## Problem Description and Symbolic Representation

The problem involves a steel shop that produces fences and doors using stainless steel and aluminum. The goal is to maximize profit given the available resources.

### Symbolic Variables:
- $x_1$ : Fences
- $x_2$ : Doors

### Objective Function:
The profit per fence is $200, and the profit per door is $100. The objective function to maximize profit ($P$) is:
\[ P = 200x_1 + 100x_2 \]

### Constraints:
1. Each fence requires 2 units of stainless steel, and each door requires 5 units. The shop has 400 units available:
\[ 2x_1 + 5x_2 \leq 400 \]
2. Each fence requires 10 units of aluminum, and each door requires 1 unit. The shop has 500 units available:
\[ 10x_1 + x_2 \leq 500 \]
3. Non-negativity constraints, as the number of fences and doors cannot be negative:
\[ x_1 \geq 0, x_2 \geq 0 \]

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'Fences'), ('x2', 'Doors')],
    'objective_function': '200*x1 + 100*x2',
    'constraints': [
        '2*x1 + 5*x2 <= 400',
        '10*x1 + x2 <= 500',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

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

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

# Objective function: Maximize profit
model.setObjective(200*x1 + 100*x2, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(2*x1 + 5*x2 <= 400, name="StainlessSteelConstraint")
model.addConstr(10*x1 + x2 <= 500, name="AluminumConstraint")

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution: Fences = {x1.varValue}, Doors = {x2.varValue}")
    print(f"Maximum profit: ${model.objVal:.2f}")
else:
    print("The model is infeasible.")
```