China
Africa
Explore chapters and articles related to this topic
Integer Programming
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
A number of other approaches have been developed and refined over the years. Cutting plane and cover inequality methods repeatedly introduce new constraints into integer problems in order to exclude non-integer extreme points from the feasible region, and then use simple LP solutions to locate the optimum, which then occurs at an integer point. Lagrangian relaxation incorporates constraints into the objective function by placing a penalty on any violated constraint. Any solution that violates a constraint has a lower value than a solution with no constraint violation. The penalties must be chosen appropriately for the given problem. The technique of column-generation is applicable to problems such as vehicle routing and workforce scheduling, in which customers or workers must be assigned to trucks or work patterns. Incomplete initial solutions are iteratively built up into complete optimal solutions.
Advances in Inverse Planning Algorithm and Strategy
Published in Siyong Kim, John Wong, Advanced and Emerging Technologies in Radiation Oncology Physics, 2018
Masoud Zarepisheh, Baris Ungun, Ruijiang Li, Yinyu Ye, Stephen Boyd, Lei Xing
Column generation is a sequential optimization technique specialized to handle problems with a large number of variables. Its objective is to find the most beneficial variable(s) at each iteration by solving an extra optimization problem, usually referred to as the “pricing” problem, and then add that variable to the pool of existing variables. This technique has been employed in IMRT and VMAT to optimize the aperture shapes (Romeijn et al., 2005; Men et al. 2010; Peng et al., 2012) and can be adopted here for SPORT. At each iteration, the pricing problem searches through all plausible aperture shapes from all angles to find the “best” station and adds that to the pool of selected stations if it provides a sufficient improvement in the objective function (e.g., the relative improvement in the objective function is more than 1%).
General introduction
Published in Adedeji B. Badiru, Handbook of Industrial and Systems Engineering, 2013
Nevertheless, the numerical results showed that neither algorithm could provide solutions quickly, so developing more efficient computational methods would be valuable. We are currently looking into the use of column generation to get improved relaxed solutions and heuristics for converting them to feasible solutions. We believe that this approach will lead to at least a 2 -fold reduction in runtimes.
The integrated lot-sizing and cutting stock problem under demand uncertainty
Published in International Journal of Production Research, 2023
Eduardo Curcio, Vinícius L. de Lima, Flávio K. Miyazawa, Elsa Silva, Pedro Amorim
Gilmore and Gomory (1961, 1963, 1965) proposed a column generation algorithm for solving the linear relaxation of a set-covering model of the deterministic cutting stock problem, which is modelled considering all the possible cutting patterns. Basically, the column generation method is an iterative procedure that considers only a subset of variables/patterns and at each iteration it solves a subproblem that gives a new cutting pattern that improves the solution of the model. The column generation solves the model considering the linear relaxation of the integer variables related to cutting patterns. Hence, the column generation gives a lower bound for the optimal solution of the standard integer problem. At the end of the column generation algorithm, to obtain a feasible solution (upper bound), we solve the model with the subset of columns generated to optimally solve the linear relaxation by considering now the integrality of the variables related to the patterns. The resulting integer linear programming model (ILP) is solved with a state-of-the-art integer linear programming solver.
A column generation-based approach for proportionate flexible two-stage no-wait job shop scheduling
Published in International Journal of Production Research, 2020
Zhi Pei, Xuefang Zhang, Li Zheng, Mingzhong Wan
Column generation is a powerful tool to solve complex optimisation problems. The detailed literature review of this technique has been stated in Section 1. The basic idea behind the proposed CG method is to start with a small subset of easily identified feasible solutions to form up the master problem. Then we obtain the restricted master problem by relaxing the integer constraints of the master problem into linear ones. Next the restricted master problem (RMP) could be solved together with its dual optimal. A key step in the CG-based algorithm is to find a column that could improve the current optimal solution of the linear relaxation. During the column seeking process, the pricing problem with the minimum reduced cost is obtained. If the minimum reduced cost is negative, then a new column should be added to the restricted master problem. Otherwise, if the minimum reduced cost is nonnegative, the algorithm terminates and the optimal solution has been attained. In general, an iterative procedure is conducted with respect to the value of the minimum reduced cost.
Branch and price for covering shipments in a logistic distribution network with a fleet of aircraft
Published in Optimization Methods and Software, 2018
Column generation is a very efficient methodology for solving linear programs with a huge number of decision variables, which works by explicitly considering only a small subset of them. One of the landmark works on column generation is the paper by Barnhart et al. [5], in which the authors discuss various application areas, alternative model formulations, as well as several computational performance-related issues. They also illustrate how column generation can be embedded within the so-called branch and price framework, which comprises a suitable extension of branch and bound for solving integer programs with a huge number of decision variables. Successful implementations of column generation methodologies have been applied in various real world applications, such as crew scheduling, vehicle routing, location planning, etc.