A Novel Approach for Solving Linear Programming Problems with Intuitionistic Fuzzy Numbers
Subject Areas : Fuzzy Optimization and Modeling Journal
1 - Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran
Keywords: Multi-Objective Optimization, linear programming, Trapezoidal intuitionistic fuzzy number, Fuzzy programming approach,
Abstract :
The literature of linear programming problem with trapezoidal intuitionistic parameters is full of solution approaches which are mainly ranking function based. Use of ranking function in the solution approaches could be a weakness as different ranking functions mag give different solutions. This paper, proposes a new solution approach without any ranking function for linear programming problem with trapezoidal intuitionistic parameters. For this aim, the trapezoidal intuitionistic fuzzy objective function is converted to a multi-objective function, and consequently, the problem is converted to a multi-objective crisp problem. As another contribution, in order to solve the obtained multi-objective problem for its efficient solutions, a new multi-objective optimization approach was developed and suited to the obtained multi-objective problem. The computational experiments of the study show the superiority of the proposed multi-objective optimization approach over the multi-objective optimization approaches of the literature.
1. Allahviranloo, T., Lotfi, F.H., Kiasary, M.Kh., Kiani, N.A., & Alizadeh, L. (2008). Solving fully fuzzy linear programming problem by the ranking function. Applied Mathematical Sciences, 2(1), 19–32.
2. Alavidoost, M.H., Babazadeh, H., & Sayyari, S.T. (2016). An interactive fuzzy programming approach for bi-objective straight and U-shaped assembly line balancing problem. Applied Soft Computing, 40, 221–235.
3. Angelove, P.P. (1997). Optimization in an intuitionistic fuzzy environment. Fuzzy Sets and Systems, 86(3), 299–306.
4. Atanassov, K.T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 87–96.
5. Bellman, R., & Zadeh, L.A. (1970). Decision making in fuzzy environment. Management Science, 17(B), 141–164.
6. Daami Remadi, F., & Moalla Frikha, H. (2023). The triangular intuitionistic fuzzy numbers CODAS method for solving green material selection problem. International Journal of Operational Research, 46 (3), 398-415.
7. Das, S. K., Mandal, T., & Edalatpanah, S. A. (2017). A mathematical model for solving fully fuzzy linear programming problem with trapezoidal fuzzy numbers. Applied Intelligence, 46 (3), 509-517.
8. Dong, J. Y., & Wan, S. P. (2018). A new trapezoidal fuzzy linear programming method considering the acceptance degree of fuzzy constraints violated. Knowledge-Based Systems, 148, 100-114.
9. Dubey, D., Chandra, S., & Mehra, A. (2012). Fuzzy linear programming under interval uncertainty based on IFS representation. Fuzzy Sets and Systems, 188(1), 68–87.
10. Dubey, D., & Mehra, A. (2011). Linear programming with triangular intuitionistic fuzzy numbers. Advances in Intelligent Systems Research, 1, 563–569.
11. Ebrahimnejad, A. (2011). Some new results in linear programs with trapezoidal fuzzy numbers: finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method. Applied Mathematical Modelling, 35, 4526–4540.
12. He, Y., He, Z., & Huang, H. (2017). Decision making with the generalized intuitionistic fuzzy power interaction averaging operators. Soft Computing, 21(5), 1129–1144.
13. Huo, Y., Qiu, P., Zhai, J., Fan, D., & Peng, H. (2017). Multi-objective service composition model based on cost-effective optimization. Applied Intelligence, 48, 651-669.
14. Kaur, A., & Kumar, A. (2012). A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers. Applied Soft Computing, 12, 1201–1213.
15. Kumar, P.S., & Hussain, R.J. (2014). Computationally simple approach for solving fully intuitionistic fuzzy real life transportation problems. International Journal of System Assurance Engineering and Management, 7, 90-101.
16. Kumar, A., Kaur, J., & Singh, P. (2011). A new method for solving fully fuzzy linear programming problems. Applied Mathematical Modelling, 35, 817–823.
17. Lai, Y.J., & Hwang, C.L. (1992). A new approach to some possibilistic linear programming problems. Fuzzy Sets and Systems, 49, 121–133.
18. Liu, X.W. (2001). Measuring the satisfaction of constraints in fuzzy linear programming, Fuzzy Sets and Systems, 122, 263–275.
19. Lotfi, F.H., Allahviranloo, T., Jondabeha, M.A., & Alizadeh, L. (2009). Solving a fully fuzzy linear programming using lexicography method and fuzzy approximate solution. Applied Mathematical Modelling, 33, 3151–3156.
20. Mahmoodirad, A., & Sanei, M. (2016). Solving a multi-stage multi-product solid supply chain network design problem by meta-heuristics. Scientia Iranica E, 23(3), 1428-1440.
21. Mahmoodirad A., Allahviranloo T., & Niroomand S. (2019). A new effective solution method for fully intuitionistic fuzzy transportation problem. Soft Computing, 23 (12), 4521-4530.
22. Maleki, H.R., & Mashinchi, M. (2004). Fuzzy number linear programming: a probabilistic approach (3). Journal of Applied Mathematics and Computing. 15(2), 333–341.
23. Maleki, H.R., Tata, M., & Mashinchi, M. (2000). Linear programming with fuzzy variables. Fuzzy Sets and Systems, 109, 21–33.
24. Mollalign Moges, D., Rangia Mushi, A., & Guta Wordofa, B. (2023).A new method for intuitionistic fuzzy multi-objective linear fractional optimization problem and its application in agricultural land allocation problem. Information Sciences, 625, 457-475.
25. Mirzaei, N., Mahmoodirad, A. & Niroomand, S. (2019). An Uncertain Multi-objective Assembly Line Balancing Problem: A Credibility-Based Fuzzy Modeling Approach. International Journal of Fuzzy Systems, 21, 2392–2404.
26. Mosallaeipour, S., Mahmoodirad, A., Niroomand, S., & Vizvari, B. (2017). Simultaneous selection of material and supplier under uncertainty in carton box industries: a fuzzy possibilistic multicriteria approach. Soft Computing, 22, 2891-2905.
27. Nagoorgani, A., & Ponnalagu, K. (2012). A new approach on solving intuitionistic fuzzy linear programming problem. Applied Mathematical Sciences, 6, 3467–3474.
28. Nayagam, V.L.G., Jeevaraj, S., & Dhanasekaran, P. (2016). An intuitionistic fuzzy multi-criteria decision-making method based on non-hesitance score for interval-valued intuitionistic fuzzy sets. Soft Computing, 21, 7077-7082.
29. Niroomand, S., Garg, H., & Mahmoodirad, A. (2020). An intuitionistic fuzzy two stage supply chain network design problem with multi-mode demand and multi-mode transportation. ISA Transactions, 107, 117–133.
30. Niroomand, S., Mahmoodirad, A., & Mosallaeipour, S. (2019). A hybrid solution approach for fuzzy multiobjective dual supplier and material selection problem of carton box production systems. Expert Systems, 36(1), 1–17.
31. Parvathi, R., & Malathi, C. (2012.a). Intuitionistic fuzzy linear optimization. Notes on Intuitionistic Fuzzy Sets, 18, 48–56.
32. Parvathi, R., & Malathi, C. (2012.b). Intuitionistic fuzzy linear programming problems. World applied Sciences Journal, 17, 1802–1807.
33. Popa, L. (2023). A new ranking method for trapezoidal intuitionistic fuzzy numbers and its application to multi-criteria decision making. International Journal of Computers Communications & Control, 18(2), 5118.
34. Salehi, M., Maleki, H. R., & Niroomand, S. (2017). A multi-objective assembly line balancing problem with worker’s skill and qualification considerations in fuzzy environment. Applied Intelligence, 48, 2137-2156.
35. Sanei, M., Mahmoodirad, A., & Molla-Alizadeh-Zavardehi, S. (2013). An Electromagnetism-like algorithm for fixed charge solid transportation problem. International journal of mathematical modelling & computations, 3(4), 345- 354.
36. Santos Arteaga, F. J., Ebrahimnejad, A., & Zabihi, A. (2021). A new approach for solving intuitionistic fuzzy data envelopment analysis Problems. Fuzzy Optimization and Modelling Journal, 2 (2), 46-57.
37. Singh, S.K., & Yadav, S.P. (2016). A novel approach for solving fully intuitionistic fuzzy transportation problem. International Journal of Operational Research, 26 (4), 460-472.
38. Suresh, M., Vengataasalam, S., & Prakash, K.A. (2014). Solving intuitionistic fuzzy linear programming problems by ranking function. Journal of Intelligent & Fuzzy Systems, 27, 3081–3087.
39. Torabi, S.A., & Hassini, E. (2008). An interactive possibilistic programming approach for multiple objective supply chain master planning. Fuzzy Sets and Systems, 159, 193–214.
40. Wan, S.P., & Dong, J.Y. (2014). Possibility linear programming with trapezoidal fuzzy numbers. Applied Mathematical Modelling, 38(5-6), 1660-1672.
41. Zimmermann, H.J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1, 45–55.