Linear Programming Calculator

The Linear Programming Calculator is designed to help you solve complex optimization problems quickly and accurately. If you are looking for solutions to problems with a linear objective function and constraints, this calculator is for you.

How to Use the Linear Programming Calculator?

• Select a Calculator

  If looking to solve a standard LP problem, choose a calculator like the Simplex Method Calculator.

• Input

  Enter your objective function and define if you're aiming to maximize or minimize. Input constraints, ensuring you use the correct inequality symbols.

• Calculation

  Click the "Calculate" button. Wait until the data is processed and the result is displayed.

• Result

  Study the result.

What Is Linear Programming?

Linear Programming (LP), also called linear optimization, is one of the most important optimization techniques that provide an optimal solution for a given linear objective function with linear constraints. The main goal of this method is to determine the values of variables that increase or decrease a given objective function to a maximum or minimum level. In this context, the objective function denotes the quantity to be optimized, and the constraints set the boundaries. Linear programming mainly includes:

• Objective Function
• Constraints
• Data
• Decision Variables

Formulation

Objective Function. It can be expressed in the following way:

$$Z=c_1x_1+c_2x_2+\ldots+c_nx_n,$$

where:

• $$$Z$$$ is the objective to be maximized or minimized.
• $$$c_1,c_2,\ldots,c_n$$$ are the coefficients of the objective function.
• $$$x_1,x_2,\ldots,x_n$$$ are the decision variables.

Constraints. These are a set of inequalities or equations:

$$a_{11}x_1+a_{12}x_2+\ldots+a_{1n}x_n\le b_1$$$$a_{21}x_1+a_{22}x_2+\ldots+a_{2n}x_n\le b_2$$$$\ldots$$$$a_{k1}x_1+a_{k2}x_2+\ldots+a_{kn}x_n\le b_k,$$

where:

• $$$a_{ij}$$$ are the coefficients of the constraints.
• $$$b_1,b_2,\ldots,b_k$$$ are the right-hand side values of the constraints.

Methods

• Graphical Method. We can use the graphical method for small LP problems with two variables:

  • Draw the constraint lines on a graph.
  • Identify the feasible region, i.e., the area where all constraints overlap.
  • Graph the objective function as a straight line.
  • Slide the objective function line towards the direction of maximization or minimization without exiting the feasible region.
  • The optimal solution is the point where the objective function leaves the feasible region last.

• Simplex Method. For larger problems, the graphical method becomes cumbersome. The Simplex Method, an iterative algorithm, can handle multiple constraints and variables. It works by moving along the edges of the feasible region formed by the constraints to find the optimal solution.

Linear Programming provides systematic methods for solving optimization problems in manufacturing, logistics, finance, etc. By mathematically formulating the objective and its constraints, methods such as the simplex and graphical can be used to determine the optimal solution.

What Calculators Does eMathHelp Offer?

Simplex Method Calculator

Efficiently solve linear programming problems using the simplex algorithm. Get your objective function's optimal solution in a matter of seconds.

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