## Problem Description and Formulation

The problem requires minimizing an objective function subject to several constraints. The objective function to be minimized is \(9.45x_0 + 4.16x_1\), where \(x_0\) represents the quantity of paper towel rolls and \(x_1\) represents the quantity of lightbulbs.

The constraints are as follows:
1. \(19x_0 + 28x_1 \geq 93\)
2. \(9x_0 - 7x_1 \geq 0\)
3. \(19x_0 + 28x_1 \leq 114\)
4. \(x_0\) and \(x_1\) must be integers.
5. Specifically, \(x_0\) is the quantity of paper towel rolls and \(x_1\) is the quantity of lightbulbs, implying \(x_0, x_1 \geq 0\).

## Gurobi Code Formulation

To solve this problem using Gurobi, we can formulate it as a mixed-integer linear programming (MILP) problem. Below is the Python code using Gurobi's API.

```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="paper_towel_rolls", vtype=gurobi.GRB.INTEGER, lb=0)
    x1 = model.addVar(name="lightbulbs", vtype=gurobi.GRB.INTEGER, lb=0)

    # Objective function: Minimize 9.45*x0 + 4.16*x1
    model.setObjective(9.45 * x0 + 4.16 * x1, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(19 * x0 + 28 * x1 >= 93, name="total_value_min")
    model.addConstr(9 * x0 - 7 * x1 >= 0, name="product_constraint")
    model.addConstr(19 * x0 + 28 * x1 <= 114, name="total_value_max")

    # Solve the model
    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Paper Towel Rolls: {x0.varValue}")
        print(f"Lightbulbs: {x1.varValue}")
        print(f"Objective: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has a non-optimal status.")

solve_optimization_problem()
```

This code defines the variables, sets up the objective function and constraints according to the problem description, and then solves the optimization problem using Gurobi. It handles the case where the model is infeasible and reports the optimal solution if one exists.