To solve this optimization problem, we first need to define the variables, objective function, and constraints using Gurobi's Python API.

## Step 1: Define the Variables
Let's define the variables for hours worked by each person: Laura, Ringo, Mary, and Dale.

## Step 2: Define the Objective Function
The objective function to maximize is: $3 \times Laura + 1 \times Ringo + 3 \times Mary + 6 \times Dale$.

## Step 3: Define the Constraints
We have multiple constraints based on productivity ratings, dollar cost per hour, work quality ratings, and likelihood to quit index. These constraints are both lower and upper bounds for various combinations of hours worked by each person.

## 4: Implement in Gurobi
Below is the Python code using Gurobi to model and solve this optimization problem:

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
laura = m.addVar(name="laura", lb=0)  # hours worked by Laura
ringo = m.addVar(name="ringo", lb=0)  # hours worked by Ringo
mary = m.addVar(name="mary", lb=0)  # hours worked by Mary
dale = m.addVar(name="dale", lb=0)  # hours worked by Dale

# Define the objective function
m.setObjective(3 * laura + ringo + 3 * mary + 6 * dale, gp.GRB.MAXIMIZE)

# Constraints
# Productivity rating constraints
m.addConstr(9 * laura + 22 * dale >= 45, "prod_rating_la_dale")
m.addConstr(13 * ringo + 31 * mary >= 98, "prod_rating_ring_mary")
m.addConstr(9 * laura + 13 * ringo + 22 * dale >= 83, "prod_rating_la_ring_dale")
m.addConstr(10 * laura + 6 * mary >= 43, "dollar_cost_la_mary")
m.addConstr(6 * mary + 18 * dale >= 68, "dollar_cost_mary_dale")

# Work quality rating constraints
m.addConstr(6 * laura + 4 * dale >= 24, "work_qual_la_dale")
m.addConstr(6 * laura + 8 * mary >= 49, "work_qual_la_mary")
m.addConstr(6 * laura + 1 * ringo + 4 * dale >= 56, "work_qual_la_ring_dale")
m.addConstr(6 * laura + 1 * ringo + 8 * mary >= 56, "work_qual_la_ring_mary")
m.addConstr(6 * laura + 1 * ringo + 4 * dale >= 32, "work_qual_la_ring_dale_2")
m.addConstr(6 * laura + 1 * ringo + 8 * mary >= 32, "work_qual_la_ring_mary_2")

# Productivity rating upper bounds
m.addConstr(13 * ringo + 31 * mary <= 424, "prod_rating_ring_mary_ub")
m.addConstr(31 * mary + 22 * dale <= 336, "prod_rating_mary_dale_ub")
m.addConstr(9 * laura + 22 * dale <= 316, "prod_rating_la_dale_ub")
m.addConstr(13 * ringo + 22 * dale <= 262, "prod_rating_ring_dale_ub")
m.addConstr(9 * laura + 31 * mary <= 390, "prod_rating_la_mary_ub")
m.addConstr(13 * ringo + 31 * mary + 22 * dale <= 289, "prod_rating_ring_mary_dale_ub")
m.addConstr(9 * laura + 13 * ringo + 22 * dale <= 126, "prod_rating_la_ring_dale_ub_2")
m.addConstr(9 * laura + 13 * ringo + 31 * mary <= 211, "prod_rating_la_ring_mary_ub")
m.addConstr(9 * laura + 13 * ringo + 31 * mary + 22 * dale <= 211, "prod_rating_all_ub")

# Dollar cost per hour upper bounds
m.addConstr(10 * laura + 6 * mary <= 266, "dollar_cost_la_mary_ub")
m.addConstr(6 * mary + 18 * dale <= 166, "dollar_cost_mary_dale_ub")
m.addConstr(5 * ringo + 6 * mary <= 170, "dollar_cost_ring_mary_ub")
m.addConstr(10 * laura + 18 * dale <= 93, "dollar_cost_la_dale_ub")
m.addConstr(5 * ringo + 18 * dale <= 221, "dollar_cost_ring_dale_ub")
m.addConstr(10 * laura + 5 * ringo + 6 * mary + 18 * dale <= 221, "dollar_cost_all_ub")

# Work quality rating upper bounds
m.addConstr(1 * ringo + 8 * mary <= 60, "work_qual_ring_mary_ub")
m.addConstr(6 * laura + 8 * mary <= 71, "work_qual_la_mary_ub")
m.addConstr(1 * ringo + 8 * mary + 4 * dale <= 63, "work_qual_ring_mary_dale_ub")
m.addConstr(6 * laura + 1 * ringo + 4 * dale <= 182, "work_qual_la_ring_dale_ub_2")
m.addConstr(6 * laura + 8 * mary + 4 * dale <= 76, "work_qual_la_mary_dale_ub")
m.addConstr(6 * laura + 1 * ringo + 8 * mary + 4 * dale <= 76, "work_qual_all_ub")

# Likelihood to quit index upper bounds
m.addConstr(21 * laura + 12 * mary <= 193, "quit_la_mary_ub")
m.addConstr(12 * mary + 31 * dale <= 251, "quit_mary_dale_ub")
m.addConstr(21 * laura + 30 * ringo + 12 * mary + 31 * dale <= 251, "quit_all_ub")

# Optimize
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Laura: ", laura.varValue)
    print("Hours worked by Ringo: ", ringo.varValue)
    print("Hours worked by Mary: ", mary.varValue)
    print("Hours worked by Dale: ", dale.varValue)
else:
    print("No optimal solution found")
```