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OPTIMIZATION PROBLEM SOLVER

Linear programming solver for two-variable optimization problems.

Build a model with an objective function, linear constraints, and optional non-negative restrictions. The lab evaluates feasible corner points so you can maximise or minimise the objective in the browser.

LINEAR PROGRAM

Objective and constraints

2 VARIABLES

OBJECTIVE FUNCTION

The model uses Z = 3x + 5y. Choose maximise for profit, output, or score problems, and minimise for cost, waste, distance, or time problems.
Include non-negative constraints: x ≥ 0 and y ≥ 0

QUICK EXAMPLE

Try maximise Z = 3x + 5y with 2x + y ≤ 18, 2x + 3y ≤ 42, 3x + y ≤ 24, and x,y ≥ 0. The solver evaluates the feasible corner points and compares objective values.
Load the sample model or clear the form before entering your own.

HOW IT WORKS

What this optimization calculator solves

LP

OBJECTIVE FUNCTION

The objective function is the quantity you want to maximise or minimise, such as profit, cost, output, time, or resource use. In a two-variable linear program it has the form Z = c₁x + c₂y.

CONSTRAINTS

Constraints describe limits such as material, labour, budget, or capacity. A constraint like 2x + 3y ≤ 42 means every feasible choice of x and y must stay on the allowed side of that boundary line.

FEASIBLE REGION

The feasible region is the overlap of all constraints. For a bounded continuous two-variable linear program, the best value occurs at one of the feasible corner points, so the lab calculates and compares those corners.

WORKED EXAMPLE

Maximise a production objective

ObjectiveMaximise Z = 3x + 5y
Constraints2x + y ≤ 18, 2x + 3y ≤ 42, 3x + y ≤ 24
MethodEvaluate feasible corner points

INTERPRETATION

If x and y represent two products, each constraint can represent a limited resource. The optimum is the feasible corner where the objective value is highest for a maximisation problem or lowest for a minimisation problem.

LIMITATIONS

This page solves continuous two-variable linear programming models. It does not solve integer, binary, nonlinear, or many-variable optimization problems.

ABOUT THIS TOOL

Optimization Lab

Build a two-variable linear programming model, enter objective coefficients and constraints, inspect feasible corner points, and compare objective values to find a maximum or minimum result.

  • Two-variable linear programming and optimization problems
  • Maximize profit, output, score, or value from constraints
  • Minimize cost, time, waste, or resource use from constraints
  • Feasible region and corner-point objective comparison

KEY FUNCTIONS

Objective function input for x and y coefficientsUp to eight linear constraints with ≤, ≥, or = operatorsMaximize or minimize objective valueOptional non-negative constraints x ≥ 0 and y ≥ 0Feasible corner points and objective valuesGraphical corner-point view for continuous two-variable models
Browse research & simulation labs →

LEARN THE METHOD

Guides for this calculator workflow

GUIDEGraphical method for two-variable linear programmingBuild constraints, identify feasible corner points, and test an objective function.Read →GUIDEPlot one, two, or multiple equationsPlot equations online, compare multiple functions, inspect tables, understand overlapping graphs, and estimate roots or intersections.Read →

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