Abstract
In this work, we deal with a global optimization problem (P) for which we look for the most preferred extreme point (vertex) of the convex polyhedron according to a new linear criterion, among all efficient vertices of a multi-objective linear programming problem. This problem has been studied for decades and a lot has been done since the 70’s. Our purpose is to propose a new and effective methodology for solving (P) using a branch and bound based technique, in which, at each node of the search tree, new customized bounds are established to delete uninteresting areas from the decision space. In addition, an efficiency test is performed considering the last simplex tableau corresponding to the current visited vertex. A comparative study shows that the proposed method outperforms the most recent and performing method dedicated to solve (P).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benson HP (1991) An all-linear programming relaxation algorithm for optimizing over the efficient set. J Global Opt 1(1):83–104
Benson HP (1993) A bisection-extreme point search algorithm for optimizing over the efficient set in the linear dependence case. J Global Optim 3(1):95–111
Benson HP (2011) An outcome space algorithm for optimization over the weakly efficient set of a multiple objective nonlinear programming problem. J Global Optim 52(3):553–574
Boland N, Charkhgard H, Savelsbergh M (2017) A new method for optimizing a linear function over the efficient set of a multiobjective integer program. Eur J Oper Res 260(3):904–919
Bolintineanu S (1993) Minimization of a quasi-concave function over an efficient set. Math Program 61(1–3):89–110
Ecker JG, Kouada IA (1975) Finding efficient points for linear multiple objective programs. Math Program 8(1):375–377
Ecker JG, Song JH (1994) Optimizing a linear function over an efficient set. J Optim Theory Appl 83(3):541–563
Fülöp J, Muu LD (2000) Branch-and-bound variant of an outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem. J Optim Theory Appl 105(1):37–54
Hamel AH, Löhne A, Rudloff B (2013) Benson type algorithms for linear vector optimization and applications. J Global Optim 59(4):811–836
Isermann H (1974) Technical note–proper efficiency and the linear vector maximum problem. Oper Res 22(1):189–191
Jorge JM (2009) An algorithm for optimizing a linear function over an integer efficient set. Eur J Oper Res 195(1):98–103
Liu Z, Ehrgott M (2018) Primal and dual algorithms for optimization over the efficient set. Optimization 67(10):1661–1686
Löhne A, Rudloff B, Ulus F (2014) Primal and dual approximation algorithms for convex vector optimization problems. J Global Optim 60(4):713–736
Philip J (1972) Algorithms for the vector maximization problem. Math Program 2(1):207–229
Piercy CA, Steuer RE (2019) Reducing wall-clock time for the computation of all efficient extreme points in multiple objective linear programming. Eur J Oper Res 277(2):653–666
Thoai NV (2000) Conical algorithm in global optimization for optimizing over efficient sets. J Global Optim 18(4):321–336
Yamamoto Y (2002) Optimization over the efficient set: overview. J Global Optim 22:285–317
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Belkhiri, H., Chergui, M.EA. & Ouaïl, F.Z. Optimizing a linear function over an efficient set. Oper Res Int J 22, 3183–3201 (2022). https://doi.org/10.1007/s12351-021-00664-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12351-021-00664-z