Optimization Methods: Secant Method for Maximizing Weekly Profit

Método de la Secante Modificada (Modified Secant Method)

Optimizing Weekly Profit in Production

This section discusses the application of the modified secant method to optimize the weekly profit for a production process. The scenario involves calculating the optimal quantity of product to maximize profit, given cost and revenue functions.

• Weekly Cost Function: The cost to produce x kilograms is given by: $CC(x) = 50 + 2.5x$ where 50 is a fixed cost and 2.5 is the variable cost per kilogram.

• Weekly Revenue Function: The revenue from selling x kilograms is: $3x + 2 \ln(x)$ where 3 is the price per kilogram and $2 \ln(x)$ represents additional revenue dependent on quantity.

• Weekly Profit Function: The profit is the difference between revenue and cost: $P(x) = (3x + 2 \ln(x)) - (50 + 2.5x)$ Simplified: $P(x) = 0.5x + 2 \ln(x) - 50$

• Optimization Objective: Find the value of x that maximizes weekly profit.

Secant Method for Root Finding

The secant method is a numerical technique used to find roots of equations, often applied to optimization problems where the derivative of the profit function is set to zero.

• Derivative of Profit Function: $P'(x) = 0.5 + \frac{2}{x}$ Set $P'(x) = 0$ to find the maximum.

• Secant Method Iteration: $x_{k+1} = x_k - \delta x \frac{f(x_k)}{f(x_k) - f(x_{k-1})}$ where $f(x)$ is the derivative of the profit function.

• Iteration Table: The table shows the iterative process for finding the optimal x:

k	x_k	f(x_k)	x_{k+1}
0	50	0.04	81.81
1	81.81	0.017	82.35
2	82.35	0.00023	82.35

Example: Starting with $x_0 = 50$, the secant method converges to $x \approx 82.35$ as the optimal quantity to maximize weekly profit.

Additional info: The secant method is commonly used in numerical analysis for root-finding and optimization, especially when the function's derivative is not easily solved analytically.