Linguistic dynamic multicriteria decision making using symbolic linguistic computing models

Original Paper
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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 2-tuple 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 time-dependent 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 2-Tuple linguistic model Virtual linguistic model 

1 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 time-consuming 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 wide-open 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 time-independent 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 2-tuple 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 2-tuple linguistic (Herrera and Martínez 2000) and the virtual linguistic (Xu 2004b) symbolic computing models. Special emphasis is put into time-dependent 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 wide-used symbolic models in LDMCDM: the 2-tuple linguistic model (Herrera and Martínez 2000), the virtual model (Xu 2004b). Section 3 presents a general resolution procedure for LDMCDM problems. Section 4 revises time-dependent 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

In many cases, due to the complexities of objects and the vagueness of the human mind, information of decision making problems cannot be assessed precisely in a quantitative form, but may be in a qualitative one. Experts use linguistic variables in natural language as a suitable way to express their assessments and to carry out uncertainty reasoning. In this situations, a better approach could be to use linguistic assessments instead of numerical values. The FLA represents qualitative aspects by means of linguistic variables (Zadeh 1975). To model linguistically the information, choosing the appropriate linguistic descriptors for the linguistic term set and their semantics is required. To do so, different possibilities have been proposed (Yager 1995). One of them consists of supplying directly the term set by considering all the terms distributed on a scale which has a defined order (Yager 1995). For example, a set of seven terms S could be:
$$\begin{aligned}&S=\{s_0: \mathrm{Extremely} \ \mathrm{Low} \ \mathrm{(EL)},s_1:\mathrm{Very} \ \mathrm{Low} \ \mathrm{(VL)},\\&s_2: \mathrm{Low} \ \mathrm{(L)},s_3: \mathrm{Medium} \ \mathrm{(M)}, s_4: \mathrm{High} \ \mathrm{(H)},\\&s_5: \mathrm{Very} \ \mathrm{High} \ (VH),s_6: \mathrm{Extremely} \ \mathrm{High} \ \mathrm{(EH)} \}, \end{aligned}$$
In these cases, it is required that in the linguistic term set there exists the following operators:
  1. 1.

    Negation: Neg\((s_{i}) = s_{j}\) such that \(j = g - i\) (\(g + 1\) is the cardinality)

     
  2. 2.

    Maximization: \(\max (s_i,s_j)= s_i \ \text {if} \ ~s_i \ge s_j\)

     
  3. 3.

    Minimization: \(\ \min (s_i,s_j)= s_i \ \text {if} \ ~s_i \le s_j\)

     
The semantics of terms are represented by fuzzy numbers defined in the interval [0, 1]. A way to characterize a fuzzy number is to use a representation based on parameters of its membership function (Herrera and Martínez 2000).

2.2 Symbolic linguistic computing models

We aforementioned that the use of linguistic variables implies processes of computing with words. To carry out these computations based on the FLA two classical computing models were developed:
  • 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.

Linguistic symbolic models extending the use of indexes includes the 2-tuple linguistic model (Herrera and Martínez 2000) and the virtual linguistic model (Xu 2004b) which are revised below.

2.3 2-Tuple linguistic model

This model was presented in Herrera and Martínez (2000) to improve the precision in processes of computing with words by avoiding the loss of information for frameworks in which the linguistic term set has an odd value of granularity, being triangular-shaped, symmetrical and uniformly distributed their membership functions.
  1. 1.

    Representation model: The 2-tuple 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 2-tuple linguistic values to facilitate linguistic computational processes.

Definition 2

(Herrera and Martínez 2000) Let \(S=\{s_0,\ldots ,s_g\}\) the set of linguistic terms, the associated 2-tuple is \(\tilde{S}= S\times [-0.5, 0.5)\) and the bijective function \(\Delta : [0,g]\rightarrow \tilde{S}\) is defined as:
$$\begin{aligned} \Delta (\beta )=\left\{ \begin{array}{ll} s_i, &{} i=\mathrm{round}(\beta ) \\ \alpha =\beta -i, &{} \alpha \in [-0.5, 0.5) \end{array}\right. \end{aligned}$$
(1)
where \(\mathrm{round}(\cdot )\) assigns to \(\beta\) the integer number \(i\in \{0,1,\ldots ,g\}\) closest to \(\beta\).

Proposition 1

Let \(S=\{s_0,\ldots ,s_g\}\) be a linguistic term set and \((s_i,\alpha )\) be a 2-tuple linguistic value. There is always a function \(\Delta ^{-1}\) such that from a 2-tuple 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 2-tuple linguistic value consists of adding a value 0 as symbolic translation.

  1. 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.

     
  2. 3.

    Time-independent Aggregation Operators Functions \(\Delta\) and \(\Delta ^{-1}\) in the fuzzy linguistic representation model with 2-tuples transform numerical values into a 2-tuples and vice versa without loss of information, therefore, numerical aggregation operators are usually extended for dealing with linguistic 2-tuples.

     
In the past few decades, many scholars have developed a variety of 2-tuple linguistic aggregation operators. Herrera and Martínez (2000) developed the 2-tuple linguistic arithmetic mean (TAM) operator, the 2-tuple linguistic weighted averaging (TWA) operator and the 2-tuple linguistic ordered weighted averaging (TOWA) operator. Merigó et al. (2010) developed a linguistic decision model with Dempster–Shafer belief structure and the belief structure-TOWA (BS-TOWA) and the BS linguistic hybrid averaging (BS-LHA) operators.

Wei (2010) defined the extended 2-tuple linguistic weighted geometric (ETWG) operator and the extended 2-tuple linguistic ordered weighted geometric (ETOWG) operator; he also proposed some harmonic aggregation operators (Wei 2011b) as well as the generalized 2-tuple weighted average (G-TWA) operator, the generalized 2-tuple linguistic ordered weighted average (GTOWA) operator, and the induced generalized 2-tuple linguistic ordered weighted average (IG-TOWA) operator (Wei 2011a). Xu and Wang (2011a) presented the 2-tuple linguistic power average (TPA) operator and the 2-tuple linguistic power ordered weighted average (TPOWA) operator. Wei and Zhao (2012) introduced the dependent 2-tuple linguistic ordered weighted averaging (DTOWA) operator and the dependent 2-tuple linguistic ordered weighted geometric (DTOWG) operator, in which the associated weights only depend on the aggregated 2-tuple linguistic arguments. Meng and Tang (2013) developed the extended 2-tuple linguistic hybrid arithmetical weighted (ET-HAW) operator, the induced extended 2-tuple linguistic hybrid arithmetical weighted (IET-HAW) operator, the extended 2-tuple linguistic hybrid geometric mean (ET-HGM) operator, and the induced extended 2-tuple linguistic hybrid geometric mean (IET-LHGM) operator. Yang (2013) proposed the quasi-arithmetic induced 2-tuple linguistic correlated averaging (QINTCA) operator, the induced 2-tuple linguistic correlated averaging (INTCA) operator and the generalized induced 2-tuple linguistic correlated averaging (GINTCA) operator.

Merigó and Gil-Lafuente (2013) presented the induced 2-tuple generalized ordered weighted averaging (TIGOWA) operator, the Quasi-TIOWA operator and the 2-tuple induced quasi-arithmetic Choquet integral aggregation. Moreover, in the field of uncertain 2-tuple linguistic information, Wang et al. (2013) introduced the interval 2-tuple linguistic correlated averaging (ITCA) operator and the interval 2-tuple linguistic correlated geometric (ITCG) operator; and Beg and Rashid (2014) developed the interval-valued 2-tuple correlated averaging (IVTCA) operator, the interval-valued 2-tuple correlated geometric (IVTCG) operator, and the generalized interval-valued 2-tuple correlated averaging (GIVTCA) operator for different variants of interval-valued 2-tuple linguistic information.

Wan (2013) designed some hybrid aggregation operators with 2-tuple linguistic information, involving the 2-tuple hybrid weighted arithmetic average (T-HWA) operator, the linguistic T-HWA (T-HLWA) operator, and the extended T-HLWA (ET-HLWA) operator.

The linguistic proportional 2-tuple 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 2-tuple linguistic extended Bonferroni mean (EBM) aggregation operators and the 2-tuple 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 2-tuple linguistic Shapley chi-square 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 2-tuple linguistic Choquet integral averaging (TCIA) operator, the 2-tuple linguistic ordered Choquet integral averaging (TOCIA) operator, and the combined 2-tuple linguistic Choquet integral averaging operator. Ju et al. (2016) gave some new Shapley 2-tuple linguistic Choquet aggregation operators for MAGDM: Shapley 2-tuple linguistic Choquet averaging operator, Shapley 2-tuple linguistic Choquet geometric operator and generalized Shapley 2-tuple linguistic Choquet averaging operator. Qin and Liu (2016) proposed the 2-tuple linguistic Muirhead mean (2TLMM) operator and the 2-tuple linguistic dual Muirhead mean (2TLDMM) operator.

Li and Liu (2015) provided some new aggregation operators of 2-tuple linguistic information based on Heronian mean. Liu et al. (2016) presented the Algebra t-norm and s-norm based 2-tuple linguistic Heronian mean operator or the Algebra t-norm and s-norm based 2-tuple linguistic weighted Heronian mean operator.

2.4 Virtual linguistic model

This model was proposed by Xu (2004b) to avoid the loss of information and increase the number of operators in processes of computing with words.
  1. 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. 2.
    Computational model: To carry out processes of computing with words, Xu presented several operations (Xu 2004b) that extend the previous ones of the 2-tuple. Let \(s_{\alpha };s_{\beta } \in \bar{S}\) be any two linguistic terms and \(\mu , \mu _1,\mu _2\in [0,1]\).
    1. (a)

      \((s_{\alpha })^\mu =s_{\alpha ^\mu }\)

       
    2. (b)

      \((s_{\alpha })^{\mu _1}\otimes (s_{\alpha })^{\mu _2} =(s_{\alpha })^{\mu _1+\mu _2}\)

       
    3. (c)

      \((s_{\alpha }\otimes s_{\beta })^{\mu }=(s_{\alpha })^{\mu }\otimes (s_{\beta })^{\mu }\)

       
    4. (d)

      \(s_{\alpha }\otimes s_{\beta }=s_{\beta }\otimes s_{\alpha }=s_{\alpha \beta }\)

       
    5. (e)

      \(s_{\alpha }\oplus s_{\beta }=s_{\alpha +\beta }\)

       
    6. (f)

      \(\mu s_{\alpha }=s_{\mu \alpha }\)

       
    7. (g)

      \((\mu _1+\mu _2)s_{\alpha }=\mu _1 s_{\alpha }\oplus \mu _2 s_{\alpha }\)

       
    8. (h)

      \((s_{\alpha }\oplus s_{\beta })=\mu s_{\alpha }\oplus \mu s_{\beta }\)

       
    9. (i)

      \((s_{\alpha })^{-1}=s_{\frac{1}{\alpha }}\)

       
    Operational laws mentioned above are defined based on a given linguistic term set, but the calculated results would not exist in the set.
     
  3. 3.

    Time-independent aggregation operators: Several time-independent 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

To gain a comprehensive understanding of a LDMCDM problem, in this section we provide its formal definition and a general procedure for solving it.
Fig. 1

Basic resolution procedure of a LDMCDM problem

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 Herrera-Viedma 2000): (1) the choice of the linguistic term set with its semantics and (2) the choice of the aggregation operator of linguistic information.

The basic resolution procedure of a LDMCDM problem, depicted in Fig. 1 was designed by extending the previous classical decision making scheme (Roubens 1997) in the following way:
  1. 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. 2.

    Phase 2: The choice of the time-independent aggregation operator of linguistic information. A proper classical linguistic aggregation operator is chosen for aggregating the linguistic assessments.

     
  3. 3.

    Phase 3: An aggregation phase that aggregates the values provided by experts to obtain a collective assessment (non-dynamic) for alternatives considered at each period.

     
  4. 4.

    Phase 4: The choice of the time-dependent 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. 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 single-period MCDM problem, it is necessary to integrate static results from each static problem to obtain a holistic dynamic assessment of alternatives. To do this, time-dependent aggregation operators should be used.

     
  6. 6.

    Phase 6: An exploitation phase of the dynamic collective assessments to rank, sort, or choose the best ones among the alternatives.

     
As a matter of fact, the selection of time-dependent aggregation operators (Phase 4) and the dynamic aggregation (Phase 5) are crucial in LDMCDM problems. The previous linguistic resolution scheme shows the necessity of linguistic computing models to operate with multi-period linguistic information and obtain accurate and understandable results. Up to now, the two symbolic linguistic computing models from Sect. 2.2 have been used in LDMCDM. In the following section time-dependent linguistic aggregation operators of the 2-tuple linguistic model and the virtual linguistic model for LDMCDM are revised due to their significance in Phase 5.

4 Time-dependent aggregation operators for LDMCDM

Linguistic computing models revised in Sect. 2.2 have been extended for dealing with linguistic time variables. Time-dependent aggregation operators have been defined to accomplish Phase 5 from resolution scheme presented in Sect. 3 due to classical time-independent aggregation operators are not suitable for fusing linguistic information gathered in multiple periods.

A wide range of conventional (time-independent) linguistic aggregation operators can be found in the literature, but few studies focus on linguistic time-dependent aggregation operators. In what follows, several linguistic time-dependent aggregation operators are reviewed.

4.1 2-Tuple time-dependent aggregation operators for LDMCDM

In this section, several linguistic 2-tuple time-dependent aggregation operators will be presented for supporting aggregation in LDMCDM.

4.1.1 2-Tuple dynamic weighted averaging aggregation operators

Definition 3

(Liu 2014; Xu 2009b) Let \(\tilde{S}=\{(s_1,\alpha _1)(t_1),\ldots ,(s_q,\alpha _q)(t_q)\}\) be a collection of q 2-tuple arguments collected from q different periods, \(T=\{(t_\lambda ) |\lambda \in (1,\ldots ,q)\}\), whose weights are given by the weighting vector \(W^T\), then the function \(\text {TDWA}: \tilde{S}^q\longrightarrow \tilde{S}\) defined as
$$\begin{aligned} \text {TDWA}(\tilde{S})= \Delta \left( \sum _{\lambda =1}^q w(t_\lambda ) \Delta ^{-1}(s_i,\alpha _i)(t_\lambda )\right) \end{aligned}$$
(2)
is called a 2-tuple Dynamic Weighted Averaging aggregation operator, TDWA.
Especially, if \(W^T=\{\frac{1}{q},\frac{1}{q},\ldots ,\frac{1}{q}\}\), the TDWA operator reduces to the TDA operator:
$$\begin{aligned} \text {TDA}(\tilde{S})= \Delta \left( \frac{1}{q}\sum _{\lambda =1}^q \Delta ^{-1}(s_i,\alpha _i)(t_\lambda )\right) \end{aligned}$$
(3)

4.1.2 2-Tuple dynamic weighted geometric aggregation operator

Definition 4

(Liu 2014) Let \(\tilde{S}=\{(s_1,\alpha _1)(t_1),\ldots ,(s_q,\alpha _q)(t_q)\}\) be a collection of q 2-tuple arguments collected from q different periods, \(T=\{(t_\lambda ) |\lambda \in (1,\ldots ,q)\}\), whose weights are given by the weighting vector \(W^T\), then the function \(\text {TDWG}: \tilde{S}^q\longrightarrow \tilde{S}\) defined as
$$\begin{aligned} \text {TDWG}(\tilde{S})= \Delta \left( \prod _{\lambda =1}^q \left( \Delta ^{-1}(s_i,\alpha _i)(t_\lambda )\right) ^{w(t_\lambda )}\right) \end{aligned}$$
(4)
is called a 2-tuple Dynamic Weighted Geometric aggregation operator, TDWG.
Especially, if \(W^T=\{\frac{1}{q},\frac{1}{q},\ldots ,\frac{1}{q}\}\), the TDWG operator reduces to the TDG operator:
$$\begin{aligned} \text {TDG}(\tilde{S})= \Delta \left( \prod _{\lambda =1}^q \left( \Delta ^{-1}(s_i,\alpha _i)(t_\lambda )\right) ^{\frac{1}{q}}\right) \end{aligned}$$
(5)

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 time-dependent aggregation operators

In this section, several virtual dynamic time-dependent aggregation operators are revised.

4.2.1 Virtual dynamic weighted averaging aggregation operators

Definition 5

(Park et al. 2010) Let \(\bar{S}=\{s_{\alpha _1}{(t_1)},\ldots ,s_{\alpha _q}{(t_q)}\}\) be a collection of q linguistic terms collected from q different periods, \(T=\{(t_\lambda ) |\lambda \in (1,\ldots ,q)\}\), whose weights are given by the weighting vector \(W^T\), then the function \(\text {VDWA}: \bar{S}^q\longrightarrow \bar{S}\) defined as
$$\begin{aligned}&\text {VDWA}(\bar{S})\nonumber \\&\quad =w(t_1)s_{1}(t_1)\oplus w(t_2)s_{2}(t_2)\oplus \cdots . \oplus w(t_q)s_q(t_q)\nonumber \\&\quad =\sum _{\lambda =1}^q w(t_\lambda )s_{\alpha }(t_\lambda ) \end{aligned}$$
(6)
is called a virtual dynamic weighted averaging aggregation operator, VDWA.

4.2.2 Virtual dynamic weighted geometric aggregation operator

Definition 6

(Xu 2009b) Let \(\bar{S}=\{s_{1}{(t_1)},\ldots ,s_{q}{(t_q)}\}\) be a collection of q linguistic terms collected from q different periods, \(T=\{(t_\lambda ) |\lambda \in (1,\ldots ,q)\}\), whose weights are given by the weighting vector \(W^T\), then the function \(\text {VDWG}: \bar{S}^q\longrightarrow \bar{S}\) defined as
$$\begin{aligned}&\text {VDWG}(\bar{S})\nonumber \\&\quad =(s_{1}(t_1))^{w(t_1)}\otimes (s_{2}(t_2))^{w(t_2)}\otimes . \cdots \otimes (s_q(t_q))^{w(t_1)}\nonumber \\&\quad =\prod _{\lambda =1}^q (s_{\alpha }(t_\lambda ))^{w(t_\lambda )} \end{aligned}$$
(7)
is called a virtual dynamic weighted geometric aggregation operator, VDWG.
Especially, if \(W^T=\{\frac{1}{q},\frac{1}{q},\ldots ,\frac{1}{q}\}\), the VDWG operator reduces to the VDG operator:
$$\begin{aligned} \text {VDG}(\bar{S})=\left( s_{1}(t_1)\otimes s_{2}(t_2)\otimes \cdots \otimes s_q(t_q)\right) ^{\frac{1}{q}} \end{aligned}$$
(8)

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 (Cid-López 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.

The main objective here is the selection of best supplier for a firm in an uncertain dynamic environment. For simplicity, we consider a fixed set of four suppliers \(A=\{a_1,a_2,a_3,a_4\}\). The set of criteria, C, includes the following elements:
  • 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.

Three experts provide assessment information on C to prioritize suppliers with respect to their green performance.
We follow the resolution scheme developed in Sect. 3 and give some calculation results to get the most desirable supplier.
  1. 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. 2.

    Phase 2: The time-independent aggregation operators selected to compute the non-dynamic collective assessment of suppliers are the TAM (Herrera and Martínez 2000) and the EAA (Xu 2004a) for the 2-tuple and virtual linguistic models respectively, since both are the linguistic extension of the conventional arithmetical averaging operator.

     
  3. 3.

    Phase 3: Using previous aggregation operators, a two-step aggregation process is accomplished: first, the collective criteria values of suppliers in each period are computed and then the non-dynamic collective assessments of suppliers in each period are computed (see Tables 2, 3). As examples, we show how to compute the non-dynamic collective assessments of supplier \(a_1\) in period \(t_1\), applying the different models:

    2-tuple linguistic model
    • Step 1: Computing the collective criteria values using the TAM. The collective \(c_1\) value is generated as:
      $$\begin{aligned}&\mathrm{TAM}((H,0),(H,0),(M,0)) \\&\quad =\mathrm{TAM}((s_4,0),(s_4,0),(s_3,0))\\&\quad =\Delta \left( \frac{1}{3} \Delta ^{-1}(s_4,0)+\Delta ^{-1}(s_4,0)+\Delta ^{-1}(s_3,0)\right) \\&\quad =\Delta \left( \frac{1}{3}\times (4+4+3)\right) =\Delta (3.67)=(s_4,- \ 0.33). \end{aligned}$$
      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.

      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 non-dynamic collective assessments for period \(t_1\). This value is generated using the previous collective criteria values as arguments for TAM:
      $$\begin{aligned}&\mathrm{TAM}((s_4,- \ 0.33),(s_2,0),(s_4,0.3),(s_4,0.3))\\&\quad =\Delta \bigg (\frac{1}{4}(\Delta ^{-1}(s_4,-0.33)+\Delta ^{-1}(s_2,0)\\&\qquad +\Delta ^{-1}(s_4,0.3) +\Delta ^{-1}(s_4,0.3)\bigg )\\&\quad =\Delta \left( \frac{1}{4} \times 14.27\right) =\Delta (3.57) =(s_4,- \ 0.43). \end{aligned}$$
    Virtual linguistic model
    • Step 1: Computing the collective criteria values using the EAA. The collective \(c_1\) value is generated as:
      $$\begin{aligned}&\mathrm{EAA}((H,0),(H,0),(M,0)) \\&\quad =\mathrm{EAA}(s_4,s_4,s_3)=\frac{1}{3}(s_4 \oplus s_4 \oplus s_3)= \frac{1}{3} s_{11} = s_{3.67}. \end{aligned}$$
      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.

      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 non-dynamic 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. 4.

    Phase 4: Once we have a non-dynamic evaluation of supplier at each period, to aggregate these multi-period linguistic data, we use different time-dependent aggregation operators in order to analyse the results obtained. For the 2-tuple 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 36).

     
Fig. 2

Linguistic set of seven terms

Table 1

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

Table 2

Non-dynamic 2-tuple 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)

Table 3

Non-dynamic virtual linguistic collective assessments of suppliers for each period

\(t_{\lambda }\)

\(a_1\)

\(a_2\)

\(a_3\)

\(a_4\)

\(t_1\)

\(s_{3.57}\)

\(s_{3.75}\)

\(s_{3.50}\)

\(s_{3.33}\)

\(t_2\)

\(s_{3.08}\)

\(s_{3.25}\)

\(s_{2.75}\)

\(s_{3.92}\)

\(t_3\)

\(s_{3.17}\)

\(s_{2.42}\)

\(s_{4.33}\)

\(s_{3.17}\)

To analyse the results obtained by prioritizing different weighting timing, different weighting vectors are used:
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).

  1. 5.

    Phase 5: By applying these operators, the collective non-dynamic 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:

    2-tuple linguistic model
    • $$\begin{aligned}&\text {TDWA}_{a}\left( (s_4,-0.43),(s_3,0.8),(s_4,-0.33)\right) \\&\quad =\Delta (0.1\times \Delta ^{-1}(s_4,-0.43)+ 0.3 \times \Delta ^{-1}(s_3,0.8)\\&\qquad + \ 0.6 \times \Delta ^{-1}(s_4,-0.33))\\&\quad =\Delta \left( 0.1\times 3.57+0.3 \times 3.08+ 0.6 \times 3.17\right) \\&\quad =\Delta (3.18)=(s_3,0.18). \\ \end{aligned}$$
    • $$\begin{aligned}&\text {TDWG}_{a}\left( (s_4,-0.43),(s_3,0.08),(s_4,-0.33)\right) \\&\quad =\Delta ((\Delta ^{-1}(s_4,-0.43))^{0.1 } \times (\Delta ^{-1}(s_3,0.8))^{0.3} \\&\qquad \times \ (\Delta ^{-1}(s_4,-0.33))^{0.6})\\&\quad = \Delta (3.57^{0.1}\times 3.08^{0.3} \times 3.17^{0.6} )\\&\quad =\Delta (3.18)=(s_3,0.18).\\ \end{aligned}$$
    Virtual linguistic model
    • $$\begin{aligned}&\mathrm{VDWA}_{a} \left( s_{3.57},s_{3.08},s_{3.67}\right) \\&\quad =0.1 s_{3.57} \oplus 0.3 s_{3.08} \oplus 0.6 s_{3.17}\\&\quad =s_{0.357} \oplus s_{0.924} \oplus s_{1.902}\\&\quad =s_{3.18}. \end{aligned}$$
    • $$\begin{aligned}&\mathrm{VTDWG}_{a}\left( s_{3.57},s_{3.08},s_{3.67}\right) \\&\quad = (s_{3.57})^{0.1} \otimes (s_{3.08})^{0.3} \otimes (s_{3.67})^{0.6}\\&\quad = s_{3.57^{0.1}} \otimes s_{3.08^{0.3}} \otimes s_{3.67^{0.6}}\\&\quad = s_{3.18}. \end{aligned}$$
     
Table 4

Dynamic 2-tuple 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)

Table 5

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

\(s_{3.27}\)

\(s_{2.78}\)

\(s_{3.47}\)

\(s_{3.46}\)

\(\mathrm{VDWA}_{a}\)

\(s_{3.18}\)

\(s_{2.70}\)

\(s_{3.78}\)

\(s_{3.41}\)

\(\mathrm{VDWG}_{a}\)

\(s_{3.18}\)

\(s_{2.68}\)

\(s_{3.70}\)

\(s_{3.39}\)

\(\mathrm{VDWA}_{d}\)

\(s_{3.39}\)

\(s_{2.87}\)

\(s_{3.36}\)

\(s_{3.49}\)

\(\mathrm{VDWG}_{d}\)

\(s_{3.38}\)

\(s_{2.85}\)

\(s_{3.33}\)

\(s_{3.48}\)

  1. 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 2-tuple model is obtained using comparison rules for 2-tuples. For two 2-tuple 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 \((s_{k},\alpha _{1})\succ (s_{l},\alpha _{2})\).

     
The order in the virtual linguistic model is obtained using its proper comparison rules. For two terms, virtual or original, \(s_{\alpha }\) and \(s_{\beta }\), if \(\alpha \succ \beta\) then \(s_{\alpha }\succ s_{\beta }\)
The order of suppliers and the best supplier are depicted in Tables 6 and 7. The different aggregation operators produce different results, however, the order for the two models is the same in all cases.
Table 6

Order of alternatives using different time-dependent 2-tuple linguistic aggregation operators

Operator

Order

Best supplier

TDA

\(a_3\succ a_4\succ a_1\succ a_2\)

\(a_3\)

TDG

\(a_3\succ a_4\succ a_1\succ a_2\)

\(a_3\)

\(\mathrm{TDWA}_a\)

\(a_3\succ a_2\succ a_1\succ a_4\)

\(a_3\)

\(\mathrm{TDWA}_d\)

\(a_4\succ a_1\succ a_3\succ a_2\)

\(a_4\)

\(\mathrm{TDWG}_a\)

\(a_3\succ a_2\succ a_1\succ a_4\)

\(a_3\)

\(\mathrm{TDWG}_d\)

\(a_4\succ a_1\succ a_3\succ a_2\)

\(a_4\)

Table 7

Order of alternatives using different time-dependent 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 2-tuple 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 2-tuple linguistic model are easy to understand, because they are represented by means of a linguistic term and a numerical value. The 2-tuple 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 2-tuple 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 2-tuple, have been extended for dealing with linguistic time variables and time-dependent 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 2-tuple 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 real-world 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 time-dependent aggregation operators and weight determination methods

Fusion of linguistic terms is an important topic in the studies of linguistic information. A wide range of time-independent aggregation operators have been proposed for 2-tuple 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 time-dependent 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.

In the LDMCDM resolution process, the dynamic problem can be seen as a collection of many classic static problems as many periods are considered in the LDMCDM problem, then their results are integrated by a dynamic weighted aggregation operator (such that defined in Sect. 4). In general, different stages have different importance, thus they should be assigned different weights. Obviously, a crucial and difficult aspect in handling properly the DMCDM problem is to determine the period weighting vector, since this step reduces one of the multiple dimensions of the complex problem. Although it can be allocated by decision maker(s) or experts directly, relying on their experience and subjective preference, some methods allocate time-sequence weights in accordance with the closeness of historical information to the latest information while others tend to emphasise time-sequences with more information in a historical period (Yang and Huang 2017). Different methods suggested in literature for obtaining the weights of periods are summarized in the following:
  1. 1.

    Poisson discrete probability distribution based method (Xu 2011).

     
  2. 2.

    Basic unit-interval monotonic function based method (Peng and Wang 2014; Xu 2008).

     
  3. 3.

    Normal distribution based method (Xu 2008).

     
  4. 4.

    Exponential distribution based method (Xu 2008).

     
  5. 5.

    Average age method (Xu 2008).

     
  6. 6.

    Arithmetic series based method (Xu and Yager 2008).

     
  7. 7.

    Geometric series based method (Xu and Yager 2008).

     
  8. 8.

    Minimum variability model based method (Xu 2009b).

     
  9. 9.

    Improved maximum entropy method (Liao et al. 2014).

     
  10. 10.

    Minimum average deviation method (Liao et al. 2014).

     
  11. 11.

    Softmax function (Torres et al. 2014).

     
  12. 12.

    Technique for order of preference by similarity to ideal solution (TOPSIS) based method (Yang and Huang 2017).

     

6.2 Potential application fields

The 2-tuple 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.

Table 8 summarizes the 2-tuple and virtual linguistic models applications in DMCDM among the whole number of applications developed. In addition, it includes DMCDM problems defined in other uncertain contexts where information is modelled by using different expressions domains based on diverse generalizations of fuzzy sets such as, intuitionistic fuzzy sets (Atanassov 1986), type n fuzzy sets (Dubois and Prade 1980), fuzzy multisets (Miyamoto 2004) and hesitant fuzzy sets (Torra 2010), among others. This is an important research issue due to similar problems may be defined under linguistic context and consequently, the symbolic linguistic models revised in this paper could be used in their solution. One special and challenging case is the hesitant fuzzy linguistic term set (Chen and Hong 2014; Chen and Tsai 2015; Lee and Chen 2015a, b) due to it provides a linguistic and computational basis to increase the richness of linguistic elicitation based on FLA and context-free grammars using comparative terms. In this way, the DLMCDM framework could manage situations in which experts hesitate between several values to assess an indicator, alternative or variable.
Table 8

Some applications based on the use of DMCDM models

Problem

2-tuple

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.

Classical MCDM models deal with a fixed set of attributes and a clear set of alternatives. Although the linguistic models reviewed in previous sections consider the changeability of decision information as they collect information and experts’s preferences over multiple periods, they do not model those situations in which the set of attributes and their weights, as well as the set of alternatives involved in problem, also change. This view of the problem corresponds with more elaborated concepts of DMCDM that emphasise in three main characteristics (Campanella and Ribeiro 2011; Zulueta et al. 2013b):
  1. 1.

    Alternatives are not fixed.

     
  2. 2.

    Criteria are not fixed.

     
  3. 3.

    The temporal profile of alternative matters for comparison with other alternatives.

     
The last characteristic is referred to the notion of feedback between inter-dependent decisions, explained below.

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 possibility-degree 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 large-scale DMCDM problems such as group e-marketplaces 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

Granularity of information may address important research on LDMCDM. On the one hand, as the decision making process is iterative, this could lead to different, presumably decreasing, degrees of uncertainty since in specific problems, same experts can assess same criteria about same alternatives more than one time. For example, an expert who decided to use a linguistic set with just three terms in the first period, might be able to better discriminate his assessments about the alternatives after an exploratory process and could prefer a linguistic term set with higher granularity in the tenth period. For such situations, it seems to be logic a fourth feature of the LDMCDM problem, added to those ones mentioned in Sect. 6.3:
  1. 4.

    The linguistic set for assessing criteria, it is not fixed.

     
Besides, another important element in modelling the granularity of information in large-scale problems could be an unknown or presumable large number of time periods, i.e., higher granularity of time.

The research on granulation of information has witnessed rich achievements in the last years since it splits a problem into simpler sub-problems, 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 2-tuple linguistic model and the virtual linguistic model and described how these classical symbolic linguistic computing models have been extended throughout time-dependent aggregation operators able to operate with multi-period linguistic information and obtain accurate and understandable results. This paper also developed a resolution procedure for LDMCDM problems in which the use of time-dependent aggregation operators for the dynamic aggregation phase is crucial.

The 2-tuple 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 2-tuple 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 time-dependent 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

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Informatics ScienceHavanaCuba
  2. 2.University of JaenJaénSpain

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