To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify the constraints provided. The objective function to minimize is: $5x_{rubber\ bands} + 7x_{monochrome\ printers} + 4x_{cans\ of\ coffee}$.

The constraints are as follows:
1. Usefulness ratings for each item.
2. Combined usefulness ratings for various combinations of items with lower bounds.
3. A constraint involving rubber bands and cans of coffee.
4. A constraint involving monochrome printers and cans of coffee.
5. Upper bounds on the combined usefulness ratings for certain combinations.

Given variables:
- $x_0$: Number of rubber bands
- $x_1$: Number of monochrome printers
- $x_2$: Number of cans of coffee

Usefulness ratings per item:
- Rubber bands: 19
- Monochrome printers: 2
- Cans of coffee: 9

Constraints:
1. $19x_0 + 9x_2 \geq 67$
2. $2x_1 + 9x_2 \geq 105$
3. $19x_0 + 2x_1 \geq 61$
4. & 5. $19x_0 + 2x_1 + 9x_2 \geq 67$ (Noted twice but essentially the same constraint)
6. $-8x_0 + 5x_2 \geq 0$
7. $-x_1 + 8x_2 \geq 0$
8. $19x_0 + 9x_2 \leq 311$
9. $19x_0 + 2x_1 \leq 174$

All variables are integers.

Now, let's write the Gurobi code to solve this optimization problem:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define the variables
rubber_bands = m.addVar(vtype=GRB.INTEGER, name="rubber_bands")
monochrome_printers = m.addVar(vtype=GRB.INTEGER, name="monochrome_printers")
cans_of_coffee = m.addVar(vtype=GRB.INTEGER, name="cans_of_coffee")

# Define the objective function
m.setObjective(5*rubber_bands + 7*monochrome_printers + 4*cans_of_coffee, GRB.MINIMIZE)

# Add constraints
m.addConstr(19*rubber_bands + 9*cans_of_coffee >= 67, name="usefulness_rubber_coffee")
m.addConstr(2*monochrome_printers + 9*cans_of_coffee >= 105, name="usefulness_printer_coffee")
m.addConstr(19*rubber_bands + 2*monochrome_printers >= 61, name="usefulness_rubber_printer")
m.addConstr(19*rubber_bands + 2*monochrome_printers + 9*cans_of_coffee >= 67, name="total_usefulness_min")
m.addConstr(-8*rubber_bands + 5*cans_of_coffee >= 0, name="rubber_coffee_constraint")
m.addConstr(-1*monochrome_printers + 8*cans_of_coffee >= 0, name="printer_coffee_constraint")
m.addConstr(19*rubber_bands + 9*cans_of_coffee <= 311, name="usefulness_rubber_coffee_max")
m.addConstr(19*rubber_bands + 2*monochrome_printers <= 174, name="usefulness_rubber_printer_max")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Rubber bands: {rubber_bands.x}")
    print(f"Monochrome printers: {monochrome_printers.x}")
    print(f"Cans of coffee: {cans_of_coffee.x}")
else:
    print("No optimal solution found.")

```