## Step 1: Define the symbolic representation of the variables
The variables are 'rubber bands', 'monochrome printers', and 'cans of coffee', which we will denote as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $5x_1 + 7x_2 + 4x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
1. $19x_1 + 9x_3 \geq 67$
2. $2x_2 + 9x_3 \geq 105$
3. $19x_1 + 2x_2 \geq 61$
4. $19x_1 + 2x_2 + 9x_3 \geq 67$
5. $19x_1 + 2x_2 + 9x_3 \geq 67$ (redundant with constraint 4)
6. $-8x_1 + 5x_3 \geq 0$
7. $-x_2 + 8x_3 \geq 0$
8. $19x_1 + 9x_3 \leq 311$
9. $19x_1 + 2x_2 \leq 174$
10. $x_1, x_2, x_3$ are integers.

## 4: Remove redundant constraints
Constraint 4 and 5 are redundant, so we remove constraint 5.

## 5: Write down the symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'rubber bands'), ('x2', 'monochrome printers'), ('x3', 'cans of coffee')],
'objective_function': '5*x1 + 7*x2 + 4*x3',
'constraints': [
    '19*x1 + 9*x3 >= 67',
    '2*x2 + 9*x3 >= 105',
    '19*x1 + 2*x2 >= 61',
    '19*x1 + 2*x2 + 9*x3 >= 67',
    '-8*x1 + 5*x3 >= 0',
    '-x2 + 8*x3 >= 0',
    '19*x1 + 9*x3 <= 311',
    '19*x1 + 2*x2 <= 174',
    'x1, x2, x3 are integers'
]
}
```

## 6: Implement the optimization problem using Gurobi
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x1 = m.addVar(name="rubber_bands", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="monochrome_printers", vtype=gp.GRB.INTEGER)
x3 = m.addVar(name="cans_of_coffee", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(5*x1 + 7*x2 + 4*x3, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(19*x1 + 9*x3 >= 67)
m.addConstr(2*x2 + 9*x3 >= 105)
m.addConstr(19*x1 + 2*x2 >= 61)
m.addConstr(19*x1 + 2*x2 + 9*x3 >= 67)
m.addConstr(-8*x1 + 5*x3 >= 0)
m.addConstr(-x2 + 8*x3 >= 0)
m.addConstr(19*x1 + 9*x3 <= 311)
m.addConstr(19*x1 + 2*x2 <= 174)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Rubber bands: ", x1.varValue)
    print("Monochrome printers: ", x2.varValue)
    print("Cans of coffee: ", x3.varValue)
else:
    print("The problem is infeasible")
```