Free online Graphing calculator of linear programming or operation research

In Mathematics, The linear programming is maximized or minimized of a linear function when subjected to various constraints(linear inequalities). We can draw any graph of inequality with the free graph drawing website www.graph2d.com. As for example, A linear programming problem is Maximize f(x,y)=3x+4y and constraints are 2x+3y<=48, x+y<=20, x>=0, y>=0. For solving this linear programming problem, type 2x+3y<=48 in the first input box, type x+y<=20 in the second input box, x>=0 in the third input box, y>=0 in the fourth input box and then the graph will drawn automatically of the inequalities. The graph of that inequalities is given belows.

Then click on the graph of intersection point to get the co-ordinates.click on the Feasible area button to get feasible area. Again click on the graph of turning point to get the co-ordinates of turning points. The graph of that feasible area is given belows.

we get the turning points (0,0),(20,0),(12,8),(0,16). so f(0,0)=3(0)+4(0)=0, f(20,0)=3(20)+4(0)=60, f(12,8)=3(12)+4(8)=68, f(0,16)=3(0)+4(16)=64 Hence Maximum value of f(x,y) is 68. For learn about piecewise function of Mathematics and graphing, visit: Graphing Piecewise function of Math.

Operation research or Linear programming is a mathematical modeling technique or analytical method of problem solving and decision making in which a linear function is minimized or maximized when subjected to various constraints that is useful in the management of organizations This technique has been useful for guiding quantitative decisions in business planning, in industrial engineering etc. Linear programming or Operation research is a method to achieve the best outcome (maximum profit or minimum cost) in a mathematical modeling whose requirements are represented by linear relationships. Linear programming is a special case of mathematical optimization or mathematical programming. Formally, linear programming is a technique for the optimization of a linear objective function, subject to linear inequality and linear equality constraints. Its Feasible area is a convex convex polytope, which is a set defined as the intersection of some half planes, each of which is defined by linear inequality. Its objective function is a real-valued linear function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the Minimum (or Maximum) value if such a point exists. For learn about Complex Analysis of Mathematics and graphing, visit: Graphing Complex Analysis of Math.

Any linear programming problem consists 1. Decision variables, 2. Objective function, 3. Constraints and 4. Non-Negative restrictions.
Decision variables : Decision variables are the unknown quantities of the linear programmming problems which are x, y.
Objective function : The Objective function is denoted by Z(x,y) which is the linear function that needs to be optimized according to the given condition to get the final solution. As for example, Z(x,y)=4x+3y.
Constraints : The constraints is the restrictions which are the mathematical conditions on the decision variables and which are linear. As forexample, 2x+y<=10,3x-2y<=15 etc.
Non-Negative restrictions : If the Decision variables are x and y, then the Non-Negative restrictions are x>=0 and y>=0.

There are 5 steps to solve the linear programming problems.
step-1: Assume the Decision variables in the linear programming problem. As for example, x and y.
step-2 : Make the Objective function of the linear programming problem and check if the function needs to be minimized or maximized. As for example, Z(x, y)=5x+10y.
step-3 : Write down all the Constraints or all the restrictions of the linear programming problems. As for example, 2x+5y<=15, 3x+4y<=20 etc.
step-4 : Ensure the Non-Negative restrictions of the decision variables. As for example, x>=0 andy>=0
step-5: Now solve the linear programming problem by using the graphical method and for helping to draw the graph by the website : graph2d.com/linear.php

1. Transportation problem in manufacturing company: Assume that there are m numbers of factories(sources) supply to n number of markets(delivery points) with some products. Linear programming techniques helps to minimizing the cost of transport between the sources and the delivery points for maximize the profit of the manufacturing company.
2. Agricultural application of farmers: Farmers apply Linear programming techniques to their work. The linear programming can be help to minimize the cost and maximize the profit in the agricultural planning to allocate limited resources such as labour, water supply,time, working capital etc and hence farmers can increase their revenue.
3. uses in the militery camps: In the militery camps involve the problems of selecting the air weapon system against gorillas to help of Linear programming techniques so as to keep them pinned down simultaneously minimize the amount of aviation gasoline used.
4. Marketing management system: Linear programming techniques can be helps in the marketing management system in analysing the effectiveness of the advertizing campaigns and time based the available advertizing media for minimizing the cost and time.
5. Apllications in the Engineering: Engineers are also use Linear programming techniques to help for solving the design and manufacturing problems. As for example, shape optimization seeks to make a shock-free airfoil with a feasible shape. linear programming techniques is an essential tool in the shape optimization.

Russian mathematician Leonid Kantorovich (1912-1986) is credited as one of the first to use linear methods for optimizing production processes in 1939. His work, however, was not widely recognized in the West due to limited communication during wartime.
Before linear programming formalized, optimization concepts were studied, especially in economics. Economists like Leonid Kantorovich and Wassily Leontief explored optimal resource allocation and input-output models, foreshadowing LP.
In 1947, American mathematician George Dantzig developed the Simplex Method, the first effective algorithm for solving LP problems. He initially developed this to aid the U.S. Air Force in optimizing resource allocation. George Dantzig introduced the concept of constraints and objective functions, providing the basis for mathematical formulations in LP. This marked the formalization of LP as a method for solving real-world problems with linear objectives and constraints.
LP quickly gained traction in operations research due to its applications in military logistics, supply chain optimization, and resource allocation, driven by Cold War-era demands.
During this period, researchers like John von Neumann and Harold Kuhn further explored LP and game theory . Von Neumann’s minimax theorem and duality theory contributed significantly to LP's theoretical foundation.
duality theory became central to LP, enabling insights into the relationship between primal and dual problems, which allowed practitioners to understand problem constraints better.
The advent of digital computers transformed LP by enabling faster and more complex computations, making the Simplex Method feasible for industrial-scale problems.
In 1984, Narendra Karmarkar introduced an alternative to the simplex method known as the interior point method. Karmarkar’s algorithm had polynomial-time complexity, providing an efficient solution to large-scale LP problems and marking a breakthrough in computational optimization.
This work emphasized that LP problems could be solved in polynomial time, leading to significant advances in computational optimization and inspiring further research into efficient algorithms.
LP techniques now contribute to machine learning and AI, as LP methods are integrated into various models for resource optimization, decision-making, and constraint satisfaction.