## Problem Description and Formulation

The problem is a classic example of a linear programming optimization problem. The construction company wants to maximize its profit by installing hardwood flooring and carpet, given the constraints on cutting and installation hours.

Let's define the decision variables:

* $H$: number of hardwood flooring units to install
* $C$: number of carpet units to install

The objective function is to maximize the total profit:

Maximize: $400H + 650C$

The constraints are:

* Cutting hours: $1H + 0.5C \leq 200$
* Installation hours: $3H + 4C \leq 400$
* Non-negativity: $H \geq 0, C \geq 0$

## Gurobi Code

```python
import gurobi

# Create a new Gurobi model
m = gurobi.Model()

# Define the decision variables
H = m.addVar(lb=0, name="Hardwood_Flooring")
C = m.addVar(lb=0, name="Carpet")

# Define the objective function
m.setObjective(400 * H + 650 * C, gurobi.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(1 * H + 0.5 * C <= 200, name="Cutting_Constraint")
m.addConstr(3 * H + 4 * C <= 400, name="Installation_Constraint")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Hardwood Flooring: {H.varValue}")
    print(f"Carpet: {C.varValue}")
    print(f"Max Profit: {m.objVal}")
else:
    print("No optimal solution found")
```

This code defines the decision variables, objective function, and constraints, and then solves the optimization problem using Gurobi. If an optimal solution is found, it prints the values of the decision variables and the maximum profit. Otherwise, it indicates that no optimal solution was found.