An adaptive method to solve multilevel multiobjective linear programming problems

• Authors: KACI Mustapha , RADJEF Sonia
• Languages of publication: EN

Abstracts

EN

This paper is a follow-up to a previous work where we defined and generated the set of all possible compromises of multilevel multiobjective linear programming problems (ML-MOLPP). We introduce a new algorithm to solve ML-MOLPP in which the adaptive method of linear programming is nested. First, we start by generating the set of all possible compromises (set of all non-dominated solutions). After that, an algorithm based on the adaptive method of linear programming is developed to select the best compromise among all the possible settlements achieved. This method will allow us to transform the initial multilevel problem into an ML-MOLPP with bonded variables. Then, apply the adaptive method which is the most efficient to solve all the multiobjective linear programming problems involved in the resolution process instead of the simplex method. Finally, all the construction stages are carefully checked and illustrated with a numerical example.

Keywords

EN

multilevel programming; multiobjective linear programming; adaptive method, sub-optimality estimate; non-dominated solutions; non-dominated facets

• Publisher: Politechnika Wrocławska. Oficyna Wydawnicza Politechniki Wrocławskiej
• Journal: Operations Research and Decisions
• Year: 2023
• Volume: 33
• Issue: 3
• Pages: 29-44

Contributors

author

KACI Mustapha

• Department of Mathematics, Faculty of Mathematics and Computer Science, University of Sciences and Technology of Oran Mohamed Boudiaf USTO-MB, El Mnaouar, BP 1505, Bir El Djir 31000, Oran, Algeria

author

RADJEF Sonia

• Department of Mathematics, Faculty of Mathematics and Computer Science, University of Sciences and Technology of Oran Mohamed Boudiaf USTO-MB, El Mnaouar, BP 1505, Bir El Djir 31000, Oran, Algeria

References

• Abo-Sinna, M. A., and Baky, I. A. Fuzzy goal programming procedure to bilevel multiobjective linear fractional programming problems. International Journal of Mathematics and Mathematical Sciences (2010), 148975.
• Anandalingam, G., and Apprey, V. Multi-level programming and conflict resolution. European Journal of Operational Research 51, 2 (1991), 233–247.
• Baky, I. A. Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach. Applied Mathematical Modelling 34, 9 (2010), 2377–2387.
• Bracken, J., and McGill, J. T. Mathematical programs with optimization problems in the constraints. Operations Research 21, 1 (1973), 37–44.
• Bracken, J., and McGill, J. T. Technical note—a method for solving mathematical programs with nonlinear programs in the constraints. Operations Research 22, 5 (1974), 1097–1101.
• Delhoum, Z. S., Radjef, S., and Boudaoud, F. Generation of efficient and epsilon-efficient solutions in multiple objective linear programming. Turkish Journal of Mathematics 42, 3 (2018), 1031–1048.
• Gabasov, R., and Kirillova, F. M. Linear Programming methods. Parts I-III, BGU Pub1ishing House, Minsk, 1977, 1978, 1980.
• Gabasov, R., Kirillova, F. M., and Prischepova, S. V. Optimal Feedback Control. Springer Berlin, Heidelberg, 1995.
• Kaci, M., and Radjef, S. A new geometric approach for sensitivity analysis in linear programming. Mathematica Applicanda 49, 2 (2022), 145–157.
• Kaci, M., and Radjef, S. The set of all the possible compromises of a multilevel multiobjective linear programming problem. Croatian Operational Research Review 13, 1 (2022), 13–30.
• Kuchta, D. Fuzzy solution of the linear programming problem with interval coefficients in the constraints. Operations Research and Decisions 15, 3-4 (2005), 35–42.
• Lachhwani, K. On solving multi-level multi objective linear programming problems through fuzzy goal programming approach. OPSEARCH 51, 4 (2014), 624–637.
• Lachhwani, K. Solving the general fully neutrosophic multi-level multiobjective linear programming problems. OPSEARCH 58, 4 (2021), 1192—1216.
• Lachhwani, K., and Dwivedi, A. Bi-level and multi-level programming problems: Taxonomy of literature review and research issues. Archives of Computational Methods in Engineering 25, 4 (2019), 847–877.
• Lu, J., Han, J., Hu, Y,. and Zhang, G. Multilevel decision-making: A survey. Information Sciences 346-347 (2016), 463–487.
• Mandal, W. A., and Islam, S. Multiobjective geometric programming problem under uncertainty. Operations Research and Decisions 27, 4 (2017), 85–109.
• Mohamed, R. H. The relationship between goal programming and fuzzy programming. Fuzzy Sets and Systems 89, 2 (1997), 215–222.
• Pieume, C. O., Marcotte, P., Fotso, L. P., and Siarry, P. Solving bilevel linear multiobjective programming problems. American Journal of Operations Research 1, 4 (2011), 214–219.
• Pramanik, S., and Roy, T.K. Fuzzy goal programming approach to multilevel programming problems. European Journal of Operational Research 176, 2 (2007), 1151–1166.
• Radjef, S., and Bibi, M. O. An effective generalization of the direct support method. Mathematical Problems in Engineering (2011), 374390.
• Radjef, S., and Bibi, M. O. An effective generalization of the direct support method in quadratic convex programming. Applied Mathematical Science 6, 31 (2012), 1525–1540.
• Sinha, S. Fuzzy programming approach to multi-level programming problems. Fuzzy Sets and Systems 136, 2 (2003), 189–202.
• Sinha, S. B., and Sinha, S. A linear programming approach for linear multi-level programming problems. Journal of the Operational Research Society 55, 3 (2017), 312–316.
• Sivri, M., Kocken, H. G., Albayrak, I., and Akin, S. Generating a set of compromise solutions of a multi objective linear programming problem through game theory. Operations Research and Decisions 29, 2 (2019), 77–88.
• Yibing, L., and Wan, Z. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial and Management Optimization 15, 3 (2019), 1213–1223.
• Yu, P. L., and Zeleny, M. The techniques of linear multi-objective programming. RAIRO Revue Française d’Automatique, Informatique et Recherche Opérationnelle 8, V3 (1974), 51–71.

Identifiers

DOI

10.37190/ord230302