To solve this linear programming problem, we will maximize the profit function $Z = 5x + 3y$, subject to the given constraints:

Constraints:

1. $x + y \leq 4$ (Machine time constraint)
2. $2x + y \leq 12$ (Raw material constraint)
3. $x \leq 6$ (Production limit for product A)
4. $x \geq 0$, $y \geq 0$ (Non-negativity constraints)

Steps to solve:

1. Identify the feasible region determined by the constraints.
2. Find the vertices of the feasible region (corner points).
3. Evaluate the profit function $Z = 5x + 3y$ at each vertex.
4. Select the vertex that yields the maximum $Z$.

Let's find the vertices of the feasible region by solving the constraints pairwise.import numpy as np import pandas as pd from scipy.optimize import linprog

Coefficients of the profit function (objective function)

c = [-5, -3] # Coefficients for maximizing (negative for minimization in linprog)

Coefficients of inequality constraints (Ax <= b)

A = [[1, 1], # Machine time constraint [2, 1], # Raw material constraint [1, 0]] # Production limit for product A b = [4, # Machine time availability 12, # Raw material availability 6] # Maximum production limit for A

Bounds for x and y (non-negativity and limits)

x_bounds = (0, None) # x >= 0 y_bounds = (0, None) # y >= 0

Solve the linear programming problem

result = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds], method='highs')

Extract the results

x_opt, y_opt = result.x # Optimal values of x and y max_profit = -result.fun # Maximum profit (negating because we minimized)

Display the results in a user-friendly format

optimal_solution = { "Variable": ["x (Product A)", "y (Product B)", "Profit"], "Optimal Value": [x_opt, y_opt, max_profit] }

optimal_df = pd.DataFrame(optimal_solution) import ace_tools as tools; tools.display_dataframe_to_user(name="Optimal Production Solution", dataframe=optimal_df) Variable Optimal Value 0 x (Product A) 4.0 1 y (Product B) 0.0 2 Profit 20.0 Variable Optimal Value 0 x (Product A) 4.0 1 y (Product B) 0.0 2 Profit 20.0successThe optimal production solution has been calculated and displayed in a table for you. You can review the number of units of Product A and Product B to produce, along with the maximum achievable profit.

Follow-Up Questions:

1. What would happen to the profit if more machine time was available?
2. How would the solution change if the profit per unit of Product B increased?
3. What are the implications if the raw material limit is relaxed?
4. Can the factory produce both products and still maximize profit under the current constraints?
5. How sensitive is the optimal solution to changes in production limits?

Tip:

Understanding the shadow price of constraints can help determine which resource is most valuable and where investing additional resources will yield the highest profit.