To solve the optimization problem described, we will use the Gurobi Python API. The problem involves maximizing an objective function subject to several constraints related to the likelihood to quit index and dollar cost per hour for three individuals: Laura, Mary, and Ringo.

First, let's outline the key components of the problem:

1. **Objective Function**: Maximize \(8 \times \text{hours worked by Laura} + 6 \times \text{hours worked by Mary} + 5 \times \text{hours worked by Ringo}\).
2. **Constraints**:
   - Likelihood to quit index constraints for each pair and all three individuals.
   - Dollar cost per hour constraints for pairs and all three individuals.
   - The number of hours worked by Laura must be a whole number, while Mary's and Ringo's can be non-integer.

Given these requirements, we will define the variables, objective function, and constraints in Gurobi. Note that we'll use `GRB.INTEGER` for Laura's hours to ensure they are whole numbers, and `GRB.CONTINUOUS` for Mary's and Ringo's hours to allow for non-integer values.

```python
from gurobipy import *

# Create a model
m = Model("Optimization_Problem")

# Define variables
laura_hours = m.addVar(lb=0, ub=100, vtype=GRB.INTEGER, name="laura_hours")
mary_hours = m.addVar(lb=0, ub=100, vtype=GRB.CONTINUOUS, name="mary_hours")
ringo_hours = m.addVar(lb=0, ub=100, vtype=GRB.CONTINUOUS, name="ringo_hours")

# Objective function: Maximize
m.setObjective(8*laura_hours + 6*mary_hours + 5*ringo_hours, GRB.MAXIMIZE)

# Constraints
# Likelihood to quit index constraints
m.addConstr(4*laura_hours + 3*ringo_hours >= 12, name="likeli_quit_lr")
m.addConstr(4*laura_hours + 11*mary_hours <= 36, name="likeli_quit_lm")
m.addConstr(11*mary_hours + 3*ringo_hours <= 47, name="likeli_quit_mr")
m.addConstr(4*laura_hours + 11*mary_hours + 3*ringo_hours <= 47, name="likeli_quit_lmr")

# Dollar cost per hour constraints
m.addConstr(6*mary_hours + 8*ringo_hours <= 48, name="dollar_cost_mr")
m.addConstr(3*laura_hours + 8*ringo_hours <= 60, name="dollar_cost_lr")
m.addConstr(3*laura_hours + 6*mary_hours + 8*ringo_hours <= 25, name="dollar_cost_lmr")
m.addConstr(3*laura_hours + 6*mary_hours + 8*ringo_hours <= 25, name="total_dollar_cost")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Laura: {laura_hours.x}")
    print(f"Hours worked by Mary: {mary_hours.x}")
    print(f"Hours worked by Ringo: {ringo_hours.x}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found")
```
```python
```