## Problem Description and Formulation

The car dealership optimization problem can be formulated as a linear programming problem. The goal is to maximize profit given certain constraints on space, capital, and the proportion of cars to total vehicles.

### Decision Variables

- \(x_c\): The number of cars to stock.
- \(x_t\): The number of trucks to stock.

### Objective Function

The objective is to maximize profit. Given that the profit per car sold is $2000 and per truck sold is $4000, the objective function can be written as:

\[ \text{Maximize:} \quad 2000x_c + 4000x_t \]

### Constraints

1. **Space Constraint**: Each car takes 30 sq ft, and each truck takes 45 sq ft, with a total of 450 sq ft available.
\[ 30x_c + 45x_t \leq 450 \]

2. **Capital Constraint**: The dealership can spend at most $800,000, with each car costing $30,000 and each truck costing $40,000.
\[ 30000x_c + 40000x_t \leq 800000 \]

3. **Car Proportion Constraint**: At least 60% of all items in stock must be cars.
\[ x_c \geq 0.6(x_c + x_t) \]
\[ 0.4x_c \geq 0.6x_t \]
\[ x_c \geq 1.5x_t \]

4. **Non-Negativity Constraints**: The number of cars and trucks cannot be negative.
\[ x_c \geq 0, \quad x_t \geq 0 \]

## Gurobi Code

```python
import gurobipy as gp

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

# Decision variables
x_c = model.addVar(name="cars", lb=0, vtype=gp.GRB.INTEGER)
x_t = model.addVar(name="trucks", lb=0, vtype=gp.GRB.INTEGER)

# Objective function: Maximize profit
model.setObjective(2000*x_c + 4000*x_t, gp.GRB.MAXIMIZE)

# Space constraint
model.addConstr(30*x_c + 45*x_t <= 450, name="space")

# Capital constraint
model.addConstr(30000*x_c + 40000*x_t <= 800000, name="capital")

# Car proportion constraint
model.addConstr(x_c >= 1.5*x_t, name="car_proportion")

# Solve the model
model.optimize()

# Print the results
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal number of cars: {x_c.varValue}")
    print(f"Optimal number of trucks: {x_t.varValue}")
    print(f"Maximum profit: {model.objVal}")
else:
    print("The model is infeasible")
```