A novel defuzzification method of SV-trapezoidal neutrosophic numbers and multi-attribute decision making: a comparative analysis

Abstract

The aim of this paper is to investigate the multiple attribute decision-making (MADM) problems where both the attribute value and attribute weight of alternatives are single-valued trapezoidal neutrosophic numbers (SVTN-numbers). Ranking of SVTN-numbers are always a necessary step in solving the MADM problems under SVTN environment, and the literature review reckoned the existence of six to seven ranking methods. After all the existing ranking methods of SVTN-numbers are examined, we firstly define the concept of centroid point and examine several useful properties of the developed concept. Then, we develop hamming ranking value and Euclidean ranking value of SVTN-numbers to compare the SVTN-numbers. Furthermore, based on the proposed ranking values, we develop a novel defuzzification method to MADM with linguistic information and give a real example deal with manufacturing company to illustrate the feasibility and effectiveness of the developed approach. Finally, we present some examples to compare the proposed method with the existing ranking results and the results verified through comparative analysis.

1 Introduction

The multiple attribute decision making (MADM) is an important part of decision making to choose the most desirable alternative from a set of alternatives according to the decision information provided by a decision maker. In MADM, since most of the decision information about attribute values is not known precisely, Zadeh (1965) has proposed the fuzzy set theory to model the uncertainty and vagueness that is nature of human thinking in the process of decision making. By added non-membership degree to fuzzy sets, Atanassov (1999) initially proposed intuitionistic fuzzy set theory as an extension of the concept of a fuzzy set. It is worth noting that the above decision-making information is based on fuzzy and intuitionistic fuzzy sets. To provide more flexibility to present imprecise data and uncertainty, Smarandache (1998) defined neutrosophic sets, which independently contain truth-membership degree, indeterminacy-membership degree and falsity-membership degree. Because of their advantages, the theories have been studied by many authors such as: on fuzzy sets (Kahraman and Otay 2019a; Rao and Shankar 2011; Yager and Zadeh 1992; Ying 2000; Zadeh 2005), on intuitionistic fuzzy sets (Liu and Li 2017; Wan et al. 2017; Wei 2010; Zhang 2017) and on neutrosophic sets in Broumi et al. (2014a, b), Kahraman and Otay (2019b), Peng et al. (2014, 2016) and Ye (2018).

However, due to uncertainty and complexity in some real situations, we often only can use a single-valued neutrosophic number that is a special neutrosophic set on the real number R. Thus, it seems to be suitable for employing in the real number R to estimate both the attribute value and attribute weight of alternatives in MADM problems. Hereby, single-valued trapezoidal neutrosophic numbers (SVTN-numbers) were defined by Deli and Şubaş (2017b) and Ye (2017, 2015). Currently, many researchers pay attention to the methods and related properties of SVTN-numbers. For example, Basset et al. (2018a) proposed a decision-making method by using Analytic Hierarchy Process for SVTN-numbers. Liu and Zhang (2018) introduced some operators on SVTN-numbers based on Maclaurin symmetric mean. Broumi et al. (2016b) developed an approach for solving shortest path problem in a network where edge weights are characterized by SVTN-numbers. Then, same authors presented a new method for solving shortest path problems in a network in which the edges length are characterized by SVTN-numbers in Broumi et al. (2016a). Biswas et al. (2016a, b) defined some operators and applied them to decision-making problems including medical representative selection problem. Deli and Şubaş (2017a) defined the values and ambiguities of the SVTN-numbers under cut sets. Liang et al. (2017a) presented new operations, an improved comparison method, and two aggregation operators to improve the methods in SVTN-numbers. Since the traditional arithmetic mean operator or geometric mean operator only deals with independent criteria, Liang et al. (2017b) introduced different operators by using entropy-weighted method. Basset et al. (2018b) developed a novel algorithm for the group decision-making problem with triangular neutrosophic additive reciprocal preference relations.

Das and Guha (2013) said that “However, after analyzing the aforementioned ranking procedures it has been observed that, for some cases, they fail to calculate the ranking results correctly. Furthermore, many of them produce different ranking outcomes for the same problem. Under these circumstances, the decision maker may not be able to carry out the comparison and recognition properly. This creates problem in practical applications”. In order to overcome these problems of the existing methods, a new method for ranking has been proposed based on centroid point of intuitionistic fuzzy numbers in Das and Guha (2013). Then, in Das and Guha (2016), a new ranking method of intuitionistic fuzzy numbers was developed by utilizing the concept of centroid point. Varghese and Kuriakose (2012) defined a formula for finding the centroid of an intuitionistic fuzzy number. Arun et al. (2016) developed a new approach to rank intuitionistic fuzzy numbers based on centroid point of intuitionistic fuzzy numbers. Hajek and Olej (2014) gave an intuitionistic fuzzy inference system of Takagi-Sugeno type which is defuzzification methods. As far as we know, however, there has been a few researches on methods of SVTN-numbers.

Therefore, the primary goal of this paper is to define the defuzzification methods of SVTN-numbers and hereby develop a simplified and an effective method for computing defuzzification methods of SVTN-numbers. The rest of this paper is organized as follows. In Sect. 2, we briefly introduced some concepts of SVTN-numbers. In Sect. 3, we presented the existing ranking methods with examples. In Sect. 4, we defined the concept of centroid point and examined several useful properties of the developed concept. Then, we proposed the hamming ranking value and Euclidean ranking value of SVTN-numbers to compare the SVTN-numbers. In Sect. 5, we developed a novel defuzzification method to MADM with linguistic information and gave a real example deal with manufacturing company to illustrate the feasibility and effectiveness of the developed approach. Finally, we presented some examples to compare the proposed method with the existing ranking results, and the results were verified through comparative analysis.

2 Preliminary

Definition 2.1

(Wang et al. 2010) Let X be a universe. A single-valued neutrosophic set A in X is characterized by a truth-membership function $T_A:X\rightarrow [0,1]$, an indeterminacy-membership function $I_A:X\rightarrow [0,1]$ and a falsity-membership function $F_A:X\rightarrow [0,1]$ as:

$$\begin{aligned} A=\{<x,(T_A(x),I_A(x),F_A(x))>:x\in X\}. \end{aligned}$$

Definition 2.2

(Şubaş 2015) Let $w_{{\tilde{a}}}, u_{{\tilde{a}}}, y_{{\tilde{a}}}\in [0,1]$ be any real number. A single-valued neutrosophic number ${\tilde{a}}=\langle ([a_{1},b_{1},c_{1},d_{1})],w_{{\tilde{a}}}),$$([a_{2},b_{2},c_{2},d_{2}],u_{{\tilde{a}}}),$$([a_{3},b_{3},c_{3},d_{3}],y_{{\tilde{a}}})\rangle$ is a special single-valued neutrosophic set on the set of real numbers R, whose truth-membership function $\mu _{{\tilde{a}}}:R\rightarrow [0,w_{{\tilde{a}}}]$, an indeterminacy-membership function $\nu _{{\tilde{a}}}:R\rightarrow [u_{{\tilde{a}}},1]$ and a falsity-membership function $\lambda _{{\tilde{a}}}:R\rightarrow [y_{{\tilde{a}}},1]$ as given by;

$$\begin{aligned}&\mu _{{\tilde{a}}}(x) = \left\{ \begin{array}{l@{\quad }l} f_{\mu l}(x) &{} (a_{1}\le x<b_{1} ) \\ w_{{\tilde{a}}} &{} ({b_{1}}\le x<{c_{1}})\\ f_{\mu r}(x) &{} (c_{1}\le x \le d_{1})\\ 0 &{} \hbox {otherwise} \end{array}\right. \\&\nu _{{\tilde{a}}}(x) = \left\{ \begin{array}{l@{\quad }l} f_{\nu l}(x) &{} (a_{2}\le x<{b_{2}} ) \\ u_{{\tilde{a}}} &{} (b_{2}\le x <c_{2})\\ f_{\nu r}(x) &{} (c_{2}\le x \le d_{2})\\ 1 &{} \hbox {otherwise} \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} \lambda _{{\tilde{a}}}(x) = \left\{ \begin{array}{l@{\quad }l} f_{\lambda l}(x) &{} (a_{3}\le x<b_{3} ) \\ y_{{\tilde{a}}} &{} (b_{3}\le x <c_{3})\\ f_{\lambda r}(x) &{} (c_{3}\le x \le d_{3})\\ 1 &{} \hbox {otherwise} \end{array}\right. \end{aligned}$$

respectively, where the functions $f_{\mu l}:[a_{1},b_{1}]\rightarrow [0,w_{{\tilde{a}}}]$, $f_{\nu r}:[c_{2},d_{2}]\rightarrow [u_{{\tilde{a}}},1]$$f_{\lambda r}:[c_{3},d_{3}]\rightarrow [y_{{\tilde{a}}},1]$ are continuous and nondecreasing, and satisfy the conditions: $f_{\mu l}(a_{1})=0, f_{\mu l}(b_{1})=w_{{\tilde{a}}}, f_{\nu r}(c_{2})=u_{{\tilde{a}}},$$f_{\nu r}(d_{2})=1$, $f_{\lambda r}(c_{3})=y_{{\tilde{a}}},$ and $f_{\lambda r}(d_{3})=1$; the functions $f_{\mu r}: [c_{1},d_{1}]\rightarrow [0,w_{{\tilde{a}}}]$, $f_{\nu l}:[{a}_{2},b_{2}]\rightarrow [u_{{\tilde{a}}},1]$ and $f_{\lambda l}:[a_{3},b_{3}]\rightarrow [y_{{\tilde{a}}},1]$ are continuous and nonincreasing, and satisfy the conditions: $f_{\mu r}(c_{1})=w_{{\tilde{a}}}$,$f_{\mu r}(d_{1})=0$, $f_{\nu l}(a_{2})=1$, $f_{\nu l}(b_{2})=u_{{\tilde{a}}}$, $f_{\lambda l}(a_{3})=1$ and $f_{\lambda l}(b_{3})=y_{{\tilde{a}}}$. $[b_{1},c_{1}],a_{1}$ and $d_{1}$ are called the mean interval and the lower and upper limits of the general neutrosophic number ${\tilde{a}}$ for the truth-membership function, respectively. $[b_{2},c_{2}],a_{2}$ and $d_{2}$ are called the mean interval and the lower and upper limits of the general neutrosophic number ${\tilde{a}}$ for the indeterminacy-membership function, respectively. $[b_{3},c_{3}],a_{3}$ and $d_{3}$ are called the mean interval and the lower and upper limits of the general neutrosophic number ${\tilde{a}}$ for the falsity-membership function, respectively. $w_{{\tilde{a}}}$, $u_{{\tilde{a}}}$ and $y_{{\tilde{a}}}$ are called the maximum truth-membership degree, minimum indeterminacy-membership degree and minimum falsity-membership degree, respectively.

Note that a single-valued neutrosophic number ${\tilde{a}}=\langle ([a_{1},b_{1},c_{1},d_{1})],w_{{\tilde{a}}}),$$([a_{2},b_{2},c_{2},d_{2}],u_{{\tilde{a}}}),$$([a_{3},b_{3},c_{3},d_{3}],y_{{\tilde{a}}})\rangle$ for $w_{{\tilde{a}}}=1, u_{{\tilde{a}}}=0, y_{{\tilde{a}}}=0$ defined in Ye (2017) as: ${\tilde{a}}=\langle [a_{1},b_{1},c_{1},d_{1}] ,$$[a_{2},b_{2},c_{2},d_{2}],$$[a_{3},b_{3},c_{3},d_{3}] \rangle$

Definition 2.3

(Deli and Şubaş 2017a) An SVTN-number ${\tilde{a}}=\langle (a_{},b_{},c_{},d_{});w_{{\tilde{a}}},u_{{\tilde{a}}}, y_{{\tilde{a}}}\rangle$ on R is characterized by a truth-membership function $T_{{\tilde{a}}}:R\rightarrow [0,1]$, an indeterminacy-membership function $I_{{\tilde{a}}}:R\rightarrow [0,1]$ and a falsity-membership function $F_{{\tilde{a}}}:R\rightarrow [0,1]$ are defined as:

$$\begin{aligned}&<x,(T_{{\tilde{a}}}(x),I_{{\tilde{a}}}(x),F_{{\tilde{a}}}(x))> \\&\quad = \left\{ \begin{array}{l@{\quad }l}<x,(\frac{(x-a_{})w_{{\tilde{a}}}}{b_{}-a_{}} ,\frac{b_{}-x + u_{{\tilde{a}}}(x-a_{})}{ b_{}-a_{}} ,\frac{b_{}-x + y_{{\tilde{a}}}(x-a_{})}{b_{}-a_{}})> ,&{} a_{}\le x<{b_{}} \\<x,(w_{{\tilde{a}}} , u_{{\tilde{a}}},y_{{\tilde{a}}})>,&{} {b_{}}\le x \le {c_{}} \\<x,(\frac{(d_{}-x)w_{{\tilde{a}}}}{d_{}-c_{}}, \frac{x-c_{}+u_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}} , \frac{x-c_{}+y_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}} )>,&{} {c_{}}< x \le d_{}\\ <x,(0,1,1) >,&{} \hbox {otherwise}. \end{array}\right. \end{aligned}$$

Note that the set of all SVTN-number-numbers on R will be denoted by $\triangle$.

For any ${\tilde{a}}=\langle ({a}_{1},b_{1},c_{1},d_{1});w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$ and ${\tilde{b}}=\langle ({a}_{2},b_{2},c_{2},d_{2});w_{{\tilde{b}}},$$u_{{\tilde{b}}},y_{{\tilde{b}}}\rangle$;

• 1. ${\tilde{a}}+{\tilde{b}}=\langle (a_{1}+a_{2},b_{1} +b_{2},c_{1}+c_{2},d_{1}+d_{2}); w_{{\tilde{a}}}\wedge w_{{\tilde{b}}},u_{{\tilde{a}}}\vee u_{{\tilde{b}}},y_{{\tilde{a}}}\vee y_{{\tilde{b}}}\rangle$

• 2. ${\tilde{a}}-{\tilde{b}}=\langle (a_{1} -d_{2},b_{1}-c_{2},c_{1}-b_{2},d_{1}-a_{2}); w_{{\tilde{a}}}\wedge w_{{\tilde{b}}},u_{{\tilde{a}}}\vee u_{{\tilde{b}}},y_{{\tilde{b}}}\vee y_{{\tilde{b}}}\rangle$

• 3. ${\tilde{a}} {\tilde{b}}$ $\, = \left\{ \begin{array}{ll} \langle (a_{1}a_{2},b_{1}b_{2},c_{1}c_{2},d_{1}d_{2}; w_{{\tilde{a}}}{\wedge } w_{{\tilde{b}}}, u_{{\tilde{a}}} {\vee } u_{{\tilde{b}}}, y_{{\tilde{a}}}{\vee } y_{{\tilde{b}}} \rangle \\ (d_{1}>0, d_{2}>0)\\ \langle (a_{1}d_{2},b_{1}c_{2},c_{1}b_{2},d_{1}a_{2}); w_{{\tilde{a}}}{\wedge } w_{{\tilde{b}}}, u_{{\tilde{a}}} {\vee } u_{{\tilde{b}}} , y_{{\tilde{a}}}{\vee } y_{{\tilde{b}}} \rangle \\ (d_{1}<0, d_{2}>0)\\ \langle (d_{1}d_{2},c_{1}c_{2},b_{1}b_{2},a_{1}a_{2}); w_{{\tilde{a}}}{\wedge } w_{{\tilde{b}}}, u_{{\tilde{a}}} {\vee } u_{{\tilde{b}}}, y_{{\tilde{a}}}{\vee } y_{{\tilde{b}}} \rangle \\ (d_{1}<0, d_{2}<0)\\ \end{array}\right.$

• 4. $\gamma {\tilde{a}}= \left\{ \begin{array}{l@{\quad }l} \langle (\gamma a_{1}, \gamma b_{1},\gamma c_{1},\gamma d_{1}); w_{{\tilde{a}}},u_{{\tilde{a}}}, y_{{\tilde{a}}} \rangle &{} (\gamma >0)\\ \langle (\gamma d_{1},\gamma c_{1}, \gamma b_{1},\gamma a_{1}); w_{{\tilde{a}}},u_{{\tilde{a}}}, y_{ {\tilde{a}}} \rangle &{} (\gamma <0)\\ \end{array}\right.$

• 5. ${\tilde{a}}^{\gamma }= \left\{ \begin{array}{l@{\quad }l} \langle ( a_{1}^{\gamma }, b_{1}^{\gamma }, c_{1}^{\gamma }, c_{1}^{\gamma });w_{{\tilde{a}}},u_{{\tilde{a}}}, y_{{\tilde{a}}} \rangle &{} (\gamma >0)\\ \langle ( c_{1}^{\gamma },c_{1}^{\gamma }, b_{1}^{\gamma }, a_{1}^{\gamma });w_{{\tilde{a}}},u_{{\tilde{a}}}, y_{{\tilde{a}}} \rangle &{} (\gamma <0)\\ \end{array}\right.$

Definition 2.4

(Şubaş 2015)  Let ${\tilde{a}}_{j}=\langle (a_{j},b_{j},c_{j},d_{j}); w_{{\tilde{a}}_{j}},u_{{\tilde{a}}_{j}},y_{{\tilde{a}}_{j}}\rangle \in \triangle \,\,(j\in I_{n})$. Then, SVTN weighted arithmetic operator, denoted by $G_{ao}$, is defined as:

$$\begin{aligned} \begin{aligned} G_{ao}({\tilde{a}}_{1},{\tilde{a}}_{2},\ldots ,{\tilde{a}}_{n})&= \bigg \langle \bigg (\sum _{j=1}^{n} a_{j} \times {w_{j}},\sum _{j=1}^{n} b_{j}\times {w_{j}},\\&\quad \sum _{j=1}^{n} c_{j}\times {w_{j}}, \sum _{j=1}^{n} d_{j}\times {w_{j}}\bigg );\\&\quad \bigwedge _{j=1}^{n}w_{{\tilde{a}}_j},\bigvee _{j=1} ^{n}u_{{\tilde{a}}_j},\bigvee _{j=1}^{n}y_{{\tilde{a}}_j}\bigg \rangle \end{aligned} \end{aligned}$$

where, $w=(w_{1},w_{2},\ldots ,w_{n})^\mathrm{T}$ is a weight vector associated with the $G_{go}$ operator, for every $j\in I_{n}$ such that, $w_{j}\in [0,1]$ and $\sum _{j=1}^{n}w_j=1.$

Table 1 Ranking for Method 1

Table 2 Ranking for Method 2

3 A brief note on existing ranking methods of SVTN-number

As mentioned in the introduction, ranking of SVTN-number is always a necessary step in solving the problems under SVTN-number environment and the literature review reckoned the existence of six to seven ranking methods. In this section, all the existing ranking methods of SVTN-numbers are examined with numerical examples.

Method 1 (Deli and Şubaş 2017a) Let $\tilde{a_1}$ and $\tilde{a_2}$ be two SVTN-numbers and $\theta \in [0,1]$. Based on weighted values and ambiguities of the SVN-numbers, the ranking order of $\tilde{a_1}$ and $\tilde{a_2}$ is defined as:

• 1. If $V_{\theta }(\tilde{a_1})>V_{\theta }(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{>}\,\tilde{a_2};$

• 2. If $V_{\theta }(\tilde{a_1})=V_{\theta }(\tilde{a_2})$, then

  • (a) If $A_{\theta }(\tilde{a_1})>A_{\theta }(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{>}\,\tilde{a_2}$

  • (b) If $A_{\theta }(\tilde{a_1})=A_{\theta }(\tilde{a_2})$, then $\tilde{a_1}$ is equal to $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{=}\,\tilde{a_2};$

where for ${\tilde{a}}_{j}=\langle (a_{j},b_{j},c_{j},d_{j}); w_{{\tilde{a}}_{j}},u_{{\tilde{a}}_{j}},y_{{\tilde{a}}_{j}}\rangle$ (j=1,2)

• 1. the weighted value $V_{\theta }(\tilde{a_j})$is calculated as:

  $$\begin{aligned}&V_{\theta }(\tilde{a_j})=\frac{a_j+2b_j+2c_j+d_j}{6}\\&\quad \bigg [\theta w_{\tilde{a_j}}^{2}+(1-\theta )(1-u_{\tilde{a_j}})^{2}+(1-\theta ) (1-y_{\tilde{a_j}})^{2}\bigg ] \end{aligned}$$

• 2. the weighted ambiguity $A_{\theta }(\tilde{a_j})$ is calculated as:

  $$\begin{aligned}&A_{\theta }(\tilde{a_j})=\frac{d_j-a_j+2c_j-2b_j}{6}\\&\quad \bigg [\theta w_{\tilde{a_j}}^{2}+(1-\theta )(1-u_{\tilde{a_j}})^{2}+(1-\theta ) (1-y_{\tilde{a_j}})^{2}\bigg ] \end{aligned}$$

Example 3.1

Let $\tilde{a_1}=\langle (3, 4, 5, 6);0.9,0.3,0.4\rangle$ and $\tilde{a_2}=\langle (5, 7, 9, 11);0.3,0.5,0.2\rangle$ be two SVTN-numbers.

Then, we can compare the SVTN-numbers $\tilde{a_1}$ and $\tilde{a_2}$ as Table 1.

Method 2 (Biswas et al. 2016a) Let $\tilde{a_1}$ and $\tilde{a_2}$ be two SVTN-numbers and $\theta \in [0,1]$. For weighted values and ambiguities of the SVTN-numbers $\tilde{a_1}=\langle [a_{11},a_{21},a_{31},a_{41}],$$[b_{11},b_{21},b_{31},b_{41}],$$[c_{11},c_{21},c_{31},c_{41}] \rangle$ and $\tilde{a_2}=\langle [a_{12},a_{22},a_{32},a_{42}],$$[b_{12},b_{22},b_{32},b_{42}],$$[c_{12},c_{22},c_{32},c_{42}] \rangle$, the ranking order of $\tilde{a_1}$ and $\tilde{a_2}$, based on $\lambda ,\mu ,\nu \in [0,1]$ is defined as:

• 1. If $V_{\lambda ,\mu ,\nu }(\tilde{a_1})>V_ {\lambda ,\mu ,\nu }(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{>}\,\tilde{a_2};$

• 2. If $V_{\lambda ,\mu ,\nu }(\tilde{a_1})=V_ {\lambda ,\mu ,\nu }(\tilde{a_2})$, then

  • (a) If $A_{\lambda ,\mu ,\nu }(\tilde{a_1})>A_ {\lambda ,\mu ,\nu }(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{>}\,\tilde{a_2}$,

  • (b) If $A_{\lambda ,\mu ,\nu }(\tilde{a_1})=A_{\lambda ,\mu ,\nu }(\tilde{a_2})$, then $\tilde{a_1}$ is equal to $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{=}\,\tilde{a_2};$

where for $\tilde{a_j}=\langle [a_{1j},a_{2j},a_{3j},a_{4j}],$$[b_{1j},b_{2j},b_{3j},b_{4j}],$$[c_{1j},c_{2j},c_{3j},c_{4j}] \rangle (j=1,2)$

• 1. The value index $V_{\lambda ,\mu ,\nu }(\tilde{a_j})$is calculated as:

  $$\begin{aligned} V_{\lambda ,\mu ,\nu }(\tilde{a_j})= & {} \frac{\lambda }{6}(a_{1j}+2a_{2j},2a_{3j}+a_{4j})\\&+\,\frac{\mu }{6} (b_{1j}+2b_{2j}+2b_{3j}+b_{4j})\\&+\,\frac{\nu }{6}(c_{1j}+2c_{2j}+2c_{3j}+c_{4j}) \end{aligned}$$

• 2. The ambiguity index $A_{\lambda ,\mu ,\nu }(\tilde{a_j})$ is calculated as:

  $$\begin{aligned} A_{\lambda ,\mu ,\nu }(\tilde{a_j})= & {} \frac{\lambda }{6}(-a_{1j}-2a_{2j},2a_{3j}+a_{4j})\\&+\,\frac{\mu }{6} (-b_{1j}-2b_{2j}+2b_{3j}+b_{4j})\\&+\,\frac{\nu }{6}(-c_{1j}-2c_{2j}+2c_{3j}+c_{4j}) \end{aligned}$$

Example 3.2

Let $\tilde{a_1}=\langle (3, 4, 5, 6);1.0,0.0,0.0\rangle$ and $\tilde{a_2}=\langle (4, 5, 6, 7);1.0,0.0,0.0\rangle$ be two SVTN-numbers.

Then, we can compare the SVTN-numbers $\tilde{a_1}$ and $\tilde{a_2}$ as Table 2.

Table 3 Ranking for Method 3

Table 4 Ranking for Method 4

Method 3 (Ye 2017) Let $\tilde{a_1}=\langle ({a}_{1},b_{1},c_{1},d_{1});w_{\tilde{a_1}},$$u_{\tilde{a_1}},y_{\tilde{a_1}}\rangle$ and $\tilde{a_2}=\langle ({a}_{2},b_{2},c_{2},d_{2});w_{\tilde{a_2}},$$u_{\tilde{a_2}},y_{\tilde{a_2}}\rangle$ be two SVTN-numbers. Then,

• 1. If $S(\tilde{a_1})< S(\tilde{a_2})$, then $\tilde{a_1}$ is smaller than $\tilde{a_2}$, denoted by $\tilde{a_1}< \tilde{a_2}$

• 2. If $S(\tilde{a_1})= S(\tilde{a_2})$;

  • (a) If $A(\tilde{a_1})< A(\tilde{a_2})$, then $\tilde{a_1}$ is smaller than $\tilde{a_2}$, denoted by $\tilde{a_1}< \tilde{a_2}$

  • (b) If $A(\tilde{a_1})= A(\tilde{a_2})$, then $\tilde{a_1}$ and $\tilde{a_2}$ are the same, denoted by $\tilde{a_1}= \tilde{a_2}$

where for ${\tilde{a}}_{j}=\langle (a_{j},b_{j},c_{j},d_{j}); w_{{\tilde{a}}_{j}},u_{{\tilde{a}}_{j}},y_{{\tilde{a}}_{j}}\rangle (j=1,2)$

• 1. The score $S(\tilde{a_j})$ is calculated as:

  $$\begin{aligned} S(\tilde{a_j})= & {} \frac{1}{12}[a_j+b_j+c_j+d_j]\nonumber \\&\times (2+w_{{\tilde{a}}_{j}}-u_{{\tilde{a}}_{j}} -y_{{\tilde{a}}_{j}}) \end{aligned}$$

  (1)

• 2. The accuracy $A(\tilde{a_j})$ is calculated as:

  $$\begin{aligned} A(\tilde{a_j})= & {} \frac{1}{12}[a_j+b_j+c_j+d_j]\\&\times (2+w_{{\tilde{a}}_{j}}-u_{{\tilde{a}}_{j}}+y_{{\tilde{a}}_{j}}) \end{aligned}$$

Example 3.3

Let $\tilde{a_1}=\langle (1, 6, 10, 12);0.5,0.1,0.3\rangle$ and $\tilde{a_2}=\langle (0, 5, 7, 15);0.8,0.9,0.7\rangle$ be two SVTN-numbers.

Then, we can compare the SVTN-numbers $\tilde{a_1}$ and $\tilde{a_2}$ as Table 3.

Method 4   (Liang et al. 2017b)  Let $\tilde{a_1}=\langle ({a}_{1},b_{1},c_{1},d_{1});w_{\tilde{a_1}},$$u_{\tilde{a_1}},y_{\tilde{a_1}}\rangle$ and $\tilde{a_2}=\langle ({a}_{2},b_{2},c_{2},d_{2});w_{\tilde{a_2}},$$u_{\tilde{a_2}},y_{\tilde{a_2}}\rangle$ be two SVTN-numbers. Then,

• 1. If $E(\tilde{a_1})<E(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{<}\,\tilde{a_2};$

• 2. If $E(\tilde{a_1})=E(\tilde{a_2})$,

  • (a) ${\bar{A}}(\tilde{a_1})<{\bar{A}}(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{<}\,\tilde{a_2};$,

  • (b) If ${\bar{A}}(\tilde{a_1})={\bar{A}}(\tilde{a_2})$,

    • i. $C(\tilde{a_1})<C(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{<}\,\tilde{a_2};$

where for ${\tilde{a}}_{j}=\langle (a_{j},b_{j},c_{j},d_{j}); w_{{\tilde{a}}_{j}},u_{{\tilde{a}}_{j}},y_{{\tilde{a}}_{j}}\rangle$ (j=1,2)

• 1. The score function $E(\tilde{a_j})$ is given as

  $$\begin{aligned} E(\tilde{a_j})= & {} \frac{(a_j+2b_j+2c_j+d_j)}{6}\\&\cdot \frac{(2+w_{{\tilde{a}}_{j}} -u_{{\tilde{a}}_{j}}-y_{{\tilde{a}}_{j}})}{3} \end{aligned}$$

• 2. Accuracy function ${\bar{A}}(\tilde{a_j})$ is given as

  $$\begin{aligned} {\bar{A}}(\tilde{a_j})= \frac{(a_j+2b_j+2c_j+d_j)}{6}\cdot (w_{{\tilde{a}}_{j}}-y_{{\tilde{a}}_{j}}) \end{aligned}$$

• 3. Certainty function $C(\tilde{a_j})$ is given as

  $$\begin{aligned} C(\tilde{a_j})= \frac{(a_j+2b_j+2c_j+d_j)}{6}\cdot (w_{{\tilde{a}}_{j}}) \end{aligned}$$

Example 3.4

Let $\tilde{a_1}=\langle (5, 10, 55, 20);0.5,0.1,0.3\rangle$ and $\tilde{a_2}=\langle (0, 5, 10, 15);0.8,0.9,0.7\rangle$ be two SVTN-numbers.

Then, we can compare the SVTN-numbers $\tilde{a_1}$ and $\tilde{a_2}$ as Table 4.

Method 5 (Liang et al. 2017a) Let $\tilde{a_1}=\langle ({a}_{1},b_{1},c_{1},d_{1});w_{\tilde{a_1}},$$u_{\tilde{a_1}},y_{\tilde{a_1}}\rangle$ and $\tilde{a_2}=\langle ({a}_{2},b_{2},c_{2},d_{2});w_{\tilde{a_2}},$$u_{\tilde{a_2}},y_{\tilde{a_2}}\rangle$ be two SVTN-numbers. Then,

Table 5 Ranking for Method 5

Table 6 Ranking for Method 6

• 1. If $E(\tilde{a_1})<E(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{<}\,\tilde{a_2};$

• 2. If $E(\tilde{a_1})=E(\tilde{a_2})$,

  • (a) ${\bar{A}}(\tilde{a_1})<{\bar{A}}(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{<}\,\tilde{a_2};$,

  • (b) If ${\bar{A}}(\tilde{a_1})={\bar{A}}(\tilde{a_2})$,

    • i. $C(\tilde{a_1})<C(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{<}\,\tilde{a_2};$

where for ${\tilde{a}}_{j}=\langle (a_{j},b_{j},c_{j},d_{j}); w_{{\tilde{a}}_{j}},u_{{\tilde{a}}_{j}},y_{{\tilde{a}}_{j}}\rangle (j=1,2)$

• 1. The score function $E(\tilde{a_j})$ based on center of gravity is given as

  $$\begin{aligned} E(\tilde{a_j})= \left\{ \begin{array}{ll} \frac{1}{3}\left( a_j+b_j+c_j+d_j-\frac{d_j.c_j-b_j.a_j}{d_j+c_j-b_j-a_j}\right) \cdot \frac{(2+w_{{\tilde{a}}_{j}}-u_{{\tilde{a}}_{j}}-y_{{\tilde{a}}_{j}})}{3}\\ \hbox {otherwise} \\ a\cdot \frac{(2+w_{{\tilde{a}}_{j}}-u_{{\tilde{a}}_{j}}-y_{{\tilde{a}}_{j}})}{3}\qquad \hbox {if} \,\, a=a_j=b_j=c_j=d_j \end{array}\right. \end{aligned}$$

• 2. Accuracy function ${\bar{A}}(\tilde{a_j})$ based on center of gravity is given as

  $$\begin{aligned} {\bar{A}}(\tilde{a_j})= \left\{ \begin{array}{ll} \frac{1}{3}\left( a_j+b_j+c_j+d_j-\frac{d_j.c_j-b_j.a_j}{d_j+c_j-b_j-a_j}\right) \cdot {(w_{{\tilde{a}}_{j}}-y_{{\tilde{a}}_{j}})} \\ \hbox {otherwise }\\ a\cdot {(w_{{\tilde{a}}_{j}}-y_{{\tilde{a}}_{j}})} \quad \hbox {if} \,\, a=a_j=b_j=c_j=d_j \end{array}\right. \end{aligned}$$

• 3. Certainty function $C(\tilde{a_j})$ based on center of gravity is given as

  $$\begin{aligned} C(\tilde{a_j})= \left\{ \begin{array}{ll} \frac{1}{3}\left( a_j+b_j+c_j+d_j-\frac{d_j.c_j-b_j.a_j}{d_j+c_j-b_j-a_j}\right) \cdot w_{{\tilde{a}}_{j}} \\ \hbox {otherwise} \\ a\cdot w_{{\tilde{a}}_{j}} \quad \hbox {if} \,\, a=a_j=b_j=c_j=d_j \end{array}\right. \end{aligned}$$

Example 3.5

Let $\tilde{a_1}=\langle (0.5, 0.10, 0.15, 0.20);0.5,0.1,0.3\rangle$ and $\tilde{a_2}=\langle (0.0, 0.5, 0.10, 0.15);0.8,0.9,0.7\rangle$ be two SVTN-numbers.

Then, we can compare the SVTN-numbers $\tilde{a_1}$ and $\tilde{a_2}$ as Table 5.

Method 6 (Ye 2015) Let $\tilde{a_1}$ and $\tilde{a_2}$ be two SVTN-numbers and $\theta \in [0,1]$. For weighted values and ambiguities of the SVTN-numbers $\tilde{a_1}=\langle [a_{11},a_{21},a_{31},a_{41}],$$[b_{11},b_{21},b_{31},b_{41}],$$[c_{11},c_{21},c_{31},c_{41}] \rangle$ and $\tilde{a_2}=\langle [a_{12},a_{22},a_{32},a_{42}],$$[b_{12},b_{22},b_{32},b_{42}],$$[c_{12},c_{22},c_{32},c_{42}] \rangle$, the ranking order of $\tilde{a_1}$ and $\tilde{a_2}$ is defined as:

• 1. If $s(\tilde{a_1})>s(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{>}\,\tilde{a_2};$

• 2. If $s(\tilde{a_1})=s(\tilde{a_2})$, then

  • (a) If $h(\tilde{a_1})>h(\tilde{a_2})$, then $\tilde{a_1}$ is bigger than $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{>}\,\tilde{a_2}$,

  • (b) If $h(\tilde{a_1})=h(\tilde{a_2})$, then $\tilde{a_1}$ is equal to $\tilde{a_2}$, denoted by $\tilde{a_1}\,\tilde{=}\,\tilde{a_2};$

where for $\tilde{a_j}=\langle [a_{1j},a_{2j},a_{3j},a_{4j}],$$[b_{1j},b_{2j},b_{3j},b_{4j}],$$[c_{1j},c_{2j},c_{3j},c_{4j}] \rangle$$(j=1,2)$

• 1. The score degrees of $\tilde{a_j}$ is given by

  $$\begin{aligned} s(\tilde{a_j})= & {} \frac{1}{3} \left( 2+\frac{a_{1j}+a_{2j}+a_{3j}+a_{4j}}{4}\right. \\&\quad -\,\frac{b_{1j}+b_{2j}+b_{3j}+b_{4j}}{4}\\&\quad \left. - \frac{c_{1j}+c_{2j}+c_{3j}+c_{4j}}{4}\right) \end{aligned}$$

• 2. The accuracy degrees of $\tilde{a_j}$ is given by

  $$\begin{aligned} a(\tilde{a_j})= & {} \frac{a_{1j}+a_{2j}+a_{3j}+a_{4j}}{4}\\&- \frac{c_{1j}+c_{2j}+c_{3j}+c_{4j}}{4} \end{aligned}$$

  .

Example 3.6

Let $\tilde{a_1}=\langle (0.1, 0.2, 0.3, 0.4);1.0,0.0,0.0\rangle$ and $\tilde{a_2}=\langle (1, 2, 3, 4);1.0,0.0,0.0\rangle$ be two SVTN-numbers.

Then, we can compare the SVTN-numbers $\tilde{a_1}$ and $\tilde{a_2}$ as Table 6.

4 Centroid and distance-measure-based approach for ranking of SVTN-number

In this subsection, we derive the centroid point of the SVTN-number. The method of ranking SVTN-number with centroid index uses the geometric center of an SVTN-number which has $X({\tilde{a}})$ value on the horizontal axis and $Y({\tilde{a}})$ value on the vertical axis.

Consider an SVTN-number of the form ${\tilde{a}}=\langle (a_{},b_{},c_{},d_{});w_{{\tilde{a}}},u_{{\tilde{a}}}, y_{{\tilde{a}}}\rangle$ on R characterized by a truth-membership function $T_{{\tilde{a}}}:R\rightarrow [0,1]$, an indeterminacy-membership function $I_{{\tilde{a}}}:R\rightarrow [0,1]$ and a falsity-membership function $F_{{\tilde{a}}}:R\rightarrow [0,1]$ is defined as:

$$\begin{aligned} {{\tilde{a}}}(x) = \left\{ \begin{array}{ll} {\tilde{a}}^L(x) ,&{}\quad a_{}\le x<{b_{}} \\ \\ {\tilde{a}}^C(x),&{}\quad {b_{}}\le x \le {c_{}}\\ \\ {\tilde{a}}^L(x),&{}\quad {c_{}}< x \le d_{}\\ \\ {\tilde{a}}^B(x),&{}\quad \hbox {otherwise}. \end{array}\right. \end{aligned}$$

where

$$\begin{aligned} {\tilde{a}}^L(x)= & {} \left( T_{{\tilde{a}}}^{L}(x),I_{{\tilde{a}}}^{L}(x),F_{{\tilde{a}}}^{L}(x)\right) \\= & {} \left( \frac{(x-a_{})w_{{\tilde{a}}}}{b_{}-a_{}} ,\frac{b_{}-x + u_{{\tilde{a}}}(x-a_{})}{ b_{}-a_{}} ,\right. \\&\left. \frac{b_{}-x + y_{{\tilde{a}}}(x-a_{})}{b_{}-a_{}>}\right) ,\\ {\tilde{a}}^C(x)= & {} \left( T_{{\tilde{a}}}^{C}(x),I_{{\tilde{a}}} ^{C}(x),F_{{\tilde{a}}}^{C}(x)\right) \\= & {} (w_{{\tilde{a}}} , u_{{\tilde{a}}},y_{{\tilde{a}}}),\\ {\tilde{a}}^R(x)= & {} \left( T_{{\tilde{a}}}^{R}(x),I_{{\tilde{a}}} ^{R}(x),F_{{\tilde{a}}}^{R}(x)\right) \\= & {} \left( \frac{(d_{}-x)w_{{\tilde{a}}}}{d_{}-c_{}},\frac{x-c_{} +u_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}} ,\right. \\&\left. \frac{x-c_{}+y_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}} \right) \\ \hbox {and}\\ {\tilde{a}}^B(x)= & {} \left( T_{{\tilde{a}}}^{B}(x),I_{{\tilde{a}}} ^{B}(x),F_{{\tilde{a}}}^{B}(x)\right) = (0,1,1), \end{aligned}$$

Here, the functions

• 1. $T_{{\tilde{a}}}^{L}:[a,b]\subseteq R \rightarrow [0,w_{{\tilde{a}}}]$, $I_{{\tilde{a}}}^{L}:[a,b]\subseteq R \rightarrow [ u_{{\tilde{a}}},1]$ and $F_{{\tilde{a}}}^{L}:[a,b]\subseteq R \rightarrow [ y_{{\tilde{a}}},1]$ are surjective and injective in $[a,b]\subseteq R$.

• 2. $T_{{\tilde{a}}}^{R}:[c,d]\subseteq R \rightarrow [0,w_{{\tilde{a}}}]$, $I_{{\tilde{a}}}^{R}:[c,d]\subseteq R \rightarrow [ u_{{\tilde{a}}},1]$ and $F_{{\tilde{a}}}^{R}:[c,d]\subseteq R \rightarrow [ y_{{\tilde{a}}},1]$ are surjective and injective in $[c,d]\subseteq R$.

Therefore, the inverse functions exist which are also of the same nature as:

• 1. ${T^{-1}}_{{\tilde{a}}}^{L}:[0,w_{{\tilde{a}}}] \rightarrow [a,b]\subseteq R$, ${I^{-1}}_{{\tilde{a}}}^{L}:[u_{{\tilde{a}}},1] \rightarrow$$[a,b]\subseteq R$ and ${F^{-1}}_{{\tilde{a}}}^{L}:[y_{{\tilde{a}}},1] \rightarrow [a,b]\subseteq R$,

• 2. ${T^{-1}}_{{\tilde{a}}}^{R}:[0,w_{{\tilde{a}}}]\rightarrow [c,d]\subseteq R$, ${I^{-1}}_{{\tilde{a}}}^{R}:[u_{{\tilde{a}}},1] \rightarrow [c,d]\subseteq R$ and ${F^{-1}}_{{\tilde{a}}}^{R}:[ y_{{\tilde{a}}},1]\rightarrow [c,d]\subseteq R$,

Then, the inverse functions can be analytically expressed as follows:

• 1. ${T^{-1}}_{{\tilde{a}}}^{L}(y)=a+\frac{(b-a)}{w_{{\tilde{a}}}}y$ ($0\le y < w_{{\tilde{a}}}$), ${I^{-1}}_{{\tilde{a}}}^{L}(y)=\frac{(a-b)y+(b-au_{{\tilde{a}}})}{1-u_{{\tilde{a}}}}$$(u_{{\tilde{a}}} \le y <1)$ and ${F^{-1}}_{{\tilde{a}}}^{L}(y)=\frac{(a-b)y+(b-ay_{{\tilde{a}}})}{1-y_{{\tilde{a}}}}$$(y_{{\tilde{a}}} \le y <1)$ ,

• 2. ${T^{-1}}_{{\tilde{a}}}^{R}(y)=d-\frac{(d-c)}{w_{{\tilde{a}}}}y$, $(0\le y <w_{{\tilde{a}}} ),$${I^{-1}}_{{\tilde{a}}}^{R}(y)=\frac{(d-c)y+(c-du_{{\tilde{a}}})}{1-u_{{\tilde{a}}}}$$(u_{{\tilde{a}}} \le y <1)$ and ${F^{-1}}_{{\tilde{a}}}^{R}(y)=\frac{(d-c)y+(c-dy_{{\tilde{a}}})}{1-y_{{\tilde{a}}}}$$(y_{{\tilde{a}}} \le y <1)$ ,

Definition 4.1

Let ${\tilde{a}}=\langle (a,b,c,d);w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$ be an SVTN-number. Then, the centroid point of the SVTN-number ${\tilde{a}}$ denoted by $C({\tilde{a}})$ is defined as: $C({\tilde{a}})= ((X({\tilde{a}})_T,Y({\tilde{a}})_T),(X({\tilde{a}})_I, Y({\tilde{a}})_I),(X({\tilde{a}})_F,Y({\tilde{a}})_F))$,

where

• 1. $X({\tilde{a}})_T=\frac{\int _{a}^{b} \frac{(x-a_{})w_{{\tilde{a}}}}{b_{}-a_{}}x\mathrm{d}x +\int _{b}^{c} w_{{\tilde{a}}}x\mathrm{d}x+ \int _{c}^{d} \frac{(d_{}-x)w_{{\tilde{a}}}}{d_{}-c_{}}x\mathrm{d}x }{\int _{a}^{b} \frac{(x-a_{})w_{{\tilde{a}}}}{b_{}-a_{}}\mathrm{d}x +\int _{b}^{c} w_{{\tilde{a}}}\mathrm{d}x+ \int _{c}^{d} \frac{(d_{}-x)w_{{\tilde{a}}}}{d_{}-c_{}}\mathrm{d}x }$

• 2. $X({\tilde{a}})_I=\frac{\int _{a}^{b} \frac{b_{}-x + u_{{\tilde{a}}}(x-a_{})}{ b_{}-a_{}} x\mathrm{d}x +\int _{b}^{c} u_{{\tilde{a}}}x\mathrm{d}x+ \int _{c}^{d} \frac{x-c_{}+u_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}}x\mathrm{d}x }{\int _{a}^{b} \frac{b_{}-x + u_{{\tilde{a}}}(x-a_{})}{ b_{}-a_{}} \mathrm{d}x +\int _{b}^{c} u_{{\tilde{a}}}\mathrm{d}x+ \int _{c}^{d} \frac{x-c_{}+u_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}}\mathrm{d}x }$

• 3. $X({\tilde{a}})_F=\frac{\int _{a}^{b}\frac{b_{}-x + y_{{\tilde{a}}}(x-a_{})}{b_{}-a_{}}x\mathrm{d}x +\int _{b}^{c} y_{{\tilde{a}}}x\mathrm{d}x+ \int _{c}^{d} \frac{x-c_{}+y_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}} x\mathrm{d}x }{\int _{a}^{b}\frac{b_{}-x + y_{{\tilde{a}}}(x-a_{})}{b_{}-a_{}}\mathrm{d}x +\int _{b}^{c} y_{{\tilde{a}}}\mathrm{d}x+ \int _{c}^{d} \frac{x-c_{}+y_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}} \mathrm{d}x }$

• 4. $Y({\tilde{a}})_T=\frac{\int _{0}^{w_{{\tilde{a}}}}\left( a+\frac{(b-a)}{w_{{\tilde{a}}}}y\right) y\mathrm{d}y -\int _{0}^{w_{{\tilde{a}}}} \left( d-\frac{(d-c)}{w_{{\tilde{a}}}}y\right) y\mathrm{d}y }{\int _{0}^{w_{{\tilde{a}}}} \left( a+\frac{(b-a)}{w_{{\tilde{a}}}}y\right) \mathrm{d}y -\int _{0}^{w_{{\tilde{a}}}} \left( d-\frac{(d-c)}{w_{{\tilde{a}}}}y\right) \mathrm{d}y }$

• 5. $Y({\tilde{a}})_I=\frac{\int _{u_{{\tilde{a}}}}^{1} \left( \frac{(a-b)y+(b-au_{{\tilde{a}}})}{1-u_{{\tilde{a}}}}\right) y\mathrm{d}y -\int _{u_{{\tilde{a}}}}^{1} \left( \frac{(d-c)y+(c-du_{{\tilde{a}}})}{1-u_{{\tilde{a}}}}\right) y\mathrm{d}y }{\int _{u_{{\tilde{a}}}}^{1}\left( \frac{(a-b)y +(b-au_{{\tilde{a}}})}{1-u_{{\tilde{a}}}}\right) \mathrm{d}y -\int _{u_{{\tilde{a}}}}^{1} \left( \frac{(d-c)y+(c-du_{{\tilde{a}}})}{1-u_{{\tilde{a}}}}\right) \mathrm{d}y }$ and

• 6. $Y({\tilde{a}})_F=\frac{\int _{y_{{\tilde{a}}}}^{1} \left( \frac{(a-b)y+(b-ay_{{\tilde{a}}})}{1-y_{{\tilde{a}}}}\right) y\mathrm{d}y -\int _{y_{{\tilde{a}}}}^{1} \left( \frac{(d-c)y+(c-\mathrm{d}y_{{\tilde{a}}})}{1-y_{{\tilde{a}}}}\right) y\mathrm{d}y }{\int _{y_{{\tilde{a}}}}^{1}\left( \frac{(a-b)y+(b-ay_{{\tilde{a}}})}{1-y_{{\tilde{a}}}}\right) \mathrm{d}y -\int _{y_{{\tilde{a}}}}^{1} \left( \frac{(d-c)y+(c-\mathrm{d}y_{{\tilde{a}}})}{1-y_{{\tilde{a}}}}\right) \mathrm{d}y }$

Proposition 4.2

Let ${\tilde{a}}=\langle (a,b,c,d);w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$ be an SVTN-number and $C({\tilde{a}})$ be the centroid point of ${\tilde{a}}$. Then, $X({\tilde{a}})_T,Y({\tilde{a}})_T,(X({\tilde{a}})_I, Y({\tilde{a}})_I),(X({\tilde{a}})_F$ and $Y({\tilde{a}})_F$ are given as:

$$\begin{aligned}&1.\quad X({\tilde{a}})_T=\frac{w_{{\tilde{a}}}}{3} \frac{a^3d-ad^3+b^3c-a^3c-b^3c+a^3c}{a^2d-db^2+b^2c-a^2c-bc^2+ac^2-ad^2} \\&2.\quad X({\tilde{a}})_I =\frac{\frac{1}{ b_{}-a_{}} ({b^3} +{2u_{{\tilde{a}}}b^3-3u_{{\tilde{a}}}a b^2}-{3ba^2+2a^3} +{ u_{{\tilde{a}}}a^3}) +{3u_{{\tilde{a}}}c^2} -{3u_{{\tilde{a}}}b^2} + \frac{1}{d_{}-c_{}}({2d^3-3c d^2} + u_{{\tilde{a}}}d^3+{c^2} - {3u_{{\tilde{a}}} d c^2+2u_{{\tilde{a}}}c^3} ) }{\frac{1}{ b-a_{}}( {-3b^2} + {3u_{{\tilde{a}}}b^2-6u_{{\tilde{a}}}ab} )- {6ba+3a^2} +3 {u_{{\tilde{a}}}a^2} )) + {6u_{{\tilde{a}}}c}-{6u_{{\tilde{a}}}b}+ \frac{1}{d_{}-c_{}}( 3d^2-6cd+{3u_{{\tilde{a}}}d^2} + {3c^2-6u_{{\tilde{a}}}dc} + {3u_{{\tilde{a}}}c^2} ) }\\&3.\quad X({\tilde{a}})_F =\frac{\frac{1}{ b_{}-a_{}} ({b^3} +{2y_{{\tilde{a}}}b^3-3y_{{\tilde{a}}}a b^2}-{3ba^2+2a^3} +{ y_{{\tilde{a}}}a^3}) +{3y_{{\tilde{a}}}c^2} -{3y_{{\tilde{a}}}b^2} + \frac{1}{d_{}-c_{}}({2d^3-3c d^2} + y_{{\tilde{a}}}d^3+{c^2} - {3y_{{\tilde{a}}} d c^2+2y_{{\tilde{a}}}c^3} ) }{\frac{1}{ b-a_{}}( {-3b^2} + {3y_{{\tilde{a}}}b^2-6y_{{\tilde{a}}}ab} )- {6ba+3a^2} +3 {y_{{\tilde{a}}}a^2} )) + {6y_{{\tilde{a}}}c}-{6y_{{\tilde{a}}}b}+ \frac{1}{d_{}-c_{}}( 3d^2-6cd+{3y_{{\tilde{a}}}d^2} + {3c^2-6y_{{\tilde{a}}}dc} + {3y_{{\tilde{a}}}c^2} ) }\\&4.\quad Y({\tilde{a}})_T=w_{{\tilde{a}}}\frac{ a+{2b- d-2c} }{{3b+3a} - {3d-3c} } \\&5.\quad Y({\tilde{a}})_I=\frac{2a+b-3au_{{\tilde{a}}} - 2d-c+3du_{{\tilde{a}}}+2b{u_{{\tilde{a}}}}^3-3b{u_{{\tilde{a}}}}^2+ a{u_{{\tilde{a}}}}^3 -2c{u_{{\tilde{a}}}}^3+3c{u_{{\tilde{a}}}}^2-d{u_{{\tilde{a}}}}^3 }{3a+3b-6au_{{\tilde{a}}}- 3d-3c+6du_{{\tilde{a}}}+3b{u_{{\tilde{a}}}}^2-6b {u_{{\tilde{a}}}}+3a{u_{{\tilde{a}}}}^2 - 3d{u_{{\tilde{a}}}}^2-3c{u_{{\tilde{a}}}}^2+6c{u_{{\tilde{a}}}}} \\&6.\quad Y({\tilde{a}})_F= \frac{2a+b-3ay_{{\tilde{a}}} - 2d-c+3dy_{{\tilde{a}}}+2b{y_{{\tilde{a}}}}^3-3b{y_{{\tilde{a}}}}^2+ a{y_{{\tilde{a}}}}^3 -2c{y_{{\tilde{a}}}}^3+3c{y_{{\tilde{a}}}}^2-d{y_{{\tilde{a}}}}^3 }{3a+3b-6ay_{{\tilde{a}}}- 3d-3c+6dy_{{\tilde{a}}}+3b{y_{{\tilde{a}}}} ^2-6b{y_{{\tilde{a}}}}+3a{y_{{\tilde{a}}}}^2 - 3d{y_{{\tilde{a}}}}^2-3c{y_{{\tilde{a}}}} ^2+6c{y_{{\tilde{a}}}}} \end{aligned}$$

Proof

• 1. For ${\tilde{a}}=\langle (a,b,c,d);w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$, the $X({\tilde{a}})_T$ is computed as:

  $$\begin{aligned} X({\tilde{a}})_T= & {} \frac{\int _{a}^{b} \frac{(x-a_{})w_{{\tilde{a}}}}{b_{}-a_{}}x\mathrm{d}x +\int _{b}^{c} w_{{\tilde{a}}}x\mathrm{d}x+ \int _{c}^{d} \frac{(d_{}-x)w_{{\tilde{a}}}}{d_{}-c_{}}x\mathrm{d}x }{\int _{a}^{b} \frac{(x-a_{})w_{{\tilde{a}}}}{b_{}-a_{}}\mathrm{d}x +\int _{b}^{c} w_{{\tilde{a}}}\mathrm{d}x+ \int _{c}^{d} \frac{(d_{}-x)w_{{\tilde{a}}}}{d_{}-c_{}}\mathrm{d}x } \\\\= & {} \frac{ \frac{w_{{\tilde{a}}}}{b_{}-a_{}}\int _{a}^{b} (x^2-a_{}x)\mathrm{d}x +w_{{\tilde{a}}}\int _{b}^{c}x\mathrm{d}x+ \frac{w_{{\tilde{a}}}}{d_{}-c_{}}\int _{c}^{d} (d_{}x-x^2)\mathrm{d}x }{\frac{w_{{\tilde{a}}}}{b_{}-a_{}}\int _{a}^{b} (x-a_{})\mathrm{d}x +w_{{\tilde{a}}}\int _{b}^{c}\mathrm{d}x+ \frac{w_{{\tilde{a}}}}{d_{}-c_{}}\int _{c}^{d} (d_{}-x)\mathrm{d}x } \\= & {} \frac{ \frac{w_{{\tilde{a}}}}{b_{}-a_{}} \left( \frac{x^3}{3}-a_{}\frac{x^2}{2}\right) |_{a}^{b} +w_{{\tilde{a}}}\frac{x^2}{2}|_{b}^{c}+ \frac{w_{{\tilde{a}}}}{d_{}-c_{}} \left( d_{}\frac{x^2}{2}-\frac{x^3}{3}\right) |_{c}^{d} }{\frac{w_{{\tilde{a}}}}{b_{}-a_{}} \left( \frac{x^2}{2}-a_{}x\right) |_{a}^{b} +w_{{\tilde{a}}}x|_{b}^{c}+ \frac{w_{{\tilde{a}}}}{d_{}-c_{}}\left( d_{}x-\frac{x^2}{2}\right) |_{c}^{d} } \\= & {} \frac{ \frac{w_{{\tilde{a}}}}{b_{}-a_{}} \left[ \left( \frac{b^3}{3}-\frac{a_{}b^2}{2}\right) - \left( \frac{a^3}{3}-\frac{a^3}{2}\right) \right] +w_{{\tilde{a}}}\left[ \frac{c^2}{2}-\frac{b^2}{2}\right] + \frac{w_{{\tilde{a}}}}{d_{}-c_{}}\left[ \left( \frac{d^3}{2}-\frac{d^3}{3}\right) - \left( \frac{dc^2}{2}-\frac{c^3}{3}\right) \right] }{\frac{w_{{\tilde{a}}}}{b_{}-a_{}} \left[ \left( \frac{b^2}{2}-a_{}b\right) - \left( \frac{a^2}{2}-a^2\right) \right] +w_{{\tilde{a}}}[c-b]+ \frac{w_{{\tilde{a}}}}{d_{}-c_{}}\left[ \left( d^2-\frac{d^2}{2}\right) -\left( d_{}c-\frac{c^2}{2}\right) \right] } \\= & {} \frac{ \frac{w_{{\tilde{a}}}}{b_{}-a_{}} \left[ \frac{2b^3-3a_{}b^2+a^3}{6}\right] +w_{{\tilde{a}}}[\frac{c^2-b^2}{2}]+ \frac{w_{{\tilde{a}}}}{d_{}-c_{}}\left[ \left( \frac{d^3-3dc^2+2c^3}{6}\right) \right] }{\frac{w_{{\tilde{a}}}}{b_{}-a_{}} \left[ \frac{b^2-2 a_{}b+a^2}{2} \right] +w_{{\tilde{a}}}[c-b]+ \frac{w_{{\tilde{a}}}}{d_{}-c_{}}\left[ \frac{d^2-2d_{}c+c^2}{2}\right] }\\= & {} \frac{w_{{\tilde{a}}}}{3}\frac{a^3d-ad^3+b^3c-a^3c-b^3c+a^3c}{a^2d-db^2+b^2c-a^2c-bc^2+ac^2-ad^2} \end{aligned}$$

• 2. For ${\tilde{a}}=\langle (a,b,c,d);w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$, the $X({\tilde{a}})_I$ is computed as:

  $$\begin{aligned} X({\tilde{a}})_I= & {} \frac{\int _{a}^{b} \frac{b_{}-x + u_{{\tilde{a}}}(x-a_{})}{ b_{}-a_{}} x\mathrm{d}x +\int _{b}^{c} u_{{\tilde{a}}}x\mathrm{d}x+ \int _{c}^{d} \frac{x-c_{}+u_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}}x\mathrm{d}x }{\int _{a}^{b} \frac{b_{}-x + u_{{\tilde{a}}}(x-a_{})}{ b_{}-a_{}} \mathrm{d}x +\int _{b}^{c} u_{{\tilde{a}}}\mathrm{d}x+ \int _{c}^{d} \frac{x-c_{} +u_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}}\mathrm{d}x } \\= & {} \frac{\frac{1}{ b_{}-a_{}} \int _{a}^{b}( b_{}-x + u_{{\tilde{a}}}(x-a_{}))x\mathrm{d}x +u_{{\tilde{a}}}\int _{b}^{c}x\mathrm{d}x + \frac{1}{d_{}-c_{}}\int _{c}^{d}(x-c_{}+ u_{{\tilde{a}}}(d_{}-x))x\mathrm{d}x }{\frac{1}{ b_{}-a_{}}\int _{a}^{b}(b_{} -x + u_{{\tilde{a}}}(x-a_{})) \mathrm{d}x +u_{{\tilde{a}}}\int _{b}^{c}\mathrm{d}x+ \frac{1}{d_{}-c_{}}\int _{c}^{d}(x-c_{}+u_{{\tilde{a}}}(d_{}-x))\mathrm{d}x } \\= & {} \frac{\frac{1}{ b_{}-a_{}} \int _{a}^{b}( bx-x^2 + u_{{\tilde{a}}} x^2-u_{{\tilde{a}}}ax)\mathrm{d}x +u_{{\tilde{a}}}\int _{b}^{c}x\mathrm{d}x+ \frac{1}{d_{}-c_{}}\int _{c}^{d}(x^2-cx+u_{{\tilde{a}}} {d}x-u_{{\tilde{a}}}x^2)\mathrm{d}x }{\frac{1}{ b_{}-a_{}}\int _{a} ^{b}(b_{}-x + u_{{\tilde{a}}}x-u_{{\tilde{a}}}a_{})) \mathrm{d}x +u_{{\tilde{a}}}\int _{b}^{c}\mathrm{d}x+ \frac{1}{d_{}-c_{}} \int _{c}^{d}(x-c_{}+u_{{\tilde{a}}}d_{}-u_{{\tilde{a}}}x)\mathrm{d}x } \\= & {} \frac{\frac{1}{ b_{}-a_{}} \left( b\frac{x^2}{2} -\frac{x^3}{3} + u_{{\tilde{a}}}\frac{x^3}{3} -u_{{\tilde{a}}}a\frac{x^2}{2}\right) |_{a}^{b} +u_{{\tilde{a}}}\frac{x^2}{2}|_{b}^{c}+ \frac{1}{d_{}-c_{}}\left( \frac{x^3}{3}-c\frac{x^2}{2} +u_{{\tilde{a}}} d\frac{x^2}{2}-u_{{\tilde{a}}}\frac{x^3}{3}\right) |_{c}^{d} }{\frac{1}{ b-a_{}}\left( bx-\frac{x^2}{2} + u_{{\tilde{a}}}\frac{x^2}{2}-u_{{\tilde{a}}}ax)\right) |_{a}^{b} +u_{{\tilde{a}}}x|_{b}^{c}+ \frac{1}{d_{} -c_{}}(\frac{x^2}{2}-cx+u_{{\tilde{a}}}d_{}x-u_{{\tilde{a}}} \frac{x^2}{2})|_{c}^{d} } \\= & {} \frac{\frac{1}{ b_{}-a_{}} ({3bx^2-2x^3} +{2u_{{\tilde{a}}}x^3-3u_{{\tilde{a}}}a x^2}) |_{a}^{b} +{3u_{{\tilde{a}}}x^2}|_{b}^{c}+ \frac{1}{d_{}-c_{}}({2x^3-3c x^2} + {3u_{{\tilde{a}}} d x^2-2u_{{\tilde{a}}}x^3} )|_{c}^{d} }{\frac{1}{ b-a_{}}( {6bx-3x^2} + {3u_{{\tilde{a}}}x^2-6u_{{\tilde{a}}}ax} ))|_{a}^{b} + {6u_{{\tilde{a}}}x} |_{b}^{c}+ \frac{1}{d_{}-c_{}} ( {3x^2-6cx+6u_{{\tilde{a}}}d_{}x} - {3u_{{\tilde{a}}}x^2} )|_{c}^{d} } \\= & {} \frac{\frac{1}{ b_{}-a_{}} ({b^3} +{2u_{{\tilde{a}}}b^3-3u_{{\tilde{a}}}a b^2}-{3ba^2+2a^3} +{ u_{{\tilde{a}}}a^3}) +{3u_{{\tilde{a}}}c^2} -{3u_{{\tilde{a}}}b^2} + \frac{1}{d_{}-c_{}}({2d^3-3c d^2} + u_{{\tilde{a}}}d^3+{c^2} - {3u_{{\tilde{a}}} d c^2+2u_{{\tilde{a}}}c^3} )}{\frac{1}{ b-a_{}}( {-3b^2} + {3u_{{\tilde{a}}}b^2-6u_{{\tilde{a}}}ab} )- {6ba+3a^2} +3 {u_{{\tilde{a}}}a^2} )) + {6u_{{\tilde{a}}}c}-{6u_{{\tilde{a}}}b}+ \frac{1}{d_{}-c_{}}( 3d^2-6cd+{3u_{{\tilde{a}}}d^2} + {3c^2-6u_{{\tilde{a}}}dc} + {3u_{{\tilde{a}}}c^2} ) } \end{aligned}$$

• 3. For ${\tilde{a}}=\langle (a,b,c,d);w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$, the $X({\tilde{a}})_F$ is computed as:

  $$\begin{aligned} X({\tilde{a}})_F= & {} \frac{\int _{a}^{b} \frac{b_{}-x + y_{{\tilde{a}}}(x-a_{})}{ b_{}-a_{}} x\mathrm{d}x +\int _{b}^{c} y_{{\tilde{a}}}x\mathrm{d}x+ \int _{c}^{d} \frac{x-c_{}+y_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}}x\mathrm{d}x }{\int _{a}^{b} \frac{b_{}-x + y_{{\tilde{a}}}(x-a_{})}{ b_{}-a_{}} \mathrm{d}x +\int _{b}^{c} y_{{\tilde{a}}}\mathrm{d}x+ \int _{c}^{d} \frac{x-c_{}+y_{{\tilde{a}}}(d_{}-x)}{d_{}-c_{}}\mathrm{d}x } \\= & {} \frac{\frac{1}{ b_{}-a_{}} \int _{a}^{b}( b_{}-x + y_{{\tilde{a}}}(x-a_{}))x\mathrm{d}x +y_{{\tilde{a}}}\int _{b}^{c}x\mathrm{d}x+ \frac{1}{d_{}-c_{}}\int _{c}^{d}(x-c_{}+y_{{\tilde{a}}}(d_{}-x))x\mathrm{d}x }{\frac{1}{ b_{}-a_{}}\int _{a}^{b}(b_{}-x + y_{{\tilde{a}}}(x-a_{})) \mathrm{d}x +y_{{\tilde{a}}}\int _{b}^{c}\mathrm{d}x+ \frac{1}{d_{}-c_{}}\int _{c}^{d}(x-c_{}+y_{{\tilde{a}}}(d_{}-x))\mathrm{d}x }\\= & {} \frac{\frac{1}{ b_{}-a_{}} \int _{a}^{b}( bx-x^2 + y_{{\tilde{a}}}x^2-y_{{\tilde{a}}}ax)\mathrm{d}x +y_{{\tilde{a}}}\int _{b}^{c}x\mathrm{d}x+ \frac{1}{d_{}-c_{}}\int _{c}^{d}(x^2-cx+y_{{\tilde{a}}} \mathrm{d}x-y_{{\tilde{a}}}x^2)\mathrm{d}x }{\frac{1}{ b_{}-a_{}}\int _{a}^{b}(b_{}-x + y_{{\tilde{a}}}x-y_{{\tilde{a}}}a_{})) \mathrm{d}x +y_{{\tilde{a}}}\int _{b}^{c}\mathrm{d}x+ \frac{1}{d_{}-c_{}}\int _{c}^{d}(x-c_{}+y_{{\tilde{a}}}d_{} -y_{{\tilde{a}}}x)\mathrm{d}x} \\= & {} \frac{\frac{1}{ b_{}-a_{}} \left( b\frac{x^2}{2} -\frac{x^3}{3} + y_{{\tilde{a}}}\frac{x^3}{3} -y_{{\tilde{a}}}a\frac{x^2}{2}\right) |_{a}^{b} +y_{{\tilde{a}}}\frac{x^2}{2}|_{b}^{c}+ \frac{1}{d_{}-c_{}}\left( \frac{x^3}{3}-c\frac{x^2}{2} +y_{{\tilde{a}}} d\frac{x^2}{2}-y_{{\tilde{a}}} \frac{x^3}{3}\right) |_{c}^{d} }{\frac{1}{ b-a_{}} \left( bx-\frac{x^2}{2} + y_{{\tilde{a}}}\frac{x^2}{2} -y_{{\tilde{a}}}ax)\right) |_{a}^{b} +y_{{\tilde{a}}}x|_{b}^{c} + \frac{1}{d_{}-c_{}}\left( \frac{x^2}{2}-cx+y_{{\tilde{a}}}d_{} x-y_{{\tilde{a}}}\frac{x^2}{2}\right) |_{c}^{d} } \\= & {} \frac{\frac{1}{ b_{}-a_{}} ({3bx^2-2x^3} +{2y_{{\tilde{a}}}x^3-3y_{{\tilde{a}}}a x^2})|_{a}^{b} +{3y_{{\tilde{a}}}x^2}|_{b}^{c}+ \frac{1}{d_{}-c_{}} ({2x^3-3c x^2} + {3y_{{\tilde{a}}} d x^2 -2y_{{\tilde{a}}}x^3} )|_{c}^{d} }{\frac{1}{ b-a_{}}( {6bx-3x^2} + {3y_{{\tilde{a}}}x^2 -6y_{{\tilde{a}}}ax} ))|_{a}^{b} + {6y_{{\tilde{a}}}x} |_{b}^{c}+ \frac{1}{d_{}-c_{}}( {3x^2-6cx+6y_{{\tilde{a}}}d_{}x} - {3y_{{\tilde{a}}}x^2} )|_{c}^{d} } \\= & {} \frac{\frac{1}{ b_{}-a_{}} ({b^3} +{2y_{{\tilde{a}}}b^3-3y_{{\tilde{a}}}a b^2}-{3ba^2+2a^3} +{ y_{{\tilde{a}}}a^3}) +{3y_{{\tilde{a}}}c^2} -{3y_{{\tilde{a}}}b^2} + \frac{1}{d_{}-c_{}}({2d^3-3c d^2} + y_{{\tilde{a}}}d^3+{c^2} - {3y_{{\tilde{a}}} d c^2+2y_{{\tilde{a}}}c^3} ) }{\frac{1}{ b-a_{}}( {-3b^2} + {3y_{{\tilde{a}}}b^2-6y_ {{\tilde{a}}}ab} )- {6ba+3a^2} +3 {y_{{\tilde{a}}}a^2} )) + {6y_{{\tilde{a}}}c}-{6y_{{\tilde{a}}}b}+ \frac{1}{d_{}-c_{}} ( 3d^2-6cd+{3y_{{\tilde{a}}}d^2} + {3c^2-6y_{{\tilde{a}}}dc} + {3y_{{\tilde{a}}}c^2} ) } \end{aligned}$$

• 4. For ${\tilde{a}}=\langle (a,b,c,d);w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$, the $Y({\tilde{a}})_T$ is computed as:

  $$\begin{aligned} Y({\tilde{a}})_T= & {} \frac{\int _{0}^{w_{{\tilde{a}}}}\left( a+\frac{(b-a)}{w_{{\tilde{a}}}}y\right) y\mathrm{d}y -\int _{0}^{w_{{\tilde{a}}}} \left( d-\frac{(d-c)}{w_{{\tilde{a}}}}y\right) y\mathrm{d}y }{\int _{0} ^{w_{{\tilde{a}}}} \left( a+\frac{(b-a)}{w_{{\tilde{a}}}}y\right) \mathrm{d}y -\int _{0}^{w_{{\tilde{a}}}} \left( d-\frac{(d-c)}{w_{{\tilde{a}}}}y\right) \mathrm{d}y }\\= & {} \frac{\int _{0}^{w_{{\tilde{a}}}}( aw_{{\tilde{a}}}+{(b-a)}y)y\mathrm{d}y -\int _{0}^{w_{{\tilde{a}}}} ( dw_{{\tilde{a}}}- {(d-c)}y)y\mathrm{d}y }{\int _{0}^{w_{{\tilde{a}}}} (aw_{{\tilde{a}}}+{(b-a)}y)\mathrm{d}y -\int _{0}^{w_{{\tilde{a}}}} ( dw_{{\tilde{a}}}-{(d-c)}y)\mathrm{d}y }\\= & {} \frac{( 3ay^2w_{{\tilde{a}}}+{2by^3-2ay^3})|_{0}^{w_{{\tilde{a}}}}- ( 3\mathrm{d}y^2w_{{\tilde{a}}}- {2\mathrm{d}y^3+2cy^3})|_{0}^{w_{{\tilde{a}}}} }{ (6ayw_{{\tilde{a}}}+{3by^2-3ay^2})|_{0}^{w_{{\tilde{a}}}} - ( 6dyw_{{\tilde{a}}}-{3\mathrm{d}y^2+3cy^2})|_{0}^{w_{{\tilde{a}}}} }\\= & {} \frac{ aw^3_{{\tilde{a}}}+{2bw^3_{{\tilde{a}}}- dw^3_{{\tilde{a}}}-2cw^3_{{\tilde{a}}}} }{{3bw^2_{{\tilde{a}}}+3aw^2_{{\tilde{a}}}} - {3dw^2_{{\tilde{a}}}-3cw^2_{{\tilde{a}}}} }\\= & {} w_{{\tilde{a}}}\frac{ a+{2b- d-2c} }{{3b+3a} - {3d-3c} }\\ \end{aligned}$$

• 5. For ${\tilde{a}}=\langle (a,b,c,d);w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$, the $Y({\tilde{a}})_I$ is computed as:

  $$\begin{aligned} Y({\tilde{a}})_I= & {} \frac{\int _{u_{{\tilde{a}}}}^{1}\left( \frac{(a-b)y +(b-au_{{\tilde{a}}})}{1-u_{{\tilde{a}}}}\right) y\mathrm{d}y -\int _{u_{{\tilde{a}}}}^{1} \left( \frac{(d-c)y+(c-du_{{\tilde{a}}})}{1-u_{{\tilde{a}}}}\right) y\mathrm{d}y }{\int _{u_{{\tilde{a}}}}^{1}\left( \frac{(a-b)y +(b-au_{{\tilde{a}}})}{1-u_{{\tilde{a}}}}\right) \mathrm{d}y -\int _{u_{{\tilde{a}}}}^{1} \left( \frac{(d-c)y+(c-du_{{\tilde{a}}})}{1-u_{{\tilde{a}}}}\right) \mathrm{d}y }\\= & {} \frac{\int _{u_{{\tilde{a}}}}^{1}(ay^2-by^2+by-au_{{\tilde{a}}}y)\mathrm{d}y -\int _{u_{{\tilde{a}}}}^{1} ( \mathrm{d}y^2-cy^2+cy-dyu_{{\tilde{a}}})\mathrm{d}y }{\int _{u_{{\tilde{a}}}}^{1}(ay-by+b-au_{{\tilde{a}}})\mathrm{d}y -\int _{u_{{\tilde{a}}}}^{1} ( \mathrm{d}y-cy+c-du_{{\tilde{a}}})\mathrm{d}y }\\= & {} \frac{(2ay^3-2by^3+3by^2-3au_{{\tilde{a}}}y^2)|_{u_{{\tilde{a}}}}^{1} - ( 2\mathrm{d}y^3-2cy^3+3cy^2-3\mathrm{d}y^2u_{{\tilde{a}}})|_{u_{{\tilde{a}}}}^{1} }{(3ay^2-3by^2+6by-6ayu_{{\tilde{a}}})|_{u_{{\tilde{a}}}}^{1} - ( 3\mathrm{d}y^2-3cy^2+6cy-6dyu_{{\tilde{a}}})|_{u_{{\tilde{a}}}}^{1} }\\= & {} \frac{2a+b-3au_{{\tilde{a}}} - 2d-c+3du_{{\tilde{a}}}+2b{u_{{\tilde{a}}}}^3-3b{u_{{\tilde{a}}}}^2+ a{u_{{\tilde{a}}}}^3 -2c{u_{{\tilde{a}}}}^3+3c{u_{{\tilde{a}}}}^2-d{u_{{\tilde{a}}}}^3 }{3a+3b-6au_{{\tilde{a}}}- 3d-3c+6du_{{\tilde{a}}}+3b{u_{{\tilde{a}}}} ^2-6b{u_{{\tilde{a}}}}+3a{u_{{\tilde{a}}}}^2 - 3d{u_{{\tilde{a}}}}^2-3c{u_{{\tilde{a}}}} ^2+6c{u_{{\tilde{a}}}}} \end{aligned}$$

• 6. For ${\tilde{a}}=\langle (a,b,c,d);w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$, the $Y({\tilde{a}})_F$ is computed as:

  $$\begin{aligned} Y({\tilde{a}})_F= & {} \frac{\int _{y_{{\tilde{a}}}}^{1}\left( \frac{(a-b)y +(b-ay_{{\tilde{a}}})}{1-y_{{\tilde{a}}}}\right) y\mathrm{d}y -\int _{y_{{\tilde{a}}}}^{1} \left( \frac{(d-c)y+(c-\mathrm{d}y_{{\tilde{a}}})}{1-y_{{\tilde{a}}}}\right) y\mathrm{d}y }{\int _{y_{{\tilde{a}}}}^{1}\left( \frac{(a-b)y+(b-ay_ {{\tilde{a}}})}{1-y_{{\tilde{a}}}}\right) \mathrm{d}y -\int _{y_{{\tilde{a}}}}^{1} \left( \frac{(d-c)y+(c-\mathrm{d}y_{{\tilde{a}}})}{1-y_{{\tilde{a}}}}\right) \mathrm{d}y }\\= & {} \frac{\int _{y_{{\tilde{a}}}}^{1}(ay^2-by^2+by-ay_{{\tilde{a}}}y)\mathrm{d}y -\int _{y_{{\tilde{a}}}}^{1} ( \mathrm{d}y^2-cy^2+cy-dyy_{{\tilde{a}}})\mathrm{d}y }{\int _{y_{{\tilde{a}}}}^{1}(ay-by+b-ay_{{\tilde{a}}})\mathrm{d}y -\int _{y_{{\tilde{a}}}}^{1} ( \mathrm{d}y-cy+c-\mathrm{d}y_{{\tilde{a}}})\mathrm{d}y }\\= & {} \frac{(2ay^3-2by^3+3by^2-3ay_{{\tilde{a}}}y^2)|_{y_{{\tilde{a}}}}^{1} - ( 2dy^3-2cy^3+3cy^2-3\mathrm{d}y^2y_{{\tilde{a}}})|_{y_{{\tilde{a}}}}^{1} }{(3ay^2-3by^2+6by-6ayy_{{\tilde{a}}})|_{y_{{\tilde{a}}}}^{1} - ( 3dy^2-3cy^2+6cy-6dyy_{{\tilde{a}}})|_{y_{{\tilde{a}}}}^{1} }\\= & {} \frac{2a+b-3ay_{{\tilde{a}}} - 2d-c+3dy_{{\tilde{a}}}+2b{y_{{\tilde{a}}}}^3-3b{y_{{\tilde{a}}}}^2+ a{y_{{\tilde{a}}}}^3 -2c{y_{{\tilde{a}}}}^3+3c{y_{{\tilde{a}}}}^2-d{y_{{\tilde{a}}}}^3 }{3a+3b-6ay_{{\tilde{a}}}- 3d-3c+6dy_{{\tilde{a}}}+3b{y_{{\tilde{a}}}} ^2-6b{y_{{\tilde{a}}}}+3a{y_{{\tilde{a}}}}^2 - 3d{y_{{\tilde{a}}}} ^2-3c{y_{{\tilde{a}}}}^2+6c{y_{{\tilde{a}}}}} \end{aligned}$$

Example 4.3

Let ${\tilde{a}}=\langle (1, 3, 5, 7);0.5,0.7,0.3\rangle$ be SVTN-numbers and $C({\tilde{a}})$ be centroid point of ${\tilde{a}}$. Then, $X({\tilde{a}})_T,Y({\tilde{a}})_T,(X({\tilde{a}}) _I,Y({\tilde{a}})_I),(X({\tilde{a}})_F$ and $Y({\tilde{a}})_F$ are given as:

• 1. $X({\tilde{a}})_T=\frac{w_{{\tilde{a}}}}{3} \frac{a^3d-ad^3+b^3c-a^3c-b^3c+a^3c}{a^2d-db^2+b^2c-a^2c-bc^2+ac^2-ad^2}=0.0658$

• 2. $X({\tilde{a}})_I =\frac{\frac{1}{ b_{}-a_{}} ({b^3} +{2u_{{\tilde{a}}}b^3-3u_{{\tilde{a}}}a b^2}-{3ba^2+2a^3} +{ u_{{\tilde{a}}}a^3}) +{3u_{{\tilde{a}}}c^2} -{3u_{{\tilde{a}}}b^2} + \frac{1}{d_{}-c_{}}({2d^3-3c d^2} + u_{{\tilde{a}}}d^3+{c^2} - {3u_{{\tilde{a}}} d c^2+2u_{{\tilde{a}}}c^3} ) }{\frac{1}{ b-a_{}}( {-3b^2} + {3u_{{\tilde{a}}}b^2- 6u_{{\tilde{a}}}ab} )- {6ba+3a^2} +3 {u_{{\tilde{a}}}a^2} )) + {6u_{{\tilde{a}}}c}-{6u_{{\tilde{a}}}b}+ \frac{1}{d_{}-c_{}}( 3d^2-6cd+{3u_{{\tilde{a}}}d^2} + {3c^2-6u_{{\tilde{a}}}dc} + {3u_{{\tilde{a}}}c^2} ) }=9.3722$

• 3. $X({\tilde{a}})_F =\frac{\frac{1}{ b_{}-a_{}} ({b^3} +{2y_{{\tilde{a}}}b^3-3y_{{\tilde{a}}}a b^2}-{3ba^2+2a^3} +{ y_{{\tilde{a}}}a^3}) +{3y_{{\tilde{a}}}c^2} -{3y_{{\tilde{a}}}b^2} + \frac{1}{d_{}-c_{}}({2d^3-3c d^2} + y_{{\tilde{a}}}d^3+{c^2} - {3y_{{\tilde{a}}} d c^2+2y_{{\tilde{a}}}c^3} ) }{\frac{1}{ b-a_{}}( {-3b^2} + {3y_{{\tilde{a}}}b^2- 6y_{{\tilde{a}}}ab} )- {6ba+3a^2} +3 {y_{{\tilde{a}}}a^2} )) + {6y_{{\tilde{a}}}c}-{6y_{{\tilde{a}}}b}+ \frac{1}{d_{}-c_{}}( 3d^2-6cd+{3y_{{\tilde{a}}}d^2} + {3c^2-6y_{{\tilde{a}}}dc} + {3y_{{\tilde{a}}}c^2} ) }=-1.6705$

• 4. $Y({\tilde{a}})_T=w_{{\tilde{a}}}\frac{ a+{2b- d-2c} }{{3b+3a} - {3d-3c} }=0.2083$

• 5. $Y({\tilde{a}})_I=\frac{2a+b-3au_{{\tilde{a}}} - 2d-c+3du_{{\tilde{a}}}+2b{u_{{\tilde{a}}}}^3-3b{u_{{\tilde{a}}}}^2+ a{u_{{\tilde{a}}}}^3 -2c{u_{{\tilde{a}}}}^3+3c{u_{{\tilde{a}}}}^2-d{u_{{\tilde{a}}}}^3 }{3a+3b-6au_{{\tilde{a}}}- 3d-3c+6du_{{\tilde{a}}}+3b{u_{{\tilde{a}}}}^2 -6b{u_{{\tilde{a}}}}+3a{u_{{\tilde{a}}}}^2 - 3d{u_{{\tilde{a}}}}^2-3c{u_{{\tilde{a}}}}^2+6c{u_{{\tilde{a}}}}}=0.0355$

• 6. $Y({\tilde{a}})_F= \frac{2a+b-3ay_{{\tilde{a}}} - 2d-c+3dy_{{\tilde{a}}}+2b{y_{{\tilde{a}}}}^3-3b{y_{{\tilde{a}}}}^2+ a{y_{{\tilde{a}}}}^3 -2c{y_{{\tilde{a}}}}^3+3c{y_{{\tilde{a}}}}^2-d{y_{{\tilde{a}}}}^3 }{3a+3b-6ay_{{\tilde{a}}}- 3d-3c+6dy_{{\tilde{a}}}+3b{y_{{\tilde{a}}}}^2 -6b{y_{{\tilde{a}}}}+3a{y_{{\tilde{a}}}}^2 - 3d{y_{{\tilde{a}}}}^2-3c{y_{{\tilde{a}}}}^2+6c{y_{{\tilde{a}}}}}= 0.2149$

Remark 4.4

Let ${\tilde{a}}=\langle ({a}_{1},b_{1},c_{1},d_{1});w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$ and ${\tilde{b}}=\langle ({a}_{2},b_{2},c_{2},d_{2});w_{{\tilde{b}}},$$u_{{\tilde{b}}},y_{{\tilde{b}}}\rangle$ be two SVTN-numbers. Then, we have $C({\tilde{a}}+{\tilde{b}})\ne C({\tilde{a}})+C({\tilde{b}})$ and $C({\tilde{a}}.{\tilde{b}})\ne C({\tilde{a}}).C({\tilde{b}})$.

Example 4.5

Let ${\tilde{a}}=\langle (4, 5, 6 ,8); 0.5 ,0.2, 0.9\rangle$ and ${\tilde{b}}=\langle (2, 5, 6, 10); 1, 0.2, 0.1\rangle$ be two SVTN-numbers. Then, we have ${\tilde{a}}+{\tilde{b}}=\langle (6, 10, 12 ,18); 1 ,0.2, 0.1\rangle$.

Now, we compute the $C({\tilde{a}})$, $C({\tilde{b}})$, $C({\tilde{a}}+{\tilde{b}})$, $C({\tilde{a}})+C({\tilde{b}})$ and $C({\tilde{a}}.{\tilde{b}})$ and $C({\tilde{a}}).C({\tilde{b}})$ as:

$$\begin{aligned} C({\tilde{a}})= & {} ((0.9142 , 0.2000) ,(0.5478, 0.1673 ) ,(0.1152 ,0.0018 ) )\\ C({\tilde{b}})= & {} ((1.7679,0.3703 ) ,(-3.7033, 0.2604) ,(-2.7262, 0.3308 ) )\\ C({\tilde{a}}+{\tilde{b}})= & {} ((3.8918, 0.3809) ,(-0.2652, )0.2181 ,(0.1658 , 0.2850 ))\\ C({\tilde{a}})+C({\tilde{b}})= & {} ((2.6822,0.5703) ,(-3.1555,0.4278),( -2.6109,0.3326))\\ C({\tilde{a}}\cdot {\tilde{b}})= & {} ((15.3572,0.3775),(788.3783,0.3467 ),(-682.4267, 0.4162)) \hbox {and}\\ C({\tilde{a}})\cdot C({\tilde{b}})= & {} ((1.6164,0.0740),(-2.0288,0.0436),(-0.3143, 0.0005)) \hbox {as}; \end{aligned}$$

Table 7 Ranking of ${\tilde{a}}$ and ${\tilde{b}}$ based on centroid point

Clearly, $C({\tilde{a}}+{\tilde{b}})\ne C({\tilde{a}})+C({\tilde{b}})$ and $C({\tilde{a}}\cdot {\tilde{b}})\ne C({\tilde{a}})\cdot C({\tilde{b}})$.

Definition 4.6

Let ${\tilde{a}}=\langle ({a}_{ },b_{ },c_{ },d_{ });w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$ be an SVTN-number and $((X({\tilde{a}})_T,Y({\tilde{a}})_T),(X({\tilde{a}}) _I,Y({\tilde{a}})_I),(X({\tilde{a}})_F,Y({\tilde{a}})_F))$ be a centroid point of the SVTN-number ${\tilde{a}}$. Then,

• 1. Hamming ranking value of ${\tilde{a}}$ denoted $H({\tilde{a}})$ is defined by

  $$\begin{aligned} H({\tilde{a}})= |X({\tilde{a}})_T-Y({\tilde{a}})_T|+ |X({\tilde{a}})_I-Y({\tilde{a}})_I|+|X({\tilde{a}})_F,Y({\tilde{a}})_F| \end{aligned}$$

• 2. Euclidean ranking value of ${\tilde{a}}$ denoted $E({\tilde{a}})$ is defined by

  $$\begin{aligned} E({\tilde{a}})= \sqrt{|X({\tilde{a}})_T-Y({\tilde{a}})_T|^2+ |X({\tilde{a}})_I-Y({\tilde{a}})_I|^2+|X({\tilde{a}})_F,Y({\tilde{a}})_F|^2} \end{aligned}$$

Definition 4.7

Let ${\tilde{a}}=\langle ({a}_{1},b_{1},c_{1},d_{1});w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$ and ${\tilde{b}}=\langle ({a}_{2},b_{2},c_{2},d_{2});w_{{\tilde{b}}},$$u_{{\tilde{b}}},y_{{\tilde{b}}}\rangle$ be two SVTN-numbers and $K\in \{{H},{E}\}$. Then, an approach of ranking ${\tilde{a}}$ and ${\tilde{b}}$ is given as:

• 1. $K({\tilde{a}})>K({\tilde{b}})$ if and only if ${\tilde{a}}>{\tilde{b}}$

• 2. $K({\tilde{a}})<K({\tilde{b}})$ if and only if ${\tilde{a}}<{\tilde{b}}$

• 3. $K({\tilde{a}})=K({\tilde{b}})$ if and only if ${\tilde{a}}={\tilde{b}}$

Example 4.8

Let ${\tilde{a}}=\langle (1, 3, 5, 7);0.5,0.7,0.3\rangle$ and ${\tilde{b}}=\langle (2,3,4,5);0.3,0.7,0.6\rangle$ be two SVTN-numbers and $C({\tilde{a}})$ and $C({\tilde{b}})$ be centroid point of ${\tilde{a}}$ and ${\tilde{b}}$, respectively. Then, $C({\tilde{a}})$ and $C({\tilde{b}})$ are computed as:

$C({\tilde{a}})= ((0.0658,0.2083),(9.3722,0.0355),(-1.6705,0.2149))$

and

$C({\tilde{b}})= ((0.3559,0.125),(0.3939,0.0216),(0.4852,0.0402))$.

Therefore, we can compare as Table 7.

Remark 4.9

Let ${\tilde{a}}=\langle ({a}_{1},b_{1},c_{1},d_{1});w_{{\tilde{a}}},$$u_{{\tilde{a}}},y_{{\tilde{a}}}\rangle$ and ${\tilde{b}}=\langle ({a}_{2},b_{2},c_{2},d_{2});w_{{\tilde{b}}},$$u_{{\tilde{b}}},y_{{\tilde{b}}}\rangle$ be two SVTN-numbers. Then, we have

• 1. $H({\tilde{a}}+{\tilde{b}})\ne H({\tilde{a}})+H({\tilde{b}})$

• 2. $E({\tilde{a}}+{\tilde{b}})\ne E({\tilde{a}})+E({\tilde{b}})$

• 3. $H({\tilde{a}}\cdot {\tilde{b}})\ne H({\tilde{a}})\cdot H({\tilde{b}})$

• 4. $E({\tilde{a}}\cdot {\tilde{b}})\ne E({\tilde{a}})\cdot E({\tilde{b}})$

Example 4.10

Let we consider Example 4.5. Then, we compute the $H({\tilde{a}})$, $H({\tilde{b}})$, $H({\tilde{a}}+{\tilde{b}})$, $H({\tilde{a}}+{\tilde{b}})$, $H({\tilde{a}}\cdot {\tilde{b}})$ and $H({\tilde{a}})\cdot H({\tilde{b}})$ as: $H({\tilde{a}})= 1.2082$, $H({\tilde{b}})= 8.4185$, $H({\tilde{a}}+{\tilde{b}})= 4.1135$, $H({\tilde{a}})+H({\tilde{b}})=9.6267$, $H({\tilde{a}}\cdot {\tilde{b}})=1485.8542$ and $H({\tilde{a}})\cdot H({\tilde{b}})=10.1714$

Clearly, $H({\tilde{a}}+{\tilde{b}})\ne H({\tilde{a}})+H({\tilde{b}})$ and $H({\tilde{a}}\cdot {\tilde{b}})\ne H({\tilde{a}})\cdot H({\tilde{b}})$.

Similarly, we compute the $E({\tilde{a}})$, $E({\tilde{b}})$, $E({\tilde{a}}+{\tilde{b}})$, $E({\tilde{a}}+{\tilde{b}})$, $E({\tilde{a}}\cdot {\tilde{b}})$ and $E({\tilde{a}})\cdot E({\tilde{b}})=$ as:

$E({\tilde{a}})= 0.8172$, $E({\tilde{b}})= 5.1972$, $E({\tilde{a}}+{\tilde{b}})= 3.5460$, $E({\tilde{a}})+E({\tilde{b}})=6.0144$, $E({\tilde{a}}\cdot {\tilde{b}})=1042.8291$ and $E({\tilde{a}})\cdot E({\tilde{b}})=4.2472$

Clearly, $E({\tilde{a}}+{\tilde{b}})\ne E({\tilde{a}})+E({\tilde{b}})$ and $E({\tilde{a}}\cdot {\tilde{b}})\ne E({\tilde{a}})\cdot E({\tilde{b}})$

5 Multi-attribute decision-making problem based on centroid points of SVNT-numbers

5.1 Problem description

Regarding a multi-attribute decision-making problem, assume $X=(A_{1},A_{2},\ldots ,A_{m})$ be a set of alternatives, $U=(u_{1},u_{2},\ldots , u_{n})$ is the attribute set, and ${\tilde{a}}_{ij}=\langle (a_{ij},b_{ij},c_{ij},d_{ij}); T^{}_{{\tilde{a}}_{ij}},I^{}_{{\tilde{a}}_{j}}, F^{}_{{\tilde{a}}_{ij}}\rangle$ is the decision-making matrix based on Table 8, where ${\tilde{a}}_{ij}$ represents the SVTN-number attribute value of the jth attribute $u_{j}$ of the ith alternative $A_{i}$ ,$1\le i\le m, 1\le j\le n$. Moreover, the attribute weight is given in the form of an SVTN-number, that is, the attribute weight vector ${\tilde{W}}=({\tilde{w}}_1,{\tilde{w}}_2,\ldots {\tilde{w}}_n)$, where ${\tilde{w}}_j=\langle (a_{j},b_{j},c_{j},d_{j});T^{} _{{\tilde{w}}_{j}},I^{}_{{\tilde{w}}_{j}},F^{}_{{\tilde{w}}_{j}}\rangle$ represents the weight of the jth attribute $u_{j}$.

Table 8 SVNT-numbers for linguistic terms of the decision matrix

Table 9 SVNT-numbers for linguistic terms of attribute weights

5.2 Decision-making steps

• Step 1 Give the linguistic decision-making matrix $[{\tilde{a}}_{ij}]_{m \times n}$ according to the attribute value of the jth attribute $u_{j}$ of the ith alternative $A_{i}$ ,$1\le i\le m, 1\le j\le n$ .

• Step 2 Convert the linguistic decision matrix $[{\tilde{a}}_{ij}]_{m \times n}$ into the SVTN-number decision matrix.

• Step 3 Give the linguistic weights of the attributes.

• Step 4 Find the weights of the attributes.

  • (a) Convert the linguistic weights of the attributes. The weight information of the attribute weight is given in the form of an SVTN-number, that is, the attribute weight vector is ${\tilde{W}}=({\tilde{w}}_1,{\tilde{w}}_2,\ldots {\tilde{w}}_n)$, where ${\tilde{w}}_j=\langle (a_{j},b_{j},c_{j},d_{j});T^{}_{{\tilde{w}}_{j}},I^{}_{{\tilde{w}}_{j}},F^{}_{{\tilde{w}}_{j}}\rangle$.

  • (b) Obtain the Hamming ranking value $H({\tilde{w}}_j)$ based on $C({\tilde{w}}_j)$, using Definition 4.1, of the SVTN-number ${\tilde{w}}_j$ (or Euclidean ranking value $E({\tilde{w}}_j)$),

  • (c) The normalized weight $w_j$ of the jth attribute $u_{j}$ can be computed as:

    $$\begin{aligned} w_j=\frac{H({\tilde{w}}_j)}{\sum _{j=1}^{n}H({\tilde{w}}_j)}\,\,\left( \hbox {or} \,\,w_j=\frac{E({\tilde{w}}_j)}{\sum _{j=1}^{n}E({\tilde{w}}_j)}\right) \end{aligned}$$

    for $1\le j\le n$.

• Step 5 Find the expected value $\hbox {EV}(A_{i})= G_{ao}(a_{i1},a_{i2},\ldots ,a_{in})$ (for $1\le i\le m)$ of ith alternative $A_{i}$, according to

  $$\begin{aligned} \begin{aligned} \hbox {EV}(A_{i})&= \bigg \langle \bigg (\sum _{j=1}^{n} a_{ij}*{w_{j}},\sum _{j=1}^{n} b_{ij}*{w_{j}}, \sum _{j=1}^{n} c_{ij}*{w_{j}},\\&\quad \sum _{j=1}^{n} d_{ij}*{w_{j}}\bigg );\bigwedge _{j=1}^{n}w_{{\tilde{a}}_j},\bigvee _{j=1}^{n}u_{{\tilde{a}}_j},\bigvee _{j=1}^{n}y_{{\tilde{a}}_j}\bigg \rangle \end{aligned} \end{aligned}$$

• Step 6 Compute the Hamming ranking value $H(\hbox {EV}(A_{i}))$ based on $C(\hbox {EV}(A_{i}))$ of the SVTN-number $\hbox {EV}(A_{i})$ for $1\le i\le m$; (or Euclidean ranking value $E(\hbox {EV}(A_{i})$),

• Step 7 Rank all alternatives $A_i$ for all i, $1\le i\le m$ by using the $H(\hbox {EV}(A_{i}))$ (or $E(\hbox {EV}(A_{i})$) and determine the best alternative.

Now, we give an example from Chan and Kumar (2007), Wang et al. (2012) and Zhang (2018) to illustrate the proposed methods.

5.2.1 Application

Suppose that the problem discussed here is concerned with a manufacturing company, search the best global suppliers for one of its most critical parts used in assembling process (adapted from Chan and Kumar 2007, Wang et al. 2012, Zhang 2018). There are potential global suppliers to be evaluated with four attributes: (1)$u_1:$ quality of the product, (2) $u_2:$ risk factor, (3) $u_3$: service performance of supplier and (4) $u_4:$ suppliers profile. An expert uses the linguistic terms shown in Table 8 to represent the characteristics of the potential global suppliers $A_i$ (i = 1,2,3,4,5) with respect to different attributes $u_i$ (i = 1,2,3,4). Also, same expert uses the linguistic terms shown in Table 9 to represent attribute weights.

Table 10 The linguistic decision matrix provided by expert

Table 11 The decision matrix provided by expert

Table 12 The linguistic weights of the attributes evaluated by the expert

Table 13 The weights of the attributes

• Step 1 We give the linguistic decision-making matrix $[{\tilde{a}}_{ij}]_{5 \times 4}$ in Table 10.

• Step 2 We converted the linguistic decision matrix $[{\tilde{a}}_{ij}]_{5 \times 4}$ into the SVTN-number decision matrix in Table 11.

• Step 3 We give the linguistic weights of the attributes in Table 12.

• Step 4 We found the weights of the attributes as:

  • (a) We converted the linguistic weights of the attributes in Table 13.

  • (b) We obtained the Hamming ranking value $H({\tilde{w}}_j)$ (for $1\le j\le 4$) of the SVTN-number ${\tilde{w}}_j$ based on $C({\tilde{w}}_j)$, using Definition 4.1 as:

    $$\begin{aligned} H({\tilde{w}}_1)=0.6440\\ H({\tilde{w}}_2)=0.3262\\ H({\tilde{w}}_3)=1.5386\\ H({\tilde{w}}_4)=0.8592 \end{aligned}$$

  • (c) The normalized weight $w_j$ of the jth attribute $u_{j}$ based on centroid point $C({\tilde{w}}_j)$ of the SVTN-number ${\tilde{w}}_j$ using Definition 4.1 can be computed as:

    $$\begin{aligned} W=(w_1,w_2,w_3,w_4)=(0.1912, 0.0968,0.4568,0.2551) \end{aligned}$$

• Step 5 We calculated the expected value $\hbox {EV}(A_{i})= G_{ao}(a_{i1},a_{i2},\ldots ,a_{in})$ (for $1\le i\le 5)$ of ith alternative $A_{i}$ as:

  $$\begin{aligned} \hbox {EV}(A_{1})= & {} \big \langle (1.6718, 11.2564, 19.2504, 40.4746); 0.4, 0.7, 1 \big \rangle \\ \hbox {EV}(A_{2})= & {} \big \langle (3.9465, 5.9136, 6.9136 ,8.7758);0.5,0.2,0.1 \big \rangle \\ \hbox {EV}(A_{3})= & {} \big \langle (4.9625, 13.7533, 19.7089, 41.8866); 0.5, 1, 0.9\big \rangle \\ \hbox {EV}(A_{4})= & {} \big \langle (5.6949, 14.5496, 20.1177, 40.7607); 0.6, 0.7 ,1 \big \rangle \\ \hbox {EV}(A_{5})= & {} \big \langle (3.6755, 6.6822, 10.4781 ,17.9275);0.5 ,1, 0.3 \big \rangle \end{aligned}$$

• Step 6 We computed the Hamming ranking value $H(\hbox {EV}(A_{i}))$ based on $C(\hbox {EV}(A_{i}))$ of the SVTN-number $\hbox {EV}(A_{i})$ for $1\le i\le 5$ as:

  $$\begin{aligned} H(\hbox {EV}(A_{1})= & {} 64.9339\\ H(\hbox {EV}(A_{2})= & {} 2.4027\\ H(\hbox {EV}(A_{3})= & {} 109.5658\\ H(\hbox {EV}(A_{4})= & {} 183.4800\\ H(\hbox {EV}(A_{5})= & {} 208.0349 \end{aligned}$$

• Step 7 We ranked all alternatives $A_i$ for all i, $1\le i\le 5$ by using the $H(\hbox {EV}(A_{i}))$ as:

  $$\begin{aligned} A_{5}> A_{4}> A_{3}> A_{1} > A_{2} \end{aligned}$$

  Similarly, we use the Euclidean ranking value $E(\hbox {EV}(A_{i})$, then we have

  $$\begin{aligned} A_{5}> A_{4}> A_{3}> A_{1} > A_{2} \end{aligned}$$

  The best alternative is $A_{5}$.

Table 14 Comparative examples of the proposed ranking process with the existing ranking process

Table 15 Comparative Examples of the proposed ranking process with the existing ranking process

5.3 Significance of the proposed method

Here, we are using for, an SVTN-number ${\tilde{a}}=\langle (a_{},b_{},c_{},d_{});w_{{\tilde{a}}},u_{{\tilde{a}}}, y_{{\tilde{a}}}\rangle$, a truth-membership function $T_{{\tilde{a}}}:R\rightarrow [0,1]$, an indeterminacy-membership function $I_{{\tilde{a}}}:R\rightarrow [0,1]$ and a falsity-membership function $F_{{\tilde{a}}}:R\rightarrow [0,1]$ to obtain a single score. If we obtain a truth-membership score, an indeterminacy-membership score and falsity-membership for an SVTN-number ranking of an SVTN-number fail if truth-membership score of $\tilde{a_1}$, truth-membership score of $\tilde{a_2}$, indeterminacy-membership score of $\tilde{a_1}$, indeterminacy-membership score of $\tilde{a_2}$ and falsity-membership score of $\tilde{a_1}$, falsity-membership score of $\tilde{a_2}$ where $\tilde{a_1}$ and $\tilde{a_2}$ are two SVTN-numbers. Getting a single score involving truth-membership, an indeterminacy-membership and falsity-membership degree overcome that situation. This new scoring method has wide applications in various fields such as: economics, decision making, game theory,...for extending equilibrium in the fuzzy sense to single-valued neutrosophic sense, these notions can be used.

5.4 Comparative analysis of the proposed method with the existing methods

In application Sect. 5.2.1, we compare the proposed method with other existing methods in the literature and comparison results are presented in Table 14.

As can be seen from Table 14, the ranking of the alternatives found by different methods is often similar. Therefore, the results are verified through comparative analysis.

The comparative results are given for the justification of the proposed method with the existing ranking in Table 15. The advantages of the proposed ranking method are:

• 1. The proposed ranking method can be applied for SVTN-numbers, which reflects the uncertainty suitably.

• 2. The proposed ranking technique based on centroid point, hamming ranking value and Euclidean ranking value is a flexible method when compared with other existing methods.

6 Conclusion

In this paper, we gave all the existing ranking methods of SVTN-numbers. Then, we firstly defined the concept of centroid point and examined several useful properties of the concept. Secondly, we developed hamming ranking value and Euclidean ranking value of SVTN-numbers to compare the SVTN-numbers. Thirdly, we proposed a novel defuzzification method to MADM with linguistic information and gave a real example deal with manufacturing company to illustrate the feasibility and effectiveness of the developed approach. Finally, we presented some examples to compare the proposed method with the existing ranking results and the results verified through comparative analysis. With the proposed method, many MADM and optimization problems of uncertain nature can be solved. In spite of existing ranking methods, no one can rank SVTN-numbers with human intuition consistently in all cases. Therefore, we pointed out the shortcoming of some recent ranking methods and introduced a novel defuzzification method for ranking SVTN-numbers. The developed hamming ranking value and Euclidean ranking value are simple and have consistent expression on the horizontal axis and vertical axis. In future, we will apply the proposed ranking technique to linear programming, game theory, medical diagnosis, and so on.