## Step 1: Define the symbolic representation of the variables
The variables are 'cans of coffee', 'rubber bands', and 'headsets'. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $9x_1 + 8x_2 + 6x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
1. $17x_1$ (usefulness rating of cans of coffee is 17, but this seems to be a parameter, not a constraint in the traditional sense)
2. $16x_2$ (usefulness rating of rubber bands is 16, similarly, a parameter)
3. $13x_3$ (usefulness rating of headsets is 13, a parameter)
4. $17x_1 + 16x_2 \geq 52$
5. $16x_2 + 13x_3 \geq 30$
6. $17x_1 + 16x_2 + 13x_3 \geq 40$ (or $\geq 40$ as given, but it seems there might be a typo and it should be considered as is)
7. $17x_1 + 16x_2 + 13x_3 \geq 40$ (duplicate of constraint 6, but ensuring clarity)
8. $6x_1 - 2x_3 \geq 0$
9. $16x_2 + 13x_3 \leq 157$

## 4: Correcting and Clarifying Constraints
- The "usefulness ratings" given as $x_0 = 17$, $x_1 = 16$, $x_2 = 13$ seem to actually refer to the coefficients of $x_1, x_2, x_3$ in the constraints rather than the variables themselves. Therefore, the correct interpretation is:
  - The usefulness rating of cans of coffee ($r_0$ for cans of coffee) is 17.
  - The usefulness rating of rubber bands is 16.
  - The usefulness rating of headsets is 13.

## 5: Formulate the Problem with Correct Understanding
Given:
- $x_1$ for 'cans of coffee'
- $x_2$ for 'rubber bands'
- $x_3$ for 'headsets'

And the resources/attributes:
- $r_0 = 17$ for 'cans of coffee'
- $r_1 = 16$ for 'rubber bands'
- $r_2 = 13$ for 'headsets'

The problem can be formulated as:
Minimize $9x_1 + 8x_2 + 6x_3$

Subject to:
- $17x_1 \geq 17 \cdot 17$ is not correct; instead, we consider direct constraints.
- $17x_1 + 16x_2 \geq 52$
- $16x_2 + 13x_3 \geq 30$
- $17x_1 + 16x_2 + 13x_3 \geq 40$
- $6x_1 - 2x_3 \geq 0$
- $16x_2 + 13x_3 \leq 157$
- $x_1, x_2, x_3$ are integers.

## 6: Symbolic Representation
```json
{
  'sym_variables': [('x1', 'cans of coffee'), ('x2', 'rubber bands'), ('x3', 'headsets')],
  'objective_function': '9*x1 + 8*x2 + 6*x3',
  'constraints': [
    '17*x1 + 16*x2 >= 52',
    '16*x2 + 13*x3 >= 30',
    '17*x1 + 16*x2 + 13*x3 >= 40',
    '6*x1 - 2*x3 >= 0',
    '16*x2 + 13*x3 <= 157',
    'x1, x2, x3 are integers'
  ]
}
```

## 7: Gurobi Code
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name="cans_of_coffee", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="rubber_bands", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="headsets", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(9*x1 + 8*x2 + 6*x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(17*x1 + 16*x2 >= 52, name="constraint_1")
    model.addConstr(16*x2 + 13*x3 >= 30, name="constraint_2")
    model.addConstr(17*x1 + 16*x2 + 13*x3 >= 40, name="constraint_3")
    model.addConstr(6*x1 - 2*x3 >= 0, name="constraint_4")
    model.addConstr(16*x2 + 13*x3 <= 157, name="constraint_5")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Cans of coffee: {x1.varValue}")
        print(f"Rubber bands: {x2.varValue}")
        print(f"Headsets: {x3.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```