1 Introduction

Fuzzy set was first introduced by Zadeh (1965). Atanassov’s intuitionistic fuzzy sets (AIFSs) (Atanassov 1994) and interval-valued fuzzy sets (IVFSs) (Atanassov 1986, 1994) are two reasonably basic developments, which were introduced to help a segment of the drawbacks of fuzzy set theory. As observed in Zadeh (1973) and recent made in Bustince and Burillo (1996); Deschrijver and Kerre (2003), the AIFS is numerically equivalent to the IVFSs. The AIFSs address one of the main ideological headways, which consult not solely to what degree a part has a place with a particular set (non-membership); moreover, to what degree this segment does not have a place with the set (non-membership function). Throughout in the last decade, the AIFS theory has been associated with aggregate intuitionistic fuzzy information (Deschrijver and Kerre 2007; Liu and Wang 2007; Turksen 1986; Xu and Cai 2010). Recently, the possibility of AIFSs is summed up by allowing the membership and non-membership capacity to expect interval values along these lines displaying the possibility of interval-valued intuitionistic fuzzy sets (IVIFSs) (Liu and Wang 2007). Wang and Liu (2013), proposed Einstein intersection, Einstein product, Einstein scalar multiplication and Einstein exponentiation and then defined new concentration and dilation of AIFSs. Wang and Zhang (2009), introduced the concept of intuitionistic trapezoidal fuzzy numbers. In Wu and Cao (2013), developed some aggregation operators based on intuitionistic trapezoidal fuzzy numbers, namely intuitionistic trapezoidal fuzzy weighted geometric (ITFWG) operator, the intuitionistic trapezoidal fuzzy ordered weighted geometric (ITFOWG) operator, the induced intuitionistic trapezoidal fuzzy ordered weighted geometric (I-ITFOWG) operator and the intuitionistic trapezoidal fuzzy hybrid geometric (ITFHG) operator. Zhang et al. (2013) developed gray relational projection method, combined gray relational analysis method and projection method under intuitionistic trapezoidal fuzzy number. Liu et al. (2017) developed the trapezoidal intuitionistic fuzzy Einstein weighted averaging (TIFEWA) operator, the trapezoidal intuitionistic fuzzy Einstein ordered weighted averaging (TIFEOWA) operator and the trapezoidal intuitionistic fuzzy Einstein hybrid averaging (TIFEHA) operator. Zhang et al. (2014) developed interval-valued dual hesitant fuzzy Einstein weighted averaging (IVDHFEWA) operator, interval-valued dual hesitant fuzzy Einstein ordered weighted averaging (IVDHFEOWA) operator, interval-valued dual hesitant fuzzy Einstein hybrid averaging (IVDHFEHA) operator, interval-valued dual hesitant fuzzy Einstein weighted geometric (IVDHFEWG) operator, interval-valued dual hesitant fuzzy Einstein ordered weighted geometric (IVDHFEOWG) operator, and interval-valued dual hesitant fuzzy Einstein hybrid geometric (IVDHFEHG) operator.

Cubic set was introduced by Jun et al. (2011), as a generalization of fuzzy set and intuitionistic fuzzy set and is characterized by membership degree and non-membership degree. The membership function is an interval fuzzy number, while non-membership function is a fuzzy set. Fahmi et al. (2018) defined the triangular cubic fuzzy number and operational laws. We developed the triangular cubic fuzzy hybrid aggregation (TCFHA) administrator to total all individual fuzzy choice structure provide by the decision makers into the aggregate cubic fuzzy decision matrix. Fahmi et al. (2017) proposed the cubic TOPSIS method and cubic gray relation analysis (GRA) method. Finally, the proposed method is used for selection in sol–gel synthesis of titanium carbide nano-powders. Fahmi et al. (2017) defined weighted average operator of triangular cubic fuzzy numbers and hamming distance of the TCFN. We develop an MCDM method approach based on an extended VIKOR method using triangular cubic fuzzy numbers (TCFNS) and multi-criteria decision-making (MCDM) method using triangular cubic fuzzy numbers (TCFNs).

Based on the above analysis in this paper, we develop trapezoidal cubic fuzzy numbers (TrCFNs), which is the generalization of trapezoidal intuitionistic fuzzy number and trapezoidal interval fuzzy number. We propose some operations based on Einstein t-norm and Einstein T-conorm for TrCFNs. We also develop score and accuracy function to compare two TrCFNs. Due to the developed operation, we propose trapezoidal cubic fuzzy Einstein weighted averaging (TrCFEWA) operator, trapezoidal cubic fuzzy Einstein ordered weighted averaging (TrCFEOWA) operator and trapezoidal cubic fuzzy Einstein hybrid weighted averaging (TrCFEHWA) operator, for aggregating a collection of trapezoidal cubic fuzzy numbers (TrCFNs).

This paper is organized as follows: In Sect. 2, we discuss some definitions and properties. In Sect. 3, we discuss the trapezoidal cubic fuzzy number (TrCFN) and operational laws. In Sect. 4, we present some Einstein operations on trapezoidal cubic fuzzy sets (TrCFSs) and analysis some desirable properties of the proposed operations. In Sect. 5, we first develop some novel arithmetic averaging operators, such as the trapezoidal cubic fuzzy Einstein weighted averaging (TrCFEWA) operator, trapezoidal cubic fuzzy Einstein ordered weighted averaging (TrCFEOWA) operator and trapezoidal cubic fuzzy Einstein hybrid weighted averaging (TrCFEHWA) operator, for aggregating a collection of trapezoidal cubic fuzzy numbers (TrCFNs). In Sect. 6, we apply the TrCFEHWA operator to multiple attribute decision making (MADM) with trapezoidal cubic fuzzy material. Section 7 gives a numerical example according to our approach. In Sect. 8, we discuss comparison analysis. In Sect. 9, we consume a conclusion.

2 Preliminaries

In this section, we give a brief review of some preliminaries.

Definition 1

(Zadeh 1965) H be a universe of discourse. Then, the fuzzy set can be defined as: \(J=\{h, \mu _J (h)|h\in H\}.\) A fuzzy set in a set H is denoted by \(\mu _J : H\rightarrow I.\) The function \(\mu _J (h)\) denoted the degree of membership of the element 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: \((\forall \mu , \eta \in I^{H})(\mu \le \eta \Leftrightarrow (\forall h\in H)(\mu (h)\le \eta (h))).\)

Definition 2

(Atanassov 1986) Let H is a non-empty set. A function \(I:H\rightarrow [I]\) is called an interval-valued fuzzy set (briefly, an IVF set) in H . Let \([I]^{H}\) stand for the set of all IVF sets in H . For every \(I\in [I]^{H}\) and \(h\in H,I(h)=[I^{-}(h), I^{+}(h)]\) is called the degree of membership of an element h to I, where \(I^{-}:H\rightarrow I\) and \(I^{+}:H\rightarrow I\) are fuzzy sets in H which are called a lower fuzzy set and an upper fuzzy set in H, respectively. For simplexes, we denote \(I=[I^{-}, I^{+}]\) . For every \(I_1 , I_2 \in [I]^{H}\), we define \(I_1 \subseteq I_2 \Leftrightarrow I_1 (h)\le I_2 (h)\) for all \(h\in H\) and \(I_1 =I_2 \Leftrightarrow I_1 (h)=I_2 (h)\forall h\in H.\)

Definition 3

(Jun et al. 2011) Let H is a non-empty set. A cubic set in H is a structure of the form \(F=\{\left\langle {h, I(h), \mu (h)} \right\rangle :h\in H\}\) in which I is an IVF set in H and \(\mu \) is a fuzzy set in H. A cubic set \(F=\{\left\langle {h, I(h), \mu (h)} \right\rangle :h\in H\}\) is simply denoted by \(F=\langle I, \mu \rangle .\) Denote by \(C^{H}\) the collection of all cubic sets in H. A cubic set \(F=\langle I, \mu \rangle \) in which \(I(h)=0\) and \(\mu (h)=1\) (resp. \(I(h)=1\) and \(\mu (h)=0\) for all \(h\in H\) is denoted by 0 (resp. 1). A cubic set \(D=\langle J, \lambda \rangle \) in which \(J(h)=0\) and \(\lambda (h)=0\) (resp. \(J(h)=1\) and \(\lambda (h)=1\)) for all \(h\in H\) is denoted by (resp. ).

Definition 4

(Jun et al. 2011) Let H is a non-empty set. A cubic set \(F=\left\langle {I, \mu } \right\rangle \) in H is said to be an internal cubic set if \(I^{-}(h)\le \mu (h)\le I^{+}(h) \quad \forall h\in H\).

Definition 5

(Jun et al. 2011) Let H is a non-empty set. A cubic set \(F=\left\langle {I, \mu } \right\rangle \) in H is said to be an external cubic set if \(\mu (h)\notin [I^{-}(h), I^{+}(h)] \quad \forall h\in H\).

3 Trapezoidal cubic fuzzy numbers

In this section, we introduce the concept of trapezoidal cubic fuzzy numbers, which is the generalization of trapezoidal intuitionistic fuzzy numbers and trapezoidal interval-valued fuzzy numbers.

Definition 6

Let Abe the trapezoidal cubic fuzzy number on the set of real numbers, its IVTrFN is defined as:

and its TrFN is

\(\Gamma _A (h)=\left\{ \begin{array}{ll} \frac{q-h+(h-p)\mu _A }{(p-q)},&{}p\le h<q \\ \mu _A&{} q\,\le h\le r \\ \frac{h-r+(s-h)\mu _A }{(s-r)}&{}\;r<h\le s \\ 0&{}h<p\hbox { or }h>s) \\ \end{array}\right. .\) Then the trapezoidal cubic fuzzy number A basically denoted by \(A \quad =\left\{ {{\begin{array}{l} \langle [(p^{-}, q^{-}, \\ r^{-}, s^{-}),(I_A^- )], \\ {[}(p^{+}, q^{+}, \\ r^{+}, s^{+}),(I_A^+ )], \\ {[}(p, q, r, s), \\ (\mu _A )]\rangle \\ \end{array} }} \right\} .\) Then A is called trapezoidal cubic fuzzy number (TrCFN).

Definition 7

Let \(A_1 =\left\{ {{\begin{array}{l} \langle [\{p_1^- (h), q_1^- (h),\\ r_1^- (h), s_1^- (h)\}; I_{A_1 }^- ], \\ {[}\{p_1^+ (h), q_1^+ (h), \\ r_1^+ (h), s_1^+ (h)\}; I_{A_1 }^+ ], \\ {[}\{p_1 (h), q_1 (h), \\ r_1 (h), s_1 (h)\}; \mu _A ]\rangle \\ |h\in H \\ \end{array} }} \right\} \) and \(A_2 =\left\{ {{\begin{array}{l} \langle [\{p_2^- (h), q_2^- (h), \\ r_2^- (h), s_2^- (h)\}; I_A^- ], \\ {[}\{p_2^+ (h), q_2^+ (h), \\ r_2^+ (h), s_2^+ (h)\}; I_{A_2 }^+ ], \\ {[}\{p_2 (h), q_2 (h), \\ r_2 (h), s_1 (h)\}; \mu _A \rangle \\ |h\in H \\ \end{array} }} \right\} \) be two trapezoidal cubic fuzzy sets; some operational laws on trapezoidal cubic fuzzy sets are defined as follows:

  1. (a)

    \(\left\{ {{\begin{array}{l} A_1 \subseteq A_2 \hbox { iff }\forall h\in H,[p_1^- (h)\ge p_2^- (h),\\ \quad q_1^- (h)\ge q_2^- (h), \\ {r_1^- (h)\ge r_1^- (h), s_1^- (h)\ge s_2^- (h), I_{A_1 }^- \ge I_{A_2 }^- ],} \\ {[p_1^+ (h)\ge p_2^+ (h), q_1^+ (h)\ge q_2^+ (h),} \\ {r_1^+ (h)\ge r_1^+ (h), s_1^+ (h)\ge s_2^+ (h), I_{A_1 }^+ \ge I_{A_2 }^+ ]\hbox { and }} \\ {[p_1 (h)\le p_2 (h), q_1 (h)\le q_2 (h),} \\ {r_1 (h)\le r_1 (h), s_1 (h)\le s_2 (h), \mu _{A_1 } \le \mu _{A_2 } ]} \\ \end{array} }} \right\} \);

  2. (b)

    \(\left\{ {{\begin{array}{l} {A_1 \cap _{T, S} A_2 =\langle T[p_1^- (h), p_2^- (h)],} \\ {T[q_1^- (h), q_2^- (h)], T[r_1^- (h), r_2^- (h)],} \\ {T[s_1^- (h), s_2^- (h)], T[I_{A_1 }^- , I_{A_2 }^- ],} \\ {T[p_1^+ (h), p_2^+ (h)], T[q_1^+ (h), q_2^+ (h)],} \\ {T[r_1^+ (h), r_2^+ (h)], T[s_1^+ (h), s_2^+ (h)],} \\ {T[I_{A_1 }^+ , I_{A_2 }^+ ], S[p_1 (h), p_2 (h)],} \\ {S[q_1 (h), q_2 (h)], S[r_1 (h), r_2 (h)],} \\ {S[s_1 (h), s_2 (h)], S[\mu _{A_1 } , \mu _{A_2 } ]\rangle } \\ \end{array} }} \right\} \);

  3. (c)

    \(\left\{ {{\begin{array}{l} {A_1 \cup _{T, S} A_2 =\langle S[p_1^- (h), p_2^- (h)],} \\ {S[q_1^- (h), q_2^- (h)], S[r_1^- (h), r_2^- (h)],} \\ {S[s_1^- (h), s_2^- (h)], S[I_{A_1 }^- , I_{A_2 }^- ],} \\ {S[p_1^+ (h), p_2^+ (h)],} \\ {S[q_1^+ (h), q_2^+ (h)], S[r_1^+ (h), r_2^+ (h)],} \\ {S[s_1^+ (h), s_2^+ (h)],} \\ {S[I_{A_1 }^+ , I_{A_2 }^+ ], T[p_1 (h), p_2 (h)],} \\ {T[q_1 (h), q_2 (h)],} \\ {T[r_1 (h), r_2 (h)],} \\ {T[s_1 (h), s_2 (h)], T[\mu _{A_1 } , \mu _{A_2 } ]\rangle } \\ \end{array} }} \right\} \);

where any pair (T, S) can be used. T denotes a \({{\varvec{t}}}\)-norm and S a so-called t-conorm dual to the t-norm T, defined by \(S(h, \tau )=1-T(1-h, 1-\tau )\) .

Example 8

Let \(A_1 =\left\{ {{\begin{array}{l} {\langle [(\{0.2, 0.4,} \\ {0.6, 0.8\}; 0.10],} \\ {[\{0.4, 0.6,} \\ {0.8, 0.10\}; 0.12)];} \\ {[\{0.3, 0.5,} \\ {0.7, 0.9\}; 0.11]\rangle } \\ \end{array} }} \right\} \) and \(A_2 =\left\{ {{\begin{array}{l} {\langle [\{0.10, 0.12,} \\ {0.14, 0.16\}; 0.20],} \\ {[\{0.12, 0.14,} \\ {0.16, 0.18\}; 0.24],} \\ {[\{0.11, 0.13,} \\ {0.15, 0.17\}; 0.22]\rangle } \\ \end{array} }} \right\} \) be two trapezoidal cubic fuzzy sets (TrCFSs)

  1. (a)

    \(\left\{ {{\begin{array}{l} {A_1 \subseteq A_2 \hbox { iff }\forall h\in H, 0.2\ge 0.10,} \\ {0.4\ge 0.12, 0.6\ge 0.14, 0.8\ge 0.16,} \\ {0.10\ge 0.20, 0.4\ge 0.12, 0.6\ge 0.14,} \\ {0.8\ge 0.16, 0.10\ge 0.18, 0.12\ge 0.24} \\ {\hbox {and }0.3\le 0.11, 0.5\le 0.13,} \\ {0.7\le 0.15, 0.9\le 0.17, 0.11\le 0.22} \\ \end{array} }} \right\} \);

  2. (b)

    \(A_1 \cap _{T, S} A_2 =\left\{ {{\begin{array}{l} {\langle T[0.2, 0.10], T[0.4, 0.12],} \\ {T[0.6, 0.14], T[0.8, 0.16],} \\ {T[0.10, 0.20], T[0.4, 0.12],} \\ {T[0.6, 0.14], T[0.8, 0.16],} \\ {T[0.10, 0.18], T[0.12, 0.24]\hbox { and}} \\ S[0.3, 0.11], S[0.5, 0.13],\\ \quad S[0.7, 0.15], \\ {S[0.9, 0.17], S[0.11, 0.22]\rangle } \\ \end{array} }} \right\} \)

  3. (c)

    \(A_1 \cup _{T, S} A_2 =\left\{ {{\begin{array}{l} {\langle S[0.2, 0.10], S[0.4, 0.12],} \\ {S[0.6, 0.14], S[0.8, 0.16],} \\ {S[0.10, 0.20], S[0.4, 0.12],} \\ {S[0.6, 0.14], S[0.8, 0.16],} \\ {S[0.10, 0.18], S[0.12, 0.24]\hbox { and}} \\ {T[0.3, 0.11], T[0.5, 0.13],} \\ T[0.7, 0.15], T[0.9, 0.17],\\ \quad T[0.11, 0.22]\rangle \\ \end{array} }} \right\} \)

Definition 9

Let \(A=\left\{ {{\begin{array}{l} {\langle \{[p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)], I_A^- \},} \\ {\{[p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)], I_A^+ \},} \\ {\{[p_A (h), q_A (h),} \\ {r_A (h), s_A (h)], \mu _A \}\rangle } \\ {|h\in H} \\ \end{array} }} \right\} \) be the trapezoidal cubic fuzzy number (TrCFN). Then, the score function S(A), accuracy function H(A), membership uncertainty index T(A) and hesitation uncertainty index G(A)of the trapezoidal cubic fuzzy number (TrCFN) A are defined by

$$\begin{aligned} S(A)= & {} \left\{ {\frac{{\begin{array}{l} {\langle [(I_A^- )[p_A^- (h)+q_A^- (h)+r_A^- (h)+s_A^- (h)]+} \\ {(I_A^+ )[p_A^+ (h)+q_A^+ (h)+r_A^+ (h)+s_A^+ (h)]]} \\ {-[p_A (h)+q_A (h)+r_A (h)+s_A (h)](\mu _A )\rangle } \\ \end{array} }}{12}} \right\} ; \\ H(A)= & {} \left\{ {\frac{{\begin{array}{l} {\langle [(I_A^- )[p_A^- (h)+q_A^- (h)+r_A^- (h)+s_A^- (h)]+} \\ {[p_A^+ (h)+q_A^+ (h)+r_A^+ (h)+s_A^+ (h)](I_A^+ )]} \\ {+[p_A (h)+q_A (h)+r_A (h)+s_A (h)](\mu _A )\rangle } \\ \end{array} }}{12}} \right\} ; \\ T(A)= & {} \left\{ {{\begin{array}{l} {\langle [(I_A^+ )[p_A^+ (h)+q_A^+ (h)+r_A^+ (h)+s_A^+ (h)]+} \\ {[p_A (h)+q_A (h)+r_A (h)+s_A (h)](\mu _A )-} \\ {[p_A^- (h)+q_A^- (h)+r_A^- (h)+s_A^- (h)](I_A^- )\rangle } \\ \end{array} }} \right\} ; \\ G(A)= & {} \left\{ {{\begin{array}{l} {\langle [(I_A^+ )[p_A^+ (h)+q_A^+ (h)+r_A^+ (h)+s_A^+ (h)]+} \\ {[p_A^- (h)+q_A^- (h)+r_A^- (h)+s_A^- (h)](I_A^- )} \\ {-[p_A (h)+q_A (h)+r_A (h)+s_A (h)](\mu _A )\rangle } \\ \end{array} }} \right\} . \\ \end{aligned}$$

Example 10

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{0.6, 0.8,} \\ {0.10, 0.12\}; 0.3];} \\ {[\{0.8, 0.10,} \\ {0.12, 0.14\}; 0.5],} \\ {[\{0.7, 0.9,} \\ {0.11, 0.13\}} \\ {; 0.4]\rangle } \\ \end{array} }} \right\} \)be the trapezoidal cubic fuzzy number (TrCFN). Then the score function S(A), accuracy function H(A), membership uncertainty index T(A) and hesitation uncertainty index G(A) of the trapezoidal cubic fuzzy number (TrCFN) A are defined by

$$\begin{aligned} S(A)= & {} \left\{ {{\begin{array}{l} {\frac{{\begin{array}{c} {\left[ {0.6+0.8+0.10+0.12} \right] (0.3)+} \\ {\left[ {0.8+0.10+0.12+0.14} \right] (0.5)} \\ {\left[ {0.7+0.9+0.11+0.13} \right] (0.4)\rangle } \\ \end{array} }}{12}} \\ {=\frac{0.306+0.58-0.376}{12}=\frac{0.886-0.376}{12}} \\ {=\frac{0.51}{12}=0.0425} \\ \end{array} }} \right\} ; \\ H(A)= & {} \left\{ {{\begin{array}{l} {\frac{{\begin{array}{c} {\left[ {0.6+0.8+0.10+0.12} \right] (0.3)+} \\ {\left[ {0.8+0.10+0.12+0.14} \right] (0.5)} \\ {+\left[ {0.7+0.9+0.11+0.13} \right] (0.4)\rangle } \\ \end{array} }}{12}} \\ {=\frac{0.306+0.58+0.376}{12}=\frac{1.262}{12}=0.1051} \\ \end{array} }} \right\} ; \\ T(A)= & {} \left\{ {{\begin{array}{l} {[0.8+0.10+0.12+0.14](0.5)+} \\ {[0.7+0.9+0.11+0.13](0.4)} \\ {-[0.6+0.8+0.10+0.12](0.3)\rangle } \\ {=0.58+0.736-0.486=0.83} \\ \end{array} }} \right\} ; \\ G(A)= & {} \left\{ {{\begin{array}{l} {\langle [[0.8+0.10+0.12+0.14](0.5)+} \\ {[0.6+0.8+0.10+0.12](0.3)-} \\ {[0.7+0.9+0.11+0.13](0.4)\rangle } \\ {=0.58+0.486-0.736=0.33} \\ \end{array} }} \right\} . \end{aligned}$$
figure a

4 Some Einstein operations on trapezoidal cubic fuzzy sets

In this section, we introduce the Einstein t-norm

$$\begin{aligned} T\left\{ {T(h, \tau )=\frac{h\tau }{1+(1-h)(1-\tau )}} \right\} \end{aligned}$$

and its dual t-conorm

$$\begin{aligned} S\left\{ {(S(h, \tau )=\frac{h+\tau }{1+h\tau })} \right\} , \end{aligned}$$

(then the generalized union d) on TrCFSs \(A_1 \) and \(A_2 \) becomes to the Einstein sum (denoted by \(A_1 +A_2 )\) on \(A_1 \) and \(A_2 \) as follows:

$$\begin{aligned} A_1 +A_2 =\left\{ {{\begin{array}{l} {\left\langle \left[ {{\begin{array}{l} \max (I_{A_1 }^- , I_{A_2 }^- )\left[ \frac{p_1^- (h)+p_2^- (h)}{1+p_1^- (h)p_2^- (h)},\right. \\ {\frac{q_1^- (h)+q_2^- (h)}{1+q_1^- (h)q_2^- (h)},} \\ \left. \frac{r_1^- (h)+r_2^- (h)}{1+r_1^- (h)r_2^- (h)}, \frac{s_1^- (h)+s_2^- (h)}{1+s_1^- (h)s_2^- (h)}\right] , \\ {\max (I_{A_1 }^+ , I_{A_2 }^+ )} \\ \left[ \frac{p_1^+ (h)+p_2^+ (h)}{1+p_1^+ (h)p_2^+ (h)}, \frac{q_1^+ (h)+q_2^+ (h)}{1+q_1^+ (h)q_2^+ (h)},\right. \\ {\frac{r_1^+ (h)+r_2^+ (h)}{1+r_1^+ (h)r_2^+ (h)},} \\ \left. \frac{s_1^+ (h)+s_2^+ (h)}{1+s_1^+ (h)s_2^+ (h)}\right] , \\ \end{array} }} \right] ,\right. } \\ {\left. \left[ {{\begin{array}{l} {\min (\mu _{A_1 } , \mu _{A_2 } )} \\ \left[ \frac{p_1 \cdot p_2 }{1+(\left( {1-p_1 (h)} \right) \left( {1-p_2 (h)} \right) )},\right. \\ {\frac{q_1 \cdot q_2 }{1+(\left( {1-q_1 (h)} \right) \left( {1-q_2 (h)} \right) )},} \\ {\frac{r_1 \cdot r_2 }{1+(\left( {1-r_1 (h)} \right) \left( {1-r_2 (h)} \right) )},} \\ \left. \frac{s_1 \cdot s_2 }{1+(\left( {1-s_1 (h)} \right) \left( {1-s_2 (h)} \right) )}\right] \\ \end{array} }} \right] \right\rangle } \\ \end{array} }} \right\} . \end{aligned}$$

By mathematical induction, we can derive the multiplication operation on trapezoidal cubic fuzzy sets (TrCFSs) as follows:

$$\begin{aligned} \lambda A=\left\{ {{\begin{array}{l} {\left\langle \left[ {{\begin{array}{l} \max (I_A^- )\left[ \frac{[1+p_A^- (h)]^{\lambda }-[1-p_A^- (h)]^{\lambda }}{[1+p_A^- (h)]^{\lambda }+[1-p_A^- (h)]^{\lambda }},\right. \\ {\frac{[1+q_A^- (h)]^{\lambda }-[1-q_A^- (h)]^{\lambda }}{[1+q_A^- (h)]^{\lambda }+[1-q_A^- (h)]^{\lambda }},} \\ {\frac{[1+r_A^- (h)]^{\lambda }-[1-r_A^- (h)]^{\lambda }}{[1+r_A^- (h)]^{\lambda }+[1-r_A^- (h)]^{\lambda }},} \\ \left. \frac{[1+s_A^- (h)]^{\lambda }-[1-s_A^- (h)]^{\lambda }}{[1+s_A^- (h)]^{\lambda }+[1-s_A^- (h)]^{\lambda }}\right] \\ \max (I_A^+ )\left[ \frac{[1+p_A^+ (h)]^{\lambda }-[1-p_A^+ (h)]^{\lambda }}{[1+p_A^+ (h)]^{\lambda }+[1-p_A^+ (h)]^{\lambda }},\right. \\ {\frac{[1+q_A^+ (h)]^{\lambda }-[1-q_A^+ (h)]^{\lambda }}{[1+q_A^+ (h)]^{\lambda }+[1-q_A^+ (h)]^{\lambda }},} \\ {\frac{[1+r_A^+ (h)]^{\lambda }-[1-r_A^+ (h)]^{\lambda }}{[1+r_A^+ (h)]^{\lambda }+[1-r_A^+ (h)]^{\lambda }},} \\ \left. \frac{[1+s_A^+ (h)]^{\lambda }-[1-s_A^+ (h)]^{\lambda }}{[1+s_A^+ (h)]^{\lambda }+[1-s_A^+ (h)]^{\lambda }}\right] \\ \end{array} }} \right] ,\right. } \\ {\left. \left[ {{\begin{array}{l} \min (\mu _A )\left[ \frac{2[p_A (h)]^{\lambda }}{[(2-p_A (h)]^{\lambda }+[p_A (h)]^{\lambda }},\right. \\ {\frac{2[p_A (h)]^{\lambda }}{[(2-p_A (h)]^{\lambda }+[p_A (h)]^{\lambda }},} \\ {\frac{2[r_A (h)]^{\lambda }}{[(2-r_A (h)]^{\lambda }+[r_A (h)]^{\lambda }},} \\ \left. \frac{2[s_A (h)]^{\lambda }}{[(2-s_A (h)]^{\lambda }+[s_A (h)]^{\lambda }}\right] \\ \end{array} }} \right] \right\rangle } \\ \end{array} }} \right\} . \end{aligned}$$

where \(\lambda \) is any positive real number.

Definition 11

Let \(A_1 =\left\{ {{\begin{array}{l} {\langle [\{p_1^- (h), q_1^- (h),} \\ {r_1^- (h), s_1^- (h)\};} \\ {I_{A_1 }^- ],} \\ {[\{p_1^+ (h), q_1^+ (h),} \\ {r_1^+ (h), s_1^+ (h)\};} \\ {I_{A_1 }^+ ],} \\ {[\{p_1 (h), q_1 (h),} \\ {r_1 (h), s_1 (h)\};} \\ {\mu _{A_1 } ]\rangle |h\in H} \\ \end{array} }} \right\} \) and \(A_2 =\left\{ {{\begin{array}{l} {\langle [\{p_2^- (h), q_2^- (h),} \\ {r_2^- (h), s_2^- (h)\};} \\ {I_{A_2 }^- ],} \\ {[\{p_2^+ (h), q_2^+ (h),} \\ {r_2^+ (h), s_2^+ (h)\};} \\ {I_{A_2 }^+ ],} \\ {[\{p_2 (h), q_2 (h),} \\ {r_2 (h), s_2 (h)\};} \\ {\mu _{A_2 } ]\rangle |h\in H} \\ \end{array} }} \right\} \) be two trapezoidal cubic fuzzy numbers (TrCFNs). Then some Einstein operations of \(A_1 \) and \(A_2 \) can be defined as:

$$\begin{aligned} A_1 +A_2 =\left\{ {{\begin{array}{l} {\left\langle \left[ {{\begin{array}{c} {\max (I_{A_1 }^- , I_{A_2 }^- )} \\ \left[ \frac{p_1^- (h)+p_2^- (h)}{1+p_1^- (h)p_2^- (h)},\right. \\ {\frac{q_1^- (h)+q_2^- (h)}{1+q_1^- (h)q_2^- (h)},} \\ {\frac{r_1^- (h)+r_2^- (h)}{1+r_1^- (h)r_2^- (h)},} \\ \left. \frac{s_1^- (h)+s_2^- (h)}{1+s_1^- (h)s_2^- (h)}\right] ; \\ \max (I_{A_1 }^+ , I_{A_2 }^+ )\left[ \frac{p_1^+ (h)+p_2^+ (h)}{1+p_1^+ (h)p_2^+ (h))},\right. \\ {\frac{q_1^+ (h)+q_1^+ (h)}{1+q_1^+ (h)q_2^+ (h))},} \\ {\frac{r_1^+ (h)+r_2^+ (h)}{1+r_1^+ (h)r_2^+ (h))},} \\ \left. \frac{s_1^+ (h)+s_2^+ (h)}{1+s_1^+ (h)s_2^+ (h))}\right] \\ \end{array} }} \right] ,\right. } \\ {\left. \left[ {{\begin{array}{l} {\min (\mu _{A_1 } , \mu _{A_2 } )} \\ \left[ \frac{p_1 (h)\cdot p_2 (h)}{(1+\left( {1-p_1 (h)} \right) \left( {1-p_2 (h)} \right) )},\right. \\ {\frac{q_1 (h)\cdot q_2 (h)}{(1+(\left( {1-q_1 (h)} \right) \left( {1-q_2 (h)} \right) )},} \\ {\frac{r_1 (h)\cdot r_2 (h)}{(1+(1-r_1 (h)\left( {1-r_2 (h)} \right) )},} \\ {\frac{s_1 (h)\cdot s_2 (h)}{(1+(\left( {1-s_1 (h)} \right) \left( {1-s_2 (h)} \right) )}} \\ \end{array} }} \right] \right\rangle } \\ \end{array} }} \right\} ; \\ \lambda A=\left\{ {{\begin{array}{l} {\left\langle \left[ {{\begin{array}{l} \max (I_A^- )\left[ \frac{[1+p_A^- (h)]^{\lambda }-[1-p_A^- (h)]^{\lambda }}{[1+p_A^- (h)]^{\lambda }+[1-p_A^- (h)]^{\lambda }},\right. \\ {\frac{[1+q_A^- (h)]^{\lambda }-[1-q_A^- (h)]^{\lambda }}{[1+q_A^- (h)]^{\lambda }+[1-q_A^- (h)]^{\lambda }},} \\ {\frac{[1+r_A^- (h)]^{\lambda }-[1-r_A^- (h)]^{\lambda }}{[1+r_A^- (h)]^{\lambda }+[1-r_A^- (h)]^{\lambda }},} \\ {\frac{[1+s_A^- (h)]^{\lambda }-[1-s_A^- (h)]^{\lambda }}{[1+s_A^- (h)]^{\lambda }+[1-s_A^- (h)]^{\lambda }}} \\ \max (I_A^+ )\left[ \frac{[1+p_A^+ (h)]^{\lambda }-[1-p_A^+ (h)]^{\lambda }}{[1+p_A^+ (h)]^{\lambda }+[1-p_A^+ (h)]^{\lambda }},\right. \\ {\frac{[1+q_A^+ (h)]^{\lambda }-[1-q_A^+ (h)]^{\lambda }}{[1+q_A^+ (h)]^{\lambda }+[1-q_A^+ (h)]^{\lambda }},} \\ {\frac{[1+r_A^+ (h)]^{\lambda }-[1-r_A^+ (h)]^{\lambda }}{[1+r_A^+ (h)]^{\lambda }+[1-r_A^+ (h)]^{\lambda }},} \\ \left. \frac{[1+s_A^+ (h)]^{\lambda }-[1-s_A^+ (h)]^{\lambda }}{[1+s_A^+ (h)]^{\lambda }+[1-s_A^+ (h)]^{\lambda }}\right] ; \\ \end{array} },} \right] \right. } \\ {\left. \left[ {{\begin{array}{l} \min (\mu _A )\left[ \frac{2[p_A (h)]^{\lambda }}{[(2-p_A (h)]^{\lambda }+[p_A (h)]^{\lambda }},\right. \\ {\frac{2[q_A (h)]^{\lambda }}{[(2-q_A (h)]^{\lambda }+[q_A (h)]^{\lambda }},} \\ {\frac{2[r_A (h)]^{\lambda }}{[(2-r_A (h)]^{\lambda }+[r_A (h)]^{\lambda }},} \\ \left. \frac{2[s_A (h)]^{\lambda }}{[(2-s_A (h)]^{\lambda }+[s_A (h)]^{\lambda }}\right] \\ \end{array} }} \right] \right\rangle } \\ \end{array} }} \right\} . \end{aligned}$$

Proposition 12

Let \(A, A_1 \) and \(A_2 \) be three trapezoidal cubic fuzzy numbers and \(\lambda , \lambda _1 , \lambda _2 >0,\) then we have:

  1. (1)

    \(A_1 +A_2 =A_2 +A_1 ,\)

  2. (2)

    \(\lambda (A_1 +A_2 )=\lambda A_2 +\lambda A_1 ,\)

  3. (3)

    \(\lambda _1 A+\lambda _2 A=(\lambda _1 +\lambda _2 )A.\)

Proof

The proof of this proposition is provided in Appendix A. \(\square \)

Remark 13

If \(\alpha _1 \le _{L_{\hbox {TrCFN}} } \alpha _2 \), then \(\alpha _1 \le \alpha _2 \), that is the total order contains the usual partial order on \(L_{\hbox {TrCFN}}\).

5 Trapezoidal cubic fuzzy arithmetic averaging operators based on Einstein operations

In this section, we introduce the concept of trapezoidal cubic fuzzy Einstein weighted averaging (TCFEWA) operator, trapezoidal cubic fuzzy Einstein ordered weighted averaging (TCFEOWA) operator and discuss some of its properties.

Definition 14

(Beliakov et al. 2007; Grabisch et al. 2009) An aggregation function \(f:[0, 1]^{n}\rightarrow [0, 1]\) is a function non-decreasing in each argument, that is \(A_j \le B_j \) for all \(j\in \{1, 2,..., n\}\) implies \(f(A_1 , A_2 ,..., A_n )\le f(B_1 , B_2 ,..., B_n )\) and satisfying \(f(0, 0,..., 0)=0\) and \(f(1, 1,..., 1)=1\).

Definition 15

\(f_{L_{\hbox {TrCFN}} }:L_{\hbox {TrCFN}}^n \rightarrow L_{\hbox {TrCFN}} \) is an aggregation function if it is monotone with respect to \(\le L_{\hbox {TrCFN}} \) and satisfies \(f_{L_{\hbox {TrCFN}} } \quad (0_{L_{\hbox {TrCFN}} },\)...\(, 0_{L_{\hbox {TrCFN}} } )=0_{L_{\hbox {TrCFN}} } \) and \(f_{L_{\hbox {TrCFN}} } (1_{L_{\hbox {TrCFN}} } ,\)...\(, 1_{L_{\hbox {TrCFN}} } )=1_{L_{\hbox {TrCFN}} } \).

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5.1 Trapezoidal cubic fuzzy Einstein weighted averaging operator

Definition 16

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\}; I_A^- ]} \\ {,[\{p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)\}; I_A^+ ]} \\ {,[\{(p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\}; \mu _A ]\rangle |} \\ {h\in H} \\ \end{array} }} \right\} \) be a collection of TrCFNs in \(L_{\hbox {TrCFN}} \) and \(\varpi =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) is the weight vector of \(A_j (j=1, 2,..., n)\) such that \(\varpi _j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\) . Then trapezoidal cubic fuzzy Einstein weighted averaging operator of dimension n is a mapping TrCFEWA \(:\;\;L_{\hbox {TrCFN}}^n \rightarrow L_{\hbox {TrCFN}} \) and TrCFEWA \((A_1 , A_2 ,..., A_n )=\varpi _1 A_1 +\varpi _2 A_2 ,..., \varpi _n A_n \) If \(\varpi =(\frac{1}{n}, \frac{1}{n},..., \frac{1}{n})^{T}\) . Then the TrCFEWA operator is reduced to trapezoidal cubic fuzzy Einstein averaging operator of dimension n , which is defined as follows: TrCFEA\((A_1 ,A_2 ,...,A_n )=\frac{1}{n}(A_1 +A_2 +...+A_n)\)

Example 17

Let \(A_1 =\left\{ {{\begin{array}{l} {\langle [0.9, 0.11,} \\ {0.13, 0.15],(0.2)} \\ {[0.11, 0.13,} \\ {0.15, 0.17],(0.4)} \\ {[0.10, 0.12,} \\ {0.14, 0.16](0.3)\rangle } \\ \end{array} }} \right\} \) and \(A_2 =\left\{ {{\begin{array}{l} {\langle [0.2, 0.4,} \\ {0.6, 0.8],(0.1)} \\ {[0.4, 0.6,} \\ {0.8, 0.10],(0.3)} \\ {[0.3, 0.5,} \\ \end{array} }} \right\} \) be a collection of TrCFNs in \(L_{\hbox {TrCFN}}\) . Then their aggregated value by using the TrCFEWA operator is also the TrCFN and \(\varpi =0.3, 0.4, 0.3\)

TrCFEWA \((A_1 , A_2,...,A_n )=\)

$$\begin{aligned}&\langle 0.2\left[ {{\begin{array}{l} {{\begin{array}{l} {\frac{\mathop {\prod }\nolimits _{j=1}^2 [1+0.9]^{0.3}[1+0.2]^{0.3}-\mathop {\prod }\nolimits _{j=1}^2 [1-0.9]^{0.3}[1-0.2]^{0.3}}{\mathop {\prod }\nolimits _{j=1}^2 [1+0.9]^{0.3}[1+0.2]^{0.3}+\mathop {\prod }\nolimits _{j=1}^2 [1-0.9]^{0.3}[1-0.s]^{0.3}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^2 [1+0.11]^{0.4}[1+0.4]^{0.4}-\mathop {\prod }\nolimits _{j=1}^2 [1-0.11]^{0.4}[1-0.4]^{0.4}}{\mathop {\prod }\nolimits _{j=1}^2 [1+0.11]^{0.4}[1+0.4]^{0.4}+\mathop {\prod }\nolimits _{j=1}^2 [1-0.11]^{0.4}[1-0.4]^{0.4}}} \\ \end{array} },} \\ {{\begin{array}{c} {\frac{\mathop {\prod }\nolimits _{j=1}^2 [1+0.13]^{0.4}[1+0.6]^{0.4}-\mathop {\prod }\nolimits _{j=1}^2 [1-0.13]^{0.4}[1-0.6]^{0.4}}{\mathop {\prod }\nolimits _{j=1}^2 [1+0.13]^{0.4}[1+0.6]^{0.4}+\mathop {\prod }\nolimits _{j=1}^2 [1-0.13]^{0.4}[1-0.6]^{0.4}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^2 [1+0.15]^{0.4}[1+0.8]^{0.4}-\mathop {\prod }\nolimits _{j=1}^2 [1-0.15]^{0.4}[1-0.8]^{0.4}}{\mathop {\prod }\nolimits _{j=1}^2 [1+0.15]^{0.4}[1+0.8]^{0.4}+\mathop {\prod }\nolimits _{j=1}^2 [1-0.15]^{0.4}[1-0.8]^{0.4}}} \\ \end{array} }} \\ \end{array} }} \right] , \\&0.4\left[ {{\begin{array}{l} {{\begin{array}{l} {\frac{\mathop {\prod }\nolimits _{j=1}^2 [1+0.11]^{0.4}[1+0.4]^{0.4}-\mathop {\prod }\nolimits _{j=1}^2 [1-0.11]^{0.4}[1-0.4]^{0.4}}{\mathop {\prod }\nolimits _{j=1}^2 [1+0.11]^{0.4}[1+0.4]^{0.4}+\mathop {\prod }\nolimits _{j=1}^2 [1-0.11]^{0.4}[1-0.4]^{0.4}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^2 [1+0.13]^{0.4}[1+0.6]^{0.4}-\mathop {\prod }\nolimits _{j=1}^2 [1-0.13]^{0.4}[1-0.6]^{0.4}}{\mathop {\prod }\nolimits _{j=1}^2 [1+0.13]^{0.4}[1+0.6]^{0.4}+\mathop {\prod }\nolimits _{j=1}^2 [1-0.13]^{0.4}[1-0.6]^{0.4}}} \\ \end{array} },} \\ {{\begin{array}{l} {\frac{\mathop {\prod }\nolimits _{j=1}^2 [1+0.15]^{0.4}[1+0.8]^{0.4}-\mathop {\prod }\nolimits _{j=1}^2 [1-0.15]^{0.4}[1-0.8]^{0.4}}{\mathop {\prod }\nolimits _{j=1}^2 [1+0.15]^{0.4}[1+0.8]^{0.4}+\mathop {\prod }\nolimits _{j=1}^2 [1-0.15]^{0.4}[1-0.8]^{0.4}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^2 [1+0.17]^{0.4}[1+0.10]^{0.4}-\mathop {\prod }\nolimits _{j=1}^2 [1-0.17]^{0.4}[1-0.10]^{0.4}}{\mathop {\prod }\nolimits _{j=1}^2 [1+0.17]^{0.4}[1+0.10]^{0.4}+\mathop {\prod }\nolimits _{j=1}^2 [1-0.17]^{0.4}[1-0.10]^{0.4}}} \\ \end{array} }} \\ \end{array} }} \right] \\&(0.2)\left[ {{\begin{array}{l} {\frac{2\mathop {\prod }\nolimits _{j=1}^2 [0.10]^{0.3}[0.3]^{0.3}}{\mathop {\prod }\nolimits _{j=1}^2 [(2-0.10]^{0.3}[(2-0.3]^{0.3}+\mathop {\prod }\nolimits _{j=1}^2 [0.10]^{0.3}[0.3]^{0.3}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^2 [0.12]^{0.3}[0.5]^{0.3}}{\mathop {\prod }\nolimits _{j=1}^2 [(2-0.12]^{0.3}[(2-0.5]^{0.3}+\mathop {\prod }\nolimits _{j=1}^2 [0.12]^{0.3}[0.5]^{0.3}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^2 [0.14]^{0.3}[0.7]^{0.3}}{\mathop {\prod }\nolimits _{j=1}^2 [(2-0.14]^{0.3}[(2-0.7]^{0.3}+\mathop {\prod }\nolimits _{j=1}^2 [0.14]^{0.3}[0.7]^{0.3}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^2 [0.16]^{0.3}[0.9]^{0.3}}{\mathop {\prod }\nolimits _{j=1}^2 [(2-0.16]^{0.3}[(2-0.9]^{0.3}+\mathop {\prod }\nolimits _{j=1}^2 [0.16]^{0.3}[0.9]^{0.3}}} \\ \end{array} }} \right] \rangle \\&=\left\{ {{\begin{array}{l} {{\begin{array}{l} {\langle [0.2\frac{1.1382-0.8312}{1.1382+0.8312}, 0.4\frac{1.3493-0.6009}{1.3493+0.6009}],} \\ {(0.2)[\frac{1.0023}{1.3471+0.5012}]\rangle } \\ {=\langle [(0.2\frac{0.307}{1.9694}, 0.4\frac{0.7484}{1.9502}],} \\ {(0.2)[\frac{1.0023}{1.8486}]\rangle } \\ \end{array} }} \\ {=\langle [0.2(0.1558); 0.4(0.3837)];} \\ {(0.2)[0.5421]\rangle } \\ {=\langle [0.0311; 0.1534]; 0.1084\rangle } \\ \end{array} }} \right\} . \end{aligned}$$

Theorem 18

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\}; I_A^- ],} \\ {[\{p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)\}; I_A^+ ],} \\ {[\{(p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\}; \mu _A ]\rangle } \\ {|h\in H} \\ \end{array} }} \right\} \) be a collection of TrCFNs in \(L_{\hbox {TrCFN}} \) . Then their aggregated value by using the TrCFEWA operator is also the TrCFN and TrCFEWA \((A_1 , A_2 ,..., A_n )=\langle \max (I_A^- )\)

$$\begin{aligned}&\left[ {{\begin{array}{l} {\frac{[\mathop {\prod }\nolimits _{j=1}^n [1+p_1^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-p_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-p_1^- (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_1^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-q_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-q_1^- (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_1^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-r_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-r_1^- (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_1^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-s_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-s_1^- (h)]^{^{\varpi }}}} \\ \end{array} }} \right] , \\&\max (I_A^+ )\left[ {{\begin{array}{l} {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-p_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-p_1^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-q_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-q_1^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-r_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-r_1^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-s_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-s_1^+ (h)]^{^{\varpi }}}} \\ \end{array} }} \right] , \\&\min (\mu _A )\left[ {{\begin{array}{l} {\frac{2\mathop {\prod }\nolimits _{j=1}^n [p_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-p_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [p_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [q_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-q_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [q_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [r_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-r_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [r_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [s_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-s_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [s_1 (h)]^{^{\varpi }}}} \\ \end{array} }} \right] \rangle \end{aligned}$$

where \(\varpi =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) is the weight vector of \(A_j (j=1, 2,..., n)\) such that \(\varpi _j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\).

Proof

The proof of this theorem is provided in “Appendix B.” \(\square \)

Proposition 19

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\}; I_A^- ]} \\ {[\{p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)\}; I_A^+ ]} \\ {[\{p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\}, \mu _A ]\rangle |} \\ {h\in H} \\ \end{array} }} \right\} \) be a collection of TrCFNs in \(L_{\hbox {TrCFN}} \) and where \(\varpi \quad =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) is the weight vector of \(A_j (j=1, 2,..., n)\) such that \(\varpi _j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\) . Then

  1. (1)

    (Idempotency): If all \(A_j , j=1, 2,..., n\) are equal, i.e., \(A_j =A,\) for all \(j=1, 2,..., n,\) then TrCFEWA \((A_1 , A_2 ,..., A_n )=A.\)

  2. (2)

    (Boundary): If

    $$\begin{aligned} p_{\min }^-= & {} \min _{1\le j\le n} p_j^- , \\ q_{\min }^-= & {} \min _{1\le j\le n} q_j^- , \quad r_{\min }^-=\min _{1\le j\le n} r_j^- , \\ s_{\min }^-= & {} \min _{1\le j\le n} s_j^- , \quad I_{\min }^- =\min _{1\le j\le n} I_j^- , \\ p_{\min }^+= & {} \min _{1\le j\le n} p_j^+ , \quad q_{\min }^+ =\min _{1\le j\le n} q_j^+ , \\ r_{\min }^+= & {} \min _{1\le j\le n} r_j^+, \quad s_{\min }^+ =\min _{1\le j\le n} s_j^+ , \\ I_{\min }^+= & {} \min _{1\le j\le n} I_j^+ , \quad p_{\max } =\max _{1\le j\le n} p_j , \\ q_{\max }= & {} \max _{1\le j\le n} q_j , \quad r_{\max } =\max _{1\le j\le n} r_j , \\ s_{\max }= & {} \max _{1\le j\le n} s_j , \quad \mu _{\max } =\max _{1\le j\le n} \mu _j , \\ p_{\max }^-= & {} \max _{1\le j\le n} p_j^- , \quad q_{\max }^- =\max _{1\le j\le n} q_j^- , \\ r_{\max }^-= & {} \max _{1\le j\le n} r_j^- , \quad s_{\max }^- =\max _{1\le j\le n} s_j^- \\ \mu _{\max }^-= & {} \max _{1\le j\le n}\mu _j^- ,\quad p_{\max }^+ =\max _{1\le j\le n} p_j^+ , \\ q_{\max }^+= & {} \max _{1\le j\le n} q_j^+ , \quad r_{\max }^+ =\max _{1\le j\le n} r_j^+ , \\ s_{\max }^+= & {} \max _{1\le j\le n} s_j^+ , \quad I_{\max }^+ =\max _{1\le j\le n} I_j^+ , \\ p_{\min }= & {} \min _{1\le j\le n} p_j , \quad q_{\min } =\min _{1\le j\le n} q_j , \\ r_{\min }= & {} \min _{1\le j\le n} r_j ,\quad s_{\min } =\min _{1\le j\le n} s_j , \\ \mu _{\min }= & {} \min _{1\le j\le n} \mu _j \end{aligned}$$

    for all \(j=1, 2,.., n,\) we can obtain that \(\left\{ {{\begin{array}{l} {\langle [p_{\min }^- (h), q_{\min }^- (h),} \\ {r_{\min }^- (h), s_{\min }^- (h)],(I_{A_j }^- )} \\ {[p_{\min }^+ (h), q_{\min }^+ (h),} \\ {r_{\min }^+ (h), s_{\min }^+ (h)],(I_{A_j }^+ )} \\ {[p_{\max } (h), q_{\max } (h),} \\ {r_{\max } (h), s_{\max } (h)],(\mu _{A_j } )\rangle |} \\ {h\in H} \\ \end{array} }} \right\} \)\(\le \) TrCFEWA \((A_1 , A_2,..., A_n )\)

    $$\begin{aligned} \le \left\{ {{\begin{array}{l} {\langle [p_{\max }^- (h), q_{\max }^- (h),} \\ {r_{\max }^- (h), s_{\max }^- (h)],(I_{A_j }^- )} \\ {[p_{\max }^+ (h), q_{\max }^+ (h),} \\ {r_{\max }^+ (h), s_{\max }^+ (h)],(I_{A_j }^+ )} \\ {[p_{\min } (h), q_{\min } (h),} \\ {r_{\min } (h), s_{\min } (h)],(\mu _{A_j } )\rangle |} \\ {h\in H} \\ \end{array} }} \right\} . \end{aligned}$$
  3. (3)

    (Monotonicity): \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\};} \\ {I_A^- ],[\{p_A^+ (h),} \\ {q_A^+ (h), r_A^+ (h),} \\ {s_A^+ (h)\}; I_A^+ ]} \\ {,[\{(p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\};} \\ {\mu _A ]\rangle |h\in H} \\ \end{array} }} \right\} \) and \(B=\left\{ {{\begin{array}{c} {\langle [p_B^- (h), q_B^- (h),} \\ {r_B^- (h), s_B^- (h)],} \\ {(I_B^- ),[p_B^+ (h),} \\ {q_B^+ (h), r_B^+ (h),} \\ {s_B^+ (h)],(I_B^+ )} \\ {[p_B (h), q_B (h),} \\ {r_B (h), s_B (h)],} \\ {(\mu _B )\rangle |h\in H} \\ \end{array} }} \right\} \) be two collection of TrCFNs in \(L_{\mathrm{TrCFN}} \) and \(A_j \le L_{\mathrm{TrCFN}} B_j\), i.e.,

    $$\begin{aligned} p_A^- (h)\le & {} p_B^- (h), \quad q_A^- (h)\le q_B^- (h),\\ r_A^- (h)\le & {} r_B^- (h), \quad s_A^- (h)\le s_B^- (h), \\ p_A^+ (h)\le & {} p_B^+ (h), \quad q_A^+ (h)\le q_B^+ (h), \\ r_A^+(h)\le & {} r_B^+ (h), \quad s_A^+ (h)\le s_B^+ (h)\, { and } \,p_A(h)\le p_B (h), \\ \quad q_A (h)\le & {} q_B (h), \quad r_A (h)\le r_B(h), \\ \quad s_A (h)\le & {} s_B (h) \end{aligned}$$

    then \(TrCFEWA(A_1 , A_2 ,..., A_n )\le TrCFEWA(B_1, B_2 ,..., B_n )\).

Proof

The proof of these propositions 1, 2, 3 is provided in “Appendix C.” \(\square \)

Corollary 20

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\};} \\ {I_A^- ],[\{p_A^+ (h),} \\ {q_A^+ (h), r_A^+ (h),} \\ {s_A^+ (h)\}; I_A^+ ]} \\ {,[\{(p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\};} \\ {\mu _A ]\rangle |h\in H} \\ \end{array} }} \right\} \) be a collection of TrCFNs in \(L_{\hbox {TrCFN}} \) and where \(\varpi \quad =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) is the weight vector of \(A_j (j=1, 2,..., n)\) such that \(\varpi _j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\). Then

TrCFWA \((A_1 , A_2 ,., A_n )\le \) TrCFWA \((A_1 , A_2 ,., A_n )\).

Proof

Omitted. \(\square \)

5.2 Trapezoidal cubic fuzzy Einstein ordered weighted averaging operator

We also develop a type of trapezoidal cubic fuzzy Einstein ordered weighted averaging (TrCFEOWA) operator.

Definition 21

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\},} \\ {I_A^- ],[\{p_A^+ (h),} \\ {q_A^+ (h), r_A^+ (h),} \\ {s_A^+ (h)\}, I_A^+ ]} \\ {[\{p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\},} \\ {\mu _A ]\rangle |h\in H}\\ \end{array}}}\right\} \) be a collection of TrCFNs in \(L_{\mathrm{TrCFN}} \) , is also the TrCFEOWA operator of dimension n, is a mapping TrCFEOWA : \(L_{\hbox {TCFN}}^n \rightarrow L_{\hbox {TCFN}} \) , that is an associated vector \(\varpi \quad =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) such that \(\varpi _j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\) . TrCFEOWA \((A_1 , A_2 ,..., A_n )=\varpi _1 A_{(\sigma )1} +\varpi _2 A_{(\sigma )2} ,..., \varpi _n A_{(\sigma )n} \), where \((\sigma (1),\sigma (2),...,\sigma (n))\) is a permutation of (1, 2, ..., n) such that \(A_{\sigma (1)} \le A_{\sigma (j-1)} \) for all \(j=2, 3,..., n(i.e.,A_{\sigma (j)} \) is the j th largest value in the collection \((A_1 , A_2 ,..., A_n ).\) If \(\varpi =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}=(\frac{1}{n}, \frac{1}{n},..., \frac{1}{n})^{T}\) , then the TrCFEOWA operator is reduced to the TrCFA operator (2) of dimension n.

Theorem 22

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\},} \\ {I_A^- ][\{p_A^+ (h),} \\ {q_A^+ (h), r_A^+ (h),} \\ {s_A^+ (h)\}, I_A^+ ]} \\ {[\{p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\},} \\ {\mu _A ]\rangle |h\in H} \\ \end{array} }} \right\} \) be a collection of trapezoidal cubic fuzzy numbers (TrCFNs) in \(L_{\hbox {TrCFN}} \). Then their aggregated value by using the TCFEOWA operator is also the TrCFN and

TrCFEOWA \((A_1 , A_2 ,..., A_n )=\)

$$\begin{aligned}&\langle \max [I_A^- ]\left[ {{\begin{array}{l} {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_{\sigma (j)}^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-p_{\sigma (j)}^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_{\sigma (j)}^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-p_{\sigma (j)}^- (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_{\sigma (j)}^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-q_{\sigma (j)}^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_{\sigma (j)}^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-q_{\sigma (j)}^- (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_{\sigma (j)}^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-r_{\sigma (j)}^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_{\sigma (j)}^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-r_{\sigma (j)}^- (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_{\sigma (j)}^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-s_{\sigma (j)}^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_{\sigma (j)}^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-s_{\sigma (j)}^- (h)]^{^{\varpi }}}} \\ \end{array} }} \right] , \\&\max [I_A^+ ]\left[ {{\begin{array}{l} {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_{\sigma (j)}^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-p_{\sigma (j)}^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_{\sigma (j)}^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-p_{\sigma (j)}^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_{\sigma (j)}^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-q_{\sigma (j)}^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_{\sigma (j)}^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-q_{\sigma (j)}^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_{\sigma (j)}^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-r_{\sigma (j)}^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_{\sigma (j)}^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-r_{\sigma (j)}^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_{\sigma (j)}^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-s_{\sigma (j)}^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_{\sigma (j)}^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-s_{\sigma (j)}^+ (h)]^{^{\varpi }}}} \\ \end{array} }} \right] , \\&\min [\mu _A ]\left[ {{\begin{array}{l} {\frac{2\mathop {\prod }\nolimits _{j=1}^n [p_{\sigma (j)} (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-p_{\sigma (j)} (h)]^{\mu }+\mathop {\prod }\nolimits _{j=1}^n [p_{\sigma (j)} (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [q_{\sigma (j)} (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-q_{\sigma (j)} (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [q_{\sigma (j)} (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [r_{\sigma (j)} (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-r_{\sigma (j)} (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [r_{\sigma (j)} (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [s_{\sigma (j)} (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-s_{\sigma (j)} (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [s_{\sigma (j)} (h)]^{^{\varpi }}}} \\ \end{array} }} \right] \rangle \end{aligned}$$

where \((\sigma (1), \sigma (2),..., \sigma (n))\) is a permutation of (1, 2, ..., n) such that \(A_{\sigma (1)} \le A_{\sigma (j-1)} \) for all \(j=2, 3,..., n, \quad \varpi \quad =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) is the weight vector of \(A_j (j=1, 2,..., n)\) such that \(\varpi _j \in [0, 1],\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1.\)

Proof

The process of this proof is the same as theorem 1. \(\square \)

Proposition 23

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\}, I_A^- ]} \\ {[\{p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)\}, I_A^+ ]} \\ {[\{p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\},} \\ {\mu _A ]\rangle |} \\ {h\in H} \\ \end{array} }} \right\} \) be a collection of TrCFNs in \(L_{\hbox {TrCFN}} \) and where \(\varpi =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) be the weighting vector of the TrCFEOWA operator, such that \(\varpi _j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\) . Then TrCFEOWA \((A_1 , A_2 ,..., A_n )\)

\(\le \) TrCFEOWA \((B_1 , B_2 ,..., B_n )\) , where \((A_1 , A_2 ,..., A_n )\) is any permutation of \((B_1 , B_2 ,..., B_n ).\)

Based on algebraic operations on trapezoidal cubic fuzzy sets (TrCFSs), defined the TrCFEOWA operator as follows:

TrCFEOWA \((A_1 , A_2 ,..., A_n )=\)

$$\begin{aligned} \left\{ {{\begin{array}{l} {\langle \{\max (I_A^- )[1-\mathop {\prod }\nolimits _{j=1}^n [1-p_{\sigma (j)}^- ],} \\ {[1-\mathop {\prod }\nolimits _{j=1}^n [1-q_{\sigma (j)}^- ]],[1-\mathop {\prod }\nolimits _{j=1}^n [1-r_{\sigma (j)}^- ]]} \\ {,[1-\mathop {\prod }\nolimits _{j=1}^n [1-s_{\sigma (j)}^- ]]\},\{\max (I_A^+ )} \\ {[1-\mathop {\prod }\nolimits _{j=1}^n [1-p_{\sigma (j)}^+ ]],[1-\mathop {\prod }\nolimits _{j=1}^n [1-q_{\sigma (j)}^+ ]],} \\ {[1-\mathop {\prod }\nolimits _{j=1}^n [1-r_{\sigma (j)}^+ ]],[1-\mathop {\prod }\nolimits _{j=1}^n [1-s_{\sigma (j)}^+ ]]\},} \\ {\min (\mu _A )\{\mathop {\prod }\nolimits _{j=1}^n p_{\sigma (j)} , \mathop {\prod }\nolimits _{j=1}^n q_{\sigma (j)} , \mathop {\prod }\nolimits _{j=1}^n r_{\sigma (j)} ,} \\ {\mathop {\prod }\nolimits _{j=1}^n s_{\sigma (j)} \}\rangle } \\ \end{array} }} \right\} \end{aligned}$$

and proposed an approach to solving group decision-making problems, where \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\}; I_A^- ]} \\ {,[\{p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)\}; I_A^+ ]} \\ {,[\{(p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\}; \mu _A ]\rangle |} \\ {h\in H} \\ \end{array} }} \right\} \) be a collection of TrCFNs in \(L_{Tr\hbox {CFN}}, (\sigma (1), \sigma (2),..., \sigma (n))\) is a permutation of (1, 2, ..., n) such that \(A_{\sigma (j)} \le A_{\sigma (j-1)} \) for all \(j=2, 3,..., n\) and where \(\varpi \quad =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) is the weight vector of the TrCFEOWA operator \(A_j (j=1, 2,..., n)\) such that \(\varpi _j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\).

Proof

Omitted. \(\square \)

Corollary 24

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\}, I_A^- ]} \\ {[\{p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)\}, I_A^+ ]} \\ {[\{p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\}, \mu _A ]\rangle |} \\ {h\in H} \\ \end{array} }} \right\} \) be a collection of TrCFNs in \(L_{\hbox {TrCFN}} \) and where

\(\varpi =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) is the weight vector of TrCFEOWA such that \(\varpi _j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\) . Then TrCFEOWA \((A_1 , A_2 ,..., A_n )\le \) TrCFEOWA \((A_1 , A_2 ,..., A_n ).\)

Proof

Omitted \(\square \)

5.3 Trapezoidal cubic fuzzy Einstein hybrid weighted averaging operator

The TrCFEWA operator weights individual the TrCFNs and the TrCFEOWA operator weights individual the ordered positions of the TrCFNs. We develop trapezoidal cubic fuzzy hybrid averaging (TrCFEHWA) operator, which weights together the given TrCFN and its well-ordered position.

Definition 25

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\}, I_A^- ]} \\ {[\{p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)\}, I_A^+ ]} \\ {[\{p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\}, \mu _A ]\rangle |} \\ {h\in H} \\ \end{array} }} \right\} \) be a collection of TrCFNs in \(L_{\mathrm{TrCFN}} \) and \(\varpi =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) is the weight vector of \(A_j (j=1, 2,..., n)\) such that \(\varpi _j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\) . Then trapezoidal cubic fuzzy Einstein hybrid weighted averaging operator of dimension n is a mapping TrCFEHWA \(:\;\;L_{\mathrm{TCFN}}^n \rightarrow L_{\mathrm{TCFN}} \) , that is an associated vector \(w =(w_1 , w_2 ,..., w_n )^{T}\) such that \(w_j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n w_j =1\).

TrCFEHWA \((A_1 , A_2 ,..., A_n )=p_1 A_{\sigma (1)} +p_2 A_{\sigma (1)} ,..., p_n A_{\sigma (1)} \) . If \(p=\theta \varpi _{\sigma (j)} +(1-\theta )w_{\sigma (j)} \) with a balancing coefficient \(\theta \in [0, 1],(\sigma (1), \sigma (2),..., \sigma (n))\) is a permutation of (1, 2, ..., n) such that \(A_{\sigma (j)} \le A_{\sigma (j-1)} \) for all \(j=2, 3,..., n(i.e., A_{\sigma (j)} \) is the j th largest value in the collection \((A_1 , A_2 ,..., A_n )\).

Theorem 26

Let \(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\}; I_A^- ]} \\ {,[\{p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)\}; I_A^+ ]} \\ {,[\{(p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\}; \mu _A ]\rangle |} \\ {h\in H} \\ \end{array} }} \right\} \) be a collection of trapezoidal cubic fuzzy numbers (TrCFNs) in \(L_{\mathrm{TrCFN}} \) and \(\varpi =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) is the weight vector of \(A_j (j=1, 2,..., n)\) such that \(\varpi _j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\) . Then their aggregated value by using the TrCFEHWA operator, which is an associated vector \(w \quad =(w_1 , w_2 ,..., w_n )^{T}\), is the weight vector of \(A_j (j=1, 2,..., n)\) such that \(w_j \in [0, 1]\) and \(\mathop {\sum }\nolimits _{j=1}^n w_j =1\) , is also the TrCFN and TrCFEWA \((A_1 , A_2 ,..., A_n )=\langle \max (I_A^- )\)

$$\begin{aligned}&\left[ {{\begin{array}{l} {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_{\sigma (j)}^- ]^{\alpha }-\mathop {\prod }\nolimits _{j=1}^n [1-p_{\sigma (j)}^- ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [1+p_{\sigma (j)}^- ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [1-p_{\sigma (j)}^- ]^{\alpha }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_{\sigma (j)}^- ]^{\alpha }-\mathop {\prod }\nolimits _{j=1}^n [1-q_{\sigma (j)}^- ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [1+q_{\sigma (j)}^- ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [1-q_{\sigma (j)}^- ]^{\alpha }}} \\ {, \frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_{\sigma (j)}^- ]^{\alpha }-\mathop {\prod }\nolimits _{j=1}^n [1-r_{\sigma (j)}^- ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [1+r_{\sigma (j)}^- ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [1-r_{\sigma (j)}^- ]^{\alpha }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_{\sigma (j)}^- ]^{\alpha }-\mathop {\prod }\nolimits _{j=1}^n [1-s_{\sigma (j)}^- ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [1+s_{\sigma (j)}^- ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [1-s_{\sigma (j)}^- ]^{\alpha }}} \\ \end{array} }} \right] , \\&\max (I_A^+ )\left[ {{\begin{array}{l} {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_{\sigma (j)}^+ ]^{\alpha }-\mathop {\prod }\nolimits _{j=1}^n [1-p_{\sigma (j)}^+ ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [1+p_{\sigma (j)}^+ ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [1-p_{\sigma (j)}^+ ]^{\alpha }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_{\sigma (j)}^+ ]^{\alpha }-\mathop {\prod }\nolimits _{j=1}^n [1-q_{\sigma (j)}^+ ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [1+q_{\sigma (j)}^+ ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [1-q_{\sigma (j)}^+ ]^{\alpha }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_{\sigma (j)}^+ ]^{\alpha }-\mathop {\prod }\nolimits _{j=1}^n [1-r_{\sigma (j)}^+ ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [1+r_{\sigma (j)}^+ ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [1-r_{\sigma (j)}^+ ]^{\alpha }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_{\sigma (j)}^+ ]^{\alpha }-\mathop {\prod }\nolimits _{j=1}^n [1-s_{\sigma (j)}^+ ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [1+s_{\sigma (j)}^+ ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [1-s_{\sigma (j)}^+ ]^{\alpha }}} \\ \end{array} }} \right] ,\\&\min (\mu _A )\left[ {{\begin{array}{l} {\frac{2\mathop {\prod }\nolimits _{j=1}^n [p_{\sigma (j)} ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [(2-p_{\sigma (j)} ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [p_{\sigma (j)} ]^{\alpha }},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [q_{\sigma (j)} ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [(2-q_{\sigma (j)} ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [q_{\sigma (j)} ]^{\alpha }},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [r_{\sigma (j)} ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [(2-r_{\sigma (j)} ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [r_{\sigma (j)} ]^{\alpha }},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [s_{\sigma (j)} ]^{\alpha }}{\mathop {\prod }\nolimits _{j=1}^n [(2-s_{\sigma (j)} ]^{\alpha }+\mathop {\prod }\nolimits _{j=1}^n [s_{\sigma (j)} ]^{\alpha }}} \\ \end{array} }} \right] \rangle . \end{aligned}$$

If \(p=\theta \varpi _{\sigma (j)} +(1-\theta )w_{\sigma (j)} \)with a balancing coefficient \(\theta \in [0, 1],(\sigma (1), \sigma (2),..., \sigma (n))\) is a permutation of (1, 2, ..., n) such that \(A_{\sigma (j)} \le A_{\sigma (j-1)} \) for all \(j=2, 3,..., n(i.e.,A_{\sigma (j)} \) is the j th largest value in the collection \((A_1 , A_2 ,..., A_n )\).

6 An approach to multiple attribute decision making with trapezoidal cubic fuzzy information

A multiple attribute decision-making (MADM) problem is to find a best compromise solution from all feasible alternatives assessed on multiple attributes. Let \(h=\{h_1 , h_2 ,..., h_n \}\) be a discrete set of alternatives and \(G=\{g_1 , g_2 ,..., g_n \}\) be the set of attributes. Suppose the rating of alternatives \(h_i \quad (i=1, 2,..., n)\) on attributes \(g_j \quad (j=1, 2,..., m)\) given by decision maker are trapezoidal cubic fuzzy numbers (TrCFNs) in \(L_{TrCFN} \) :

\(A=\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\}; I_A^- ]} \\ {,[\{p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)\}; I_A^+ ]} \\ {,[\{(p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\}; \mu _A ]\rangle |} \\ {h\in H} \\ \end{array} }} \right\} ,\quad \) where \(\left\{ {{\begin{array}{l} {[p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)](I_A^- ),} \\ {[p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)](I_A^+ )} \\ \end{array} }} \right\} \) indicates the interval-valued trapezoidal fuzzy set that the alternative \(h_i \) satisfies the attribute \(g_j \) and \(\left\{ {{\begin{array}{l} {[p_A (h), q_A (h),} \\ {r_A (h), s_A (h)](\mu _A )} \\ \end{array} }} \right\} \) indicates the trapezoidal fuzzy set that the alternative \(h_i \) does not satisfy the attribute \(g_j \) . Hence, a multiple attribute decision-making (MADM) problem can be concisely expressed in the trapezoidal cubic fuzzy decision matrix \(D=(A_{ij} )_{m\times n} =\left\{ {{\begin{array}{l} {\langle [\{p_A^- (h), q_A^- (h),} \\ {r_A^- (h), s_A^- (h)\}; I_A^- ]} \\ {,[\{p_A^+ (h), q_A^+ (h),} \\ {r_A^+ (h), s_A^+ (h)\}; I_A^+ ]} \\ {,[\{(p_A (h), q_A (h),} \\ {r_A (h), s_A (h)\}; \mu _A ]\rangle |} \\ {h\in H} \\ \end{array} }} \right\} \).

Next, we shall apply the TrCFEHWA operator to deal with the multiple attribute decision-making (MADM) problem, which involves the following steps.

  1. Step 1:

    Get the normalized trapezoidal cubic fuzzy decision matrix. The following normalization formula \(E^{k}=(\beta _{ij}^k )_{n\times 1} .\) Accordingly, we attain the normalized trapezoidal cubic fuzzy decision matrix \(E=(\beta _{ij} )_{n\times m} \) .

  2. Step 2.

    Utilize the TrCFEHWA operator to aggregate all the rating values \(\beta _{ij} \quad (j=1, 2,..., m)\) of the i th line and get the overall rating value \(\beta _{ij} \) corresponding to the alternative \(h_i \quad (i=1, 2,..., n)\), i.e., \(\beta _{ij} \quad =\) TrCFHWA \((\beta _{i1} , \beta _{i2} ,..., \beta _{im} )\), \((i=1, 2,..., n),\) where \(\varpi \quad =(\varpi _1 , \varpi _2 ,..., \varpi _n )^{T}\) is the attribute weight vector of \(g_j \quad (j=1, 2,..., m)\) such that \(\varpi _j \in [0, 1]\), \((j=1, 2,..., m)\) and \(\mathop {\sum }\nolimits _{j=1}^n \varpi _j =1\). \(w=(w_1 , w_2 ,..., w_n )^{T}\) is the associated vector of the TrCFEHWA operator, such that \(w_j \in [0, 1], j=1, 2,..., n\) and \(\mathop {\sum }\nolimits _{j=1}^n w_j =1.\)

  3. Step 3.

    Rank the order of all alternatives. Utilize the method in definition 8 to rank the overall rating values \(\beta _i \quad (i=1, 2,..., n)\) and rank all the alternatives \(h_i \quad (i=1, 2,..., n)\) in accordance with \(\beta _i \quad (i=1, 2,..., n)\) in ascending order.

  4. Step 4:

    Lastly, we choice the most appropriate alternative(s) with the smallest largely rating value.

7 Illustrative example

In this section, a multiple attribute decision-making (MADM) problem involves the prioritization of a set of propulsion systems used to illustrate the developed operator.

The propulsion system selection is based on the study that has been conducted for the selection of propulsion system of a double-ended passenger ferry to operate across the Lahore in Karachi with the aim of reducing the journey time in highly congested seaway traffic. Propulsion system alternatives are given the set of alternatives \(A=\{A_1 , A_2 , A_3 , A_4 \}\)

\(A_1 :\) :

Conventional propeller and high lift rudder;

\(A_2 :\) :

Get-up-and-go,

\(A_3 :\) :

Cycloidal propeller,

\(A_4 :\) :

Outmoded

The selection decision is made on the basis of one objective and four subjective attributes, which are the following:

\(C_1 :\) :

Investment cost (IC);

\(C_2 :\) :

Reparation and maintenance expenditures;

\(C_3 :\) :

Maneuverability (MV);

\(C_4 :\) :

Vibration and noise (VN).

where the attribute weight vector is \(\mu =(0.24, 0.35, 0.41)\) . Therefore, trapezoidal cubic fuzzy multiple attribute decision making (MADM) problem is to choose the appropriate propulsion system from among three alternatives.

Assume that the decision maker uses the linguistic terms to represent the evaluating values of the alternatives with respect to different attributes, respectively, as shown in Table 1. (Here the relationship between the linguistic terms and the corresponding TrCFNs in \(L_{\mathrm{TrCFN}} \) as shown in Tables 2, 3, 4.

Table 1 Linguistic terms
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Table 2 Decision matrix 2
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Table 3 Decision matrix 3
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Table 4 Decision matrix 4
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Table 5 Expert decision matrix 5
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Table 6 Expert decision matrix 6
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Table 7 Expert decision matrix 7
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Step 1: Get the normalized trapezoidal cubic fuzzy decision matrix. The following normalization formula \(E^{k}=(\beta _{ij}^k )_{n\times 1}\).

Table 8 Normalized trapezoidal cubic fuzzy decision matrix 8
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Table 9 Normalized trapezoidal cubic fuzzy decision matrix 9
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Table 10 Normalized trapezoidal cubic fuzzy decision matrix 10
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Henceforth, we acquire the standardized trapezoidal cubic fuzzy decision matrix \(E=(\beta _{ij} )_{n\times m} \) .

Step 2. Utilize the TrCFEHWA operator to total all the rating values \(\beta _{ij}(j=1, 2,..., m)\) of the i th line and get the general rating value \(\beta _{ij} \) comparing to the alternative \(h_i\)

Table 11 Utilize the TrCFEHWA operator
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$$\begin{aligned} w=(0.25, 0.25, 0.25, 0.25) \end{aligned}$$
Table 12 Aggregation operator
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Step 3. Rank the request of all choices. Use the technique in definition 8 to rank the general rating values \(\beta _i \quad (i=1, 2,..., n)\) and rank every one of the alternatives \(h_i \quad (i=1, 2,..., n)\) in accordance with \(\beta _i \quad (i=1, 2,..., n)\) in ascending order. At long last, we select the most alluring alternative(s) with the littlest general rating esteem. \(A_1=0.1771, A_2 =0.2022\),

$$\begin{aligned} A_3 =0.1977, A_4 =0.1709. \end{aligned}$$

Step 4: Ranking \(A_2>A_3>A_1 >A_4 \) and \(A_2 \) is the best solution.

Table 13 Interval valued intuitionistic trapezoidal fuzzy number
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figure b

8 Comparison analyses

In way to verify the sagacity and efficiency of the proposed approach, a comparative study is driven overshadowing the methods of interval-valued intuitionistic trapezoidal fuzzy number (Wu and Liu 2013), intuitionistic trapezoidal fuzzy number (Liu et al. 2017) and triangular cubic fuzzy number (Fahmi et al. 2018), which are special cases of trapezoidal cubic fuzzy numbers (TrCFNs), to the related expressive example.

8.1 A comparison analysis with the existing MCDM interval-valued intuitionistic trapezoidal fuzzy number

(Wu and Liu 2013) Step 1: According to the decision information given in the interval-valued intuitionistic trapezoidal fuzzy number decision matrix \(R^{k}=(r_{ij}^k)\)

Step 2. Utilize the IVTrFHWA operator to total all the rating values \(\beta _{ij} (j=1, 2,..., m)\) of the i th line and get the general rating value \(\beta _{ij} \) comparing to the alternative \(h_i .\) Utilize the decision information given in matrix R, and we get: (0.2, 0.3, 0.2, 0.3)

Table 14 Utilize the IVTrFHWA operator
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Table 15 Intuitionistic trapezoidal fuzzy number
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Step 3. Find the score value

$$\begin{aligned} S(r_1 )= & {} -0.0406, \quad S(r_2 )=-0.0864, \\ S(r_3 )= & {} 0.0585, \quad S(r_4 )=-0.0412. \end{aligned}$$
Table 16 Overall preference value
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Table 17 Triangular cubic fuzzy number
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Table 18 Using the TCFHA operator
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Table 19 Comparison analysis with existing methods
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Step 4. Rank all the alternatives \(A_i (i=1,2,3,4)\) in accordance with the scores \(S(r_i )\) of the overall preference values \(r_i \) : \(A_3>A_2>A_4 >A_1 \), and thus, the most desirable alternative is \(A_3 \).

8.2 A comparison analysis with the existing MCDM intuitionistic trapezoidal fuzzy number

(Liu et al. 2017) Step 1. According to the decision information given in the trapezoidal intuitionistic fuzzy decision matrix \(R^{k}=(r_{ij}^k )\), and the TIFEWA operator to derive the individual overall preference trapezoidal intuitionistic fuzzy values \(r_i^k \) of the alternative \(A_i \), we get:

Step 2. Utilize the individual overall preference trapezoidal intuitionistic fuzzy values \(r_i^k \) of the alternative \(A_i \) and we get

Step 3. Calculate the scores \(S(r_i )\) of the overall trapezoidal intuitionistic fuzzy preference values \(r_i \)

$$\begin{aligned} S(r_1 )= & {} 0.0932, \quad S(r_2 )=0.0365, \\ S(r_3 )= & {} 0.0523, \quad S(r_4 )=0.0201. \end{aligned}$$

Step 4. Rank all the alternatives \(A_i (i=1,..., n)\) in accordance with the scores \(S(r_i )\) of the overall preference values \(r_i \):

\(A_1>A_3>A_2 >A_4 ,\) and thus the most desirable alternative is \(A_1\).

8.3 A comparison analysis with the existing MCDM method triangular cubic fuzzy number

(Fahmi et al. 2018) Step 1: According to the decision information given in the triangular cubic fuzzy number decision matrix \(R^{k}=(r_{ij}^k )\),

By step 2 using the TCFHA Operator to aggregate all the decision matrices into single collective decision matrix with triangular cubic fuzzy ratings. Consider

Step 3. To find the ranking order of the alternatives, use the score function

$$\begin{aligned} S(r_1 )= & {} 0.0021, \quad S(r_2 )=0.0316,\\ S(r_3 )= & {} 0.0025, \quad S(r_4 )=0.0331. \end{aligned}$$

Step 4. Rank all the alternatives \(A_i (i=1, 2, 3, 4)\) in accordance with the scores \(S(r_i )\) of the overall preference values \(r_i \) : \(A_4>A_2>A_3 >A_1 ,\) and thus the most desirable alternative is \(A_2 \) (Tables 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19).

9 Conclusion

In this paper, we define the new concept of trapezoidal cubic fuzzy number and Hamming distance. We develop three arithmetic averaging operators, that is trapezoidal cubic fuzzy Einstein weighted averaging (TrCFEWA) operator, trapezoidal cubic fuzzy Einstein ordered weighted averaging (TrCFEOWA) operator and trapezoidal cubic fuzzy Einstein hybrid weighted averaging (TrCFEHWA) operator, for gathering cubic fuzzy data. The TrCFEHWA operator simplifies both the TrCFEWA and TrCFEOWA operators. Furthermore, we originate the relationship between the current aggregation operators and suggested operators and establish many properties of these operators. We apply on the TrCFEHWA operator to multiple attribute decision making with fuzzy material. Finally, a numerical example is providing to demonstrate the submission of the established approach. 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. These operators can be applied to many other fields, such as information fusion, data mining, pattern recognition, triangular cubic linguistic fuzzy VIKOR method and trapezoidal cubic linguistic fuzzy VIKOR method, which may be the possible topic for the future research.