To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints in terms of these variables.

Let's denote:
- $x_1$ as the number of hours Plant Alpha is run,
- $x_2$ as the number of hours Plant Beta is run.

The objective is to minimize the total cost, which can be represented by the objective function: $40x_1 + 100x_2$.

The constraints are:
1. Produce at least 100 tons of cement: $3x_1 + 5x_2 \geq 100$
2. Produce at least 80 tons of stucco: $2x_1 + 4x_2 \geq 80$

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'hours Plant Alpha is run'), ('x2', 'hours Plant Beta is run')],
    'objective_function': '40*x1 + 100*x2',
    'constraints': ['3*x1 + 5*x2 >= 100', '2*x1 + 4*x2 >= 80']
}
```

Given this symbolic representation, we can now formulate the problem in Gurobi Python code to find the optimal production plan that minimizes costs while meeting demand requirements.

```python
from gurobipy import *

# Create a new model
m = Model("Young_Bucks_Production")

# Define variables
x1 = m.addVar(name='hours_Plant_Alpha', vtype=GRB.CONTINUOUS, lb=0)
x2 = m.addVar(name='hours_Plant_Beta', vtype=GRB.CONTINUOUS, lb=0)

# Set the objective function
m.setObjective(40*x1 + 100*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(3*x1 + 5*x2 >= 100, name='cement_demand')
m.addConstr(2*x1 + 4*x2 >= 80, name='stucco_demand')

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution found: Run Plant Alpha for {x1.x} hours and Plant Beta for {x2.x} hours.")
else:
    print("No optimal solution found.")
```