- Label propagation
- Language identification
- Language modeling
- Language translation
- Large margin classifiers
- Latent Dirichlet allocation
- Latent semantic analysis
- Layer-wise relevance propagation
- Learning from imbalanced datasets
- Learning from noisy data
- Learning to rank
- Lexical analysis
- Linear algebra
- Linear discriminant analysis
- Linear dynamical systems
- Linear programming
- Linear regression
- Linear-quadratic-Gaussian control
- Link analysis
- Link prediction
- Local binary patterns
- Local feature extraction
- Locality-sensitive hashing
- Logical inference
- Logical reasoning
- Logistic regression
- Long short-term memory networks
- Low-rank matrix completion
- Low-rank matrix factorization
What is Linear programming
Introduction to Linear Programming
Linear programming is a mathematical optimization technique that is used in solving different types of complex problems that could be expressed as linear equations. It is used in a wide range of fields and industries such as economics, engineering, management, science and technology, and manufacturing, among others. Linear programming is concerned with finding the optimal solution for a given problem with a number of linear constraints using mathematical models and algorithms.
History of Linear Programming
The history of linear programming can be traced back to the early 1930s when the mathematician Leonid Kantorovich first introduced the concept of linear programming as a method for economic planning and allocation of resources in the Soviet Union. However, it was not until the late 1940s and early 1950s that the concept gained widespread recognition and started being used in different applications.
The Basic Concepts of Linear Programming
The basic concepts of linear programming include the objective function, the decision variables, the constraints, and the feasible region.
Objective Function:The objective function is the mathematical expression that defines the goal or objective of the problem. The goal of the problem could be to minimize or maximize a certain value. The objective function is usually expressed as a linear combination of the decision variables. For example, a company may want to minimize the cost of shipment or maximize the profit of a product.
Decision Variables:The decision variables are the unknown quantities that the problem is trying to solve. They are usually represented by algebraic symbols and could be any real number. The decision variables are the variables that the algorithm will manipulate to find the optimal solution. For example, in a manufacturing environment, the decision variables could be the quantity of raw materials used, the number of machines required, or the capacity of the production line.
Constraints:The constraints are the linear equations that define the limitations or bounds of the problem. They are used to restrict the decision variables to a certain range of values. For example, in a manufacturing environment, the constraints could be the maximum capacity of the machines, the minimum quality standards, or the maximum quantity of waste produced.
Feasible Region:The feasible region is the region of values that satisfy all the constraints and are feasible solutions to the problem. The feasible region could be a point, a line, or a region in a multi-dimensional space.
Linear Programming Models
Linear programming models are mathematical schemes of how a linear programming problem should be defined and solved. The three common types of linear programming models include the transportation, the assignment, and the production models.
The Transportation Model:The transportation model is normally used in logistics and transportation management. It deals with the problem of transporting goods from different sources to different destinations while minimizing transportation costs. The objective function of the transportation model is usually the total transportation cost, and the constraints are represented by the supply and demand requirements.
The Assignment Model:The assignment model is used in personnel and job assignment. It deals with the problem of assigning different tasks or jobs to available personnel while minimizing the total assignment cost. The objective function of the assignment model is usually the total assignment cost, and the constraints are represented by the availability and compatibility of the personnel.
The Production Model:The production model is used in manufacturing and production operations. It deals with the problem of optimizing the production process given limited resources. The objective function of the production model is usually the profit or the cost, and the constraints are represented by the resource availability and production requirements.
Linear Programming Algorithms
Linear programming algorithms are mathematical procedures used to solve linear programming problems. The algorithms work by manipulating the decision variables iteratively while checking the constraints and objective function. The three common types of linear programming algorithms include the simplex method, the interior point method, and the primal-dual method.
The Simplex Method:The simplex method is a popular linear programming algorithm that is used in solving problems with a few decision variables. The approach works by selecting a decision variable that has a non-zero coefficient in the objective function and then using it to eliminate a constraint. The process continues iteratively until an optimal solution is reached. The simplex method has been widely used in different industries such as finance, transportation, and manufacturing.
The Interior Point Method:The interior point method is a linear programming algorithm that is used in solving problems with a large number of decision variables. The approach works by transforming the problem into an unconstrained problem that can be solved using Newton's method. The interior point method is used in different applications such as chemical engineering, telecommunications, and astronomy.
The Primal-Dual Method:The primal-dual method is a linear programming algorithm that is used in solving problems with a large number of constraints. The approach works by introducing a dual problem that is formed by dualizing the constraints and then using a solver to find the optimal solution. The primal-dual method is used in different applications such as network flow optimization, scheduling, and electricity pricing.
Conclusion
Linear programming is a crucial tool that helps in solving complex problems in different industries and fields. Its impact has been felt in logistics, manufacturing, finance, engineering, and telecommunications, among others. The use of linear programming models and algorithms has significantly improved efficiency, productivity, and quality of production in different sectors. The development of the simplex method, the interior point method, and the primal-dual method has provided different approaches for solving linear programming problems. However, the choice of the algorithm will depend on the number of decision variables, the number of constraints, and the complexity of the objective function.