To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify or clarify the constraints provided. The objective function is clear: minimize \(8 \times \text{hours worked by Peggy} + 2 \times \text{hours worked by Mary} + 3 \times \text{hours worked by Jean}\).

The variables are:
- \(P\) = hours worked by Peggy
- \(M\) = hours worked by Mary
- \(J\) = hours worked by Jean

Given constraints:
1. Dollar cost per hour for Peggy: $12
2. Productivity rating for Peggy: 2
3. Dollar cost per hour for Mary: $9
4. Productivity rating for Mary: 1
5. Dollar cost per hour for Jean: $1
6. Productivity rating for Jean: 10
7. \(9M + J \geq 7\)
8. \(12P + 9M + J \geq 7\)
9. \(2P + M \geq 10\)
10. \(M + 10J \geq 13\)
11. \(2P + 10J \geq 16\)
12. \(2P + M + 10J \geq 16\)
13. \(-8P + 5M \geq 0\)
14. \(8M - 7J \geq 0\)
15. \(2P + M \leq 32\)

All hours worked can be fractional.

Let's express these constraints in a more Gurobi-friendly format and then write the Python code to solve this problem.

### Gurobi Code

```python
from gurobipy import *

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

# Define variables
P = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="hours_worked_by_Peggy")
M = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="hours_worked_by_Mary")
J = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="hours_worked_by_Jean")

# Objective function
m.setObjective(8*P + 2*M + 3*J, GRB.MINIMIZE)

# Constraints
m.addConstr(9*M + J >= 7, "dollar_cost_MJ")
m.addConstr(12*P + 9*M + J >= 7, "total_dollar_cost")
m.addConstr(2*P + M >= 10, "productivity_PM")
m.addConstr(M + 10*J >= 13, "productivity_MJ")
m.addConstr(2*P + 10*J >= 16, "productivity_PJ")
m.addConstr(2*P + M + 10*J >= 16, "total_productivity")
m.addConstr(-8*P + 5*M >= 0, "constraint_13")
m.addConstr(8*M - 7*J >= 0, "constraint_14")
m.addConstr(2*P + M <= 32, "max_productivity_PM")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Peggy: {P.x}")
    print(f"Hours worked by Mary: {M.x}")
    print(f"Hours worked by Jean: {J.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```
```