Optimization Methods: Secant Method for Maximizing Weekly Profit

Método de la Secante Modificada (Secant Method for Optimization)

Introduction to the Secant Method

The secant method is a numerical technique used to find roots of equations, often applied in optimization problems to maximize or minimize functions. In this context, it is used to determine the optimal quantity of product (in kilograms) to maximize weekly profit, given cost and income functions.

• Optimization involves finding the value of a variable that maximizes or minimizes a function, such as profit.

• Secant Method is an iterative algorithm that approximates the root of a function by using two initial guesses and updating them based on the function's values.

Problem Statement

The weekly cost for producing a certain product is given by:

• Fixed cost: 50 units

• Variable cost: 2.5 units per kilogram produced

The weekly income from sales is:

• Income per kilogram: 3 units

• Additional income: 2 units times the logarithm of the quantity produced

Let x be the quantity (in kg) produced and sold each week. The goal is to determine the optimal value of x that maximizes weekly profit.

Formulas and Equations

• Weekly Cost Function: $CC(x) = 50 + 2.5x$

• Weekly Income Function: $I(x) = 3x + 2 \ln(x)$

• Weekly Profit Function: $P(x) = I(x) - CC(x) = 3x + 2 \ln(x) - (50 + 2.5x)$

• Profit Maximization: Set the derivative of the profit function to zero to find the optimal x:

$\frac{d}{dx}P(x) = 3 + \frac{2}{x} - 2.5 = 0$

• Solve for x using the secant method.

Secant Method Iterative Formula

The secant method updates the estimate of the root using: $x_{k+1} = x_k - \frac{f(x_k) \cdot (x_k - x_{k-1})}{f(x_k) - f(x_{k-1})}$

• Here, $f(x)$ is the derivative of the profit function.

• Initial guesses: $x_0 = 50$, $x_1 = 81.81$ (from the notes).

Example Calculation

• Iteration 0: $x_0 = 50$

• Iteration 1: $x_1 = 81.81$

• Iteration 2: $x_2 = 82.35$

Table of Iterations:

k	x_k	f(x_k)
0	50	-17.17
1	81.81	0.28
2	82.35	0.00023

• The secant method converges to $x \approx 82.35$ kg, which is the optimal weekly production to maximize profit.

Summary of Steps

1. Define cost and income functions.

2. Formulate the profit function.

3. Set the derivative of the profit function to zero.

4. Apply the secant method to solve for the optimal value of x.

5. Interpret the result: produce and sell approximately 82.35 kg per week for maximum profit.

Applications

• This method is widely used in economics, business, and engineering to optimize production, minimize costs, or maximize revenue.

• Numerical methods like the secant method are essential when analytical solutions are difficult or impossible.