## Step 1: Define the symbolic variables
Let's define the symbolic variables as follows:
- $x_1$ = number of full-time employees
- $x_2$ = number of part-time employees

## Step 2: Formulate the objective function
The objective is to minimize the weekly wage bill. The wage for full-time employees is $800 per week, and for part-time employees, it is $400 per week. Therefore, the objective function can be written as:
\[ \text{Minimize:} \quad 800x_1 + 400x_2 \]

## 3: Define the constraints
Based on the problem description, we have the following constraints:
1. The projects require at least 60 employees: $x_1 + x_2 \geq 60$
2. At least 40 employees have to be full-time: $x_1 \geq 40$
3. The number of full-time employees should be at least a third of the number of part-time employees: $x_1 \geq \frac{1}{3}x_2$
4. The weekly wage bill should be below $30,000: $800x_1 + 400x_2 \leq 30000$

## 4: Convert constraints into standard form for LP
All constraints are already in a suitable form for linear programming.

## 5: Symbolic representation
The symbolic representation of the problem is:
```json
{
'sym_variables': [('x1', 'full-time employees'), ('x2', 'part-time employees')],
'objective_function': '800*x1 + 400*x2',
'constraints': [
    'x1 + x2 >= 60',
    'x1 >= 40',
    'x1 >= 1/3*x2',
    '800*x1 + 400*x2 <= 30000'
]
}
```

## 6: Gurobi code in Python
Now, let's implement this linear programming problem using Gurobi in Python:
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(lb=0, name="full-time_employees", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(lb=0, name="part-time_employees", vtype=gurobi.GRB.INTEGER)

    # Objective function: minimize 800*x1 + 400*x2
    model.setObjective(800*x1 + 400*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x1 + x2 >= 60, name="total_employees")
    model.addConstr(x1 >= 40, name="min_full-time_employees")
    model.addConstr(x1 >= (1/3)*x2, name="full-time_to_part-time_ratio")
    model.addConstr(800*x1 + 400*x2 <= 30000, name="wage_bill")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Full-time employees: {x1.varValue}")
        print(f"Part-time employees: {x2.varValue}")
        print(f"Minimum wage bill: ${model.objVal:.2f}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```