Some Interval Neutrosophic Linguistic Maclaurin Symmetric Mean Operators and Their Application in Multiple Attribute Decision Making

Abstract

There are many practical decision-making problems in people’s lives, but the information given by decision makers (DMs) is often unclear and how to describe this information is of critical importance. Therefore, we introduce interval neutrosophic linguistic numbers (INLNs) to represent the less clear and uncertain information and give their operational rules and comparison methods. In addition, since the Maclaurin symmetric mean (MSM) operator has the special characteristic of capturing the interrelationships among multi-input arguments, we further propose an MSM operator for INLNs (INLMSM). Furthermore, considering the weights of attributes are the important parameters and they can influence the decision results, we also propose a weighted INLMSM (WINLMSM) operator. Based on the WINLMSM operator, we develop a multiple attribute decision making (MADM) method with INLNs and some examples are used to show the procedure and effectiveness of the proposed method. Compared with the existing methods, the proposed method is more convenient to express the complex and unclear information. At the same time, it is more scientific and flexible in solving the MADM problems by considering the interrelationships among multi-attributes.

1. Introduction

The unclear set (FS) theory was put forward by Zadeh [1] in 1965. In this theory, the membership degree (MD) $T(x)$ is used to describe fuzzy information and it has also been widely used in practice. However, the inadequacies of FS are evident. For example, it is difficult to express the non-membership degree (NMD) $F(x)$. In order to fix this problem, Intuitionistic FS (IFS) was proposed by Atanassov [2] in 1986. It is made up of two parts: MD and NMD. IFS is an extension and development of Zadeh’ FS and Zadeh’ FS is a special case of IFS [3]. IFS needs to meet two conditions: (1) $T(x),F(x)\in [0,1]$; (2) $0\le T(x)+F(x)\le 1$ [2]. Subsequently, the IFS theory was further extended such as Zadeh [4] proposed interval IFS (IIFS). Zwick et al. [5] put forward the triangular IFS while Zeng and Li [6] defined trapezoidal IFS. However, under some circumstances due to the limited cognitive ability of the DMs, they may hesitate in the two choices for accuracy and uncertainty. Since they choose both of them at the same time, this can produce an imprecise or contradictory evaluation result. Therefore, Smarandache [7,8] introduced a concept called neutrosophic set (NS), which included MD, NMD, and indeterminacy membership degree (IMD) in a non-standard unit interval [9]. Clearly, the NS is the generalization of FS and IFS. Furthermore, Wang [10] proposed the definition of interval NS (INS) which uses the standard interval to express the function of MD, IMD, and NMD. Broumi and Smarandache [11] presented the correlation coefficient of INS.

When dealing with the MADM problems with qualitative information, it is difficult for DMs to describe their own ideas with precise values. Generally, DMs ordinarily uses some linguistic terms (LTs) like “excellent”, “good”, “bad”, “very bad”, or “general” to indicate their evaluations. For example, when we look at a person’s height, we usually describe him as “high” or “very high” by visual inspection, but we will not give the exact value. In order to easily process the qualitative information, Herrera and Herrera-Viedma [12] proposed the LTs to deal with this kind of information instead of numerical computation. However, because LT such as “high” is not with MD, or we can think its MD is 1, which means LTs cannot describe the MD and NMD. Therefore, in order to facilitate DMs to describe the MD and NMD for one LT, Liu and Chen [13] defined the linguistic intuitionistic fuzzy number (LIFN), which combined the advantages of intuitionistic fuzzy numbers (IFNs) and linguistic variables (LVs). Therefore, LIFN can fully express the complex fuzzy information and there is a good prospect in MADM. After that, Ye [14] came up with the single-valued neutrosophic linguistic number (SVNLN). The most striking feature of the SVNLN is that it used LTs to describe the MD, IMD, and NMD. Sometimes, the three degrees are not expressed in a single real number, but is expressed in intervals [15]. And then, Ye [16] defined an interval neutrosophic linguistic set (INLS) and INLNs. INLNs is used to represent three values of MD, IMD, and NMD in the form of intervals. Clearly, INLS is a generalization of FS, IFS, NS, INS, LIFN, and SVNLN. It is general and beneficial for describing practical problems.

The aggregation operators (AOs) are an efficient way to handle MADM problems [17,18]. Many AOs are proposed for achieving some special functions. Yager [19] employed the ordered weighted average (OWA) operator for MADM. Bonferroni [20] proposed the Bonferroni mean (BM) operator, which can capture the correlation between input variables very well. Then BM operators have been extended to process different uncertain information such as IFS [21,22], interval-valued IFS [23], q-Rung Orthopai Fuzzy set [24], and Multi-valued Ns [25]. In addition, Beliakov [26] presented the Heronian mean (HM) operators, which have the same feature as the BM (i.e., they can capture the interrelationship between input parameters). Some HM operators have been proposed [27,28,29,30]. Furthermore, Yu [31] gave the comparison of BM with HM. However, since the BM operator and the HM operator can only reflect the relationship between any two parameters, they cannot process the MADM problems, which require the relationship for multiple inputs. In order to solve this shortcoming, Maclaurin [32] proposed the MSM operator, which has prominent features of capturing the relationship among multiple input parameters. Afterward, Qin and Liu [33] developed some MSM operators for uncertain LVs. Liu and Qin [34] developed some MSM for LIFNs. Liu and Zhang [35] proposed some MSM operators for single valued trapezoidal neutrosophic numbers.

Since the INLNs are superior to other ways of expressing complex uncertain information [16] and the MSM has good flexibility and adaptability, it can capture the relationship among multiple input parameters. However, now the MSM cannot deal with INLNs. Therefore, the objectives of this paper are to extend the MSM and weighted MSM (WMSM) operators to INLNs and to propose the INLMSM operator and the WINLMSM operator, to prove some properties of them and discuss some special cases, to propose a MADM approach with INLNs, and show the advantages of the proposed approach by comparing with other studies.

In Section 2 of this paper, we introduce some basic concepts about NS, INS, INLS, and MSM. In Section 3, we introduce the INLN and its operations including a new scoring function and a comparison method of INLN. In Section 4, we introduce an operator of INLMSM. Additionally, in order to improve flexibility, we propose the INLGMSM operator based on the GMSM operator. Furthermore, we develop the WINLMSM operator and the WINLGMSM operator to compare with operators that lack weight. Afterwards, we use examples to prove our theories. In Section 5, we give a MADM method for INLNs. In Section 6, we provide an example to demonstrate the effectiveness of the proposed method. Lastly, we provide the conclusions.

2. Preliminaries

In this section, we will introduce some existing definitions and basic concepts in order to understand this study.

2.1. The NS and INS

Definition 1 [7,8,9].

Let$X$be a space of points (objects) with a generic element in$X$denoted by$x$. A NS$A$in$X$is expressed by a MD$T_A(x)$, an IMD$I(x)$, and a NMD$F_A(x)$.

Then a NS$A$is denoted below.

$$A=\left\{\langle x,T_A(x),I_A(x),F_A(x)\rangle |x\in X\right\}$$

(1)

$T_A(x)$, $I(x)$, and$F_A(x)$are real standard or non-standard subsets of$\left]{}^{-}0,1^{+}\right[$. That is

$$T_A:X\to \left]{}^{-}0,1^{+}\right[; I_A:X\to \left]{}^{-}0,1^{+}\right[; F_A:X\to \left]{}^{-}0,1^{+}\right[$$

With the condition${}^{-}0\le T_A(x)+I_A(x)+F_A(x)\le 3^{+}$.

Definition 2 [10,11].

Let$X$be a space of points (objects) with a generic element in$X$denoted by$x$. For convenience, the lower and upper ends of T, I, F are expressed as$T_A^L(x)$, $T_A^U(x)$, $I_A^L(x)$, $I_A^U(x)$, $F_A^L(x)$, and$F_A^U(x)$. An INS$A$in$X$is defined below.

$$A=\left\{x,\langle \left[T_A^L(x),T_A^U(x)\right],\left[I_A^L(x),I_A^U(x)\right],\left[F_A^L(x),F_A^U(x)\right]\rangle |x\in X\right\}$$

(2)

For each point$x$in$X$, we have that$\left[T_A^L(x),T_A^U(x)\right]\subseteq \left[0,1\right]$, $\left[I_A^L(x),I_A^U(x)\right]\subseteq \left[0,1\right]$, $\left[F_A^L(x),F_A^U(x)\right]\subseteq [0,1]$, and$0\le T_A^U(x)+I_A^U(x)+F_A^U(x)\le 3$.

Definition 3 [10,11].

An INS$A$is contained in the INS$B$, $A\subseteq B$, if and only if$T_A^L(x)\le T_B^L(x)$, $T_A^U(x)\le T_B^U(x)$, $I_A^L(x)\ge I_B^L(x)$, $I_A^U(x)\ge I_B^U(x)$, $F_A^L(x)\ge F_B^L(x)$, and$F_A^U(x)\ge F_B^U(x)$. If$A\subseteq B$and$A\supseteq B$, then$A=B$.

2.2. LVs

Definition 4 [36,37].

Let$S=\{s_i|i=0,1,\dots ,l,l\in N^{*}\}$be a LT set (LTS) where$N^{*}$is a set of positive integers and$s_i$represents LV.

Because the LTS is convenient and efficient, it is widely used by DMs in decision making. For instance, when we evaluate the production quality, we can set l = 9, then $S$ is given below.

$$\begin{array}{l}S=\{s_0=extremelybad,s_1=verybad,s_2=bad,s_3=slightlybad,s_4=fair,s_5=slightlygood,\\ s_6=good,s_7=verygood,s_8=extremelygood\}\end{array}$$

To relieve the loss of linguistic information in operations, Xu [38,39] extended LTS S to continuous LTS $\overline{S}=\{s_{\theta }|0\le \theta \le l\}$. About the characteristics of LTS, please refer to References [38,39,40].

Definition 5 [13].

Let$s_{\alpha }$and$s_{\beta }$be any two LVs in$\overline{S}$. The related operations can be defined below.

$$s_{\alpha }\oplus s_{\beta }=s_{\alpha +\beta -\frac{\alpha \cdot \beta }{l}}$$

(3)

$$\lambda s_{\alpha }=s_{l-l\cdot {(1-\frac{\alpha }{l})}^{\lambda }},\lambda >0$$

(4)

$$s_{\alpha }\otimes s_{\beta }=s_{\frac{\alpha \cdot \beta }{l}}$$

(5)

$${(s_{\alpha })}^{\lambda }=s_{l\cdot {(\frac{\alpha }{l})}^{\lambda }},\lambda >0$$

(6)

2.3. MSM Operator

Definition 6 [15,32].

Let$x_i(i=1,2,\dots ,n)$be the set of the non-negative real number. An MSM operator of dimension$n$is a mapping$MS{M}^{(m)}:{(R^{+})}^n\to R^{+}$and it can be defined below.

$$MS{M}^{(m)}(x_1,\dots ,x_n)={\left(\frac{{\displaystyle {\sum }_{1\le i_1<\dots <i_m\le n}{\displaystyle {\prod }_{j=1}^m{x}_{i_j}}}}{C_n^m}\right)}^{\frac{1}{m}}$$

(7)

where$(i_1,i_2,\dots ,i_m)$traverses all the m-tuple combination of$(1,2,\dots ,n)$and$C_n^m=\frac{n!}{m!(n-m)!}$is the binomial coefficient. In addition,$x_{i_j}$refers to$i_j$th element in a particular arrangement.

There are some properties of the $MS{M}^{(m)}$ operator, which are defined below.

(1)

Idempotency. If $x_i=x$ for each $i$, and then $MS{M}^{(m)}(x,x,\dots ,x)=x$;

(2)

Monotonicity. If $x_i<=y_i$ for all $i$, $MS{M}^{(m)}(x_1,x_2,...,x_n)\le MS{M}^{(m)}(y_1,y_2,...,y_n)$;

(3)

Boundedness. $\min \{x_1,x_2,...x_n\}\le MS{M}^{(m)}\{x_1,x_2,...x_n\}\le \max \{x_1,x_2,...x_n\}.$

Furthermore, the $MS{M}^{(m)}$ operator would degrade some particular forms when $m$ takes some special values, which are shown as follows.

• When $m=1$, the $MS{M}^{(m)}$ operator would become the average operator.

  $$MS{M}^{(1)}(x_1,x_2,...x_n)=\left(\frac{{\displaystyle {\sum }_{1\le i_1\le n}x{i}_1}}{C_n^1}\right)=\frac{{\displaystyle \sum _{i=1}^{n}xi}}{n}$$

  (8)
• When $m=2$, the $MS{M}^{(m)}$ operator would become the following BM operator ($p=q=1$).

  $$MS{M}^{(2)}(x_1,\dots ,x_n)={\left(\frac{{\displaystyle {\sum }_{1\le i_1<i_2\le n}{\displaystyle {\prod }_{j=1}^2{x}_{i_j}}}}{C_n^2}\right)}^{\frac{1}{2}}={(\frac{2{\sum }_{1\le i_1<i_2\le n}x{i}_{1}x{i}_2}{n(n-1)})}^{\frac{1}{2}}={\left(\frac{{\displaystyle {\sum }_{i.j=1,i\ne j}^{n}xixj}}{n(n-1)}\right)}^{\frac{1}{2}}=B{M}^{1,1}(x_1,...,x_n)$$

  (9)
• When $m=n$, the $MS{M}^{(m)}$ operator would become the geometric mean.

  $$MS{M}^{(n)}(x_1,\dots ,x_n)={({\displaystyle {\prod }_{j=1}^n{x}_j})}^{\frac{1}{n}}$$

  (10)

Definition 7 [15].

Let$x_i(i=1,2,\dots ,n)$be the set of non-negative real numbers and$p_1,p_2,\dots ,p_m\ge 0$. A generalized MSM operator of dimension$n$is a mapping$GMS{M}^{(m,p_1,p_2,...,p_m)}:{(R^{+})}^n\to R^{+}$and it is defined below.

$$GMS{M}^{(m,p_1,p_2,...,p_m)}(x_1,\dots ,x_n)={(\frac{{\displaystyle {\sum }_{1\le i_1<\dots <i_m\le n}{\displaystyle {\prod }_{j=1}^m{x}_{i_j}^{p_j}}}}{C_n^m})}^{\frac{1}{p_1+p_2+...p_m}}$$

(11)

where$(i_1,i_2,\dots ,i_m)$traverses all the m-tuple combination of$(1,2,\dots ,n)$and$C_n^m=\frac{n!}{m!(n-m)!}$is the binomial coefficient.

There are some properties of the $GMS{M}^{(m,P_1,P_2,...,P_m)}$ operator below.

(1)

Idempotency. If $x_i=x$ for each $i$, and then $GMS{M}^{(m,P_1,P_2,...,P_m)}(x,x,...,x)=x$;

(2)

Monotonicity. If $x_i\le y_i$ for all $i$, ${GMSM}^{(m,p_1,p_2,...,p_m)}(x_1,x_2,...,x_n)\le GMS{M}^{(m,P_1,P_2,...,P_m)}(y_1,y_2,\dots ,y_n)$;

(3)

Boundedness. $\min \{x_1,x_2,...x_n\}\le {GMSM}^{(m,p_1,p_2,...,p_m)}\{x_1,x_2,...x_n\}\le \max \{x_1,x_2,...x_n\}.$

In addition, the $GMS{M}^{(m,P_1,P_2,...,P_m)}$ operator would degrade to some particular forms when m takes some special values, which are shown below.

• When $m=1$, we have the formula below.

  $$GMS{M}^{(1,P_1)}(x_1,x_2,...x_n)={(\frac{{\displaystyle {\sum }_{1\le i_1\le n}{x}_{i_1}{}^{p_1}}}{C_n^1})}^{\frac{1}{p_1}}={(\frac{{\displaystyle \sum _{i=1}^n{x}_i{}^{p_1}}}{n})}^{\frac{1}{p_1}}$$

  (12)
• When $m=2$, the $GMS{M}^{(m,P_1,P_2,...,P_m)}$ operator would become the following BM operator.

  $$GMS{M}^{(2,p_1,p_2)}(x_1,\dots ,x_n)={(\frac{{\displaystyle {\sum }_{1\le i_1<i_2\le n}{{\displaystyle x}}_{i_1}^{p_1}{{\displaystyle x}}_{i_2}^{p_2}}}{C_n^2})}^{\frac{1}{p_1+p_2}}={(\frac{2{\displaystyle {\sum }_{1\le i<j\le n}{{\displaystyle x}}_i^{p_1}{{\displaystyle x}}_j^{p_2}}}{n(n-1)})}^{\frac{1}{p_1+p_2}}={(\frac{{\displaystyle \sum _{i.j=1,i\ne j}^n{{\displaystyle x}}_i^{p_1}{{\displaystyle x}}_j^{p_2}}}{n(n-1)})}^{\frac{1}{p_1+p_2}}=B{M}^{p_1,p_2}$$

  (13)
• When $m=n$, the $MS{M}^{(m)}$ operator would become the following formula.

  $$GMS{M}^{(n,p_1,p_2,...,p_n)}(x_1,\dots ,x_n)={({\displaystyle {\prod }_{j=1}^n{x}_j^{p_j}})}^{\frac{1}{p_1+p_2+...p_n}}$$

  (14)
• When $p_1=p_2=...=p_m=1$, the $GMS{M}^{(m,P_1,P_2,...,P_m)}$ operator would degenerate to the MSM operator and the parameter is $m$ below.

  $$GMS{M}^{(m,1,1,...,1)}(x_1,\dots ,x_n)={(\frac{{\displaystyle {\sum }_{1\le i_1<...<i_m\le n}{\displaystyle {\prod }_{j=1}^m{{\displaystyle x}}_{ij}^1}}}{C_n^m})}^{\frac{1}{m}}=MS{M}^{(m)}(x_1,\dots ,x_n).$$

  (15)

3. INLNs and Operations

Definition 8 [16,41].

Let$X$be a finite universal set. An INLS in$X$is defined by the equation below.

$$A=\left\{x,\langle s_{\theta (x)},\left[T_A(x),I_A(x),F_A(x)\right]\rangle |x\in X\right\}$$

(16)

where$s_{\theta (x)}\in \overline{S}$, $T_A(x)=\left[T_A^L(x),T_A^U(x)\right]\subseteq [0,1]$, $I_A(x)=\left[I_A^L(x),I_A^U(x)\right]\subseteq [0,1]$, $F_A(x)=\left[F_A^L(x),F_A^U(x)\right]\subseteq [0,1]$represent the MD, the IMD, and the NMD of the element$x$in$X$to the LV$s_{\theta (x)}$, respectively, with the condition$0\le T_A^U(x)+I_A^U(x)+F_A^U(x)\le 3$for any$x\in X$.

Then the seven tuple $\langle s_{\theta (x),}(\left[T_A^L(x),T_A^U(x)\right],\left[I_A^L(x),I_A^U(x)\right],\left[F_A^L(x),F_A^U(x)\right])\rangle$ in $A$ is called an INLN. For convenience, an INLN can be represented as $a=\langle s_{\theta (a)},(\left[T^L(a),T^U(a)\right],\left[I^L(a),I^U(a)\right],\left[F^L(a),F^U(a)\right])\rangle$.

Then we introduced the operational rules of operators of INLNs.

Definition 9 [16,37,42].

Let$a_1=\langle s_{\theta (a_1)},(\left[T^L(a_1),T^U(a_1)\right],\left[I^L(a_1),I^U(a_1)\right],\left[F^L(a_1),F^U(a_1)\right])\rangle$and$a_2=\langle s_{\theta (a_2)},(\left[T^L(a_2),T^U(a_2)\right],\left[I^L(a_2),I^U(a_2)\right],\left[F^L(a_2),F^U(a_2)\right])\rangle$be two INLNs and$\lambda \ge 0$. Then the operation of the INLNs can be expressed by the equation below.

$$a_1\oplus a_2=\langle s_{\theta (a_1)+\theta (a_2)},(\left[T^L(a_1)+T^L(a_2)-T^L(a_1)\times T^L(a_2),T^U(a_1)+T^U(a_2)-T^U(a_1)\times T^U(a_2)\right],\left[I^L(a_1)\times I^L(a_2),I^U(a_1)\times I^U(a_2)\right],\left[F^L(a_1)\times F^L(a_2),F^U(a_1)\times F^U(a_2)\right])\rangle$$

(17)

$$I^U(a_1)+I^U(a_2)-I^U(a_1)\times I^U(a_2)],[F^L(a_1)+F^L(a_2)-F^L(a_1)\times F^L(a_2),a_1\otimes a_2=\langle s_{\theta (a_1)\times \theta (a_2)},([T^L(a_1)\times T^L(a_2),T^U(a_1)\times T^U(a_2)],[I^L(a_1)+I^L(a_2)-I^L(a_1)\times I^L(a_2),F^U(a_1)+F^U(a_2)-F^U(a_1)\times F^U(a_2)])\rangle$$

(18)

$$\lambda a_1=\langle s_{\lambda \times \theta (a_1)},(\left[1-{(1-T^L(a_1))}^{\lambda },1-{(1-T^U(a_1))}^{\lambda }\right],\left[{(I^L(a_1))}^{\lambda },{(I^U(a_1))}^{\lambda }\right],\left[{(F^L(a_1))}^{\lambda },{(F^U(a_1))}^{\lambda }\right])\rangle (\lambda >0)$$

(19)

$$a_1^{\lambda }=s_{{\theta }^{\lambda }(a_1)},(\left[{(T^L(a_1))}^{\lambda },{(T^U(a_1))}^{\lambda }\right],\left[1-{(1-I^L(a_1))}^{\lambda },1-{(1-I^U(a_1))}^{\lambda }\right],\left[1-{(1-F^L(a_1))}^{\lambda },1-{(1-F^U(a_1))}^{\lambda }\right])\rangle ,(\lambda >0)$$

(20)

Example 1.

Let$a_1=\langle s_3,(\left[0.1,0.2\right],\left[0.2,0.3\right],\left[0.4,0.5\right])\rangle$and$a_2=\langle s_4,(\left[0.3,0.5\right],\left[0.3,0.4\right],\left[0.5,0.6\right])\rangle$be two INLNs and$S=\{s_0=verybad,s_1=bad,s_2=slightlybad,s_3=fair,s_4=slightlygood,$$s_5=good,s_6=verygood\}$, then we have the equations below.

$$\begin{array}{l}{a}_1\oplus a_2=\langle s_{3+4},(\left[0.1+0.3-0.1\times 0.3,0.2+0.5-0.2\times 0.5\right],\left[0.2\times 0.3,0.3\times 0.4\right],\left[0.4\times 0.5,0.5\times 0.6\right]\rangle \\ =\langle s_7,(\left[0.37,0.6\right],\left[0.06,0.12\right],\left[0.2,0.3\right]\rangle \end{array}$$

$$\begin{array}{l}{a}_1\otimes a_2=\langle s_{3\times 4},([0.1\times 0.3,0.2\times 0.5],\left[0.2+0.3-0.2\times 0.3,0.3+0.4-0.3\times 0.4\right],\\ \left[0.4+0.5-0.4\times 0.5,0.5+0.6-0.5\times 0.6\right])\rangle \\ =\langle s_{12},([0.03,0.1],\left[0.44,0.58\right],\left[0.7,0.8\right])\rangle \end{array}$$

As seen from the above examples, these results are not reasonable because they exceed the range of LTS. In order to overcome these limitations, we will improve these operations by Definition 10.

Definition 10.

Let$a_1=\langle s_{\theta (a_1)},(\left[T^L(a_1),T^U(a_1)\right],\left[I^L(a_1),I^U(a_1)\right],\left[F^L(a_1),F^U(a_1)\right])\rangle$and$a_2=\langle s_{\theta (a_2)},(\left[T^L(a_2),T^U(a_2)\right],\left[I^L(a_2),I^U(a_2)\right],\left[F^L(a_2),F^U(a_2)\right])\rangle$be two INLNs and$\lambda \ge 0$. Then the operations of the INLNs can be defined by the equations below.

$$a_1\oplus a_2=\langle s_{\theta (a_1)+\theta (a_2)-\frac{\theta (a_1)\cdot \theta (a_2)}{l}},(\left[T^L(a_1)+T^L(a_2)-T^L(a_1)\times T^L(a_2),T^U(a_1)+T^U(a_2)-T^U(a_1)\times T^U(a_2)\right],\left[I^L(a_1)\times I^L(a_2),I^U(a_1)\times I^U(a_2)\right],\left[F^L(a_1)\times F^L(a_2),F^U(a_1)\times F^U(a_2)\right])\rangle$$

(21)

$$a_1\otimes a_2=\langle s_{\frac{\theta (a_1)\times \theta (a_2)}{l}},([T^L(a_1)\times T^L(a_2),T^U(a_1)\times T^U(a_2)],[I^L(a_1)+I^L(a_2)-I^L(a_1)\times I^L(a_2),I^U(a_1)+I^U(a_2)-I^U(a_1)\times I^U(a_2)],[F^L(a_1)+F^L(a_2)-F^L(a_1)\times F^L(a_2),F^U(a_1)+F^U(a_2)-F^U(a_1)\times F^U(a_2)])\rangle$$

(22)

$$\lambda a_1=\langle s_{l-l\cdot {(1-\frac{\theta (a_1)}{l})}^{\lambda }},(\left[1-{(1-T^L(a_1))}^{\lambda },1-{(1-T^U(a_1))}^{\lambda }\right],\left[{(I^L(a_1))}^{\lambda },{(I^U(a_1))}^{\lambda }\right],\left[{(F^L(a_1))}^{\lambda },{(F^U(a_1))}^{\lambda }\right])\rangle ,(\lambda >0)$$

(23)

$$a_1^{\lambda }=s_{l\cdot {(\frac{\theta (a_1)}{l})}^{\lambda }},(\left[{(T^L(a_1))}^{\lambda },{(T^U(a_1))}^{\lambda }\right],\left[1-{(1-I^L(a_1))}^{\lambda },1-{(1-I^U(a_1))}^{\lambda }\right],\left[1-{(1-F^L(a_1))}^{\lambda },1-{(1-F^U(a_1))}^{\lambda }\right])\rangle ,(\lambda >0).$$

(24)

Based on the operational rules above, the above example is recalculated as follow.

Example 2.

Let$a_1=\langle s_3,(\left[0.1,0.2\right],\left[0.2,0.3\right],\left[0.4,0.5\right])\rangle$and$a_2=\langle s_4,(\left[0.3,0.5\right],\left[0.3,0.4\right],\left[0.5,0.6\right])\rangle$be two INLNs and$S=\{s_0=verybad,s_1=bad,s_2=slightlybad,s_3=fair,s_4=slightlygood,$$s_5=good,s_6=verygood\}$, then we have the equations below.

$$\begin{array}{l}{a}_1\oplus a_2=\langle s_{3+4-\frac{3\text{×}4}{6}},(\left[0.1+0.3-0.1\times 0.3,0.2+0.5-0.2\times 0.5\right],\left[0.2\times 0.3,0.3\times 0.4\right],\left[0.4\times 0.5,0.5\times 0.6\right]\rangle \\ =\langle s_5,(\left[0.37,0.6\right],\left[0.06,0.12\right],\left[0.2,0.3\right]\rangle \end{array}$$

$$\begin{array}{l}{a}_1\otimes a_2=\langle s_{\frac{3\text{×}4}{6}},([0.1\times 0.3,0.2\times 0.5],\left[0.2+0.3-0.2\times 0.3,0.3+0.4-0.3\times 0.4\right]\\ ,\left[0.4+0.5-0.4\times 0.5,0.5+0.6-0.5\times 0.6\right])\rangle \\ =\langle s_2,([0.03,0.1],\left[0.44,0.58\right],\left[0.7,0.8\right])\rangle \end{array}$$

From the above example, the results are more reasonable than the previous ones.

In the following definitions, a new scoring function and a comparison method of INLN are described.

Definition 11. [37].

Let$a=\langle s_{\theta (a)},(\left[T^L(a),T^U(a)\right],\left[I^L(a),I^U(a)\right],\left[F^L(a),F^U(a)\right])\rangle$be an INLN. Then the score function of a can be expressed by the equation below.

$$S(a)=\alpha \cdot \frac{\theta (a)}{6}\left[0.5(T^U(a)+1-F^L(a))+\alpha I^U(a)\right]+(1-\alpha )\cdot \frac{\theta (a)}{6}\left[0.5(T^L(a)+1-F^U(a))+\alpha I^L(a)\right]$$

(25)

where the values of$\alpha \in [0,1]$reflect the attitudes of the decision makers.

Definition 12. [37].

Let$a$and$b$be two INLNs. Then the INLN comparison method can be expressed by the statements below.

$$\mathrm{If} S(a)>S(b), \mathrm{then} a\succ b;$$

(26)

$$\mathrm{If} S(a)=S(b), \mathrm{then} a\sim b;$$

(27)

$$\mathrm{If} S(a)\text{<}(b), \mathrm{then} a\prec b;$$

(28)

4. Some Interval Neutrosophic Linguistic MSM Operators

In this section, we will propose INLMSM operators and INLGMSM operators.

4.1. The INLMSM Operators

Definition 13.

Let$a_i={\langle s}_{{{\displaystyle \theta }}_i},([T^L(a_i),T^U(a_i)],[I^L(a_i),I^U(a_i)],[F^L(a_i),F^U(a_i)])\rangle$$(i=1,2,...,n)$be a set of INLNs. Then the INLMSM operator:${\Omega }^n\to \Omega$is shown below.

$$INLMS{M}^{(m)}(a_1,\dots ,a_n)={(\frac{{}_{1\le i_1<...<i_m\le n}^{\oplus }(\underset{j=1}{\overset{m}{{\displaystyle \otimes }}}{a}_{i_j})}{C_n^m})}^{\frac{1}{m}}$$

(29)

$\Omega$is a set of INLNs and$m=1,2,...,n$.

According to the operational laws of INLNs in Definition 10, we can get the expression of the INLMSM operator shown below.

Theorem 1.

Let$a_i={\langle s}_{{{\displaystyle \theta }}_i},([T^L(a_i),T^U(a_i)],[I^L(a_i),I^U(a_i)],[F^L(a_i),F^U(a_i)])\rangle$$(i=1,2,...,n)$be a set of INLNs and$m=1,2,...,n$. Then the value aggregated from Definition 13 is still an INLN.

$$INLMS{M}^{(m)}(a_1,\dots ,a_n)=\langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle {\prod }_{j=1}^m\left(\frac{{\theta }_{i_j}(k)}{l}\right)}\right)}^{\frac{1}{C_n^m}}}\right)}^{\frac{1}{m}}},(\left[{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^L{}_{i_j(k)}}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}},{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^U{}_{i_j(k)}}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}}\right],\left[1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^L{}_{i_j(k)}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^U{}_{i_j(k)}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}\right],\left[1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^L{}_{i_j(k)}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^U{}_{i_j(k)}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}\right]\rangle$$

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where$k=1,2,...{{\displaystyle C}}_n^m$, $a_{i_j(k)}$is the$i_j$th element of$k$th permutation.

Proof. 

Because

$$a_{i_j(k)}=\langle s_{\theta i_j(k)},((T^L{{\displaystyle i}}_{{}_j}(k),T^U{{\displaystyle i}}_{{}_j}(k)),(I^L{{\displaystyle i}}_{{}_j}(k),I^U{{\displaystyle i}}_{{}_j}(k)),(F^L{{\displaystyle i}}_{{}_j}(k),F^U{{\displaystyle i}}_{{}_j}(k)))\rangle (j=1,2,...,m)$$

$$\Rightarrow \underset{j=1}{\overset{m}{{\displaystyle \otimes }}}{{\displaystyle a}}_{{{\displaystyle i}}_j(k)}=\langle s_{l\cdot {\displaystyle {\prod }_{j=1}^m\left(\frac{\theta i_j(k)}{l}\right)}},(\left[{\displaystyle \prod _{j=1}^m{T}^L{{\displaystyle i}}_j(k)},{\displaystyle \prod _{j=1}^m{T}^U{{\displaystyle i}}_{{}_j}(k)}\right],$$

$$\left[1-{\displaystyle \prod _{j=1}^m\left(1-I^L{}_{i_j(k)}\right)},1-{\displaystyle \prod _{j=1}^m\left(1-I^U{}_{i_j(k)}\right)}\right],\left[1-{\displaystyle \prod _{j=1}^m\left(1-F^L{}_{i_j(k)}\right)},1-{\displaystyle \prod _{j=1}^m\left(1-F^U{}_{i_j(k)}\right)}\right])\rangle$$

$$\Rightarrow {}_{1\le i_1<...<i_m\le n}^{\oplus }\left(\underset{j=1}{\overset{m}{{\displaystyle \otimes }}}a{i}_j\right)=\langle s_{l-l\cdot {\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle {\prod }_{j=1}^m\left(\frac{\theta i_j(k)}{l}\right)}\right)}},$$

$$(\left[1-{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^L{}_{i_j(k)}}\right)},1-{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^U{}_{i_j(k)}}\right)}\right],$$

$$[{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^L{}_{i_j(k)}\right)}\right)},{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^U{}_{i_j(k)}\right)}\right)}],$$

$$[{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^L{}_{i_j(k)}\right)}\right)},{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^U{}_{i_j(k)}\right)}\right)}])\rangle$$

$$\Rightarrow {\left(\frac{{}_{1\le i_1<...<i_m\le n}^{\oplus }\left(\underset{j=1}{\overset{m}{{\displaystyle \otimes }}}{a}_{i_j}\right)}{{{\displaystyle C}}_n^m}\right)}^{\frac{1}{m}}=\langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle {\prod }_{j=1}^m\left(\frac{\theta i_j(k)}{l}\right)}\right)}^{\frac{1}{C_n^m}}}\right)}^{\frac{1}{m}}},$$

$$(\left[{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^L{}_{i_j(k)}}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}},{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^U{}_{i_j(k)}}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}}\right],$$

$$[1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^L{}_{i_j(k)}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^U{}_{i_j(k)}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}],$$

$$[1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^L{}_{i_j(k)}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^U{}_{i_j(k)}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}])\rangle$$

Therefore, Theorem 1 is kept. ☐

Property 1.

Let$x_i={\langle s}_{{\alpha }_i},([T^L(x_i),T^U(x_i)],[I^L(x_i),I^U(x_i)],[F^L(x_i),F^U(x_i)])\rangle$$(i=1,2,...,n)$and$y_i={\langle s}_{{\beta }_i},([T^L(y_i),T^U(y_i)],[I^L(y_i),I^U(y_i)],[F^L(y_i),F^U(y_i)])\rangle$$(i=1,2,...,n)$be sets of INLNs. There are four properties of$INLMS{M}^{(m)}$operator, which is shown below.

(1) 

Idempotency. If the INLNs$x_i=x={\langle s}_{{{\displaystyle \theta }}_x},([T^L{}_x,T^U{}_x],[I^L{}_x,I^U{}_x],[F^L{}_x,F^U{}_x])\rangle$for each$i(i=1,2,...,n)$and then$INLMS{M}^{(m)}=x=\langle s_{{\theta }_x},(T_x,I_x,F_x)\rangle$.

(2) 

Commutativity. If$x_i$is a permutation of$y_i$for all i$(i=1,2,...,n)$and then$INLMS{M}^{(m)}(x_1,x_2,...,x_n)=INLMS{M}^{(m)}(y_1,y_2,...,y_n)$.

(3) 

Monotonicity. If${\alpha }_i\le {\beta }_i$, $T^L(x_i)\le T^L(y_i)$, $T^U(x_i)\le T^U(y_i)$, $I^L(x_i)\ge I^L(y_i)$, $I^U(x_i)\ge I^U(y_i)$, $F^L(x_i)\ge F^L(y_i)$and$F^U(x_i)\ge F^U(y_i)$for all i$(i=1,2,...,n)$, then$x_i\le y_i$and$INLMS{M}^{(m)}(x_1,x_2,...,x_n)\le INLMS{M}^{(m)}(y_1,y_2,...,y_n)$.

(4) 

Boundedness.$\min \{x_1,x_2,...x_n\}\le {INLMSM}^{(m)}\{x_1,x_2,...x_n\}\le \max \{x_1,x_2,...x_n\}.$.

Proof. 

1

If each $a_i=x$, then we get the equation below.

$$INLMS{M}^{(m)}(x,x,...,x)=\langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle {\prod }_{j=1}^m\left(\frac{{\theta }_x}{l}\right)}\right)}^{\frac{1}{C_n^m}}}\right)}^{\frac{1}{m}}},(\left[{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^L{}_x}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}},{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^U{}_x}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}}\right],\left[1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^L{}_x\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^U{}_x\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}\right],\left[1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^L{}_x\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^U{}_x\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}\right]\rangle =\langle s_{{\theta }_x},(Tx,Ix,Fx)\rangle =x.$$

2

This property is clear and it is now omitted.

3

If ${\alpha }_i\le {\beta }_i,$ $T^L(x_i)\le T^L(y_i)$, $T^U(x_i)\le T^U(y_i)$, $I^L(x_i)\ge I^L(y_i)$, $I^U(x_i)\ge I^U(y_i)$, $F^L(x_i)\ge F^L(y_i)$ and $F^U(x_i)\ge F^U(y_i)$ for all $i$, according to Theorem 1.

Since

$${\displaystyle \prod _{j=1}^m{\alpha }_i}\le {\displaystyle \prod _{j=1}^m{\beta }_i}, {\displaystyle \prod _{j=1}^m{T}^L(x_i)}\le {\displaystyle \prod _{j=1}^m{T}^L(y_i)}, {\displaystyle \prod _{j=1}^m{T}^U(x_i)}\le {\displaystyle \prod _{j=1}^m{T}^U(y_i)}, {\displaystyle \prod _{j=1}^m{I}^L(x_i)}\ge {\displaystyle \prod _{j=1}^m{I}^L(y_i)}, {\displaystyle \prod _{j=1}^m{I}^U(x_i)}\ge {\displaystyle \prod _{j=1}^m{I}^U(y_i)}, {\displaystyle \prod _{j=1}^m{F}^L(x_i)}\ge {\displaystyle \prod _{j=1}^m{F}^L(y_i)}, {\displaystyle \prod _{j=1}^m{F}^U(x_i)}\ge {\displaystyle \prod _{j=1}^m{F}^U(y_i)}$$

$$\mathrm{then} l\cdot {\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m\frac{{\alpha }_i}{l}}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}}\le l\cdot {\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m\frac{{\beta }_i}{l}}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}},$$

$${\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^L(x_i)}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}}\le {\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^L(y_i)}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}},$$

$${\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^U(x_i)}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}}\le {\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{T}^U(y_i)}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}},$$

$$1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^L(x_i)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}\ge 1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^L(y_i)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},$$

$$1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^U(x_i)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}\ge 1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-I^U(y_i)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},$$

$$1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^L(x_i)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}\ge 1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^L(y_i)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},$$

$$1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-F^U(x_i)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}\ge 1-{(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{(1-{\displaystyle \prod _{j=1}^m(1-F^U(y_i))})}^{\frac{1}{{{\displaystyle C}}_n^m}}})}^{\frac{1}{m}}.$$

Therefore, we can get the following conclusion.

$$INLMS{M}^{(m)}(x_1,x_2,...,x_n)\le INLMS{M}^{(m)}(y_1,y_2,...,y_n)$$

4

According to the idempotency, let $\min \{x_1,x_2,...x_n\}=x_a=INLMS{M}^{(m)}(x_a,x_a,...,x_a)$ and $\max \{x_1,x_2,...x_n\}=x_b=INLMS{M}^{(m)}(x_b,x_b,...,x_b)$. According to the monotonicity, if $x_a\le x_i$ and $x_b\ge x_i$ for all i, then we have $x_a=INLMS{M}^{(m)}(x_a,x_a,...,x_a)\le INLMS{M}^{(m)}(x_1,x_2,...,x_n)$ and

$$INLMS{M}^{(m)}(x_1,x_2,...,x_n)\le x_b=INLMS{M}^{(m)}(x_b,x_b,...,x_b).$$

Therefore, we can get the conclusion below.

$$\min \{x_1,x_2,...x_n\}\le {INLMSM}^{(m)}\{x_1,x_2,...x_n\}\le \max \{x_1,x_2,...x_n\}.$$

Furthermore, the $INLMS{M}^{(m)}$ operator would degrade to some particular forms when m takes some special values.

(1) When $m=1$, we have the formula below.

$$\begin{gathered} INLMS{M}^{(1)}(x_1,x_2,...x_n)=\left(\frac{{{\displaystyle \oplus }}_{i=1}^n{x}_i}{C_n^1}\right)= \\ \langle s_{l\cdot \left(1-{\displaystyle {\prod }_{k=1}^n{\left(1-\frac{k}{l}\right)}^{\frac{1}{n}}}\right)},(\left[1-{{\displaystyle {\prod }_{k=1}^n(1-T^L{}_k)}}^{\frac{1}{n}},1-{{\displaystyle {\prod }_{k=1}^n(1-T^U{}_k)}}^{\frac{1}{n}}\right], \\ \left[{\displaystyle {\prod }_{k=1}^n{(I^L{}_k)}^{\frac{1}{n}}},{\displaystyle {\prod }_{k=1}^n{(I^U{}_k)}^{\frac{1}{n}}}\right],\left[{\displaystyle {\prod }_{k=1}^n{(F^L{}_k)}^{\frac{1}{n}}},{\displaystyle {\prod }_{k=1}^n{(F^U{}_k)}^{\frac{1}{n}}}\right])\rangle \end{gathered}$$

(31)

(2) When $m=2$, we have the formula below.

$$\begin{gathered} INLMS{M}^{(2)}(x_1,x_2,...x_n)= \\ \langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-\left(\frac{\theta i_1(k)}{l}\right)\cdot \left(\frac{\theta i_2(k)}{l}\right)\right)}^{\frac{1}{C_n^2}}}\right)}^{\frac{1}{2}}},(\left[{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^2}\left(1-T^L{}_{i_1(k)}\cdot T^L{}_{i_2(k)}\right)}}^{\frac{1}{{{\displaystyle C}}_n^2}}\right)}^{\frac{1}{2}},{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^2}\left(1-T^U{}_{i_1(k)}\cdot T^U{}_{i_2(k)}\right)}}^{\frac{1}{{{\displaystyle C}}_n^2}}\right)}^{\frac{1}{2}}\right], \\ \left[1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-\left(1-I^L{{\displaystyle i}}_{{}_1}(k)\right)\cdot \left(1-I^L{{\displaystyle i}}_{{}_2}(k)\right)\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{2}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-\left(1-I^U{{\displaystyle i}}_{{}_1}(k)\right)\cdot \left(1-I^U{{\displaystyle i}}_{{}_2}(k)\right)\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{2}}\right], \\ \left[1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-\left(1-F^L{{\displaystyle i}}_{{}_1}(k)\right)\cdot \left(1-F^L{{\displaystyle i}}_{{}_2}(k)\right)\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{2}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-\left(1-F^U{{\displaystyle i}}_{{}_1}(k)\right)\cdot \left(1-F^U{{\displaystyle i}}_{{}_2}(k)\right)\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{2}}\right]\rangle \end{gathered}$$

(32)

(3) When $m=n$, the $INLMS{M}^{(m)}$ operator would reduce to the following form.

$$\begin{gathered} INLMS{M}^{(n)}(x_1,\dots ,x_n)= \\ \langle s_{l\cdot {\left({\displaystyle {\prod }_{j=1}^n\left(\frac{{\theta }_j}{l}\right)}\right)}^{\frac{1}{n}}},(\left[{\left({\displaystyle {\prod }_{j=1}^n{T}^L{}_j}\right)}^{\frac{1}{n}},{\left({\displaystyle {\prod }_{j=1}^n{T}^U{}_j}\right)}^{\frac{1}{n}}\right],\left[1-{\left({\displaystyle \prod _{j=1}^n\left(1-I^L{}_j\right)}\right)}^{\frac{1}{n}},1-{\left({\displaystyle \prod _{j=1}^n\left(1-I^U{}_j\right)}\right)}^{\frac{1}{n}}\right],\left[1-{\left({\displaystyle \prod _{j=1}^n\left(1-F^L{}_j\right)}\right)}^{\frac{1}{n}},1-{\left({\displaystyle \prod _{j=1}^n\left(1-F^U{}_j\right)}\right)}^{\frac{1}{n}}\right])\rangle ☐ \end{gathered}$$

(33)

☐

Definition 14.

Let$a_i={\langle s}_{{{\displaystyle \theta }}_i},([T^L(a_i),T^U(a_i)],[I^L(a_i),I^U(a_i)],[F^L(a_i),F^U(a_i)])\rangle$$(i=1,2,...,n)$be a set of INLNs. Then the INLGMSM operator:${\Omega }^n\to \Omega$is shown below.

$$INLGMS{M}^{(m,p_1,p_2,...,p_m)}(a_1,\dots ,a_n)={\left(\frac{{}_{1\le i_1<...<i_m\le n}^{\oplus }\left(\underset{j=1}{\overset{m}{{\displaystyle \otimes }}}{{\displaystyle a}}_{i_j}^{p_j}\right)}{C_n^m}\right)}^{\frac{1}{p_1+p_2+...+p_m}},$$

(34)

$\Omega$is a set of INLNs and$m=1,2,...,n$.

According to the operational laws of INLNs in Definition 10, we can get the expression of the INLMSM operator shown below.

Theorem 2.

Let$a_i={\langle s}_{{{\displaystyle \theta }}_i},([T^L(a_i),T^U(a_i)],[I^L(a_i),I^U(a_i)],[F^L(a_i),F^U(a_i)])\rangle$$(i=1,2,...,n)$be a set of INLNs and$m=1,2,...,n$. Then the value aggregated from Definition 14 is still an INLN.

$$\begin{gathered} INLGMS{M}^{(m,p_1,p_2,...,p_m)}(a_1,\dots ,a_n)=\langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle {\prod }_{j=1}^m{\left(\frac{\theta i_j(k)}{l}\right)}^{p_j}}\right)}^{\frac{1}{C_n^m}}}\right)}^{\frac{1}{p_1+p_2+...+p_m}}}, \\ ([{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{\left(T^L{}_{i_j(k)}\right)}^{p_j}}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{p_1+p_2+...+p_m}}, \\ {\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{\left(T^U{}_{i_j(k)}\right)}^{p_j}}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{p_1+p_2+...+p_m}}], \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m{\left(1-I^L{}_{i_j(k)}\right)}^{p_j}}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{p_1+p_2+...+p_m}}, \\ 1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m{\left(1-I^U{}_{i_j(k)}\right)}^{p_j}}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{p_1+p_2+...+p_m}}], \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m{\left(1-F^L{}_{i_j(k)}\right)}^{p_j}}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{p_1+p_2+...+p_m}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m{\left(1-F^U{}_{i_j(k)}\right)}^{p_j}}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{p_1+p_2+...+p_m}}])\rangle \end{gathered}$$

(35)

where$k=1,2,...{{\displaystyle C}}_n^m$, $a_{i_j(k)}$is the$i_{j}th$element of$k_{j}th$permutation. Therefore, Theorem 2 is kept. The process of proof is similar to Theorem 1 and is now omitted.

Property 2.

Let$x_i={\langle s}_{{\alpha }_i},([T^L(x_i),T^U(x_i)],[I^L(x_i),I^U(x_i)],[F^L(x_i),F^U(x_i)])\rangle$$(i=1,2,...,n)$and$y_i={\langle s}_{{\beta }_i},([T^L(y_i),T^U(y_i)],[I^L(y_i),I^U(y_i)],[F^L(y_i),F^U(y_i)])\rangle$$(i=1,2,...,n)$be two sets of INLNs. There are four properties of$INLGMS{M}^{(m,p_1,p_2,...,p_m)}$operator shown as follows.

1

Idempotency. If the INLNs $x_i=x={\langle s}_{{{\displaystyle \theta }}_x},([T^L{}_x,T^U{}_x],[I^L{}_x,I^U{}_x],[F^L{}_x,F^U{}_x])\rangle$ for each $i(i=1,2,...,n)$ and then $INLGMS{M}^{(m,p_1,p_2,...,p_m)}=x=\langle s_{{\theta }_x},(T_x,I_x,F_x)\rangle$.

2

Commutativity. If $x_i$ is a permutation of $y_i$ for all I $(i=1,2,...,n)$, and then $INLGMS{M}^{(m,p_1,p_2,...,p_m)}(x_1,x_2,...,x_n)=INLGMS{M}^{(m,p_1,p_2,...,p_m)}(y_1,y_2,...,y_n)$.

3

Monotonicity. If ${\alpha }_i\le {\beta }_i,$ $T^L(x_i)\le T^L(y_i)$, $T^U(x_i)\le T^U(y_i)$, $I^L(x_i)\ge I^L(y_i)$, $I^U(x_i)\ge I^U(y_i)$, $F^L(x_i)\ge F^L(y_i)$ and $F^U(x_i)\ge F^U(y_i)$ for all $i$ $(i=1,2,...,n)$, then $x_i\le y_i$ and $INLGMS{M}^{(m,p_1,p_2,...,p_m)}(x_1,x_2,...,x_n)\le INLGMS{M}^{(m,p_1,p_2,...,p_m)}(y_1,y_2,...,y_n)$.

4

Boundedness. $\min \{x_1,x_2,...x_n\}\le INLGMS{M}^{(m,p_1,p_2,...,p_m)}\{x_1,x_2,...x_n\}\le \max \{x_1,x_2,...x_n\}.$

The proofs are similar to Property 1, which are now omitted.

Furthermore, the $INLGMS{M}^{(m,p_1,p_2,...,p_m)}$ operator would degrade to some particular forms when m takes some special values.

(1) When $m=1$, we have the following formula.

$$\begin{gathered} INLGMS{M}^{(1)}(x_1,x_2,...,x_n)={\left(\frac{{{\displaystyle \oplus }}_{i=1}^{n}x{i}^{P_1}}{C_n^1}\right)}^{{}_{\frac{1}{P_1}}}= \\ \langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^n{\left(1-{\left(\frac{k}{l}\right)}^{p_{{}_1}}\right)}^{\frac{1}{n}}}\right)}^{\frac{1}{P_1}}},(\left[{\left(1-{{\displaystyle {\prod }_{k=1}^n\left(1-{\left(T^L{}_k\right)}^{p_1}\right)}}^{\frac{1}{n}}\right)}^{\frac{1}{P_1}},{\left(1-{{\displaystyle {\prod }_{k=1}^n\left(1-{\left(T^U{}_k\right)}^{p_1}\right)}}^{\frac{1}{n}}\right)}^{\frac{1}{P_1}}\right], \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^n{\left(1-{\left(1-I^L{{\displaystyle i}}_{{}_1}(k)\right)}^{p_1}\right)}^{\frac{1}{n}}}\right)}^{\frac{1}{p_1}},1-{\left(1-{\displaystyle {\prod }_{k=1}^n{\left(1-{\left(1-I^U{{\displaystyle i}}_{{}_1}(k)\right)}^{p_1}\right)}^{\frac{1}{n}}}\right)}^{\frac{1}{p_1}}], \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^n{\left(1-{\left(1-F^L{{\displaystyle i}}_{{}_1}(k)\right)}^{p_1}\right)}^{\frac{1}{n}}}\right)}^{\frac{1}{p_1}},1-{\left(1-{\displaystyle {\prod }_{k=1}^n{\left(1-{\left(1-F^U{{\displaystyle i}}_{{}_1}(k)\right)}^{p_1}\right)}^{\frac{1}{n}}}\right)}^{\frac{1}{p_1}}])\rangle \end{gathered}$$

(36)

(2) When $m=2$, we have the following formula.

$$\begin{gathered} INLMS{M}^{(2)}(x_1,x_2,...,x_n)= \\ \langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-{\left(\frac{\theta i_1(k)}{l}\right)}^{p_{{}_1}}\cdot {\left(\frac{\theta i_2(k)}{l}\right)}^{p_{{}_2}}\right)}^{\frac{1}{C_n^2}}}\right)}^{\frac{1}{p_1+p_2}}}, \\ (\left[{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^2}\left(1-{\left(T^L{}_{i_1(k)}\right)}^{P_1}\cdot {\left(T^L{}_{i_2(k)}\right)}^{P_2}\right)}}^{\frac{1}{{{\displaystyle C}}_n^2}}\right)}^{\frac{1}{p_1+p_2}},{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^2}\left(1-{\left(T^U{}_{i_1(k)}\right)}^{P_1}\cdot {\left(T^U{}_{i_2(k)}\right)}^{P_2}\right)}}^{\frac{1}{{{\displaystyle C}}_n^2}}\right)}^{\frac{1}{p_1+p_2}}\right] \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-{\left(1-I^L{{\displaystyle i}}_{{}_1}(k)\right)}^{P_1}\cdot {\left(1-I^L{{\displaystyle i}}_{{}_2}(k)\right)}^{P_2}\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{p_1+p_2}}, \\ 1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-{\left(1-I^U{{\displaystyle i}}_{{}_1}(k)\right)}^{P_1}\cdot {\left(1-I^U{{\displaystyle i}}_{{}_2}(k)\right)}^{P_2}\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{p_1+p_2}}], \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-{\left(1-F^L{{\displaystyle i}}_{{}_1}(k)\right)}^{P_1}\cdot {\left(1-F^L{{\displaystyle i}}_{{}_2}(k)\right)}^{P_2}\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{p_1+p_2}}, \\ 1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-{\left(1-F^U{{\displaystyle i}}_{{}_1}(k)\right)}^{P_1}\cdot {\left(1-F^U{{\displaystyle i}}_{{}_2}(k)\right)}^{P_2}\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{p_1+p_2}}])\rangle \end{gathered}$$

(37)

When $m=2$, the $INLGMS{M}^{(m,p_1,p_2,...,p_m)}$ operator would reduce to the BM for INLNs (INLGBM) operator.

(3) When $m=n$, the $INLMS{M}^{(m)}$ operator would reduce to the form below.

$$\begin{gathered} INLGMS{M}^{(n,p_1,p_2,...,p_m)}(a_1,\dots ,a_n)= \\ \langle s_{l\cdot {\left({\displaystyle {\prod }_{j=1}^n{\left(\frac{{\theta }_{{}_j}(k)}{l}\right)}^{p_j}}\right)}^{\frac{1}{p_1+p_2+...+p_{{}_n}}}}, \\ ([{\left({\displaystyle \prod _{j=1}^n{\left(T^L{}_{i_j(k)}\right)}^{p_j}}\right)}^{\frac{1}{p_1+p_2+...+p_{{}_n}}},{\left({\displaystyle \prod _{j=1}^n{\left(T^U{}_{i_j(k)}\right)}^{p_j}}\right)}^{\frac{1}{p_1+p_2+...+p_{{}_n}}}], \\ [1-{\left({\displaystyle \prod _{j=1}^m{\left(1-I^L{}_{i_j(k)}\right)}^{p_j}}\right)}^{\frac{1}{p_1+p_2+...+p_{{}_n}}},1-{\left({\displaystyle \prod _{j=1}^m{\left(1-I^U{}_{i_j(k)}\right)}^{p_j}}\right)}^{\frac{1}{p_1+p_2+...+p_m}}], \\ [1-{\left({\displaystyle \prod _{j=1}^m{\left(1-F^L{}_{i_j(k)}\right)}^{p_j}}\right)}^{\frac{1}{p_1+p_2+...+p_n}},1-{\left({\displaystyle \prod _{j=1}^m{\left(1-F^U{}_{i_j(k)}\right)}^{p_j}}\right)}^{\frac{1}{p_1+p_2+...+p_{{}_n}}}])\rangle \end{gathered}$$

(38)

4.2. Some Weighted INLMSM Operators

We will introduce two operators, which are the weighted forms of the INLMSM operator and INLGMSM operator.

Definition 15.

Let$a_i={\langle s}_{{{\displaystyle \theta }}_i},([T^L(a_i),T^U(a_i)],[I^L(a_i),I^U(a_i)],[F^L(a_i),F^U(a_i)])\rangle$$i(i=1,2,...,n)$be a set of INLNs. Let$\omega =({\omega }_1,{\omega }_2,...,{\omega }_n)$T is the weight vector and satisfies${\displaystyle {\sum }_{i=1}^n{\omega }_i=1}$with${\omega }_i>0(i=1,2,...,n)$. Each${\omega }_i$represents the importance of$a_i$. Then the WINLMSM operator:${\Omega }^n\to \Omega$is defined below.

$$WINLMS{M}^{(m)}(a_1,\dots ,a_n)={\left(\frac{{}_{1\le i_1<...<i_m\le n}^{\oplus }\left(\underset{j=1}{\overset{m}{{\displaystyle \otimes }}}\left(n{\omega }_{i_j}\right)a_{i_j}\right)}{C_n^m}\right)}^{\frac{1}{m}},$$

(39)

$\Omega$is a set of INLNs and$m=1,2,...,n$.

According to the operational laws of INLNs in Definition 10, we can get the expression of the WINLMSM operator, which is shown below.

Theorem 3.

Let$a_i={\langle s}_{{{\displaystyle \theta }}_i},([T^L(a_i),T^U(a_i)],[I^L(a_i),I^U(a_i)],[F^L(a_i),F^U(a_i)])\rangle$$i(i=1,2,...,n)$be a set of INLNs and$m=1,2,...,n$, then the value aggregated from Definition 15 is still a WINLMSM operator.

$$\begin{gathered} WINLMS{M}^{(m)}(a_1,\dots ,a_n)=\langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle {\prod }_{j=1}^m\left(1-{\left(1-\frac{\theta i_j(k)}{l}\right)}^{n\cdot {\omega }_{i_j}}\right)}\right)}^{\frac{1}{C_n^m}}}\right)}^{\frac{1}{m}}}, \\ ([{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m\left(1-{\left(1-T^L{}_{i_j\left(k\right)}\right)}^{n\cdot {\omega }_{ij}}\right)}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}}, \\ {\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m\left(1-{\left(1-T^U{}_{i_j(k)}\right)}^{n\cdot {\omega }_{ij}}\right)}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}}], \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-{\left(I^L{}_{i_j(k)}\right)}^{n\cdot {\omega }_{ij}}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}, \\ 1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-{\left(I^U{}_{i_j(k)}\right)}^{n\cdot {\omega }_{ij}}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}], \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-{\left(F^L{}_{i_j(k)}\right)}^{n\cdot {\omega }_{ij}}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-{\left(F^U{}_{i_j(k)}\right)}^{n\cdot {\omega }_{ij}}\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}])\rangle \end{gathered}$$

(40)

where$k=1,2,...,{{\displaystyle C}}_n^m$, $a_{i_j(k)}$is the$i_j$th element of $k$th permutation. The process of proof is similar to Theorem 1. Now it is omitted.

Property 3.

Let$x_i={\langle s}_{{\alpha }_i},([T^L(x_i),T^U(x_i)],[I^L(x_i),I^U(x_i)],[F^L(x_i),F^U(x_i)])\rangle$$(i=1,2,...,n)$and$y_i={\langle s}_{{\beta }_i},([T^L(y_i),T^U(y_i)],[I^L(y_i),I^U(y_i)],[F^L(y_i),F^U(y_i)])\rangle$$(i=1,2,...,n)$be sets of INLNs. There are some properties of the$WINLMS{M}^{(m)}$operator as shown below.

1 

Reducibility. When$\omega =(\frac{1}{n},\frac{1}{n},...,\frac{1}{n})$, then$WINLMS{M}^{(m)}(a_1,a_2,...,a_n)=INLMS{M}^{(m)}(a_1,a_2,...,a_n)$.

2 

Monotonicity. If${\alpha }_i\le {\beta }_i,$$T^L(x_i)\le T^L(y_i)$, $T^U(x_i)\le T^U(y_i)$, $I^L(x_i)\ge I^L(y_i)$, $I^U(x_i)\ge I^U(y_i)$, $F^L(x_i)\ge F^L(y_i)$and$F^U(x_i)\ge F^U(y_i)$for all$i$$(i=1,2,...,n)$, then$x_i\le y_i$and$WINLMS{M}^{(m)}(x_1,x_2,...,x_n)\le WINLMS{M}^{(m)}(y_1,y_2,...,y_n)$.

3 

Boundedness.$\min \{x_1,x_2,...x_n\}\le {WINLMSM}^{(m)}\{x_1,x_2,...x_n\}\le \max \{x_1,x_2,...x_n\}.$

Proof. 

1

If $\omega =(\frac{1}{n},\frac{1}{n},...,\frac{1}{n})$, then $WINLMS{M}^{(m)}(a_1,a_2,...,a_n)=$

$$\begin{gathered} \langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle {\prod }_{j=1}^m\left(1-{\left(1-\frac{\theta i_j(k)}{l}\right)}^{n\cdot \frac{1}{n}}\right)}\right)}^{\frac{1}{C_n^m}}}\right)}^{\frac{1}{m}}}, \\ ([{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m\left(1-\left(1-T^L{}_{i_j(k)}\right)\right)}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}},{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m\left(1-\left(1-T^U{}_{i_j(k)}\right)\right)}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{m}}], \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-\left(I^L{}_{i_j(k)}\right)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-\left(I^U{}_{i_j(k)}\right)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}] \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-\left(F^L{}_{i_j(k)}\right)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}},1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle \prod _{j=1}^m\left(1-\left(F^U{}_{i_j(k)}\right)\right)}\right)}^{\frac{1}{{{\displaystyle C}}_n^m}}}\right)}^{\frac{1}{m}}])\rangle \\ =INLMS{M}^{(m)}(a_1,a_2,...,a_n). \end{gathered}$$

2

The proofs of Monotonicity and Boundedness are similar to Property 1, which are now omitted.

Furthermore, the WINLMSM^(m) operator would degrade a particular form when m takes some special values.

(1) When $m=1$, we have the formula below.

$$\begin{gathered} WINLMS{M}^{(1)}(a_1,\dots ,a_n)= \\ \langle s_{l\cdot \left(1-{\displaystyle {\prod }_{i=1}^n{\left(1-\frac{{\theta }_i}{l}\right)}^{{\omega }_i}}\right)},([\left(1-{\displaystyle {\prod }_{i=1}^n{\left(1-T_i^L\right)}^{{\omega }_i}}\right),\left(1-{\displaystyle {\prod }_{i=1}^n{\left(1-T_i^U\right)}^{{\omega }_i}}\right)], \\ [{\displaystyle {\prod }_{i=1}^n{\left(I_i^L\right)}^{{\omega }_i}},{\displaystyle {\prod }_{i=1}^n{\left(I_i^U\right)}^{{\omega }_i}}],[{\displaystyle {\prod }_{i=1}^n{\left(F_i^L\right)}^{{\omega }_i}},{\displaystyle {\prod }_{i=1}^n{\left(F_i^U\right)}^{{\omega }_i}}])\rangle \end{gathered}$$

(41)

(2) When $m=2$, we have the formula below.

$$\begin{gathered} WINLMS{M}^{(2)}(a_1,\dots ,a_n)=\langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-\left(1-{\left(1-\frac{\theta i_1(k)}{l}\right)}^{n\cdot {\omega }_{i_1}}\right)\cdot \left(1-{\left(1-\frac{\theta i_2\left(k\right)}{l}\right)}^{n\cdot {\omega }_{i_2}}\right)\right)}^{\frac{1}{C_n^2}}}\right)}^{\frac{1}{2}}}, \\ \begin{array}{l}([{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^2}\left(1-\left(1-{\left(1-T_{i_1(k)}^L\right)}^{n\cdot {\omega }_{i_1}}\right)\cdot \left(1-{\left(1-T_{i_2\left(k\right)}^L\right)}^{n\cdot {\omega }_{i_2}}\right)\right)}}^{\frac{1}{{{\displaystyle C}}_n^2}}\right)}^{\frac{1}{2}},\\ {\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^2}\left(1-\left(1-{\left(1-T_{i_1(k)}^U\right)}^{n\cdot {\omega }_{i_1}}\right)\cdot \left(1-{\left(1-T_{i_2(k)}^U\right)}^{n\cdot {\omega }_{i_2}}\right)\right)}}^{\frac{1}{{{\displaystyle C}}_n^2}}\right)}^{\frac{1}{2}}],\end{array} \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-\left(1-{\left(I_{i_1(k)}^L\right)}^{n\cdot {\omega }_{i_1}}\right)\cdot \left(1-{\left(I_{i_2(k)}^L\right)}^{n\cdot {\omega }_{i_2}}\right)\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{2}}, \\ 1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-\left(1-{\left(I_{i_1(k)}^U\right)}^{n\cdot {\omega }_{i_1}}\right)\cdot \left(1-{\left(I_{i_2(k)}^U\right)}^{n\cdot {\omega }_{i_2}}\right)\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{2}}], \\ [1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-\left(1-{\left(F_{i_1(k)}^L\right)}^{n\cdot {\omega }_{i_1}}\right)\cdot \left(1-{\left(F_{i_2(k)}^L\right)}^{n\cdot {\omega }_{i_2}}\right)\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{2}}, \\ 1-{\left(1-{\displaystyle {\prod }_{k=1}^{C_n^2}{\left(1-\left(1-{\left(F_{i_1(k)}^U\right)}^{n\cdot {\omega }_{i_1}}\right)\cdot \left(1-{\left(F_{i_2(k)}^U\right)}^{n\cdot {\omega }_{i_2}}\right)\right)}^{\frac{1}{{{\displaystyle C}}_n^2}}}\right)}^{\frac{1}{2}}])\rangle \end{gathered}$$

(42)

(3) When $m=n$, we have the formula below.

$$\begin{gathered} WINLMS{M}^{(n)}(a_1,\dots ,a_n)=\langle s_{l\cdot {\left({\displaystyle {\prod }_{j=1}^n\left(1-{\left(1-\frac{{\theta }_j}{l}\right)}^{n\cdot {\omega }_{{}_j}}\right)}\right)}^{\frac{1}{n}}}, \\ ([{\left({\displaystyle \prod _{j=1}^n\left(1-{\left(1-T_j^L\right)}^{n\cdot {\omega }_j}\right)}\right)}^{\frac{1}{n}},{\left({\displaystyle \prod _{j=1}^n\left(1-{\left(1-T_j^U\right)}^{n\cdot {\omega }_j}\right)}\right)}^{\frac{1}{n}}], \\ [1-{\left({\displaystyle \prod _{j=1}^n\left(1-{\left(I_j^L\right)}^{n\cdot {\omega }_j}\right)}\right)}^{\frac{1}{n}},1-{\left({\displaystyle \prod _{j=1}^n\left(1-{\left(I_j^U\right)}^{n\cdot {\omega }_j}\right)}\right)}^{\frac{1}{n}}], \\ [1-{\left({\displaystyle \prod _{j=1}^n\left(1-{\left(F_j^L\right)}^{n\cdot {\omega }_j}\right)}\right)}^{\frac{1}{n}},1-{\left({\displaystyle \prod _{j=1}^n\left(1-{\left(F_j^U\right)}^{n\cdot {\omega }_j}\right)}\right)}^{\frac{1}{n}}])\rangle . \end{gathered}$$

(43)

☐

Definition 16.

Let$a_i={\langle s}_{{{\displaystyle \theta }}_i},([T^L(a_i),T^U(a_i)],[I^L(a_i),I^U(a_i)],[F^L(a_i),F^U(a_i)])\rangle$$i(i=1,2,...,n)$be a set of INLNs. Let$\omega =({\omega }_1,{\omega }_2,...,{\omega }_n)$T is the weight vector and it satisfies${\displaystyle {\sum }_{i=1}^n{\omega }_i=1}$with${\omega }_i>0(i=1,2,...,n)$. Each${\omega }_i$represents the importance of$a_i$. Then the WINLGMSM operator:${\Omega }^n\to \Omega$is defined below.

$$WINLGMS{M}^{(m,p_1,p_2,...p_m)}(a_1,\dots ,a_n)={\left(\frac{{}_{1\le i_1<...<i_m\le n}^{\oplus }\left(\underset{j=1}{\overset{m}{{\displaystyle \otimes }}}{\left({{\displaystyle n{\omega }_{i_j}\cdot a}}_{i_j}^{}\right)}^{p_j}\right)}{C_n^m}\right)}^{\frac{1}{p_1+p_2+...+p_m}}$$

(44)

$\Omega$is a set of INLNs and$m=1,2,...,n$.

According to the operational laws of INLNs in Definition 10, we can get the expression of WINLMSM operator shown below.

Theorem 4.

Let$a_i={\langle s}_{{{\displaystyle \theta }}_i},([T^L(a_i),T^U(a_i)],[I^L(a_i),I^U(a_i)],[F^L(a_i),F^U(a_i)])\rangle$$i(i=1,2,...,n)$be a set of INLNs and$m=1,2,...,n$. Then the value aggregated from Definition 16 is still an WINLGMSM.

$$\begin{gathered} WINLGMS{M}^{(m,p_1,p_2,...,p_m)}(a_1,\dots ,a_n)=\langle s_{l\cdot {\left(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{\left(1-{\displaystyle {\prod }_{j=1}^m{\left(1-{\left(1-\frac{{\theta }_{i_j}(k)}{l}\right)}^{n\cdot {\omega }_{i_j}}\right)}^{p_j}}\right)}^{\frac{1}{C_n^m}}}\right)}^{\frac{1}{p_1+p_2+...+p_m}}}, \\ ([{\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{\left(1-{\left(1-T^L{{\displaystyle i}}_{{}_j}(k)\right)}^{n\cdot {\omega }_{i_j}}\right)}^{p_j}}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{p_1+p_2+...+p_m}}, \\ {\left(1-{{\displaystyle {\prod }_{k=1}^{C_n^m}\left(1-{\displaystyle \prod _{j=1}^m{\left(1-{\left(1-T^U{{\displaystyle i}}_{{}_j}(k)\right)}^{n\cdot {\omega }_{i_j}}\right)}^{p_j}}\right)}}^{\frac{1}{{{\displaystyle C}}_n^m}}\right)}^{\frac{1}{p_1+p_2+...+p_m}}], \\ [1-{(1-{\displaystyle {\prod }_{k=1}^{C_n^m}{(1-{\displaystyle \prod _{j=1}^m{(1-{(I^L{{\displaystyle i}}_{}}^{}}^{}}}^{}}}^{} \end{gathered}$$