What is linear programming?

Linear programming is a mathematical optimization technique used to find the maximum or minimum value of a linear function subject to linear inequality constraints.

What is the objective function?

The objective function is the linear equation representing the quantity you want to maximize (like profit) or minimize (like cost), written as $Z = c_1 x + c_2 y$.

What is the feasible region?

The feasible region is the set of all coordinate points $(x, y)$ that satisfy all the linear inequality constraints simultaneously.

What is the Fundamental Theorem of Linear Programming?

The theorem states that if a linear programming problem has an optimal solution, it must occur at one of the corner points (vertices) of the feasible region.

How does the graphical method work?

The graphical method plots all constraint lines on a coordinate plane, identifies the feasible region, locates its corner points, and evaluates the objective function at each corner to find the optimal value.

What is an inconsistent or infeasible problem?

An infeasible problem occurs when no points satisfy all constraints simultaneously, meaning there is no feasible region and no solution.

What is an unbounded solution?

An unbounded solution occurs when the feasible region is infinite in the direction of optimization, allowing the objective function to grow without limit.

What are decision variables?

Decision variables are the unknown quantities ($x$ and $y$) that you solve for to optimize the objective function.

What are non-negativity constraints?

Non-negativity constraints require that decision variables cannot be negative ($x \geq 0, y \geq 0$), representing real-world limitations (like producing a positive quantity of goods).

Can this solver handle more than two variables?

This graphical solver is designed for 2-variable systems. Problems with three or more variables require algebraic methods like the Simplex algorithm.

What is the Simplex method?

The Simplex method is an algebraic algorithm that solves linear programming problems by moving along the edges of the feasible region from vertex to vertex to find the optimal value.

What is sensitivity analysis in linear programming?

Sensitivity analysis determines how changes in the coefficients of the objective function or constraints affect the optimal solution.

math calculator

Linear Programming & Simplex Solver

Optimize objective functions subject to linear inequality constraints using graphical corner points.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Optimization Goal

Objective function Z = Ax + By

Multiplier A

Multiplier B

Constraint 1: a₁x + b₁y ≤ c₁

a₁ multiplier

b₁ multiplier

Limit Constant c₁

Constraint 2: a₂x + b₂y ≤ c₂

a₂ multiplier

b₂ multiplier

Limit Constant c₂

Results

Optimal MAX Value Z

18.0000

Optimal Coordinates (x*, y*)(x = 2.000, y = 3.000)

Feasible Region Graphic

Blue line: Constraint 1 | Green line: Constraint 2 | Orange dot: Optimal point

Feasible Corner Points

Origin (0.00, 0.00)Z = 0.000

C1 y-intercept (0.00, 4.00)Z = 16.000

C2 x-intercept (4.00, 0.00)Z = 12.000

C1 & C2 Intersection (2.00, 3.00)Z = 18.000

Step-by-Step Optimization Steps

Formulate the objective function: \[Z = 3x + 4y\]

Formulate constraint boundaries:

\[(1)\quad 1x + 2y \leq 8\]

\[(2)\quad 3x + 2y \leq 12\]

Identify the intersection points of the boundary equations:

Solve the intersection of Constraint 1 and 2:

\[1x + 2y = 8\]\[3x + 2y = 12\] \implies x = 2.000,\; y = 3.000

Evaluate valid corner vertices (lying in the first quadrant and satisfying all inequalities):

$Z(0.00, 0.00) = 3(0.00) + 4(0.00) = 0.0000$

$Z(0.00, 4.00) = 3(0.00) + 4(4.00) = 16.0000$

$Z(4.00, 0.00) = 3(4.00) + 4(0.00) = 12.0000$

$Z(2.00, 3.00) = 3(2.00) + 4(3.00) = 18.0000$

The maximum value of the objective function is **$18.0000$**, which occurs at coordinates **$(2.000, 3.000)$**.

Formula

Linear Programming formulas

Z = c_1 x + c_2 y \quad \to \text{Optimize}, \quad a_i x + b_i y \leq c_i

The optimal coordinates always lie on a boundary vertex (corner point) of the convex feasible region polygon.

Step by step

How the calculation works

1Define the optimization goal (Maximize vs. Minimize) and enter the objective parameters.
2Define the inequalities of the resource boundary constraints.
3The graphical solver calculates all coordinate vertices, shades the feasible region, and highlights the optimal point.

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math calculator

Linear Programming & Simplex Solver

Optimize objective functions subject to linear inequality constraints using graphical corner points.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Optimization Goal

Objective function Z = Ax + By

Multiplier A

Multiplier B

Constraint 1: a₁x + b₁y ≤ c₁

a₁ multiplier

b₁ multiplier

Limit Constant c₁

Constraint 2: a₂x + b₂y ≤ c₂

a₂ multiplier

b₂ multiplier

Limit Constant c₂

Results

Optimal MAX Value Z

18.0000

Optimal Coordinates (x*, y*)(x = 2.000, y = 3.000)

Feasible Region Graphic

Blue line: Constraint 1 | Green line: Constraint 2 | Orange dot: Optimal point

Feasible Corner Points

Origin (0.00, 0.00)Z = 0.000

C1 y-intercept (0.00, 4.00)Z = 16.000

C2 x-intercept (4.00, 0.00)Z = 12.000

C1 & C2 Intersection (2.00, 3.00)Z = 18.000

Step-by-Step Optimization Steps

Formulate the objective function: \[Z = 3x + 4y\]

Formulate constraint boundaries:

\[(1)\quad 1x + 2y \leq 8\]

\[(2)\quad 3x + 2y \leq 12\]

Identify the intersection points of the boundary equations:

Solve the intersection of Constraint 1 and 2:

\[1x + 2y = 8\]\[3x + 2y = 12\] \implies x = 2.000,\; y = 3.000

Evaluate valid corner vertices (lying in the first quadrant and satisfying all inequalities):

$Z(0.00, 0.00) = 3(0.00) + 4(0.00) = 0.0000$

$Z(0.00, 4.00) = 3(0.00) + 4(4.00) = 16.0000$

$Z(4.00, 0.00) = 3(4.00) + 4(0.00) = 12.0000$

$Z(2.00, 3.00) = 3(2.00) + 4(3.00) = 18.0000$

The maximum value of the objective function is **$18.0000$**, which occurs at coordinates **$(2.000, 3.000)$**.

Formula

Linear Programming formulas

Z = c_1 x + c_2 y \quad \to \text{Optimize}, \quad a_i x + b_i y \leq c_i

The optimal coordinates always lie on a boundary vertex (corner point) of the convex feasible region polygon.

Step by step

How the calculation works

1Define the optimization goal (Maximize vs. Minimize) and enter the objective parameters.
2Define the inequalities of the resource boundary constraints.
3The graphical solver calculates all coordinate vertices, shades the feasible region, and highlights the optimal point.

Related guides

Learn more about this calculation

Guide