## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'rubber gloves' and 'bottles of ibuprofen', which can be represented symbolically as $x_1$ and $x_2$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to maximize is $5 \times \text{rubber gloves} + 2 \times \text{bottles of ibuprofen}$, which translates to $5x_1 + 2x_2$.

## 3: Translate the constraints into symbolic notation
The constraints given are:
1. The total combined sustainability score from rubber gloves and bottles of ibuprofen must be no less than 15: $4x_1 + 2x_2 \geq 15$.
2. $1 \times \text{rubber gloves} + (-5) \times \text{bottles of ibuprofen} \geq 0$: $x_1 - 5x_2 \geq 0$.
3. The total combined sustainability score from rubber gloves and bottles of ibuprofen must be no more than 22: $4x_1 + 2x_2 \leq 22$.
4. $x_1$ and $x_2$ must be non-negative integers.

## 4: Express the problem in the required JSON format
```json
{
    'sym_variables': [('x1', 'rubber gloves'), ('x2', 'bottles of ibuprofen')],
    'objective_function': '5*x1 + 2*x2',
    'constraints': [
        '4*x1 + 2*x2 >= 15',
        'x1 - 5*x2 >= 0',
        '4*x1 + 2*x2 <= 22',
        'x1 >= 0',
        'x2 >= 0',
        'x1 % 1 == 0',
        'x2 % 1 == 0'
    ]
}
```

## 5: Implement the problem using Gurobi
To solve this problem using Gurobi, we will use Python. We need to ensure that $x_1$ and $x_2$ are integers, which can be achieved by setting them as integer variables in Gurobi.

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # rubber gloves
    x2 = model.addVar(name="x2", vtype=gurobi.GRB.INTEGER)  # bottles of ibuprofen

    # Objective function: maximize 5*x1 + 2*x2
    model.setObjective(5*x1 + 2*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(4*x1 + 2*x2 >= 15, name="sustainability_score_min")
    model.addConstr(x1 - 5*x2 >= 0, name="rubber_gloves_vs_ibuprofen")
    model.addConstr(4*x1 + 2*x2 <= 22, name="sustainability_score_max")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"rubber gloves: {x1.varValue}")
        print(f"bottles of ibuprofen: {x2.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```