## Symbolic Representation

To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. Let's denote:

- \(x_1\) as the number of storefront signs
- \(x_2\) as the number of street signs

The objective is to maximize profit, where the profit per storefront sign is $400 and per street sign is $120. Therefore, the objective function can be represented as:

\[ \text{Maximize:} \quad 400x_1 + 120x_2 \]

The constraints based on the available time for each activity are:

- Cutting: \(2x_1 + x_2 \leq 100\)
- Printing: \(x_1 + 0.5x_2 \leq 50\)
- Assembly: \(2x_1 + 0.8x_2 \leq 60\)

Also, \(x_1 \geq 0\) and \(x_2 \geq 0\), as the number of signs cannot be negative.

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'storefront signs'), ('x2', 'street signs')],
    'objective_function': '400*x1 + 120*x2',
    'constraints': [
        '2*x1 + x2 <= 100',
        'x1 + 0.5*x2 <= 50',
        '2*x1 + 0.8*x2 <= 60',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

# Create a new model
model = gp.Model("Sign_Shop_Optimization")

# Define variables
x1 = model.addVar(name="storefront_signs", lb=0, vtype=gp.GRB.CONTINUOUS)
x2 = model.addVar(name="street_signs", lb=0, vtype=gp.GRB.CONTINUOUS)

# Objective function: Maximize profit
model.setObjective(400*x1 + 120*x2, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(2*x1 + x2 <= 100, name="cutting_constraint")
model.addConstr(x1 + 0.5*x2 <= 50, name="printing_constraint")
model.addConstr(2*x1 + 0.8*x2 <= 60, name="assembly_constraint")

# Solve the model
model.optimize()

# Print the results
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution found. Storefront signs: {x1.varValue}, Street signs: {x2.varValue}")
    print(f"Maximum profit: ${400*x1.varValue + 120*x2.varValue:.2f}")
else:
    print("No optimal solution found.")
```