Simplex algorithm for hesitant fuzzy linear programming problem with hesitant cost coefficient

Document Type : Research Paper

Authors

1 Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 3619995161-316, Tel-Fax No:+9823-32300235, Shahrood, Iran.

2 Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

3 Center of Excellence of Soft Computing and Intelligent Information Processing (SCIIP), Ferdowsi University of Mashhad, Mashhad, Iran.

Abstract

In the real world, in most cases, such as industry, management, and even in daily life, we encounter optimization and decision-making problems that require the opinions of experts and masters on the problem to be able to make the best decision. In these cases, it is necessary to use an optimization problem with hesitant fuzzy parameters.
There are few studies on hesitant fuzzy linear programming (HFLP) problems. Therefore,
in this paper, we  consider such problems.
Especially, we study HFLP problems with hesitant cost coefficients. For this purpose,
we propose the simplex  method to solve the introduced optimization problems and draw a flowchart of the proposed  method.
Finally, by solving two illustrative examples with hesitant fuzzy information, we examine the applicability of the proposed method.

Keywords


[1] J. C. R. Alcantud, Ranked hesitant fuzzy sets for multi-criteria multi-agent decisions, Expert Systems with Applications, 209 (2022), 118276. DOI:10.1016/j.eswa.2022.118276.
[2] J. C. R. Alcantud, V. Torra, Decomposition theorems and extension principles for hesitant fuzzy sets, Information Fusion, 41 (2018), 48-56.
[3] A. Baykasoglu, K. Subulan, Constrained fuzzy arithmetic approach to fuzzy transportation problems with fuzzy decision variables, Expert Systems with Applications, 81 (2017), 193-222.
[4] S. K. Bharati, Solving optimization problems under hesitant fuzzy environment, Life Cycle Reliability and Safety Engineering, 7 (2018), 127-136.
[5] J. J. Buckley, T. Feuring, Evolutionary algorithm solution to fuzzy problems: Fuzzy linear programming, Fuzzy Sets and Systems, 109 (2000), 35-53.
[6] A. Ebrahimnejad, Some new results in linear programming problems with fuzzy cost coefficients, Walailak Journal of Science and Technology, 10(2) (2013), 191-199.
[7] R. Ezzati, E. Khorram, R. Enayati, A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem, Applied Mathematical Modelling, 39(12) (2015), 3183-3193.
[8] H. Garg, R. Krishankumar, K. S. Ravichandran, Decision framework with integrated methods for group decisionmaking under probabilistic hesitant fuzzy context and unknown weights, Expert Systems with Applications, 200 (2022), 117082. DOI:10.1016/j.eswa.2022.117082.
[9] R. R. Gasimov, K. Yenilmez, Solving fuzzy linear programming problems with linear membership function, Turkish Journal of Mathematics, 26 (2002), 375-396.
[10] J. W. Gong, H. C. Liu, L. Yin, An integrated multi-criteria decision making approach with linguistic hesitant fuzzy sets for E-learning website evaluation and selection, Applied Soft Computing, 102 (2021), 107118.DOI:10.1016/j.asoc.2021.107118.
[11] X. Gou, P. Xiao, D. Huang, F. Deng, Probabilistic double hierarchy linguistic alternative queuing method for real economy development evaluation under the perspective of economic financialization, Economic Research-Ekonomska Istrazivanja, 34(1) (2021), 3225-3244.
[12] X. Gou, Z. Xu, H. Liao, Hesitant fuzzy linguistic entropy and cross-entropy measures and alternative queuing method for multiple criteria decision making, Information Sciences, 388 (2017), 225-246.
[13] X. Gou, Z. Xu, W. Zhou, Interval consistency repairing method for double hierarchy hesitant fuzzy linguistic preference relation and application in the diagnosis of lung cancer, Economic Research-Ekonomska Istrazivanja, 34(1) (2020), 1-20.
[14] X. Gou, Z. Xu, W. Zhou, E. Herrera-Vidma, The risk assessment of construction project investment based on prospect theory with linguistic preference orderings, Economic Research-Ekonomska Istrazivanja, 34(1) (2021), 709-731.
[15] C. Jin, J. Mi, F. Li, M. Liang, A novel probabilistic hesitant fuzzy rough set based multi-criteria decision-making method, Information Sciences, 608 (2022), 489-516.
[16] B. Kizielewicz, A. Shekhovtsov, W. Salabun, How to make decisions with uncertainty using hesitant fuzzy sets?,Intelligent and Fuzzy Systems, (2022), 763-771.
[17] R. Krishankumaar, A. R. Mishra, X. Gou, K. S. Ravichandran, New ranking model with evidence theory under probabilistic hesitant fuzzy context and unknown weights, Neural Computing and Applications, 34(3) (2022), 1-15.
[18] F. H. Lotfi, T. Allahviranloo, M. A. Jondabeh, L. Alizadeh, Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution, Applied Mathematical Modelling, 33(7) (2009), 3151-3156.
[19] H. R. Maleki, Ranking functions and their applications to fuzzy linear programming, Far East Journal of Mathematical Sciences, 4(3) (2002), 283-301.
[20] H. R. Maleki, M. Tata, M. Mashinchi, Linear programming with fuzzy variables, Fuzzy Sets and Systems, 109 (2000), 21-33.
[21] A. R. Mishra, S. M. Chen, P. Rani, Multiattribute decision making based on Fermatean hesitant fuzzy sets and modified VIKOR method, Information Sciences, 607 (2022), 1532-1549.
  [22] S. H. Nasseri, E. Ardil, A. Yazdani, R. Zaefarian, Simplex method for solving linear programming problems with fuzzy numbers, World Academy of Science, Engineering and Technology, 10 (2005), 284-288.
[23] H. M. Nehi, H. R. Maleki, M. Mashinchi, Solving fuzzy number linear programming problem by lexicographic ranking function, Italian Journal of Pure and Applied Mathematics, 15 (2004), 9-20.
[24] M. Ranjbar, S. Effati, Symmetric and right-hand-side hesitant fuzzy linear programming, IEEE Transactions on Fuzzy Systems, 28(2) (2020), 215-227.
[25] M. Ranjbar, S. Effati, S. M. Miri, Fully hesitant fuzzy linear programming with hesitant fuzzy numbers, Engineering Applications of Artificial Intelligence, 114 (2022), 105047. DOI:10.1016/j.engappai.2022.105047.
[26] M. Ranjbar, S. M. Miri, S. Effati, Hesitant fuzzy numbers with (α, k)-cuts in compact intervals and applications, Expert Systems with Applications, 151 (2020), 113363. DOI:10.1016/j.eswa.2020.113363.
[27] M. Ranjbar, S. M. Miri, S. Effati, Arithmetic operations and ranking of hesitant fuzzy numbers by extension principle, Iranian Journal of Fuzzy Systems, 19(10) (2022), 97-114.
[28] H. Tanaka, T. Okuda, K. Asai, On fuzzy mathematical programming, Journal of Cybernetics, 3 (1974), 37-46.
[29] V. Torra, Hesitant fuzzy sets, International Journal of Intelligent Systems, 25 (2010), 529-539.
[30] V. Torra, Y. Narukawa, On hesitant fuzzy sets and decision, The 18-th, IEEE International Conference on Fuzzy Systems, Jeju Island Korea, (2009), 1378-1382.
[31] N. Van Hop, Solving linear programming problems under fuzziness and randomness environment using attainment values, Information Sciences, 177(14) (2007), 2971-2984.
[32] X. Wang, E. E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy Sets and Systems, 118 (2001), 375-385.
[33] H. C. Wu, Optimality conditions for linear programming problems with fuzzy coefficients, Computers and Mathematics with Applications, 55(12) (2008), 2807-2822.
[34] P. Wu, L. Zhou, L. Martinez, An integrated hesitant fuzzy linguistic model for multiple attribute group decisionmaking for health management center selection, Computers and Industrial Engineering, 171 (2022), 108404.DOI:10.1016/j.cie.2022.108404.
[35] M. Xia, Z. Xu, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning, 52(3) (2011), 395-407.
[36] Z. Xu, M. Xia, Distance and similarity measures for hesitant fuzzy sets, Information Sciences, 181 (2011), 2128-2138.
[37] R. R. Yager, A procedure for ordering fuzzy subsets of the unit interval, Information Sciences, 24 (1981), 143-161.
[38] R. Zhang, X. Gou, Z. Xu, A multi-attribute decision-making framework for Chinese Medicine medical diagnosis with correlation measures under double hierarchy hesitant fuzzy linguistic environment, Computers and Industrial Engineering, 156(5) (2021), 107243. DOI:10.1016/j.cie.2021.107243.
[39] R. Zhang, Z. Xu, X. Gou, ELECTRE II method based on the cosine similarity to evaluate the performance of financial logistics enterprises under double hierarchy hesitant fuzzy linguistic environment, Fuzzy Optimization and Decision Making, (2022). DOI:10.1007/s10700-022-09382-3.
[40] X. Zhang, Z. Xu, X. Xing, Hesitant fuzzy programming technique for multidimensional analysis of hesitant fuzzy preferences, OR Spectrum, 38(3) (2016), 789-817.
[41] T. Zhou, Z. Chen, X. Ming, Multi-criteria evaluation of smart product-service design concept under hesitant fuzzy linguistic environment: A novel cloud envelopment analysis approach, Engineering Applications of Artificial Intelligence, 115 (2022), 105228. DOI:10.1016/j.engappai.2022.105228.
[42] H. J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1 (1978), 45-55.
[43] H. J. Zimmermann, Fuzzy mathematical programming, Computers and Operations Research, 10 (1992), 291-298.