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Formulation of LPP
Formulation of LPP Linear programming (LP) is a mathematical optimization technique used to find the best solution to a set of constraints while maxi...
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Formulation of LPP Linear programming (LP) is a mathematical optimization technique used to find the best solution to a set of constraints while maxi...
Linear programming (LP) is a mathematical optimization technique used to find the best solution to a set of constraints while maximizing or minimizing a obiettivo function. The process involves transforming the problem into an equivalent linear programming (LPP), which can be solved by an optimizer.
Key elements of an LPP include:
Decision variables: Variables that the decision maker can control or influence.
Constraints: Inequalities or equalities that limit the range of possible values for the decision variables.
Objective function: A function that represents the desired outcome, such as maximizing profit or minimizing cost.
Coefficients: Numerical values associated with the constraints and objective function.
Graphical method: A visual tool that helps identify the optimal solution by representing the constraints and objective function as a graph.
Steps involved in formulating an LPP:
Identify the decision variables: Determine the variables that the decision maker needs to consider and control.
Identify the objective function: Define the function that measures the performance of the decision maker based on the chosen variables.
Set up the constraints: Formulate inequalities or equalities that represent the limitations on the decision variables.
Determine the coefficients: Calculate the numerical values associated with the objective function and the constraints.
Graph the constraints and objective function: Use a graphing software or plot the points manually to visualize the problem.
Find the optimal solution: Use an optimization algorithm to find the values of the decision variables that optimize the objective function while satisfying the constraints.
Example:
Consider a company that produces two types of products, A and B. The company has two resources, A and B, which have limited amounts. It also has a production capacity of 10 units of product A and 15 units of product B. The profit per unit of product A is 10 dollars, while the profit per unit of product B is 15 dollars.
The company's objective function is to maximize its profit, which can be expressed as:
Maximize Profit = 10A + 15B
The company's constraints are:
A + B ≤ 10
2A + 3B ≤ 15
These constraints represent the limited resources available for production.
Using the graphical method, the company can find the optimal solution to the LPP, which is A = 5 and B = 5. This solution will maximize the company's profit