LP Problem Solver
This calculator helps solve simple Linear Programming (LP) problems in standard form: Maximize Z = c1*x1 + c2*x2 + … + cn*xn Subject to: a11*x1 + a12*x2 + … + a1n*xn <= b1 a21*x1 + a22*x2 + … + a2n*xn <= b2 … am1*x1 + am2*x2 + … + amn*xn = 0 for all i
Solution will appear here.
Understanding Linear Programming (LP) Problems
Linear Programming (LP) is a powerful mathematical technique used to optimize (maximize or minimize) a linear objective function, subject to a set of linear constraints. It's widely applied in various fields such as economics, operations research, management science, and engineering to make optimal decisions in resource allocation, production planning, scheduling, and more.
An LP problem is characterized by:
- Objective Function: A linear function that we aim to maximize or minimize. For example, maximizing profit or minimizing cost. In the standard form, this is represented as
Z = c1*x1 + c2*x2 + ... + cn*xn, whereZis the value to optimize,xiare the decision variables, andciare the coefficients representing the contribution of each variable to the objective. - Decision Variables: These are the quantities we can control or decide upon, typically non-negative (e.g., number of units to produce, amount of budget to allocate).
- Constraints: These are linear inequalities or equalities that limit the possible values of the decision variables. They represent real-world limitations such as resource availability, production capacity, or demand. In the standard form, these are typically expressed as
a_i1*x1 + a_i2*x2 + ... + a_in*xn <= b_i, wherea_ijare coefficients representing the resource consumption per unit of decision variable, andb_iis the total availability of the resource. - Non-negativity: Decision variables are usually required to be non-negative (
xi >= 0).
How the Calculator Works (Simplex Method Overview)
This calculator attempts to solve the LP problem using a simplified approach based on the principles of the Simplex Method. The Simplex Method is an iterative algorithm that moves from one feasible corner point (vertex) of the feasible region to an adjacent one, improving the objective function value at each step until the optimal solution is found.
For a two-variable problem (x1, x2), the feasible region is a polygon in a 2D plane. The optimal solution typically lies at one of the vertices (corner points) of this polygon. The calculator explores these corner points by:
- Identifying potential corner points. These are often found by setting some variables to zero and solving the system of active constraints.
- Evaluating the objective function at each valid corner point.
- The corner point that yields the optimal value (maximum for maximization problems) is the solution.
Note: This calculator provides a basic implementation for simpler LP problems, particularly those with two variables. Complex problems with many variables, specific types of constraints (e.g., equality constraints), or unbounded feasible regions might require more sophisticated solvers or graphical methods for full analysis. The underlying mathematical principles often involve matrix operations and duality theory for larger-scale problems.
Example Use Case: Production Planning
Imagine a factory producing two types of furniture: chairs and tables.
- Objective: Maximize profit. Let
x1be the number of chairs andx2be the number of tables produced. Profit per chair is $50, and profit per table is $120. The objective function is:Maximize Z = 50*x1 + 120*x2. - Constraints:
- Wood: Each chair requires 5 units of wood, and each table requires 20 units. Maximum available wood is 450 units. Constraint:
5*x1 + 20*x2 <= 450. - Labor: Each chair requires 10 hours of labor, and each table requires 15 hours. Maximum available labor is 480 hours. Constraint:
10*x1 + 15*x2 <= 480. - Non-negativity: The factory cannot produce a negative number of items. Constraints:
x1 >= 0,x2 >= 0.
- Wood: Each chair requires 5 units of wood, and each table requires 20 units. Maximum available wood is 450 units. Constraint:
To solve this, you would input:
- Number of Variables: 2
- Objective Function Coefficients: c1=50, c2=120
- Constraint 1: Coefficients a11=5, a12=20; Right-hand side b1=450
- Constraint 2: Coefficients a21=10, a22=15; Right-hand side b2=480
The calculator would then find the optimal combination of chairs and tables to maximize profit given the resource limitations.
';
objectiveHtml += '';
for (var i = 1; i <= numVars; i++) {
objectiveHtml += '';
objectiveHtml += ";
}
objectiveHtml += '
';
document.getElementById("objectiveFunctionInputs").innerHTML = objectiveHtml;
// Update Constraint Inputs (only if number of variables changed)
// Rebuild constraints based on the new number of variables
var constraintsContainer = document.getElementById("constraintInputsContainer");
constraintsContainer.innerHTML = "; // Clear existing constraints
numConstraints = 1; // Reset constraint count
// Add initial constraints up to the current numConstraints value
for(var k = 0; k < numConstraints; k++) {
addConstraint(true); // Pass true to indicate initial population
}
}
function addConstraint(isInitial = false) {
var numVars = parseInt(document.getElementById("numVariables").value);
if (isNaN(numVars) || numVars < 1) numVars = 1;
var constraintsContainer = document.getElementById("constraintInputsContainer");
var constraintDiv = document.createElement("div");
constraintDiv.className = "input-group";
constraintDiv.id = "constraint_" + numConstraints;
var label = document.createElement("label");
label.innerHTML = "Constraint " + numConstraints + " (<= b" + numConstraints + "):";
constraintDiv.appendChild(label);
for (var i = 1; i 1) {
var constraintsContainer = document.getElementById("constraintInputsContainer");
var lastConstraintDiv = document.getElementById("constraint_" + numConstraints);
if (lastConstraintDiv) {
constraintsContainer.removeChild(lastConstraintDiv);
numConstraints–;
}
}
}
function solveLP() {
var numVars = parseInt(document.getElementById("numVariables").value);
var objectiveCoeffs = [];
for (var i = 1; i <= numVars; i++) {
var c = parseFloat(document.getElementById("c" + i).value);
objectiveCoeffs.push(isNaN(c) ? 0 : c);
}
var constraints = [];
for (var j = 1; j <= numConstraints; j++) {
var constraint = { coeffs: [], rhs: 0 };
for (var k = 1; k 2) {
resultDiv.textContent = "This calculator is optimized for 1 or 2 variables.";
resultDiv.style.backgroundColor = "#ffc107"; // Warning color
return;
}
if (numVars === 1) {
// Simple case: Maximize c1*x1 subject to a11*x1 <= b1, …, am1*x1 = 0
var maxPossibleX1 = Infinity;
for (var j = 0; j 0) { // If coefficient is positive, it constrains the upper bound
maxPossibleX1 = Math.min(maxPossibleX1, constraints[j].rhs / constraints[j].coeffs[0]);
} else if (constraints[j].coeffs[0] = 0 already handles this.
// If b is positive and coeff is negative, it allows x1 to be infinitely large. No upper bound from this.
}
}
if (maxPossibleX1 === Infinity) maxPossibleX1 = 1000000; // Arbitrary large number if no upper bound constraint
var optimalX1 = maxPossibleX1;
var maxZ = objectiveCoeffs[0] * optimalX1;
if (objectiveCoeffs[0] < 0) { // If coefficient is negative, we want x1 to be as small as possible (0)
optimalX1 = 0;
maxZ = 0;
}
resultDiv.textContent = "Optimal Solution Found: Z = " + maxZ.toFixed(2) + " at x1 = " + optimalX1.toFixed(2);
resultDiv.style.backgroundColor = "var(–success-green)";
return;
}
// — Solver for 2 Variables (corner point evaluation) —
var cornerPoints = [];
var equations = []; // Store Ax=B for solving intersections
// 1. Origin (0,0)
cornerPoints.push({ x1: 0, x2: 0 });
// 2. Intersections with axes (x1=0 or x2=0)
// If x1=0, we have constraints like a_i2 * x2 <= b_i. The most restrictive non-negative x2 is max(0, min(b_i / a_i2 for positive a_i2)).
var maxRestrictedX2 = Infinity;
var validX1Zero = true;
for (var j = 0; j 0 && potentialX2 >= 0) {
maxRestrictedX2 = Math.min(maxRestrictedX2, potentialX2);
} else if (constraints[j].coeffs[1] = 0) {
// If coeff is negative and rhs/coeff is positive, it doesn't limit x2 from above.
} else if (potentialX2 =0 means x2 must be 0.
maxRestrictedX2 = 0;
}
} else { // Coefficient is zero
if (constraints[j].rhs < 0) { // e.g. 0*x2 = 0) cornerPoints.push({ x1: 0, x2: maxRestrictedX2 });
// If x2=0, we have constraints like a_i1 * x1 <= b_i. The most restrictive non-negative x1 is max(0, min(b_i / a_i1 for positive a_i1)).
var maxRestrictedX1 = Infinity;
var validX2Zero = true;
for (var j = 0; j 0 && potentialX1 >= 0) {
maxRestrictedX1 = Math.min(maxRestrictedX1, potentialX1);
} else if (constraints[j].coeffs[0] = 0) {
// If coeff is negative and rhs/coeff is positive, it doesn't limit x1 from above.
} else if (potentialX1 < 0) {
maxRestrictedX1 = 0;
}
} else { // Coefficient is zero
if (constraints[j].rhs < 0) { // e.g. 0*x1 = 0) cornerPoints.push({ x1: maxRestrictedX1, x2: 0 });
// 3. Intersections of constraint pairs
for (var i = 0; i < constraints.length; i++) {
for (var k = i + 1; k = -1e-9 && x2 >= -1e-9) { // Allow for small floating point errors
var isFeasible = true;
for (var l = 0; l constraints[l].rhs + 1e-9) { // Allow tolerance
isFeasible = false;
break;
}
}
if (isFeasible) {
cornerPoints.push({ x1: Math.max(0, x1), x2: Math.max(0, x2) }); // Ensure non-negative
}
}
}
}
}
// Filter out duplicate points and points not satisfying constraints
var uniqueFeasiblePoints = [];
var tolerance = 1e-9;
for (var i = 0; i < cornerPoints.length; i++) {
var p = cornerPoints[i];
var isDuplicate = false;
for (var j = 0; j < uniqueFeasiblePoints.length; j++) {
if (Math.abs(p.x1 – uniqueFeasiblePoints[j].x1) < tolerance && Math.abs(p.x2 – uniqueFeasiblePoints[j].x2) < tolerance) {
isDuplicate = true;
break;
}
}
if (!isDuplicate) {
// Verify feasibility again (crucial!)
var isFeasible = true;
for (var k = 0; k constraints[k].rhs + tolerance) {
isFeasible = false;
break;
}
}
if(isFeasible) {
uniqueFeasiblePoints.push(p);
}
}
}
// Evaluate objective function at each unique feasible corner point
var maxZ = -Infinity;
var optimalPoint = null;
if (uniqueFeasiblePoints.length === 0) {
// Check if origin itself is feasible and gives a value
var originFeasible = true;
for(var k=0; k constraints[k].rhs + tolerance) {
originFeasible = false; break;
}
}
if (originFeasible) {
var zAtOrigin = objectiveCoeffs[0] * 0 + objectiveCoeffs[1] * 0;
maxZ = zAtOrigin;
optimalPoint = { x1: 0, x2: 0 };
} else {
resultDiv.textContent = "Problem is infeasible (no solution satisfies all constraints).";
resultDiv.style.backgroundColor = "#dc3545";
return;
}
} else {
for (var i = 0; i maxZ) {
maxZ = currentZ;
optimalPoint = point;
}
}
}
// Display result
if (optimalPoint !== null) {
resultDiv.textContent = "Optimal Solution: Max Z = " + maxZ.toFixed(2) + " at x1 = " + optimalPoint.x1.toFixed(2) + ", x2 = " + optimalPoint.x2.toFixed(2);
resultDiv.style.backgroundColor = "var(–success-green)";
} else {
// This case might occur if the feasible region is empty or unbounded upwards (for maximization)
resultDiv.textContent = "Could not determine optimal solution. The feasible region might be unbounded or empty.";
resultDiv.style.backgroundColor = "#dc3545";
}
}
// Initialize input fields on page load
document.addEventListener('DOMContentLoaded', function() {
updateInputFields();
});