1 Introduction
Nowadays, as the society continues developing, practical problems and actual scenarios of both human nature and real-world situations, as well as uncertainty, vagueness, inconsistency and imprecision, seem to be prevalent. Therefore, fuzzy set theory (Zadeh, 1965) has been utilized in various fields with imprecise information. Although the fuzzy set theory is a useful tool for modelling problems, including uncertainty information, it can be too difficult in some cases. To avoid this difficulty, recently, intuitionistic fuzzy sets (Atanassov, 1986), type-2 fuzzy sets (Mizumoto and Tanaka, 1976), type-n fuzzy sets (Rickard et al., 2008) and extensions of fuzzy sets, which express information in different ways, have been defined and researched widely. Also, as a generalization of fuzzy sets, Torra and Narukawa (2009) and Torra (2010) introduced the concept of hesitant fuzzy sets which allows the membership of an element of a set to be represented by several possible values. The relationships among hesitant fuzzy sets were also discussed. Many studies on hesitant fuzzy sets have been conducted: Xu and Xia (2011a) and Li et al. (2015) developed some distance measures for hesitant fuzzy sets under similarity measures, Xu and Xia (2011b) introduced some distance and correlation measures on hesitant fuzzy sets, including desired properties in detail, Wei (2012) developed a few prioritized aggregation operators for hesitant fuzzy sets, and then applied decision making problems in which the attributes are in a different priority level and so on. Since these studies still cannot provide all original data information for the decision making problems, Deli and Karaaslan (2021) introduced the generalized trapezoidal hesitant fuzzy numbers on R. As a consequence, a large number of studies have been conducted by the following authors: Ali et al. (2023), Atanassov (2000), Yager (2008), Xia and Xu (2011), Deli (2021, 2020), Anusha et al. (2023), Liao et al. (2014), Wei (2012).
As the classical sets, fuzzy sets and the generalization of the collected information for the values of the alternatives based on criteria of aggregation operators are useful to convert the whole data into a single value. Therefore, the aggregation operators have great importance and significance in solving MADM problems in the whole set theory. For example, Wan (2013) developed a new decision method based on power average operators of fuzzy numbers. Aydemir and Yilmaz Gündüz (2020) defined some operational laws of q-rung orthopair fuzzy sets and then proposed Dombi prioritized weighted aggregation operators. Zhao and Wei (2013) proposed the intuitionistic fuzzy Einstein hybrid averaging operator and intuitionistic fuzzy Einstein hybrid geometric operator. Verma and Sharma (2014) introduced the trapezoid fuzzy linguistic prioritized weighted average operators and developed an approach to multiple attribute group decision-making trapezoid fuzzy linguistic information. Liu et al. (2016) developed the intuitionistic trapezoidal fuzzy prioritized ordered weighted aggregation operator, then proposed the prioritized multi-criteria decision-making problems under intuitionistic trapezoidal fuzzy information. Jiang (2018) developed some models for interval intuitionistic trapezoidal fuzzy multiple attribute decision making problems in which the attributes are in different priority level and with some prioritized aggregation operators. Fahmi et al. (2019, 2021) defined some new operation laws for trapezoidal cubic hesitant fuzzy numbers and then developed some new aggregation operators. Liang et al. (2017) proposed some new prioritized aggregation operator, and then some desired properties of the new aggregation operators were studied. Liu et al. (2017) proposed two prioritized aggregation operators for hesitant intuitionistic fuzzy linguistic sets, and then, based on these aggregation operators, an approach for multi-attribute decision-making was developed under the hesitant intuitionistic fuzzy linguistic sets. Verma (2017) combined the idea of generalized mean and prioritized weighted average operators. Recently, some authors introduced several prioritized aggregation operators within different mathematical structures (Akram et al., 2020; Garg and Rani, 2023; Jana et al., 2020; Kumar and Chen, 2022; Liu and Gao, 2020; Wang et al., 2022).
1.1 Novelty
Some methods have been using generalized trapezoidal hesitant fuzzy (GTHF) numbers such as proposed by Deli and Karaaslan (2021). However, each of these methods can work well in a specific situation when the attributes have the same priority, but can also generate undesirable decision-making results when the attributes have the same priority. Therefore, inspired by the ideal of prioritized aggregation operators (Yager, 2008), we developed an approach to solve a multi-attribute decision-making (MADM) problems with GTHF-numbers in which the attributes are in a different priority level.
1.2 Motivation and Contribution
The motivation and contributions of the study are as follows:
-
1. Prioritized operators of GTHF-numbers are introduced to propose a new method for solving MADM problems with GTHF-numbers in which the attributes are in a different priority level.
-
2. The main aim is to develop GTHF-numbers-prioritized weighted average operator and GTHF-prioritized weighted geometric operator for MADM problems with GTHF-numbers.
-
3. A method is constructed with an algorithm for MADM problems with GTHF-numbers.
-
4. An example for application is presented to demonstrate the effectiveness and advantage of the proposed method.
1.3 Paper
Structure
The remainder of the paper is organized as follows:
-
• ↓ In Section 2, we give a brief introduction on some basic definitions and propositions.
-
• ↓ In Section 3, we introduce some GTHF-numbers prioritized operators and discuss their desirable properties.
-
• ↓ In Section 4, we develop an MADM method and then initiate an example in which the attributes are in a different priority level.
-
• ↓ In Section 5, we give a comparison with some existing methods.
-
• ↪ In Section 6, we propose a conclusion.
2 Preliminaries
In the following, we briefly describe some basic concepts and basic operational laws related to hesitant fuzzy sets and generalized hesitant trapezoidal fuzzy numbers.
Definition 1 (Zadeh, 1965).
Definition 2 (Wang, 2009).
Let ${\eta _{\tilde{a}}}\in [0,1]$ and $a,b,c,d\in R$ such that $a\leqslant b\leqslant c\leqslant d$. Then, a trapezoidal fuzzy number (TF-number) $\tilde{a}=\langle (a,b,c,d);{\eta _{\tilde{a}}}\rangle $ is a special fuzzy set on the real number set R, whose membership function is defined as
\[ {\mu _{\tilde{a}}}(x)=\left\{\begin{array}{l@{\hskip4.0pt}l}(x-a){\eta _{\tilde{a}}}/(b-a),\hspace{1em}& a\leqslant x\leqslant b,\\ {} {\eta _{\tilde{a}}},\hspace{1em}& b\leqslant x\leqslant c,\\ {} (d-x){\eta _{\tilde{a}}}/(d-c),\hspace{1em}& c\leqslant x\leqslant d,\\ {} 0,\hspace{1em}& \text{otherwise}.\end{array}\right.\]
Definition 3 (Wang et al., 2006).
Let $\tilde{a}=\langle (a,b,c,d);{\eta _{\tilde{a}}}\rangle $ be a TF-number with it’s membership function ${\mu _{\tilde{a}}}(x)$. Centroid point of $\tilde{a}$, denoted by ${\tilde{a}^{\ast }}$, is computed as:
Definition 4 (Torra, 2010).
Definition 5 (Yager, 2008).
Let $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes and let a prioritization between the attribute expressed by the linear order ${G_{1}}\succ {G_{2}}\succ {G_{3}}\succ \cdots \succ {G_{n}}$ indicate that attribute ${G_{j}}$ has a higher priority than ${G_{k}}$, if $j\lt k$. The value ${G_{j}}(x)$ is the performance of any alternative x under attribute ${G_{j}}$, and satisfies ${G_{j}}(x)\in [0,1]$. Then $PA({G_{1}}(x),{G_{2}}(x),\dots ,{G_{n}}(x))$, called the prioritized average operator, is defined as:
where ${w_{j}}=\frac{{T_{j}}}{{\textstyle\sum _{i=1}^{n}}{T_{i}}}$, ${T_{i}}={\textstyle\prod _{i=1}^{i-1}}{G_{i}}(x)$, ${T_{j}}={\textstyle\prod _{j=1}^{j-1}}{G_{j}}(x)\hspace{2.5pt}\text{such that}\hspace{2.5pt}{T_{1}}=1$.
Definition 6 (Wei, 2012).
Let ${\xi ^{j}}$ $(j\in \{1,2,\dots ,n\})$ be a collection of HFEs. Then,
-
1. The hesitant fuzzy prioritized weighted average operator of the set ${\xi ^{j}}$ $(j\in \{1,2,\dots ,n\})$ is defined as:where ${T_{i}}={\textstyle\prod _{k=1}^{i-1}}{\textstyle\sum _{{h_{1}^{k}}\in {\xi ^{k}}}}{h_{1}^{k}}$, ${T_{j}}={\textstyle\prod _{k=1}^{j-1}}{\textstyle\sum _{{h_{1}^{k}}\in {\xi ^{k}}}}{h_{1}^{k}}$ $(i,j=2,\dots ,n)$, ${T_{1}}=1$.
(3)
\[\begin{aligned}{}\textit{HFPWAO}\big({\xi ^{1}},{\xi ^{2}},\dots ,{\xi ^{n}}\big)& ={\underset{j=1}{\overset{n}{\bigoplus }}}\bigg(\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}{\xi ^{j}}\bigg)\\ {} & =\bigcup \limits_{{h_{1}^{1}}\in {\xi ^{1}},{h_{1}^{2}}\in {\xi ^{2}},\dots ,{h_{1}^{n}}\in {\xi ^{n}}}\Bigg\{1-{\prod \limits_{j=1}^{n}}{\big(1-{h_{1}^{j}}\big)^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}}}\Bigg\},\end{aligned}\] -
2. The hesitant fuzzy prioritized weighted geometric operator of the set ${\xi ^{j}}$ $(j\in \{1,2,\dots ,n\})$ is defined as:where ${T_{i}}={\textstyle\prod _{k=1}^{i-1}}{\textstyle\sum _{{h_{1}^{k}}\in {\xi ^{k}}}}{h_{1}^{k}}$, ${T_{j}}={\textstyle\prod _{k=1}^{j-1}}{\textstyle\sum _{{h_{1}^{k}}\in {\xi ^{k}}}}{h_{1}^{k}}$ $(i,j=2,\dots ,n)$, ${T_{1}}=1$.
(4)
\[\begin{aligned}{}\textit{HFPWGO}\big({\xi ^{1}},{\xi ^{2}},\dots ,{\xi ^{n}}\big)& ={\underset{j=1}{\overset{n}{\bigotimes }}}{\big({\xi ^{j}}\big)^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}}}\\ {} & =\bigcup \limits_{{h_{1}^{1}}\in {\xi ^{1}},{h_{1}^{2}}\in {\xi ^{2}},\dots ,{h_{1}^{n}}\in {\xi ^{n}}}\Bigg\{{\prod \limits_{j=1}^{n}}{\big({\xi ^{n}}\big)^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{j}}}}}\Bigg\},\end{aligned}\]
Theorem 1 (Wei, 2012).
Let ${\xi ^{j}}$ and ${\acute{\xi }^{j}}$ $(j\in \{1,2,\dots ,n\})$ be two sets of HFEs.
Definition 7 (Deli and Karaaslan, 2021).
Let $\mathbb{R}$ be a set of real numbers such that $a\leqslant b\leqslant c\leqslant d$. Then, a generalized hesitant trapezoidal fuzzy number (GTHF-number), denoted by ℏ, is defined as:
(${h_{i}}$ is set of some values in $[0,1]$, $i\in \{1,2,\dots ,n\}$) which is a special hesitant fuzzy set on the real number set $\mathbb{R}$, whose membership functions are defined as
In the paper, for purposes of focusing on GTHF- numbers, note that the set of all GTHF-numbers on ${\mathbb{R}^{+}}$ will be denoted by Θ.
Definition 8 (Deli and Karaaslan, 2021).
Let $\hslash =\langle (a,b,c,d);\xi \rangle $, ${\hslash ^{1}}=\langle ({a_{1}},{b_{1}},{c_{1}},{d_{1}});{\xi ^{1}}\rangle $, ${\hslash ^{2}}=\langle ({a_{2}},{b_{2}},{c_{2}},{d_{2}});{\xi ^{2}}\rangle \in \Theta $ and $\gamma \ne 0$ be any real number. Then,
-
1. ${\hslash ^{1}}\oplus {\hslash ^{2}}=\big\langle ({a_{1}}+{a_{2}},{b_{1}}+{b_{2}},{c_{1}}+{c_{2}},{d_{1}}+{d_{2}});{\textstyle\bigcup _{{h_{1}^{1}}\in {\xi ^{1}},{h_{1}^{2}}\in {\xi ^{2}}}}\{{\xi _{1}^{1}}+{h_{1}^{2}}-{h_{1}^{1}}.{h_{1}^{2}}\}\big\rangle $;
-
2. ${\hslash ^{1}}\odot {\hslash ^{2}}=\big\langle ({a_{1}}{a_{2}},{b_{1}}{b_{2}},{c_{1}}{c_{2}},{d_{1}}{d_{2}});{\textstyle\bigcup _{{h_{1}^{1}}\in {\xi ^{1}},{h_{1}^{2}}\in {\xi ^{2}}}}\{{h_{1}^{1}}.{h_{1}^{2}}\}\big\rangle $;
-
3. $\gamma \hslash =\big\langle (\gamma a,\gamma b,\gamma c,\gamma d);{\textstyle\bigcup _{h\in \xi }}\{1-{(1-h)^{\gamma }}\}\big\rangle \hspace{1em}(\gamma \geqslant 0)$;
-
4. ${(\hslash )^{\gamma }}=\big\langle ({a^{\gamma }},{b^{\gamma }},{c^{\gamma }},{d^{\gamma }});{\textstyle\bigcup _{h\in \xi }}\{{h^{\gamma }}\}\big\rangle \hspace{1em}(\gamma \geqslant 0)$.
Definition 9.
Deli and Karaaslan (2021) Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $, $j\in {I_{n}}$ be a set of GTHF-numbers. Then,
where $w={({w_{1}},{w_{2}},\dots ,{w_{n}})^{T}}$ is the weight vector of ${\hslash ^{j}},j\in {I_{n}}$ such that ${w_{j}}\in [0,1]$ and ${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$.
-
1. The hesitant fuzzy weighted geometric operator of the set ${\hslash _{j}}$ $(j\in \{1,2,\dots ,n\})$ is defined as:\[\begin{aligned}{}{H_{w}^{G}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})={\underset{j=1}{\overset{n}{\bigotimes }}}{{\hslash _{j}}^{{w_{j}}}}& =\Bigg\langle \Bigg({\prod \limits_{j=1}^{n}}{a_{j}^{{w_{j}}}},{\prod \limits_{j=1}^{n}}{b_{j}^{{w_{j}}}},{\prod \limits_{j=1}^{n}}{c_{j}^{{w_{j}}}},{\prod \limits_{j=1}^{n}}{d_{j}^{{w_{j}}}}\Bigg);\\ {} & \hspace{1em}\bigcup \limits_{{h_{1}^{1}}\in {\xi ^{1}},{h_{1}^{2}}\in {\xi ^{2}},\dots ,{h_{1}^{n}}\in {\xi ^{n}}}\Bigg\{{\prod \limits_{j=1}^{n}}{{h_{1}^{j}}^{{w_{j}}}}\Bigg\}\Bigg\rangle ;\end{aligned}\]
-
2. The hesitant fuzzy weighted average operator of collection ${\hslash _{j}}$ $(j\in \{1,2,\dots ,n\})$ is defined as:\[\begin{aligned}{}& {H_{w}^{A}}\big({\hslash ^{1}},{\hslash ^{2}},\dots ,{\hslash ^{n}}\big)\\ {} & \hspace{1em}={\underset{j=1}{\overset{n}{\bigoplus }}}{w_{j}}.{\hslash ^{j}}=\Bigg\langle \Bigg({\sum \limits_{j=1}^{n}}{w_{j}}.{a_{j}},{\sum \limits_{j=1}^{n}}{w_{j}}.{b_{j}},{\sum \limits_{j=1}^{n}}{w_{j}}.{c_{j}},{\sum \limits_{j=1}^{n}}{w_{j}}.{d_{j}}\Bigg);\\ {} & \hspace{1em}\bigcup \limits_{{h_{1}^{1}}\in {\xi ^{1}},{h_{1}^{2}}\in {\xi ^{2}},\dots ,{h_{1}^{n}}\in {\xi ^{n}}}\Bigg\{1-{\prod \limits_{j=1}^{n}}{\big(1-{h_{1}^{j}}\big)^{{w_{j}}}}\Bigg\}\Bigg\rangle ,\end{aligned}\]
3 GTHFN-Prioritized Aggregation Operators
In this section, we developed GTHFN-prioritized average operator and GTHF-prioritized geometric operator and examined some desired properties such as idempotency, boundedness and monotonicity in detail.
Definition 10.
Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $ $(j=1,2,\dots ,n)$ be a set of GTHF-numbers on $[0,1]\subseteq \mathbb{R}$, $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes such that there is a prioritization as in the Definition 5. Then, the GTHFN-prioritized average operator, denoted by ${\digamma _{A}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})$, is defined by
where
(9)
\[\begin{aligned}{}{\digamma _{A}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})=& {\underset{j=1}{\overset{n}{\bigoplus }}}\bigg(\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}{\hslash _{j}}\bigg),\end{aligned}\]
\[\begin{aligned}{}& {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n)\end{aligned}\]
and
\[\begin{aligned}{}& {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n).\end{aligned}\]
Theorem 2.
Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $ $(j=1,2,\dots ,n)$ be a set of GTHF-numbers on $[0,1]\subseteq \mathbb{R}$, $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes such that there is a prioritization as in the Definition 5. Then, their aggregated value by using the ${\digamma _{A}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})$ operator is also a GTHF-number, and
where
(10)
\[\begin{aligned}{}& {\digamma _{A}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})={\underset{j=1}{\overset{n}{\bigoplus }}}\bigg(\frac{{T_{j}}{\hslash _{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}\bigg)\\ {} & \hspace{1em}=\Bigg\langle \Bigg({\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}{a_{j}},{\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}{b_{j}},{\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}{c_{j}},{\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}{d_{j}}\Bigg);\\ {} & \hspace{2em}\bigcup \limits_{{h_{1}^{1}}\in {\xi ^{1}},{h_{1}^{2}}\in {\xi ^{2}},\dots ,{h_{1}^{n}}\in {\xi ^{n}}}\Bigg\{1-{\prod \limits_{j=1}^{n}}{\big(1-{h_{1}^{j}}\big)^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}}}\Bigg\}\Bigg\rangle ,\end{aligned}\]
\[\begin{aligned}{}& {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n)\end{aligned}\]
and
\[\begin{aligned}{}& {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n).\end{aligned}\]
It can be easily proved that the ${\digamma _{A}}$ operator has the following properties.
Theorem 3 (Idempotency).
Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $ $(j=1,2,\dots ,n)$ be a set of GTHF-numbers on $[0,1]\subseteq \mathbb{R}$, $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes such that there is a prioritization as in the Definition 5. If ${\hslash _{j}}=\hslash (\hslash =\langle (a,b,c,d);{\xi ^{j}}\rangle \hspace{2.5pt}(j=1,2,\dots ,n))$ for all $j\in \{1,2,\dots ,n\}$, then
where
\[\begin{aligned}{}& {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n)\end{aligned}\]
and
\[\begin{aligned}{}& {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n).\end{aligned}\]
Proof.
For ${\hslash _{j}}=\hslash $ for all $j\in \{1,2,\dots ,n\}$, and by definition of ${\digamma _{A}}$ operator, we have
where
(12)
\[\begin{aligned}{}& {\digamma _{A}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})={\underset{j=1}{\overset{n}{\bigoplus }}}\bigg(\frac{{T_{j}}{\hslash _{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}\bigg)\\ {} & \hspace{1em}=\Bigg\langle \Bigg({\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}{a_{j}},{\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}{b_{j}},{\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}{c_{j}},{\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}{d_{j}}\Bigg);\\ {} & \bigcup \limits_{{h_{1}^{1}}\in {\xi ^{1}},{h_{1}^{2}}\in {\xi ^{2}},\dots ,{h_{1}^{n}}\in {\xi ^{n}}}\Bigg\{1-{\prod \limits_{j=1}^{n}}{\big(1-{h_{1}^{j}}\big)^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}}}\Bigg\}\Bigg\rangle \\ {} & \hspace{1em}=\Bigg\langle \Bigg({\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}a,{\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}b,{\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}c,{\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}d\Bigg);\\ {} & \hspace{2em}\bigcup \limits_{{h_{1}^{1}}\in {\xi ^{1}},{h_{1}^{2}}\in {\xi ^{2}},\dots ,{h_{1}^{n}}\in {\xi ^{n}}}\Bigg\{1-{\prod \limits_{j=1}^{n}}{\big(1-{h_{1}^{j}}\big)^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}}}\Bigg\}\Bigg\rangle ,\Bigg({\sum \limits_{j=1}^{n}}\frac{{T_{j}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}=1\Bigg)\\ {} & \hspace{1em}=\big\langle (a,b,c,d);{\xi ^{j}}\big\rangle (j=1,2,\dots ,n)\}\rangle \\ {} & \hspace{1em}=\hslash ,\end{aligned}\]
\[\begin{aligned}{}& {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n)\end{aligned}\]
and
\[\begin{aligned}{}& {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n).\end{aligned}\]
This completes the proof. □Theorem 4 (Boundedness).
Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $ $(j=1,2,\dots ,n)$ be a set of GTHF-numbers on $[0,1]\subseteq \mathbb{R}$, $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes such that there is a prioritization as in the Definition 5. Let
then
\[\begin{aligned}{}{\hslash ^{-}}& =\Big\langle \Big(\underset{j\in \{1,2,\dots ,n\}}{\min }\{{a_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\min }\{{b_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\min }\{{c_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\min }\{{d_{j}}\}\Big);\\ {} & \hspace{1em}\underset{j\in \{1,2,\dots ,n\}}{\min }\Big\{\underset{{h_{1}^{j}}\in {\xi ^{j}}}{\min }\big\{{h_{1}^{j}}\big\}\Big\}\Big\rangle \end{aligned}\]
and
\[\begin{aligned}{}{\hslash ^{+}}& =\Big\langle \Big(\underset{j\in \{1,2,\dots ,n\}}{\max }\{{a_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\max }\{{b_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\max }\{{c_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\max }\{{d_{j}}\}\Big);\\ {} & \hspace{1em}\underset{j\in \{1,2,\dots ,n\}}{\max }\Big\{\underset{{h_{1}^{j}}\in {\xi ^{j}}}{\max }\big\{{h_{1}^{j}}\big\}\Big\}\Big\rangle .\end{aligned}\]
If
(13)
\[\begin{aligned}{}& \textit{``}{a_{j}}\leqslant {a_{i}},{b_{j}}\leqslant {b_{i}},{c_{j}}\leqslant {c_{i}},{d_{j}}\leqslant {d_{i}}\hspace{2.5pt}\textit{and}\hspace{2.5pt}{h_{1}^{j}}\leqslant {h_{1}^{i}}\hspace{2.5pt}\textit{for all}\hspace{2.5pt}{h_{1}^{j}}\in {\xi ^{j}},{h_{1}^{i}}\in {\xi ^{i}}\textit{''}\\ {} & \hspace{1em}\Rightarrow \textit{``}{\hslash _{j}}=\big\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\big\rangle \leqslant {\hslash _{i}}=\big\langle ({a_{i}},{b_{i}},{c_{i}},{d_{i}});{\xi ^{i}}\big\rangle \textit{''}\end{aligned}\]Proof.
We have $\frac{{\textstyle\sum _{j=1}^{n}}{T_{j}}}{{\textstyle\sum _{i=1}^{n}}{T_{i}}}=1$, since
\[\begin{aligned}{}& \underset{j\in \{1,2,\dots ,n\}}{\min }\{{a_{j}}\}\leqslant \{{a_{j}}\}\leqslant \underset{j\in \{1,2,\dots ,n\}}{\max }\{{a_{j}}\},\hspace{1em}\underset{j\in \{1,2,\dots ,n\}}{\min }\{{b_{j}}\}\leqslant \{{b_{j}}\}\leqslant \underset{j\in \{1,2,\dots ,n\}}{\max }\{{b_{j}}\},\\ {} & \underset{j\in \{1,2,\dots ,n\}}{\min }\{{c_{j}}\}\leqslant \{{c_{j}}\}\leqslant \underset{j\in \{1,2,\dots ,n\}}{\max }\{{c_{j}}\},\hspace{1em}\underset{j\in \{1,2,\dots ,n\}}{\min }\{{d_{j}}\}\leqslant \{{d_{j}}\}\leqslant \underset{j\in \{1,2,\dots ,n\}}{\max }\{{d_{j}}\},\end{aligned}\]
and
\[ \underset{{h_{1}^{j}}\in {\xi ^{j}}}{\min }\big\{{h_{1}^{j}}\big\}\leqslant {h_{1}^{j}}\leqslant \underset{{h_{1}^{j}}\in {\xi ^{j}}}{\max }\big\{{h_{1}^{j}}\big\}.\]
Let ${\digamma _{A}}$ be an operator. Then, by using the Theorem 3.2, it yields that
□Theorem 5 (Monotonicity).
Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $ $(j=1,2,\dots ,n)$ and ${\acute{\hslash }_{j}}=\langle ({\acute{a}_{j}},{\acute{b}_{j}},{\acute{c}_{j}},{\acute{d}_{j}});{\acute{xi}^{j}}\rangle $ $(j=1,2,\dots ,n)$ be two sets of GTHF-numbers on $[0,1]\subseteq \mathbb{R}$, $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes such that there is a prioritization as in the Definition 5. If ${\hslash _{j}}\leqslant {\acute{\hslash }_{j}}$ for all $j\in \{1,2,\dots ,n\}$ based on equation (13), then
where
(16)
\[ {\digamma _{A}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})\leqslant {\digamma _{A}}({\acute{\hslash }_{1}},{\acute{\hslash }_{2}},\dots ,{\acute{\hslash }_{n}}).\]
\[\begin{aligned}{}& {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n),\\ {} & {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n),\\ {} & {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{\acute{h}_{1}^{k}}\in {\acute{xi}^{k}}}\frac{{\acute{h}_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{\acute{h}_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n)\end{aligned}\]
and
\[\begin{aligned}{}& {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{\acute{h}_{1}^{k}}\in {\acute{xi}^{k}}}\frac{{\acute{h}_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{\acute{h}_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n).\end{aligned}\]
Definition 11.
Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $ $(j=1,2,\dots ,n)$ be a collection of GTHF-numbers on $[0,1]\subseteq \mathbb{R}$, $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes such that there is a prioritization as in the Definition 5. Then, the GTHFN-prioritized geometric operator, denoted by ${\digamma _{G}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})$, is defined by
where
(17)
\[ \begin{array}{r@{\hskip4.0pt}l}{\digamma _{G}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})=& {\hslash _{1}^{\frac{{T_{1}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}}}\otimes {\hslash _{2}^{\frac{{T_{2}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}}}\otimes \cdots \otimes {\hslash _{1}^{\frac{{T_{2}}}{{\textstyle\textstyle\sum _{i=1}^{n}}{T_{i}}}}},\end{array}\]
\[\begin{aligned}{}& {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n)\end{aligned}\]
and
\[\begin{aligned}{}& {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n).\end{aligned}\]
Theorem 6.
Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $ $(j=1,2,\dots ,n)$ be a set of GTHF-numbers on $[0,1]\subseteq \mathbb{R}$, $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes such that there is a prioritization as in the Definition 5. Then, their aggregated value by using the ${\digamma _{G}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})$ operator is also a GTHF-number, and
where
(18)
\[\begin{aligned}{}{\digamma _{G}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})& ={\hslash _{1}^{\frac{{T_{1}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{j}}}}}\bigotimes {\hslash _{2}^{\frac{{T_{2}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{j}}}}}\bigotimes \dots \bigotimes {\hslash _{n}^{\frac{{T_{n}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{j}}}}}\\ {} & =\Bigg\langle \big({a_{j}^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{j}}}}},{b_{j}^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{j}}}}},{c_{j}^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{j}}}}},{d_{j}^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{j}}}}}\big);\\ {} & \hspace{2em}\bigcup \limits_{{\gamma _{1}}\in {\hslash _{1}},{\gamma _{2}}\in {\hslash _{2}},\dots ,{\gamma _{n}}\in {\hslash _{n}}}\Bigg\{{\prod \limits_{j=1}^{n}}{({\gamma _{j}})^{\frac{{T_{j}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{j}}}}}\Bigg\}\Bigg\rangle ,\end{aligned}\]
\[\begin{aligned}{}& {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n)\end{aligned}\]
and
\[\begin{aligned}{}& {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n).\end{aligned}\]
It can be easily proved that the ${\digamma _{G}}$ operator has the following properties.
Theorem 7 (Idempotency).
Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $ $(j=1,2,\dots ,n)$ be a set of GTHF-numbers on $[0,1]\subseteq \mathbb{R}$, $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes such that there is a prioritization as in the Definition 5. If ${\hslash _{j}}=\hslash $ for all $j\in \{1,2,\dots ,n\}$, then
where
\[\begin{aligned}{}& {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n)\end{aligned}\]
and
\[\begin{aligned}{}& {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n).\end{aligned}\]
Theorem 8 (Boundedness).
Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $ $(j=1,2,\dots ,n)$ be a set of GTHF-numbers on $[0,1]\subseteq \mathbb{R}$, $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes such that there is a prioritization as in the Definition 5. Let
then
where
\[\begin{aligned}{}{\hslash ^{-}}& =\Big\langle \Big(\underset{j\in \{1,2,\dots ,n\}}{\min }\{{a_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\min }\{{b_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\min }\{{c_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\min }\{{d_{j}}\}\Big);\\ {} & \hspace{1em}\underset{j\in \{1,2,\dots ,n\}}{\min }\Big\{\underset{{h_{1}^{j}}\in {\xi ^{j}}}{\min }\big\{{h_{1}^{j}}\big\}\Big\}\Big\rangle \end{aligned}\]
and
\[\begin{aligned}{}{\hslash ^{+}}& =\Big\langle \Big(\underset{j\in \{1,2,\dots ,n\}}{\max }\{{a_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\max }\{{b_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\max }\{{c_{j}}\},\underset{j\in \{1,2,\dots ,n\}}{\max }\{{d_{j}}\}\Big);\\ {} & \hspace{1em}\underset{j\in \{1,2,\dots ,n\}}{\max }\Big\{\underset{{h_{1}^{j}}\in {\xi ^{j}}}{\max }\big\{{h_{1}^{j}}\big\}\Big\}\Big\rangle .\end{aligned}\]
If
(20)
\[\begin{aligned}{}& \textit{``}{a_{j}}\leqslant {a_{i}},{b_{j}}\leqslant {b_{i}},{c_{j}}\leqslant {c_{i}},{d_{j}}\leqslant {d_{i}}\hspace{2.5pt}\textit{and}\hspace{2.5pt}{h_{1}^{j}}\leqslant {h_{1}^{i}}\hspace{2.5pt}\textit{for all}\hspace{2.5pt}{h_{1}^{j}}\in {\xi ^{j}},{h_{1}^{i}}\in {\xi ^{i}}\textit{''}\\ {} & \hspace{1em}\Rightarrow \textit{''}{\hslash _{j}}=\big\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\big\rangle \leqslant {\hslash _{i}}=\big\langle ({a_{i}},{b_{i}},{c_{i}},{d_{i}});{\xi ^{i}}\big\rangle \textit{''}\end{aligned}\](21)
\[ {\hslash ^{-}}\leqslant {\digamma _{G}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})\leqslant {\hslash ^{+}},\]
\[\begin{aligned}{}& {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n)\end{aligned}\]
and
\[\begin{aligned}{}& {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n).\end{aligned}\]
Theorem 9 (Monotonicity).
Let ${\hslash _{j}}=\langle ({a_{j}},{b_{j}},{c_{j}},{d_{j}});{\xi ^{j}}\rangle $ $(j=1,2,\dots ,n)$ and ${\acute{\hslash }_{j}}=\langle ({\acute{a}_{j}},{\acute{b}_{j}},{\acute{c}_{j}},{\acute{d}_{j}});{\acute{xi}^{j}}\rangle $ $(j=1,2,\dots ,n)$ be two sets of GTHF-numbers on $[0,1]\subseteq \mathbb{R}$, $G=\{{G_{j}}:j\in \{1,2,\dots ,n\}\}$ be a set of attributes such that there is a prioritization as in the Definition 5. If ${\hslash _{j}}\leqslant {\acute{\hslash }_{j}}$ for all $j\in \{1,2,\dots ,n\}$ based on equation (20), then
where
(22)
\[ {\digamma _{G}}({\hslash _{1}},{\hslash _{2}},\dots ,{\hslash _{n}})\leqslant {\digamma _{G}}({\acute{\hslash }_{1}},{\acute{\hslash }_{2}},\dots ,{\acute{\hslash }_{n}}),\]
\[\begin{aligned}{}& {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n),\\ {} & {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{h_{1}^{k}}\in {\xi ^{k}}}\frac{{h_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{h_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n),\\ {} & {T_{1}}=1,{T_{j}}={\prod \limits_{k=1}^{j-1}}\sum \limits_{{\acute{h}_{1}^{k}}\in {\acute{xi}^{k}}}\frac{{\acute{h}_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{\acute{h}_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(j=2,\dots ,n)\end{aligned}\]
and
\[\begin{aligned}{}& {T_{1}}=1,{T_{i}}={\prod \limits_{k=1}^{i-1}}\sum \limits_{{\acute{h}_{1}^{k}}\in {\acute{xi}^{k}}}\frac{{\acute{h}_{1}^{k}}({d_{k}^{2}}-2{c_{k}^{2}}+2{b_{k}^{2}}-{a_{k}^{2}}+{d_{k}}{c_{k}}-{a_{k}}{b_{k}})+3({c_{k}^{2}}-{b_{k}^{2}})}{3{\acute{h}_{1}^{k}}({d_{k}}-{c_{k}}+{b_{k}}-{a_{k}})+6({c_{k}}-{b_{k}})}\\ {} & \hspace{1em}(i=2,\dots ,n).\end{aligned}\]
4 An Approach for GTHFPA Operator to MADM
4.1 Decision-Making Steps
In this section, we shall utilize the GTHFPA operators to MADM. To do this, we develop an algorithm which is presented in Fig. 1.
Definition 12.
Let $U=\{{U_{1}},{U_{2}},\dots ,{U_{m}}\}$ be a set of alternatives, $E=\{{E_{1}}={\textstyle\bigcup _{r=1}^{{k_{1}}}}\{{e_{1r}}\}$, ${E_{2}}={\textstyle\bigcup _{r=1}^{{k_{2}}}}\{{e_{1r}}\},\dots ,$ ${E_{n}}={\textstyle\bigcup _{r=1}^{{k_{n}}}}\{{e_{1r}}\}:{k_{1}},{k_{2}},\dots $, ${k_{n}}\in {Z^{+}}\}$ be a set of attributes. Here, there is a prioritization between the attributes expressed by the linear order ${E_{1}}\succ {E_{2}}\succ \cdots \succ {E_{n}}$ that indicates attribute ${E_{j}}$ has a higher priority than ${E_{k}}$, if $j\lt k$. Suppose that GTHF sub-decision matrix for sub-attribute set ${E_{j}}={\textstyle\bigcup _{r=1}^{{k_{j}}}}\{{e_{1r}}\}$ be ${H^{j}}={({x_{ir}^{j}})_{m\times {k_{j}}}}$ ($i=1,2,\dots ,m$, $r=1,2,\dots ,{k_{j}}$), where ${x_{ir}^{j}}$ is in the form of GTHF-number based on the alternative ${U_{i}}$ and sub-attribute ${e_{jr}}$.
Now, we develop an approach for MADM problems under the GTHFPA operator (or GTHFPG operator) with GTHF-numbers as the following algorithm.
Algorithm:
Step 1.
Construct the GTHF sub-decision matrix for sub-attribute set ${E_{j}}={\textstyle\bigcup _{r=1}^{{k_{j}}}}\{{e_{jr}}\}$ as ${H^{j}}={({x_{ir}^{j}})_{m\times {k_{j}}}}={(\langle ({a_{ir}^{j}},{b_{ir}^{j}},{c_{ir}^{j}},{d_{ir}^{j}});{\xi _{ir}^{j}}\rangle )_{m\times {k_{j}}}}$ $(i=1,2,\dots ,m,r=1,2,\dots ,{k_{j}})$ for decision.
Step 2.
Compute overall values matrix
for $i\in \{1,2,\dots ,m\}$ and $j\in \{1,2,\dots ,n\}$ where ${A_{ij}}$ denotes evaluation of the alternative ${U_{i}}$ with respect to the attribute ${E_{j}}$,
(23)
\[\begin{aligned}{}{A_{ij}}& =\big\langle ({a_{ij}},{b_{ij}},{c_{ij}},{d_{ij}});{\xi ^{ij}}\big\rangle =\textit{GTHFPA}\big({x_{i1}^{j}},{x_{i2}^{j}},\dots ,{x_{i{k_{j}}}^{j}}\big)\\ {} & =\frac{{T_{i1}^{J}}}{{\textstyle\textstyle\sum _{r=1}^{{k_{j}}}}{T_{ir}^{J}}}{x_{i1}^{j}}\bigoplus \frac{{T_{i2}^{J}}}{{\textstyle\textstyle\sum _{r=1}^{{k_{j}}}}{T_{ir}^{J}}}{x_{i2}^{j}}\bigoplus \cdots \bigoplus \frac{{T_{i{k_{j}}}^{J}}}{{\textstyle\textstyle\sum _{r=1}^{{k_{j}}}}{T_{ir}^{J}}}{x_{i{k_{j}}}^{j}}\end{aligned}\]where
and
(24)
\[\begin{aligned}{}& {T_{ir}^{j}}={\prod \limits_{p=1}^{r-1}}\sum \limits_{{h_{ip}}\in {\xi _{ir}^{j}}}\frac{{h_{ip}}({d_{ip}^{2}}-2{c_{ip}^{2}}+2{b_{ip}^{2}}-{a_{ip}^{2}}+{d_{ip}}{c_{ip}}-{a_{ip}}{b_{ip}})+3({c_{ip}^{2}}-{b_{ip}^{2}})}{3{h_{ip}}({d_{ip}}-{c_{ip}}+{b_{ip}}-{a_{ip}})+6({c_{ip}}-{b_{ip}})}\\ {} & \hspace{1em}(i=1,2,\dots ,m,r=2,\dots ,{k_{j}})\end{aligned}\]Step 3.
Find ${A_{i}}$ $(i=1,2,\dots ,m)$ by agregating all GTHF-numbers under ${A_{ij}}$ $(j=1,2,\dots ,n)$ using the GTHFPA operator as follows:
where
and
(26)
\[\begin{aligned}{}{A_{i}}& =\textit{GTHFPA}({A_{i1}},{A_{i2}},\dots {A_{in}})=\big\langle ({a_{i}},{b_{i}},{c_{i}},{d_{i}});{\xi ^{i}}\big\rangle \\ {} & =\frac{{T_{i1}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{ij}}}{A_{i1}}\bigoplus \frac{{T_{i2}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{ij}}}{A_{i2}}\bigoplus \dots \bigoplus \frac{{T_{in}}}{{\textstyle\textstyle\sum _{j=1}^{n}}{T_{ij}}}{A_{in}},\end{aligned}\](27)
\[\begin{aligned}{}& {T_{ij}}={\prod \limits_{p=1}^{j-1}}\sum \limits_{{h_{ip}}\in {\xi ^{ij}}}\frac{{h_{ip}}({d_{ip}^{2}}-2{c_{ip}^{2}}+2{b_{ip}^{2}}-{a_{ip}^{2}}+{d_{ip}}{c_{ip}}-{a_{ip}}{b_{ip}})+3({c_{ip}^{2}}-{b_{ip}^{2}})}{3{h_{ip}}({d_{ip}}-{c_{ip}}+{b_{ip}}-{a_{ip}})+6({c_{ip}}-{b_{ip}})}\\ {} & \hspace{1em}(i=1,2,\dots ,m,j=2,\dots ,n)\end{aligned}\]Step 4.
Step 5.
Rank all the alternatives ${U_{i}}$ $(i=1,2,,\dots ,m)$ and select the best one(s). Here if $s({A_{s}})\gt s({A_{t}})\Rightarrow {U_{s}}\gt {U_{t}}$ $(s,t\in \{1,2,\dots ,m\})$.
5 An Application Case for Turkey
In this section, we present a MADM problem which is a software selection problem adapted/inspirated from Çelikbilek and Tüysüz (2016), Yuan et al. (2018). The application of the proposed approach is for evaluating renewable energy under subjective perspective with linguistic scales.
5.1 Case Study
According to the plan of renewable energy development in Turkey, the Turkish government aims to reduce the country’s dependence on imported energy. Having insufficient quantities of domestic oil and natural gas resources to support demand, the best guarantee of security of energy supply is clearly to maintain a diversity of energy sources and to focus on renewable energy sources, which the country has abundantly. There are five alternatives which are given in Table 1: Geothermal energy ${U_{1}}$, Solar energy ${U_{2}}$, Biomass energy ${U_{3}}$ and Wind energy ${U_{4}}$. The most common attributes for renewable energy evaluation involve ${E_{1}}$ economic, ${E_{2}}$ technical and ${E_{3}}$ environmental. While many national and international companies invest in these resources for a new source of income, states encourage these investments both to produce clean energy and to utilize their own resources in energy production. Therefore, our priority is environment ${E_{3}}$. Then we evaluate wind power plants as they will have a significant contribution to the country’s economy since it is a new source of income ${E_{1}}$, and finally, ${E_{2}}$ technical effect. That is, in this case, there is a strict prioritization of parameters ${E_{3}}\gt {E_{1}}\gt {E_{2}}$, here > indicates preference.
Table 1
Renewable energy resource alternatives.
| Symbol | The renewable energy resource |
| ${U_{1}}$ | Geothermal energy |
| ${U_{2}}$ | Solar energy |
| ${U_{3}}$ | Biomass energy |
| ${U_{4}}$ | Wind energy |
Table 2
GTHF-numbers for linguistic terms.
| Linguistic terms | Linguistic values of GTHF-numbers | Score |
| Absolutely Low (AL) | $\langle (0.1,0.2,0.3,0.5);\{0.1,0.2,0.4\}\rangle $ | 0, 0338 |
| Very very Low (L) | $\langle (0.2,0.3,0.4,0.6);\{0.1,0.3,0.5\}\rangle $ | 0, 0585 |
| Very Low (VL) | $\langle (0.3,0.4,0.5,0.6);\{0.2,0.4,0.7\}\rangle $ | 0, 0780 |
| Fairly Low (FL) | $\langle (0.1,0.4,0.5,0.6);\{0.2,0.4,0.6\}\rangle $ | 0, 0880 |
| Low (L) | $\langle (0.4,0.5,0.7,0.8);\{0.2,0.4,0.7\}\rangle $ | 0, 1560 |
| Medium (M) | $\langle (0.5,0.7,0.8,0.9);\{0.3,0.5,0.6\}\rangle $ | 0, 1657 |
| Fairly High (FH) | $\langle (0.3,0.5,0.6,0.8);\{0.2,0.5,0.9\}\rangle $ | 0, 1760 |
| High (H) | $\langle (0.1,0.4,0.8,0.9);\{0.1,0.3,0.5\}\rangle $ | 0, 1920 |
| Very High (VH) | $\langle (0.2,0.5,0.6,0.7);\{0.7,0.8,0.9\}\rangle $ | 0, 2240 |
| Very Very High (VVH) | $\langle (0.6,0.7,0.8,0.9);\{0.6,0.8,0.9\}\rangle $ | 0, 2300 |
| Absolutely High (AH) | $\langle (0.4,0.5,0.7,0.9);\{0.3,0.7,0.9\}\rangle $ | 0, 2818 |
Table 3
Linguistic assessment of the renewable energy alternatives based on GTHF-number.
| Primary criteria | Secondary criteria | Alternatives | Evaluation values |
| ${E_{1}}$-Economical criterion | ${e_{11}}$-Service life | ${U_{1}}$ | {(L)} |
| ${U_{2}}$ | {(FL)} | ||
| ${U_{3}}$ | {(H)} | ||
| ${U_{4}}$ | {(AH)} | ||
| ${e_{12}}$-Investment cost | ${U_{1}}$ | {(H)} | |
| ${U_{2}}$ | {(FH)} | ||
| ${U_{3}}$ | {(VH)} | ||
| ${U_{4}}$ | {(AH)} | ||
| ${e_{13}}$-Operation and maintenance cost | ${U_{1}}$ | {(AH)} | |
| ${U_{2}}$ | {(L)} | ||
| ${U_{3}}$ | {(VVH)} | ||
| ${U_{4}}$ | {(FL)} | ||
| ${E_{2}}$-Technical criterion | ${e_{21}}$-Availability | ${U_{1}}$ | {(M)} |
| ${U_{2}}$ | {(VVH)} | ||
| ${U_{3}}$ | {(H)} | ||
| ${U_{4}}$ | {(L)} | ||
| ${e_{22}}$-Capacity | ${U_{1}}$ | {(M)} | |
| ${U_{2}}$ | {(L)} | ||
| ${U_{3}}$ | {(H)} | ||
| ${U_{4}}$ | {(VH)} | ||
| ${e_{23}}$-Resource density | ${U_{1}}$ | {(FH)} | |
| ${U_{2}}$ | {(H)} | ||
| ${U_{3}}$ | {(VH)} | ||
| ${U_{4}}$ | {(VVH)} | ||
| ${e_{24}}$-Efficiency | ${U_{1}}$ | {(AL) } | |
| ${U_{2}}$ | {(VL)} | ||
| ${U_{3}}$ | {(FH)} | ||
| ${U_{4}}$ | {(VVH)} | ||
| ${E_{3}}$-Environmental criterion | ${e_{31}}$-Air pollution | ${U_{1}}$ | {(AL)} |
| ${U_{2}}$ | {(AH)} | ||
| ${U_{3}}$ | {(H)} | ||
| ${U_{4}}$ | {(VH)} | ||
| ${e_{32}}$-Noise pollution | ${U_{1}}$ | {(M)} | |
| ${U_{2}}$ | {(VL)} | ||
| ${U_{3}}$ | {(AL)} | ||
| ${U_{4}}$ | {(VVH)} |
Also, description of subattributes for attributes ${E_{1}}$, ${E_{2}}$ and ${E_{3}}$ is presented in Table 3 as linguistic assessment of the renewable energy alternatives based on GTHF-numbers, is given as:
-
${E_{1}}:$ Economical attribute contains ${e_{11}}$ = ervice life, ${e_{12}}$ = investment cost and ${e_{13}}$ = operation and maintenance cost. There is a strict prioritization between parameters ${e_{11}}\gt {e_{12}}\gt {e_{13}}$, here > indicates preferred to.
-
${E_{2}}:$ Technical attribute contains ${e_{21}}$ = availability, ${e_{22}}$ = capacity, ${e_{23}}$ = resource density and ${e_{24}}$ = efficiency. There is a strict prioritization between parameters ${e_{21}}\gt {e_{22}}\gt {e_{23}}\gt {e_{24}}$, here > indicates preference.
-
${E_{3}}:$ Environmental attribute contains ${e_{31}}$ = air pollution and ${e_{32}}$ = noise pollution. There is a strict prioritization between parameters ${e_{31}}\gt {e_{32}}$, here > indicates preference.
Moreover, experts are selected from a variety of departments of faculty of engineering to increase the objectivity of the results as much as possible. Experts give their evaluation information by GTHF-numbers for linguistic terms shown in Table 2.
Then, in order to select/rank a renewable energy resource, we utilize the GTHFPA operator to develop an approach to multiple-attribute decision-making problems with GTHF information, which can be expressed by using the following algorithm:
Algorithm:
Step 1.
We constructed three GTHF sub-decision matrices for sub-attribute set ${E_{j}}={\textstyle\bigcup _{r=1}^{{k_{j}}}}\{{e_{jr}}\}$ as ${H^{j}}={({x_{ir}^{j}})_{4\times {k_{j}}}}$ $(i=1,2,3,4;j=1,2,3;r={k_{1}},{k_{2}},{k_{3}};{k_{1}}=1,2,3;{k_{2}}=1,2,3,4;{k_{3}}=1,2)$ for decision in Table 4–6.
Table 4
GTHF sub-decision matrices ${({x_{ir}^{1}})_{5\times 3}}$.
| Economical | Service life | Investment cost |
| ${u_{1}}$ | $\langle (0.2,0.3,0.4,0.6);\{0.1,0.3,0.5\}\rangle $ | $\langle (0.1,0.4,0.8,0.9);\{0.1,0.3,0.5\}\rangle $ |
| ${u_{2}}$ | $\langle (0.1,0.4,0.5,0.6);\{0.2,0.4,0.6\}\rangle $ | $\langle (0.3,0.5,0.6,0.8);\{0.2,0.5,0.9\}\rangle $ |
| ${u_{3}}$ | $\langle (0.1,0.4,0.8,0.9);\{0.1,0.3,0.5\}\rangle $ | $\langle (0.2,0.5,0.6,0.7);\{0.7,0.8,0.9\}\rangle $ |
| ${u_{4}}$ | $\langle (0.4,0.5,0.7,0.9);\{0.3,0.7,0.9\}\rangle $ | $\langle (0.4,0.5,0.7,0.9);\{0.3,0.7,0.9\}\rangle $ |
| Maintenance cost | ||
| ${u_{1}}$ | $\langle (0.4,0.5,0.7,0.9);\{0.3,0.7,0.9\}\rangle $ | |
| ${u_{2}}$ | $\langle (0.4,0.5,0.7,0.8);\{0.2,0.4,0.7\}\rangle $ | |
| ${u_{3}}$ | $\langle (0.6,0.7,0.8,0.9);\{0.6,0.8,0.9\}\rangle $ | |
| ${u_{4}}$ | $\langle (0.1,0.4,0.5,0.6);\{0.2,0.4,0.6\}\rangle $ |
Table 5
GTHF sub-decision matrices ${({x_{ir}^{2}})_{5\times 4}}$.
| Technical | Availability | Capacity |
| ${u_{1}}$ | $\langle (0.5,0.7,0.8,0.9);\{0.3,0.5,0.6\}\rangle $ | $\langle (0.5,0.7,0.8,0.9);\{0.3,0.5,0.6\}\rangle $ |
| ${u_{2}}$ | $\langle (0.6,0.7,0.8,0.9);\{0.6,0.8,0.9\}\rangle $ | $\langle (0.4,0.5,0.7,0.8);\{0.2,0.4,0.7\}\rangle $ |
| ${u_{3}}$ | $\langle (0.1,0.4,0.8,0.9);\{0.1,0.3,0.5\}\rangle $ | $\langle (0.1,0.4,0.8,0.9);\{0.1,0.3,0.5\}\rangle $ |
| ${u_{4}}$ | $\langle (0.4,0.5,0.7,0.8);\{0.2,0.4,0.7\}\rangle $ | $\langle (0.2,0.5,0.6,0.7);\{0.7,0.8,0.9\}\rangle $ |
| Resource density | Efficiency | |
| ${u_{1}}$ | $\langle (0.3,0.5,0.6,0.8);\{0.2,0.5,0.9\}\rangle $ | $\langle (0.1,0.2,0.3,0.5);\{0.1,0.2,0.4\}\rangle $ |
| ${u_{2}}$ | $\langle (0.1,0.4,0.8,0.9);\{0.1,0.3,0.5\}\rangle $ | $\langle (0.3,0.4,0.5,0.6);\{0.2,0.4,0.7\}\rangle $ |
| ${u_{3}}$ | $\langle (0.2,0.5,0.6,0.7);\{0.7,0.8,0.9\}\rangle $ | $\langle (0.3,0.5,0.6,0.8);\{0.2,0.5,0.9\}\rangle $ |
| ${u_{4}}$ | $\langle (0.6,0.7,0.8,0.9);\{0.6,0.8,0.9\}\rangle $ | $\langle (0.6,0.7,0.8,0.9);\{0.6,0.8,0.9\}\rangle $ |
Table 6
GTHF sub-decision matrices ${({x_{ir}^{3}})_{5\times 2}}$.
| Environmental | Air pollution | Noise pollution |
| ${u_{1}}$ | $\langle (0.1,0.2,0.3,0.5);\{0.1,0.2,0.4\}\rangle $ | $\langle (0.5,0.7,0.8,0.9);\{0.3,0.5,0.6\}\rangle $ |
| ${u_{2}}$ | $\langle (0.4,0.5,0.7,0.9);\{0.3,0.7,0.9\}\rangle $ | $\langle (0.3,0.4,0.5,0.6);\{0.2,0.4,0.7\}\rangle $ |
| ${u_{3}}$ | $\langle (0.1,0.4,0.8,0.9);\{0.1,0.3,0.5\}\rangle $ | $\langle (0.1,0.2,0.3,0.5);\{0.1,0.2,0.4\}\rangle $ |
| ${u_{4}}$ | $\langle (0.2,0.5,0.6,0.7);\{0.7,0.8,0.9\}\rangle $ | $\langle (0.6,0.7,0.8,0.9);\{0.6,0.8,0.9\}\rangle $ |
Table 7
The decision makers’ evaluation of the alternatives with respect to criteria.
| Economical | Technical | |
| ${u_{1}}$ | $\langle (0.191,0.321,0.469,0.654);\{0.107,0.318,0.524\}\rangle $ | $\langle (0.459,0.662,0.770,0.906);\{0.1,0.3,0.5\}\rangle $ |
| ${u_{2}}$ | $\langle (0.159,0.426,0.533,0.651);\{0.2,0.420,0.697\}\rangle $ | $\langle (0.497,0.627,0.801,0.910);\{0.2,0.5,0.9\}\rangle $ |
| ${u_{3}}$ | $\langle (0.152,0.437,0.769,0.868);\{0.286,0.475,0.655\}\rangle $ | $\langle (0.112,0.414,0.801,0.907);\{0.7,0.8,0.9\}\rangle $ |
| ${u_{4}}$ | $\langle (0.363,0.488,0.675,0.863);\{0.288,0.673,0.881\}\rangle $ | $\langle (0.409,0.552,0.720,0.824);\{0.3,0.7,0.9\}\rangle $ |
| Environmental | ||
| ${u_{1}}$ | $\langle (0.141,0.251,0.351,0.541);\{0.123,0.238,0.9\}\rangle $ | |
| ${u_{2}}$ | $\langle (0.369,0.469,0.637,0.806);\{0.270,0.627,0.859\}\rangle $ | |
| ${u_{3}}$ | $\langle (0.1,0.366,0.715,0.832);\{0.1,0.284,0.484\}\rangle $ | |
| ${u_{4}}$ | $\langle (0.324,0.562,0.662,0.762);\{0.672,0.8,0.9\}\rangle $ |
Step 3.
We found ${A_{i}}$ $(i=1,2,\dots ,m)$ by aggregated all GTHF-numbers under ${A_{ij}}$ $(j=1,2,\dots ,n)$ by using Eqs. (26)–(28) in Table 8.
Table 8
${A_{i}}(i=1,2,3,4)$ by aggregated all GTHF-numbers.
| ${A_{1}}$ | $\langle (0.2271,0.3665,0.5053,0.6835);\{0.1468,0.3409,0.5417\}\rangle $ |
| ${A_{2}}$ | $\langle (0.2470,0.4698,0.5966,0.7183);\{0.2610,0.5001,0.7479\}\rangle $ |
| ${A_{3}}$ | $\langle (0.1398,0.4282,0.7730,0.8751);\{0.2450,0.4363,0.6229\}\rangle $ |
| ${A_{4}}$ | $\langle (0.3703,0.5135,0.6855,0.8405);\{0.3812,0.6727,0.8656\}\rangle $ |
Step 4.
Step 5.
All the alternatives ${U_{i}}$ $(i=1,2,3,4)$ are ranked and the best one(s) is selected. Here, if $s({A_{4}})\gt s({A_{3}})\gt s({A_{2}})\gt s({A_{1}})\Rightarrow {U_{4}}\gt {U_{3}}\gt {U_{2}}\gt {U_{1}}$. Thus, the most environmentally friendly renewable energy source is ${U_{4}}$.
6 Comparison and Analysis Discussion
When a decision analyst collects the data or information using GTHF-numbers, no prioritization operator can handle this information or data. The above defined GTHFN-prioritized aggregation operators are the only tool to solve this kind of information and help the decision analyst to make a decision. To demonstrate the effectiveness of the proposed method, a comparison of the decision-making results of the MADM methods based on prioritized aggregation operators of existing presented by Wei (2012), Wan et al. (2015) and Liang et al. (2017) is carried out, as shown in Table 10.
Table 10
The results from the different operators.
| Methods | Ranking of alternatives | The optimal alternative |
| The method in Wei (2012) | ${U_{4}}\gt {U_{2}}\gt {U_{3}}\gt {U_{1}}$, | ${U_{4}}$ |
| The method in Wan et al. (2015) | ${U_{4}}\gt {U_{3}}\gt {U_{1}}\gt {U_{2}}$, | ${U_{4}}$ |
| The method in Liang et al. (2017) | ${U_{4}}\gt {U_{3}}\gt {U_{2}}\gt {U_{1}}$, | ${U_{4}}$ |
| Proposed (GTHFNPA)Method | ${U_{4}}\gt {U_{3}}\gt {U_{2}}\gt {U_{1}}$, | ${U_{4}}$ |
| Proposed (GTHFNPG) Method | ${U_{4}}\gt {U_{3}}\gt {U_{2}}\gt {U_{1}}$, | ${U_{4}}$ |
7 Conclusion
GTHF-numbers are very useful for expressing ill-known quantities. Since aggregation operators play a vital role in decision-making, this paper investigates the prioritized MADM problems in which the attribute values are in the form of GTHF-numbers. Firstly, we introduced two aggregation techniques called generalized trapezoidal hesitant fuzzy prioritized weighted average operator and generalized trapezoidal hesitant fuzzy prioritized weighted geometric operator for aggregating the generalized trapezoidal hesitant fuzzy information. Next, we discussed some basic properties of the developed operators, namely idempotency, boundary and monotonicity. In addition, two approaches for multiple-attribute decision-making under the generalized trapezoidal hesitant fuzzy environments are developed. Finally, a practical study has been conducted to demonstrate the proposed MADM method in more practicality and effectiveness, since it considers prioritization relationships among attributes. Meanwhile, the prioritized operators of GTHF-numbers provide a new tool of information fusion for solving decision problems under hesitant environments. Further, the extensions of hesitant prioritized aggregation operators model to the MADM problems under other fuzzy environments would also be studied in the near future. In future, we plan to extend our research work to TOPSIS, ARAS, ELECTRE, WASPAS, MABAC, EDAS, QUALIFLEX, and so on. Obtaining the data and writing in this study as GTHF-numbers is the most common problem. Therefore, we will try to solve this problem in future.