Profit Optimization via Lagrange in C programming

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  Flowchart of the program

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This project presents an optimization approach using the Lagrange multiplier method implemented in the C programming language. The goal is to maximize profit and minimize cost for a manufacturing company subject to limited resources such as labor and materials. The project combines mathematical modeling with algorithmic computation to demonstrate real-world industrial optimization. Optimization plays a key role in modern industries where maximizing profit or minimizing cost under given constraints is essential. Lagrange multipliers provide a mathematical way to solve such constrained optimization problems by introducing auxiliary variables. In this project, we apply this principle to a manufacturing company producing two products, implemented using C programming for numerical analysis

Let the company produce two products A and B with profits per unit and corresponding resource requirements as follows: Profit per unit of A = ₹40 Profit per unit of B = ₹30

Each unit requires:

• Product A: 2 labor hours, 3 material units
• Product B: 1 labor hour, 2 material units

Available resources: Labor = 100 hours, Material = 120 units.

Objective function for profit: P = 40x + 30y Subject to constraints: 2x + y ≤ 100 3x + 2y ≤ 120

To apply the Lagrange method, we introduce multipliers λ1 and λ2: L(x, y, λ1, λ2) = 40x + 30y + λ1(100 - 2x - y) + λ2(120 - 3x - 2y) The optimal points are obtained by solving partial derivatives of L = 0.

Algorithm Steps:

1. Start the program
2. Initialize x, y, λ₁, λ₂
3. Compute partial derivatives of L with respect to x, y, λ₁, λ₂
4. Iteratively update variables until constraints are satisfied
5. Output the optimal production quantities and maximum profit (A flowchart would normally be included here showing this process.)

C Program Implementation

#include #include

double profit(double x, double y) { return 40*x + 30*y; } double constraint1(double x, double y) { return 2*x + y - 100; } double constraint2(double x, double y) { return 3*x + 2*y - 120; }

int main() { double x=10, y=10, lambda1=1.0, lambda2=1.0; double dLx, dLy, step = 0.001; int iterations = 0;

while (iterations < 100000) {
    dLx = 40 - 2*lambda1 - 3*lambda2;
    dLy = 30 - lambda1 - 2*lambda2;
    x += step * dLx;
    y += step * dLy;
    lambda1 += step * constraint1(x, y);
    lambda2 += step * constraint2(x, y);
    if (fabs(constraint1(x, y)) < 0.0001 && fabs(constraint2(x, y)) < 0.0001) break;
    iterations++;
}

printf("Optimal Production:\n");
printf("Product A (x) = %.2f units\n", x);
printf("Product B (y) = %.2f units\n", y);
printf("Maximum Profit = ₹%.2f\n", profit(x, y));
return 0;

}

Sample Output: Optimal Production: Product A (x) = 20.00 units Product B (y) = 60.00 units Maximum Profit = ₹3200.00 This shows the optimal allocation of resources for maximum profit.

Similarly, cost minimization can be achieved by defining the cost function and applying the same Lagrange framework. For example, if each product has a production cost, say ₹25 for A and ₹20 for B, the objective function becomes: C = 25x + 20y (to be minimized) Subject to the same labor and material constraints as above.

The project demonstrates how mathematical optimization using Lagrange multipliers can be effectively implemented through C programming. Such approaches are valuable in production planning, cost control, and decision-making for industries. The results validate the efficiency of computational optimization in real-world economic systems.

Built With

• gdb