Linguistic dynamic multicriteria decision making using symbolic linguistic computing models
 18 Downloads
Abstract
In fuzzy environments, decision information is more suitable to be expressed in linguistic labels than exact numerical values. A linguistic dynamic multicriteria decision making (LDMCDM) problem consists of a finite set of periods in which a set of experts express their evaluations about a finite set of alternatives on a linguistic term set, to select the best alternative of the problem. To deal with linguistic information in LDMCDM, two main symbolic linguistic computing models have been applied: the 2tuple and the virtual linguistic computing models. In this contribution, we review these symbolic computing models and also propose a resolution scheme for solving LDMCDM problems. Special emphasis is put into timedependent aggregation operators due to they are crucial in this type of problems. Thereafter we apply them to a green supplier selection problem to stress the suitability of the proposed resolution scheme, and to analyse the results obtained with both models mainly in terms of representation of linguistic outcomes as well as their interpretability and accuracy. Eventually, some challenges are introduced for further research.
Keywords
Linguistic decision making Dynamic decision making 2Tuple linguistic model Virtual linguistic model1 Introduction
Decision making is the procedure for finding the best alternative from a feasible set. Decision making problems considering multiple criteria are called multicriteria decision making (MCDM) problems. In many MCDM problems, due to the complexity of objects and the vagueness of human thinking, experts may have difficulties in evaluating objects with specific numerical values. Experts often use linguistic descriptors to express their assessments of those difficulties because of time pressure, lack of knowledge or data, and their limited expertise related to the problem domain. This decision making case is called a linguistic MCDM (LMCDM) problem.
The fuzzy linguistic approach (FLA) (Zadeh 1975) provides a direct way to manage uncertainty and represents qualitative aspects of problems by means of linguistic variables. The use of linguistic information implies to operate with linguistic variables, i.e., processes of computing with words (Zadeh 1996). There are two basic linguistic computing models to accomplish them: the semantic model based on the extension principle (Degani and Bortolan 1988) and the symbolic model based on ordinal scales (Yager 1995). The former makes operations on the fuzzy numbers that support the semantics of the linguistic terms (Carrasco et al. 2012; Massanet et al. 2014). The latter makes computations on the indexes of the linguistic terms and obtains results closer to the cognitive model of human beings, but it needs a linguistic approximation of the final computed result which is mostly timeconsuming and computationally complex (Herrera and Martínez 2000; Ho and Wechler 1990); and consequently presents drawbacks related to its lack of precision (Rodríguez and Martínez 2013). To overcome these drawbacks, different symbolic approaches based on the FLA have been proposed (Herrera and Martínez 2000; Wang and Hao 2006; Xu 2004b) which take the advantages of operating without loss of information and computational simplicity by avoiding the use of membership functions.
Decision making is a common process in daily life as in business and industry. However, with the increasing complexity of the socioeconomic environment, the time pressures and the lack of accurate information, it is less likely that experts can analyse and assess all the relevant elements of a problem in a single moment.
Continual equilibrium is not usual in decision making problems, criteria may change in importance as circumstances change while alternatives being considered often change as well. The dynamic feature of MCDM needs to be given more consideration in the development of decision models and processes. In a dynamic MCDM (DMCDM) problem the decision information is usually collected from different periods and/or refers to different moments in time (Xu 2009b). As a series of choices taken over time to achieve some overall goal, one decision may be dynamic at different levels or degrees. Alternatives usually unfold over time in an iterative process where current valuations may depend on external events and conditions as well as on experience from previous choices or feedback.
In the granular computing discipline (Pedrycz and Chen 2011, 2015a, b), special significance is given to the Zadeh’s fundamental definition of the granulation of an object which leads to a collection of its granules, with a granule being a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality (Zadeh 1997). In this seminal paper, the granulation of time into years, months, days, hours and minutes is considered a familiar example for understanding the hierarchical nature of granulation. Pedrycz considered time as an important and omnipresent variable that is subjected to granulation. The size of information granules (time intervals) and their meaning could vary quite significantly depending on the nature of the problem and requirements of decision makers. Time granulation in decision making helps to focus on the most suitable level of detail (Pedrycz 2013). From this perspective, the DMCDM problem consists of selecting the best alternatives from a set of available ones, but considering time granulation.
DMCDM models enable decision makers to change their strategic decisions periodically, merging past with present information, without losing past information, which may improve the quality and consistency of the decision making process. As in MCDM, in DMCDM when the decision information collected in the multiple periods is represented by means of linguistic variables, we are in presence of a linguistic DMCDM (LDMCDM) problem. However, due to LDMCDM are more complex than LMCDM, achievements on the former issue are less than on the latter and how to deal with the LDMCDM problems with linguistic information is an interesting and wideopen meaningful research topic. Linguistic information aggregation is an essential process in LMCDM and LDMCDM. While in LMCDM, an aggregation phase is required for obtaining a collective performance value for the alternatives, in LDMCDM this step is made in each period, then an additional aggregation phase is demanded for integrating results from each LMCDM problem that compounds the LDMCDM problem. Classical aggregation operators from linguistic computing models can only be used to deal with timeindependent arguments. However, if time is taken into account, then the aggregation operators should reflect this fact.
By far, two symbolic models have mainly been used to deal with LDMCDM problems: the 2tuple linguistic model (Herrera and Martínez 2000) and the virtual linguistic model (Xu 2004b). Hence, in this contribution we review the LDMCDM problems in which the decision information gathered in the multiple periods is represented and managed using the 2tuple linguistic (Herrera and Martínez 2000) and the virtual linguistic (Xu 2004b) symbolic computing models. Special emphasis is put into timedependent aggregation operators in these models due to they are crucial in LDMCDM. The aim of this paper is to study and analyse the influence of these different symbolic computing models to cope with dynamic linguistic information. To do so, we apply them to a green supplier selection problem.
The rest of the paper is structured as follows. Section 2 reviews in short the representation and the operational laws of the most wideused symbolic models in LDMCDM: the 2tuple linguistic model (Herrera and Martínez 2000), the virtual model (Xu 2004b). Section 3 presents a general resolution procedure for LDMCDM problems. Section 4 revises timedependent aggregation operators required to accomplish aggregation processes. Section 5 presents a case study conducted on a green supplier selection problem solved by the previous linguistic computing models, whose results are analysed considering different features. Section 6 provides some potential directions with insights for LDMCDM, which are still open issues and need more research efforts. Finally, some conclusions are drawn in Sect. 7.
2 Symbolic linguistic computing models: basic concepts and main solutions
Here we review briefly the FLA and the two symbolic classical computing models mainly used in LDMCDM.
2.1 Fuzzy linguistic approach
 1.
Negation: Neg\((s_{i}) = s_{j}\) such that \(j = g  i\) (\(g + 1\) is the cardinality)
 2.
Maximization: \(\max (s_i,s_j)= s_i \ \text {if} \ ~s_i \ge s_j\)
 3.
Minimization: \(\ \min (s_i,s_j)= s_i \ \text {if} \ ~s_i \le s_j\)
2.2 Symbolic linguistic computing models

Semantic model (Degani and Bortolan 1988): This model computes with linguistic terms by means of operations associated to their membership functions based on the Extension Principle. So, the obtained results are fuzzy numbers that usually do not match with the initial linguistic terms.

Symbolic model (Yager 1995): This model uses the ordered structure of the linguistic terms set \(S=\{s_0, \ldots ,s_g\}\) where \(s_i<s_j\) if \(i<j\), to operate. The results are numeric values that will be approximated to a numeric value that indicates the index of the associated linguistic term.
2.3 2Tuple linguistic model
 1.
Representation model: The 2tuple linguistic model represents the information by means of a pair of values \((s_i,\alpha )\), where s is a linguistic term with syntax and semantics, and \(\alpha\) is a numerical value that represents the symbolic translation.
Definition 1
(Herrera and Martínez 2000) The symbolic translation is a numerical value assessed in \([0.5,0.5)\) that supports the difference of information between a counting of information \(\beta\) assessed in the interval of granularity [0, g] of the term set S and the closest value in \(\{0,\ldots ,g\}\) which indicates the index of the closest linguistic term in S.
It defines transformation functions between numerical values and 2tuple linguistic values to facilitate linguistic computational processes.
Definition 2
Proposition 1
Let \(S=\{s_0,\ldots ,s_g\}\) be a linguistic term set and \((s_i,\alpha )\) be a 2tuple linguistic value. There is always a function \(\Delta ^{1}\) such that from a 2tuple linguistic value, it returns its equivalent numerical value \(\beta \in [0,g]\) as \(\Delta ^{1}(s_i,\alpha )= i+\alpha\).
Remark 1
The conversion of a linguistic term into 2tuple linguistic value consists of adding a value 0 as symbolic translation.
 2.
Computational model: It is based on the functions \(\Delta\) and \(\Delta ^{1}\). In Herrera and Martínez (2000) different operations as comparison, negation and aggregation operators were defined.
 3.
Timeindependent Aggregation Operators Functions \(\Delta\) and \(\Delta ^{1}\) in the fuzzy linguistic representation model with 2tuples transform numerical values into a 2tuples and vice versa without loss of information, therefore, numerical aggregation operators are usually extended for dealing with linguistic 2tuples.
Wei (2010) defined the extended 2tuple linguistic weighted geometric (ETWG) operator and the extended 2tuple linguistic ordered weighted geometric (ETOWG) operator; he also proposed some harmonic aggregation operators (Wei 2011b) as well as the generalized 2tuple weighted average (GTWA) operator, the generalized 2tuple linguistic ordered weighted average (GTOWA) operator, and the induced generalized 2tuple linguistic ordered weighted average (IGTOWA) operator (Wei 2011a). Xu and Wang (2011a) presented the 2tuple linguistic power average (TPA) operator and the 2tuple linguistic power ordered weighted average (TPOWA) operator. Wei and Zhao (2012) introduced the dependent 2tuple linguistic ordered weighted averaging (DTOWA) operator and the dependent 2tuple linguistic ordered weighted geometric (DTOWG) operator, in which the associated weights only depend on the aggregated 2tuple linguistic arguments. Meng and Tang (2013) developed the extended 2tuple linguistic hybrid arithmetical weighted (ETHAW) operator, the induced extended 2tuple linguistic hybrid arithmetical weighted (IETHAW) operator, the extended 2tuple linguistic hybrid geometric mean (ETHGM) operator, and the induced extended 2tuple linguistic hybrid geometric mean (IETLHGM) operator. Yang (2013) proposed the quasiarithmetic induced 2tuple linguistic correlated averaging (QINTCA) operator, the induced 2tuple linguistic correlated averaging (INTCA) operator and the generalized induced 2tuple linguistic correlated averaging (GINTCA) operator.
Merigó and GilLafuente (2013) presented the induced 2tuple generalized ordered weighted averaging (TIGOWA) operator, the QuasiTIOWA operator and the 2tuple induced quasiarithmetic Choquet integral aggregation. Moreover, in the field of uncertain 2tuple linguistic information, Wang et al. (2013) introduced the interval 2tuple linguistic correlated averaging (ITCA) operator and the interval 2tuple linguistic correlated geometric (ITCG) operator; and Beg and Rashid (2014) developed the intervalvalued 2tuple correlated averaging (IVTCA) operator, the intervalvalued 2tuple correlated geometric (IVTCG) operator, and the generalized intervalvalued 2tuple correlated averaging (GIVTCA) operator for different variants of intervalvalued 2tuple linguistic information.
Wan (2013) designed some hybrid aggregation operators with 2tuple linguistic information, involving the 2tuple hybrid weighted arithmetic average (THWA) operator, the linguistic THWA (THLWA) operator, and the extended THLWA (ETHLWA) operator.
The linguistic proportional 2tuple power average operator was introduced by Jiang et al. (2015) to aggregate linguistic values of unbalanced linguistic term sets considering the relationship among the aggregated values. Besides, the 2tuple linguistic extended Bonferroni mean (EBM) aggregation operators and the 2tuple linguistic partition Bonferroni mean (PBM) aggregation operators were developed in Dutta and Guha (2015b, c), respectively, for dealing with group decision making problems.
Some authors have also considered the interactive phenomenon among experts or attributes. Lin et al. (2015) proposed a generalized interval 2tuple linguistic Shapley chisquare averaging operator for facility location selection in group decision making environments. So et al. (2016) introduced some linguistic aggregation operators with conservation of interaction between criteria, which include the 2tuple linguistic Choquet integral averaging (TCIA) operator, the 2tuple linguistic ordered Choquet integral averaging (TOCIA) operator, and the combined 2tuple linguistic Choquet integral averaging operator. Ju et al. (2016) gave some new Shapley 2tuple linguistic Choquet aggregation operators for MAGDM: Shapley 2tuple linguistic Choquet averaging operator, Shapley 2tuple linguistic Choquet geometric operator and generalized Shapley 2tuple linguistic Choquet averaging operator. Qin and Liu (2016) proposed the 2tuple linguistic Muirhead mean (2TLMM) operator and the 2tuple linguistic dual Muirhead mean (2TLDMM) operator.
Li and Liu (2015) provided some new aggregation operators of 2tuple linguistic information based on Heronian mean. Liu et al. (2016) presented the Algebra tnorm and snorm based 2tuple linguistic Heronian mean operator or the Algebra tnorm and snorm based 2tuple linguistic weighted Heronian mean operator.
2.4 Virtual linguistic model
 1.
Representation model: This model extends the discrete term set S to a continuous linguistic term set \(\bar{S} = \{{s_\alpha  s_0 < s_\alpha \le s_f, \alpha \in [0,f]}\}\), where, if \(s_\alpha \in S\), \(s_\alpha\) is called an original linguistic term, otherwise, \(s_\alpha\) is called virtual linguistic term which does not have assigned any semantics (Xu 2004b; Xu and Wang 2017).
 2.Computational model: To carry out processes of computing with words, Xu presented several operations (Xu 2004b) that extend the previous ones of the 2tuple. Let \(s_{\alpha };s_{\beta } \in \bar{S}\) be any two linguistic terms and \(\mu , \mu _1,\mu _2\in [0,1]\).Operational laws mentioned above are defined based on a given linguistic term set, but the calculated results would not exist in the set.
 (a)
\((s_{\alpha })^\mu =s_{\alpha ^\mu }\)
 (b)
\((s_{\alpha })^{\mu _1}\otimes (s_{\alpha })^{\mu _2} =(s_{\alpha })^{\mu _1+\mu _2}\)
 (c)
\((s_{\alpha }\otimes s_{\beta })^{\mu }=(s_{\alpha })^{\mu }\otimes (s_{\beta })^{\mu }\)
 (d)
\(s_{\alpha }\otimes s_{\beta }=s_{\beta }\otimes s_{\alpha }=s_{\alpha \beta }\)
 (e)
\(s_{\alpha }\oplus s_{\beta }=s_{\alpha +\beta }\)
 (f)
\(\mu s_{\alpha }=s_{\mu \alpha }\)
 (g)
\((\mu _1+\mu _2)s_{\alpha }=\mu _1 s_{\alpha }\oplus \mu _2 s_{\alpha }\)
 (h)
\((s_{\alpha }\oplus s_{\beta })=\mu s_{\alpha }\oplus \mu s_{\beta }\)
 (i)
\((s_{\alpha })^{1}=s_{\frac{1}{\alpha }}\)
 (a)
 3.
Timeindependent aggregation operators: Several timeindependent aggregation operators have been developed applying previous operational laws. Xu (2004b) defined the linguistic geometric averaging (LGA) operator, linguistic weighted geometric averaging (LWGA) operator, linguistic ordered weighted geometric averaging (LOWGA) operator and linguistic hybrid geometric averaging (LHGA) operator. Also he introduced the extended geometric mean (EGM), the extended arithmetical averaging (EAA), the extended OWA and extended OWG operators (Xu 2004a); as well as the extended weighted arithmetic averaging (EWAA) operator (Xu 2005a); and the extended induced OWG (EIOWG) operator (Xu 2005b).
For the uncertain linguistic information under the virtual representation model, Xu presented the uncertain linguistic geometric mean (ULGM) operator, uncertain linguistic weighted geometric mean (ULWGM) operator, uncertain linguistic OWG (ULOWG) operator and induced ULOWG operator, the uncertain LOWG and the induced uncertain LOWG operators (Xu 2006a); as well as some induced uncertain linguistic OWA operators (Xu 2006b). Additionally, he introduced the linguistic correlated averaging (LCA) operator linguistic correlated geometric (LCG) operator (Xu 2009a). Merigó and Casanovas (2010) provided the linguistic ordered weighted averaging distance (LOWAD) operator, and also Xu and Wang (2011b) developed other linguistic distance operators, such as linguistic weighted distance (LWD) operator and the ordered LWD (LOWD) operator. Liu and Su developed the trapezoid fuzzy linguistic hybrid ordered weighted averaging (TFLHOWA) operator (Liu and Su 2010).
Zhang et al. (2015) defined a series of linguistic aggregation operators considering the confidence levels of the aggregated arguments, such as the confidence linguistic weighted averaging (CLWA) operator, the confidence linguistic ordered weighted averaging (CLOWA) operator, and the confidence generalized linguistic ordered weighted averaging (CGLOWA) operator.
3 Resolution procedure for linguistic dynamic multicriteria decision making problems
A LDMCDM problem consists of a finite set of periods in which a set of experts express their evaluations about a finite set of alternatives on a linguistic term set, to select the best alternative of the problem. Let \(T=\{(t_\lambda ) \lambda \in (1,\ldots ,q)\}\), a discrete set of q periods. At every period, \(A(t_\lambda )=\{a_i(t_\lambda )i\in (1,\ldots ,m)\}\) a discrete set of m feasible alternatives, \(E(t_\lambda )=\{e_k(t_\lambda )k\in (1,\ldots ,p\}\) the set of experts assessing the alternatives according to the set of criteria \(C(t_\lambda )=\{c_j(t_\lambda )j\in (1,\ldots ,n\}\). In this scenario, periods, experts and criteria, may have different importance. Hence, weights of periods across time are given by the vector \(W^T=(w^T_\lambda \lambda \in (1,\ldots ,q)), w^T_\lambda \in [0,1]\) with \(\sum ^{q}_{\lambda =1}w^T_\lambda =1\); while the weights of experts and criteria are respectively given by the vectors \(W^C(t_\lambda )=(w^C_j(t_\lambda ) j\in (1,\ldots ,n)), w^C_j(t_\lambda )\in [0,1]\) with \(\sum ^{n}_{j=1}w^C_j(t_\lambda )=1\) and \(W^E(t_\lambda )=(w^E_k(t_\lambda )k\in (1,\ldots ,p)), w^E_k(t_\lambda )\in [0,1]\) with \(\sum ^{p}_{k=1}w^E_k(t_\lambda )=1\). The assessment provided by expert \(e_k(t_\lambda )\in E(t_\lambda )\) about alternative \(a_i(t_\lambda )\in A(t_\lambda )\) according to criterion \(c_j(t_\lambda )\in C(t_\lambda )\) is represented by \(x_{ijk}(t_\lambda )\).
A solution scheme of a decision making problem consists of two phases (Roubens 1997): (1) an aggregation phase that obtains collective valuations of each alternative and (2) an exploitation phase that obtains the solution set of alternatives of the problem. The use of linguistic information in decision making modifies the previous scheme by introducing two new steps (Herrera and HerreraViedma 2000): (1) the choice of the linguistic term set with its semantics and (2) the choice of the aggregation operator of linguistic information.
 1.
Phase 1: The choice of the linguistic term set with its semantics. It establishes the linguistic expression domain in which experts provide their linguistic assessments about alternatives according to their knowledge.
 2.
Phase 2: The choice of the timeindependent aggregation operator of linguistic information. A proper classical linguistic aggregation operator is chosen for aggregating the linguistic assessments.
 3.
Phase 3: An aggregation phase that aggregates the values provided by experts to obtain a collective assessment (nondynamic) for alternatives considered at each period.
 4.
Phase 4: The choice of the timedependent aggregation operator of linguistic information. A proper linguistic aggregation operator is chosen for aggregating the linguistic assessments collected from different periods. As time is taken into account, for example, the argument information may be collected at different periods, then the aggregation operators and their associated weights should not be kept constant. Clearly, one key point in aggregation operator is to determine its associated weights. The appropriateness of the operator and weights vectors, depends on each LDMCDM problem.
 5.
Phase 5: An aggregation phase that aggregates the collective assessment of alternatives at each period to obtain a dynamic collective assessment for each alternative. Due to the time dimension should be added into the common singleperiod MCDM problem, it is necessary to integrate static results from each static problem to obtain a holistic dynamic assessment of alternatives. To do this, timedependent aggregation operators should be used.
 6.
Phase 6: An exploitation phase of the dynamic collective assessments to rank, sort, or choose the best ones among the alternatives.
4 Timedependent aggregation operators for LDMCDM
Linguistic computing models revised in Sect. 2.2 have been extended for dealing with linguistic time variables. Timedependent aggregation operators have been defined to accomplish Phase 5 from resolution scheme presented in Sect. 3 due to classical timeindependent aggregation operators are not suitable for fusing linguistic information gathered in multiple periods.
A wide range of conventional (timeindependent) linguistic aggregation operators can be found in the literature, but few studies focus on linguistic timedependent aggregation operators. In what follows, several linguistic timedependent aggregation operators are reviewed.
4.1 2Tuple timedependent aggregation operators for LDMCDM
In this section, several linguistic 2tuple timedependent aggregation operators will be presented for supporting aggregation in LDMCDM.
4.1.1 2Tuple dynamic weighted averaging aggregation operators
Definition 3
4.1.2 2Tuple dynamic weighted geometric aggregation operator
Definition 4
Remark 2
It is noteworthy that if a linguistic argument is (\(s_0,0\)) the aggregation result of TDWG and TDG will be (\(s_0,0\)).
4.2 Virtual timedependent aggregation operators
In this section, several virtual dynamic timedependent aggregation operators are revised.
4.2.1 Virtual dynamic weighted averaging aggregation operators
Definition 5
4.2.2 Virtual dynamic weighted geometric aggregation operator
Definition 6
Remark 3
It is noteworthy that if a linguistic argument is \(s_0\) the aggregation result of VDWG and VDG will be \(s_0\).
5 A LDMCDM problem resolution
The supplier selection problem is usually modelled in a linguistic context. That is why we can find in specialized literature, diverse applications of linguistic models for solving the supplier selection problem (CidLópez et al. 2016; Jimenez and Zulueta 2017; Karsak and Dursun 2015; Qin and Liu 2016; Oliveira et al. 2017; Wen et al. 2016; You et al. 2015).
For a better understanding on how LDMCDM problems are solved using symbolic linguistic computing models, this section presents a green supplier selection problem in a changeable environment (Jimenez and Zulueta 2017) which is tackled following the resolution scheme developed in Sect. 3, and using the two symbolic linguistic computing models reviewed in Sect. 2.2.
In the green supplier selection, the enterprises providing environmentally friendly products and services have extra credit and recognition for their concern on sustainable economic development, and the positive environmental management performance. In the real decision contexts general and green criteria might vary over time, new ones might be considered, or existing ones could turn into irrelevant in different market conditions. Consequently, decisions about suppliers take place in a dynamic environment, where the final decision is taken at the end of some exploratory process. In such cases, beneficial exploration of the problem can be performed throughout LDMCDM approaches.

Green capability (\(c_1\)): The ability to prepare, produce and deliver green products based on environmental standards.

Price (\(c_2\)): The total cost of products offered as the price.

Environmental management system (\(c_3\)): Applying any environmental management systems.

Green design (\(c_4\)): A systematic method to reduce the environmental impact of products and processes.
 1.
Phase 1: The linguistic term set selected to be used by experts is that of the Sect. 2.1. Its semantic can be seen in Figure 2. Using this linguistic term set, experts express their assessments about suppliers in three periods. The input data of LDMCDM is represented in Table 1.
 2.
Phase 2: The timeindependent aggregation operators selected to compute the nondynamic collective assessment of suppliers are the TAM (Herrera and Martínez 2000) and the EAA (Xu 2004a) for the 2tuple and virtual linguistic models respectively, since both are the linguistic extension of the conventional arithmetical averaging operator.
 3.
Phase 3: Using previous aggregation operators, a twostep aggregation process is accomplished: first, the collective criteria values of suppliers in each period are computed and then the nondynamic collective assessments of suppliers in each period are computed (see Tables 2, 3). As examples, we show how to compute the nondynamic collective assessments of supplier \(a_1\) in period \(t_1\), applying the different models:
2tuple linguistic modelVirtual linguistic model Step 1: Computing the collective criteria values using the TAM. The collective \(c_1\) value is generated as:Here, for \(\Delta (3.67)\), 3.67 is a real number which represents the equivalent linguistic information and only appear in operations. But the use of \(\Delta\) function supports the syntactical dimension and allows to obtain a linguistic value that, in the semantical dimension, uses the real number \(\beta\) to represent its information.$$\begin{aligned}&\mathrm{TAM}((H,0),(H,0),(M,0)) \\&\quad =\mathrm{TAM}((s_4,0),(s_4,0),(s_3,0))\\&\quad =\Delta \left( \frac{1}{3} \Delta ^{1}(s_4,0)+\Delta ^{1}(s_4,0)+\Delta ^{1}(s_3,0)\right) \\&\quad =\Delta \left( \frac{1}{3}\times (4+4+3)\right) =\Delta (3.67)=(s_4, \ 0.33). \end{aligned}$$
In similar way, collective values for the rest of criteria are obtained: (\(s_2\), 0) for \(c_2\), (\(s_4\), 0.3) for \(c_3\) and (\(s_4\), 0.3) for \(c_4\).
 Step 2: Computing the nondynamic collective assessments for period \(t_1\). This value is generated using the previous collective criteria values as arguments for TAM:$$\begin{aligned}&\mathrm{TAM}((s_4, \ 0.33),(s_2,0),(s_4,0.3),(s_4,0.3))\\&\quad =\Delta \bigg (\frac{1}{4}(\Delta ^{1}(s_4,0.33)+\Delta ^{1}(s_2,0)\\&\qquad +\Delta ^{1}(s_4,0.3) +\Delta ^{1}(s_4,0.3)\bigg )\\&\quad =\Delta \left( \frac{1}{4} \times 14.27\right) =\Delta (3.57) =(s_4, \ 0.43). \end{aligned}$$
 Step 1: Computing the collective criteria values using the EAA. The collective \(c_1\) value is generated as:Here, \(s_{11}\) and \(s_{3.67}\) are virtual terms, i.e., not original linguistic terms that appear in operations. They will be used in successive operations but they lack of syntax and semantics.$$\begin{aligned}&\mathrm{EAA}((H,0),(H,0),(M,0)) \\&\quad =\mathrm{EAA}(s_4,s_4,s_3)=\frac{1}{3}(s_4 \oplus s_4 \oplus s_3)= \frac{1}{3} s_{11} = s_{3.67}. \end{aligned}$$
In similar way collective values for the rest of criteria are obtained: \(s_2\) for \(c_2\), \(s_{4.33}\) for \(c_3\) and \(s_{4.33}\) for \(c_4\).
 Step 2: Computing the nondynamic collective assessments for period \(t_1\). This value is generated using the previous collective criteria values as arguments for TAM:$$\begin{aligned} \mathrm{EAA}(s_{3.67},s_2,s_{4.3},s_{4.3})= \frac{1}{4} (s_{3.67} \oplus s_2 \oplus s_{4.3} \oplus s_{4.3})= \frac{1}{4}s_{14.27}= s_{3.57}. \end{aligned}$$

 4.
Phase 4: Once we have a nondynamic evaluation of supplier at each period, to aggregate these multiperiod linguistic data, we use different timedependent aggregation operators in order to analyse the results obtained. For the 2tuple linguistic model, the TDA , TDG, TDWA and TDWG are utilized (see Definitions 3, 4), while for the virtual linguistic model, the VDA, VDG, VDWA and VDWG are utilized (see Definitions 3–6).
Linguistic decision matrices for the three periods
\(t_{\lambda }\)  \(e_k\)  \(a_i\)  \(c_{1}\)  \(c_{2}\)  \(c_{3}\)  \(c_{4}\) 

\(t_1\)  \(e_1\)  \(a_1\)  H  VL  H  VH 
\(a_2\)  H  H  VL  M  
\(a_3\)  VL  VH  EH  M  
\(a_4\)  M  VH  L  M  
\(e_2\)  \(a_1\)  H  L  VH  M  
\(a_2\)  H  H  L  VL  
\(a_3\)  VL  M  VH  VH  
\(a_4\)  L  EL  M  L  
\(e_3\)  \(a_1\)  M  M  H  VH  
\(a_2\)  VH  L  EL  M  
\(a_3\)  M  L  H  H  
\(a_4\)  EH  H  EH  H  
\(t_2\)  \(e_1\)  \(a_1\)  VL  L  VH  M 
\(a_2\)  M  M  VH  VL  
\(a_3\)  H  M  H  L  
\(a_4\)  M  VH  H  M  
\(e_2\)  \(a_1\)  L  VL  EH  L  
\(a_2\)  L  H  VH  VH  
\(a_3\)  L  VL  VH  VL  
\(a_4\)  L  EH  VL  VH  
\(e_3\)  \(a_1\)  L  M  EH  H  
\(a_2\)  EH  EL  VL  M  
\(a_3\)  EH  EL  M  L  
\(a_4\)  H  VH  H  H  
\(t_3\)  \(e_1\)  \(a_1\)  EH  H  M  VH 
\(a_2\)  EL  EH  VL  VL  
\(a_3\)  M  L  EH  VH  
\(a_4\)  L  M  L  H  
\(e_2\)  \(a_1\)  M  VL  VL  EH  
\(a_2\)  H  H  EH  EH  
\(a_3\)  VH  VH  H  L  
\(a_4\)  L  M  EH  L  
\(e_3\)  \(a_1\)  H  M  L  EH  
\(a_2\)  M  VH  M  EH  
\(a_3\)  VH  VH  EH  EH  
\(a_4\)  H  L  VH  H 
Nondynamic 2tuple linguistic collective assessments of suppliers for each period
\(t_{\lambda }\)  \(a_1\)  \(a_2\)  \(a_3\)  \(a_4\) 

\(t_1\)  (\(s_4\), \(\) 0.43)  (\(s_3\), \(\) 0.25)  (\(s_4\), \(\) 0.50)  (\(s_3\), 0.33) 
\(t_2\)  (\(s_3\), 0.08)  (\(s_3\), 0.25)  (\(s_3\), \(\) 0.25)  (\(s_4\), \(\) 0.08) 
\(t_3\)  (\(s_3\), 0.17)  (\(s_2\), 0.42)  (\(s_4\), 0.33)  (\(s_3\), 0.17) 
Nondynamic virtual linguistic collective assessments of suppliers for each period
\(t_{\lambda }\)  \(a_1\)  \(a_2\)  \(a_3\)  \(a_4\) 

\(t_1\)  \(s_{3.57}\)  \(s_{3.75}\)  \(s_{3.50}\)  \(s_{3.33}\) 
\(t_2\)  \(s_{3.08}\)  \(s_{3.25}\)  \(s_{2.75}\)  \(s_{3.92}\) 
\(t_3\)  \(s_{3.17}\)  \(s_{2.42}\)  \(s_{4.33}\)  \(s_{3.17}\) 
 TDWA\(_{a}\) and TDWG\(_{a}\):

In these cases we will use a vector \(W_T=(0.1,0.3,0.6)\) for reward the earlier the better (ascending importance).
 TDWA\(_{d}\) and TDWG\(_{d}\):

In these cases we will use a vector \(W_T=(0.6,0.3,0.1)\) for reward the later the better (descending importance).
 5.
Phase 5: By applying these operators, the collective nondynamic assessments for the suppliers at each period are aggregated to obtain a dynamic collective assessment for each supplier (see Tables 4, 5).
As examples, we show how to compute the dynamic collective assessments of supplier \(a_1\) by applying the weighted averaging and geometric aggregation operator for the two different models:
2tuple linguistic modelVirtual linguistic model $$\begin{aligned}&\text {TDWA}_{a}\left( (s_4,0.43),(s_3,0.8),(s_4,0.33)\right) \\&\quad =\Delta (0.1\times \Delta ^{1}(s_4,0.43)+ 0.3 \times \Delta ^{1}(s_3,0.8)\\&\qquad + \ 0.6 \times \Delta ^{1}(s_4,0.33))\\&\quad =\Delta \left( 0.1\times 3.57+0.3 \times 3.08+ 0.6 \times 3.17\right) \\&\quad =\Delta (3.18)=(s_3,0.18). \\ \end{aligned}$$
 $$\begin{aligned}&\text {TDWG}_{a}\left( (s_4,0.43),(s_3,0.08),(s_4,0.33)\right) \\&\quad =\Delta ((\Delta ^{1}(s_4,0.43))^{0.1 } \times (\Delta ^{1}(s_3,0.8))^{0.3} \\&\qquad \times \ (\Delta ^{1}(s_4,0.33))^{0.6})\\&\quad = \Delta (3.57^{0.1}\times 3.08^{0.3} \times 3.17^{0.6} )\\&\quad =\Delta (3.18)=(s_3,0.18).\\ \end{aligned}$$
 $$\begin{aligned}&\mathrm{VDWA}_{a} \left( s_{3.57},s_{3.08},s_{3.67}\right) \\&\quad =0.1 s_{3.57} \oplus 0.3 s_{3.08} \oplus 0.6 s_{3.17}\\&\quad =s_{0.357} \oplus s_{0.924} \oplus s_{1.902}\\&\quad =s_{3.18}. \end{aligned}$$
 $$\begin{aligned}&\mathrm{VTDWG}_{a}\left( s_{3.57},s_{3.08},s_{3.67}\right) \\&\quad = (s_{3.57})^{0.1} \otimes (s_{3.08})^{0.3} \otimes (s_{3.67})^{0.6}\\&\quad = s_{3.57^{0.1}} \otimes s_{3.08^{0.3}} \otimes s_{3.67^{0.6}}\\&\quad = s_{3.18}. \end{aligned}$$

Dynamic 2tuple linguistic collective assessments for suppliers by using different aggregation operators
Operator  \(a_1\)  \(a_2\)  \(a_3\)  \(a_4\) 

TDA  (\(s_3\), 0.28)  (\(s_3\), \(\) 0.19)  (\(s_4\), \(\) 0.47)  (\(s_3\), 0.47) 
TDG  (\(s_3\), 0.27)  (\(s_3\), \(\) 0.22)  (\(s_3\), 0.47)  (\(s_3\), 0.46) 
\(\text {TDWA}_{a}\)  (\(s_3\), 0.18)  (\(s_3\), \(\) 0.30)  (\(s_4\), \(\) 0.22)  (\(s_3\), 0.41) 
\(\text {TDWG}_{a}\)  (\(s_3\), 0.18)  (\(s_3\), \(\) 0.32)  (\(s_4\), \(\) 0.30)  (\(s_3\), 0.39) 
\(\text {TDWA}_{d}\)  (\(s_3\), 0.39)  (\(s_3\), \(\) 0.13)  (\(s_3\), 0.36)  (\(s_3\), 0.49) 
\(\text {TDWG}_{d}\)  (\(s_3\), 0.38)  (\(s_3\), \(\) 0.15)  (\(s_3\), 0.33)  (\(s_3\), 0.48) 
Dynamic virtual linguistic collective assessments for suppliers by using different aggregation operators
Operator  \(a_1\)  \(a_2\)  \(a_3\)  \(a_4\) 

VDA  \(s_{3.28}\)  \(s_{2.81}\)  \(s_{3.53}\)  \(s_{3.47}\) 
VDG  \(s_{3.27}\)  \(s_{2.78}\)  \(s_{3.47}\)  \(s_{3.46}\) 
\(\mathrm{VDWA}_{a}\)  \(s_{3.18}\)  \(s_{2.70}\)  \(s_{3.78}\)  \(s_{3.41}\) 
\(\mathrm{VDWG}_{a}\)  \(s_{3.18}\)  \(s_{2.68}\)  \(s_{3.70}\)  \(s_{3.39}\) 
\(\mathrm{VDWA}_{d}\)  \(s_{3.39}\)  \(s_{2.87}\)  \(s_{3.36}\)  \(s_{3.49}\) 
\(\mathrm{VDWG}_{d}\)  \(s_{3.38}\)  \(s_{2.85}\)  \(s_{3.33}\)  \(s_{3.48}\) 
 6.Phase 6: In the exploitation phase, the dynamic collective assessments are used to order suppliers and choose the best one among them. The order in the 2tuple model is obtained using comparison rules for 2tuples. For two 2tuple linguistic values, \((s_{k},\alpha _{1})\) and \((s_{l},\alpha _{2})\), the comparison rules are as follows:

if \(k < l\) then \((s_{k},\alpha _{1})\prec (s_{l},\alpha _{2})\).
 if \(k = l\) then

if \(\alpha _{1} = \alpha _{2}\) then \((s_{k},\alpha _{1})\) = \((s_{l},\alpha _{2})\);

if \(\alpha _{1} < \alpha _{2}\) then \((s_{k},\alpha _{1})\prec (s_{l},\alpha _{2})\);

if \(\alpha _{1}> \alpha _{2}\) then \((s_{k},\alpha _{1})\succ (s_{l},\alpha _{2})\).


Order of alternatives using different timedependent 2tuple linguistic aggregation operators
Operator  Order  Best supplier 

TDA  \(a_3\succ a_4\succ a_1\succ a_2\)  \(a_3\) 
TDG  \(a_3\succ a_4\succ a_1\succ a_2\)  \(a_3\) 
\(\mathrm{TDWA}_a\)  \(a_3\succ a_2\succ a_1\succ a_4\)  \(a_3\) 
\(\mathrm{TDWA}_d\)  \(a_4\succ a_1\succ a_3\succ a_2\)  \(a_4\) 
\(\mathrm{TDWG}_a\)  \(a_3\succ a_2\succ a_1\succ a_4\)  \(a_3\) 
\(\mathrm{TDWG}_d\)  \(a_4\succ a_1\succ a_3\succ a_2\)  \(a_4\) 
Order of alternatives using different timedependent virtual linguistic aggregation operators
Operator  Order  Best supplier 

VDA  \(a_3\succ a_4\succ a_1\succ a_2\)  \(a_3\) 
VDG  \(a_3\succ a_4\succ a_1\succ a_2\)  \(a_3\) 
VDWA\(_a\)  \(a_3\succ a_2\succ a_1\succ a_4\)  \(a_3\) 
VDWA\(_d\)  \(a_4\succ a_1\succ a_3\succ a_2\)  \(a_4\) 
VDWG\(_a\)  \(a_3\succ a_2\succ a_1\succ a_4\)  \(a_3\) 
VDWG\(_d\)  \(a_4\succ a_1\succ a_3\succ a_2\)  \(a_4\) 
5.1 Discussion
Once the two symbolic linguistic models have been applied to the green supplier selection problem, it is interesting to analyse and compare them despite both models provide equal order results.
When problems are tackled in dynamic linguistic environments, some issues should be discussed to understand the suitability of the model selected for making decisions. For green supplier selection problems as well as for dynamic linguistic decision problems in general, different features are crucial to analyse the resolution procedure and results, such as (1) the representation of linguistic results, (2) the interpretability of results and (3) the accuracy.
Regarding the first feature, using the virtual linguistic model, the result obtained is a virtual term that does not have assigned any semantics either proper linguistic syntax, that is, the output is not really linguistic. In contrast, the results obtained by 2tuple linguistic model have assigned fuzzy semantics and syntax. Therefore, they keep a fuzzy representation of the linguistic information, whereas that the virtual linguistic model do not have any fuzzy representation. Therefore, it is not possible to represent graphically their virtual semantics.
The second feature is related to the first one. The virtual linguistic model obtains values difficult to understand, because they are not linguistic. The appearance of virtual terms without syntax either semantics limits the interpretability of results of this computational model. Therefore, this model also needs an approximation process, implying lack of accuracy, if results of operations are virtual terms (and they will usually be virtual ones) and the problem looks for interpretable final results in the original linguistic term set. Otherwise, they can be used for ranking purposes. But, the obtained results are not always linguistic terms in the linguistic term set designated a priori, i.e, the decision results cannot be represented in a natural language way. As we can see in Table 4, results obtained with 2tuple linguistic model are easy to understand, because they are represented by means of a linguistic term and a numerical value. The 2tuple linguistic model is based on the fuzzy linguistic approach and it keeps a syntax and fuzzy semantics in its representation. Its computational model provides accurate and understandable results for human beings.
The lack of a clear representation of syntax and semantics of the virtual model has derived further researches and recently Xu and Wang (2017) extended the theoretical foundation of this model by proposing a syntactical rule that generates virtual linguistic terms by a symbolic transformation, and a semantic rule that presents the semantics of virtual linguistic terms by means of linguistic modifiers. Although this new representation model enables for solving problems in a “from words to words” procedure, it has not been widely used yet.
Regarding the third feature, the computational model of the virtual linguistic model is accurate in any term set, because it does not use semantics in the term set. Besides, the results obtained in the computing processes can be values that are out of the universe of discourse of the linguistic variable. The 2tuple linguistic computational model is accurate when the labels are symmetrically distributed around a medium label. The results obtained in the computing processes are values within the universe of discourse of the linguistic variable.
Both computing models, virtual and 2tuple, have been extended for dealing with linguistic time variables and timedependent aggregation operators have been defined for fusing linguistic information gathered in multiple periods. These reasons suggest that the most suitable symbolic linguistic computing model to deal with linguistic information in DMCDM problems is the 2tuple linguistic model due to it provides desirable features: its results are closer to human natural language and it provides interpretability and understandability.
6 Future research directions
The management of linguistic information in realworld dynamic problems is always difficult and complex. The concept of LDMCDM facilitates dealing with such a type of problem. In this paper, we have shown different symbolic computing models concepts, extensions and tools to handle LDMCDM problems. Although great progress in LDMCDM has been achieved, there are still remaining issues to be tackled which are pointed out in this section.
6.1 Linguistic timedependent aggregation operators and weight determination methods
Fusion of linguistic terms is an important topic in the studies of linguistic information. A wide range of timeindependent aggregation operators have been proposed for 2tuple linguistic model and virtual model in literature as we have shown in Sects. 2.3 and 2.4. Although solutions for aggregating linguistic information provided in different periods are less common, the proposal of new timedependent aggregation operators should always be justified, as well as their usefulness and difference with the existing ones should be explained considering the features of LDMCDM problems.
 1.
Poisson discrete probability distribution based method (Xu 2011).
 2.
 3.
Normal distribution based method (Xu 2008).
 4.
Exponential distribution based method (Xu 2008).
 5.
Average age method (Xu 2008).
 6.
Arithmetic series based method (Xu and Yager 2008).
 7.
Geometric series based method (Xu and Yager 2008).
 8.
Minimum variability model based method (Xu 2009b).
 9.
Improved maximum entropy method (Liao et al. 2014).
 10.
Minimum average deviation method (Liao et al. 2014).
 11.
Softmax function (Torres et al. 2014).
 12.
Technique for order of preference by similarity to ideal solution (TOPSIS) based method (Yang and Huang 2017).
6.2 Potential application fields
The 2tuple and the virtual linguistic models have been applied to a wide variety of applications which are mainly evaluation or selection processes for different aims based on decision making and decision analysis problems.
Some applications based on the use of DMCDM models
Problem  2tuple  Virtual  Other 

Project risk evaluation  Liu (2014)  Zulueta et al. (2013a)  
Enterprise performance evaluation  Ai and Yang (2014)  
Human resources performance appraisal  Dong and Guo (2013)  
Quality of service evaluation  Torres et al. (2014)  
Zhang et al. (2011)  
Option investment selection  Xu (2009b)  Park et al. (2010)  Dong and Guo (2013) 
Wei (2009)  
Xu (2009b)  
Xu (2011)  
Supplier and subcontractor selection  Lin et al. (2008)  
Peng and Wang (2014)  
Zulueta et al. (2013b)  
Landing place selection  Campanella and Ribeiro (2011)  
Zhu et al. (2013)  
Dynamic group decision making  Dutta and Guha (2015a) 
6.3 Modelling complex decision making situations
The complex features of decision making problems in general, also apply for LDMCDM including, for instance, to deal with missing values, to manage multigranular linguistic information and heterogeneous information to represent variety of uncertainties.
 1.
Alternatives are not fixed.
 2.
Criteria are not fixed.
 3.
The temporal profile of alternative matters for comparison with other alternatives.
6.4 Modelling the feedback
In many practical situations (Campanella and Ribeiro 2011; Zulueta et al. 2013a, b), different successive decisions may be related between each other. For instance, in employees performance appraisal, decision makers may be interested into giving some advantage to personnel who got good results in previous periods. Therefore, decisions are constrained by earlier decisions then the solution of dynamic decision making is a continuous and progressive process. Zhu et al. (2013), according to this logic continuity of decision results at adjoining periods, defined the concept of retain ability. They developed a method that first aggregates the attribute value to construct a possibilitydegree matrix using the Triangular Fuzzy Language Weighted Averaging operator, then calculates the dynamic ordering vector based on the concept of the Advantage Retention Degree at different periods, which is used to rank alternatives. Although this method recognizes the notion of feedback in decision making, a more general framework for DMCDM which is expression domain independent, was introduced by Campanella and Ribeiro (2011). It includes three main tasks: (1) rating the alternatives regarding each criterion, (2) obtaining a dynamic assessment considering performance of alternatives then ranking them and (3) updating an historical set composed by alternatives with higher dynamic assessment. Here again, the definition of a suitable aggregation operator is a key element in the dynamic assessment function, due to its associativity property can highly modify the computing cost and very different results may be obtained attending to the type of reinforcement (Ribeiro et al. 2010; Yager and Rybalov 1998) supported by the aggregation operator.
This DMCDM model is suitable for more complex DMCDM problems including large and strongly changeable sets of alternatives, criteria and experts. For instance, DMCDM problems defined in urgent situations (or urgent computing environments), where is required that stakeholders manage a great amount of information about experts’ preferences and it is important to discard irrelevant alternatives in each iteration with time constraints. Also in largescale DMCDM problems such as group emarketplaces or social media including a large group of individuals, may be necessary the advantages of this other view of DMCDM for decreasing computational costs of selecting best alternatives. Decision making related to moving targets (driving any type of vehicle, for example) is made based on continuous decisions thought to depend upon the temporal derivative of the expected utility, where the expected utility is the effective value of a future reward. This could be another possible scenario for DMCDM models application.
6.5 Information granulation
 4.
The linguistic set for assessing criteria, it is not fixed.
The research on granulation of information has witnessed rich achievements in the last years since it splits a problem into simpler subproblems, and this might help also in developing more robust LDMCDM approaches.
7 Conclusions
The use of linguistic information enhances the reliability and flexibility of DMCDM approaches in uncertain environment while keeps the interpretability of results using linguistic computing models. This contribution reviewed the 2tuple linguistic model and the virtual linguistic model and described how these classical symbolic linguistic computing models have been extended throughout timedependent aggregation operators able to operate with multiperiod linguistic information and obtain accurate and understandable results. This paper also developed a resolution procedure for LDMCDM problems in which the use of timedependent aggregation operators for the dynamic aggregation phase is crucial.
The 2tuple linguistic and the virtual linguistic models were applied to a green supplier selection problem under LDMCDM context, to carry out a comparative analysis among them and verify how the resolution scheme works. The discussion found that both outputs are consistent, but the 2tuple linguistic model provides more accurate and understandable results because they are represented by means of a linguistic term and a numerical value while keeps the syntax and fuzzy semantics.
Some future directions in researching LDMCDM were proposed, including the development of new timedependent aggregation operations and the modelling of feedback among periods and as well as different types of information granulation in the problem. These issues may reinforce the application of LDMCDM models to a wide range of real problems to obtain more accurate models, closer to real decision scenarios. This study would be helpful for researchers to identify further interesting topics and for increasing the usability and effectiveness of symbolic linguistic models in solving LDMCDM problems.
Notes
References
 Ai F, Yang J (2014) Approaches to dynamic multiple attribute decision making with 2tuple linguistic information. J Intell Fuzzy Syst 27(6):2715–2723. https://doi.org/10.3233/IFS131094 MathSciNetMATHGoogle Scholar
 Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96CrossRefMATHGoogle Scholar
 Beg I, Rashid T (2014) Aggregation operators of intervalvalued 2tuple linguistic information. Int J Intell Syst 29(7):634–667. https://doi.org/10.1002/int.21650 CrossRefGoogle Scholar
 Campanella G, Ribeiro R (2011) A framework for dynamic multiplecriteria decision making. Decis Support Syst 52(1):52–60CrossRefGoogle Scholar
 Carrasco R, Noz FM, Sánchez J, Liébana F (2012) A model for the integration of efinancial services questionnaires with SERVQUAL scales under fuzzy linguistic modeling. Expert Syst Appl 39(14):11,535–11,547CrossRefGoogle Scholar
 Chen S, Hong J (2014) Multicriteria linguistic decision making based on hesitant fuzzy linguistic term sets and the aggregation of fuzzy sets. Inf Sci 286(Supplement C):63–74. https://doi.org/10.1016/j.ins.2014.06.020 CrossRefGoogle Scholar
 Chen S, Tsai B (2015) Autocratic decision making using group recommendations based on intervals of linguistic terms and likelihoodbased comparison relations. IEEE Trans Syst Man Cybern Syst 45(2):250–259. https://doi.org/10.1109/TSMC.2014.2356436 CrossRefGoogle Scholar
 CidLópez A, Hornos M, Carrasco R, HerreraViedma E (2016) Applying a linguistic multicriteria decisionmaking model to the analysis of ict suppliers’ offers. Expert Syst Appl 57(Supplement C):127–138. https://doi.org/10.1016/j.eswa.2016.03.025 CrossRefGoogle Scholar
 Degani R, Bortolan G (1988) The problem of linguistic approximation in clinical decision making. Int J Approx Reason 2(2):143–162. https://doi.org/10.1016/0888613X(88)901053 CrossRefGoogle Scholar
 Dong Q, Guo Y (2013) Multiperiod multiattribute decisionmaking method based on trend incentive coefficient. Int Trans Oper Res 20:141–152MathSciNetCrossRefMATHGoogle Scholar
 Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Kluwer, DordrechtMATHGoogle Scholar
 Dutta B, Guha D (2015a) Decision makers’ opinions changing attitudedriven consensus model under linguistic environment and its application in dynamic MAGDM problems. Springer International Publishing, Cham, pp 73–95. https://doi.org/10.1007/9783319168296_4 Google Scholar
 Dutta B, Guha D (2015b) A model based on linguistic 2tuples for dealing with heterogeneous relationship among attributes in multiexpert decision making. IEEE Trans Fuzzy Syst 23(5):1817–1831. https://doi.org/10.1109/TFUZZ.2014.2379291 CrossRefGoogle Scholar
 Dutta B, Guha D (2015) Partitioned bonferroni mean based on linguistic 2tuple for dealing with multiattribute group decision making. Appl Soft Comput 37:166–179. https://doi.org/10.1016/j.asoc.2015.08.017 CrossRefGoogle Scholar
 Herrera F, HerreraViedma E (2000) Linguistic decision analysis: steps for solving decision problems under linguistic information. Fuzzy Sets Syst 115:67–82MathSciNetCrossRefMATHGoogle Scholar
 Herrera F, Martínez L (2000) A 2tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 8(6):746–752CrossRefGoogle Scholar
 Ho N, Wechler W (1990) Hedge algebras: an algebraic approach to structures of sets of linguistic domains of linguistic truth variable. Fuzzy Sets Syst 35(3):281–293CrossRefMATHGoogle Scholar
 Jiang L, Liu H, Cai J (2015) The power average operator for unbalanced linguistic term sets. Inf Fusion 22:85–94. https://doi.org/10.1016/j.inffus.2014.06.002 CrossRefGoogle Scholar
 Jimenez G, Zulueta Y (2017) A 2tuple linguistic multiperiod decision making approach for dynamic green supplier selection. Revis DYNA 84(202):199–206. https://doi.org/10.15446/dyna.v84n202.58032 CrossRefGoogle Scholar
 Ju Y, Liu X, Wang A (2016) Some new shapley 2tuple linguistic choquet aggregation operators and their applications to multiple attribute group decision making. Soft Comput 20(10):4037–4053. https://doi.org/10.1007/s0050001517403 CrossRefMATHGoogle Scholar
 Karsak EE, Dursun M (2015) An integrated fuzzy mcdm approach for supplier evaluation and selection. Comput Ind Eng 82(Supplement C):82–93. https://doi.org/10.1016/j.cie.2015.01.019 CrossRefGoogle Scholar
 Lee L, Chen S (2015a) Fuzzy decision making and fuzzy group decision making based on likelihoodbased comparison relations of hesitant fuzzy linguistic term sets. J Intell Fuzzy Syst 29(3):1119–1137. https://doi.org/10.3233/IFS151715 MathSciNetCrossRefMATHGoogle Scholar
 Lee L, Chen S (2015b) Fuzzy decision making based on likelihoodbased comparison relations of hesitant fuzzy linguistic term sets and hesitant fuzzy linguistic operators. Inf Sci 294(Supplement C):513–529. https://doi.org/10.1016/j.ins.2014.09.061 MathSciNetCrossRefMATHGoogle Scholar
 Li Y, Liu P (2015) Some heronian mean operators with 2tuple linguistic information and their application to multiple attribute group decision making. Technol Econ Dev Econ 21(5):797–814. https://doi.org/10.3846/20294913.2015.1055614 CrossRefGoogle Scholar
 Liao H, Xu Z, Xu J (2014) An approach to hesitant fuzzy multistage multicriterion decision making. Kybernetes 43(9):1447–1468MathSciNetCrossRefGoogle Scholar
 Lin J, Zhang Q, Meng F (2015) An approach for facility location selection based on optimal aggregation operator. Knowl Based Syst 85:143–158. https://doi.org/10.1016/j.knosys.2015.05.001 CrossRefGoogle Scholar
 Lin Y, Lee P, Ting H (2008) Dynamic multiattribute decision making model with grey number evaluations. Expert Syst Appl 35:1638–1644CrossRefGoogle Scholar
 Liu P, Su Y (2010) The multipleattribute decision making method based on the TFLHOWA operator. Comput Math Appl 60(9):2609–2615. https://doi.org/10.1016/j.camwa.2010.08.087 MathSciNetCrossRefMATHGoogle Scholar
 Liu X, Tao Z, Chen H, Zhou L (2016) A magdm method based on 2tuple linguistic heronian mean and new operational laws. Int J Uncertainty Fuzziness Knowl Based Syst 24(04):593–627. https://doi.org/10.1142/S0218488516500288 MathSciNetCrossRefMATHGoogle Scholar
 Liu Y (2014) A method for 2tuple linguistic dynamic multiple attribute decision making with entropy weight. J Intell Fuzzy Syst 27(4):1803–1810. https://doi.org/10.3233/IFS141147 MathSciNetMATHGoogle Scholar
 Massanet S, Riera J, Torrens J, HerreraViedma E (2014) A new linguistic computational model based on discrete fuzzy numbers for computing with words. Inf Sci 258:277–290. https://doi.org/10.1016/j.ins.2013.06.055 MathSciNetCrossRefMATHGoogle Scholar
 Meng F, Tang J (2013) Extended 2tuple linguistic hybrid aggregation operators and their application to multiattribute group decision making. Int J Comput Intell Syst 4(2):1–14Google Scholar
 Merigó J, Casanovas M (2010) Linguistic weighted aggregation under confidence levels. Decis Mak Distance Meas Linguist Aggreg Oper 12(3):190–198Google Scholar
 Merigó J, GilLafuente A (2013) Induced 2tuple linguistic generalized aggregation operators and their application in decisionmaking. Inf Sci 236:1–16MathSciNetCrossRefMATHGoogle Scholar
 Merigó J, Casanovas M, Martínez L (2010) Linguistic aggregation operators for linguistic decision making based on the Dempster–Shafer theory of evidence. Int J Uncertainty Fuzziness Knowledge Based Syst 18(3):287–304. https://doi.org/10.1142/S0218488510006544 MathSciNetCrossRefMATHGoogle Scholar
 Miyamoto S (2004) Fuzzy decision making based on likelihoodbased comparison relations of hesitant fuzzy linguistic term sets and hesitant fuzzy linguistic operators. Int J Intell Syst 19(7):639–652CrossRefGoogle Scholar
 de Oliveira MouraSantos L, Osiro L, PalmaLima R (2017) A model based on 2tuple fuzzy linguistic representation and analytic hierarchy process for supplier segmentation using qualitative and quantitative criteria. Expert Syst Appl 79(Supplement C):53–64. https://doi.org/10.1016/j.eswa.2017.02.032 Google Scholar
 Park J, Kwun Y, Koo J (2010) Dynamic linguistic weighted averaging operators applied to decision making. In: Proceedings of the 2010 IEEE IEEM, pp 921–925Google Scholar
 Pedrycz W (2013) Granular computing: analysis and design of intelligent systems. CRC Press, Boca RatonCrossRefGoogle Scholar
 Pedrycz W, Chen S (2011) Granular computing and intelligent systems: design with information granules of higher order and higher type. Springer, HeidelbergCrossRefGoogle Scholar
 Pedrycz W, Chen S (2015a) Granular computing and decisionmaking: interactive and iterative approaches. Springer, HeidelbergCrossRefGoogle Scholar
 Pedrycz W, Chen S (2015b) Information Granularity, big data, and computational intelligence. Springer, HeidelbergCrossRefGoogle Scholar
 Peng D, Wang H (2014) Dynamic hesitant fuzzy aggregation operators in multiperiod decision making. Kybernetes 43(5):715–736MathSciNetCrossRefGoogle Scholar
 Qin J, Liu X (2016) 2tuple linguistic muirhead mean operators for multiple attribute group decision making and its application to supplier selection. Kybernetes 45(1):2–29. https://doi.org/10.1108/K1120140271 MathSciNetCrossRefGoogle Scholar
 Ribeiro R, Pais T, Simoes L (2010) Benefits of fullreinforcement operators for spacecraft target landing. Stud Fuzziness Soft Comput 257:353–367CrossRefMATHGoogle Scholar
 Rodríguez R, Martínez L (2013) An analysis of symbolic linguistic computing models in decision making. Int J Gen Syst 42(1):121–136MathSciNetCrossRefMATHGoogle Scholar
 Roubens M (1997) Fuzzy sets and decision analysis. Fuzzy Sets Syst 90(2):199–206. https://doi.org/10.1016/S01650114(97)000870 MathSciNetCrossRefMATHGoogle Scholar
 So H, N H, C H (2016) Development of some linguistic aggregation operators with conservation of interaction between criteria and their application in multiple attribute group decision problems. TOP 24(3):635–664. https://doi.org/10.1007/s1175001604125 MathSciNetCrossRefGoogle Scholar
 Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529–539. https://doi.org/10.1002/int.v25:6 MATHGoogle Scholar
 Torres R, Salas R, Astudillo H (2014) Timebased hesitant fuzzy information aggregation approach for decisionmaking problems. Int J Intell Syst 29(6):579–595. https://doi.org/10.1002/int.21658 CrossRefGoogle Scholar
 Wan S (2013) 2Tuple linguistic hybrid arithmetic aggregation operators and application to multiattribute group decision making. Knowl Based Syst 45(Supplement C):31–40. https://doi.org/10.1016/j.knosys.2013.02.002 CrossRefGoogle Scholar
 Wang J, Hao J (2006) A new version of 2tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 14(3):435–445. https://doi.org/10.1109/TFUZZ.2006.876337 CrossRefGoogle Scholar
 Wang J, Wang D, Zhang H, Chen X (2013) Multicriteria group decision making method based on interval 2tuple linguistic information and choquet integral aggregation operators. Soft Comput 19(2):389–405CrossRefMATHGoogle Scholar
 Wei G (2009) Some geometric aggregating operator and their application to dynamic multiple attribute decision making in intuitionistic fuzzy setting. Int J Uncertainty Fuzziness Knowl Based Syst 17(02):179–196. https://doi.org/10.1142/S0218488509005802 CrossRefMATHGoogle Scholar
 Wei G (2010) A method for multiple attribute group decision making based on the etwg and etowg operators with 2tuple linguistic information. Expert Syst Appl 37:7895–7900CrossRefGoogle Scholar
 Wei G (2011) Some generalized aggregating operators with linguistic information and their application to multiple attribute group decision making. Comput Ind Eng 61(1):32–38. https://doi.org/10.1016/j.cie.2011.02.007 CrossRefGoogle Scholar
 Wei G (2011b) Some harmonic aggregation operators with 2tuple linguistic assessment information and their application to multiple attribute group decision making. Int J Uncertainty Fuzziness Knowl Based Syst 19(6):977–998. https://doi.org/10.1142/S0218488511007428 MathSciNetCrossRefMATHGoogle Scholar
 Wei G, Zhao X (2012) Some dependent aggregation operators with 2tuple linguistic information and their application to multiple attribute group decision making. Expert Syst Appl 39(5):5881–5886. https://doi.org/10.1016/j.eswa.2011.11.120 CrossRefGoogle Scholar
 Wen X, Yan M, Xian J, Yue R, Peng A (2016) Supplier selection in supplier chain management using choquet integralbased linguistic operators under fuzzy heterogeneous environment. Fuzzy Optim Decis Mak 15(3):307–330. https://doi.org/10.1007/s1070001592282 MathSciNetCrossRefGoogle Scholar
 Xu Y, Wang H (2011) Approaches based on 2tuple linguistic power aggregation operators for multiple attribute group decision making under linguistic environment. Appl Soft Comput 11(5):3988–3997. https://doi.org/10.1016/j.asoc.2011.02.027 CrossRefGoogle Scholar
 Xu Y, Wang H (2011b) Distance measure for linguistic decision making. Syst Eng Proc 1:450–456CrossRefGoogle Scholar
 Xu Z (2004a) EOWA and EOWG operators for aggregating linguistic labels based on linguistic preference relations. Int J Uncertainty Fuzziness Knowl Based Syst 12(06):791–810. https://doi.org/10.1142/S0218488504003211 MathSciNetCrossRefMATHGoogle Scholar
 Xu Z (2004b) A method based on linguistic aggregation operators for group decision making with linguistic preference relations. Inf Sci 166(1–4):19–30. https://doi.org/10.1016/j.ins.2003.10.006 MathSciNetCrossRefMATHGoogle Scholar
 Xu Z (2005a) An approach to group decision making based on incomplete linguistic preference relations. Int J Inf Technol Decis Mak 4(1):153–160. https://doi.org/10.1142/S0219622005001349 CrossRefGoogle Scholar
 Xu Z (2005b) Extended eiowg operator and ita use in group decision making based on multiplicative linguistic preference relations. Am J Appl Sci 2(3):633–643CrossRefGoogle Scholar
 Xu Z (2006a) An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations. Decis Support Syst 41(2):488–499. https://doi.org/10.1016/j.dss.2004.08.011 CrossRefGoogle Scholar
 Xu Z (2006b) Induced uncertain linguistic OWA operators applied to group decision making. Inf Fusion 7(2):231–238. https://doi.org/10.1016/j.inffus.2004.06.005 CrossRefGoogle Scholar
 Xu Z (2008) On multiperiod multiattribute decision making. Knowl Based Syst 21(2):164–171CrossRefGoogle Scholar
 Xu Z (2009a) Correlated linguistic information aggregation. Int J Uncertainty Fuzziness Knowl Based Syst 17(5):633–647. https://doi.org/10.1142/S0218488509006182 MathSciNetCrossRefMATHGoogle Scholar
 Xu Z (2009b) Multiperiod multiattribute group decisionmaking under linguistic assessments. Int J Gen Syst 38(8):823–850. https://doi.org/10.1080/03081070903257920 MathSciNetCrossRefMATHGoogle Scholar
 Xu Z (2011) Approaches to multistage multiattribute group decision making. Int J Inf Technol Decis Mak 10(01):121–146. https://doi.org/10.1142/S0219622011004257 CrossRefMATHGoogle Scholar
 Xu Z, Wang H (2017) On the syntax and semantics of virtual linguistic terms for information fusion in decision making. Inf Fusion 34(C):43–48. https://doi.org/10.1016/j.inffus.2016.06.002 CrossRefGoogle Scholar
 Xu Z, Yager R (2008) Dynamic intuitionistic fuzzy multiattribute decision making. Int J Approx Reason 48(1):246–262CrossRefMATHGoogle Scholar
 Yager R (1995) An approach to ordinal decision making. Int J Approx Reason 2:237–261. https://doi.org/10.1016/0888613X(94)000352 MathSciNetCrossRefMATHGoogle Scholar
 Yager R, Rybalov A (1998) Full reinforcement operators in aggregation techniques. Syst Man Cybern Part B Cybern IEEE Trans 28(6):757–769CrossRefGoogle Scholar
 Yang W (2013) Induced Choquet integrals of 2tuple linguistic information. Int J Uncertainty Fuzziness Knowl Based Syst 21(02):175–200. https://doi.org/10.1142/S0218488513500104 MathSciNetCrossRefMATHGoogle Scholar
 Yang Z, Huang L (2017) Dynamic stochastic multiattribute decisionmaking that considers stochastic variable variance characteristics under timesequence contingency environments. Math Probl Eng. https://doi.org/10.1155/2017/7126856 MathSciNetGoogle Scholar
 You X, You J, Liu H, Zhen L (2015) Group multicriteria supplier selection using an extended VIKOR method with interval 2tuple linguistic information. Expert Syst Appl 42(4):1906–1916. https://doi.org/10.1016/j.eswa.2014.10.004 CrossRefGoogle Scholar
 Zadeh L (1975) The concept of a linguistic variable and its application to approximate reasoningI. Inf Sci 8(3):199–249. https://doi.org/10.1016/00200255(75)900365 MathSciNetCrossRefMATHGoogle Scholar
 Zadeh L (1996) Fuzzy logic = computing with words. Trans Fuzzy Syst 4(2):103–111. https://doi.org/10.1109/91.493904 CrossRefGoogle Scholar
 Zadeh LA (1997) Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets Syst 90(2):111–127. https://doi.org/10.1016/S01650114(97)000778 MathSciNetCrossRefMATHGoogle Scholar
 Zhang C, Su W, Zeng S, Zhang L (2015) Linguistic weighted aggregation under confidence levels. Math Probl Eng 2015:1–8MathSciNetGoogle Scholar
 Zhang L, Zou H, Yang F (2011) A dynamic web service composition algorithm based on topsis. J Netw 6(9):1296–1304Google Scholar
 Zhu Q, Li H, Yu M (2013) Dynamic multiattribute decision making based on advantage retention degree. J Inf Comput Sci 10(04):1105–1119CrossRefGoogle Scholar
 Zulueta Y, Martell V, Martínez L (2013a) A dynamic multiexpert multicriteria decision making model for risk analysis. Lectu Notes Comput Sci Lectu Notes Artif Intelli Mexico 8265:132–143Google Scholar
 Zulueta Y, Martínez J, Bello R, Martínez L (2013b) A discrete time variable index for supporting dynamic multicriteria decision making. Int J Uncertainty Fuzziness Knowl Based Syst 22(1):1–22MathSciNetCrossRefMATHGoogle Scholar