Group Decision-Making Method Under Hesitant Interval Neutrosophic Uncertain Linguistic Environment

Abstract

Motivated by the idea of interval neutrosophic uncertain linguistic sets and hesitant fuzzy sets, this paper proposed the concept of hesitant interval neutrosophic uncertain linguistic sets (HINULSs) and hesitant interval neutrosophic uncertain linguistic elements (HINULEs) and then proposed some basic operational rules, properties, the score, accuracy, and certainty functions for HINULEs. Further, based on these operational rules, we defined some aggregation operators, such as hesitant interval neutrosophic uncertain linguistic (HINUL) prioritized weighted averaging average (HINULPWA) operator, HINUL prioritized weighted geometric (HINULPWG) operator, and generalized HINUL prioritized weighted aggregation (GHINULPWA) operator to aggregate HINULSs, and some desired properties of these operators are investigated. A group decision-making method based on GHINULPWA operator is developed to handle multiple criteria group decision-making problems, in which criteria values take the form of HINULEs and there exist prioritized relations among the criteria. Finally, a numerical example about investment selections is given to show the advantages of the proposed method.

Appendices

Appendix 1: Proof of Theorem 2

We prove Eq. (35) by the use of mathematical induction.

• (1) For $m = 2$, we have

  $$\begin{aligned} \frac{{T_{1} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}{\mathfrak{N}}_{1} = \bigcup\nolimits_{{u_{1} \in {\mathfrak{N}}_{1} }} {\left\langle {\left[ {f^{* - 1} \left( {\frac{{T_{1} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right)} \right),f^{* - 1} \left( {\frac{{T_{1} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)} \right],\left( {t\left( {u_{1} } \right),i\left( {u_{1} } \right),f\left( {u_{1} } \right)} \right)} \right\rangle } \hfill \\ \frac{{T_{2} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}{\mathfrak{N}}_{2} = \bigcup\nolimits_{{u_{2} \in {\mathfrak{N}}_{2} }} {\left\langle {\left[ {f^{* - 1} \left( {\frac{{T_{2} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right)} \right),f^{* - 1} \left( {\frac{{T_{2} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)} \right],\left( {t\left( {u_{2} } \right),i\left( {u_{2} } \right),f\left( {u_{2} } \right)} \right)} \right\rangle } \hfill \\ \end{aligned}$$

  Then

  $$\begin{aligned} {\rm HINULPWA}\left( {{\mathfrak{N}}_{1} ,{\mathfrak{N}}_{2} } \right) = & \frac{{T_{1} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}{\mathfrak{N}}_{1} \oplus \frac{{T_{2} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}{\mathfrak{N}}_{2} \\ & = \bigcup\limits_{{u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} }} \begin{aligned} \left\langle {\left[ {f^{* - 1} \left( {\frac{{T_{1} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + \frac{{T_{2} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right)} \right),f^{* - 1} \left( {\frac{{T_{1} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{1} )}} } \right) + \frac{{T_{2} }}{{\sum\nolimits_{j = 1}^{2} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)} \right]} \right., \hfill \\ \left( {\left[ \begin{aligned} \frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} t_{1}^{L} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} t_{2}^{L} }}{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} }}, \hfill \\ \frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} t_{1}^{U} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} t_{2}^{U} }}{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} }} \hfill \\ \end{aligned} \right]} \right., \hfill \\ \left[ \begin{aligned} \frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} i_{1}^{L} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} i_{2}^{L} }}{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} }}, \hfill \\ \frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} i_{1}^{U} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} i_{2}^{U} }}{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} }} \hfill \\ \end{aligned} \right], \hfill \\ \left. {\left. {\left[ \begin{aligned} \frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} f_{1}^{L} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} f_{2}^{L} }}{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} }}, \hfill \\ \frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} f_{1}^{U} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} f_{2}^{U} }}{{\left( {f^{*} \left( {s_{{\vartheta (u_{1} )}} } \right) + f^{*} \left( {s_{{\tau (u_{1} )}} } \right)} \right)T_{1} + \left( {f^{*} \left( {s_{{\vartheta (u_{2} )}} } \right) + f^{*} \left( {s_{{\tau (u_{2} )}} } \right)} \right)T_{2} }} \hfill \\ \end{aligned} \right]} \right)} \right\rangle \hfill \\ \end{aligned} \\ \end{aligned}$$

• (2) Let us assume that Eq. (35) is true for $m = b$, then

  $$\begin{aligned} {\text{HINULPWA}}\left( {{\mathfrak{N}}_{1} ,{\mathfrak{N}}_{2} , \ldots ,{\mathfrak{N}}_{b} } \right) = & \frac{{T_{1} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}{\mathfrak{N}}_{1} \oplus \frac{{T_{2} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}{\mathfrak{N}}_{2} \oplus \cdots \oplus \frac{{T_{b} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}{\mathfrak{N}}_{b} \\ & = \bigcup\limits_{{u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{b} \in {\mathfrak{N}}_{b} }} {\left\langle {\left[ {f^{* - 1} \left( {\sum\limits_{v = 1}^{b} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {\sum\limits_{v = 1}^{b} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)} } \right)} \right]} \right.} \\ \left( {\left[ {\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{L} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{U} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right., \\ \left[ {\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{L} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{U} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right], \\ \left. {\left. {\left[ {\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{L} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{U} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right)} \right\rangle \\ \end{aligned}$$

• (3) When $m = b + 1,$ then

  $$\begin{aligned} {\text{HINULPWA}}\left( {{\mathfrak{N}}_{1} ,{\mathfrak{N}}_{2} , \ldots ,{\mathfrak{N}}_{b + 1} } \right) = & \frac{{T_{1} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}{\mathfrak{N}}_{1} \oplus \frac{{T_{2} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}{\mathfrak{N}}_{2} \oplus \cdots \oplus \frac{{T_{b} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}{\mathfrak{N}}_{b} \oplus \frac{{T_{b + 1} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}{\mathfrak{N}}_{b + 1} \\ & = \bigcup\limits_{{u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{b} \in {\mathfrak{N}}_{b} }} {\left\langle \begin{aligned} \left[ {f^{* - 1} \left( {\sum\limits_{v = 1}^{b} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {\sum\limits_{v = 1}^{b} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)} } \right)} \right] \hfill \\ \left\{ \begin{aligned} \left[ {\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{L} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{U} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right], \hfill \\ \left[ {\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{L} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{U} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right], \hfill \\ \left[ {\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{L} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{U} }}{{\sum\nolimits_{v = 1}^{b} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right] \hfill \\ \end{aligned} \right\} \hfill \\ \end{aligned} \right\rangle } \\ \end{aligned}$$
  $$\oplus \bigcup\limits_{{u_{b + 1} \in {\mathfrak{N}}_{b + 1} }} {\left\langle \begin{aligned} f^{* - 1} \left( {\left[ {\frac{{T_{b + 1} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{b + 1} )}} } \right),\frac{{T_{b + 1} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{b + 1} )}} } \right)} \right]} \right), \hfill \\ \left[ {t_{b + 1}^{L} ,t_{b + 1}^{U} } \right],\left[ {i_{b + 1}^{L} ,i_{b + 1}^{U} } \right],\left[ {f_{b + 1}^{L} ,f_{b + 1}^{U} } \right] \hfill \\ \end{aligned} \right\rangle }$$
  $$= \bigcup\limits_{{u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{b + 1} \in {\mathfrak{N}}_{b + 1} }} \begin{aligned} \left\langle {\left[ {f^{* - 1} \left( {\sum\limits_{v = 1}^{b + 1} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {\sum\limits_{v = 1}^{b + 1} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)} } \right)} \right]} \right. \hfill \\ \left( {\left[ {\frac{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right., \hfill \\ \left[ {\frac{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right], \hfill \\ \left. {\left. {\left[ {\frac{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{b + 1} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right)} \right\rangle \hfill \\ \end{aligned}$$

Hence Eq. (35) is true for all $m.$

According to (1)–(3), Eq. (35) is kept.

Appendix 2: Proof of Theorem 3

According to the operational laws defined for HINULNs, we have

$$\zeta {\mathfrak{N}}_{v} = \bigcup\limits_{{u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{m} \in {\mathfrak{N}}_{m} }} {\left\langle \begin{aligned} \left[ {f^{* - 1} \left( {\zeta \left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right)} \right),f^{* - 1} \left( {\zeta \left( {f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)} \right)} \right] \hfill \\ \left( {\left[ {\left( {t_{v}^{L} } \right),\left( {t_{v}^{U} } \right)} \right],\left[ {\left( {i_{v}^{L} } \right),\left( {i_{v}^{U} } \right)} \right],\left[ {\left( {f_{v}^{L} } \right),\left( {f_{v}^{U} } \right)} \right]} \right) \hfill \\ \end{aligned} \right\rangle }$$

According to Theorem (2), we have

$$\begin{aligned} {\text{HINULPWA}}\left( {a{\mathfrak{N}}_{1} ,a{\mathfrak{N}}_{2} , \ldots ,a{\mathfrak{N}}_{m} } \right) \hfill \\ = \bigcup\limits_{{u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{m} \in {\mathfrak{N}}_{m} }} \begin{aligned} \left\langle {\left[ {f^{* - 1} \left( {\sum\nolimits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}af^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {\sum\nolimits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}af^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)} } \right)} \right]} \right. \hfill \\ \left( {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right],} \right. \hfill \\ \left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right], \hfill \\ \left. {\left. {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right)} \right\rangle \hfill \\ \end{aligned} \hfill \\ \end{aligned}$$

$$\begin{aligned} = \bigcup\limits_{{u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{m} \in {\mathfrak{N}}_{m} }} \begin{aligned} \left\langle {\left[ {f^{* - 1} \left( {a\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {a\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)} } \right)} \right]} \right. \hfill \\ \left( {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right., \hfill \\ \end{aligned} \hfill \\ \left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right], \hfill \\ \left. {\left. {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right)} \right\rangle \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{aHINULPWA}}\left( {{\mathfrak{N}}_{1} ,{\mathfrak{N}}_{2} , \ldots ,{\mathfrak{N}}_{m} } \right) = & \left( a \right).\bigcup\limits_{{u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{m} \in {\mathfrak{N}}_{m} }} \begin{aligned} \left\langle {\left[ {f^{* - 1} \left( {\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)} } \right)} \right]} \right. \hfill \\ \left( {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right],} \right. \hfill \\ \end{aligned} \\ \left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right], \\ \left. {\left. {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right)} \right\rangle \\ \end{aligned}$$

$$= \bigcup\limits_{{u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{m} \in {\mathfrak{N}}_{m} }} \begin{aligned} \left\langle {\left[ {f^{* - 1} \left( {a\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {a\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)} } \right)} \right]} \right. \hfill \\ \left( {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right., \hfill \\ \left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right], \hfill \\ \left. {\left. {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right)} \right\rangle \hfill \\ \end{aligned}$$

Hence, ${\text{HINULPWA}}\left( {a{\mathfrak{N}}_{1} ,a{\mathfrak{N}}_{2} , \ldots ,a{\mathfrak{N}}_{m} } \right) = {\text{aHINULPWA}}\left( {{\mathfrak{N}}_{1} ,{\mathfrak{N}}_{2} , \ldots ,{\mathfrak{N}}_{m} } \right).$

Appendix 3: Proof of Theorem 4

According to Definition (7)

$$\begin{aligned} {\text{HINULPWA}}\left( {{\mathfrak{N}}_{1} \oplus \rho_{1} ,{\mathfrak{N}}_{2} \oplus \rho_{2} , \ldots ,{\mathfrak{N}}_{m} \oplus \rho_{m} } \right) \hfill \\ = \bigcup\limits_{\begin{subarray}{l} u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{m} \in {\mathfrak{N}}_{m} , \\ k_{1} \in \rho_{1} ,k_{2} \in \rho_{2} , \ldots ,k_{m} \in \rho_{m} , \end{subarray} } \begin{aligned} \left\langle {\left[ {f^{* - 1} \left( {\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right) + \left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)} } \right)} \right]} \right. \hfill \\ \left( {\left[ \begin{aligned} \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)t_{v}^{L} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)t_{v}^{L} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }}, \hfill \\ \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)t_{v}^{U} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)t_{v}^{U} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }} \hfill \\ \end{aligned} \right]} \right., \hfill \\ \end{aligned} , \hfill \\ \left[ \begin{aligned} \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)i_{v}^{L} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)i_{v}^{L} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }}, \hfill \\ \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)i_{v}^{U} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)i_{v}^{U} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }} \hfill \\ \end{aligned} \right], \hfill \\ \left. {\left. {\left[ \begin{aligned} \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)f_{v}^{L} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)f_{v}^{L} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }}, \hfill \\ \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)f_{v}^{U} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)f_{v}^{U} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }} \hfill \\ \end{aligned} \right]} \right)} \right\rangle \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{HINULPWA}}\left( {{\mathfrak{N}}_{1} ,{\mathfrak{N}}_{2} , \ldots ,{\mathfrak{N}}_{m} } \right) \oplus {\text{HINULPWA}}\left( {\rho_{1} ,\rho_{2} , \ldots ,\rho_{m} } \right) \hfill \\ = \bigcup\limits_{{u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{m} \in {\mathfrak{N}}_{m} }} \begin{aligned} \left\langle {\left[ {f^{* - 1} \left( {\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)} } \right)} \right]} \right. \hfill \\ \left( {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} t_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right., \hfill \\ \left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} i_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right], \hfill \\ \left. {\left. {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} f_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right)} \right\rangle \hfill \\ \end{aligned} \hfill \\ \end{aligned}$$

$$\begin{aligned} \oplus \bigcup\limits_{{k_{1} \in \rho_{1} ,k_{2} \in \rho_{2} , \ldots ,k_{m} \in \rho_{m} }} \begin{aligned} \left\langle {\left[ {f^{* - 1} \left( {\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)} } \right)} \right]} \right. \hfill \\ \left( {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} t_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} t_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right., \hfill \\ \left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} i_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} i_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }}} \right], \hfill \\ \end{aligned} \hfill \\ \left. {\left. {\left[ {\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} f_{v}^{L} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }},\frac{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} f_{v}^{U} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }}} \right]} \right)} \right\rangle \hfill \\ \end{aligned}$$

$$\begin{aligned} = \bigcup\limits_{\begin{subarray}{l} u_{1} \in {\mathfrak{N}}_{1} ,u_{2} \in {\mathfrak{N}}_{2} , \ldots ,u_{m} \in {\mathfrak{N}}_{m} , \\ k_{1} \in \rho_{1} ,k_{2} \in \rho_{2} , \ldots ,k_{m} \in \rho_{m} , \end{subarray} } \begin{aligned} \left\langle {\left[ {f^{* - 1} \left( {\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right)} \right) + \left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right)} \right)} } \right),f^{* - 1} \left( {\sum\limits_{v = 1}^{m} {\left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {\frac{{T_{v} }}{{\sum\nolimits_{j = 1}^{m} {T_{j} } }}f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)} } \right)} \right]} \right. \hfill \\ \left( {\left[ \begin{aligned} \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)t_{v}^{L} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)t_{v}^{L} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }}, \hfill \\ \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)t_{v}^{U} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)t_{v}^{U} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }} \hfill \\ \end{aligned} \right]} \right., \hfill \\ \end{aligned} , \hfill \\ \left[ \begin{aligned} \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)i_{v}^{L} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)i_{v}^{L} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }}, \hfill \\ \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)i_{v}^{U} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)i_{v}^{U} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }} \hfill \\ \end{aligned} \right], \hfill \\ \left. {\left. {\left[ \begin{aligned} \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)f_{v}^{L} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)f_{v}^{L} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }}, \hfill \\ \frac{{\sum\nolimits_{v = 1}^{m} {\left( {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right)f_{v}^{U} + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)f_{v}^{U} } \right)T_{v} } }}{{\sum\nolimits_{v = 1}^{m} {\left( {f^{*} \left( {s_{{\vartheta (u_{v} )}} } \right) + f^{*} \left( {s_{{\tau (u_{v} )}} } \right)} \right) + \left( {f^{*} \left( {s_{{\vartheta (k_{v} )}} } \right) + f^{*} \left( {s_{{\tau (k_{v} )}} } \right)} \right)T_{v} } }} \hfill \\ \end{aligned} \right]} \right)} \right\rangle \hfill \\ \end{aligned}$$

Thus, ${\text{HINULPWA}}\left( {{\mathfrak{N}}_{1} \oplus \rho_{1} ,{\mathfrak{N}}_{2} \oplus \rho_{2} , \ldots ,{\mathfrak{N}}_{m} \oplus \rho_{m} } \right) = {\text{HINULPWA}}\left( {{\mathfrak{N}}_{1} ,{\mathfrak{N}}_{2} , \ldots ,{\mathfrak{N}}_{m} } \right) \oplus {\text{HINULPWA}}\left( {\rho_{1} ,\rho_{2} , \ldots ,\rho_{m} } \right).$