Duality theory in Atanassov’s intuitionistic fuzzy mathematical programming problems: Optimistic, pessimistic and mixed approaches

  • Vishnu SinghEmail author
  • Shiv Prasad Yadav
  • Sujeet Kumar Singh
S.I.: MOPGP 2017


Linear programming problems in fuzzy environment have been investigated by many researchers in the recent years. Some researchers have solved these problems by using primal-dual method with linear and exponential membership functions. These membership functions are particular form of the reference functions. In this paper, we introduce a pair of primal-dual PPs in Atanassov’s intuitionistic fuzzy environment (AIFE) in which the membership and non-membership functions are taken in the form of the reference functions and prove duality results in AIFE by using an aspiration level approach with different view points, viz., optimistic, pessimistic and mixed. Since fuzzy and AIF environments cause duality gap, we propose to investigate the impact of membership functions governed by reference functions on duality gap. This is specially meaningful for fuzzy and AIF programming problems, when the primal and dual objective values may not be bounded. Finally, the duality gap obtained by the approach has been compared with the duality gap obtained by existing approaches.


Fuzzy sets IF linear programming Primal-dual problems Duality approach Duality gap 



The first author is thankful to the Ministry of Human Resource and Development(MHRD), Govt. of India, India for providing financial grant. The authors would like to thank the Editor-in-Chief and anonymous referees for various suggestions which have led to an improvement in both the quality and clarity of the paper.


  1. Alidaee, B., & Wang, H. (2012). On zero duality gap in surrogate constraint optimization: The case of rational-valued functions of constraints. Applied Mathematical Modelling, 36, 4218–4226.CrossRefGoogle Scholar
  2. Angelov, P. P. (1997). Optimization in an intuitionistic fuzzy environment. Fuzzy Sets and Systems, 86, 299–306.CrossRefGoogle Scholar
  3. Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 87–96.CrossRefGoogle Scholar
  4. Atanassov, K. T. (1989). More on intuitionistic fuzzy sets. Fuzzy Sets and Systems, 33, 37–45.CrossRefGoogle Scholar
  5. Atanassov, K. T. (1994). New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems, 61, 137–142.CrossRefGoogle Scholar
  6. Aggarwal, A., Dubey, D., Chandra, S., & Mehra, A. (2012). Application of Atanassov’s intuitionistic fuzzy set theory to matrix games with fuzzy goals and fuzzy payoffs. Fuzzy Information and Engineering, 4, 401–414.CrossRefGoogle Scholar
  7. Aggarwal, A., & Khan, I. (2016). On solving Atanassov’s I-fuzzy LPPs: Some variants of Angelov’s model. Opsearch, 53, 375–389.CrossRefGoogle Scholar
  8. Aggarwal, A., Mehra, A., & Chandra, S. (2012). Application of linear programming with intuitionistic fuzzy sets to matrix games with intuitionistic fuzzy goals. Fuzzy Optimization and Decision making, 11, 465–480.CrossRefGoogle Scholar
  9. Bector, C. R., & Chandra, S. (2002). On duality in linear programming under fuzzy environment. Fuzzy Sets and Systems, 125, 317–325.CrossRefGoogle Scholar
  10. Bellman, R. E., & Zadeh, L. A. (1970). Decision making in a fuzzy environment. Management Sciences, 17, 141–164.CrossRefGoogle Scholar
  11. Dubey, D., Chandra, S., & Mehra, A. (2012). Fuzzy linear programming under interval uncertainity based on intuitionistic fuzzy set representation. Fuzzy Sets and Systems, 188, 68–87.CrossRefGoogle Scholar
  12. Gupta, S. K., & Dangar, D. (2010). Duality in fuzzy quadratic programming with exponential membership functions. Fuzzy Information and Engineering, 4, 337–346.CrossRefGoogle Scholar
  13. Gupta, P., & Mehlawat, M. K. (2009). Bector-Chandra type duality in fuzzy linear programming with exponential membership functions. Fuzzy Sets Systems, 160, 3290–3308.CrossRefGoogle Scholar
  14. Kumar, M. (2014). Applying weakest t-norm based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers to evaluate realibility of PCBA fault. Applied Soft Computing, 23, 387–406.CrossRefGoogle Scholar
  15. Luhandjula, M. K., & Rangoaga, M. J. (2014). An approach for solving a fuzzy multiobjective programming problem. European Journal of Operational Research, 232, 249–255.CrossRefGoogle Scholar
  16. Mahdavi-Amiri, N., & Nasseri, S. H. (2006). Duality in fuzzy number linear programming by use of certain linear ranking function. Applied Mathematics and Computation, 180, 206–216.CrossRefGoogle Scholar
  17. Mahdavi-Amiri, N., & Nasseri, S. H. (2007). Duality results and a dual simplex method for LPPs with trapezoidal fuzzy variables. Fuzzy Sets and Systems, 158, 1961–1978.CrossRefGoogle Scholar
  18. Ramik, J. (2005). Duality in fuzzy linear programming: some new concepts and results. Fuzzy Optimization and Decision making, 4, 25–39.CrossRefGoogle Scholar
  19. Ramik, J. (2006). Duality in fuzzy linear programming with possibility and necessity relations. Fuzzy Sets and Systems, 157, 1283–1302.CrossRefGoogle Scholar
  20. Rodder, W., & Zimmermann, H.-J. (1980). Duality in fuzzy linear programming. In A.V. Fiacco, K.O. Kortanek (Eds.), Extremal methods and system analysis (pp. 415–429). Berlin.Google Scholar
  21. Singh, V., & Yadav, S. P. (2018). Modeling and optimization of multi-objective programming problems in intuitionistic fuzzy environment:Optimistic, pessimistic and mixed approaches. Expert Systems with Applications, 102, 143–157.CrossRefGoogle Scholar
  22. Verdegay, J. L. (1984). A dual approach to solve the fuzzy linear programming problem. Fuzzy Sets and Systems, 14, 131–141.CrossRefGoogle Scholar
  23. Wu, H. C. (2003). Duality theory in fuzzy linear programming problems with fuzzy coefficients. Fuzzy Optimization and Decision Making, 2, 61–73.CrossRefGoogle Scholar
  24. Zadeh, L. A. (1965). Fuzzy Set. Information and Control, 8, 338–353.CrossRefGoogle Scholar
  25. Zimmermann, H.-J. (2001). Fuzzy set theory and its applications (4th ed.). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Vishnu Singh
    • 1
    Email author
  • Shiv Prasad Yadav
    • 1
  • Sujeet Kumar Singh
    • 2
  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia
  2. 2.The Logistics Institute-Asia Pacific, National University of SingaporeSingaporeSingapore

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