Linear Programming
Linear programming (LP) is a mathematical tool that statisticians use to achieve optimal conditions within the available constraints. The term linear implies that we are using variables in the first degree, while programming is the operations we perform to achieve the desired optimal solutions.
On forming and subjecting linear objective functions to the constraints, we form linear inequalities-which we can now maximize and minimize to achieve optimum values. Note that an objective function is the actual mathematical equation we want to maximize or minimize based on the given constraints. It is usually represented in the form f(x,y)=ax+by+c.
Linear programming is widely applied in manufacturing, agriculture, and business, etc., to design systems that help obtain the best output using the available resources.
What is the difference between linear and non-linear programming?
The primary difference between linear and non-linear programming is that linear programming operations involve first-order objective functions while non-linear programming deals with higher degree objective functions. And that translates to different techniques of solving the problems.
In non-linear programming, the main method used is called the decision tree models. However, other simpler techniques like substitution and use of the method of Langrage's multipliers come in handy in special cases.
Decision tree model for non-linear programming
As the name suggests, this is a tree-like structure on which we model the expected outcomes, results, or the cost of resources. It's a technique used to develop and present control statements for the desired objective function.
Linear programming Solution Methods
There are three ey solution approaches for linear programming problems. They are:
- Optimization-simplex
- Graphical method
- Software programming tools like Open solver and R
In this post, we will discuss the first two.
Optimization-Simplex technique
The optimization-simplex method is perhaps the most common technique for solving linear programming problems. The idea is iterating through the objective function until you obtain the most feasible solution. And that involves a series of steps as outlined below:
- Come up with the suitable objective function to represent the problem at hand-using the constraints and variables given.
- Add a "slack variable" to the inequalities and replace the inequality signs with an equal sign—to form an equation.
- Come up with your first simplex tableau and place your inequality at the bottom row. Also, place each constraint in the equation on its own row. Next, form an augmented matrix by filling the table with the appropriate values.
- Check out the most negative value in your bottom row—that represents our pivot column.
- Quotients: To get quotients, divide those entries on your far most right column with those in the first column. Be keen not to include those on the bottom column. The least quotient in the row becomes our pivot element.
- Pivoting: Making all others within that column zero.
- End or continue iterating: The process ends if you get all positive values in the bottom row. If not, then iterate from step four.
- Identifying the solution: What does the final simplex tableau represent?
Graphical method
The graphical method is used explicitly in optimizing two-variable objective functions in linear programming. That is because we only use two axes in the X-Y plane.
The process is pretty straightforward:
- Subject inequalities to the given constraints
- Plot them on the graph
- Use the inequalities to identify the most feasible region represented by the different inequalities.
- All points within the feasible region represent optimal solutions for the objective function.
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