To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify or clarify the constraints provided. The objective function aims to maximize \(4 \times\) hours worked by Bobby \(+ 5 \times\) hours worked by Paul \(+ 3 \times\) hours worked by Ringo.

Given variables:
- Hours worked by Bobby (\(B\))
- Hours worked by Paul (\(P\))
- Hours worked by Ringo (\(R\))

Attributes/Resources:
- Dollar cost per hour for each: Bobby (\$5), Paul (\$1), Ringo (\$7)
- Likelihood to quit index for each: Bobby (4), Paul (9), Ringo (5)

Constraints:
1. \(5B + 7R \geq 16\)
2. \(9P + 5R \geq 18\)
3. \(4B + 9P \geq 12\)
4. \(4B + 9P + 5R \geq 15\)
5. \(5B + 1P \leq 45\)
6. \(1P + 7R \leq 47\)
7. \(5B + 1P + 7R \leq 26\)
8. (Duplicate of 7, so we ignore it)
9. \(4B + 9P \leq 52\)
10. \(9P + 5R \leq 37\)
11. \(4B + 9P + 5R \leq 37\)

All hours worked must be integers.

The objective function to maximize is: \(4B + 5P + 3R\)

Here's how we can express this problem in Gurobi Python code:

```python
from gurobipy import *

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

# Define variables
B = m.addVar(vtype=GRB.INTEGER, name="Hours_worked_by_Bobby")
P = m.addVar(vtype=GRB.INTEGER, name="Hours_worked_by_Paul")
R = m.addVar(vtype=GRB.INTEGER, name="Hours_worked_by_Ringo")

# Objective function: Maximize 4B + 5P + 3R
m.setObjective(4*B + 5*P + 3*R, GRB.MAXIMIZE)

# Constraints
m.addConstr(5*B + 7*R >= 16, "Min_dollar_cost_B_R")
m.addConstr(9*P + 5*R >= 18, "Min_quit_index_P_R")
m.addConstr(4*B + 9*P >= 12, "Min_quit_index_B_P")
m.addConstr(4*B + 9*P + 5*R >= 15, "Min_total_quit_index")
m.addConstr(5*B + 1*P <= 45, "Max_dollar_cost_B_P")
m.addConstr(1*P + 7*R <= 47, "Max_dollar_cost_P_R")
m.addConstr(5*B + 1*P + 7*R <= 26, "Max_total_dollar_cost")
m.addConstr(4*B + 9*P <= 52, "Max_quit_index_B_P")
m.addConstr(9*P + 5*R <= 37, "Max_quit_index_P_R")
m.addConstr(4*B + 9*P + 5*R <= 37, "Max_total_quit_index")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Bobby works {B.x} hours")
    print(f"Paul works {P.x} hours")
    print(f"Ringo works {R.x} hours")
    print(f"Objective value: {m.objVal}")
else:
    print("No optimal solution found")
```