Computational Enhancement in the Application of the Branch and Bound Method for Linear Integer Programs and Related Models (original) (raw)
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A Note on Branch and Bound Algorithm for Integer Linear Programming
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In branch and bound algorithm for integer linear programming the usual approach is incorporating dual simplex method to achieve feasibility for each sub-problem. Although one can also employ the phase 1 simplex method but the simplicity and easy implementation of the dual simplex method bounds the users to use it. In this paper a new technique for handling sub-problems in branch and bound method has been presented, which is an efficient alternative of dual simplex method.
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Integer linear programming (ILP) problems are harder to solve than linear programming (LP) problems. It doesn't work if try to round off the results of LP problems and claim they are the optimum solution. The branch-and-bound (B&B) is the popular method to solve ILP problems. In this paper, we propose a revised B&B, which is demonstrated to be more efficient most of time. This method is extraordinarily useful when facing ILP problems with large differences between constraints and variables. It could reduce the number of constraint and work efficiently when handling ILP problems with many constraints and less variables. Even if the ILP problems have fewer constraints but many variables, we suggest using duality concept to interchange variables with constraints. Then, the revised B&B could be used to compute results very quickly.
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This paper describes an experimental code that has been developed to solve zero-one mixed integer linear programs. The experimental code uses a primal{dual interior point method to solve the linear programming subproblems that arise in the solution of mixed integer linear programs by the branch and bound method. Computational results for a number of test problems are provided.
Improved branch and bound algorithm with a comparison between different LP solvers
2010 9th IEEE/IAS International Conference on Industry Applications, INDUSCON 2010, 2010
This paper presents a comparison of the effectiveness of different linear programming (LP) solvers when used with a branch-and-bound algorithm to find optimal solutions to mixed integer linear problems (MIP). We compare the performance of an improved implementation of the branch-and-bound algorithm with three different LP solvers in order to evaluate each one. We also compare our implementation with other available solvers, to obtain multiple optimal solutions when they exist. The comparison between those different situations was useful in order to prove the robustness and the efficiency of the developed algorithm.
This paper presents a branch and bound (B&B) algorithm for the 0-1 mixed integer knapsack problem with linear multiple choice constraints. The formulation arose in an application to transportation management for allocating funds to highway improvements. Several model properties are developed and utilized to design a B&B solution algorithm. The algorithm solves at each node of the B&B tree a linear relaxation using an adaptation of an existing algorithm for the linear multiple choice knapsack problem. The special relationship between the parent and children subproblems is exploited by the algorithm. This results in high e ciency and low storage space requirements. The worst case complexity of the algorithm is analyzed and computational results that demonstrate its e ciency in the average case are reported.
Computational results with a branch-and-bound algorithm for the general knapsack problem
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In this paper, a branch-and-bound procedure is presented for treating the general knapsack problem. The fundamental notion of the procedure involves a variation of traditional branching strategies as well as the incorporation of penalties in order to improve bounds. Substantial computational experience has been obtained, the results of which would indicate the feasibility of the procedure for problems of large size.