Linear Programming & Constraints Explorer

Set up an objective function and linear constraints, then visualize the feasible region, evaluate corner points, and find the optimal solution. Supports sensitivity analysis and integer programming.

Controls

Presets

Objective

Z =x +y

Constraints (2 active)

x +y

Non-negativity constraints x ≥ 0 and y ≥ 0 are always included.

Integer Programming Mode (restrict to integer solutions)

Feasible Region

Feasible regionCorner pointOptimalC1C2

Results

\text{Maximize } Z = 5x + 4y

Optimal Solution

x^* = 3.00, \quad y^* = 1.50

Z^* = 21.00

Corner Points

Point	x	y	Z
★ A	3.00	1.50	21.00
B	4.00	0.00	20.00
C	0.00	3.00	12.00
D	0.00	0.00	0.00

Sensitivity Analysis

Coefficient of x can range from 7.00 to 11.00

Coefficient of y can range from 7.33 to 14.00

Shadow Prices

C1: 0.7500

C2: 0.5000

Step-by-Step Solution

Maximize Z = 5x + 4y

Subject to:

6x + 4y ≤ 24

1x + 2y ≤ 6

x ≥ 0, y ≥ 0

Found 4 corner point(s) in the feasible region.

Evaluating Z at each corner point:

(3.00, 1.50): Z = 5(3.00) + 4(1.50) = 21.00

(4.00, 0.00): Z = 5(4.00) + 4(0.00) = 20.00

(0.00, 3.00): Z = 5(0.00) + 4(3.00) = 12.00

(0.00, 0.00): Z = 5(0.00) + 4(0.00) = 0.00

Optimal solution: (3.00, 1.50) with Z = 21.00

Sensitivity Analysis:

Coefficient of x can range from 7.00 to 11.00

Coefficient of y can range from 7.33 to 14.00

Shadow price for constraint 1: 0.7500

Shadow price for constraint 2: 0.5000

Reference Guide

Linear Programming Setup

A linear program has an objective function to optimize and a set of linear inequality constraints that define what solutions are allowed.

Standard form

\text{Maximize/Minimize } Z = ax + by

Subject to constraints

a_i x + b_i y \le c_i, \quad x \ge 0, \; y \ge 0

Feasible Region

Each constraint defines a half-plane. The feasible region is the intersection of all half-planes, including the non-negativity constraints. It forms a convex polygon (or is empty or unbounded).

Key property

The feasible region of a linear program is always convex. If it is bounded, it is a convex polygon with a finite number of vertices.

Corner Point Theorem

If the feasible region is bounded, the optimal value of the objective function occurs at one of the corner points (vertices) of the feasible region.

Algorithm

Find all corner points by intersecting constraint boundary lines
Evaluate the objective function Z at each corner point
Select the corner with the largest (or smallest) value of Z

Sensitivity Analysis

Sensitivity analysis asks how much the objective coefficients or constraint bounds can change before the optimal vertex shifts.

Allowable increase/decrease

Each objective coefficient has a range over which the current optimal solution remains optimal.

Shadow price

The shadow price of a constraint tells you how much the optimal Z value changes per unit increase in that constraint's right-hand side. Non-binding constraints have a shadow price of zero.