What is Linear Programming? Methods and Problems

Linear programming is a mathematical technique used to identify the optimal solution within a model that contains linear relationships. It helps maximise or minimise a specific objective, such as profit, cost, or output, while considering various constraints like time, budget, or resource availability.

In simple terms, linear programming helps answer questions like:

• How can a company maximise profit using limited resources?
• What is the most cost-effective way to manufacture products?
• How should a business allocate budget across different channels?
• What combination of variables produces the best result?

Linear programming focuses on optimising an objective function while satisfying a set of constraints.

Key Idea

Linear programming solves problems where:

• The objective is linear (for example, maximise profit)
• Constraints are linear equations or inequalities
• Decision variables are continuous values

For instance, a manufacturer may want to determine how many units of product A and product B to produce to maximise profits while considering labour and material constraints.

To understand what is linear programming, it is important to break down its key components.

1. Decision Variables

Decision variables represent the quantities we want to determine.

Example:
Let:
x = number of units of product A
y = number of units of product B

These variables help define the solution.

2. Objective Function

The objective function defines what needs to be optimised — either maximised or minimised.

Example:

Maximise:
Z = 40x + 30y

Here, Z represents the total profit. This means each unit of product A contributes ₹40 towards profit, while each unit of product B contributes ₹30. The goal of linear programming is to determine the values of x and y that maximise the value of Z, while satisfying all given constraints.

3. Constraints

Constraints represent limitations such as budget, labour, raw materials, or time.

Example:

2x + y ≤ 100 (labour constraint)
x + 3y ≤ 90 (material constraint)

Constraints ensure the solution is realistic and feasible.

4. Non-negativity Restriction

Decision variables cannot be negative.

x ≥ 0
y ≥ 0

Negative production quantities do not make practical sense.

Linear programming is widely used because it helps organisations make optimal decisions using limited resources.

Benefits of Linear Programming

• Improves efficiency in resource allocation
• Helps maximise profit or minimise cost
• Supports strategic planning
• Enables data-driven decision making
• Simplifies complex decision problems
• Provides clear quantitative solutions

Professionals in operations, supply chain, finance, analytics, and engineering frequently rely on linear programming problems to optimise performance.

Linear programming models operate under certain assumptions:

1. Proportionality

The contribution of each decision variable is directly proportional to its value.

2. Additivity

The total value of the objective function equals the sum of individual contributions.

3. Divisibility

Decision variables can take fractional values.

4. Certainty

All coefficients are known with certainty and remain constant.

Formulating linear programming problems involves converting real-life scenarios into mathematical expressions.

Example Problem

A company produces two products: A and B.

Profit per unit:

Product A = ₹50
Product B = ₹70

Constraints:

Product A requires 2 hours of labour
Product B requires 3 hours of labour

Total available labour = 120 hours

Material constraint:

Product A requires 4 units of material
Product B requires 2 units of material

Total material available = 160 units

Step 1: Define Decision Variables

Let:

x = units of product A
y = units of product B

Step 2: Objective Function

Maximise Profit:

Z = 50x + 70y

Step 3: Constraints

Labour constraint:

2x + 3y ≤ 120

Material constraint:

4x + 2y ≤ 160

Non-negativity constraint:

x ≥ 0
y ≥ 0

There are different approaches used to solve linear programming problems depending on complexity.

1. Graphical Method

The graphical method is one of the simplest ways to solve linear programming problems involving two variables.

*Calcworkshop.com

Steps:

1. Plot constraints on a graph
2. Identify the feasible region
3. Find corner points
4. Evaluate the objective function
5. Select optimal value

The feasible region represents all possible solutions satisfying constraints.

The optimal solution lies at one of the vertices (corner points).

Advantages

• Easy to understand
• Visual representation
• Suitable for small problems

Limitation

Cannot be used for more than two variables.

2. Simplex Method

The linear programming simplex method is a mathematical procedure used to solve complex problems involving multiple variables.

Developed by George Dantzig in 1947, the simplex method systematically evaluates feasible solutions until the optimal solution is found.

How the Simplex Method Works

The simplex method:

• Converts inequalities into equations
• Uses iterative calculations
• Moves from one feasible solution to another
• Improves the objective function value at each step
• Stops when the optimal solution is reached

The simplex method is widely used in:

• Logistics optimisation
• Production planning
• Financial modelling
• Portfolio optimisation

3. Interior Point Method

The interior point method is another approach used to solve large-scale linear programming problems efficiently.

Unlike the simplex method, which moves along edges of the feasible region, the interior point method moves through interior points.

This method is often used in machine learning optimisation models.

Understanding various types of linear programming problems helps in applying appropriate methods.

1. Maximisation Problems

The objective is to maximise value.

Example:

Maximise profit
Maximise output
Maximise efficiency

2. Minimisation Problems

The objective is to minimise value.

Example:

Minimise cost
Minimise risk
Minimise delivery time

3. Transportation Problems

Used to minimise transportation cost between supply and demand locations.

Widely used in logistics and supply chain management.

4. Assignment Problems

Used to assign resources to tasks efficiently.

Example:

Assign employees to projects
Assign machines to production tasks

5. Diet Problems

Used to determine an optimal diet plan at minimum cost while meeting nutrition requirements.

Linear programming is widely applied across industries.

1. Business and Finance

Companies use linear programming to:

• Maximise profits
• Minimise operational costs
• Optimise investment portfolios
• Allocate budgets

Example:
Determining the best combination of products to maximise revenue.

2. Supply Chain Management

Logistics companies use linear programming to:

• Optimise transportation routes
• Minimise delivery time
• Reduce fuel costs
• Manage warehouse inventory

3. Manufacturing Industry

Manufacturers use linear programming to:

• Optimise production schedules
• Allocate labour efficiently
• Reduce waste
• Maximise output

4. Marketing Optimisation

Marketing teams use linear programming to:

• Allocate advertising budget
• Optimise campaign performance
• Maximise ROI

5. Healthcare Industry

Hospitals use linear programming to:

• Schedule staff shifts
• Allocate medical resources
• Optimise patient care systems

6. Agriculture Planning

Farmers use linear programming to:

• Determine crop combinations
• Maximise yield
• Optimise fertiliser usage
• Manage irrigation planning

Let us consider a simple linear programming problem.

A company manufactures chairs and tables.

Profit:

Chair = ₹20
Table = ₹30

Constraints:

A chair requires 3 units of wood
A table requires 5 units of wood

Total wood available = 90 units

Labour:

Chair requires 2 hours
The table requires 4 hours

Total labour hours = 80

Step 1: Define variables

x = number of chairs
y = number of tables

Step 2: Objective function

Maximise:

Z = 20x + 30y

Step 3: Constraints

3x + 5y ≤ 90
2x + 4y ≤ 80

x ≥ 0
y ≥ 0

Solving the above constraints using the graphical or simplex method gives the optimal solution at:

x = 20 chairs
y = 6 tables

Maximum Profit:

Z = 20(20) + 30(6)
Z = 400 + 180
Z = ₹580

Therefore, the company should produce 20 chairs and 6 tables to achieve the maximum profit of ₹580, while satisfying both wood and labour constraints.

Linear programming provides several advantages:

• Helps with optimal decision-making
• Improves resource utilisation
• Reduces operational cost
• Provides a structured mathematical approach
• Useful in solving complex problems
• Widely applicable across industries

Despite its advantages, linear programming has certain limitations.

1. Assumes Linear Relationships

Real-world relationships may not always be linear.

2. Requires Accurate Data

Incorrect data leads to incorrect results.

3. Ignores Qualitative Factors

Human behaviour and market uncertainty cannot always be quantified.

4. Complexity in Large Models

Large problems require advanced computational tools.

Knowledge of linear programming is highly valuable for professionals working in:

• Data analytics
• Business analytics
• Operations research
• Supply chain management
• Financial modelling
• Artificial intelligence
• Machine learning

As organisations increasingly rely on optimisation techniques, professionals with strong analytical and problem-solving skills are in high demand.

Many advanced management programmes now include optimisation techniques such as the linear programming simplex method to build decision-making expertise.

Several software tools help solve linear programming problems efficiently.

Popular Tools & Libraries

• Microsoft Excel Solver: Best for beginners and small-scale business problems.
• Python (SciPy, PuLP, Pyomo): The standard for data scientists and developers.
• MATLAB: Preferred in engineering and academic research for complex matrix calculations.
• Gurobi/CPLEX: High-performance industrial solvers used for massive, multi-variable datasets.
• R Programming (lpSolve): Frequently used for statistical optimization.
• LINDO & SAS: Specialized tools for operations research and advanced analytics

These tools automate calculations and help solve large-scale problems quickly.

Here’s the difference between linear programming and non-linear programming

Here is a quick step-by-step process:

• Identify decision variables
• Write the objective function
• Identify constraints
• Apply the non-negativity condition
• Choose a solving method
• Interpret results

With the increasing use of AI, big data, and analytics, optimisation techniques such as linear programming continue to evolve.

Future applications include:

• Smart cities optimisation
• AI decision models
• Energy optimisation
• Automated logistics
• Predictive resource allocation

As industries embrace digital transformation, understanding what is linear programming becomes increasingly important for professionals aiming to stay competitive.

Linear programming is one of the most powerful mathematical tools used for optimisation and decision-making. From business operations to artificial intelligence, its applications span multiple industries.

Understanding linear programming problems, learning the linear programming simplex method, and applying optimisation techniques can significantly improve strategic planning and operational efficiency.

Whether you are a student, analyst, or business professional, mastering linear programming can enhance your ability to solve complex problems and make smarter, data-driven decisions.

As organisations continue to rely on analytics-driven insights, linear programming remains a foundational concept that supports innovation, efficiency, and competitive advantage.