Aggregation operators on cubic linguistic hesitant fuzzy numbers and their application in group decision-making

Abstract

In this paper, we define linguistic hesitant fuzzy sets. We introduce cubic linguistic fuzzy sets and their operational laws. We propose a cubic linguistic hesitant fuzzy sets and their operational laws. We describe aggregation operators for cubic linguistic hesitant fuzzy sets which extended the generalized cubic linguistic hesitant fuzzy averaging (geometric) operator, generalized cubic linguistic hesitant fuzzy weighted averaging (GLCHFWA) operator, generalized cubic linguistic hesitant fuzzy weighted geometric (GCHFWG) operator, generalized cubic linguistic hesitant fuzzy ordered weighted average (GCLHFOWA) operator, generalized cubic linguistic hesitant fuzzy ordered weighted geometric (GCLHFOWG) operator, generalized cubic linguistic hesitant fuzzy hybrid averaging (GCLHFHA) operator and generalized cubic linguistic hesitant fuzzy hybrid geometric (GCLHFHG) operator. Moreover, we relate these aggregation operators to develop an approach to multiple-attribute group decision-making with cubic linguistic-hesitant fuzzy information. Finally, the overall ranking of an alternative is obtained. This example is presented to illustrate the feasibility and effectiveness of the proposed approach.

1 Introduction

In mathematics, fuzzy sets (uncertain sets) are somewhat like sets whose elements have degrees of membership. Fuzzy sets were introduced independently by (Zadeh 1965). Chen (1998) introduced the new method for dealing with fuzzy opinion aggregation in group decision-making problems. Chen et al. (2001) present a new method for generating fuzzy rules from numerical data for handling fuzzy classification problems based on the fuzzy subsethood values between decisions to be made. Wang and Chen (2008) present a new approach for evaluating students’ answer scripts using fuzzy numbers associated with degrees of confidence of the evaluator. Chen and Tanuwijaya (2011) proposed the automatic clustering algorithm to partition the universe of discourse into different lengths of intervals. Chen and Chang (2011) proposed a GA-based weight-learning algorithm to automatically learn the optimal weights of the antecedent variables of the fuzzy rules. Chen et al. (2012) proposed method consider the areas of the positive side, the areas of the negative side and the centroid values of generalized fuzzy numbers as the factors for calculating the ranking scores of generalized fuzzy numbers with different left heights and right heights. Chiclana et al. (2002) based on the geometric mean and the OWA operator. Dubois and Prade (1980) introduced the enthusiasm for a theory that challenges the traditional reliance on two-valued logic and classical set theory as a basis for scientific inquiry. Dyckhoff and Pedrycz (1984) proposed as connective operators for fuzzy set theory which easily allow for modelling the degree of compensation.

In fuzzy set theory, the membership of an element to a fuzzy set is a single value between zero and one. However, in reality, it may not always be true that the degree of non-membership of an element in a fuzzy set is equal to 1 minus the membership degree because there may be some hesitation degree. Therefore, a generalization of fuzzy sets was proposed by Atanassov (1986) as intuitionistic fuzzy sets (IFS) which incorporate the degree of hesitation called hesitation margin (and is defined as 1 minus the sum of membership and non-membership degrees, respectively). The notion of defining an intuitionistic fuzzy set as the generalized fuzzy set is quite interesting and useful in many application areas Lindahl et al. (2010). The hybrid geometric averaging (HGA), the hybrid quadratic averaging (HQA), the generalized ordered weighted averaging (GOWA) operator and the weighted generalized mean (WGM). Liu et al. (2017) introduced the interaction PBM (IFIPBM) operator for intuitionistic fuzzy numbers (IFNs), the weighted interaction PBM (IFWIPBM) operator for IFNs, the interaction PGBM (IFIPGBM) operator for IFNs and the weighted interaction PGBM (IFWIPGBM) operator for IFNs. Peng et al. (2015) based on the developed intuitionistic hesitant fuzzy cross-entropy proposed for solving multi-criteria decision-making (MCDM) problems within an intuitionistic hesitant fuzzy environment.

Chen and Chang (2015) presented a new similarity measure between Atanassov’s intuitionistic fuzzy values. The method proposed by Chen et al. (2016) uses the evidential reasoning methodology to aggregate each decision-maker’s decision matrix and the weights of the attributes to get the aggregated decision matrix of each decision-maker. Liu et al. (2017) proposed the interaction PBM (IFIPBM) operator for intuitionistic fuzzy numbers (IFNs), the weighted interaction PBM (IFWIPBM) operator for IFNs, the interaction PGBM (IFIPGBM) operator for IFNs and the weighted interaction PGBM (IFWIPGBM) operator for IFNs.

These consist of intuitionistic fuzzy sets (IFSs), interval-valued fuzzy sets (IVFSs), hesitant fuzzy sets (HFSs) 2010. Chen et al. (2013) developed approach to GDM based on interval-valued hesitant preference relations in order to consider the differences of opinions between individual decision makers. Jun et al. (2012) introduced the cubic set. Torra (2010) originally provided the idea of HFSs that allows membership grade to be the finite set of possible values between 0 and 1. IVFSs (1975) permits the membership grade of an element to closed subinterval of the [0, 1]. Jun et al. (2012) defined cubic sets which include an IVFs (1975) with the fuzzy set (1965). Chen et al. (2013) presented a structure which generalizes the idea of HFS (2010) to IVHFS that make possible the membership grade of an element into many realizable interval numbers. Xia and Xu (2011) developed a series of aggregation operators for hesitant fuzzy information. Xia et al. (2013) proposed to determine the aggregation weight vectors. Xia and Xu (2013) introduced the hesitant multiplicative set. Xu and Da (2003) introduced the induced ordered weighted geometric averaging (IOWGA) operator, generalized induced ordered weighted averaging (GIOWA) operator, hybrid weighted averaging (HWA) operator. Xu and Yager (2006) develop some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator. Xu (2007), and Xu and Chen (2007) introduced the different intuitionistic fuzzy set. Yager (1988) introduced ordered weighted aggregation (OWA) operator and investigates the properties of this operator. Yager (2004) provided a new class of operators called the generalized OWA (GOWA) operators. Zhang (2013) developed a series of aggregation operators for interval-valued intuitionistic hesitant fuzzy information. Zhao et al. (2010) developed some new generalized aggregation operators, such as generalized intuitionistic fuzzy weighted averaging operator, generalized intuitionistic fuzzy ordered weighted averaging operator, generalized intuitionistic fuzzy hybrid averaging operator, generalized interval-valued intuitionistic fuzzy weighted averaging operator, generalized interval-valued intuitionistic fuzzy ordered weighted averaging operator, and generalized interval-valued intuitionistic fuzzy hybrid average operator. Zhou et al. (2016) introduced the intuitionistic hesitant linguistic hybrid averaging (IHLHA) operator and the intuitionistic hesitant linguistic hybrid geometric (IHLHG) operator.

The elements in HFLSs are called hesitant fuzzy linguistic numbers (HFLNs). That is to say, for one object, an HFLS is reduced to an HFLN, which can be considered as a special case of HFLSs. For example, $\left\langle s_{2},(0.3,0.4,0.5)\right\rangle$, is an HFLN and 0.3, 0.4, and 0.5 are the possible membership degrees to the linguistic term $s_{2}$. HFLSs have enabled great progress in describing linguistic information and to some extent may be considered an innovative construct. The main advantage of HFLSs is that they can describe two fuzzy attributes of an object: a linguistic term and a hesitant fuzzy element (HFE). The former provides an evaluation value, such as “excellent” or “good”. The latter describes the hesitancy for the given evaluation value and denotes the membership degrees associated with the specific linguistic term. Jun et al. (2012) introduced the cubic set, which are the generalizations of fuzzy sets and intuitionistic fuzzy sets.

Despite having a bulk of related literature on the problem under consideration, the following aspects related to cubic linguistic hesitant fuzzy numbers (CLHFNs) and their aggregation operators motivated the researchers to carry it an in-depth inquiry into the current study.

The main advantages of the proposed operators are these aggregation operators provided more accurate and precious result as compared to the above-mentioned operators.

We generalized the concept of cubic linguistic hesitant fuzzy numbers (CLHFNs), intuitionistic linguistic fuzzy sets and introduce the concept of cubic linguistic hesitant fuzzy numbers. If we take only one element in the membership degree of the cubic linguistic hesitant fuzzy number, i.e., instead of the interval we take a fuzzy number, then we get intuitionistic linguistic fuzzy numbers, similarly, if we take membership degree as the fuzzy number and non-membership degree equal to zero, then we get fuzzy numbers.

The objectives of the study include:

Propose cubic linguistic hesitant fuzzy number, operational laws, score value and accuracy value of CLHFNs.

Propose six aggregation operators, namely generalized cubic linguistic hesitant fuzzy weighted averaging operator, generalized cubic linguistic hesitant fuzzy weighted geometric operator, generalized cubic linguistic hesitant fuzzy ordered weighted averaging operator, generalized cubic linguistic hesitant fuzzy ordered weighted geometric operator, generalized cubic linguistic hesitant fuzzy hybrid averaging operator and generalized cubic linguistic hesitant fuzzy hybrid geometric operator.

Establish MADM program approach-based cubic linguistic hesitant fuzzy numbers.

Provide illustrative examples of MADM program.

To testify the application of the developed method, we apply for the cubic linguistic hesitant fuzzy numbers in the decision-making.

The initial decision matrix is composed of LVs. To fully consider the randomness and ambiguity of the linguistic term, we convert LVs into the cubic linguistic hesitant fuzzy numbers, and the decision matrix is transformed into the cubic linguistic hesitant fuzzy matrix.

The operator can fully express the uncertainty of the qualitative concept and cubic linguistic hesitant fuzzy operators can capture the interdependencies among any multiple inputs or attributes by a variable parameter. The aggregation operators can take into account the importance of the attribute weights. Nevertheless, sometimes, for some MAGDM problems, the weights of the attributes are important factors for the decision process.

Moreover, there are many multiple-attribute group decision-making (MAGDM) problems since that the approximations of the attribute ideals are cubic linguistic hesitant fuzzy sets; it is, therefore, necessary to give some aggregation techniques to aggregate the cubic linguistic hesitant fuzzy information. However, we are conscious that the standing aggregation skills struggle in surviving with group decision-making problems with cubic linguistic hesitant fuzzy information. Therefore, we, in the current paper, suggest a sequence of aggregation operators for aggregating the cubic linguistic hesitant fuzzy information and consider some properties of these operators, then based on these aggregation operators, we develop an approach to MAGDM with cubic linguistic hesitant fuzzy information. Moreover, we use a numerical example to show the application of the developed approach.

The best parts of this paper are organized as follows. In Sect. 2, we review the concept and properties of the cubic set, the basic linguistic hesitant fuzzy sets, and linguistic hesitant fuzzy elements briefly. In Sect. 3, we present cubic linguistic fuzzy sets and cubic linguistic fuzzy elements. In Sect. 4, we present cubic linguistic hesitant fuzzy sets and cubic linguistic hesitant fuzzy elements. In Sect. 4, we exhibit of cubic linguistic hesitant fuzzy sets and cubic linguistic hesitant fuzzy elements. In Sect. 5, we exhibit a series of aggregation operators for cubic linguistic hesitant fuzzy information and observe the connections among these aggregation operators. Section 6 develops an approach to group decision-makings with cubic linguistic hesitant fuzzy data. In Sect. 7, the application of the developed approach in group decision-making problems is shown by an illustrative example. In Sect. 8, we discuss in comparison analysis. Finally, we give the conclusions in Sect. 9.

2 Preliminaries

Hence, we recall some basic definitions of fuzzy set theory, intuitionistic fuzzy set and cubic set theory, which we will use throughout in this paper.

Definition 1

(Zadeh 1965) Let H be a universe of discourse. The idea of fuzzy set was presented by Zadeh and defined as follows: $J=\{\ddot{h},\Gamma _{J}(\ddot{h} )|\ddot{h}\in H\}$. A fuzzy set in a set $\ddot{h}$ is defined $\Gamma _{J}:H\rightarrow I,$ is a membership function, $\Gamma _{J}(\ddot{h})$ denoted the degree of membership of the element $\ddot{h}$ to the set H, where $I=[0,1]$. The collection of all fuzzy subsets of H is denoted by $I^{H}$. Define a relation on $I^{H}$ as follows: $(\forall \Gamma ,\eta \in I^{H})(\Gamma \le \eta \Leftrightarrow (\forall \ddot{h}\in \ddot{h} )(\Gamma (\ddot{h})\le \eta (\ddot{h}))).$

Definition 2

(Atanassov and Gargov 1989) By an interval number we mean a closed subinterval $\ddot{h}=[\ddot{h}^{-},\ddot{h}^{+}]$ of I, where $0\le \ddot{h}^{-}\le \ddot{h}^{+}\le 1$. The interval number $[\ddot{h}^{-},\ddot{h}^{+}]$ with $\ddot{h}^{-}=\ddot{h}^{+}$ is denoted by $\ddot{h}$. [I] is the set of all interval numbers. Let us define what is known as elaborate minimum (briefly, $r\min$), by the symbols “$\succeq$”, “$\preceq$” and “$=$” in case of two elements in [I]. Consider two interval numbers $\ddot{h} _{1}=[\ddot{h}_{1},\ddot{h}_{1}]$ and $\ddot{h}_{2}=[\ddot{h}_{2},\ddot{h} _{2}].$ Then $r\min \{\ddot{h}_{1},\ddot{h}_{2}\}=[\min \{\ddot{h}_{1},\ddot{h} _{2}\},\min \{\ddot{h}_{1},\ddot{h}_{2}\}],r\max \{\ddot{h}_{1},\ddot{h} _{2}\}=[\max \{\ddot{h}_{1},\ddot{h}_{2}\},\max \{\ddot{h}_{1},\ddot{h}_{2}\}],\ddot{h}_{1}\succeq \ddot{h}_{2}$ if and only if $\ddot{h}_{1}\succeq \ddot{h}_{2}$ and $\ddot{h}_{1}\succeq \ddot{h}_{2}$ and similarly we may have $\ddot{h}_{1}\preceq \ddot{h}_{2}$ and $\ddot{h}_{1}=\ddot{h}_{2}$. To say $\ddot{h}_{1}\succ \ddot{h}_{2}$ (resp. $\ddot{h}_{1}\preceq \ddot{h}_{2}$ and $\ddot{h}_{1}\ne \ddot{h}_{2}$ ) we mean $\ddot{h}_{1}\succeq \ddot{h}_{2}$ and (resp. $\ddot{h}_{1}\preceq \ddot{h}_{2}$ and $\ddot{h}_{1}\ne \ddot{h}_{2}$). Let $\ddot{h}_{i}\in [I]$ where $i\in \delta$, then $r\inf \left( \ddot{h}\right) =[inf_{i\in \Lambda }\ddot{h}_{i}^{-}, \inf _{i\in \delta }\ddot{h}_{i}^{+}],r\sup \ddot{h}=[\sup _{i\in \delta }\ddot{h}_{i}^{-}, \sup _{i\in \delta }\ddot{h} _{i}^{+}]$.

Definition 3

(Atanassov and Gargov 1989) For a family $\{A_{i}|i\in \delta \}$ of IVF sets in H where $\delta$ is an index set, the union $G=\cup _{i\in \delta }A_{i}$ and the intersection $F=\cap _{i\in \delta }A_{i},$ are defined as follows: $G(\ddot{h})=\left( \mathop {{\displaystyle \bigcup }}\nolimits _{i\in \delta }A_{i}\right) ( \ddot{h})=r\sup _{i\in \delta }A_{i}(\ddot{h})$ and $F(\ddot{h})=\left( \mathop {{\displaystyle \bigcap }}\nolimits _{i\in \delta }A_{i}\right) (\ddot{h})=r\inf _{i\in \delta }A_{i}(\ddot{h})$ for all $\ddot{h}\in H$, respectively. For a point $p\in H$ and $a=[a^{-},a^{+}]\in {[} I]$ with $a^{+}>0$, the IVF set which takes the value a at p and 0 somewhere in $\ddot{h}$ is called an interval-valued fuzzy point (briefly, an IVF point) and is denoted by $a_{p}$ . The set of all IVF points in $\ddot{h}$ is denoted by IVFP(H). For any $a\in {[} I]$ and $\ddot{h}\in H$, the IVF point $a_{\ddot{h}}$ is said to belong to an IVF set A in H, denoted by $a_{\ddot{h}}\in A$, if $A( \ddot{h})\succeq a$. It can be easily shown that $A=\cup \{a_{\ddot{h}}|a_{ \ddot{h}}\in A\}$.

Definition 4

(Atanassov and Gargov 1989) An Atanassov intuitionistic fuzzy set on H is a set $A=\cup \{a_{\ddot{h}}|a_{\ddot{h}}\in H\}$. The membership and non-membership function, $\Gamma _{J}$ and $\eta _{j}$, are given by, respectively, $\Gamma _{J}(\ddot{h}):\ddot{h}\rightarrow {[} 0,1],\ddot{h} \in H\rightarrow \ \Gamma _{J}(\ddot{h})\in {[} 0,1];\eta _{J}(\ddot{h}): \ddot{h}\rightarrow {[} 0,1],\ddot{h}\in H\rightarrow \ \eta _{J}(\ddot{h })\in {[} 0,1]\$and $0\le \Gamma _{J}(\ddot{h})+\eta _{J}(\ddot{h} )\le 1$ for all $\ddot{h}\in H.\pi _{J}(\ddot{h})=1-\Gamma _{J}(\ddot{ h})-\eta _{J}(\ddot{h}).$

Definition 5

(Jun et al. 2012) Let H be a nonempty set. By a cubic set in H we mean a structure $F=\{\ddot{h},\alpha (\ddot{h}),\beta (\ddot{h}):\ddot{h}\in H\}$ in which $\alpha$ is an IVF set in H and $\beta$ is a fuzzy set in H. A cubic set $F=\{\ddot{h},\alpha (\ddot{h}),\beta (\ddot{h}):\ddot{h}\in H\}$ is simply denoted by $F=\langle \alpha ,\beta \rangle .$ The collection of all cubic sets in $\ddot{h}$ is denoted by $C^{H}.$ A cubic set $F=\langle \alpha ,\beta \rangle$ in which $\alpha (\ddot{h})=0$ And $\beta ( \ddot{h})=1$ (resp. $\alpha (\ddot{h})=1$ And $\beta (\ddot{h})=0$ for all $\ddot{h}\in H$ is denoted by 0 (resp. 1). A cubic set $D=\langle \lambda ,\xi \rangle$ in which $\lambda (\ddot{h})=0$ and $\xi (\ddot{h})=0$ (resp.$\lambda (\ddot{h})=1$ and $\xi (\ddot{h})=1$) for all $\ddot{h}\in H$ is denoted by 0 (resp. 1).

Definition 6

(Jun et al. 2012) Let H be a non empty set. A cubic set $F=(C,\lambda )$ in H is said to be an internal cubic set if $C^{-}(\ddot{h})\le \lambda ( \ddot{h})\le C^{+}(\ddot{h})$ for all $\ddot{h}\in H.$

Definition 7

(Jun et al. 2012) Let H be a non empty set. A cubic set $F=(C,\lambda )$ in H is said to be an external cubic set if $\lambda (\ddot{h})\notin (C^{-}(\ddot{h}),C^{+}(\ddot{h}))$ for all $\ddot{h}\in H.$

2.1 Linguistic hesitant fuzzy sets

In this subsection, linguistic hesitant fuzzy sets and some results are studied.

Definition 8

Let X be a fixed set, a linguistic hesitant fuzzy set (LHFS) on X is given in terms of a function that when applied to X returns a subset of [0, 1]. We express the LHFS by a mathematical as follows: $E=\{s_{\theta (x))},\ddot{h}_{E}(x)|x\in X\},$ where $[s_{\theta (x))},\ddot{h}_{E}(x)]$ is a set of some values in [0, 1], denoting the possible hesitant fuzzy set of the element $x\in X$ to the set E. For convenience, called $\ddot{h}= \ddot{h}_{E}(x)$ a hesitant fuzzy element (HFE) and H the set of all HFEs. Given three LHFEs represented by $\ddot{h},\ddot{h}_{1}$ and $\ddot{h}_{2}$, defined some operations on them, which can be described as follows:

$$\begin{aligned} \ddot{h}^{c}= & \, \left\{ s_{\theta (x))},1-{\tilde{\Omega }}|{\tilde{\Omega }}\in \ddot{h} \right\} ,\\ \ddot{h}_{1}\cup \ddot{h}_{2}= & \, \left\{ s_{\theta (x_{1})\cup \theta (x_{2})}, {\tilde{\Omega }}_{1}\vee {\tilde{\Omega }}_{2}|{\tilde{\Omega }}_{1}\in \ddot{h}_{1}, {\tilde{\Omega }}_{2}\in \ddot{h}_{2}\right\} ,\\ \ddot{h}_{1}\cap \ddot{h}_{2}= & \, \left\{ s_{\theta (x_{1})\cap \theta (x_{2})}, {\tilde{\Omega }}_{1}\wedge {\tilde{\Omega }}_{2}|{\tilde{\Omega }}_{1}\in \ddot{h} _{1},{\tilde{\Omega }}_{2}\in \ddot{h}_{2}\right\} . \end{aligned}$$

Furthermore, to aggregate linguistic hesitant fuzzy information, we defined some new operations on the LHFEs $\ddot{h},\ddot{h}_{1}$ and $\ddot{h }_{2}$ as follows.

Example 1

Let $\ddot{h}=\{s_{2},0.4\},\ddot{h}_{1}=[s_{1},0.5]$ and $\ddot{h} _{2}=[s_{3},0.5]$ be any three LFNs.

$$\begin{aligned} \ddot{h}^{c}= & \, \left\{ s_{2},1-0.4|{\tilde{\Omega }}\in \ddot{h}\right\} =\left\{ s_{2},0.6|{\tilde{\Omega }}\in \ddot{h}\right\} ,\\ \ddot{h}_{1}\cup \ddot{h}_{2}= & \, \left\{ s_{1\cup 3},0.5\cup 0.5|{\tilde{\Omega }} _{1}\in \ddot{h}_{1},{\tilde{\Omega }}_{2}\in \ddot{h}_{2}\right\} \\= & \, \left\{ s_{1\cup 3},0.5| {\tilde{\Omega }}_{1}\in \ddot{h}_{1},{\tilde{\Omega }}_{2}\in \ddot{h}_{2}\right\} ,\\ \ddot{h}_{1}\cap \ddot{h}_{2}= & \, \left\{ s_{1\cap 3},0.5\cap 0.5|{\tilde{\Omega }} _{1}\in \ddot{h}_{1},{\tilde{\Omega }}_{2}\in \ddot{h}_{2}\right\} \\= & \, \left\{ s_{1\cap 3},0.5| {\tilde{\Omega }}_{1}\in \ddot{h}_{1},{\tilde{\Omega }}_{2}\in \ddot{h}_{2}\right\} . \end{aligned}$$

Definition 9

For an LHFE $\ddot{h}$, $s(\ddot{h})=$ $\mathop {{\sum }}\nolimits _{{\tilde{\Omega }}\in \ddot{h}}s_{\theta (x)}{\tilde{\Omega }}/\#\ddot{h}$ is called the score function of $\ddot{h}$, where $\#\ddot{h}$ is the number of the elements in $\ddot{h}$. For two LHFEs $\ddot{h}_{1}$ and $\ddot{h}_{2}$, if $s(\ddot{h} _{1})>s(\ddot{h}_{2})$, then $\ddot{h}_{1}>\ddot{h}_{2}$; if $s(\ddot{h} _{1})=s(\ddot{h}_{2})$, then $\ddot{h}_{1}=\ddot{h}_{2}$. If $s_{\theta (x_{1})}>s_{\theta (x_{2})},$ then $s_{1}>s_{2}.$

3 Cubic linguistic fuzzy sets

In this subsection, we developed cubic linguistic fuzzy sets and basic results are defined.

Definition 10

Let X be an ordinary nonempty set. The cubic linguistic fuzzy set A in X is an object that has the form $A=\langle \{s_{\theta (x)},[u_{A}^{-}(x),u_{A}^{+}(x)],v_{A}(x))]|x\in X)\}\rangle$ where $s_{\theta (x)}:X\rightarrow {[} 0;1],$ $\langle {[} u_{A}^{-}:X\rightarrow {[} 0;1],u_{A}^{+}:X\rightarrow {[} 0;1]]$ and $v_{A}:X\rightarrow {[} 0;1]\rangle .$ For CLS $A=\langle \{x,s_{\theta (x)}[u_{A}^{-}(x),u_{A}^{+}(x)],v_{A}(x))]|x\in X)\}\rangle$ and are, respectively, called the IVLFS and the linguistic fuzzy set of the element $x\in U$ to A. The pair $\langle \{x,s_{\theta (x)},[u_{A}^{-}(x),u_{A}^{+}(x)],v_{A}(x))]|x\in X)\}\rangle$ in A is the cubic linguistic fuzzy number and for convenience each CFN can be simply denoted by $\alpha = \langle s_{\theta (x)},[u_{\alpha }^{-},u_{\alpha }^{+}],v_{\alpha }\rangle$ where $s_{\theta (x)}=s_{\theta (x)}\subset {[} 0,1],$ $u_{A} =[u_{A}^{-},u_{A}^{+}]\subset {[} 0,1],$ $v_{A}=[v_{A}]\subset {[} 0,1].$ Let A be the set of all CLFNs.

Definition 11

Let $h=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} u^{-}, \\ u^{+}], \\ v\rangle \end{array} \right\} ,h_{1}=\left\{ \begin{array}{c} s_{\theta (h_{1})}, \\ \langle {[} u_{1}^{-}, \\ u_{1}^{+}], \\ v_{1}\rangle \end{array} \right\}$ and $h_{2}=\left\{ \begin{array}{c} s_{\theta (h_{2})}, \\ \langle {[} u_{2}^{-}, \\ u_{2}^{+}], \\ v_{2}\rangle \end{array} \right\}$ be any three CLFNs. Then $\alpha ^{c}=\langle v_{\alpha },s_{\theta (\alpha )},[u_{\alpha }^{-},u_{\alpha }^{+}]\rangle ,$

$$\begin{aligned} \alpha _{1}\cup \alpha _{2}= & \, \left\langle s_{\theta (\alpha _{1})\cup }{}_{\theta (\alpha _{2})},\max \left[ u_{\alpha _{1}}^{-}\cup u_{\alpha _{2}}^{-},u_{\alpha _{1}}^{+}\cup u_{\alpha _{2}}^{+}\right] ,\min \left[ v_{\alpha _{1}}\cap v_{\alpha _{2}}\right] \right\rangle ,\\ \alpha _{1}\cap \alpha _{2}= & \, \left\langle s_{\theta (\alpha _{1})\cap \theta (\alpha _{2})},\min \left[ u_{\alpha _{1}}^{-}\cap u_{\alpha _{2}}^{-},u_{\alpha _{1}}^{+}\cap u_{\alpha _{2}}^{+}\right] ,\max \left[ v_{\alpha _{1}}\cup v_{\alpha _{2}}\right] \right\rangle ,\\ \alpha _{1}\oplus \alpha _{2}= & \, \left\{ \left\langle s_{\theta (\alpha _{1})+\theta (\alpha _{2})},\left[ u_{\alpha _{1}}^{-}+u_{\alpha _{2}}^{-}-u_{\alpha _{1}}^{-}u_{\alpha _{2}}^{-}\right] ,\left[ u_{\alpha _{1}}^{+}+u_{\alpha _{2}}^{+}-u_{\alpha _{1}}^{+}u_{\alpha _{2}}^{+}\right] ,v_{\alpha _{1}}v_{\alpha _{2}}\right\rangle \right\} \\ \alpha _{1}\otimes \alpha _{2}= & \, \left\{ \left\langle s_{\theta (a_{1})\times \theta (a_{2})},\left[ u_{\alpha _{1}}^{-}u_{\alpha _{2}}^{-}\right] ,\left[ u_{\alpha _{1}}^{+}u_{\alpha _{2}}^{+}\right] ,\left[ v_{\alpha _{1}}+v_{\alpha _{2}}-v_{\alpha _{1}}v_{\alpha _{2}}\right] \right\rangle \right\} ,\\ \lambda \alpha= & \, \left\{ \left\langle s_{\lambda \times \theta (a)},[1-(1-u_{\alpha }^{-})^{\lambda }],[1-(1-u_{\alpha }^{+})^{\lambda }],(\upsilon _{\alpha })^{\lambda }]\right\rangle \right\} ,\\ \alpha ^{\lambda }= & \, \left\{ \left\langle s_{\theta ^{\lambda }(a)},[(u_{\alpha }^{-})^{\lambda },(u_{\alpha }^{+})^{\lambda }],[1-(1-\upsilon _{\alpha })^{\lambda }]\right\rangle \right\} . \quad \lambda >0. \end{aligned}$$

Example 2

Let $\alpha =$ $\left\langle \begin{array}{c} s_{2},[0.3, \\ 0.5], \\ 0.4 \end{array} \right\rangle ,$ $\alpha _{1}=$ $\left\langle \begin{array}{c} s_{2},[0.2, \\ 0.4], \\ 0.3 \end{array} \right\rangle$ and $\alpha _{2}=$ $\left\langle \begin{array}{c} s_{3},[0.1, \\ 0.4], \\ 0.2, \end{array} \right\rangle$ be any three CLFNs. Then

$$\begin{aligned} \alpha ^{c}= & \, \left\langle 0.6,s_{2},[0.7,0.5]\right\rangle ,\\ \alpha _{1}\cup \alpha _{2}= & \, \left\langle s_{2\cup }{}_{2},[0.2\cup 0.1,0.4\cup 0.4],0.3\cap 0.2\right\rangle ,\\ \alpha _{1}\cap \alpha _{2}= & \, \left\langle s_{2\cup }{}_{2},[0.2\cap 0.1,0.4\cap 0.4],0.3\cup 0.2\right\rangle ,\\ \alpha _{1}\oplus \alpha _{2}= & \, \left\{ \begin{array}{l} \Big \langle s_{2+2},[0.2+0.1-(0.2)(0.1), \\ 0.4+0.4-(0.4)(0.4)],(0.3)(0.2)\Big \rangle \\ =\left\langle s_{4},[0.28,0.64],0.06\right\rangle \end{array} \right\} ,\\ \alpha _{1}\otimes \alpha _{2}= & \, \left\{ \begin{array}{l} \Big \langle s_{2\times 2},[(0.2)(0.1),(0.4)(0.4)], \\ 0.3+0.2-(0.3)(0.2)\Big \rangle =\Big \langle s_{4},[(0.02,0.16],0.44\Big \rangle \end{array} \right\} ,\\ \lambda= & \, 0.25,0.25,0.25,0.25\\ \lambda \alpha= & \, \left\{ \begin{array}{l} \Big \langle s_{0.25\times 2},[1-(1-0.3)^{0.25}],[1-(1-0.5)^{0.25}],(0.4)^{0.25}]\Big \rangle \\ =\Big \langle s_{0.5},[0.0853,0.1591],0.7952]\Big \rangle \end{array} \right\} ,\\ \alpha ^{\lambda }= & \, \left\{ \begin{array}{l} \Big \langle s_{2^{0.25}},[(0.3)^{0.25},(0.5)^{0.25}],1-(1-0.4)^{0.25}]\Big \rangle \\ =\Big \langle s_{1.1892},[(0.7400,0.8408],0.1198]\Big \rangle \end{array} \right\} ,\quad \lambda >0.\\ \end{aligned}$$

Definition 12

Let $\alpha =\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} u^{-}, \\ u^{+}], \\ v\rangle \end{array} \right\}$ be any CLFNs. Then the score function is $S(\alpha )=\langle \frac{s_{\theta (\alpha )}\{[u_{\alpha }^{-}+u_{\alpha }^{+}]-v_{\alpha }\}}{ 3}\rangle .$

Theorem 1

Let $\alpha =\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} u^{-}, \\ u^{+}], \\ v\rangle \end{array} \right\} ,\alpha _{1}=\left\{ \begin{array}{c} s_{\theta (h_{1})}, \\ \langle {[} u_{1}^{-}, \\ u_{1}^{+}], \\ v_{1}\rangle \end{array} \right\}$ and $\alpha _{2}=\left\{ \begin{array}{c} s_{\theta (h_{2})}, \\ \langle {[} u_{2}^{-}, \\ u_{2}^{+}], \\ v_{2}\rangle \end{array} \right\}$ be any three CLFNs. Then the score function $S(\alpha )=\langle \frac{s_{\theta (\alpha )}\{[u_{\alpha }^{-}+u_{\alpha }^{+}]-v_{\alpha }\}}{ 3}\rangle ,$ accuracy function $h(\alpha )=\langle \frac{s_{\theta (\alpha )}\{[u_{\alpha }^{-}+u_{\alpha }^{+}]+v_{\alpha }\}}{3}\rangle$ gave an order relation between two CLFNs $\alpha _{1}$ and $\alpha _{2}.$

• 1. If $s(\alpha _{1})>s(\alpha _{2})$, then $s(\alpha _{1})>s(\alpha _{1})$.

• 2. If $s(\alpha _{1})=s(\alpha _{2}),$ then the following hold.

• (a) If $\ddot{h}(\alpha _{1})>\ddot{h}(\alpha _{2})$, then $\alpha _{1}>\alpha _{1}$.

• (b) If $\ddot{h}(\alpha _{1})=\ddot{h}(\alpha _{2})$, then $\alpha _{1}=\alpha _{2}$.

• (c) If $\ddot{h}(\alpha _{1})>\ddot{h}(\alpha _{1})$, then $\alpha _{1}>\alpha _{2}$.

4 Cubic linguistic hesitant fuzzy sets

In the following, we exhibit the concept of cubic linguistic hesitant fuzzy sets, which certification the interval value linguistic hesitant fuzzy set the element to be a set of certain thinkable cubic linguistic fuzzy numbers. The motivation is that when essential the interval value linguistic hesitant fuzzy set of an element, but because we have some possible cubic linguistic fuzzy numbers.

Definition 13

Let X be a fixed set and $c=\{x,s_{\theta (\alpha )},\ddot{h}_{c}(x)|x\in X\},$ where $\ddot{h}_{c}(x)=\langle s_{\theta (\alpha )},[\Gamma _{\alpha }^{-},\Gamma _{\alpha }^{+}],\Gamma _{\alpha }\rangle$ denoting the interval-value linguistic hesitant fuzzy set and linguistic hesitant fuzzy set $\ddot{h}=\ddot{h}_{c}(x)$ a cubic linguistic hesitant fuzzy element. If $\alpha$ $\in \ddot{h}$, then $\alpha =\langle s_{\theta (\alpha )},[\Gamma _{\alpha }^{-},\Gamma _{\alpha }^{+}],\Gamma _{\alpha }\rangle =\langle s_{\theta (\alpha )},[\Gamma _{\alpha }^{-},\Gamma _{\alpha }^{+}],\Gamma _{\alpha }\rangle .$

Definition 14

Let X be a fixed set and is cubic linguistic hesitant fuzzy set on X is given in terms of a function that when applied to X returns a subset of $\Omega$. To be effectively comprehended, we express the CLHFS by a mathematical symbol $E=\{x,\ddot{h}_{E}(x)|x\in X\},$ where $s_{E(x)}\ddot{h}_{E}(x)$ is a set of some CHFNs in $\Omega$, denoting the possible intervals and non-membership of the element $x\in X$ to the set E. For convenience, we call $\ddot{h}=s_{E(x)}\ddot{h}_{E}(x)$ cubic linguistic hesitant fuzzy element and $\ddot{h}$ the set of all CLHFEs. If $\alpha$ $\in \ddot{h}$, then $\alpha$ is CHFN and it is given by $\alpha =\langle s_{\theta (\alpha )},[\Gamma _{\alpha }^{-},\Gamma _{\alpha }^{+}],\Gamma _{\alpha }\rangle =\langle s_{\theta (\alpha )},[\Gamma _{\alpha }^{-},\Gamma _{\alpha }^{+}],\Gamma _{\alpha }\rangle .$

For any $\alpha$ $\in \ddot{h}$, if $\alpha$ is a real number in [0, 1], then $\ddot{h}$ reduces to a linguistic hesitant fuzzy element. If $\alpha$ $\in \ddot{h}$ is a cubic hesitant fuzzy number, then $\ddot{h}$ reduces to cubic linguistic hesitant fuzzy element. Therefore, LHFEs and CLHFEs are the special cases of CLHFEs.

Definition 15

Let $h=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\} ,h_{1}=\left\{ \begin{array}{c} s_{\theta (h_{1})}, \\ \langle {[} \Gamma _{1}^{-}, \\ \Gamma _{1}^{+}], \\ \Gamma _{1}\rangle \end{array} \right\}$ and $h_{2}=\left\{ \begin{array}{c} s_{\theta (h_{2})}, \\ \langle {[} \Gamma _{2}^{-}, \\ \Gamma _{2}^{+}], \\ \Gamma _{2}\rangle \end{array} \right\}$ be three CLHFEs. Then we have

$$\begin{aligned}&\ddot{h}^{c}=\{\alpha ^{c}|\alpha \in \ddot{h}\}=\left\langle v_{\alpha },[u_{\alpha }^{-},u_{\alpha }^{+}],s_{\theta (\alpha )}\right\rangle ,\\&\ddot{h}_{1}\cup \ddot{h}_{2}=\alpha _{1}\cup \alpha _{2}\\&\quad =\left\langle s_{\theta (\alpha _{1})\cup }{}_{\theta (\alpha _{2})},\max \left[ u_{\alpha _{1}}^{-}\cup u_{\alpha _{2}}^{-},u_{\alpha _{1}}^{+}\cup u_{\alpha _{2}}^{+}\right] ,\min \left[ v_{\alpha _{1}}\cap v_{\alpha _{2}}\right] \right\rangle ,\\&\ddot{h}_{1}\cap \ddot{h}_{2}=\alpha _{1}\cap \alpha _{2}\\&\quad =\left\langle s_{\theta (\alpha _{1})\cap \theta (\alpha _{2})}{},\min \left[ u_{\alpha _{1}}^{-}\cap u_{\alpha _{2}}^{-},u_{\alpha _{1}}^{+}\cap u_{\alpha _{2}}^{+}\right] ,\max \left[ v_{\alpha _{1}}\cup v_{\alpha _{2}}\right] \right\rangle ,\\&\ddot{h}_{1}\oplus \ddot{h}_{2}=\alpha _{1}\oplus \alpha _{2}\\&\quad =\left\{ \begin{array}{l} \Bigg \langle s_{\theta (\alpha _{1})+\theta (\alpha _{2})},\mathop {{\displaystyle \bigcup }}\limits _{\mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-},}\left[ u_{\alpha _{1}}^{-}+u_{\alpha _{2}}^{-}-u_{\alpha _{1}}^{-}u_{\alpha _{2}}^{-}\right] , \\ \mathop {{\displaystyle \bigcup }}\limits _{\mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+}}\left[ u_{\alpha _{1}}^{+}+u_{\alpha _{2}}^{+}-u_{\alpha _{1}}^{+}u_{\alpha _{2}}^{+}\right] ,\mathop {{\displaystyle \bigcup }}\limits _{v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2}}v_{\alpha _{1}}v_{\alpha _{2}}\Bigg \rangle \end{array} \right\} ,\\&\ddot{h}_{1}\otimes \ddot{h}_{2}=\alpha _{1}\otimes \alpha _{2}\\&\quad =\left\{ \begin{array}{l} \Bigg \langle s_{\theta (a_{1})\times \theta (a_{2})},\mathop {{\displaystyle \bigcup }}\limits _{\mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-},}[u_{\alpha _{1}}^{-}u_{\alpha _{2}}^{-}], \\ \mathop {{\displaystyle \bigcup }}\limits _{\mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+}}\left[ u_{\alpha _{1}}^{+}u_{\alpha _{2}}^{+}\right] ,\mathop {{\displaystyle \bigcup }}\limits _{v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2}}\left[ v_{\alpha _{1}}+v_{\alpha _{2}}-v_{\alpha _{1}}v_{\alpha _{2}}\right] \Bigg \rangle \end{array} \right\} ,\\&\lambda \alpha =\left\{ \begin{array}{l} \Bigg \langle s_{\lambda \times \theta (a)},\mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-}}\left[ 1-(1-u_{\alpha }^{-})^{\lambda }\right] , \\ \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{+}\in \Gamma ^{+}}\left[ 1-(1-u_{\alpha }^{+})^{\lambda }\right] ,\mathop {{\displaystyle \bigcup }}\limits _{\upsilon \in \Gamma }\left[ (\upsilon _{\alpha })^{\lambda }\right] \Bigg \rangle \end{array} \right\} ,\\&\alpha ^{\lambda }=\left\{ \begin{array}{l} \Bigg \langle s_{\theta ^{\lambda }(a)},\mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+}}\left[ (u_{\alpha }^{-})^{\lambda },(u_{\alpha }^{+})^{\lambda }\right] , \\ \mathop {{\displaystyle \bigcup }}\limits _{\upsilon \in \Gamma }\left[ 1-(1-\upsilon _{\alpha })^{\lambda }\right] \Bigg \rangle \end{array} \right\} .\quad \lambda >0. \end{aligned}$$

Example 3

Let $\alpha =$ $\left\langle \begin{array}{c} s_{1},[0.5, \\ 0.7], \\ 0.6 \end{array} \right\rangle ,$ $\alpha _{1}=$ $\left\langle \begin{array}{c} s_{2},[0.4, \\ 0.6], \\ 0.5 \end{array} \right\rangle$ and $\alpha _{2}=$ $\left\langle \begin{array}{c} s_{3},[0.7, \\ 0.9], \\ 0.8 \end{array} \right\rangle$ be any three CLHFNs. Then

$$\begin{aligned} \alpha ^{c}= & \, \left\langle 0.4,s_{1},[0.5,0.3]\right\rangle ,\\ \alpha _{1}\cup \alpha _{2}= & \, \left\langle s_{2\cup }{}_{2},[0.4\cup 0.7,0.6\cup 0.9],0.5\cap 0.8\right\rangle ,\\ \alpha _{1}\cap \alpha _{2}= & \, \left\langle s_{2\cap }{}_{2},[0.4\cap 0.7,0.6\cap 0.9],0.5\cup 0.8\right\rangle ,\\ \alpha _{1}\oplus \alpha _{2}= & \, \left\{ \begin{array}{l} \Big \langle s_{2+2},\left[ 0.4+0.7-(0.4)(0.7),0.6+0.9-(0.6)(0.9)\right] , \\ (0.5)(0.8)\Big \rangle =\Big \langle s_{4},[0.82,0.96],0.4\Big \rangle \end{array} \right\} \\ \alpha _{1}\otimes \alpha _{2}= & \, \left\{ \begin{array}{l} \Big \langle s_{2\times 2},\left[ (0.4)(0.7),(0.6)(0.9)\right] ,0.5+0.8-(0.5)(0.8)\Big \rangle \\ =\Big \langle s_{4},[(0.28,0.54],0.9\Big \rangle \end{array} \right\} ,\\ \lambda= & \, 0.25,0.25,0.25,0.25\\ \lambda \alpha= & \, \left\{ \begin{array}{l} \Big \langle s_{0.25\times 1},[1-(1-0.5)^{0.25}],\Big [1-(1-0.7)^{0.25}\Big ],(0.6)^{0.25}\Big ]\Big \rangle \\ =\Big \langle s_{0.25},[0.1591,0.2599],0.8891]\Big \rangle \end{array} \right\} ,\\ \alpha ^{\lambda }= & \, \left\{ \begin{array}{l} \Big \langle s_{1^{0.25}},\Big [(0.5)^{0.25},(0.7)^{0.25}],1-(1-0.6)^{0.25}\Big ]\Big \rangle \\ =\Big \langle s_{1},[(0.8408,0.9146],0.2047]\Big \rangle \end{array} \right\} ,\quad \lambda >0. \end{aligned}$$

Theorem 2

Let $h=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\} ,h_{1}=\left\{ \begin{array}{c} s_{\theta (h_{1})}, \\ \langle {[} \Gamma _{1}^{-}, \\ \Gamma _{1}^{+}], \\ \Gamma _{1}\rangle \end{array} \right\}$ and $h_{2}=\left\{ \begin{array}{c} s_{\theta (h_{2})}, \\ \langle {[} \Gamma _{2}^{-}, \\ \Gamma _{2}^{+}], \\ \Gamma _{2}\rangle \end{array} \right\}$ be three CLHFEs and $\lambda ,\lambda _{1},\lambda _{2}\ge 0$, then we have that

• 1. $\ddot{h}_{1}\cup \ddot{h}_{2}=(\ddot{h}_{1}\cap \ddot{h}_{2})^{c},$

• 2. $\ddot{h}_{1}\cap \ddot{h}_{2}=(\ddot{h}_{1}\cup \ddot{h}_{2})^{c}$,

• 3. $(\ddot{h}^{c})^{\lambda }=(\lambda \ddot{h})^{c},$

• 4. $\lambda (\ddot{h}^{c})=(\ddot{h}^{\lambda })^{c}$,

• 5. $\ddot{h}_{1}\oplus \ddot{h}_{2}=(\ddot{h}_{1}\otimes \ddot{h}_{2})^{c},$

• 6. $\ddot{h}_{1}+\ddot{h}_{2}=\ddot{h}_{2}+\ddot{h}_{1},$

• 7. $\ddot{h}_{1}\otimes \ddot{h}_{2}=\ddot{h}_{2}\otimes \ddot{h}_{1},$

• 8. $\lambda (\ddot{h}_{1}\oplus \ddot{h}_{2})=\lambda \ddot{h}_{1}\oplus \lambda \ddot{h}_{2};\lambda \ge 0$

• 9. $\lambda _{1}\ddot{h}_{1}+\ddot{h}_{2}a_{1}=(\ddot{h}_{1}+\ddot{h} _{2})a_{1};\lambda _{1},\lambda _{2}\ge 0;$

• 10. $\ddot{h}_{1}\otimes \ddot{h}_{2}=(\ddot{h}_{1})^{\lambda _{1}+\lambda _{2}},\lambda _{1},\lambda _{2}\ge 0;$

• 11. $\ddot{h}_{1}\otimes \ddot{h}_{2}=(\ddot{h}_{1}\otimes \ddot{h} _{2})^{\lambda _{1}},\lambda _{1}\ge 0;$

• 12. $\ddot{h}_{1}\otimes \ddot{h}_{2}=(\ddot{h}_{1}\oplus \ddot{h}_{2})^{c}$ .

Proof

• 1. $\left\{ \begin{array}{l} \ddot{h}_{1}\cup \ddot{h}_{2}={\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}}\Bigg \langle \left\{ v_{\alpha _{1}},[u_{\alpha _{1}}^{-},u_{\alpha _{1}}^{+}]|\alpha _{1}\in \ddot{h}\right\} \cup \\ \left\{ v_{\alpha _{2}},\left[ u_{\alpha _{2}}^{-},u_{\alpha _{2}}^{+}\right] |\alpha _{2}\in \ddot{h}\right\} = \\ {\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}}\{\upsilon _{\alpha _{1}}\cup \upsilon _{\alpha _{2}},[u_{\alpha _{1}}^{-}\cap u_{\alpha _{2}}^{-},u_{\alpha _{1}}^{+}\cap u_{\alpha _{2}}^{+}], \\ s_{\theta (\alpha _{1})\cap \theta (\alpha _{2})}\}=(\ddot{h}_{1}\cap \ddot{h }_{2})^{c} \end{array} \right\} ;$

• 2. $\left\{ \begin{array}{l} \ddot{h}_{1}\cap \ddot{h}_{2}={\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}}\Bigg \langle \left\{ v_{\alpha _{1}},[u_{\alpha _{1}}^{-},u_{\alpha _{1}}^{+}]|\alpha _{1}\in \ddot{h}\right\} \\ \cap \left\{ v_{\alpha _{2}},[u_{\alpha _{2}}^{-},u_{\alpha _{2}}^{+}]|\alpha _{2}\in \ddot{h}\right\} \\ ={\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}}\Bigg \{\upsilon _{\alpha _{1}}\cap \upsilon _{\alpha _{2}},\Bigg [u_{\alpha _{1}}^{-}\cup u_{\alpha _{2}}^{-},u_{\alpha _{1}}^{+}\cup u_{\alpha _{2}}^{+}\Bigg ], \\ s_{\theta (\alpha _{1})\cup \theta (\alpha _{2})}\Bigg \}=(\ddot{h}_{1}\cup \ddot{h }_{2})^{c} \end{array} \right\} ;$

• 3. $\left\{ \begin{array}{l} (\ddot{h}^{c})^{\lambda }={\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }}\Bigg \langle \left\{ v_{\alpha },[u_{\alpha }^{-},u_{\alpha }^{+}],s_{\theta (\alpha )}|\alpha \in \ddot{h}\right\} = \\ {\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }}\Bigg \{1-(1-v_{\alpha })^{\lambda },(u_{\alpha }^{-})^{\lambda },(u_{\alpha }^{+})^{\lambda },s_{(\theta (\alpha ))^{\lambda }}\Bigg \} \\ =(\lambda \ddot{h})^{c} \end{array} \right\} ;$

• 4. $\left\{ \begin{array}{l} \lambda (\ddot{h}^{c})={\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }}\lambda \left\{ v_{\alpha },[u_{\alpha }^{-},u_{\alpha }^{+}],|\alpha \in \ddot{h}\right\} = \\ {\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }}\lambda \{1-(1-v_{\alpha })^{\lambda },\Big [(u_{\alpha }^{-})^{\lambda },(u_{\alpha }^{+})^{\lambda },s_{(\theta (\alpha ))^{\lambda }}\Big ], \\ |\alpha \in \ddot{h}\}=(\ddot{h}^{\lambda })^{c} \end{array} \right\} ;$

• 5. $\left\{ \begin{array}{l} \ddot{h}_{1}\oplus \ddot{h}_{2}={\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}}\Bigg \langle \Bigg \{v_{\alpha _{1}},\Big [u_{\alpha _{1}}^{-},u_{\alpha _{1}}^{+}\Big ], \\ s_{\theta (\alpha _{1})}|\alpha _{1}\in \ddot{h}\Bigg \}\oplus \Bigg \{v_{\alpha _{2}},[u_{\alpha _{2}}^{-},u_{\alpha _{2}}^{+}],s_{\theta (\alpha _{2})}|\alpha _{2}\in \ddot{h}\Bigg \}= \\ {\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}}\Bigg \langle \Bigg \{\left[ v_{\alpha _{1}}+v_{\alpha _{2}}-v_{\alpha _{1}}v_{\alpha _{2}}\right] , \\ {[} u_{\alpha _{1}}^{-}u_{\alpha _{2}}^{-},u_{\alpha _{1}}^{+}u_{\alpha _{2}}^{-}],s_{\theta (\alpha _{1}),}s_{\theta (\alpha _{2}),} \\ |\alpha _{1}\in \ddot{h},\alpha _{1}\in \ddot{h}\Bigg \}=(\ddot{h}_{1}\otimes \ddot{h}_{2})^{c} \end{array} \right\} ;$

• 6. $\left\{ \begin{array}{l} \ddot{h}_{1}\oplus \ddot{h}_{2}=\Bigg \langle \Bigg \{v_{\alpha _{1}},[u_{\alpha _{1}}^{-},u_{\alpha _{1}}^{+}],s_{\theta (\alpha _{1}),}|\alpha _{1}\in \ddot{h}\Bigg \}\otimes \\ \Bigg \{v_{\alpha _{2}},[u_{\alpha _{2}}^{-},u_{\alpha _{2}}^{+}],s_{\theta (\alpha _{2}),}|\alpha _{2}\in \ddot{h}\Bigg \}=\Bigg \langle \Bigg \{s_{\theta (\alpha _{1})+\theta (\alpha _{2}),} \\ \left[ u_{\alpha _{1}}^{-}+u_{\alpha _{2}}^{-}-u_{\alpha _{1}}^{-}u_{\alpha _{2}}^{-},u_{\alpha _{1}}^{+}+u_{\alpha _{2}}^{+}-u_{\alpha _{1}}^{+}u_{\alpha _{2}}^{+}\right] \left[ v_{\alpha _{1}}v_{\alpha _{2}}\right] \Bigg \}^{c}, \\ |\alpha _{1}\in \ddot{h},\alpha _{1}\in \ddot{h}\}=(\ddot{h}_{1}\oplus \ddot{ h}_{2})^{c} \end{array} \right\} ;$

• 7. $\left\{ \begin{array}{l} \ddot{h}_{1}+\ddot{h}_{2}=\Bigg \{s_{\theta (\alpha _{1})+\theta (\alpha _{2}),}{\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}} \\ \left[ u_{\alpha _{1}}^{-}+u_{\alpha _{2}}^{-}-u_{\alpha _{1}}^{-}u_{\alpha _{2}}^{-},u_{\alpha _{1}}^{+}+u_{\alpha _{2}}^{+}-u_{\alpha _{1}}^{+}u_{\alpha _{2}}^{+}\right] , \\ {[} v_{\alpha _{1}}v_{\alpha _{2}}]\Bigg \}=\Bigg \{s_{\theta (\alpha _{2})+\theta (\alpha _{1}),}{\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}} \\ \left[ u_{\alpha _{2}}^{-}+u_{\alpha _{1}}^{-}-u_{\alpha _{2}}^{-}u_{\alpha _{1}}^{-},u_{\alpha _{2}}^{+}+u_{\alpha _{1}}^{+}-u_{\alpha _{2}}^{+}u_{\alpha _{1}}^{+}\right] , \\ \left[ v_{\alpha _{2}}v_{\alpha _{1}}\right] \Bigg \}=\ddot{h}_{2}+\ddot{h}_{1} \end{array} \right\} ;$

• 8. $\left\{ \begin{array}{l} \lambda (\ddot{h}_{1}\oplus \ddot{h}_{2})=\lambda \Bigg \{s_{\theta (\alpha _{1}),}s_{\theta (\alpha _{2}),}{\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}} \\ \left[ u_{\alpha _{1}}^{-}+u_{\alpha _{2}}^{-}-u_{\alpha _{1}}^{-}u_{\alpha _{2}}^{-},u_{\alpha _{1}}^{+}+u_{\alpha _{2}}^{+}-u_{\alpha _{1}}^{+}u_{\alpha _{2}}^{+}\right] ,\left[ v_{\alpha _{1}}v_{\alpha _{2}}\right] \Bigg \} \\ =\Bigg \{s_{\lambda (\theta (\alpha _{1})+\theta (\alpha _{2}),} \\ {\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}}\Bigg [1-\left( 1-u_{\alpha _{1}}^{-}-u_{\alpha _{2}}^{-}+u_{\alpha _{1}}^{-}u_{\alpha _{2}}^{-}\right) ^{\lambda }, \\ 1-\left( 1-u_{\alpha _{1}}^{+}+u_{\alpha _{2}}^{+}-u_{\alpha _{1}}^{+}u_{\alpha _{2}}^{+}\right) ^{\lambda }\Bigg ],(v_{\alpha _{1}}v_{\alpha _{2}})^{\lambda }\} \\ ={\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}}\Bigg \{\Bigg [1-(1-u_{\alpha _{1}}^{-})^{\lambda }(1-u_{\alpha _{2}}^{-})^{\lambda }, \\ 1-(1-u_{\alpha _{1}}^{+})^{\lambda }(1-u_{\alpha _{2}}^{+})^{\lambda }\Bigg ],(v_{\alpha _{1}})^{\lambda }(v_{\alpha _{2}})^{\lambda }\Bigg \} \end{array} \right\}$

• 9. $\left\{ \begin{array}{l} \lambda \ddot{h}_{1}\oplus \lambda \ddot{h}_{2}=\Bigg \{s_{\theta (\alpha _{1})),}{\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}} \\ {[} 1-(1-u_{\alpha _{1}}^{-})^{\lambda }(1-u_{\alpha _{1}}^{+})^{\lambda },(v_{\alpha _{1}})^{\lambda }\Bigg \}\oplus \\ \left\{ s_{\theta (\alpha _{2}),}\left[ 1-(1-u_{\alpha _{2}}^{-})^{\lambda }(1-u_{\alpha _{2}}^{+})^{\lambda }\right] ,(v_{\alpha _{2}})^{\lambda }\right\} \\ =\Bigg \{s_{(\theta (\alpha _{1}))^{\lambda }}+s_{(\theta (\alpha _{2}))^{\lambda },} \\ {\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-},\mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2}}} \\ {[} 1-(1-u_{\alpha _{1}}^{-})^{\lambda }+1-(1-u_{\alpha _{2}}^{-})^{\lambda }- \\ (1-(1-u_{\alpha _{1}}^{-})^{\lambda }(1-(1-u_{\alpha _{2}}^{-})^{\lambda }, \\ \Bigg [ 1-(1-u_{\alpha _{1}}^{+})^{\lambda }+1-(1-u_{\alpha _{2}}^{+})^{\lambda }-(1-(1-u_{\alpha _{1}}^{+})^{\lambda } \\ (1-(1-u_{\alpha _{2}}^{+})^{\lambda }\Bigg ],(v_{\alpha _{1}})^{\lambda }(v_{\alpha _{2}})^{\lambda }|\alpha _{1}\in \ddot{h},\alpha _{2}\in \ddot{h} \Bigg \}= \\ \left[ s_{(\theta (\alpha _{1}))^{\lambda }}-s_{(\theta (\alpha _{2}))^{\lambda }}\right] ,{\displaystyle \mathop {{\displaystyle \bigcup }}\limits _{\begin{array}{c} \mu _{1}^{-}\in \Gamma _{1}^{-},\mu _{2}^{-}\in \Gamma _{2}^{-}, \\ \mu _{1}^{+}\in \Gamma _{1}^{+},\mu _{2}^{+}\in \Gamma _{2}^{+},v_{1}\in \Gamma _{1},v_{2}\in \Gamma _{2} \end{array}}} \\ \Bigg [ 1-(1-u_{\alpha _{1}}^{-})^{\lambda }(1-u_{\alpha _{2}}^{-})^{\lambda }, \\ (1-(1-u_{\alpha _{1}}^{+})^{\lambda }(1-u_{\alpha _{2}}^{+})^{\lambda }\Bigg ],(v_{\alpha _{1}})^{\lambda }(v_{\alpha _{2}})^{\lambda } \\ |\alpha _{1}\in \ddot{h},\alpha _{2}\in \ddot{h}\}=\lambda \ddot{h} _{1}\oplus \lambda \ddot{h}_{2}. \end{array} \right\}$

Thus, we have that $\lambda (\ddot{h}_{1}\oplus \ddot{h}_{2})=\lambda \ddot{h}_{1}\oplus \lambda \ddot{h}_{2}.$ $\square$

Definition 16

For an CLHFE $\ddot{h}$, $s(\ddot{h})=$ $\mathop {{ \sum }}\nolimits _{{\tilde{\Omega }}\in \ddot{h}}{\tilde{\Omega }}/\#\ddot{h}$ is called the score function of $\ddot{h}$, where $\#\ddot{h}$ is the number of the elements in $\ddot{h}$. For two LHFEs $\ddot{h}_{1}$ and $\ddot{h}_{2}$, if $s(\ddot{h}_{1})>s(\ddot{h}_{2})$ , then $\ddot{h}_{1}>\ddot{h}_{2}$, if $s(\ddot{h}_{1})=s(\ddot{h}_{2})$, then $\ddot{h}_{1}=\ddot{h}_{2}$.

5 Aggregation cubic linguistic hesitant fuzzy information

In the current section, we exhibit a series of operators for aggregating the cubic linguistic hesitant fuzzy information and investigate some desired properties of these operators.

5.1 The GCLHFWA and GCLHFWG operators

Definition 17

Assume $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\} (i=1,2,\ldots ,n)$ are the collections of CLHFEs and let $\ddot{\Psi }=( \ddot{\Psi }_{1},\ddot{\Psi }_{2},\ldots ,\ddot{\Psi }_{n})^{T}$ be the weight vector of CLHFEs $\ddot{h}_{i}(i=1,2,\ldots ,n),$ where $\ddot{\Psi }_{i}\in {[} 0,1]$, $\mathop {{ \sum }}\nolimits _{i=1}^{n}\ddot{\Psi }_{i}=1.$ A generalized cubic linguistic hesitant fuzzy weighted averaging operator is a mapping $\ddot{h}^{n}\rightarrow \ddot{h}$ such that $\hbox {GCLHFWA}_{\lambda }(\ddot{h} _{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})=\left( \mathop {{ \bigoplus }}\nolimits _{i=1}^{n}(\ddot{ \Psi }_{i}\ddot{h}_{i}^{\lambda })\right) ^{\frac{1}{\lambda }}$with $\lambda >0.$

Theorem 3

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ $(i=1,2,\ldots ,n)$ be a collection of CLHFEs. Then the aggregated value is calculated using the GCLHFWA operator is also a CLHFE and

$$\begin{aligned}&\hbox {GCLHFWA} \,(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})\\&\quad =\left\langle \begin{array}{c} \Bigg [ s_{\mathop {{ \prod }}\limits _{i=1}^{n}\theta (\alpha _{i})^{\ddot{\Psi } _{i}}},\mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }\Bigg [\left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{-})^{\lambda })^{\ddot{\Psi }_{i}}\right) ^{\frac{1}{\lambda }}, \\ \left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{+})^{\lambda })^{ \ddot{\Psi }_{i}}\right) ^{\frac{1}{\lambda }}\Bigg ],1-\left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\nu _{\alpha _{i}})^{\lambda })^{\ddot{\Psi } _{i}}\right) ^{\frac{1}{\lambda }} \end{array} \right\rangle . \end{aligned}$$

Definition 18

Assume $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\} (i=1,2,\ldots ,n)$ are the collections of CLHFEs and let $\ddot{\Psi }=( \ddot{\Psi }_{1},\ddot{\Psi }_{2},\ldots ,\ddot{\Psi }_{n})^{T}$ be the weight vector of CLHFEs $\ddot{h}_{i}(i=1,2,\ldots ,n),$ where $\ddot{\Psi }_{i}\in {[} 0,1]$, $\mathop {{ \sum }}\nolimits _{i=1}^{n}\ddot{\Psi }_{i}=1.$ A generalized cubic linguistic hesitant fuzzy weighted geometric operator is a mapping $\ddot{h}^{n}\rightarrow \ddot{h}$ defined as follows:

$$\begin{aligned} \mathrm{GLCHFWG}_{\lambda }(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})=\frac{1}{ \lambda }\left( \mathop {{ \bigotimes }}\limits _{i=1}^{n}(\lambda \ddot{h}_{i})^{\ddot{ \Psi }_{i}}\right) . \end{aligned}$$

Theorem 4

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ $(i=1,2,\ldots ,n)$ be a collection of CLHFEs. Then the aggregated value is calculated using the GCLHFWG operator is also a CLHFE and

$$\begin{aligned}&\hbox {GCLHFWG}\, (\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})\\&\quad =\left\langle \begin{array}{c} \Bigg [ \mathop {{ \prod }}\limits _{i=1}^{n}s_{\theta (\alpha _{i})^{\ddot{\Psi } _{i}}},\mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }\Bigg [1-\left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{-})^{\lambda })^{\ddot{\Psi }_{i}}\right) ^{\frac{1}{\lambda }}, \\ 1-\left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{+})^{\lambda })^{ \ddot{\Psi }_{i}}\right) ^{\frac{1}{\lambda }}\Bigg ],\left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\nu _{\alpha _{i}})^{\lambda })^{\ddot{\Psi } _{i}}\right) ^{\frac{1}{\lambda }} \end{array} \right\rangle . \end{aligned}$$

Example 4

Suppose that $\ddot{h}_{1}=\left\langle \begin{array}{c} \{s_{2},[0.2, \\ 0.5],0.3\}, \\ \{s_{3},[0.4, \\ 0.8],0.6\}, \\ \{s_{1},[0.8, \\ 0.12],0.10\} \end{array} \right\rangle ,\ddot{h}_{2}=\left\langle \begin{array}{c} \{s_{2}[0.6, \\ 0.8],0.7\}, \\ \{s_{1},[0.9, \\ 0.15],0.13\}, \\ \{s_{3}[0.7, \\ 0.17],0.13\} \end{array} \right\rangle$ and $\ddot{h}_{3}=\left\langle \begin{array}{c} \{s_{3},[0.14, \\ 0.29],0.20\}, \\ \{s_{1},[0.23, \\ 0.25],0.24\}, \\ \{s_{2}[0.28, \\ 0.44],0.38\} \end{array} \right\rangle$ are three CLHFEs, and $\mathring{\tau }=\{0.2,0.3,0.5\}$ is their weight vector. Then, by Definition above, we can obtain

$$\begin{aligned}&\left\{ \begin{array}{l} \text {GCLHFWA}(\ddot{h}_{1},\ddot{h}_{2},\ddot{h}_{3})=\Bigg \{\Bigg \langle s_{1.5971},[0.6187,0.4595],0.6989\Bigg \rangle , \\ \Bigg \langle s_{1.9896},[0.7691,0.3339],0.7365\Bigg \rangle , \\ \Bigg \langle s_{1.7617},[0.2365,0.5042],0.3467\Bigg \rangle \Bigg \} \end{array} \right\} ;\\&\left\{ \begin{array}{l} \text {GCLHFWG}(\ddot{h}_{1},\ddot{h}_{2},\ddot{h}_{3})=\Bigg \{\Bigg \langle s_{1.5971},[0.3813,0.5405],0.4196\Bigg \rangle , \\ \Bigg \langle s_{1.9896},[0.2309,0.6661],0.2635\Bigg \rangle , \\ \Bigg \langle s_{1.7617},[0.7635,0.4958],0.6533\Bigg \rangle \Bigg \} \end{array} \right\} . \end{aligned}$$

Theorem 5

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ be a collection of CLHFEs, $\lambda >0$ and let $\ddot{\Psi }=( \ddot{\Psi }_{1},\ddot{\Psi }_{2},\ldots ,\ddot{\Psi }_{n})^{T}$ is the weight vector of CLHFEs $\ddot{h}_{i}(i=1,2,\ldots ,n)$ with $\ddot{\Psi }_{i}\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n}\ddot{\Psi }_{i}=1.$ Then the GCLHFWA operator GCLHFW $(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})$ is monotonically increasing with respect to the parameter $\lambda .$

Proof

We can obtain that $(s_{\mathop {{ \prod }}\nolimits _{i=1}^{n}\theta (\alpha _{i})^{\ddot{ \Psi }_{i}}}),\left( \begin{array}{c} 1- \\ (\mathop {{ \prod }}\nolimits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{-})^{\lambda })^{\ddot{\Psi } _{i}} \end{array} \right) ^{\frac{1}{\lambda }}$ and $\left( \begin{array}{c} 1- \\ (\mathop {{ \prod }}\nolimits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{+})^{\lambda })^{\ddot{\Psi } _{i}} \end{array} \right) ^{\frac{1}{\lambda }}$are monotonically increasing with respect to the parameter $\lambda$. Furthermore, $\left( \begin{array}{c} 1- \\ (\mathop {{ \prod }}\nolimits _{i=1}^{n}(1-(\upsilon _{\alpha _{i}})^{\lambda })^{\ddot{\Psi } _{i}} \end{array} \right) ^{\frac{1}{\lambda }}$ are monotonically decreasing with respect to the parameter $\lambda .$ Therefore, by

$$\begin{aligned} \left\{ \begin{array}{l} s(\text {GCLHFWA}(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n}))=\frac{1}{3} \Bigg [s_{\mathop {{ \prod }}\limits _{i=1}^{n}\theta (\alpha _{i})^{\ddot{\Psi }_{i}}}\Bigg ], \\ \mathop {{ \sum }}\limits _{\alpha _{1}\in \ddot{h}_{1},\alpha _{2}\in \ddot{h} _{2},\ldots ,\alpha _{n}\in \ddot{h}_{n}}\mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }\left( 1-\left( \mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{-})^{\lambda })^{\ddot{\Psi }_{i}}\right. \right) ^{\frac{1}{\lambda }}-1+ \\ \left( 1-\left( \mathop {{ \prod }}\limits _{i=1}^{n}(1-(\upsilon _{\alpha _{i}})^{\lambda })^{ \ddot{\Psi }_{i}}\right. \right) ^{\frac{1}{\lambda }}+\left( 1-\left( \mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{+})^{\lambda })^{\ddot{\Psi }_{i}}\right. \right) ^{\frac{1}{\lambda }}\} \\ \times \left( \mathop {{ \prod }}\limits _{i=1}^{n}\#\ddot{h}_{i}\right) ^{-1} \end{array} \right\} \end{aligned}$$

is monotonically increasing with respect to the parameter $\lambda$, which implies that GCLHFWA$(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})$ is monotonically increasing with respect to the parameter $\lambda$. $\square$

Theorem 6

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ be a collection of CLHFEs, $\lambda >0$ and let $\ddot{\Psi }=( \ddot{\Psi }_{1},\ddot{\Psi }_{2},\ldots ,\ddot{\Psi }_{n})^{T}$ is the weight vector of CLHFEs $\ddot{h}_{i}(i=1,2,\ldots ,n)$ with $\ddot{\Psi }_{i}\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n}\ddot{\Psi }_{i}=1.$ Then the GCLHFWG operator GCLHFWG $(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})$ is monotonically decreasing with respect to the parameter $\lambda .$

Proof

We can obtain that $s_{\mathop {{ \prod }}\nolimits _{i=1}^{n}\theta (\alpha _{i})^{\ddot{ \Psi }_{i}}},\left( 1-(\mathop {{ \prod }}\nolimits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{-})^{\lambda })^{\ddot{\Psi }_{i}}\right) ^{\frac{1}{\lambda }}$ and $\left( 1-(\mathop {{ \prod }}\nolimits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{+})^{\lambda })^{ \ddot{\Psi }_{i}}\right) ^{\frac{1}{\lambda }}$are monotonically increasing with respect to the parameter $\lambda$. Furthermore, $\left( 1-(\mathop {{ \prod }}\nolimits _{i=1}^{n}(1-(\upsilon _{\alpha _{i}})^{\lambda })^{\ddot{ \Psi }_{i}}\right) ^{\frac{1}{\lambda }}$ are monotonically decreasing with respect to the parameter $\lambda .$ Therefore, by

$$\begin{aligned}&s(\hbox {GCLHFWG}(\ddot{h}_{1}, \ddot{h}_{2},\ldots ,\ddot{h}_{n}))\\&\quad =\left\{ \begin{array}{l} \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }\frac{1}{3}\left[ s_{\mathop {{ \prod }}\limits _{i=1}^{n}\theta (\alpha _{i})^{\ddot{\Psi } _{i}}}\right] ,\mathop {{ \sum }}\limits _{\alpha _{1}\in \ddot{h}_{1},\alpha _{2}\in \ddot{h} _{2},\ldots ,\alpha _{n}\in \ddot{h}_{n}} \\ \left( 1-\left( \mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{-})^{\lambda })^{ \ddot{\Psi }_{i}}\right. \right) ^{\frac{1}{\lambda }}-1+\left( 1-\left( \mathop {{ \prod }}\limits _{i=1}^{n}(1-(\upsilon _{\alpha _{i}})^{\lambda })^{\ddot{ \Psi }_{i}}\right. \right) ^{\frac{1}{\lambda }} \\ +\left( 1-\left( \mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{i}}^{+})^{\lambda })^{ \ddot{\Psi }_{i}}\right. \right) ^{\frac{1}{\lambda }}\}\times \left( \mathop {{ \prod }}\limits _{i=1}^{n}\#\ddot{h}_{i}\right) ^{-1} \end{array} \right\} \end{aligned}$$

is monotonically increasing with respect to the parameter $\lambda$, which implies that GCLHFWG$(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})$ is monotonically increasing with respect to the parameter $\lambda$. $\square$

Theorem 7

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\} (i=1,2,\ldots ,n)$ be a collection of CLHFEs having the weight vector $\ddot{\Psi }=(\ddot{\Psi }_{1},\ddot{\Psi }_{2},\ldots ,\ddot{\Psi }_{n})^{T}$ such that $\ddot{\Psi }_{i}\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n}\ddot{\Psi } _{i}=1,$ $\lambda >0;$ then GCLHFWG$(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h} _{n})\le G$CLHFWA$(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})$.

5.2 The GCLHFOWA and GCLHFOWG operators

Definition 19

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ be a collection of CLHFEs. A generalized cubic linguistic hesitant fuzzy ordered weighted averaging operator is denoted and defined by the mapping $\ddot{h}^{n}\rightarrow \ddot{h}$, such that GCLHFOWA$(\ddot{h} _{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})=\left( \mathop {{ \bigoplus }}\nolimits _{i=1}^{n}( \mathring{\tau }_{i}\ddot{h}_{\sigma (i)}^{\lambda })\right) ^{\frac{1}{ \lambda }}\$with $\lambda >0$ where $\mathring{\tau }=(\mathring{\tau }_{1}, \mathring{\tau }_{2},\ldots ,\mathring{\tau }_{n})^{T}$ is the aggregation-associated vector of CLHFEs, such that $\mathring{\tau }_{i}\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n}\mathring{\tau }_{i}=1.$

Theorem 8

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ $(i=1,2,\ldots ,n)$ be a collection of CLHFEs. Then the aggregated value is calculated using the GCLHFOWA operator is also a CLHFE and

$$\begin{aligned}&\mathrm{GCLHFOWA}(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})\\&\quad =\left\langle \begin{array}{l} \Bigg \{s_{\left( \mathop {{ \prod }}\limits _{i=1}^{n}\theta (\alpha _{i})^{\mathring{\tau }_{i}}\right) ^{ \frac{1}{\lambda }}}, \\ \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }\left[ \begin{array}{l} \left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{\sigma (i)}}^{-})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }}, \\ \left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{\sigma (i)}}^{+})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }} \end{array} \right] \\ \left( \mathop {{ \prod }}\limits _{i=1}^{n}((\upsilon _{\alpha _{\sigma (i)}})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }}|\alpha _{\sigma (1)},\alpha _{\sigma (2)},\ldots ,\alpha _{\sigma (n)}\in \ddot{h}_{\alpha _{\sigma (n)}}\Bigg \} \end{array} \right\rangle . \end{aligned}$$

Definition 20

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\} (i=1,2,\ldots ,n)$ be a collection of CLHFEs. A generalized cubic linguistic hesitant fuzzy ordered weighted geometric operator is denoted and defined by the mapping $\ddot{h}^{n}\rightarrow \ddot{h}$, such that

$$\begin{aligned} \text {GCLHFOWG}(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})=\frac{1}{\lambda }\left( \mathop {{ \bigotimes }}\nolimits _{i=1}^{n}(\lambda \ddot{h}_{\sigma (i)})^{ \mathring{\tau }_{i}}\right) \quad \text { with }\lambda >0, \end{aligned}$$

where $\mathring{\tau }=(\mathring{\tau }_{1},\mathring{\tau }_{2},\ldots , \mathring{\tau }_{n})^{T}$ is the aggregation-associated vector of CLHFEs, such that $\mathring{\tau }_{i}\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n} \mathring{\tau }_{i}=1.$

Theorem 9

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ $(i=1,2,\ldots ,n)$ be a collection of CLHFEs. Then the aggregated value is calculated using the GCLHFOWG operator is also a CLHFE and

$$\begin{aligned}&\hbox {GCLHFOWG}(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})\\&\quad =\left\langle \begin{array}{l} \Bigg \{s_{\left( \mathop {{ \prod }}\limits _{i=1}^{n}\theta (\alpha _{i})^{\mathring{\tau }_{i}}\right) ^{ \frac{1}{\lambda }}},\mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma } \\ \left[ \begin{array}{l} 1-\left( (1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(1-(\mu _{\alpha _{\sigma (i)}}^{-})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }},\\ 1-\left( (1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(1-(\mu _{\alpha _{\sigma (i)}}^{+})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }} \end{array} \right], \\ \left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\upsilon _{\alpha _{\sigma (i)}})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }}|\alpha _{\sigma (1)},\alpha _{\sigma (2)},\ldots ,\alpha _{\sigma (n)}\in \ddot{h} _{\alpha _{\sigma (n)}}\Bigg \} \end{array} \right\rangle . \end{aligned}$$

Remark 1

The GCLHFWA and GCLHFWG operators weigh only the cubic linguistic hesitant fuzzy arguments. However, the GCLHFOWA and GCLHFOWG operators weigh the ordered positions of the cubic linguistic hesitant fuzzy arguments instead of weighting the cubic linguistic hesitant fuzzy arguments themselves. The prominent characteristic of the GCLHFOWA and GCLHFOWG operators is the reordering step in which the input arguments are rearranged in descending order; in particular, cubic linguistic hesitant fuzzy argument $\ddot{h}_{i}$ is not associated with a particular weight $\mathring{\tau }_{i}$, but rather a weight $\mathring{\tau }_{i}$ is associated with a particular ordered position i of the cubic linguistic hesitant fuzzy arguments.

Example 5

Suppose that $\ddot{h}_{1}=\left\langle \begin{array}{c} \{s_{3},[0.3, \\ 0.6],0.4\}, \\ \{s_{3},[0.2, \\ 0.4],0.3\}, \\ \{s_{2},[0.1, \\ 0.3],0.2\} \end{array} \right\rangle ,\ddot{h}_{2}=\left\langle \begin{array}{c} \{s_{1},[0.5, \\ 0.10],0.7\}, \\ \{s_{1},[0.12, \\ 0.20],0.16\}, \\ \{s_{3},[0.1, \\ 0.10],0.7\} \end{array} \right\rangle$ and $\ddot{h}_{3}=\left\langle \begin{array}{c} \{s_{2},[0.23, \\ 0.30],0.22\}, \\ \{s_{1},[0.24, \\ 0.28],0.27\}, \\ \{s_{1},[0.30, \\ 0.36],0.34\} \end{array} \right\rangle$ are three CLHFEs and $\mathring{\tau }=\{0.2,0.4,0.4\}$ is their weight vector. Then we can define score function

$$ \left\{ \begin{array}{c} s(\ddot{h}_{1})=\frac{s_{2.5508},[\{0.6+1.1\}]-0.9\}}{3} =\frac{s_{2.5508}[1.7-0.9]}{3}=s_{0.6802} \end{array} \right\} ; \left\{ s(\ddot{h}_{2})=\frac{s_{1.5518},[\{1.62+0.4\}]-1.56\}}{3}=\frac{ s_{1.5518}[0.46]}{3}=s_{0.2379}\right\} ; \left\{ s(\ddot{h}_{3})=\frac{s_{1.1486},\{0.77+0.94\}-0.84}{3}=\frac{ 1.71-0.84}{3}=\frac{s_{1.1486}(0.87)}{3}=s_{0.3330}\right\} .$$

Theorem 10

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ $(i=1,2,\ldots ,n)$ be a collection of CLHFEs, $\lambda >0$ and let $\mathring{\tau }=(\mathring{\tau }_{1},\mathring{\tau }_{2},.\ldots ,\mathring{\tau } _{n})^{T}$ be the aggregation-associated vector such that $\mathring{\tau } _{i}\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n}\mathring{\tau }_{i}=1$, then the GCLHFOWA operator is monotonically increasing with respect to the parameter $\lambda$.

Theorem 11

Assume $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ $(i=1,2,\ldots ,n)$ are the collections of CLHFEs, $\lambda >0$ and let $\mathring{\tau }=(\mathring{\tau }_{1},\mathring{\tau }_{2},\ldots ,\mathring{ \tau }_{n})^{T}$ be the aggregation-associated vector such that $\mathring{ \tau }_{i}\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n}\mathring{\tau }_{i}=1,$ then the GCLHFOWG operator is monotonically decreasing with respect to the parameter $\lambda$.

Theorem 12

Assume $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ are the collections of CLHFEs, $\lambda >0$ and let $\mathring{ \tau }=(\mathring{\tau }_{1},\mathring{\tau }_{2},\ldots ,\mathring{\tau }_{n})^{T}$ be the aggregation-associated vector such that $\mathring{\tau }_{i}\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n}\mathring{\tau }_{i}=1$ then GCLHFOWG $(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})\le$ GCLHFOWA $(\ddot{h}_{1}, \ddot{h}_{2},\ldots ,\ddot{h}_{n}).$

5.3 The GCLHFHA and GCLHFHG operators

Definition 21

For a collection of CLHFEs $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ $(i=1,2,\ldots ,n)$, $\mathring{\tau }=(\mathring{\tau }_{1},\mathring{ \tau }_{2},\ldots ,\mathring{\tau }_{n})^{T}$ is the weight vector of CLHFEs, with $\mathring{\tau }_{i}$ $\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n} \mathring{\tau }_{i}=1$ and n is the balancing coefficient which plays a role of balance. Then we define the following aggregation operators, which are all based on the mapping $\ddot{h}^{n}\rightarrow \ddot{h}$ with an aggregation-associated vector $\mathring{\tau }=(\mathring{\tau }_{1}, \mathring{\tau }_{2},\ldots ,\mathring{\tau }_{n})^{T}$such that $\mathring{\tau }_{i}$ $\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n}\mathring{\tau }_{i}=1.$ The generalized cubic linguistic hesitant fuzzy hybrid averaging operator is GCLHFHA$(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})=\left( \mathop {{ \bigoplus }}\nolimits _{i=1}^{n}\left( \mathring{\tau }_{i}\ddot{h}_{\sigma (i)}\right) \right) ^{\frac{1}{\lambda }}$, where $\ddot{h}_{\sigma (i)}$ is the largest ith of $\ddot{h}_{k}=(n\mathring{\tau }_{k}\ddot{h}_{\sigma (i)})$ $(k=1,2,\ldots ,n).$

Theorem 13

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ $(i=1,2,\ldots ,n)$ be a collection of CLHFEs. Then the aggregated value calculated using the GCLHFHA operator is also a CLHFE and

$$\begin{aligned}&\hbox {GCLHFHA }(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})\\&\quad \left\langle \begin{array}{l} \Bigg \{s_{\left( \mathop {{ \prod }}\limits _{i=1}^{n}\theta (\alpha _{i})^{\mathring{\tau }_{i}}\right) ^{ \frac{1}{\lambda }}},\mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma } \\ \left[ \begin{array}{l} \left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{\sigma (i)}}^{-})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }},\\ \left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\mu _{\alpha _{\sigma (i)}}^{+})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }} \end{array} \right], \\ \left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(1-\upsilon _{\alpha _{\sigma (i)}})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }} \\ |\alpha _{\sigma (1)},\alpha _{\sigma (2)},\ldots ,\alpha _{\sigma (n)}\in \ddot{ h}_{\alpha _{\sigma (n)}}\Bigg \} \end{array} \right\rangle . \end{aligned}$$

Definition 22

For a collection of CLHFEs $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ $(i=1,2,\ldots ,n)$, $\mathring{\tau }=(\mathring{\tau }_{1},\mathring{ \tau }_{2},\ldots ,\mathring{\tau }_{n})^{T}$ is the weight vector of CLHFEs, with $\mathring{\tau }_{i}$ $\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n} \mathring{\tau }_{i}=1,$ and n is the balancing coefficient which plays a role of balance. Then we define the following aggregation operators, which are all based on the mapping $\ddot{h}^{n}\rightarrow \ddot{h}$ with an aggregation-associated vector $\mathring{\tau }=(\mathring{\tau }_{1}, \mathring{\tau }_{2},\ldots ,\mathring{\tau }_{n})^{T}$ such that $\mathring{\tau }_{i}$ $\in {[} 0,1]$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n}\mathring{\tau }_{i}=1.$ The generalized cubic linguistic hesitant fuzzy hybrid geometric operator is GCLHFHG$(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})=\frac{1}{\lambda } \left( \mathop {{ \bigotimes }}\nolimits _{i=1}^{n}(\lambda \ddot{h}_{\sigma (i)})^{ \mathring{\tau }_{i}}\right) ,$ where $\ddot{h}_{\sigma (i)}$ is the largest ith of $\ddot{h}_{k}=(\ddot{h}_{\sigma (i)})^{n\mathring{\tau }_{k}}$ $(k=1,2,\ldots ,n).$

Theorem 14

Let $\ddot{h}_{i}=\left\{ \begin{array}{c} s_{\theta }, \\ \langle {[} \Gamma ^{-}, \\ \Gamma ^{+}], \\ \Gamma \rangle \end{array} \right\}$ $(i=1,2,\ldots ,n)$ be a collection of CLHFEs. Then the aggregated value is calculated using the GCLHFHG operator is also a CLHFE and

$$\begin{aligned}&\hbox {GCLHFHG }(\ddot{h}_{1},\ddot{h}_{2},\ldots ,\ddot{h}_{n})\\&\quad =\left\langle \begin{array}{l} \Bigg \{s_{\left( \mathop {{ \prod }}\limits _{i=1}^{n}\theta (\alpha _{i})^{\mathring{\tau }_{i}}\right) ^{ \frac{1}{\lambda }}},\mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma } \\ \left[ \begin{array}{l} 1-\left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(1-\mu _{\alpha _{\sigma (i)}}^{-})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }},\\ 1-\left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(1-\mu _{\alpha _{\sigma (i)}}^{+})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }} \end{array} \right] ,\\ \left( 1-\mathop {{ \prod }}\limits _{i=1}^{n}(1-(\upsilon _{\alpha _{\sigma (i)}})^{\lambda })^{\mathring{\tau }_{i}}\right) ^{\frac{1}{\lambda }}|\alpha _{\sigma (1)},\alpha _{\sigma (2)},\ldots ,\alpha _{\sigma (n)}\in \ddot{h} _{\alpha _{\sigma (n)}}\Bigg \} \end{array} \right\rangle . \end{aligned}$$

Example 6

Suppose that $\ddot{h}_{1}=\left\langle \begin{array}{c} s_{3},\{[0.3, \\ 0.6],0.4\}, \\ \{s_{4},[0.2, \\ 0.4],0.3\}, \\ \{s_{1},[0.1, \\ 0.3],0.2\} \end{array} \right\rangle ,\ddot{h}_{2}=\left\langle \begin{array}{c} \{s_{2},[0.5, \\ 0.8],0.7\}, \\ \{s_{3},[0.12, \\ 0.16],0.14\}, \\ \{s_{1},[0.9, \\ 0.19],0.15\} \end{array} \right\rangle$ and $\ddot{h}_{3}=\left\langle \begin{array}{c} \{s_{3},[0.23, \\ 0.31],0.22\}, \\ \{s_{2},[0.24, \\ 0.28],0.27\}, \\ \{s_{2},[0.30, \\ 0.36],0.34\} \end{array} \right\rangle$ are three CLHFEs and $\mathring{\tau }=\{0.2,0.4,0.4\}$ is their weight vector. Then we can define score function

$$\begin{aligned}&\left\{ s(\ddot{h}_{1})=\frac{s_{2.1689},[\{0.6+1.3\}-0.9]}{3}=\frac{ s_{2.1689},[1.9-0.9]}{3}=\frac{s_{2.1689}\times 1}{3}=s_{0.7229}\right\} ;\\&\left\{ s(\ddot{h}_{2})=\frac{s_{1.7826},\{1.52+1.15\}-0.99}{3}=\frac{ s_{1.7826,}[2.67-0.99]}{3}=\frac{s_{1.7826}\times {[} 1.68]}{3} =s_{0.9982}\right\} ;\\&\left\{ s(\ddot{h}_{3})=\frac{s_{2.1689},\{0.77+0.95\}-0.83}{3}=\frac{ s_{2.1689},[1.72-0.83]}{3}=\frac{s_{2.1689}\times {[} 0.89]}{3} =s_{0.6434}\right\} . \end{aligned}$$

6 An approach to multiple-attribute group decision-making with cubic linguistic hesitant fuzzy information

In this section, we consume the suggested cubic linguistic hesitant fuzzy aggregation operators to develop an approach to multiple-attribute group decision-making with cubic linguistic hesitant fuzzy information. First, a multiple-attribute group decision-making with cubic linguistic hesitant fuzzy information can be described as follows. Let $Y= \{Y_{1},Y_{2},\ldots ,Y_{m}\}$ be a set of m alternatives, $G=\{G_{1},G_{2},\ldots ,G_{n}\}$ a gathering of n attributes, whose weight vector is $\ddot{\Psi }=(\ddot{\Psi }_{1},\ddot{\Psi }_{2},\ldots ,\ddot{\Psi } _{n})^{T}$, with $\ddot{\Psi }_{i}\in {[} 0,1]$, $i=1,2,\ldots ,n$ and $\mathop {{ \sum }}\nolimits _{i=1}^{n}\ddot{\Psi }_{i}=1$ and let $D=\{D_{1},D_{2},\ldots ,D_{l} \}$ be a set of l decision-makers, whose weight vector is $\mathring{\tau } =(\mathring{\tau }_{1},\mathring{\tau }_{2},\ldots ,\mathring{\tau }_{n})^{T}$, with $\mathring{\tau }_{k}\in {[} 0,1],k=1,2,\ldots ,l$ and $\mathop {{ \sum }}\nolimits _{k=1}^{l}\mathring{\tau }_{k}=1$. Let $R^{k}=(r_{ij}^{k})_{m \times n}$ be the cubic linguistic hesitant fuzzy decision matrix, where $r_{ij}^{k}=\{\gamma _{ij}^{k}|\gamma _{ij}^{k}\in r_{ij}^{k}\}=\left\langle \begin{array}{c} s_{(\theta (\gamma _{ij}^{k})}, \\ {[}\Gamma _{\gamma _{ij}^{k}}^{-}, \\ \Gamma _{\gamma _{ij}^{k}}^{+}{]}, \\ \Gamma _{\gamma _{ij}^{k}} \end{array} \right\rangle$ is a CLHFE given by the decision-maker $D_{k}\in D$, where $s_{(\theta (\gamma _{ij}^{k})},[\Gamma _{\gamma _{ij}^{k}}^{-},\Gamma _{\gamma _{ij}^{k}}^{+}]$ indicates the possible interval value linguistic hesitant fuzzy set range that the alternative $Y_{i}\in Y$ satisfies the attribute $G_{j}\in G$, while $\Gamma _{\gamma _{ij}^{k}}$ indicates the possible linguistic hesitant fuzzy set range that the alternative $Y_{i}\in Y$ does not satisfy the attribute $G_{j}\in G$.

In general, there are benefit attributes (i.e., the bigger the attribute values, the better) and cost attributes (i.e., the smaller the attribute values, the better) in a multiple-attribute group decision-making problem. In such cases, we transform the attribute values of cost type into the attribute values of benefit type. That is, normalize the cubic linguistic hesitant fuzzy decision matrix $R^{k}=(r_{ij}^{k})_{m\times n}$ a corresponding cubic linguistic hesitant fuzzy decision matrix $A_{ij}^{k}=a_{ij}^{k}=\{\alpha _{ij}^{k}|\alpha _{ij}^{k}\in A_{ij}^{k}\}=\left\langle \begin{array}{l} s_{(\theta (\gamma _{ij}^{k})}, \\ {[} \Gamma _{\gamma _{ij}^{k}}^{-}, \\ \Gamma _{\gamma _{ij}^{k}}^{+}], \\ \Gamma _{\gamma _{ij}^{k}} \end{array} \right\rangle ,$ where

$$\begin{aligned} a_{ij}^{k}=\left\{ \begin{array}{l} r_{ij}^{k}\quad \text { for benefit attribute},G_{j} \\ (r_{ij}^{k})^{c}\quad \text { for cost attribute},G \end{array} \right. \ \ \end{aligned}$$

$i=1,2,\ldots ,m,$ $j=1,2,\ldots ,n,k=1,2,\ldots ,l$, where $r_{ij}^{k}$ is the complement of $(r_{ij}^{k})^{c}$ such that $(r_{ij}^{k})^{c}=\{\gamma _{ij}^{k}|\gamma _{ij}^{k}\in r_{ij}^{k}\}=\left\langle \begin{array}{c} s_{(\theta (\gamma _{ij}^{k})}, \\ {[} \Gamma _{\gamma _{ij}^{k}}^{-}, \\ \Gamma _{\gamma _{ij}^{k}}^{+}], \\ \Gamma _{\gamma _{ij}^{k}} \end{array} \right\rangle .$

In the following, we utilize the proposed operators to develop an approach to multiple-attribute group decision-making with cubic linguistic hesitant fuzzy information, which includes the following steps.

Step 1: Reconstruct the decision matrix $(R_{ij}^{k})= (r_{ij}^{k})_{m\times n}$ into a normalized matrix $A_{ij}^{k}=a_{ij}^{k}=\{ \alpha _{ij}^{k}|\alpha _{ij}^{k}\in A_{ij}^{k}\}=\left\langle \begin{array}{c} s_{(\theta (\gamma _{ij}^{k})}, \\ {[} \Gamma _{\gamma _{ij}^{k}}^{-}, \\ \Gamma _{\gamma _{ij}^{k}}^{+}], \\ \Gamma _{\gamma _{ij}^{k}} \end{array} \right\rangle .$

Step 2: Utilize the GCLHFWA operator

$$\begin{aligned} \left\{ \begin{array}{l} a_{ij}=\text {GCLHFWA}(a_{ij}^{1},a_{ij}^{2},\ldots ,a_{ij}^{l})=\Bigg \{s_{((\theta (\alpha _{ij)}^{k})^{\mathring{\tau }_{i}})^{\frac{1}{\lambda }}}, \\ \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }\Bigg [\left( 1-\mathop {{ \prod }}\limits _{k=1}^{l}(1-(\mu _{\alpha _{ij)}^{k}}^{-})^{\lambda })^{\mathring{\tau }_{k}}\right) ^{\frac{1}{\lambda }}, \\ 1-\left( 1-\mathop {{ \prod }}\limits _{k=1}^{l}(1-(\mu _{\alpha _{ij}^{k}}^{+})^{\lambda })^{\mathring{\tau }_{k}}\right) ^{\frac{1}{\lambda }}, \\ 1-\left( 1-\mathop {{ \prod }}\limits _{k=1}^{l}(1-(1-\upsilon _{\alpha _{ij}^{k}})^{\lambda })^{\mathring{\tau }_{k}}\right) ^{\frac{1}{\lambda }} \\ |\alpha _{\sigma (1)},\alpha _{\sigma (2)},\ldots ,\alpha _{\sigma (n)}\in \ddot{ h}_{\alpha _{\sigma (n)}}\Bigg \} \end{array} \right\} \end{aligned}$$

or the GCLHFWG operator

$$\begin{aligned} \left\{ \begin{array}{c} a_{ij}=\text {GCLHFWG}(a_{ij}^{1},a_{ij}^{2},\ldots ,a_{ij}^{l})=\Bigg \{s_{((\theta (\alpha _{ij)}^{k})^{\mathring{\tau }_{i}})^{\frac{1}{\lambda }}}, \\ \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }\Bigg [1-\left( 1-\mathop {{ \prod }}\limits _{k=1}^{l}(1-(\mu _{\alpha _{ij}^{k}}^{-})^{\lambda })^{\mathring{\tau }_{k}}\right) ^{\frac{1}{\lambda } }, \\ 1-\left( 1-\mathop {{ \prod }}\limits _{k=1}^{l}(1-(1-\mu _{\alpha _{ij)}^{k}}^{+})^{\lambda })^{\mathring{\tau }_{k}}\right) ^{\frac{1}{\lambda }}, \\ \left( 1-\mathop {{ \prod }}\limits _{k=1}^{l}(1-(\upsilon _{\alpha _{ij}^{k}})^{\lambda })^{\mathring{\tau }_{k}}\right) ^{\frac{1}{\lambda }} \\ |\alpha _{\sigma (1)},\alpha _{\sigma (2)},\ldots ,\alpha _{\sigma (n)}\in \ddot{ h}_{\alpha _{\sigma (n)}}\Bigg \}. \end{array} \right\} \end{aligned}$$

to aggregate all the individual cubic linguistic hesitant fuzzy decision matrix $(A_{ij}^{k}) = (\alpha _{ij}^{k})_{m\times n}$ into the collective cubic linguistic hesitant fuzzy decision matrix

$$\begin{aligned} A_{ij}^{k}=a_{ij}^{k}=\{\alpha _{ij}^{k}|\alpha _{ij}^{k}\in A_{ij}^{k}\}=\left\langle \begin{array}{c} s_{(\theta (\gamma _{ij}^{k})}, \\ {[} \Gamma _{\gamma _{ij}^{k}}^{-}, \\ \Gamma _{\gamma _{ij}^{k}}^{+}], \\ \Gamma _{\gamma _{ij}^{k}} \end{array} \right\rangle \end{aligned}$$

Step 3: Utilize the GCLHFHA operator

$$\begin{aligned} \left\{ \begin{array}{l} \alpha _{i}=\text {GCLHFHA}(a_{i1},a_{i2},\ldots ,a_{in})=\Bigg \{s_{((\theta (\alpha _{i\sigma (j)})^{\xi _{i}})^{\frac{1}{\lambda }}}, \\ \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }\left( (1-\mathop {{ \prod }}\limits _{j=1}^{n}\left( 1-(\mu _{\alpha _{i\sigma (j)}}^{-})^{\lambda }\right) ^{\xi _{i}}\right) ^{\frac{1}{\lambda }}, \\ \left( (1-\mathop {{ \prod }}\limits _{j=1}^{n}\left( 1-(\mu _{\alpha _{i\sigma (j)}}^{+})^{\lambda }\right) ^{\xi _{i}}\right) ^{\frac{1}{\lambda }}, \\ 1-\left( 1-\mathop {{ \prod }}\limits _{j=1}^{n}\left( 1-(1-\upsilon _{\alpha _{i\sigma (j)}})^{\lambda }\right) ^{\xi _{i}}\right) ^{\frac{1}{\lambda }} \\ |\alpha _{\sigma (1)},\alpha _{\sigma (2)},\ldots ,\alpha _{\sigma (n)}\in \ddot{ h}_{\alpha _{\sigma (n)}}\Bigg \} \end{array} \right\} \end{aligned}$$

or the GCLHFHG operator

$$\begin{aligned} \left\{ \begin{array}{l} \alpha _{i}=\text {GCLHFHG}(a_{i1},a_{i2},\ldots ,a_{in}) =\Bigg \{s_{\mathop {{ \prod }}\limits _{j=1}^{n}((\theta (\alpha _{i\sigma (j)})^{\ddot{\xi }_{i}})^{\frac{1 }{\lambda }}}, \\ \mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }[1-\left( (1-\mathop {{ \prod }}\limits _{j=1}^{n}\left( 1-\left( 1-\mu _{\alpha _{i\sigma (j)}}^{-}\right) ^{\lambda }\right) ^{\ddot{\xi }_{i}}\right) ^{\frac{1}{\lambda }}, \\ 1-\left( (1-\mathop {{ \prod }}\limits _{j=1}^{n}\left( 1-\left( 1-\mu _{\alpha _{i\sigma (j)}}^{+}\right) ^{\lambda }\right) ^{\ddot{\xi }_{i}}\right) ^{\frac{1}{\lambda }}, \\ \left( 1-\mathop {{ \prod }}\limits _{j=1}^{n}\left( 1-\left( \upsilon _{\alpha _{i\sigma (j)}}\right) ^{\lambda }\right) ^{\ddot{\xi }_{i}}\right) ^{\frac{1}{\lambda }}|\alpha _{\sigma (1)},\alpha _{\sigma (2)},\ldots ,\alpha _{\sigma (n)}\in \ddot{h} _{\alpha _{\sigma (n)}}\Bigg \} \end{array} \right\} \end{aligned}$$

to aggregate all the preference values $a_{ij}(j=1,2,\ldots ,n)$ in the ith line of A and then derive the collective overall preference value $A_{ij}^{k}=a_{ij}^{k}=\{\alpha _{ij}^{k}|\alpha _{ij}^{k}\in A_{ij}^{k}\}=\left\langle \begin{array}{c} s_{(\theta (\gamma _{ij}^{k})}, \\ {[} \Gamma _{\gamma _{ij}^{k}}^{-}, \\ \Gamma _{\gamma _{ij}^{k}}^{+}], \\ \Gamma _{\gamma _{ij}^{k}} \end{array} \right\rangle$ $(i=1,2,\ldots ,m)$ of the alternative $Y_{i}(i=1,2,\ldots ,m),$ where $\ddot{\xi }=(\ddot{\xi }_{1},\ddot{\xi }_{2},\ldots ,\ddot{\xi }_{n})^{T}$ is the associated weight vector of the GCLHFHA (or GCLHFHG) operator, with $\ddot{\xi }_{j}\in {[} 0,1],$ $j=1,2,\ldots ,n,$ and $\mathop {{ \sum }}\nolimits _{j=1}^{n} \xi _{j}=1.$

Step 4: Calculate the score values $s(a_{i})(i=1,2,\ldots ,m)$ of $a(i=1,2,\ldots ,m)$:

$$\begin{aligned} s(a_{i})=\mathop {{\displaystyle \bigcup }}\limits _{\mu ^{-}\in \Gamma ^{-},\mu ^{+}\in \Gamma ^{+},v\in \Gamma }\frac{\frac{1}{3}\mathop {{ \sum }}\limits _{\alpha _{i}\in \alpha _{i}}s_{((\theta (\alpha _{\gamma _{ij}^{k}})},([\mu _{\gamma _{ij}^{k}}^{-}+\mu _{\gamma _{ij}^{k}}^{+}]-\upsilon _{\gamma _{ij}^{k}})}{ \#(a_{i})}. \end{aligned}$$

Step 5:In this step, we calculate the values of the score function $s(a_{i})$ based on Definition 12.

7 The application of the developed approach in group decision-making problems

Let us consider a factory which intends to select a new site for new buildings. Three alternatives $(C_{i}=1,2,3)$ are available and the three decision-makers $(i=1,2,3)$ consider three criteria to decide which site to choose: (1) $G_{1}$(price); (2) $G_{2}$ (location) and (3) $G_{3}$ (environment). The weight vector of the decision-makers $D_{k}(k=1,2,3),$ $\mathring{\tau }=(0.34,0.26,0.40)^{T}.$ The DMs evaluate these alternatives using the linguistic term set $S=f,$ $s_{0}=$ extremely poor(EP); $s_{1}=$ very poor(VP); $s_{2}=$ poor(P); $s_{3}=$ medium(M); $s_{4}=$ good : $s_{5}=$ high. After the data acquisition and statistical treatment, the ratings of the alternatives with respect to attributes can be represented by CLVs shown in Tables 1, 2 and 3. Suppose the decision-makers

Step 1: We transform the cubic linguistic hesitant fuzzy decision matrices three

Table 1 Cubic linguistic hesitant fuzzy decision matrix

Table 2 Cubic linguistic hesitant fuzzy decision matrix

Table 3 Cubic linguistic hesitant fuzzy decision matrix

Step 2: $\lambda =1,\mathring{\tau }=(0.34,0.26,0.40)^{T}$ (Table 4).

Table 4 Utilize the GCLHFWA operator

or the GCLHFWG operator (Table 5).

Table 5 Utilize the GCLHFWG operator

Step 3: Utilize the GCHFHA operator and $\xi =(0.25,0.25,0.25,0.25)^{T}$ (Table 6).

Table 6 Utilize the GCHFHA operator

or the GCLHFHG operator (Table 7).

Table 7 Utilize the GCLHFHG operator

where $\xi =(\xi _{1},\xi _{2},\ldots ,\xi _{n})^{T}$ is the associated weight vector of the GCHFHA (or GCHFHG) operator, with $\xi _{j}\in {[} 0,1],j=1,2,\ldots ,n,$ and $\mathop {{ \sum }}\nolimits _{j=1}^{n}$ $=1$.

Step 4: Calculate the score values $s(a_{i})(i=1,2,\ldots ,m)$ of $a(i=1,2,\ldots ,m)$:

$$\begin{aligned} s(a_{i})=\frac{\frac{1}{3}[s_{((\theta (\alpha _{\gamma _{ij}^{k}})},\mathop {{ \sum }}\limits _{\alpha _{i}\in \alpha _{i}}([\mu _{\gamma _{ij}^{k}}^{-}+\mu _{\gamma _{ij}^{k}}^{+}]-\upsilon _{\gamma _{ij}^{k}})}{ \#(a_{i})}. \end{aligned}$$

GCHFHA score function

$$\begin{aligned} s(a_{1})=s_{0.1268},s(a_{2})=s_{0.1207},s(a_{3})=s_{0.1147},s(a_{4})=s_{0.1583}. \end{aligned}$$

GCHFHG score function

$$\begin{aligned} s(a_{1})=s_{0.0946},s(a_{2})=s_{0.1023},s(a_{3})=s_{0.1033},s(a_{4})=s_{0.1258}. \end{aligned}$$

Step 5: According to the value of the score function, the ranking of the candidates can be confirmed,

i.e., $s(a_{4})>s(a_{1})>s(a_{2})>s(a_{3})$, so $s(a_{4})$ is the best alternatives.

i.e., $s(a_{4})>s(a_{3})>s(a_{2})>s(a_{1})$, so $s(a_{4})$ is the best alternatives. The results of the GCHFHA score value and GCHFHG score value of the numerical example in Table 8.

Table 8 The results of the GCHFHA and GCHFHG score values

8 Comparison analysis

To verify the validity and effectiveness of the proposed approach, a comparative study is conducted using the methods of intuitionistic hesitant fuzzy number Peng et al. (2015) and hesitant intuitionistic linguistic fuzzy number Zhou et al. (2016), which are special cases of CLHFNs, to the same illustrative example.

8.1 A comparison analysis of the existing MCDM intuitionistic hesitant fuzzy number with our proposed methods

An intuitionistic hesitant fuzzy number can be considered as a special case of cubic linguistic hesitant fuzzy numbers when there is the only element in membership and non-membership degree Peng et al. (2015). For comparison, the intuitionistic hesitant fuzzy number can be transformed to the cubic linguistic hesitant fuzzy numbers by calculating the average value of the membership and nonmembership degrees. After transformation, the intuitionistic hesitant fuzzy number is given in Table 9.

Table 9 Intuitionistic hesitant fuzzy decision matrix

Step 1: Calculate the intuitionistic hesitant fuzzy weighted averaging (IHFWA) operator and $\omega =(0.2,0.3,0.2,0.3)^{T}$ (Table 10).

Table 10 IHFWA operator

Step 2: Calculate the score function

$$\begin{aligned} s(a_{1})=-0.1321,s(a_{2})=-0.0583,s(a_{3})=-0.1984,s(a_{4})=-0.1534. \end{aligned}$$

Step 3: Find the ranking $s(a_{3})>s(a_{4})>s(a_{1})>s(a_{2}).$

The ranking of all alternatives $s(a_{3})>s(a_{4})>s(a_{1})>s(a_{2})$ and $s(a_{3})$ is the best selection. Obviously, the ranking is derived from the method proposed by Peng et al. (2015), which is different from the result of the proposed method. The main reasons are that intuitionistic hesitant fuzzy number only consider the hesitant fuzzy number, membership degrees of an element and nonmembership degrees, which may result in information intuitionistic hesitant fuzzy number are not equal.

8.2 A comparison analysis of the existing MCDM hesitant intuitionistic linguistic fuzzy number with our proposed methods

Hesitant intuitionistic linguistic fuzzy number can be considered as a special case of cubic linguistic hesitant fuzzy numbers when there is the only element in membership and non-membership degrees Zhou et al. (2016). For comparison, the hesitant intuitionistic linguistic fuzzy number can be transformed to the cubic linguistic hesitant fuzzy numbers by calculating the average value of the membership and nonmembership degrees. After transformation, the hesitant intuitionistic linguistic fuzzy number is given in Table 11.

Table 11 Hesitant intuitionistic linguistic fuzzy decision matrix

Step 1: Calculate the hesitant intuitionistic fuzzy weighted averaging (HIFWA) operator and $\omega =(0.3,0.3,0.2,0.2)^{T}$ (Table 12).

Table 12 HILFWA operator

Step 2: Calculate the score function

$s(a_{1})=-0.4153,s(a_{2})=0.5564,s(a_{3})=-0.4445,s(a_{4})=-0.3902.$

Step 3: Find the ranking $s(a_{2})>s(a_{3})>s(a_{1})>s(a_{4}).$

The ranking of all alternatives $s(a_{2})>s(a_{3})>s(a_{1})>s(a_{4})$ and $s(a_{2})$ is the best selection. Obviously, the ranking derived from the method proposed by Zhou et al. (2016) is different from the result of the proposed method. The main reasons are that hesitant intuitionistic linguistic fuzzy number only consider the linguistic number, membership degrees of an element and nonmembership degrees, which may result in information hesitant intuitionistic linguistic fuzzy number are not equal (Table 13).

Table 13 Comparison analysis with existing methods

The following advantages of our proposal can be summarized on the basis of the above comparison analyses. Cubic linguistic hesitant fuzzy number (CLHFN) are very suitable for illustrating uncertain or fuzzy information in MCDM problems because the membership and non-membership degrees can be two sets of several possible values, which cannot be achieved by intuitionistic hesitant fuzzy number and hesitant intuitionistic linguistic fuzzy number. On the bases of basis operations, aggregation operators and comparison method of cubic linguistic hesitant fuzzy number (CLHFN) can be also used to process intuitionistic hesitant fuzzy number and hesitant intuitionistic linguistic fuzzy number after slight adjustments, because cubic linguistic hesitant fuzzy number (CLHFN) can be considered as the generalized form of intuitionistic hesitant fuzzy number and hesitant intuitionistic linguistic fuzzy number. The defined operations of cubic linguistic hesitant fuzzy number (CLHFN) give us more accurate than the existing operators.

9 Conclusion

In this paper, we initiate the concept of cubic linguistic hesitant fuzzy numbers and defined some operational laws. The concept of cubic linguistic hesitant fuzzy number is the generalization of cubic number, intuitionistic linguistic hesitant fuzzy numbers, cubic linguistic hesitant fuzzy numbers, and interval-valued hesitant linguistic fuzzy numbers. As investigated in introduction that cubic linguistic hesitant fuzzy numbers become cubic fuzzy number when removing the linguistic hesitant numbers form it, cubic linguistic hesitant fuzzy numbers become intuitionistic linguistic hesitant fuzzy numbers if we take only fuzzy number instead of interval in the membership degree, cubic linguistic hesitant fuzzy number become interval-valued linguistic hesitant fuzzy number if we remove non-membership degree from it and cubic linguistic hesitant fuzzy number becomes linguistic hesitant fuzzy numbers if we remove the non-membership degree and take fuzzy number instead of interval in the membership degree. The cubic linguistic hesitant fuzzy information is more abundant and flexible than cubic sets, ILHFSs, LHFS, IVLHFSs. We introduce cubic linguistic fuzzy sets and cubic linguistic fuzzy numbers, their operational law. We propose a cubic linguistic hesitant fuzzy sets and cubic linguistic hesitant fuzzy elements, their operational laws. We describe aggregation operators for cubic linguistic hesitant fuzzy sets which extended the generalized cubic linguistic hesitant fuzzy averaging (geometric) operator, generalized cubic linguistic hesitant fuzzy weighted averaging (GLCHFWA) operator, generalized cubic linguistic hesitant fuzzy weighted geometric (GCHFWG) operator, generalized cubic linguistic hesitant fuzzy ordered weighted average (GCLHFOWA) operator, generalized cubic linguistic hesitant fuzzy ordered weighted geometric (GCLHFOWG) operator, generalized cubic linguistic hesitant fuzzy hybrid averaging (GCLHFHA) operator and generalized cubic linguistic hesitant fuzzy hybrid geometric (GCLHFHG) operator. Moreover, we relate these aggregation operators to develop an approach to multiple-attribute group decision-making with cubic linguistic hesitant fuzzy information. At last, a numerical example is used to illustrate the validity of the exhibit approach in group decision-making problems. In group decision-making problems, because the experts usually come from different specialty fields and have different backgrounds and levels of knowledge, they usually have diverging opinions.