Optimization Methods: Secant Method for Cost Analysis

Método de la Secante Modificado (Modified Secant Method)

Introduction to Optimization in Business Context

Optimization methods are essential in business statistics for determining the best values of variables to maximize profit or minimize cost. The secant method is a numerical technique used to find roots of equations, which can be applied to solve cost minimization or profit maximization problems.

Problem Statement: Cost and Income Analysis

The problem involves calculating the optimal quantity of product (in kilograms) to produce weekly, given the following:

• Fixed weekly cost: $50

• Variable cost per kg: $2.5

• Weekly income from sales: $3x + 2 \ln(x)

• x: Number of kilograms sold each week

The goal is to determine the value of x that maximizes profit, using the secant method.

Formulas and Equations

• Weekly Cost Function:

  • $CC(x) = 50 + 2.5x$

• Weekly Income Function:

  • $I(x) = 3x + 2 \ln(x)$

• Profit Function:

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

  • Simplified: $P(x) = 0.5x + 2 \ln(x) - 50$

Secant Method for Root Finding

The secant method is used to find the value of x where the profit function is maximized (or where its derivative equals zero). The iterative formula is:

• $x_{k+1} = x_k - \frac{f(x_k) \cdot (x_k - x_{k-1})}{f(x_k) - f(x_{k-1})}$

• Where $f(x)$ is the function whose root we seek (e.g., the derivative of the profit function).

In the notes, the secant method is applied to the profit function to find the optimal production quantity.

Step-by-Step Application

• Initial guesses: $x_0 = 50$, $x_1 = 81.81$

• Function values: $f(x_0)$, $f(x_1)$ calculated for each iteration

• Iterative updates: $x_{k+1}$ calculated using the secant formula

Tabular Summary of Iterations

The table below summarizes the secant method iterations:

k	x_k	f(x_k)	x_{k+1}
0	50	-17.17	81.81
1	81.81	0.28	82.35
2	82.35	0.00023	82.35

Interpretation: The optimal production quantity converges to approximately 82.35 kg per week.

Example Application

Suppose a business wants to maximize weekly profit given the cost and income functions above. By applying the secant method, the optimal production quantity is found to be about 82.35 kg, where the profit is maximized.

Additional info: The secant method is a useful numerical tool in business statistics for optimization problems, especially when analytical solutions are difficult to obtain.