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Integer linear programming for mining systems
Published in Amit Kumar Gorai, Snehamoy Chatterjee, Optimization Techniques and their Applications to Mine Systems, 2023
Amit Kumar Gorai, Snehamoy Chatterjee
Integer linear programming (ILP) solves linear optimization models containing variables that are restricted to integer values. ILP also provides a mechanism for handling nonlinearities such as fixed costs within an LP format. In ILP, at least one of the variables is restricted to integer values. ILP problems can be classified into different types, as demonstrated in Figure 5.1.Pure Integer Programming (PIP): In PIP, all decision variables are restricted to integer values.Mixed Integer Programming (MIP): In MIP, some but not all of the decision variables are restricted to integer values.Binary Integer Programming (BIP): In BIP, all decision variables are restricted to binary values, i.e., 0 or 1.Integer Knapsack (IK): The knapsack problem is a problem in combinatorial optimization in which finding an optimal solution is done from a finite set of alternatives.
Integer Programming
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
Branch-and-bound algorithms are widely considered to be the most effective methods for solving medium-sized general integer programming problems. These algorithms make no assumptions about the structure of a problem except that the objective function and the constraints must be linear. Even these restrictions can be relaxed without changing the basic framework of the technique.
Voltage/Var
Published in James A. Momoh, Adaptive Stochastic Optimization Techniques with Applications, 2015
The integer programming technique is a mathematical optimization of feasibility region where the decision variables can take only integer values. The most common approach is branch and bound technique. The procedure includes initialization, branching, bounding, fathoming, and optimality test.
An automated topology optimization interpreter that generates space frames
Published in Mechanics Based Design of Structures and Machines, 2023
David H. Myszka, Camden L. Ives, James J. Joo
Density-based, structural TO methods discretize the design domain into a mesh of Ne elements and the density of each element serves as the Ne design variables (Sigmund and Maute 2013). The problem was initially formulated as a nonlinear integer problem, where each element e is constrained to a material density ρe of either 1 (solid material) or 0 (void) (Bendsøe and Kikuchi 1988). This integer-based optimization was found to be ill-conditioned in structural compliance related problems (Sigmund and Petersson 1998). Additionally, large-scale integer programming problems are non-convex and computationally expensive as they require a systematic and potentially exhaustive search. To establish a tractable process, Bendsøe (1989) introduced an artificial density approach, which expresses the density as continuous variables between 0 and 1 and includes a penalty function for intermediate values. The method with most widespread adoption is called Solid Isotropic Microstructure with Penalization (SIMP), which penalizes the stiffness tensor of the material to the density with a power larger than 1. Other penalization and interpolation methods include Rational Approximation of Material Properties (RAMP) and the hyperbolic sine function (SINH) (Sigmund and Maute 2013). Beyond the density-based methods, other TO methods include hard-kill, such as Evolutionary Structural Optimization and boundary variation methods, such as level-set (Deaton and Grandhi 2014).
Fixed-point iterative approach for solving linear Diophantine systems with bounds on the variables
Published in Cyber-Physical Systems, 2022
Haocheng Yu, Luyao Yang, Jinyu Dai, Baoping Jiang, Zhengtian Wu, Shuxian Zhu
When solving linear Diophantine systems, if we can rewrite the problem in the form of an integer programming problem, then it can be easily solved using the fixed-point iterative approach. Therefore, on the basis of the algorithm proposed by Aardal et al. [12,13], we transform linear Diophantine equations into an integer programming problem. Aardal et al. proved that Problem (2) has an integer solution if and only if an integer solution exists in the new problem that is transformed above. Then, the fixed-point iteration method is used to determine whether integer points are present in the new problem. One of the most commonly used methods in solving integer programming problems is the branch-and-bound method. It has the advantage of fast computation speed to obtain the optimal solution. The results show that compared with the branch-and-bound method, our method is effective and feasible, and it exhibits advantages in dealing with high-dimensional problems.
Two heuristic methods based on decomposition to the integrated multi-agent supply chain scheduling and distribution problem
Published in Optimization Methods and Software, 2022
In 1962, Benders [5] proposed a type of decomposition method which was called the Benders decomposition to solve integer and mixed-integer complicated problems. Solving integer programming problems is much harder than linear problems due to integer variables. That is why most combinatorial problems are in the class of NP. In the Benders algorithm, based on dual theory, we split the original problem into two problems: the master problem (MP) and the sub-problem (SP). First, the master problem is solved, without any or few constraints. Then we put solutions of the master problem to the sub-problem. The dual sub-problem (DSP) is formulated and dual variables of sub-problem are obtained. Now, after an iteration of the algorithm, the termination condition of the algorithm has to be checked. If the termination conditions are satisfied, the algorithm is stopped and otherwise, an optimal Benders cut is added to the master problem. Again the master problem is solved and this process is repeated. Each time the conditions are not satisfied, a cut is added to the master problem. Recently, Rahmaniani et al. [26] has presented a comprehensive review of the published papers of Benders decomposition method. They categorize this approach based on solution generation, decomposition strategy, solution procedure, and cut generation. Also, some applications of this approach that have been used in previous papers, such as production planning and routing are presented.