To solve this problem, we will use the Gurobi optimization library in Python. We need to define the variables, constraints, and objective function according to the given problem.

```python
from gurobipy import *

# Define the model
model = Model("Work_Hours_Optimization")

# Define the variables
Jean_hours = model.addVar(name="Jean_hours", lb=0)
Bill_hours = model.addVar(name="Bill_hours", lb=0)
Ringo_hours = model.addVar(name="Ringo_hours", lb=0)
Dale_hours = model.addVar(name="Dale_hours", lb=0)
Paul_hours = model.addVar(name="Paul_hours", lb=0)

# Define the constraints
model.addConstr(Jean_hours >= 0, name="Jean_non_negative")
model.addConstr(Bill_hours >= 0, name="Bill_non_negative")
model.addConstr(Ringo_hours >= 0, name="Ringo_non_negative")
model.addConstr(Dale_hours >= 15, name="Dale_minimum")
model.addConstr(Paul_hours >= 0, name="Paul_non_negative")

# Add constraints for work quality ratings
model.addConstr(Jean_hours + Ringo_hours + Paul_hours >= 32, name="Quality_1")
model.addConstr(Bill_hours + Dale_hours + Paul_hours <= 76, name="Cost_1")
model.addConstr(Ringo_hours + Dale_hours + Paul_hours <= 20, name="Cost_2")

# Add constraints for likelihood to quit
model.addConstr(Jean_hours + Ringo_hours <= 154, name="Quit_1")
model.addConstr(Bill_hours + Dale_hours <= 187, name="Quit_2")

# Define the objective function (minimize total hours)
obj = Jean_hours + Bill_hours + Ringo_hours + Dale_hours + Paul_hours
model.setObjective(obj, GRB.MINIMIZE)

# Add constraints for dollar cost per hour
model.addConstr(89 * Jean_hours + 24 * Bill_hours <= 63, name="Cost_per_hour_1")
model.addConstr(71 * Bill_hours + 20 * Ringo_hours + 49 * Dale_hours <= 76, name="Cost_per_hour_2")

# Add constraints for likelihood to quit index
model.addConstr(112 * Jean_hours + 55 * Ringo_hours <= 164, name="Quit_index_1")
model.addConstr(101 * Bill_hours + 148 * Dale_hours <= 187, name="Quit_index_2")

# Optimize the model
model.optimize()

```python 
```