Linear Programming

A mathematical technique called linear programming (LP) is used to identify the optimal result (maximum or minimum) from a collection of linear equations and inequalities. It is frequently applied to resource optimisation issues in a variety of sectors, including operations research, manufacturing, finance, and logistics.

 

Table of Content

• Key Terminologies
• Mathematical Formulation
• Visual Approach to LP Solving
• Algebraic Methods
• Types of LP Problems
• Common Errors and Pitfalls
• Solved Examples
• Practice Questions
• Real-Life Applications
• Conclusion
• FAQs on Linear Programming

 

Key Terminologies

• The purpose of the objective

A linear function that must be minimised or maximised.

For instance, maximise Z=3x+4y.

• Limitations

Equations or linear inequalities that specify bounds or constraints.

For instance, x+2y≤100

• Region of Feasibility

the shared area on a graph where every constraint is present. Every point in this area satisfies every constraint.

• The Best Option

The best (maximum or minimum) value of the objective function is provided by the point in the feasible region.

 

Mathematical Formulation

• Use decision variables (such as x and y) to translate the problem statement into mathematical terms.

• Build the objective function according to what has to be optimised.

• Based on the limitations of the problem, write constraints.

• When the values represent quantities, such as money or items, include non-negativity conditions (e.g., x≥0, y≥0).

 

Visual Approach to LP Solving

• On a graph, depict each linear constraint as a straight line.

• Determine which feasible region satisfies all constraints and shade it.

• Find the feasible region's vertices, or corner points.

• Determine the objective function's value at every corner.

• The ideal solution is the point with the highest (or lowest) value.

 

Algebraic Methods

• The Simplex Method is a tabular technique that moves from one vertex to another to evaluate every potential algebraic solution.

• Slack variables were added in order to apply the Simplex method by transforming inequalities into equations.

• Utilised when graphical representation is impractical.
  
  

 

Types of LP Problems

• Maximization Problems
  Common in profit and output optimization.
  
  

• Minimization Problems
  Often found in cost-reduction scenarios.
  
  

• Balanced Problems
  Where resources match exactly with needs.
  
  

• Unbalanced Problems
  Involve excess or shortage of resources.

 

Common Errors and Pitfalls

• Failing to recognise all of the problem's constraints.

• Making inaccurate graph plots or shading areas that are not feasible.

• Omitting the non-negativity conditions.

• Assessing the objective function at points that are not corners.

• Ignoring redundant or inconsistent constraints.

 

Solved Examples

Example 1: Profit Maximization

A company makes two items A and B:

• A needs 1 hour labor, 2 units material.

• B needs 2 hours labor, 1 unit material.

• Max labor = 6 hours, material = 4 units.

• Profit: A = ₹3, B = ₹4
  
  

Let x = units of A, y = units of B

• Objective Function: Maximize Z=3x+4y

• Constraints:
  x+2y≤6
  2x+y≤4
  X≥0, y≥0
  
  

Plot the constraints, find feasible regions, evaluate Z at corner points.

 

Other Common Examples

• Diet Problem: Minimize cost while satisfying nutritional requirements.

• Transportation Cost Optimization: Minimize delivery costs from multiple warehouses to various destinations.

• Investment Planning: Allocate funds to get maximum returns under risk constraints.

 

Practice Questions

• Convert real-world scenarios (manufacturing, scheduling, budgeting) into LP models.

• Graphically solve 5 maximization problems with two variables.

• Graphically solve 5 minimization problems with two variables.

• Modify constraints and observe changes in the feasible region.

• Practice evaluating objective functions at feasible region corner points.

 

Real-Life Applications

• Supply Chain: Route planning, inventory control, and transport optimization.

• Finance: Risk-return analysis and portfolio optimization.

• Manufacturing: Decide optimal product mix given limited raw materials.

• Human Resource Allocation: Allocate staff or shifts to maximize efficiency or output.
  
  

Conclusion

A strong and methodical technique for resolving practical optimisation issues is linear programming. It enables decision-makers to determine the most effective outcomes under a specific set of constraints by converting real-world situations into mathematical models. Whether the goal is cost reduction, profit maximisation, or efficient resource allocation, linear programming provides planning clarity and accuracy. Finding the right answers requires a thorough grasp of problem formulation, graph constraints, and corner point evaluation. Gaining proficiency in linear programming can greatly improve analytical and decision-making abilities due to its numerous applications in sectors such as manufacturing, finance, logistics, and human resources.

 

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Frequently Asked Questions on Linear Programming

1. What is meant by linear programming?

Linear Programming is a mathematical method used to determine the best possible outcome - such as maximum profit or minimum cost - under given constraints. It involves optimizing a linear objective function subject to linear inequalities or equations.

 

2. What are the 7 requirements of linear programming?

The seven essential requirements are:

• Objective function must be linear

• Constraints must be linear

• Non-negativity of variables

• Finite choices (bounded region)

• Decision variables must be controllable

• Independence of constraints

• Feasible solutions must exist
  
  

3. What are the three types of linear programming?

The three main types of linear programming problems are:

• Maximization problems (e.g., maximize profit)

• Minimization problems (e.g., minimize cost)

• Balanced/unbalanced problems depending on whether available resources match the requirements
  
  

4. What is linear programming in real life?

In real life, linear programming is used in industries for optimizing resources. Examples include scheduling employees, maximizing crop yields, minimizing transportation costs, and allocating budgets effectively in business or manufacturing.

 

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