Open Journal of Optimization, 2019, 8, 100-111 https://www.scirp.org/journal/ojop ISSN Online: 2325-7091 ISSN Print: 2325-7105 DOI: 10.4236/ojop.2019.83009 Sep. 25, 2019 100 Open Journal of Optimization A Lagrange Relaxation Based Approach to Solve a Discrete-Continous Bi-Level Model Zaida E. Alarcón-Bernal 1 , Ricardo Aceves-García 2 1 Department of Biomedical Systems Engineering, Universidad Nacional Autónoma de México, Mexico City, Mexico 2 Department of Systems Engineering, Universidad Nacional Autónoma de México, Mexico City, Mexico Abstract In this work we propose a solution method based on Lagrange relaxation for discrete-continuous bi-level problems, with binary variables in the leading problem, considering the optimistic approach in bi-level programming. For the application of the method, the two-level problem is reformulated using the Karush-Kuhn-Tucker conditions. The resulting model is linearized taking advantage of the structure of the leading problem. Using a Lagrange relaxa- tion algorithm, it is possible to find a global solution efficiently. The algo- rithm was tested to show how it performs. Keywords Bi-Level Programming, Lagrange Relaxation, Discrete-Continous Linear Bilevel 1. Introduction The use of optimization techniques has been a fundamental part of solving prob- lems. However, the most common mathematical programming models are often unsuitable for situations involving more than one goal and more than one deci- sion-maker. In most cases, decisions about a given process are the result of an interaction between the preferences of a group of individuals, so deciding on a single crite- rion seems insufficient, particularly when the decision-making process is applied to complex organizational environments. Consequently, it is more realistic to seek the achievement of several objectives at the same time. This implies that problems must be solved according to a crite- rion that satisfies all decision-makers as a whole. As a result, contradictory and incommensurable criteria will commonly appear. This situation has led to the How to cite this paper: Alarcón-Bernal, Z.E. and Aceves-García, R. (2019) A Lagrange Relaxation Based Approach to Solve a Dis- crete-Continous Bi-Level Model. Open Jour- nal of Optimization, 8, 100-111. https://doi.org/10.4236/ojop.2019.83009 Received: August 22, 2019 Accepted: September 22, 2019 Published: September 25, 2019 Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access