An Innovative Approach to Solve Fuzzy Linear Fractional Programming Problems

Karthick Sivakumar; Saraswathi Appasamy; Dragan Pamucar

doi:10.14513/actatechjaur.00768

Authors

Karthick Sivakumar Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Chengalpattu 603203, Tamilnadu, India // Department of Mathematics, Department of Science and Humanities, Mohamed Sathak A.J College of Engineering, Siruseri, Chennai 603103, Tamilnadu, India
Saraswathi Appasamy Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Chengalpattu 603203, Tamilnadu, India
Dragan Pamucar Department of Operations Research and Statistics, Faculty of Organizational Sciences, University of Belgrade, Jove Ilića 154, 11010 Belgrade, Serbia // Department of Applied Mathematical Science, College of Science and Technology, Korea University, Sejong 30019, Republic of Korea // School of Engineering and Technology, Sunway University, 47500 Selangor Darul Ehsan, Malaysia

DOI:

https://doi.org/10.14513/actatechjaur.00768

Keywords:

Fuzzy Linear fractional programming, Parametric form, Equality constraint, LU decomposition, Optimal solution

Abstract

Linear Fractional Programming is a mathematical optimization approach that addresses problems involving the optimization of a linear fractional objective function subject to linear constraints. We discussed approach to solving the fuzzy LFPP with and without equality constraints. We have solved this problem without converting it from fuzzy to crisp. First, we changed FLFPP into the FLPP. This problem was converted into parametric form then we solved it using the LU decomposition method to obtain the solution. We presented a numerical example with a real-life application for the simplicity of presenting the algorithm. While most researchers solve FLFPPs using the ranking function method, this method reduces the efficiency of the fuzzy problem. Linear fractional problems with inequality constraints sometimes do not have an optimal solution by using the LU decomposition method. Therefore, we conclude that there is a limitation for this LU decomposition method only for LFP problems with equality constraints.

Downloads

Download data is not yet available.

References

Bellman, R. E., Zadeh, L. A., "Decision-marking in a fuzzy environment", Management Science, 17 (4) pp. 141-164, 1970. https://doi.org/10.1287/mnsc.17.4.B141

Zadeh, L.A., "Fuzzy Sets", Information and Control, 8 (3) pp. 338-353, 1965. https://doi.org/10.1016/S0019-9958(65)90241-X

A.Charnes and W.W.Cooper," Programs with linear fractional functions", Naval Research Logistics Quarterly, 9, pp.181-196, 1962. https://doi.org/10.1002/nav.3800090303

Zimmermann, H.J, Fuzzy Set Theory and Its Applications, (4th Ed.).2015, 45, 124–131.

J.J. Buckley, T. Feuring, "Evolutionary algorithm solution to fuzzy problems: fuzzy linear programming", Fuzzy Sets and Systems, 109, pp.35-53, 2000. https://doi.org/10.1016/S0165-0114(98)00022-0

Hira, D. S., Operations research, S. Chand Publishing,1992.

Ganesan, K., Veeramani, P.,"Fuzzy linear programs with trapezoidal fuzzy numbers", Ann Oper Res,Vol. 143, pp. 305-315, 2006. https://doi.org/10.1007/s10479-006-7390-1

Safaei, "A new method for solving fully fuzzy linear fractional programming with a triangular fuzzy number", Journal of Applied Mathematics and Computing, 3, pp. 273-281, 2014.

P. Pandian, M. Jayalakshmi, "On solving linear fractional programming problems", Modern Applied Science, 7, pp.90-100, 2013.

Pop, B., Stancu-Minasian, I.M., "A method of solving fully fuzzified linear fractional programming problems", Journal of Applied Mathematics and Computing, 27, 227-242, 2008. https://doi.org/10.1007/s12190-008-0052-5

B. Stanojevic, I.M. Stancu-Minasian,"Evaluating fuzzy inequalities and solving fully fuzzified linear fractional programs", Yugoslav Journal of Operations Research, 22, pp. 41–50, 2012. http://elib.mi.sanu.ac.rs/files/journals/yjor/43/yujorn43p41-50.pdf

Veeramani, C., Sumathi, M., "Fuzzy mathematical programming approach for solving fuzzy linear fractional programming problem", RAIRO-Operations Research, 48 (1) pp.109-122, 2014. https://doi.org/10.1109/FUZZ-IEEE.2013.6622568

S.Muruganandam and P. Ambika,"Harmonic Mean Technique to Solve Multi-Objective Fuzzy Linear Fractional Programming Problems", Global Journal of Pure and Applied Mathematics, 13 (10) pp. 7321-7329, 2017.

S. Muruganandam and P. Ambika, "Solving Linear Fractional Programming Problem using LU Decomposition Method", Annals of Pure and Applied Mathematics, 13 (1) pp.89-97, 2017.

S. Muruganandam and P. Ambika, "Solving fuzzy integer linear fractional programming problem", Global Journal for Research Analysis, 7 (4) pp. 103–105, 2018.

R Srinivasan, "On solving fuzzy linear fractional programming in material aspects", Materials Today, proceedings 21, pp.155-157,2020.

Sadeghi, H., Moslemi, F., "A multiple objective programming approach to linear bilevel multi-follower programming.", AIMS Mathematics, 4, pp. 763-778, 2019. https://doi.org/10.3934/math.2019.3.763

Kumar, J. Kaur, P. Singh, "A new method for solving fully fuzzy linear programming problem", Applied Mathematical Model, 35, pp.817-823, 2011.

K. Swarup, "Linear fractional functional programming", Operation Research, 13, pp. 1029-1036, 1965.

T. Loganathan and K. Ganesan, "A solution approach to fully fuzzy linear fractional programming problems", Journal of Physics: Conference Series, pp.13-77, 2019.

T.Loganathan and K.Ganesan, Solution of fully Linear Fractional Programming Problem-A Simple Approach, IOP Conference Series: Materials Science and Engineering, pp. 11-30, 2021.

S.K. Das and T.Mandal,"A new model for solving fuzzy linear fractional programming problem with ranking function", Journal of Applied Research on Industrial Engineering, 4 (2) pp.89-96, 2017.

Das, S. K., S. A. Edalatpanah," A General Form of Fuzzy Linear Fractional Programs with Trapezoidal Fuzzy Numbers", International Journal of Data Envelopment Analysis and Operations Research, 2 (1) pp.16-19, 2016.

Das, S. K., Mandal, T., Edalatpanah, S. A.,"A new approach for solving fully fuzzy linear fractional programming problems using the multi-objective linear programming", RAIRO-Operations Research, 51 (1) pp.285-297, 2017. https://doi.org/10.1051/ro/2016022

ChiranjitChangdar, G.S. Mahapatra, Rajat Kumar Pal, "An efficient genetic algorithm for multi-objective solid travelling salesman problem under fuzziness", Swarm and Evolutionary Computation, 15, pp.27-37, 2014. https://doi.org/10.1016/j.swevo.2013.11.001

Basha, J., Nandhini, S., "A fuzzy-based solution to multiple objective LPP.", AIMS Mathematics, 8 pp. 7714-7730, 2023.

Gilany, Mahmoud, and Doaa khalil Ibrahim. ,"Traveling-wave-based fault-location scheme for multiend-aged underground cable system.", IEEE Transactions on power delivery, 22 (1) pp. 82-89, 2006. https://doi.org/10.1109/TPWRD.2006.881473

Malik, Manisha, and S. K. Gupta., "An Application of Fully Intuitionistic Fuzzy Multi-objective Linear Fractional Programming Problem in E-education System." ,International Journal of Fuzzy Systems,24 (8) pp. 3544-3563, 2022. https://doi.org/10.1007/s40815-022-01344-5

Khalifa, Hamiden Abd El-Wahed, and Pavan Kumar. "A goal programming approach for multi-objective linear fractional programming problem with LR possibilistic variables.", International Journal of System Assurance Engineering and Management, 13 (4) pp. 2053-2061, 2022. https://doi.org/10.1007/s13198-022-01612-0

Singh, Sujeet Kumar, and Shiv Prasad Yadav., "Scalarizing fuzzy multi-objective linear fractional programming with application. Computational and Applied Mathematics, 41 (3) pp 93, 2022. https://doi.org/10.1007/s40314-022-01714-4

Rizk-Allah, Rizk M., and Mahmoud A. Abo-Sinna., "A comparative study of two optimization approaches for solving a bi-level multi-objective linear fractional programming problem.", Opsearch, 58 (2) pp.374-402,2021. https://doi.org/10.1007/s12597-020-00477-9

Peric, Tunjo, Josip Matejas, and Zoran Babic., "Advantages, sensitivity and application efficiency of the new iterative method to solve multi-objective linear fractional programming problem.", Central European Journal of Operations Research, pp. 1-17, 2023. https://doi.org/10.1007/s10100-023-00834-4

Das, Sapan Kumar, and S. A. Edalatpanah. ,"Optimal solution of neutrosophic linear fractional programming problems with mixed constraints.", Soft Computing, 26 (17) pp. 8699-8707, 2022. https://doi.org/10.1007/s00500-022-06998-3

Stanojević, Bogdana. "Extension principle-based solution approach to full fuzzy multi-objective linear fractional programming." SoftComputing, 26 (11) pp. 5275-5282, 2022. https://doi.org/10.1007/s00500-021-06634-3

Bhatia, Tanveen Kaur, Amit Kumar, and Mahesh Kumar Sharma. "Mehar approach to solve fuzzy linear fractional transportation problems.",Soft Computing, 26 (21) pp. 11525-11551, 2022. https://doi.org/10.1007/s00500-022-07122-1

Farnam, Madineh, and Majid Darehmiraki., "Hesitant Fuzzy Linear Fractional Programming Problem." In Progress in Intelligent Decision Science: Proceeding of IDS 2020, pp. 864-872. Springer International Publishing, 2021.

Bajaj, Rakesh Kumar, Saurabh Srivastava, and Abhishek Guleria., "Solving Multi-objective Linear Fractional Programming Problem Utilizing (α-Cut in Triangular Intuitionistic Fuzzy Setup." In Real Life Applications of Multiple Criteria Decision Making Techniques in Fuzzy Domain, pp. 351-368. Singapore: Springer Nature Singapore, 2022.

Nayak, Suvasis, and Sujit Maharana. ,"An efficient fuzzy mathematical approach to solve multi-objective fractional programming problem under fuzzy environment.", Journal of Applied Mathematics and Computing, pp.1-27, 2023.

Borza, M., and A. S. Rambely., "An approach based on α-cuts and max-min technique to linear fractional programming with fuzzy coefficients."Iranian Journal of Fuzzy Systems 19 (1) pp. 153-169, 2022.

Dubois, Didier, and Henri Prade. ,"Ranking fuzzy numbers in the setting of possibility theory", Information sciences, 30 (3) pp. 183-224, 1983.

Okada, Shinkoh, "Fuzzy shortest path problems incorporating interactivity among paths" Fuzzy Sets and Systems, 142 (3) pp. 335-357, 2004.

Ali, A., Ahmadini, H., Ahmad, F., "Solving intuitionistic fuzzy multiobjective linear programming problem under neutrosophicenvironment", AIMS Mathematics, 6 pp. 4556–4580, 2021.

Das, Sapan Kumar, Tarni Mandal, and S. A. Edalatpanah,"A new approach for solving fully fuzzy linear fractional programming problems using the multi-objective linear programming.", RAIRO-operations research, 51 (1) pp. 285-297, 2017.

Alemohammad, H., Rashidinia, J., & Ghatee, M,"A novel interval type-2 fuzzy chance-constrained programming model for capacitated multi-product multi-period multi-objective supply chain design.", Applied Soft Computing, 52, pp. 1342-1357, 2017.

Chen, J. C., H. C. Lai, and S. Schaible, "Complex fractional programming and the Charnes-Cooper transformation.", Journal of Optimization Theory and Applications., 126, pp. 203-213, 2005.

Sapan Kumar Das, S.A. Edalatpanah, T. Mandal, "A proposed model for solving fuzzy linear fractional programming problem: Numerical Point of View", Journal of Computational Science, 25, 2018, pp. 367-375, 2018. https://doi.org/10.1016/j.jocs.2017.12.004

Ming Ma, "Menahem Friedman and Abraham Kandel, A new approach fuzzy arithmetic", Fuzzy sets and systems, 108, pp.83-90, 1999. https://doi.org/10.1016/S0165-0114(97)00310-2

Mohammadi, M., Zolfaghari, S., & Karimi, B.,"A hybrid model for reliable green supplier selection in a closed-loop supply chain under uncertainty.", Journal of Cleaner Production, Vol.215, pp.992-1006, 2019.

Rezaei, J., Wang, J., & Tavasszy, L. (2018). ,"Linking supplier development to supplier segmentation using a sustainability-based multi-criteria decision-making approach.", International Journal of Production Research, 56 (1-2) 534-547, 2018.

Karthick, S., Saraswathi, A., A method for solving Fuzzy Linear Fractional Programming Problem . Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 32 (1) pp.59-69, 2025.

P. Umamaheswari and K. Ganesan, "A Solution Approach to Fuzzy Nonlinear Programming Problems", International Journal of Pure and Applied Mathematics, 113 (13) pp.291–300, 2017.

Prasad, R., Maiti, I., Das, S. K., Pramanik, S., and Mandal, T. "Fuzzy Goal Programming Approach for Solving Linear Fractional Programming Problems with Fuzzy Conditions." SSRN Electronic Journal, 2025. https://doi.org/10.2139/ssrn.5048302

Sivakumar, K., Appasamy, S. (2024). Fuzzy mathematical approach for solving multi-objective fuzzy linear fractional programming problem with trapezoidal fuzzy numbers. Mathematical Modelling of Engineering Problems, 11 (1) pp. 255-262. https://doi.org/10.18280/mmep.110128

Karthick, S., Saraswathi, A. and Baranidharan, B., Neutrosophic Linear Fractional Programming Problem using Denominator Objective Restriction Method. Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 31 (2) pp.89-101, 2024. https://online.watsci.org/abstract_pdf//2024v31/v31n2b-pdf/2.pdf?utm_source=chatgpt.com

Karthick, S., Saraswathi, A. and S.A. Edalapatnah, Fuzzy linear fractional programming problem using the lexicography method, Vojnotehničkiglasnik, 72 (3) 2024. https://doi.org/10.5937/vojtehg72-50429

Broumi, Said. “An Efficient Approach for Solving Time-Dependent Shortest Path 635 Problem under FermateanNeutrosophic Environment”, Neutrosophic Sets and Systems, 63 (1) 636 pp. 1-6, 2024. https://doi.org/10.5281/zenodo.1234567

Alamin, A., Rahaman, M., & Mondal, S. P. (2025). Geometric Approach for Solving First Order Non-Homogenous Fuzzy Difference Equation. Spectrum of Operational Research, 2 (1) 61-71. https://doi.org/10.31181/sor2120257

Ali, S., Naveed, H., Siddique, I., & Zulqarnain, R. M. (2024). Extension of Interaction Geometric Aggregation Operator for Material Selection Using Interval-Valued Intuitionistic Fuzzy Hypersoft Set. Journal of Operations Intelligence, 2 (1) 14-35. https://doi.org/10.31181/jopi21202410

Bouraima, M. B., Qian, S., Qiu, Y., Badi, I., & Sangaré-Oumar, M. M. (2025). Addressing Human Capital Development Challenges in Developing Countries Using an Interval-spherical Fuzzy Environment. Management Science Advances, 2 (1) 59-68. https://doi.org/10.31181/msa2120258

Ézsiás, L., Tompa, R., & Fischer, S. (2024). Investigation of the possible correlations between specific characteristics of crushed stone aggregates. Spectrum of Mechanical Engineering and Operational Research, 1 (1) 10-26. https://doi.org/10.31181/smeor1120242

Fatima, A., Ashraf, S., & Jana, C. (2024). Approach to Multi-Attribute Decision Making Based on Spherical Fuzzy Einstein Z-Number Aggregation Information. Journal of Operations Intelligence, 2 (1) 179-201. https://doi.org/10.31181/jopi21202411

Fischer, S. (2025). Investigation of the Settlement Behavior of Ballasted Railway Tracks Due to Dynamic Loading. Spectrum of Mechanical Engineering and Operational Research, 2 (1) 24-46. https://doi.org/10.31181/smeor21202528

Fischer, S., & Kocsis Szürke, S. (2023). Detection process of energy loss in electric railway vehicles. Facta Universitatis, Series: Mechanical Engineering, 21 (1) 81-99. https://doi.org/10.22190/FUME221104046F

Gökalp, Y., & Eti, S. (2025). Priority Strategy Development with Intuitionistic Fuzzy DEMATEL Method for Reducing Energy Costs in Hospitals. Journal of Soft Computing and Decision Analytics, 3 (1) 26-32. https://doi.org/10.31181/jscda31202548

Hamid Ch, T., Abid, M., Naeem, K., & Zulqarnain, R. M. (2025). Correlation Measures for q-rung Orthopair m-polar Fuzzy Sets with Application to Pattern Recognition. Decision Making Advances, 3 (1) 139–163. https://doi.org/10.31181/dma31202560

Imran, R., Ullah, K., Ali, Z., & Akram, M. (2024). A Multi-Criteria Group Decision-Making Approach for Robot Selection Using Interval-Valued Intuitionistic Fuzzy Information and Aczel-Alsina Bonferroni Means. Spectrum of Decision Making and Applications, 1 (1) 1-32. https://doi.org/10.31181/sdmap1120241

Petchimuthu, S., Palpandi, B., P, P., & M, F. B. (2025). Sustainable Urban Innovation and Resilience: Artificial Intelligence and q-Rung Orthopair Fuzzy ExpoLogarithmic Framework. Spectrum of Decision Making and Applications, 2 (1) 242-267. https://doi.org/10.31181/sdmap21202526

Ullah, F., & Waqar Shah, S. (2024). Optimizing Multi-attributive Decision-making: A Novel Approach through Matrix Theory and Fuzzy Hypersoft Sets. Spectrum of Engineering and Management Sciences, 2 (1) 100-109. https://doi.org/10.31181/sems2120248r

Wang, H., Zhao, W., & Zheng, J. (2024). Improved q-Rung Orthopair Fuzzy WASPAS Method Based on Softmax Function and Frank operations for Investment Decision of Community Group-Buying Platform. Journal of Soft Computing and Decision Analytics, 2(1), 188-212. https://doi.org/10.31181/jscda21202442

Vidhya, K, Saraswathi A, (2023). A novel method for finding the shortest path with two objectives under Trapezoidal intuitionistic fuzzy arc costs. International Journal of Analysis and applications, 21 121-121. https://doi.org/10.28924/2291-8639-21-2023-121

Vidhya, K., and A. Saraswathi (2021). An improved A* search algorithm for the shortest path under interval-valued Pythagorean fuzzy environment. Granular Computing, 8 (2) 241-25. https://doi.org/10.1007/s41066-022-00326-1(0123456789().,-volV)(0123456789().

P. Yuvashri, and A.Saraswathi(2024). A Novel Approach for Multi-objective Linear Programming Model Under Spherical Fuzzy Environment and Its Application. Journal of Intelligent and Fuzzy Systems, 46 (2) 3259–3280.https://doi.org/10.3233/JIFS-233441

Saraswathi, A (2019),A fuzzy-trapezoidal DEMATEL approach method for solving decision making problems under uncertainty. AIP Conference Proceedings, 2112 (1) 020076-1-12. https://doi.org/10.1063/1.5112261

Fischer, S., Harangozó, D., Németh, D., Kocsis, B., Sysyn, M., Kurhan, D., & Brautigam, A. (2024). Investigation of heat-affected zones of thermite rail welding. Facta Universitatis, Series: Mechanical Engineering, 22 (4) 689-710. https://doi.org/10.22190/FUME221217008F

Jafar, M. N., Muniba, K., & Saqlain, M. (2023). Enhancing Diabetes Diagnosis through an Intuitionistic Fuzzy Soft Matrices-Based Algorithm. Spectrum of Engineering and Management Sciences, 1 (1) 73-82. https://doi.org/10.31181/sems1120238u

Kalaiselvan, K. (2025). Fuzzy Inventory Implementation of Minimum Value unused Storing Profitable by Executing Python. Decision Making Advances, 3(1), 18–30. https://doi.org/10.31181/dma31202524

Mahmood, T., & Rehman, U. (2023). Bipolar complex fuzzy subalgebras and ideals of BCK/BCI-algebras. Journal of Decision Analytics and Intelligent Computing, 3 (1) 47–61. https://doi.org/10.31181/jdaic10021042023m

Mahmoodirad, A., & Niroomand, S. (2023). A heuristic approach for fuzzy fixed charge transportation problem. Journal of Decision Analytics and Intelligent Computing, 3 (1) 139–147. https://doi.org/10.31181/jdaic10005092023m

Sing, P., Rahaman, M., & Sankar, S. P. M. (2024). Solution of Fuzzy System of Linear Equation Under Different Fuzzy Difference Ideology. Spectrum of Operational Research, 1 (1) 64-74. https://doi.org/10.31181/sor1120244