This calculator helps find the optimal solution for a linear programming problem by evaluating the objective function at the vertices of the feasible region.
Understanding Linear Optimization
Linear optimization, also known as linear programming (LP), is a mathematical method used to determine the best possible outcome or decision in a given situation, subject to a set of linear constraints. It's widely used in various fields like business, economics, engineering, and operations research to maximize profits, minimize costs, allocate resources efficiently, and optimize production schedules.
The Core Components:
Objective Function: This is the linear function you want to maximize (e.g., profit) or minimize (e.g., cost). It's typically expressed as \( Z = c_1x_1 + c_2x_2 + … + c_nx_n \), where \(c_i\) are the coefficients and \(x_i\) are the decision variables.
Decision Variables: These are the unknown quantities that we need to determine the optimal values for (e.g., the number of units of product A and product B to produce).
Constraints: These are limitations or restrictions that the decision variables must satisfy. They are expressed as linear inequalities or equalities (e.g., \( a_{11}x_1 + a_{12}x_2 \le b_1 \)).
Feasible Region: The set of all possible solutions that satisfy all the constraints. Graphically, this is the region where all constraint lines intersect.
Optimal Solution: The point within the feasible region that yields the maximum or minimum value for the objective function. A fundamental theorem of linear programming states that if an optimal solution exists, it will occur at one of the vertices (corner points) of the feasible region.
How This Calculator Works:
This calculator simplifies the process of finding the optimal solution for a linear programming problem with two decision variables (e.g., \(x\) and \(y\)) and a set of linear constraints. The steps involved are:
Input Parsing: It takes the coefficients of the objective function, the coefficients of the constraints, the right-hand side values of the constraints, and the type of each constraint (less than or equal to, greater than or equal to, or equal to).
Vertex Calculation: The calculator identifies the vertices of the feasible region. For a 2D problem, this involves finding the intersection points of the boundary lines of the constraints. Some intersections might lie outside the feasible region and are discarded.
Feasibility Check: For each potential vertex, the calculator verifies if it satisfies all the given constraints.
Objective Function Evaluation: For every vertex that is confirmed to be feasible, the calculator plugs its coordinates into the objective function to calculate the value of \(Z\).
Optimal Solution Determination: The calculator then compares the \(Z\) values for all feasible vertices and identifies the one that maximizes (or minimizes, if that was the objective) the objective function.
Limitations:
This calculator is designed for problems with two decision variables and a limited number of constraints. For more complex problems with many variables or constraints, specialized software or algorithms like the Simplex method are typically used.
Example Use Case: Product Mix Optimization
Suppose a company manufactures two products, Product A and Product B.
Product A requires 1 hour of Machine Time and 2 hours of Assembly.
Product B requires 3 hours of Machine Time and 1 hour of Assembly.
The company has a maximum of 12 hours of Machine Time and 10 hours of Assembly time available per day.
The profit for Product A is $5 per unit, and for Product B is $8 per unit.
The objective is to maximize profit.
To use this calculator for this scenario:
Objective Function Coefficients: Enter 5,8 (for \(5x + 8y\), where \(x\) is units of A and \(y\) is units of B).
Constraint Coefficients Matrix: Enter [[1,2],[3,1]] (for 1A+2B, 3A+1B).
Constraint Values: Enter 12,10 (for <= 12 hours machine time, <= 10 hours assembly time).
Constraint Types: Enter leq,leq.
The calculator will then determine the number of units of Product A and Product B to produce to achieve maximum daily profit.
function calculateLinearOptimization() {
var objectiveCoefficientsStr = document.getElementById("objectiveCoefficients").value;
var constraintMatrixStr = document.getElementById("constraintMatrix").value;
var constraintValuesStr = document.getElementById("constraintValues").value;
var constraintTypesStr = document.getElementById("constraintTypes").value;
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
// — Input Parsing and Validation —
var objectiveCoefficients;
try {
objectiveCoefficients = JSON.parse("[" + objectiveCoefficientsStr + "]");
if (!Array.isArray(objectiveCoefficients) || objectiveCoefficients.length === 0) {
throw new Error("Invalid objective coefficients format.");
}
// Ensure all are numbers
if (!objectiveCoefficients.every(num => typeof num === 'number' && isFinite(num))) {
throw new Error("Objective coefficients must be valid numbers.");
}
} catch (e) {
resultDiv.innerHTML = "Error parsing objective coefficients: " + e.message;
return;
}
var constraintMatrix;
try {
constraintMatrix = JSON.parse(constraintMatrixStr);
if (!Array.isArray(constraintMatrix)) {
throw new Error("Constraint matrix must be a JSON array.");
}
// Basic structure check
for (var i = 0; i typeof num === 'number' && isFinite(num))) {
throw new Error("All constraint coefficients must be valid numbers.");
}
}
} catch (e) {
resultDiv.innerHTML = "Error parsing constraint matrix: " + e.message;
return;
}
var constraintValues;
try {
constraintValues = JSON.parse("[" + constraintValuesStr + "]");
if (!Array.isArray(constraintValues) || constraintValues.length !== constraintMatrix.length) {
throw new Error("Constraint values must be a comma-separated list matching the number of constraints.");
}
// Ensure all are numbers
if (!constraintValues.every(num => typeof num === 'number' && isFinite(num))) {
throw new Error("Constraint values must be valid numbers.");
}
} catch (e) {
resultDiv.innerHTML = "Error parsing constraint values: " + e.message;
return;
}
var constraintTypes;
try {
constraintTypes = constraintTypesStr.split(',').map(type => type.trim().toLowerCase());
if (constraintTypes.length !== constraintMatrix.length) {
throw new Error("Constraint types must match the number of constraints.");
}
var validTypes = ['leq', 'geq', 'eq'];
if (!constraintTypes.every(type => validTypes.includes(type))) {
throw new Error("Invalid constraint type found. Use 'leq', 'geq', or 'eq'.");
}
} catch (e) {
resultDiv.innerHTML = "Error parsing constraint types: " + e.message;
return;
}
// Ensure we only handle 2 variables for this simplified vertex method
if (objectiveCoefficients.length !== 2) {
resultDiv.innerHTML = "This calculator currently supports only problems with exactly two decision variables.";
return;
}
var xCoeff = objectiveCoefficients[0];
var yCoeff = objectiveCoefficients[1];
// — Vertex Finding Logic (for 2 variables) —
var vertices = [];
var constraintLines = [];
// Add axes as implicit constraints if not already present and assuming non-negativity
// For simplicity here, we will find intersections first, then check feasibility.
// A more robust solution would incorporate x>=0, y>=0 from the start.
for (var i = 0; i line.a !== 0 && line.b === 0);
var hasYConstraint = constraintLines.some(line => line.a === 0 && line.b !== 0);
if (!hasXConstraint) constraintLines.push({ a: 1, b: 0, c: 0, type: 'geq' }); // x >= 0
if (!hasYConstraint) constraintLines.push({ a: 0, b: 1, c: 0, type: 'geq' }); // y >= 0
// Find intersection points of all pairs of constraint lines
for (var i = 0; i < constraintLines.length; i++) {
for (var j = i + 1; j < constraintLines.length; j++) {
var line1 = constraintLines[i];
var line2 = constraintLines[j];
// Solve the system of linear equations:
// a1*x + b1*y = c1
// a2*x + b2*y = c2
var det = line1.a * line2.b – line2.a * line1.b;
if (det !== 0) { // Lines are not parallel
var x = (line2.b * line1.c – line1.b * line2.c) / det;
var y = (line1.a * line2.c – line2.a * line1.c) / det;
vertices.push({ x: x, y: y });
} else {
// Lines are parallel. If they are the same line, it doesn't create a new vertex.
// If they are distinct parallel lines, they don't intersect.
}
}
}
// Filter vertices to include only those within the feasible region
var feasibleVertices = [];
for (var k = 0; k < vertices.length; k++) {
var vertex = vertices[k];
var isFeasible = true;
// Check against original constraints
for (var l = 0; l line.c + 1e-9) { // Add tolerance for floating point errors
isFeasible = false;
break;
}
if (line.type === 'geq' && value 1e-9) {
isFeasible = false;
break;
}
}
if (isFeasible) {
// Check for duplicate vertices (due to floating point inaccuracies)
var isDuplicate = false;
for(var m=0; m < feasibleVertices.length; m++) {
if (Math.abs(feasibleVertices[m].x – vertex.x) < 1e-9 && Math.abs(feasibleVertices[m].y – vertex.y) < 1e-9) {
isDuplicate = true;
break;
}
}
if (!isDuplicate) {
feasibleVertices.push(vertex);
}
}
}
// Handle case where feasible region might be unbounded or empty
if (feasibleVertices.length === 0) {
// Could be unbounded or no feasible solution.
// A simple check for unboundedness is complex here. Let's assume for now if no vertices, no solution.
// Or, if the objective function can be increased indefinitely within constraints.
// For this basic calculator, we report no solution found or potentially unbounded if constraints are loose.
// A quick check: if constraints allow infinite growth along an axis where objective is positive.
// This is a simplification. Proper unboundedness check is more involved.
var potentiallyUnbounded = false;
// Check if constraints *could* allow infinite positive x or y
// Example: if x+y =0, y>=0 -> bounded
// Example: if x >= 5, y >= 5 -> unbounded for maximization
// Let's just report "No feasible vertices found. The feasible region might be empty or unbounded."
resultDiv.innerHTML = "No feasible vertices found. The feasible region might be empty or unbounded.";
return;
}
// Evaluate objective function at each feasible vertex
var maxObjectiveValue = -Infinity;
var minObjectiveValue = Infinity;
var optimalVertices = []; // Store vertices that achieve the optimal value
for (var n = 0; n maxObjectiveValue) {
maxObjectiveValue = objectiveValue;
optimalVertices = [vertex]; // Reset optimal vertices
} else if (Math.abs(objectiveValue – maxObjectiveValue) < 1e-9) {
optimalVertices.push(vertex); // Add vertex if it yields the same max value
}
if (objectiveValue 0) {
var resultHtml = "Optimal Solution Found:";
if (optimalVertices.length === 1) {
resultHtml += "Variable X = " + optimalVertices[0].x.toFixed(4) + ", Variable Y = " + optimalVertices[0].y.toFixed(4);
} else {
// Multiple optimal solutions exist
resultHtml += "Multiple Optimal Solutions Exist:";
optimalVertices.forEach(function(v) {
resultHtml += " X = " + v.x.toFixed(4) + ", Y = " + v.y.toFixed(4) + "";
});
resultHtml += "All points on the line segment connecting these vertices are also optimal.";
}
resultHtml += "Maximum Objective Value (Z) = " + maxObjectiveValue.toFixed(4);
resultDiv.innerHTML = resultHtml;
} else {
// This case should ideally be caught by feasibleVertices.length === 0 check,
// but as a fallback:
resultDiv.innerHTML = "Could not determine an optimal solution. Check constraints.";
}
}