Introduction

In this blog post, we look into the symbolic and numerical exploration of revenue, cost, and profit functions. The provided Python code defines symbolic functions and employs numerical methods to evaluate and visualize them. We will break down the code, emphasizing the mathematical concepts involved.

Symbolic Definitions

# Define the variable symbolically
x = symbols('x')

# Define the revenue and cost functions symbolically
revenue_function = -x**2 + 8*x
cost_function = 2*x + 5
profit_function = revenue_function - cost_function

Here, \(x\) is the symbolic variable representing quantity. Symbolic expressions are used to define revenue (\(R(x)\)), cost (\(C(x)\)), and profit (\(P(x)\)) functions.

Numeric Evaluation

# Lambdify the symbolic functions for numerical evaluation
revenue_function_numeric = lambdify(x, revenue_function, 'numpy')
cost_function_numeric = lambdify(x, cost_function, 'numpy')
profit_function_numeric = lambdify(x, profit_function, 'numpy')

The lambdify function converts symbolic expressions into numerical functions, facilitating the evaluation of these functions for specific values of (x).

Generating Data

# Generate x values
x_values = np.linspace(0, 5, 100)

# Generate y values for revenue, cost, and profit functions
revenue_values = revenue_function_numeric(x_values)
cost_values = cost_function_numeric(x_values)
profit_values = profit_function_numeric(x_values)

A range of \(x\) values is generated, and corresponding \(y\) values are computed for revenue, cost, and profit functions.

Break-Even Analysis

# Find the break-even quantity 
break_even_quantity = sp.solve(revenue_function - cost_function, x)
break_even_quantity = break_even_quantity[0]

The break-even quantity is determined by solving the equation \(R(x) - C(x) = 0\). The resulting \(x\) value signifies the quantity at which revenue equals cost.

Optimization Analysis

# Find the quantity that maximizes revenue
maximize_revenue_quantity = sp.solve(sp.diff(revenue_function, x), x)
maximize_revenue_quantity = maximize_revenue_quantity[0]

# Find the quantity that maximizes profit
maximize_profit_quantity = sp.solve(sp.diff(profit_function, x), x)
maximize_profit_quantity = maximize_profit_quantity[0]

Critical points, where the derivative is zero, are computed to identify the quantity that maximizes revenue and profit.

Equations for Critical Points and Derivatives

The critical points and derivatives are given by:

  • For maximizing revenue: \(\frac{dR}{dx} = 0 \Rightarrow x = {}\)

  • For maximizing profit: \(\frac{dP}{dx} = 0 \Rightarrow x = {}\)

Plotting

# Plot the revenue, cost, and profit functions
plt.plot(x_values, revenue_values, label='Revenue Function: $R(x) = {}$'.format(sp.latex(revenue_function)))
plt.plot(x_values, cost_values, label='Cost Function: $C(x) = {}$'.format(sp.latex(cost_function)))
plt.plot(x_values, profit_values, label='Profit Function $P(x) = {}$'.format(sp.latex(profit_function)))

Matplotlib is used to visualize the revenue, cost, and profit functions. LaTeX formatting is employed for clear mathematical representation.

Highlighting Points

# Mark break-even point
plt.scatter(break_even_quantity, revenue_function_numeric(break_even_quantity), color='red', label='Break-Even Point')

# Mark quantity that maximizes revenue
plt.scatter(maximize_revenue_quantity, revenue_function_numeric(maximize_revenue_quantity), color='green', label='Maximize Revenue Point')

# Mark quantity that maximizes profit
plt.scatter(maximize_profit_quantity, revenue_function_numeric(maximize_profit_quantity) - cost_function_numeric(maximize_profit_quantity), color='purple', label='Maximize Profit Point')

Key points, such as break-even, maximum revenue, and maximum profit, are highlighted on the plot.

Finalizing the Plot

# Add labels and legend
plt.xlabel('Quantity $(x)$')
plt.ylabel('Value')
plt.title('Revenue, Cost, and Profit Functions')
plt.legend(loc='lower center', bbox_to_anchor=(0.5, -0.3), ncol=3)

# Show the plot
plt.grid(True)
plt.show()

Labels, a title, and a legend are added for clarity, and the plot is displayed.

Summary

This analysis provides insights into the behavior of revenue, cost, and profit functions, for making informed decisions based on mathematical models.