Skip to main content

Advertisement

Springer Nature Link
Log in

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

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

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

References

  1. Zadeh, L.A.: Fuzzy sets. Inf Control 8(3), 338–356 (1965)

    MATH  Google Scholar 

  2. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. 8(3), 199–249 (1975)

    MathSciNet  MATH  Google Scholar 

  3. Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25(6), 529–539 (2010)

    MATH  Google Scholar 

  4. Zhu, B.: Decision Method for Research and Application Based on Preference Relation. Southeast University, Nanjing (2014)

    Google Scholar 

  5. Pang, Q., Wang, H., Xu, Z.S.: Probabilistic linguistic linguistic term sets in multi-attribute group decision making. Inf. Sci. 369, 128–143 (2016)

    Google Scholar 

  6. Ren, Z.L., Xu, Z.S., Wang, H.: Normal wiggly hesitant fuzzy sets and their application to environmental quality evaluation. Knowl.-Based Syst. 159, 286–297 (2018)

    Google Scholar 

  7. Liu, P.D., Xu, H.X., Geng, Y.S.: Normal wiggly hesitant fuzzy linguistic power Hamy mean aggregation operators and their application to multi-attribute decision-making. Comput. Ind. Eng. 140, 1–21 (2020)

    Google Scholar 

  8. Narayanamoorthy, S., Ramya, L., Baleanu, D., Kureethara, J.V., Annapoorani, V.: Application of normal wiggly dual hesitant fuzzy sets to site selection for hydrogen underground storage. Int. J. Hydrog. Energy 44(54), 28874–28892 (2019)

    Google Scholar 

  9. Liu, P.D., Zhang, P.: A normal wiggly hesitant fuzzy MABAC method based on CCSD and prospect theory for multiple attribute decision making. Int. J. Intell. Syst. 36(1), 447–477 (2021)

    Google Scholar 

  10. Ramya, L., Narayanamoorthy, S., Kalaiselvan, S., Kureethara, J.V., Annapoorani, V., Kang, D.: A congruent approach to normal wiggly interval-valued hesitant pythagorean fuzzy set for thermal energy storage technique selection applications. Int. J. Fuzzy Syst. 23(6), 1581–1599 (2021)

    Google Scholar 

  11. Zindani, D., Maity, S.R., Bhowmik, S.: Extended TODIM method based on normal wiggly hesitant fuzzy sets for deducing optimal reinforcement condition of agro-waste fibers for green product development. J. Clean. Prod. 301, 1–22 (2021)

    Google Scholar 

  12. Xia, J.Y., Pi, Z.Y., Fang, W.G.: Predicting war outcomes based on a fuzzy influence diagram. Int. J. Fuzzy Syst. 23(4), 984–1002 (2021)

    Google Scholar 

  13. Maeng, J.L., Malone, M., Cornell, D.: Student threats of violence against teachers: prevalence and outcomes using a threat assessment approach. Teach. Teach. Educ. 87, 1–11 (2020)

    Google Scholar 

  14. Ma, S.D., Zhang, H.Z., Yang, G.Q.: Target threat level assessment based on cloud model under fuzzy and uncertain conditions in air combat simulation. Aerosp. Sci. Technol. 67, 49–53 (2017)

    Google Scholar 

  15. Lee, H., Choi, B.J., Kim, C.O., Kim, J.S., Kim, J.E.: Threat evaluation of enemy air fighters via neural network-based Markov chain modeling. Knowl.-Based Syst. 116, 49–57 (2017)

    Google Scholar 

  16. Carling, R.L.: Naval situation assessment using a real-time knowledge-based system. Nav. Eng. J. 111(3), 108–113 (2010)

    Google Scholar 

  17. Xu, Y.J., Wang, Y.C., Miu, X.D.: Multi-attribute decision making method for air target threat evaluation based on intuitionistic fuzzy sets. J. Syst. Eng. Electron. 23(6), 891–897 (2012)

    Google Scholar 

  18. Wang, Y., Liu, S.Y., Niu, W., Liu, K., Liao, Y.: Threat assessment method based on intuitionistic fuzzy similarity measurement reasoning with orientation. China Commun. 11(6), 119–128 (2014)

    Google Scholar 

  19. Zhang, K., Kong, W.R., Liu, P.P., Shi, J., Lei, Y., Zou, J.: Assessment and sequencing of air target threat based on intuitionistic fuzzy entropy and dynamic VIKOR. J. Syst. Eng. Electron. 29(2), 305–310 (2018)

    Google Scholar 

  20. Kong, D.P., Chang, T.Q., Wang, Q.D., Sun, H.Z., Dai, W.J.: A threat assessment method of group targets based on interval-valued intuitionistic fuzzy multi-attribute group decision-making. Appl. Soft Comput. 67, 350–369 (2018)

    Google Scholar 

  21. Lu, Y., Lei, X., Zhou, Z., Song, Y.: Approximate reasoning based on IFRS and DS theory with its application in threat assessment. IEEE Access. 8, 160558–160568 (2020)

    Google Scholar 

  22. Gao, Y., Li, D.S., Zhong, H.: A novel target threat assessment method based on three-way decisions under intuitionistic fuzzy multi-attribute decision making environment. Eng. Appl. Artif. Intell. 87, 1–8 (2020)

    Google Scholar 

  23. Goguen, J.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–147 (1967)

    MathSciNet  MATH  Google Scholar 

  24. Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Kluwer Academic, New York (1980)

    MATH  Google Scholar 

  25. Turksen, I.B.: Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst. 20(2), 191–210 (1986)

    MathSciNet  MATH  Google Scholar 

  26. Zeng, W.Y., Guo, P.: Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship. Inf. Sci. 178(5), 1334–1342 (2008)

    MathSciNet  MATH  Google Scholar 

  27. Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)

    MATH  Google Scholar 

  28. Guo, K.H.: Knowledge measure for atanassov’s intuitionistic fuzzy sets. IEEE Trans. Fuzzy Syst. 24(5), 1072–1078 (2016)

    MathSciNet  Google Scholar 

  29. Atanassov, K., Gargov, G.: Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31, 343–349 (1989)

    MathSciNet  MATH  Google Scholar 

  30. Zhu, B., Xu, Z., Xia, M.: Dual hesitant fuzzy sets. J. Appl. Math. 2012, 2607–2645 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Zhu, B., Xu, Z.S.: Probability-hesitant fuzzy sets and the representation of preference relations. Technol. Econ. Dev. Econ. 24(3), 1029–1040 (2018)

    MathSciNet  Google Scholar 

  32. Xu, Z.S., Zhou, W.: Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optim. Decis. Making 16(4), 481–503 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Ding, J., Xu, Z.S., Zhao, N.: An interactive approach to probabilistic hesitant fuzzy multi-attribute group decision making with incomplete weight information. J. Intell. Fuzzy Syst. 32(3), 2523–2536 (2017)

    MATH  Google Scholar 

  34. Zhou, W., Xu, Z.S.: Expected hesitant VaR for tail decision making under probabilistic hesitant fuzzy environment. Appl. Soft Comput. 60, 297–311 (2017)

    Google Scholar 

  35. Zhang, S., Xu, Z.S., He, Y.: Operations and integrations of probabilistic hesitant fuzzy information in decision making. Inf. Fusion 38, 1–11 (2017)

    Google Scholar 

  36. Li, J., Wang, J.Q.: Multi-criteria outranking methods with hesitant probabilistic fuzzy sets. Cogn. Comput. 9(5), 611–625 (2017)

    Google Scholar 

  37. Song, C.Y., Xu, Z.S., Zhao, H.: New correlation coefficients between probabilistic hesitant fuzzy sets and their applications in cluster analysis. Int. J. Fuzzy Syst. 21(2), 355–368 (2019)

    MathSciNet  Google Scholar 

  38. Hao, Z.N., Xu, Z.S., Zhao, H., Su, Z.: Probabilistic dual hesitant fuzzy set and its application in risk evaluation. Knowl.-Based Syst. 127, 16–28 (2017)

    Google Scholar 

  39. Rodriguez, R.M., Martinez, L., Herrera, F.: Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst. 20(1), 109–119 (2012)

    Google Scholar 

  40. Joshi, D.K., Beg, I., Kumar, S.: Hesitant probabilistic fuzzy linguistic sets with applications in multi-criteria group decision making problems. Mathematics 6(4), 1–20 (2018)

    MATH  Google Scholar 

  41. Zhai, Y.L., Xu, Z.S., Liao, H.C.: Probabilistic linguistic vector-term set and its application in group decision making with multi-granular linguistic information. Appl. Soft Comput. 49, 801–816 (2016)

    Google Scholar 

  42. Farhadinia, B., Herrera-Viedma, E.: Entropy measures for hesitant fuzzy linguistic term sets using the concept of interval-transformed hesitant fuzzy elements. Int. J. Fuzzy Syst. 20(7), 2122–2134 (2018)

    Google Scholar 

  43. Liao, H.C., Qin, R., Gao, C.Y., Wu, X.L., Hafezalkotob, A., Herrera, F.: Score-HeDLiSF: A score function of hesitant fuzzy linguistic term set based on hesitant degrees and linguistic scale functions: An application to unbalanced hesitant fuzzy linguistic MULTIMOORA. Inf. Fusion 48, 39–54 (2019)

    Google Scholar 

  44. Tang, M., Liao, H.C.: Managing information measures for hesitant fuzzy linguistic term sets and their applications in designing clustering algorithms. Inf. Fusion 50, 30–42 (2019)

    Google Scholar 

  45. Singh, A., Beg, I., Kumar, S.: Analytic hierarchy process for hesitant probabilistic fuzzy linguistic set with applications to multi-criteria group decision-making method. Int. J. Fuzzy Syst. 22(5), 1596–1606 (2020)

    Google Scholar 

  46. Yu, D.: Triangular hesitant fuzzy set and its application to teaching quality evaluation. J. Inf. Comput. Sci. 10(7), 1925–1934 (2013)

    Google Scholar 

  47. Yager, R.R.: On ordered weighted averaging aggregation operators in multicriteria decisionmaking. IEEE Trans. Syst. Man Cybern. 18(1), 183–190 (1988)

    MathSciNet  MATH  Google Scholar 

  48. Yuan, X.H., Li, H.X., Sun, K.B.: The cut sets, decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets. Sci. China-Inf. Sci. 54(1), 91–110 (2011)

    MathSciNet  MATH  Google Scholar 

  49. Yuan, X.H., Li, H.X., Lee, E.S.: Three new cut sets of fuzzy sets and new theories of fuzzy sets. Comput. Math. Appl. 57(5), 691–701 (2009)

    MathSciNet  MATH  Google Scholar 

  50. Rotte, R., Schmidt, C.M.: On the production of victory: empirical determinants of battlefield success in modern war. Defence Peace Econ. 14(3), 175–192 (2003)

    Google Scholar 

  51. Dong, J., Wu, G.W., Yang, T.T., Jiang, Z.: Battlefield situation awareness and networking based on agent distributed computing. Phys. Commun. 33, 178–186 (2019)

    Google Scholar 

  52. Brathwaite, K.J.H., Konaev, M.: War in the city: Urban ethnic geography and combat effectiveness. J. Strateg. Stud. 45(1), 33–68 (2022)

    Google Scholar 

  53. Wang, H.T., Song, L.H., Zhang, G.M., Chen, H.: Timetable-aware opportunistic DTN routing for vehicular communications in battlefield environments. Future Gener. Comput. Syst. Int. J. Esci. 83, 95–103 (2018)

    Google Scholar 

  54. Lin, K., Xia, F.Z., Li, C.S., Wang, D., Humar, I.: Emotion-aware system design for the battlefield environment. Inf. Fusion 47, 102–110 (2019)

    Google Scholar 

Download references

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
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Mengqi Chen
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Weiguo Fang
    View author publications

    You can also search for this author inPubMed Google Scholar

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

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-022-01371-3

Keywords