## Problem Description and Formulation

The problem is a classic example of a linear programming problem. The goal is to maximize the profit by determining the optimal number of star, circle, and snowflake layouts to use, given the constraints on the available workstations, servers, and switches.

Let's define the decision variables:

* $x_1$: number of star layouts
* $x_2$: number of circle layouts
* $x_3$: number of snowflake layouts

The objective function to maximize is:

* Profit = $2231x_1 + 3434x_2 + 8621x_3$

The constraints are:

* $40x_1 + 20x_2 + 323x_3 \leq 3000$ (workstations)
* $10x_1 + 12x_2 + 122x_3 \leq 400$ (servers)
* $2x_1 + 5x_2 + 41x_3 \leq 200$ (switches)
* $x_1, x_2, x_3 \geq 0$ (non-negativity)

## Gurobi Code

```python
import gurobipy as gp

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

# Define the decision variables
x1 = m.addVar(name="star_layouts", lb=0, vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="circle_layouts", lb=0, vtype=gp.GRB.INTEGER)
x3 = m.addVar(name="snowflake_layouts", lb=0, vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(2231*x1 + 3434*x2 + 8621*x3, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(40*x1 + 20*x2 + 323*x3 <= 3000, name="workstations")
m.addConstr(10*x1 + 12*x2 + 122*x3 <= 400, name="servers")
m.addConstr(2*x1 + 5*x2 + 41*x3 <= 200, name="switches")

# Solve the model
m.optimize()

# Print the results
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Star layouts: {x1.varValue}")
    print(f"Circle layouts: {x2.varValue}")
    print(f"Snowflake layouts: {x3.varValue}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found.")
```