```json
{
  "sym_variables": [
    ("x1", "couches"),
    ("x2", "beds")
  ],
  "objective_function": "200*x1 + 400*x2",
  "constraints": [
    "15*x1 + 20*x2 <= 300",
    "x2 >= 0.5*(x1 + x2)",
    "300*x1 + 600*x2 <= 8000",
    "x1 >= 0",
    "x2 >= 0"
  ]
}
```

```python
import gurobipy as gp
from gurobipy import GRB

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

# Create variables
couches = m.addVar(vtype=GRB.INTEGER, name="couches")
beds = m.addVar(vtype=GRB.INTEGER, name="beds")

# Set objective function
m.setObjective(200*couches + 400*beds, GRB.MAXIMIZE)

# Add constraints
m.addConstr(15*couches + 20*beds <= 300, "space_constraint")
m.addConstr(beds >= 0.5*(couches + beds), "bed_ratio_constraint")
m.addConstr(300*couches + 600*beds <= 8000, "budget_constraint")
m.addConstr(couches >= 0, "non_negativity_couches")
m.addConstr(beds >= 0, "non_negativity_beds")


# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print(f"Optimal profit: ${m.objVal}")
    print(f"Number of couches: {couches.x}")
    print(f"Number of beds: {beds.x}")
elif m.status == GRB.INFEASIBLE:
    print("The model is infeasible.")
else:
    print(f"Optimization ended with status {m.status}")

```
