Solving the binary integer bi-level linear programming problem

This thesis will introduce a historical perspective of the development of work in the field of multi-level linear programming. It will then proceed to extend the theoretical work of the mixed integer bi-level linear programming problem to encompass the binary integer bi-level linear programming problem. An algorithm will be developed to solve this particular problem using a preference function to determine the choice of branching in a branch and bound tree. Computational results will be compiled and the implications discussed

A Two-Level Approach to Large Mixed-Integer Programs with Application to Cogeneration in Energy-Efficient Buildings

We study a two-stage mixed-integer linear program (MILP) with more than 1 million binary variables in the second stage. We develop a two-level approach by constructing a semi-coarse model (coarsened with respect to variables) and a coarse model (coarsened with respect to both variables and constraints). We coarsen binary variables by selecting a small number of pre-specified daily on/off profiles. We aggregate constraints by partitioning them into groups and summing over each group. With an appropriate choice of coarsened profiles, the semi-coarse model is guaranteed to find a feasible solution of the original problem and hence provides an upper bound on the optimal solution. We show that solving a sequence of coarse models converges to the same upper bound with proven finite steps. This is achieved by adding violated constraints to coarse models until all constraints in the semi-coarse model are satisfied. We demonstrate the effectiveness of our approach in cogeneration for buildings. The coarsened models allow us to obtain good approximate solutions at a fraction of the time required by solving the original problem. Extensive numerical experiments show that the two-level approach scales to large problems that are beyond the capacity of state-of-the-art commercial MILP solvers

An Effective Branch-and-cut algorithm in Order to Solve the Mixed Integer Bi-level Programming

[EN] In this paper, a new branch-and-cut algorithm for mixed integer bi-level programming is proposed. For achieving this purpose, a historical perspective of the development of enumeration methods in the field of bi-level linear programming is considered. Then, we present some obstacles for using branch and bound method based on them, and an algorithm is developed to solve for mixed integer bi-level problem. Finally, we use a preference function to determine the choice of branching and specialized cuts in a branch and cut tree. Computational results are reported and compared favorably to those of previous methods and then implications discussed. The results show that not only the proposed algorithm can find high quality solutions for solving a number of the problems, but also it is competitive with other famous published algorithms.Rahmani, A.; Yousefikhoshbakht, M. (2017). An Effective Branch-and-cut algorithm in Order to Solve the Mixed Integer Bi-level Programming. International Journal of Production Management and Engineering. 5(1):1-10. doi:10.4995/ijpme.2017.6512SWORD1105

Resolution Method for Mixed Integer Linear Multiplicative-Linear Bilevel Problems Based on Decomposition Technique

I In this paper, we propose an algorithm base on decomposition technique for solving the mixed integer linear multiplicative-linear bilevel problems. In fact, this algorithm is an application of the algorithm given by G. K. Saharidis et al for the case in which the first level objective function is linear multiplicative. We use properties of quasi-concave of bilevel programming problems and decompose the initial problem into two subproblems named RM P and SP . The lower and upper bound provided from the RM P and SP are updated in each iteration. The algorithm converges when the difference between the upper and lower bound is less than an arbitrary tolerance. In conclusion, some numerical examples are presented in order to show the efficiency of algorithm

Multilevel decision-making: A survey

© 2016 Elsevier Inc. All rights reserved. Multilevel decision-making techniques aim to deal with decentralized management problems that feature interactive decision entities distributed throughout a multiple level hierarchy. Significant efforts have been devoted to understanding the fundamental concepts and developing diverse solution algorithms associated with multilevel decision-making by researchers in areas of both mathematics/computer science and business areas. Researchers have emphasized the importance of developing a range of multilevel decision-making techniques to handle a wide variety of management and optimization problems in real-world applications, and have successfully gained experience in this area. It is thus vital that a high quality, instructive review of current trends should be conducted, not only of the theoretical research results but also the practical developments in multilevel decision-making in business. This paper systematically reviews up-to-date multilevel decision-making techniques and clusters related technique developments into four main categories: bi-level decision-making (including multi-objective and multi-follower situations), tri-level decision-making, fuzzy multilevel decision-making, and the applications of these techniques in different domains. By providing state-of-the-art knowledge, this survey will directly support researchers and practical professionals in their understanding of developments in theoretical research results and applications in relation to multilevel decision-making techniques

Multi-parametric programming : novel theory and algorithmic developments

Imperial Users onl

Optimal Design of the Annual Influenza Vaccine

Seasonal influenza is a major public health concern, and the first line of defense is the flu shot. Antigenic drifts and the high rate of influenza transmission require annual updates to the flu shot composition. The World Health Organization recommends which flu strains to include in the annual vaccine based on surveillance and epidemiological analysis. There are two critical decisions regarding the flu shot design. One is its composition; currently, three strains constitute the flu shot, and they influence vaccine effectiveness. Another critical decision is the timing of the composition decisions, which affects the flu shot production. Both of these decisions have to be made under uncertainty many months before the flu season starts. We quantify the trade offs involved through multistage stochastic mixed-integer programs that determine the optimal flu shot composition and its timing in a stochastic and dynamic environment. Our first model takes the view of a social planner, and optimizes strain selections based on a production plan that is provided by the flu shot manufacturers. It also incorporates risk-sensitivity through mean-risk models. Our second model relaxes the exogenous production planning assumption and, hence, provides a more accurate representation of the hierarchical decision mechanism between a social planner, who selects the flu shot strains, and the manufacturers, who make the flu shot available. We derive structural properties of both models, and calibrate them using publicly available data. The flu shot strains are updated based on clinical, virological and immunological surveillance. In the virological surveillance, hemagglutinin inhibition assays are used to identify antigenic properties of the influenza viruses. However, this serology assay is labor-intensive and time-consuming. As an alternative, pairwise amino acid sequence comparison of influenza strains is used in statistical learning models to identify positions that cause antigenic variety. The performance of these models is evaluated by cross validation. In Chapter 5, we formulate cross validation as a bilevel program where an upper-level program chooses the model variables to minimize the out-of-sample error, and lower-level problems in each fold optimize in-sample errors according to their training data set by selecting the regression coefficients of the chosen model variables. We provide an extensive computational study using clinical data, and identify amino acid positions that significantly contribute to antigenic variety of influenza strains