To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify the constraints provided. The objective function aims to minimize the total cost calculated as $4.5 \times$ hours worked by Jean + $9.58 \times$ hours worked by Dale + $5.64 \times$ hours worked by Paul.

The constraints include various combinations of dollar costs per hour and productivity ratings for each worker, along with some additional linear constraints involving the hours worked by each individual.

Here is a breakdown before we proceed to coding:

1. **Objective Function**: Minimize $4.5J + 9.58D + 5.64P$, where $J$ represents hours worked by Jean, $D$ by Dale, and $P$ by Paul.
2. **Constraints**:
   - Dollar cost per hour constraints for each worker individually are given but seem to be more about providing individual costs rather than direct constraints on the optimization problem since they don't directly limit the variables unless combined in some way.
   - Minimum total dollar cost constraints: $6D + 10P \geq 32$, $16J + 6D \geq 54$, and $16J + 6D + 10P \geq 54$.
   - Minimum total productivity rating constraints: $23D + 20P \geq 41$ and $12J + 23D + 20P \geq 41$.
   - Additional linear constraints: $6D - 7P \geq 0$, $6J - 5P \geq 0$.
   - Maximum total dollar cost constraints: $6D + 10P \leq 169$ and $16J + 10P \leq 148$.
   - Maximum productivity rating constraint: $12J + 23D \leq 116$.

Now, let's write the Gurobi code to solve this problem:

```python
from gurobipy import *

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

# Define variables
J = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Jean")
D = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Dale")
P = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Paul")

# Objective function
m.setObjective(4.5*J + 9.58*D + 5.64*P, GRB.MINIMIZE)

# Constraints
m.addConstr(6*D + 10*P >= 32, "min_cost_dale_paul")
m.addConstr(16*J + 6*D >= 54, "min_cost_jean_dale")
m.addConstr(16*J + 6*D + 10*P >= 54, "min_cost_all")
m.addConstr(23*D + 20*P >= 41, "min_productivity_dale_paul")
m.addConstr(12*J + 23*D + 20*P >= 41, "min_productivity_all")
m.addConstr(6*D - 7*P >= 0, "additional_constraint_1")
m.addConstr(6*J - 5*P >= 0, "additional_constraint_2")
m.addConstr(6*D + 10*P <= 169, "max_cost_dale_paul")
m.addConstr(16*J + 10*P <= 148, "max_cost_jean_paul")
m.addConstr(12*J + 23*D <= 116, "max_productivity_jean_dale")

# Optimize the model
m.optimize()

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