## Step 1: Define the optimization problem and its components
The problem is a linear programming problem where we need to minimize the objective function: $4x_0 + 7x_1 + 8x_2$, where $x_0$, $x_1$, and $x_2$ represent the hours worked by Bobby, Mary, and Ringo, respectively.

## 2: List all the constraints
The constraints given are:
- $3x_0 \leq 112$
- $11x_0 \leq 114$
- $13x_0 \leq 140$
- $2x_1 \leq 112$
- $3x_1 \leq 114$
- $5x_1 \leq 140$
- $16x_2 \leq 112$
- $13x_2 \leq 114$
- $15x_2 \leq 140$
- $3x_0 + 2x_1 \geq 14$
- $2x_1 + 16x_2 \geq 30$
- $3x_0 + 16x_2 \geq 28$
- $3x_0 + 2x_1 + 16x_2 \geq 28$
- $11x_0 + 13x_2 \geq 37$
- $3x_1 + 13x_2 \geq 32$
- $11x_0 + 3x_1 + 13x_2 \geq 32$
- $13x_0 + 5x_1 \geq 44$
- $13x_0 + 5x_1 + 15x_2 \geq 44$
- $-4x_0 + 7x_1 \geq 0$
- $3x_0 + 16x_2 \leq 80$
- $3x_1 + 13x_2 \leq 103$
- $11x_0 + 13x_2 \leq 53$
- $11x_0 + 3x_1 + 13x_2 \leq 99$
- $x_0, x_1, x_2$ are integers.

## 3: Convert the problem into Gurobi code
We will use the Gurobi Python API to model and solve this problem.

```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="hours_worked_by_Bobby", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="hours_worked_by_Mary", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="hours_worked_by_Ringo", vtype=gurobi.GRB.INTEGER)

    # Objective function: Minimize 4*x0 + 7*x1 + 8*x2
    model.setObjective(4*x0 + 7*x1 + 8*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(3*x0 <= 112)
    model.addConstr(11*x0 <= 114)
    model.addConstr(13*x0 <= 140)
    model.addConstr(2*x1 <= 112)
    model.addConstr(3*x1 <= 114)
    model.addConstr(5*x1 <= 140)
    model.addConstr(16*x2 <= 112)
    model.addConstr(13*x2 <= 114)
    model.addConstr(15*x2 <= 140)
    model.addConstr(3*x0 + 2*x1 >= 14)
    model.addConstr(2*x1 + 16*x2 >= 30)
    model.addConstr(3*x0 + 16*x2 >= 28)
    model.addConstr(3*x0 + 2*x1 + 16*x2 >= 28)
    model.addConstr(11*x0 + 13*x2 >= 37)
    model.addConstr(3*x1 + 13*x2 >= 32)
    model.addConstr(11*x0 + 3*x1 + 13*x2 >= 32)
    model.addConstr(13*x0 + 5*x1 >= 44)
    model.addConstr(13*x0 + 5*x1 + 15*x2 >= 44)
    model.addConstr(-4*x0 + 7*x1 >= 0)
    model.addConstr(3*x0 + 16*x2 <= 80)
    model.addConstr(3*x1 + 13*x2 <= 103)
    model.addConstr(11*x0 + 13*x2 <= 53)
    model.addConstr(11*x0 + 3*x1 + 13*x2 <= 99)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Bobby: {x0.varValue}")
        print(f"Hours worked by Mary: {x1.varValue}")
        print(f"Hours worked by Ringo: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```