To solve the given optimization problem, we need to formulate it as a linear programming problem. The objective is to minimize the cost function `4.11 * hours_worked_by_Bill + 4.49 * hours_worked_by_Bobby` subject to several constraints.

The variables are:
- `hours_worked_by_Bill`: The number of hours worked by Bill.
- `hours_worked_by_Bobby`: The number of hours worked by Bobby.

Given information:
- Dollar cost per hour for Bill: $4
- Likelihood to quit index for Bill: 1
- Dollar cost per hour for Bobby: $9
- Likelihood to quit index for Bobby: 3

Constraints:
1. Total combined dollar cost per hour must be at least $38.
2. Total combined likelihood to quit index must be at least 46.
3. `-2 * hours_worked_by_Bill + 2 * hours_worked_by_Bobby >= 0`
4. Total combined dollar cost per hour should not exceed $87.
5. Total combined likelihood to quit index should not exceed 94.

Since the problem allows for non-integer (fractional) amounts of hours worked by both Bill and Bobby, we do not need to consider integer constraints for these variables.

Here is how we can formulate this in Gurobi Python code:

```python
from gurobipy import *

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

# Define the variables
hours_worked_by_Bill = m.addVar(lb=0, name="hours_worked_by_Bill")
hours_worked_by_Bobby = m.addVar(lb=0, name="hours_worked_by_Bobby")

# Objective function: Minimize 4.11 * hours_worked_by_Bill + 4.49 * hours_worked_by_Bobby
m.setObjective(4.11 * hours_worked_by_Bill + 4.49 * hours_worked_by_Bobby, GRB.MINIMIZE)

# Constraints
# 1. Total combined dollar cost per hour must be at least $38
m.addConstr(4 * hours_worked_by_Bill + 9 * hours_worked_by_Bobby >= 38, name="min_dollar_cost")

# 2. Total combined likelihood to quit index must be at least 46
m.addConstr(1 * hours_worked_by_Bill + 3 * hours_worked_by_Bobby >= 46, name="min_quit_index")

# 3. -2 * hours_worked_by_Bill + 2 * hours_worked_by_Bobby >= 0
m.addConstr(-2 * hours_worked_by_Bill + 2 * hours_worked_by_Bobby >= 0, name="hours_balance")

# 4. Total combined dollar cost per hour should not exceed $87
m.addConstr(4 * hours_worked_by_Bill + 9 * hours_worked_by_Bobby <= 87, name="max_dollar_cost")

# 5. Total combined likelihood to quit index should not exceed 94
m.addConstr(1 * hours_worked_by_Bill + 3 * hours_worked_by_Bobby <= 94, name="max_quit_index")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Bill: {hours_worked_by_Bill.x}")
    print(f"Hours worked by Bobby: {hours_worked_by_Bobby.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```

```python
```