Linear programming is a mathematical technique that is used to optimize the allocation of limited resources among competing demands. The goal of linear programming is to find the best solution to a problem, given a set of constraints and a set of objectives.
It is widely used in business, engineering, and science to make decisions about production planning, inventory management, transportation, and resource allocation. In this article we discussed what is linear programming and how it works.
What is linear programming?
At its core, linear programming involves creating a mathematical model of a problem, which consists of a set of decision variables, a set of constraints, and an objective function.
Decision variables are the things that we want to optimize, such as the number of units to produce or the number of resources to allocate. Constraints are the limitations that we must work within, such as limited resources or production capacity.
Finally, the objective function is the measure of success that we want to maximize or minimize, such as profit or cost.
The process of solving a linear programming problem involves several steps. First, we define the decision variables and the objective function. Next, we identify the constraints and write them as linear equations or inequalities.
Then, we graph the constraints on a coordinate plane and find the feasible region, which is the set of all possible solutions that satisfy the constraints. Finally, we use an optimization algorithm to find the optimal solution within the feasible region.
Let’s walk through an example of how linear programming works in practice.
Suppose we own a factory that produces two types of products: A and B. Each unit of product A requires two hours of labor and one unit of raw material, while each unit of product B requires three hours of labor and two units of raw material.
We have 500 hours of labor and 300 units of raw material available per week. Product A sells for $10 per unit and product B sells for $15 per unit. How many units of each product should we produce to maximize our profit?
To solve this problem using linear programming, we first define our decision variables. Let x be the number of units of product A that we produce, and let y be the number of units of product B that we produce. Our objective function is to maximize our profit, which is given by:
- Profit = 10x + 15y
- Next, we identify our constraints. We have a constraint on labor hours: 2x + 3y ≤ 500
- And we have a constraint on raw materials: X + 2y ≤ 300
- Finally, we have constraints that restrict the number of units we can produce: X ≥ 0 and Y ≥ 0
- We graph these constraints on a coordinate plane and find the feasible region, which is shown in the figure below.
Feasible Region
The feasible region is the shaded area in the figure. It represents all possible combinations of x and y that satisfy our constraints.
The next step is to use an optimization algorithm to find the optimal solution within the feasible region. One common algorithm is the simplex method, which involves iteratively improving the objective function until we reach the optimal solution.
Using the simplex method, we find that the optimal solution is to produce 100 units of product A and 50 units of product B. This will result in a profit of $1750 per week.
Linear programming is a powerful tool that can help businesses and organizations make more informed decisions.
By modeling complex problems in a mathematical framework, we can find the best solution given limited resources and competing demands. With modern software tools, linear programming is more accessible than ever, and it can be applied to a wide range of real-world problems.
Conclusion
Linear programming is a valuable mathematical tool that can help businesses and organizations optimize their decision-making processes. By creating a mathematical model of a problem and using optimization algorithms, we can find the best solution given a set of constraints and objectives.
Linear programming has applications in a wide range of fields, including business, engineering, and science. With the increasing availability of software tools, linear programming is becoming more accessible and easier to use than ever before.
By incorporating linear programming into our decision-making processes, we can make more informed and efficient decisions, leading to improved outcomes and increased profitability.