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Linear Programming
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
There are two primary approaches that we can use to ensure that the artificial variables are not in the final solution. One method, commonly called the Big-M method, achieves this end by creating a modified objective function with huge negative coefficients −M on the artificial variables. In our example, the modified objective function would be zM= x1+3x2+0s1−MR1−MR2
Modeling Processes and Operations with Linear Programming
Published in Robert M. Peart, R. Bruce Curry, Agricultural Systems Modeling and Simulation, 2018
Since artificial variables are added for solution purposes only, they must be removed from the problem. One technique used to “force” artificial variables out of the solution to a linear programming problem is called the Big M method.
Adaptive relay co-ordination scheme for radial microgrid
Published in International Journal of Ambient Energy, 2022
Belwin J. Brearley, R. Raja Prabu, K. Regin Bose, V. Sankaranarayanan
In reference Gupta, Gnana Swathika, and Hemamalini (2015), relay co-ordination is done to minimise the operating time of relay using Big M method and dual simplex method. In Big M method artificial variables are involved, but it is not so in the dual simplex method. In reference Bedekar, Bhide, and Kale (2011), the optimal TMS and Plug Setting Multiplier (PSM) are identified for the co-ordination of relays using Linear Programming Problems (LPP) namely, revised simplex, dual simplex, two-phase simplex and Big-M method. In reference Kida and Gallego (2016), optimal TMS is obtained by applying mixed-integer linear programming and treating TMS as a discrete variable. This technique is very complex, since its dealing with discrete variables. In all these conventional optimisation techniques suggested for co-ordination of relays, weighting factors are chosen by trial and error method. Hence there is more possibility to converge at local minima and hence the relays will not be co-ordinated properly. Also in case of large networks with an increased number of relays, providing an optimal solution by conventional methods is very much complex.
New mathematical and constraint programming models for U-type assembly line balancing problems with assignment restrictions
Published in Engineering Optimization, 2022
ARs require the addition of new constraints that are difficult to model for the mathematical program of the ALBP. These constraints include logical expressions, such as either–or constraints. The big-M method is the most commonly used method for modelling logical constraints. Also, today's commercial software (e.g. IBM ILOG CPLEX) offers a new modelling method called indicator constraints (ICs) for more user-friendly modelling of logical constraints. Modelling with ICs could be more robust and accurate numerically than traditional mathematical modelling methods such as Big M.
Automated job shop scheduling with dynamic processing times and due dates using project management and industry 4.0
Published in Journal of Industrial and Production Engineering, 2021
Parsa Kianpour, Deepak Gupta, Krishna Kumar Krishnan, Bhaskaran Gopalakrishnan
This study proposes the mixed integer linear programming (MILP) model as the solution approach. This exact model was proposed for solving small and medium size problems. Considering the objective function (5), two decision variables have been multiplied to each other, which makes the equation quadratic. In order to use MILP, the objective function should be linearized first. Since one of the variables is continuous and the other is binary, the Big M method is used to linearize the objective function. Solving the MILP problem needs the combination of a continuous optimization algorithms (e.g. interior-point method) and an in-depth search algorithm (e.g. branch and bound (B&B)). The key steps of designing branch and bound algorithm are selection of lower bound, the usage of the initial solution as an upper bound and the branching method [34]. In some cases, the model can be solved globally after only few branches, while in other cases, the solver may fail to find a solution in a reasonable time. The depth-first search algorithm is used in this study. A similar approach was used in [35]. In this algorithm, jobs are assigned in a forward manner starting from the first position on the selected machine. In search tree, each node in each level (e.g. level ) indicates a partial schedule in which jobs are scheduled. Each parent node is partitioned into the child nodes corresponding to the assignment of unscheduled jobs to the first available sequence of the machine. With this branching mechanism, redundant scheduled could be generated. The method presented in [36] is used to avoid redundancy, which decreases the number of nodes generated in B&B algorithm significantly.