Notes from Toppers
Linear Programming
Introduction
- Introduction to linear programming:
- Linear programming is a method for solving optimization problems that involve linear functions.
- It is used in a wide variety of applications, including economics, business management, and engineering.
- Graphical method of solving linear programming problems:
- The graphical method is a simple method for solving linear programming problems involving two variables.
- It involves graphing the constraints and objective function and finding the point of intersection that gives the optimal solution.
Linear Programming
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Standard Form to a Canonical form:
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The standard form of a linear programming problem is a mathematical representation of the problem that is used for solving it using the simplex method. It is used for converting real-world applications to mathematical representations that can be easily solved.
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Solving linear programming problems using simplex method:
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The simplex method is an iterative method for solving linear programming problems.
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It starts at a feasible solution and moves to adjacent feasible solutions until it reaches the optimal solution.
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Duality in Linear Programming:
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Duality in linear programming refers to the relationship between a linear programming problem and its dual problem.
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The dual problem is formed by interchanging the roles of the variables and constraints in the original problem.
Special Cases of Linear Programming Problems
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Transportation problems:
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Transportation problems are linear programming problems that involve the transportation of goods from one location to another.
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The objective is to minimize the transportation cost while satisfying the demand at each destination.
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Assignment problems:
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Assignment problems are linear programming problems that involve assigning tasks to people or resources.
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The objective is to minimize the total cost or time required to complete the tasks.
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Travelling salesman problem:
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The travelling salesman problem is a linear programming problem that involves finding the shortest route for a salesperson who visits a set of cities and returns to the starting city.
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The objective is to minimize the total distance traveled by the salesperson.
Applications of Linear Programming
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In economics and business management:
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Linear programming is used in economics to solve problems such as resource allocation, production planning, and capital budgeting.
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It is also used in business management to solve problems such as inventory control, scheduling, and facility location.
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In the optimization of production and distribution:
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Linear programming is used in the optimization of production and distribution systems.
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It is used to determine the optimal production levels, the optimal distribution routes, and the optimal inventory levels.
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In the design of transportation systems:
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Linear programming is used in the design of transportation systems.
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It is used to determine the optimal routes for transportation vehicles, the optimal schedules, and the optimal number of vehicles.
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In the planning of resource allocation:
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Linear programming is used in the planning of resource allocation.
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It is used to determine the optimal allocation of resources to different projects or activities.
Mathematical Concepts Used in Linear Programming
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Linear inequalities:
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Linear inequalities are mathematical expressions that involve linear functions.
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They are used to represent the constraints in linear programming problems.
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Convex sets:
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Convex sets are sets that have the property that any two points in the set can be connected by a straight line that lies entirely within the set.
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Convex sets are important in linear programming because the feasible region of a linear programming problem is a convex set.
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Hyperplanes:
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Hyperplanes are flat, multi-dimensional surfaces that divide space into two half-spaces.
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Hyperplanes are important in linear programming because they are used to represent the constraints in linear programming problems.
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Simplexes:
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Simplexes are polytopes that have n + 1 vertices in n dimensions.
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Simplexes are important in linear programming because the feasible region of a linear programming problem is a simplex.
Solving Techniques for Linear Programming
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The graphical method:
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The graphical method is a simple method for solving linear programming problems involving two variables.
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It involves graphing the constraints and objective function and finding the point of intersection that gives the optimal solution.
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The simplex method:
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The simplex method is an iterative method for solving linear programming problems.
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It starts at a feasible solution and moves to adjacent feasible solutions until it reaches the optimal solution.
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The revised simplex method:
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The revised simplex method is a variant of the simplex method that is more efficient for solving large-scale linear programming problems.
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It involves using a different set of pivot rules to select the variables that enter and leave the basis.
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Duality theory:
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Duality theory is a mathematical theory that relates a linear programming problem to its dual problem.
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Duality theory can be used to solve linear programming problems by solving their dual problems.
Extensions and Generalizations of Linear Programming
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Nonlinear programming:
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Nonlinear programming is a generalization of linear programming that allows for nonlinear functions in the objective function and/or the constraints.
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Nonlinear programming problems are more difficult to solve than linear programming problems, but they can be used to model a wider range of real-world applications.
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Integer programming:
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Integer programming is a generalization of linear programming that requires the variables to be integers.
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Integer programming problems are more difficult to solve than linear programming problems, but they can be used to model a wider range of real-world applications.
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Dynamic programming:
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Dynamic programming is a technique for solving optimization problems that involve multiple stages.
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Dynamic programming can be used to solve linear programming problems by breaking them down into a sequence of smaller problems that can be solved more easily.
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Game theory:
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Game theory is a branch of mathematics that studies the interactions between decision-makers.
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Game theory can be used to solve linear programming problems by modeling the interactions between the decision-makers as a game.