Skip to main content
SpringerLink
Log in

A computational algorithm for the solution of fully fuzzy multi-objective linear programming problem

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

The present paper defines a new distance function between two trapezoidal fuzzy (TrF) numbers which satisfies all the properties of metric and a degree of deviation between two TrF numbers. The proposed degree of deviation has been used to solve a fully fuzzy multi-objective linear programming problem (FMOLPP). The proposed algorithm uses converting a FMOLPP into crisp linear programming problem and then to get Pareto-optimal solution to the problem. Further, with Pareto-optimal solutions a balance Pareto optimal solution of the problem using fuzzy optimization technique has been obtained with minimum deviation degree in decision variables. The developed computational algorithm has been implemented on an example for the validity of the proposed algorithm.

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

Fig. 1

Similar content being viewed by others

References

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

    Article  MATH  Google Scholar 

  2. Zimmermann HJ (1978) Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst 1:45–55

    Article  MathSciNet  MATH  Google Scholar 

  3. Kim K, Park KS (1990) Ranking fuzzy numbers with index of optimisim. Fuzzy Sets Syst 35:143–150

    Article  Google Scholar 

  4. Liou TS, Wang MJ (1992) Ranking fuzzy numbers with integral value. Fuzzy Sets Syst 50:247–255

    Article  MathSciNet  MATH  Google Scholar 

  5. Jimenez M (1996) Ranking fuzzy numbers through the comparison of their expected interval. Int J Uncertain Fuzzy Knowl Base 4:379–388

    Article  MATH  Google Scholar 

  6. Cheng CH (1998) A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets Syst 95:307–317

    Article  MathSciNet  MATH  Google Scholar 

  7. Tran L, Duckstein L (2002) Comparison of fuzzy numbers using a fuzzy distance measure. Fuzzy Sets Syst 130:331–341

    Article  MathSciNet  MATH  Google Scholar 

  8. Chu TC, Tsao CT (2002) Ranking fuzzy numbers with an area between the centroid point and original point. Comput Math Appl 43:111–117

    Article  MathSciNet  MATH  Google Scholar 

  9. Abbasbandy S, Asady B (2006) Ranking of fuzzy numbers by sign distance. Inf Sci 176:2405–2416

    Article  MathSciNet  MATH  Google Scholar 

  10. Asady B, Zendehnam A (2007) Ranking fuzzy numbers by distance minimization. Appl Math Model 31:2589–2598

    Article  MATH  Google Scholar 

  11. He X, Deng H (2011) An area based approach to ranking fuzzy numbers in fuzzy decision making. J Comput Inf Syst 7(9):3333–3342

    Google Scholar 

  12. Rao PPB, Shankar NR (2011) A ranking fuzzy numbers with distance method using circumcenter of centroids an index of modality. Adv Fuzzy Syst. https://doi.org/10.1155/2011/178308

    MathSciNet  MATH  Google Scholar 

  13. Kumar A, Singh P, Kaur A, Kaur P (2010) Ranking generalized trapezoidal fuzzy numbers based on rank, mode, divergence and spread. Turk J Fuzzy Syst 1(2):141–152

    Google Scholar 

  14. Nejad AM, Mashinchi M (2011) Ranking fuzzy numbers based on the areas on the left and right sides of fuzzy number. Comput Math Appl 61:431–442

    Article  MathSciNet  MATH  Google Scholar 

  15. Bakar ASA, Mohamad D, Sulaiman NH (2012) Distance based ranking fuzzy numbers. Adv Comput Math Appl 1(3):146–150

    Google Scholar 

  16. Jahantigh MA, Hajighasemi S (2012) Ranking of generalized fuzzy numbers using distance measure and similarity measure. Int J Ind Math 4(4):405416

    Google Scholar 

  17. Grzegorzewski P (2002) Nearest interval approximation of a fuzzy number. Fuzzy Sets Syst 130:321–330

    Article  MathSciNet  MATH  Google Scholar 

  18. Brunelli M, Mezet J (2013) How different are ranking methods for fuzzy numbers. A numerical study. Int J Approx Reason 54:627–638

    Article  MathSciNet  MATH  Google Scholar 

  19. Yu VF, Chi HTX, Shen CW (2013) Ranking fuzzy numbers based on epsilon-deviation degree. Appl Soft Comput 13:3621–3627

    Article  Google Scholar 

  20. Zhang F, Ignatius J, Lim CP, Zhao Y (2014) A new method for ranking fuzzy numbers and its application to group decision making. Appl Math Model 38:1563–1582

    Article  MathSciNet  Google Scholar 

  21. Yu VF, Dat LQ (2014) An improved ranking method for fuzzy numbers with integral values. Appl Soft Comput 14:603–608

    Article  Google Scholar 

  22. Garcia JCF, Harnandez G (2014) A method for solving linear programming model with interval type-2 fuzzy constraints. Pesquisa Operacional 34(1):73–89

    Article  Google Scholar 

  23. Cheng H, Hung W, Zhou Q, Cai J (2013) Solving fuzzy multi objective linear programming problems using deviation degree measures and weighted max-min method. Appl Math Model 37:6855–6869

    Article  MathSciNet  Google Scholar 

  24. Bellman RE, Zadeh LA (1970) Decision making in a fuzzy environment. Manag Sci 17(4):141–164

    Article  MathSciNet  MATH  Google Scholar 

  25. Lee ES, LI RJ, (1993) Fuzzy multi objective programming and compromise programming with Pareto-Optimum. Fuzzy Sets Syst 53:275–288

  26. Jimenez M, Bilbo A (2009) Pareto-optimal solutions in fuzzy multi-objective linear programming. Fuzzy Sets Syst 160:2714–2721

    Article  MathSciNet  MATH  Google Scholar 

  27. Allahviranloo T, Alizadeh L (2008) Solving fully fuzzy linear programming problems by the ranking function. Appl Math Sci 2(1):19–32

    MathSciNet  MATH  Google Scholar 

  28. Lofti FH, Allahviranloo T, Jondabeha MA, Alizadeh L (2009) Solving fully fuzzy linear programming using lexicography method and fuzzy approximate solution. Appl Math Model 33:3151–3156

    Article  MathSciNet  MATH  Google Scholar 

  29. Kumar A, Kaur J, Singh P (2011) A new method for solving fully fuzzy linear programming problems. Appl Math Model 35:817–823

    Article  MathSciNet  MATH  Google Scholar 

  30. Ezzati R, Khorram E, Enatyati R (2013) A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem. Appl Math Model 39:3183. https://doi.org/10.1016/j.apm.2013.03.014

    Article  MathSciNet  Google Scholar 

  31. Campos L, Verdegay JL (1989) linear programming problems and ranking of fuzzy numbers. Fuzzy Sets Syst 32:1–11

  32. Sharma U, Aggarwal S (2017) Solving fully fuzzy multi-objective linear programming problem using nearest interval approximation of fuzzy number and interval programming. Int. J. Fuzzy Syst pp 1–12

  33. Tarabia AM, Kassem MA, El-Badry NM (2017) A modified approach for solving a fuzzy multi-objective programming problem. In: Applied Informatics 4(1). Springer Berlin Heidelberg

  34. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96

    Article  MATH  Google Scholar 

  35. Bharati SK, Singh SR (2014) Solving multi-objective linear programming problems using intuitionistic fuzzy optimization method: a comparative study. Int J Model Optim 4:10–16

    Article  Google Scholar 

  36. Bharati SK, Nishad AK, Singh SR (2014) Solution of multi-objective linear programming problems in intuitionistic fuzzy environment. Adv Intel Syst Comput 236:161–171

    Article  Google Scholar 

  37. Bharati SK, Singh SR (2014) Intuitionistic fuzzy optimization technique in agricultural production planning: a small farm holder perspective. Int J Comput Appl 89:25–31

    Google Scholar 

  38. 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. https://doi.org/10.1007/13198-017-0613-9

  39. 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(23):30–35

    Google Scholar 

  40. Malhotra R, Bharati SK (2016) Intuitionistic fuzzy two stage multiobjective transportation problems. Adv Theor Appl Math 11(3):305–316

    Google Scholar 

  41. Cadenas JM, Verdegay JL (2000) Using ranking functions in multiobjective fuzzy linear programming. Fuzzy Sets Syst 111(1):47–53

    Article  MathSciNet  MATH  Google Scholar 

  42. Kahraman C, Kaya I (2008) Fuzzy multiple objective linear programming. Fuzzy Multi-Criteria Decision Making, 325–337

  43. Dubey D, Mehra A (2014) A bipolar approach in fuzzy multi-objective linear programming. Fuzzy Sets Syst 246:127–141

    Article  MathSciNet  MATH  Google Scholar 

  44. Chen HK, Chou HW (1996) Solving multiobjective linear programming problemsa generic approach. Fuzzy Sets Syst 82(1):35–38

    Article  Google Scholar 

  45. Guu SM, Wu YK (1997) Weighted coefficients in two-phase approach for solving the multiple objective programming problems. Fuzzy Sets Syst 85(1):45–48

    Article  MathSciNet  MATH  Google Scholar 

  46. Luhandjula MK (1982) Compensatory operators in fuzzy linear programming with multiple objectives. Fuzzy Sets Syst 8(3):245–252

    Article  MathSciNet  MATH  Google Scholar 

  47. Chanas S (1989) Fuzzy programming in multiobjective linear programming-a parametric approach. Fuzzy Sets Syst 29(3):303–313

    Article  MathSciNet  MATH  Google Scholar 

  48. 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), December 28–30, (2012), pp 151–159

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kamala Nehru College, University of Delhi, Delhi, 110049, India

    S. K. Bharati

  2. Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, 221005, India

    Abhishekh & S. R. Singh

Authors
  1. S. K. Bharati
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abhishekh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. R. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. K. Bharati.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bharati, S.K., Abhishekh & Singh, S.R. A computational algorithm for the solution of fully fuzzy multi-objective linear programming problem. Int. J. Dynam. Control 6, 1384–1391 (2018). https://doi.org/10.1007/s40435-017-0355-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-017-0355-1

Keywords

Search

Navigation