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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
Linear Programming (LP) is a branch of optimization that minimizes (or maximizes) a linear objective function subject to linear equality and linear inequality constraints (Cococcioni and Fiaschi, 2021). The LP is a widely applied technique both in theoretical and real-world contexts, especially in engineering applications. The simplex algorithm is one of its most used algorithms, which was proposed by Kantorovič and Dantzig in the 1940s. Dantzig independently developed formulation of general linear programming during 1946–1947 to use for the US Air Force planning problems (Dantzig and Thapa, 1997). At the same time, Dantzig (1948) invented the simplex algorithm for solving the linear programming problem efficiently. Subsequently, he planned for solving the industrial and business problems. The simplex method has been applied to solve linear problems in different fields like agriculture, human resources and manufacturing decision making, and so on, to optimize maximum profit and minimize cost. The same method is used in Excel solver to solve linear programming problems. Most researchers in this field (Kurtz, 1992; Taha, 2008; Ezema and Amakon, 2012) applied linear programming in the allocation of scarce resources in the manufacturing industry to boost the output.
Linear Programming
Published in Anindya Ghosh, Prithwiraj Mal, Abhijit Majumdar, Advanced Optimization and Decision-Making Techniques in Textile Manufacturing, 2019
Anindya Ghosh, Prithwiraj Mal, Abhijit Majumdar
LP methods are some of the most popular, versatile, and commonly used approaches that provide optimal solutions to the problems with linear objective and constraint functions. This chapter presents a detailed application of graphical and simplex methods of solving linear programs with suitable examples related to textile manufacturing, along with coding in MATLAB® language. In the graphical method, initially a feasible region is identified in a two-dimensional plane, after satisfying all of the constraints simultaneously. Then, by a trial-and-error method, a point is located in the feasible region whose coordinate gives the optimum value. Though the graphical method is easy to understand, it can only be used when the number of variables in a linear problem is limited to two. Simplex methods are found to be very efficient in solving LP problems of any magnitude. The simplex method begins with a basic feasible solution and at each iteration it projects an improved solution over the earlier step. A solution is considered optimum when there is no further improvement. Simplex methods require formulation of the LP in a standard form in which all the inequality constraints are converted into equality constraints by using slack, surplus, or artificial variables. Then key operations at each iteration are performed to obtain the optimal solution. Published works on application of LP in textile manufacturing are also illustrated in this chapter.
Linear Programming
Published in Shyam S. Sablani, M. Shafiur Rahman, Ashim K. Datta, Arun S. Mujumdar, Handbook of Food and Bioprocess Modeling Techniques, 2006
This important theorem indicates that the search for an optimal solution to a LP problem can be confined to the evaluation of the objective function at a finite set of vertex and corner points of the feasible region of the problem. Indeed, the simplex method, which is an efficient, successful and widely used algorithm to solve LP problems, searches for an optimal solution by proficiently iterating from one vertex or corner point to another, and the value of the objective function increases (in a maximum problem, or decreases in a minimum problem) with each iteration, until it converges to its optimal value. A presentation of the (relatively simple) simplex algorithm is beyond the scope of the current chapter and can be found elsewhere.4,5 Next, an example of a minimum-optimization problem, also known as a food-blending problem, is discussed. This problem was chosen to highlight the principles; therefore, the complexity of a typical nutritional formulation was circumvented for the sake of simplicity.
Blocking OMARS designs and definitive screening designs
Published in Journal of Quality Technology, 2023
To obtain orthogonal blocking arrangements for a given OMARS design, we adopt a mixed integer linear programming approach. Linear programming (LP) is a common method to determine the values of a set of decision variables so as to optimize a particular linear objective function, while satisfying a set of linear constraints. When all the variables in the solution are required to be integer, the method is called integer linear programming (ILP). It is called mixed integer linear programming (MILP) if only some of the variables are required to be integer. In design of experiments, ILP was used earlier by Bulutoglu and Margot (2008) to classify orthogonal arrays, by Sartono et al. (2015a) and Vo-Thanh et al. (2018) to block given regular and nonregular orthogonal designs, by Capehart et al. (2011) to construct regular two-level split-plot designs, by Sartono et al. (2015b) to construct more general orthogonal fractional factorial split-plot designs, by Núñez Ares and Goos (2019) to identify trend-robust run orders for standard experimental designs, and by Vo-Thanh et al. (2020) to find optimal row-column arrangements of two-level orthogonal designs. In this section, we modify the MILP approach of Sartono et al. (2015a) to find good blocking arrangements for OMARS designs, and the special case of DSDs.
A review of different optimisation techniques for solving single and multi-objective optimisation problem in power system and mostly unit commitment problem
Published in International Journal of Ambient Energy, 2021
D. V. N. Ananth, K. S. T. Vineela
Linear and non-linear programming: The linear programming (LP) is a mathematical programming solving technique for either maximising or minimising te objective function based on optimisation pattern represented in a linear relation format. Simplex method is a type of LP problem is based on polytope edges of visualisation for attaining optimal results. LP in power systems is widely used in economic load sharing, unit commitment, distributed generator optimal placing and sharing, reactive power control and loss minimisation. In order to solve power balancing, reliability and cost equations here mostly the variables are continuous and discrete objective function where the constraints involved are mostly non-linear, non-linear programming methods are widely used. All the linear programming optimal problems can also be solved using non-linear programming methods also. The basic disadvantage of non-linear programming methods is long computational time and very large number of decision variables. Many software tools like MATLAB, MATPOWER, DIGISOFT, CPLEX etc., are used for solving these.
An effective computational attempt for solving fully fuzzy linear programming using MOLP problem
Published in Journal of Industrial and Production Engineering, 2019
Linear programming (LP) is one of the most frequently applied operations research techniques in real-world problems, which is used as a quantitative tool for optimal allocation of limited resources among competing activities. Fuzzy set theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally incorporate the imprecision related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. The fundamental challenge in FLP is to construct an optimization model that can produce the optimal solution with imprecise data. Therefore, a large number of researchers have devoted their efforts to the area of FLP problems [6,12–22].