Linguistic intuitionistic fuzzy Hamacher aggregation operators and their application to group decision making

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Abstract

Linguistic intuitionistic fuzzy variables (LIFVs) can efficiently denote the qualitative preferred and non-preferred cognitions of decision makers. This paper researches group decision making with linguistic intuitionistic fuzzy information. To do this, several Hamacher operational laws on LIFVs are defined. To derive the comprehensive evaluating values of alternatives, several linguistic intuitionistic fuzzy Hamacher aggregation operators are proposed, including the linguistic intuitionistic fuzzy Hamacher weighted average operator, the linguistic intuitionistic fuzzy Hamacher weighted geometric mean operator, the linguistic intuitionistic fuzzy Hamacher ordered weighted average operator, the linguistic intuitionistic fuzzy Hamacher ordered weighted geometric mean operator, the linguistic intuitionistic fuzzy Hamacher hybrid weighted average operator, and the linguistic intuitionistic fuzzy Hamacher hybrid weighted geometric mean operator. Then, several of their desirable properties are researched to guarantee the rationality. Methods for determining the weights of criteria, decision makers as well as the ordered positions are offered, respectively. After that, a procedure for group decision making with linguistic intuitionistic fuzzy information is provided. Finally, a group decision-making problem is offered to illustrate the application of the new results.

Keywords

Group decision making Linguistic intuitionistic fuzzy variable Hamacher t-norm and t-conorm Aggregation operator 

1 Introduction

Zadeh’s fuzzy set theory (Zadeh 1965) is an important and powerful tool for decision making with fuzzy information. At present, fuzzy decision-making theory is still a hot researching topic because of its advantages in expressing fuzzy, hesitant and inconsistent information. Following the characteristics of fuzzy sets for representing information, Zadeh’s fuzzy sets can be classified into two types: type-1 fuzzy sets (Zadeh 1965) and type-2 fuzzy sets (Zadeh 1975). Note that type-2 fuzzy sets are more powerful than type-1 fuzzy sets to handle uncertain information. Mendel et al. (2006), Chen and Lee (2010) and Lee and Chen (2008) researched decision making with interval type-2 fuzzy sets, respectively. More researches about the applications of fuzzy decision making are discussed in Horng et al. (2005), Chen and Kao (2013), Chen and Hong (2014), Chen (1996) and Chen et al. (2014), and decision making based on granular computing is studied in Pedrycz and Chen (2011, 2015a, b).

Later, Atanassov (1983) noted that the sum of the membership and non-membership degrees is not necessarily equal to one when the decision maker (DM) makes a judgment. To denote such situations, Atanassov (1983) introduced the concept of intuitionistic fuzzy sets (IFSs) that are composed by a membership degree, a non-membership degree and a hesitancy degree. Following the work of Atanassov (1983), many decision-making methods with intuitionistic fuzzy information are proposed. For example, intuitionistic fuzzy decision making based on aggregation operators are researched in Atanassov (1983), Liu and Chen (2016), Liu et al. (2017), Xu (2007, 2010), Xu and Yager (2006), Yu (2013) and Zeng and Su (2011), intuitionistic fuzzy decision making based on entropy and similarity measures are studied in Burillo and Bustince (1996), Li et al. (2012), Li and Cheng (2002), Liang and Shi (2003), Meng and Chen (2016) and Szmidt and Kacprzyk (2001). Besides the theoretical aspect, the applications of intuitionistic fuzzy decision making have also received considerable attentions from researchers, such as medical diagnosis (De et al. 2001), homeopathic drug selection (Kharal 2009), pattern recognition (Chen and Chang 2015), selecting flexible manufacturing system (Liao and Xu 2014), selecting the most appropriate supplier (Meng et al. 2017), and selecting the locations for shopping center (Xu et al. 2011). Although IFSs can conveniently denote the DMs’ preferred and non-preferred recognitions, the requirement of offering the accurate judgements is still difficult that cannot express the uncertainty of the DMs. Therefore, Atanassov and Gargov (1989) further introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs) that use an interval in [0, 1] to denote the membership, non-membership and hesitancy degrees, respectively. IVIFSs endow the DMs with more flexibility to express their judgements and further extend the application of fuzzy sets. To facilitate the application, Xu and Chen (2007) further introduced the concept of interval-valued intuitionistic fuzzy variables (IVIFVs) and introduced them for decision making. After the original work of Xu and Chen (2007), the theory and application of IVIFVs are largely developed (Meng et al. 2013; Meng and Tan 2015; Meng and Chen 2015; Qiu and Li 2017; Wang and Chen 2017; Wei et al. 2011; Xu 2010; Xu and Cai 2015; Zhao et al. 2017). Taking the advantages of intuitionistic fuzzy sets for denoting the judgements of the DMs, several extending forms are developed, such as triangular intuitionistic fuzzy variables (Zhang and Nan 2013), trapezoidal intuitionistic fuzzy variables (Wang and Zhang 2009), and interval-valued intuitionistic hesitant fuzzy variables (Zhang 2013).

Different from the above-mentioned quantitative fuzzy sets, linguistic variables (Zadeh 1975) can well express the qualitative judgements of the DMs and facilitate the DMs to express their recognitions in complex decision-making problems. To calculate the comprehensive evaluating values of alternatives, many computing with word (CW) models are proposed, such as type-1 fuzzy CW model (Zadeh 1996), type-2 fuzzy CW model (Türkşen 2002), 2-tuple linguistic fuzzy CW model (Herrera and Martínez 2000), continuous linguistic CW model (Xu 2004b), proportional 2-tuple linguistic fuzzy CW model (Wang and Hao 2006), granular CW model (Cabrerizo et al. 2013), discrete fuzzy CW model (Massanet et al. 2014), and multi-granular CW model (Morente-Molinera et al. 2017). Note that the two-tuple linguistic model (Herrera and Martínez 2000) and continuous linguistic CW model (Xu 2004b) are two most widely used methods that can avoid information loss and distortion. Since these two models are defined on the subscripts of linguistic variables, it makes the computation easily. Similar to the analysis for IFSs (Atanassov 1983), it is too restrictive to only use a linguistic variable to express a qualitative judgement. Thus, Xu (2004a) introduced interval linguistic variables to express the qualitative uncertainty of the DMs, and Rodríguez et al. (2012) proposed hesitant fuzzy linguistic term sets to indicate the qualitative hesitancy of the DMs. Taking the advantages of intuitionistic fuzzy sets and linguistic variables, Zhang et al. (2017) proposed linguistic intuitionistic fuzzy variables to denote the qualitative preferred and non-preferred judgments. This paper continues to research decision making with linguistic intuitionistic fuzzy information, and introduces a method for group linguistic intuitionistic fuzzy decision making based on the Hamacher t-norm and t-conorm. To do this, the rest part of this paper is organized as follows:

Section 2 first recalls several basic concepts about linguistic variables. Then, it offers several linguistic intuitionistic Hamacher operational laws. Section 3 defines two types of the linguistic intuitionistic fuzzy Hamacher aggregation operators: the linguistic intuitionistic fuzzy Hamacher mathematical average operators and the linguistic intuitionistic fuzzy Hamacher geometric mean operators. Several important cases and desirable properties are studied. Section 4 first studies how to determine the weights of criteria, DMs as well as the ordered positions. Then, an algorithm for group decision making with linguistic intuitionistic fuzzy information is offered. Section 5 offers a practical decision-making problem to show the application of the new method.

2 Basic concepts

To denote the positive and negative information of the DMs simultaneously, Atanassov (1983) introduced the concept of intuitionistic fuzzy sets (IFSs) as follows:

Definition 1

(Atanassov 1983) An IFS A on X = {x1, x2, …, x n } is expressed as \(A=\{ \langle x,{\mu _A}(x),{v_A}(x)\rangle |x \in X\}\), where \({\mu _A}(x) \in [0,1]\) and \({v_A}(x) \in [0,1]\) are respective of the membership and non-membership degrees of the element \(x \in X\) with the condition \({\mu _A}(x)+{v_A}(x) \leq 1\).

To denote simply, Xu (2007) introduced the concept of intuitionistic fuzzy values (IFVs), where an IFV is denoted as \(\tilde {\alpha }=(\mu ,v)\) under the conditions of \(0 \leq \mu ,v \leq 1\) and \(\mu +v \leq 1\).

Archimedean t-norm and t-conorm (Klement et al. 2000) is an important pair of dual binary operations that represents two families of operational laws. As present, most of intuitionistic fuzzy decision-making methods are based on two special types of Archimedean t-norms and t-conorms: Algebraic t-norm and t-conorm and Einstein t-norm and t-conorm. Note that these two types of Archimedean t-norms and t-conorms are two special cases of Hamacher t-norm and t-conorm (Hamacher 1978), which are defined as follows:
  1. (i)

    \(T_{\gamma }^{H}(x,y)=\frac{{xy}}{{\gamma +(1 - \gamma )(x+y - xy)}},\)

     
  2. (ii)

    \(S_{\gamma }^{H}(x,y)=\frac{{x+y - xy - (1 - \gamma )xy}}{{\gamma +(1 - \gamma )(x+y - xy)}},\)

     
where \(x,y \in [0,1]\) and \(\gamma>0\).

Zadeh (1975) noted the limitations of quantitative fuzzy sets and introduced linguistic variables to denote the qualitative judgements of the DMs, such as “bad”, “fair”, and “good”. Later, Herrera and Martínez (2000) introduced linguistic term sets to express linguistic variables, denoted by S = {s i | i = 0, 1, …, 2t}, where t is a positive integer. Any term s i indicates a possible value for a linguistic variable. Furthermore, the linguistic terms in S have the properties: (1) the set is ordered: s i  > s j , if i > j; (2) maximum operator: max(s i , s j ) = s i , if s i  ≥ s j ; (3) minimum operator: min(s i , s j ) = s i , if s i  ≤ s j ; (4) a negation operator: neg(s i ) = s j such that j = 2t − i (Herrera and Martínez 2000). To preserve information, Xu (2004b) extended the discrete linguistic term set S to the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). For any \({s_\alpha } \in {S_{\text{c}}}\), if \({s_\alpha } \in S\), then it is called an original linguistic term. Otherwise, it is called a virtual linguistic term.

On the basis of IFVs and linguistic variables, Zhang et al. (2017) introduced linguistic intuitionistic fuzzy variables (LIFVs) as follows:

Definition 2

(Zhang et al. 2017) A LIFV \(\tilde {s}\) on the continuous linguistic term set \({S_c}=\{ {s_\alpha }|\alpha \in [0,2t]\}\) is expressed as \(\tilde {s}=({s_\alpha },{s_\beta })\), where \({s_\alpha }\) and \({s_\beta }\) are respective of the qualitative preferred and non-preferred degrees with the condition \({s_\alpha } \oplus {s_\beta } \leq {s_{2t}}\).

Next, we introduce Hamacher t-norm and t-conorm for LIFVs and present the following several linguistic intuitionistic fuzzy Hamacher operational laws:

Definition 3

Let \({\tilde {s}_1}=({s_{{\alpha _1}}},{s_{{\beta _1}}})\) and \({\tilde {s}_2}=({s_{{\alpha _2}}},{s_{{\beta _2}}})\) be any two LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, some of their Hamacher operational laws are defined as follows:
  1. (i)

    \({\tilde {s}_1}{ \oplus _H}{\tilde {s}_2}=f(\tilde {s}_{1}^{\prime }{ \oplus _H}\tilde {s}_{2}^{\prime })=f\left( {\left( {{s_{\frac{{\alpha _{2}^{\prime }+\alpha _{1}^{\prime } - \alpha _{2}^{\prime }\alpha _{1}^{\prime } - (1 - \gamma )\alpha _{2}^{\prime }\alpha _{1}^{\prime }}}{{1 - (1 - \gamma ){{\alpha ^{\prime}}_2}\alpha _{1}^{\prime }}}}},{s_{\frac{{{{\beta ^{\prime}}_2}\beta _{1}^{\prime }}}{{\gamma +(1 - \gamma )\left( {{{\beta ^{\prime}}_2}+{{\beta ^{\prime}}_1} - {{\beta ^{\prime}}_2}{{\beta ^{\prime}}_1}} \right)}}}}} \right)} \right);\)

     
  2. (ii)

    \({\tilde {s}_1}{ \otimes _H}{\tilde {s}_2}=f\left( {\tilde {s}^{\prime}_{1}{ \otimes _H}\tilde {s}^{\prime}_{12}} \right)=f\left( {\left( {{s_{\frac{{\alpha ^{\prime}_{12}\alpha ^{\prime}_{11}}}{{\gamma +(1 - \gamma )\left( {\alpha ^{\prime}_{12}+\alpha ^{\prime}_{11} - \alpha ^{\prime}_{12}\alpha ^{\prime}_{11}} \right)}}}},{s_{\frac{{\beta ^{\prime}_{12}+\beta ^{\prime}_{11} - \beta ^{\prime}_{12}\beta ^{\prime}_{11} - (1 - \gamma )\beta ^{\prime}_{12}\beta ^{\prime}_{11}}}{{1 - (1 - \gamma )\beta ^{\prime}_{12}\beta ^{\prime}_{11}}}}}} \right)} \right);\)

     
  3. (iii)

    \(\lambda {\tilde {s}_1}=f\left( {\lambda \tilde {s}^{\prime}_{11}} \right)=f\left( {\left( {{s_{\frac{{{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{11}} \right)}^\lambda } - {{(1 - \alpha ^{\prime}_{11})}^\lambda }}}{{{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{11}} \right)}^\lambda }+(\gamma - 1){{(1 - \alpha ^{\prime}_{11})}^\lambda }}}}},{s_{\frac{{\gamma \beta ^{\prime\lambda }_{11}}}{{{{\left( {1+(\gamma - 1)(1 - \beta ^{\prime}_{11})} \right)}^\lambda }+(\gamma - 1)\beta ^{\prime}_{11}}}}}} \right)} \right),\quad 0<\lambda \leq 1;\)

     
  4. (iv)

    \(\tilde{s}_{1}^{\lambda } = f\left( {\tilde{s}^{\prime\lambda }_{1} } \right) = f\left( {\left( {s_{{\frac{{\gamma \alpha^{\prime\lambda }_{1} }}{{\left( {1 + (\gamma - 1)(1 - \alpha ^{\prime\lambda}_{1} )} \right) + (\gamma - 1)\alpha ^{\prime\lambda}_{1} }}}} ,s_{{\frac{{\left( {1 + (\gamma - 1)\beta ^{\prime\lambda }_{1} } \right) - (1 - \beta ^{\prime\lambda }_{1} )}}{{\left( {1 + (\gamma - 1)\beta ^{\prime\lambda}_{1} } \right) + (\gamma - 1)(1 - \beta ^{\prime\lambda}_{1} ) }}}} } \right)} \right),\quad 0{\text{ }} < \lambda \le 1,\)

     
where \(\tilde {s}^{\prime}_{11}=({s_{\alpha ^{\prime}_{11}}},{s_{\beta ^{\prime}_{11}}})\) with \(\left\{ \begin{gathered} \alpha ^{\prime}_{11}=\frac{{{\alpha _1}}}{{2t}} \hfill \\ \beta ^{\prime}_{11}=\frac{{{\beta _1}}}{{2t}} \hfill \\ \end{gathered} \right.\), \(\tilde {s}^{\prime}_{12}=({s_{\alpha ^{\prime}_{12}}},{s_{\beta ^{\prime}_{12}}})\) with \(\left\{ \begin{gathered} \alpha ^{\prime}_{12}=\frac{{{\alpha _2}}}{{2t}} \hfill \\ \beta ^{\prime}_{12}=\frac{{{\beta _2}}}{{2t}} \hfill \\ \end{gathered} \right.\), and \(f\) is a transferred function such that \(\tilde {s}=f(\tilde {s}')={s_{2t}} \odot \tilde {s}'=({s_{2t \times \alpha ^{\prime}_{11}}},{s_{2t \times \beta ^{\prime}_{11}}})\) for any LIFV \(\tilde {s}=({s_\alpha },{s_\beta })\) with \(\tilde {s}^{\prime}_{11}=({s_{\alpha ^{\prime}_{11}}},{s_{\beta ^{\prime}_{11}}})=\left( {{s_{\frac{\alpha }{{2t}}}},{s_{\frac{\beta }{{2t}}}}} \right)\).

Remark 1

When \(\gamma =1\), the linguistic intuitionistic fuzzy Hamacher operational laws listed in Definition 3 degenerate to the following linguistic intuitionistic fuzzy algebraic operational laws:
  1. (i)

    \({\tilde {s}_1}{ \oplus _A}{\tilde {s}_2}=f(\tilde {s}^{\prime}_{11}{ \oplus _A}\tilde {s}^{\prime}_{12})=f(({s_{\alpha ^{\prime}_{12}+\alpha ^{\prime}_{11} - \alpha ^{\prime}_{12}\alpha ^{\prime}_{11}}},{s_{\beta ^{\prime}_{12}\beta ^{\prime}_{11}}}));\)

     
  2. (ii)

    \({\tilde {s}_1}{ \otimes _A}{\tilde {s}_2}=f(\tilde {s}^{\prime}_{11}{ \otimes _A}\tilde {s}^{\prime}_{12})=f(({s_{\alpha ^{\prime}_{12}\alpha ^{\prime}_{11}}},{s_{\beta ^{\prime}_{12}+\beta ^{\prime}_{11} - \beta ^{\prime}_{12}\beta ^{\prime}_{11}}}));\)

     
  3. (iii)

    \(\lambda {\tilde {s}_1}=f(\lambda \tilde {s}^{\prime}_{11})=f(({s_{1 - {{(1 - \alpha ^{\prime\lambda }_{11})}}}},{s_{\beta ^{\prime\lambda }_{11}}})),\quad 0 \leq \lambda \leq 1;\)

     
  4. (iv)

    \({\tilde {s}_1}^{\lambda }=f(\tilde {s}^{\prime\lambda }_{11})=f(({s_{\alpha ^{\prime\lambda }_{11}}},{s_{1 - {{(1 - \beta ^{\prime\lambda }_{11})}}}})),\quad 0 \leq \lambda \leq 1,\)

     
where the notations are as shown in Definition 3.
Furthermore, when \(\gamma =2\), the linguistic intuitionistic fuzzy Hamacher operational laws listed in Definition 3 degenerate to the following linguistic intuitionistic fuzzy Einstein operational laws:
  1. (i)

    \({\tilde {s}_1}{ \oplus _E}{\tilde {s}_2}=f(\tilde {s}^{\prime}_{11}{ \oplus _E}\tilde {s}^{\prime}_{12})=f\left( {\left( {{s_{\frac{{\alpha ^{\prime}_{12}+\alpha ^{\prime}_{11}}}{{1+\alpha ^{\prime}_{12}\alpha ^{\prime}_{11}}}}},{s_{\frac{{\beta ^{\prime}_{12}\beta ^{\prime}_{11}}}{{2 - \left( {\beta ^{\prime}_{12}+\beta ^{\prime}_{11} - \beta ^{\prime}_{12}\beta ^{\prime}_{11}} \right)}}}}} \right)} \right);\)

     
  2. (ii)

    \({\tilde {s}_1}{ \otimes _E}{\tilde {s}_2}=f(\tilde {s}^{\prime}_{11}{ \otimes _E}\tilde {s}^{\prime}_{12})=f\left( {\left( {{s_{\frac{{\alpha ^{\prime}_{12}\alpha ^{\prime}_{11}}}{{2 - \left( {\alpha ^{\prime}_{12}+\alpha ^{\prime}_{11} - \alpha ^{\prime}_{12}\alpha ^{\prime}_{11}} \right)}}}},{s_{\frac{{\beta ^{\prime}_{12}+\beta ^{\prime}_{11}}}{{1+\beta ^{\prime}_{12}\beta ^{\prime}_{11}}}}}} \right)} \right);\)

     
  3. (iii)

    \(\lambda {\tilde {s}_1}=f(\lambda \tilde {s}^{\prime}_{11})=f\left( {\left( {{s_{\frac{{{{\left( {1+\alpha ^{\prime\lambda }_{11}} \right)}} - {{(1 - \alpha ^{\prime\lambda }_{11})} }}}{{{{\left( {1+\alpha ^{\prime\lambda }_{11}} \right)}}+{{(1 - \alpha ^{\prime\lambda }_{11})}}}}}},{s_{\frac{{2\beta ^{\prime\lambda }_{11}}}{{{{\left( {1+(1 - \beta ^{\prime\lambda }_{11})} \right)}}+\beta ^{\prime\lambda }_{11}}}}}} \right)} \right),\quad 0 \leq \lambda \leq 1;\)

     
  4. (iv)

    \({\tilde {s}_1}^{\lambda }=f(\tilde {s}^{\prime\lambda }_{11})=f\left( {\left( {{s_{\frac{{2\alpha ^{\prime\lambda }_{11}}}{{{{\left( {1+(1 - \alpha ^{\prime\lambda }_{11})} \right)}}+\alpha ^{\prime\lambda }_{11}}}}},{s_{\frac{{{{\left( {1+\beta ^{\prime\lambda }_{11}} \right)}} - {{(1 - \beta ^{\prime\lambda }_{11})}}}}{{{{\left( {1+\beta ^{\prime\lambda }_{11}} \right)}}+{{(1 - \beta ^{\prime\lambda }_{11})}}}}}}} \right)} \right),\quad 0 \leq \lambda \leq 1,\)

     
where the notations are as shown in Definition 3.

3 Linguistic intuitionistic fuzzy Hamacher aggregation operators

Based on the linguistic intuitionistic fuzzy Hamacher operational laws offered in Definition 3, this section defines several linguistic intuitionistic fuzzy Hamacher aggregation operators.

Definition 4

Let \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), \(i=1,2, \ldots ,n\), be a set of LIFVs on the continuous linguistic term set\({S_c}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, the linguistic intuitionistic fuzzy Hamacher weighted average (LIFHWA) operator is defined as follows:
$${\text{LIFHWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})={( \oplus _{{i=1}}^{n})_H}({w_i}{\tilde {s}_i})=f({( \oplus _{{i=1}}^{n})_H}({w_i}\tilde {s}^{\prime}_{1i})),$$
(1)
and the linguistic intuitionistic fuzzy Hamacher weighted geometric mean (LIFHWGM) operator is defined as follows:
$${\text{LIFHWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})={( \otimes _{{i=1}}^{n})_H}{({\tilde {s}_i})^{{w_i}}}=f({( \otimes _{{i=1}}^{n})_H}{(\tilde {s}^{\prime}_{1i})^{{w_i}}}),$$
(2)
where \(\tilde {s}^{\prime}_{1i}=({s_{\alpha ^{\prime}_{1i}}},{s_{\beta ^{\prime}_{1i}}})=({s_{{\alpha _i}/2t}},{s_{{\beta _i}/2t}})\) and \(w=({w_1},{w_2}, \ldots ,{w_n})\) is a weighting vector such that \(\sum\nolimits_{{i=1}}^{n} {{w_i}} =1\) and \({w_i} \geq 0\) for all \(i=1,2, \ldots ,n\).

Theorem 1

Let \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), \(i=1,2, \ldots ,n\), be a set of LIFVs on the continuous linguistic term set\({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, their aggregation value using the LIFHWA operator is also a LIFV on \({S_{\text{c}}}\), denoted by
$${\text{LIFHWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\frac{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}} - \Pi _{{i=1}}^{n}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}},{s_{\frac{{\gamma \Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)(1 - \beta ^{\prime}_{1i})} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}}} \right),$$
(3)
where \(\tilde {s}^{\prime}_{1i}=({s_{\alpha ^{\prime}_{1i}}},{s_{\beta ^{\prime}_{1i}}})=({s_{{\alpha _i}/2t}},{s_{{\beta _i}/2t}})\) and \(w=({w_1},{w_2}, \ldots ,{w_n})\) is a weighting vector such that\(\sum\nolimits_{{i=1}}^{n} {{w_i}} =1\) and \({w_i} \geq 0\) for all \(i=1,2, \ldots n\).

Proof

We adopt the mathematical induction on n to show Eq. (3). When n = 1, the conclusion obviously holds. Suppose that Eq. (3) holds for n = k, namely
$$\begin{aligned} {\text{LIFHWA}}({{\tilde {s}}_1},{{\tilde {s}}_2}, \ldots ,{{\tilde {s}}_k}) & =\left( {{s_{\frac{{\Pi _{{i=1}}^{k}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}} - \Pi _{{i=1}}^{k}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{k}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{k}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}},{s_{\frac{{\gamma \Pi _{{i=1}}^{k}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{k}{{\left( {1+(\gamma - 1)(1 - \beta ^{\prime}_{1i})} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{k}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}}} \right) \\ & =f\left( {{s_{\frac{{\Pi _{{i=1}}^{k}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}} - \Pi _{{i=1}}^{k}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{k}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{k}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}}},{s_{\frac{{\gamma \Pi _{{i=1}}^{k}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{k}{{\left( {1+(\gamma - 1)(1 - \beta ^{\prime}_{1i})} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{k}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}}}} \right). \\ \end{aligned}$$
Let n = k + 1, we derive
$$\begin{aligned} & {\text{LIFHWA}}({{\tilde {s}}_1},{{\tilde {s}}_2}, \ldots ,{{\tilde {s}}_{k+1}})=f(({( \oplus _{{i=1}}^{k})_H}({w_i}\tilde {s}^{\prime}_{1i})){ \oplus _H}({w_{k+1}}\tilde {s}^{\prime}_{1{k+1}})) \\ & \quad =f\left( {\left( {{s_{\frac{{\Pi _{{i=1}}^{k}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}} - \Pi _{{i=1}}^{k}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{k}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{k}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}}},{s_{\frac{{\gamma \Pi _{{i=1}}^{k}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{k}{{\left( {1+(\gamma - 1)(1 - \beta ^{\prime}_{1i})} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{k}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}}}} \right){ \oplus _H}\left( {{s_{\frac{{{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1{k+1}}} \right)}^{{w_{k+1}}}} - {{(1 - \alpha ^{\prime}_{1{k+1}})}^{{w_{k+1}}}}}}{{{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1{k+1}}} \right)}^{{w_{k+1}}}}+(\gamma - 1){{(1 - \alpha ^{\prime}_{1{k+1}})}^{{w_{k+1}}}}}}}},{s_{\frac{{\gamma {{\left( {\beta ^{\prime}_{1{k+1}}} \right)}^{^{{{w_{k+1}}}}}}}}{{{{\left( {1+(\gamma - 1)(1 - \beta ^{\prime}_{1{k+1}})} \right)}^{{w_{k+1}}}}+(\gamma - 1){{\left( {\beta ^{\prime}_{1{k+1}}} \right)}^{{w_{k+1}}}}}}}}} \right)} \right) \\ \end{aligned}$$
Let \({\eta _i}={(1+(\gamma - 1)\alpha ^{\prime}_{1i})^{{w_i}}}\), \({\mu _i}={\left( {1 - \alpha ^{\prime}_{1i}} \right)^{{w_i}}}\), \({\kappa _i}={(1+(\gamma - 1)(1 - \beta ^{\prime}_{1i}))^{{w_i}}}\), and \({\nu _i}={(\beta ^{\prime}_{1i})^{{w_i}}}\) for all i = 1, 2, …, k + 1. Then, we have
$$\begin{gathered} {\text{LIFHWA}}({{\tilde {s}}_1},{{\tilde {s}}_2}, \ldots ,{{\tilde {s}}_{k+1}})=f\left( {\left( {{s_{\frac{{\Pi _{{i=1}}^{k}{\eta _i} - \Pi _{{i=1}}^{k}{\mu _i}}}{{\Pi _{{i=1}}^{k}{\eta _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\mu _i}}}}},{s_{\frac{{\gamma \Pi _{{i=1}}^{k}{\nu _i}}}{{\Pi _{{i=1}}^{k}{\kappa _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\nu _i}}}}}} \right){ \oplus _H}\left( {{s_{\frac{{{\eta _{k+1}} - {\mu _{k+1}}}}{{{\eta _{k+1}}+(\gamma - 1){\mu _{k+1}}}}}},{s_{\frac{{\gamma {\nu _{k+1}}}}{{{\kappa _{k+1}}+(\gamma - 1){\nu _{k+1}}}}}}} \right)} \right) \hfill \\ =f\left( {\left( {{s_{\frac{{\frac{{\Pi _{{i=1}}^{k}{\eta _i} - \Pi _{{i=1}}^{k}{\mu _i}}}{{\Pi _{{i=1}}^{k}{\eta _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\mu _i}}}+\frac{{{\eta _{k+1}} - {\mu _{k+1}}}}{{{\eta _{k+1}}+(\gamma - 1){\mu _{k+1}}}} - \frac{{\Pi _{{i=1}}^{k}{\eta _i} - \Pi _{{i=1}}^{k}{\mu _i}}}{{\Pi _{{i=1}}^{k}{\eta _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\mu _i}}}\frac{{{\eta _{k+1}} - {\mu _{k+1}}}}{{{\eta _{k+1}}+(\gamma - 1){\mu _{k+1}}}} - (1 - \gamma )\frac{{\Pi _{{i=1}}^{k}{\eta _i} - \Pi _{{i=1}}^{k}{\mu _i}}}{{\Pi _{{i=1}}^{k}{\eta _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\mu _i}}}\frac{{{\eta _{k+1}} - {\mu _{k+1}}}}{{{\eta _{k+1}}+(\gamma - 1){\mu _{k+1}}}}}}{{1 - (1 - \gamma )\frac{{\Pi _{{i=1}}^{k}{\eta _i} - \Pi _{{i=1}}^{k}{\mu _i}}}{{\Pi _{{i=1}}^{k}{\eta _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\mu _i}}}\frac{{{\eta _{k+1}} - {\mu _{k+1}}}}{{{\eta _{k+1}}+(\gamma - 1){\mu _{k+1}}}}}}}}} \right.} \right., \hfill \\ \left. {\left. {{s_{\frac{{\frac{{\gamma \Pi _{{i=1}}^{k}{\nu _i}}}{{\Pi _{{i=1}}^{k}{\kappa _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\nu _i}}}\frac{{\gamma {\nu _{k+1}}}}{{{\kappa _{k+1}}+(\gamma - 1){\nu _{k+1}}}}}}{{\gamma +(1 - \gamma )\left( {\frac{{\Pi _{{i=1}}^{k}{\nu _i}}}{{\Pi _{{i=1}}^{k}{\kappa _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\nu _i}}}+\frac{{\gamma {\nu _{k+1}}}}{{{\kappa _{k+1}}+(\gamma - 1){\nu _{k+1}}}} - \frac{{\gamma \Pi _{{i=1}}^{k}{\nu _i}}}{{\Pi _{{i=1}}^{k}{\kappa _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\nu _i}}}\frac{{\gamma {\nu _{k+1}}}}{{{\kappa _{k+1}}+(\gamma - 1){\nu _{k+1}}}}} \right)}}}}} \right)} \right) \hfill \\ =f\left( {\left( {{s_{\frac{{\left( {\Pi _{{i=1}}^{k}{\eta _i} - \Pi _{{i=1}}^{k}{\mu _i}} \right)\left( {{\eta _{k+1}}+(\gamma - 1){\mu _{k+1}}} \right)+\left( {{\eta _{k+1}} - {\mu _{k+1}}} \right)\left( {\Pi _{{i=1}}^{k}{\eta _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\mu _i}} \right) - \left( {\Pi _{{i=1}}^{k}{\eta _i} - \Pi _{{i=1}}^{k}{\mu _i}} \right)\left( {{\eta _{k+1}} - {\mu _{k+1}}} \right) - (1 - \gamma )\left( {\Pi _{{i=1}}^{k}{\eta _i} - \Pi _{{i=1}}^{k}{\mu _i}} \right)\left( {{\eta _{k+1}} - {\mu _{k+1}}} \right)}}{{\left( {\Pi _{{i=1}}^{k}{\eta _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\mu _i}} \right)\left( {{\eta _{k+1}}+(\gamma - 1){\mu _{k+1}}} \right) - (1 - \gamma )\left( {\Pi _{{i=1}}^{k}{\eta _i} - \Pi _{{i=1}}^{k}{\mu _i}} \right)\left( {{\eta _{k+1}} - {\mu _{k+1}}} \right)}}}}} \right.} \right., \hfill \\ \left. {\left. {{s_{\frac{{{\gamma ^2}\Pi _{{i=1}}^{{k+1}}{\nu _i}}}{{\gamma \left( {\Pi _{{i=1}}^{k}{\kappa _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\nu _i}} \right)\left( {{\kappa _{k+1}}+(\gamma - 1){\nu _{k+1}}} \right)+(1 - \gamma )\left( {\Pi _{{i=1}}^{k}\gamma {\nu _i}\left( {{\kappa _{k+1}}+(\gamma - 1){\nu _{k+1}}} \right)+\gamma {\nu _{k+1}}\left( {\Pi _{{i=1}}^{k}{\kappa _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\nu _i}} \right) - {\gamma ^2}\Pi _{{i=1}}^{{k+1}}{\nu _i}} \right)}}}}} \right)} \right) \hfill \\ =f\left( {\left( {{s_{\frac{{\Pi _{{i=1}}^{{k+1}}{\eta _i}+(\gamma - 1){\mu _{k+1}}\Pi _{{i=1}}^{k}{\eta _i} - \Pi _{{i=1}}^{k}{\mu _i}{\eta _{k+1}} - (\gamma - 1)\Pi _{{i=1}}^{{k+1}}{\mu _i}+\Pi _{{i=1}}^{{k+1}}{\eta _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\mu _i}{\eta _{k+1}} - \Pi _{{i=1}}^{k}{\eta _i}{\mu _{k+1}} - (\gamma - 1)\Pi _{{i=1}}^{{k+1}}{\mu _i} - \Pi _{{i=1}}^{{k+1}}{\eta _i}+\Pi _{{i=1}}^{k}{\eta _i}{\mu _{k+1}}+\Pi _{{i=1}}^{k}{\mu _i}{\eta _{k+1}} - \Pi _{{i=1}}^{{k+1}}{\mu _i} - (1 - \gamma )\Pi _{{i=1}}^{{k+1}}{\eta _i}+(1 - \gamma )\Pi _{{i=1}}^{k}{\eta _i}{\mu _{k+1}}+(1 - \gamma )\Pi _{{i=1}}^{k}{\mu _i}{\eta _{k+1}} - (1 - \gamma )\Pi _{{i=1}}^{{k+1}}{\mu _i}}}{{\Pi _{{i=1}}^{{k+1}}{\eta _i}+(\gamma - 1)\Pi _{{i=1}}^{k}{\eta _i}{\mu _{k+1}}+(\gamma - 1)\Pi _{{i=1}}^{k}{\mu _i}{\eta _{k+1}}+(\gamma - 1)\Pi _{{i=1}}^{{k+1}}{\mu _i} - (1 - \gamma )\Pi _{{i=1}}^{{k+1}}\eta +(1 - \gamma )\Pi _{{i=1}}^{k}{\eta _i}{\mu _{k+1}}+(1 - \gamma )\Pi _{{i=1}}^{k}{\mu _i}{\eta _{k+1}}+(1 - \gamma )\Pi _{{i=1}}^{{k+1}}{\mu _i}}}}}} \right.} \right. \hfill \\ \left. {\left. {{s_{\frac{{\gamma \Pi _{{i=1}}^{{k+1}}{\nu _i}}}{{\gamma \Pi _{{i=1}}^{{k+1}}{\kappa _i}+\gamma (\gamma - 1)\Pi _{{i=1}}^{k}{\kappa _i}{\nu _{k+1}}+\gamma (\gamma - 1){\kappa _{k+1}}\Pi _{{i=1}}^{k}{\nu _i}+\gamma {{(\gamma - 1)}^2}\Pi _{{i=1}}^{{k+1}}{\nu _i}+(1 - \gamma )\gamma \Pi _{{i=1}}^{k}{\nu _i}{\kappa _{k+1}} - {{(1 - \gamma )}^2}\gamma \Pi _{{i=1}}^{{k+1}}{\nu _i}+(1 - \gamma )\gamma {\nu _{k+1}}\Pi _{{i=1}}^{k}{\kappa _i} - \gamma {{(1 - \gamma )}^2}\Pi _{{i=1}}^{{k+1}}{\nu _i} - (1 - \gamma )\gamma \Pi _{{i=1}}^{{k+1}}{\nu _i}}}}}} \right)} \right) \hfill \\ =f\left( {\left( {{s_{\frac{{r\left( {\Pi _{{i=1}}^{{k+1}}{\eta _i} - \Pi _{{i=1}}^{{k+1}}{\mu _i}} \right)}}{{\gamma \Pi _{{i=1}}^{{k+1}}{\eta _i}+\gamma (\gamma - 1)\Pi _{{i=1}}^{{k+1}}{\mu _i}}}}}} \right.,\left. {{s_{\frac{{{\gamma ^2}\Pi _{{i=1}}^{{k+1}}{\nu _i}}}{{\gamma \Pi _{{i=1}}^{{k+1}}{\kappa _i}+\gamma (\gamma - 1)\Pi _{{i=1}}^{{k+1}}{\nu _i}}}}}} \right)} \right) \hfill \\ =f\left( {\left( {{s_{\frac{{\Pi _{{i=1}}^{{k+1}}{\eta _i} - \Pi _{{i=1}}^{{k+1}}{\mu _i}}}{{\Pi _{{i=1}}^{{k+1}}{\eta _i}+(\gamma - 1)\Pi _{{i=1}}^{{k+1}}{\mu _i}}}}}} \right.,\left. {{s_{\frac{{\gamma \Pi _{{i=1}}^{{k+1}}{\nu _i}}}{{\Pi _{{i=1}}^{{k+1}}{\kappa _i}+(\gamma - 1)\Pi _{{i=1}}^{{k+1}}{\nu _i}}}}}} \right)} \right). \hfill \\ \end{gathered}$$
Thus, we derive
$$\begin{aligned} {\text{LIFHWA}}({{\tilde {s}}_1},{{\tilde {s}}_2}, \ldots ,{{\tilde {s}}_{k+1}})&=f\left( {\left( {{s_{\frac{{\Pi _{{i=1}}^{{k+1}}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}} - \Pi _{{i=1}}^{{k+1}}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{{k+1}}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{{k+1}}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}}},{s_{\frac{{\gamma \Pi _{{i=1}}^{{k+1}}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{{k+1}}{{\left( {1+(\gamma - 1)(1 - \beta ^{\prime}_{1i})} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{{k+1}}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}}}} \right)} \right) \hfill \\ &=\left( {{s_{\frac{{\Pi _{{i=1}}^{{k+1}}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}} - \Pi _{{i=1}}^{{k+1}}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{{k+1}}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{{k+1}}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}},{s_{\frac{{\gamma \Pi _{{i=1}}^{{k+1}}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{{k+1}}{{\left( {1+(\gamma - 1)(1 - \beta ^{\prime}_{1i})} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{{k+1}}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}}} \right), \hfill \\ \end{aligned}$$
and Eq. (3) holds.
On the other hand, following
$$\frac{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1i})}^{{w_i}}} - \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}},\frac{{\gamma \Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)(1 - \beta ^{\prime}_{1i}))}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}}} \in [0,1],$$
we can easily derive that Eq. (3) is a LIFV.

Theorem 2

Let \({\tilde {s}_i}=\left( {{s_{{\alpha _i}}},{s_{{\beta _i}}}} \right)\), \(i=1,2, \ldots ,n\), be a set of LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, their aggregation value using the LIFHWGM operator is also a LIFV on \({S_{\text{c}}}\), denoted by
$${\text{LIFHWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\frac{{\gamma \Pi _{{i=1}}^{n}{{\left( {\alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)(1 - \alpha ^{\prime}_{1i})} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {\alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}},{s_{\frac{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\beta ^{\prime}_{1i}} \right)}^{{w_i}}} - \Pi _{{i=1}}^{n}{{\left( {1 - \beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\beta ^{\prime}_{1i}} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {1 - \beta ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}}} \right),$$
(4)
where \(\tilde {s}^{\prime}_{1i}=({s_{\alpha ^{\prime}_{1i}}},{s_{\beta ^{\prime}_{1i}}})=({s_{{\alpha _i}/2t}},{s_{{\beta _i}/2t}})\) and \(w=({w_1},{w_2}, \ldots ,{w_n})\) is a weighting vector such that \(\sum\nolimits_{{i=1}}^{n} {{w_i}} =1\) and \({w_i} \geq 0\) for all \(i=1,2, \ldots ,n\).

Proof

Following the proof of Theorem 1, we can easily obtain the conclusions.

Remark 2

When \(\gamma =1\), the LIFHWA operator reduces to the linguistic intuitionistic fuzzy algebraic weighted average (LIFAWA) operator
$${\text{LIFAWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{(1 - \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}) \cdot 2t}},{s_{(\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}) \cdot 2t}}} \right),$$
and the LIFHWGM operator reduces to the linguistic intuitionistic fuzzy algebraic weighted geometric mean (LIFAWGM) operator
$${\text{LIFAWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=({s_{(\Pi _{{i=1}}^{n}{{(\alpha ^{\prime}_{1i})}^{{w_i}}}) \cdot 2t}},{s_{(1 - \Pi _{{i=1}}^{n}{{(1 - \beta ^{\prime}_{1i})}^{{w_i}}}) \cdot 2t}}).$$
When \(\gamma =2\), the LIFHWA operator reduces to the linguistic intuitionistic fuzzy Einstein weighted average (LIFEWA) operator
$${\text{LIFEWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\frac{{\Pi _{{i=1}}^{n}{{(1+\alpha ^{\prime}_{1i})}^{{w_i}}} - \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+\alpha ^{\prime}_{1i})}^{{w_i}}}+\Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}} \cdot 2t}},{s_{\frac{{2\Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(1 - \beta ^{\prime}_{1i}))}^{{w_i}}}+\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}}} \cdot 2t}}} \right),$$
and the LIFHWGM operator reduces to the linguistic intuitionistic fuzzy Einstein weighted geometric mean (LIFEWGM) operator
$${\text{LIFEWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\frac{{2\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(1 - \beta ^{\prime}_{1i})} \right)}^{{w_i}}}+\Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}},{s_{\frac{{\Pi _{{i=1}}^{n}{{(1+\alpha ^{\prime}_{1i})}^{{w_i}}} - \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+\alpha ^{\prime}_{1i})}^{{w_i}}}+\Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}} \cdot 2t}}} \right).$$

Next, we study several desirable properties of the LIFHWA and LIFHWGM operators to ensure their rationality. First, we offer the following order relationship between LIFVs.

Let \(\tilde {s}=({s_\alpha },{s_\beta })\) be a LIFV on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, there is a corresponding interval linguistic variable \(\bar {s}=[{s_\alpha },{s_{2t - \beta }}]\). Following the median and deviation of intervals, we offer an order relationship between LIFVs.

Definition 5

Let \(\tilde {s}=({s_\alpha },{s_\beta })\) be a LIFV on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, the median of \(\tilde {s}\) is defined as \(M(\tilde {s})=\frac{{\alpha +2t - \beta }}{2}\), and the deviation of \(\tilde {s}\) is \(D(\tilde {s})=\frac{{2t - \beta - \alpha }}{2}\).

Let \({\tilde {s}_1}=({s_{{\alpha _1}}},{s_{{\beta _1}}})\) and \({\tilde {s}_2}=({s_{{\alpha _2}}},{s_{{\beta _2}}})\) be any two LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, their order relationship is defined as follows:

  1. (i)

    If \(M({\tilde {s}_1})<M({\tilde {s}_2})\), then \({\tilde {s}_1}<{\tilde {s}_2}\);

     
  2. (ii)

    If \(M({\tilde {s}_1})=M({\tilde {s}_2})\), then \(\left\{ {\begin{array}{*{20}{c}} {D({{\tilde {s}}_1})>D({{\tilde {s}}_2}) \Rightarrow {{\tilde {s}}_1}<{{\tilde {s}}_2}} \\ {D({{\tilde {s}}_1})=D({{\tilde {s}}_2}) \Rightarrow {{\tilde {s}}_1}={{\tilde {s}}_2}} \end{array}} \right.\).

     

Definition 6

Let \(\tilde {s}_{i}^{1}=({s_{\alpha _{i}^{1}}},{s_{\beta _{i}^{1}}})\) and \(\tilde {s}_{i}^{2}=({s_{\alpha _{i}^{2}}},{s_{\beta _{i}^{2}}})\), i = 1, 2, …, n, be any two collections of LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). \(\tilde {s}_{i}^{1}\) and \(\tilde {s}_{i}^{2}\) are said to be comonotonic if
$$\tilde {s}_{{(1)}}^{1} \leq \tilde {s}_{{(2)}}^{1} \leq \cdots \leq \tilde {s}_{{(n)}}^{1} \Leftrightarrow \tilde {s}_{{(1)}}^{2} \leq \tilde {s}_{{(2)}}^{2} \leq \cdots \leq \tilde {s}_{{(n)}}^{2},$$
where \(( \cdot )\) denotes a permutation on N = {1, 2, …, n}.

Property 1

(Commutativity) Let \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), i = 1, 2, …, n, be a collection of LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\), and let \({\tilde {s}_{(i)}}=({s_{{\alpha _{(i)}}}},{s_{{\beta _{(i)}}}})\) be a permutation of \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), i = 1, 2, …, n, then
$${\text{LIFHWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})={\text{LIFHWA}}({\tilde {s}_{(1)}},{\tilde {s}_{(2)}}, \ldots ,{\tilde {s}_{(n)}}),$$
$${\text{LIFHWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})={\text{LIFHWGM}}({\tilde {s}_{(1)}},{\tilde {s}_{(2)}}, \ldots ,{\tilde {s}_{(n)}}).$$

Proof

From Definition 4, it is easy to derive the conclusion.

Property 2

(Idempotency) Let \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), i = 1, 2, …, n, be a collection of LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\), and let \(w=({w_1},{w_2}, \ldots ,{w_n})\) be a weighting vector such that \(\sum\nolimits_{{i=1}}^{n} {{w_i}} =1\) and \({w_i} \geq 0\) for all \(i=1,2, \ldots ,n\). If all LIFVs \({\tilde {s}_i}\), i = 1, 2, …, n, are equal, i.e.,\({\tilde {s}_i}=\tilde {s}=({s_\alpha },{s_\beta })\) for all \(i=1,2, \ldots ,n\), then
$${\text{LIFHWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\tilde {s},$$
$${\text{LIFHWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\tilde {s}.$$

Proof

Following Eq. (3), we get
$$\begin{aligned} {\text{LIFHWA}}({{\tilde {s}}_1},{{\tilde {s}}_2}, \ldots ,{{\tilde {s}}_n})&=\left( {{s_{\frac{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}} - \Pi _{{i=1}}^{n}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1i}} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}},{s_{\frac{{\gamma \Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)(1 - \beta ^{\prime}_{1i})} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1i}} \right)}^{{w_i}}}}} \cdot 2t}}} \right) \hfill \\ &=\left( {{s_{\frac{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\alpha '} \right)}^{{w_i}}} - \Pi _{{i=1}}^{n}{{\left( {1 - \alpha '} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\alpha '} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {1 - \alpha '} \right)}^{{w_i}}}}} \cdot 2t}},{s_{\frac{{\gamma \Pi _{{i=1}}^{n}{{\left( {\beta '} \right)}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)(1 - \beta ')} \right)}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {\beta '} \right)}^{{w_i}}}}} \cdot 2t}}} \right) \hfill \\ &=\left( {{s_{\frac{{{{\left( {1+(\gamma - 1)\alpha '} \right)}^{\sum\nolimits_{{i=1}}^{n} {{w_i}} }} - {{\left( {1 - \alpha '} \right)}^{\sum\nolimits_{{i=1}}^{n} {{w_i}} }}}}{{{{\left( {1+(\gamma - 1)\alpha '} \right)}^{\sum\nolimits_{{i=1}}^{n} {{w_i}} }}+(\gamma - 1){{\left( {1 - \alpha '} \right)}^{\sum\nolimits_{{i=1}}^{n} {{w_i}} }}}} \cdot 2t}},{s_{\frac{{\gamma {{\left( {\beta '} \right)}^{\sum\nolimits_{{i=1}}^{n} {{w_i}} }}}}{{{{\left( {1+(\gamma - 1)(1 - \beta ')} \right)}^{\sum\nolimits_{{i=1}}^{n} {{w_i}} }}+(\gamma - 1){{\left( {\beta '} \right)}^{\sum\nolimits_{{i=1}}^{n} {{w_i}} }}}} \cdot 2t}}} \right) \hfill \\ &=\left( {{s_{\frac{{\gamma \alpha '}}{\gamma } \cdot 2t}},{s_{\frac{{\gamma \beta '}}{\gamma } \cdot 2t}}} \right)=({s_\alpha },{s_\beta }), \hfill \\ \end{aligned}$$
where \(\tilde {s}=\left( {{s_{\frac{\alpha }{{2t}}}},{s_{\frac{\beta }{{2t}}}}} \right)\) and \(\tilde {s}^{\prime}_{1i}=\left( {{s_{\frac{{{\alpha _i}}}{{2t}}}},{s_{\frac{{{\beta _i}}}{{2t}}}}} \right)=\left( {{s_{\alpha ^{\prime}_{1i}}},{s_{\beta ^{\prime}_{1i}}}} \right)\), i = 1, 2, …, n.

Similarly, we can show \({\text{LIFHWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\tilde {s}\).

Property 3

(Boundary) Let \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), i = 1, 2, …, n, be a collection of LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\), and let \(w=({w_1},{w_2}, \ldots ,{w_n})\) be a weighting vector such that \(\sum\nolimits_{{i=1}}^{n} {{w_i}} =1\) and \({w_i} \geq 0\) for all \(i=1,2, \ldots ,n\). Let \({\tilde {s}^ - }=\left( {{s_{\mathop {\hbox{min} }\limits_{{i \in N}} {\alpha _i}}},{s_{\mathop {\hbox{max} }\limits_{{i \in N}} {\beta _i}}}} \right)\) and \({\tilde {s}^+}=\left( {{s_{\mathop {\hbox{max} }\limits_{{i \in N}} {\alpha _i}}},{s_{\mathop {\hbox{min} }\limits_{{i \in N}} {\beta _i}}}} \right)\), then
$${\tilde {\alpha }^ - } \leq {\text{LIFHWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n}) \leq {\tilde {\alpha }^+},$$
$${\tilde {\alpha }^ - } \leq {\text{LIFHWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n}) \leq {\tilde {\alpha }^+},$$
where N = {1, 2, …, n}.

Proof

We consider the function \(f(x)=\frac{{1+(\gamma - 1)x}}{{1 - x}}\), where \(0 \leq x<1\). One can check that f is an increasing function for \(f'(x)=\frac{\gamma }{{{{(1 - x)}^2}}}>0\). Thus,
$$\frac{{1+(\gamma - 1){{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}} \leq \frac{{1+(\gamma - 1){{\alpha ^{\prime}}_i}}}{{1 - {{\alpha ^{\prime}}_i}}} \leq \frac{{1+(\gamma - 1){{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}},$$
(5)
where \({s_{\frac{{{\alpha _i}}}{{2t}}}}={s_{\alpha ^{\prime}_{1i}}}\) for all i = 1, 2, …, n.
Following Eq. (5), we have
$$\begin{gathered} {\left( {\frac{{1+(\gamma - 1){{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}}} \right)^{{w_i}}} \leq {\left( {\frac{{1+(\gamma - 1){{\alpha ^{\prime}}_i}}}{{1 - {{\alpha ^{\prime}}_i}}}} \right)^{{w_i}}} \leq {\left( {\frac{{1+(\gamma - 1){{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}}} \right)^{{w_i}}} \hfill \\ \Rightarrow \Pi _{{i=1}}^{n}{\left( {\frac{{1+(\gamma - 1){{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}}} \right)^{{w_i}}} \leq \Pi _{{i=1}}^{n}{\left( {\frac{{1+(\gamma - 1){{\alpha ^{\prime}}_i}}}{{1 - {{\alpha ^{\prime}}_i}}}} \right)^{{w_i}}} \leq \Pi _{{i=1}}^{n}{\left( {\frac{{1+(\gamma - 1){{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}}} \right)^{{w_i}}} \hfill \\ \Rightarrow \frac{{1+(\gamma - 1){{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}} \leq \Pi _{{i=1}}^{n}{\left( {\frac{{1+(\gamma - 1){{\alpha ^{\prime}}_i}}}{{1 - {{\alpha ^{\prime}}_i}}}} \right)^{{w_i}}} \leq \frac{{1+(\gamma - 1){{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}} \hfill \\ \Rightarrow \gamma +\frac{{\gamma {{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}} \leq \Pi _{{i=1}}^{n}{\left( {\frac{{1+(\gamma - 1){{\alpha ^{\prime}}_i}}}{{1 - {{\alpha ^{\prime}}_i}}}} \right)^{{w_i}}}+\gamma - 1 \leq \gamma +\frac{{\gamma {{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}} \hfill \\ \Rightarrow \frac{1}{{\gamma +\frac{{\gamma {{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}}}} \leq \frac{1}{{\Pi _{{i=1}}^{n}{{\left( {\frac{{1+(\gamma - 1){{\alpha ^{\prime}}_i}}}{{1 - {{\alpha ^{\prime}}_i}}}} \right)}^{{w_i}}}+\gamma - 1}} \leq \frac{1}{{\gamma +\frac{{\gamma {{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{{1 - {{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}}}} \hfill \\ \Rightarrow \frac{{1 - {{\hbox{max} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{\gamma } \leq \frac{{\Pi _{{i=1}}^{n}{{(1 - {{\alpha ^{\prime}}_i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1){{\alpha ^{\prime}}_i})}^{{w_i}}}+\Pi _{{i=1}}^{n}(\gamma - 1){{(1 - {{\alpha ^{\prime}}_i})}^{{w_i}}}}} \leq \frac{{1 - {{\hbox{min} }_{i \in N}}{{\alpha ^{\prime}}_i}}}{\gamma } \hfill \\ \Rightarrow 1 - \mathop {\hbox{max} }\limits_{{i \in N}} \alpha ^{\prime}_{1i} \leq \frac{{\gamma \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1i})}^{{w_i}}}+\Pi _{{i=1}}^{n}(\gamma - 1){{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}} \leq 1 - \mathop {\hbox{min} }\limits_{{i \in N}} \alpha ^{\prime}_{1i} \hfill \\ \Rightarrow \mathop {\hbox{min} }\limits_{{i \in N}} \alpha ^{\prime}_{1i} \leq 1 - \frac{{\gamma \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1i})}^{{w_i}}}+\Pi _{{i=1}}^{n}(\gamma - 1){{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}} \leq \mathop {\hbox{max} }\limits_{{i \in N}} \alpha ^{\prime}_{1i} \hfill \\ \Rightarrow \mathop {\hbox{min} }\limits_{{i \in N}} \alpha ^{\prime}_{1i} \leq \frac{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1i})}^{{w_i}}} - \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1i})}^{{w_i}}}+\Pi _{{i=1}}^{n}(\gamma - 1){{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}} \leq \mathop {\hbox{max} }\limits_{{i \in N}} \alpha ^{\prime}_{1i} \hfill \\ \Rightarrow \mathop {\hbox{min} }\limits_{{i \in N}} {\alpha _i} \leq \frac{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1i})}^{{w_i}}} - \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1i})}^{{w_i}}}+\Pi _{{i=1}}^{n}(\gamma - 1){{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}} \cdot 2t \leq \mathop {\hbox{max} }\limits_{{i \in N}} {\alpha _i} \hfill \\ \end{gathered}$$
(6)
Furthermore, we consider the function \(f(x)=\frac{{1+(\gamma - 1)\left( {1 - x} \right)}}{x}\), where \(0<x \leq 1\). One can check that f is a decreasing function for \(f'(x)= - \frac{\gamma }{{{x^2}}}<0\). Thus,
$$\frac{{1+(\gamma - 1)(1 - {{\hbox{max} }_{i \in N}}{{\beta ^{\prime}}_i})}}{{{{\hbox{max} }_{i \in N}}{{\beta ^{\prime}}_i}}} \leq \frac{{1+(\gamma - 1)(1 - {{\beta ^{\prime}}_i})}}{{{{\beta ^{\prime}}_i}}} \leq \frac{{1+(\gamma - 1)(1 - {{\hbox{min} }_{i \in N}}{{\beta ^{\prime}}_i})}}{{{{\hbox{min} }_{i \in N}}{{\beta ^{\prime}}_i}}},$$
(7)
where \({s_{\frac{{{\beta _i}}}{{2t}}}}={s_{\beta ^{\prime}_{1i}}}\) for all i = 1, 2, …, n.
Following Eq. (7), we obtain
$$\begin{gathered} {\left( {\frac{{1+(\gamma - 1)\left( {1 - {{\hbox{max} }_{i \in N}}{{\beta ^{\prime}}_i}} \right)}}{{{{\hbox{max} }_{i \in N}}{{\beta ^{\prime}}_i}}}} \right)^{{w_i}}} \leq {\left( {\frac{{1+(\gamma - 1)(1 - {{\beta ^{\prime}}_i})}}{{{{\beta ^{\prime}}_i}}}} \right)^{{w_i}}} \leq {\left( {\frac{{1+(\gamma - 1)\left( {1 - {{\hbox{min} }_{i \in N}}{{\beta ^{\prime}}_i}} \right)}}{{{{\hbox{min} }_{i \in N}}{{\beta ^{\prime}}_i}}}} \right)^{{w_i}}} \hfill \\ \Rightarrow \Pi _{{i=1}}^{n}{\left( {\frac{{1+(\gamma - 1)\left( {1 - {{\hbox{max} }_{i \in N}}{{\beta ^{\prime}}_i}} \right)}}{{{{\hbox{max} }_{i \in N}}{{\beta ^{\prime}}_i}}}} \right)^{{w_i}}} \leq \Pi _{{i=1}}^{n}{\left( {\frac{{1+(\gamma - 1)(1 - {{\beta ^{\prime}}_i})}}{{{{\beta ^{\prime}}_i}}}} \right)^{{w_i}}} \leq \Pi _{{i=1}}^{n}{\left( {\frac{{1+(\gamma - 1)\left( {1 - {{\hbox{min} }_{i \in N}}{{\beta ^{\prime}}_i}} \right)}}{{{{\hbox{min} }_{i \in N}}{{\beta ^{\prime}}_i}}}} \right)^{{w_i}}} \hfill \\ \Rightarrow \frac{1}{{{{\hbox{max} }_{i \in N}}{{\beta ^{\prime}}_i}}} - (\gamma - 1) \leq \Pi _{{i=1}}^{n}{\left( {\frac{{1+(\gamma - 1)(1 - {{\beta ^{\prime}}_i})}}{{{{\beta ^{\prime}}_i}}}} \right)^{{w_i}}} \leq \frac{1}{{{{\hbox{min} }_{i \in N}}{{\beta ^{\prime}}_i}}} - (\gamma - 1) \hfill \\ \Rightarrow \frac{1}{{{{\hbox{max} }_{i \in N}}{{\beta ^{\prime}}_i}}} \leq \Pi _{{i=1}}^{n}{\left( {\frac{{1+(\gamma - 1)(1 - {{\beta ^{\prime}}_i})}}{{{{\beta ^{\prime}}_i}}}} \right)^{{w_i}}}+(\gamma - 1) \leq \frac{1}{{{{\hbox{min} }_{i \in N}}{{\beta ^{\prime}}_i}}} \hfill \\ \Rightarrow \mathop {\hbox{min} }\limits_{{i \in N}} \beta ^{\prime}_{1i} \leq \frac{1}{{\Pi _{{i=1}}^{n}{{\left( {\frac{{1+(\gamma - 1)(1 - \beta ^{\prime}_{1i})}}{{\beta ^{\prime}_{1i}}}} \right)}^{{w_i}}}+(\gamma - 1)}} \leq \mathop {\hbox{max} }\limits_{{i \in N}} \beta ^{\prime}_{1i} \hfill \\ \Rightarrow \mathop {\hbox{min} }\limits_{{i \in N}} \beta ^{\prime}_{1i} \leq \frac{{\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)(1 - \beta ^{\prime}_{1i}))}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}}} \leq \mathop {\hbox{max} }\limits_{{i \in N}} \beta ^{\prime}_{1i} \hfill \\ \Rightarrow \mathop {\hbox{min} }\limits_{{i \in N}} {\beta _i} \leq \frac{{\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)(1 - \beta ^{\prime}_{1i}))}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}}} \cdot 2t \leq \mathop {\hbox{max} }\limits_{{i \in N}} {\beta _i}. \hfill \\ \end{gathered}$$
(8)
Equations (6) and (8) show that
$$\begin{gathered} \frac{{{{\hbox{min} }_{i \in N}}{\alpha _i}+2t - {{\hbox{max} }_{i \in N}}{\beta _i}}}{2} \leq 2t\left( {1+\frac{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1i})}^{{w_i}}} - \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1i})}^{{w_i}}}+\Pi _{{i=1}}^{n}(\gamma - 1){{(1 - \alpha ^{\prime}_{1i})}^{{w_i}}}}}} \right. \hfill \\ - \left. {\frac{{\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)(1 - \beta ^{\prime}_{1i}))}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1i})}^{{w_i}}}}}} \right) \leq \frac{{{{\hbox{max} }_{i \in N}}{\alpha _i}+2t - {{\hbox{min} }_{i \in N}}{\beta _i}}}{2}. \hfill \\ \end{gathered}$$
Similarly, we can derive
$$\begin{gathered} \frac{{{{\hbox{min} }_{i \in N}}{\alpha _i}+2t - {{\hbox{max} }_{i \in N}}{\beta _i}}}{2} \leq 2t\left( {1+\frac{{\Pi _{{i=1}}^{n}{{(\alpha ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)(1 - \alpha ^{\prime}_{1i}))}^{{w_i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{(\alpha ^{\prime}_{1i})}^{{w_i}}}}}} \right. \hfill \\ - \left. {\frac{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\beta ^{\prime}_{1i})}^{{w_i}}} - \Pi _{{i=1}}^{n}{{(1 - \beta ^{\prime}_{1i})}^{{w_i}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\beta ^{\prime}_{1i})}^{{w_i}}}+\Pi _{{i=1}}^{n}(\gamma - 1){{(1 - \beta ^{\prime}_{1i})}^{{w_i}}}}}} \right) \leq \frac{{{{\hbox{max} }_{i \in N}}{\alpha _i}+2t - {{\hbox{min} }_{i \in N}}{\beta _i}}}{2}. \hfill \\ \end{gathered}$$

Property 4

(Comonotonicity) Let \(\tilde {s}_{i}^{1}=({s_{\alpha _{i}^{1}}},{s_{\beta _{i}^{1}}})\) and \(\tilde {s}_{i}^{2}=({s_{\alpha _{i}^{2}}},{s_{\beta _{i}^{2}}})\), i = 1, 2, …, n, be any two collections of LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\), and let \(w=({w_1},{w_2}, \ldots ,{w_n})\) be a weighting vector such that \(\sum\nolimits_{{i=1}}^{n} {{w_i}} =1\) and \({w_i} \geq 0\) for all \(i=1,2, \ldots ,n\). If \(\tilde {s}_{i}^{1}\) and \(\tilde {s}_{i}^{2}\) are comonotonic, namely \(\tilde {s}_{{(1)}}^{1} \leq \tilde {s}_{{(2)}}^{1} \leq \cdots \leq \tilde {s}_{{(n)}}^{1} \Leftrightarrow \tilde {s}_{{(1)}}^{2} \leq \tilde {s}_{{(2)}}^{2} \leq \cdots \leq \tilde {s}_{{(n)}}^{2}\) and \(\tilde {s}_{{(i)}}^{1} \leq \tilde {s}_{{(i)}}^{2}\) for all \(i=1,2, \ldots ,n\), where \(( \cdot )\) denotes a permutation on N = {1, 2, …, n}. Then,
$${\text{LIFHWA}}(\tilde {s}_{1}^{1},\tilde {s}_{2}^{1}, \ldots ,\tilde {s}_{n}^{1}) \leq {\text{LIFHWA}}(\tilde {s}_{1}^{2},\tilde {s}_{2}^{2}, \ldots ,\tilde {s}_{n}^{2}),$$
$${\text{LIFHWGM}}(\tilde {s}_{1}^{1},\tilde {s}_{2}^{1}, \ldots ,\tilde {s}_{n}^{1}) \leq {\text{LIFHWGM}}(\tilde {s}_{1}^{2},\tilde {s}_{2}^{2}, \ldots ,\tilde {s}_{n}^{2}).$$

Proof

Property 1 shows that
$${\text{LIFHWA}}(\tilde {s}_{1}^{1},\tilde {s}_{2}^{1}, \ldots ,\tilde {s}_{n}^{1})={\text{LIFHWA}}(\tilde {s}_{{(1)}}^{1},\tilde {s}_{{(2)}}^{1}, \ldots ,\tilde {s}_{{(n)}}^{1})\quad {\text{and}}$$
$${\text{LIFHWA}}(\tilde {s}_{1}^{2},\tilde {s}_{2}^{2}, \ldots ,\tilde {s}_{n}^{2})={\text{LIFHWA}}(\tilde {s}_{{(1)}}^{2},\tilde {s}_{{(2)}}^{2}, \ldots ,\tilde {s}_{{(n)}}^{2}),$$
where \(( \cdot )\) denotes a permutation on N = {1, 2, …, n}.
Because \(\tilde {s}_{{(i)}}^{1} \leq \tilde {s}_{{(i)}}^{2}\) for all \(i=1,2, \ldots ,n\), Property 3 shows that
$$\frac{{1+(\gamma - 1)\alpha _{{_{{(i)}}}}^{{\prime 1}}}}{{1 - \alpha _{{_{{(i)}}}}^{{\prime 1}}}} \leq \frac{{1+(\gamma - 1)\alpha _{{_{{(i)}}}}^{{\prime 2}}}}{{1 - \alpha _{{_{{(i)}}}}^{{\prime 2}}}}\quad {\text{and}}\quad \frac{{1+(\gamma - 1)(1 - \beta _{{_{{(i)}}}}^{{\prime 2}})}}{{\beta _{{_{{(i)}}}}^{{\prime 2}}}} \leq \frac{{1+(\gamma - 1)(1 - \beta _{{_{{(i)}}}}^{{\prime 1}})}}{{\beta _{{_{{(i)}}}}^{{\prime 1}}}},$$
where \(\left\{ \begin{gathered} \alpha _{i}^{{\prime 1}}=\frac{{\alpha _{i}^{1}}}{{2t}},\beta _{i}^{{\prime 1}}=\frac{{\beta _{i}^{1}}}{{2t}} \hfill \\ \alpha _{i}^{{\prime 2}}=\frac{{\alpha _{i}^{2}}}{{2t}},\beta _{i}^{{\prime 2}}=\frac{{\beta _{i}^{2}}}{{2t}} \hfill \\ \end{gathered} \right.\) for all \(i=1,2, \ldots ,n\).
Following the proof of Property 3, we derive
$${\text{LIFHWA}}(\tilde {s}_{{(1)}}^{1},\tilde {s}_{{(2)}}^{1}, \ldots ,\tilde {s}_{{(n)}}^{1}) \leq {\text{LIFHWA}}(\tilde {s}_{{(1)}}^{2},\tilde {s}_{{(2)}}^{2}, \ldots ,\tilde {s}_{{(n)}}^{2})\quad {\text{and}}\quad {\text{LIFHWGM}}(\tilde {s}_{{(1)}}^{1},\tilde {s}_{{(2)}}^{1}, \ldots ,\tilde {s}_{{(n)}}^{1}) \leq {\text{LIFHWGM}}(\tilde {s}_{{(1)}}^{2},\tilde {s}_{{(2)}}^{2}, \ldots ,\tilde {s}_{{(n)}}^{2}).$$
Thus,
$${\text{LIFHWA}}(\tilde {s}_{1}^{1},\tilde {s}_{2}^{1}, \ldots ,\tilde {s}_{n}^{1}) \leq {\text{LIFHWA}}(\tilde {s}_{1}^{2},\tilde {s}_{2}^{2}, \ldots ,\tilde {s}_{n}^{2})\quad {\text{and}}\quad {\text{LIFHWGM}}(\tilde {s}_{1}^{1},\tilde {s}_{2}^{1}, \ldots ,\tilde {s}_{n}^{1}) \leq {\text{LIFHWGM}}(\tilde {s}_{1}^{2},\tilde {s}_{2}^{2}, \ldots ,\tilde {s}_{n}^{2}).$$

Different from the LIFHWA and LIFHWGM operators, we next define the linguistic intuitionistic fuzzy Hamacher ordered weighted average (LIFHOWA) operator and the linguistic intuitionistic fuzzy Hamacher ordered weighted geometric mean (LIFHOWGM) operator.

Definition 7

Let \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), \(i=1,2, \ldots ,n\), be a set of LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, the LIFHOWA operator is defined as follows:
$${\text{LIFHOWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})={( \oplus _{{i=1}}^{n})_H}({\omega _i}{\tilde {s}_{(i)}})=f({( \oplus _{{i=1}}^{n})_H}({\omega _i}\tilde {s}^{\prime}_{1{(i)}})),$$
(9)
and the LIFHOWGM operator is defined as follows:
$${\text{LIFHOWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})={( \otimes _{{i=1}}^{n})_H}{({\tilde {s}_{(i)}})^{{\omega _i}}}=f({( \otimes _{{i=1}}^{n})_H}{(\tilde {s}^{\prime}_{1{(i)}})^{{\omega _i}}}),$$
(10)
where \(\tilde {s}^{\prime}_{1i}=({s_{\alpha ^{\prime}_{1i}}},{s_{\beta ^{\prime}_{1i}}})=({s_{{\alpha _i}/2t}},{s_{{\beta _i}/2t}})\), \({\tilde {s}_{(i)}}\) is the ith smallest value of \({\tilde {s}_i}\), \(i=1,2, \ldots ,n\), and \({\omega _i}\) is the weight of the ith ordered position such that \(\sum\nolimits_{{i=1}}^{n} {{\omega _i}} =1\) and \({\omega _i} \geq 0\) for all\(i=1,2, \ldots ,n\).

Theorem 3

Let \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), \(i=1,2, \ldots ,n\), be a set of LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, their aggregation value using the LIFHOWA operator is a LIFV on \({S_{\text{c}}}\), denoted by
$${\text{LIFHOWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\frac{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}} - \Pi _{{i=1}}^{n}{{\left( {1 - \alpha ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\alpha ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {1 - \alpha ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}}}} \cdot 2t}},{s_{\frac{{\gamma \Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)(1 - \beta ^{\prime}_{1{(i)}})} \right)}^{{\omega _i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}}}} \cdot 2t}}} \right),$$
(11)
and their aggregation value using the LIFHOWGM operator is also a LIFV on \({S_{\text{c}}}\), denoted by
$${\text{LIFHOWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\frac{{\gamma \Pi _{{i=1}}^{n}{{\left( {\alpha ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)(1 - \alpha ^{\prime}_{1{(i)}})} \right)}^{{\omega _i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {\alpha ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}}}} \cdot 2t}},{s_{\frac{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\beta ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}} - \Pi _{{i=1}}^{n}{{\left( {1 - \beta ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\beta ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {1 - \beta ^{\prime}_{1{(i)}}} \right)}^{{\omega _i}}}}} \cdot 2t}}} \right),$$
(12)
where \(\tilde {s}^{\prime}_{1i}=({s_{\alpha ^{\prime}_{1i}}},{s_{\beta ^{\prime}_{1i}}})=({s_{{\alpha _i}/2t}},{s_{{\beta _i}/2t}})\), \({\tilde {s}_{(i)}}\) is the ith smallest value of \({\tilde {s}_i}\), \(i=1,2, \ldots ,n\), and \({\omega _i}\) is the weight of the ith ordered position such that \(\sum\nolimits_{{i=1}}^{n} {{\omega _i}} =1\) and \({\omega _i} \geq 0\) for all \(i=1,2, \ldots ,n\).

To consider the weights of IFVs as well as that of ordered positions, we further define the linguistic intuitionistic fuzzy Hamacher hybrid weighted average (LIFHHWA) operator and the linguistic intuitionistic fuzzy Hamacher hybrid weighted geometric mean (LIFHHWGM) operator as follows:

Definition 8

Let \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), \(i=1,2, \ldots ,n\), be a set of LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, the LIFHHWA operator is defined as follows:
$${\text{LIFHHWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})={\left( { \oplus _{{i=1}}^{n}} \right)_H}\left( {\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}{{\tilde {s}}_{(i)}}} \right)=f\left( {{{\left( { \oplus _{{i=1}}^{n}} \right)}_H}\left( {\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}\tilde {s}^{\prime}_{1{(i)}}} \right)} \right),$$
(13)
and the LIFHHWGM operator is defined as follows:
$${\text{LIFHHWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})={\left( { \otimes _{{i=1}}^{n}} \right)_H}{({\tilde {s}_{(i)}})^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}=f\left( {{{\left( { \otimes _{{i=1}}^{n}} \right)}_H}{{(\tilde {s}^{\prime}_{1{(i)}})}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}} \right),$$
(14)
where \(\tilde {s}^{\prime}_{1i}=({s_{\alpha ^{\prime}_{1i}}},{s_{\beta ^{\prime}_{1i}}})=({s_{{\alpha _i}/2t}},{s_{{\beta _i}/2t}})\), \({w_{(i)}}{\tilde {s}_{(i)}}\) is the ith smallest value of \({w_i}{\tilde {s}_i}\), \(i=1,2, \ldots ,n\), for the LIFHHWA operator, \({\tilde {s}_{(i)}}^{{{w_{(i)}}}}\) is the ith smallest value of \({\tilde {s}_i}^{{{w_i}}}\), \(i=1,2, \ldots ,n\), for the LIFHHWA operator, and \(w=({w_1},{w_2}, \ldots ,{w_n})\) and \(\omega =({\omega _1},{\omega _2}, \ldots ,{\omega _n})\) are two weighting vectors such that \(\sum\nolimits_{{i=1}}^{n} {{w_i}} =\sum\nolimits_{{i=1}}^{n} {{\omega _i}} =1\) and \({w_i},{\omega _i} \geq 0\) for all \(i=1,2, \ldots ,n\).

Theorem 4

Let \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), \(i=1,2, \ldots ,n\), be a set of LIFVs on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, their aggregation value using the LIFHHWA operator is a LIFV on \({S_{\text{c}}}\), denoted by
$${\text{LIFHHWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\frac{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1{(i)}})}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}} - \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1{(i)}})}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)\alpha ^{\prime}_{1{(i)}})}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1{(i)}})}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}} \cdot 2t}},{s_{\frac{{\gamma \Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1{(i)}})}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}}{{\Pi _{{i=1}}^{n}{{(1+(\gamma - 1)(1 - \beta ^{\prime}_{1{(i)}}))}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1{(i)}})}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}} \cdot 2t}}} \right),$$
(15)
and their aggregation value using the LIFHHGM operator is also a LIFV on \({S_{\text{c}}}\), denoted by
$${\text{LIFHHWGM}}\left( {{{\tilde {s}}_1},{{\tilde {s}}_2}, \ldots ,{{\tilde {s}}_n}} \right)=\left( {{s_{\frac{{\gamma \Pi _{{i=1}}^{n}{{\left( {\alpha ^{\prime}_{1{(i)}}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)(1 - \alpha ^{\prime}_{1{(i)}})} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {\alpha ^{\prime}_{1{(i)}}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}} \cdot 2t}},{s_{\frac{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\beta ^{\prime}_{1{(i)}}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}} - \Pi _{{i=1}}^{n}{{\left( {1 - \beta ^{\prime}_{1{(i)}}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(\gamma - 1)\beta ^{\prime}_{1{(i)}}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}+(\gamma - 1)\Pi _{{i=1}}^{n}{{\left( {1 - \beta ^{\prime}_{1{(i)}}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}} \cdot 2t}}} \right),$$
(16)
where the notations are as shown in Definition 8.

Remark 3

When \(\gamma =1\), the LIFHHWA operator reduces to the linguistic intuitionistic fuzzy algebraic hybrid weighted average (LIFAHWA) operator
$${\text{LIFAHWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\left( {1 - \Pi _{{i=1}}^{n}{{(1 - \alpha ^{\prime}_{1{(i)}})}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}} \right) \cdot 2t}},{s_{\left( {\Pi _{{i=1}}^{n}{{(\beta ^{\prime}_{1{(i)}})}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}} \right) \cdot 2t}}} \right)$$
and the LIFHHWGM operator reduces to the linguistic intuitionistic fuzzy algebraic hybrid weighted geometric mean (LIFAHWGM) operator
$${\text{LIFAHWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\left( {\Pi _{{i=1}}^{n}{{\left( {\alpha ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}} \right) \cdot 2t}},{s_{\left( {1 - \Pi _{{i=1}}^{n}{{\left( {1 - \beta ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}} \right) \cdot 2t}}} \right).$$
When \(\gamma =2\), the LIFHHWA operator reduces to the linguistic intuitionistic fuzzy Einstein hybrid weighted average (LIFEHWA) operator
$${\text{LIFEHWA}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\frac{{\Pi _{{i=1}}^{n}{{\left( {1+\alpha ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}} - \Pi _{{i=1}}^{n}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+\alpha ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}+\Pi _{{i=1}}^{n}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}} \cdot 2t}},{s_{\frac{{2\Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(1 - \beta ^{\prime}_{1i})} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}+\Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}} \cdot 2t}}} \right),$$
and the LIFHHWGM operator reduces to the linguistic intuitionistic fuzzy Einstein hybrid weighted geometric mean (LIFEHWGM) operator
$${\text{LIFEHWGM}}({\tilde {s}_1},{\tilde {s}_2}, \ldots ,{\tilde {s}_n})=\left( {{s_{\frac{{2\Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+(1 - \beta ^{\prime}_{1i})} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}+\Pi _{{i=1}}^{n}{{\left( {\beta ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}} \cdot 2t}},{s_{\frac{{\Pi _{{i=1}}^{n}{{\left( {1+\alpha ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}} - \Pi _{{i=1}}^{n}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}}{{\Pi _{{i=1}}^{n}{{\left( {1+\alpha ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}+\Pi _{{i=1}}^{n}{{\left( {1 - \alpha ^{\prime}_{1i}} \right)}^{\frac{{{w_{(i)}}{\omega _i}}}{{\sum\nolimits_{{i=1}}^{n} {{w_{(i)}}{\omega _i}} }}}}}} \cdot 2t}}} \right).$$

Similarly, we can easily derive the linguistic intuitionistic fuzzy algebraic ordered weighted average (LIFAOWA) operator, the linguistic intuitionistic fuzzy algebraic ordered weighted geometric mean (LIFAOWGM) operator, the linguistic intuitionistic fuzzy Einstein ordered weighted average (LIFEOWA) operator, and the linguistic intuitionistic fuzzy Einstein ordered weighted geometric mean (LIFEOWGM) operator.

Remark 4

Similar to the LIFHWA and LIFHWGM operators, the LIFHOWA, LIFHOWGM, LIFHHWA and LIFHHWGM operators all satisfy the properties: Commutativity, Idempotency, Boundary, and Comonotonicity. Because the proofs are similar to that for the LIFHWA and LIFHWGM operators, we no longer list them.

4 A method for group decision making with linguistic intuitionistic fuzzy information

This section contains two parts. The first part studies how to determine the weights of criteria, DMs, and ordered positions, and the second part offers an algorithm for group decision making with linguistic intuitionistic fuzzy information.

4.1 The weighting information

Without loss of generality, let A = {a1, a2, …, a m } be the set of alternatives, let C = {c1, c2, …, c n } be the set of criteria, and let E={e1, e2, …, e q } be the set of the DMs. Assume that \(\tilde {s}_{{ij}}^{k}=({s_{\alpha _{{ij}}^{k}}},{s_{\beta _{{ij}}^{k}}})\) be the LIFV of the alternative a i for the criterion c j offered by the DM e k on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\), where i = 1, 2, …, m; j = 1, 2, …, n; k = 1, 2, …, q. By \({\tilde {S}^k}={(\tilde {s}_{{ij}}^{k})_{m \times n}}\), we denote the LIFV matrix \({\tilde {S}^k}={(\tilde {s}_{{ij}}^{k})_{m \times n}}\) provided by the DM e k , k = 1, 2, …, q.

In group decision making, how to determine the weights of DMs is a crucial issue (Chen and Huang 2003; Chen and Chung 2006; Chen and Chien 2011; Tsai et al. 2008, 2012). Next, we introduce a new method to ascertain the weights of the DMs based on similarity measure.

Definition 9

Let \({\tilde {S}^k}={(\tilde {s}_{{ij}}^{k})_{m \times n}}\) and \({\tilde {S}^l}={(\tilde {s}_{{ij}}^{l})_{m \times n}}\) be any two LIFV matrices on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, their similarity measure matrix \(E={(e_{{ij}}^{{kl}})_{m \times n}}\) is defined as follows:
$$e_{{ij}}^{{kl}}=\frac{{2(\alpha _{{ij}}^{k}\alpha _{{ij}}^{l}+\beta _{{ij}}^{k}\beta _{{ij}}^{l})}}{{{{(\alpha _{{ij}}^{k})}^2}+{{(\alpha _{{ij}}^{l})}^2}+{{(\beta _{{ij}}^{k})}^2}+{{(\beta _{{ij}}^{l})}^2}}},$$
(17)
where i = 1, 2, …, m; j = 1, 2, …, n.

Definition 10

Let \({\tilde {S}^k}={(\tilde {s}_{{ij}}^{k})_{m \times n}}\), k = 1, 2, …, q, be a set of LIFV matrices on the continuous linguistic term set \({S_{\text{c}}}=\{ {s_\alpha }|\alpha \in [0,2t]\}\). Then, the weight of the DM e k is defined as follows:
$${w_{{e_k}}}=\frac{{\sum\nolimits_{{l=1,l \ne k}}^{q} {\sum\nolimits_{{i=1}}^{m} {\sum\nolimits_{{j=1}}^{n} {e_{{ij}}^{{kl}}} } } }}{{\sum\nolimits_{{k=1}}^{q} {\sum\nolimits_{{k=1,k \ne l}}^{q} {\sum\nolimits_{{i=1}}^{m} {\sum\nolimits_{{j=1}}^{n} {e_{{ij}}^{{kl}}} } } } }},$$
(18)
where k = 1, 2, …, q.

Let \({\tilde {T}^k}={(\tilde {t}_{{ij}}^{k})_{m \times n}}\) be the weighted LIFV matrix of \({\tilde {S}^k}\), k = 1, 2, …, q, where \(\tilde {t}_{{ij}}^{k}={w_{{e_k}}}\tilde {s}_{{ij}}^{k}=({s_{\mu _{{ij}}^{k}}},{s_{v_{{ij}}^{k}}})\) or \(\tilde {t}_{{ij}}^{k}={(\tilde {s}_{{ij}}^{k})^{{w_{{e_k}}}}}=({s_{\mu _{{ij}}^{k}}},{s_{v_{{ij}}^{k}}})\) for all i = 1, 2, …, m; j = 1, 2, …, n. For each pair of (i, j), we rank \(\tilde {t}_{{ij}}^{1},\tilde {t}_{{ij}}^{2}, \ldots ,\tilde {t}_{{ij}}^{n}\) in an increasing order, where \(\tilde {t}_{{ij}}^{{(1)}} \leq \tilde {t}_{{ij}}^{{(2)}} \leq \cdots \leq \tilde {t}_{{ij}}^{{(n)}}\) with \(( \cdot )\) being a permutation of \(Q=\{ 1,2, \ldots ,q\}\).

We calculate \(d_{{ij}}^{{(k)}}=\left\{ \begin{gathered} \frac{{2t - \mu _{{ij}}^{{(k)}}+v_{{ij}}^{{(k)}}}}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k \leq {\text{mid(}}q{\text{)}} \hfill \\ \frac{{2t+\mu _{{ij}}^{{(k)}} - v_{{ij}}^{{(k)}}}}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k>{\text{mid(}}q{\text{)}} \hfill \\ \end{gathered} \right.\), k = 1, 2, …, q, where \({\text{mid(}}q{\text{)=}}\left\{ {\begin{array}{*{20}{c}} {\tfrac{q}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} q{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}{\kern 1pt} {\kern 1pt} {\text{an}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{even}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{number}}{\kern 1pt} {\kern 1pt} } \\ {\tfrac{{q+1}}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} q{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}{\kern 1pt} {\kern 1pt} {\text{an}}{\kern 1pt} {\kern 1pt} {\text{odd}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{number}}} \end{array}} \right.\) for each pair of (i, j). Furthermore, let \({d^{(k)}}=\sum\nolimits_{{i=1}}^{m} {\sum\nolimits_{{j=1}}^{n} {d_{{ij}}^{{(k)}}} }\), k = 1, 2, …, q. Then, the weights of the ordered positions are defined as follows:
$${\omega _k}=\frac{{1/{d^{(k)}}}}{{\sum\nolimits_{{k=1}}^{q} {1/{d^{(k)}}} }},$$
(19)
where k = 1, 2, …, q.

Definition 11

Let \(\tilde {S}={({\tilde {s}_{ij}})_{m \times n}}={(({s_{{\mu _{ij}}}},{s_{{v_{ij}}}}))_{m \times n}}\) be a comprehensive LIFV matrix. Then, the weights of criteria are defined as follows:
$${w_{{c_i}}}=\frac{{\sum\nolimits_{{i=1}}^{{m - 1}} {\sum\nolimits_{{h=i+1}}^{m} {\left( {\left| {\frac{{{\mu _{ij}}+2t - {v_{ij}}}}{2} - \frac{{{\mu _{hj}}+2t - {v_{hj}}}}{2}} \right|+\left| {\frac{{2t - {\mu _{ij}} - {v_{ij}}}}{2} - \frac{{2t - {\mu _{hj}} - {v_{hj}}}}{2}} \right|} \right)} } }}{{\sum\nolimits_{{j=1}}^{n} {\sum\nolimits_{{i=1}}^{{m - 1}} {\sum\nolimits_{{h=i+1}}^{m} {\left( {\left| {\frac{{{\mu _{ij}}+2t - {v_{ij}}}}{2} - \frac{{{\mu _{hj}}+2t - {v_{hj}}}}{2}} \right|+\left| {\frac{{2t - {\mu _{ij}} - {v_{ij}}}}{2} - \frac{{2t - {\mu _{hj}} - {v_{hj}}}}{2}} \right|} \right)} } } }},$$
(20)
where i = 1, 2, …, n.

Equation (20) shows that the bigger the difference of alternatives’ LIFVs for the same criterion, the bigger the weight of this criterion will be. Especially, when the LIFVs of all alternatives for the same criterion are equal, then the weight of this criterion is zero. Note that such criterion has not any influence on the ranking of alternatives.

Let \(\tilde {T}={({\tilde {t}_{ij}})_{m \times n}}\) be the weighted LIFV matrix of \(\tilde {S}\), where \({\tilde {t}_{ij}}={w_{{c_j}}}{\tilde {s}_{ij}}=({s_{{\mu _{ij}}}},{s_{{v_{ij}}}})\) or \({\tilde {t}_{ij}}={\tilde {s}_{ij}}^{{{w_{{c_j}}}}}=({s_{{\mu _{ij}}}},{s_{{v_{ij}}}})\) for all i = 1, 2, …, m; j = 1, 2, …, n. For each i = 1, 2, …, m, we rank \({\tilde {t}_{ij}},{\tilde {t}_{ij}},...,{\tilde {t}_{ij}}\) in an increasing order, where \({\tilde {t}_{i(1)}} \leq {\tilde {t}_{i(2)}} \leq ... \leq {\tilde {t}_{i(n)}}\) with \(( \cdot )\) being a permutation of \(N=\{ 1,2,...,n\}\).

We calculate \({d_{i(j)}}=\left\{ \begin{gathered} \frac{{2t - {\mu _{i(j)}}+{v_{i(j)}}}}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j \leq {\text{mid(}}n{\text{)}} \hfill \\ \frac{{2t+{\mu _{i(j)}} - {v_{i(j)}}}}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j>{\text{mid(}}n{\text{)}} \hfill \\ \end{gathered} \right.,\) j = 1, 2, …, n, where \({\text{mid(}}n{\text{)=}}\left\{ {\begin{array}{*{20}{c}} {\tfrac{n}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}{\kern 1pt} {\kern 1pt} {\text{an}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{even}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{number}}{\kern 1pt} {\kern 1pt} } \\ {\tfrac{{n+1}}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}{\kern 1pt} {\kern 1pt} {\text{an}}{\kern 1pt} {\kern 1pt} {\text{odd}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{number}}} \end{array}} \right.\) for each i = 1, 2, …, m. Furthermore, let \({d_{(j)}}=\sum\nolimits_{{i=1}}^{m} {{d_{i(j)}}}\), j = 1, 2, …, n. Then, the weights of the ordered positions are defined as follows:
$${\omega _{^{j}}}=\frac{{1/{d_{(j)}}}}{{\sum\nolimits_{{j=1}}^{n} {1/{d_{(j)}}} }},$$
(21)
where j = 1, 2, …, n.

4.2 An algorithm

Following the defined aggregation operators and methods for determining weights, the subsection offers an algorithm for group decision making with linguistic intuitionistic fuzzy information.

Step 1: With respect to the individual LIFV matrices \({\tilde {S}^k}={(\tilde {s}_{{ij}}^{k})_{m \times n}}\), k = 1, 2, …, q, Eq. (18) is used to determine the DMs’ weights \({w_{{e_k}}}\), k = 1, 2, …, q. Furthermore, Eq. (19) is adopted to determine the ordered positions’ weights \({\omega _k}\), k = 1, 2, …, q;

Step 2: The LIFHHWA or LIFHHWGM operator is used to calculate the comprehensive LIFV matrix \(\tilde {S}={({\tilde {s}_{ij}})_{m \times n}}\), where \({\tilde {s}_{ij}}=({s_{{\mu _{ij}}}},{s_{{v_{ij}}}})\) for all i = 1, 2, …, m; j = 1, 2, …, n;

Step 3: Eq. (20) is applied to determine the criteria’s weights \({w_{{c_j}}}\), j = 1, 2, …, n. Furthermore, Eq. (21) is adopted to determine the ordered positions’ weights \({\omega _{^{j}}}\), j = 1, 2, …, n;

Step 4: The LIFHHWA or LIFHHWGM operator is adopted to calculate the comprehensive LIFVs \({\tilde {s}_i}=({s_{{\alpha _i}}},{s_{{\beta _i}}})\), i = 1, 2, …, m;

Step 5: The median and deviation for LIFVs are applied to rank the comprehensive LIFVs \({\tilde {s}_i}\), i = 1, 2, …, m, and then the ranking of alternatives a1, a2, …, a m is obtained.

5 A case study

A shipyard plans to select a steel supplier. After preliminary screening, four steel companies are selected as the possible suppliers, namely A = {a1, a2, a3, a4}. To select the best one, three DMs E = {e1, e2, e3} are invited to compare these four steel companies following the criteria: c1: quality of products, c2: scale, c3: reputation, c4: price, and c5: logistics speed. To avoid influencing each other, the DMs are required to provide their preferences in anonymity. Let S = {s0: extremely bad, s1: too bad, s2: very bad, s3: bad, s4: a little bad, s5: fair, s6: a little good, s7: good, s8: very good, s9: too good, s10: extremely good} be the given predefined linguistic term set. The DMs can offer their preferred and non-preferred qualitative judgements simultaneously. Suppose that the individual LIFV matrices are offered as shown in Tables 1, 2 and 3:

Table 1

Individual LIFV matrix \({\tilde {S}^1}\) offered by the DM e1

 

c 1

c 2

c 3

c 4

c 5

a 1

(s5, s4)

(s5, s3)

(s6, s2)

(s4, s5)

(s6, s3)

a 2

(s7, s2)

(s3, s6)

(s5, s3)

(s3, s6)

(s5, s4)

a 3

(s6, s3)

(s4, s4)

(s4, s5)

(s5, s4)

(s7, s2)

a 4

(s4, s4)

(s6, s3)

(s3, s6)

(s7, s2)

(s5, s4)

Table 2

Individual LIFV matrix \({\tilde {S}^2}\) offered by the DM e2

 

c 1

c 2

c 3

c 4

c 5

a 1

(s6, s2)

(s4, s5)

(s6, s3)

(s5, s3)

(s6, s2)

a 2

(s5, s3)

(s3, s6)

(s5, s4)

(s3, s6)

(s5, s3)

a 3

(s4, s5)

(s5, s4)

(s7, s2)

(s4, s4)

(s4, s5)

a 4

(s3, s6)

(s7, s2)

(s5, s4)

(s6, s3)

(s3, s6)

Table 3

Individual LIFV matrix \({\tilde {S}^3}\) offered by the DM e3

 

c 1

c 2

c 3

c 4

c 5

a 1

(s3, s6)

(s7, s2)

(s5, s4)

(s4, s4)

(s4, s5)

a 2

(s4, s5)

(s5, s4)

(s7, s2)

(s5, s4)

(s6, s3)

a 3

(s7, s2)

(s4, s4)

(s4, s5)

(s5, s4)

(s7, s2)

a 4

(s5, s4)

(s6, s3)

(s3, s6)

(s3, s6)

(s7, s2)

To rank these four steel companies and select the best one, the following procedure is offered:

Step 1: Following Eq. (18), the weights of the DMs are \({w_{{e_1}}}=0.34,{w_{{e_2}}}=0.33,{w_{{e_3}}}=0.33\). Furthermore, the individual weighted LIFV matrix \({\tilde {T}^k}={(\tilde {t}_{{ij}}^{k})_{4 \times 5}}\) is derived, where \(\tilde {t}_{{ij}}^{k}={w_{{e_k}}}\tilde {s}_{{ij}}^{k}=({s_{\mu _{{ij}}^{k}}},{s_{v_{{ij}}^{k}}})\) for all i = 1, 2, 3, 4; j = 1, 2, 3, 4, 5; k = 1, 2, 3. Taking the individual LIFV matrix \({\tilde {S}^1}\), for example, the weighted LIFV matrix \({\tilde {T}^1}={(\tilde {t}_{{ij}}^{1})_{4 \times 5}}\) is listed as follows:
$${\tilde {T}^1}=\left( {\begin{array}{*{20}{c}} {({s_{{{2.4776}}}},{s_{{{6.7265}}}})}&{({s_{{{2.4776}}}},{s_{{{5.7501}}}})}&{({s_{{{3.2735}}}},{s_{{{4.5208}}}})}&{({s_{{{1.8157}}}},{s_{{{7.5224}}}})}&{({s_{{{3.2735}}}},{s_{{{5.7501}}}})} \\ {({s_{{{4.2499}}}},{s_{{{4.5208}}}})}&{({s_{{{1.2565}}}},{s_{{{8.1843}}}})}&{({s_{{{2.4776}}}},{s_{{{5.7501}}}})}&{({s_{{{1.2565}}}},{s_{{{8.1843}}}})}&{({s_{{{2.4776}}}},{s_{{{6.7265}}}})} \\ {({s_{{{3.2735}}}},{s_{{{5.7501}}}})}&{({s_{{{1.8157}}}},{s_{{{6.7265}}}})}&{({s_{{{1.8157}}}},{s_{{{7.5224}}}})}&{({s_{{{2.4776}}}},{s_{{{6.7265}}}})}&{({s_{{{4.2499}}}},{s_{{{4.5208}}}})} \\ {({s_{{{1.8157}}}},{s_{{{6.7265}}}})}&{({s_{{{3.2735}}}},{s_{{{5.7501}}}})}&{({s_{{{1.2565}}}},{s_{{{8.1843}}}})}&{({s_{{{4.2499}}}},{s_{{{4.5208}}}})}&{({s_{{{2.4776}}}},{s_{{{6.7265}}}})} \end{array}} \right).$$

Following Eq. (19), the weights of the ordered positions are \({\omega _1}=0.24,{\omega _2}=0.27,{\omega _3}=0.48\).

Step 2: We adopt the LIFHHWA operator with \(\gamma =0.1\) to calculate the comprehensive LIFV matrix \(\tilde {S}={({\tilde {s}_{ij}})_{4 \times 5}}\), where
$$\tilde {S}=\left( {\begin{array}{*{20}{c}} {({s_{{{4.8958}}}},{s_{{{3.5560}}}})}&{({s_{{{6.0575}}}},{s_{{{2.6647}}}})}&{({s_{{{5.7957}}}},{s_{{{2.5576}}}})}&{({s_{{{4.5230}}}},{s_{{{3.6187}}}})}&{({s_{{{5.5944}}}},{s_{{{3.0052}}}})} \\ {({s_{{{6.0817}}}},{s_{{{2.6501}}}})}&{({s_{{{4.1161}}}},{s_{{{4.8554}}}})}&{({s_{{{6.1863}}}},{s_{{{2.5708}}}})}&{({s_{{{4.1161}}}},{s_{{{4.8554}}}})}&{({s_{{{5.5315}}}},{s_{{{3.2039}}}})} \\ {({s_{{{6.1789}}}},{s_{{{2.6647}}}})}&{({s_{{{4.2706}}}},{s_{{{4.0000}}}})}&{({s_{{{5.1056}}}},{s_{{{3.7702}}}})}&{({s_{{{4.5606}}}},{s_{{{4.0000}}}})}&{({s_{{{6.0027}}}},{s_{{{2.8843}}}})} \\ {({s_{{{3.9722}}}},{s_{{{4.3594}}}})}&{({s_{{{6.2878}}}},{s_{{{2.6936}}}})}&{({s_{{{3.5993}}}},{s_{{{5.3761}}}})}&{({s_{{{6.1949}}}},{s_{{{2.7174}}}})}&{({s_{{{5.9580}}}},{s_{{{2.9288}}}})} \end{array}} \right).$$
Step 3: On the basis of Eq. (20), the weights of the criteria are \({w_{{c_1}}}=0.21,{w_{{c_2}}}=0.25,{w_{{c_3}}}=0.27,{w_{{c_4}}}=0.21,{w_{{c_5}}}=0.05\). We calculate the comprehensive weighted LIFV matrix \(\tilde {T}={({\tilde {t}_{ij}})_{4 \times 5}}\) of \(\tilde {S}\), where
$$\tilde {T}=\left( {\begin{array}{*{20}{c}} {({s_{{{1.6355 }}}},{s_{{{7.3617}}}})}&{({s_{{{2.6722}}}},{s_{{{6.1413}}}})}&{({s_{{{2.6315}}}},{s_{{{5.8186}}}})}&{({s_{{{1.4648}}}},{s_{{{7.3832}}}})}&{({s_{{{0.5849}}}},{s_{{{9.0171}}}})} \\ {({s_{{{2.3650}}}},{s_{{{6.5310}}}})}&{({s_{{{1.4595}}}},{s_{{{7.9644}}}})}&{({s_{{{2.9425}}}},{s_{{{5.8339}}}})}&{({s_{{{1.2745}}}},{s_{{{8.2080}}}})}&{({s_{{{0.5718}}}},{s_{{{9.0895}}}})} \\ {({s_{{{2.4356}}}},{s_{{{6.5464}}}})}&{({s_{{{1.5382}}}},{s_{{{7.3722}}}})}&{({s_{{{2.1463}}}},{s_{{{7.0211}}}})}&{({s_{{{1.4833}}}},{s_{{{7.6672}}}})}&{({s_{{{0.6782}}}},{s_{{{8.9694}}}})} \\ {({s_{{{1.1960}}}},{s_{{{7.9328}}}})}&{({s_{{{2.8567}}}},{s_{{{6.1732}}}})}&{({s_{{{1.3026}}}},{s_{{{8.1518}}}})}&{({s_{{{2.4742}}}},{s_{{{6.5687}}}})}&{({s_{{{0.6672}}}},{s_{{{8.9873}}}})} \end{array}} \right)$$

With \({\tilde {t}_{ij}}={w_{{c_j}}}{\tilde {s}_{ij}}=({s_{{\mu _{ij}}}},{s_{{v_{ij}}}})\) for all i = 1, 2, 3, 4; j = 1, 2, 3, 4, 5.

On the basis of Eq. (21), the weights of the ordered positions are \({w_1}=0.11,{w_2}=0.12,{w_3}=0.12,{w_4}=0.34,{w_5}=0.3\).

Step 4: We adopt the LIFHHWA operator with \(\gamma =0.1\) to calculate the comprehensive LIFVs
$${\tilde {s}_1}=\left( {{s_{{{2.3818}}}},{s_{{{6.3093}}}}} \right),\,{\tilde {s}_2}=\left( {{s_{{{2.3323}}}},{s_{{{6.6177}}}}} \right),\,{\tilde {s}_3}=\left( {{s_{{{2.0489}}}},{s_{{{7.0295}}}}} \right),\,{\tilde {s}_4}=\left( {{s_{{{2.2813}}}},{s_{{{6.8146}}}}} \right)$$

Step 5: With respect to comprehensive LIFVs, their medians are \(M({\tilde {s}_1})={{3.0362,}}\;M({\tilde {s}_2})={{2.8573}},\;M({\tilde {s}_3})={{2.5097}},\) \(M({\tilde {s}_4})=2.7333\). Thus, the ranking of alternatives is\({a_1} \succ {a_2} \succ {a_4} \succ {a_3}\), namely the first supplier a1 is the best choice.

The above ranking results are obtained using the LIFHHWA operator with \(\gamma =0.1\). When different values of γ are used, the associated comprehensive LIFVs and ranking orders are offered as shown in Table 4.

Table 4

Ranking results based on the LIFHHWA operator with respect to different values of γ

The values of γ

The comprehensive LIFVs of a1

The comprehensive LIFVs of a2

The comprehensive LIFVs of a3

The comprehensive LIFVs of a4

Ranking orders

γ = 0.1

(s2.3818, s6.3093)

(s2.3323, s6.6177)

(s2.0489, s7.0295)

(s2.2813, s6.8146)

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

γ = 0.2

(s2.3223, s6.4575)

(s2.2816, s6.7468)

(s1.9963, s7.1442)

(s2.2070, s6.9590)

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

γ = 0.5

(s2.1610, s6.7999)

(s2.1522, s7.0340)

(s1.8632, s7.4117)

(s2.0089, s7.3024)

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

γ = 1

(s1.9747, s7.1573)

(s1.9874, s7.3458)

(s1.6909, s7.7152)

(s1.7852, s7.6638)

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

γ = 2

(s1.6953, s7.6255)

(s1.7206, s7.7697)

(s1.4396, s8.1120)

(s1.4883, s8.1033)

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

γ = 5

(s1.2533, s8.2971)

(s1.2852, s8.3874)

(s1.0515, s8.6720)

(s1.0405, s8.7162)

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

γ = 10

(s0.9283, s8.7615)

(s0.9492, s8.8314)

(s0.7689, s9.0522)

(s0.7349, s9.1120)

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

Table 4 shows that different comprehensive LIFVs are obtained with respect to different values of γ, but the same ranking order is obtained.

In this example, when the LIFHHWGM operator with respect to different values of γ is adopted, the associated ranking results are listed as shown in Table 5.

Table 5

Ranking results based on the LIFHHWGM operator with respect to different values of γ

The values of γ

The comprehensive LIFVs of a1

The comprehensive LIFVs of a2

The comprehensive LIFVs of a3

The comprehensive LIFVs of a4

Ranking orders

γ = 0.1

(s8.1822, s1.0167)

(s7.6335, s1.6297)

(s7.9758, s1.2612)

(s7.9469, s1.3005)

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

γ = 0.2

(s8.2363, s0.9988)

(s7.7176, s1.5864)

(s8.0273, s1.2435)

(s8.0171, s1.2748)

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

γ = 0.5

(s8.3742, s0.9517)

(s7.9352, s1.4690)

(s8.1855, s1.1767)

(8.1995, s1.1945)

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

γ = 1

(s8.5830, s0.9061)

(s8.1832, s1.3208)

(s8.3750, s1.0912)

(s8.3991, s1.0995)

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

γ = 2

(s8.7823, s0.8174)

(s8.4744, s1.1403)

(8.6198, s0.9651)

(s8.6485, s0.9634)

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

γ = 5

(s9.0995, s0.6446)

(s8.9050, s0.8480)

(s8.9945, s0.7427)

(s9.0139, s0.7356)

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

γ = 10

(s9.3333, s0.4979)

(s9.2038, s0.6319)

(s9.2600, s0.5655)

(s9.2707, s0.5564)

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

Table 5 shows that different comprehensive LIFVs as well as different ranking orders are obtained with respect to different values of γ, but all of them show that the first supplier a1 is the best choice.

6 Conclusions

To express the qualitative preferred and non-preferred judgements of the DMs, this paper developed a method for group decision making with linguistic intuitionistic fuzzy information. The new method overcomes the following limitations of Zhang et al. method (2017): (1) Zhang et al. method adopted linguistic scale functions that convert linguistic variables into real numbers, in which exists information loss (Herrera and Martínez 2000), while the new method is defined on the continuous linguistic term set that avoids information loss; (2) Zhang et al. method cannot deal with group decision making with linguistic intuitionistic fuzzy information; (3) Zhang et al. method only studied the weights of the criteria, but disregarded the weights of the ordered positions.

The main contributions include: (1) several linguistic intuitionistic fuzzy Hamacher operational laws are defined; (2) a ranking order of LIFVs is offered; (3) two types of linguistic intuitionistic fuzzy Hamacher hybrid aggregation operators are proposed; (4) several of their desirable properties are studied; (5) a method for determining the weights of the DMs, criteria, and the ordered positions is proposed, respectively; (6) an algorithm for group decision making with linguistic intuitionistic fuzzy information is developed; (7) a practical group decision-making problem about selecting the best supplier is offered.

However, this paper only researched group decision making with linguistic intuitionistic fuzzy information based on the Hamacher t-norm and t-conorm. We will continue to study linguistic intuitionistic fuzzy decision making and propose new methods based on other aggregation operators as well as linguistic intuitionistic fuzzy entropy. Furthermore, we will study the application of the new method in some other fields, including the evaluating machine tool, social media, evaluation of professor in a university, and economic production problem.

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (nos. 71571192, and 71671188), the National Social Science Foundation of China (no. 16BJY119), the Innovation-Driven Project of Central South University (no. 2018CX039), the Major Project for National Natural Science Foundation of China (no. 71790615), the State Key Program of National Natural Science of China (no. 71431006), the Projects of Major International Cooperation NSFC (no. 71210003), and the Hunan Province Foundation for Distinguished Young Scholars of China (no. 2016JJ1024).

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of BusinessCentral South UniversityChangshaChina

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