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Big M Method
In operations research, the Big M method is a method of solving linear programming problems using the simplex algorithm. The Big M method extends the simplex algorithm to problems that contain "greater-than" constraints. It does so by associating the constraints with large negative constants which would not be part of any optimal solution, if it exists. Algorithm The simplex algorithm is the original and still one of the most widely used methods for solving linear maximization problems. It is obvious that the points with the optimal objective must be reached on a vertex of the simplex which is the shape of feasible region of an LP (linear program). Points on the vertex of the simplex are represented as a basis. So, to apply the simplex algorithm which aims improve the basis until a global optima is reached, one needs to find a feasible basis first. The trivial basis (all problem variables equal to 0) is not always part of the simplex. It is feasible if and only if all the constr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex Algorithm
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial ''cones'', and these become proper simplices with an additional constraint. The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function. History George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator. During 1946, his colleague challenged him to mechanize the planning process to distract him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of Wassily Leontief, however, at tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operations Research
Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods to improve management and decision-making. Although the term management science is sometimes used similarly, the two fields differ in their scope and emphasis. Employing techniques from other mathematical sciences, such as mathematical model, modeling, statistics, and mathematical optimization, optimization, operations research arrives at optimal or near-optimal solutions to decision-making problems. Because of its emphasis on practical applications, operations research has overlapped with many other disciplines, notably industrial engineering. Operations research is often concerned with determining the extreme values of some real-world objective: the Maxima and minima, maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost). Originating in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slack Variable
In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it into an equality constraint. A non-negativity constraint on the slack variable is also added. Slack variables are used in particular in linear programming. As with the other variables in the augmented constraints, the slack variable cannot take on negative values, as the simplex algorithm requires them to be positive or zero. * If a slack variable associated with a constraint is ''zero'' at a particular candidate solution, the constraint is binding there, as the constraint restricts the possible changes from that point. * If a slack variable is ''positive'' at a particular candidate solution, the constraint is non-binding there, as the constraint does not restrict the possible changes from that point. * If a slack variable is ''negative'' at some point, the point is infeasible (not allowed), as it does not satisfy the constraint. Slack variables are also used i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Row Reductions
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855). To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: * Swapping two rows, * Multiplying a row by a nonzero number, * Adding a multiple of one row to another row. Using these operations, a matrix can always be transformed into an upper triangular matrix (possibly bordered by rows or columns of zeros), and in fact one that is in row echelon form. Once all of the leading ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Integer Linear Optimization
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0–1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems. If some decision variables are not discrete, the problem is known as a mixed-integer programming problem. Canonical and standard form for ILPs In integer linear programming, the ''canonical form'' is distinct from the ''standard form''. An integer linear program in canonical form is expressed thus (note that it is the \mathbf vector which is to be decided): : \begin & \underset && \mathbf^\mathrm \mathbf\\ & \text && A \mathbf \le \mathbf, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two Phase Method (linear Programming)
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and the only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures. Mathematics The number 2 is the second natural number after 1. Each natural number, including 2, is constructed by succession, that is, by adding 1 to the previous natural number. 2 is the smallest and the only even prime number, and the first Ramanujan prime. It is also the first superior highly composite number, and the first colossally abundant number. An integer is determined to be even if it is divisible by two. When written in base 10, all multiples of 2 will end in 0, 2, 4, 6, or 8; more generally, in any even base, even numbers will end with an even digit. A digon is a polygon with two sides (or edges) and two vertices. Two distinct points in a plane are always sufficient to define a unique line in a nontrivi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karush–Kuhn–Tucker Conditions
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. Similar to the Lagrange approach, the constrained maximization (minimization) problem is rewritten as a Lagrange function whose optimal point is a global maximum or minimum over the domain of the choice variables and a global minimum (maximum) over the multipliers. The Karush–Kuhn–Tucker theorem is sometimes referred to as the saddle-point theorem. The KKT conditions were originally named after Harold W. Kuhn and Albert W. Tucker, who first published the conditions in 1951. Later scholars discovered that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Optimization
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints are not linear equalities or the objective function is not a linear function. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) of an objective function over a set of unknown real variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear. Definition and discussion Let ''n'', ''m'', and ''p'' be positive integers. Let ''X'' be a subset of ''Rn'' (usually a box-constrained one), let ''f'', ''gi'', and ''hj'' be real-valued functions on ''X'' for each ''i'' in and each ''j'' in , with at least one of ''f'', ''gi'', and ''hj'' being nonlinear. A nonlinear programming problem is an optimization problem of the form : \begin \text & f(x) \\ \text ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dublin City University
Dublin City University (abbreviated as DCU) () is a Third-level education in the Republic of Ireland, university based on the Northside, Dublin, Northside of Dublin, Republic of Ireland, Ireland. Created as the ''National Institute for Higher Education, Dublin'' in 1975, it enrolled its first students in 1980, and was elevated to university status (along with the NIHE Limerick, now the University of Limerick) in September 1989 by Act of the Oireachtas, statute. In September 2016, DCU completed the process of incorporating four other Dublin-based educational institutions: the Church of Ireland College of Education, All Hallows College, Mater Dei Institute of Education and St Patrick's College, Dublin, St Patrick's College. As of 2020, the university has 17,400 students and over 80,000 alumni. In addition, the university has around 1,200 online distance education students studying through DCU Connected. There were 1,690 staff in 2019. Notable members of the academic staff inc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
