Study of Linear Programming and Its Applications

Abstract

The linear programming technique is a crucial tool applicable to a wide range of real-world scenarios, from optimising production processes to maximising profits in business and revenue, from planning flight paths to transporting oil from refineries to cities, to discovering affordable diets that fulfil daily nutritional needs. The paper covers the basics of linear programming, its formulation, its applications, its advantages, its limitations and various issues that it can address.

Introduction

Linear Programming (LP) is a branch of operations research that optimizes decision-making by finding the best possible values for system variables. It’s used extensively in sectors like production, finance, transportation, military, and even non-profit organizations.

Key Concepts:

• Introduced by George B. Dantzig in 1947.

• "Linear" refers to linear relationships between variables.

• "Programming" refers to systematic planning to reach an optimal goal.

Core Components:

• Decision Variables: Quantities to be determined.

• Objective Function: The function to be maximized or minimized.

• Constraints: Restrictions or limitations on the variables.

• Non-negativity: Variables cannot be negative.

Formulation Steps:

1. Define decision variables.

2. Formulate the objective function.

3. Identify constraints as linear inequalities or equations.

4. Include non-negativity conditions.

5. Analyze feasible solutions.

6. Choose the optimal solution.

Mathematical Form:

Maximize or minimize
    Z = c?x? + c?x? + ... + c?x?
Subject to constraints
    a??x? + a??x? + ... + a??x? ≤, ≥ or = b?
With x? ≥ 0

Types of LP Applications:

• Allocation Problems: Resource distribution.

• Assignment Problems: Assigning tasks to resources.

• Transportation Problems: Minimizing shipping costs.

Examples:

1. Production Allocation – Maximizing profit from manufacturing pencils, erasers, and sharpeners.

2. Product Mix – Maximizing profit from chairs and tables within resource limits.

3. Media Selection – Maximizing audience reach under advertising budget and media availability constraints.

Assumptions in LP:

• Certainty, Additivity, Proportionality, Divisibility, Multiplicativity.

Applications:

• Business: Production, marketing, manpower, capital budgeting.

• Industry: Transportation, blending, inventory control.

• Government/Military: Planning, logistics, defense strategies.

• Airlines/Railways: Route and resource optimization.

Advantages:

• Optimal use of resources.

• Helps adjust plans based on constraints.

• Identifies production bottlenecks.

• Offers practical, adaptable solutions.

Limitations:

• Real-world problems may not be linear.

• Doesn’t handle time or uncertainty.

• Large problems can be computationally intensive.

• Assumes fixed and known parameters.

• Handles only single-objective problems (multi-objective problems require different methods).

Conclusion

Linear Programming is an outstanding quantitative technique of decision-making. This technique plays a vital role in the fields of uncertainty, viz., business and commerce. Every company, regardless of its size, has access to resources, money, labour, and equipment however, the supply may be restricted. On the contrary, there will not be any necessity for management tools like linear programming if the supply of these resources remains unrestricted. Due to restricted availability of resources, management must make a decision how to best allocate its resources to maximize profit, reduce loss, or make the utmost use of production capacity to the highest extent. Many problems can be overcome by the linear programming technique.

References

[1] Mohan C., Deep K., Optimization Techniques, New Age Publication,2017 [2] Gupta P.K., Mohan M., Linear Programming and Theory of games, Sultan Chand & Sons, 1998 [3] Natarajan A.M, Balasubramani P.,Tamilarasi A., Operations Research ,Person Education,2007 [4] Sharma J.K., Operations Research Theory and Applications, 4th Edition, Macmillan India 4th, 2009 [5] S. Kanti, Gupta P.K., Operation Research, Sultan Chand & Sons, 2010 [6] Sharma S. D., Operations Research (Theory, Methods & Applications 2014, KNRN Publications

Copyright

Copyright © 2025 Chetna Rani Gupta(Asst. Prof. in Mathematics, Multani Mal Modi College, Patiala). This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.