"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 are represented by linear relationships."
A mathematical optimization technique used to maximize or minimize a linear objective function, subject to a set of linear constraints.
Linear Programming: The foundation of optimization that seeks to maximize (or minimize) a linear objective function subject to linear inequality or equality constraints.
Simplex Algorithm: An iterative computational procedure that is used to solve linear programming problems by identifying the optimal solution.
Duality Theory: The relationship between primal and dual linear programming problems that helps in obtaining additional information and determining optimal solutions.
Sensitivity Analysis: The study of the effect of changes in the coefficients of the objective function and/or the constraints on the optimal solution.
Integer Programming: A linear programming problem that has additional constraints that require some or all decision variables to be integers.
Network Optimization: A linear programming problem that involves finding the shortest path, the maximum flow, or the minimum spanning tree in a network.
Transportation Problem: A linear programming problem that deals with the allocation of goods from a set of supply nodes to a set of demand nodes.
Assignment Problem: A linear programming problem that deals with assigning jobs to workers or machines in the most efficient way.
Game Theory: The study of conflict and cooperation among rational decision-makers using mathematical models that involve interaction, strategy, and payoff.
Stochastic Programming: An extension of linear programming that deals with a system under uncertainty using probability models.
Simplex Method: It is the most commonly used technique for solving linear programming problems. It involves finding the optimal solution by iteratively improving the candidate solution until no further improvement is possible.
Dual simplex method: It is a variation of the Simplex method where the dual problem of the original linear programming problem is solved. It is useful for problems where the number of constraints is much larger than the number of decision variables.
Two-phase method: It is a method used to solve linear programming problems that have artificial variables in the initial tableau. These artificial variables help in identifying the feasible region and are eliminated in the second phase of the method.
Interior point method: It is a method that uses algorithms to efficiently solve linear programming problems by exploring the feasible region of the problem. It is useful for solving large-scale linear programming problems with a large number of decision variables.
Branch and Bound Method: It is an algorithm used to solve integer programming problems. It involves dividing the feasible region into smaller sub-regions and solving each sub-region using the Simplex method.
Cutting plane method: It is a technique used to solve integer programming problems by adding additional constraints to the problem to limit the feasible region.
Genetic algorithm: It is a heuristic method used to solve linear programming problems by adapting the biological principles of natural selection and survival of the fittest.
Decomposition method: It is a method that takes advantage of the structure of a linear programming problem to break it down into smaller sub-problems that can be solved independently. This technique is useful for solving large-scale linear programming problems.
Parametric programming: It is a method that analyzes changes in an objective function or constraints while keeping other variables constant. This technique is useful for sensitivity analysis and decision-making under uncertainty.
Stochastic programming: It is a technique used to solve linear programming problems with uncertain parameters. It involves modeling uncertainties as probability distributions and finding a solution that minimizes the expected value of the objective function.
"Linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints."
"Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality."
"Its objective function is a real-valued affine (linear) function defined on this polyhedron."
"A linear programming algorithm finds a point in the polytope where this function has the smallest (or largest) value if such a point exists."
"Linear programs are problems that can be expressed in standard form as: 'Find a vector x that maximizes c^T x subject to A x ≤ b and x ≥ 0.'"
"The components of x are the variables to be determined."
"c and b are given vectors."
"A is a given matrix."
"The function whose value is to be maximized (x ↦ c^T x in this case) is called the objective function."
"The constraints A x ≤ b and x ≥ 0 specify a convex polytope over which the objective function is to be optimized."
"Linear programming can be applied to various fields of study."
"Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing."
"It is widely used in mathematics and, to a lesser extent, in business, economics, and some engineering problems."
"It has proven useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design."
"Linear programming (LP), also called linear optimization..."
"Linear programming is a special case of mathematical programming (also known as mathematical optimization)."
"Linear programming can be expressed in standard form as... x that maximizes c^T x subject to A x ≤ b and x ≥ 0."
"Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces..."
"A linear programming algorithm finds a point in the polytope where this function has the smallest (or largest) value if such a point exists."