2 and 3.

• Case 1: Consider the influence of parameter θ on the decision-making process.

Table 2 Ranking results using different θ in Case 1

Table 3 Ranking results using different LFSs in Case 2

In this case, we consider the influence of parameter θ on decision results, and therefore we only consider the situation that f ^∗(θ _i ) = F ₄(θ _i ) , g ^∗(θ _i ) = F ₁(θ _i ), whereas θ takes different values.

From the results in Table 2, we can find that the ranking results of the five alternatives generally were consistent, and a ₃ was the worst alternative. The ranking sequences of alternatives a ₁ and a ₄ were inconsistent when θ takes different values. When θ took different values, the comprehensive evaluation values of each alternative were as shown in Fig. 3. As we can see from Fig. 3 and Table 2, when 0.1 ≤ θ ≤ 1.1, the best alternative was a ₄, and the comprehensive value of alternative a ₁ increased. When θ = 1.2, the comprehensive value of alternative a ₁ was equal to 1, and the comprehensive value of alternative a ₄ was smaller than 1; at this moment, a ₁ was the best alternative. The inconsistency of rankings may be caused by the different attitudes of DMs toward losses. As noted in the “Procedures of the Linguistic Z-Numbers MCDM Method” section, parameter θ represents the attenuation factor of losses. In the TODIM method, when θ < 1, the effect of loss is magnified. Conversely, when θ > 1, the effect of loss is reduced. If the value of θ is smaller, the optimal alternative obtained is usually a minimum loss alternative; if the value of θ is larger, the optimal alternative is usually a greater income alternative, and even the optimal alternative may produce losses under some criteria. Therefore, comparing alternative a ₁ with alternative a ₄, the loss of alternative a ₄ is lower, and the income of alternative a ₄ is also lower. Therefore, if the DMs are risk averse, then they will choose alternative a ₄; if the DMs prefer risk, then they will choose alternative a ₁.

Fig. 3

Comprehensive evaluation values of each alternative

• Case 2: Consider the influence of LFS on the decision-making process.

In this case, we consider only the situation that θ = 1, f ^∗(θ _i ) = F ₄(θ _i ), and g ^∗(θ _i ) takes values from F ₁(θ _i ) to F ₄(θ _i ). Analyses of other cases are similar.

Table 3 shows four kinds of different semantic environments in which a combination of different LFSs are used. These four situations are suitable for different DMs and complex decision-making environments, especially when the DMs’ semantic cognition is considered. From the results in Table 3, we can find that the ranking results of the five alternatives were almost totally different. The reason for this may be that DMs’ preferences and attitudes toward loss are manifested by their use of different LFSs to represent evaluation information. For example, when f ^∗(θ _i ) = F ₄(θ _i ), according to the curve of Formula (5) in Fig. 1, when evaluating for the good aspect, DMs’ sensitivity regarding the gap between “good” and “very  good” was greater than the gap between “slightly  good” and “good.” Conversely, when evaluating for the bad aspect, DMs were more sensitive to the gap between “very  poor” and “poor” than to the gap between “poor” and “slightly  poor.” When g ^∗(θ _i ) = F ₁(θ _i ), according to the curve of Formula (2) in Fig. 1, the evaluation scale of linguistic information is averaged. That is, the evaluation information given by DMs is not affected by subjective cognition.

In summary, Table 3 presents the ranking results obtained by using different LFSs to deal with the two components of linguistic Z-numbers. The two components of linguistic Z-numbers can be represented by different LFSs because they represent evaluation preferences according to different aspects. One is the evaluation preference regarding external things and the other concerns the specific DM. Using different LFSs to handle the first component of linguistic Z-numbers emphasizes the preferences of DMs when giving evaluation on external things. Using different LFSs to handle the second component of linguistic Z-numbers emphasizes the degrees of confidence in the evaluation information given by the DM. Therefore, DMs can flexibly choose the LFSs that suit their preferences, thus ensuring the rationality and accuracy of results.

Comparative Analysis and Discussion

To verify the feasibility of the proposed decision-making approach based on linguistic Z-numbers, we conducted a comparative analysis based on the same illustrative example.

This comparative analysis included two cases. One incorporated the methods outlined by Kang et al. [29] and Yaakob and Gegov [31], which we compared to the proposed method using Z-numbers. In the second case, we compared the method introduced by Wang et al. [38] to the proposed approach using linguistic information.

• Case 1: The proposed method was compared with other methods using Z-numbers.

In the method by Kang et al. [29], first, the two components of Z-numbers were transformed into corresponding triangular fuzzy numbers, respectively. Then, the triangular fuzzy numbers were converted to crisp values. Through the multiplication operation, we transformed the Z-numbers into corresponding crisp values. Finally, we ranked the alternatives by the value priority weight of each alternative. According to this method, the priority weight values of alternatives were as follows: p ₁ = 0.7182, p ₂ = 0.4304, p ₃ = 0.5188, p ₄ = 0.5152, and p ₅ = 0.5408.

The method by Yaakob and Gegov [31] has two main steps (transformation of evaluation and ranking obtained through the method of technique for order preference by similarity to an ideal solution (TOPSIS)). First, we transformed the evaluation information represented in the form of Z-numbers into trapezoidal fuzzy number according to the converting method [31]. Then, we used an extended TOPSIS method to rank the alternatives. The closeness coefficients of alternatives were cc ₁ = 0.1821, cc ₂ = 0.1491, cc ₃ = 0.1415, cc ₄ = 0.1646, and cc ₅ = 0.1754.

Table 4 summarizes the ranking results according to different methods in Case 1, based on the criteria weights obtained by the proposed method with f ^∗(θ _i ) = F ₄(θ _i ), g ^∗(θ _i ) = F ₁(θ _i ), and θ = 1.

Table 4 Ranking results using different methods in Case 1

Table 5 shows differences in the rankings. The reasons for these inconsistencies follow:

• (1) The inconsistency of ranking results between the proposed method and that of Kang et al. [29] was the order of a ₁, a ₄, and a ₅. The operations of the method by Kang et al. [29] were based on transforming the linguistic terms into triangular fuzzy numbers. These operations are simple, but inevitably they result in loss and distortion of information. The present study’s newly defined operations of linguistic Z-numbers were based on different LFSs and were applicable to different semantic environments. DMs could deal directly with linguistic Z-numbers, and the proposed method took the interactivity of criteria into consideration. Therefore, the decision-making results obtained by the proposed method were more convincing and more in line with the actual situation.

• (2) The final ranking using the method by Yaakob and Gegov [31] was almost totally different from the proposed approach. First, the transformation from Z-numbers to trapezoidal fuzzy numbers indeed avoided complex calculations; however, it caused loss and distortion of the original evaluation. The extended TOPSIS method developed to rank alternatives actually was based on trapezoidal fuzzy numbers and not on Z-numbers. Thus, it was based on the assumption that DMs are completely rational. In real decision-making processes, however, DMs are not fully rational. Thus, the proposed method based on bounded rationality could address these behaviors more appropriately.

• Case 2: The proposed method was compared with other methods using linguistic information.

Table 5 Transformed evaluation information

To apply the method by Wang et al. [38], the reliability measure of the evaluation information must be changed into the same value. Usually, we believe the linguistic information is completely reliable when handling MCDM problems with linguistic information. Therefore, we transformed the reliability measure of the evaluation information in Table 1 into the highest level. The first component of the linguistic Z-number remained unchanged. The transformed information is shown in Table 5.

The method by Wang et al. [38] has three phases. First, we transformed the linguistic terms into clouds. Then we aggregated the transformed evaluation information by the cloud-weighted arithmetic averaging operator and ranked the alternatives through the average values of the cloud score functions. The average values of alternatives’ cloud score functions were S ₁ = 5.9545, S ₂ = 4.2288, S ₃ = 5.5533, S ₄ = 4.5902, and S ₅ = 4.506.

Table 6 summarizes the ranking results according to different methods for Case 2, based on the criteria weights obtained by the proposed method with f ^∗(θ _i ) = F ₄(θ _i ), g ^∗(θ _i ) = F ₁(θ _i ), and θ = 0.5.

Table 6 Ranking results using different methods for Case 2

As depicted in Table 6, the ranking results obtained by Wang et al.’s [38] method generally coincided with those obtained by the proposed method. The best alternative was a ₁, and the worst was a ₂. The inconsistence was reflected in the ranking of alternatives a ₃ and a ₄. In the method by Wang et al. [38], the reliability of decision-making information was not considered. In contrast, the proposed method provided a new representation for real-life decision-making information. It not only described the decision information as flexibly and accurately as the linguistic terms but also characterized the reliability of the information. The method by Wang et al. [38] was based on the operators of the cloud model, whereas the proposed method was based on the classic TODIM method and Choquet integral, which is more approximate to real decision-making environments. Moreover, the calculation complexity of the proposed method was relatively lower than that by Wang et al. [38].

On the basis of this comparative analysis, the proposed method for solving MCDM problems using linguistic Z-numbers has the following advantages:

• (1) Linguistic Z-numbers are more flexible for expressing cognitive information, and the representation of linguistic Z-numbers is simple. It is more convenient to transform daily information into linguistic Z-numbers. Moreover, it is easy for DMs to give evaluation information directly in the form of linguistic Z-numbers.

• (2) The operations of linguistic Z-numbers are defined on the basis of LFSs, which can yield different results using different LFSs f ^∗ and g ^∗. Thus, DMs can flexibly select LFSs f ^∗ and g ^∗ depending on their preferences and the actual semantic environment. Furthermore, the ranking results of our comparative analysis verified the feasibility and validity of the defined operations of linguistic Z-numbers.

• (3) The proposed method was based on the classical TODIM method and Choquet integral. Compared with traditional methods based on complete rationality (namely, the DMs are fully rational when evaluating alternatives), the proposed method not only extended the classical TODIM method to linguistic Z-numbers decision environments but also considered the bounded rational of DMs. Compared with methods based on the TODIM method, the proposed method inherited the features of the TODIM method and considered the interactivity of criteria. Moreover, the proposed method did not need to transform the evaluation information into fuzzy numbers or clouds. Thus, the calculation process was straightforward and the calculation complexity was relatively low.

Conclusions

Linguistic Z-numbers have applied the advantages of linguistic term sets and Z-numbers. They can flexibly express cognitive information as well as effectively characterize the reliability of information. Therefore, it is of great significance to study MCDM methods with linguistic Z-numbers. This paper introduced the definitions and operations of linguistic Z-numbers as well as a comparison method and distance measure for linguistic Z-numbers. Then, considering the bounded rational of DMs and the interactivity of criteria, we presented an extended TODIM method based on the Choquet integral for linguistic Z-numbers MCDM problems. Finally, an illustrative example and comparative analysis showed the application of the proposed approach.

The main contribution of this study is a proposed method that simply and reliably represents human cognition while also taking into account the interactivity of criteria and the cognition toward loss of DMs. The defined linguistic Z-number is an effective tool for presenting cognitive information and considering the reliability of the information. The developed MCDM approach combines the advantages of Choquet integral and TODIM, which is more feasible and practical than other methods. The proposed method is fairly flexible to use. Results may change by using different LFSs according to the DMs’ preferences and actual semantic situations.

In future research, studies on linguistic Z-numbers must be extended to handle more practical problems in other areas, such as artificial intelligence, image recognition, and recommendation systems. This paper listed only four possible LFSs as examples, and future LFS studies should explore more functions that satisfy both the definition of LFS and reflect the cognitive preference of DMs. In the TODIM method, the value of θ usually equals 1; thus, this paper also assumed that θ equaled 1 and discussed situations in which θ took different values. Future research should try to use the genetic algorithm [66] or qualitative flexible [67, 68] approach to obtain the optimal value of θ.