A novel linguistic decision-making method based on the voting model for large-scale linguistic decision making

Abstract

The notable characteristic of large-scale linguistic decision-making problems is that there are so many decision makers who provide linguistic assessments by using fuzzy linguistic representation models. In real-world applications, fuzzy linguistic terms mean different things for different people, and linguistic assessments based on different linguistic representation models may be simultaneous in the same large-scale linguistic decision-making problems. To this end, a novel linguistic decision-making method based on the voting model is proposed in the paper to deal with multi-linguistic assessments provided by decision makers. In large-scale linguistic decision process, evaluation-based voting is defined and multi-linguistic decision matrix is designed to represent multi-linguistic assessments provided by decision makers by using different linguistic representation models, and properties of the decision matrix are analyzed to show that linguistic assessments based on different linguistic representation models can be simultaneously represented. Based on multi-linguistic decision matrix, a new linguistic decision-making framework is developed to deal with large-scale linguistic decision-making problems with multi-linguistic assessments, in which normalization of multi-linguistic decision matrix and trust degrees of linguistic terms are contained, and more important, based on trust degrees of linguistic terms and 2-tuple fuzzy linguistic aggregation operators, an uniform fusion method of multi-linguistic assessments is proposed to aggregate multi-linguistic assessments of large-scale linguistic decision-making problems. Finally, user experiences of shared bikes, which are a large-scale linguistic decision-making problem in real-world applications, are employed to show the new decision-making framework and the uniform fusion method of multi-linguistic assessments, and furthermore, compared with existing linguistic decision-making methods analyzed in the example, it seems that multi-linguistic decision matrix and the uniform fusion method are useful and effective tools to deal with large-scale linguistic decision-making problems with multi-linguistic assessments.

1 Introduction

For any decision-making problems, how to reasonably evaluate and express assessments of alternatives with respect to criteria is the first and important step. Because natural language is the expression way closest to one’s cognitive process and expression habits in our daily lives, and more important natural language or fuzzy linguistic terms provide a more direct way to represent imprecise or uncertain information, linguistic decision-making (LDM) problems have been paid close attention in recent years, in which assessments of alternatives provided by decision makers are represented by fuzzy linguistic terms or complex linguistic expressions (Li et al. 2020; Malhotra and Gupta 2020; Morente-Molinera et al. 2015; Rodríguez et al. 2016, 2013, 2014; Martinez and Herrera 2012). Up to now, many fuzzy linguistic representation models have been proposed to deal with complex LDM problems, such as considering the natural meanings ordering of natural languages and continuity of linguistic terms, 2-tuple fuzzy linguistic representation model (2-TLM) (Herrera and Martínez 2000) is utilized to represent aggregation of single-linguistic assessments of alternatives, and theoretical advantages of 2-TLM are computational simplicity, continuous linguistic information, no loss information, accuracy and understandability in dealing with LDM problems (Martinez and Herrera 2012; Martínez et al. 2015). To describe hesitancy of decision makers among linguistic terms, hesitant fuzzy linguistic term set (HFLTS) (Rodríguez et al. 2012) is proposed to overcome limitation of single-linguistic assessments and provide a great flexible representations of linguistic assessments in LDM problems. Considering probability distribution on linguistic terms, probabilistic linguistic term set (PLTS) (Pang et al. 2016) is presented to represent different importance or preference of linguistic terms in hesitant environment, which overcomes limitation of the same importance of linguistic terms in HFLTS. Combining the finite linguistic terms with discrete fuzzy number, the fuzzy linguistic model based on discrete fuzzy numbers (FLM-DFN) (Massanet et al. 2014) is developed to represent different importance or preference of linguistic terms in possible environment of LDM process.

Intuitively, 2-TLM, HFLTS, PLTS and FLM-DFN aim to express different imprecise or uncertain linguistic information, provide us tools to more reasonable express linguistic assessments of alternatives than using single-linguistic term in $S=\{s_0, \cdots , s_g\}$ and obtain acceptable decision-making results. However, they also produce a significant challenge in decision-making process. For a LDM problem, because fuzzy linguistic terms mean different things for different decision makers, decision maker $d_{i}$ maybe provide single-linguistic assessments of alternatives, but decision maker $d_{i'} (i'\ne i)$ maybe utilize 2-TLM, HFLTS, PLTS or FLM-DFN to assess alternatives according to his/her knowledge level (social context or experience); this means that multi-linguistic assessments based on these linguistic representation models are simultaneous in the LDM problem. In the framework of decision matrix, how to distinguish multi-linguistic assessments of alternatives is a significant challenge. In addition, aggregation of multi-linguistic assessments is a problem because no any linguistic aggregation operator can be utilized to simultaneously aggregate them. The situation is very common in large-scale linguistic decision environments, such as in social networks and e-democracy, huge amounts of users take part in decision making and provide linguistic assessments of alternatives based on fuzzy linguistic representation models (Ding and Palomares 2020; Zhong and Xu 2020; Zhang et al. 2017; Labella et al. 2018; Rodriguez et al. 2018), and multi-linguistic assessments instead of one kind of linguistic assessments may be more suitable to large-scale linguistic decision-making problems (LS-MLDM).

1.1 Related works

All the time, representation of linguistic assessments is an important issue in LDM problems. After computing with words proposed by Zadeh (1996), many basic linguistic representation models have been proposed to represent linguistic assessments of alternatives. In recent years, the increasing complexity of real-world LDM problems drives researchers to consider generalization linguistic representation model or complex linguistic expression (such as “between fair and good,” “at least good”) to naturally and freely assess alternatives with respect to criteria (Wang et al. 2018).

In Wu et al. (2020), based on a set whose elements are some subsets of a fixed linguistic term set $S=\{s_0, \cdots , s_g\}$ and distribution information of the elements of the set, a flexible linguistic expression (FLE) is design to generalize or improve the construction of existing hesitant linguistic expressions. Formally, if linguistic terms of an element are consecutive in S and distribution information of the element is 1, then the FLE is reduced to HFLTS of S. If each element of the FLE is a single element subset of S, then the FLE may be reduced to PLTS, proportional HFLTS (Chen et al. 2016) or possibility distribution for HFLTS (Wu and Xu 2016) according to different distribution information. Till now, many kinds of hesitant linguistic expressions have been proposed to represent linguistic assessments of alternatives in hesitant linguistic decision environments (Wei et al. 2018; Liu et al. 2019; Gou et al. 2017), and it seems that FLE is an uniform representation of existing hesitant linguistic expressions with distribution information on linguistic terms (Wu et al. 2021; Chao et al. 2021).

In Dong et al. (2009), the numerical scale function from linguistic term set to real number set is designed and the numerical scale model of linguistic term set is proposed to extend 2-TLM under the numerical scale; formally, the numerical scale function transforms each linguistic term into the numerical index of linguistic term, and then operations, aggregations and preference relations on linguistic terms can be transformed into computations on numerical indexes of linguistic terms. By setting different numerical scale functions or interval numerical scale functions, it seems that the numerical scale model of linguistic term set provides a connection framework among different versions of 2-TLM, such as the proportional 2-TLM and the hesitant unbalanced linguistic model based on a linguistic hierarchy (Dong et al. 2016, 2013; Herrera et al. 2008; Wang and Hao 2006).

In Li et al. (2017), considering “fuzzy linguistic terms mean different things for different people,” a personalized individual semantics model of linguistic terms is developed, which aims to personalize individual semantics by means of an interval numerical scale and 2-TLM, i.e., different people provides different numerical scale function or interval numerical scale function to transform each linguistic term into numerical index. Intuitively, the personalized individual semantics model provides a connection framework among different versions of 2-TLM and the hesitant unbalanced linguistic model based on a linguistic hierarchy in large-scale linguistic decision environments (Li et al. 2018, 2019).

In Gou et al. (2017), double hierarchy linguistic term set is proposed to express the real thoughts of experts and handle complex linguistic information in large-scale linguistic decision environments. Formally, double hierarchy linguistic term set consists of two simple linguistic hierarchies, in which the first hierarchy linguistic term set is the main linguistic hierarchy and the second hierarchy is the linguistic feature or detailed supplementary of each linguistic term in the first hierarchy linguistic term set. By combining simple linguistic terms with linguistic feature, double hierarchy linguistic term set extends many linguistic representation and expresses complex linguistic information.

To our knowledge, many scholars concentrate to develop or generalize new linguistic representation models to represent and aggregate more complex linguistic assessments than classical representation models in large-scale linguistic decision environments. Theoretically, it seems that new linguistic representation model has more complex forms than classical model, and it always means more complex analysis, computation and difficult processing in large-scale linguistic decision making. In fact, 2-TLM, HFLTS, PLTS or FLM-DFN has many interesting and important advantages, such as simple forms, easy interpretation, accuracy, understandability, and computational simplicity; these advantages maybe lose in the new linguistic representation model due to adding new uncertain information.

1.2 Our contribution

In the paper, the voting model (also called as district-based election [37]) is employed in the decision process when evaluation information is provided by a huge amounts of decision makers, accordingly multi-linguistic decision matrix is designed to represent multi-linguistic assessments of alternatives. In a new LS-MLDM framework, an uniform fusion method is developed to simultaneously fuse multi-linguistic assessments of alternatives. Major contributions of the paper are summarized as follows:

• 1. The voting model is employed in large-scale decision process, which can help decision makers or users providing linguistic assessments of alternatives according to their expert knowledge and linguistic representation models. Then, multi-linguistic decision matrix is designed to represent different linguistic assessments of alternatives provided by different users, and properties show that multi-linguistic assessments of alternatives can be represented and distinguished in the multi-linguistic decision matrix;

• 2. A new decision-making framework to deal with LS-MLDM problems is developed, and the refined step can help decision makers providing linguistic assessments of alternatives and multi-linguistic decision matrix can uniformly represent multi-linguistic assessments of alternatives;

• 3. An uniform fusion method is proposed to aggregate multi-linguistic assessments, where trust degrees of linguistic terms can be computed in the multi-linguistic decision matrix and the uniform fusion method is presented by using trust degrees of linguistic terms. Compared with existing linguistic aggregation operators, the uniform fusion method is an useful and effective tool to deal with LS-MLDM problems.

The rest of the paper is structured as follows: In Sect. 2, basic fuzzy linguistic representation models and linguistic decision matrix are briefly reviewed. In Sect. 3, the voting model in large-scale linguistic decision-making environment is analyzed and multi-linguistic decision matrix is proposed to represent multi-linguistic assessments of alternatives, and then several interesting and important properties of the matrix are discussed. In Sect. 4, a new decision-making framework and main decision resolution scheme are presented to deal with LS-MLDM problems, and then based on normalization of multi-linguistic decision matrix and trust degrees of linguistic terms, an uniform fusion method is proposed to aggregate multi-linguistic assessments of alternatives. In Sect. 5, user experience of shared bikes as a LS-MLDM problem is employed to show the proposed method and make comparative analysis with existing linguistic decision-making methods. Sect. 6 represents conclusion and future works.

2 Preliminaries

In the section, basic fuzzy linguistic representation models are briefly reviewed and linguistic decision matrix of LDM problems is recalled.

2.1 Basic fuzzy linguistic representation model

Fuzzy linguistic representation model is the main issue of computing with words; in Yan et al. (2019), computing with words is divided into three categories, i.e., via fuzzy sets of linguistic values, via fuzzy logic or algebra on the set of linguistic values and via an ordered structure of linguistic values. In LDM methods, computing with words via an ordered structure of linguistic values is also called as the symbolic model of fuzzy linguistic representation (Herrera and Martínez 2000), 2-TLM is the foundation of the symbolic model; formally, 2-TLM is consisted by a pair of elements $(s_{j}, \alpha )$, where linguistic term $s_{j}$ is in an primary linguistic term set $S=\{s_0, \ldots , s_g\}$ and $\alpha \in [-0.5, 0.5)$ is a numerical value that represents the value of the symbolic translation.

Definition 1

Herrera and Martínez (2000) Let $\beta \in [0, g]$ be the result of an aggregation of the indices of a set of linguistic terms assessed in an primary linguistic term set $S=\{s_0, s_1,$ $\ldots , s_g\}$, i.e., the result of a symbolic aggregation operation. Let $j=$round$(\beta )$ and $\alpha =\beta -j$ be two values such that $j\in [0, g]$ and $\alpha \in$ $[-0.5, 0.5)$, then $\alpha$ is called a symbolic translation.

where round$(\cdot )$ is the usual rounding operation. Theoretically, the symbolic translation $\alpha$ of linguistic term $s_{j}$ supports “the difference of information” between a counting of information $\beta$ and the closest linguistic term $s_{j}\in S (j=$ round$(\beta ))$. Furthermore, 2-TLM provides transformation between numerical values of [0, g] and 2-tuple linguistic terms $(s_{j}, \alpha )$ on S:

$$\begin{aligned}&\varDelta : [0, g] \longrightarrow S\times [-0.5, 0.5), \beta \mapsto \varDelta (\beta )=(s_j, \alpha ). \end{aligned}$$

(1)

$$\begin{aligned}&\varDelta ^{-1}: S\times [-0.5, 0.5)\longrightarrow [0, g], \nonumber \\&\quad (s_j, \alpha )\mapsto \varDelta ^{-1}(s_j, \alpha )=\beta =j+\alpha . \end{aligned}$$

(2)

In the paper, all 2-tuple linguistic terms on S are denoted by $TS=$ $\{(s_j, \alpha )|s_j\in S,\alpha$ $\in [-0.5, 0.5)\}$. The new decision-making framework (Martínez et al. 2015) of LDM problems based on 2-TLM and its generalizations, such as multi-granular linguistic terms, the linguistic hierarchy and unbalanced linguistic assessments, have been widely studied (Malhotra and Gupta 2020; Morente-Molinera et al. 2015; Zhang et al. 2017; Dong et al. 2016, 2013).

Due to the complexity of decision-making problems, decision makers maybe hesitate to provide several linguistic terms instead of single-linguistic term in decision processes, HFLTS is proposed to describe hesitancy of decision makers among different linguistic terms, and up to now, it has become an important linguistic representation model and been widely utilized in LDM problems.

Definition 2

Rodríguez et al. (2012) Let $S=\{s_{0}, \ldots , s_{g}\}$ be a linguistic term set, an HFLTS, $H_{S}$, is an ordered finite subset of the consecutive linguistic terms of S.

According to Definition 2, an HFLTS on S is formally expressed by $H_{S}=\{s_{k},$ $s_{k+1}, \ldots , s_{k+t}\}$ such that $k\in \{0, 1, \ldots , g\}$ and $k+t\le g$, Compared with single-linguistic term, HFLTS provides different forms to represent linguistic assessments of alternatives. In addition, let $H^{1}_{S}$ and $H^{2}_{S}$ be two HFLTSs on S, the following basic operations on $H^{1}_{S}$ and $H^{2}_{S}$ are always used to manage hesitant fuzzy linguistic assessments in LDM problems: (1) lower bound: $H_{S^{-}} = min(s_{i}) = s_{j}$, $s_{i }\in H_{S}$ and $s_{i}\ge s_{j} \forall i$; (2) upper bound: $H_{S^{+}} = max(s_{i}) = s_{j}$, $s_{i }\in H_{S}$ and $s_{i} \le s_{j} \forall i$; (3) complement: $H^{c}_{S} = S - H_{S} = \{s_{i}|s_{i}\in S$ and $s_{i} \nsubseteq H_{S}\}$; (4) union: $H^{1}_{S}\cup H^{2}_{S} =\{s_{i}|s_{i}\in H^{1}_{S}$ or $s_{i}\in H^{2}_{S}\}$; (5) intersection: $H^{1}_{S}\cap H^{2}_{S} =\{s_{i}|s_{i}\in H^{1}_{S}$ and $s_{i}\in H^{2}_{S}\}$; (6) envelope: $env(H_{S}) = [H_{S^{-}}, H_{S^{+}}]$. In applications, HFLTS overcomes limitation of the number of linguistic terms and provides a flexible form to represent hesitancy among different linguistic terms. Now HFLTS decision methods have become hot in LDM (Liao and Xu 2015; Krishankumar et al. 2021; Wu et al. 2019; Wu and Xu 2018; Xu et al. 2017; Liao et al. 2015).

In hesitant fuzzy linguistic term set $H_{S}=\{s_{k}, s_{k+1}, \ldots , s_{k+t}\}$, each linguistic term $s_{k+t'} (t'\le t)$ has the same importance; however, the situation does not always happen in real-world LDM problems. In fact, decision makers may prefer some of linguistic terms in $H_{S}$, i.e., linguistic terms of $H_{S}$ may be with probabilities, possibilities, belief degrees or membership degrees (Pei et al. 2012). PLTS is proposed to describe HFLTS with probability distribution.

Definition 3

Pang et al. (2016) Let $S=\{s_0, \ldots , s_g\}$ be linguistic term set. Then, $L(p)=\{L^{k}(p^{k})|L^{k}\in S, p^{k}\ge 0, k=1, \ldots ,$ $\sharp L(p), \sum ^{\sharp L(p)}_{k=1}p^{k}\le 1\}$ is used to indicate a probabilistic linguistic term set, where $L^{k}(p^{k})$ is linguistic term $L^{k}$ associated with probability $p^{k}$ and $\sharp L(p)$ is the amount of all linguistic terms in L(p).

In brief, a PLTS can be described as $L(p)=\{s_{k}(p_{k}), \ldots , s_{k+t}(p_{k+t})\}$ such that each $p_{k+t'}\in [0, 1]$ and $\sum ^{t}_{t'=0}p_{k+t'}\le 1$. If $\sum ^{t}_{t'=0}p^{k+t'}=1$, then L(p) is a PLTS with the complete information of probabilistic distribution. If $\sum ^{t}_{t'=0}p^{k+t'}<1$, then L(p) is a PLTS with the non-complete information of probabilistic distribution. Now PLTS decision methods, including operations, aggregation operators and consensus-based decision, have been studied (Bai et al. 2017; Feng et al. 2020; Jiang and Liao 2020; Wu et al. 2019; Liu and Teng 2019; Mao et al. 2019; Chen et al. 2018, 2019, 2018; Chen and Chin 2017; Yang et al. 2019). By considering subjective evaluations of linguistic terms, FLM-DFN provides a flexible way to manage multi-granular linguistic terms in LDM.

Definition 4

Massanet et al. (2014) Let finite chain $L_{n}=\{0, \ldots , n\}$. We call a subjective evaluation to discrete fuzzy number belonging to the partially ordered set $A^{L_{n}}_{1}$.

Formally, a fuzzy subset $A: R\longrightarrow [0, 1]$ is called as a discrete fuzzy number if its support is finite (Voxman 2001), i.e., there exist $x_{1}, \ldots , x_{n}\in R$ with $x_{1}\le \ldots \le x_{n}$ such that $supp(A)=\{x_{1}, \ldots , x_{n}\}$ and there are s and t with $1\le s\le t\le n$ such that 1) $A(x_{i})=$1 for any i with $s\le i\le t$ (core); 2) $A(x_{i})\le A(x_{j})$ for each i, j with $1\le i\le j$ $\le s$; 3) $A(x_{i})\ge A(x_{j})$ for each i, j with $t\le i\le j$ $\le n$. According to Definition 4 and a bijective mapping between $L_{n}$ and $S=\{s_0, \ldots , s_n\}$, a FLM-DFN on S is described as $A=\{\mu _{k}/s_{k}, \ldots , \mu _{k+t'}/s_{k+t'}, \ldots , \mu _{k+t}/s_{k+t}\}$, in which each $\mu _{k+t'}\in [0, 1]$ and there exists $t'$ such that $\mu _{k}\le \mu _{k+1}\le \cdots , \mu _{k+t'}=1$ and $\mu _{k+t'}\ge \ldots \ge \mu _{k+t}$. In fact, FLM-DFN can be utilized to describe subjective evaluations of decision makers by membership degrees on linguistic terms, and FLM-DFN decision methods, property analysis or aggregation operators of FLM-DFNs have been studied (Rieraa et al. 2015; Riera and Torrens 2014, 2015; Ma et al. 2019; Roszkowska and Kacprzak 2016; Zhao et al. 2019).

2.2 Linguistic decision matrix

In LDM methods, linguistic assessments of alternatives with respect to criteria are uniformly represented in alternatives-criteria decision matrix, i.e., rows of the matrix are alternatives and columns are criteria, elements of matrix are linguistic assessments provided by the decision maker, i.e., the decision maker employs a fuzzy linguistic representation model to represent linguistic assessments of alternatives. Formally, a LDM problem is that decision makers $M=\{d_{1}, \ldots , d_{m}\}$ are asked to assess a set of alternatives $A=\{a_{1}, \ldots , a_{n}\}$ with respect to $C=\{c_{1}, \ldots , c_{r}\}$, where linguistic assessments of alternatives are selected from linguistic term set $S=\{s_{0}, \ldots , s_{g}\}$, and then alternatives-criteria decision matrix provided by decision maker $d_{i} (i=1, \ldots , m)$ is

If $e_{jk}^{i}\in S (j\in \{1, \ldots , n\}, k\in \{1, \ldots , r\}, i\in \{1, \ldots , m\})$, then the decision maker $d_{i}$ uses single-linguistic term to assess the alternative $a_{j}$ with respect to the criterion $c_{k}$. If $e_{jk}^{i}\in TS$, then $d_{i}$ uses 2-TLM on S to assess $a_{j}$ with respect to $c_{k}$. If $e_{jk}^{i}$ is an HFLTS (PLTS or FLM-DFN) on S, then $d_{i}$ uses HFLTS (PLTS or FLM-DFN) to assess $a_{j}$ with respect to $c_{k}$. Theoretically, limitation of alternatives-criteria decision matrix is higher-dimensional matrix and computational complexity. In addition, it is difficult to represent, distinguish and aggregate linguistic assessments when 2-TLM, HFLTS, PLTS and FLM-DFN are simultaneous in the alternatives-criteria decision matrix.

To overcome limitations of alternatives-criteria decision matrix, Pei et al proposed alternatives-linguistic terms decision matrix to represent linguistic assessments of alternatives with respect to each criterion (Pei et al. 2019), which is inspired by “every decision maker votes or assigns t (true or 1) or f (false or 0) on linguistic terms according to alternative with respect to the criterion,” i.e.,

where $m_{jl}^{ik}\in \{0, 1\} (l\in \{0, \ldots , g\})$, $m_{jl}^{ik}=1$ means that $d_{i}$ uses linguistic term $s_{l}\in S$ to assess alternative $a_{j}$ with respect to the criterion $c_{k}$, $m_{jl}^{ik}=0$ means that $d_{i}$ does not use $s_{l}$ to assess alternative $a_{j}$. Intuitively, alternatives-linguistic terms decision matrix $D_{ik}$ is the refinement of alternatives-criteria decision matrix, and its advantages are reduction of dimension and distinction of linguistic assessments in distinct, partial unknown or hesitant linguistic environments. However, in many large-scale situations, huge amounts of decision makers employ different fuzzy linguistic representation models to represent linguistic assessments; according to their knowledge level (social context or experience), it is highly desirable to develop more effective linguistic decision matrix to represent, distinguish and handle multi-linguistic assessments of alternatives in LS-MLDM problems.

3 Multi-linguistic decision matrix derived by the voting model

In the section, the voting model is district-based election, which is used in large-scale linguistic decision-making environment. Then, evaluation-based voting is defined, multi-linguistic decision matrix is proposed, and its several interesting and important properties are discussed to show multi-linguistic assessments of alternatives can be represented in the decision matrix.

3.1 District-based election

A voting mechanism is analogous to a valuation in mathematical logic, in which a semantics is an assignment of ’true’ or ’false.’ By extending ’true’ or ’false’ into [0, 1] representing the degree of confidence or scepticism of the agent, a voting mechanism for fuzzy logic has been proposed to computing with words (Lawry 1998). Recently, many researchers extensively focus on online social voting and many interesting new features of online social voting have been discussed, which are widely applied in recommendation systems, electoral systems, personalized advertising, market research or public opinion analysis (Wang et al. 2020; Yang and Liang 2017).

Definition 5

Filos-Ratsikas et al. (2020) A district-based election is defined as a tuple $\varepsilon =({\mathcal {M}}, {\mathcal {N}}, D,$ W, V, f), where ${\mathcal {M}}$ is a set of m alternatives, ${\mathcal {N}}$ is a set of n voters, D is a set of k districts that define a partition of the set of voters, $W=(w_{d})_{d\in D}$ is a weight vector consisting of a weight $w_{d}>0$ for each district $d\in D$, $V=(v_{1}, \ldots , v_{n})$ is a valuation profile for the n voters such that $v_{i}=(v_{ij})_{j\in M}$ contains the valuation of voter i for all alternatives and f is a voting rule that maps a valuation (sub)profile to a single alternative in ${\mathcal {M}}$.

Formally, a district-based election models a social electoral system, through which a set of voters are asked to vote over a set of alternatives; that is, the set of voters is partitioned into districts and each district holds a local election over all alternatives according to some voting rule. District $d\in D$ in Definition 5 contains $n_{d}$ voters such that $\sum _{d\in D}n_{d}=n$, for each voter $i\in {\mathcal {N}}$, $d(i)\in D$ denotes i belongs to district d. The winner of each local election is awarded a weight that depends on the district, and the overall winner is chosen to be the alternative with the highest weight; that is, for every district $d\in D$, a local election between its members takes place, and the winner of this election is the alternative $j_{d}=f(v_{d})$ that gets elected according to the voting rule f, where $v_{d}=(v_{i})_{i:d(i)=d}$ denotes the valuation sub-profile of the voters that belong to district d so that $V=\bigcup _{d\in D}v_{d}$. The outcome of the district-based election $\varepsilon$ is an alternative [37]

$$\begin{aligned} w(\varepsilon )\in arg max_{j\in {\mathcal {M}}}\left\{ \sum _{d\in D}w_{d}\cdot 1\{j=j_{d}\}\right\} . \end{aligned}$$

Intuitively, the voting rule f has a local effect within each district d and a global effect over the whole district-based election, i.e., for a given valuation profile V, the social welfare of the voters for alternative $j\in {\mathcal {M}}$ is defined as the total value that the voters have for alternative j,

$$\begin{aligned} SW(j|V)=\sum _{i\in {\mathcal {N}}}v_{ij}. \end{aligned}$$

In large-scale linguistic decision-making environment, a district-based election can be used to normalize the evaluation process provided by decision makers and obtain accurate linguistic assessments of alternatives.

• In district-based election, partitions of the set of voters ${\mathcal {N}}$ aim to distinguish the local voters, which have the same voting rule. In large-scale linguistic decision-making environment, districts D can be used to cluster decision makers or users which have the similar knowledge level (social context or experience) or use the same fuzzy linguistic representation model to evaluate alternatives with respect to criteria;

• In district-based election, the overall winner is dependent on the local valuation $j_{d}=f(v_{d})$ as well as the overall valuation $w(\varepsilon )$, in which the local valuation $j_{d}=f(v_{d})$ is limited in district $d\in D$ such that the unit-sum representation $\sum _{j\in {\mathcal {M}}}v_{ij}=1$ for every voter $i\in {\mathcal {N}}$ and the total value of every alternative $j\in {\mathcal {M}}$ is SW(j|V). In large-scale linguistic decision-making environment, the local valuation and the total value of every alternative can be used to aggregate multi-linguistic assessments of alternatives based on linguistic representation models in LS-MLDM problems.

Accordingly, an evaluation mechanism can be formally defined to model the evaluation process provided by decision makers.

Definition 6

An evaluation-based voting is defined as a tuple $\varepsilon =(A, M, \overline{M}, W,$ V, f), where

• $A=\{a_{1}, \ldots , a_{n}\}$ is a set of n alternatives.

• $M=\{d_{1}, \ldots , d_{m}\}$ is a set of m decision makers or users.

• $\overline{M}=\{M_{1}, \ldots , M_{p}\}$ is partitions of the set of M. Each decision maker or user belongs to a partition $M_{p'}$, every partition $M_{p'}$ contains $m_{p'}$ decision makers or users such that $\sum _{M_{p'}\in \overline{M}}m_{p'}=m$.

• $W=(w_{p'})_{M_{p'}\in \overline{M}}$ is a weight vector consisting of a weight $w_{p'}>0$ for every partition $M_{p'}\in \overline{M}$ and $\sum _{M_{p'}\in \overline{M}}w_{p'}=1$.

• $V=(v_{1}, \ldots , v_{m})$ is an evaluation profile for the m decision makers or users such that $v_{m'}=(v_{m'n'})_{a_{n'}\in A}$ contains the evaluation information of the decision maker or user $d_{m'}$ for all alternatives.

• f is a voting rule that maps an evaluation (sub)profile to a single alternative in A. In decision-making environment, the voting rule f consists of a set $C=\{c_{1}, \ldots , c_{r}\}$ of r criteria which are utilized by decision makers or users to evaluate alternatives, the domain and constraint of evaluation of decision makers or users and so on.

In Definition 6, different evaluation profiles and voting rules correspond to different evaluation information of alternatives represented by different fuzzy linguistic model.

• 1. In classical fuzzy decision methods, partitions of the set of decision makers or users M are not considered, in other words, $p=1$ and $w_{p'}=1$ in the evaluation-based voting for classical fuzzy decision making. In addition, the domain of evaluation is [0, 1] and each evaluation with respect to $c_{r'}\in C$ is fuzzy subset on [0, 1], $v_{m'n'}$ has the form

  $$\begin{aligned} v_{m'}= & {} (v_{m'n'})_{a_{n'}\in A}\\= & {} ((\mu ^{m'}_{11}, \ldots , \mu ^{m'}_{1r}); \ldots ; (\mu ^{m'}_{n1}, \ldots , \mu ^{m'}_{nr})), \end{aligned}$$

  and the evaluation information of the decision maker or user $d_{m'}$ for alternative $a_{n'}$ with respect to criteria $C=\{c_{1}, \ldots , c_{r}\}$ is $a_{n'}=f(v_{m'})=(\mu ^{m'}_{n'1}, \ldots , \mu ^{m'}_{n'r})$. In classical fuzzy decision-making methods, $v_{m'}$ is constructed by fuzzy decision matrix, i.e.,

  

• 2. In LDM methods, $p=1$ and $w_{p'}=1$ in the evaluation-based voting, the domain of evaluation is linguistic term set $S=\{s_{0}, \ldots , s_{g}\}$ and each evaluation with respect to $c_{r'}\in C$ is a linguistic term, $v_{m'n'}$ has the form

  $$\begin{aligned} v_{m'}= & {} (v_{m'n'})_{a_{n'}\in A}\\= & {} ((e^{m'}_{11}, \ldots , e^{m'}_{1r}); \ldots ; (e^{m'}_{n1}, \ldots , e^{m'}_{nr})). \end{aligned}$$

  the linguistic evaluation $v_{m'}$ provided by decision maker $d_{m'}$ is represented by linguistic decision matrix Eq. (3).

• 3. In fuzzy linguistic multiset decision method (Pei et al. 2019), the domain of evaluation is the space $\{0, 1\}^{g+1}$ with respect to linguistic term set $S=\{s_{0}, \ldots , s_{g}\}$ and each evaluation with respect to every $c_{r'}\in C$ is an vector in the space $\{0, 1\}^{g+1}$, then $v_{m'n'}$ has the form

  $$\begin{aligned} v_{m'}= & {} (v_{m'n'})_{a_{n'}\in A}=(((e^{m'1}_{10}, \ldots , e^{m'1}_{1g}),\ldots ,\nonumber \\&(e^{m'r}_{10}, \ldots , e^{m'r}_{1g})); \ldots ; \nonumber \\&((e^{m'1}_{n0}, \ldots , e^{m'1}_{ng}),\ldots , (e^{m'r}_{n0}, \ldots , e^{m'r}_{ng}))). \end{aligned}$$

  (5)

  which is represented by alternatives-linguistic terms decision matrix.

3.2 Multi-linguistic decision matrix

Here, a new voting rule f is employed to obtain linguistic evaluation information in large-scale linguistic decision-making environment, i.e., the domain of evaluation is the space $[0, 1]^{g+1}$ with respect to linguistic term set $S=\{s_{0}, \ldots , s_{g}\}$ and each evaluation is an vector on the space $[0, 1]^{g+1}$, compared with Eq. (5), elements in $v_{m'}$ are extended from $\{0, 1\}$ to [0, 1] and the linguistic evaluation $v_{m'}$ of all alternatives provided by the decision maker or user $d_{m'}$ is represented by multi-linguistic decision matrix, i.e.,

where $m_{jl}^{ik}\in [0, 1] (l=0, \ldots , g)$, $m_{jl}^{ik}=1$ means that the decision maker $d_{i}$ confidently use linguistic term $s_{l}\in S$ to assess alternative $a_{j}$ with respect to the criterion $c_{k}$, $m_{jl}^{ik}\in (0, 1)$ means that the decision maker $d_{i}$ with degree of confidence $m_{jl}^{ik}$ use linguistic term $s_{l}\in S$ to assess alternative $a_{j}$ with respect to the criterion $c_{k}$ and $m_{jl}^{ik}=0$ means that linguistic term $s_{l}$ is not suitable to assess alternative $a_{j}$. The following properties show that multi-linguistic decision matrix is an extension of alternatives-linguistic terms decision matrix, and PLTS and FLM-DFN assessments of alternatives can also be represented by multi-linguistic decision matrix $D_{ik}$.

Proposition 1

In multi-linguistic decision matrix $D_{ik}$ Eq. (6), let $D_{ik}^{j}=\{m_{jl}^{ik}$ | $m_{jl}^{ik}>0, l=0, 1, \ldots , g\}$. Then, for decision maker $d_{i}$ and the criterion $c_{k}$,

• 1. linguistic assessment of alternative $a_{j}$ is distinct or single-linguistic term if and only if $|D_{ik}^{j}|=1$ and $\sum ^{g}_{l=0}m_{jl}^{ik}=1$;

• 2. linguistic assessment of alternative $a_{j}$ is partial unknown if and only if $D_{ik}^{j}=\emptyset$ or $\sum ^{g}_{l=0}m_{jl}^{ik}=0$;

• 3. linguistic assessment of alternative $a_{j}$ is an HFLTS if and only if there exists natural numbers l and t such that $l+t\le g$ and $\sum ^{t}_{t'=0}m_{j(l+t')}^{ik}=|D_{ik}^{j}|>1$.

Proof

According to explanation of multi-linguistic decision matrix Eq. (6), the decision maker $d_{i}$ with degree of confidence $m_{jl}^{ik}$ uses linguistic term $s_{l}\in S$ to assess alternative $a_{j}$ with respect to the criterion $c_{k}$ if and only if $m_{jl}^{ik}\in D_{ik}^{j}$.

• (1) $|D_{ik}^{j}|=1$ means that linguistic assessment of alternative $a_{j}$ is a single-linguistic term $s_{l}\in S$ with degree of confidence $m_{jl}^{ik}$, $\sum ^{g}_{l=0}m_{jl}^{ik}=1$ means $m_{jl}^{ik}=1$, i.e., linguistic assessment of alternative $a_{j}$ is distinct or single-linguistic term if and only if $|D_{ik}^{j}|=1$ and $\sum ^{g}_{l=0}m_{jl}^{ik}=1$.

• (2) $D_{ik}^{j}=\emptyset$ or $\sum ^{g}_{l=0}m_{jl}^{ik}=0$ means each $m_{jl}^{ik}=0$ and vice versa, i.e., each linguistic term $s_{l}\in S$ is not suitable to assess alternative $a_{j}$ or linguistic assessment of alternative $a_{j}$ is partial unknown.

• (3) $\sum ^{t}_{t'=0}m_{j(l+t')}^{ik}=|D_{ik}^{j}|>1$ means that the set of consecutive linguistic terms $\{s_{l},$ $s_{l+1},\ldots , s_{l+t}\}$ with degree of confidence $\{m_{j(l)}^{ik}, m_{j(l+1)}^{ik}, \ldots , m_{j(l+t)}^{ik}\}$ is the linguistic assessment of alternative $a_{j}$ with respect to the criterion $c_{k}$, $\sum ^{t}_{t'=0}m_{j(l+t')}^{ik}$ $=|D_{ik}^{j}|$ means each $m_{j(l+t')}^{ik}=1$ and vice versa, i.e., HFLTS $\{s_{l},$ $s_{l+1}, \ldots ,s_{l+t}\}$ is used to assess alternative $a_{j}$ with respect to $c_{k}$. $\square$

Proposition 2

In multi-linguistic decision matrix Eq. (6), let $D_{ik}^{j}=\{m_{jl}^{ik}|$ $m_{jl}^{ik}>0, l=0, 1, \ldots , g\}$. Then for decision maker $d_{i}$ and the criterion $c_{k}$,

• 1. linguistic assessment of alternative $a_{j}$ is a PLTS if and only if $|D_{ik}^{j}|>1$ and $\sum ^{g}_{l=0}m_{jl}^{ik}\le 1$;

• 2. linguistic assessment of alternative $a_{j}$ is a FLM-DFN if and only if $|D_{ik}^{j}|>1$, $\sum ^{g}_{l=0}m_{jl}^{ik}\in (1, g)$ and there exists $m_{jl}^{ik}=1$ such that $m_{j0}^{ik}\le m_{j1}^{ik}\le \cdots \le m_{jl}^{ik}$ and $m_{jl}^{ik}\ge m_{j(l+1)}^{ik}\ldots \ge m_{jg}^{ik}$.

Proof

• (1) $|D_{ik}^{j}|>1$ means that more than one linguistic term is utilized to assess alternative $a_{j}$ with respect to the criterion $c_{k}$ and vice versa, $\sum ^{g}_{l=0}m_{jl}^{ik}\le 1$ means each $m_{jl}^{ik}\in [0, 1)$. According to Definition 3, all degrees of confidence in $D_{ik}^{j}$ satisfy probability distribution on linguistic terms, i.e., degree of confidence can be considered as probability of linguistic term and linguistic assessment of alternative $a_{j}$ is a PLTS.

• (2) $\sum ^{g}_{l=0}m_{jl}^{ik}\in (1, g)$ means that there exists $m_{jl}^{ik}$ such that $0<m_{jl}^{ik}<1$, i.e., all degrees of confidence in $D_{ik}^{j}$ satisfy membership degrees of linguistic terms. In addition, there exists $m_{jl}^{ik}=1$ such that $m_{j0}^{ik}\le m_{j1}^{ik}\le \cdots \le m_{jl}^{ik}$ and $m_{jl}^{ik}\ge m_{j(l+1)}^{ik}\ldots \ge m_{jg}^{ik}$ satisfies Definition 4; hence, degree of confidence can be considered as membership degree of linguistic term and $D_{ik}^{j}$ is a FLM-DFN. $\square$

Properties 1 and 2 show that multi-linguistic decision matrix $D_{ik}$ is a flexible and effective tool to represent linguistic assessments of alternatives when decision makers utilize several fuzzy linguistic representation models to evaluate alternatives.

Example 1

A customer $d_{i}$ evaluates quality ($c_{k}$) of five clothes by online product reviews, clothes are jacket (J), trousers (T), shirt (S), trousers suit (TS) and underdress (U), the primary linguistic term set is $S=\{s_0$ (extremely poor), $s_1$ (very poor), $s_2$ (poor), $s_3$ (slightly poor), $s_4$ (fair), $s_5$ (slightly good), $s_6$ (good), $s_7$ (very good), $s_8$ (extremely good)$\}$. According to life habit and experience, $d_{i}$ is familiar with jacket and trousers, but partly knows or does not know about shirt, trousers suit and underdress, hence multi-linguistic decision matrix provided by $d_{i}$ with respect to $c_{k}$ may be as follows:

According to Properties 1 and 2 , due to $|D_{ik}^{1}|=1$ and $\sum ^{8}_{l=0}m_{1l}^{ik}=1$, linguistic assessment of J provided by $d_{i}$ with respect to $c_{k}$ is a single-linguistic term $s_{6}$. Due to $\sum ^{8}_{l=0}m_{2l}^{ik}=|D_{ik}^{2}|=2$, linguistic assessment of T provided by $d_{i}$ with respect to $c_{k}$ is HFLTS $\{s_{4}, s_{5}\}$. Due to $\sum ^{8}_{l=0}m_{5l}^{ik}=0$, i.e., linguistic assessment of U provided by $d_{i}$ with respect to $c_{k}$ is partial unknown. Due to $|D_{ik}^{3}|=3>1$ and $\sum ^{8}_{l=0}m_{3l}^{ik}=0.9< 1$, linguistic assessment of S provided by $d_{i}$ with respect to $c_{k}$ is PLTS $\{s_{5}(0.5), s_{6}(0.2), s_{7}(0.2)\}$. Due to $|D_{ik}^{4}|=5>1$, $\sum ^{8}_{l=0}m_{4l}^{ik}=3.6\in (1, 8)$ and $m_{46}^{ik}=1$, i.e., linguistic assessment of TS provided by $d_{i}$ with respect to $c_{k}$ is FLM-DFN $\{0.5/s_{4}, 0.7/s_{5}, 1/s_{6}, 0.8/s_{7}, 0.6/s_{8}\}$.

4 The linguistic decision-making framework of LS-MLDM problems

In the section, a new linguistic decision-making framework is designed to deal with LS-MLDM problems, in which evaluation-based voting and the multi-linguistic decision matrix are utilized to normalize the decision process and represent linguistic assessments provided by decision makers in large-scale linguistic decision-making environments. In addition, main decision resolution scheme of the framework is analyzed and trust degrees of linguistic terms in the multi-linguistic decision matrix are developed; more important an uniform fusion method of multi-linguistic assessments is proposed according to trust degrees of linguistic terms.

4.1 The linguistic decision-making phases of LS-MLDM problems

Fig. 1

The linguistic decision-making phases of classical LDM problems

Formally, classical linguistic decision-making framework of LDM problems is mainly consisted of four phases (Martínez et al. 2015):

• The choice of the linguistic term set with its semantics, which provides linguistic expression domain for decision makers to evaluate alternatives, and linguistic assessments based on a fuzzy linguistic representation model are uniformly represented in alternatives-criteria linguistic decision matrix;

• The choice of linguistic aggregation operators, which is chosen for aggregating linguistic assessments based on linguistic representation model. Theoretically, different linguistic aggregation operators obtain different results;

• Aggregation phase, which obtains linguistic collective assessments by using the chosen linguistic aggregation operator;

• Exploitation phase, which obtains a ranking on linguistic aggregation results to choose the best alternatives. In real-world applications, different fuzzy linguistic representation models are corresponding to different rankings and obtain the different best solution of LDM problem.

The classical linguistic decision-making framework of LDM problems is pictured in Fig.  1 (Labella et al. 2018). Compared with LS-MLDM problems, because there are a huge amounts of decision makers and fuzzy linguistic terms mean different things for different decision makers, in decision process different fuzzy linguistic representation models may be employed by decision makers to assess alternatives with respect to criteria, a new linguistic decision-making framework of LS-MLDM problems is designed in the paper, which is pictured in Fig. 2 and mainly consisted of the following five phases:

• 1. The choice of the linguistic term set with its semantics, which is the primary linguistic expression domain and helps decision makers to provide linguistic assessments of alternatives;

• 2. The choice of voting rule f in evaluation-based voting (Definition 6) and partition the set of decision makers according to knowledge level (social context or experience), then multi-linguistic decision matrix can be constructed, in which all multi-linguistic assessments of alternatives provided by the decision maker are uniformly represented;

• 3. Normalization of multi-linguistic decision matrix, which aims to transform multi-linguistic assessments into the same linguistic representation and avoid undesirable consequences flowing from different fuzzy linguistic representation models;

• 4. Aggregation phase, in which linguistic aggregation operators are chosen for aggregating the normalized linguistic assessments;

• 5. Exploitation phase, in which a ranking on linguistic aggregation results is obtained and selection of the best alternatives can be finished.

Fig. 2

The new linguistic decision-making phases of LS-MLDM problems

Comparing Fig. 1 with Fig. 2, it can be noticed that the first phase in the classical linguistic decision-making framework is refined by (1) and (2) in the new linguistic decision-making framework. In fact, considering a huge amounts of decision makers with different knowledge levels (social context or experience), in evaluation-based voting, they maybe use different voting rule f to provide and obtain more suitable or accurate linguistic assessments of alternatives than one kind of linguistic representation model. According to Properties 1 and 2 , the following one-to-one mappings can be designed to formally describe linguistic assessments of alternative provided by decision makers with respect to criterion and distinguish the function relationship between fuzzy linguistic representations and multi-linguistic decision matrices:

• Single assessment of alternative and the function relationship between linguistic term set and multi-linguistic decision matrix:

  $$\begin{aligned}&F_{H}: M\times C\times S\times A \longrightarrow \{0, 1\}, \nonumber \\&\quad (d_{i}, c_{k}, s_{l}, a_{j})\longmapsto m_{j1}^{ik}\in \{0, 1\}, \end{aligned}$$

  (7)

  in which $F_{H}(d_{i}, c_{k}, s_{l}, a_{j})=0$ means that decision maker $d_{i}$ with respect to criterion $c_{k}$ does not utilize linguistic term $s_{l}$ to assess alternative $a_{j}$, $F_{H}(d_{i}, c_{k}, s_{l}, a_{j})=1$ means that decision maker $d_{i}$ utilizes linguistic term $s_{l}$ to assess $a_{j}$ with respect to criterion $c_{k}$. Single-linguistic decision matrix provided by decision maker $d_{i}$ can be constructed by using values $\{F_{H}(d_{i}, c_{k}, s_{l}, a_{j})|s_{l}\in S, a_{j}\in A\}$ and single-linguistic assessment of alternative $a_{j}$ is equal to $\sum ^{g}_{l=0}F_{H}(d_{i}, c_{k}, s_{l}, a_{j})=1$;

• HFLTS assessment of alternative and the function relationship between HFLTSs and multi-linguistic decision matrix:

  $$\begin{aligned}&F_{H}: M\times C\times S\times A \longrightarrow \{0, 1\},\nonumber \\&\quad (d_{i}, c_{k}, s_{l}, a_{j})\longmapsto m_{j1}^{ik}\in \{0, 1\}, \end{aligned}$$

  (8)

  in which $F_{H}(d_{i}, c_{k}, s_{l}, a_{j})=0$ means that decision maker $d_{i}$ does not utilize linguistic term $s_{l}$ to assess alternative $a_{j}$, $F_{H}(d_{i}, c_{k}, s_{l}, a_{j})=1$ means that decision maker $d_{i}$ utilizes linguistic term $s_{l}$ to assess alternative $a_{j}$. HFLTS linguistic decision matrix provided by decision maker $d_{i}$ can be constructed by using values $\{F_{H}(d_{i}, c_{k}, s_{l}, a_{j})|s_{l}\in S, a_{j}\in A\}$ and HFLTS assessment of alternative $a_{j}$ is equal to $\sum ^{g}_{l=0}F_{H}(d_{i}, c_{k}, s_{l}, a_{j})=N$, which is a natural number in (1, g);

• PLTS assessment of alternative and the function relationship between PLTSs and multi-linguistic decision matrix:

  $$\begin{aligned}&F_{H}: M\times C\times S\times A \longrightarrow [0, 1],\nonumber \\&\quad (d_{i}, c_{k}, s_{l}, a_{j})\longmapsto m_{j1}^{ik}\in [0, 1], \end{aligned}$$

  (9)

  in which $F_{H}(d_{i}, c_{k}, s_{l}, a_{j})=m_{j1}^{ik}$ means that decision maker $d_{i}$ utilize linguistic term $s_{l}$ to assess alternative $a_{j}$ with probability $m_{j1}^{ik}$. PLTS linguistic decision matrix provided by decision maker $d_{i}$ can be constructed by using values $\{F_{H}(d_{i}, c_{k}, s_{l}, a_{j})|s_{l}$ $\in S, a_{j}\in A\}$ and PLTS assessment of alternative $a_{j}$ is equal to $0<\sum ^{g}_{l=0}F_{H}$ $(d_{i}, c_{k}, s_{l}, a_{j})\le 1$;

• FLM-DFN assessment of alternative and the function relationship between FLM-DFNs and multi-linguistic decision matrix:

  $$\begin{aligned}&F_{H}: M\times C\times S\times A \longrightarrow [0, 1],\nonumber \\&\quad (d_{i}, c_{k}, s_{l}, a_{j})\longmapsto m_{j1}^{ik}\in [0, 1], \end{aligned}$$

  (10)

  in which $F_{H}(d_{i}, c_{k}, s_{l}, a_{j})=m_{j1}^{ik}$ means that decision maker $d_{i}$ utilize linguistic term $s_{l}$ to assess alternative $a_{j}$ with membership degree $m_{j1}^{ik}$. FLM-DFN linguistic decision matrix provided by decision maker $d_{i}$ can be constructed by using values $\{F_{H}(d_{i}, c_{k}, s_{l}, a_{j})|s_{l}$ $\in S, a_{j}\in A\}$ and FLM-DFN assessment of alternative $a_{j}$ is equal to $1<\sum ^{g}_{l=0}F_{H}(d_{i},$ $c_{k}, s_{l}, a_{j})\le g$;

• Multi-linguistic assessment of alternative and the function relationship between multi-linguistic representations and multi-linguistic decision matrix:

  $$\begin{aligned}&F_{H}: M\times C\times S\times A \longrightarrow [0, 1], \nonumber \\&\quad (d_{i}, c_{k}, s_{l}, a_{j})\longmapsto m_{j1}^{ik}\in [0, 1], \end{aligned}$$

  (11)

  in which $F_{H}(d_{i}, c_{k}, s_{l}, a_{j})=m_{j1}^{ik}$ means that decision maker $d_{i}$ utilize linguistic term $s_{l}$ to assess alternative $a_{j}$ with degree of confidence $m_{j1}^{ik}$. Multi-linguistic decision matrix provided by decision maker $d_{i}$ can be constructed by using values $\{F_{H}(d_{i}, c_{k}, s_{l}, a_{j})|s_{l}$ $\in S, a_{j}\in A\}$ and multi-linguistic assessment of alternative $a_{j}$ is equal to $0\le \sum ^{g}_{l=0}F_{H}(d_{i},$ $c_{k}, s_{l}, a_{j})\le g$.

4.2 Normalization of multi-linguistic decision matrix

Theoretically, normalization of multi-linguistic decision matrix is equal to normalization of multi-linguistic assessments. Here, normalization of multi-linguistic assessments is to transform degree of confidence on linguistic terms into trust degrees of linguistic terms in multi-linguistic assessments of alternatives.

Definition 7

In multi-linguistic decision matrix $D_{ik}$, for alternative $a_{j} (j=1, \ldots , n)$, trust degree of linguistic term $s_{l} (l=0, \ldots , g)$ provided by decision maker $d_{i}$ with respect to criterion $c_{k}$ is defined as

$$\begin{aligned} u^{ik}_{jl}=\left\{ \begin{array}{ll} \frac{m^{ik}_{jl}}{\sum ^{g}_{l=0}m^{ik}_{jl}},&{} \text{ if }\ \sum ^{g}_{l=0}m^{ik}_{jl}>0,\\ m^{ik}_{jl}=0,&{} \text{ if }\ \sum ^{g}_{l=0}m^{ik}_{jl}=0.\end{array} \right. \end{aligned}$$

(12)

Based on trust degree $u^{ik}_{jl}$ of linguistic term $s_{l} (l=0, \ldots ,$ g), multi-linguistic decision matrix $D_{ik}$ can be converted into linguistic decision matrix $D'_{ik}$ with trust degrees of linguistic terms.

$$\begin{aligned} D'_{ik}= \left( \begin{array}{ccc} u^{ik}_{10} &{} \ldots &{} u^{ik}_{1g} \\ \vdots &{} \vdots &{} \vdots \\ u^{ik}_{n0} &{} \ldots &{} u^{ik}_{ng} \\ \end{array} \right) =\left( \begin{array}{ccc} \frac{m_{10}^{ik}}{\sum ^{g}_{l=0}m^{ik}_{1l}} &{} \ldots &{} \frac{m_{1g}^{ik}}{\sum ^{g}_{l=0}m^{ik}_{1l}} \\ \vdots &{} \vdots &{} \vdots \\ \frac{m_{n0}^{ik}}{\sum ^{g}_{l=0}m^{ik}_{nl}} &{} \ldots &{} \frac{m_{ng}^{ik}}{\sum ^{g}_{l=0}m^{ik}_{nl}} \\ \end{array} \right) \end{aligned}$$

(13)

Proposition 3

In trust degree linguistic decision matrix $D'_{ik}$, trust degrees of each alternative $a_{j} (j=1, \ldots , n)$ are such that $0\le u^{ik}_{jl}\le 1$ and $\sum ^{g}_{l=0}u^{ik}_{jl}=1.$

Proof

For each trust degree $u^{ik}_{jl}$ of linguistic term $s_{l} (l=0, \ldots , g)$, according to multi-linguistic decision matrix Eq. 6, we have

$$\begin{aligned} 0\le u^{ik}_{jl}=\frac{m^{ik}_{jl}}{\sum ^{g}_{l=0}m^{ik}_{jl}}\le \frac{m^{ik}_{jl}}{m^{ik}_{jl}}=1. \end{aligned}$$

For each alternative $a_{j}$ of linguistic decision matrix $D'_{ik}$, we have

$$\begin{aligned} \sum ^{g}_{l=0}u^{ik}_{jl}=\sum ^{g}_{l=0}\frac{m^{ik}_{jl}}{\sum ^{g}_{l=0}m^{ik}_{jl}}=\frac{\sum ^{g}_{l=0}m^{ik}_{jl}}{\sum ^{g}_{l=0}m^{ik}_{jl}}=1. \end{aligned}$$

$\square$

In the following, the relationship between degree of confidence $m_{jl}^{ik}$ in $D_{ik}$ Eq. (6) and trust degree $u^{ik}_{jl}$ of linguistic term $s_{l}$ in $D'_{ik}$ Eq. (13) is analyzed.

Proposition 4

For any $D_{ik}$, $D'_{ik}$ and $D_{ik}^{j}=\{m_{jl}^{ik}|$ $m_{jl}^{ik}>0, l=0, 1, \ldots , g\}$.

• 1. If linguistic assessment of alternative $a_{j}$ is distinct or single-linguistic term, then $u^{ik}_{jl}=m^{ik}_{jl}$ for any $l=0, \ldots , g$;

• 2. If linguistic assessment of alternative $a_{j}$ is hesitant linguistic term set, then $u^{ik}_{jl}=\frac{m_{jl}^{ik}}{|D_{ik}^{j}|}\le m^{ik}_{jl}$;

• 3. If linguistic assessment of alternative $a_{j}$ is probabilistic linguistic term set and $m^{ik}_{jl}\ne 0$, then $u^{ik}_{jl}=\frac{m^{ik}_{jl}}{\sum ^{g}_{l=0}m^{ik}_{jl}}\ge m_{jl}^{ik}$;

• 4. If linguistic assessment of alternative $a_{j}$ is discrete fuzzy number on linguistic term set S and $m^{ik}_{jl}\ne 0$, then $u^{ik}_{jl}=\frac{m^{ik}_{jl}}{\sum ^{g}_{l=0}m^{ik}_{jl}}<m^{ik}_{jl}$.

Proof

• 1) According to Property 1 (1), if linguistic assessment of alternative $a_{j}$ is distinct or single-linguistic term, then $\sum ^{g}_{l=0}m_{jl}^{ik}=1$. Based on Eq. (12), $u^{ik}_{jl}=\frac{m^{ik}_{jl}}{\sum ^{g}_{l=0}m^{ik}_{jl}}=m^{ik}_{jl}$ for any $l=0, \ldots , g$.

• 2) According to Property 1 (3), if linguistic assessment of alternative $a_{j}$ is hesitant linguistic term set, then $\sum ^{g}_{l=0}m_{jl}^{ik}=|D_{ik}^{j}|>1$ and $u^{ik}_{jl}=\frac{m_{jl}^{ik}}{|D_{ik}^{j}|}$, it is obvious that $u^{ik}_{jl}=m^{ik}_{jl}=0$ when $m^{ik}_{jl}=0$ and $u^{ik}_{jl}<m^{ik}_{jl}$ when $m^{ik}_{jl}=1$.

• 3) According to Property 2 (1), if linguistic assessment of alternative $a_{j}$ is probabilistic linguistic term set, then $\sum ^{g}_{l=0}m_{jl}^{ik}\le 1$, when $m^{ik}_{jl}\ne 0$, $u^{ik}_{jl}=\frac{m^{ik}_{jl}}{\sum ^{g}_{l=0}m^{ik}_{jl}}\ge m^{ik}_{jl}$, especially, when $\sum ^{g}_{l=0}m_{jl}^{ik}= 1$, then $u^{ik}_{jl}=m_{jl}^{ik}$.

• 4) According to Property 2 (2), if linguistic assessment of alternative $a_{j}$ is discrete fuzzy number on linguistic term set S, then $\sum ^{g}_{l=0}m_{jl}^{ik}\in (1, g)$ and $u^{ik}_{jl}=\frac{m^{ik}_{jl}}{\sum ^{g}_{l=0}m^{ik}_{jl}}< m^{ik}_{jl}$ if $m^{ik}_{jl}\ne 0$. $\square$

Example 2

Continues Example 1. Based on Eq. (13), multi-linguistic decision matrix $D_{ik}$ is converted into trust degree linguistic decision matrix $D'_{ik}$, i.e.,

4.3 Aggregation and exploitation of LS-MLDM

According to Fig.  2 and trust degree linguistic decision matrix $D'_{ik}$ Eq. (13), linguistic aggregation operators can be selected to aggregate linguistic assessments with trust degrees, and then exploitation of LS-MLDM can be carried out.

4.3.1 Aggregation of multi-linguistic assessments

Theoretically, Definition 7 and Property 3 show that trust degree $u^{ik}_{jl}$ of linguistic term $s_{l} (l=0, \ldots , g)$ can be employed as weights of linguistic terms to aggregate multi-linguistic assessments of alternatives, i.e., for decision maker $d_{i}$ and criterion $c_{k}$, multi-linguistic assessments of alternatives in $D'_{ik}$ (Eq. (13)) can be aggregated by

$$\begin{aligned} E^{ik}_{j}=\varDelta \left( \sum ^{g}_{l=0}(u^{ik}_{jl}\times l)\right) =\varDelta (\frac{\sum ^{g}_{l=0}(m^{ik}_{jl}\times l)}{\sum ^{g}_{l=0}m^{ik}_{jl}})=(s^{ik}_{l'}, \alpha ^{ik}_{j}), \nonumber \\ \end{aligned}$$

(14)

where $\varDelta$ is transformation from numerical values [0, g] to 2-TLMs on S defined by Eq. (1), $l'\in \{0, 1, \ldots , g\}$ and $(s^{ik}_{l'}, \alpha ^{ik}_{j})$ is a 2-tuple linguistic term on $S=\{s_0, \ldots , s_g\}$. Based on Eq. (14), multi-linguistic assessments of alternatives in $D'_{ik}$ are uniformly represented by 2-TLM $E^{ik}_{j}=(s^{ik}_{l'}, \alpha ^{ik}_{j})$, such as in Example 2, multi-linguistic assessments of five clothes provided by customer $d_{i}$ with respect to quality are uniformly represented by 2-TLMs $(s_{6}, 0)$ of J, $(s_{5}, -0.5)$ of T, $(s_{6}, -0.4)$ of S, $(s_{6}, 0.08)$ of TS and $(s_{0}, 0)$ of U, respectively. Accordingly, multi-linguistic assessments provided by decision maker $d_{i}$ with respect to all criteria are represented by 2-tuple fuzzy linguistic decision matrix, i.e.,

Based on the 2-tuple fuzzy linguistic decision matrix $D_{i}$, if weights $(w_{1}, \ldots ,$ $w_{m})$ of decision makers are considered, then 2-tuple linguistic assessment of $a_{j}$ provided by all decision makers with respect to criterion $c_{k}$ is

$$\begin{aligned} E^{k}_{j}=\varDelta \left( \sum ^{m}_{i=1}(w_{i}\varDelta ^{-1}(E^{ik}_{j}))\right) =(s^{k}_{l'}, \alpha ^{k}_{j}), \end{aligned}$$

(16)

in which $s^{k}_{l'}\in S=\{s_0, s_1, \ldots , s_g\}$ and $\alpha ^{k}_{j}\in [-0.5, 0.5)$. If weights $(\omega _{1}, \ldots , \omega _{r})$ of criteria are considered, then 2-tuple linguistic assessment of $a_{j}$ provided by decision maker $d_{i}$ with respect to criteria is

$$\begin{aligned} E^{i}_{j}=\varDelta \left( \sum ^{r}_{k=1}(\omega _{k}\varDelta ^{-1}(E^{ik}_{j}))\right) =(s^{i}_{l'}, \alpha ^{i}_{j}), \end{aligned}$$

(17)

in which $s^{i}_{l'}\in S=\{s_0, s_1, \ldots , s_g\}$ and $\alpha ^{i}_{j}\in [-0.5, 0.5)$. Accordingly, the final aggregation of a LS-MLDM problem can be divided into two categories, i.e.,

• 1. First aggregating linguistic assessments of decision makers then criteria:

  $$\begin{aligned} R(a_{j})=\varDelta \left( \sum ^{r}_{k=1}(\omega _{k}\varDelta ^{-1}(E^{k}_{j}))\right) =(s_{l'}, \alpha _{j}), \end{aligned}$$

  (18)

  in which, $E^{k}_{j}$ is decided by Eq. (16).

• 2. First aggregating linguistic assessments of criteria then decision makers:

  $$\begin{aligned} R(a_{j})=\varDelta \left( \sum ^{m}_{i=1}(w_{i}\varDelta ^{-1}(E^{i}_{j}))\right) =(s_{l'}, \alpha _{j}), \end{aligned}$$

  (19)

  in which, $E^{i}_{j}$ is decided by Eq. (17).

Where $R(a_{j})=(s_{l'}, \alpha _{j})$ on $S=\{s_0, \ldots , s_g\}$.

4.3.2 Exploitation of alternatives

Exploitation phase of LS-MLDM problems is depended on linguistic aggregation result $(s_{l'}, \alpha _{j})$ calculated by Eqs. (18) or (19), formally, for any two alternatives $a_{j}$ and $a_{j'}$, $a_{j}\preceq a_{j'}$ ($a_{j'}$ prefer to $a_{j}$) if and only if $R(a_{j})=(s_{l'}, \alpha _{j})\preceq R(a_{j'})=(s_{l''}, \alpha _{j'})$ if and only if $\varDelta ^{-1}(s_{l'},$ $\alpha _{j})\le \varDelta ^{-1}(s_{l''}, \alpha _{j'})$ (Herrera and Martínez 2000), and the best alternatives is

$$\begin{aligned} R_{max}=\{a_{j}|R(a_{j})=max\{R(a_{1}), \ldots , R(a_{n})\}\}. \end{aligned}$$

(20)

According to Fig. 2 and Eqs. (6)-(20), Algorithm 1 can be designed to carry out LS-MLDM problems.

5 Case study

Shared economy is significant in Chinese market, which has continuously permeated and changed ourselves traditional lifestyle over the past several years. As a most representative form of shared economy, shared bikes are improving the efficiency of transport and solving the problem of scarcity of transportation resources (Tian et al. 2018). In the section, user experience of shared bikes is utilized to show the proposed method and compare with decision making methods with 2-TLMs (Herrera and Martínez 2000), HFLTSs (Rodríguez et al. 2012), PLTSs (Pang et al. 2016) and FLM-DFNs (Massanet et al. 2014), respectively.

Table 1 User experiences of shared bikes with respect to $c_{1}, c_{2}, c_{3}, c_{4}, c_{5}$ and $c_{6}$

5.1 Multi-linguistic assessments of user experience of shared bikes

Shared bikes are useful and effective schemes to solve short trips, many shared bikes are rapidly appeared over the past several years, such as “Meituanbike” $(a_1)$, “hellobike” $(a_2)$ and “green orange” $(a_3)$ in Chengdu city. Because users of shared bikes may be anyone of individuals with different knowledge level, user experiences of shared bikes become a LS-MLDM problem. In the example, “Fund Security” ($c_{1}$), “Riding Safety” ($c_{2}$), “Riding Convenience” ($c_{3}$), “Riding Comfort” ($c_{4}$), “Perfect Emergency Response Mechanism” ($c_{5}$) and “Sustainability” ($c_{6}$) (Tian et al. 2018) are criteria to finish user experiences of $\{a_1, a_2, a_3, a_4\}$, linguistic terms $S_{6}=\{s_0$ (very poor), $s_1$ (poor), $s_2$ (slightly poor), $s_3$ (fair), $s_4$ (slightly good), $s_5$ (good), $s_6$ (very good)$\}$ are utilized by users to evaluate user experiences of shared bikes $\{a_1, a_2, a_3, a_4\}$ with respect to criteria $\{c_1, c_2, c_3, c_4, c_{5}, c_{6}\}$. According to the evaluation-based voting, users are partitioned by users’ knowledge level (social context or experience), users in the same partition have the similar knowledge level (social context or experience) and utilize the same fuzzy linguistic representation model to provide evaluation information about user experiences of shared bikes. In the example, three partitions (students, office workers and citizens) are employed, and every partition consists of more than 20 users to provide evaluation information. For the sake of simplicity, one student $d_{1}$, office worker $d_{2}$ and citizen $d_{3}$ are selected from partitions, respectively, and their evaluation information about user experiences of shared bikes is shown follows:

1) User experiences of shared bikes provided by student $d_{1}$ are

where $D_{1i} (i=1, 2, 3, 4, 5, 6)$ provided by student $d_{1}$ is linguistic assessments of user experiences of shared bikes with respect to criterion $c_{i}$.

2) User experiences of shared bikes provided by office worker $d_{2}$ are

3) User experiences of shared bikes provided by citizen $d_{3}$ are

Accordingly, $d_{1}$, $d_{2}$ and $d_{3}$ provide multi-linguistic decision matrices to express their user experiences of shared bikes. Using Algorithm 1, user experience of each shared bike can be obtained as follows:

• (1) Normalization of user experiences of shared bikes. Using Eq.(12), each $D_{ik}$ is converted into trust degree linguistic decision matrix $D'_{ik} (i=1, 2, 3,$ $k=1, 2, 3, 4, 5, 6)$, for example,

  

• (2) Aggregation of user experiences of shared bikes in each $D'_{ik} (i=1, 2, 3,$ $k=1, 2, 3, 4, 5, 6)$. Using Eq.(14), 2-tuple fuzzy linguistic decision matrices $D_{1}$, $D_{2}$ and $D_{3}$ of $d_{1}$, $d_{2}$ and $d_{3}$ can be obtained, respectively.

  

  Suppose users $d_{1}$, $d_{2}$ and $d_{3}$ (or criteria $c_{1}, c_{2}, c_{3},$ $c_{4}, c_{5}$ and $c_{6}$) with the same weight information, according to Eqs. (16)–(19), user experience of each shared bike can be obtained, which are shown in Tables 1 (considering weights of $d_{1}$, $d_{2}$ and $d_{3}$) and 2 (considering weights of $c_{1}, c_{2}, c_{3}, c_{4}, c_{5}$ and $c_{6}$).

Table 2 User experiences of shared bikes provided by $d_{1}$, $d_{2}$ and $d_{3}$

Table 3 The LDM methods to evaluate user experiences provided by $d_{1}$, $d_{2}$ and $d_{3}$

Table 4 Decision-making table provided by $d_{1}$, $d_{2}$ and $d_{3}$ with respect to criterion $c_{6}$

(3) Exploitation of user experiences of shared bikes provided by student $d_{1}$, office worker $d_{2}$ and citizen $d_{3}$ with respect to the six criteria. According to Table 1 or 2, $R(a_{1})=max \{R(a_{1}), R(a_{2}), R(a_{3})\}=(s_{5}, 0.48)$ due to $(s_{5}, 0.48)>(s_{5}, 0.47)>(s_{5}, -0.12)$, i.e., user experience of “meituanbike” is the best one in $\{a_1, a_2, a_3\}$.

5.2 Comparative analyses

In the subsection, the proposed LS-MLDM method for user experiences of shared bikes is compared with LDM methods with 2-TLM, HFLTS, PLTS and FLM-DFN assessments of alternatives. Because user experiences of shared bikes provided by $d_{1}$, $d_{2}$ and $d_{3}$ are multi-linguistic decision matrices, LDM methods with 2-TLM, HFLTS, PLTS and FLM-DFN assessments of alternatives can partly carry out user experiences of shared bikes. Table 3 points out that the LDM methods can be utilized to evaluate user experiences of shared bikes with respect to $c_{1}, c_{2}, c_{3}, c_{4}, c_{5}$ and $c_{6}$.

Based on Table 3, user experiences of shared bikes are carried out as follows:

(1) For criterion $c_{6}$ (Sustainability), $d_{1}$, $d_{2}$ and $d_{3}$ provide single-linguistic terms to assess user experiences, hence 2-tuple fuzzy linguistic aggregation operator with weights of $d_{1}$, $d_{2}$ and $d_{3}$ can be utilized to obtain user experience of each shared bike with respect to $c_{6}$, where decision-making table provided by $d_{1}$, $d_{2}$ and $d_{3}$ is shown in Table 4. By using LDM method in Herrera and Martínez (2000), user experiences of shared bikes provided by $d_{1}$, $d_{2}$ and $d_{3}$ with respect to $c_{6}$ are shown as follows:

Table 5 Decision-making table provided by $d_{1}$, $d_{2}$ and $d_{3}$ with respect to criterion $c_{3}$

Table 6 Decision-making table provided by $d_{1}$, $d_{2}$ and $d_{3}$ with respect to criterion $c_{4}$

(2) For criteria $c_{3}$ (Riding Convenience) and $c_{4}$ (Riding Comfort), $d_{1}$ and $d_{2}$ provides HFLTSs to evaluate user experiences, $d_{3}$ provides single-linguistic terms to evaluate user experience, here the LDM method with HFLTSs (Rodríguez et al. 2012) can be utilized to obtain user experiences of shared bikes with respect to $c_{3}$ and $c_{4}$, i.e., according to hesitant linguistic decision-making tables provided by $d_{1}$, $d_{2}$ and $d_{3}$ shown in Tables 5 and 6 , for each shared bike, $H(a_{j})=[H_{min}(a_{j}), H_{max}(a_{j})] (j=1, 2, 3, 4)$ is employed to aggregate user experiences provided by $d_{1}$, $d_{2}$ and $d_{3}$ with respect to $c_{3}$ and $c_{4}$, in which

$$\begin{aligned} H_{min}(a_{j})= & {} max\{H^{d_{1}}_{S^{-}}(a_{j}), H^{d_{2}}_{S^{-}}(a_{j}), H^{d_{3}}_{S^{-}}(a_{j})\},\\ H_{max}(a_{j})= & {} min\{H^{d_{1}}_{S^{+}}(a_{j}), H^{d_{2}}_{S^{+}}(a_{j}), H^{d_{3}}_{S^{+}}(a_{j})\}. \end{aligned}$$

For example, for shared bike $a_{1}$ with respect to $c_{3}$, $H_{min}(a_{1})=max\{s_{5}, s_{5}, s_{5}\}=s_{5}$ and $H_{max}(a_{1})=min\{s_{5}, s_{6}, s_{5}\}=s_{5}$, hence $H(a_{1})=[s_{5}, s_{5}]$. Others are shown in Tables 5 and 6.

Table 7 Decision-making table provided by $d_{1}$, $d_{2}$ and $d_{3}$ with respect to criterion $c_{2}$

Table 8 Decision-making table provided by $d_{1}$, $d_{2}$ and $d_{3}$ with respect to criterion $c_{5}$

(3) For criteria $c_{2}$ (Riding Safety) and $c_{5}$ (Perfect Emergency Response Mechanism), $d_{1}$ provides FLM-DFNs to evaluate user experiences of $\{a_{1}, a_{3}, a_{4}\}$ and PLTSs to evaluate user experience of $a_{2}$. However, $d_{2}$ and $d_{3}$ provide single-linguistic terms to evaluate user experiences of shared bikes, and linguistic decision-making tables provided by $d_{1}$, $d_{2}$ and $d_{3}$ are shown in Tables 7 and 8 . Considering the single-linguistic term is FLM-DFN with membership degree 1 or PLTS with probability 1, here user experiences of shared bikes with respect to $c_{2}$ and $c_{5}$ can be carried out by using LDM methods with PLTSs or FLM-DFNs, respectively, i.e.,

a) Using the LDM method with FLM-DFNs and aggregation function on the finite chain $S=\{s_0, s_1, \ldots , s_6\}$ (Massanet et al. 2014), i.e., $F(x_{1}, x_{2}, x_{3})=max\{min\{x_{1}, x_{2},$ $x_{3}\},$ $max\{x_{1}, x_{2}, x_{3}\}-3\}$, user experiences of $a_{1}$ and $a_{3}$ with respect to $c_{2}$ are

$$\begin{aligned} H_{c_{2}}(a_{1})= & {} F(\{0.4/s_{4}, 1/s_{5}, 0.6/s_{6}\}, \{1/s_{6}\},\\ \{1/s_{6}\})= & {} \{0.4/s_{4}, 1/s_{5}, 0.6/s_{6}\},\\ H_{c_{2}}(a_{3})= & {} F(\{0.3/s_{4}, 1/s_{5}, 0.7/s_{6}\}, \{1/s_{6}\},\\ \{1/s_{5}\})= & {} \{0.3/s_{4}, 1/s_{5}\}. \end{aligned}$$

User experiences of $a_{1}$, $a_{2}$ and $a_{3}$ with respect to $c_{5}$ are

$$\begin{aligned} H_{c_{5}}(a_{1})= & {} F(\{0.2/s_{4}, 1/s_{5}, 0.7/s_{6}\}, \{1/s_{6}\},\\ \{1/s_{6}\})= & {} \{0.2/s_{4}, 1/s_{5}, 0.7/s_{6}\},\\ H_{c_{5}}(a_{2})= & {} F(\{0.5/s_{4}, 1/s_{5}, 0.6/s_{6}\}, \{1/s_{5}\},\\ \{1/s_{5}\})= & {} \{0.5/s_{4}, 1/s_{5}\},\\ H_{c_{5}}(a_{3})= & {} F(\{0.5/s_{4}, 1/s_{5}, 0.7/s_{6}\}, \{1/s_{6}\},\\ \{1/s_{6}\})= & {} \{0.5/s_{4}, 1/s_{5}, 0.7/s_{6}\}. \end{aligned}$$

b) Using the LDM method with PLTSs and the PLWA operator (Pang et al. 2016)

$$\begin{aligned} Z_{j}(w)= & {} w_{1}L_{j1}(p)\oplus \ldots \oplus w_{n}L_{jn}(p)\\= & {} \cup _{L^{k}_{j1}\in L_{j1}(p)}\{w_{1}p^{k}_{j1}L^{k}_{j1}\}\oplus \ldots \oplus \cup _{L^{k}_{jn}\in L_{jn}(p)}\{w_{n}p^{k}_{jn}L^{k}_{jn}\}, \end{aligned}$$

user experiences of $a_{2}$ with respect to $c_{2}$ are

$$\begin{aligned} Z^{c_{2}}_{2}\!\!\!\!&\left( \left\{ \frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right\} \right) \!=\!\frac{1}{3}\{s_{6}(0.4), s_{5}(0.4), s_{4}(0.2)\}\oplus \frac{1}{3}\{s_{5}(1),\\&s_{5}(0), s_{5}(0)\}\oplus \frac{1}{3}\{s_{6}(1), s_{6}(0),\\&s_{6}(0)\}=\{s_{4.47}, s_{0.67}, s_{0.27}\}. \end{aligned}$$

Table 9 Decision-making table provided by $d_{1}$, $d_{2}$ and $d_{3}$ with respect to criterion $c_{1}$

(4) For criterion $c_{1}$ (Fund Security), $d_{1}$, $d_{2}$ and $d_{3}$ provide single-linguistic term, HFLTS, PLTS or FLM-DFN to evaluate user experiences of shared bikes, respectively, and linguistic decision-making table provided by $d_{1}$, $d_{2}$ and $d_{3}$ is shown in Table 9. Similarly, user experiences of shared bikes can be obtained as follows: User experience of $a_{1}$ is

$$\begin{aligned}&Z^{c_{1}}_{1}(\{\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\})=\frac{1}{3}\{s_{5}(0.8), s_{4}(0.2)\}\oplus \frac{1}{3}\{s_{6}(0.5),\\&\quad s_{5}(0.5)\}\oplus \frac{1}{3}\{s_{5}(1), s_{5}(0)\}=\{s_{4}, s_{1.1}\}, \end{aligned}$$

here HFLTS $\{s_{5}, s_{6}\}$ provided by $d_{2}$ is equal to PLTS $\{s_{6}(0.5), s_{5}(0.5)\}$. User experiences of $a_{2}$ and $a_{3}$ are

$$\begin{aligned}&H_{c_{1}}(a_{2})=F(\{0.4/s_{4}, 1/s_{5}, 0.6/s_{6}\}, \{1/s_{5}\},\\&\{1/s_{5}\})=\{0.4/s_{4}, 1/s_{5}\},\\&H_{c_{1}}(a_{3})=F(\{0.3/s_{4}, 1/s_{5}, 0.7/s_{6}\}, \\&\{1/s_{5}, 1/s_{6}\}, \{1/s_{5}\})=\{0.3/s_{4}, 1/s_{5}\}, \end{aligned}$$

here HFLTS $\{s_{5}, s_{6}\}$ and $\{s_{5}\}$ provided by $d_{2}$ and $d_{3}$ are equal to FLM-DFNs $\{1/s_{5}, 1/s_{6}\}$ and $\{1/s_{5}\}$, respectively.

Table 10 Assessment results of user experiences of shared bikes with respect to criteria

Table 11 Comparing analysis of LDM methods in carrying out LDM problems

Based on the above-mentioned steps (1)-(4), assessment results of user experiences of shared bikes with respect to criteria $c_{1}, c_{2}, c_{3}, c_{4}, c_{5}$ and $c_{6}$ are shown in Table 10. Accordingly, it seems that we can do nothing, because linguistic assessment results of user experiences in Table 10 are very different; for example, user experience of $a_{1}$ with respect to $c_{1}$ is PLTS $\{s_{4}, s_{1.1}\}$, user experiences of $a_{1}$ with respect to $c_{2}$ and $c_{5}$ are FLM-DFNs $\{0.4/s_{4}, 1/s_{5}, 0.6/s_{6}\}$ and $\{0.2/s_{4}, 1/s_{5}, 0.7/s_{6}\}$, respectively, user experiences of $a_{1}$ with respect to $c_{3}$ and $c_{4}$ are HFLTSs $[s_{5}, s_{5}]$ and $[s_{5}, s_{6}]$, respectively, user experience of $a_{1}$ with respect to $c_{6}$ is 2-TLM $(s_{5}, 0.33)$. Up to now, there is no any linguistic aggregation operator can be utilized to simultaneously aggregate them.

Summary, Table 11 shows differences among LDM methods, in which $M_{1}$, $M_{2}$, $M_{3}$ and $M_{4}$ are LDM methods with 2-TLMs, HFLTSs, PLTSs and FLM-DFNs, $M_{5}$ is the proposed LS-MLDM method with multi-linguistic assessments. It is obvious that $M_{1}$, $M_{2}$, $M_{3}$ and $M_{4}$ are useful and effective tools for dealing with LDM problems with 2-TLMs, HFLTSs, PLTSs and FLM-DFNs, respectively. However, they are invalid when linguistic assessments of alternatives are multi-linguistic terms. In LS-MLDM environment, multi-linguistic assessments widely exist because a huge amounts of decision makers employ several fuzzy linguistic representation models to assess alternatives. Unlike $M_{1}$, $M_{2}$, $M_{3}$ and $M_{4}$, the proposed LS-MLDM method ($M_{5}$) is based on multi-linguistic decision matrix and trust degrees of linguistic terms, which can be utilized to not only handle LDM problems with 2-TLMs, HFLTSs, PLTSs or FLM-DFNs, but also carry out LS-MLDM problems.

6 Conclusions and future works

LS-MLDM problems are widely existed in large-scale linguistic decision-making environments because decision makers maybe utilize several fuzzy linguistic representation models to evaluate alternatives with respect to criteria. To deal with LS-MLDM problems, evaluation-based voting is defined and multi-linguistic decision matrix as well as trust degrees of linguistic terms are developed in the paper, and the new decision-making framework of LS-MLDM problems is designed. Theoretically, multi-linguistic decision matrix and trust degrees of linguistic terms aim to uniformly represent multi-linguistic assessments of alternatives and normalize multi-linguistic decision matrix; based on trust degrees of linguistic terms, an uniform fusion method of multi-linguistic assessments is proposed to aggregate multi-linguistic assessments of alternatives with respect to criterion. Comparing analysis in user experience of shared bikes shows that the proposed method is an useful and effective tool to carry out not only classical LDM problems but also LS-MLDM problems.

In reality, huge amounts of decision makers maybe produce many problems besides multi-linguistic assessments of alternatives, such as conflict assessments of alternatives. To deal with the problem, the study of consensus-based decisions in the past 10 years has become a consensus issue of LDM, which is to guide decision makers to reach a consensus before making a decision and obtain acceptable linguistic decision results of alternatives (Labella et al. 2018; Rodriguez et al. 2018; Li and Dong 2019). In multi-linguistic decision matrices, multi-linguistic assessments of alternatives provided by decision makers may be conflict, “how to analyze consensus in LS-MLDM problems” will be our future works. In addition, decision makers maybe utilize multi-granular linguistic terms or numerical and linguistic information to assess alternatives with respect to criteria, “how to manage, transform and deal with numerical, linguistic or multi-granular linguistic assessments of alternatives” may be interesting and important issues in LS-MLDM problems.