To solve this optimization problem using Gurobi, we need to translate the given objective function and constraints into a format that can be understood by the Gurobi solver. The objective is to minimize the value of \(2 \times\) hours worked by Bill \(+ 9 \times\) hours worked by Jean \(+ 4 \times\) hours worked by Paul.

The constraints include various limitations on dollar cost per hour and productivity ratings for each worker, as well as combined requirements for different subsets of workers. Additionally, there are constraints that ensure the total number of hours worked by each individual does not result in a non-integer value.

Here's how we can represent this problem using Gurobi in Python:

```python
from gurobipy import *

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

# Define variables
hours_Bill = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Bill")
hours_Jean = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Jean")
hours_Paul = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Paul")

# Objective function
m.setObjective(2*hours_Bill + 9*hours_Jean + 4*hours_Paul, GRB.MINIMIZE)

# Constraints
# Dollar cost per hour constraints
m.addConstr(7*hours_Jean + 4*hours_Paul >= 13, "dollar_cost_Jean_Paul")
m.addConstr(8*hours_Bill + 4*hours_Paul >= 11, "dollar_cost_Bill_Paul")
m.addConstr(8*hours_Bill + 7*hours_Jean + 4*hours_Paul >= 11, "dollar_cost_all")

# Productivity rating constraints
m.addConstr(hours_Jean + 3*hours_Paul >= 6, "productivity_Jean_Paul")
m.addConstr(hours_Bill + 3*hours_Paul >= 12, "productivity_Bill_Paul")
m.addConstr(hours_Bill + hours_Jean + 3*hours_Paul >= 12, "productivity_all")

# Additional constraints
m.addConstr(-3*hours_Bill + 8*hours_Jean >= 0, "constraint_Bill_Jean")
m.addConstr(-2*hours_Bill + 6*hours_Paul >= 0, "constraint_Bill_Paul")

# Upper bound constraint for dollar cost per hour from Jean and Paul
m.addConstr(7*hours_Jean + 4*hours_Paul <= 25, "upper_bound_dollar_cost_Jean_Paul")

# Optimize the model
m.optimize()

# Print results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")
print(f"Objective: {m.objVal}")
```