Soft Computing

, Volume 23, Issue 1, pp 77–84 | Cite as

Solution of multiobjective linear programming problems in interval-valued intuitionistic fuzzy environment

  • S. K. BharatiEmail author
  • S. R. Singh
Methodologies and Application


The present paper gives a new computational algorithm for the solution of multiobjective linear programming (MOLP) problem in interval-valued intuitionistic fuzzy (IV-IF) environment. In MOLP problem which occurs in agricultural production planning, industrial planning and waste management. The parameters involved in real-life MOLP problems are impure, and several pioneer works have been done based on fuzzy or intuitionistic fuzzy sets for its compromise solutions. But many times the degree of membership and non-membership for certain element is not defined in exact numbers, so we observe another important kind of uncertainty. Thus fixed values of membership and non-membership cannot handle such uncertainty involved in real-life MOLP problem. Atanassov and Gargov first identified it and presented concept of IV-IF sets which is characterized by sub-intervals of unit interval. In this paper, we study IV-IF sets and develop a new computational method for the solution of real-life MOLP problems based on IV-IF sets. Further, the developed method has been presented in the form of a computational algorithm and implemented on a production problem, and solutions are compared with other existing methods.


Multiobjective programming Interval-valued intuitionistic fuzzy sets Positive deal solution Pareto optimal solution 


Compliance with ethical standards

Conflict of interest

Authors of this paper declare that we have no conflict of interest.


  1. Angelov PP (1997) Optimization in an intuitionistic fuzzy environment. Fuzzy Sets Syst 86:299–306MathSciNetCrossRefzbMATHGoogle Scholar
  2. Atanassov T (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96CrossRefzbMATHGoogle Scholar
  3. Atanassov KT, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bellman RE, Zadeh LA (1970) Decision-making in a fuzzy environment. Manag Sci 14:141–164MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bharati SK, Malhotra R (2017) Two stage intuitionistic fuzzy time minimizing transportation problem based on generalized Zadeh’s extension principle. Int J Syst Assur Eng Manag 8:1–8CrossRefGoogle Scholar
  6. Bharati SK, Singh SR (2014) Solving multi objective linear programming problems using intuitionistic fuzzy optimization method: a comparative study. Int J Model Optim 4:10CrossRefGoogle Scholar
  7. Bharati SK, Singh SR (2015) A note on solving a fully intuitionistic fuzzy linear programming problem based on sign distance. Int J Comput Appl 119:30–35Google Scholar
  8. Bharati SK, Nishad AK, Singh SR (2014) Solution of multi-objective linear programming problems in intuitionistic fuzzy environment. In: Proceedings of the second international conference on soft computing for problem solving (SocProS 2012), vol 236. Advances in Intelligent Systems and Computing, pp 161–171Google Scholar
  9. Bharati, SK, Abhishekh, Singh SR (2017) A computational algorithm for the solution of fully fuzzy multi-objective linear programming problem. Int J Dyn Control.
  10. Chanas S, Kuchta D (1996) A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets Syst 82:299–305MathSciNetCrossRefGoogle Scholar
  11. Chinneck JW, Ramadan K (2000) Linear programming with interval coefficients. J Oper Res Soc 5:209–220CrossRefzbMATHGoogle Scholar
  12. De SK, Biswas RA, Ray R (2000) Some operations on intuitionistic fuzzy sets. Fuzzy Sets Syst 114:474–487MathSciNetCrossRefGoogle Scholar
  13. Dubey D, Mehra A (2011) Linear programming with triangular intuitionistic fuzzy number, EUSFLAT-LFA 2011, vol 1. Advances in Intelligent Systems Research, Atlantis Press, pp 563–569Google Scholar
  14. Dubey D, Chandra S, Mehra A (2012) Fuzzy linear programming under interval uncertainty based on IFS representation. Fuzzy Sets Syst 188:68–87MathSciNetCrossRefzbMATHGoogle Scholar
  15. Garg A, Singh SR (2010) Optimization under uncertainty in agricultural production planning. iconcept Pocket J Comput Intell Financ Eng 1:1–12Google Scholar
  16. Hwang CL, Chen SJ (1992) Fuzzy multiple attribute decision making: methods and applications. Springer, BerlinzbMATHGoogle Scholar
  17. Ishibuchi H, Tanaka H (1990) Multiobjective programming in optimization of the interval objective function. Eur J Oper Res 48:219–225CrossRefzbMATHGoogle Scholar
  18. Itoh T, Ishii H, Nanseki T (2003) Fuzzy crop planning problem under uncertainty in agriculture management. Int J Prod Econ 81–82:555–558CrossRefGoogle Scholar
  19. Jana, B, Roy TK (2007) Multiobjective intuitionistic fuzzy linear programming and its application in transportation model. NIFS-13-1-34-51, pp 1–18Google Scholar
  20. Jiuping X (2011) A kind of fuzzy multi-objective linear programming problems based on interval valued fuzzy sets. J Syst Sci Complex 14:149–158MathSciNetzbMATHGoogle Scholar
  21. Lee ES, Li RJ (1993) Fuzzy multi objective programming and compromise programming with Pareto-Optimum. Fuzzy Sets Syst 53:275–288CrossRefzbMATHGoogle Scholar
  22. Li DF (2010) Linear programming method for MADM with interval valued intuitionistic fuzzy sets. Expert Syst Appl 37:5939–5945CrossRefGoogle Scholar
  23. Malhotra R, Bharati SK (2016) Intuitionistic fuzzy two stage multiobjective transportation problems. Adv Theor Appl Math 11:305–316Google Scholar
  24. Mondal TK, Samanta SK (2012) Generalized intuitionistic fuzzy set. J Fuzzy Math 10:839–861MathSciNetzbMATHGoogle Scholar
  25. Nishad AK, Bharati SK, Singh SR (2014) A new centroid method of ranking for intuitionistic fuzzy numbers. In: Proceedings of the second international conference on soft computing for problem solving (SocProS 2012), vol 236. Advances in Intelligent Systems and Computing, pp 151–159Google Scholar
  26. Parvathi R, Malathi C (2012) Intuitionistic fuzzy linear optimization. Notes Intuit Fuzzy Sets 18:48–56zbMATHGoogle Scholar
  27. Parvathi R, Parvathi CM (2011) Intuitionistic fuzzy linear programming problems. World Appl Sci J 17:1802–1807zbMATHGoogle Scholar
  28. Shaocheng T (1994) Interval number and fuzzy number linear programming. Fuzzy Sets Syst 66:301–306MathSciNetCrossRefGoogle Scholar
  29. Su JS (2007) Fuzzy linear programming with interval valued fuzzy set and ranking. Int J Contemp Math Sci 2:393–410MathSciNetCrossRefzbMATHGoogle Scholar
  30. Tanaka HK, Asai K (1984) Fuzzy linear programming problems with fuzzy numbers. Fuzzy Sets Syst 139:1–10MathSciNetCrossRefzbMATHGoogle Scholar
  31. Zangiababdi M, Maleki HR (2013) Fuzzy goal programming technique to solve multiobjective transportation problem with some nonlinear membership functions. Iran J Fuzzy Syst 10:61–74MathSciNetzbMATHGoogle Scholar
  32. Zimmermann HJ (1978) Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst 1:45–55MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Kamala Nehru CollegeUniversity of DelhiDelhiIndia
  2. 2.Department of Mathematics, Institute of ScienceBanaras Hindu UniversityVaranasiIndia

Personalised recommendations