Skip to main content

Advertisement

SpringerLink
Log in

Lexicography minimum solution of fuzzy relation inequalities: applied to optimal control in P2P file sharing system

  • Original Article
  • Published:
International Journal of Machine Learning and Cybernetics Aims and scope Submit manuscript

Abstract

A peer-to-peer (P2P) file sharing system can be reduced into a system of addition-min fuzzy relation inequalities. Concept of lexicography minimum solution is introduced and applied to such system. It is found that the unique lexicography minimum solution can be selected from the minimal solution set of the corresponding fuzzy relation inequalities. However it is difficult to find the minimal solution set which is probably infinite. In order to avoid such difficulty, we propose a so-called Circulation Algorithm to find the unique lexicography minimum solution. The algorithm is developed step by step and illustrated by a numerical application example.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Zadeh LA (1965) Fuzzy sets. Inform. Control 8:338–353

    Article  MATH  Google Scholar 

  2. Wang XZ, Xing HJ, Li Y, Hua Q, Dong CR, Pedrycz W (2015) A study on relationship between generalization abilities and fuzziness of base classifiers in ensemble learning. IEEE Trans Fuzzy Syst 23(5):1638–1654

    Article  Google Scholar 

  3. Wang XZ, Ashfaq RAR, Fu AM (2015) Fuzziness based sample categorization for classifier performance improvement. J Intell Fuzzy Syst 29(3):1185–1196

    Article  MathSciNet  Google Scholar 

  4. Wang XZ (2015) Uncertainty in Learning from Big Data-Editorial. J Intell Fuzzy Syst 28(5):2329–2330

    Article  Google Scholar 

  5. Wang R, Kwong S, Wang XZ, Jiang QS (2015) Segment based decision tree induction with continuous valued attributes. IEEE Trans Cybern 45(7):1262–1275

    Article  Google Scholar 

  6. He YL., Wang XZ, Huang JZX (2016) Fuzzy nonlinear regression analysis using a random weight network. Inf Sci, in press. doi: 10.1016/j.ins.2016.01.037

  7. Sanchez E (1972) Equations de relations floues. Thèse Biologie Humaine, Marseille, France

    Google Scholar 

  8. Sanchez E (1976) Resolution of composite fuzzy relation equations. Inf Control 30:38–48

    Article  MathSciNet  MATH  Google Scholar 

  9. Sanchez E (1977) Solutions in composite fuzzy relation equations: Application to medical diagnosis in Brouwerian logic. In: Gupta MM, Saridis GN, Gaines BR (eds) Fuzzy automata and decision processes. North-Holland, Amsterdam, pp 221–234

    Google Scholar 

  10. Czogala E, Drewniak J, Pedrycz W (1982) Fuzzy relation equations on a finite set. Fuzzy Sets Syst 7:89–101

    Article  MathSciNet  MATH  Google Scholar 

  11. Wang PZ, Sessa S, Di Nola A, Pedrycz W (1984) How many lower solutions does a fuzzy relation equation have? BUSEFAL 18:67–74

    MATH  Google Scholar 

  12. Li X, Ruan D (2000) Novel neural algorithms based on fuzzy \(\delta\) rules for solving fuzzy relation equations: part III. Fuzzy Sets Syst 109:355–362

    Article  MATH  Google Scholar 

  13. Chen L, Wang PP (2002) Fuzzy relation equations (I): the general and specialized solving algorithms. Soft Comput 6:428–435

    Article  MATH  Google Scholar 

  14. Luoh L, Wang W-J, Liaw Y-K (2003) Matrix-pattern-based computer algorithm for solving fuzzy relation equations. IEEE Trans Fuzzy Syst 11(1):100–108

    Article  Google Scholar 

  15. Loetamonphong J, Fang S-C (1999) An efficient solution procedure for fuzzy relation equations with max-product composition. IEEE Trans Fuzzy Syst 7:441–445

    Article  Google Scholar 

  16. Li P, Fang S-C (2011) On the unique solvability of fuzzy relational equations. Fuzzy Decis Mak Optim 10:115–124

    Article  MathSciNet  MATH  Google Scholar 

  17. Peeva K, Kyosev Y (2007) Algorithm for solving max-product fuzzy relational equations. Soft Comput 11(7):593–605

    Article  MATH  Google Scholar 

  18. De Baets B, De Meyer H (2003) On the existence and construction of T-transitive closures. Inf Sci 152:167–179

    Article  MathSciNet  MATH  Google Scholar 

  19. De Baets B (2000) Analytical solution methods for fuzzy relational equations. In: Dubois D, Prade H (Eds.) Fundamentals of fuzzy sets. In: Handb. Fuzzy Sets Ser., vol.1, Kluwer Academic Publishers, pp 291–340

  20. Di Nola A, Sessa S, Pedrycz W, Sanchez E (1989) Fuzzy relation equations and their applications to knowledge engineering. Kluwer Academic Publishers, Boston

    Book  MATH  Google Scholar 

  21. Loia V, Sessa S (2005) Fuzzy relation equations for coding/decoding processes of images and videos. Infor Sci 171:145–172

    Article  MathSciNet  MATH  Google Scholar 

  22. Nobuhara H, Bede B, Hirota K (2006) On various eigen fuzzy sets and their application to image reconstruction. Inf Sci 176:2988–3010

    Article  MathSciNet  MATH  Google Scholar 

  23. Nobuhara H, Pedrycz W, Sessa S, Hirota K (2006) A motion compression/ reconstruction method based on max t-norm composite fuzzy relational equations. Inf Sci 176:2526–2552

    Article  MATH  Google Scholar 

  24. Di Nola A, Russo C (2007) Lukasiewicz transform and its application to compression and reconstruction of digital images. Inf Sci 177:1481–1498

    Article  MATH  Google Scholar 

  25. Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New York

    MATH  Google Scholar 

  26. Dubois D, Prade H (Eds.) (2000) Fundamentals of fuzzy sets, Handb. Fuzzy Sets Ser., vol.1, Kluwer Academic Publishers

  27. Peeva K (2013) Resolution of fuzzy relational equations—method, algorithm and software with applications. Inf Sci 234:44–63

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang PZ, Zhang DZ, Sanchez E, Lee ES (1991) Latticized linear programming and fuzzy relation inequalities. J Math Anal Appl 159(1):72–87

    Article  MathSciNet  MATH  Google Scholar 

  29. Li P, Fang S-C (2009) Latticized linear optimization on the unit interval. IEEE Trans Fuzzy Syst 17:1353–1365

    Article  Google Scholar 

  30. Li H, Wang Y (2013) A matrix approach to latticized linear Programming with fuzzy-relation inequality constraints. IEEE Trans Fuzzy Syst 21:781–788

    Article  Google Scholar 

  31. Fang S-C, Li G (1999) Solving fuzzy relation equations with a linear objective function. Fuzzy Sets Syst 103:107–113

    Article  MathSciNet  MATH  Google Scholar 

  32. Wu Y-K, Guu S-M, Liu JY-C (2002) An accelerated approach for solving fuzzy relation equations with a linear objective function. IEEE Trans Fuzzy Syst 10(4):552–558

    Article  Google Scholar 

  33. Ghodousian A, Khorram E (2006) Solving a linear programming problem with the convex combination of the max-min and the max-average fuzzy relation equations. Appl Math Comput 180:411–418

    MathSciNet  MATH  Google Scholar 

  34. Wu Y-K, Guu S-M (2005) Minimizing a linear function under a fuzzy max-min relational equation constraint. Fuzzy Sets Syst 150:147–162

    Article  MathSciNet  MATH  Google Scholar 

  35. Guo F-F, Pang L-P, Meng D, Xia Z-Q (2013) An algorithm for solving optimization problems with fuzzy relational inequality constraints. Inf Sci 252:20–31

    Article  MathSciNet  MATH  Google Scholar 

  36. Chang C-W, Shieh B-S (2013) Linear optimization problem constrained by fuzzy max-min relation equations. Inf Sci 234:71–79

    Article  MathSciNet  MATH  Google Scholar 

  37. Li P, Liu Y (2014) Linear optimization with bipolar fuzzy relational equation constraints using the Łukasiewicz triangular norm. Soft Comput 18(7):1399–1404

    Article  MATH  Google Scholar 

  38. Lu J, Fang S-C (2001) Solving nonlinear optimization problems with fuzzy relation equations constraints. Fuzzy Sets Syst 119:1–20

    Article  MathSciNet  Google Scholar 

  39. Hassanzadeh R, Khorram E, Mahdavi I, Mahdavi-Amiri N (2011) A genetic algorithm for optimization problems with fuzzy relation constraints using max-product composition. Appl Soft Comp 11:551–560

    Article  Google Scholar 

  40. Khorram E, Ezzati R, Valizadeh Z (2012) Solving nonlinear multi-objective optimization problems with fuzzy relation inequality constraints regarding Archimedean triangular norm compositions. Fuzzy Optim Decis Mak 11:299–335

    Article  MathSciNet  MATH  Google Scholar 

  41. Yang J-H, Cao B-Y (2005) Geometric programming with fuzzy relation equation constraints. In: Proceedings of IEEE International Conference on Fuzzy Systems, pp 557–560

  42. Cao B-Y (2012) The more-for-less paradox in fuzzy posynomial geometric programming. Inf Sci 211:81–92

    Article  MathSciNet  MATH  Google Scholar 

  43. Yang X-P (2015) Linear programming method for solving semi-latticized fuzzy relation geometric programming with max-min composition. Int J Uncertain Fuzziness Knowl Based Syst 23:781–804

    Article  MathSciNet  Google Scholar 

  44. Zhou X-G, Yang X-P, Cao B-Y (2016) Posynomial geometric programming problem subject to max-min fuzzy relation equations. Inf Sci 328:15–25

    Article  Google Scholar 

  45. Zhou XG, Ahat R (2011) Geometric programming problem with single-term exponents subject to max-product fuzzy relational equations. Math Comp Model 53:55–62

    Article  MathSciNet  MATH  Google Scholar 

  46. Molai A A (2012) The quadratic programming problem with fuzzy relation inequality constraints. Comput Ind Eng 62:256–263

    Article  Google Scholar 

  47. Molai AA (2014) A new algorithm for resolution of the quadratic programming problem with fuzzy relation inequality constraints. Comput Ind Eng 72:306–314

    Article  Google Scholar 

  48. Zhong Y, Zhou X-G, Wu M-Y (2016) A comment on “The quadratic programming problem with fuzzy relation inequality constraints”. Comput Ind Eng 95:10–15

    Article  Google Scholar 

  49. Li J-X., Yang S-J (2012) Fuzzy relation inequalities about the data transmission mechanism in bittorrent-like peer-to-peer file sharing systems. In: Proceedings of the 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery, FSKD, pp 452–456

  50. Yang S-J (2014) An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition-min composition. Fuzzy Sets Syst 255:41–51

    Article  MathSciNet  MATH  Google Scholar 

  51. Yang X-P, Zhou X-G, Cao B-Y (2016) Min-max programming problem subject to addition-min fuzzy relation inequalities. IEEE Trans Fuzzy Syst 24(1):111–119

    Article  Google Scholar 

  52. Yang X-P, Zhou X-G, Cao B-Y (2015) Single-variable term semi-latticized fuzzy relation geometric programming with max-product operator. Inf Sci 325:271–287

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to express our appreciation to the editor and the anonymous reviewers for their valuable comments, which have been very helpful in improving the paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, Guangdong, 521041, China

    Xiao-Peng Yang

  2. School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, 510006, China

    Xiao-Peng Yang, Xue-Gang Zhou & Bing-Yuan Cao

  3. School of Computer and Information Engineering, Hanshan Normal University, Chaozhou, Guangdong, 521041, China

    Geng-Zhong Zheng

  4. Guangzhou Vocational College of Science and Technology, Guangzhou, Guangdong, 510550, China

    Bing-Yuan Cao

Authors
  1. Xiao-Peng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Geng-Zhong Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xue-Gang Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bing-Yuan Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bing-Yuan Cao.

Additional information

Supported by the Innovation and Building Strong School Project of Colleges of Guangdong Province (2015KQNCX094); Special funds of public welfare research and capacity building of Guangdong Province (2015A010103015); Natural Science Foundation of Guangdong Province (2014A030307014); High Level Talents in Colleges of Guangdong Province (Guangdong Finance Education [2013] No. 246).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, XP., Zheng, GZ., Zhou, XG. et al. Lexicography minimum solution of fuzzy relation inequalities: applied to optimal control in P2P file sharing system. Int. J. Mach. Learn. & Cyber. 8, 1555–1563 (2017). https://doi.org/10.1007/s13042-016-0527-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13042-016-0527-x

Keywords

Search

Navigation