If you are an engineer or a student of engineering, you may know what it means to optimize.
To get the best possible result, it is important to find the best way to do things.
In linear programming, you can use a basic solution to find the best solution.
But what is a basic solution, and why is it so important for engineers to know about them? In this article, I'll talk about what basic solutions are, why they're important in engineering, and how they can be used to get the best results in different situations.
So buckle up and get ready to dive into the world of basic solutions, where I'll break down the mysteries and show you how powerful this technique can be.
Basic Solutions in Linear Programming
Formal definition:
A solution to a linear program model consisting of m equations in n variables is obtained by solving for m variables in terms of the remaining (n m) variables and setting the (n m) variables equal to zero.
A basic solution in linear programming is a way to solve a linear programming problem that meets certain technical requirements.
In particular, a vector x is a basic solution for a polyhedron if the vectors {ai : xi = 0} are linearly independent.
This means that the columns of A that have variables xi that are not zero are linearly independent.
A basic solution with nonnegative components is called a basic feasible solution (BFS) (BFS).
A BFS meets all of the rules that define a polyhedron.
Each BFS is a corner of the polyhedron of feasible solutions from a geometric point of view.
To find a basic solution, you must set n-m variables that aren't basic to zero and solve the m variables that are basic.
It is possible for different bases to lead to the same basic solution, which means that there may be more than one way to solve the same problem.
The Simplex Method is an iterative process that moves from one BFS to the next BFS until it finds the best BFS.
After using the simplex method to find a BFS, we can tell if the solution is the best one by seeing if any other BFSs nearby give a better value for the objective function.
If there is no such BFS, then the current BFS is the best one.
Linear Programming Model
A linear programming model involves three main components: decision variables, an objective function, and constraints.
Both the objective function and the constraints must be linear functions, and the decision variables must be continuous.
The objective function is used to either increase or decrease a number that represents profit, cost, number of products made, etc.
Constraints are limits or restrictions on the total amount of a certain resource that is needed to do the tasks that will determine the level of success in the decision variables.
In addition, some linear programs require that all decision variables be nonnegative.
In linear programming models, you can also use integer and binary variables.
Binary variables can only have a value of 0 or 1, so they can only have a value of 0 or 1.
The Simplex Method
One of the most used ways to solve linear programming problems is the Simplex Method.
Basic solutions are important in the simplex method because they correspond to the corner points of the feasible region, and the simplex method moves from one corner to another until an optimal solution is found.
The simplex method is a quick way to find the best answer to a linear programming problem by using the properties of basic solutions.
To use the simplex method to find the best BFS, we need to find a basis B for the constraint matrix A and solve the system Ax = b with all variables other than the basis set to zero.
The resulting values for the basic variables form a BFS.
If there exists an optimal solution, then there exists an optimal BFS.
The Simplex Method moves from one BFS to an adjacent BFS until it reaches an optimal BFS by using pivot procedures.
Comparison between Basic Solutions and Feasible Solutions
The difference between a basic solution and a feasible solution is that a basic solution doesn't have to meet any conditions.
In particular, it must have vectors that are linearly independent and have non-zero values for xi, and x must be less than 0.
On the other hand, a feasible solution is any point that fits within the limits of the problem.
But not all feasible solutions are basic feasible solutions.
Basic feasible solutions (BFSs) are only those that match the corners of the polyhedron of feasible solutions.
Back to Basics: Unlocking the Power of Basic Solutions in Engineering
Still hard to understand? Let me change the point of view a bit:
Are you sick of using complicated methods and algorithms to solve hard problems? Do you wish there was a simpler, more straightforward way to deal with your linear program model problems?
Well, don't worry, because the answer is here: solve for m variables in terms of the remaining (n m) variables, and set the (n m) variables to zero.
Who needs algorithms that sound fancy when you can go back to the basics? So put away your calculators and let's start learning about simple solutions.
Okay, that was just a joke made to look like a TV ad.
Now let's go back to the explanation.
Basic Solution Linear Programming
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Use cases
| Used in: | Description: |
|---|---|
| Allocation of Resources: | Basic solution can be used in resource allocation problems, where the goal is to divide limited resources among competing needs. For instance, a company might need to divide its budget between different departments or projects. By using basic solutions, they can figure out the best way to use their resources to make the most money or spend as little as possible. |
| Planning for Production: | In production planning, the basic solution can be used to figure out the best mix of products to make in order to make the most money. Companies can find the best production mix that brings in the most money and costs the least by using the basic solution. |
| Scheduling: | Basic solution can be used to figure out how to schedule tasks or jobs so that they can be done in the most efficient way. For example, a company may need to plan the work hours of its employees to make sure they have enough workers when business is busy. By using a basic solution, they can figure out the best way to schedule things so that there is as little downtime as possible and as much work gets done as possible. |
| Management of the Supply Chain: | In supply chain management, the goal is to make sure that goods and services move as smoothly as possible from the supplier to the customer. For example, a business may need to figure out the best routes for transporting goods so that costs are kept to a minimum and goods are delivered on time. By using basic solutions, they can find the best plan for managing the supply chain that keeps costs low and keeps customers happy. |
| Portfolio Optimization: | In portfolio optimization, where the goal is to find the best mix of investments to make the most money while taking the least risk, basic solutions can be used. For example, an investment firm may need to figure out the best mix of stocks, bonds, and other securities to help their clients reach their investment goals. By using a simple solution, they can find the best way to mix their portfolios so that they get the best returns while taking the least amount of risk. |
Conclusion
In conclusion, the idea of a basic solution is very important in the field of engineering and can be used in many different ways.
By knowing what a basic solution is and what it does in linear programming, we can improve solutions, cut costs, and make them more efficient.
But it's important to remember that basic solution is not a one-size-fits-all solution, even though it is a powerful tool.
To get the best results, each problem needs to be carefully looked at and thought about.
As engineers, we need to keep looking into how basic solutions and other optimization techniques can help us make progress and come up with new ideas.
So, let's recognize the power of simple solutions and keep pushing the limits of what's possible by using new techniques and strategies.
Links and references
Books:
- Linear Programming by Vasek Chvatal
- Modeling and Solving Linear Programming with R by Jose M. Sallan
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