Introduction to Linear Programming: Meaning, Importance, and Limitations

Introduction to Linear Programming 

Time and resources are finite, and everyone wants to use them efficiently. By using optimization techniques, companies can solve supply chain problems within very less time. Optimizing is made simple with linear programming. It is the best method for performing linear optimization based on a few simple assumptions. Linear functions are known as objective functions. There is no doubt that relationships in the real world are challenging. However, the use of linear programming makes it easier to analyze such relationships. Many industries use linear programming, including energy, telecommunications, transportation, and manufacturing.  

What Is Linear Programming? 

Linear Programming, abbreviated as LP, is a simple method that uses a linear function to describe complicated real-world relationships. There is a linear relationship between the elements in this mathematical model. The linear programming technique involves optimizing a linear function to achieve the best results. A linear function is comprised of linear equality and inequality constraints. The meaning of linear programming is one can maximize or minimize a linear function under linear constraints. 

What Is the Role of Linear Programming in Decision Making? 

Linear programming aims to determine the feasible region and optimize the solution to obtain a function’s highest or lowest value. Linear programming analyzes various inequalities in a scenario and determines the best value that can be obtained under the given constraints. When dealing with linear programming, it is essential to consider some of the following assumptions when making a decision: 

  • Numerical representation of the total number of constraints is required 
  • A linear relationship should exist between constraints and objective function 
  • Optimization of the linear function (the objective function) is necessary. 

Basic Terminologies Used in Linear Programming 

  1. Constraints: An inequality that specifies how variable values are limited during a problem is called a constraint. Applied mathematics techniques function best when all constraints are linear inequalities. 
  2. Decision Variables: Identifying the decision variables is the first step in solving any problem. The decision variables X and Y in any given linear equation help the decision maker to optimize the problem.
  3. Objective Function: It is defined as the process of making a decision. It is important to mention the objective function quantitatively in a problem.
  4. Non-negativity restriction: Decision variables in linear programs should always have non-negative values. In other words, decision variables should have values greater than or equal to zero.
  5. Optimal solution: In general, an optimal solution maximizes profit or minimizes cost as the objective function. This is the best value for the objective function.
  6. Feasible solution: A feasible solution meets all the constraints for the decision variables. A feasible LP problem has at least one feasible solution.
  7. Infeasible solution: A infeasible solution is one in which all constraints are not satisfied or one in which there is no optimal solution. LP problems are infeasible if no solution meets all constraints. 

The Process To Define an LP Problem 

A generic linear programming problem is defined as follows: 

Step 1: Determining the decision variables. 

Step 2: Defining the objective function and deciding whether the function should be minimized or maximized. 

Step 3: Explaining the constraints. 

Step 4: Ensuring each decision variable is greater than or equal to 0 (Non-negative restraint). 

Step 5: Using either the simplex or graphical methods to solve the linear programming problem. 

A linear programming problem must have linear functions for the decision variables, objective functions, and constraints. The linear program is defined as one that satisfies all three conditions. 

Importance of Linear Programming 

Some of the advantages of linear programming problems are, 

  1. A linear program can provide insight into a business problem and helps solve multidimensional problems.
  2. Linear programming techniques provide practical solutions for problems having operating constraints.
  3. A linear programming technique improves the quality of decisions by calculating the cost and profit of different things. The user’s approach to decision-making becomes more objective with this technique.
  4. It helps in maximizing the use of productive resources through linear programming.
  5. This technique is most useful for highlighting bottlenecks in production processes.
  6. As conditions change, linear programming also helps re-evaluate a basic plan. 

Applications of Linear Programming 

Linear Programming is widely used in all fields, including agriculture, engineering, manufacturing, energy, logistical, and supply chain activities. 

  • Engineering – Using linear programming, engineers optimize resources in the manufacturing and production industries to maximize profits. 
  • Logistics – Transportation and logistics companies use linear programming to calculate the shortest routes, travel times, and pricing strategies. 
  • Agriculture – Engineering can use linear programming in agriculture to determine what crops to grow and how much to grow to increase revenue. 
  • Food Industry – To keep away from noncommunicable diseases, linear programming can help health managers and nutritionists plan dietary needs. 
  • Energy Sector – This technique is also applied in the energy sector to optimize the electric load, the shortest distribution lines, and the electrical power grid design. 
  • Programming and Data Science – Data Scientists encounter linear programming problems regularly. Data science developers can benefit from linear programming.  
  • Linear Programming in Python – Any linear programming problems can be solved by using PuLP, one of the libraries in python ecosystems, to optimize any problems. 

Limitations of Linear Programming 

  • It is important to have an objective that is clearly defined and quantifiable. In real life, however, such objectives are not achievable, such as maximizing sales, profit, or minimizing costs 
  • Time and uncertainty are not taken into account in linear programming models 
  • Often, in real-life situations, parameters that appear in the model are neither known nor constant 
  • An organization cannot take into account factors such as human behavior, weather conditions, and the stress of employees, which can adversely affect its performance 
  • In real life, problems arise with multiple objectives rather than one single objective.

Conclusion 

By using this technique, we can reduce or maximize a linear function subject to multiple constraints. In addition to several business planning applications, this technique can direct quantitative judgments in industrial engineering and, to a lesser extent, in social and physical sciences. 

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