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Normal Wiggly Probabilistic Hesitant Fuzzy Set and Its Application in Battlefield Threat Assessment

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Abstract

For decision-making in a highly uncertain environment, experts may be unable to accurately express or even neglect some deep latent uncertain information, leading to decision bias. This study extends the normal wiggly hesitant fuzzy set (NWHFS) to propose a novel fuzzy set, the normal wiggly probabilistic hesitant fuzzy set (NWPHFS), with the aim of further mining the probability hesitation fuzzy information hidden in expert assessment and improving the limitation pertaining to the inability of NWHFS in containing membership. To establish the theoretical basis for operationalizing NWPHFS, we put forward a preliminary theoretical and methodological framework of NWPHFS, which formulates the concept of cut set and decomposition theorem of NWPHFS, defines a score function of NWPHFS, and clarifies the basic properties and operations of NWPHFS. Finally, two aggregation operators of NWPHFS are proposed. The proposed NWPHFS method can comprehensively consider the subjective preference of membership importance and the potential preference of membership itself in expert assessment so as to mine the real preference of experts entirely, thus enhancing the rationality of decision-making. Furthermore, to demonstrate its application and superiority, NWPHFS is applied to multi-criteria battlefield threat assessment in a hypothetical war scenario, and compared with other types of fuzzy sets. Our study shows that NWPHFS can more comprehensively excavate and utilize the deep uncertainty information in expert assessment, and has great potential in applying to practical decision-making scenarios, such as battlefield threat assessment, a complex military decision-making problem heavily dependent on expert judgment.

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Abbreviations

HFS:

Hesitant fuzzy set

PHFS:

Probabilistic hesitant fuzzy set

NWHFS:

Normal wiggly hesitant fuzzy set

NWPHFS:

Normal wiggly probabilistic hesitant fuzzy set

NWPHFE:

Normal wiggly probabilistic hesitant fuzzy element

GPWAA:

General normal wiggly probabilistic hesitant fuzzy weighted arithmetic average operator

GPWGA:

General normal wiggly probabilistic hesitant fuzzy weighted geometric average operator

WAA:

Normal wiggly hesitant fuzzy weighted arithmetic average operator

WGA:

Normal wiggly hesitant fuzzy weighted geometric average operator

PHFWA:

Probabilistic hesitant fuzzy weighted averaging operator

PHFWG:

Probabilistic hesitant fuzzy weighted geometric operator

HFWA:

Hesitant fuzzy weighted averaging operator

HFWG:

Hesitant fuzzy weighted geometric operator

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Author information

Authors and Affiliations

  1. School of Economics and Management, Beihang University, Beijing, 100191, China

    Jingyang Xia, Mengqi Chen & Weiguo Fang

  2. Key Laboratory of Complex System Analysis, Management and Decision (Beihang University), Ministry of Education, Beijing, 100191, China

    Weiguo Fang

Authors
  1. Jingyang Xia
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  2. Mengqi Chen
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  3. Weiguo Fang
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Corresponding author

Correspondence to Weiguo Fang.

Appendix

Appendix

Proof of corollary 1:

$$\begin{gathered} nwh_{1} (p) \oplus nwh_{2} (p) \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{1} (p)}} {\left[ {r_{{1_{l} }} + r_{{2_{k} }} - r_{{1_{l} }} r_{{2_{k} }} } \right](p_{{1_{l} }} p_{{2_{k} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \oplus\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} } \right]} } (p_{{1_{l} }} p_{{2_{k} }} )} \right\rangle \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{1} (p)}} {\left[ {r_{{2_{k} }} + r_{{1_{l} }} - r_{{2_{k} }} r_{{1_{l} }} } \right](p_{{2_{k} }} p_{{1_{l} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \oplus\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} } \right]} } (p_{{2_{k} }} p_{{1_{l} }} )} \right\rangle \hfill \\ = nwh_{2} (p) \oplus nwh_{1} (p) \hfill \\ \end{gathered}$$
$$\begin{gathered} (nwh_{1} (p) \oplus nwh_{2} (p)) \oplus nwh_{3} (p) \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{2} (p)}} {\left[ {r_{{1_{l} }} + r_{{2_{k} }} - r_{{1_{l} }} r_{{2_{k} }} } \right](p_{{1_{l} }} p_{{2_{k} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)),\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \oplus\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} } \right]} } (p_{{1_{l} }} p_{{2_{k} }} )} \right\rangle \oplus nwh_{3} (p) \hfill \\ = \left\langle \begin{gathered} \bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{2} (p), \, r_{{3_{m} }} \in h_{3} (p)}} {\left[ {r_{{1_{l} }} + r_{{2_{k} }} - r_{{1_{l} }} r_{{2_{k} }} + r_{{3_{m} }} - r_{{3_{m} }} (r_{{1_{l} }} + r_{{2_{k} }} - r_{{1_{l} }} r_{{2_{k} }} )} \right](p_{{1_{l} }} p_{{2_{k} }} p_{{3_{m} }} ), \, } \hfill \\ \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)),\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{3_{m} }} \in \eta (h_{3} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \oplus\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \oplus\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{3_{m} }} } \right]} (p_{{1_{l} }} p_{{2_{k} }} p_{{3_{m} }} ) \hfill \\ \end{gathered} \right\rangle \hfill \\ = \left\langle \begin{gathered} \bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{2} (p), \, r_{{3_{m} }} \in h_{3} (p)}} {\left[ {r_{{2_{k} }} + r_{{3_{m} }} - r_{{2_{k} }} r_{{3_{m} }} + r_{{1_{l} }} - r_{{1_{l} }} (r_{{2_{k} }} + r_{{3_{m} }} - r_{{2_{k} }} r_{{3_{m} }} )} \right](p_{{2_{k} }} p_{{3_{m} }} p_{{1_{l} }} ), \, } \hfill \\ \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{3_{m} }} \in \eta (h_{3} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \oplus\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \oplus\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{3_{m} }} } \right]} (p_{{1_{l} }} p_{{2_{k} }} p_{{3_{m} }} ) \hfill \\ \end{gathered} \right\rangle \hfill \\ = \left\langle {\bigcup\limits_{{r_{{2_{k} }} \in h_{2} (p), \, r_{{3_{m} }} \in h_{3} (p)}} {\left[ {r_{{2_{k} }} + r_{{3_{m} }} - r_{{2_{k} }} r_{{3_{m} }} } \right](p_{{2_{k} }} p_{{3_{m} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{3_{m} }} \in \eta (h_{3} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \oplus\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{3_{m} }} } \right]} } (p_{{2_{k} }} p_{{3_{m} }} )} \right\rangle \oplus nwh_{1} (p) \hfill \\ = (nwh_{2} (p) \oplus nwh_{3} (p)) \oplus nwh_{1} (p) = nwh_{1} (p) \oplus (nwh_{2} (p) \oplus nwh_{3} (p)) \hfill \\ \end{gathered}$$
$$\begin{gathered} \lambda (nwh_{1} (p) \oplus nwh_{2} (p)) \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{1} (p)}} {\left[ {1 - (1 - (1 - r_{{1_{l} }} )(1 - r_{{2_{k} }} ))^{\lambda } } \right](p_{{1_{l} }} p_{{2_{k} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\lambda \left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \oplus\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} } \right]} } (p_{{1_{l} }} p_{{2_{k} }} )} \right\rangle \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{1} (p)}} {\left[ {1 - (1 - r_{{1_{l} }} )^{\lambda } (1 - r_{{2_{k} }} )^{\lambda } } \right](p_{{1_{l} }} p_{{2_{k} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)),\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\lambda \left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \oplus\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} } \right]} } (p_{{1_{l} }} p_{{2_{k} }} )} \right\rangle \hfill \\ = \left\langle \begin{gathered} \bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{1} (p)}} {\left[ {1 - (1 - r_{{1_{l} }} )^{\lambda } + 1 - (1 - r_{{2_{k} }} )^{\lambda } - (1 - (1 - r_{{1_{l} }} )^{\lambda } )(1 - (1 - r_{{2_{k} }} )^{\lambda } )} \right](p_{{1_{l} }} p_{{2_{k} }} ), \, } \hfill \\ \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\left[ {\lambda\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \oplus \lambda\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} } \right]} (p_{{1_{l} }} p_{{2_{k} }} ) \hfill \\ \end{gathered} \right\rangle \hfill \\ = \lambda nwh_{2} (p) \oplus \lambda nwh_{1} (p) \hfill \\ \end{gathered}$$
$$\begin{gathered} nwh_{1} (p) \otimes nwh_{2} (p) \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{1} (p)}} {\left[ {r_{{1_{l} }} r_{{2_{k} }} } \right](p_{{1_{l} }} p_{{2_{k} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)),\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \otimes\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} } \right]} } (p_{{1_{l} }} p_{{2_{k} }} )} \right\rangle \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{1} (p)}} {\left[ {r_{{2_{k} }} r_{{1_{l} }} } \right](p_{{2_{k} }} p_{{1_{l} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)),\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \otimes\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} } \right]} } (p_{{2_{k} }} p_{{1_{l} }} )} \right\rangle \hfill \\ = nwh_{2} (p) \otimes nwh_{1} (p) \hfill \\ \end{gathered}$$
$$\begin{gathered} (nwh_{1} (p) \otimes nwh_{2} (p)) \otimes nwh_{3} (p) \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{2} (p)}} {\left[ {r_{{1_{l} }} r_{{2_{k} }} } \right](p_{{1_{l} }} p_{{2_{k} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \otimes\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} } \right]} } (p_{{1_{l} }} p_{{2_{k} }} )} \right\rangle \otimes nwh_{3} (p) \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{2} (p), \, r_{{3_{m} }} \in h_{3} (p)}} {\left[ {r_{{1_{l} }} r_{{2_{k} }} r_{{3_{m} }} } \right](p_{{1_{l} }} p_{{2_{k} }} p_{{3_{m} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{3_{m} }} \in \eta (h_{3} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \otimes\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \otimes\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{3_{m} }} } \right]} } (p_{{1_{l} }} p_{{2_{k} }} p_{{3_{m} }} )} \right\rangle \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{2} (p), \, r_{{3_{m} }} \in h_{3} (p)}} {\left[ {r_{{2_{k} }} r_{{3_{m} }} r_{{1_{l} }} } \right](p_{{2_{k} }} p_{{3_{m} }} p_{{1_{l} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p)), \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{3_{m} }} \in \eta (h_{3} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \otimes\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{3_{m} }} \otimes\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} } \right]} } (p_{{2_{k} }} p_{{3_{m} }} p_{{1_{l} }} )} \right\rangle \hfill \\ = (nwh_{2} (p) \otimes nwh_{3} (p)) \otimes nwh_{1} (p) = nwh_{1} (p) \otimes (nwh_{2} (p) \otimes nwh_{3} (p)) \hfill \\ \end{gathered}$$
$$\begin{gathered} (nwh_{1} (p) \otimes nwh_{2} (p))^{\lambda } \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p), \, r_{{2_{k} }} \in h_{1} (p)}} {\left[ {r_{{1_{l} }} r_{{2_{k} }} } \right]^{\lambda } (p_{{1_{l} }} p_{{2_{k} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p)),\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \otimes\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} } \right]^{\lambda } } } (p_{{1_{l} }} p_{{2_{k} }} )} \right\rangle \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p)}} {\left[ {r_{{1_{l} }} } \right]^{\lambda } (p_{{1_{l} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} } \right]^{\lambda } } } (p_{{1_{l} }} )} \right\rangle \otimes \left\langle {\bigcup\limits_{{r_{{2_{k} }} \in h_{1} (p)}} {\left[ {r_{{2_{k} }} } \right]^{\lambda } (p_{{2_{k} }} ), \, \bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} \in \eta (h_{2} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{2_{k} }} } \right]^{\lambda } } } (p_{{2_{k} }} )} \right\rangle \hfill \\ = nwh_{1} (p)^{\lambda } \otimes nwh_{2} (p)^{\lambda } \hfill \\ \end{gathered}$$
$$\begin{gathered} (nwh_{1} (p)^{{\lambda_{1} }} )^{{\lambda_{2} }} \hfill \\ = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p)}} {\left[ {r_{{1_{l} }}^{{\lambda_{1} }} } \right]^{{\lambda_{2} }} (p_{{1_{l} }} ),\bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }}^{{\lambda_{1} }} } \right]^{{\lambda_{2} }} } } (p_{{1_{l} }} )} \right\rangle = \left\langle {\bigcup\limits_{{r_{{1_{l} }} \in h_{1} (p)}} {\left[ {r_{{1_{l} }}^{{\lambda_{1} \lambda_{2} }} } \right](p_{{1_{l} }} ),\bigcup\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }} \in \eta (h_{1} (p))}} {\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r}_{{1_{l} }}^{{\lambda_{1} \lambda_{2} }} } \right]} } (p_{{1_{l} }} )} \right\rangle \hfill \\ = nwh_{1} (p)^{{\lambda_{1} \lambda_{2} }} \hfill \\ \end{gathered}$$

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Xia, J., Chen, M. & Fang, W. Normal Wiggly Probabilistic Hesitant Fuzzy Set and Its Application in Battlefield Threat Assessment. Int. J. Fuzzy Syst. 25, 145–167 (2023). https://doi.org/10.1007/s40815-022-01371-3

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