Simplex Method Calculator
:root {
–primary-blue: #004a99;
–success-green: #28a745;
–light-background: #f8f9fa;
–border-color: #dee2e6;
–text-color: #343a40;
–label-color: #495057;
}
body {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
line-height: 1.6;
color: var(–text-color);
background-color: #ffffff;
margin: 0;
padding: 20px;
display: flex;
justify-content: center;
align-items: flex-start;
min-height: 100vh;
}
.loan-calc-container {
background-color: var(–light-background);
padding: 30px;
border-radius: 8px;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1);
max-width: 800px;
width: 100%;
}
h1, h2 {
color: var(–primary-blue);
text-align: center;
margin-bottom: 25px;
}
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margin-bottom: 30px;
padding: 20px;
background-color: #ffffff;
border: 1px solid var(–border-color);
border-radius: 5px;
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.input-group {
margin-bottom: 15px;
display: flex;
align-items: center;
gap: 10px;
flex-wrap: wrap;
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margin-bottom: 5px;
font-weight: 500;
color: var(–label-color);
min-width: 150px; /* Ensure labels have consistent width */
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padding: 10px;
border: 1px solid var(–border-color);
border-radius: 4px;
box-sizing: border-box;
font-size: 1rem;
min-width: 200px; /* Minimum width for input fields */
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transition: background-color 0.3s ease;
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button:hover {
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margin-top: 25px;
padding: 20px;
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color: white;
border-radius: 5px;
text-align: center;
font-size: 1.4rem;
font-weight: bold;
min-height: 50px;
display: flex;
justify-content: center;
align-items: center;
box-shadow: 0 2px 10px rgba(40, 167, 69, 0.3);
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#result.error {
background-color: #dc3545; /* Red for errors */
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margin-top: 20px;
margin-bottom: 10px;
border-bottom: 2px solid var(–primary-blue);
padding-bottom: 5px;
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padding: 2px 5px;
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/* Responsive adjustments */
@media (max-width: 768px) {
.input-group {
flex-direction: column;
align-items: stretch;
}
.input-group label {
min-width: auto;
margin-bottom: 8px;
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.input-group input[type="number"],
.input-group textarea {
min-width: 100%;
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.loan-calc-container {
padding: 20px;
}
}
Simplex Method Calculator
Understanding the Simplex Method
What is the Simplex Method?
The Simplex Method is a fundamental algorithm in linear programming used to find the optimal solution (maximum or minimum value) of a linear objective function, subject to a set of linear constraints. Developed by George Dantzig in 1947, it is an iterative process 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 reached.
How it Works (Conceptual Overview)
- Standard Form: The linear programming problem is first converted into standard form. This involves ensuring all constraints are equalities and all variables are non-negative. For inequality constraints (like =), slack or surplus variables are introduced to convert them into equalities. For minimization problems, the objective function is converted to maximization by multiplying by -1.
- Initial Basic Feasible Solution: An initial basic feasible solution is found. This is often obtained by setting decision variables to zero and introducing artificial variables if necessary to form an identity matrix in the constraint matrix.
- Tableau Construction: The problem is represented in a tabular format called the simplex tableau. This tableau contains coefficients of the objective function, constraints, and the right-hand side values.
- Optimality Test: The current solution is checked for optimality. For maximization, if all coefficients in the objective function row (the 'Cj – Zj' row) are non-negative, the current solution is optimal. For minimization, all coefficients should be non-positive.
- Entering Variable Selection: If not optimal, an entering variable (a non-basic variable to enter the basis) is chosen. For maximization, this is typically the variable with the most negative coefficient in the objective row.
- Leaving Variable Selection: A leaving variable (a basic variable to exit the basis) is determined using the minimum ratio test. Ratios are calculated by dividing the RHS values by the corresponding positive coefficients in the entering variable's column. The row with the minimum non-negative ratio indicates the leaving variable.
- Pivoting: The tableau is updated using row operations (Gaussian elimination) to create a new basic feasible solution. The entering variable replaces the leaving variable in the basis.
- Iteration: Steps 4-7 are repeated until the optimality condition is met.
Use Cases
The Simplex Method is widely used in various fields, including:
- Operations Research: Resource allocation, production planning, scheduling.
- Economics: Portfolio optimization, market equilibrium analysis.
- Management Science: Logistics, supply chain management, optimizing business operations.
- Engineering: Design optimization, control systems.
While this calculator provides a simplified interface, the underlying principles are powerful for solving complex optimization problems.
Limitations
The standard Simplex Method can be computationally intensive for problems with a very large number of variables and constraints. Variations like the Revised Simplex Method are often used in practice. Degeneracy (where a basic variable has a value of zero) can sometimes lead to cycling, although specific rules (like Bland's rule) can prevent this.
// — Helper Functions —
function parseCoefficients(inputString, numVars) {
var coeffs = Array(numVars).fill(0);
if (!inputString || inputString.trim() === "") return coeffs;
var terms = inputString.match(/([+-]?\s*\d*\.?\d*)\s*(x\d+)?/g);
if (!terms) return coeffs;
for (var i = 0; i index 0
var value;
if (valueStr === "+" || valueStr === "") value = 1;
else if (valueStr === "-") value = -1;
else value = parseFloat(valueStr.replace(/\s/g, "));
if (variableIndex !== -1) {
if (variableIndex < numVars) {
coeffs[variableIndex] = value;
} else {
console.warn(`Variable x${variableIndex + 1} is out of expected range (1-${numVars}). Ignoring.`);
}
} else {
// This case handles constants if they were allowed, but for LP objective/constraints, it's usually variables.
// For simplicity, we ignore standalone numbers here.
}
}
return coeffs;
}
function getNumVariables(objectiveString) {
var matches = objectiveString.match(/x\d+/g);
if (!matches) return 0;
var maxVarNum = 0;
for (var i = 0; i maxVarNum) {
maxVarNum = num;
}
}
return maxVarNum;
}
// — Simplex Logic (Simplified – Actual implementation is complex) —
// NOTE: A full, robust Simplex method implementation is extremely complex and beyond the scope
// of a single script for demonstration without external libraries.
// This function will parse inputs and return a placeholder message or basic error.
// A real implementation would involve matrix operations, pivoting, etc.
function calculateSimplex() {
var objectiveFunctionStr = document.getElementById("objectiveFunction").value;
var constraintsStr = document.getElementById("constraints").value;
var constraintTypesStr = document.getElementById("constraintTypes").value;
var rhsValuesStr = document.getElementById("rhsValues").value;
var objectiveType = document.getElementById("objectiveType").value;
var resultDiv = document.getElementById("result");
resultDiv.innerText = "Calculating…";
resultDiv.className = ""; // Reset class
try {
var numVars = getNumVariables(objectiveFunctionStr);
if (numVars === 0) {
throw new Error("No variables found in the objective function.");
}
var objectiveCoeffs = parseCoefficients(objectiveFunctionStr, numVars);
var constraints = constraintsStr.trim().split('\n');
var constraintTypes = constraintTypesStr.trim().split('\n');
var rhsValues = rhsValuesStr.trim().split('\n');
if (constraints.length !== constraintTypes.length || constraints.length !== rhsValues.length) {
throw new Error("Number of constraints, types, and RHS values must match.");
}
var constraintMatrix = [];
var rhsVector = [];
for (var i = 0; i < constraints.length; i++) {
var constraintCoeffs = parseCoefficients(constraints[i], numVars);
constraintMatrix.push(constraintCoeffs);
var rhsVal = parseFloat(rhsValues[i]);
if (isNaN(rhsVal)) {
throw new Error(`Invalid RHS value: ${rhsValues[i]}`);
}
rhsVector.push(rhsVal);
}
// — Placeholder for actual Simplex Algorithm —
// A real implementation would now:
// 1. Convert to standard form (add slack/surplus/artificial variables).
// 2. Construct the initial simplex tableau.
// 3. Perform iterations (pivot operations) until optimality is reached.
// 4. Extract the solution from the final tableau.
// For demonstration, we'll just confirm parsing and return a message.
console.log("Objective Coefficients:", objectiveCoeffs);
console.log("Constraint Matrix:", constraintMatrix);
console.log("RHS Vector:", rhsVector);
console.log("Constraint Types:", constraintTypes);
console.log("Objective Type:", objectiveType);
resultDiv.innerText = "Simplex calculation logic requires a full implementation.";
resultDiv.className = "error"; // Indicate it's not a real result yet
// Simulate a successful (but fake) outcome for demonstration purposes
// In a real app, this would be the actual computed optimal value and variable assignments.
// Example: If objective was "3×1 + 5×2" and optimal is x1=1, x2=4, then value = 3*1 + 5*4 = 23
var exampleOptimalValue = 23.5; // Replace with actual calculation
var exampleVariables = { x1: 2.1, x2: 3.4 }; // Replace with actual calculation
// If you had a real solver, you'd update the result like this:
// resultDiv.innerText = `Optimal Solution Found: ${objectiveType === 'maximize' ? 'Max' : 'Min'} = ${optimalValue}`;
// Add variable details as needed.
// For now, just a placeholder message.
// resultDiv.innerText = "Placeholder: Actual Simplex computation is complex.";
} catch (error) {
resultDiv.innerText = "Error: " + error.message;
resultDiv.className = "error";
console.error(error);
}
}