Optimization and Modeling: Weekly Cost and Revenue Analysis

Método de los Siguiente Modulo (Method of Successive Modules)

Optimizing Weekly Production Cost and Revenue

This section explores the mathematical modeling of weekly production costs and revenues for a business scenario, focusing on optimizing the quantity produced to maximize profit. The approach uses algebraic expressions and iterative methods to find the optimal solution.

• Weekly Cost Function: The cost to produce a certain quantity of product is modeled as: $CC(x) = 50 + 2.5x$ where $x$ is the number of kilograms produced.

• Weekly Revenue Function: The revenue from selling the product is given by: $Ingreso(x) = 3x + 2 \ln(x)$ where $x$ is the number of kilograms sold.

• Profit Function: The weekly profit is calculated as: $Utilidad(x) = Ingreso(x) - CC(x) = 3x + 2 \ln(x) - (50 + 2.5x)$ Simplifying: $Utilidad(x) = 0.5x + 2 \ln(x) - 50$

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

Iterative Solution: Successive Module Method

The method of successive modules is used to iteratively approach the optimal value of $x$. This involves updating the estimate of $x$ based on the derivative of the profit function.

• Update Formula: The iterative update is given by: $x_{k+1} = x_k - \delta x \cdot OC(x_k)$ where $OC(x_k)$ is the derivative of the profit function with respect to $x$ at iteration $k$, and $\delta x$ is a step size.

• Derivative Calculation: The derivative of the profit function is: $OC(x) = \frac{d}{dx} Utilidad(x) = 0.5 + \frac{2}{x}$

• Iteration Table: The table below shows the values of $x$ and $OC(x)$ for each iteration.

k	x_k	OC(x_k)	x_{k+1}
0	50	0.54	81.81
1	81.81	0.52	82.35
2	82.35	0.52	82.35

Example: If the initial guess is $x_0 = 50$, the method converges to $x \approx 82.35$ after a few iterations, indicating the optimal production quantity for maximum profit.

Applications: This modeling approach is commonly used in business statistics and operations research to optimize production, pricing, and resource allocation.

Additional info: The iterative method shown is a basic form of numerical optimization, often used when analytical solutions are difficult or when the function is complex.