Linear Programming

The mathematical method used to find the best possible output is called linear programming.

 

• It is used in situations involving linear relationships, such as the maximum profit or minimum cost in a business.
  
   
• It helps make decisions, such as maximizing profit or minimizing cost, using equations and inequalities.
  
   
• Furthermore, it is used in many fields to solve real-life problems.

Objective function, constraints, and decision variables are the components of linear programming. We will learn about them one by one.

 

• Objective Function: This is the main goal you want to achieve. For example, you might want to make the most money or spend the least.
  
   
• Constraints: These are the limits or rules you have to follow, like how much money, time, or materials you can use.
  
   
• Decision Variables: These are the choices you can make or the unknowns in the problem. For example, how many products to make or how many hours to work.

Formulating the problem using the given data is the first step for solving linear programming problems. The steps given below are used to solve linear programming problems.

 

Step 1: Identify the Decision Variables. Decide what choices you can make. For example, how many items to produce. These are your decision variables.


Step 2: Define the Objective Function. Decide what you want to achieve, like maximize profit or minimize cost. Write it as a simple equation using your decision variables.


Step 3: List the Constraints. Write down all the limits or rules, such as how much money, time, or materials you have.


Step 4: Make Sure Variables Are Non-Negative. You cannot produce negative items, so all decision variables must be zero or more. 


Step 5: Solve the Problem. Use a method like the graphical method (for small problems) or the simplex method (for bigger problems) to find the best solution.

Linear programming can be used to achieve the best result when we have limitations. These two methods mentioned below are used to solve linear programming:

 

• Graphical Method
  
   
• Simplex Method

The constraints (conditions or limits) are drawn on the graph in this method. Then we have to look for the common area where all the constraints are true. This is called feasible region. We can check the corners of the area to see where we can get the best result.


Example: Rohan has  $10 with which he wants to buy pencils worth $2 a piece and erasers worth $1 each. Rohan wants to buy at most 7 items in total. 


Solution: 

 

1. Let x be the number of pencils and y be the number of erasers.


2. Write the constraints. 


   \(2x + 1y \le 10\) (money limit).


    \(x + y \le 7\) (items limit).


    \(x \ge 0, \quad y \ge 0\), because the items cannot be negative.


3. Draw these lines on a graph.


4. Shade the area that fits all the constraints.


5. Try the corner points and calculate the cost.


6. Evaluate the objective function at each corner point of the feasible region and select the point giving the optimal value. Select the point that gives the lowest or highest value. The selection of this point depends on the objective.

The simplex method checks only one possible answer at a time. It helps in solving bigger problems. It keeps improving the answer until it can’t get better.


For example:


Imagine you have red and blue Lego bricks and want to build the tallest tower, but you can only use a certain number of each color. The simplex method helps you figure out how many red and blue bricks to use to make the tower as tall as possible.


The method uses a table called the simplex tableau to keep everything organized. At each step, it swaps one variable for another (called pivoting) to improve the solution. Think of it like trying different Lego bricks one at a time to see which combination gives the tallest tower. At the end, it finds the best possible solution.

Linear programming helps find the best possible outcome when working within limits. These tips and tricks simplify solving problems, making it easier for students to plan, analyze, and make smart decisions quickly and accurately.

 

• Read carefully and know what to maximize or minimize.
  
   
• Clearly define the choices you can control (x, y, z).
  
   
• Remember, ≤ means “at most” and ≥ means “at least.”
  
   
• Organize equations in a table for the simplex method.
  
   
• Ensure it satisfies all constraints and gives the optimal value.

Linear programming is a branch of mathematics that focuses on finding the best possible outcome in situations where certain conditions or limits must be considered. It helps in making optimal decisions while working within these constraints. Here are some real-life applications:

• Healthcare: In hospitals, It helps schedule shifts for staff, ensuring patients receive care and staff are not overworked.
  
   
• Aerospace: In aerospace, linear programming helps design flight paths, fuel use, and payload distribution to maximize efficiency and safety.
  
   
• Coding / Software Development: In coding, linear programming helps optimize algorithms and resource allocation, like distributing tasks in cloud computing or scheduling processes.
  
   
• Robotics: In robotics, linear programming helps to plan robot movements and task allocation to complete jobs efficiently without wasting energy.
  
   
• Computer Graphics & Animation: In graphics and animation, linear programming helps optimize rendering, resource use, and motion planning, ensuring realistic visuals with minimal computation.