## Step 1: Define the symbolic representation of the problem
The problem involves three variables: smoke detectors, packs of paper, and black pens, which we can denote as $x_0$, $x_1$, and $x_2$ respectively. The objective function to minimize is $6x_0 + 2x_1 + 5x_2$. The constraints are as follows:

- $8.84x_0 \leq 167$
- $2.88x_0 \leq 185$
- $8.91x_0 \leq 85$
- $13.06x_1 \leq 167$
- $14.55x_1 \leq 185$
- $6.67x_1 \leq 85$
- $12.85x_2 \leq 167$
- $7.18x_2 \leq 185$
- $7.17x_2 \leq 85$
- $13.06x_1 + 12.85x_2 \geq 45$
- $8.84x_0 + 12.85x_2 \geq 31$
- $8.84x_0 + 13.06x_1 + 12.85x_2 \geq 50$
- $2.88x_0 + 7.18x_2 \geq 61$
- $14.55x_1 + 7.18x_2 \geq 27$
- $2.88x_0 + 14.55x_1 + 7.18x_2 \geq 27$
- $6.67x_1 + 7.17x_2 \geq 15$
- $8.91x_0 + 6.67x_1 + 7.17x_2 \geq 15$
- $-x_1 + 2x_2 \geq 0$
- $3x_0 - 2x_2 \geq 0$
- $14.55x_1 + 7.18x_2 \leq 73$
- $2.88x_0 + 14.55x_1 \leq 161$
- $8.91x_0 + 6.67x_1 \leq 34$
- $8.91x_0 + 6.67x_1 + 7.17x_2 \leq 77$
- $x_0, x_1, x_2$ are integers.

## Step 2: Convert the problem into a Gurobi code
We will use Gurobi's Python API to model and solve this problem.

```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="smoke detectors", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="packs of paper", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="black pens", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(6 * x0 + 2 * x1 + 5 * x2, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(8.84 * x0 <= 167)
m.addConstr(2.88 * x0 <= 185)
m.addConstr(8.91 * x0 <= 85)
m.addConstr(13.06 * x1 <= 167)
m.addConstr(14.55 * x1 <= 185)
m.addConstr(6.67 * x1 <= 85)
m.addConstr(12.85 * x2 <= 167)
m.addConstr(7.18 * x2 <= 185)
m.addConstr(7.17 * x2 <= 85)
m.addConstr(13.06 * x1 + 12.85 * x2 >= 45)
m.addConstr(8.84 * x0 + 12.85 * x2 >= 31)
m.addConstr(8.84 * x0 + 13.06 * x1 + 12.85 * x2 >= 50)
m.addConstr(2.88 * x0 + 7.18 * x2 >= 61)
m.addConstr(14.55 * x1 + 7.18 * x2 >= 27)
m.addConstr(2.88 * x0 + 14.55 * x1 + 7.18 * x2 >= 27)
m.addConstr(6.67 * x1 + 7.17 * x2 >= 15)
m.addConstr(8.91 * x0 + 6.67 * x1 + 7.17 * x2 >= 15)
m.addConstr(-x1 + 2 * x2 >= 0)
m.addConstr(3 * x0 - 2 * x2 >= 0)
m.addConstr(14.55 * x1 + 7.18 * x2 <= 73)
m.addConstr(2.88 * x0 + 14.55 * x1 <= 161)
m.addConstr(8.91 * x0 + 6.67 * x1 <= 34)
m.addConstr(8.91 * x0 + 6.67 * x1 + 7.17 * x2 <= 77)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Smoke detectors: ", x0.varValue)
    print("Packs of paper: ", x1.varValue)
    print("Black pens: ", x2.varValue)
else:
    print("The model is infeasible")
```

## Step 3: Provide the symbolic representation
The symbolic representation of the problem is:

```json
{
    'sym_variables': [
        ('x0', 'smoke detectors'), 
        ('x1', 'packs of paper'), 
        ('x2', 'black pens')
    ], 
    'objective_function': '6*x0 + 2*x1 + 5*x2', 
    'constraints': [
        '8.84*x0 <= 167', 
        '2.88*x0 <= 185', 
        '8.91*x0 <= 85', 
        '13.06*x1 <= 167', 
        '14.55*x1 <= 185', 
        '6.67*x1 <= 85', 
        '12.85*x2 <= 167', 
        '7.18*x2 <= 185', 
        '7.17*x2 <= 85', 
        '13.06*x1 + 12.85*x2 >= 45', 
        '8.84*x0 + 12.85*x2 >= 31', 
        '8.84*x0 + 13.06*x1 + 12.85*x2 >= 50', 
        '2.88*x0 + 7.18*x2 >= 61', 
        '14.55*x1 + 7.18*x2 >= 27', 
        '2.88*x0 + 14.55*x1 + 7.18*x2 >= 27', 
        '6.67*x1 + 7.17*x2 >= 15', 
        '8.91*x0 + 6.67*x1 + 7.17*x2 >= 15', 
        '-x1 + 2*x2 >= 0', 
        '3*x0 - 2*x2 >= 0', 
        '14.55*x1 + 7.18*x2 <= 73', 
        '2.88*x0 + 14.55*x1 <= 161', 
        '8.91*x0 + 6.67*x1 <= 34', 
        '8.91*x0 + 6.67*x1 + 7.17*x2 <= 77'
    ]
}
```