A variables neighborhood search algorithm for solving fuzzy quadratic programming problems using modified Kerre’s method

Abstract

To solve a fuzzy optimization problem, we need to compare fuzzy numbers. Here, we make use of our recently proposed modified Kerre’s method as an effective approach for comparison of LR fuzzy numbers. Using our new results on LR fuzzy numbers, we show that to compare two LR fuzzy numbers, we do not need to compute the fuzzy maximum of two numbers directly. We propose a new variable neighborhood search approach for solving fuzzy number quadratic programming problems by using the modified Kerre’s method. In our algorithm, a local search is performed using descent directions, found by solving five crisp mathematical programming problems. In several available methods, a fuzzy optimization problem is converted to a crisp problem, but in our proposed approach, using our modified Kerre’s method, the fuzzy optimization problem is solved directly, without changing it to a crisp program. We give some examples to compare the performance of our proposed algorithm with some available methods and show the effectiveness of our proposed algorithm by using the nonparametric statistical sign test.

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

References

  1. Abualigah LM, Hanandeh ES (2015) Applying genetic algorithms to information retrieval using vector space model. Int J Comput Sci Eng Appl 5:19–28

    Google Scholar 

  2. Abualigah LM, Khader AT (2017) Unsupervised text feature selection technique based on hybrid particle swarm optimization algorithm with genetic operators for the text clustering. J Supercomput 73:4773–4795

    Article  Google Scholar 

  3. Abualigah LM, Khader AT, Al-Betar MA (2016) Unsupervised feature selection technique based on genetic algorithm for improving the text clustering. In: 2016 7th international conference on computer science and information technology, IEEE, pp 1–6

  4. Abualigah LM, Khader AT, Al-Betar MA, Hanandeh ES (2017a) A new hybridization strategy for Krill Herd algorithm and harmony search algorithm applied to improve the data clustering. In: First EAI international conference on computer science and engineering, pp 1–10

  5. Abualigah LM, Khader AT, Hanandeh ES, Gandomi AH (2017b) A novel hybridization strategy for Krill Herd algorithm applied to clustering techniques. Appl Soft Comput 60:423–435

    Article  Google Scholar 

  6. Abualigah LM, Khader AT, Al-Betar MA, Alomari OA (2017c) Text feature selection with a robust weight scheme and dynamic dimension reduction to text document clustering. Expert Syst Appl 84:24–36

    Article  Google Scholar 

  7. Abualigah LM, Khader AT, Hanandeh ES (2018a) Hybrid clustering analysis using improved Krill Herd algorithm. Appl Intel 48:4047–4071

    Article  Google Scholar 

  8. Abualigah LM, Khader AT, Hanandeh ES (2018b) A combination of objective functions and hybrid Krill Herd algorithm for text document clustering analysis. Eng Appl Artif Intel 73:111–125

    Article  Google Scholar 

  9. Abualigah LM, Khader AT, Hanandeh ES (2018c) A novel weighting scheme applied to improve the text document clustering techniques. Innov Comput Optim Appl 741:305–320

    Google Scholar 

  10. Abualigah LM, Khader AT, Hanandeh ES (2018d) A new feature selection method to improve the document clustering using particle swarm optimization algorithm. J Comput Sci 25:456–466

    Article  Google Scholar 

  11. Abualigah LM, Khader AT, Hanandeh ES (2018e) A hybrid strategy for Krill Herd algorithm with harmony search algorithm to improve the data clustering. Intel Decis Technol 12:3–14

    Article  Google Scholar 

  12. Adamo JM (1980) Fuzzy decision trees. Fuzzy Sets Syst 4:207–219

    MathSciNet  MATH  Article  Google Scholar 

  13. Ammar E, Khalifa HA (2003) Fuzzy portfolio optimization: a quadratic programming approach. Chaos Solitons Fractals 18:1045–1054

    MathSciNet  MATH  Article  Google Scholar 

  14. Back T, Schwefel HP (1993) An overview of evolutionary algorithms for parameter optimization. Evol Comput 1:1–23

    Article  Google Scholar 

  15. Bellman RE, Zadeh LA (1970) Decision making in a fuzzy environment. Manag Sci 17:141–164

    MathSciNet  MATH  Article  Google Scholar 

  16. Bertsimas D, Tsitsiklis JN (1997) Introduction to linear optimization, 3rd edn. Athena Scientific, Belmont

    Google Scholar 

  17. Buckely JJ, Jowers LJ (2007) Monte Carlo method in fuzzy optimization. Springer, New York

    Google Scholar 

  18. Chiua CH, Wang WJ (2002) A simple computation of MIN and MAX operations for fuzzy numbers. Fuzzy Sets Syst 126:273–276

    MathSciNet  Article  Google Scholar 

  19. Cruz C, Silva RC, Verdegay JL (2011) Extending and relating different approaches for solving fuzzy quadratic problems. Fuzzy Optim Decis Making 10:193–210

    MathSciNet  MATH  Article  Google Scholar 

  20. Dinkelbach W (1967) On non-linear fractional programming. Manag Sci 13:492–498

    MathSciNet  Google Scholar 

  21. Dubois D, Prade H (1983) Ranking fuzzy numbers in the setting of possibility theory. Inf Sci 30:183–224

    MathSciNet  MATH  Article  Google Scholar 

  22. Freund JE, Walpole RE (1980) Mathematical statistics. Prentice Hall, Prentice

    Google Scholar 

  23. Gabr WI (2014) Quadratic and nonlinear programming problems solving and analysis in fully fuzzy environment. Alex Eng J 54:457–472

    Article  Google Scholar 

  24. Ghanbari R, Ghorbani-Moghadam K, Mahdavi-Amiri N (2018) A variable neighborhood search algorithm for solving fuzzy number linear programming problems using modified Kerre’s method. IEEE Trans Fuzzy Syst https://doi.org/10.1109/TFUZZ.2018.2876690

  25. Hansen P, Mladenović N (1997) A variable neighborhood search. Comput Oper Res 24:1097–1100

    MathSciNet  MATH  Article  Google Scholar 

  26. Kheirfam B (2011) A method for solving fully fuzzy quadratic programming problems. Acta Universitatis Appulensis 27:69–76

    MathSciNet  MATH  Google Scholar 

  27. Kheirfam B, Verdegay JL (2012) Strict sensitivity analysis in fuzzy quadratic programming. Fuzzy Sets Syst 198:99–111

    MathSciNet  MATH  Article  Google Scholar 

  28. Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and application. Prentice Hall, Prentice

    Google Scholar 

  29. Liu ST (2004) Fuzzy geometric programming approach to a fuzzy machining economics model. Int J Prod Res 42:3253–3322

    MATH  Article  Google Scholar 

  30. Liu ST (2006) Optimization of a machining economics model with fuzzy exponents and coefficients. Int J Prod Res 44:3083–3187

    MATH  Article  Google Scholar 

  31. Liu ST (2009) Quadratic programming with fuzzy parameters: a membership function approach. Chaos Solitons Fractals 40:237–245

    MathSciNet  MATH  Article  Google Scholar 

  32. Luozhong G, Xuegang Z, Weijun L, Kun W. (2009) First-order optimality conditions for fuzzy number quadratic programming with fuzzy coefficients. In: Sixth international conference on fuzzy systems and knowledge discovery, pp 286–290

  33. Mahdavi-Amiri N, Nasseri SH (2006) Duality in fuzzy number linear programming by use of certain linear ranking function. Appl Math Comput 180:206–216

    MathSciNet  MATH  Google Scholar 

  34. Mahdavi-Amiri N, Nasseri SH, Yazdani A (2009) Fuzzy primal simplex algorithm for solving fuzzy linear programming problems. Iran J Oper Res 1:68–84

    Google Scholar 

  35. Maleki HR, Tata M, Mashinchi M (2000) Linear programming with fuzzy variables. Fuzzy Sets Syst 109:21–33

    MathSciNet  MATH  Article  Google Scholar 

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

    Article  Google Scholar 

  37. Nasseri SH (2008) Fuzzy nonlinear optimization. J Nonlinear Sci Appl 1:230–235

    MathSciNet  MATH  Article  Google Scholar 

  38. Nguyen HT, Walker EA (2000) A first course in fuzzy logic. Chapman & Hall, London

    Google Scholar 

  39. Saber YM, Alsharari F (2018) Generalized fuzzy ideal closed sets on fuzzy topological spaces in Sostak sense. Int J Fuzzy Logic Intel Syst 18:161–166

    Article  Google Scholar 

  40. Silva RC, Verdegay JL, Yamakami A (2007) Two-phase method to solve fuzzy quadratic programming problems. In: IEEE international conference fuzzy systems, London, UK, pp 1–6

  41. Wang X, Kerre EE (1996) On the classification and the dependencies of the ordering methods, fuzzy logic foundations and industrial applications. Int Ser Intel Technol 8:73–90

    MATH  Article  Google Scholar 

  42. Wang X, Kerre EE (2001) Reasonable properties for the ordering of fuzzy quantities (II). Fuzzy Sets Syst 118:387–405

    MathSciNet  MATH  Article  Google Scholar 

  43. Wasserman L (2006) All of nonparametric statistics. Springer, New York

    Google Scholar 

  44. Yager RR (1980) On choosing between fuzzy subsets. Kybernetes 9:151–154

    MATH  Article  Google Scholar 

  45. Yager RR (1981) A procedure for ordering fuzzy subsets of the unit interval. Inf Sci 24:143–161

    MathSciNet  MATH  Article  Google Scholar 

  46. Zhong Y, Zhou X, Wu MY (2016) A comment on The quadratic programming problem with fuzzy relation inequality constraints. Comput Ind Eng 95:10–15

    Article  Google Scholar 

  47. Zimmermann HJ (2001) Fuzzy set theory and its applications. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Reza Ghanbari.

Ethics declarations

Conflict of interest

Authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by V. Loia.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ghanbari, R., Ghorbani-Moghadam, K. & Mahdavi-Amiri, N. A variables neighborhood search algorithm for solving fuzzy quadratic programming problems using modified Kerre’s method. Soft Comput 23, 12305–12315 (2019). https://doi.org/10.1007/s00500-019-03771-4

Download citation

Keywords

  • Quadratic programming problem
  • Modified Kerre’s method
  • Ranking function